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Submitted to Microw. Opt. Technol. Lett. Microwave gyroscope – novel rotation sensor G.G.Karapetyan Yerevan Physics Institute, Yerevan, Armenia High performance microwave gyroscope (MG) is theoretically developed for the first time to our knowledge. MG is based on Sagnac effect in microwave ring resonator (RR), where a specially tailored phase shifter (PS) on the basis of surface acoustic waves is inserted. Due to that the beat frequency becomes proportional to square (or cubic) root upon rotation rate, and therefore hugely increases. In the result MG has few order higher sensitivity and dynamic range, than state-of-the-art laser gyros, so it can serve as an advanced rotation sensor in navigation, and fundamental sciences. 1. INTRODUCTION Since the advent of laser three main types of optical rotation sensors have been under development. These include ring laser gyroscope (RLG), ring resonator gyroscope (RRG), and fiber optics gyroscope (FOG) [1-3]. State-of-the-art RLG is now the most sensitive device among them [4,5] and is used extensively in inertial navigation systems for aircraft, and in fundamental physics and geophysics [6]. All types of gyros are based on Sagnac effect, discovered in 1913 [7]. In Sagnac interferometer two light beams propagate in opposite directions around a common path. Rotating interferometer effectively shortens the optical path traveled by one of the beams, while lengthening the other on the same value δL, given by v/S2LΩ=δ . (1) Here S -is the area enclosed by that path, v -is the speed of light there, Ω -is rotation angular frequency. In the case of FOG this change of path length causes phase difference between two counter propagating beams (having the same frequency), which is detected by interfering them outside the path. However in RLG and RRG light propagates in RR so change of optical path length causes the change of RR resonant frequency. Thus, counter propagating beams in RR have different frequencies, and produce the beat frequency 2 δf, which is direct proportional to change of path length vL/fS4L/Lf2f2 Ω−=δ−=δ , (2) where f – is frequency of light. As it is seen from (2) beat frequency is direct proportional also to operating frequency, so a microwave gyroscope with operating frequency around 1 GHz will have about on 5 orders lower beat frequency, and therefore sensitivity than RLG with the same dimensions. Because of that till now microwave gyros have not been considering as a possible rotation sensors at all. Now we use a novel method, called the method of phase shifting enabling to increase considerably the beat frequency in rotating RR. Due to that a possibility of creating an advanced rotation sensor with very high performance - microwave gyroscope arises. 2. PRINCIPAL EXPRESSIONS Let us consider a microwave RR, with perimeter L. Its resonant frequencies f 0 are determined from the condition that path length is multiple to the wavelength, or that along that path the phase of microwaves is changed on an integer 2 π, i.e. mv/Lf0= , (3) where m – is an integer. Suppose that a frequency dependent PS, which adds the phase ϕ(f) to the waves with frequency f is inserted into RR. Then the resonant frequencies f 1 of that RR satisfy to another condition m2/)f(v/Lf1 1 =πϕ+ . (4) When RR is rotated path length of waves are changed on a small value ΔL, which causes the shift Δf of resonant frequency, determined from the equation m2/)ff(v/)ff)(LL(1 1 =πΔ+ϕ+Δ+Δ+ . (5) Let us assume that ϕ(f) is a quadratic function in vicinity of point f=f 1. Then ϕ(f1+Δf) can be expressed as an expansion in Taylor series with 3 terms: 2/ff)f()ff(2 1 1 Δϕ′′+Δϕ′+ϕ=Δ+ϕ , (6) where primes mean derivatives in respect to the frequency with arguments f 1. Substituting (6) in (5) we have [] 0v/Lf4fv/)LL(22f12=Δπ+Δϕ′+Δ+π+Δϕ′′ . (7) This square equation determines the shift of RR resonant frequency stipulated by the shift of its length. The solutions of (7) is written as ()( )ϕ′′ Δϕ′′π−±−=Δ /v/Lf4AAf2/1 12, (8) where A=2 π(L+ΔL)/v+ϕ′. Analyzing (8) one can conclude that shift of RR resonant frequency is sharply increased when ϕ′→ −2πL/v, becoming proportional to square root of ΔL (see Fig.1, and Fig.2). By this it is necessary that ΔL and ϕ′′ have opposite signs, because if ϕ′′ΔL>0 then (8) gives imaginary value of Δf, which indicates on the absence of oscillations in RR. Therefore in rotating RR with PS one of counter propagating waves is disappeared, but another one is splitted on two waves propagating in the same directions. Below we will assume that condition ϕ′′ΔL< 0 is satisfied. Thus appropriately installed negative value of PFD can strongly change the functional dependence between Δf and ΔL, considerably increasing Δf. The range of PFD, where it takes place is determined from (8) by taking into account the condition A2<<f1ϕ′′ΔL /v, which leads to ()2/1 1 v/\|L\|f\|v/L2\| Δϕ′′<π+ϕ′ . (9) By satisfying this condition the shifts of RR resonant frequency are () ( ) v/\|\|/S8f v/Lf4f2/1 12/1 1 ϕ′′Ωπ±=ϕ′′Δπ−±≈Δ , (10) that produce the beat frequency 2 Δf, considerably surpassing the beat frequency (2) in RR without PS, because of its proportionality to square root upon rotation rate. By analogy a PS with cubic function ϕ(f) in vicinity of point f=f 1 can be considered. Then ϕ′′=0, and Δf is determined by third derivative ϕ′′′ () ()3/12 1 13/1 1 v/Sf24f v/Lf12f ϕ′′′Ωπ−−≈ϕ′′′Δπ−≈Δ . (11) Contrary to previous case, both of counter-propagating waves exist here. One of them has positive shift of resonant frequency, another one- negative. Beat frequency produced by them is however higher than that in previous case because of its proportionality to cubic root upon rotation rate. Required PS can be designed on the basis of surface acoustic waves (SAW) delay lines [8] (with bi-directional amplifier to compensate the losses). Since SAW wavelength λ is around a micron the change s in path traveled by SAW for example on 0.1 mm causes the phase shift ∼s/λ∼100. To obtain required by (9) value of PFD it is necessary that this phase shift took place in frequency interval Δf∼(v/L)(s/λ). Thus required PFD can be obtained by tailoring appropriate chirp low of SAW transducers. Following this approach we designed a PS on the basis of SAW chirped delay line. Its transducers have both linear and quadratic chirp, providing the quadratic function ϕ(f) in vicinity of resonant frequency 600 MHz, having ϕ′= −0.09424 1/MHz, and ϕ′′=0.0001 1/MHz2. 3. DISCUSSION Let us compare the performances of proposed MG and optical gyros. Numerical evaluations of (10) by substituting L=3 m, v=2 ⋅108 m/s, ϕ′′=0.0001 1/MHz2, f1=600 MHz is presented on Fig. 3 by solid line. The beat frequency in conventional MG without PS, determined from (2) is plotted by dashed line. By dotted line it is presented the beat frequency in RLG having the same dimensions. It is seen that after inserting of PS, the beat frequency, and therefore sensitivity of MG increases on 7…9 order, and surpasses by this a few order the beat frequency in RLG. Dynamic range of MG also is much larger than that of RLG, because change of rotation rate in region for example 1010 causes the change of beat frequency only in region 105. Therefore proposed MG is an advanced rotation sensor, that being created can replace existing RLG and FOG in inertial navigation systems because of its higher performance, and possible lower cost. Another application of MG can be monitoring of Earth rotation. As it follows from (10) Earth rotation angular velocity 15 grad/hour causes the beat frequency of MG (Earth Sagnac frequency) about 800 Hz, meanwhile the world largest laser gyros – Canterbury C-2 produces Earth Sagnac frequency only 79 Hz [9]. Moreover, beat frequency 800 Hz can be increased further by increasing of MG dimensions, and operating frequency. Thus MG can be used also in geophysics for monitoring Earth rotation angular velocity with very high precision. 4. CONCLUSIONS In conclusion we proposed an advanced rotation sensor - MG, where appropriately tailored SAW chirped delay line is inserted. Due to that beat frequency becomes proportional to square (or cubic) root upon rotation rate so is hugely increased, surpassing even the beat frequency in RLG with the same dimensions. In the result MG has higher sensitivity and dynamic range than state-of-the-art RLG. Being created proposed MG can be used in inertial navigation systems, and fundamental sciences. REFERENCES 1. A.H.Rosenthal, Regenerative circulatory multiple-beam interferometry for the study of light-propagation effects, J. Opt. Soc. Am. Vol. 52, (1962), 1143-1148. 2. S.Ezekiel, and S.R.Balsamo, Passive ring resonator laser gyroscope, Appl. Physl. Lett, Vol. 30, (1977), pp. 478-483. 3. H.Lefevre, The Fibre Optic Gyroscope, Norwood, MA, Artech House, 1993. 4. T.L.Gustavsen, P.Bouyer, and M.A.Kasevich, Precision rotation measurements with an atom interferometer gyroscope, Phys. Rev. Lett. Vol. 78, (1997), pp.2046-2049. 5. G.E.Stedman, L.Ziyec, A.D.McGregor, and H.R.Bilger, Harmonic analysis in a precision ring laser with back-scatter induced pulling, Phys.Rev. Vol. A 51, ( 1995), pp. 4944-4958. 6. G.E.Stedman, Ring laser tests of fundamental physics and geophysics, Reports Progr. Phys., Vol.60, (1997), pp. 615-683. 7. E.J.Post, Sagnac effect, Rev. Modern Phys. Vol. 39, (1967), pp.475-493. 8. Acoustic Surface Waves, Editted by A.A.Oliner, Spring-Verlag Berlin-Heidelberg-New York 1978. 9. U.Shreiber, M.Shneider at all, Preliminary results from a large ring laser gyroscope for fundamental physics and geophysics, Symposium gyro technology, Stuttgart, 1997. 024681012 -0,095 -0,0945 -0,094 -0,0935 PFD, 1/MHzShift of frequency, kHz Fig.1 Frequency shift in RR versus PFD of phase shifter, when ΔL=30 nm, L=3 m. The solid line corresponds to ϕ′′=0.01 1/MHz2, the dotted line ϕ′′=0.1 1/MHz2. 012345 02 04 06 08 01 0 0 1 2 0 Change of RR length, nmShift of frequency, kHz Fig.2 Frequency shift versus the shift of RR length, when ϕ′′= −0.1 1/MHz2, L=3 m. The solid line corresponds to ϕ′= −0.09424 1/MHz, the dotted line ϕ′= −0.1 1/MHz. 1,E-101,E-071,E-041,E-011,E+021,E+05 1,0E-08 1,0E-05 1,0E-02 1,0E+01Rotation Rate, deg/hourBeat Frequency, Hz Fig.3 Beat frequency in conventional MG (the dashed line), and in MG with PS (the solid line). L=3m, v=2x108m/s, f1=600 MHz, ϕ′′=0.0001 1/MHz2. The dotted line corresponds to RLG having L=3m, v=3x108m/s, f1=6x1014 Hz.
arXiv:physics/0011015v1 [physics.class-ph] 7 Nov 2000Biot-Savart-like law in electrostatics M´ ario H. Oliveira and Jos´ e A.Miranda∗ Laborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica, Universidade Federal de Pernambuco, Recife, PE 50670-901 B razil (February 2, 2008) Abstract The Biot-Savart law is a well-known and powerful theoretica l tool used to calculate magnetic ﬁelds due to currents in magnetostatics . We extend the range of applicability and the formal structure of the Biot- Savart law to elec- trostatics by deriving a Biot-Savart-like law suitable for calculating electric ﬁelds. We show that, under certain circumstances, the tradi tional Dirichlet problem can be mapped onto a much simpler Biot-Savart-like p roblem. We ﬁnd an integral expression for the electric ﬁeld due to an arb itrarily shaped, planar region kept at a ﬁxed electric potential, in an otherw ise grounded plane. As a by-product we present a very simple formula to com pute the ﬁeld produced in the plane deﬁned by such a region. We illustrate t he usefulness of our approach by calculating the electric ﬁeld produced by planar regions of a few nontrivial shapes. ∗Corresponding author e-mail:jme@lftc.ufpe.br 1I. INTRODUCTION The Biot-Savart law [1–4] is one of the most basic relations i n electricity and magnetism. It allows one to determine the total magnetic ﬁeld Bat a given point in space as the superposition of inﬁnitesimal contributions dB, caused by the ﬂow of current Ithrough an inﬁnitesimal path segment ds, oriented in the same direction as the current. If ˆ ris the position unit-vector pointing from the element of length to the observation point P, then the total ﬁeld contribution due to a closed circuit CatPis given by a closed lineintegral [1–4] B=µ0I 4π/contintegraldisplay C(ds×ˆ r) r2, (1) where µ0is the permeability of free space and the path of integration is along the wire. The usefulness of the Biot-Savart law goes far beyond its bas ic textbook applications. Equation (1) is a much studied integral form which arises in v arious interesting physical problems involving topological defects [5], magnetic ﬂuid s [6], amphiphilic monolayers [7], type-I superconductors [8] and the n-body problem of celestial mechanics [9]. In the frame- work of classical electrodynamics the Biot-Savart law prov ides a useful technique to calculate the magnetic ﬁeld generated by current carrying wires. Equa tion (1) is quite general, valid for nonplanar current loops of arbitrary shape. Obviously, the presence of a cross-product in the formula may introduce some diﬃculties in practical ca lculations. Ultimately, the range of problems to which equation (1) can be applied is limi ted primarily by the diﬃculty experienced in performing the integrations. These diﬃcult ies are less serious if the loop Cis ﬂat and if the observation point lies in the plane of the loop. It has been recently shown [10] that the magnitude of the magnetic ﬁeld due to an arbitrarily shaped (not self-intersecting), planar, current carrying wire at a point lying on the wire’s p lane can be written as B=µ0I 4π/contintegraldisplay Cdθ r. (2) 2Equation (2) is very simple and compact, expressing Bin terms of the wire shape r=r(θ), where θdenotes the polar angle. As shown in reference [10], this bas ic result expands the degree of applicability of the Biot-Savart law allowing exa ct, closed form solutions for a whole new set of elementary problems. Based on the general nature of the Biot-Savart law, its conne ction to recent physics research topics [6–9], and its success in performing magnet ic ﬁeld calcultions, we felt moti- vated to investigate the following question: is it possible to formulate a Biot-Savart-like law in electrostatics ? If so, what sort of electrostatic proble m could be more easily solved by such an approach? In this work we address these issues and sho w, for the ﬁrst time, that it is indeed possible to formulate a Biot-Savart-type law in th e realm of electrostatics. The formulation we propose is suitable to calculate the elec tric ﬁeld due to an arbitrary shaped, planar region maintained at a ﬁxed scalar potencial V, with the rest of the plane held at zero potential. We show that the calculation of the el ectric ﬁeld produced by such a region is analogous to the evaluation of the magnetic ﬁeld du e to a ﬂat current carrying wire of the same shape (ﬁgure 1). We point out that the “brute-forc e” calculation of the electric ﬁeld due to such an arbitrarily shaped region, kept at a ﬁxed p otential, by directly applying conventional boundary-value techniques [11] may be quite c hallenging. The nontrivial na- ture of the problem comes from the fact that we do not know the s urface charge distribution in advance. Our eletrostatic Biot-Savart law provides a muc h simpler way to perform this nontrivial electric ﬁeld calculation, and allows one to bor row many of the standard proce- dures ordinarily used in the corresponding magnetostatic s ituation. Therefore, complicated electrostatic problems may have straightforward solution s, if solved by the Biot-Savart-like approach we develop here. We stress that, even though our der ivation and new results (8) and (9) presented below are quite simple, they have not been d erived in any standard elec- tromagnetism book or journal publication. This work comes t o ﬁll this gap, oﬀering a new 3and simple tool to perform electric ﬁeld calculations. As a by-product, we obtain a very compact expression for the e lectric ﬁeld in the plane of the ﬂat sheet (see equation (9)), which is as simple as its m agnetic ﬁeld counterpart given by equation (2). Finally, we illustrate our results by explicitly calculating the electric ﬁeld produced by a constant potential, ﬂat region that has th e shape of a regular, n-sided polygon, for observation points located along its axis of sy mmetry. The calculation for the ﬁeld produced in the plane of regions having other peculi ar shapes (elliptical, spiral, regularly undulating) is also presented. II. ELECTRIC FIELD CALCULATION Consider an arbitrarily shaped, two-dimensional region, l ocated in the x-yplane, kept at a ﬁxed potential V while the rest of the plane is held at zero potential (see ﬁgure 2). The shape of the ﬂat plate is determined by a closed boundary curv eC. We want to calculate the electric ﬁeld caused by this charge conﬁguration at a giv en observation point P located by the vector x E(x) =−∇Φ(x), (3) where Φ( x) denotes the electric scalar potential. Since the scalar potential is speciﬁed everywhere on the x-yplane, we have to solve a Dirichlet problem [11]. We apply Dirichlet boundary condit ions, i.e., GD(x,x′)\|z′=0= 0, where GD(x,x′) denotes the Green’s function for Dirichlet conditions, an d the primed vector x′locates the charge distribution points. Using Green’s theo rem and the fact that the volume charge density ρ(x′) = 0, the electrostatic potential for z >0 can be expressed in terms of the value of the potential on the plate Φ( x′) =V[11] Φ(x) =−1 4π/integraldisplay Φ(x′)/parenleftBigg∂GD ∂n′/parenrightBigg da′, (4) 4where ˆn′=−ˆz′is the outward unit normal and da′is an inﬁnitesimal area element of the plate. By employing the method of images [11], we can easily ﬁnd the G reen’s function by considering the potential in xdue to a unit point charge located at a point in the region z >0, plus the potential of an image charge placed in a symmetric position in the lower-half plane z <0. Through this procedure we calculate ( ∂GD/∂n′) = (−∂GD/∂z′)\|z′=0explicitly, and rewrite equation (4) as Φ(x) =V 2πΩ(x), (5) where Ω(x) =/integraldisplay(−ˆ z)·(x′−x) \|x−x′\|3da′(6) is the solid angle subtended by the surface (ﬂat plate of area a′) spanning the loop Cas seen from x. The sign convention established for the solid angle is the f ollowing: Ω( x) is positive when the observation point Pviews the “inner” side of the ﬂat plate, in other words, Ω(x)>0 if the unit normal points away from P. This sign convention is the same as the one adopted in references [3,4]. By substituting equations (5) and (6) into the electric ﬁeld equation (3) and using the well-known relation [12] ∇Ω(x) =/contintegraldisplay Cds′×(x−x′) \|x−x′\|3, (7) we ﬁnally obtain E(x) =V 2π/contintegraldisplay C(x−x′)×ds′ \|x−x′\|3, (8) whereds′is an element of length of the integration path C. The direction of the integration around Cis determined by the direction of the outward unit normal via the right-hand 5rule. Notice that the cross-product in (8) has a reversed ord er in comparison to the one that appears in the magnetostatic case (1). This distinctio n is necessary to give the correct direction for the electric ﬁeld vector at the observation po intP. Equation (8) is our central result. It allows the calculatio n of the electric ﬁeld through a Biot-Savart-type law. Notice that (8) is written in terms of a line integral, so to calculate the electric ﬁeld we just need to take into account the contribut ions coming from the boundary contour C. This result makes the solution of the electrostatic proble m much easier. Under the circumstances studied in this work, the traditional Dir ichlet problem can be mapped onto a Biot-Savart-like problem, a mapping that can simplif y considerably the computation of the electric ﬁeld in many problems of interest. As it was in the magnetic case [10], a much simpler expression for the ﬁeld can be obtained if we concentrate our attention in the calculation of the electric ﬁeld at observation points that lie in the plane of the charge distribution. Usin g the same arguments as those presented in reference [10] it can be easily shown that E(x) =V 2π/contintegraldisplay Cdθ rˆ z, (9) where r=\|x−x′\|. Equation (9) is a surprisingly simple result. Note that thi s line integral expression (9) works for ﬂat charge distributions of any bou ndary shape, including those described by curves Cwhich are not single valued functions of θ. In addition, it is valid for observation points Plocated either inside or outside the loop C. To illustrate our approach, in the next section we calculate the electric ﬁeld due to spec iﬁc planar charge conﬁgurations of some representative shapes. III. ILLUSTRATIVE EXAMPLES In this section we illustrate the usefuness the the Biot-Sav art-like law (8) by discussing a class of electrostatic problem that is trivially solved by using (8) or (9), but that would 6require a much more involved solution otherwise. We start by applying equation (8) to calculate the electric ﬁeld due to a thin, ﬂat plate, which ha s the shape of a regular n-sided polygon, inscribed in a circle of radius a(see ﬁgure 3). Consider that the ﬂat plate is located in the x-yplane with its center at the origin, and that it is maintained at a ﬁxed potential V. In the plane z= 0 the region outside the plate is held at zero potential. We w ish to compute the electric ﬁeld at a point Pin the z-axis. As discussed in section 2, by inspecting equation (8) we noti ce that in order to calcu- late the electric ﬁeld due to this ﬂat conﬁguration, we just n eed to take into account the contributions coming from the boundary contour C. Let us analyze this point a little more carefully: the total electric ﬁeld at point Pin the z-axis can be written as the superposition of the ﬁeld due to ntriangles obtained by joining the center of circle to the ver tices of the polygon (ﬁgure 3). If we sum all the contributions from these various triangular paths, we will be left with the integration around the contour C. This happens because the sense of integration along their common sides is opposite for two a djacent triangles, making the contributions from the common sides to cancel. Therefore, a fter performimg the integration around all the triangles, the only nonzero contributions co me from the nstraight edges that deﬁne the plate’s boundary contour C. With these considerations in mind we turn to the ﬁeld calculation itself. When the ﬁeld contributions of the nsides are summed vectorially, the horizontal ( x-y plane) components add to zero. By symmetry, only the vertica l components located along thez-axis will survive. From ﬁgure 3 we verify that the total elec tric ﬁeld at Pis given by E=n Encosα, (10) where Enis the net ﬁeld due to just one of the ntriangles that compose the polygon and cos α=acos (π/n)//radicalBig a2cos2(π/n) +z2. Considering the fact that the only nonzero contribution from such triangular path comes from the singl e edge of the polygon, we employ 7equation (8) to get En=V πasin (π/n)/radicalBig [a2cos2(π/n) +z2] [a2+z2]. (11) Using equations (10) and (11) we obtain the total electric ﬁe ld along the z-axis E=V 2π/braceleftBiggna2sin (2π/n) [a2cos2(π/n) +z2]√ a2+z2/bracerightBigg ˆ z. (12) This result is quite handy since it provides, all at once, the calculation of the electric ﬁeld along the z-axis due to any regular polygon of nsides (equilateral triangle, square, pentag- onal, etc.). We point out that our electric ﬁeld result (12) agrees with the equivalent (but conceptually distinct) formula for the magnetic ﬁeld at the axis of an n-sized polygonal, current carrying circuit, previously obtained in referenc e [13]. It is worth mentioning that result (12) can also be used compu te the magnitude of the electric ﬁeld along the axis passing through the center of a circular conducting plate of radius a. This can be easily obtained by taking the limit n→ ∞ in equation (12) yielding E=V a2 [a2+z2]3/2ˆ z. (13) This limit agrees with reference [14], in which the potentia l calculation is done solely for the circular plate, by employing traditional boundary-value p roblem techniques. We conclude this section by calling the attention of the read er to the fact that the calculation of the electric ﬁeld at observation points lyin g in the plane of ﬂat plates of various shapes can be obtained, with great facility, by dire ctly using our expression (9). For example, the magnitude of the electric ﬁeld in the plane o f elliptical, spiral shaped and harmonically deformed circular plates (see ﬁgure 4), kept a t a ﬁxed potential V, are readily obtained by (9). Table 1 displays the values of the electric ﬁ eld at observation points P, for these three characteristic ﬂat regions. So, despite the bou ndary curve especiﬁc geometry, theEﬁeld calculation may be promptly performed by using our Biot -Savart-like approach. 8In practical terms the diﬃculty of having a complicated boun dary shape is not a very serious obstacle in order to compute Ein closed form, as long as the related integrations are not terribly hard to handle. IV. CONCLUDING REMARKS In this work we show that, under certain circumstances, comp licated electrostatic prob- lems may have straightforward solutions, if solved by a Biot -Savart-like approach. We consider the general situation in which a ﬂat region of arbit rary shape is kept at a ﬁxed potential V, in a otherwise grounded plane. The objective is to compute t he electric ﬁeld due to this nontrivial charge distribution at a given point i n space. At ﬁrst glance, such a calculation looks very involved, mainly because we have no prior knowledge about the precise charge distribution on the planar region. We derive d a Biot-Savart-like law suitable to deal with such electrostatic situation, allowing the cal culation of the electric ﬁeld to be done in a simple fashion. This Biot-Savart-like law is written in terms of a line integ ral along the boundary contour deﬁned by the nonzero potential region, so that the calculat ion does not require information about the charge distribution in the bulk. As a by-product, w e show that for observation points located in the z= 0 plane, the Biot-Savart-like expression can be written in a very simple and compact form, as was the case for similar magnetic ﬁeld calculations recently published in reference [10]. We illustrate our results by ca lculating the electric ﬁeld along the axis of a polygonal ﬂat region of nsides, kept at a constant potential V. In addition, we calculate the magnitude of the electric ﬁeld in the plane o f charged plates presenting nontrivial boundary geometries such as elliptical, spiral and regularly undulating borders. In summary, we show that it is possible to deﬁne a Biot-Savart -like in electrostatics. Our new approach provides an alternative and simple technique t o solve a class of complicated 9boundary value problems in electrostatics. ACKNOWLEDGMENTS This work was supported by CNPq and FINEP (Brazilian Agencie s). 10REFERENCES [1] Halliday D, Resnick R and Walker J 1997 Fundamentals of Physics (New York: John Wiley & Sons) pp 729-732 [2] Griﬃths D J 1989 Introduction to Electrodynamics (New Jersey: Prentice Hall) pp 207- 211 [3] Jackson J D 1975 Classical Electrodynamics (New York: John Wiley & Sons) pp 169-173 [4] Eyges L 1980 The Classical Electromagnetic Field (New York: Dover Publications, Inc.) pp 117-123 [5] Kroener E 1955 Dislocations and the Biot-Savart Law Proc. Phys. Soc. London A 68 53-55 [6] Jackson D P, Godstein R E and Cebers A O 1994 Hydrodynamics of Fingering Insta- bilities in Dipolar Fluids Phys. Rev. E 50298-307 [7] Kessler D A and Levine H 1991 Maximal Dendrite Size in Monolayer Systems Phys. Rev. Lett. 673121-3123 [8] Goldstein R E, Jackson D J and Dorsey A T 1996 Current-loop Model for the Interme- diate State of Type-I Superconductors Phys. Rev. Lett. 763818-3821 [9] Buck G 1998 Most Smooth Closed Space Curves Contain Approximate Soluti ons of the n-Body Problem Nature 39551-53 [10] Miranda J A 2000 Magnetic Field Calculation for Arbitrarily Shaped Planar W iresAm. J. Phys. 68254-258 [11] See reference [3], pp 42-45 and reference [4], pp 82-87 [12] See, for instance, reference [4], p 122 11[13] Chirgwin H B, Plumpton C and Kilmister C W 1972 Elementary Electromagnetic The- ory, Volume 2, (New York: Pergamon Press) p 215, problem 3 [14] See reference [3], p 79, problem 2.3 [15] Gradshteyn I S and Ryzhik I M 1994 Table of Integrals, Series, and Products (New York: Academic Press) pp 907-913 12Figure Captions Figure 1: Two equivalent systems: (a) arbitrarily shaped, planar wir e carryng a steady current I, and (b) a ﬂat charge distribution of the same shape, maintai ned at a ﬁxed potential V, with zero potential in the rest of the plane. In (a) the magne tic ﬁeld Bcan be calculated with the help of the usual Biot-Savart law (equation (1)), wh ile in (b) the calculation of the electric ﬁeld Emay be performed with the help of an electrostatic Biot-Sava rt-like law (equation (8)). Figure 2: Charged ﬂat plate located in the x-yplane, bounded by the curve Cand kept at a ﬁxed potential V. The potential is set to zero in the region of the plane z= 0 outside the curve C. Vector x′locates the source point, while vector xrefers to the ﬁeld point. The observation point is denoted by P. Figure 3: Schematic view of a regular n-sized polygonal plate, kept at a potential V. The plate lies in the x-yplane, while the observation point Pis located along the z-axis. Each individual triangle deﬁnes the angle 2 π/n. The direction of integration around Cis determined by the outward unit-normal ˆ n=−ˆ zvia the right-hand rule. Figure 4: (a) Elliptical plate, presenting major axis 2 aand minor axis 2 b, centered at point P; (b) Harmonically deformed plate (Perturbed Circle) r(θ)=a[1 +ǫcos(nθ)] , centered at P, forn= 5 and ǫ= 0.5; (c) Logarithmic spiral plate. The magnitude of the electr ic ﬁelds at observation point Pdue to these shapes are listed in Table 1. 13TABLES Boundary Shape Parametric Equation Electric Field Magnitude Elliptical r(θ) =ab √ a2sin2θ+b2cos2θE=2V πaG/parenleftbigg/radicalBig 1−a2 b2/parenrightbigg Perturbed Circle r(θ) =a[1 +ǫcos (nθ)] E=V a√ 1−ǫ2 Spiral r(θ) =qepθE=V 2πq/bracketleftBig 1−e−2πp p/bracketrightBig TABLE I. Electric ﬁeld in the plane of ﬂat regions presenting nontrivial boundary shapes. 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arXiv:physics/0011016v1 [physics.data-an] 7 Nov 2000Nonlinear limits to the information capacity of optical ﬁber communications Partha P Mitra Jason B Stark Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07 974 February 2, 2008 The exponential growth in the rate at which information can b e communicated through an optical ﬁber is a key element in the so called infor mation revolution. However, like all exponential growth laws, there are physic al limits to be consid- ered. The nonlinear nature of the propagation of light in opt ical ﬁber has made these limits diﬃcult to elucidate. Here we obtain basic insi ghts into the limits to the information capacity of an optical ﬁber arising from t hese nonlinearities. The key simpliﬁcation lies in relating the nonlinear channe l to a linear channel with multiplicative noise, for which we are able to obtain an alytical results. In fundamental distinction to the linear additive noise case, the capacity does not grow indeﬁnitely with increasing signal power, but has a max imal value. The ideas presented here have broader implications for other nonline ar information channels, such as those involved in sensory transduction in neurobiol ogy. These have been often examined using additive noise linear channel models, and as we show here, nonlinearities can change the picture qualitatively. The classical theory of communications ` a laShannon [1] was developed mostly in the con- text of linear channels with additive noise, which was adequ ate for electromagnetic propagation through wires and cables that have until recently been the ma in conduits for information ﬂow. Fading channels or channels with multiplicative noise have been considered, for example in the context of wireless communications [2], although such c hannels remain theoretically less tractable than the additive noise channels. However, with t he advent of optical ﬁber commu- nications we are faced with a nonlinear propagation channel that poses major challenges to our understanding. The diﬃculty resides in the fact that the input output relationship of an optical ﬁber channel is obtained by integrating a nonlinear partial diﬀerential equation and may not be represented by an instantaneous nonlinearity. Ch annels where the nonlinearities in the input output relationship are not instantaneous are in g eneral ill understood, the optical ﬁber simply being a case of current relevance. The understan ding of such nonlinear channels with memory are of fundamental interest, both because commu nication rates through optical ﬁber are increasing exponentially and we need to know where t he limits are, and also because understanding such channels may give us insight elsewhere, such as into the design principles of neurobiological information channels at the sensory per iphery. The capacity of a communication channel is the maximal rate a t which information may be transferred through the channel without error. The capacit y can be written as a product of two conceptually distinct quantities, the spectral bandwidth Wand the maximal spectral eﬃciency 1which we will denote C. In the classic capacity formula for the additive white Gaus sian noise channel with an average power constraint, C=Wlog(1+ S/N) [1], the spectral bandwidth W, which has dimensions of inverse time, multiplies the dimens ionless maximal spectral eﬃciency C= log(1+ S/N). Here SandNare the signal and noise powers respectively. It is instruct ive to examine this formula in the context of an optical ﬁber. Since the maximal spectral eﬃciency is logarithmic in the signal to noise ratio (SNR), it can never b e too large in a realistic situation, so that the capacity is principally determined by the bandwidt hW. In the case of an optical ﬁber, the intrinsic loss mechanisms of light propagating through silica fundamentally limits Wto a maximum of about 50 THz[3] corresponding to a wavelength range of about 400 nm(1.2−1.6µ). This is to be compared with current systems where the total ba ndwidth is limited to about 15THz. If the channel was linear, the maximal spectral eﬃciency wo uld be C= log(1+ S/N), Sbeing input light intensity and Nthe intensity of ampliﬁed spontaneous emission noise in the system. An output SNR of say 100 (i.e. 20dB), would then yield a spectral eﬃciency of 6.6, which for a 50 THz channel would correspond to a capacity of 330 Tbit/sec . The channel, of course, is not linear; how do the nonlinearities impact the s pectral eﬃciency of the ﬁber? The basic conclusion of the present work is that the impact is sev ere and qualitative. As shown in Fig.1, the eﬀect is a saturation and eventual decline of spec tral eﬃciency as a function of input signal power, in complete contrast with the linear channel c ase. We now proceed to motivate and discuss this result. It is widely recognised that nonlinearities impair the chan nel capacity. However, estimation of the impact of the nonlinearities on channel capacity has r emained ad hoc from an information theory perspective. Here we obtain what appears to be the ﬁrs t systematic estimates (Fig.1) for the maximal spectral eﬃciency of an optical ﬁber channel as a function of the relevant parameters. In basic distinction to the linear channel, our considerations indicate that the maximal spectral eﬃciency does not grow indeﬁnitely with si gnal power, but reaches a maximum of several bits and eventually declines , as illustrated in Figure 1. It is to be noted that current systems use a binary signalling scheme which limits the achi evable spectral eﬃciency a priori to 1 bit, and to reach the higher spectral eﬃciencies predicted by the theory, multi-bit signalling schemes would have to be used. Since the spectral eﬃciencies of current systems are already approaching 1 bit, it is clear that the limits discussed here will be of practical relevance in the future. Although a number of nonlinearities are present in light pro pagation in a ﬁber, we concen- trate on the most important one for ﬁber communications, nam ely the dependence of the refrac- tive index (and therefore the propagation velocity of light ) on the light intensity, n=n0+n2I. This nonlinearity is weak, but its eﬀects accumulate due to t he long propagation distances involved in ﬁbre communications, and is responsible for the eﬀects considered here. Three principle physical parameters characterising the propaga tion are of interest: the group veloc- ity dispersion β∼10ps2/km, the propagation loss α∼0.2dB/km and the strength of the nonlinear refractive index, usually expressed in terms of t he parameter γ∼1/W/km . The propagation loss is compensated by interposing optical amp liﬁers into the system. Each am- pliﬁer also injects spontaneous emission noise into the sys tem with strength I1=aGhν∆ν[5], withGbeing the ampliﬁer gain, hthe Planck’s constant, νand ∆ νbeing the centre frequency and frequency bandwidth of light respectively. Here ‘ a’ is a numerical constant (which we as- sume to be 2). For nsspans of ﬁber interspersed with ampliﬁers that make the tota l channel gain unity, the eﬀects of absorption may be accounted for sim ply by redeﬁning the system length in terms of an eﬀective length, Leff∼ns/α. If the nonlinearity were absent ( γ= 0), we would have obtained, for the maximal spectral eﬃciency, C0= log(1 + I/In),Ibeing the input power and In=nsI1being the total additive noise power. Note that C0declines logarithmically with system length, and would eventually vanish for inﬁnite ly long systems. Note also that although spectral eﬃciency is dimensionless, it is often wr itten for convenience with the “units” bits/sec/Hz. For a variety of reasons, the principal one being limitation s in the electronic bandwidth, it 2is impractical to modulate the full optical bandwidth at onc e. Instead, current attempts towards achieving maximal information throughput involve so calle d Wavelength Division Multiplexing (WDM) [3], where the whole optical bandwidth is broken up int o disjoint frequency bands (“channels”) each of which is modulated separately. We conﬁ ne our attention to such systems (which from an information theory perspective corresponds to the “multi-user” case) [6], though we also comment on the ideal case of utilising the full optica l bandwidth for a single data stream (the “single user” case). Quantitatively, the single user c ase is expected to have larger maximal spectral eﬃciencies, though we will argue that it shows the s ame qualitative behaviour as the multi-user case. The diﬀerence between the two reside in the fact that in the multi-user case, each channel is an independent information stream, and appe ars as an additional source of noise to every other channel due to nonlinear mixing. The nonlinear propagation eﬀects in the evolution of the ele ctric ﬁeld amplitude involve a cubic term in the electric ﬁeld. In a WDM system, the nonline arities are classiﬁed by the ﬁeld amplitudes participating in this cubic term for the evo lution of the ﬁeld amplitude of a given channel: self phase modulation denotes the case wher e all three ﬁelds belong to the same channel, cross phase modulation where two ﬁelds belong to a diﬀerent channel and one to the same channel, and four wave mixing denotes the case whe re all three amplitudes belong to diﬀerent channels. Out of these terms, four wave mixing gi ves rise to additive noise to the channel of interest and will not be considered further in this paper. One reason for this is that four wave mixing is strongly suppressed by dispersio n when the channel spacings are substantial. Its eﬀects can be accounted for by augmenting t he additive noise term in the subsequent considerations. We also neglect self phase modu lation eﬀects, since these eﬀects are deterministic for the given channel and in principle cou ld be reduced by using nonlinear precompensation. Finally, we are left with cross phase modu lation, which appears to be the principle source of nonlinear capacity impairment in the mu ltiuser case for realistic parameter ranges. A further reason for our focus on cross phase modulat ion is that it gives rise to multiplicative noise, which gives rise to qualitatively ne w eﬀects in the channel capacity. We model the propagation channel in the presence of cross pha se modulation by means of alinear Schroedinger equation with a random potential ﬂuctuating both in space and time. This is easily justiﬁed starting from the nonlinear Schroed inger equation description commonly used to describe light propagation in single mode optical ﬁb res [4]. Cross phase modulation arises from terms in the equation where the ﬁeld intensity in the nonlinear refractive index is approximated by the sum of the ﬁeld intensities in the channe ls other than the one for which the propagation is being studied. Therefore, if only cross phas e modulation eﬀects were retained, the propagation equation for the ﬁeld amplitude in channel ithen becomes i∂zEi=β2 2∂2 tEi+V(z, t)Ei, (1) where V(z, t) =−2γ/summationtext j/negationslash=i\|Ej(z, t)\|2, the sum being taken over the other channels. Since independent streams of information are transmitted in the o ther channels, V(z, t) appears as a random noise term. Notice that the nonlinear propagation e quation has now been reduced to a linear Schroedinger equation with a stochastic potenti al, so that the nonlinear channel has become a channel with multiplicative noise. We now need a n adequate model for the stochastic properties of V(z, t). If the dispersion is substantial, we propose that V(z, t) may be approximated by a Gaussian stochastic process short rang e correlated in both space and time. Since V is obtained by adding a large number of diﬀerent channels, each of which is short range correlated in time ( τ∼1/B, where Bis the channel bandwidth), we can expect V to have a correlation time of approximately 1 /B. Dispersion causes the channels to travel at diﬀerent speeds, thus causing Vto be short range correlated in space as well, with a correlat ion length related to the dispersion length. Since Vis a sum of intensities, it has nonzero mean, so we deﬁne δV(z, t) =V(z, t)− ∝angbracketleftV∝angbracketright, where ∝angbracketleftV∝angbracketrightdenotes the average value of V. Removing a 3constant from the potential causes an overall phase shift in dependent of space and time, which is irrelevant to the present considerations. The parameter of interest in the following is the integrated strength of the ﬂuctuating ﬁeld,η=/integraltextdz∝angbracketleftδV(z,0)δV(0,0)∝angbracketright. In order to estimate η, we consider a simpliﬁed propagation model for the channels other than the one of interest, in whic h nonlinearities are neglected, and stochastic bit streams at the inputs to the channels are p ropagated forward with constant group velocities. The group velocity diﬀerence between two channels separated by a spacing ∆λisD∆λ. In this model with ncother channels evenly spaced by ∆ λaround the channel of interest, each with intensity Iand bandwidth B, we obtain η= 2 ln( nc/2)(γI)2/(BD∆λ). Here Dis the dispersion parameter D=−2πcβ/λ2. Although this is a simpliﬁed model for the other channels, numerical simulations of propagation including the nonlinearities and dispersion for the side channels show that the estimate of ηis accurate. Note that the denominator in the expression of ηis the inverse of the dispersion length LDfor the given channel spacing. This form for ηfollows from assuming that Leff>> L D, since in this limit the integral deﬁning ηis cut oﬀ by LD. If on the other hand, Leff≤LD, the integral would be cut oﬀ by Leff, so that one would have to replace LDbyLeffin the equation forη. The ﬂuctuation strength scales with the logarithm of the nu mber of channels rather than the total number since channels at larger spacings are suppr essed proportionately to channel spacing. This suppression due to dispersion leads to the log arithmic factor via a sum of the form/summationtext j1/(∆λj)∝/summationtext j1/j. Within the model under consideration, the propagation down the ﬁber is given in terms of a propagator U(t, t′;L) obtained by integrating the stochastic Schroedinger equa tion. For simplicity, we model the ampliﬁer noise as an additive term w ith strength Inas deﬁned earlier. The channel is speciﬁed in terms of a relation between the inp ut and output electric ﬁeld amplitudes, Eout(t) =/integraltextdt′U(t, t′;L)Ein(t′) +n(t). Since Uis stochastic, due to the underlying stochasticity of V(z, t), the model corresponds to a channel with multiplicative no ise. It is still intractable in terms of an exact capacity computation , but an analytic lower bound may now be obtained. This bound is based on the following informa tion theoretic result (E.Telatar, private communications): the capacity Cof a channel with input Xand output Yrelated by a conditional distribution p(Y\|X) and an input power constraint E(\|\|X\|\|2) =Psatisﬁes the inequalities C=max p(X)I(X, Y)≥I(XG, Y)≥I(XG, YG) Here I(XG, Y) is the mutual information when p(X) is chosen to be pG(X), a Gaussian satisfying the power constraint; I(XG, YG) is the mutual information of a pair ( XG, YG) with the same second moments as the pair ( X, Y). The ﬁrst inequality is trivial since pG(X) is not necessarily the optimal input distribution. A proof of the second inequality is outlined i n the methods section. The quantity I(XG, YG) for the channel deﬁned above may be computed from knowledge of the correlators ∝angbracketleftEin(t)E∗ in(t′)∝angbracketright,∝angbracketleftEout(t)E∗ out(t′)∝angbracketrightand∝angbracketleftEout(t)E∗ in(t′)∝angbracketright. The ﬁrst is deﬁned a priori through the assumption of bandlimited Gaussian white noise input with a power constraint. The second follows from the ﬁrst using the unitarity of U. The third correlator requires computation of the average propagator ∝angbracketleftU∝angbracketright, where the average is over realisations of V(z, t). For a Gaussian, delta-correlated V, we obtain ∝angbracketleftU(t, t′;L)∝angbracketright= exp( −ηL/2)U0(t−t′;L) (see methods), where U0 is the propagator for V= 0. Assembling these results, we ﬁnally obtain an analytic e xpression for a lower bound CLBto the channel capacity of the stochastic Schroedinger equa tion model: CLB=ncBln(1 +e−(I I0)2I In+ (1−e−(I I0)2)I) (2) where I0is given by I0=/radicaltp/radicalvertex/radicalvertex/radicalbtBD∆λ 2γ2ln(nc/2)Leff. (3) 4The fundamental departure from a linear channel in the above capacity expression is the appearance of an intensity scale I0, which governs the onset of nonlinear eﬀects. To obtain an idea about the value of I0, consider the parameter values B= 40GHz,D= 20ps/nm/km , ∆λ= 1nm,γ= 1/W/km ,nc= 100, Leff≈ns/α= 100 km. Then I0= 32mW. Examination of Eq.3 shows that the intensity scale I0at which nonlinearities set in shows reasonable dependence on all relevant parameters, namely it increases with increa ses in the dispersion, the bandwidth and the channel spacing, but decreases with increasing syst em length and number of channels. The most striking feature of Eq.2 is that instead of increasi ng logarithmically with signal intensity like in the linear case, the capacity estimate act ually peaks and then declines beyond a certain input intensity. From Eq.2, it is easily derived th at the maximum value is given approximately by Cmax≈2 3ncBln(2I0/In), the maximum being achieved for an intensity Imax≈ (I2 0In/2)1/3. The reason for this behaviour is that if we consider any part icular channel, the signal in the other channels appear as noise in the channel of interest, due to the nonlinearities. This ‘noise’ power increases with the ‘signal’ strength, th us causing degradation of the capacity at large ‘signal’ strength. The behaviour of Eq.2 is graphic ally illustrated in Fig.1, where the spectral eﬃciency (bits transmitted per second per unit ban dwidth) is shown as a function of input power. It is of interest to note that if the input intensity is kept ﬁx ed, the capacity bound declines exponentially with the system length. This is only to be expe cted, since the correlations of the electric ﬁeld should decay exponentially due to the ﬂuct uating potential in the propagation equation. On the other hand, the maximal spectral eﬃciency g iven by Cmaxdeclines only logarithmically in system length, in parallel with the beha viour for linear channels. It can therefore be inferred that if the input power was adjusted wi th system length instead of being kept ﬁxed, the decline of spectral eﬃciency with system leng th will be logarithmic. Finally, we present qualitative arguments as to why the sing le user case is expected to show the same non-monotonicity of spectral eﬃciency with th e input signal intensities. In the multi-user case, the noise power as eﬀectively generated by cross phase modulation grows as I3since it involves three signal photons. In the single user ca se, the cubic nonlinearity is a deterministic process that does not necessarily degrade ch annel capacity. However, subleading processes which involve two signal and one spontaneous nois e photon still scale superlinearly in signal intensity, as I2In. Therefore, one should still observe the same behaviour of t he eﬀective noise power overwhelming the signal at large signal intensi ties. Thus, we would still expect the spectral eﬃciency to decline at large input intensity, thou gh not as rapidly in the multi-user (WDM) case. Methods Gaussian bound to the channel capacity Proof of the inequality I(XG, Y)≥I(XG, YG): deﬁne p(X, Y) as the product pG(X)p(Y\|X), andpG(X, Y) to be the joint Gaussian distribution having the same secon d moments as p(X, Y). Also deﬁne pG(Y) to be the corresponding marginal of pG(X, Y). I(XG, Y) =/integraldisplay dXdY p (X, Y) log(p(X, Y) pG(X)p(Y)) (4) =/integraldisplay dXdY p (X, Y)[log(pG(X, Y) pG(X)pG(Y))−log(pG(X, Y) p(X, Y)p(Y) pG(Y))] (5) (6) 5Since p(X, Y) and pG(X, Y) share second moments, the ﬁrst term on the RHS is I(XG, YG). The second term may be simpliﬁed using the convexity of the lo garithm, ∝angbracketleftlog(f)∝angbracketright ≤log(∝angbracketleftf∝angbracketright) to obtain I(XG, Y)≥I(XG, YG)−log[/integraldisplay dXdY p G(X, Y)p(Y) pG(Y)] (7) ≥I(XG, YG) (8) The second inequality follows by ﬁrst performing the integr al over X, and noting that log(/integraltextdY p(Y)) = log(1) = 0. Derivation of the average propagator ∝angbracketleftU∝angbracketright: This can be done by resumming the perturbation series exactl y for ∝angbracketleftU∝angbracketright, for delta correlated V(z, t). Alternatively, in the path integral formalism [7], ∝angbracketleftU(t, t′;L)∝angbracketright=U0(t−t′;L)∝angbracketleft∝angbracketleftexp(i/integraldisplayL 0dzV(z, t(z))∝angbracketright∝angbracketright, (9) where the average is taken over Vas well as over paths t(z) satisfying t(0) = t,t(L) = t′. The result in the paper follows by performing the Gaussian a verage over V. Since φ=/integraltextL 0dzV(z, t(z)) is a linear combination of Gaussian variables, it is also G aussian distributed and satisﬁes ∝angbracketleftexp(iφ)∝angbracketright= exp( −∝angbracketleftφ2∝angbracketright/2). The result follows by noting that for delta correlated V,∝angbracketleftφ2∝angbracketrightis a constant given by ηL. The delta correlations need to be treated carefully, this c an be done by smearing the delta functions slightly and leads to the deﬁnition of ηgiven earlier in the paper. References [1] Shannon, C. E. A mathematical theory of communications. ,Bell Syst. Tech. J. ,27, p. 379- 423, p. 623-656 (1978). [2] Biglieri E., Proakis, J. & Shamai S., Fading channels: In formation-theoretic and commu- nications aspects., Information Theory Transactions 44:6 p. 2619-2692 (1998). [3] Glass, A.M. et al., Advances in Fiber Optics. Bell Labs Technical Journal 5, p. 168 (2000). [4] Agrawal, G. P., Nonlinear Fiber Optics , Academic Press, Inc., San Diego, 1995. [5] Agrawal, G. P., Fiber-Optic Communication Systems , John Wiley & Sons, Inc., New York, 1992, pp. 334. [6] Cover, T. M. & Thomas, J. A. Information Theory , John Wiley & Sons, Inc., New York, 1991. [7] Feynman, R. P. & Hibbs, R. A., Quantum Mechanics and Path Integrals , McGraw-Hill, New York, 1965. 6Acknowledgements We gratefully acknowledge discussions with E. Telatar, R. S lusher, A. Chraplyvy, G. Foschini and other members of the ﬁber capacity modelling group at Bel l Laboratories. We would also like to thank D. R. Hamann and R. Slusher for careful readings of the manuscript. Figure Captions Figure 1. The curves in Fig.1 represent lower bounds to the spectral eﬃ ciency for a homoge- neous length of ﬁber for a multi-user WDM system, given analy tically by Eq.2. Although the curves represent lower bounds, we argue in the text that t he true capacity shows the same qualitative non-monotonic behaviour with respect to input signal powers. The spectral eﬃciencies displayed in the ﬁgure correspond to th e capacity per unit bandwidth, C=C/(nδν). Here δνincludes both the channel bandwidths and the inter-channel spac- ing. The parameters used for the ﬁgure are nc= 100, Leff= 100 km,D= 20ps/nm/km , δν= 1.5Bwhere B= 10GHz is the individual channel width. The two continuous curves correspond to γ= 1/W/km andγ= 0.1/W/km , the lower curve corresponding to γ= 1. The spontaneous noise strength Inis computed from the formula In=aGhνB as explained in the text, with a= 2,G= 1000, ν= 200 THz. The dotted curve represents the spectral eﬃciencies of the corresponding linear channe ls given by γ= 0. 710−310−210−1100012345 Input power density (mW/GHz)Spectral efficiency (bits/sec/Hz)
arXiv:physics/0011017v1 [physics.flu-dyn] 8 Nov 2000FluidParticleAccelerationsinFullyDevelopedTurbulenc e A. LaPorta, Greg A.Voth,AliceM. Crawford, JimAlexander, a nd Eberhard Bodenschatz LaboratoryofAtomicand SolidStatePhysics,Laboratoryof NuclearStudies CornellUniversity,Ithaca, NY 14853-2501 November5,2000 The motion of ﬂuid particles as they are pushed along erratic trajectories by ﬂuctuating pressure gra- dients is fundamental to transport and mixing in tur- bulence. It is essential in cloud formation and atmo- spheric transport[1, 2], processes in stirred chemical reactorsand combustion systems[3],and in the indus- trial production of nanoparticles[4]. The perspective of particle trajectories has been used successfully to describe mixing andtransportin turbulence[3, 5],but issues of fundamental importance remain unresolved. One such issue is the Heisenberg-Yaglom prediction of ﬂuid particle accelerations[6, 7], based on the 1941 scaling theory of Kolmogorov[8, 9] (K41). Here we report acceleration measurements using a detector adapted from high-energy physics to track particles in a laboratorywaterﬂow at Reynoldsnumbers up to 63,000. WeﬁndthatuniversalK41scalingoftheaccel- erationvarianceisattainedathighReynoldsnumbers. Ourdatashowstrongintermittency—particlesareob- served with accelerations of up to 1,500 times the ac- celeration of gravity (40 times the root mean square value). Finally,we ﬁnd thataccelerationsmanifestthe anisotropyofthelargescaleﬂowatallReynoldsnum- bersstudied. In principle, ﬂuid particle trajectories are easily mea- sured by seeding a turbulent ﬂow with minute tracer par- ticles and following their motions with an imaging sys- tem. In practice this can be a very challengingtask since we must fully resolve particle motions which take place on times scales of the order of the Kolmogorov time, τη= (ν/ǫ)1/2where νis the kinematic viscosity and ǫ is the turbulentenergydissipation. This is exempliﬁedin Fig. 1, which shows a measured three-dimensional, timeresolved trajectory of a tracer particle undergoing vio- lent accelerations in our turbulent water ﬂow, for which τη= 0.3 ms. Theparticle entersthe detectionvolumeon the upperright,is pushedtothe left bya burstof acceler- ation and comesnearly to a stop beforebeing rapidly ac- celerated (1200timesthe accelerationof gravity)upward in acork-screwmotion. Thistrajectoryillustratesthedif - ﬁculty in following tracer particles—a particle’s acceler - ationcan gofromzeroto30 timesits rmsvalueandback to zero in fractions of a millisecond and within distances ofhundredsofmicrometers. Conventional detector technologies are effective for low Reynolds number ﬂows[10, 11], but do not provide adequate temporal resolution at high Reynolds numbers. However, the requirements are met by the use of silicon strip detectors as optical imaging elements in a particle tracking system. The strip detectors employed in our ex- periment(SeeFig.2a)weredevelopedtomeasureparticle tracks in the vertex detector of the CLEO III experiment operating at the Cornell Electron Positron Collider[12]. When applied to particle tracking in turbulence (See Fig. 2b) each detector measures a one-dimensional pro- jection of the image of the tracer particles. Using a data acquisition system designed for the turbulence experi- ment, several detectorscan be simultaneouslyread out at upto 70,000framespersecond. The acceleration of a ﬂuid particle, a+, in a turbulent ﬂow isgivenbythe Navier-Stokesequations, a+=−/vector∇p ρ+ν∇2u (1) where pis thepressure, ρistheﬂuiddensity,and uisthe velocity ﬁeld. In fully developed turbulence the viscousdamping term is small comparedto the pressure gradient term[13, 14] and therefore the acceleration is closely re- latedtothe pressuregradient. Ourmeasurementofthedistributionofaccelerationsis shown in Figure 3, where the probability density func- tion of a normalized acceleration component is plotted at three Reynolds numbers. All of the distributions have a stretched exponential shape, in which the tails extend much furtherthan they would for a Gaussian distribution with the same variance. This indicates that accelerations manytimesthermsvaluearenotasrareasonemightex- pect,i.e., the acceleration is extremely intermittent. The acceleration ﬂatness, shown in the inset to Fig. 3, char- acterizesthe intermittencyof the acceleration,and would be 3 for a Gaussian distribution. These ﬂatness values are consistent with direct numerical simulation (DNS) at low Reynolds number[14] and exceed 60 at the highest Reynoldsnumbers. ThepredictionbyHeisenbergandYaglomforthevari- ance of an acceleration component based on K41 theory is /angb∇acketleftaiaj/angb∇acket∇ight=a0ǫ3/2ν−1/2δij, (2) where a0isauniversalconstantwhichisapproximately1 in a model assuming Gaussian ﬂuctuations[6, 7, 15, 13]. However, DNS has found that a0depends on ǫ. Con- ventionally this is expressed in terms of the Taylor mi- croscale Reynolds number, Rλ, which is related to the conventional Reynolds number by Rλ= (15Re)1/2and is proportional to ǫ1/6. Using this notation, DNS results indicate a0∼R1/2 λforRλ<250[14],withatendencyto leveloffas Rλapproaches470[16]. Our measurement of the Kolmogorov constant a0is shown in Fig. 4 for eight orders of magnitude of scal- ing in accelerationvariance. We ﬁnd a0to be anisotropic andto dependsigniﬁcantlyonthe Reynoldsnumber. The a0values for both components increase as a function of Reynoldsnumberup to Rλ≈500, abovewhich they are approximatelyconstant. Thetrendin a0isconsistentwith DNS results in the range 140≤Rλ≤470[14, 17, 16]. However,theconstantvalueof a0athighReynoldsnum- ber suggests that K41 scaling becomes valid at higher Reynolds numbers. Weak deviations from the K41 scal- ingsuchas the a0∼R0.135 λpredictionofthe multifractal modelbyBorgas[18]cannotberuledoutbyourmeasure- ments.The acceleration variance is larger for the transverse component than for the axial component at all values of the Reynoldsnumber. Thisis shown in the inset to Fig. 4 where the ratio of the Kolmogorov constants for the ax- ial and transverse acceleration components is plotted as a function of Reynolds number. The anisotropy is large at low Reynolds number and diminishes to a small value atRλ= 970. This observation tends to conﬁrm recent experimental results which indicate that anisotropy may persisttomuchhigherReynoldsnumbersthanpreviously believed[19, 20]. In summary, our measurements indicate that the Heisenberg-Yaglom scaling of acceleration variance is observed for 500≤Rλ≤970. At lower Reynolds number,ourmeasurementsareconsistentwiththeanoma- lousscalingobservedinDNS[14,16]. Ourmeasurements show that the anisotropy of the large scales affects the acceleration components even at Rλ≈1000. It is im- possible to say on the basis of these measurements if the anisotropy will persist as the Reynolds number ap- proaches inﬁnity. We found the acceleration distribution tobeveryintermittent,withextremelylargeacceleration s often arising in vortical structures such as the one shown in Fig.1. Ourresultshaveimmediateapplicationforthedevelop- mentofLagrangianstochasticmodels,someofwhichuse a0directly as a model constant. These models are being developed and used to efﬁciently simulate mixing, par- ticulate transport, and combustion in practical ﬂows with varying Reynolds numbers[3, 21, 22]. Our research also has surprisingimplicationsforeverydayphenomena. For instance, a mosquito ﬂying on a windy day (wind speed 18 km/h and an altitude of 1 meter) would experiencean rmsaccelerationof 15 m/s2. But giventhe extremelyin- termittent nature of the acceleration, our mosquito could expectto experienceaccelerationsof 150 m /s2(15times the acceleration of gravity) every 15 seconds. This may explain why, under windy conditions, a mosquito would prefer to cling to a blade of grass rather than take part in therollercoasterridethroughtheEarth’sturbulentbound - arylayer[23]. 2Acknowledgments This research is supported by the Physics Division of the National Science Foundation. We thank Reginald Hill, Mark Nelkin, Stephen B. Pope, Eric Siggia, and Zell- man Warhaft for stimulating discussions and suggestions throughout the project. We also thank Curt Ward, who assisted in the initial development of the strip detector. EBand ALP aregratefulforsupportfromthe Instituteof TheoreticalPhysicsat the Universityof California, Santa Barbara,wherepartsofthemanuscriptwerewritten. References [1] Vaillancourt, P. A. and Yau, M. K. Review of particle-turbulence interactions and consequences for cloud physics. B. Am. Meteorol. Soc. 81, 285– 298(2000). [2] Weil,J.C.,Sykes,R.I.,andVenkatram,A. Evaluat- ingair-qualitymodels: Reviewandoutlook. J.Appl. Meteorol. 31,1121–1145(1992). [3] Pope, S. B. Lagrangian PDF methods for turbulent ﬂows.Annu.Rev.FluidMech. 26,23–63(1994). [4] Pratsinis,SotirisE.,andSrinivas,V. Particleforma- tioningases,areview. PowderTechnol. 88,267–273 (1996). [5] Shraiman,B. I. andSiggia,E. D. Scalar turbulence. Nature405,639–646(2000). [6] Heisenberg, W. Zur statistichen theorie der turbu- lenz.Zschrf. Phys. 124,628–657(1948). [7] Yaglom, A. M. On the acceleration ﬁeld in a turbu- lentﬂow. C. R.Akad.URSS 67,795–798(1949). [8] Kolmogorov, A. N. The local structure of turbu- lence in incompressible viscous ﬂuid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305(1941). [9] Kolmogorov,A. N. Dissipation of energyin the lo- cally isotropic turbulence. Dokl. Akad. Nauk SSSR 31,538–540(1941).[10] Virant, M. and Dracos, T. 3D PTV and its applica- tiononLagrangianmotion. Meas. Sci.andTechnol. 8,1539–1552(1997). [11] Ott, S. and Mann, J. An experimental investiga- tion of relative diffusion of particle pairs in three- dimensional turbulent ﬂow. J. Fluid Mech. 422, 207–223(2000). [12] Skubic,P.,et.al., TheCLEOIIIsilicontracker. Nucl Instrum.Meth.A 418,40–51(1998). [13] Batchelor, G. K. Pressure ﬂuctuations in isotropic turbulence. Proc. Cambridge Philos. Soc. 47, 359– 374(1951). [14] Vedula,P.andYeung,P.K. Similarityscalingofac- celeration and pressure statistics in numerical sim- ulations of isotropic turbulence. Phys. Fluids 11, 1208–1220,(1999). [15] Obukhov,A.M.andYaglom,A.M. Themicrostruc- ture of turbulent ﬂow. Frikl. Mat. Mekh. 15(3) (1951). translated in National Advisory Commit- teeforAeronautics(NACA),TM1350,Washington, DC(1953). [16] Gotoh, T. and Fukayama, D. Pressure spectrum in homogeneous turbulence. Submitted to Phys. Rev. Lett.(2000). [17] Gotoh,T.andRogallo,R.S. Intermittancyandscal- ingofpressureatsmallscalesinforcedisotropictur- bulence.J. FluidMech. 396,257–285(1999). [18] Borgas,M.S. ThemultifractalLagrangiannatureof turbulence. Phil. Trans. R. Soc. Lond.A 342(1665), 379–411,(1993). [19] Kurien,S. and Sreenivasan,K. R. Anisotropicscal- ingcontributionstohigh-orderstructurefunctionsin high-Reynolds-numberturbulence. Phys.Rev.E 62, 2206–2212(2000). [20] Shen, X. and Warhaft, Z. The anisotropy of the smale scale structure in high reynolds number (Rλ∼1000)turbulentshear ﬂow. Phys. Fluids 12, 2976–2989(2000). 3[21] Reynolds, A. M. A second-order Lagrangian stochastic modelfor particletrajectoriesin inhomo- geneous turbulence. Q. J. R. Meterorol. Soc. 125, 1735–1746(1999). [22] Sawford, B. L. and Yeung, P. K. Eulerian accelera- tionstatisticsasadiscriminatorbetweenLagrangian stochastic modelsin uniformshear ﬂow. Phys. Flu- ids12,2033–2045(2000). [23] Bidlingmayer, W. L., Day, J. F., and Evans, D. G. Effect of wind velocity on suction trap catches of someFloridamosquitos. J.Am.MosquitoContr. 11, 295–301(1995). [24] Voth, G. A., Satyanarayan, K., and Bodenschatz, E. Lagrangian acceleration measurements at large Reynolds numbers. Phys. Fluids 10, 2268–2280 (1998). This paper reports a constant value of a0at veryhighReynoldsnumber.However,ournewmea- surements indicate that the sensor used in this ex- periment failed to resolve the ﬁnest time and length scalesoftheturbulencebecauseofhighnoiselevels. The correct scaling was obtained,but the numerical values of the acceleration variance and dissipation wereinaccurate. [25] La Porta, A., Voth, G. A., Moisy, F., and Boden- schatz, E. Using cavitation to measure statistics of low-pressure events in large-Reynolds-number tur- bulence.Phys.Fluids 12,1485–1496(2000). [26] Sreenivasan, K. R. On the universality of the Kolmogorov constant. Phys. Fluids 7, 2778–2784 (1995). 4z 1mm x y (m/s )2acceleration scale 0 12,000Figure 1: M EASURED PARTICLE TRAJECTORY The 3- dimensional time-resolved trajectory of a 46 micrometer diameter particle in a turbulent water ﬂow at Reynolds number 63,000 ( Rλ= 970). A sphere marks the mea- sured position of the particle in each of 300 frames taken every 0.014 ms ( ≈τη/20). The shading indicates the acceleration magnitude, with the maximum value of 12,000m/ s2correspondingto approximately30 standard deviations. The turbulence is generated between coaxial counter-rotatingdisks[24,25]inaclosedﬂowchamberof volume 0.1 m3with rotation rates ranging from 0.15 Hz to 7.0 Hz, giving rms velocity ﬂuctuation ˜uin the range 0.018m/s<˜u <0.87m/s. Measurements are made in an 8mm3volume at the center of the apparatus where the mean velocity is zero and the ﬂow is nearly homo- geneous but not isotropic. As a result of a mean stretch- ing of the ﬂow along the propeller axis the rms ﬂuctu- ations are 1/3 larger for the transverse velocity compo- nents than for the axial component. The energy dissipa- tionwasdeterminedfrommeasurementsofthetransverse second order structure function and the Kolmogorov re- lation DNN=4 3C1(ǫr)2/3withC1= 2.13[26]. The dissipation was found to be related to the rms velocity ﬂuctuation by ǫ= ˜u3/Lwith an energy injection scale L= (71 ±7) mm. UsingthedeﬁnitionoftheTaylormi- croscale Reynoldsnumber Rλ= (15˜uL/ν)1/2the range of Reynoldsnumbersaccessible is 140≤Rλ≤970, (in terms of the classical Reynolds number 1300 ≤Re≤ 63,000). At the highest Reynolds number the system is characterizedby Kolmogorovdistanceand time scales of η= 18µmandτη= 0.3 ms,respectively. 5Current(a) L1'illumination L2L3L3' L1X det.Z det.(b)Light Spots Figure2: A PPARATUS (a)Schematicrepresentationofthe CLEO III strip detector[12], in which grey bars indicate sense strips which collect chargecarriersfreedbyoptical radiation. The 511 strips allow measurement of the one dimensional projection of the light striking the detector. The detector may be read out 70,000 times per second. (b) A combination of lenses (L1, L2, L3, L3’) is used to image the active volume onto a pair of strip detectors which are oriented to measure the xandycoordinates. Anotherdetectorassemblymaybeplacedontheopposite port(L1’)to measure yandz. Theﬂow isilluminatedby a6Wargonionlaserbeamorientedat 45◦withrespectto the two viewports. The optics image (46±7)µm diam- etertransparentpolystyrenesphereswhichhavea density of 1.06 g/cm3. Particle positions are measured with ac- curacy0.1strips, correspondingto 0.7 µm intheﬂow.-20 0 2010 0 10-1 10-2 10-3 10-4 10-5 10-60 500 1000050100probability a /<a >R = 200λ λR = 970λR = 690 2 1/2 Figure 3: A CCELERATION DISTRIBUTION . Prob- ability density functions of the transverse accelera- tion normalized by its standard deviation at different Reynolds numbers. The acceleration is measured from parabolic ﬁts over 0.75τηsegments of each trajec- tory. The solid line is a parameterization of the high- est Reynolds number data using the function P(a) = Cexp/parenleftbig −a2//parenleftbig (1 +\|aβ/σ\|γ)σ2/parenrightbig/parenrightbig , with β= 0.539,γ= 1.588,σ= 0.508andthedashedlineisaGaussiandistri- bution with the same variance. The inset shows the ﬂat- ness of the acceleration distribution, ( /angb∇acketlefta4/angb∇acket∇ight//angb∇acketlefta2/angb∇acket∇ight2, evalu- atedusing 0.5τηparabolicﬁts) asa functionof Rλ. 6a0 Rλ0 500 10000246 (a )0y (a )0x/ 0 500 10001.01.21.41.6 Rλ Figure 4: a0AS A FUNCTION OF Rλ. Open red circles indicatea transversecomponentandopenredsquaresthe axial component of the acceleration variance. DNS data isrepresentedbybluetriangles[14]andgreencircles[16] . The error bars represent random and systematic errors in the measurement of the acceleration variance. There is an additionaluncertaintyof 15% in the overall scaling of the vertical axis for the experimental data due to the un- certainty in the measuredvalue of the energydissipation. The degree to which the 45 µm diameter tracer particles follow the ﬂow was investigated by measuring the accel- erationvarianceasa functionof particlesize anddensity. The results, to be published elsewhere, conﬁrm that the accelerationvarianceofthe 45µmparticlesiswithinafew percentofthezeroparticlesizelimit. Theinsetshowsthe ratioofthe a0valuesfortransverseandaxialcomponents oftheacceleration. 7
arXiv:physics/0011018v1 [physics.flu-dyn] 8 Nov 2000Pressure determinations for incompressible ﬂuids and magnetoﬂuids Brian T. Kress and David C. Montgomery Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755-3528, USA To appear in Journal of Plasma Physics Abstract Certain unresolved ambiguities surround pressure determi nations for incompressible ﬂows, both Navier-Stokes and magnetohydro dynamic. For uniform-density ﬂuids with standard Newtonian viscous ter ms, taking the divergence of the equation of motion leaves a Poisson equati on for the pressure to be solved. But Poisson equations require bounda ry condi- tions. For the case of rectangular periodic boundary condit ions, pressures determined in this way are unambiguous. But in the presence o f “no-slip” rigid walls, the equation of motion can be used to infer both D irichlet and Neumann boundary conditions on the pressure P, and thus amounts to an over-determination. This has occasionally been recogni zed as a prob- lem, and numerical treatments of wall-bounded shear ﬂows us ually have built in some relatively ad hoc dynamical recipe for dealing with it, often one which appears to “work” satisfactorily. Here we conside r a class of solenoidal velocity ﬁelds which vanish at no-slip walls, ha ve all spatial derivatives, but are simple enough that explicit analytica l solutions for Pcan be given. Satisfying the two boundary conditions separa tely gives two pressures, a “Neumann pressure” and a “Dirichlet pressu re” which diﬀer non-trivially at the initial instant, even before any dynamics are implemented. We compare the two pressures, and ﬁnd that in pa rticu- lar, they lead to diﬀerent volume forces near the walls. This suggests a reconsideration of no-slip boundary conditions, in which t he vanishing of the tangential velocity at a no-slip wall is replaced by a loc al wall-friction term in the equation of motion. 1 Introduction It has long been the case that pressure determinations for in compressible ﬂows, both Navier-Stokes and magnetohydrodynamic (MHD), are kno wn to be highly 1non-local. Taking the divergence of the equation of motion ∂v ∂t+v· ∇v=j×B ρc− ∇P+ν∇2v, (1) and using ∇ ·v= 0 leaves us with a Poisson equation for the pressure P, which is said to function as an equation of state: ∇2P=−∇ ·(v· ∇v−j×B ρc) (2) Here,v=v(x,t) is the ﬂuid velocity ﬁeld as a function of position and time, Bis the magnetic ﬁeld, j=c∇×B/4πis the electric current density, cis the speed of light,νis the kinematic viscosity, assumed spatially uniform and c onstant, and Pis the pressure normalized to ρthe mass density, also spatially uniform. (1) and (2) are written for MHD. Their Navier-Stokes equivalent s can be obtained simply by dropping the terms containing Bandj. If we are to solve (2) for P, boundary conditions are required. In the im- mediate neighborhood of a stationary “no-slip” boundary, b oth the terms on the left of (1) vanish and we are left with the following equat ion for ∇Pas a boundary condition: ∇P=ν∇2v+j×B ρc(3) We now focus on the Navier-Stokes case, where the magnetic te rms dis- appear from (3), for simplicity. All the complications of MH D are illustrated by this simpler case. It is apparent that (3) must apply to all components of ∇P, and that while the normal component of ∇Pis enough to determine P through Neumann boundary conditions, the tangential compo nents of (3) at the wall equally well determine Pthrough Dirichlet boundary conditions. This is a problem which some inventive procedures have been propo sed to resolve, usually by some degree of “pre-processing” or various dynam ical recipes which seem to lead to approximately no-slip velocity ﬁelds after a few time steps (e.g. [Gresho 1991], [Roache 1982] and [Canuto et al. 1988] ). It is not our purpose to review or critique these recipes, but rather to focus on a s et of velocity ﬁelds, related to Chandrasekhar-Reid functions [Chandrasekhar 1 961], for which (2) is explicitly soluble at a level where the Neumann or Dirichl et conditions can be exactly implemented. In §2, we explore the diﬀerence between the two pres- sures so arrived at. Then in §3, we propose a replacement for the long-standing practice of demanding that all components of a solenoidal vvanish at material walls, in favor of a replacement by a wall friction term for wh ich the above mathematical diﬃculty is no longer present. Of course, simi lar statements and options will apply to all comparable incompressible MHD pro blems. 2 Pressure determinations Restricting attention at present to the Navier-Stokes case , we consider two- dimen- sional, solenoidal, velocity ﬁelds obtained from th e following stream 2function: ψ(x,y) =Ckλcos(kx)[cos(λy) +Akλcosh(ky)] (4) The hyperbolic cosine term in (4) contributes a potential ﬂo w velocity compo- nent to vwhich makes it possible to demand that vobey two boundary con- ditions: the vanishing of both components at rigid walls [Ch andrasekhar 1961]. The function in (4) is even in xandy, but can obviously be converted into an odd or mixed one by the appropriate trigonometric substitut ions. The velocity ﬁeld v=∇ψ×ˆezhas onlyxandycomponents and is periodic inx, with an arbitrary wavenumber k.Ckλis a normalizing constant, and the constantsλandAkλcan systematically be found numerically to any desired accuracy so that both components of vvanish at symmetrically placed no-slip walls aty=aandy=−a. In fact, for given k, an inﬁnite sequence of such pairs ofλandAkλcan be determined straightforwardly. Thus any such v, or superposition thereof, is not only solenoidal, but has bo th components zero aty=±a, and all spatial derivatives exist. Moreover, the “source” term, or ∇ ·(v· ∇v), from the right hand side of (2), is of a relatively simple na ture for such a v, since every term in it can be written as a product of exponent ials of kx,λyandky. It is straightforward to ﬁnd an inhomogeneous solution for P, which then is the same for all boundary conditions for a given vof the form stated. To this inhomogeneous part of Pmust be added a solution of Laplace’s equation. This can be chosen so that the total Pmay satisfy either the normal component of (3) at the walls, or the tangential component of it, but not both. The determination involves only simple but tedious algebra . We illustrate, in ﬁgure 1, an arrow plot of the velocity ﬁeld g iven by choosing k=π/2,λ= 2.6424 andAkλ=.3499, in units of a= 1. The two pressures re- sulting from the satisfaction of the normal and tangential c omponents of (3) can best be compared by comparing their respective values of ∇P, sincePitself is indeterminate up to an additive constant in both cases. In ﬁg ure 2, we display, as an arrow plot, the diﬀerence between the pressure gradients associated with the velocity ﬁeld shown in ﬁgure 1. We have rewritten (1)-(3) in d imensionless units for this purpose, with the kinematic viscosity being replac ed by the reciprocal of a Reynolds number, which may be deﬁned as Re= (/angb∇acketleftv2/angb∇acket∇ight/(k2+λ2))1/2/ν. Here, the angle brackets refer to the mean of v2taken over the 2-D box, containing one period in the xdirection and from y=−atoy=a. The value of Reused to construct ﬁgure 2 is Re= 2293, with the dimensionless version of Ckλ= 5000 in (4). The two pressures are similar but not identical. In ﬁgure 3, a fractional measure of the diﬀerence between the “Neumann pressure”PNand the “Dirichlet pressure” PDis exhibited as a contour plot of the scalar ratio (∇PD− ∇PN)2 /angb∇acketleft(∇PN)2/angb∇acket∇ight(5) There is no absolute signiﬁcance to the numerical value of th is ratio. It initially increases with Reapproaching a maximum of about 2% near the wall for Re∼> 10. It is considered interesting however, that the fraction al diﬀerence is nearly x-independent where it is largest. That occurs formally bec ause the algebra 3−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1−0.8−0.6−0.4−0.2024681 XY Figure 1: Velocity ﬁeld: v=∇ψ×ˆezusingψfrom (4) with k=π/2,λ= 2.6424 andAkλ=.3499 reveals it to be dominated by a term which varies as cosh (4 ky)−cos(4kx) in a region where ky∼>1. It is amusing but perhaps not signiﬁcant to superpose the vel ocity ﬁeld from (4) with a parabolic plane Poiseuille ﬂow of a larger amplitu de. The resulting ﬂow ﬁeld is shown in ﬁgure 4, and it bears a striking but perhap s not signiﬁcant similarity to the ﬂow patterns seen in two-dimensional plan e Poiseuille ﬂow [Jones & Montgomery 1994] when linear stability thresholds are approached. The pressure gradient diﬀerence for this case will be fracti onally smaller than in ﬁgure 3, since pure parabolic plane Poiseuille ﬂow is a rar e case where the two pressures happen to agree, and it quantitatively domina tes the pressures determined from equation (2) in this example. 3 Discussion and a possible modiﬁcation An alternative to the no-slip condition is the “Navier” boun dary condition [Lamb 1932]: the slip velocity at the wall surface is taken to be proportional to the rate of shear at the wall. This may be expressed ∆ V=Ls˙γwhere ∆Vis the slip velocity of the ﬂuid at the wall, ˙ γis the rate of shear at the wall and Lsis a 4−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1−0.8−0.6−0.4−0.2024681grad PN − grad PD XY Figure 2: The diﬀerence between pressure gradients: ∇PN− ∇PDwithRe= 2293 constant with the dimensions length. Molecular dynamic sim ulations of Newto- nian liquids under shear [Thompson & Robbins 1990] have show n this to be the case under some circumstances. In fact recent work [Thompso n & Troian 1997] has shown that, in cases where the shear rate is large, there i s a nonlinear relationship between Lsand ˙γ. We note that the velocity ﬁeld shown in ﬁgure 1 does not lead to one which obeys the Navier boundary condition, after an initial time s tep, where the ﬂuid has been allowed to slip at the wall. If the velocity ﬁeld dete rmined by (4) is advanced in time using (1) with the “Neumann pressure”, the p roportionality between the slip velocity and the rate of shear at the wall, af ter the initial time step, varies sinusoidally with x. It is to be stressed that we are concerned here only with initi al conditions, not with circumstances under which initial slip velocities might be coaxed dy- namically into vanishing after some time. It is diﬃcult to see in what sense the velocity ﬁeld obtained f rom (4) might be an unacceptable one from the point of view of the Navier-St okes or MHD descriptions. It seems to have all the properties that are th ought to be rele- vant. The family of functions of the same x-periodicity in (4 ) can be shown to be orthogonal, and is a candidate for a complete set, in whi ch any vmight 5−1.6−1.2−0.8−0.4 00.40.81.21.6 2−0.8−0.6−0.4−0.200.20.40.60.81( grad PD − grad PN )2 / ( mean square grad PN ) 0.001 0.0010.005 0.0050.009 0.0090.013 0.0130.017 0.0170.021 0.021 XY Figure 3: Normalized mean square pressure gradient diﬀeren ce: ( ∇PD− ∇PN)2//angb∇acketleft(∇PN)2/angb∇acket∇ightwithRe= 2293. Note that the fractional diﬀerence between the two values of ∇Pis signiﬁcant only near the wall. be expanded, when supplemented by ﬂux-bearing functions of y alone. The mathematical question of which if any velocity ﬁelds, which are both solenoidal and vanish at the wall, would lead to Neumann and Dirichlet pr essures that were in agreement with each other, must remain open. Indeed, the question of whether there are any, without some degree of “pre-processi ng,” must remain open. This is an unsatisfactory situation for ﬂuid mechanic s and MHD, in our opinion, even if it is a not unfamiliar one. The search for alt ernatives seems mandatory. One alternative that may be explored is one that seemed some t ime ago, in a rather diﬀerent context [Shan & Montgomery 1994a,b], to ha ve worked well enough for MHD. Namely, we may think of replacing the require ment of the vanishing of the tangential velocity at a rigid wall with a wa ll friction term, added to the right hand side of (1), of the form −v τ(x)(6) where the coeﬃcient 1 /τ(x) vanishes in the interior of the ﬂuid and rises sharply to a large positive value near the wall. The region over which it is allowed 6−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1−0.8−0.6−0.4−0.2024681 XY Figure 4: Velocity ﬁeld shown in ﬁgure 1 plus parabolic plane Poiseuille ﬂow: v=∇ψ×ˆez+ 2(y2−1)ˆexwithCkλ= 1 to rise should be smaller than the characteristic thickness of any boundary layer that it might be intended to resolve, but seems otherwi se not particularly restrictive. Such a term provides a mechanism for momentum l oss to the wall and constrains the tangential velocity to small values, but does not force it to zero. The Dirichlet boundary condition disappears in favor of a relation that permits the time evolution of the tangential components of v, while demanding thatPbe determined solely by the Neumann condition (the normal co mponent of (3) only). In a previous MHD application [Shan & Montgomer y 1994a,b] dealing with rotating MHD ﬂuids, the scheme seemed to perfor m acceptably well, but was not intensively tested or benchmarked sharply against any of the better understood Navier-Stokes ﬂows. This comparison see ms worthy of future attention. The work of one of us (D.C.M.) was supported by hospitality in the Fluid Dynamics Laboratory at the Eindhoven University of Technol ogy in the Nether- lands. A preliminary account of this work was presented oral ly at a meeting of the American Physical Society [Kress & Montgomery 1999]. 7References [Batchelor 1967] Batchelor, G. K. 1967An Introduction to Fluid Mechanics. Cambridge Univ. Press. [Canuto et al. 1988]Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 1988Spectral Methods in Fluid Mechanics. Springer. [Chandrasekhar 1961] Chandrasekhar, S. 1961Hydrodynamic and Hydro- magnetic Stability. Oxford Univ. Press, p.634 [Gresho 1991] Gresho, P. M. 1991 Incompressible ﬂuid dynamics: some fun- damental formulation issues. Annu. Rev. Fluid Mech. 23, 413–453. [Kress & Montgomery 1999] Kress, B.T. & Montgomery D.C. 1999 Bull. Am. Phys. Soc. 44, No.8, p.85. [Lamb 1932] Lamb, H. 1967Hydrodynamics. Dover, NY, p.576 [Roache 1982] Roache, P. J. 1982Computational Fluid Dynamics. Hermosa Publishers. [Jones & Montgomery 1994] Jones, W. B. & Montgomery, D. C. 1994 Fi- nite amplitude steady states of high Reynolds number 2-D cha nnel ﬂow. Phys- ica D73, 227–243. [Shan & Montgomery 1994a] Shan, X. & Montgomery, D.C. 1994a Magne- tohydrodynamic stabilization through rotation. Phys. Rev. Letters 73, 1624– 1627. [Shan & Montgomery 1994b] Shan, X. & Montgomery, D.C. 1994b Rotat- ing magnetohydrodynamics. J. Plasma Physics 52, 113–128. [Thompson & Robbins 1990] Thompson, P.A. & Robbins, M.O. 1990 Shear ﬂow near solids: epitaxial order and ﬂow boundary condition s.Phys. Rev.A 41, 6830–6837. [Thompson & Troian 1997] Thompson, P.A. & Troian S.M. 1997 A general boundary condition for liquid ﬂow at solid surfaces. Nature 389, 360–362. 8
arXiv:physics/0011019v1 [physics.atom-ph] 9 Nov 2000Driving superﬂuidity with photoassociation Matt Mackie Helsinki Institute of Physics, University of Helsinki, PL 9 , FIN-00014 Helsingin yliopisto, Finland Eddy Timmermans T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Robin Cˆ ot´ e and Juha Javanainen Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3046, USA Abstract We theoretically examine photoassociation of a two-compon ent Fermi degen- erate gas. Our focus is on adjusting the atom-atom interacti on, and thereby increaing the critical temperature of the BCS transition to the superﬂuid state. In order to avoid spontaneous decay of the molecules, the photoassoci- ating light must be far-oﬀ resonance. Very high light intens ities are therefore required for eﬀective control of the BCS transition. Typeset using REVT EX 1As the ﬁeld of quantum degenerate vapors emerges from its bur geoning adolescence, attention is increasingly shifting from Bose [1] to Fermi [2 ] systems. Beyond degeneracy itself, much of this eﬀort is concentrated on observing the B CS transition to a superﬂuid state. However, such investigations are currently at an imp asse: the lowest temperature that has been achieved is about a third of the Fermi temperatu re [3], whereas the formation of Cooper pairs requires temperatures colder by at least an o rder of magnitude [4]. Rather than ﬁnesse the experiments to lower the temperature of the g as further, a more fruitful approach might be to adjust the atom-atom interaction so as t oraisethe value of the critical temperature. Possible means for adjustment include the mag netic-ﬁeld-induced Feshbach resonance [5], rf microwave ﬁelds [6], dc electric ﬁelds [7] , and photoassociation [8,9]. In fact, the Feshbach resonance has been recently applied in this man ner to induce condensation in the otherwise incondensable85Rb [10], and was additionally explored (theoretically) for its usefulness in spurring the superﬂuid transition [11]. The purpose of this paper is to investigate the utility of pho toassociation for inducing the BCS transition. We [12,13,9] and others [14] have earlie r written down ﬁeld theories for photoassociation of bosons. In the present case we consi der instead a binary mixture of fermionic atoms, given by the ﬁelds φ±(r), photoassociating into a bosonic molecule, given by the ﬁeld ψ(r). The fermions would typically be two states with diﬀerent zcomponents of angular momentum in the same atom. As a result of the Pauli exc lusion principle, there is nos-wave photoassociation for two atoms in the same internal st ate, but such a restriction does not apply to two diﬀerent spin components. We thus have a model Hamiltonian density governing photoass ociation, H ¯h=−φ† +¯h∇2 2mφ+−φ† −¯h∇2 2mφ−+ψ†/bracketleftBigg −¯h∇2 4m+δ−1 2iγs/bracketrightBigg ψ −/bracketleftBig Dψ†φ+φ−+D∗φ† −φ† +ψ/bracketrightBig +4π¯ha mφ† −φ† +φ+φ−, (1) wheremis the mass of an atom and δis the detuning of the laser from the threshold of photodissociation. The detuning is positive when the photo dissociation (inverse of photoas- sociation) channel is open. The coupling strength for photo association is D. It may be 2deduced implicitly from Refs. [12,13], and is discussed exp licitly in Ref. [9]. Either way, we have \|D(r)\|= lim v→0/radicaltp/radicalvertex/radicalvertex/radicalbtπ¯h2Γ(r) vµ2. (2) Here Γ( r) is the photodissociation rate that a light with the intensi ty prevailing at rwould cause in a nondegenerate gas of molecules, given that the las er is tuned in such a way that the relative velocity of the molecular fragments (atoms) is v, andµ=m/2 is the reduced mass of two atoms. Because of the statistics, there is a facto r of√ 2 diﬀerence in Eq. (2) from the corresponding expression for identical bosons. Finall y, we have included an interspecies collisional interaction governed by the s-wave scattering length ain the Hamiltonian. Under ordinary circumstances photoassociation by absorpt ion of a photon leads to a molecular state that is unstable against spontaneous emiss ion. There is no particular rea- son why spontaneous break-up of a primarily photoassociate d molecule would deposit the ensuing atoms back to the degenerate Fermi gases. A spontane ously decaying molecule is considered lost for our purposes. Correspondingly, we add t o the Hamiltonian a nonhermi- tian term proportional to the spontaneous emission rate of t he molecular state γs. The Heisenberg equation of motion for the molecular ﬁeld ψis i˙ψ=/bracketleftBigg −¯h∇2 4m+δ−1 2iγs/bracketrightBigg ψ− Dφ+φ−. (3) We assume that the detuning δis the largest frequency parameter in the problem, and solve Eq. (3) adiabatically for the ﬁeld ψ. In the process we keep the imaginary part in the energy, and obtain ψ≃/bracketleftbiggD δ+iγsD 2δ2/bracketrightbigg φ+φ−. (4) Inserting into Eq. (1), we ﬁnd an eﬀective Hamiltonian densi ty for fermions only, H ¯h≃ −φ† +¯h∇2 2mφ+−φ† −¯h∇2 2mφ−+4π¯ha mφ† −φ† +φ+φ− +/bracketleftBigg −\|D\|2 δ−iγs\|D\|2 2δ2/bracketrightBigg φ† −φ† +φ+φ−. (5) 3Let us ﬁrst ignore the decay term ∝γs. Equation (5) displays an added contact interac- tion between the two spin species, as if from the s-wave scattering length ¯a=−\|D\|2m 4πδ¯h. (6) The interaction is attractive if the detuning is positive. B ut an attractive interaction is exactly what is needed for the BCS transition. To simplify ma tters we assume here that the collisional interaction in the absence of light ∝ais too weak for experiments on the BCS transition, and ignore the native collisions altogether. The critical temperature for the BCS transition is [4] Tc=TFexp/bracketleftBigg −π 2kF\|¯a\|/bracketrightBigg =TFexp/bracketleftBigg −2π2¯hδ kFm\|D\|2/bracketrightBigg . (7) HerekF= (3π2ρ)1/3is the Fermi wave number for the total density of atoms ρ, andTF= ¯h2k2 F/2mkBis the corresponding Fermi temperature. Finally, using ( ρ/2)2forφ† −φ† +φ+φ−, we ﬁnd the loss rate per atom due to spontaneous emission from photoassociated molecules, 1 τ=γs\|D\|2ρ 2δ2. (8) To estimate practical experimental numbers, we ﬁrst note th at the rate of photoassoci- ation in a nondegenerate sample at temperature Tis [15,16] R=λ3 Dρe−¯hδ kBTΓ≡ρ/parenleftbiggI ¯hω/parenrightbigg κ. (9) HereλD=/radicalBig 2π¯h2/µkBTis the thermal deBroglie wavelength, Iis the intensity (W cm−2) of photoassociating light, and κ(cm5) is the photoassociation rate coeﬃcients. There may be statistics dependent numerical factors in Eq. (9). Howev er, in the current literature such factors are usually ignored, and we write Eq. (9) accordingl y. Using Eq. (9), a calculation or a measurement of the photoass ociation rate in a thermal sample may be converted into a prediction of eﬀective scatte ring length, transition tem- perature, and lifetime in a degenerate Fermi-Dirac gas. We e xpress the results in terms of 4λ=λ/2π, wavelength of photoassociating light divided by 2 π,ǫR= ¯h/(2mλ2), familiar pho- ton recoil frequency, and a characteristic intensity for th e given photoassociation transition, I0. This gives ¯a λ= 0.0140077I I0ǫR δ, (10) Tc TF= exp/bracketleftBigg −36.24781 (λ3ρ)1/3δ ǫRI0 I/bracketrightBigg , (11) ǫRτ= 4δ2 ǫRγsI0 I1 λ3ρ. (12) The obscure numerical factors, powers of 2 and π, are there because we want to use the characteristic intensity for photoassociation deﬁned in R ef. [9]. For instance, if the photoas- sociation rate coeﬃcient κis known at a temperature Tand detuning δ, the critical intensity is I0=√π√ ¯hδc¯h4 2κm2(kBT)3/2λ2e−¯hδ/k BT. (13) Detailed microscopic calculations (or measurements) of ph otoassociation rates are sparse, but they exist for the fermionic isotope6Li of lithium [17–20]. Let us consider an example already discussed in Ref. [9], transitions to the triplet vi brational state v′= 79 with the binding energy 1 .05 cm−1. The characteristic intensity is then I0= 9.8 mW cm−2, the wave- length isλ= 671 nm, and the recoil frequency is ǫR= 63.3×2πkHz. We take the decay rate of the molecular state to be twice the spontaneous decay rate of the corresponding atom, so thatγs= 12×2πMHz. In our estimate we assume λ3ρ= 1, corresponding to the density ρ= 8.21×1014cm−3that is high but not unreasonable. It would then take the inte rmediate detuningδ= 2×2π1014Hz and the intensity I= 460 MW cm−2to makeTc= 0.1TFand τ= 10 s. The intensity came out very high for a continuous-wave laser , so it seems that the only potential candidate for experiments is a tightly focused, p owerful CO 2laser. Our formalism, though, is based on the assumption that the laser is close to a photoassociating resonance. We need to amend the calculations to give meaningful estimat es for the CO 2laser, whose 5electric ﬁeld is in practice direct current compared to the m olecular transition frequencies involved. To this eﬀect we ﬁrst note that in an ordinary two-level syste m one may carry out perturbation theory both within the rotating-wave approxi mation, and in the quasistatic limit without the rotating-wave approximation as well. The result is that the quasistatic results are obtained from the near-resonance formulas by re placing the detuning with the molecular transition frequency, δ→ω0, and multiplying the intensity by two, I→2I. Applying this substitution to the scattering length, at λ3ρ= 1 we ﬁnd that the intensity required for Tc= 0.1TFagain becomes 460 MW cm−2. With the same substitutions, the lifetime would be about 20 s. However, as the frequency of the CO2laser is 1/16 of the resonance frequency for photoassociation, the phase space for spontaneously emitted photons is reduced, and the actual rate of spontaneous emission woul d be reduced by an extra factor of at least 162∼300. It is clear that spontaneous emission is not an issue wit h CO 2laser excitation. Up to this point we have only considered photoassociation wi th one molecular state, the triplet state with vibrational quantum number v′= 79. Now, in lithium as well as in other alkali atoms, most of the transition strength for dipole tra nsitions starting from the ground state is in the Dlines. Just a few electronic states in a molecule then inheri t most of the transition strength for photoassociation. We only conside r the singlet and triplet excited manifolds in the6Li dimer, for which calculations of the photoassociation ma trix elements exist for all vibrational states [17–20]. It turns out that t he triplet state v′= 79 carries about the factor 0 .07 of the total transition strength for photoassociation of low-energy atoms. As one should obviously add the changes of the scattering lengt hs due to all molecular states, in our CO 2laser example the intensity also gets multiplied by 0 .07 and becomes 30 MW cm−2. It is, in principle, possible to tailor the scattering lengt h by oﬀ-resonant photoassociation, and thereby eﬀect the BCS transition in a low-temperature Fe rmi gas of, say,6Li vapor. The required laser intensities, however, are high. As in the cas e of coherent photoassociation [9], the problem is not so much that the matrix elements for photoa ssociation are weak, but 6that the primarily photoassociated molecules tend to decay spontaneously and the sample is lost. To avoid spontaneous emission, one has to go very far oﬀ resonance, which leads to challenging requirements on laser intensity. In pursuit of BCS transition by means of oﬀ- resonant photoassociation, it might be worthwhile to try an d look for other ways of getting around the spontaneous emission. 7REFERENCES [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198-201 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J . van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. L ett.75, 3969-3973 (1995); C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Re v. Lett. 78, 985-989 (1997). [2] B. DeMarco and D. S. Jin, Science 285, 1703-1706 (1999). [3] M. J. Holland, B. DeMarco, and D. S. Jin, Phys. Rev. A 61, 053610 (2000) (6 pages). [4] H. T. C. Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 76, 10-13 (1996); M. Houbiers, H. T. C. Stoof, R. Ferwerda, W. I. McAlex ander, C. A. Sackett, and R. G. Hulet, Phys. Rev. A 56, 4864-4878 (1997). [5] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Sto of, Phys. Rev. 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arXiv:physics/0011020v1 [physics.comp-ph] 9 Nov 2000Life in Silico - Simulation of Complex Systems by Enzymatic Computation Gerhard Mack and Jan W¨ urthner II. Institut f¨ ur Theoretische Physik, Universit¨ at Hambu rg September 28, 2000 abstract We describe software and a language for quasibiological com puta- tions. Its theoretical basis is a uniﬁed theory of complex (a daptive) systems where all laws are regularities of relations between things or agents, and dy- namics is made from “atomic constituents” called enzymes. T he notion is abstracted from biochemistry. The software can be used to si mulate physi- cal systems as well as basic life processes. Systems can be co nstructed and manipulated by mouse click and there is an automatic transla tion of all op- erations into a LISP-like scripting language, so that one ma y compose code by mouse click. 1 Introduction It is a fruitful idea to learn from nature how to do informatio n processing by abstracting from what happens in the living cell. In 1959, R. Feynman gave a visionary talk describing the poss ibility of building computers that were “submicroscopic”. [11]. More recently, L.M. Adleman demonstrated the feasibility o f carrying out computations at the molecular level by solving a standard NP -hard graph problem (similar to the travelling salesman) using molecul es of DNA, and standard protocols and biochemical enzymes [1] Here we are not interested in doing real chemistry in a bucket , but in mathematical abstractions which incorporate the basic log ic of biochemical processes in the living cell and can be put on a computer. This was a dream 1also of workers on Artiﬁcial Life [16]. But we may infer from A dleman’s work thatlife in silico will be useful not only for simulating life processes, but fo r other complex problems as well. Starting point is a philosophical principle, The human mind thinks about relations between things or agents. This tells us how to model parts of the world. A uniﬁed theory o f complex (adaptive) systems was built upon this principle [21]. Here we describe its implementation in software. As in category theory [19], the relations are directed binar y relations, and it is regarded as their constitutive property that they c an be composed - think of friend of a friend, brother in law, next nearest neighbor . There are also some distinguished relations - the identities 1Xof objects Xwith themselves - they play a basic role similar to the number 0 in a rithmetics. Motivated by physics, a principle of locality is added by sin gling out some of the relations as direct relations, called links. All others are composed from them and possibly their “adjoints” - i.e. relations in the op posite direction. Links between objects form networks. Their properties are l aid down in the mathematical deﬁnition of a system , given below. Basically it makes precise the notion of structure . We consider dynamics in discrete time. It is required to be lo cal. Dy- namics is also built from “atomic constituents”, called enz ymes. They are composed from very simple basic moves, including in particu lar composition of two links to a single link (friend of a friend becomes frien d), making or deletion of adjoint links (reciprocation) and making or ﬁnd ing copies.1 In contrast with automata theory [30], system s theory is supposed to be self contained. Everything that is used should be provided f or by the axioms or be constructed with them. There is no other information bu t structure. But in the software implementation, we will not push this poi nt of view to the extreme. We will admit numerical or text data that reside inside objects and links. Regard them as coding for structure. For more disc ussion on the relation between structural and numerical descriptions se e ref. [21]. The whole theory may be regarded as a kind of universal chemistry where general objects substitute for atoms and molecules, and mor e general links for chemical bonds and spacial proximity. 1They are very simple examples of graph transformations [26] 22 Systems According to the pioneer of general systems theory, L. van Be rtalanﬀy [3], a system is a set of units with relationships between them. And according to F. Jacob [15], every object considered in biology is a system of systems . We precisize to2 Deﬁnition 1 (System )Asystem Sis a model of a part of the world as a network of objects X, Y, ... (which represent things or agents) with arrows f, g, ... which represent directed relations between them. One writes f:X/mapsto→Yfor a relation from a source Xto atarget Y. The arrows are characterized by axiomatic properties as fol lows: 1.composition . Arrows can be composed. If f:X/mapsto→Yandg:Y/mapsto→Z are arrows, then the arrow g◦f:X/mapsto→Z is deﬁned. The composition is associative, i.e. (h◦g)◦f=h◦(g◦f). 2.adjoint . To every arrow f:X/mapsto→Ythere is a unique arrow f∗: Y/mapsto→Xin the opposite direction, called the adjoint of f.f∗∗=fand (g◦f)∗=f∗◦g∗. 3.identity . To every object Xthere is a unique arrow 1X:X/mapsto→Xwhich represents the identity of a thing or agent with itself. 1X=1∗ X,and 1Y◦f=f=f◦1X for every arrow f:X/mapsto→Y. 4.locality : Some of the arrows are declared direct (or fundamental); they are called links. All arrows fcan be made from links by composition and adjunction, f=bn◦...◦b1,(n≥0)where biare links or adjoints of links; the empty product (n= 0)represents the identity. 5.composites: The objects Xare either atomic orsystem s. In the latter case,Xis said to have internal structure, and the objects of the system Xare called its constituents . 2Axioms 1,3 are those of a category [19]. Therefore system s are categories with a ∗-operation and a notion of locality 36.non-selﬁnclusion: Asystem cannot be its own object or constituent of an object etc. Ultimately, constituents of ... of constitue nts are atomic. ASemiadditive system satisﬁes the same axioms, except that arrows may be composed from links and their adjoints with the help of two operations, ◦ and⊕. The ⊕-operation adds parallel arrows. It makes the set S(X, Y)of all arrows with given source Xand target Yinto an additive (=commutative) semigroup. The distributive law holds (f1⊕f2)◦(g1⊕g2) =f1◦g1⊕f2◦g1⊕f1◦g2⊕f2◦g2 An arrow ois a zero arrow if f⊕o=ffor all f. It is understood that arrows are modulo zero arrows. The rationale behind the ⊕operation is that it should be possible to interpret two parallel links as a single link, and similarly for arrows . Remark 1 In a Semiadditive system , the set S(X, Y)can be extended to an additive group if and only if f⊕h=g⊕himplies f=g, whatever is h. In the following, we will sometimes refer to composition wit h◦asmultipli- cation , and to ⊕asaddition of links. In addition to the axioms, it is assumed that constituents, c onstituents of constituents etc. of objects of Sare not objects of S. This assumption is subject to being weakened, but weakening it may require ad justment or redesign of software. It would be interesting to weaken also the axiomatic property of non-selﬁnclusion. But this is a very subtle oper ation, cp. the discussion in section 8.1 We found it convenient to generalize the axiomatic notion of identity arrow by introducing identity links as a special type of link which may connect objects that are identical in the sense of indistinguishabl e. Deﬁnition 2 (Subsystems) Asubsystem S1of asystem Sis generated by a set of objects in Sand a set of links in Sbetween these objects. Its arrows are all arrows in Sthat can be composed from these links and their adjoints. Theboundary ofS1consists of the links in Swith target in S1which are not links or adjoints of links in S1. Theenvironment ofS1is the system generated by the objects of Snot in S1and the links between them. 4In the software, subsystems can be speciﬁed by marks in two di ﬀerent ways, either by marking the (inward) links in its boundary, or by ma rking the links which belong to it. In the description of the software design, we use the followi ng Nomenclature 1 Valence The links with target Xare called the Valences ofX Radical An object Xtogether with all its valences is called a Radical .Given some link b, the radical which contains its source (object) is called th e source radical ofb. Membrane Marks on links which identify the boundary of a subsystem are called Membranes Path APathfromXtoYis a sequence b1, ..., b n, where biare links or adjoints of links, Xis the source of b1,Yis the target of bn, and the target of biis the source of bi+1fori= 1, ..., n −1. The empty path(n= 0)fromXtoXis identiﬁed with 1X. Paths are special subsystems. Marks on links which belong to a certain subsyst em are calledpath-links . In simulations of biological organisms and their parts, mem branes can be used as models of cell membranes or envelopes of organs. Path -links are used to model blood vessels and other transport channels, an d neural nets. Attachig enzymes can convert terminal objects of neural net s to sensors and eﬀectors, and objects within blood vessels to pumps. And cop y processes can make organisms grow. Any system with all its interna can be copied by purely local processes, induced e.g. by propagating shocks , cp. section 8.1. 2.1 Basic equations Insystem s theory, basic equations are often of the form l=1X, where l:X/mapsto→Xis the arrow speciﬁed by some path. Maxwell’s equations have this form [21]. Validity of constraints of this form can distinguish betwee n modes of being [21]. For instance, space time is distiguished by havi ng a Lorentzian geometry,3and material bodies are in space and obey conservation laws. 3Diﬀerential geometry is a father of system s theory, with parallel transporters as arrows 52.2 Isomorphic systems The notion of isomorphic systems is somwhat subtle. Basical ly, two system s S1andS2are isomorphic if there exists a structure preserving map F:S1/mapsto→ S2which maps objects into objects and links into links, and who se inverse exists and has the same property. This induces a map of arrows finto arrows F(f). We require of a structure preserving map Fthat source and target of F(f) are the images of source and target of f, and4 F(f◦g) = F(f)◦F(g), (1) F(1X) =1F(X). (2) F(f∗) = F(f)∗(3) There are also anti-isomorphisms. They relate complementa ry shapes. For an antiisomorphism, eq.(1) is replaced by F(f◦g) =F(g)◦F(f), (4) and source and target are interchanged. It is NOT required that the internal structure of correspond ing objects matches when they are not atomic. Nonatomic objects are rega rded as black boxes. The internal structure of black boxes (nonatomic obj ects) is declared irrelevant when one does not distinguish isomorphic system s, and so is the distinction between atomic and nonatomic objects. The only usage of the internal structure is in constructing links of the system and their composi- tion◦. One does not look into black boxes anymore once they are in pl ace and connected. We speak of a strong isomorphism if it IS demanded that corresponding nonatomic objects are strongly isomorphic system s. (This recursive deﬁni- tion makes sense because of the axiom of non-selﬁnclusion.) 3 Dynamical systems We consider dynamics in discrete time. A deterministic dyna mics of ﬁrst order shall determine a system St+1at time t+1 and links between objects 4These are the axiomatic properties of functors of a category [19], with the added demand that the map is local in the sense that it maps links to l inks, and preserves the ∗-operation. Anti-isomorphisms are invertible contravari ant functors 6XinSt+1and objects YinStfrom a given a system Stat time t. If there is a link between X∈St+1andY∈St,Xis said to be descendent of Y, and Yis ancestor of X. It is required that 1. Every object X∈St+1is descendent of at least one object Y. Con- versely, {Y1, ..., Y n}can only be the set of ancestors of some Xif the system generated by Y1, ..., Y nand identity links between them is con- nected. If (n >1) we say that there is fusion. An object may have more than one descendent ( e.g. copies or translations). 2. The dynamics is local in the sense that the isomorphism cla ss (resp. strong isomorphism class) of a subsystem of St+1depends only on the system generated by the objects in a 1-neighborhood of its ancestor s and the links between them. A deterministic dynamics determines Stup to isomorphism, a strongly de- terministic dynamics determines it up to strong isomorphis m, i.e. including the internal structure of nonatomic objects. A dynamics is c alled universal if it is well deﬁned for every system Stwhatever. The idea of enzymatic computation is ﬁrst, to build dynamics from “atomic constituents” (elementary moves) which we call enzymes. second , to incorporate the information about the dynamics into the initial state by attaching enzymes to links and objects of St. The enzymes may act conditionally. To this end they are combi ned with predicates which examine the structure of a neigborhood in t he system. A predicate pdetermines a boolean function p.evaluate(r,v) whereris a radical, and vis (pointer to) a valence, typically of r. The pair (predicate, enzyme) is also called a mechanism . The enzymes are abstractions not only of the biochemical enzymes [2] which govern life processes i n cells, but of any agent of change. Biochemical enzymes show speciﬁcity [2], i .e. they come with their predicates inseparably build in. We shall occasi onally speak of enzymes when we really mean mechanisms. There are several kinds of basic enzymes, i.e. elementary mo ves. motion An arrow becomes promoted to link. Either 7a) The composite of two (or more) links becomes a link, i.e. in direct relations become direct. example 1: Friend of a friend becomes friend. example 2: Equations of motion of fundamental physics like M axwells equations [21, 22] or motion of particles in space.[Transpo rt of material involves motion of particles in space (or an eﬀective decrip tion of that in terms of algebraic values)]. example 3: catalysis, ﬁgure 1. example 4: logical deductions, cp. ref.[21, 27] . b) The adjoint of a link which was not a link becomes a link. example: reciprocation The converse - adjoint of a link loses the status of a link - is a lso subsumed under motion. Motion is reversible. growth An object has two or more descendents - e.g copies which are ma de or taken from the environment. Fusion of objects is also subs umed under growth, and so is the making of an object with internal s tructure from a subsystem. Growth is reversible if descendents are co nnected by identity links (cp. section 2 after deﬁnition 1). Removal of the identity link would be subsumed under death, s. below. death An object may disappear with its links, links may disappear t ogether with their adjoints. Death is usually irreversible. cognition A new link is made between objects X1,X2with matching inter- nal structure. Locality requires that X1,X2are connected by a path of short length in St, i.e. there is a preexisting relation between them. But the new link need not be the arrow determined by the preexi st- ing path. example: a chemical bond may be established when there was spacial proximity before. Links made this way are called cognitive links . Carl Anne Bert Bert AnneCarl A A B BC C Figure 1: Catalysis in chemistry and elsewhere. A catalyst Cbinds molecules A andB. First a substrate-enzyme complex is built, where AandBbind toC. Next the composite arrow from AtoBbecomes fundamental. 8enzyme management Enzymes are attached to objects and links of St+1, or their action somewhere at time t+ 1 is speciﬁed in other ways. The making of cognitive links only makes sense if the dynamic s determines future system s up to strong isomorphism, because the internal structure of nonatomic objects matters. Cognitive links can be of severa l types f.fis a boolean function (“matching structure”) of two objects, ty pically determined by a predicate pwith the property that p(r, v) depends only on the object of radical rand the source object of valence v. Its return value is also sup- posed to be obtained by enzymatic computation (with access t o the internal structure of objects which are system s). The most important example of matching structure are isomor phisms and antiisomorphism. The latter implement the lock key mechanism which is re- sponsible for speciﬁcity in biochemistry [2]. Determining isomorphisms of graphs is an NP-hard problem. But this is not very relevant he re because in practise one will encounter graphs with special properties . Matching struc- tures can also be detected by neural nets, here implemented a s subsystems made with path-links. We will subsume the making of identity links between indisti nguishable atomic objects under cognition as well, and regard such iden tity links as prototypical examples of cognitive links. A description of the most important enzymes will be given lat er. So far we talked about deterministic dynamics. In a stochast ic dynamics, enzymes act with certain probabilities. 3.1 Concurrency The action of enzymes at diﬀerent locations will not commute in general. This results in what is known to computer scientists working on parallel computing as the concurrency problem . Petri nets are a well know example [25]. In the spirit of biology, one may just ignore this probl em, admitting some randomness or indeterminacy in the dynamics, e.g. thro ugh a hidden dependence on an ordering of the valences of a radical. But it is useful to have the option of specifying a well deﬁned deterministic dynami cs forsystem s. We propose to achieve this by a generalization of Jacobi swee ps. Let us ﬁrst recall the notion of a Jacobi sweep. Consider a gri d made of nodes (objects) connected by links, where the nodes and/or t he links carry some data. Given a cost function whose arguments are the afor ementioned 9data and which is a sum of contributions which depend only on a neighbor- hood of individual nodes or links. One may attempt its minimi zation by relaxation. One makes sweeps through the grid, updating dat a at individual nodes or links such that the cost function is minimized under the constraint that all other data remain frozen. In a Jacobi sweep (as oppos ed to Gauss Seidel sweeps)[13], the data after sweep t+1 are determined solely by the data after sweep t. As a result it does not matter in which order the nodes and links of the grid are visited - the eﬀect of the individual upd ating operations will commute. We may proceed in the same way, except that we divide the makin g of thesystem St+1at time tfrom the system Stat time tinto several steps which take place at times t+1 4,t+1 2,t+3 4andt+ 1. In intermediate steps, there will be links connecting objects in Stwith objects in St+ 1. We call them “time links” We may divide enzymes into classes according to whether they act at timet+1 4,t+1 2ort+1. The intermediate step at time t+3 4was introduced for pedagogical reasons only. The enzymes which act at time t+1 4will be called “object-making”, those at time t+1 2“link-making”. The link-making enzymes may also put marks on newly made links to indicate inf ormation on their adjoints. As explained in section 3, our processes are such that every o bjectYin St+1is descendent of (at least) one object XinSt. Ignoring the attachment of enzymes to St+1at ﬁrst, the course of events is like this (cp. ﬁgure 2 for an example) . t+1 4Make descendents Y1, ..., Y n∈Stofobjects X∈St, if any; ( n≤2). Connect them by time links. When ( n >1) make identity links between identical copies Y1, ..., Y nif demanded by an enzyme. t+1 2Make linksthat will end up in St+1. Every link-making enzyme may contribute some link or links, in a manner which depends on it s neigh- borhood in St. For pedagogical reasons, the links are made at time t+1 2to connect one object in Stand one object in St+1, and at time t+3 4the ends in Stof such links are lifted to St+1by composition with a time link. t+ 1 Depending on the marks on the newly made links in St+1, adjointness relations are established among links, or new fundamental a djoints are 10t 9q - - -- t+1 4 9q - - --M6 ?6 ?6 ?6 ?6 ?6 ? t+1 2 9q - - --MM6 ?6 ?6 ?6 ?6 ?6 ?II kI K t+ 1 9q - - --M6 ?6 ?6 ?6 ?6 ?6 ?9q - - -- - Figure 2: splitfork dynamics, concurrent version made. The operation is local in that it involves only the indi vidual pairs of sets S(Y, Y′) andS(Y′, Y) of links between given objects Y, Y′. So far we neglected to say how the enzymes are put into the system St+1. Object making enzymes are attached to descendents or ident ity links between them in step t+1 4. Link-making enzymes are attached to links in stept+1 2. Dynamics of this kind is well deﬁned because all operations w ithin anyone of the steps commute. In the end, the time links may be removed (if desired). This st ep is not shown in ﬁgure 2. Unfortunately the use of Jacobi sweeps is less convenient th an admitting a hidden dependence of the order in which enzymes at neighbou ring locations act on an ordering of valences in radicals. This is so because the composition of chains of enzymes which act one after another is not a simpl e matter anymore. 114 The System Class Laboratory The System Class Laboratory (SyCL) is a software package com posed of three parts core package is a C++class library which encodes the basic functionality demanded by the axioms of a semi-additive system and enzymatic dy- namics on it. presentation package includes a graphic interface. Using it one can con- struct and manipulate such system s by mouse-click, invoking the con- ditional action of basic enzymes and of compound enzymes mad e from them. The presentation package also includes to translate system s to and from XML , i.e. text. interpreter and parser package is based on a LISP-like scripting languag e. One can write programs in this language and run them. Alterna tively, operations carried out by mouse click on the graphic interfa ce are au- tomatically recorded as commands of the scripting language . In this way one may compose programs by mouse click. 5 The class library 5.1 Basic Data Types In principle, system theory is self-contained5. There are no data in system s other than their structure, and no states of any part of a System other than its structure. The miracle is how much can be modeled with so l ittle building material. In practice, it is nevertheless convenient to admit data ins ide objects and links which can be thought to code for internal structure in s ome way. (Links with internal structure can be thought of as system ’s with two interfaces) The data inside links are instances of a class AlgebraicValue . They admit algebraic operations. They can be added, multiplied and mul tiplied with real numbers in a manner which is described below in subsecti on 5.2. Their equality can also be ascertained. The class AlgebraicValue is derived from a 5in contrast with automata theory [30] 12classValue which does not have the algebraic operations as methods. Ob- jects can contain arbitrary Value s. Multiplication and addition of algebraic values that reside inside links is invoked when links are com posed using the axiomatic ◦and⊕-operations. Several diﬀerent classes are derived from AlgebraicValue, including the class Predicate. We mention it separately be- cause of its important role. Enzymes may also be attached to o bjects and links in the guise of EnzymaticValue, also derived from Alge braicValue. The other data types correspond with Nomenclature 1, except for the following implementational detail. The adjoint of a link bwith source X which is not itself a link is nevertheless included in the lis t of Valences of the radical containing X, and is marked as a “virtual valence”. This is technically convenient because the target of bcan be addressed as source of its adjoint. The basic data types are Value, AlgebraicValue contain text or numerical data with operations as described Objects may contain a singly linked list of Values . They can be copied. Valences contain a pointer to their source radical , a pointer to their ad- joints, and possibly: a singly linked list of AlgebraicValue ’s, a singly linked list of Membrane s, a singly linked list of Paths. By deﬁnition, the adjoint is a valence of the source radical, possibly virtual. Composition ◦of a valence with valences of its source radical is deﬁned. Parallel valences may be added to a single valence. Radicals contain an object, a doubly linked list of valences, and poss ibly a position in 3-dimensional space. Predicates determine a boolean function which takes as argument a pair (r,v) , wherevis a pointer to valence (maybe NULL) and ris a radical (typically the target of v). Predicates can be composed with junctors “and, or, not”. Among the predicates are keys which permit to pass membranes selectively. Membranes are attached to valences. They mark the boundary of a sub- system. They carry a string or a void* pointer as code to distinguish them. 13PathLinks contain a reference to a valence and data to identify the prec ur- sor, successor and adjoint pathLink. A valence knows the pat hLinks that pass through it, cp. above. System A connected nonempty system is identiﬁed by a pointer rootto one of its radicals, and possibly by the code of a membrane whi ch bounds it. Fromroot one can access the source radicals of its valences, and so on. All the information on the system is in the radicals which can be accessed in this way. Enzyme Enzymes have a method Radical & act(r,v) which takes as its arguments a radical rand a pointer vto a pointer v to a valence (of r).vmay be NULL. The action is on a neighborhood of r.vmay be changed to point to a diﬀerent valence, therefore a pair (Radical &, Valence ) is eﬀectively returned, which can be used for input to another enzyme. One may deﬁne a path distance dbetween objects XandYby use of paths {b1, ..., b n}.dis the minimal value of namong all the paths from XtoY. 5.2 AlgebraicValues The implementation of algebraic operations with Algebraic Values makes es- sential use of object oriented design features like inherit ance and virtuality. [12]. We assume that the reader is familiar with these. The algebraic operations satisfy the usual laws of associat ivity and dis- tributivity. Multiplication may be noncommutative, and in one special case (the max-plus-”algebra”), subtraction is undeﬁned. There is an inheritance tree which speciﬁes how diﬀerent cla sses of alge- braic values are derived from the base class AlgebraicValue . Each class has its (virtual) methods for multiplication with members of th e same class from the right or left, for addition, and for multiplication with real numbers α. In some cases (like strings) the multiplication with real αdoes nothing if α/ne}ationslash= 0, and converts to a zero-element otherwise. Given any two c lasses A,B of algebraic values, there is a unique class C, the “ least common ancestor“, such that AandBare derived from C, but not from any class that is derived fromC. The result of right multiplication ∗= is computed by invoking the 14multiplication-method of C. Addition and left multiplication are treated in the same way. The inheritance tree is shown in ﬁgure 3. AlgebraicValue StringMatrixDoubleAffineTransformation Int TrueForAllValencesMap KeyValue AffineFrame AntiKeyEnzymeValue PredicateMaxPlus Figure 3: The class diagram for diﬀerent types of values. There is one type of algebraic values of interest where the co ndition of remark 1 is violated, the max-plus ortimetable “algebra”. Its data are real numbers (or real matrices), a◦b=a+b(addition of real numbers) and a⊕b=max(a, b). For matrices, the operations are entry-wise. This “algeb ra” is useful in so called discrete event systems [4], and in optimization problems where one requires the maximum of errors which are given by re al return values of local cost functions. There may be a (singly linked) list of AlgebraicValues in a li nk. In this case, multiplication is element by element in the list. Values in Objects need not be of type AlgebraicValue . But for instances ηof useful data types, multiplication with AlgebraicValue Afrom the left is deﬁned and is associative (i.e. ( A∗B)∗η=A∗(B∗η)). The multipli- cation may be trivial, i.e. A∗η=ηfor all A. In case ηis itself of type AlgebraicValue, multiplication is deﬁned as described abo ve. Deﬁnition 3 (Parallel Transport ) Given a link bwith AlgebraicValue Aand the value ηof its source object Y,A∗ηis regarded as a potential value of the target object Xof the link b, and the operation is called the parallel transport ofηfromYtoXalong the link b. 15The parallel transport along links may be composed to parall el transport along paths {b0, ..., b n}. Parallel transport along 1Yis the trivial map η/mapsto→η. The notion comes from lattice gauge theory of elementary par ticles [6] and general relativity. Remark 2 Because of associativity, the result of the parallel transp ort de- pends actually only on the arrow deﬁned by the path. It may happen that the result of parallel transport of some ty pe of value fromYtoXis independent of the chosen path from YtoX. In this case the type of value is called an invariant . Invariants have a global meaning in a connected system , because they may be parallel transported from any Yto any Xin a unique way. Important examples are i) quantities of any chemical substance which i s transferred from object to object by diﬀusion, ii) payments in an economy. Counterexamples are any kind of “non-ideal communication” , such as communication of humans in natural language. Utterances in natural lan- guage do not generally have a globally well deﬁned meaning - o ne’s owl is the other’s nightingale. Gossip is an example. When the message comes back to speaker Xwho send it out, it may have undergone dramatic change. So the result of communication around some loop is not the same a s along the identity 1X. 5.3 Enzymes We proceed to describe the most important micro-enzymes, be ginning with motion : It involves either multiplication of two links, or promoti on of a virtual adjoint of a link to the status of a link. We have enzym esV ML for left multiplication and AMR for right multiplication. Given a V ML - enzymeewith attached predicate p,e.act(r,m) will go through all valences vofr, and multiplies those vfrom the left with the adjoint of mfor which p.evaluate(r,v) returns TRUE. An illustration is found below. In this ﬁgure, virtual valences are indicated as dotted lines. The AMR -enzyme is similar. oam m v6 ?-VML-am◦v6 ?-I Roam m v6 ?-VML-am◦v6 ?I R 16Alternatively, motion may involve the promotion of the adjo int of a link to the status of link. There are enzymes to do that, one of them is the MAD -enzyme. It makes adjoints of all valences into links which f ulﬁll the condition set by a predicate. RFU acts in the same way, except that the argument vis replaced by its adjoint, and the source of vsubstitutes for r. --MAD- --RFU- -- To change a valences status from fundamental to virtual, an e nzyme MV V is used. Growth : The most important enzyme is CPO which copies an object, and links original and copy by an identity link. (Another ver sion searches for the copy in the environment). The object may be composite, i. e. asystem . CPO appears in the ﬁrst micro-enzyme in the splitFork enzyme as s hown in ﬁgure 6 in section 8.1. In the ﬁgure there, some predicates are indicated by a string-code {?...}enclosed in braces. Their meaning is as follows ?nLKvis not a link (i.e. is a virtual valence) ?iADadjoint of vis a link, negation ? nAD. ?iIDvis an identity link. ?hLSdetect a triangular structure. There are also enzymes like SY A which make objects of subsystems whose boundary is marked by membranes. Death : links are removed by RTV orRMV enzymes which kill the particular valence given as argument, or all valences which fulﬁll a condition. (The RMV -enzyme operates in ﬁgure 6 to remove identity links.) A removal of a radical with all its valences and their adjoint s by the DEL-enzyme. Enzyme management enzymes: Among them there are presentation en- zymes. They have relatives in cell biology [2]. They hand arg uments (r,v) to another enzyme. For instance, for a PRS-enzymee, (“present Source”) e.act(r,v) , presents (s, adjoint(v)) , (s=source of v) to be acted on by the next enzyme following ein an enzyme chain. Predicates may make the presentation conditional. There are also quantors such as fALLwhich invokes the action e.act(r,v) of an enzyme efor all valences vof a radical rwhich fulﬁll a condition; e is appended as tail of an enzyme chain with head fALL. ( AforAll Quantor exists also for predicates.) 17Unless one uses the concurrent version of the dynamics (cf. s ection 3.1), the present implementation for single processor machines o ﬀers the option of reporting an enzyme to an agenda together with the argument p air(r,v) it will act on, instead of attaching it to the valence vor the object of r in the system . There are enzymes to do that. A Sag-enzyme (“source- to-agenda”) in an enzyme chain reports the whole chain toget her with (s, NULL) ,sthe source of its v-argument. Membranes : The “push membrane” enzymes PIM andPOM sur- rounds a radical by a membrane (attached to all its valences) or pushes the radical inside resp. outside a membrane that goes throug h one of its valences resp. its adjoint. The membrane’s code is speciﬁed by a predicate. Paths: There are enzymes to make or prolong paths along speciﬁed va - lences. 6 Graphic Interface There is a graphic interface which can be used to construct system s and manipulate them by use of enzymes. All this is done by mouse cl ick, except that numerical or text-data need to be entered from the key bo ard when they are not copied from somewhere. All operations are recorded and translated into code by use o f the LISP- like scripting language which is described in the next secti on. In this way, programs which construct system s or code for processes on system s can be composed by mousclick. In detail one may - Position objects or delete them - Link them by valences (with or without adjoint link) or dele te links. - Enter data ( Value s) into objects or links. - Select basic enzymes and accompanying predicates from lis ts. - Build composite enzymes as chains of basic “micro-enzymes ”. - Invoke conditional application of enzymes to radicals rwith or without a selected valence v. Among the enzymes there are some which - create a membrane around an object, or push a membrane beyon d other objects. - create paths or prolong them - create copies of individual objects, of the whole system, o r of a subsystem 18bounded by a membrane. - induce sweeps through the whole system [or through a part of it which is bounded by a membrane or marked by path-Links], applying the tail of an aFRK -enzyme to every radical in it once. - induce sweeps which are reﬂected at the locus of maximal pat h distance, assemble data from all the objects and links of the system and report them to the system ’ root. The computation of maxima of real data at all objects is a sample application. - induce diﬀusion within the system or a subsystem. Coordinates are assigned to each object. A panel allows to ro tate and translate the coordinate system. Diﬀerent windows admit di ﬀerent views on a system and its dynamical behavior. 6.1 Permanence: XML Methods are provided to translate a system intoXML and conversely to construct a system fromXML -code. The methods can be invoked through the graphic interface. In this subsection we assume the reader is familiar with XML . An intro- duction to XML can be found in [18]. The dtd ( document type deﬁnition ) listed in [31] deﬁnes tags and their possible attributes for the systems elements. An internal i d number is as- signed to each object and valence to let a valence remember it s source, target and adjoint valence. The<system> tag encloses anything else inside the document. The <object> tag carries an objects id number and a position as attributes . <value> tags may be enclosed. The <valence> tag carries a valences id number, the id numbers of its source and target objects and of its adjoint valence as attributes. Again, <value> tags may be enclosed. The <value> tag carries the value type and the value content as attribute s. The way XML is designed it allows future extensions while ass uring up- and downward compatibility. 197 The Scripting Language If a particular kind of complex problem occurs very often, it might be worthwhile to express instances of the problem as sentences in a simple language[12]. Solving a problem then means to interpret the sentence, for which an interpreter is needed. Based on a certain grammar which deﬁnes a simple language, th e interpreter represents and interprets sentences in that language. To be eﬃcient, charac- ter strings which make up the sentences are transformed into trees of objects, which are easy to handle by the interpreter. This transforma tion is done by a software called parser . The grammar used for our purposes is very similar to the one us ed by the list processing language LISP . expression ::= list \| atom list ::= ‘(’ expression* ‘)’ atom ::= ‘a’ \| ‘b’ \| ... \| ‘z’ \| ‘+’ \| ‘-’ \| ‘’ \| ‘/’ \| ‘$’ \| { ‘a’ \| ‘b’ \| ... \| ‘z’ } \| ‘ quote ’ expression In object oriented software design, a well known interprete r pattern (sug- gested in ref.[12]) has shown to work ﬁne in most cases. To imp lement a parser and interpreter for the SyCL program package, this pa ttern is used in combination with a composite pattern [12] and extended by one more ab- straction in the way shown in ﬁgure 4. TheElement class is abstract. It provides polymorphism by assuring tha t every object is of type Element . A Element can thus either be an object of typeList, which consists of further Elements, or an object of type Atom , where Atom is abstract again. The following classes are derived from cl ass Atom. A list can consist of elements of any kind. To evaluate a list, the ﬁrst element has to be a function, or an enzyme, where the following elemen ts serve as parameters, which have to be evaluable unless they are quote d (i.e. they start with a quote). 20value : int value : int valuevaluevalueDoubleAtom StringAtom FunctionAtom ObjectAtom ObjectList Enzymevalue : double value : textAtomElement Eval() : Elementnext top IntAtom ValenceAtom EnzymeAtom Valence Figure 4: The class diagram for the SyCL parser. Implemented Functions The commands, this interpreter can execute are listed below . Enzymes can be applied as well, taking a radical or a list of a radical and a valence as a parameter. (adjoint v) returns the adjoint valence to valence v (append expr li) appends the expression exprto the list liand returns this list. The original list remains unchanged! (car li) returns the ﬁrst element of the list li (cdr li) returns the list liwithout its ﬁrst element (connectto a x y z) searches the system for an object at the position (x, y, z ) and assignes the variable ato it. (delete a) removes element afrom the variable list and deletes it (compare remove ). (edit a b) changes the value of an object or valence ato the SyCL value b. (enzyme expr) deﬁnes a new enzyme from the expression expr.exprhas to be of the form xxxx=enz1enz2enz3.... and must be quoted in order not to be executed as a command! The newly deﬁned enzyme is automa tically inserted into the enzymelist as well as the enzyme menu from t he GUI. 21(enzymes) returns the list of enzymes. (eval expr) evaluates the expression expr. (equal a b) or (eq a b) returns 1 if aandbare equal, an empty list oth- erwise. (for (n i0 i1) ( expr1 expr2 expr 3 ... )) evaluates the expressions expr1, expr2 andexpr3 ... with the variable ntaking integer values from i0 toi1 (funlist) returns the list of implemented functions. (greaterthan a b) or (gth a b) returns 1 if ais greater than b, an empty list otherwise. (if expr ( expr1 expr2 expr3 ... )) evaluates the expressions expr1, expr2 andexpr3 ... ifexpris true. (length li) returns the number of elements of the list list. (lessthan a b) or (lth a b) returns 1 if ais less than b, an empty list otherwise. (list a b ... ) returns a list with the parameters a, b, ... als elements. (load ﬁlename) loads and evaluates commands in a ﬁle ﬁlename . This can be done more comforably with a ﬁle-select box via the GUI open ﬁle menu entry. (minus a b) (- a b) returns the diﬀerence of aand it b. (newo a) creates a new SyCL object of SyCL value aand returns an object element. (notequal a b) or (neq a b) returns an empty list if aandbare equal, 1 otherwise. (nth n li) returns the nth element of a list li. Counting starts at zero! (nthval n a) returns the nth valence of the valences directed toward the object a(counting from 0). (oblist) returns the list of variables. (poso a x y z) repositions the object designated by variable aat (x, y, z ). (plus a b) (+ a b) returns the sum of aandb. (rand n) returns an integer random number between zero (inclusive) a ndn (exclusive). (remove a) removes element afrom variable list (compare delete). (replacenth n li expr) replaces the nth element of a list libyexprand returns this list. The original list liremains unchanged! (run ’(e v)) lets the enzyme eact on the source of valence v. (set a b) same as setq, but evaluates aﬁrst. (setl l v a b) aandbhave to be object elements. An unidirectional valence 22is created between aandbwith a SyCL values v. The valence element is assigned to the variable l. (seto a [’int \|’real\|’bool \|’enz] [’vector m \|’covector n \|’matrix m n] b... [x [y [z]]]) creates a SyCL object with a SyCL value b(evaluated) and assigns the object element to the variable a. Creating a vector of dimen- sion m, a covector of dimension n, or a matrix of dimension mxnrequires mxnelements b!x, y, z can be appended to specify coordinates. (setq a b) creates an expression, sets it to b(evaluated) and assigns it to the variable a. (setv v v0 v1 a b) aandbhave to be object elements. A bidirectional valence is created between aandbwith SyCL values v0andv1for the valence and its adjoint. The valence element is assigned to the varia blev. (times a b) or (* a b) returns the product of aandb. (valences a) returns the number of valences directed toward an object a. The following script may serve as an example of motion along a valence vbetween two objects aandc: (seto a 1) (seto b 2 100) (seto c 3 0 100) (setv v 1 1 a b) (setq w (adjoint v)) (setv m 1 1 a c) (setq q (adjoint m)) (enzyme ’move=_AMR_RMV{?nAD}_PRS_RFU) (move ’(a m)) The last command invokes the action of the above deﬁned enzym emovewith object aand valence mas parameters. 8 Some basic Processes 8.1 Enzymatic Copying Let us call a link bidirectonal if its adjoint is also a link, unidirectional other- wise. Note that the terms are used to characterize individua l links, not pairs (link, adjoint link). 23- -- - =⇒ -+- s + - s -- Figure 5: Replication fork dynamics. There exists an enzyme, call it the splitFork-enzyme, whose continued conditional application to a ﬁnite system Swhose links are all bidirectional, produces after some ﬁnite time two copies of S. This is a generalization of the asymmetrical replication fo rk mechanism which the living cell uses to copy DNA [2]. It works not only fo r chains of pairs of bidirectional links (which mimick the double helix of DNA), but for anysystem whose links are all bidirectional, independent of its topol ogy. [All the interna like (possibly overlapping) internal stru cture of constituent objects, membranes and pathLinks marking various subsyste ms are copied as well]. The operation on chains is shown in ﬁgure 5 The descr iption of the action sXof the splitFork-enzyme to the radical of object Xis as follows. 1. A copy X′ofXis made. 2. The links incident on Xother than loops are distributed among Xand its copy as follows: - bidirectional links with target XgetX′as their target - unidirectional links with target Xretain Xas their target - bidirectional links with source Xretain Xas their source - unidirectional links with source XgetX′as their source The loops X/mapsto→Xremain in place and get a copy X′/mapsto→X′. 3. The adjoints of formerly unidirectional links are promot ed to the status of links. One may imagine that the copy X′is connected by a pair of adjoint bidirec- tional identity links to Xto begin with. Then the management of the links can be interpreted as instances of motion - multiplication w ith identity links and creation of fundamental adjoints. In the end, the identi ty links between XandX′are killed. This decomposition of the splitfork-enzyme int o an action of micro-enzymes is illustrated in ﬁgure 6. 24_VML{?iAD} _MAD _PRS _RFU __RMV{?iID}_CPO _AMR{?nLK} Figure 6: One step within a sweep of the SplitFork enzyme: sFRK = CPO AMR{?nLK}VML{?iAD}MADPRSRFURMV{?iID}Sag{?nAD} Deﬁne a fork at Xas a pair of links, both unidirectional, one with target Xand the other with source X. Theorem 1 (Universal copy constructor) LetS0be obtained from a ﬁnite connected system Swhose links are all bidirectional by action of sX0at some X0∈S. For t >0, letStbe obtained from St−1by action of sXfor all objects Xsuch that there is a fork at X. Stis well deﬁned for t≥0. For suﬃciently large t, it is independent of t and consists of two disconnected copies of S. “Once replication has started, it continues until the entir esystem has been duplicated”. Upon substituting “genome” for “ system “ this becomes a quote from a genetics text book [17]. This theorem was ﬁrst demonstrated in [20]. The fact that sXis mathematically well deﬁned is somewhat subtle. It rests on a theorem, proven in [21], that a system can be speciﬁed up to isomorphism by enumerating its arrows, which of them can be c omposed 25and which arrow is the result. One must also indicate which ar rows are links and what are their adjoints. But nothing need be said about ob jects - they can be reconstructed. Using this, the above theorem can be pr oven, with the understanding that the phrase “two copies of S” means “two system s, both isomorphic to S”. The isomorphism class of Sdoes not retain the information about the internal structure of non-atomic obj ects (black boxes ). But this information can be retained by the copies. To retain it, one uses the universal copy constructor to copy objects of Swhich are themselves system s, to copy their non-atomic constituents, and so on. This doe s not continue ad inﬁnitum by the axiom of non-self-inclusion. The fact that the dynamics is well deﬁned rests on the fact tha tsXsY= sYsXfor the kind of system s that occur as St. The action of the splitFork-enzyme is quite robust against e rrors due to computer failures which mimic local mutations. But there is one exception which leads to catastrophic results of the type of Down’s syn drome - a third copy is made of part of the system . This happens when a fundamental adjoint gets lost (or added) “at the wrong moment”, cf. [20].6 There is a more ﬂexible (but not quite as axiomatic) version o f the splitFork-enzyme which can be used to restrict copying to su bsystems bounded by membranes. It employs membranes in place of the prototypi cal membrane made by unidirectional links. 8.2 Digestion In animals, building blocks of newly made structures must ﬁr st be prepared by degrading ingested organic material by metabolic proces ses. Here we describe an enzyme whose continues action dXat an object X of a connected system degrades its internal structure in the manner shown in ﬁgure 8.2. dXconsists of consecutive steps. 1. (Death) The far side of all triangles of 3 links with tip Xis removed, together with their adjoints. 2. (Motion) If bis a link from Y/ne}ationslash=XtoXandb′is a link from Y′toY thenb◦b′becomes a link and b′ceases to exist as a link. 3. Fundamental loops X/mapsto→Xare removed. 6In man, Down’s syndrome is caused by the presence of three cop ies of chromosome no. 21 instead of the usual two 26X ⇒X ⇒X ⇒X Figure 7: Digestion enzyme attacks at X Actually the 3rd step can be dispensed with when step 2 operat es also forY=X. 8.3 Reaction Diﬀusion Restrict attention to those values of objects which can be ad ded and sub- tracted. Consider a radical rwith object Xand its valences vwith source objects Yv. There is an enzyme which does the following. For every valence vwith source Yv(or those which fulﬁll the condition set by a predicate) it takes a fraction α∈Rof the original value ξofX, subtracts it from ξ, parallel transports it to Yvalongvand adds the result to the value ηvofYv. Conversely, it takes a fraction αof the original value ηv, subtracts it from ηv, parallel transports it to Xand adds the result to ξ. In the most important examples, the parallel transport is t rivial (i.e. ξ,η are invariants). This models diﬀusion. One may add loops X/mapsto→Xwhich eﬀect nonlinear maps of the values of Xas “parallel transports”. This models reactions, e.g. chem ical reactions. 8.4 Shock fronts The splitFork dynamics is an example of the mechanism of prop agating shock fronts. It is an abstraction of the most essential feature of shock fronts in nature.7In the splitFork dynamics, there is at any time a bipartite bo undary between the part of the system which has not yet been copied, and the two 7Shock fronts occur in nature as a consequence of nonlinearit y in excitable media [7] when the passage of an excitatory event is followed by a dead t ime. It has the consequence 27copies of the rest. The two boundaries with the two copies hav e opposite orientation. They are made of the unidirectional fork prong s. The excitation (sFRK -action) is restricted to radicals which have forks, and it c an only pass through bidirectional links into the part of the system which has not been copied yet. Similarly, a single boundary made of unidirectional links m ay be made to propagate unidirectionally. There is an enzyme aFRK to do that. It generates sweeps through the system. Making an enzyme chain with head aFRK and some tail will make the tail act on every radical rof thesystem . This may be used to induce relaxation sweeps, for instance. 8.5 Cellular growth Cell replication can be achieved by realization of the follo wing idea: A cell Agets triggered to replicate, for instance by a certain value of a gradient to a neighboring cell B. A copy CofAis made and placed between A andBin order to smoothen the gradient. Therefore the fundamenta l links between AandBare replaced by new fundamental links between BandC. Furthermore Cobtains new fundamental links to all neighbors, AandBare both connected to (see ﬁgure 8). 8.6 Assembly of data Consider a ﬁnite system with a distinguished radical root. We describe another shock wave mechanism, call it assemblyFork, which c an be used to assemble data from the system at root. First one sends out a shock wave which builds a spanning tree - details are given below. Take any data which are of type AlgebraicValue . They may be attached to objects or links. The assemblyFork-dynamics computes th e sum of these algebraic values, parallel transported along the branches of the tree, and deposits it at root. For instance, the data maybe sets of strings8, as occur in encodings of that the excitation can only propagate in the outward direct ion because a renewed exci- tation is prohibited during the dead time. Forest ﬁres are a w ell known and much studied example [9]. Natural neurons also have a dead time after ﬁrin g. 8sets become lists by lexicographic ordering 28_AMR {?nAD\|\|?nLK}_MVV _MAD_PRS_RFUfALL{?hLS} (_MAV) _CPO Figure 8: Cell replication in a system theoretic diagram: cell = MVV fALL {?hLS}(MAV)CPOAMR{?nAD\|\|?nLK}MADPRSRFU data in XML , where ◦is concatenation of strings, and ⊕is disjoint union of sets. No loss of information is involved in taking the sum ( disjoint union) in this case. Or the data may be the (real) elements of a max-plus -algebra, where addition amounts to taking the maximum, and multiplication is addition of real numbers. The maximum of all the data is computed in this c ase. If the data are invariants (cf. earlier) then, by deﬁnition, it would make no diﬀerence if parallel transport were along any path to root other than the path along the tree.9 When a shock proceeds up along any of the branches of the tree, a leaf will eventually be reached where no prolongation of the bran ch is possible without creating a loop. Then the shock gets reﬂected and pro pagates down the tree, taking data from the leaf along via parallel transp ort (cf. section 5.2). At any node of the tree, the shock has to wait until the re ﬂected shocks 9More generally, if all links are unitary, i.e. if b∗=b−1, the results diﬀer at most by a gauge transformation, cf. [21]. 29from all the branches are in. Then the data which they transpo rt are all added up and the result is transported further down the tree. Themax-plus version of this mechanism is important when one does op- timization by relaxation. One needs to know the maximum of a m easure of the local error in order to know when to stop. Let us ﬁnally describe how the spanning tree is made and marke d. In principle, the order of the valences in the valence list of a r adical is irrelevant. But we may either make an exception for the purpose of the asse mblyFork- mechanism, distinguishing the ﬁrst valence as a “principal port”. Or we may use paths to mark the branches of the tree. Suppose we do th e ﬁrst; generalization will be obvious. Given a distinguished radical root, thesystem decomposes into shells labeled by path-distance10n≥0 fromroot, and links between adjacent shells. The shells are like the shells of an onion. The n-th shell is a subsystem which contains all objects with path distance nfromroot, and the links between them. A shock emanating from rootpropagates from shell to shell in order of increasing n. When starting at shell 0, the tree has no link. When the shock reaches shell n, there is a membrane = boundary between shellnandn+ 1, and the tree nwill have been constructed up to shell n. Now one selects in some arbitrary way links from the boundary such that each object Xof the n+ 1st shell is source of exactly one such link. This can be done by local operations. The adjoint of the unique lin k with source Xis made principal port of X. In this way the tree has been grown to shell n+1. Now one continues. In a ﬁnite system , there are ﬁnitely many shells. So the process comes to an end. 9 Outlook There exists also an (exterior) diﬀerential calculus and ge ometry on system s [8, 21]. It is intermediate between standard exterior diﬀer ential calculus and geometry on manifolds and non-commutative diﬀerential calculus and geometry [5]. It satisﬁes d2= 0 and the Leibniz rule, the algebra of functions is commutative, and derivatives of functions are ordinary ﬁ nite diﬀerence 10path distance was deﬁned in subsection 5.1 30derivatives. It has not been implemented in software yet, bu t we plan to do so in the future. Another line of current investigation is multiscale analys is [13]. By deﬁ- nition, genuinely complex systems show “emergent” propert ies that are not shared by their subsystems with few objects. The basic idea o f multiscale analysis, including multigrid methods[13] and the renorma lization group [29], is that although a genuinely complex system can not be unders tood as a whole by studying reasonably small subsystems in isolation, a com plexity reduction is often possible by doing so. To this end one introduces new o bjects which represent subsystems, but one retains only that part of the i nformation on their internal structure (including functionality, i.e. e nzymes) that is rele- vant for the cooperation that causes emergent - i.e. nonloca l - phenomena. A related plan is known in information science under the name ofintegration [10]. Some theoretical considerations concerning this wer e presented in [21] and in [28]. Relaxation sweeps can be implemented by enzymat ic compu- tation, and it is straightforward to extend them to system s which have a multilevel structure like multigrids. The challenge is to e xtend the domain of applicability of existing multigrid and renormalizatio n group technology, including in particular disordered systems. Finally it is a challenge to invent an “enzymatic game of life ”, and to implement within the present frame as much of biochemical an d larger scale processes that constitute life. This would be in the spirit o f work on Artiﬁcial life [16]. 9.1 Towards an infome project A much more challenging and long range plan would be an infome project which amounts to mapping into software what is known about st ructure and function in the living cell in such a way that processes that c onstitute life can be simulated in silico . After the genome project and the envisaged proteome project [32] which is supposed to classify all proteins that a human body makes in its lifetime and their function, this is a logical ne xt step. Work on immunology along similar lines is in progress [23]; t he establish- ment of a data base has been the proposed. For a discussion of t he cognition problems that are of interest in immunology see [24]. 3110 Acknowledgement We thank the Deutsche Forschungsgemeinschaft for ﬁnancial support, and the German Israel foundation for a travel grant. We would lik e to express our thanks to Achi Brandt, Irun Cohen, Martin Meier-Schelle rsheim, Dirk Rathje, Sorin Solomon, and York Xylander for numerous enlig htening dis- cussions. We also thank Daniel L¨ ubbert, Bleicke Holm, Marc us Speh and York Xylander for their collaboration in earlier stages of t his project. References [1] L.M.Adleman, Molecular Computation of Solutions to Combinatorial Problems , Science 2661021 (1994) [2] B. Alberts, D. Bray, J. Lewis, M. Raﬀ, K. Roberts, J. Watso n. Molecular Biology of The Cell. Garland Publishing, Inc. 1989. 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1The 1st Law of Thermodynamics in Chemical Reactions I. A. Stepanov Latvian University , Rainis Bulv. 19 , Riga, LV-1586 , Latvia Abstract In the previous papers of the author it has been shown that the 1st law of thermodynamics in chemical reactions is the following one: ΔU=ΔQ+PΔV+ i∑µiΔNi In the present paper this theory was developed and it has been shown that the 1st law of thermodynamics in chemical reactions has the following form: ΔC=-ΔU+ΔA -ΔU=ΔQ where ΔC is the change in the chemical energy, ΔU is the change in the internal energy. Internal energy is the energy of thermal motion of molecules. 1. Introduction Earlier it was supposed that the 1st law of thermodynamics in chemical reactions is the following one: ΔU=ΔQ-PΔV+ i∑µiΔNi Recently it has been shown [1-6] that it must have the following form:2ΔU=ΔQ+PΔV+ i∑µiΔNi In the present paper this theory is developed. 2. Theory The internal energy of substance is its full energy. However, chemical reaction is the result of change only in the chemical energy. In substances there is the chemical energy which is potential energy and during chemical reactions turns to mechanical work, heat or electricity. Chemical energy is the energy of external electronic shells. Let's denote the change in the chemical energy ΔC. Chemical energy can turn only to the energy of thermal motion of molecules ΔW and to work ΔA. Then the first law of thermodynamics for chemical reactions can be written as ΔC=-ΔW+ΔA -ΔW=ΔQ Let's denote ΔW by ΔU. Then the first law of thermodynamics will look like ΔC=-ΔU+ΔA -ΔU=ΔQ The key idea of this paper is that chemical reaction is due to the change only in the chemical energy. Other types of energy remain constant. On sees that i n chemical reactions the internal energy of substance is the energy of thermal motion of molecules but not its whole energy. Pay attention that the heat of chemical reaction is the change in the internal energy of the system in isochoric and isobaric cases. Earlier it was supposed that the heat of chemical reaction in isobaric case is the difference in the enthalpy.3Let's derive the second law of t hermodynamics for chemical reactions. δC is not an exact differential because it is the sum of an exact and not an exact differential. But δC/T will be an exact differential because dU/T+δA/T is an exact differential. Therefore, one can introduce a new function, let's call it chemical entropy SC. Then dSC ≥ δC/T=-dU/T+δA/T (1) Let's suppose that A is the work of expansion: δA=PdV . It is work done by chemical system. Then, if P and T are constant d(-U+PV-TS C)≤0 The Gibbs energy in chemical reaction is G=-U+PV-TS C (2) Correspondingly, one can show that the Helmholtz energy in chemical reactions is F=-U-TSC From (1) it follows that in reversible processes TdSC=-dU+δA (3) According to (2) dG=-dU+PdV+VdP-TdS C-SCdT (4) Introducing (3) to (4), taking into account that δA=PdV one gets dG=VdP-S CdT Whence (∂G/∂T)P=-SC (5) and (∂G/∂P)T=V There are numerous articles where dependence ΔG° on T was measured, for example [7-12]. It is the following one:4ΔG° =a+bT (6) where a and b are constants. Accuracy of (6) is very high. Eq. (6) is true in temperature intervals a few hundreds grades, d ΔG°/dT=b=const. But (∂ΔG°/∂T)P =-ΔS° can not be constant because (∂ΔS/∂T)P=(CP2 - CP1 )/T where CPi are heat capacities of the products and of the reactants. If ΔS°is constant then CP2=CP1 which is an absurd. Using (6), it is impossible to calculate ΔCP. 3. Experimental Check and Discussion In [11] the following reaction was given: CdS(sol)=Cd(liq)+S(liq), T=640-690 K ΔG0=-164295+31,37T −+310 J/mol (7) Eq. (7) can be written in the form ΔG0=a+bT+cT2+δ (8) where cT2+δ≤Δ, Δ is the absolute mistake of (7). From (8) it is easy to estimate ΔCP. For (7) Δ CP<0,9 J/mol⋅K. Using [13], it is possible to calculate that ΔCP=8,5 J/mol⋅K. In [12] the following reaction was considered: 0,95Fe+1/2O 2=Fe0,95O 1120<T<1320 K and the change in the Gibbs energy for it: ΔG0=-266358+63,10T −+629 J/mol5One obtains from (8) that Δ CP<0,95 J/mol⋅K. Using [13], one may calculate that ΔCP=8,5 J/mol⋅K. It is not surprising: the change in the entropy of chemical reaction will be ΔC/T, not ΔQ/T. ΔC depends on the temperature not so as ΔQ does. For chemical reactions (∂ΔG°/∂T)P≠-ΔS° hence it is another evidence that thermodynamics of chemical processes is not such as that of physical ones. From (6) and (5) it follows that (∂ΔG°/∂T)P=-ΔSC0 =const It is not surprising that ΔSC0 =const and ( ∂ΔSC0 /∂T)P≠ΔCP/T. The reason is that ( ∂ΔC/∂T)P≠CP and δC/T≠dS. One can draw the following conclusions. The traditional thermodynamics is available only for description of physical processes, for description of chemical reactions another thermodynamics is necessary. Physical and chemical phenomena are qualitatively different things. The 1st and the 2nd laws of thermodynamics, Gibbs and Helmholtz energies for chemical reactions have another form than these for physical processes . References 1. I. A. Stepanov , DEP VINITI, No 37-B96 , (1996). Available from VINITI , Moscow. 2. I. A. Stepanov , DEP VINITI, No 3387-B98. (1998). Available from VINITI , Moscow. 3. I. A. Stepanov , 7th European Symposium on Thermal Analysis and Calorimetry . Aug. 30 - Sept. 4. Balatonfuered , Hungary. 1998. Book of Abstracts. P. 402-403.64. I. A. Stepanov , The Law of Conservation of Energy in Chemical Reactions. - http://ArXiv.org/abs/physics/0010052 . 5. I. A. Stepanov , The Heats of Reactions. Calorimetry and Van't-Hoff . 1. - http://ArXiv.org/abs/physics/0010054. 6. I. A. Stepanov , The Heats of Dilution. Calorimetry and Van't-Hoff . - http://ArXiv.org/abs/physics/0010075 . 7. A. Nasar and M. Shamsuddin, Therm. Acta, 1992, 205, 157. 8. M. Shamsuddin and A. Nasar, High Temp. Sci., 1989, 28, Spec.Vol ., 245. 9. H. Chou, H. S. Chen, W. C. Fang and P. L. Trevor, J. Electrochem Soc. , 1992, 139, 3545. 10. A. Azad, R. Sudha and O. Sreedharan, Therm. Acta, 1992, 194, 129. 11. A. Nasar and M. Shamsuddin, Therm. Acta, 1992, 197, 373. 12. V. A. Levitski and T. N. Rezuhina, Zhurnal Fizicheskoy Khimii, 1963, 37, 1135. [English translation: J. Phys. Chem. USSR ] 13. I. Barin, Thermochemical Data of Pure Substances , VCH, Weinheim, 1989.
Poster for Vienna 10-14.11.2000Nakhmanson 1/7The J. S. Bell's theorem is based on the following assertion: if P a is a probability of result a measured on the particle 1 in the point A having a condition (e.g. angle of analyzer) α , and P b is a probability of result b measured on the particle 2 in the distant point B having a condition β , then β has no influence on the P a , and vice versa. Here Bell and others saw the indispensable requirement of local realism and "separability". Mathematically it can be written as Pab(λ1i,λ2i,α,β) = P a(λ1i,α)×Pb(λ2i,β) (Bell) , (1) where P ab is the probability of the joined result ab , and λ1i and λ2i are hidden parameters of particles 1 and 2 in an arbitrary local-realistic theory. Under the influence of Bell's theorem and the experiments following it and showing, that forentangled particles the condition (1) is no longer valid, some scientists rejectlocality. In this case an instantaneous action at a distance is possible, and one canwrite Pab(λ1i,λ2i,α,β) = P a(λ1i,α,β)×Pb(λ2i,β,α) (non-locality) . (2) In principle such a relation permits a description of any correlation between a and b, particularly correlation predicted by QM and observed in experiments. But in the frame of local realism the condition (1) is not indispensable. Instead, one canwrite ∗∗∗∗ Pab(λ1i,λ2i,α,β) = P a(λ1i,α,β´)×Pb(λ2i,β,α´) (forecast) , (3) where α´ and β´ are the conditions of measurements in points A and B , respectively, as they can be forecast by particles at the moment of their parting. If the forecast is good enough, i.e., α´ ≈ α and β´ ≈ β , then (3) practically coincides with (2) and has all its advantages plus locality. Strictly speaking we suppose the particles possess some kind of consciousness. This idea has old tradition and seems very natural as compared with "many ∗ Raoul Nakhmanson Frankfurt am Main, GermanyE-mail: Nakhmanson@t-online.deVery promising hole in Bell's theorem R. Nakhmanson* Through the hole in Bell’s theorem we can communicate with matter.John Stewart Bell (1928-1990 ) "Imagination is more important thanknowledge.Knowledge islimited.Imagination encirclesthe world." - Albert Einstein (1879-1955)Poster for Vienna 10-14.11.2000Nakhmanson 2/7worlds" or "nonlocality". Such a "hidden parameter" missed by Bell can forecast future and provide EPR-correlation. The idea leads to an alternative local-realistic interpretation of quantum mechanics (QM): In space there is a particle, its wave function is a product of itsconsciousness processing the known information about the world to optimize theparticle’s behavior. In other words, the wave function is not in the space but in theconsciousness of the particle and is its strategy. This explains why it does notaffect other particles being in the same 3d-space. If there are many particles, theirdistribution, e.g. interference fringes, looks like a product of real wave in realspace. If two or more particles have a common strategy they are "entangled" as long as they can forecast the future at the moment of their parting. The new-comingunlooked-for circumstances allow them gradually to cut off and forget the oldpartnership. If the particle receives new information, it corrects the strategy, that is, the so- called "collapse" of wave function occurs, local and instant "FAPP". Wavefunction gives only a distribution of priorities. Taking this into account theparticle makes its choice. The optimal tactics of proportional proving of allpossibilities is randomization of this choice. Etc., including explanation of allQM-paradoxes. Can this idea be proved? There are two possibilities: 1. To destroy informational channels of particles. So, in 1982 Aspect et al. [1] in Paris switched analyzers very fast to cut off any subluminal informationalcontact between EPR-particles. They have confirmed the results of previous"static" experiments. Because of technical limitations they used periodicalswitching instead of a random one being desirable to prevent a forecast of theanalyzer state. Sixteen years later Weihs et al. [2] in Innsbruck used the randomgenerator - with the same result. These works, of course, were only first attempts in the complex, quaking, and provocative field where particles have much more experience as overweening"aggressors" impeding their life. For example, Weihs et al. wrote: "Selection of ananalyzer direction has to be completely unpredictable, which necessitates aphysical random number generator. A pseudo-random-number generator cannot beused, since its state at any time is predetermined" (p.5039). As a "physical randomnumber generator" they used a "light-emitting diode illuminating a beamsplitterwhose outputs are monitored by photomultipliers" (p.5041). But particleseveryday contact with "physical" generators and nevertheless forecast future. Ifparticles possess consciousness such "physical" generators are rather pseudo-Poster for Vienna 10-14.11.2000Nakhmanson 3/7random-number ones too. Perhaps a good "human" pseudo-random generator is for experiments preferable because it belongs to another civilization... 2. A more polite and interesting possibility is informational contact with particles, an attempt to speak with them. Fig. 1 introduces into the field. Fig. 1( a) shows a "black box" which is tested by the linearly polarized light beam. Inside ofthe box the beam meets a thick transparent glass plate fixed at the Brewster angleso that all photons pass the box. The glass plate manifest itself physically only by space shift ∆z and time delay ∆t of output photons. Further there is a movable mirror ("traffic divarication") which is controlled by the experimenter to turn ornot to turn the beam. Such a control is a brutal one like a traffic barrier closing oneof two branches of the road. /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 ∆z ∆tb( ) S /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 ∆z ∆tc( ) SM/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 /i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0/i0 ∆z ∆ta( ) S Fig. 1. ( a ) - "brutal" control, ( b ) and ( c ) - informational control . In Figs. 1( b) and 1( c) the mirror is semi-transparent and immovable, and the thick glass plate is divided into eight thin plates, two of them being thicker thanthe remaining six. As before, the content of the black box manifests itself physically only by the same space shift ∆z and time delay ∆t . But if the photons are intelligent and know English and Morse code, they can read the messages,namelyPoster for Vienna 10-14.11.2000Nakhmanson 4/7 • — • • • • — • = REF (reflect) in Fig. 1( b) , — • • • • • — • = THR (through) in Fig. 1( c) , and follow the instructions. Such a control is an informational one like traffic signs on the road. It is important to emphasize that the idea of informational experiments with particles, as it seems, has never been publicly discussed, and all experiments madewith particles up to now cannot be considered as informational ones even inretrospect, that is, the revision of their results would not enable us to make anyconclusion relating to this idea. 321 321 321 321 161 161 161 161 161 161 81 8181 81 818181 81 41 41 41 41414141 414141 21 21 100 0 00 0 000 000 0000 000 00 00 00 00 00 00 0 Fig. 2. Binary-tree experiment. Circles stand for beamsplitters, rectangles denote informational cells. Ciphers show the probability of detecting theparticle in the case of the most rapid formation of a rigid, conservativeconditioned reflex. Fig. 2 shows the scheme of "binary-tree" experiment which does not require the particle’s knowledge of English, Morse code, etc. The initial beam of microobjects(particles, atoms) enters into a system of beamsplitters (shown by circles). TheyPoster for Vienna 10-14.11.2000Nakhmanson 5/7can be semi-transparent mirrors for photons, crystals for electrons or neutrons, Stern-Gerlach apparatuses for atoms, etc. Fig. 2 shows only five floors ofbeamsplitters, but there can be as many as experimentally feasible. According topresent-day theoretical ideas and practical experience, each of the output beamshas the same intensity, namely, 1/32 of intensity of the initial beam (realbeamsplitters may have, of course, some absorption, but here it is not a matter ofprinciple). Into each of the right branches of the binary tree (e.g. corresponding to reflection for photons) is introduced an "informational cell" (shown by rectangles),which is a device leaving unchanged the intensity of the beam passing it, butoffering some information to particles. For polarized photons such a cell may beagain a set of transparent plates fixed at the Brewster angle, and the informationcan be coded, say, by differences in the materials of plates, their thickness, anddistances between them (as in Figs. 1( b), 1( c)) . The information in each subsequent floor is a sequel of that in the preceding floor. As a whole it can bemusic and some course of studies of language for following communication likeones being developed for the project "Search for Extra-Terrestrial Intelligence(SETI)". The commonly accepted point of view is that the introduction of information cells will not change the uniform probability distribution of particles in the outputbeams. But if particles have consciousness, and are able to notice the informationoffered to them, they may become interested in it. After a number of floors, theparticles should notice that the information is offered only in the right branches,and should prefer to choose their passing through the following beamsplitters. Inother words, particles could develop a "conditioned reflex", of essentially the samekind as in behavior experiments on living beings. Such an inquisitiveness of particles should lead to a change of their distribution in the output branches. For example, if the conditioned reflex appears immediatelyafter comparison of right and left branches and particles have no interest to the leftones anymore, the distribution of probability to find the particle in differentbranches of the binary tree is like the one shown in Fig. 2 by ciphers. Deviation from the uniform distribution of particles in the output beams will mean that the particles at least recognize the information offered and have aninterest in it. Such an interest is thought to be an inherent attribute of eachconsciousness. This, however, still does not mean that the particles understandthis information: people of modern times were interested in ancient hieroglyphicsymbols long before they learned how to interpret them. To establish adeciphering stage, one can, starting from some floor of a binary tree, introducesome specific "requests" into the information cells. For example, one can "ask"Poster for Vienna 10-14.11.2000Nakhmanson 6/7particles to choose a left channel after the next beamsplitter rather than a right one. Because between the output branches of the binary tree and the trajectories of theparticles there is a one-to-one interrelation, the honoring of such requests caneasily be detected by an experimenter. However, the possibilities of an experimenttypified in Fig. 2 are not exhausted by this. Purposefully choosing direction ateach subsequent beamsplitter, the particle, in its turn, can send information to theexperimenter using "right" and "left" as a binary code. For example, extreme leftand extreme right trajectories in Fig. 2 present 00000 = 0 and 11111 = 31,respectively. 12 34«««D24D23 SI S M Fig. 3 Informational experiment with a single atom. 1, 2, 3, and 4 are the energy levels; D23 and D24 are detectors, S is the source of light, M is the modulator, SI is the source of information . The experiments illustrated in Fig. 1 and Fig. 2 can be called "coordinate- impulse" ones to distinguish them from the "energy-time" experiment whosescheme is shown in Fig. 3 . Here a four-level quantum system, e.g., an atom, withone low (1), one high (2), and two intermediate (3,4) energy levels is pumped by intensive radiation inducing the 1 →2 transition, so that the atom does not stay in the state 1 but immediately is translated into the state 2. From it, the atom makes atransition spontaneously to the state 3 or 4, and later makes a transition to the state1 completing the cycle. The radiation corresponding to some of the transitions 2→3, 2→4, 3→1, and 4 →1 are detected (in Fig. 3 two detectors are shown). Besides, there is an informational action on the atom, e.g., by modulation of light coming from the source S . The modulator M is controlled by the source ofPoster for Vienna 10-14.11.2000Nakhmanson 7/7information SI , which, in turn, is connected with one or more detectors to close the feedback loop. The feedback works in such a way as to stimulate a channel and rate of transitions, in the case of Fig. 3, the 2 →4→1 transitions. The source SI sends a message, e.g., one line of a page or a measure of a music, only if it receives a signal from detector D24 . Each next message continues the previous one, i.e., is the next line or the next measure. If the atom has a consciousness and is interested in the information being proposed, it develops a conditioned reflex and will prefer the 2 →4 transition to the 2→3 one. Besides, the rates of both 2 →4 and 4 →1 transitions must increase. All this can be registered by the experimenter. To be sure that the effectis connected with information, one can make a control experiment to cut off thefeedback or/and to use some "trivial" information, etc. Like with the scheme of Fig. 2, in the last case one may hope to observe not only an interest of a quantum object to receive a new information, but deciphering italso, as well as the sending of messages from the object to the experimenter beingcoded in states of the atom and time intervals between the states. In 1876 A. G. Bell invented telephone for communication between people. Eighty five years later J. S. Bell left a hole in his theorem to give us possibility tocommunicate with matter. References [1] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982). [2] G. Weihs, T. Jennewein, Ch. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998). See also: http://xxx.lanl.gov/pdf/physics/0004047 and http://xxx.lanl.gov/pdf/physics/0005042 . Note: This poster was submitted for the conference in commemoration of John S. Bell “Quantum [Un]speakables”, Vienna, Austria, 10-14 Nov. 2000
arXiv:physics/0011023v1 [physics.comp-ph] 13 Nov 2000Characterisation of the probabilistic travelling salesma n problem Neill E. Bowler1,Thomas M. Fink2, Robin C. Ball3 1,3Department of Physics, University of Warwick, Coventry, CV 4 7AL. 2Theory of Condensed Matter, Cavendish Laboratory, Cambrid ge, CB3 0HE. 1N.E.Bowler@warwick.ac.uk2tmf20@cus.cam.ac.uk3R.C.Ball@warwick.ac.uk (February 23, 2013) A new stochastic optimisation technique has enabled us to ex plore the probabilistic travelling salesman problem (PTSP) with large numbers of cities (hithe rto unobtainable). In this problem, the salesman must chose the ‘a-priori’ order in which to visi t the cities, before learning that certain cities do not need to be visited, and can be pruned from the tou r. The objective is to minimise the expected length of the pruned tour. We ﬁnd the mean length of t he pruned tour, for n≫1, follows E(Lpruned ) =√np(0.872−0.105p)f(np) where nis the number of cities, pis the probability of a city needing a visit, and f(np)→1 asnp→ ∞. The mean length of the a-priori tour is found to beE(La−priori) =/radicalbign pβ(p) where β(p) =1 1.25−0.82ln(p)is measured for 0 .05≥p≥0.6. Scaling arguments and indirect measurements suggest that β(p) tends towards a constant for p <0.03. I. INTRODUCTION The travelling salesman problem (TSP) is to ﬁnd the shortest tour around a number of cities, in which each city is visited once. For small numbers of cities this is an easy task, but the problem is NP-complete, meaning it is believed that there is no algorithm which can solve the problem in a time polynomial in the number of cities. The TSP is a paradigm amongst NP-complete problems. Consideration of the travelling salesman problem be- gan with Beardwood et al. [1]. They showed that in the limit of large numbers of cities nwhich are randomly distributed on the unit square, the optimal tour length (LTSP) follows [2] E(LTSP) =βTSP√n+α (1.1) where βTSPandαare constants. Numerical simulation [3] gives βTSP= 0.7211(3) and α= 0.604(5) as estimates when n≥50. Signiﬁcant divergence from this behaviour is found for n≤10, but numerical estimates can be found quickly (see appendix). The probabilistic travelling salesman problem (PTSP), introduced by Jaillet [4,5], is an extension of the travel- ling salesman problem (and hence of NP-complete prob- lems) to optimisation in the face of unknown data. Whereas all of the cities in the TSP must be visited once, in the PTSP each city only needs to be visited with some probability, p. One ﬁrst decides upon the order in which the cities are to be visited, the ‘a-priori’ tour. Subse- quently, it is revealed which cities need to be visited, and those which do not need to be visited are skipped to leave a ‘pruned tour’. The objective is to chose an a-priori tour which minimises the expected length of the pruned tour. It has been claimed [6] that, in the limit of large n, the PTSP strategy is as good as constructing a TSP tour on the cities requiring a visit, the re-optimisation strategy . Our results in ﬁg. 3 show that this is false. However, Jaillet [4] has shown thatlim n→∞/parenleftbiggE(Lpruned )√np/parenrightbigg =βpruned (p) (1.2) where βTSP< βpruned (p)< Min (0.9204,βTSP√p).(1.3) Jaillet [4] demonstrated various interesting properties of the PTSP, including an exact formula for the expected length of the pruned tour. If each of the ncities only needs to be visited with some (given) probability p, then the expected length of the pruned tour is E(Lpruned ) =p2n−2/summationdisplay r=0(1−p)rL(r) t (1.4) where L(r) tis the sum of the distances between each city and the following rthstop on the a-priori tour L(r) t=n/summationdisplay j=1d(j,(j+r)modn). (1.5) The expected length is averaged with respect to instances of the visiting list∗. Jaillet’s closed form expression for the expected tour length renders the PTSP to some ex- tent accessible as a standard (but still NP complete) opti- misation problem, and provides some check on the PTSP results by stochastic optimisation methods. One attempt to solve the PTSP using a standard op- timisation approach was taken by Laporte et al. [7] who introduced the use of integer linear stochastic program- ming. Although use of algorithms which may exactly solve the PTSP are useful, they are always very limited ∗i.e. the list of cities which actually need to be visited 1in the size of problem which may be attempted. Further- more, the stochastic programming algorithm even fails to solve the PTSP on certain occasions, thus the accuracy of any statistics concerning these results is dubious. Two studies have used heuristics to solve the PTSP [6,8]. Neither of these studies used global search heuris- tics, and were very restricted in the problem size at- tempted. The reason why a global search was not at- tempted by either author is due to a lack of computa- tional power. Equation 1.4 takes O(n2) time to calculate. Thus, to solve a 100 city problem for the PTSP would takeO(10,000) times longer than it would to solve a 100 city problem for the TSP. It should be noted, however, that it is only possible to make this comparison is due to the relative simplicity of the PTSP. For many more stochastic optimisation problems, standard optimisation techniques are not applicable. The stochastic aspect of the PTSP crucially distin- guishes it from the TSP. For stochastic optimisation problems the diﬃculty is not in the optimisation of a par- ticular function, but in the evaluation of that function. In the PTSP we are seeking to minimise the expected pruned tour length. In a general optimization context, we are seeking to minimise a given function, normally referred to as the cost function. When solving the TSP, heuristics are commonly used. One takes an initial tour, and considers making a series of improvements to that tour. Since improvements are often small changes (of- ten called a move), the eﬀect of this change can be cal- culated quickly and easily. For stochastic optimisation problems, this is not generally the case, and one may well be forced to use estimates of the eﬀect of a partic- ular move. This eﬀects the acceptance probability of a move, rendering ordinary optimisation methods ineﬀec- tive. New search methods are required which take into account the stochastic nature of the problem. The topic of stochastic optimisation problems was in- troduced by Robbins et al[9], and a basic heuristic for its solution. Since then, a number of alternative tech- niques have been proposed. We will restrict our atten- tion to global search heuristics, having in mind diﬃcult problems (for which exact algorithms would be inappro- priate) where we want to ﬁnd the global minimum of the problem. There are currently three such approaches to this problem, all based (loosely) on simulated annealing. Firstly, there is stochastic comparison [10] which is a development of a technique introduced by Yan and Mukai [11]. One takes a set of random problem instances†for both the initial and proposed conﬁguration, and com- pares these instances. If, for any of the instances the pro- posed conﬁguration has a larger value of the cost function than the initial conﬁguration, then the move is rejected. †In the case of the PTSP, an instance would be a particular list of cities that needed to be visitedAlthough it has been demonstrated that this method can converge to the global optimum, it lacks in one major area. As the number of instances considered becomes large, and one is more and more strenuous about which moves to accept, it becomes more and more likely that one rejects a favourable move (since the ﬂuctuations in the estimate of the change in the cost function will of- ten be much larger than the energy diﬀerence). Hence, not only does the algorithm become ineﬃcient at large numbers of instances, it also is incapable of supporting a quench. A second approach is based on a simple modiﬁed simu- lated annealing approach. This was ﬁrst was proposed by Bulgak and Sanders [12]. The principle behind this ap- proach is to estimate the cost function (using a variable number of instances) and accept all moves on the basis of the estimate of the cost function, using a standard simulated annealing acceptance rule. A theoretical anal- ysis of these methods has been provided by Gutjahr and Pﬂug [13] and Gelfand and Mitter [14]. Gutjahr et al. showed that provided the uncertainty in the estimates of the cost function at step nwere kept to below O(n−γ) (γan arbitrary constant greater than one), then the gen- eral convergence result [15] for simulated annealing still applies. Stochastic annealing, recently introduced by two of us, [16] diﬀers from the other approaches in two impor- tant areas. Firstly, the general concept of the other ap- proaches is one of minimising the eﬀect of noise in the cost-function evaluations. It is entirely natural, howeve r, that the noise from the inexact evaluations of the cost- function can be used as a positive eﬀect, rather than as something to be minimised. Secondly, this approach can be easily modiﬁed to give an exact reproduction of the simulated annealing algorithm. Although this possibility is not ruled out by the other approaches, we know of no attempts to perform a thermal annealing of the system. Our approach is as follows. We take rinstances of the problem, and from these estimate the average change in the cost-function (length in the case of the PTSP). We chose to accept all moves for which our average of the change in cost-function is negative (i.e. the move shortens the tour). If we assume that the average of these estimates is Gaussianly distributed about the exact change in the cost-function, with standard deviationσ√r, then it follows that the acceptance probability is [16] PG A→B=1 2/parenleftbigg 1−erf√r∆µ√ 2σ/parenrightbigg (1.6) where ∆ µis the exact diﬀerence in the cost-function be- tween states AandB. In a thermodynamic system which follows the Boltzmann distribution, the acceptance prob- abilities obey PA→B PB→A=e−β∆µ(1.7) where β=1 kBT. Stochastic annealing does not satisfy eq. 2FIG. 1. Typical near optimal a-priori PTSP tours with n= 300 for p= 0.1 and p= 0.5, respectively. 1.7 exactly, but closely approximates it. Using equation 1.6 and expanding in ∆ µwe obtain ln/parenleftBig PA→B PB→A/parenrightBig =1−erf√r∆µ√ 2σ 1+erf√r∆µ√ 2σ ≃ −βG∆µ−4−π 48(βG∆µ)3−. . .(1.8) where βG=√ 8r√πσ(1.9) identiﬁes the equivalent eﬀective temperature. The small coeﬃcient of the cubic term in eq. 1.8 makes this a rather good approximation to true thermal selection. Increasing sample size rmeans that we are more stringent about not accepting moves that increase the cost-function, equiva- lent to lowering the temperature. II. FORM OF THE OPTIMAL TOUR Optimal a-priori PTSP tours for small presemble an angular sort, as can be seen from ﬁg. 1. An angular sort is where cities are sorted by their angle with respect to the centre of the square. Bertsimas [6] proposed that an angular sort is optimal as p→0; we can show this to be false, since a space-ﬁlling curve algorithm is generally superior to an angular sort as n→ ∞. Such an algorithm was introduced by Bartholdi et al. [17] using a technique based on a Sierpinski curve. For the angular sort with np≫1,the probability of two cities being nearest neighbours on the pruned tour will be vanishingly small for cities which are separatedfrom each other by a large angle on the a-priori tour. This means that only cities that are separated by a small angle contribute signiﬁcantly to eq. 1.4. Thus for an n city tour, we may approximate L(r) t=Lon (2.1) where Lois some fraction of the side of the unit square, since cities which are sorted with respect to angle will be unsorted with respect to radial distance. This leads to E(Lang)≃Lonp2n−2/summationdisplay r=0(1−p)r. (2.2) Fornp≫1, we ﬁnd E(Lang)→Lonp. (2.3) By contrast it has been shown [6] that E(Lτsf) E(ΣTSP)≤O(log(np)) (2.4) where E(Lτsf) is the expected length of a tour generated by the Sierpinski curve. E(ΣTSP) is the re-optimisation strategy, whereupon discovering the actual city visiting list, one re-optimises for a TSP tour (i.e. E(ΣTSP) = βTSP√np). Hence, E(Lτsf) is given by E(Lτsf)≤O(βTSP√np log (np)) (2.5) which leads to E(Lτsf) E(Lang)≤O(log(np)√np). (2.6) 3So, for np≫1 and n→ ∞, the space ﬁlling curve is al- ways superior to the angular sort, demonstrating that the angular sort is generally never optimal. To summarise, it has been demonstrated that an angular sort is not op- timal for the PTSP. III. SCALING ARGUMENTS From diagrams of near-optimal PTSP tours such as ﬁg. 1, we propose that the tour behaves diﬀerently on diﬀerent length scales. The tour is TSP-like on the larger length scales, but resembles a sort at smaller distances. We may construct such a tour and use scaling arguments to analyse both the pruned and a-priori tour lengths of the optimal tour. Consider dividing the unit square into a series of ‘blobs’, each blob containing 1 /pcities so that of order one city requires a visit. The number of such blobs is given by N≃np (3.1) and for these to approximately cover the unit square their typical linear dimension ξmust obey Nξ2∼1. (3.2) Since each blob is visited of order once by a pruned tour, we can estimate the expected pruned tour length to be E(Lpruned )∼Nξ∼√np (3.3) which we will see below is veriﬁed numerically. We can similarly estimate the a-priori tour length to be ntimes the distance between two cities in the same blob. Thus, the expected a-priori tour length is E(La−priori)∼nξ∼/radicalbiggn p(3.4) which we will see is more diﬃcult to conﬁrm numerically. IV. COMPUTATIONAL RESULTS FOR THE PTSP Using the stochastic annealing approach of Fink and Ball [16], we have investigated near optimal PTSP tours for a range of diﬀerent numbers of cities, and various values of p. Eﬀective temperatures in the range kT= 0.07− −0.01 were used, giving sample sizes in the range r= 2− −500. Between 10 and 80 diﬀerent random city conﬁgurations were optimised (80 conﬁgurations for n= 30, 40 conﬁgurations for n= 60, 20 conﬁgurations for n= 90 and 10 conﬁgurations for n≥120 cities). Figure 2 shows the master curve for the expected pruned tour length divided by√np. The shift factors have a linear ﬁt, suggesting that11.051.11.151.2 0 50 100 150 200Pruned tour length npp=0.05 p=0.1 p=0.2 p=0.4 p=0.5 p=0.6 0.80.820.840.860.88 00.10.20.30.40.50.60.7Shift factor βpruned(p) pβ pruned (p) * np FIG. 2. The master curve for the pruned tour length di- vided by βpruned (p)√np. The data follows a smooth curve forn >30. The shift factors follow a linear relationship, suggesting thatE(Lpruned )√np(0.872−0.105p)=f(np). 11.021.041.061.081.11.12 0 50 100 150 200<Pruned tour length> npp=0.05 p=0.1 p=0.2 p=0.4 p=0.5 p=0.6<Re-optimised tour length> FIG. 3. The expected pruned tour length divided by the expected re-optimised tour length. This indicates the im- provement one would expect from re-optimisation. E(Lpruned )√np(a−bp)=f(np) (4.1) forn≫1, where a= 0.872±0.002,b= 0.105±0.005 andf(np)→1 for large np. Figure 3 shows the expected pruned tour length divided by the expected re-optimised tour length. This demonstrates that the pruned tour can be at least 10% worse than the re-optimisation strategy. However, the data for the shift factors (ﬁg. 2) indicates that the PTSP strategy can be no more than0.872 0.767−1 = 14%(±1%) worse than the re-optimisation strategy. The master curve for the a-priori tour length is shown in ﬁg. 4. The a-priori tour length does not show the ex- pected behaviour, as we would hope βa−priori(p) to tend towards a constant as p→0. The shift factors exhibit a slight but signiﬁcant departure from linearity with p. In particular and in potential conﬂict with our scaling arguments it is not clear whether βa−priori(p) tends to a non-zero constant as p→0. 40.9511.051.11.151.2 020406080100120140160180A-priori tour length npp=0.05 p=0.1 p=0.2 p=0.4 p=0.5 p=0.6 0.20.30.40.50.60.7 00.10.20.30.40.50.60.7Shift factor βa-priori(p) pβ a-priori (p) * n/p FIG. 4. The master curve for the a-priori tour length di- vided by/radicalbign pβa−priori(p). The shift factors, inset, are ex- pected to tend towards a constant for p→0. The slight, but signiﬁcant, deviation from linear suggests that this might not be the case. V. THE LIMITING CASE P→0 The above computational results for the a-priori tour length do not conﬁrm the predictions made by the scal- ing arguments. The scaling arguments should apply in the limits np≫1 and p≪1. Investigation of p≪1 is available using a simple adjustment to our previous method. For a 3 city TSP tour, it does not matter in which order the cities are visited. For a 4 city TSP tour, there are 3 distinct orders in which the cities may be visited. Thus, 4 city tours are the smallest at which it matters in which order the cities are visited. If we only generate visiting lists which contain 4 cities this will be (approximately) equivalent to choosing p=4 n. Thus 4 city tours provide an eﬃcient way in which we can probe very small values ofp. For this situation, the scaling arguments predict that E(La−priori)∼n 2. (5.1) Simulations in this regime were performed for N= 12−−210. 100 diﬀerent random city conﬁgurations were used for N <30, 20 conﬁgurations were used for N≤90 and 10 conﬁgurations for N≥120. Figure 5 shows a linear-log plot of the shift factors for the a-priori tour length and the equivalent measurement for the 4-city data/parenleftBig n 2∗Length/parenrightBig . The two sets of data do lie very close to each other (as one might expect them to) but are not exactly the same. The saturation ofn 2∗Lengthfor large n suggests that the proposed scaling behaviour is correct for small p. To summarise E(La−priori) =/radicalbiggn pβa−priori(p) (5.2) where11.522.533.544.5 00.5 11.5 22.5 33.5 44.51 ln(1/p)p=0.05-0.6 p=4/n β (p) a-priori FIG. 5. Shift factors for a-priori tours at small pcompared to the equivalent measurement of 4 city tours at large n. The diamond points show1 βa−priori (p)does not appear to saturate within the accessible range of p. The stars show matching behaviour, with saturation at larger ncorresponding to in- accessible p, suggesting that E(La−priori) =β/radicalbign pfor small p. An a-priori tour, and proposed move We take one paticular visiting list and calculate the estimated length change from that FIG. 6. When estimating the expected length change due to a move, we randomly select realisations of the visiting li st, and make our estimate from these. βa−priori(p)/braceleftbigg=1 1.25−0.82ln(p)p <0.03 =β0 p >0.03.(5.3) VI. NOTES ON ALGORITHM IMPLEMENTATION Stochastic annealing was implemented with a combina- tion of the 2-opt and 1-shift move-sets [18]. Both move- sets work similarly to that which would be expected for the deterministic case. One must sample a number of dif- ferent visiting lists from which the expected pruned tour length change can be estimated. Once a visiting list has been revealed, the length change is determined entirely locally. One need not generate the entire visiting list, but rather determine the set of nearest cities that will specify the length change (see ﬁg. 6). For the PTSP, the location of the nearest cities on the visiting list to the move is determined from a simple Poisson distribution. When using stochastic optimisation, the only variable over which we have control is the sample size (in the 53.544.555.566.577.58 0102030405060708090100Pruned tour length Percent way throughSlow annealing Fast annealing FIG. 7. The pruned tour length for a fast and a slow an- nealing in raveraged over 10 runs. If the transition were simple ﬁrst order, then the ﬁnal length would be the same in both cases. They are not, and the transition is unhelpful in annealing. The fast annealing does show a sharp transition, but this occurs at diﬀerent points in the annealing, thus the average (plotted) does not show a sharp transition. PTSP, the number of visiting lists we generate). How- ever, this does not necessarily imply that r, the sample size, should be a monotonically increasing function. The temperature in simulated annealing is analogous toσ√r (see eq. 1.9). Thus it is important that we attempt to anneal the system with respect toσ√rrather than just r. In the particular case of the PTSP σ, as measured, decreases on average with decreasing temperature. This can be a helpful eﬀect, since it means that rneed not be as large as might be expected to achieve a low tem- perature. However, the variation in σcan dominate the behaviour ofσ√r, thus it may be necessary to transiently reduce rto avoid a sharp drop in T. If one continuously increases the sample size, then a phase transition is seen in pruned tour length (see ﬁg. 7). This is a transition to a non-optimal state for the system, and thus is not of use in optimisation. The system was annealed using a monotonically increasing r, from r= 130 to r= 370 for 500,000 and 5,000,000 Monte Carlo steps with n= 300 andp= 0.05 using 10 runs. The ﬁnal pruned tour length of the fast and slow runs were Lfast= 3.79±0.01 and Lslow= 3.73±0.01. This demonstrates that it is a tran- sition to a non-optimal state. VII. CONCLUSION We have considered the PTSP as a diﬃcult stochastic optimisation problem and introduced a crossover scal- ing interpretation of the PTSP where the a-priori tour is TSP-like on large scales, and a sort on shorter length scales. This gives behaviour which agrees with the com- putational results, with the pruned tour length given by eq. 4.1. The a-priori tour length is more subtle in nature thanthe pruned tour length. We introduced 4-city tours to probe the a-priori tour length as p→0. This allows us to conclude that the crossover scaling interpretation gives the correct behaviour for p <0.03, as given by eq. 5.2. Direct conﬁrmation of this result is left as a future challenge. We have also demonstrated stochastic annealing to be a robust and eﬀective stochastic optimisation technique. It has been noted that for a Gaussian error distribution σ√ris the eﬀective temperature. It is this quantity which needs to be controlled, rather than simply r. By intro- ducing an acceptance probability it is possible to simulate a truly thermal system, provided the error distribution is known. This is a topic for future work. NEB would like to thank BP Amoco & EPSRC for the support of a CASE award during this research. APPENDIX: THE LENGTH OF A TSP TOUR FOR SMALL NUMBERS OF CITIES Numerical estimates of the length of a TSP tour for n≤10 are given below [1] J. Beardwood, J. H. Halton, and J. M. Hammersley, Pro- ceedings of the Cambridge Philosophical Society 55, 299 (1959). [2] J. M. Steele, Annals of Probability 9, 365 (1981). [3] J. Lee and M. Y. Choi, Phys. Rev. E 50, R651 (1994). [4] P. Jaillet, Ph.D. thesis, M.I.T., 1985. [5] P. Jaillet, Operations research 36, 929 (1988). [6] D. J. Bertsimas, Ph.D. thesis, M.I.T., 1988. [7] G. Laporte, F. V. Louveaux, and H. Mercure, Operations research 42 No. 3 , 543 (1994). [8] F. A. Rossi and I. Gavioli, in Advanced school on stochas- tics in combinatorial optimization , edited by G. An- dreatta, F. Mason, and P. Seraﬁni (World Scientiﬁc (Sin- gapore), ADDRESS, 1987), pp. 214–227. [9] H. Robbins and S. Munro, The annals of mathematical statistics 22, 400 (1951). [10] W. B. Gong, Y. C. Ho, and W. Zhai, in Proceedings of the 31st IEEE conference on decision and control (IEEE, PO Box 1331, Piscataway, NJ, 1992), pp. 795–802. [11] D. Yan and H. Mukai, SIAM journal on control and op- timization 30 No. 3 , 594 (1992). [12] A. A. Bulgak and J. L. Sanders, in Proceedings of the 1988 Winter Simulation Conference (IEEE, PO Box 1331, Piscataway, NJ, 1988), pp. 684–690. [13] W. J. Gutjahr and G. C. Pﬂug, Journal of global opti- mization 8, 1 (1996). [14] S. B. Gelfand and S. K. Mitter, J. Optimization Theory and Applications 62, 49 (1989). 6Number of cities n Number of instances I Average tour length σ/√ I−1 2 100000 1.043429 0.002 3 100000 1.564702 0.002 4 5000 1.889601 0.006 5 5000 2.123484 0.006 6 5000 2.311458 0.005 7 5000 2.472799 0.005 8 5000 2.616990 0.005 9 5000 2.740075 0.005 10 5000 2.862946 0.005 TABLE I. The length of the optimal TSP tour for ncities. [15] S. Geman and D. Geman, IEEE Proc. pattern analysis and machine intelligence (PAMI) 1984, 721 (1984). [16] T. M. Fink and R. C. Ball, submitted to Science (unpub- lished). [17] J. J. Bartholdi and L. K. Blatzman, Operations Research Lett.1, 121 (1982). [18] S. Lin, Bell Systems Technological Journal 44, 2245 (1965). 7
arXiv:physics/0011024v1 [physics.atom-ph] 13 Nov 2000Threshold fragmentation under dipole forces Thomas Pattard and Jan M. Rost Max-Planck-Institute for the Physics of Complex Systems, N ¨ othnitzer Str. 38, D-01187 Dresden, Germany (February 2, 2008) Abstract The threshold law for N−body fragmentation under dipole forces is formu- lated. It emerges from the energy dependence of the normaliz ation of the cor- related continuum wave function for Nfragments. It is shown that the dipole threshold law plays a key role in understanding all threshol d fragmentation phenomena since it links the classical threshold law for lon g-range Coulomb interactions to the statistical law for short-range intera ctions. Furthermore, a tunnelling mechanism is identiﬁed as the common feature wh ich occurs for all three classes of interactions, short-range, dipole and Coulomb. PACS numbers: 34.50s, 3.65Sq, 32.80Ge Typeset using REVT EX 1In 1949 Wigner derived a threshold law for the break-up of two quantum particles under short-range and long-range (Coulomb) forces [1]. Using exc lusively classical mechanics Wan- nier formulated in 1953 a threshold law for the break-up of a t wo-electron atom into three charged particles [2]. Corresponding threshold laws for fo ur charged fragments have been published in the meantime [3]. More than 15 years after Wigne r’s paper O’Malley provided the two-body threshold law for dipole interactions [4] thro ugh the analysis of the normaliza- tion constant of the continuum ﬁnal state wavefunction in a s imilar way as Wigner obtained his threshold laws. Wigner’s as well as Wannier’s threshold law have been conﬁrmed by a number of experiments [5]. This is also true for the statisti cal threshold law for break-up into multiple fragments under short-range forces: Derived from simply counting the avail- able states of free motion in the continuum at the respective energy it has been used to successfully interprete experimental results and it has be en shown to be compatible with Wigner’s law for short-range forces [6]. Summarizing the situation one can say that three types of thr eshold laws, Wigner’s, Wannier’s and the statistical one, have been derived by diﬀe rent means for diﬀerent situa- tions. Here, we will show that a connection between these thr eshold laws exists: It is the threshold law for N-body break-up under dipole forces. We wi ll derive it in the following by generalizing O’Malley’s two-body approach to an arbitrary number of particles. This allows us to formulate the threshold law for N−body break-up under dipole interactions. Moreover, for short-range interactions the statistical law is direct ly recovered from the general dipole threshold law. Finally, we discuss a semiclassical tunnell ing interpretation which provides insight into the mechanism of threshold fragmentation on th e one hand side and clariﬁes the connection with the (semi-)classical threshold law for lon g-range Coulomb forces. Our starting point is the N-particle Schr¨ odinger equation in hyperspherical coordinates (r,/vector ω), whererrepresents the hyperradius r= (/summationtext i/vector r2 i)1/2. The/vector riare mass scaled Jacobi coordinates and the set of angles /vector ωdenotes the usual geometric directions of the /vector riin conﬁguration space as well as the so called hyperangles whic h describe the relative lengths ri=\|/vector ri\|, e.g., tanω1=r1/r2[7]. Writing the total wavefunction as ψ(r,/vector ω) =r(D−1)/2Ψ(r,/vector ω) (1) withD= 3N−3, we obtain the Schr¨ odinger equation /parenleftBigg∂2 ∂r2−Λ2−JD r2−VLR−VSR+k2/parenrightBigg Ψ(r,/vector ω) = 0, (2) where the energy has been expressed through the wavenumber k= (2mE)1/2/¯h. The Jaco- bian factor JD=1 4(D−1)(D−3) is a consequence of the transformation Eq. (1) and Λ2 is the Laplace operator on the D-dimensional unit sphere [7]. In Eq. (2) we have split the potential into the long-range part VLR(r,/vector ω) = 2C(/vector ω)/r2and the short-range part deﬁned by the property lim k→0k−2VSR(r/k,/vector ω ) = 0 (3) for all ﬁnite r. Since we are interested in the threshold region k→0 we can scale r=R/k and divide Eq. (2) by k2. In the limit k→0 the short-range potential vanishes due to 2Eq. (3). Hence, the angular problem in /vector ωbecomes independent of rand the wavefunction can be written as Ψ( R,/vector ω) =/summationtext juj(R)Φj(/vector ω), where Φ jis an eigenfunction of (Λ2+ 2C(/vector ω)−λj)Φj(/vector ω) = 0 (4) with eigenvalue λj. The remaining radial problem represents the diﬀerential e quation for a Bessel function if we insert the eigenvalue λj(for reasons of clarity we continue using unscaled coordinates): /parenleftBigg∂2 ∂r2−ν2 j−1 4 r2+k2/parenrightBigg uj(r). (5) From the eﬀective radial potential Veff(r) =1 4(D−1)(D−3) +λj r2≡ν2 j−1 4 r2(6) follows νj= [1 4(D−2)2+λj]1/2(7) forνj>0. If (D−2)2<−4λjthenνj=i¯νjbecomes imaginary. In this case the dipole potential is so attractive due to the negative value of λjthat it overcomes the repulsive D−dependent part. With Eq. (5) we have reduced the problem of ﬁnding the thresho ld behavior of multi- particle break-up under dipole forces to the corresponding problem for two particles solved by O’Malley [4]. The only diﬀerence is that the strength of th e dipole potential Eq. (7) depends now on the dimension of the problem, D = (3N-3), i.e. t he number of particles N, and on the dynamics in the other than radial degrees of freedo m through the eigenvalue λj. The solution uj(r) to Eq. (5) is a linear combination of Bessel functions J±νj(kr). The energy dependence of the threshold cross section can be e xtracted from the energy dependence of the normalization constant of uj(r). According to [4] it is given by σj∝k2νj, ν j>0 (8a) ¯σj∝[sinh2(¯νjπ/2) + cos2(¯νjlnk+δj)]−1, νj=i¯νj (8b) From Eq. (8) it is clear that the threshold cross section is do minated by the ‘partial wave’ with the lowest eigenvalue λ0ifν0≥0. In the case of a net attractive dipole-potential νj=i¯νjthe threshold cross section will be a superposition σ∝/summationtextaj¯σjwhere theajas well as theδjin Eq. (8b) are determined by the short-range part of the pote ntial and all partial waves contribute for which the eigenvalue λjis suﬃciently negative to yield an imaginary νj. For strong enough attractive dipole forces, the threshold cross section will approach a constant value. However, the latter case will be the excepti on for many-particle systems since the repulsive centrifugal barrier ( D−2)2∝N2grows much faster with the number N of particles than the eigenvalue λ0∝N. The dipole threshold law Eq. (8a) contains also the behavior for short-range forces, C(/vector ω) = 0. Then ν0=1 2(D−2) and therefore with D= 3N−3 3σs∝k3N−5. (9) This is exactly the statistical threshold law, derived unde r the assumption that the frag- mented particles are completely free and that they occupy a p hase space volume SEonly restricted by total energy conservation, SE=/integraldisplay K3N−4δ(E−K2/2)dK/integraldisplay /vector ωd/vector ω∝k3N−5. (10) Here, we have used again hyperspherical coordinates, this t ime in momentum space where the hypermomentum is given by K= (/summationtext i/vectork2 i)1/2and the set of angles /vector ωrefers to the Jacobi momenta /vectorkiin the same way as for the coordinates, described above. The s tatistical threshold law states that the cross section close to thresho ld varies according to the available ﬁnal states in the continuum given by their phase space volum eσs∝ SEwith the same energy dependence as Eq. (9). Of course, Wigner’s law σ∝√ EforN= 2 is reproduced by Eq. (9). It seems that the other extreme of interaction, namely charg ed fragments which exert mutual forces through the Coulomb potential VLR= 2C(r,/vector ω)/reven at very large distances, is also compatible with the dipole law of Eq. (8a) since Wanni er’s threshold law for this case is again a power law σ∝kβ. The exponent for two escaping electrons and the remaining (charged) core of the atom, e.g., may be expressed in the form [8] β=1 4 /parenleftBigg 1 +8 C∗d2C dω2/parenrightBigg1/2 −1 , (11) where tanω=r1/r2, the ratio between the two electron-nucleus distances, ω∗=π/4 and C∗=C(ω∗). However, this similarity is misleading for two reasons. F irstly, Wannier’s law is purely classical. It contains the stability properties of a single classical orbit (with /vector r1=−/vector r2, denoted as ’*’ in Eq. (11)). Secondly, the radial Coulomb pot ential for this orbit is attractive . However, the power law Eq. (8a) belongs to an eﬀectively repulsive dipole potential. On the other hand, the threshold behavior of fragmentation u nder repulsive Coulomb forces is an exponential law rather than a power law, e.g. for electron detachment of negative ions by electron impact [9]. There, as a result of the (semi)- classical tunnelling process under the repulsive barrier, the dominant energy variation near t hreshold is given by a Gamow factor σ∝exp[−2Γ(k)], (12) where Γ is the imaginary tunnelling action, Γ≡iφ=/integraldisplay (−p2)1/2dr=/integraldisplayrt ri(−k2+ 2C∗/r)1/2dr. (13) This tunnelling process leads through the energy scaling of the homogeneous Coulomb po- tential toσ∝exp(−a/k). Nevertheless, the qualitatively diﬀerent threshold laws f or repulsive Coulomb and dipole/short-range interactions may be semiclassically i nterpreted with the same tunnelling mechanism Eq. (12). In the dipole case the tunnelling action is given by 4Γ =/integraldisplayrt ri[−k2+ (ν2 j−1 4)/r2]1/2drk→0−→ −νjlnk, (14) which is logarithmically divergent for k→0. Hence, despite the exponential form of the Gamow factor Eq. (12), inserting the action from Eq. (14) lea ds exactly to the power law of Eq. (8a). We conclude that for most interactions, the thre shold mechanism can be in- terpreted as a tunnelling process. For two fragments and hig her angular momentum, the tunnelling mechanism has already been proposed by Rau [6]. H owever, as shown here, it actually holds for allshort-range forces, and for repulsive dipole and Coulomb forces. The only exception are attractive dipole forces with a relative ly complicated threshold behavior Eq. (8b) and attractive Coulomb forces with the well known, c lassically derived, power law behavior. A systematic classiﬁcation of threshold laws acc ording to the nature of the re- spective interaction is presented in Table I. However, one s hould keep in mind that within the enormous size of the parameter space covered by Table I, t here are always exceptional cases [10]. In summary we have provided the threshold law for fragmentat ion ofNparticles under dipole forces. Furthermore, it has been shown that this thre shold law links the statistical law for short-range forces to the corresponding law for long-ra nge repulsive Coulomb forces by a semiclassical tunnelling mechanism. Overlooking the thre shold fragmentation for all types of interactions (Table I) a relatively simple principle emerg es which governs this fragmentation: the balance between potential and kinetic energy for large i nterparticle distances, i.e. for large hyperradius r. The quantum mechanical kinetic energy scales as r−2and has the corresponding, dipole-like, long-range characteristics . Hence, for short-range potentials the kinetic energy dominates the threshold behavior. Indeed, a statistical approach, counting simply available states of free particles, is suﬃcient to de scribe this situation, see Eq. (10). Although the r−2behavior of the kinetic energy is intrinsically quantum mec hanical, the ¯h-dependence of the kinetic energy can be scaled out in the abs ence of a potential and the threshold problem can be solved semiclassically by a tun nelling trajectory (Eq. (14)). The other extreme is the Coulomb potential. With its r−1range it reaches further than the kinetic energy. Hence, the threshold behavior is decide d by properties of the potential only (essentially its relative curvature, see Eq. (11)). Fi nally, the subtle case of a dipole potentialr−2is left. Here, both parts, kinetic and potential energy, con tribute on the same footing. Consequently, the threshold law for N−body break-up as it has been derived in Eq. (8) exhibits a complicated behavior. However, since the dipole interaction ‘interpolates’, roughly speaking, between short-range and Coulomb potenti als, understanding its threshold dynamics allows one to solve the general problem of N−body threshold fragmentation under arbitrary forces, originally formulated by Wigner for two p articles [1]. It is a pleasure to dedicate this article to Martin Gutzwille r. His work on semiclassical theory has been a great inspiration, even more has he himself been inspiring for all of us who have been lucky enough to exchange ideas with him. 5TABLES TABLE I. Overview of the threshold laws for N-body fragmenta tion under diﬀerent interactions interaction type energy dependence of equation mechanism threshold cross section (see text) short range, V(r≫1)∝r−α,α >2 power law (9) semiclassical (tunnelling) dipole, repulsive power law (8a) semiclassicala(tunnelling) V(r≫1)∝r−2attractive oscillating (8b) quantum Coulomb, repulsive exponential law (12) semiclassical (tunnelling) V(r≫1)∝r−1attractive power law (11) classical aincludes the quantum eigenvalue of the angular equation Eq. (4), see text. 6REFERENCES [1] E. P. Wigner, Phys. Rev. 73, 1002 (1949). [2] G. H. Wannier, Phys. Rev. 90, 817 (1953). [3] H. Klar and W. Schlecht, J. Phys. B 9, 1699 (1976); P. Grujic, J. Phys. B 15, 1913 (1981); 16, 2567 (1983); K. A. Poelstra, J. M. Feagin, and H. Klar, J. Phy s. B27, 781 (1994); M. Yu. Kuchiev and V. N. Ostrovsky, Phys. Rev. A 58, 321 (1998). [4] T. F. O’Malley, Phys. Rev. 137, A1668 (1965). [5] J. W. McGowan and E. M. Clarke, Phys. Rev. 167, 43 (1968); S. Cevjanovi´ c and F. H. Read, J. Phys. B 7, 1841 (1974); J. A. R. Samson and G. C. Angel, Phys. Rev. Lett. 61, 1584 (1988); D. Calabrese, A. M. Covington, J. S. Thompson, R. W. Marawar, and J. W. Farley, Phys. Rev. A 54, 2797 (1996); R. Wehlitz, T. Pattard, M. T. Huang, I. A. Sellin, J. Burgd¨ orfer, and Y. Azuma, Phys. Rev. A 61, 030704(R) (2000). [6] A. R. .P. Rau, Comm. At. Mol. Phys. 14, 285 (1984). [7] J. D. Louck, J.Mol.Spec. 4, 298 (1960). [8] J. M. Rost, Phys. Reports 297, 271 (1998). [9] L. H. Andersen, D. Mathur, H. T. Schmidt, and L. Vejby-Chr istensen, Phys. Rev. Lett. 74, 892 (1995); L. H. Andersen, M. J. Jensen, H. B. Pedersen, L. V ejby-Christensen, N. Djuric, Phys. Rev. A 58, 2819 (1998); J. M. Rost, Phys. Rev. Lett. 82, 1652, (1999). [10] J. H. Macek and W. Ihra, Phys. Rev. A 55, 2024 (1997); W. Ihra, F. Mota-Furtado, and P. F. O’Mahony, Phys. Rev. A 55, 4263 (1997). 7
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arXiv:physics/0011026v1 [physics.flu-dyn] 14 Nov 2000Air entrainment through free-surface cusps Jens Eggers Universit¨ at Gesamthochschule Essen, Fachbereich Physik , 45117 Essen, Germany In many industrial processes, such as pouring a liquid or coating a rotating cylinder, air bubbles are entrapped in - side the liquid. We propose a novel mechanism for this phe- nomenon, based on the instability of cusp singularities tha t generically form on free surfaces. The air being drawn into the narrow space inside the cusp destroys its stationary sha pe when the walls of the cusp come too close. Instead, a sheet emanates from the cusp’s tip, through which air is entrained . Our analytical theory of this instability is conﬁrmed by ex- perimental observation and quantitative comparison with n u- merical simulations of the ﬂow equations. fluid air fluid fluid air fluidjet jet air mantlea b FIG. 1. aCross section of the stationary air-ﬂuid interface produced by a thin (1mm) stream of viscous oil poured into a deep pool of the same ﬂuid. The position of the cusp is marked by a circle. bA hollow cylinder of air forms after the cusp has become unstable at a slightly higher ﬂow rate. (Photograph by Itai Cohen) Air bubbles are a ubiquitous presence in ﬂuid ﬂow, ap-pearing when pouring a liquid into a beaker, when beat- ing an egg, or in river streams. This aeration is often desirable, for example to promote chemical reactions [1], yet in many industrial processes entrainment of air bub- bles is detrimental to the quality of the product, render- ing the ﬂow unsteady. For example, it is the single most important factor limiting the speed at which paints or coatings can be applied to a solid surface [2,3]. But in spite of its importance, no general understanding of air entrainment exists, except for the rather special circum- stance that the free surface conspires to enclose an air bubble from all sides, as was recently found for a dis- turbed water jet [4]. Instead, here we argue that the presence of cusp singularities on the free surface results in a generic mechanism for air entrainment. In recent years it has emerged that free surfaces are rather susceptible to the formation of sharp tips or cusps [3,5–7]. This is true in particular for viscous ﬂuids, where shear stresses are strong compared with surface tension forces [7]. Examples are drop impact on a surface [10], jets impinging on a pool of liquid [1], and the coating of a pre-wetted solid cylinder [3]. Once the air enters these narrow passages, the gas ﬂow serves to destroy the original structure to let the air penetrate below the sur- face. What is surprising about this novel mechanism is that the forces the gas ﬂow exerts on the ﬂuid plays the crucial role, even though the viscosity of the air may be smaller by many orders of magnitude than that of the ﬂuid. Other instabilities may occur at a three-phase boundary, for example when the solid to be coated is dry, a problem studied in [2,8]. In the presence of surfactant, still another mechanism for the instability of a cusp has been suggested by Siegel [9]. For purposes of illustration, consider the particular ex- ample of a two-dimensional cusp that forms when a thin stream of a viscous silicone oil is poured into a container of the same ﬂuid. Since the falling liquid drags other ﬂuid away from the surface, a dip is produced around the ﬂuid stream. Increasing the ﬂow rate above a critical value, this dip is no longer smooth, but a singular point on the surface is approached with two vertical tangents. A cross section of this cusped proﬁle is shown as a black silhouette in Fig.1a, the outer wall of the free surface end- ing in a vertical tangent at the cusp point. For clarity, the lighting is chosen such that the free surface appears opaque, so the falling jet is only indicated symbolically to guide the eye, but not visible directly. Increasing the ﬂow rate still further, there is a second critical value where the stationary proﬁle of Fig.1a ceases to exist and a sheet of air shoots out from the tip of the 1cusp. The bottom picture shows this dynamical structure 1/60th of a second after the cusp has become unstable. A thin air sheet now forms the wall of a transparent ﬂuid cylinder. The details of this dynamical structure, such as the bell-shaped opening at its lower rim, is not the subject of this paper, but only the instability of the static shape leading to it. The cylinder eventually grows to about ten times the length shown, and is unstable to the formation of bubbles at its lower end, so the liquid pool quickly becomes contaminated by bubbles of a broad variety of sizes. xy h(y) viscosity ηviscosity air λη fluid FIG. 2. The local shape of the cusp, cut perpendicular to the sheet of air. The variable y is the distance from the tip. Since the air sheet near the cusp is of micron size, the curvature with which it is wrapped around the impinging jet is of no consequence and the cusp can be viewed as a two-dimensional object. In this spirit, Joseph et al. [6] performed experiments with a two-dimensional ﬂow produced by two counter-rotating cylinders submerged below the surface of a very viscous liquid. If the cylinders are placed suﬃciently close to each other, a cusp forms in a symmetrical position between the cylinders [7]. Letting the distance between the rollers going to zero, Jeong and Moﬀatt found a family of exact solutions to this problem, obeying the local scaling form h(y) =κ−3/4H(yκ1/2), (1) where κis the curvature at the tip (see Fig.2). The struc- ture of the solution (1) is typical for ﬂows near singular- ities [11–13], which involve very small scales. Physically (1) means that the shape of the interface is independent of scale, up to a rescaling of the axes. The other crucial property of singularities is that their shape is universal, i.e. independent of the particular type of ﬂow that gener- ates the cusp. Thus our theory, based on the stability of such a singular structure, will be equally general. Indeed,Antonovskii [14] discovered yet another class of exact so- lutions, but where the cusp is formed on the surface of a circular bubble. A local analysis reveals that the scaling function H(ξ) =/radicalbig aξ(ξ+/radicalbig 2/a) is identical to the one in [7] except for a diﬀerent value of the numerical constant a, conﬁrming the expectation that the ﬂow on small scales is universal, independent of the particular features of the driving ﬂow. In all solutions, the tip curvature κgrows exponen- tially with the capillary number Ca=ηU/γ, where η is the ﬂuid viscosity, γthe surface tension, and Uis a typical velocity scale of the external ﬂow. Thus as the strength of the driving ﬂow increases relative to the sur- face tension, the size of the tip may easily reach micro- scopic dimensions [7] if the eﬀect of the air is not taken into account. Without it, stable solutions are thus pre- dicted to exist for all capillary numbers, in disagreement with experiment. Moﬀatt suggested [15] that this is be- cause all previous analyses neglect the viscosity ληof the air being drawn into the cusp by the ﬂow u(0) y(y) parallel to the cusp surface. The air entering a narrow space and having to escape again generates a so-called lubrication pressure plub(y) inside the cusp, whose derivative with respect to the distance yfrom the cusp is p′ lub= 3ληu(0) y(y)/h2(y) (2) by Reynolds’ theory [16]. Since the cusp narrows like h(y)∼y3/2, the lubrication pressure pushes the walls apart according to plub∼y−2, just as it would keep sep- arated to narrowly spaced mechanical parts. Figure 3 proves by direct numerical simulation that this is enough to destroy the stationary solution found forλ= 0. We use a boundary integral code [17,18], opti- mized to resolve the cusp between two merging cylin- ders [19], neglecting the ﬂuid inertia. Starting from Antonovskii’s solution with κ0= 104,λis increased in steps of 2 .5·10−5, pushing the interface forward,but only every fourth proﬁle is shown. At λ= 5.5·10−4, no more stationary solution is found, but instead air enters the ﬂuid forming a narrow sheet, as seen in Fig. 1 and ob- served in earlier experiments [20] An important conse- quence is that in a physically correct description which incorporates the eﬀect of the air (or some other gas at- mosphere), molecular dimensions are never reached, so continuum theory remains valid throughout [21]. To describe the inﬂuence of the air analytically, note that the extra transverse velocity ﬁeld u(λ) x(y) generated by the air pressure can simply be added to u(0) x(y) as given in [7], since Stokes’ equation is linear. Geometri- cally, the cusp looks like a two-dimensional crack enter- ing the ﬂuid, a problem well studied in linear elasticity [22]. Borrowing Muskhelishvili’s result, we can now write u(λ) x(y) as 2u(λ) x(y) =/integraldisplay∞ 0p(y′)m(y′/y)dy′, (3) m(x) = (1 /2π)ln((1 +√x)/(1−√x)). −0.01 −0.005 0 0.005 0.01−1.73−1.68−1.63 3 3.5 4 4.5−3.5−3−2.5 λ) κ)log10( log10( FIG. 3. A boundary integral simulation of a bubble in the ﬂow proposed by Antonovskii for ǫ= 5 and Ca= 0.0992 [14]. The undisturbed bubble radius is used to non-dimensionaliz e all lengths in the problem. As λis increased, the tip is pushed forward, but becomes narrower at a given y. The lowest pro- ﬁle is non-stationary. The inset shows the critical value of λ beyond which there is no more stationary solution for a given curvature. But our free-surface problem is of course non-linear, since the free surface has to follow the streamlines of the ﬂow, which are modiﬁed by u(λ) x. Namely, the inverse slope of the interface is h′= (u(0) x+u(λ) x)/u(0) y, (4) where u(0) xandu(0) yare known [7] and u(λ) xis calculated fromhas outlined above. Note that while u(0) xhas to point inward towards the cusp, u(λ) xresults from the lu- brication pressure and points away from the cusp. Thus, at a given distance yfrom the tip, h′becomes smaller and the channel narrows . Owing to (2) the lubrication pressure is increased, further increasing u(λ) x, so this non- linear feedback eventually destroys the cusp solution, as seen in Fig. 3. It is extremely useful to recast equations (2)-(4) in the scaling variable ξ=yκ1/2, cf. (1). First, from (4) and sinceu(0) yis a constant up to logarithmic corrections, u(0) x must scale like κ−3/4κ1/2=κ−1/4. From (2) plubis esti- mated as plub∼λκ, and thus u(λ) x∼λκ1/2from integrat- ing once. The two opposing velocities become compara- ble at some critical value of the parameter r=λκ3/4. Thus (2)-(4) can be recast in similarity variables, leadingto an integral equation for the correction Hc(ξ) to the unperturbed surface proﬁle H(ξ): Hc(ξ) =−3rξ u(0) y(ξ/κ1/2)/integraldisplay∞ 0u(0) y(η/κ1/2)M(η/ξ) [H(η) +Hc(η)]2dη(5) where M′(η) =m(η). It is a simple matter to solve (5) numerically, giving increasingly large corrections Hc(ξ) to the proﬁle as ris raised. Since Hcis negative, the denominator in the integrand of (5) decreases , leading to a further increase in the absolute magnitude of the correction, in accordance with the qualitative argument given above. Owing to this nonlinear feedback, a solu- tion ceases to exist above a critical value of r, which has a weak dependence on κdue to the logarithmic dependence ofu(0) yon its argument. Hence for the ﬂow parameters of Fig. 3, we predict that the cusp becomes unstable when the curvature reaches a critical value of κcr≈0.45λ−4/3. This approximation is hardly distinguishable from the re- sult of the full solution of (5), which in the insert of Fig. 3 is seen to be in good agreement with numerical simula- tions for various values of λ. Because of the relationship between curvature and capillary number, this translates into the anticipated critical value Cacrabove which sta- tionary solutions are no longer found. At low viscosities, the capillary number never even reaches the critical value for the formation of a cusp, so an unperturbed water jet does not entrain air [4]. In conclusion, we have incorporated the eﬀect of an outer ﬂuid like air into the theoretical description of a cusp. This allows for the ﬁrst quantitative description of air entrainment through surface singularities. A descrip- tion of the resulting sheet of air and its stability remains to be done. Other future challenges include the analysis of its close three-dimensional relatives, namely jet forma - tion out of bubble tips “tip-streaming” [23], Taylor cones “electric jets” [24], or spouts formed by planar interfaces “selective withdrawal” [25]. ACKNOWLEDGMENTS I am very grateful to Keith Moﬀatt for pointing out this problem to me, and to Itai Cohen for donating his experimental pictures. Thanks are also due to Todd Dupont for help with the numerics, and to Howard Stone for very useful discussions. [1] A.K. Bin, Chem. Eng. Sc. 48, 3585 (1993). [2] P.G. Simpkins and V.J. Kuck, Nature 403, 641 (2000). [3] B. Bolton and S. Middleman, Chem. Eng. Sc. 35, 597 (1980). 3[4] C.D. Ohl, H.N. Oguz and A. Prosperetti, Phys. Fluids 12, 1710 (2000). [5] S. Richardson, J. Fluid Mech. 58, 475 (1968). [6] D.D. Joseph, J. Nelson and M. Renardy, Y. Renardy, J. Fluid Mech. 223, 383 (1991). [7] J.-T. Jeong and H. K. Moﬀatt, J. Fluid Mech. 241, 1 (1992). [8] I.N. Veretennikov, A.E. Indeikina and H.-C. Chang, talk presented at the ICTAM 2000 meeting in Chicago, IL, 1 September 2000. [9] M. Siegel, SIAM J. Appl. Math 59, 1998 (1999). [10] D.A. Weiss and A.L. Yarin, J. Fluid Mech. 385, 229 (1999). [11] J. Eggers, Rev. Mod. Phys. 69, 865 (1997). [12] X.D. Shi, M.P. Brenner and S.R. Nagel, Science 265, 219, (1994). [13] B.W. Zeﬀ, B. Kleber and J. Fineberg, D.P. Lathrop, Na- ture403, 401 (2000). [14] L.K. Antonovskii, J. Fluid Mech. 327, 325 (1996). [15] H.K. Moﬀatt, priv. comm. (1995). [16] O. Reynolds, Philos. Trans. R. Soc. London 177, 157 (1886). [17] J.M. Rallison and A. Acrivos, J. Fluid Mech. 89, 191 (1978). [18] C. Pozrikidis, J. Fluid Mech. 357, 29 (1998). [19] J. Eggers, J.R. Lister and H.A. Stone, J. Fluid Mech. 401, 293 (1999). [20] H.K. Moﬀatt ( priv. comm. ) reports a thin but stable sheet of air penetrating the ﬂuid from the cusp. At its lower end, it undergoes a three-dimensional instability and decays into tiny bubbles. [21] Y.D. Shiikhmurzaev, J. Fluid Mech. 359, 313 (1998). [22] N.I. Muskhelishvili, Some Basic Problems of the Mathe- matical Theory of Elasticity , (P. Noordhoﬀ, 1953). [third edition] [23] J.D. Sherwood, J. Fluid Mech. 144, 281 (1984). [24] A.G.Bayley, Electrostatic Spraying of Liquids , (Wiley, New York , 1988). [25] T.J. Singler and J.F. Geer, Phys. Fluids A 5, 1156 (1993). 4
arXiv:physics/0011027v1 [physics.ao-ph] 14 Nov 2000PIPAER PIPAER-IEEC-TN-1100/2100 ESTEC Contract No. 14071/99/NL/MM PARIS Interferometric Processor Analysis and Experimenta l Results Theoretical Feasibility Analysis G. Ruﬃni, F. Soulat IEEC-CSIC Research Unit Gran Capit` a, 2-4, 08034 Barcelona, Spain August 31, 2000Contents 1 Abstract 4 2 Introduction 6 3 State-of-the-Art, Review 7 3.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 The goal of this research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Background and Applicable Models 10 4.1 Statistics of the ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 The Fresnel-Huygens-Kirchhoﬀ integral for the ﬁeld . . . . . . . . . . . . . 11 4.3 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3.1 Corrections to the Geometrical Optics approximation : frequency de- pendence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 The WAF zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Simulation Tools: fresnel and speckles 17 5.1fresnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.1.1 Random ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.1.2 Elfouhaily spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 5.1.3 Field calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 5.2speckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2.1 Specular point determination . . . . . . . . . . . . . . . . . . . . . . 19 5.2.2 Field computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Coherence and Structure Functions 23 6.1 The coherence and structure functions, and the coherenc e time . . . . . . . 23 6.2 Doppler spread of the reﬂected signal . . . . . . . . . . . . . . . . . . . . . . 32 7 Statisitical Properties of the Reﬂected Fields 38 7.1 Analysis of ﬁeld correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 8 Simulation experiments with winding number and GIP 46 8.1 The uses of winding number . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 9 Recommendations for Post-processing Procedures 49 9.1 The PIP instrument: architecture and data products . . . . . . . . . . . . . 49 210 Conclusions and Future Work 51 10.1 What we have found: PIP’s superior performance . . . . . . . . . . . . . . . 51 10.2 What we’d now like to know: future work . . . . . . . . . . . . . . . . . . . 52 11 Acknowlegments 53Chapter 1 Abstract Abstract Several experimental results show that it is possible to ext ract useful phase information from reﬂected GPS signals over the oceans. In this work we beg in the development of the theoretical background to account for these results and ful ly understand the phenomena involved. This information will then be used to deﬁne and car ry out new experiments to evaluate the feasibility of using the phase from reﬂected GPS signals for altimetric purposes and the advantages of using interferometric combi nations of the signals at diﬀerent frequencies—the PIP concept. We focus on the coherence properties of the signals, includi ng the PIP interferometric combination of phases in the diﬀerent frequencies. In this w ork we will concentrate on a static, 8 m high receiver (at least in regards to the simulati ons), and an inﬁnitely removed static source. As the ocean moves, the received ﬁeld will pic k up a random phase. We want to understand the behavior of this phase, as the goal is t o carry out altimetric mea- surements using phase ranging. We will also keep in mind that this random phase carries geophysical information (intuitively, the bigger the sign iﬁcant wave height, the larger the phase excursions). Our simulations are based on the Fresnel integral and use sim ulated Gaussian oceans using the Elfouhaily et al. spectrum. The simulation tool, F RESNEL, is capable of gen- erating time series of the reﬂected ﬁeld at diﬀerent frequen cies, and then analyzing their properties. This software is written in IDL. The most important point we need to answer is whether the sign al can be tracked in the aforementioned situations. As we show, the PIP combination of the signals helps clean the signal from noise—the more correlated the signals at diﬀere nt frequencies the more eﬀective is the PIP mechanism. The following questions are speciﬁcal ly addressed: •What is the reﬂected ﬁeld spectrum? This is determined by the orbital ocean motion, for a static receiver. We also discuss the moving case and sho w that it depends on the system gain. •Does the winding number accumulate, or does it average to zer o? Winding number refers to phase accumulation. Large phase excursions (many cycles) are seen in our simulations. Our simulations show, however, that there isn ’t a preferred direction for phase winding. This means that altimetric phase measuremen ts can be accurate. •What are the theoretical values for the average values of the interferometric ﬁelds? How do the ﬁelds correlate across frequencies? We show that t he most important 4factor is the relation between signiﬁcant wave height and th e (real or synthetic) elec- tromagnetic wavelength. •What is the coherence time of the signals (the reﬂected phase )? We show it is longest for the PIP combination. For rough seas, the correlation bet ween the ﬁelds (and therefore interferometric coherence) in diﬀerent frequen cies disappears, and the co- herence time goes to zero even for the interferometric combi nations. In calmer ocean conditions, however, our results indicate that the interfe rometric combination remains coherent while the individual signals lose coherence rapid ly. •What is the structure function of the reﬂected phase? We see a good ﬁt with a random walk model, with a drift rate proportional to wind speed. •In what ways is the PIP interferometric signal diﬀerent and, presumably, superior to the original ones? Coherence, a basic element for altimetri c purposes. Finally, we discuss the robustness of altimetric phase meas urements after low-pass ﬁl- tering the PIP combination. We show that low-pass ﬁltered PI P data can provide more robust and accurate altimetric measurements to detect slow -varying geophysical signals in the ocean. 5Chapter 2 Introduction This is the sum of Technical Notes PIPAER-IEEC-TN-1100 and P IPAER-IEEC-TN-2100, Theoretical Feasibility Analysis. The inputs to this WP are 1. GMV/IEEC Proposal GMVSA1123/99 (GMV) 2. Relevant literature Theoutputs are 1. This technical note 2. Recommendations for experiments and post-processing pr ocedures And the Tasks : 1. Review the relevant documentation 2. Establish the worst and best case scenarios for the concep t applicability 3. Issue recommendations for experiments and post-process ing procedures 4. Prepare this technical note In addition we analyze some issues relating to the architect ure of the proposed PIP instrument. 6Chapter 3 State-of-the-Art, Review 3.1 Historical overview The PIP idea can be traced back to the PARIS concept [ Mart´ ın-Neira, 1993 ], where the use of GNSS signals as sources of opportunity for bistatic al timetry was ﬁrst proposed. The goal of the present research is to assess the possibiliti es of using the phase information of the reﬂected signal from a PARIS GNSS system for altimetri c purposes. In particular, we will study the merits of using the phase from a dual frequen cy GNSS system in an interferometric fashion. Other uses of reﬂected phase data can also be envisioned (surely the temporal behavior of phase ﬂuctuations and other charac teristics contain geophysical information about the sea surface) but we will not explicitl y investigate this possibility in what follows. Rather, we will, for now, content ourselves wi th providing the background for discussion of these ideas in future work. This is a natura l byproduct of our interest in phase altimetry, as we will see. That the GPS signal reﬂected from the ocean surface can retai n coherence under some conditions has been demonstrated in the past. In [ Auber et al., 1994 ] we have the ﬁrst report of a GPS receiver “locking” onto a sea-reﬂected signa l. Motivated by this somewhat surprising event, tracking of reﬂected signals was later se eked and achieved for extended periods of time in a series of experiments with ﬂights (over t he Chesapeake Bay and the Eastern Shore of Virginia) at an altitude of up to 5500 m [ Garrison et al., 1996 ]. This was accomplished using an Ashtech Z-12 oﬀ-the-shelf receiv er with a LHCP nadir looking antenna, and it was observed that carrier lock was obtained. This particular receiver would not have been able to track otherwise. On the other hand, several groups have to date successfully a nalyzed reﬂected data in order to correlate the direct and reﬂected signals with cl ean code replicas. In the ex- periment carried out by ESA in September 1997 two receivers w ere employed, operating independently of each other, one tracking the direct signal with an up-looking antenna, the other recovering the reﬂected signal via a down-looking LHC P 9 dB helical antenna over an 18 m bridge near Rotterdam (The Netherlands), the “Zeeland B rug”. After ampliﬁcation, each of the signals was sent to a diﬀerent GPS receiver (GEC Pl essey builder kit 2). The receivers down-converted the signals to IF (4.309 MHz, with a bandwidth of 1.9 MHz). The IF signals were sent directly to a high-performance samp ling card, which sampled the data at 6.25 MHz with 2 MHz bandwidth. They were then cross-co rrelated with replicas of the signal adjusted for Doppler due to satellite motion and w ith 20 ms duration (20 C/A code periods). Using the correlation delay data from the dir ect and reﬂected signal, this 7group solved for the height of the bridge over mean surface (w hich depended on the tide) as well as a hardware bias constant via a least squares ﬁtting procedure. In addition, the receivers recorded standard RINEX ﬁles, including SNR. The GPS signals were deemed usable if the SNR of the reﬂected signal was greater than 6 to 9 dB, and if the geome- try was favorable (visibility and multi-path were rather co mplicated in some geometrical conﬁgurations, which were then excluded). Signiﬁcant Wave Height during the experiment was around 1.4 meters (equivalent to a height standard devia tionσζof about 35 cm), and the altimetric performance was rather poor, as only the C/A c ode could be used for corre- lation (in general, a 1% of the chip length is assumed to be the optimal precision for one measurement—this is 3 m for C/A code). A very good descriptio n and further details can be found in [ Caparrini, 1998 ]. The IEEC Earth Sciences Department has been able to reproduc e the analysis of the Bridge Experiment data with similar results. The analysis t ools for the generation of the GPS signals and their correlation with the received ones hav e been developed in Matlab. Another aspect of the Bridge Experiment analysis carried ou t by ESA was the attempt to extract phase information from the data. The ﬁrst attempt of phase processing was an evaluation of Doppler frequency based on zero-crossings co unting [ Caparrini, 1998 ]. The approach was to ﬁrst multiply the signal by an appropriate cl ear replica of the PRN code of a satellite in view. It was then expected that a peak in the F ourier transform of the resulting signal would appear at the IF plus the Doppler freq uency. The direct approach was not deemed feasible, however, due to the limited length o f the signals (only 10 ms long), which did not allow for enough spectral resolution. The sign al multiplied by the replica was instead ﬁrst ﬁltered with a rather narrow ﬁlter in order to el iminate all but the expected Doppler-shifted component. After that, frequency was esti mated using the zero-crossings count for each 10 ms stream of data, with rather good results f or both the direct and the reﬂected signal. This suggested that the reﬂected signal st ill contained a certain degree of coherence. A second study was then devoted to the measuremen t of the relative tide height (that is, the diﬀerence in the height measurements at diﬀere nt times through the diﬀerence in the number of zero-crossings between the direct and the re ﬂected signals). The reasoning was that if the phase could be measured at all, it would be only up to a phase bias. If the bias is assumed to be constant over some short measurement ti mes, then the change in the phase between those times would be related directly to the ch ange in delay between the direct and the reﬂected signals. This calculation, however , could not be carried through because the data was too noisy and the intervals between data measurements too far apart in time. 3.2 The goal of this research As we have seen, several experimental results show that it is possible to extract useful phase information from reﬂected signals. In the analysis sketche d below we will begin to develop the theoretical background to account for these results. Th e goal of this work is to continue this development and analysis of past work to fully understa nd the phenomena involved. This information will then be used to deﬁne and carry out new e xperiments to evaluate the feasibility of using the phase from reﬂected GPS signals for altimetric purposes. We will focus on the coherence of the signals, including the i nterferometric combination of phases in the diﬀerent frequencies. In this work we will co ncentrate on a static, 8 m high receiver (at least in regards to the simulations), and an inﬁ nitely removed static source. As 8the ocean moves, the received ﬁeld will pick up a random phase . We want to understand the behavior of this phase, as the goal is to carry out altimet ric measurements using phase ranging. We will also keep in mind that this random phase carr ies geophysical information (intuitively, the bigger the signiﬁcant wave height, the la rger the phase excursions). The PIP concept concept consists in combining two phases fro m the signals at diﬀerent frequencies. Imagine a very smooth and slowly changing ocea n. By combining the two phases we obtain a slowly changing interferometric phase. I n some sense we would expect that this interferometric phase will be less sensitive to sm all ripples in the ocean and thus provide a more robust ranging tool for altimetry. Several implementations of this concept can be envisioned: a static receiver on a coast or bridge to monitor tides or ﬂoods as well as sea state. Or deplo yment on boats, aircraft and spacecraft for global altimetric or oceanographic measure ments. Carrier altimetric signals over the oceans, even when seen by a fast moving Low Earth Orbi ter, will mostly be low frequency. A strong topographic signal over the oceans woul d be a slope of 1 meter every 100 km. As seen by the LEO (traveling at 7 km/s, say), this tran slates into 0.35 Hz in L1, a rise of 5 cycles in 100/7 seconds. In L25, this becomes 0.01 H z. On the other hand, the roughness and motion of the sea will ind uce high frequency “jitter” on the received signal. This is a nuisance for altim etry, and this is where we expect the PIP concept to bring added value, as the interferometric combination should be less sensitive to these eﬀects. It is appropriate to recall here t he old ado, “What is noise to some is signal to others.” This jitter may contain very usefu l information. In the author’s mind, the most important point we need to answ er is whether the signal can be tracked in the aforementioned situations. As we will s ee, the PIP concept will help clean the signal from noise. For the most part, we will focus i n this work on the low altitude, static receiver situation. The moving cases at higher altit udes will be left for future work, as we will discuss in the last Chapter. In this report we will address, among others, the following q uestions: 1. What is the reﬂected ﬁeld spectrum? 2. Does the winding number (to be described below) accumulat e, or does it average to zero? 3. What are the theoretical values for the average values of t he interferometric ﬁelds? That is, how do the ﬁelds correlate across frequencies? 4. What is the coherence time of the signals (the reﬂected pha se)? 5. What is the structure function of the reﬂected phase? 6. In what ways is the interferometric signal diﬀerent and, p resumably, superior to the original ones? 9Chapter 4 Background and Applicable Models The Rayleigh criterion deﬁnes a surface to be smooth if σζ< λ/(8cos θ), where σζis the surface height standard deviation, θis the incidence angle with respect to the normal to the surface. A more stringent condition is provided by the Fr aunhofer criterion, which is used to deﬁne the far-ﬁeld distance of an antenna, σζ< λ/(32cos θ). A surface becomes smooth under two conditions: σζ∼0 orθ∼90o. The eﬀective roughness of the surface is therefore σζcosθ[Beckmann et al., 1963 ]. Several aspects must be considered to understand the scatte ring phenomenon. We will examine two limiting situation models. In one case, we can as sume that the surface is made up of many independent mirrors (specular points) whose contributions to the ﬁeld add up incoherently. The other limit is that of a surface whic h deviates from ﬂatness only slightly. These two cases must be treated diﬀerently. The go al in both cases will be to understand the behavior of the received phase and modulus. L et us keep in mind in the back of our heads that, at the end, we should look at the relative phase between the two available wavelengths, which should be a statistically bet ter behaved quantity than either of the two phases. 4.1 Statistics of the ﬁeld In this subsection we brieﬂy review some of the results in [ Beckmann et al., 1963 ], concern- ing the sum U=reiψ=n/summationdisplay j=1eiφj. (4.1) This is relevant to understand what happens when we sum the ﬁe lds in a rough reﬂection situation. We state the result for two illuminating conditi ons: 1. A uniformly distributed phase from −πtoπ. 2. A normal phase distribution with standard deviation σ. In the ﬁrst case the resulting phase has again a uniformly dis tributed phase, but the modulus has the so-called Rayleigh distribution p(r) =2r nexp(−r2/n)—and therefore has a non-zero average (the average is proportional to n1/2). In the second case we ﬁnd that the components 10of the resulting phasor are normally distributed (which is a quite general result following directly from the Central Limit Theorem) and that /angb∇acketleftr2/angb∇acket∇ight=n2e−σ2+n(1−e−σ2). (4.2) In this beautiful expression we can see the coherent and inco herent contributions to the ﬁeld modulus. Beckmann summarizes: Outside a narrow cone (or wedge) about the direction of specu lar reﬂection, the amplitude of the ﬁeld scattered by a rough surface is alwa ys Rayleigh- distributed; if the surface is very rough, and grazing incid ence is excluded, the amplitude of the scattered ﬁeld is Rayleigh-distributed ev erywhere. The specularly scattered ﬁeld is composed of a coherent comp onent and a random, Hoyt- distributed component. When the surface is very rough, the l atter becomes incoherent and the former vanishes. In fact, if the surface height distribu tion is normal with deviation σζ, then /angb∇acketleftr2/angb∇acket∇ight=n2e−(4πσζcosθ/λ)2+n(1−e−(4πσζcosθ/λ)2). (4.3) Thus, we see that the magnitude of the reﬂected ﬁeld should de pend mainly on the ratio of signiﬁcant wave height to wavelength. We will obtain a rel ated result through more sophisticated analysis below. Some veriﬁcation of these ideas via simulations can be found in [Daout et al., 1999 ], where the authors, using a 1-D bistatic scattering simulato r, reproduce the mentioned aspects of the phase statistical distribution. The surface is there simulated by facets, and the Kirchhoﬀ theory for the scattering ﬁeld from each facet i s used to accumulate the rays with regard to their phase. The phase histograms they show ve rify very clearly the ideas just discussed. In particular, that the signal is very coher ent at low elevations. 4.2 The Fresnel-Huygens-Kirchhoﬀ integral for the ﬁeld This is basically the scalar Kirchhoﬀ approximation, valid for surfaces with large radii of curvature compared to wavelength. Let the incoming ﬁeld be d escribed by1 Uo(p) =eikr r. (4.4) The Fresnel integral for the scattered ﬁeld is (see [ Born & Wolf, 1993 , p. 380, eq. 17]) U(p) =−i 4π/integraldisplay R ·eik(r+s) rs(/vector q·ˆn)dS. (4.5) The vector /vector q= (/vector q⊥,qz) is the scattering vector : the vector normal to the plane that would specularly reﬂect the rays in the direction we are looking. T his vector is a function of the incoming and outgoing unit vectors /vector niand/vector ns,/vector q=k(ˆni−ˆns). The scattering vector is related to the specular angle βthrough cos β= ˆz·ˆq/q. Note, as an aside, that in the nadir case all wind direction dependence disappears (surfa ce anisotropy). Changing wind direction is akin to performing a rotation in the surface, an d the integral is invariant under rotations in the nadir case. The nadir case is special becaus eR[/vector q(/vector x)] =/vector q(R[/vector x]). 1Here we do a scalar treatment, think of Uas a component of the ﬁeld. 114.3 Geometrical Optics This is the process that dominates in specular scattering wh en the frequency is large (i.e., the wavelength is small compared to the wave height and to the surface correlation length). The case of L-band scattering from the ocean is at best border line “high-frequency”, and we are carrying out some separate studies within the framewo rk of GNSS-OPPSCAT2to understand the relevance of this approximation in such circ umstances. To this end, we have developed the speckles software tool, written in IDL (Interactive Data Language). In Ge- ometrical Optics the surface height standard deviation is a ssumed to be at least of the order of the electromagnetic wavelength. According to the analys is in [Beckmann et al., 1963 ], the resulting wave will be largely incoherent. There is an im portant ingredient in our sit- uation, however, and that is because we have a Woodward Ambig uity Function (WAF) at our disposal to select a given surface patch, as we will see sh ortly. This means that we can ﬁlter ﬁeld contributions from a rather small area. In the Geometrical Optics approximation to the Kirchhoﬀ the ory for electromagnetic scattering, the physical picture can be understood in terms of a specular point model. That is, the ﬁeld at the receiver is the superposition of the ﬁ elds generated by a number of ”mirrors” (not ﬂat mirrors, thought, rather parabolic caps ) on the scattering surface which are oriented in the correct manner. The radiation from each s pecular point is as coherent as the incoming radiation, but there is no coherence in the ph ase relationship between the radiation out-coming from the diﬀerent mirrors if the su rface height distribution has a large range. The result is that there is power at the receive r, but that the phase of the received signal is randomly distributed and the power is sim ply proportional to the number of scatterers. The behavior of the resulting phase will depe nd on the geometry, on the number and distribution of mirrors, and also on the temporal variation of these quantities. The goal of this section is to understand this relationship a nd quantify it. It is useful here to stop and give a physical picture of the spe cular point model—one of the two ingredients in Geometrical Optics. We will treat her e the radiation coming from each mirror separately, as was mentioned above. Since the si ze of each mirror is small compared to the Fresnel zone, we can use the Fraunhofer appro ximation (plane waves all the way through) to estimate the out-coming ﬁeld. Recall that the Fresnel integral for the scattered ﬁeld is U(p) =−i 4π/integraldisplay R ·eik(r+s) rs(/vector q·ˆn)dS. (4.6) In the far ﬁeld (both emitter and receiver are very far compar ed to the size of the scatterer, i.e, the size of the scatterer is much smaller that the Fresne l zone), and since our integral is near a specular point with ˆ n≈ˆq, we ﬁnd that U(p) =−i 4πeik(r′+s′) r′s′/integraldisplay R ·e−i/vector q·/vector r(/vector q·ˆn)dS, (4.7) see Figure 4.1 for a pictorial deﬁnition of the involved vect ors. Here all we have done is to approximate k(r+s)≈k(r′+s′)−/vector q·/vector r. Now, let us focus on a single specular point: imagine that there is only one specular point on the surface. This means /vector q·ˆn=q. To perform the integral let us use a coordinate system in which t hezaxis is parallel to /vector q. In 2ESA Contract 13461/99/NL/GD, Utilization of Scatterometr y Using Sources of Opportunity. 12r 0z x i r´ rk Figure 4.1: Vector geometry for scattering. this coordinate system the tangent plane at the specular pla ne is therefore parallel to the x-yplane. Hence, dS=d2x /vector n·ˆz=d2x. We need to compute I=q/integraldisplay R ·e−iqz(x,y)d2x, (4.8) where U=−i 4πeik(r′+s′) r′s′I. (4.9) Now we use the stationary phase approximation. The idea is th at as qgets larger the integral gets contributions only very near the specular poi nt—the contributions farther out cancel out. Then, I≈I0=qRspece−iqzspec/integraldisplay e−iq 2∇2 ijz\|specxixjd2x (4.10) =qRspece−i/vector q·/vector rspec2π −iqdet1/2/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg (4.11) =iRspece−i/vector q·/vector rspec2π det1/2/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg. (4.12) This result, that the square root of an inverse determinant f ollows from a Gaussian integral is a much used result in Quantum Field Theory (see for instanc e, Quantum Field Theory by L. H Ryder, Cambridge Univ Press 1996 for a derivation). It can be seen to follow from extending/integraldisplay e−αx2dx=/radicalbiggπ α to higher dimensions. Now, it is not hard to show that det1/2/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg−1 =√r1r2— the determinant just yields the products of the radii of curv ature at the specular point: 13the determinant is invariant under coordinate transformat ions of the surface, so we can use a coordinate system along the principal directions—the surface is then described as a simple parabola. The result is then immediate (the radius o f curvature is the inverse of the second derivative in such coordinates and the matrix b ecomes diagonal in such a coordinate system, so the computation of the determinant ju st yields the product of the diagonal terms). Finally, U0(p) =−i 4πeik(r′+s′) r′s′·iRspece−i/vector q·/vector r 2π det1/2/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg (4.13) =Rspec 2e−ik(rspec+sspec) rspecsspec√r1r2 (4.14) The coeﬃcient Rspecdepends on the Fresnel coeﬃcients and on the local geometry— we will discuss this below. The cross section is given by the rat io of the resulting ﬁeld squared divided by the ﬁeld squared at the surface (given above) time s 4πr2. The result is σ=4πr2\|U\|2 \|Uo\|2=\|Rspec\|2πr1r2. (4.15) For all this to work, some requirements have to be met on top of those from the Kirchhoﬀ approximation. Basically, this is a high frequency approxi mation— qhas to be large. Let us look at things in 1D, along one of the radii of curvature. Th e integral is essentially of the form J=/integraldisplaylx −lxdxeiqx2/rx(4.16) =/radicalBig rx/q/integraldisplaylx√ q/rx −lx√ q/rxeiu2du. (4.17) For the integral to become√π, we need lx/radicalbig q/rxto be large—of the order of 10. The expression for the incoherent sum of all the specular points is now immediate from Equation 4.14, given a large height deviation. This expression for th e ﬁeld is also useful for a not-so- incoherent sum of the ﬁelds—we will return to this point belo w. 4.3.1 Corrections to the Geometrical Optics approximation : frequency dependence In this section we add a comment on the next order corrections to the Geometrical Optics approximation. This is not a crucial section for the work at h and and can be taken as a small aside, but we believe it will be useful in future work. In reality the surface is not a parabola and other terms appea r. These are frequency dependent corrections to the ﬁeld. In order to obtain the cor rections we need to expand the integrand as a Taylor series beyond the well-behaved par abolic term. The basic idea for this type of computation is that I(b) =/integraldisplay e−ax2+bxdx=e−b2/4a/radicalBig π/a, and thatdn dbnI(b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle b=0=/integraldisplay e−ax2+bxxndx. 14From this we can compute/integraldisplay e−ax2x4dx=12 (4a)2/radicalBig π/a. The expansion is I=q/integraldisplay R ·e−iqz(x,y)d2/vector x=I0+I4+...,. (4.18) where I4=qRspece−iqzspec−iq 4!∇4 ijklz/vextendsingle/vextendsingle/vextendsingle spec/integraldisplay xixjxkxle−iq 2∇2 ijz\|specxixjd2/vector x. (4.19) Now, I4=qRspece−iqzspec−iq 4!∇4 ijklz/vextendsingle/vextendsingle/vextendsingle spec/integraldisplay xixjxkxle−iq 2∇2 ijz\|specxixjd2/vector x(4.20) =qRspece−iqzspec−iq 4!∇4 ijklz/vextendsingle/vextendsingle/vextendsingle spec+ /parenleftbigg −iq∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg−1/parenleftbigg −iq∇2 klz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg−12π −iqdet1/2/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg (4.21) and I4=Rspece−iqzspec−2π√r1r2 q4!Rspece−iqzspec∇4 ijklz/vextendsingle/vextendsingle/vextendsingle spec/parenleftbigg ∇2 ijz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg−1/parenleftbigg ∇2 klz/vextendsingle/vextendsingle/vextendsingle spec/parenrightbigg−1 (4.22) In a future implementation of speckles (see discussion below), we would like to include these second order eﬀects. These eﬀects can also be accounte d for theoretically using, e.g., Gaussian statistics. At the end the ﬁnal expression mu st depend only on the usual parameters. Note that the I4has units of length. Thus, aside from constants, these higher order corrections may be of the formσζ l·q, say. In this last expression we see that this correction is indeed inversely proportional to freque ncy, as it should, since Geometric Optics is a high frequency limit. 4.4 The WAF zone If the signal is ﬁltered by cross-correlation with several C /A (or P) code periods at the delay and frequencies corresponding to the specular zone, f or instance, only the surface patch in the ﬁrst Delay and Doppler zone will contribute (let us call it the “WAF zone” to associate as well as distinguish this concept from the con cept of Fresnel zone). This is described by the equation SNR(τ,fc) =1 kTSBD·λ2 (4π)3/integraldisplayPtGtGr R2 1R2 2σ0 rtχ2(/vector ρ,t,δτ,δf )dA. which is a slight reﬁnement of a similar equation in [ Zavorotny et al., 1999 ] and which can be found in the review [ Ruﬃni et al., 1999 ]. The important thing to keep in mind is that the support in the integrand, the WAF zone, is the intersecti on of four spatial zones: 1. The receiver antenna footprint. 152. The annulus zone deﬁned by the Λ2function in χ2. 3. The Doppler zone deﬁned by the \|S\|2function in χ2. 4. The scattering cross section coeﬃcient σ0. The WAF, through its support, selects a given portion of the s urface from which power is measured. To get a feeling for these numbers, the area involved in the re ﬂection is of the order of 10 km2for an aircraft and 1000 km2for a Low Earth Orbiter. For h >> cτ , where τis the chip length (1 µs for C/A code) and hthe receiver height), the one-WAF (or one-chip here) area can be approximated by AWAF= 2πhcτ/ cos2θ.Note that AWAFincreases only linearly with altitude. We can add another element at this point. In the above discuss ion, the WAF is assumed to result from choosing a given delay and Doppler in the reﬂec ted signal. However, as time passes the signal reﬂected by a given surface element cannot be characterized by a ﬁxed Delay and Doppler: both will change as the situation evolves . It is clear, however, that a more sophisticated WAF algorithm can be devised that focus es on that speciﬁc surface patch. This observation opens the door to longer integratio n times, which can be useful for altimetric purposes. Figure 4.2: Nadir case ﬁrst chip radius size for C/A and P code s. This is for a static situ- ation, no Doppler ﬁltering aﬀects the result (based on the si mple formula s=√2∗h∗τc), where τcis the chip length in meters (300 m for C/A code). 16Chapter 5 Simulation Tools: fresnel and speckles These are two modeling tools we have developed to understand , via a computer simulation, some of the characteristics of the reﬂected ﬁeld. fresnel models the reﬂected ﬁeld (just the carrier part after removing the source time-dependence) us ing an ocean model and direct integration of the Fresnel-Huygens-Kirchhoﬀ integral des cribed above. We have carried out several types of simulations: a slowly vertically moving re ceiver, a static receiver, a realistic ocean and a so-called chaotic ocean. We will focus here on the static realistic ocean case. speckles has been developed with two goals in mind. We will eventually attempt to compute the reﬂected ﬁeld using a GO method: ﬁnding ﬁrst th e specular points and integrating just around these to sum up the resulting ﬁeld. W e will just outline here some of its characteristics and initial results. We have fou nd that this is very diﬃcult to do (in general, comparisons with the Fresnel integral are po or). We have seen, however that the number of specular points correlates reasonably we ll with the reﬂected Fresnel ﬁeld, specially in rough ocean simulated conditions. We are attempting to understand this phenomenon better, and to study its implications: GO, i n some form of another, is presently the leading model used for the analysis of reﬂecte d power in GPSR. It is a simple method, and experimental results to date seem to validate it . We would like to understand its somewhat puzzling high performance, since L-band is not really high-frequency in the ocean surface case. That is, GO is not, strictly speaking app licable in the whole range of ocean conditions with an L-band instrument—the waveleng th in neither large of short enough. Nonetheless, some semi-empirical modiﬁcations of this model are currently being successfully used. We would like to understand this success and, more importantly, its limitations. We will brieﬂy mention some of this work, altho ugh it is not central to the study. 5.1fresnel This routine ( written in Interactive Data Language—IDL) in tegrates the ﬁeld from the ocean surface below, using nadir incidence. The receiver is hovering a few meters over the water. The ocean model is Gaussian (we use a routine from B . Chapron using the Elfouhaily spectrum) [ Elfouhaily et al., 1997 ], “chaotic” (random heights, moving) and a non-Gaussian simulations are planned for future work. The s oftware simulates the reﬂected complex ﬁeld at L1, L2 and L5 frequencies. The goals of this pa rt of the work were to show 17that the simulation is possible, to compare the behavior of t he ﬁeld in the diﬀerent cases, and to assess the potential of the PIP idea. 5.1.1 Random ocean The random ocean is generated by choosing a height standard d eviation. The code assigns a random height with the desired characteristic to each 10 cm by 10 cm square of the reﬂecting surface. In addition, a velocity random ﬁeld is ch osen, and each square moves up and down with a chosen period. 5.1.2 Elfouhaily spectrum The procedure to compute the ocean state, based on the IDL rou tine kindly supplied by B. Chapron, is, starting from the energy spectrum, to generate a plausible random spectrum. To do this, the square root of the energy spectrum is taken, an d a uniformly random phase appended to it. The appropriate time dependence is also adde d to this random phase— using the dispersion relation for gravity ocean waves. Then , the inverse Fourier transform is taken. Using this procedure, a moving, random ocean with t he appropriate spectral characteristics is generated. Using a uniform random phase distribution in the diﬀerent frequencies means that the resulting ocean is Gaussian. Thi s can be changed, of course, but we have not attempted to analyze the subtleties of non-un iform phase distributions in this work. Generally the sea state is characterized by its power spectr um. As the sea evolves in a random process, the spectral density of sea elevations is ob tained by the Fourier transform of the autocorrelation function of the elevations ζ(x,y). The spectral density can also be estimated by the Fourier transform of one realization of ζ(x,y) [Blackman and Tukey, 1958]: F(kx,ky) =\|TF[ζ(x,y)]\|2 If we determine the matrix φof random phases uniformly distributed between 0 and 2 π, the sea height at the point r= (x,y) is: ζ(r) =TF−1[/radicalBig F(kx,ky)eiφ] Thus the probability density functions of heights and slope s are Gaussian. For this study we use the uniﬁed spectrum of [ Elfouhaily et al., 1997 ]. This is just a consequence of the Central Limit Theorem applied to sums of harmonics. 5.1.3 Field calculation The ﬁeld is calculated as a complex number using the Fresnel i ntegral expression: U(p) =−i 4π/integraldisplay R ·eik(r+s) rs(/vector q·ˆn)dS. (5.1) To be precise, we Fraunhofer-expand the part of the integran d corresponding to sthe distance to the inﬁnitely far transmitter. That is, we write s=ks′−/vectorkin·/vector xin the phase part of the integrand, where /vector xdenotes the position of the scattering point. Thus, the integrand becomes, U(p) =−i 4πeiks′ s′/integraldisplay R ·eikr−/vectorkin·/vector x r(/vector q·ˆn)dS. (5.2) 18We actually throw out all the s′dependence (in eﬀect assuming an incoming plane wave, as well as on the Fresnel coeﬃcient ) and obtain U(p) =−i 4π/integraldisplay R ·eikr−/vectorkin·/vector x r(/vector q·ˆn)dS. (5.3) This means that the outgoing ﬁeld would be a plane wave if the s urface were ﬂat, with unit modulus. The ocean surface is divided into squares 10 cm wide (or less) , and an area of up to 200x200 meters is integrated. For the most part we will conce ntrate on the ﬁrst chip zone, with radius r=√2∗h∗τc. The resulting ﬁeld is a complex number in this scalar treatme nt,U(p(t)) =reiφ(t). In order to evaluate the phase as the receiver moves up we take δφi= ln(w(ti)/w(ti−1)) (5.4) withw(t) =eiφ(t). Summing the δφiyields the overall phase. This method is related to the deﬁnition of winding number of a curve around the origin in the complex plane: I=/contintegraldisplay γdz z=/contintegraldisplay γd(lnz) =/summationdisplay ilnzi zi−1. (5.5) A full cycle, 360 degrees, is equivalent to 19 (L1), 24 (L2), 2 5 (L5), 86 (L12) or 568 (L25) cm. We will return to this deﬁnition shortly. 5.2speckles As mentioned above, this code has been developed to study the use of Geometric Optics for the study of ocean surface reﬂections in L-band. It will also be used to study specular point statistics, using both Gaussian and, eventually, non-Gaus sian models. Taking a Gaussian ocean model, we determine the specular points distribution and compute the reﬂected ﬁeld using geometrical optics. We are to compare this to straight integration of Fresnel integral. Then the analysis will be compared with a non-linear ocean mo del. 5.2.1 Specular point determination We consider the incident radiation as a plane wave normal to a Gaussian surface and a receiver at ( xr,yr,zr). The point ( x,y,ζ) of the surface is considered to be a specular point if the slopes follow rather clear geometrical conditions (s ee for instance [D.E.Freund 1997]): dζ dx=x−xr zr−ζ(5.6) dζ dy=y−yr zr−ζ. (5.7) Our approach consists in the detection of slope variations a round the expected specular slopes deﬁned above. We compute at each point P of the surface , and for both directions xandy, the expected specular slope S—i.e. the slope needed to reﬂe ct the incident wave to the receiver—and the slopes of the facets located just bef ore (slope S[before]) and after (slope S[after]) the point P. The condition for P to be a speck le is that (S[before]-S) and 19(S -S[after]) must be of the same sign. This determination is attractive because we doesn’t need to deﬁne any tolerance on the specular slopes. We are neg lecting saddle point by this approch (saddle points are not well deﬁned in discrete space s). Then an analysis is made on the radius of curvature at the spec ular point. The Kirchhoﬀ approximation leads to a condition on the radius of curvatur e at the specular facet, /radicalBig q√r1r2>10. (5.8) The product of the principal radii of curvature can be comput ed using the derivatives of Figure 5.1: Fresnel contours for the 10 by 10 surface patch un der and 8 meter high receiver. U10=8 m/s. The colored contours represent the oce an instantaneous topography, with higher areas represented brighter. The line pairs are i so-lambda delay curves for L1 and L2 (4, 10 and 20 lambdas are shown). 20ζ(r), see [Barrick, 1968]: \|r1r2\|=/parenleftBig 1 +ζ2 x+ζ2 y/parenrightBig2 /vextendsingle/vextendsingle/vextendsingleζxxζyy−ζ2xy/vextendsingle/vextendsingle/vextendsingle The following Figures (5.2 to 5.4) present the specular poin t position (circles) for three diﬀerent times on the moving surface. The receiver is locate d at 8 meters above the surface, which has a 5 m by 5 m dimension and 1 cm of resolution. The stars correspond to the specular points that are under the curvature condition. The specular distribution is greatly modiﬁed within 10 ms. Figure 5.2: Specular points position for Time = 0 s Figure 5.3: Specular points position for Time = 10 ms 5.2.2 Field computation The idea is to compute the Fresnel ﬁeld with the contribution of scatterers only. We integrate on the vicinity of each mirror. The result is then c ompared to the straight Fresnel 21Figure 5.4: Specular points position for Time = 20 ms computation. To date these comparisons have not been very su ccessful. It appears that in order for the Geometric Optics approach to work at this lev el (not after substantial averaging) we need to impose more stringent frequency or sur face scale characteristics and/or take into account higher order eﬀects. If the main con tribution is from specular points, as we suspect, the key may be to calculate the ﬁeld fro m each scatterer to higher order than the stationary phase approximation. Work in this area is planned for the future. 22Chapter 6 Coherence and Structure Functions 6.1 The coherence and structure functions, and the coher- ence time Another question that needs to be answered is how the sea surf ace motion and varying geometry aﬀect the coherence of the reﬂected signal. There a re several models for the sea spectrum, and these can be used to try to extract this info rmation. Consider the toy model of one-dimensional scattering, with static, line d up, receiver and transmitter, with the scatterer (a single facet) also in line but now movin g up and down with average speed v. It is easy to see that the correlation of the direct and the sc attered signal is zero unless the coherent integration time is less than the charac teristic time of the surface at GPS wavelengths. The coherent integration time is roughly that time for which the RMS phase error is 1 radian [ Thomson et al., 1990 ]. More precisely, let the coherence function be deﬁned by c(T) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 T/integraldisplayT 0eiφ(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (6.1) where φ(t) is the error phase. Then, the coherence time is the time Tit takes for /angb∇acketleftC(T)2/angb∇acket∇ightto drop to 0.5—say [ Thomson et al., 1990 ]. Note that for long times the coherence function can also be related to the magnitude of the zero-frequency sp ectral component of the (windowed) normalized ﬁeld associated to the error phase (t hat is, in our simulations, the Fourier transform of the normalized ﬁeld). We will get back t o this point below. The coherence time of the GPS signal is about 2–3 hours. What a bout the reﬂected signal? If the WAF zone and the sea surface are ﬁxed, we should expect the same. If we include time dependence (and we ignore all motions), we can r ewrite the equation on the statistics of the signal by U(t) =reiψ=/summationdisplay jeiφj−iωt=e−iωt/summationdisplay jeiφj, (6.2) so the resulting ﬁeld is as coherent as the incident one. But i n real situations neither of this is the case. Both the ocean surface and the WAF patch are movin g (although this second eﬀect can be corrected in principle by the use of SAR techniqu es). The characteristic time of the surface can roughly be deﬁned as the time it takes the su rface to move a wavelength— in the present case λ/v. This thought experiment is useful to understand the limita tion in coherent integration time for a moving surface like the oc ean. An advantage of using 23a synthetic wavelength using two GNSS frequencies is that th is coherent integration time should be longer. We will verify in a moment that this is the ca se. An associated concept to the coherence function is the structure function of the phase. This is deﬁned by s(T) =/angb∇acketleft(φ(t+T)−φ(t))2/angb∇acket∇ightt. (6.3) It gives us a measure of how tied up are two points in the phase a s we vary their temporal separation. If this phase drift can be approximated by a rand om walk stochastic process [Herring, 1990 ], s(T) =d·T (6.4) we can use the associated “Geophysically Induced Phase drif t” (GIP) rate√ das a geo- physical parameter if it correlates well to U10 in our models . This parameters has units of cycle per square root second. We will check this below (but le t us anticipate that it does). The coherence time and the structure function are related. I n fact, /angb∇acketleftC(T)2/angb∇acket∇ight=1 T2/integraldisplayT 0/integraldisplayT 0/angb∇acketleftei(φ(t)−φ(t′))/angb∇acket∇ightdt dt′. (6.5) If we assume Gaussian statistics for the phase ﬂuctuations [ Thomson et al., 1990 ] we can simplify things considerably, /angb∇acketleftC(T)2/angb∇acket∇ight=1 T2/integraldisplayT 0/integraldisplayT 0e−s(t′−t)/2dt dt′. (6.6) =2 T/integraldisplayT 0/parenleftbigg 1−τ T/parenrightbigg e−s(τ)/2dτ. (6.7) For a random walk process, then, /angb∇acketleftC(T)2/angb∇acket∇ight=2 T/integraldisplayT 0/parenleftbigg 1−τ T/parenrightbigg e−d·τ/2dτ. (6.8) See Figure 6.1 for a graph of this function for various drift r ates. Note that in this model /angb∇acketleftC(T)2/angb∇acket∇ightdecreases to zero as time increases. This fact alerts us that there is something amiss: we see in our simulations larg e phase excursions (certainly much larger than a cycle), but there is coherence left at larg e times—see Figures 6.2, 6.3, 6.4 and 6.5. How is this possible? The Gaussian model for phase ﬂu ctuations does not really apply to our situation, however. Gaussian means that phase e xcursions probabilities are bell-shaped, of course. Are they? In our simulations the pha se of the reﬂected signal gathers about a point in the complex plane—the ﬁelddoes seem to have a Gaussian distribution about an average point. The cumulative phase does not have to have such a distribution, however. In fact, it could conceivably indulge in arbitrari ly large excursions while spending more time in that average ﬁeld point: this would still result in a Gaussian ﬁeld distribution. However, the probability distribution for such a phase hist ory would not be a Gaussian distribution (centered about 0, say). It would consist of a s eries of Gaussian-like humps spaced one cycle apart, with the one about zero the largest. I n this situation we could well have a very long coherence time (since the phase modulo 2 πcould be a rather narrow Gaussian distribution about a point) while the structure fu nction may show high drift. This type of behavior is evident in our simulations when plotting the ﬁeld points (see Figure 6.6). Note that the interferometric signal has a stronger average ﬁeld value. See Figure 6.7 for a histogram of the L25 cumulative phase illustrating the mult iple-hump distribution that we alluded to. 24Figure 6.1: /angb∇acketleftC(T)2/angb∇acket∇ightfor a random walk with drift rates of√ d=0.5, 1.0, 2.0 and 3.0 cycles per square root second. 25Figure 6.2: This is a plot with the structure function (left) and the coherence function for the case U10=6 m/s, 43 m side 10 cm resolution simulation a t 8 m height. This is 0.4 meters height standard deviation. From top to bottom, L1, L2 , L5 and L12, L25 ﬁelds. The drift rates are 4.2, 2.1, 1.1, 6.2, 2.5 cycles per square r oot second. Coherence times 0.04, 0.06, 0.08, 0.09 and 0.3 seconds. 26Figure 6.3: This is a plot with the structure function (left) and the coherence function for the case U10=1 m/s. From top to bottom, L1, L2, L5 and L12, L25 ﬁ elds. The drift rates are 1.9, 0.9, 0.8, 1.4, 1.3 cycles per square root second. Sim ulation size is 19 m side, 10 cm resolution, and height standard deviation is 3 cm. 27Figure 6.4: This is a plot with the structure function (left) and the coherence function for the case U10=3 m/s. From top to bottom, L1, L2, L5 and L12, L25 ﬁ elds. The drift rates are 0.8, 1.0, 1.0, 1.0, 1.5 cycles per square root second. Sim ulation size is 19 m side, 10 cm resolution, and height standard deviation is 8 cm, receiver height 8 m. 28Figure 6.5: This is a plot with the structure function (left) and the coherence function for the case U10=4 m/s. From top to bottom, L1, L2, L5 and L12, L25 ﬁ elds. The drift rates are 3.7, 2.1, 1.6, 2.1, 1.8 cycles per square root second. Sim ulation size is 19 m side, 10 cm resolution, and height standard deviation is 12 cm, receive r height 8 m. See next Figure for further analysis of this particular simulation. 29Figure 6.6: Refer to the previous ﬁgure. This is a plot of the ﬁ eld in the complex plane for L 2, L5and L 25, and it serves to illuminate the diﬀerence between the struc ture function and the coherence function. The ﬁeld mean for L 2is (-0.06,-0.1), for L 5is (-0.1,-0.1), and for L 25(-0.6,-0.20), much larger. The standard deviation is about 1 for all of them. In this 19 meter simulation, U10=4 m/s, height standard deviation w as 12 cm, receiver height 8 m. 30Figure 6.7: In these two plots we depict the phase and phase hi stogram for the L25 ﬁeld in the previous ﬁgure. The highest peak corresponds to the in itial zero phase. 31Notice that the above discussion is relevant to understand t he coherence properties that were found in the Bridge Experiment reﬂected signal (integr ation times where of 10 ms, short enough for the static approximation to work in that geo metry). In longer time scales the accumulated phase will pick up a random component, but if there is an overall phase change (due for instance, to a tide) it should be distinguish able from the random part (which should not add nor subtract to the overall phase on the average). 6.2 Doppler spread of the reﬂected signal Another thought experiment is useful. Imagine that the surf ace height is varying as a function of time as ζ(t) =αt+Ar(t), where αis a constant and the second term adds a random, noise-like di splacement between −AandA. We can think of this as the eﬀect of a tide superimposed on the wave mo- tion on the sea. The scattered phase will not be coherent if A i s large enough, but the accumulated phase will certainly contain information abou tα. What happens if there are many scatterers, how fast is the residual phase varying? Thi s will give us a feel for the Phase Lock Loop (PLL) bandwidth required for tracking. It is not hard to see that the bandwidth required will be less than 100 Hz (assuming that th e sea surface vertical speed is not greater than 10 m/s), in a static geometry (or if a stati c-WAF is used). The largest frequency present in the signal cannot be greater than the co ntribution due to the fastest moving surface patch—the result is always the linear combin ation of such contributions, U(t)∼reiψ(t)=/summationtextn j=1eiφj(t). Now, if the maximal orbital velocity for a wave is 10 m/s (see Figure 6.8, this implies a Doppler of 2 ×10/λ≈100 Hz (remember that the signal is bouncing oﬀ the surface). Thus, a sampling rate of at least 20 0 Hz is required to retrieve the phase of the reﬂected signal (even if the SNR is high enoug h). In our simulations we have seen that it is safe to sample the signal at such rate. Here is another useful question: given that a receiving LEO i s moving at, say, 7 km/s, would it be possible to integrate coherently (correlate) th e direct and reﬂected signals for 1 ms, if the LEO antenna has a footprint of 10 meters? Note that this “eye on the sea” is moving very fast over the surface, and this is introducing a r andom, time-dependent phase to the signal. The answer is no. But if the footprint is enlarg ed enough, yes. This may be an important problem for the PIP, but there are ways throug h which it could be ﬁxed. One obvious one is to use a WAF that sticks to a given surface pa tch, or that it moves slowly over the sea surface. Some of these ideas are discusse d within the context of the GNSS-OPPSCAT project (Report WP3320). To get an idea of the i nduced Doppler, we can reason that the scattered ﬁeld is just a sum of the type F=R(t)eiΦ(t)≈N/summationdisplay i=1ri(t)eiφi(t), which already looks like a Fourier sum if we approximate ri(t) as constant and the exponent by its ﬁrst derivative. Thus, we can guess that the maximum fr equency in the spectrum is simply Doppler due to the fastest moving patch on the ocean. A s an upper bound we can use 10 m/s—see Figure 6.9. Multiplying times 2 and dividing b yλwe obtain a scattering Doppler of about 100 Hz. 32To be a bit more precise, let us write F=/integraldisplay AWAFeiqzzd2x=/integraldisplay AWAFeiqz(z0+˙zt)d2x, using the simpler backscattering expression in the Fraunho fer zone. Notice the restriction of the integral to the (static) WAF zone. Let us now rewrite th e transform of the ﬁeld: ˜F(ω) =/integraldisplay Feiωtdt=/integraldisplay eiqzzo/integraldisplay ei(ωt−qz˙z)dtdx ∼/integraldisplay A(q˙z≡ω)e−iqzzod2x, if we assume that vertical motions take over a “long enough” t ime compared to the time scale associated to the frequency we are interested in. This is saying that the spectral component at a given frequency is proportional to the area on the surface moving at the right speed to produce that frequency. We can extend this reasoning to the moving-receiver case. Th ere are two limiting situations we can think about. One is the “high gain” situati on. Let us ﬁrst discuss this situation—we will get back to the low gain situation at the en d. Assume for now the ocean is frozen. In this situation the WAF zone is very small ( due to antenna or processing gain) and coherence time is deﬁnitely inﬂuenced by the fact t hat as the receiver moves the contributing surface is changing. If the WAF zone is smaller than the correlation length of the ocean (certainly an extreme case, but never mind), the co herence time is limited by l/v, where vis the receiver speed and lis the correlation length of the surface—assuming large enough height deviations. To be more precise, if we use a repl ica with ﬁxed Doppler and Delay, the “eye on the sea” will move over the ocean at a speed s imilar to the receiver’s. This will introduce additional bandwidth in the carrier if t he ocean is rough—even if is frozen. This bandwidth will be mainly proportional to ocean surface rugosity, not to ocean surface motion. It can be very large (kHz), depending on the r eceiver speed and the WAF area size. Following the above reasoning, and keeping in min d that the WAF zone is now a moving ﬁlter on the ocean surface, F(t) =/integraldisplay AWAF(t)eiqzz(/vector x)d2x=/integraldisplay ΞWAF(/vector x−/vector vt)eiqzz(/vector x)d2x =/integraldisplay ΞWAF(/vector x)eiqzz(/vector x+/vector vt)d2x =/integraldisplay AWAFeiqzz(/vector x+/vector vt)d2x Now let us look at ˙F(t): ˙F(t) =/vector v·/integraldisplay AWAF/vector∇/parenleftBig eiqzz(/vector x+/vector vt)/parenrightBig d2x=/vector v·/integraldisplay ∂AWAFeiqzz(/vector x+/vector vt)ˆn dl (6.9) where the last is an integral over the boundary of the WAF area and/vector nis the normal to the boundary. The result illustrates the “edge eﬀect”, i.e. , that the change is just due to the change at the boundary of the WAF area. We have used a 2D v ersion of Green’s theorem in the last step. This equation says that the change i n the ﬁeld is proportional to the variation of the ﬁeld contribution at the edges of the WAF area (properly mapped in the velocity direction). The faster the velocity and the lar ger the diﬀerence, the larger the 33rate of change of the ﬁeld. To relate this to the spectrum, not e that this will contribute a high frequency componene to the spectrum. Whether this is a relatively large or small contribution depends on the total ﬁeld, which is proportion al to the total area. Thus, for large WAF zones this high frequency eﬀect contributes a s mall portion of the total spectrum. If we allow for a moving surface, z=z(/vector x,t) it is readily seen that the complete result is the sum of two distinct eﬀects: ˙F(t) =/vector v·/integraldisplay ∂AWAFeiqzz(/vector x+/vector vt,t)ˆn dl+/integraldisplay ΞWAF(/vector x−/vector vt)∂t/parenleftBig eiqzz(/vector x,t)/parenrightBig d2x. (6.10) Let us now return to F(t) =/integraldisplay AWAF(t)eiqzz(/vector x)d2x=/integraldisplay ΞWAF(/vector x−/vector vt)eiqzz(/vector x)d2x (6.11) and compute the Fourier transform: ˜F(ω) =/integraldisplay /integraldisplay∞ −∞dt eiωtΞWAF(/vector x−/vector vt)eiqzz(/vector x)d2x. (6.12) To evaluate this, let us assume for simplicity and without lo ss of generality that ΞWAFis a box of size Smoving along in the xdirection with velocity vx: ΞWAF(/vector x−/vector vt) = Ξx(x−vxt)·Ξy(y). (6.13) Then/integraldisplay∞ −∞dt eiωtΞWAF(/vector x−/vector vt) = 2Ξy(y)eiωx/v x wsin(ωS/vx). (6.14) Hence, ˜F(ω) =2sin(ωS/vx) ω/integraldisplay Ξy(y)eiωx/v xeiqzz(/vector x)d2x. (6.15) The characteristic time is roughly given by S/vx, as can be seen from the multiplying sinc. Frequencies higher than vx/Sare supressed by this factor. The power in frequencies small er than this are modulated by the horizontal roughness of exp iqzz(/vector x) in the scale deﬁned by vx/ωin the direction of motion: if the surface does not vary in tha t scale, there will be little power at the frequency vx/S. In this approximation we have implicitly assumed a very smal l WAF in relation to the geometry, since we have worked in the Fraunhofer approximat ion. This means that all the points in the reﬂecting patch have the same geometry and gene rate the same geometrically induced Doppler. In recent experimental conditions this is hardly the case, and the Doppler of the reﬂected signal is dominated by the Doppler amplitude of the WAF and glistening zones. If the glistening zone is large enough, then radiatio n will be received from the entire WAF. The Doppler span of the WAF zone determines then the Dopp ler spread of the received ﬁeld. This is the basic mechanism for Delay-Dopple r mapping and the basis for the relationship betweeen waveforms and the characteristi cs of the sea surface. Even if we restrict the signal to the ﬁrst chip in the C/A code, there wil l be quite a bit of geometric Doppler. This second source of Doppler is in principle remov able by many means: high antenna gain or processing gain. By any of these means we can r educe the patch to the Fraunhofer zone mantaining SNR. It will then become increas ingly important to use SAR 34Ti(ms)Ocean motion (Hz) Geometric (Hz) Edge (Hz) 1 200 1000 1 5 200 200 7 10 200 100 14 20 200 50 28 50 200 20 70 Table 6.1: Back of the envelope calculations for Doppler spr ead induced by ocean motion, geometry and edge eﬀects. Note that above 20 ms edge eﬀects fo rce the use of focusing. techniques to focus and eliminate the edge eﬀect we just disc ussed. We plan to revisit these calculations extending them to the Fresnel zone in future wo rk. The ultimate limit to the Doppler width and therefore to the c oherent integration time of the signal is ocean motion–that cannot be anticipated and compensated for. The size of the ﬁrst Fresnel zone for a receiver at 350 km is abo ut 2√ 2hλor about 1 km. In this narrow area, the geometric Doppler spans about 200 Hz edge to edge (the receiver looking down and is moving along at 7 km/s) and the correspond ing integration time is 5 ms. This Doppler is of the same order of magnitude of ocean ind uced spread. Integration times longer than 10 ms will certainly begin to be sensitive t o ocean motion. The edge eﬀect we discussed earlier will also start to play a role at lo nger integration times : for a 1 km WAF (roughly corresponding to 5 ms integration time) the c orresponding frequency is of 7 Hz— still too small to make spotlight processing necessa ry. This may be the best way to retrieve ocean induced Doppler spread. See Table-6.1 for a summary of these ideas. We have two diﬀerent new ways to extract sea-state informati on from the Carrier band- width, by playing with the eye location. One the one hand, we c an use a static WAF to measure vertical velocities—this will entail using SAR tec hniques from moving platforms. On the other, we can measure surface roughness by letting the WAF drift over the surface at a speed of choice. This is just a matter of tuning the matche d ﬁlter appropriately, and it could even be done from a static platform. 35Figure 6.8: The ocean surface and vertical velocity for a win d ofU10 = 8 m/s, Elfouhaily spectrum. The contour denotes the zero value in each case. 36Figure 6.9: Ocean heights and vertical velocities using the Elfouhaily et al. spectrum for wind speeds of U10=1,3,5,..,17 m/s. Note that due to limitat ions in the simulation size values for U10 above 12 m/s suﬀer from saturation. 37Chapter 7 Statisitical Properties of the Reﬂected Fields 7.1 Analysis of ﬁeld correlations In this section we address the question of the correlation be tween the ﬁelds at diﬀerent frequencies (say L1andL2and the future L5) for diﬀerent sea conditions. An approach to this problem is to rewrite the Fresnel integral using the “zo ne” concept. That is, we classify areas in the surface according to their distance to the recei ver. We assume we are in the Fraunhofer-emitter zone, so the distance to the emitter pla ys no role in this discussion. The idea is then to rewrite the ﬁeld as a sum of ﬁeld contributions from equal delay ( r)zones: Uq=q/integraldisplayrmax rmineiqrA′(r)dr. This idea can be found in [ Berry, 1972 ]. The function A′(r) is an “areal” density function that takes into account how much area contributes to each zon e. It’s exact expression is not of immediate concern. In the case of a ﬂat surface it just b ecomes one. In order to extend the integration over the whole r-axis, let us simply extend the deﬁnition of A′(r) to be zero outside rminandrmax. We then have, Uq=q/integraldisplay∞ −∞eiqrA′(r)dr. We can now read: the ﬁeld is the Fourier transform of A′(r). And the question about the relationship of the ﬁeld between frequencies becomes a ques tion on the relationship between the Fourier components of this areal function at diﬀerent fr equencies. Let us forget for now the meaning of the areal function, and ju st think in terms of the Fourier transform of a function with limited support, since we are looking at the Fourier transform of a function with support L≡rmin−rmax. The larger the support of the function, the more small-scale structure can be found in the Fourier transform—everything else the same. We are accostumed to thinking about this fact i n the opposite domain: a function with a lot of small time-scale structure will have a large bandwidth—a large support in the frequncy domain. Intuitively, the larger the support two given nearby freque ncy components will end up sampling the pulse at more separated regions. It is useful to think about two sine functions of slightly diﬀerent frequencies, running side by side. As a rule of thumb, we expect that the 38correlation between these Fourier components will be sensi tive to the correlation function of the (areal) function at a distance (∆ λ)∗L/λmean. This is because ∆ λis the sampling distance diﬀerence gained per cycle, and ∗L/λmeanis the number of cycles available in a region of size L. Therefore, before carrying out any calculation we can say t hat if the areal function is correlated at the distance dictated by thi s (maximal possible) frequency separation, then we will see correlation in the ﬁelds. It hel ps to look at the separation between the equi-delay contours in the surface of integrati on. If this separation is smaller than the correlation length of the surface, then high correl ation between the ﬁelds is possible. There is an extra consideration, however. We have been assum ing that the separation between contours is constant, but this is not the case. That i s, we have been implicitly assuming that the contours are ﬁxed on the surface, and all we have considered is the correlation between this contours. But in fact, as the ocean moves, the contours move, especially with large seas. This introduces an additional s ource of decorrelation, which gets worse with larger seas. The time-correlation between the ﬁe lds is given by /angb∇acketleftUq1U∗ q2/angb∇acket∇ightt=q1q2/integraldisplay /angb∇acketleftA′ 1(r)A′ 2(r′)/angb∇acket∇ightteiq1r−iq2r′drdr′. This is a diﬃcult beast to deal with, so we will change strateg y. The correlation between the ﬁelds can also be calculated in a nother manner. It will be useful to approximate things entirely in the Fraunhofer zon e, i.e., with both transmitter and receiver in the far ﬁeld, assuming Gaussian statistics. The ﬁeld in the nadir case is given, in 1-D and up to a constant, by U=q/integraldisplay Aeiqz(x)dx. (7.1) Now, \|U\|2=q2/integraldisplay A/integraldisplay Aeiq(z(x)−z(x′))dxdx′, (7.2) and /angb∇acketleft\|U\|2/angb∇acket∇ight=q2/integraldisplay A/integraldisplay A/angb∇acketlefteiq(z(x)−z(x′))/angb∇acket∇ightdxdx′. (7.3) Now, if we assume Gaussian statistics, with P(z(x)−z(x′)) =1 2πσ2/radicalbig 1−ρ2(x−x′)exp/bracketleftBigg −z(x)2−2ρ(x−x′)z(x)z(x′) +z2(x′) 2σ2(1−ρ2(x−x′))/bracketrightBigg , (7.4) we can carry out these calculations explicitly. Here σis the height standard deviation from the (zero) mean, and ρis the correlation function of the surface. It is healthy to keep in mind that in the gaussian case, there are the only para meters, together with the frequencies, that can appear in the ﬁnal expressions. If we f urther assume a Gaussian correlation function with correlation length l, the answer to any question must be expressed in terms of dimensionally meaningful expressions contanta ining, σ,l,λ1andλ2. For the case at hand, the key result is ([ Beckmann et al., 1963 ], p. 190) /angb∇acketlefteiq1z(x)+q2z(x′)/angb∇acket∇ight= exp/bracketleftbigg −1 2σ2(q2 1+ 2ρ(x−x′)q1q2+q2 2)/bracketrightbigg . (7.5) This implies, for instance, /angb∇acketleftU1U∗ 2/angb∇acket∇ight=q1q2/integraldisplay A/integraldisplay Aexp/bracketleftbigg −1 2σ2(q2 1−2ρ(x−x′)q1q2+q2 2)/bracketrightbigg dxdx′. (7.6) 39In the following we use a simple approximation, ρ(u) = 1 − \|u\|/l,\|u\| ≤l, (7.7) else zero. In this simple case we model a Gaussian correlatio n function by a linear approx- imation. For simplicity we will also work in the 1-D case. The ﬁrst step now is to deﬁne u=x−x′, and v=x+x′. The Jacobian of this change of variables is 1 /2. The integral becomes (let Abe a 1-D area from −LtoL) /angb∇acketleftU1U∗ 2/angb∇acket∇ight=q1q2/integraldisplay√ 2L −√ 2L/integraldisplay√ 2L−u −√ 2L+ududv 2exp/bracketleftbigg −1 2σ2(q2 1−2ρ(u)q1q2+q2 2)/bracketrightbigg (7.8) =√ 2Lq1q2/integraldisplay√ 2L −√ 2Lduexp/bracketleftbigg −1 2σ2(q2 1+ 2ρ(u)q1q2+q2 2)/bracketrightbigg . (7.9) Now, using our simpliﬁed correlation function model (and we will henceforth assume that L > l), we ﬁnd /angb∇acketleftU1U∗ 2/angb∇acket∇ight= 2√ 2L/parenleftbiggl σ2/parenleftBig 1−e−σ2q1q2/parenrightBig e−σ2∆q2/2+q1q2(√ 2L−l)e−σ2(q2 1+q2 2)/2/parenrightbigg .(7.10) For L-band, q= 2kis about 60 per meter. If σis greater than 0.1 m, σ2∗q2is greater than 40, and all the exponential terms in this expression van ish, except for exp[ −σ2∆q2/2]. That is, for σ >0.1 m, /angb∇acketleftU1U∗ 2/angb∇acket∇ight ∼2√ 2Ll σ2e−σ2∆q2/2. (7.11) It also follows that C(U1,U2) =/angb∇acketleftU1U∗ 2/angb∇acket∇ight/radicalBig /angb∇acketleftU1U∗ 1/angb∇acket∇ight/angb∇acketleftU2U∗ 2/angb∇acket∇ight∼e−σ2∆q2/2. (7.12) Figure 7.1: The coherency factor C(U1,U2) (dashed) and C(U2,U5) for the case l= 5 and L= 20. There is not much sensitivity in either case to Lorl. 40According to this result, the critical parameter in any reas onable ocean state is simply the signiﬁcant wave height. Imagine for instance an ocean wi th a very large correlation length and large height standard deviation. This is mirror- like ocean with a global up- down displacement. It is quite clear that ih this example /angb∇acketleftU1U∗ 2/angb∇acket∇ightwill be zero, since this product will be a number in the complex plane with a reasonabl e magnitude and a rather random phase. The average of such a complex number is zero. The general result, relevant for smaller signiﬁcant wave he ights, is C(U1,U2) =l σ2/parenleftBig 1−e−σ2q1q2/parenrightBig e−σ2∆q2/2+q1q2(√ 2L−l)e−σ2(q2 1+q2 2)/2 /radicalbigg/parenleftBig l σ2/parenleftBig 1−e−σ2q2 1/parenrightBig +q2 1(√ 2L−l)e−σ2q2 1/parenrightBig /parenleftBig l σ2/parenleftBig 1−e−σ2q2 2/parenrightBig +q2 2(√ 2L−l)e−σ2q2 2/parenrightBig. (7.13) We show a plot for the case l= 5 and L= 20 in Figure 7.1. Although we did not discuss it, it is rather immediate that th e average ﬁeld for the pure ﬁelds are governed again by exp [ −σ2q2/2]—this can be read oﬀ Equation 7.5 assuming a zero correlation function to infer /angb∇acketleftexp [iqz(x)]/angb∇acket∇ight= exp [ −σ2q2/2]. We have checked this trend in our data. In general, the interf erometric combination average ﬁelds are larger. For instance, for a σζ= 79 cm simulation (50 seconds at 0.0025 temporal resolution, 44 m size, spatial resolution 10cm, U1 0=1100cmpers)—see Figure 7.4, the L1, L2, L5, L12, L25 average ﬁelds were: L1: (+0 .03,−0.01) L2: (+0 .00,−0.06) L5: (−0.04,+0.03) L12: ( −0.02,+0.09) L25: ( −0.12,+0.07) while for U10=4 ms, σζ= 12 cm simulation, 19 m side (see Figures 6.5 , 6.6 and 6.7 for more details about this simulation) L1: (+0 .05,−0.03) L2: (−0.06,−0.12) L5: (−0.04,+0.10) L12: ( −0.09,+0.12) L25: ( −0.63,+0.20) This is where the real strength of PIP becomes evident. Moreo ver, low-pass ﬁltering (see next section) does not change these numbers much—the averag e ﬁeld is robust. The Fraunhofer approximation has given us a useful means to u nderstand the correlation properties of the signals in diﬀerent frequencies. We shoul d note, however, that in the GPS case we are not in the Fraunhofer zone—we are in the WAF zone, w hich is is usually substantially larger. Understanding all the implications of the extension to the WAF zone involves the analysis of the “beast” in Equation 7.1 which is substantially more complicated by the factors inthe areal function. This is left for future w ork. 7.2 Filtering Another way to understand and take advantadge of the coheren ce proporties is to focus on the low frequency aspects of the reﬂected signals for the s tatic case. As we mentioned 41above, in the static situation the coherence function of the scattered phase is related to the zeroth Fourier component of the ﬁeld—see Equation 6.1. Filt ering the signal to retrieve slow varying geophysical signals is thus a plausible approach. A s an example, we show in Figure 7.3 the ﬁelds with and without ﬁltering at 0.5 Hz (relevant fo r the case of a LEO). The result is quite good, as can be seen. This behavior is also exp ected from the analysis above of the average ﬁeld above, of course. The interferometric co mbination is again superior, especially L 25. For the satellite of aircraft case, good modeling of the rec eiver position and ﬁltering can be used to remove all but the slowly varying geop hysical signals from the phase drift. Important points: ﬁltering doesn not seem to aﬀect the ﬁeld m ean value (a very desirable result). It does signiﬁcantly reduce the scatter, however. For instance, in Figure 7.2, we have the ﬁelds for a 3 m/s wind. The standard deviations of the ﬁeld go from 0.87, 0.82, 0.79, 0.73, 0.68 to 0.51, 0.51, 0.49, 0.48, 0.29 (for L1, L2, L 5, L12, L25) after 0.5 Hz ﬁltering of the ﬁelds. See also the results for higher wind speed is Fig ures 7.3 and 7.4. The mean ﬁeld remains virtually unchanged after this process. 42Figure 7.2: These are plots of the ﬁelds and phases for the cas e U10=3m/s ( σζ= 18 cm), with an ocean patch of 43 m, resolution 10 cm. The top is the ori ginal time series, the second is with a ﬁeld ﬁlter at 0.5 Hz. Temporal resolution in a ll the simulations is of 0.0025 seconds. 43Figure 7.3: These are plots of the ﬁelds and phases for the cas e U10=6 m/s ( σζ= 42 cm), with an ocean patch of 43 m, resolution 10 cm. The top is the ori ginal time series, the second is with a ﬁeld ﬁlter at 0.5 Hz. 44Figure 7.4: These are plots of the ﬁelds and phases for the cas e U10=11m/s ( σζ= 79 cm), with an ocean patch of 43 m, resolution 10 cm. The top is th e original time series, the second is with a ﬁeld ﬁlter at 0.5 Hz. See the next ﬁgure for the corresponding (unﬁltered) coherence and strucuture functions. This is complete simul ation, with the ocean going from ﬂat to moving to ﬂat. Observe the picked up winding number. Fi ltering at 0.5 Hz did not entirely cure the problem in this case. 45Chapter 8 Simulation experiments with winding number and GIP Consider the following scenario. The receiver is static. In itially the ocean is calm, totally ﬂat. Then a disturbance gradually appears, peaks, and slowl y disappears. The ocean ﬁnal state is the same as the original state: total ﬂatness. We hav e run several simulations with this scenario. The result is that the winding number of the ﬁe ld history is non-zero. That is, the ﬁeld can loop several times arond zero. The ﬁrst example is in Figure 8.1 and accompanying Figure 8.2 . How is this possible? As can be seen in the example in Figure 8.3, the sum of zero-win ding-number curves can result in a winding number 1 curve. Since the resulting total ﬁeld is a linear combination of such ﬁelds, all that we need to show is how the moving ocean c an generate such winding number zero curves. This is easy: the only requirements are t hat a) the initial and ﬁnal phase be the same, with the phase moving in between, and b) tha t the ﬁeld magnitude increase or decrease before returning to its original value . It is possible to imagine examples based on GO reasoning (a usefull approximation, at any rate) to obtain this behaviour. Suppose, for instance, that the distance from a specular poi nt to the receiver decreases and then increases, while the radius of curvature decreases the n increases. It is easy to see that this yields a clockwise loop in phasor space (with the usual a ngular convention used in the complex plane). We have seen the phase dispersion eﬀect, very much related to winding number (which is a special case of the ﬁrst when initial and ﬁnal states are t he same), in all wind conditions, with U10=2 m/s and up. At ﬁrst sight this poses a signiﬁcant pr oblem in the use of phase for altimetry. How can we use a “deﬁting” measure for altimet ry? It is now very important to characterize the statistical and geophysical propertie s of this phase drift (Geophysically Induced Phase drift, or GIP for short). 8.1 The uses of winding number Winding number, or more generally, phase dispersion under a moving ocean is certainly related to sea state characteristics. An interesting Gedan ken1: imagine a satellite tracking a reﬂection over a smooth region, then rough, then smooth aga in. The winding number induced will be related to the roughness encountered. We hav e already seen that, in general, 1Gedanken (thought) experiments are little mental exercise s for exploring hypothetical experiments. The terminology comes from Einstein’s work, who is said to have e ngaged in them even when he was very young. 46Figure 8.1: Simulation: 20 meters side, res=10cm, max ocean 4 m/s U10 with a peak σζ of 12 cm, sine fourth modulation. Note that in the bottom row w e present the FFT power of the normalized ﬁeld. The zero component is basically the c oherence function mentioned earlier. The interferometric component shows a higher degr ee of coherence. the phase drifts without a speciﬁc chirality in our simulati ons. It remains to be seen if non- gaussian eﬀects in the surface can lead to chirality, or pref erred rotation direction. If this is the case, this phenomenon could conceivably be used for ge ophysical measurements. We have already seen that GIP is related to wind speed in our simu lations. 47Figure 8.2: In this Figure we illustrate the L2 ﬁeld phasor in Figure 8.1, which has a winding number of -1. Figure 8.3: Winding number magic: four curves with winding n umber 0 yield a winding number 1 curve after being added. The trick can be done with 2 c urves. In this case they all turn in the same direction. Changing that will result in a n ellipse. 48Chapter 9 Recommendations for Post-processing Procedures Based on the work carried out so far, we are prepared to outlin e some of the possible incarnations of the PIP concept. What we must emphasize is th at the strength lies in the noise cancellation in the interferometric combination. Fi ltering is best carried out after this combination. 9.1 The PIP instrument: architecture and data products Based on our results, we have the following suggestions. 1. Use two wideband (200 Hz for the static receiver, possibly more for the moving case) PLL’s to extract the phase in each frequency. 2. Combine the phases and low-pass ﬁlter to recover altimetr ic trends. The normalized interferometric ﬁeld is more coherent, so it can be used to tr ack slow-varying altimetric changes (such as a tide). We have seen that the interferometr ic ﬁels is a more robust source, with a better deﬁned phase. 3. Use phase drift as a geophysical parameter More generally , the phase and power time series are probably rich information source of sea state. To be more precise, the following recipes for PIP implementa tion can be conceived: One, the most natural one perhaps, is the Single PLL PIP (SPIP) 1. Multiply the two signals, form the interferometric signa l. 2. Filter from 0 to 0.5 Hz 3. Use a single PLL to extract the phase. The disadvantadge is that the ﬁeld magnitude will oscillate quite a bit. This may not be important, given that a ﬁlter will be used. An alternative wo uld be to used two PLLs (MPIP) 1. Extract the phase in L1 and in L2 2. Combine the two phases 493. Filter from 0 to 0.5 Hz. This is probably similar to the previous case, but it takes th e ﬁeld magnitude out of the picture from the very beginning—probably not a good thing. F inally, a non-contender is to 1. Filter from 0 to 0.5 Hz 2. Extract the phase in L1 and in L2 3. Combine the two phases This is a bad option, it makes no use of the jitter cancellatio n across frequencies. We have focused on the static case. For other situations we ju st need to work with excess phase and carry out the same ﬁltering process. 50Chapter 10 Conclusions and Future Work 10.1 What we have found: PIP’s superior performance 1. The mean ﬁeld is largest for the PIP case L 25for all sea conditions. In general the mean ﬁeld is proportional to exp [ −σ2q2/2], where q= 4π/λandλis the pure or interferometric wavelength. 2. The coherence integration time is consequently substant ially larger for the interfer- ometric combination, especially L 25(if the mean sea level scatter is very large the eﬀect disappears, of course). The interferometric phase is coherent in a reasonable range of sea conditions, unlike the single frequency phases . This implies that the PIP interferometric combination is superior to extract altime tric low frequency trends in the phase, as we discussed at the beginning. 3. The phase in our simulations behaves like a random walk, an d the drift rate is directly related to sea state: in general, the larger the wind speed, t he higher the drift rate. This implies that a system capable of tracking the phase of th e reﬂected signals can provide, aside from altimetric measurements, sea state inf ormation. 4. Filtering the received ﬁeld does not aﬀect its mean value, while decreasing its scatter. It can probably be used for altimetric purposes, as it remove s noise but leaves the slow-varying geophysical signals alone. 5. No long-term trends in phase drift chirality have been det ected in our simulations with Gaussian ocean models based on [ Elfouhaily et al., 1997 ]. We have performed simulations up to 2 minutes long. This is good news for our alt imetric eﬀorts. It seems that despite the phase drift present, and the winding number it can lead to, in practice this eﬀect should not introduce systematic eﬀects on altime tric determination. It is important to check that the simulations are correctly imita ted by nature, of course. Among other things, non-Gaussian eﬀects in the sea could con ceivably lead to real drifting. The single most important parameter in the correlation betw een the ﬁelds at diﬀerent frequencies is the signiﬁcant wave height. This has to be com pared to the synthetic wave- length. For rough seas, the correlation bethween the ﬁelds ( and therefore interferometric coherence) in diﬀerent frequencies disappears, and the coh erence time goes to zero even for the interferometric combinations. In calmer ocean conditi ons, however, our results indicate that the interferometric combination remains coherent. 5110.2 What we’d now like to know: future work Due to the limited scope of this study we were not able to cover all the interesting aspects of this problem. Future theoretical work includes the exten sion of the present results to the case in which the receiver is far away from the ground and is mo ving. We would also like to understand the impact of changing the ocean statistics fr om gaussian to non-gaussian. Another important aspect that deserves further research is to extend the analysis to a bistatic situation. So far, our simulations have only consi dered the monostatic case. We intend to continue this work within the scope of PARIS- α: 1. Examine more realistic situations (higher, faster). Thi s includes, in particular, un- derstanding the issues that will arise in the aircraft and LE O scenario, and it will require larger simulations—in fact it may impossible to sim ulate the ocean with the required size and resolution, as the WAF zone are increases l inearly with height. 2. Understand the theoretical issues involving the correla tions and Doppler spread of the reﬂected ﬁelds without using the Fruanhofer approximat ion for the receiver. 3. Extend the analysis to the bistatic situation. 4. Carry out a detailed study of phase drift versus wind speed . 5. Look deeper at the advantages (which we have seen and are ex pected) of using even closer frequencies. This may result in recomendations for G ALILEO, for instance. 6. Understand the impact of non-Gaussian ocean eﬀects. 7. Understand better the possible sources of chirality in GI D. This is important to assure altimetric accuracy. 8. Obtain and analyze data for the static situation. 9. Understand the eﬀectiveness of GO or its higher order corr ections. It is rather clear that experimental work is needed in this ﬁe ld. The most important experiment for the concept at this point, is a redo of the brid ge experiment: a static receiver over the moving ocean, using both L 1and L 2. The desired product from such an experiment from the point of view of the present work, woul d time-series’ of the ﬁelds. MEATEX campaigns will also provide aircraft data which will be very useful to understand some of these issues. 52Chapter 11 Acknowlegments This is a document produced for ESA under ESTEC Contract No. 1 4071/99/NL/MM. 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arXiv:physics/0011029v1 [physics.chem-ph] 15 Nov 2000Benchmark ab initio energy proﬁles for the gas-phase S N2 reactions Y−+ CH 3X→CH3Y + X−(X,Y = F,Cl,Br). Validation of hybrid DFT methods Srinivasan Parthiban, Glˆ enisson de Oliveira∗, and Jan M.L. Martin† Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, IL-76100 Reh .ovot, Israel. (J. Phys. Chem. A manuscript JP0031000; revised October 31, 2000) Abstract The energetics of the gas-phase S N2 reactions Y−+ CH 3X− →CH3Y + X−(where X,Y = F, Cl, Br), were studied using (variants on) the r ecent W1 and W2 ab initio computational thermochemistry methods. These cal- culations involve CCSD and CCSD(T) coupled cluster methods , basis sets of up to spd fgh quality, extrapolations to the one-particle basis set limi t, and contributions of inner-shell correlation, scalar relativ istic eﬀects, and (where relevant) ﬁrst-order spin-orbit coupling. Our computatio nal predictions are in excellent agreement with experimental data where these h ave small error bars; in a number of other instances re-examination of the ex perimental data may be in order. Our computed benchmark data (including case s for which experimental data are unavailable altogether) are used to a ssess the quality of a number of semiempirical compound thermochemistry scheme s such as G2 theory, G3 theory, and CBS-QB3, as well as a variety of densit y functional theory methods. Upon applying some modiﬁcations to the leve l of theory ∗Present address: Chemistry Department, Pensacola Christi an College, 250 Brent Lane, Pensacola, FL 32503 †Author to whom correspondence should be addressed. Email: comartin@wicc.weizmann.ac.il 1used for the reference geometry (adding diﬀuse functions, r eplacing B3LYP by the very recently proposed mPW1K functional [Lynch, B.J. ; Fast, P.L.; Harris, M.; Truhlar, D.G. J. Phys. Chem. A 2000,104, 4811]), the com- pound methods appear to perform well. Only the ’half-and-ha lf’ functionals BH&HLYP and mPWH&HPW91, and the empirical mPW1K functional , con- sistently ﬁnd all required stationary points; the other fun ctionals fail to ﬁnd a transition state in the F/Br case. BH&HLYP and mPWH&HPW91 s ome- what overcorrect for the tendency of B3LYP (and, to a somewha t lesser extent, mPW1PW91) to underestimate barrier heights. The Becke97 an d Becke97-1 functionals perform similarly to B3LYP for the problem unde r study, while the HCTH and HCTH-120 functionals both appear to underestimate central bar- riers. HCTH underestimates complexation energies; this pr oblem is resolved in HCTH-120. mPW1K appears to exhibit the best performance o f the func- tionals considered, although its energetics are still infe rior to the compound thermochemistry methods. mPW1K, however, appears to be ver y suitable for generating reference geometries for more elaborate thermo chemical methods in kinetics applications. 2I. INTRODUCTION Due to the central importance of bimolecular nucleophilic s ubstitution (S N2) reactions in organic chemistry,1,2the prototype S N2 reactions Y−+ CH 3X− →CH3Y + X−(X,Y = F ,Cl,and Br) (1) have aroused considerable interest in the past three decade s. (Halomethanes have also received considerable attention in the area of atmospheric chemistry in connection with global warming3and ozone layer destruction.4) Both theoretical and experimental studies (see Refs.5,6for reviews) indicate that the preferred gas phase reaction pathway involves a backside attack of the halide ion, Y−, at the carbon atom followed by the familiar ‘Walden inversion’ of the CH 3group. The resulting reaction proﬁle (Figure 1) exhibits tw o local minima, i.e. entry and exit channel ion-molecule complexes Y−· · ·CH3X and YCH 3· · ·X−, connected by a central transition state [Y · · ·CH3· · ·X]−, which has D3handC3vsymmetries in the identity (X=Y) and nonidentity (X /ne}ationslash=Y) cases, respectively. Although the qualitative form of this reaction pathway is widely accepted for substit ution reactions in the gas phase, there is still considerable uncertainty about the exact ene rgetics. B¨ ohme et al.7and Brauman et al.8were the ﬁrst to investigate the gas-phase S N2 re- actions experimentally. Brauman and coworkers concluded t hat the measurements were best explained by a double-well potential with a central bar rier. Subsequent experimental studies9–12for a series of anionic nucleophiles with alkyl halides reve aled that changes in the nucleophile, leaving group, and alkyl moiety leads to a w ide variation of reaction rate constants; their observed variation was attributed to the c entral barrier height. The double-well S N2 potential energy surface also ﬁnds abundant theoretical s upport fromab initio calculations, which are currently one of the most useful too ls for evaluating reaction potential energy proﬁles. For instance, Chandras ekhar et al.13presented a compre- hensive examination of Cl/Cl identity S N2 reaction at the HF/6-31G* level, and Tucker and Truhlar14examined the S N2 reactions at the MP2/6-31G* level. Wladkowski et al.15studied the F/F identity S N2 reaction by large-scale coupled cluster theory involving single and dou- ble excitation operators with an a posteriori a quasiperturbative treatment of the eﬀects of 3connected triple substitutions (CCSD(T)) (see Refs.16,17for reviews). Later, Radom, Pross and coworkers have carried out ab initio molecular orbital calculations at the G2(+) level of theory for the back-side identity18and nonidentity19SN2 reactions. These authors have also investigated the identity front-side S N2 reactions with retention of conﬁguration.20The G2(+) theory is essentially G2 theory carried out from MP2/6 -31+G* (rather than MP2/6- 31G) geometries and employing scaled HF/6-31+G (rather t han HF/6-31G) zero-point energies. For the bromine and iodine containing systems, th ese authors employed Hay- Wadt21relativistic eﬀective core potentials (RECPs). Botschwin a and coworkers examined the stationary points of the potential surface for the F/Cl n onidentity S N2 reaction22and for the Cl/Cl identity S N2 reaction23by means of large-scale CCSD(T) calculations. Finally, a referee brought a very recent large-scale coupled cluster study by Schmatz et al.24to our attention. Despite the well known successes (e.g.25,26) of the increasingly popular DFT (density functional theory) methods,27their performance for transition state structures and reac - tion barrier heights leaves something to be desired. For ins tance, Durant28found that the B3LYP, B3P86 and B3PW91 functionals all systematically und erestimated barrier heights, while only the Becke half-and-half/Lee, Yang and Parr (BH&H LYP)29functional predicted transition state barrier heights reasonably well — despite the fact that its performance for thermochemical and other properties is generally substant ially poorer than that of B3LYP and B3PW91. Baker et al.30arrived at a similar conclusion stating that the currently a vail- able density functionals are unable to provide a correct des cription of the transition states. For the prototype S N2 reactions (Cl/Cl and Cl/Br) considered here, Radom and cow orkers31 found that the popular B3LYP32,33exchange-correlation functional signiﬁcantly underes- timated the overall and central barrier heights compared to the G2(+) and experimental results. Nevertheless, the size of the systems involved in kinetic an d mechanistic problems of organic and organometallic interest often makes DFT the onl y practical option. As a mat- ter of fact, our group has recently reported DFT studies of th e mechanism of competitive intramolecular C-C and C-H bond activation in rhodium(I) pi ncer complexes34and of the 4Heck reaction.35 Aside from BH&HLYP, better performance for barrier heights has been claimed for a number of newer exchange-correlation functionals. For exa mple, Adamo and Barone36found that their mPW1PW91 (modiﬁed Perdew-Wang 1991 1-parameter hybrid exchange with Perdew-Wang 1991 correlation37) at least correctly predicts a positive overall barrier for the Cl/Cl identity S N2 reaction, although it is still being underestimated. Very recently, Truhlar and coworkers38proposed a new hybrid model called the modiﬁed Perdew-Wang 1 -parameter model for kinetics (mPW1K). In this empirical functional, t he coeﬃcient Xfor admixture of ”exact” Hartree-Fock exchange VXC=XVX,HF+ (1−X)VX,mPW 1+VC,PW 91 (2) (where X=1/4 for standard mPW1PW91) was determined (using the fairl y small 6-31+G basis set) by minimizing the average deviation from a set of 4 0 barrier heights (20 forward, 20 reverse) obtained from a combination of experiment and th eory (see Ref.38for details). (Note that the Walden inversion, or for that matter cationic or anionic reactions of any kind, were not part of the parametrization set.) It was found that mPW1K reduced the mean unsigned error in reaction barrier heights by a factor o f 2.4 over mPW1PW91 and by a factor of 3 over B3LYP. Theoretical models such as Transition State Theory (TST)39and Rice-Ramsperger- Kassel-Marcus (RRKM)40theory were also employed to examine the S N2 reactions. Results from such studies (see41and references therein) suggested that the assumption of st atistical behavior in ion-molecule intermediate complexes is not val id. This ”nonstatistical” behavior has been documented for several halide-methyl halide react ions and a thorough discussion is given by Hase.5Classical trajectory simulations performed by Hase and cow orkers42ques- tioned the basic assumptions of statistical theories and fo und that the trajectory calculations are very useful in interpreting the kinetics and dynamics of SN2 reactions. Despite the enormous amount of work in the past, there are sti ll signiﬁcant gaps in the experimental data for the gas-phase S N2 reactions and, even where data are available, the results often possess large uncertainties. Recently, t wo computational thermochemistry 5methods known as W1 and W2 (Weizmann-1 and Weizmann-2) theor y43have been developed in our laboratory. These are free of parameters derived from experiment and on average can claim ‘benchmark accuracy’ (deﬁned in Ref.43as a mean absolute error of 1 kJ/mol, or 0.25 kcal/mol) for molecular total atomization energies (TAEs) of ﬁrst-and second-row compounds. The primary objective of the present study is to o btain high-quality energetic data for reaction (1) by means of W1 and W2 theory. Using these benchmark data, we shall then examine the performance of various DFT methods an dab initio computational thermochemistry methods such as G1,44G2,45G3,46and CBS-QB347theories. II. COMPUTATIONAL METHODS All calculations were carried out on the 4-processor Compaq ES40 of our research group, and on the 12-CPU SGI Origin 2000 of the Faculty of Chemistry. Energetics for the gas-phase stationary points for all six s urfaces (i.e. F/F, Cl/Cl, Br/Br, F/Cl, F/Br, and Cl/Br) were obtained by means of the W1′method described in Refs.43,48. The W1′method48is a minor variation on W1 theory43that exhibits improved accuracy for second-row systems at no additional computational cost . For a detailed description and theoretical and empirical arguments for each step, see Ref.43; we shall merely summarize the main points here for the sake of clarity. The basis sets em ployed are mostly Dunning’s augmented correlation consistent n-tuple zeta49–51(aug-cc-pV nZ) basis sets; for second-row atoms high-exponent dandffunctions were added (denoted ’+2d’ or ’+2d1f’) as recom- mended in Ref.52for accommodating inner polarization. Since the standard a ug-cc-pV nZ basis sets for bromine53already contain quite high-exponent dfunctions in order to describe the 3dorbitals, no ’inner polarization’ functions were deemed to be necessary on Br. We may distinguish the following six components in the ‘bottom -of-the-well’ TAE at the W1′ level: •The SCF component of the TAE is obtained using the aug-cc-pVD Z+2d, aug-cc- pVTZ+2d, and aug-cc-pVQZ+2d1f basis sets, and extrapolate d to the inﬁnite-basis limit using the geometric expression54A+B·C−L, where the ‘cardinal number’ 6L={2,3,4}for these three basis sets. (It is identical to the maximum an gular mo- mentum present for nonhydrogen atoms. Regular cc-pV nZ basis sets were used on hydrogen atoms throughout.) •The CCSD (coupled cluster with all singles and doubles55) valence correlation contri- bution to TAE is obtained using the aug-cc-pVTZ+2d and aug-c c-pVQZ+2d1f basis sets, then extrapolated to the inﬁnite basis limit using the expression A+B/L3.22. •The (T) connected triple excitations component56of TAE was computed using the aug- cc-pVDZ+2d and aug-cc-pVTZ+2d basis sets, and extrapolate d to the inﬁnite basis limit using the expression A+B/L3.22. •The inner-shell correlation contribution was computed as t he diﬀerence between CCSD(T)/MTsmall43values with and without constraining the inner-shell orbit als to be doubly occupied. The very deep-lying chlorine (1s) and bromine (1s,2s,2p) or- bitals were doubly occupied throughout; the ‘inner-shell c orrelation’ thus represents carbon (1s), chlorine (2s,2p) and bromine (3s,3p,3d) corre lation. (Basis set superposi- tion error, BSSE, can be an issue for inner-shell correlatio n energies in heavier element systems;57our experience43suggests that BSSE in the W1/W2 inner shell correlation contributions largely cancels with basis set incompletene ss.) •The scalar relativistic contribution was computed as expec tation values of the one- electron Darwin and mass-velocity (DMV) operators58,59for the ACPF/MTsmall (av- eraged coupled pair functional60) wave function, with all inner-shell electrons correlated except for chlorine (1s) and bromine (1s,2s,2p). •The spin-orbit contribution to TAE, in the present case of al l-closed-shell systems, is nothing more than the sum of the atomic ﬁne structure correct ions. Where our computational hardware permitted (in practice, f or F/F, Cl/Cl, and the Br/Br transition state), we also carried out even more deman ding W2h calculations. In W2 theory, the same steps occur as above, except that the three v alence basis sets are aug-cc- pVTZ+2d1f, aug-cc-pVQZ+2d1f, and aug-cc-pV5Z+2d1f (with L=3,4, and 5, respectively) 7and that the extrapolation formula61used for the CCSD and (T) steps is simply A+B/L3. The W2h variant62indicates, in this particular case, the use of unaugmented c c-pVnZ basis sets on carbon. The largest basis set CCSD step was carried ou t using the direct algorithm of Lindh, Sch¨ utz, and Werner.63All these calculations were performed using MOLPRO 98.164 and a driver for the W1/W2 calculations65written in MOLPRO’s scripting language. Ref- erence geometries were obtained primarily using the B3LYP32,33density functional method, which employs the Lee-Yang-Parr33correlation functional in conjugation with a hybrid ex- change functional ﬁrst proposed by Becke.32 A number of lower-level procedures were validated against t he W1′and W2h re- sults. These include the following set of DFT exchange-corr elation functionals and ba- sis sets: B3LYP/cc-pVTZ(+X), BH&HLYP/cc-pVTZ(+X), mPW1P W91/cc-pVTZ(+X), mPW1K/6-31+G, mPW1K/cc-pVDZ(+X), mPW1K/cc-pVTZ(+X), w here (+X) indicates that diﬀuse functions are included only for halogens. BH&HL YP29is essentially the B3LYP method, with the exception that the fraction of HF exchange i s 50% (H&H denotes ”half and half”). Analogous to BH&HLYP, we have also performed mPWH&H PW91.66In addition, we carried out calculations using the standard G1, G2, G3, and C BS-QB3 model chemistries. The MP2/6-31G and B3LYP/6-311G(2d,d,p) levels of theory ( used for the reference ge- ometries in G346and CBS-QB3,47respectively) fail to ﬁnd stationary points for several of the ion-molecule complexes in the nonidentity cases (becau se of the absence of diﬀuse func- tions). Therefore, we have deﬁned, by analogy with Radom and coworkers,18–20G3(+) and CBS-QB3(+) model chemistries where MP2/6-31+G* and B3LYP/ 6-311+G(2d,d,p) refer- ence geometries, respectively, are used. The G2(+) results quoted in the tables are taken from Radom and coworkers.18,19All of these calculations were carried out using Gaussian 98 rev. A767or trivial modiﬁcations thereof. Following the recommenda tions in Ref.68, larger grids than the default were used in the DFT calculations if ne cessary, speciﬁcally a pruned (99,590) grid for integration and gradients, and a pruned (5 0,194) grid for the solution of the coupled perturbed Kohn-Sham equations. Finally, following very recent suggestions in the literatu re69that some of these function- als may perform better for transition states, some calculat ions using the novel B97 (Becke- 81997),70B97-1 (reparametrized Becke-1997),71HCTH (Hamprecht-Cohen-Tozer-Handy),71 and HCTH-120 (reparametrization of HCTH including anions a nd weakly interacting systems)72functionals were carried out by means of a slightly modiﬁed v ersion of NWCHEM 3.3.1.73 III. RESULTS AND DISCUSSION A. Reference Geometries Reference geometries for the W1′calculations were mostly obtained at the B3LYP/cc- pVTZ+1 level, where the ‘+1’ signiﬁes the addition of a high- exponent dfunction on second- row elements.74(The Br basis set already includes high-exponent dfunctions to cover the (3 d) orbital.) It was previously shown26that B3LYP/cc-pVTZ geometries for stable molecules are generally within a few thousandths of an ˚A from experiment, as well that the use of B3LYP/cc-pVTZ+1 rather than much costlier CCSD(T)/cc-pVQ Z+1 reference geometries insigniﬁcantly aﬀects computed energies.43(Modiﬁcations of popular computational ther- mochemistry methods that use DFT reference geometries incl ude variants75,76of G2 theory, G3//B3LYP,77and CBS-QB3.47) In some cases where B3LYP fails to locate the required stationary point, we used mPW1K/cc-pVTZ(+X) reference geo metries. For the W2h cal- culations, CCSD(T)/cc-pVQZ+1 reference geometries were u sed. The geometries of all the structures involved in the present study calculated at vari ous levels of theory are provided in the Supplementary Material. B. Energetics In order to assess the accuracy of W1′and W2h results, we consider ﬁrst the total atomization energies (TAEs) of CH 3X and electron aﬃnities (EAs) of X−. A summary of our computed results and their components for the reactants /products of the S N2 reactions is presented in Table 1. The ﬁnal energies presented in the last column of the Table correspond to EAs of X (X = F, Cl and Br) and TAEs without zero-point vibrat ional energy (ZPVE) 9of CH 3X. The inner-shell correlation contributions are all posit ive and the largest is 1.49 kcal/mol for CH 3Br. The core correlation contribution for TAE(CH 3X) is found to increase in the order F <Cl<Br, while for EA(X) it increases in the order Cl <F<Br. The importance of Darwin and mass-velocity corrections increa ses, as expected, with increasing atomic number (Z) of X and its contribution becomes substant ial when X = Br. It is perhaps more pertinent for our purposes to examine the r elative energies (with respect to reactants) of the ion-molecule complexes and tra nsition state structures. Table 2 presents W1′and W2h results for identity reactions and only W1′results for nonidentity reactions. W2h calculations for the nonidentity reactions are extremely expensive as the reaction intermediates are less symmetric. Moreover, the s ize of the bromine atoms prevents us from performing a W2h calculation on the identity Br−· · ·CH3Br ion-molecule complex. Likewise, we could not obtain the core correlation contribu tions for the Br−· · ·CH3Br ion- molecule complex at the W1′level of theory. It was previously established78that the inclusion of connected triple excitations in CCSD(T) is absolutely ne cessary for reliable core correlation contributions: the n3N4CPU time dependence of the (T) step dominates the required CP U time for Cl and Br. Both the size of the halogen atoms and the re duced symmetry prevented us from performing core-correlation calculations for noni dentity S N2 reactions, except for the F/Cl nonidentity case. From Table 2, it can be seen that the ﬁnal W1′and W2h energy values for the iden- tity reactions are very close to each other. Considering the very close agreement between W1′and W2h results, the conclusion is warranted that the result s from W1′theory can be used as reference values to compare the results from other methods when W2h results are not available. As a general observation, the core contri butions for the transition state structures are noticeably larger than for the ion-molecule complexes (see Table 2). Although the core-correlation contribution is small in absolute ter ms, its relative contribution to the likewise small overall barrier heights can be substantial. For example, it is 0.36 kcal/mol for [F · · ·CH3· · ·F]−, while the total energy is -0.37 kcal/mol. Likewise, the cor e correlation contributions for [Cl · · ·CH3· · ·Cl]−and [Br · · ·CH3· · ·Br]−are nearly 10% and 25% of the overall barrier (relative to reactants), respectively. At the W2h level, the core correlation 10contribution to the total energy increases slightly. Scala r relativistic eﬀects exhibit simi- lar trends as those of core correlation, but the eﬀects are fa irly small. Only Cl−· · ·CH3Br, [Cl· · ·CH3· · ·Br]−and Br−· · ·CH3Cl exhibit noticeable scalar relativistic contributions, due to the presence of the heavy halogen Br. Among the identity reactions only [F · · ·CH3· · ·F]−has a transition state below the reac- tants energy level. In the nonidentity case all the transiti on state structures lie below the reactants. The computed ﬁnal heats of formation (∆ H◦ f) of CH 3X in kcal/mol are compared with experiment in Table 3. Both W1′and W2h values are presented after accounting for ZPVEs and thermal corrections calculated at the B3LYP/cc-pVTZ+1 level. At this level the ZPVEs, after scaling by 0.985,43are found to be 24.16, 23.26 and 22.89 kcal/mol, respectivel y, for CH3F, CH 3Cl and CH 3Br. The corresponding thermal corrections are -1.92, -1.89 and -3.68 kcal/mol. The computed ∆ H◦ fvalue for CH 3Cl lies within the experimental error bar: the experimental value for CH 3F is a crude estimate ( ±7 kcal/mol) and our computed value is certainly more reliable. Our calculated value for CH 3Br is slightly outside the experimental error bar: some of the discrepancy could be due to the limitat ions of the scalar relativistic treatment. As shown by Bauschlicher,79the simple DMV correction starts to exhibit minor deﬁciencies for third-row compounds; for ﬁrst- and second- row compounds, it is in excellent agreement with more rigorous treatments.80,81 Also included in Table 3 are calculated electron aﬃnities of X (X = F, Cl and Br) in eV together with experimental results. Using a similar approa ch, but with even larger spdfghi basis sets as well as full CI corrections, we were able82to reproduce the experimental EAs of the 1st- and 2nd-row atoms to within ±0.001 eV on average. The presently calculated W2h results of F and Cl EAs diﬀer by only about 0.001 eV from the se benchmark values (EA(F)=3.403 eV and EA(Cl)=3.611 eV), and the W2h results fo r F, Cl and Br are all within 0.003 eV of experiment. Although the W1′values diﬀer about 0.01 eV for F and Cl and 0.02 eV for Br, this is comparable to the W1/W2 target accu racy (0.25 kcal/mol on average). The performance of the W1′and W2h methods for the reactants and products is obviously encouraging for the study of the problem at hand. 11C. S N2 Reactions The reaction mechanism with the double-well potential ener gy surface for the gas-phase SN2 reactions is shown in Figure 1. Obviously, the energy proﬁl e is symmetric for the identity reactions (Figure 1a), and asymmetric for the noni dentity reactions (Figure 1b). The complexation energy (∆ Hcomp), central barrier (∆ H‡cent), and overall activation barrier relative to the separated reactants (∆ H‡ovr) are deﬁned in Figure 1. In the nonidentity case, the following additional quantities are deﬁned in Figure 1b : overall enthalpy change for the reaction (∆ Hovr) and the central enthalpy diﬀerence ∆ Hcentbetween product and reactant ion-molecule complexes, 3and1. D. Identity Reactions Complexation energies (∆ Hcomp), overall barrier heights (∆ H‡ovr) and central barriers (∆H‡cent) obtained from W1′and W2h methods are compared in Table 4 with DFT, Gn, and CBS-QB3 methods together with available experimental v alues. It should be emphasized that the experimental data for the S N2 reactions are insuﬃcient and the available data are subject to large uncertainties. T herefore, it would be appropriate to analyze the performance of various methods with respect t o Wn methods. First of all, note that the mPW1K/6-31+G* ∆ Hcomp(13.55 kcal/mol) for the F/F case is very close to the W1′and W2h results (13.66 and 13.72 kcal/mol, respectively). m PW1K/cc-pVDZ(+X) and mPW1K/cc-pVTZ(+X) methods however predict lower ∆ Hcompvalues. In fact, the B3LYP, B97, HCTH-120, mPW1PW91, mPWH&HPW91 and mPW1K metho ds all predict roughly 1 kcal/mol lower complexation energies, while BH&H LYP and B97-1 agree well with W1′and W2h. (The HCTH ∆ Hcompis much lower than the others, vide infra .) G2 and CBS- QB3 values are close to the Wn results while the G3 method pred icts higher complexation energy compared to the Wn methods. Inclusion of diﬀuse funct ions for the Gn and CBS-QB3 reference geometries (i.e. Gn(+) and CBS-QB3(+)) increase s the ∆ Hcompvalue by 0.4–0.6 kcal/mol. A comparison of overall barrier heights is presented in the t hird column of Table 4. Both 12W1′(-0.37 kcal/mol) and W2h (-0.34 kcal/mol) theories predict negative barrier heights in the F/F case and the values are very close. The CBS-QB3(+) res ult is in excellent agreement therewith; all G ntheories predict barrier heights that are lower by 1 kcal/mo l, with further lowering seen at the G2(+) and G3(+) levels. Among the DFT met hods considered, only mPW1K/6-31+G* and HCTH/cc-pVDZ(+X) fortuitously predict overall barrier heights close to the Wn results: basis set extension for mPW1K leads t o positive overall barrier heights, which are likewise found for the ”half and half” fun ctionals. B3LYP, B97(-1) and HCTH-120 all signiﬁcantly underestimate the barrier, mPW1 PW91 to a lesser extent. For the chlorine identity gas-phase S N2 reactions, fairly accurate experimental values are available and are presented in Table 4. The experimental values reported by Li and coworkers11correspond to the standard state. Hence, thermal correctio ns and ZPVEs are subtracted from experimental values in order to compare wit h the ”bottom of the well” calculated values. It is noteworthy that the W1′(10.54 kcal/mol) and W2h (10.94 kcal/mol) complexation energies are in good agreement with the experi mental value (10.53 kcal/mol). CBS-QB3 results are also in agreement with the Wn and experim ental values, while those from DFT calculations are less satisfactory as they are abou t 1 kcal/mol lower. Also note that the G1 and G2 methods reproduce the complexation energy well, while G3 results are 0.5 kcal/mol higher than the Wn and experimental values. The overall barrier height for the Cl/Cl reaction is found to be 3.07 and 2.67 kcal/mol at the W1′and W2h levels of theory. Note ﬁrst that the experimental val ue (2.90 kcal/mol) is very close and lies between the W1′and W2h values. The ∆ H‡ovrvalue calculated at the mPW1K/6-31+G, and CBS-QB3 levels of theory as well as the G2 (+) value by Radom et al.18and the CCSD(T)/ spdfg value by Botschwina23agree well with the W2h result. The mPW1K exchange-correlation functional with the cc-pVDZ(+ X) and cc-pVTZ(+X) basis sets predict somewhat higher ∆ H‡ovrvalues, while the G1, G2MP2 and G3(+) values are about 1 kcal/mol lower. B3LYP, B97, B97-1, and HCTH-120 all p redict a negative overall barrier for the Cl/Cl system, in disagreement with all other methods considered and with experiment. BH&HLYP performs moderately well, while mPWH& HPW91 predicts a larger barrier height (4.50 kcal/mol) than W n. The central barrier values presented in the last 13column of the Table 4 reveal that the agreement between Wn the ories (13.61 kcal/mol) and experiment (13.66 kcal/mol) is excellent. The G2MP2, G2 (+) and CBS-QB3 methods also reproduce the central barriers very well. As expected f rom the overall barrier heights, the DFT results are less satisfactory, except for mPW1K/cc- pVDZ(+X) and mPW1K/cc- pVTZ(+X) which are in good agreement with the Wn values. For the bromine identity S N2 reaction, W1′theory predicts 10.03 kcal/mol for ∆ Hcomp. Note that the G2(+) value is in close agreement with W1′theory. The reported experimental value (11.34 ±0.4 kcal/mol) agrees fairly well. Most DFT levels of theory c onsidered suggest a complexation energy about 1 kcal/mol lower than the W1′value, except mPW1K/6-31+G* which is higher (12.78 kcal/mol, probably an artifact of the small basis set); B97 and B97- 1 which closely bracket the W1′value; HCTH-120 which is close to the W1′value (see below); and HCTH which is 2.5 kcal/mol lower than the latter. The complexation energies for X−· · ·CH3X are found to decrease in the order F >Cl>Br. This trend was noted previously by Radom and coworkers,18who attributed it to the electronegativities of the halogens. The overall barrier height for the Br/Br reaction is found to be 1.02 and 0.77 kcal/mol at the W1′and W2h levels of theory. Of the various exchange-correlati on functionals consid- ered, only BH&HLYP, mPWH&HPW91, and mPW1K ﬁnd positive barr iers (as do the G n theories). It should be pointed out that the DFT results for t his system display appreciable basis set sensitivity: for instance, the mPW1K/6-31+G* ove rall barrier has the wrong sign. It is interesting to note that the complexation energy deriv ed from the experimental overall (1.73 kcal/mol83) and central (11.68 kcal/mol84) barrier heights is 9.95 kcal/mol while the reported experimental complexation energy (11.34 kcal/mo l)11is inconsistent with the de- rived value. In fact, the derived value is in excellent agree ment with the W1′value (10.03 kcal/mol). This clearly suggests that the experimental dat a should be re-examined. The performance of both B97 and B97-1 for the identity reacti ons is quite similar to that of B3LYP. While the ‘pure DFT’ HCTH functional appears to yie ld markedly better over- all barrier heights, this comes at the expense of signiﬁcant ly underestimated complexation energies (and severely overestimated ion-molecule distan ces, see Supplementary Material). 14It was previously noted85that HCTH severely underestimates interaction energies of H- bonded complexes; this was ascribed to the absence of anions and H-bonded dimers in the original HCTH parametrization set. A reparametrization72against an enlarged sample of high-quality ab initio energies, denoted HCTH-120, eliminates this particular pr oblem.85For the identity S N2 reactions, we ﬁnd that complexation energies (and ion-mol ecule distances) are dramatically improved compared to HCTH: no correspondi ng improvement is however seen for the central barrier heights, and the overall barrie r heights deteriorate accordingly. Overall, the DFT methods are less satisfactory for barrier h eight calculations. Although the performance of mPW1K/6-31+G* method for F/F and Cl/Cl re actions was excellent, it is not the ultimate low cost method for barrier heights as i t has predicted a negative barrier for the Br/Br system. This behavior illustrates the inadequacy of the 6-31+G* basis set for Br: the more extended correlation consistent basis s ets with the mPW1K exchange- correlation functional do predict the sign correctly. In ad dition, Gn(+) and CBS-QB3(+) provide an acceptable account of reaction energetics. E. Nonidentity Reactions A comparison of computed and experimental complexation ene rgies for the nonidentity SN2 reactions is provided in Table 5. For the F−· · ·CH3Cl ion-molecule complex we could ﬁnd a stationary point neither at the MP2/6-31G* level of the ory used for the G2 and G3 reference geometries nor at the B3LYP/6-311G(2 d, d, p) level used for the CBS-QB3 reference geometries; at these levels of theory, the optimization lea ds to Cl−· · ·CH3F even if the initial geometries were chosen to correspond to F−· · ·CH3Cl. Addition of diﬀuse functions to the basis set for the reference geometry remedies the problem. S imilarly, in the F/Br case only the Br−· · ·CH3F complex is found as a stationary point at the MP2/6-31G* lev el of theory, and the transition state and second ion-molecule complex on ly appear when diﬀuse functions are added to the basis set. Furthermore, and regardless of th e basis set employed, none of the DFT functionals except mPW1K, mPWH&HPW91, and BH&HLYP ﬁ nd a transition state or a F−· · ·CH3Br complex. (The CBS-QB3 method is not deﬁned for Br and hence 15no CBS-QB3 data are presented for the F/Br and Cl/Br ion-mole cule complexes.) Table 5 also presents large-scale CCSD(T) energetics for the F/Cl22and Cl/Br24cases reported by Botschwina and coworkers. Available experimental values a re presented at the end of the Table with uncertainties in parentheses. Examination of Table 5 indicates that the ∆ Hcompvalues strongly depend on the nucle- ophile (Y−), decreasing in the order F−>Cl−>Br−. They also depend on the leaving group (X−), in the order CH 3F<CH3Cl<CH3Br. Similar observations were made earlier by Radom and coworkers.19 Comparison of the complexation energies obtained from vari ous methods with W1′theory indicates that all DFT results for Cl−· · ·CH3F are lower by 1 kcal/mol. The only available experimental value86for Cl−· · ·CH3F (∆Ho= 11.41 kcal/mol) has an uncertainty of 2.01 kcal/mol. Comparison of this value with the calculated valu es suggest that more accurate measurements are in order. For Cl−· · ·CH3Br and Br−· · ·CH3Cl, rather more accurate high- pressure mass spectrometry data are available (12.54 and 11 .01 kcal/mol). The W1′values (11.91 and 10.32 kcal/mol) are very close to the experimenta l results, considering the exper- imental uncertainty of 0.4 kcal/mol. While the mPW1K/6-31+ G* values for Cl−· · ·CH3Br and Br−· · ·CH3Cl are fortuitously within the experimental error bars, the other DFT meth- ods predict lower values. Also note that G2(+) predicts comp lexation energies close to W1′ and experiment for Cl−· · ·CH3Br, while the Br−· · ·CH3Cl value is small. A complete as- sessment of CBS-QB3 is not possible as it could not be applied to the bromine-containing systems. Like for the identity reactions, complexation energies are signiﬁcantly underestimated (and ion-molecule distances overestimated by up to 0.3 ˚A: see Supplementary Material) by HCTH, and this problem is mostly remedied by HCTH-120. B97 an d especially B97-1 appear to represent an improvement over B3LYP for the complexation energies. Calculated overall reaction enthalpies, central enthalpy diﬀerences between reactant and product ion-molecule complexes, overall barrier heights, and central barrier heights for the nonidentity S N2 reactions are presented with available experimental resu lts in Table 6. It needs to be reemphasized that all values are “bottom-of-the -well” (i.e, zero-point exclusive): 16that is, the experimental values are presented after subtra cting the ZPVEs (scaled by 0.985) and thermal corrections obtained using the B3LYP method. The overall reaction enthalpies of the three nonidentity re actions, viz F/Cl, F/Br, and Cl/Br, calculated at the W1′level are -32.65, -41.43 and -8.56 kcal/mol, respectively. The corresponding experimental values are available and are pr esented in Table 6. The exper- imental value for the F/Cl reaction (-33.34 kcal/mol) is in c lose agreement with the W1′ value. B3LYP, B97(-1), mPW1PW91, Gn, and CCSD(T)/ spdfg results are all in close agreement with the W1′value, but mPW1K, ”half and half” and CBS-QB3 theories pre- dict 3-5 kcal/mol higher exothermicity. For the F/Br reacti on, the mPW1K/6-31+G* and mPW1PW91/cc-pVTZ(+X) methods yield overall reaction enth alpies which are quite close to the W1′result. The experimental result (-40.20 kcal/mol) is in goo d agreement with the best calculated values considering the uncertainty of 1 kca l/mol. The G ntheories predict exothermicities below, and mPW1K/cc-pV nZ(+X) above, the W1′value. Concerning the Cl/Br nonidentity reaction, the reported experimental val ue (-6.86 kcal/mol) diﬀers from the W1′value by 2 kcal/mol. As expected, the very recent CCSD(T)/ spdfgh results of Botschwina and coworkers24are in close agreement with our predictions. Our results sug gest that the Cl/Br experimental data may need to be reconsidered . Note the signiﬁcant basis set dependence in the mPW1K results, which illustrates the inad equacy of the 6-31+G* basis set. As a general observation, 50:50 admixture of HF exchange in t he DFT theories increases the magnitude of the overall reaction enthalpy, and the incr ease is greater in BH&HLYP than in mPWH&HPW91. Performance of B97 and B97-1 for the overall r eaction enthalpies is similar to B3LYP, while HCTH and HCTH-120 represent underes timates in absolute value. At the W1′level of theory, the calculated central barrier (∆ H‡cent) for the F/Cl system is 2.89 kcal/mol. G n(+), CBS-QB3(+), BH&HLYP, mPWH&HPW91, and mPW1K all reproduce the W1′value moderately well, while B3LYP, mPW1PW91, B97, B97-1, H CTH, and HCTH-120 all underestimate the central barriers. The ex perimental central barrier for the F/Cl system is nearly 4 kcal/mol higher than the calculat ed values. Judging from the performance of the various methods for the identity S N2 reactions, it is almost certain that 17the experimental F/Cl central barrier is in error and unambi guous new measurements are in order. For the F/Cl and Cl/Br systems, the B3LYP, mPW1PW91 , B97, B97-1, HCTH, and HCTH-120 central barriers are all underestimated, whil e these exchange-correlation functionals ﬁnd no barrier at all for the F/Br case. Like for t he identity case, mPW1K/cc- pVTZ(+X) and BH&HLYP/cc-pVTZ(+X) central barriers agree w ell with the benchmark ab initio values, although the basis set sensitivity of particularly the Cl/Br results argues against using small basis sets like 6-31+G. Several studies have reported experimental overall barrie r heights (∆ H‡ovr), but only for the Cl/Br system, and the experimental data range from -0.61 to -1.83 kcal/mol. To our knowledge, no experimental data are available for the F/Cl a nd F/Br systems. For the Cl/Br system, the theoretical values span a range from -1.17 to -6. 60 kcal/mol. Nevertheless, it is worth noting that the W1′value (-1.82 kcal/mol) for the Cl/Br system is in excellent agreement with the experimental overall barrier height rep orted by Caldwell and coworkers9 (-1.83 kcal/mol, after accounting for ZPVE and thermal corr ections). Some caution should be exercised as the W1′value does not include the core correlation contribution. A lso note that G2 theory (-1.82 kcal/mol) reproduces the W1′value very well. Except for mPW1K/cc- pVTZ(+X), mPWH&HPW91/cc-pVTZ(+X), and BH&HLYP/cc-pVTZ( +X), all the DFT methods perform poorly, consistent with the preceding disc ussion. IV. CONCLUSIONS A benchmark study using the W1′and W2h methods has been carried out for the po- tential surface of the gas-phase SN2 reactions Y−+ CH 3X− →CH3Y + X−. A number of more approximate (and less expensive) methods — both comp ound models (like G2/G3 theory and CBS-QB3) and density functional methods — have be en applied in an attempt to assess their performance for barrier heights in SN2 reactions. We arrive at the following conclusions. (1)Our best calculations are in excellent agreement with exper iment for the ∆ H◦ fvalues of the methyl halides (where available) and the electron aﬃn ities of the halogens. Where 18accurate experimental data are available for the title reac tions (e.g. for the Cl/Cl case), our best calculations agree with experiment to within overlapp ing uncertainties. Our calculations however suggest that more reliable experimental data are in order for most of the reactions considered. (2)The nonidentity S N2 reactions and F/F identity reaction possess transition st ate struc- tures below the reactants energy while Cl/Cl and Br/Br trans ition structures are above the reactants energy. The complexation energies for identity S N2 reactions are found to increase in the order Br <Cl<F while the barrier heights follow the order F <∼Br<Cl. The complexation energies for the nonidentity S N2 reactions indicate that the ∆ Hcompstrongly depend on the nucleophile and leaving group. (3)The B3LYP, and to a lesser extent, mPW1PW91 exchange-correl ation functionals sys- tematically underestimate barrier heights and, in the F/Br case, are not even able to locate the correct stationary points on the potential surface. The latter problem is remedied by using the corresponding ‘half-and-half’ functionals BH&H LYP and mPWH&HPW91, which however appear to somewhat overcorrect the barrier height. The B97 and B97-1 function- als perform similarly to B3LYP for the problem under study. T he ‘pure DFT’ HCTH and HCTH-120 functionals both underestimate central barrier h eights; HCTH in addition under- estimates complexation energies (and severely overestima tes ion-molecule distances), which are however well reproduced by HCTH-120. Overall, the mPW1K functional appears to put in the best performance of all DFT methods considered, es pecially when using extended basis sets. (4)The performance of G2(+), G3(+), and CBS-QB3(+) methods for the energetics still appears to be superior to the DFT methods. (The ‘(+)’ stands f or the addition of diﬀuse functions to the basis set used in obtaining the reference ge ometries; this is mandatory to get transition states at all in the F/Br and Cl/Br cases.) The limitations for transition states of the B3LYP exchange-correlation functional sugge st its replacement — at least for kinetics applications — by mPW1K in thermochemistry method s that employ DFT reference geometries, e.g. G3B3, CBS-QB3, and W1 theory. (In addition , larger basis sets than 6- 31+G should deﬁnitely be considered for the Br compounds.) 19The present calculations illustrate the power of state-of- the-science theoretical methods in providing both qualitative and quantitative informatio n regarding the reaction energetics. In the absence of accurate experimental data, our high quali ty results should be useful to future experimental and theoretical studies. ACKNOWLEDGMENTS SP and GdO acknowledge Postdoctoral Fellowships from the Fe inberg Graduate School (Weizmann Institute). JM is the incumbent of the Helen and Mi lton A. Kimmelman Career Development Chair. The authors would like to thank Mark Iron for editorial assistance. This research was supported by the Minerva Foundation, Munich, G ermany, and by the Tashtiyot Program of the Ministry of Science (Israel). Supporting Information Available: Calculated geometries of species involved in the SN2 reactions are available on the World Wide Web at the Uniform Resource Locator (URL) http://theochem.weizmann.ac.il/web/papers/sn2.html. This material is also available free of charge via the Internet at http://pubs.acs.org. 20REFERENCES 1Shaik, S.S.; Schlegel, H.B.; Wolfe, S. Theoretical Aspects of Physical Organic Chemistry: The S N2 Mechanism ; Wiley: New York, 1992. 2Ingold, C.K. Structure and Mechanism in Organic Chemistry ; Cornell University Press: Ithaca, 1969. 3Wuebbles, D.J.; Edmonds, J. Primer on Greenhouse Gases ; Lewis Publishers: Michigan, 1991. 4WMO (World Meteorological Organization), Scientiﬁc Asses sment of Ozone Depletion: 1998 (Global Ozone Research and Monitoring Project Report N o. 44, WMO, Geneva, 1999). 5Hase, W.L. Science 1994,266, 998 6Chabinyc, M.L.; Craig, S.L.; Regan, C.K.; Brauman, J.I. Science 1998,279, 1882 7(a) B¨ ohme, D.K.; Young, L.B. J. Am. Chem. 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Ref. Data 17, supplement 1, 1988. 2792Graul, S.T.; Bowers, M.T. J. Am. Chem. Soc. 1994,116, 3875 93Knighton, W.B.; Bognar, J.A.; O’Connor, P.M.; Grimsrud, E. P.J. Am. Chem. Soc. 1993, 115, 12079 94Hu, W.-P.; Truhlar, D.G. J. Am. Chem. Soc. 1995,117, 10726 28TABLES TABLE 1. Components of computed electron aﬃnities of X and to tal atomization energies (kcal mol−1) of CH 3X (X = F, Cl and Br). SCF CCSD (T) Core Spin-Orbit Scalar Rel. Final Species limit limit limit corr. splitting eﬀects Energy W1′ F−30.21 44.76 4.17 0.17 -0.39 -0.26 78.66 Cl−58.35 24.16 2.28 0.02 -0.84 -0.34 83.63 Br−58.37 21.76 1.98 0.31 -3.51 -0.90 78.02 CH3F 319.57 97.39 5.41 1.12 -0.47 -0.37 422.65 CH3Cl 303.73 86.14 5.27 1.19 -0.93 -0.42 394.98 CH3Br 292.99 85.48 5.25 1.45 -3.60 -0.79 380.78 W2h F−30.08 44.71 4.15 0.17 -0.39 -0.26 78.46 Cl−58.33 23.82 2.26 0.02 -0.84 -0.34 83.25 Br−58.31 21.49 1.91 0.31 -3.51 -0.90 77.62 CH3F 319.82 96.89 5.34 1.14 -0.47 -0.37 422.34 CH3Cl 303.90 86.03 5.28 1.21 -0.93 -0.42 395.07 CH3Br 293.14 85.45 5.21 1.49 -3.60 -0.79 380.91 29TABLE 2. Components of relative energies (kcal mol−1) of ion-molecule complexes and transi- tion state structures with respect to reactants. SCF CCSD (T) Core Scalar Rel. Final Species limit limit limit corr. eﬀects Energy W1′ F−· · ·CH3F -10.63 -2.54 -0.56 0.09 -0.02 -13.66 [F· · ·CH3· · ·F]−8.23 -6.23 -2.65 0.36 -0.08 -0.37 Cl−· · ·CH3Cl -8.08 -1.93 -0.62 0.08 0.01 -10.54 [Cl· · ·CH3· · ·Cl]−7.81 -2.62 -2.37 0.35 -0.10 3.07 Br−· · ·CH3Br -7.55 -1.92 -0.65 0.09 -10.03a [Br· · ·CH3· · ·Br]−5.71 -2.43 -2.36 0.26 -0.16 1.02 W2h F−· · ·CH3F -11.16 -2.17 -0.45 0.07 -0.02 -13.72 [F· · ·CH3· · ·F]−8.58 -6.61 -2.58 0.35 -0.08 -0.34 Cl−· · ·CH3Cl -8.01 -2.39 -0.65 0.10 0.01 -10.94 [Cl· · ·CH3· · ·Cl]−8.55 -3.63 -2.46 0.33 -0.11 2.67 [Br· · ·CH3· · ·Br]−6.57 -3.41 -2.38 0.21 -0.21 0.77 W1′ F−· · ·CH3Cl -13.67 -1.16 -0.76 0.18 -0.02 -15.43 [F· · ·CH3· · ·Cl]−-10.39 -0.84 -1.60 0.36 -0.07 -12.54 Cl−· · ·CH3F -7.08 -2.03 -0.42 0.03 0.00 -9.51 F−· · ·CH3Br -15.08 -1.24 -0.72 0.03 -17.01a [F· · ·CH3· · ·Br]−-13.85 -1.02 -1.43 -0.07 -16.37a Br−· · ·CH3F -6.63 -1.69 -0.24 0.05 -8.51a Cl−· · ·CH3Br -8.89 -2.02 -0.59 -0.41 -11.91a [Cl· · ·CH3· · ·Br]−2.24 -2.69 -2.28 -0.57 -3.30a Br−· · ·CH3Cl -7.38 -2.03 -0.45 -0.46 -10.32a aW1′−Core Correlation 30TABLE 3. Calculated and experimental heats of formation (kc al mol−1) of CH 3X and electron aﬃnities (eV) of X (X = F, Cl and Br) Heats of Formation Electron Aﬃnity Species W1′W2h Experiment Species W1’ W2h Experiment CH3F -57.06 -56.75 -56(7)87F 3.411 3.402 3.401 190(4)88 CH3Cl -20.14 -20.23 -20.00(50)87Cl 3.627 3.610 3.612 69(6)88 CH3Br -8.50 -8.63 -8.20(19)89Br 3.383 3.366 3.363 583(44)89 31TABLE 4. Comparison of complexation energies (∆ Hcomp) of the ion-molecule complexes, over- all barrier heights relative to reactants (∆ H‡ovr), and central barriers (∆ H‡cent) of identity S N2 reactions, X−+ CH 3X− →XCH 3+ X−, calculated at various levels of theory. All values in kcal/mol. X Method ∆Hcomp ∆H‡ovr ∆H‡cent F W1′13.66 -0.37 13.29 W2h 13.72 -0.34 13.38 B3LYP/cc-pVTZ(+X) 12.72 -2.58 10.15 BH&HLYP/cc-pVTZ(+X) 13.22 1.31 14.53 mPW1PW91/cc-pVTZ(+X) 12.49 -0.95 11.55 mPWH&HPW91/cc-pVTZ(+X) 12.77 2.60 15.38 B97/cc-pVDZ(+X) 12.48 -2.47 10.01 B97-1/cc-pVDZ(+X) 13.21 -3.29 9.92 HCTH/cc-pVDZ(+X) 9.87 -0.60 9.27 HCTH-120/cc-pVDZ(+X) 12.39 -4.20 8.18 mPW1K/6-31+G* 13.55 -0.30 13.26 mPW1K/cc-pVDZ(+X) 12.63 0.36 13.00 mPW1K/cc-pVTZ(+X) 12.66 1.69 14.36 G1 13.01 -1.37 11.64 G2 13.34 -1.15 12.19 G2MP2 13.41 -0.63 12.78 G3 14.23 -1.97 12.26 CBS-QB3 13.46 -0.85 12.61 G3(+) 14.59 -2.68 11.90 CBS-QB3(+) 14.15 -0.52 13.63 G2(+)a13.81 -1.86 11.95 CCSD(T)/ spd fc13.73 -0.92 12.81 Cl W1′10.54 3.07 13.61 W2h 10.94 2.67 13.61 B3LYP/cc-pVTZ(+X) 9.50 -0.48 9.02 BH&HLYP/cc-pVTZ(+X) 9.67 3.17 12.84 mPW1PW91/cc-pVTZ(+X) 9.59 1.23 10.82 mPWH&HPW91/cc-pVTZ(+X) 9.69 4.50 14.19 B97/cc-pVDZ(+X) 10.10 -0.66 9.44 B97-1/cc-pVDZ(+X) 10.74 -1.46 9.28 HCTH/cc-pVDZ(+X) 7.91 1.45 9.36 HCTH-120/cc-pVDZ(+X) 9.96 -1.93 8.03 mPW1K/6-31+G* 9.75 3.20 12.95 mPW1K/cc-pVDZ(+X) 9.65 3.63 13.28 mPW1K/cc-pVTZ(+X) 9.64 3.66 13.30 G1 10.52 1.79 12.31 G2 10.77 3.06 13.83 32G2MP2 10.89 2.74 13.63 G3 11.15 1.79 12.95 CBS-QB3 10.65 2.47 13.12 G3(+) 11.04 1.80 12.83 CBS-QB3(+) 10.69 2.40 13.09 G2(+)a10.71 3.01 13.72 CCSD(T)/ spd fgb2.75 Experiment 10.53(40)d2.90e13.66(2.01)f Br W1′−core 10.03 1.02i10.79 W2h 0.77 B3LYP/cc-pVTZ(+X) 9.06 -2.42 6.64 BH&HLYP/cc-pVTZ(+X) 9.04 1.25 10.29 mPW1PW91/cc-pVTZ(+X) 9.21 -1.03 8.18 mPWH&HPW91/cc-pVTZ(+X) 9.19 2.22 11.40 B97/cc-pVDZ(+X) 9.62 -2.29 7.33 B97-1/cc-pVDZ(+X) 10.24 -3.02 7.22 HCTH/cc-pVDZ(+X) 7.56 -0.70 6.86 HCTH-120/cc-pVDZ(+X) 9.73 -4.06 5.68 mPW1K/6-31+G* 12.78 -1.95 10.83 mPW1K/cc-pVDZ(+X) 9.34 0.68 10.02 mPW1K/cc-pVTZ(+X) 9.16 1.38 10.54 G1 9.68 1.11 10.78 G2 9.85 1.52 11.38 G2MP2 9.83 1.83 11.66 G2(+)a−ECP 10.17 1.48 11.65 Experiment 11.34(40)d1.73g11.68h aG2(+) values are from Ref.18;bCCSD(T)/ spdfg values are from Ref.23;cRef.15 Experimental values:dFrom Ref.11,eFrom Ref.90,fFrom Ref.10,gFrom Ref.83,hFrom Ref.84 iCore contribution included 33TABLE 5. Comparison of complexation energies (∆ Hcomp, kcal/mol) of the ion-molecule complexes for the nonidentity S N2 reactions, calculated at various levels of theory. Method F−· · ·CH3Cl Cl−· · ·CH3F F−· · ·CH3Br Br−· · ·CH3F Cl−· · ·CH3Br Br−· · ·CH3Cl W1′−core 15.43a9.51a17.01 8.51 11.91 10.32 B3LYP/cc-pVTZ(+X) 15.37 8.09 7.18 10.24 8.42 BH&HLYP/cc-pVTZ(+X) 15.39 8.42 16.65 7.45 10.23 8.56 mPW1PW91/cc-pVTZ(+X) 15.06 8.13 7.27 10.32 8.54 mPWH&HPW91/cc-pVTZ(+X) 15.03 8.32 16.28 7.42 10.30 8.65 B97/cc-pVDZ(+X) 15.43 8.49 7.69 10.66 9.13 B97-1/cc-pVDZ(+X) 16.26 9.05 8.21 11.32 9.74 HCTH/cc-pVDZ(+X) 12.66 6.56 5.82 8.56 7.00 HCTH-120/cc-pVDZ(+X) 15.42 8.48 7.69 10.78 9.01 mPW1K/6-31+G* 15.32 8.79 17.63 9.12 12.97 10.56 mPW1K/cc-pVDZ(+X) 14.52 8.46 16.25 7.62 10.41 8.68 mPW1K/cc-pVTZ(+X) 14.97 8.25 16.30 7.37 10.27 8.60 G1 9.58 8.35 11.17 9.10 G2 9.68 8.42 11.35 9.36 G2MP2 9.71 8.40 11.37 9.45 G3 10.03 CBS-QB3 9.33 G3(+) 16.34 9.97 CBS-QB3(+) 15.85 9.51 G2(+)b15.62 9.64 16.74 8.56 11.47 9.64 CCSD(T)/largec16.07 9.75 11.31 9.71 Experiment. 11.41(2.01)d12.54(40)e11.01 (40)e aCore contribution included bG2(+) values are from Ref.19, cF/Cl: CCSD(T)/ spdfg values from Ref.22; Cl/Br: CCSD(T)/ spdfgh values from Ref.24. Experimental values:dFrom Ref.86,eFrom Ref.11 34TABLE 6. Comparison of overall reaction enthalpies (∆ Hovr), central enthalpy diﬀerences between reactant and product ion-molecule complexes (∆ Hcent), overall barrier heights (∆ H‡over) and central barrier heights (∆ H‡cent) for exothermic Y−+ CH 3X− →YCH 3+ X−reactions, calculated at various levels of theory. All values in kcal/m ol. Y,X Method ∆Hovr ∆Hcent ∆H‡over ∆H‡cent F/Cl W1′-32.65 -26.73 -12.54 2.89 B3LYP/cc-pVTZ(+X) -32.77 -25.49 -14.69 0.67 BH&HLYP/cc-pVTZ(+X) -37.02 -30.05 -12.86 2.53 mPW1PW91/cc-pVTZ(+X) -33.08 -26.15 -13.43 1.63 mPWH&HPW91/cc-pVTZ(+X) -36.45 -29.73 -11.51 3.52 B97/cc-pVDZ(+X) -32.90 -25.95 -14.70 0.73 B97-1/cc-pVDZ(+X) -33.13 -25.92 -15.60 0.66 HCTH/cc-pVDZ(+X) -30.77 -24.67 -11.95 0.71 HCTH-120/cc-pVDZ(+X) -30.58 -23.64 -15.14 0.27 mPW1K/6-31+G* -36.59 -30.07 -13.02 2.30 mPW1K/cc-pVDZ(+X) -34.74 -28.68 -11.95 2.57 mPW1K/cc-pVTZ(+X) -35.50 -28.78 -11.97 3.01 G1 -30.62 G2 -31.59 G2MP2 -32.25 G3 -33.00 CBS-QB3 -35.21 G3(+) -32.86 -26.50 -14.04 2.30 CBS-QB3(+) -35.15 -28.81 -13.77 2.07 G2(+)a-31.44 -25.46 -12.63 2.98 CCSD(T)/ spd fgb-32.34 -26.36 -11.84 3.89 Experiment -33.34(72)c7.52(1.20)d F/Br W1′−core -41.43 -32.93 -16.37 0.64 B3LYP/cc-pVTZ(+X) -40.78 BH&HLYP/cc-pVTZ(+X) -46.13 -36.93 -16.33 0.32 mPW1PW91/cc-pVTZ(+X) -41.65 mPWH&HPW91/cc-pVTZ(+X) -45.90 -37.05 -15.42 0.85 B97/cc-pVDZ(+X) -40.41 B97-1/cc-pVDZ(+X) -40.68 HCTH/cc-pVDZ(+X) -38.18 HCTH-120/cc-pVDZ(+X) -37.96 mPW1K/6-31+G* -42.35 -33.84 -16.92 0.71 mPW1K/cc-pVDZ(+X) -44.76 -36.13 -16.08 0.18 mPW1K/cc-pVTZ(+X) -44.70 -35.77 -15.76 0.54 G1 -40.87 G2 -40.02 35G2MP2 -40.35 G2(+)a-39.47 -31.29 -15.90 0.84 Experiment -40.20(96)c Cl/Br W1′−core -8.56 -6.97 -1.82 8.61 B3LYP/cc-pVTZ(+X) -8.01 -6.19 -5.25 4.99 BH&HLYP/cc-pVTZ(+X) -9.11 -7.43 -2.15 8.08 mPW1PW91/cc-pVTZ(+X) -8.57 -6.79 -3.99 6.33 mPWH&HPW91/cc-pVTZ(+X) -9.45 -7.79 -1.17 9.14 B97/cc-pVDZ(+X) -7.52 -5.99 -5.06 5.60 B97-1/cc-pVDZ(+X) -7.54 -5.97 -5.83 5.48 HCTH/cc-pVDZ(+X) -7.41 -5.85 -3.18 5.38 HCTH-120/cc-pVDZ(+X) -7.38 -5.62 -6.52 4.26 mPW1K/6-31+G* -5.75 -3.34 -3.12 9.85 mPW1K/cc-pVDZ(+X) -10.02 -8.30 -6.60 3.81 mPW1K/cc-pVTZ(+X) -9.20 -7.53 -1.88 8.38 G1 -10.25 -8.18 -3.45 7.72 G2 -8.43 -6.44 -1.82 9.53 G2MP2 -8.11 -6.18 -1.67 9.70 G2(+)a-8.04 -6.21 -1.71 9.76 CCSD(T)/ spd fghb-8.53 -6.93 -2.33 8.98 Experiment -6.86(72)c-1.83(5)e -1.69(33)f -1.52g -1.11h -0.61i aG2(+) values are from Ref.19 bRef.22(F/Cl) and Ref.24(Cl/Br) b′Table 2 of Ref.24 Experimental values:cFrom Ref.91,dFrom Ref.84,eFrom Ref.9,fFrom Ref.92,gFrom Ref.93, hFrom Ref.90,iFrom Ref.94 36FIGURES FIG. 1. Schematic representation of potential energy surfa ce for the (a) identity and (b) non- identity S N2 reactions 37∆H cent✢ ✢ ∆H comp∆H ovr✢ ✢ ∆H cent∆H ovr 312Y− + CH3X YCH3 + X− ∆H compY CH 3 X− Y− CH3X YCH3 X−∆H cent✢ ✢ ∆H ovr✢ ✢ ∆H compX− + CH3X XCH3 + X− X CH 3 X− XCH3 X−X− CH3X 1 1'2a. b.arXiv:physics/0011029v1 [physics.chem-ph] 15 Nov 2000Benchmark ab initio energy proﬁles for the gas-phase S N2 reactions Y−+ CH 3X→CH3Y + X−(X,Y = F,Cl,Br). Validation of hybrid DFT methods Supplementary Data Srinivasan Parthiban, Glˆ enisson de Oliveira∗, and Jan M.L. Martin† Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, IL-76100 Reh .ovot, Israel. (J. Phys. Chem. A manuscript JP0031000; revised October 31, 2000) ∗Present address: Chemistry Department, Pensacola Christi an College, 250 Brent Lane, Pensacola, FL 32503 †Author to whom correspondence should be addressed. Email: comartin@wicc.weizmann.ac.ilTABLES TABLE I. Calculated and experimental geometries ( ˚A , degree) of CH 3X (X=F, Cl, and Br). Species Method/Basis Set r(C −X) r(C −H) /negationslashXCH CH3F CCSD(T)/cc-pVQZ+1 1.382 1.089 108.9 B3LYP/cc-pVTZ(+X) 1.392 1.090 108.7 BH&HLYP/cc-pVTZ(+X) 1.372 1.082 108.8 mPW1PW91/cc-pVTZ(+X) 1.380 1.090 108.9 mPWH&HPW91/cc-pVTZ(+X) 1.362 1.083 109.1 B97/cc-pVDZ(+X) 1.399 1.102 108.7 B97-1/cc-pVDZ(+X) 1.398 1.102 108.7 HCTH/cc-pVDZ(+X) 1.400 1.103 108.8 HCTH-120/cc-pVDZ(+X) 1.402 1.104 108.7 mPW1K/6-31+G* 1.374 1.087 108.7 mPW1K/cc-pVDZ(+X) 1.378 1.093 108.8 mPW1K/cc-pVTZ(+X) 1.367 1.085 109.0 MP2/6-31G* 1.390 1.092 109.1 MP2/6-31+Ga1.407 1.090 108.0 B3LYP/6-311G(2d,d,p) 1.388 1.093 109.2 B3LYP/6-311+G(2d,d,p) 1.396 1.091 108.6 Expt.b1.383 1.086 108.8 CH3Cl CCSD(T)/cc-pVQZ+1 1.783 1.085 108.4 B3LYP/cc-pVTZ(+X) 1.796 1.085 108.3 BH&HLYP/cc-pVTZ(+X) 1.779 1.077 108.4 mPW1PW91/cc-pVTZ(+X) 1.776 1.085 108.5 mPWH&HPW91/cc-pVTZ(+X) 1.761 1.079 108.6 B97/cc-pVDZ(+X) 1.808 1.097 108.1 B97-1/cc-pVDZ(+X) 1.808 1.097 108.0 HCTH/cc-pVDZ(+X) 1.793 1.098 108.4 HCTH-120/cc-pVDZ(+X) 1.796 1.098 108.4 mPW1K/6-31+G 1.772 1.084 108.8 mPW1K/cc-pVDZ(+X) 1.768 1.090 108.5 mPW1K/cc-pVTZ(+X) 1.765 1.081 108.6 MP2/6-31G* 1.777 1.088 108.9 MP2/6-31+Ga1.780 1.089 108.9 B3LYP/6-311G(2d,d,p) 1.803 1.087 108.2 B3LYP/6-311+G(2d,d,p) 1.803 1.087 108.2 Expt.c1.776 1.085 108.6 CH3Br CCSD(T)/cc-pVQZ+1 1.944 1.084 107.8 B3LYP/cc-pVTZ(+X) 1.957 1.083 107.6 BH&HLYP/cc-pVTZ(+X) 1.938 1.076 107.8 mPW1PW91/cc-pVTZ(+X) 1.936 1.084 107.9 mPWH&HPW91/cc-pVTZ(+X) 1.917 1.078 108.0 B97/cc-pVDZ(+X) 1.959 1.097 107.6B97-1/cc-pVDZ(+X) 1.959 1.097 107.6 HCTH/cc-pVDZ(+X) 1.952 1.097 107.9 HCTH-120/cc-pVDZ(+X) 1.956 1.097 107.8 mPW1K/6-31+G 1.925 1.083 108.2 mPW1K/cc-pVDZ(+X) 1.927 1.089 107.9 mPW1K/cc-pVTZ(+X) 1.922 1.079 108.0 MP2/6-31G* 1.947 1.086 107.9 MP2/6-31+Ga1.954 1.088 108.0 Expt.d1.934 1.082 107.7 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1995,117, 2024. Experimental values: bFrom Egawa, T.; Yamamoto, S.; Nakata, M.; Kuchitsu, K.; J. Mol. Struct. 1987,156, 213. cFrom Jensen, T.; Brodersen, S.; Guelachvili, G.; J. Mol.Spectrosc. 1981,88, 378. dFrom Graner, G.; J. Mol.Spectrosc. 1981,90, 394.TABLE II. Geometries ( ˚A , degree) of ion-molecule complexes X−· · ·CH3X (X=F, Cl, and Br) of the SN 2identity reactions. Species Method/Basis Set r(X · · ·C) r(C −X) r(C −H) /negationslashHCX X=F CCSD(T)/cc-pVQZ+1 2.494 1.432 1.082 109.1 B3LYP/cc-pVTZ(+X) 2.588 1.447 1.082 108.4 BH&HLYP/cc-pVTZ(+X) 2.581 1.418 1.075 108.8 mPW1PW91/cc-pVTZ(+X) 2.575 1.428 1.083 108.9 mPWH&HPW91/cc-pVTZ(+X) 2.567 1.404 1.077 109.2 B97/cc-pVDZ(+X) 2.643 1.453 1.095 108.3 B97-1/cc-pVDZ(+X) 2.613 1.453 1.095 108.3 HCTH/cc-pVDZ(+X) 2.794 1.453 1.096 108.4 HCTH-120/cc-pVDZ(+X) 2.668 1.462 1.096 108.2 mPW1K/6-31+G 2.572 1.421 1.080 108.5 mPW1K/cc-pVDZ(+X) 2.579 1.425 1.087 108.8 mPW1K/cc-pVTZ(+X) 2.571 1.411 1.079 109.1 MP2/6-31G* 2.426 1.439 1.083 109.5 MP2/6-31+Ga2.628 1.456 1.084 107.7 B3LYP/6-311G(2d,d,p) 2.420 1.453 1.084 110.0 B3LYP/6-311+G(2d,d,p) 2.565 1.455 1.084 108.2 X=Cl CCSD(T)/cc-pVQZ+1 3.123 1.860 1.080 107.8 B3LYP/cc-pVTZ(+X) 3.191 1.846 1.080 107.9 BH&HLYP/cc-pVTZ(+X) 3.203 1.818 1.073 108.2 mPW1PW91/cc-pVTZ(+X) 3.163 1.817 1.081 108.5 mPWH&HPW91/cc-pVTZ(+X) 3.166 1.796 1.075 108.7 B97/cc-pVDZ(+X) 3.198 1.855 1.094 107.7 B97-1/cc-pVDZ(+X) 3.168 1.857 1.093 107.6 HCTH/cc-pVDZ(+X) 3.375 1.838 1.094 108.1 HCTH-120/cc-pVDZ(+X) 3.241 1.848 1.095 107.9 mPW1K/6-31+G 3.181 1.809 1.080 108.7 mPW1K/cc-pVDZ(+X) 3.182 1.804 1.086 108.6 mPW1K/cc-pVTZ(+X) 3.167 1.801 1.077 108.7 MP2/6-31G* 3.158 1.812 1.084 109.0 MP2/6-31+Ga3.270 1.810 1.085 108.8 B3LYP/6-311G(2d,d,p) 3.146 1.861 1.082 107.6 B3LYP/6-311+G(2d,d,p) 3.187 1.854 1.083 107.8 X=Br CCSD(T)/cc-pVQZ+1 3.277 2.023 1.079 106.9 B3LYP/cc-pVTZ(+X) 3.321 2.016 1.079 106.9 BH&HLYP/cc-pVTZ(+X) 3.361 1.980 1.072 107.4 mPW1PW91/cc-pVTZ(+X) 3.290 1.982 1.080 107.5 mPWH&HPW91/cc-pVTZ(+X) 3.313 1.953 1.074 107.9 B97/cc-pVDZ(+X) 3.320 2.013 1.093 107.0 B97-1/cc-pVDZ(+X) 3.281 2.014 1.093 106.9 HCTH/cc-pVDZ(+X) 3.471 2.007 1.094 107.2 HCTH-120/cc-pVDZ(+X) 3.332 2.022 1.094 106.9mPW1K/6-31+G 3.215 1.960 1.079 108.0 mPW1K/cc-pVDZ(+X) 3.304 1.968 1.086 107.6 mPW1K/cc-pVTZ(+X) 3.312 1.960 1.076 107.8 MP2/6-31G* 3.196 1.992 1.083 107.7 MP2/6-31+Ga3.395 1.988 1.084 107.8 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1995,117, 2024.TABLE III. Geometries ( ˚A , degree) of ion-molecule complexes, Y−· · ·CH3X, of the SN 2 non-Identity reactions (complex 1). Y/X Method/Basis Set r(Y · · ·C) r(C −X) r(C −H) /negationslashHCX F/Cl B3LYP/cc-pVTZ(+X) 2.443 1.908 1.076 106.1 BH&HLYP/cc-pVTZ(+X) 2.494 1.853 1.070 107.4 mPW1PW91/cc-pVTZ(+X) 2.462 1.861 1.078 107.3 mPWH&HPW91/cc-pVTZ(+X) 2.489 1.825 1.072 108.1 B97/cc-pVDZ(+X) 2.467 1.913 1.090 106.0 B97-1/cc-pVDZ(+X) 2.445 1.916 1.090 105.9 HCTH/cc-pVDZ(+X) 2.565 1.903 1.091 106.5 HCTH-120/cc-pVDZ(+X) 2.459 1.931 1.090 105.6 mPW1K/6-31+G 2.487 1.846 1.077 107.8 mPW1K/cc-pVDZ(+X) 2.495 1.837 1.083 107.8 mPW1K/cc-pVTZ(+X) 2.486 1.833 1.074 108.0 MP2/6-31+Ga2.616 1.832 1.083 108.5 B3LYP/6-311+G(2d,d,p) 2.410 1.931 1.078 105.5 CCSD(T)/ spd fgb2.502 1.853 1.080 107.6 F/Br BH&HLYP/cc-pVTZ(+X) 2.392 2.052 1.068 105.2 mPWH&HPW91/cc-pVTZ(+X) 2.404 2.007 1.070 106.4 mPW1K/6-31+G 2.411 2.022 1.075 106.2 mPW1K/cc-pVDZ(+X) 2.360 2.045 1.081 105.2 mPW1K/cc-pVTZ(+X) 2.389 2.022 1.072 106.0 MP2/6-31+Ga2.528 2.028 1.081 106.8 Cl/Br B3LYP/cc-pVTZ(+X) 3.112 2.024 1.079 106.7 BH&HLYP/cc-pVTZ(+X) 3.149 1.986 1.072 107.3 mPW1PW91/cc-pVTZ(+X) 3.088 1.989 1.080 107.4 mPWH&HPW91/cc-pVTZ(+X) 3.113 1.958 1.074 107.8 B97/cc-pVDZ(+X) 3.134 2.021 1.093 106.8 B97-1/cc-pVDZ(+X) 3.098 2.022 1.093 106.8 HCTH/cc-pVDZ(+X) 3.269 2.012 1.094 107.1 HCTH-120/cc-pVDZ(+X) 3.132 2.031 1.093 106.8 mPW1K/6-31+G 3.070 1.966 1.079 107.9 mPW1K/cc-pVDZ(+X) 3.114 1.974 1.085 107.5 mPW1K/cc-pVTZ(+X) 3.109 1.966 1.076 107.8 MP2/6-31G* 3.092 1.992 1.082 107.6 MP2/6-31+Ga3.199 1.992 1.084 107.7 CCSD(T)/ spd fc3.095 1.986 1.082 107.5 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1996,118, 6273. bFrom Schmatz, S.; Botschwina, P.; Stoll, H.; Int. J. Mass Spectrom. 2000,201, 277. cFrom Botschwina, P.; Horn, M.; Seeger, S.; Oswald, R.; Ber. Bunsenges. Phys. Chem. 1997,101, 387.TABLE IV. Geometries ( ˚A , degree) of ion-molecule complexes, YCH 3· · ·X−, of the SN 2 non-identity reactions (complex 3). Y/X Method/Basis Set r(X · · ·C) r(C −Y) r(C −H) /negationslashHCY F/Cl B3LYP/cc-pVTZ(+X) 3.286 1.425 1.085 108.8 BH&HLYP/cc-pVTZ(+X) 3.268 1.401 1.077 109.0 mPW1PW91/cc-pVTZ(+X) 3.246 1.409 1.085 109.2 mPWH&HPW91/cc-pVTZ(+X) 3.230 1.389 1.079 109.4 B97/cc-pVDZ(+X) 3.320 1.432 1.098 108.7 B97-1/cc-pVDZ(+X) 3.268 1.432 1.098 108.7 HCTH/cc-pVDZ(+X) 3.528 1.431 1.099 108.7 HCTH-120/cc-pVDZ(+X) 3.358 1.436 1.099 108.7 mPW1K/6-31+G 3.241 1.403 1.083 108.8 mPW1K/cc-pVDZ(+X) 3.252 1.407 1.089 109.0 mPW1K/cc-pVTZ(+X) 3.235 1.394 1.081 109.3 MP2/6-31G* 3.227 1.415 1.087 109.4 MP2/6-31+Ga3.255 1.438 1.086 108.0 B3LYP/6-311G(2d,d,p) 3.299 1.418 1.089 109.4 B3LYP/6-311+G(2d,d,p) 3.271 1.430 1.087 108.7 CCSD(T)/ spd fgb3.188 1.418 1.086 108.9 F/Br B3LYP/cc-pVTZ(+X) 3.497 1.421 1.085 108.8 BH&HLYP/cc-pVTZ(+X) 3.482 1.398 1.078 109.0 mPW1PW91/cc-pVTZ(+X) 3.451 1.406 1.086 109.2 mPWH&HPW91/cc-pVTZ(+X) 3.433 1.386 1.080 109.3 B97/cc-pVDZ(+X) 3.518 1.428 1.098 108.7 B97-1/cc-pVDZ(+X) 3.458 1.428 1.098 108.7 HCTH/cc-pVDZ(+X) 3.746 1.428 1.099 108.8 HCTH-120/cc-pVDZ(+X) 3.565 1.433 1.100 108.7 mPW1K/6-31+G 3.367 1.401 1.083 108.8 mPW1K/cc-pVDZ(+X) 3.445 1.404 1.090 109.0 mPW1K/cc-pVTZ(+X) 3.443 1.391 1.081 109.3 MP2/6-31G* 3.315 1.414 1.088 109.5 MP2/6-31+Ga3.457 1.435 1.087 108.0 Cl/Br B3LYP/cc-pVTZ(+X) 3.405 1.839 1.081 108.0 BH&HLYP/cc-pVTZ(+X) 3.416 1.813 1.074 108.3 mPW1PW91/cc-pVTZ(+X) 3.368 1.812 1.082 108.5 mPWH&HPW91/cc-pVTZ(+X) 3.369 1.791 1.076 108.7 B97/cc-pVDZ(+X) 3.396 1.850 1.094 107.8 B97-1/cc-pVDZ(+X) 3.354 1.851 1.094 107.7 HCTH/cc-pVDZ(+X) 3.588 1.832 1.095 108.2 HCTH-120/cc-pVDZ(+X) 3.440 1.842 1.095 108.0 mPW1K/6-31+G 3.274 1.809 1.080 108.6 mPW1K/cc-pVDZ(+X) 3.381 1.800 1.087 108.6 mPW1K/cc-pVTZ(+X) 3.371 1.797 1.077 108.7 MP2/6-31G* 3.257 1.812 1.084 109.1MP2/6-31+Ga3.457 1.807 1.085 108.9 CCSD(T)/ spd fc3.318 1.820 1.083 108.2 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1996,118, 6273. bFrom Schmatz, S.; Botschwina, P.; Stoll, H.; Int. J. Mass Spectrom. 2000,201, 277. cFrom Botschwina, P.; Horn, M.; Seeger, S.; Oswald, R.; Ber. Bunsenges. Phys. Chem. 1997,101, 387.TABLE V. Geometries ( ˚A , degree) of the XCH 3X−transition structures, (X=F, Cl, and Br) of the SN 2identity reactions (complex 2). Species Method/Basis Set r(X· · ·C) r(C −H) X=F CCSD(T)/cc-pVQZ+1 1.808 1.071 B3LYP/cc-pVTZ(+X) 1.854 1.070 BH&HLYP/cc-pVTZ(+X) 1.823 1.062 mPW1PW91/cc-pVTZ(+X) 1.824 1.071 mPWH&HPW91/cc-pVTZ(+X) 1.797 1.064 B97/cc-pVDZ(+X) 1.852 1.085 B97-1/cc-pVDZ(+X) 1.848 1.084 HCTH/cc-pVDZ(+X) 1.875 1.085 HCTH-120/cc-pVDZ(+X) 1.872 1.086 mPW1K/6-31+G 1.807 1.070 mPW1K/cc-pVDZ(+X) 1.810 1.077 mPW1K/cc-pVTZ(+X) 1.804 1.066 MP2/6-31G* 1.778 1.076 MP2/6-31+Ga1.837 1.074 B3LYP/6-311G(2d,d,p) 1.830 1.074 B3LYP/6-311+G(2d,d,p) 1.862 1.073 X=Cl CCSD(T)/cc-pVQZ+1 2.305 1.070 B3LYP/cc-pVTZ(+X) 2.355 1.069 BH&HLYP/cc-pVTZ(+X) 2.332 1.061 mPW1PW91/cc-pVTZ(+X) 2.310 1.070 mPWH&HPW91/cc-pVTZ(+X) 2.290 1.064 B97/cc-pVDZ(+X) 2.349 1.083 B97-1/cc-pVDZ(+X) 2.344 1.083 HCTH/cc-pVDZ(+X) 2.374 1.084 HCTH-120/cc-pVDZ(+X) 2.372 1.084 mPW1K/6-31+G 2.313 1.069 mPW1K/cc-pVDZ(+X) 2.303 1.076 mPW1K/cc-pVTZ(+X) 2.295 1.065 MP2/6-31G* 2.308 1.072 MP2/6-31+Ga2.317 1.073 B3LYP/6-311G(2d,d,p) 2.366 1.072 B3LYP/6-311+G(2d,d,p) 2.364 1.072 X=Br CCSD(T)/cc-pVQZ+1 2.461 1.071 B3LYP/cc-pVTZ(+X) 2.511 1.069 BH&HLYP/cc-pVTZ(+X) 2.488 1.062 mPW1PW91/cc-pVTZ(+X) 2.464 1.070 mPWH&HPW91/cc-pVTZ(+X) 2.442 1.064 B97/cc-pVDZ(+X) 2.502 1.084 B97-1/cc-pVDZ(+X) 2.496 1.084 HCTH/cc-pVDZ(+X) 2.533 1.085 HCTH-120/cc-pVDZ(+X) 2.530 1.085mPW1K/6-31+G 2.430 1.069 mPW1K/cc-pVDZ(+X) 2.459 1.076 mPW1K/cc-pVTZ(+X) 2.447 1.066 MP2/6-31G* 2.444 1.073 MP2/6-31+Ga2.480 1.074 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1995,117, 2024.TABLE VI. Geometries ( ˚A , degree) of the YCH 3X−transition structures, of the SN 2 non-identity reactions (complex 2). Y/X Method/Basis Set r(Y · · ·C) r(C · · ·X) r(C −H) /negationslashHCX F/Cl B3LYP/cc-pVTZ(+X) 2.143 2.091 1.070 99.0 BH&HLYP/cc-pVTZ(+X) 2.065 2.107 1.062 97.2 mPW1PW91/cc-pVTZ(+X) 2.072 2.086 1.071 97.8 mPWH&HPW91/cc-pVTZ(+X) 2.016 2.088 1.064 96.7 B97/cc-pVDZ(+X) 2.143 2.091 1.085 98.8 B97-1/cc-pVDZ(+X) 2.139 2.086 1.085 98.8 HCTH/cc-pVDZ(+X) 2.190 2.098 1.086 99.2 HCTH-120/cc-pVDZ(+X) 2.208 2.077 1.086 100.0 mPW1K/6-31+G 2.063 2.080 1.070 98.0 mPW1K/cc-pVDZ(+X) 2.047 2.083 1.076 97.4 mPW1K/cc-pVTZ(+X) 2.029 2.089 1.066 97.0 MP2/6-31+Ga2.016 2.142 1.073 95.6 B3LYP/6-311+G(2d,d,p) 2.173 2.080 1.074 99.7 CCSD(T)/ spd fgb2.030 2.121 1.072 96.3 F/Br BH&HLYP/cc-pVTZ(+X) 2.174 2.181 1.063 100.0 mPWH&HPW91/cc-pVTZ(+X) 2.105 2.175 1.065 99.2 mPW1K/6-31+G 2.114 2.181 1.070 99.2 mPW1K/cc-pVDZ(+X) 2.179 2.145 1.078 100.9 mPW1K/cc-pVTZ(+X) 2.129 2.168 1.067 99.7 MP2/6-31+Ga2.108 2.242 1.075 97.9 Cl/Br B3LYP/cc-pVTZ(+X) 2.416 2.451 1.069 91.8 BH&HLYP/cc-pVTZ(+X) 2.388 2.433 1.062 91.6 mPW1PW91/cc-pVTZ(+X) 2.369 2.409 1.070 91.7 mPWH&HPW91/cc-pVTZ(+X) 2.342 2.391 1.064 91.5 B97/cc-pVDZ(+X) 2.408 2.445 1.084 91.7 B97-1/cc-pVDZ(+X) 2.403 2.439 1.084 91.7 HCTH/cc-pVDZ(+X) 2.488 2.467 1.085 91.3 HCTH-120/cc-pVDZ(+X) 2.435 2.469 1.085 91.8 mPW1K/6-31+G 2.313 2.421 1.069 90.1 mPW1K/cc-pVDZ(+X) 2.363 2.401 1.076 91.8 mPW1K/cc-pVTZ(+X) 2.349 2.396 1.065 91.6 MP2/6-31G* 2.336 2.419 1.072 90.5 MP2/6-31+G*a2.371 2.430 1.073 91.4 CCSD(T)/ spd fc2.354 2.422 1.072 91.2 aFrom Glukhovtsev, M.N.; Pross, A.; Radom, L.; J. Am. Chem. Soc. 1996,118, 6273. bFrom Schmatz, S.; Botschwina, P.; Stoll, H.; Int. J. Mass Spectrom. 2000,201, 277. cFrom Botschwina, P.; Horn, M.; Seeger, S.; Oswald, R.; Ber. Bunsenges. Phys. Chem. 1997,101, 387.
arXiv:physics/0011030v1 [physics.chem-ph] 15 Nov 2000Correlation consistent valence basis sets for use with the Stuttgart-Dresden-Bonn relativistic eﬀective core poten tials: the atoms Ga–Kr and In-Xe. Jan M.L. Martin* and Andreas Sundermann Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, IL-76100 Reh .ovot, Israel. E-mail:comartin@wicc.weizmann.ac.il (J. Chem. Phys. MS A0.09.107; Received Sept. 14, 2000; Revised February 2, 2 008) Abstract We propose large-core correlation-consistent pseudopote ntial basis sets for the heavy p-block elements Ga–Kr and In–Xe. The basis sets ar e of cc-pVTZ and cc-pVQZ quality, and have been optimized for use with the large-core (valence-electrons only) Stuttgart-Dresden-Bonn relati vistic pseudopotentials. Validation calculations on a variety of third-row and fourt h-row diatomics suggest them to be comparable in quality to the all-electron cc-pVTZ and cc-pVQZ basis sets for lighter elements. Especially the SDB -cc-pVQZ basis set in conjunction with a core polarization potential (CPP) yields excellent agreement with experiment for compounds of the later heavy p -block elements. For accurate calculations on Ga (and, to a lesser extent, Ge) compounds, ex- plicit treatment of 13 valence electrons appears to be desir able, while it seems inevitable for In compounds. For Ga and Ge, we propose correl ation con- sistent basis sets extended for (3d) correlation. For accur ate calculations on organometallic complexes of interest to homogenous cataly sis, we recommend a combination of the standard cc-pVTZ basis set for ﬁrst- and second-row elements, the presently derived SDB-cc-pVTZ basis set for h eavier p-block el- ements, and for transition metals, the small-core [6s5p3d] Stuttgart-Dresden 1basis set-RECP combination supplemented by (2 f1g) functions with expo- nents given in the Appendix to the present paper. Typeset using REVT EX 2I. INTRODUCTION AND THEORETICAL BACKGROUND The two major factors that determine the quality of a wavefun ction-based electronic structure calculation are the quality of the one-particle b asis set and that of the n-particle correlation treatment. Thanks to great progress in electron correlation methods (n otably in the area of coupled cluster theory [1]), the n-particle problem is to a large ext ent solved, leaving the 1-particle basis set as the main factor that determines the quality of an electronic structure calculation. Abundant research has been carried out on basis set converge nce and the development of extended basis sets for ﬁrst- and second-row systems (see e.g. [2] for a review): we note in particular the ANO (atomic natural orbital [3]) basis sets o f Alml¨ of and Taylor, the WMR (Widmark-Malmqvist-Roos, or averaged ANO [4]) basis sets o f the eponymous group, and the correlation consistent ( cc) basis sets of Dunning and coworkers [5,6]. Due to their rela tive compactness in terms of Gaussian primitives, the ccbasis sets have become very popular for benchmark wavefunction-based ab initio calculations: to a lesser extent, the same holds true for DFT (density functional theory [7]) calculations. Basis set convergence of the dynamical correlation energy i n conventional electronic struc- ture calculations is known to be very slow. This is less of an i ssue for DFT calculations [8–12]: as a rule basis set convergence appears to be reached for basi s sets of spdfquality and cer- tainly for basis sets of spdfg quality. Standard basis sets of such quality are readily ava ilable for ﬁrst- and second-row compounds: in addition, ANO and WMR basis sets are available for the ﬁrst-row transition metals [13] and ccbasis sets for the third-row main group elements [14]. Our group has recently become involved in a number of mechani stic studies by means of DFT methods (e.g. on competitive CC/CH activation by Rh(I ) pincer complexes [15,16] and on Pd(0/II) and Pd(II/IV) catalyzed mechanisms of the He ck reaction [17]) that involve second-row transition metals and fourth-row main group ele ments. Generally, one is limited to basis set/ECP (eﬀective core potential) combinations of approximately valence double- 3zeta quality. If one wants to establish basis set convergenc e for a given property, one is forced to optimize basis sets ad hoc (as we have done [17]), wh ich is however not necessarily the most elegant solution. Given that present-day DFT metho ds are less than ideal for the treatment of transition states [18–20], calibration calcu lations using coupled cluster methods are in order (at least for some small model systems) — and here the basis set issue becomes even more important. It is well known that for such heavy elements, relativistic e ﬀects cannot gratuitously be ne- glected without paying a heavy toll in terms of reliability. The theory of relativistic electronic structure methods has been reviewed in detail by Pyykk¨ o [21 ] and most recently by Reiher and Hess [22]. For systems in the size range of interest to org anometallic chemists, four- component all-electron relativistic calculations are pre sently out of the question, and even quasirelativistic calculations are very costly: conseque ntly, by far the most commonly em- ployed alternative has been the application of relativisti c eﬀective core potentials (RECPs). A useful ‘fringe beneﬁt’ of the latter is that they reduce the number of electrons that need to be treated, and hence, indirectly, the overall size of the ba sis set and cost of the calculation. The theory and practice of ECPs have been reviewed repeatedl y (e.g. [23–25]), most recently by Dolg [26]. Several ECP families are available fo r the range of the periodic table of interest to us, such as the LANL (Los Alamos National Labor atory) ECPs of Hay and Wadt [27], the CEP (Consistent Eﬀective Potential) family o f Stevens, Basch, and coworkers [28], the Ermler-Christensen family [29], and the Stuttgar t-Dresden-Bonn (SDB) energy- consistent pseudopotentials [30]. The purpose of this paper is to present and validate valence b asis sets for RECPs of a quality comparable to that of the cc-pVTZ and cc-pVQZ correl ation-consistent basis sets for lighter elements, to be used in conjunction with the latt er. In selecting the underlying RECP, we have opted for the SDB pseudopotentials for the foll owing methodological and pragmatic reasons (some, but not all, of which are satisﬁed f or the other popular ECPs): •compact mathematical form 4•ready availability in the commonly used quantum chemistry p ackages Gaussian 98 [31] and MOLPRO 2000 [32] •consistent treatment of relativistic eﬀects in all relevan t rows of the periodic table •independence of the ECP on the valence basis set •availability of core polarization potentials (CPPs) [33], since we were planning to use ‘large core’ potentials for the main group elements •availability of extended valence basis sets (speciﬁcally, [6s5p3d] contractions) for the transition metals. In Appendix I, we shall present optimize d [2f1g] polarization func- tions for those valence basis sets, to be used in conjunction with the presently derived SDB-cc-pVTZ basis sets for third- and fourth-row elements, and standard cc-pVTZ basis sets for ﬁrst-and second-row elements. To our knowledge, the only published example so far of a ‘corr elation consistent’ basis set based on an ECP is the work of Bauschlicher [34], who publishe d cc-pV nZ (n=T,Q,5) basis sets for indium, optimized for (5 s,5p,4d) correlation, to be used in conjunction with a small-core SDB pseudopotential. In this paper and a subseq uent application study [35], benchmark calculations on a number of In compounds were pres ented that clearly support the idea that the development of SDB-based correlation cons istent basis sets is warranted. In the next section, we shall describe the procedure by which the valence basis sets were optimized. In the following section, we shall present valid ation calculations with these basis sets on a variety of diatomic molecules. Conclusions are pre sented in a ﬁnal section. II. GENERATION OF BASIS SETS All electronic structure calculations were carried using M OLPRO2000 [32] running on a Compaq ES40 at the Weizmann Institute of Science. Basis sets were carried out by means of an adaptation of the DOMIN program by P. Spellucci [36], whic h is an implementation of the BFGS (Broyden-Fletcher-Goldfarb-Shanno) variable-metr ic method. Numerical derivatives 5of order two, four, and six were used: the lower orders until a n approximate minimum was reached, after which the optimization was reﬁned using the h igher orders. For the third-row main group elements, we employed the SDB ps eudopotentials denoted by the SDB group as ECP28MWB [37], i.e. large-core (1s2s2p3s 3p3d) energy-consistent pseudopotentials obtained from quasirelativistic Wood-B oring [38] calculations. For the fourth-row main group elements, we employed the ECP46MWB se t [37], i.e. large-core (1s2s2p3s3p3d4s4p4d). Unless indicated otherwise, all Hartree-Fock calculation s were carried out using proper symmetry and spin eigenfunctions. A. Valence spbasis sets In order to obtain an idea as to the size of the required spset for the valence orbitals, we carried out the following numerical experiment for the Se at om: a valence SCF calculation was carried out using the complete (26s17p) part of the all-e lectron cc-pV5Z basis set [14] added to the ECP28MWB pseudopotential. Then all primitives with coeﬃcients below 10−5 were discarded, leaving us with a (16s13p) primitive set at t he expense of only 0.38 micro- hartree in energy. Raising the ‘cutoﬀ’ to 10−4reduced the primitive set to (13s11p), and raises the energy by another 3 microhartree. Raising the cut oﬀ by another order of magni- tude reduces the primitive set to (12s9p), at the expense of a n additional 13 microhartree. Applying the same sequence of cutoﬀs to the (21s16p) primiti ves in the all-electron cc-pVQZ basis set leads to (14s11p), (13s9p), and (11s7p), respecti vely: from the (20s13p) primitives of the all-electron cc-pVTZ basis set we obtain in the same ma nner (12s9p) for a 10−4cutoﬀ, and (10s7p) for a 10−3cutoﬀ. Similar patterns were observed for other third-row e lements: the bottom line appears to be that 3–4 more sprimitives are required than pprimitives. We subsequently attempted to minimize (( k+ 4)skp) basis sets ( k=6–10) directly at the SCF level. However, the Hessian for some of the higher-ex ponent sfunctions is ex- tremely ﬂat, and as a result no reliable optimization can be c arried out. Considering the fact that, for instance, the outer (13s11p) exponents of the all-electron cc-pV5Z basis set 6display roughly even-tempered sequences ζk=αβk−1except for the outermost four primitives of every symmetry, we adopted the compromise solution of opt imizing the four outermost primitives of each symmetry without restriction, but const raining the remainder to follow an even-tempered sequence. This leads to an optimization pr oblem with twelve parameters in all (eight independent exponents, plus one α, βpair each for sandp). In this manner, we were able to obtain (10s6p) through (14s10 p) primitive sets. For Ga, Ge, and As, multiple minima were invariably found, with a solution that exhibits a ‘gap’ between the 3rd and 4th (or 4th and 5th) outermost primi tive being marginally lower in energy than a solution where nosuch gaps were present. (This behavior is particularly noticeable for the sprimitives.) Carrying out 4-parameter optimizations with purely even- tempered (14s10p) basis set quickly reveals the cause: as ζincreases, the coeﬃcients are initially positive, but then decay and change sign as the hig her exponent primitives ensure the proper inner shape of the orbital. The energy is rather in sensitive to the location — or even the presence — of the primitive near the crossing point, and especially with smaller sets of primitives, a marginal gain in energy might be obtain ed from a solution with an additional primitive in the very high exponent region rathe r than in the ‘crossing’ region. Since for application in correlated calculations, the pres ence of a gap in the outer part of the exponent sequence is clearly undesirable, we have delib erately chosen the most ‘even- tempered’ solution even where it was not the global minimum. Similar phenomena were observed for In–I: and likewise, we o btained the most ‘even- tempered’ primitive valence sets up to (14s10p). B. Addition of higher angular momentum functions Parameters for added higher angular momentum functions wer e then optimized at the CISD level. At ﬁrst even-tempered sequences of up to four (3 d)-type functions were added, followed by up to three additional (4 f)-type functions and up to two additional (5 g)-type functions. For the third-row main group elements, these opt imizations progressed unevent- fully. Not surprisingly, the dexponents diﬀer somewhat from those obtained by Dunning and 7coworkers for all-electron basis sets: in the latter, the dfunctions do double-duty as angular correlation functions for the (4 s,4p) orbitals and as (3 d) primitives, while in our case they solely take on the former role. For the fandgfunctions, the similarity is greater. In terms of energetic increments, the familiar ‘correlation consis tent’ (2d1f) and (3d2f1g) groupings of functions with similar energy lowerings emerge. In the fourth row, the convergence pattern of the dexponents is somewhat peculiar, in that for instance for Te and I, the energy lowerings for the 2n d and 3rd (3 d) function are similar. This is caused by the rather low-lying (5 d) orbital, which also causes a somewhat peculiar (2 d) exponent pattern for Te. We shall return to this point short ly. C. Deﬁnition of the ﬁnal contracted basis sets We carried out an analysis similar to that of Dunning and cowo rkers, in that we for instance completely contracted the porbital in a (14 s10p4d3f2g) basis set, then optimized even-tempered sequences of added pprimitives. The optimum sandpexponents revealed similar trends. In terms of contracting our Se basis set for c orrelation, however, they un- equivocally suggest that the 2nd and 4th outermost (s) and (p ) primitives be decontracted for a valence triple zeta basis set, and the 2nd–4th outermost pr imitives for a valence quadruple zeta basis set. (From here on, we shall be counting primitive s starting from the ‘outermost’, i.e. smallest and most diﬀuse, exponent.) By comparison, in the Dunning all-electron case these were the 1st and 3rd, and 1st–3rd primitives, respecti vely. However, our outermost ( sp) primitives are considerably more diﬀuse than theirs, by vir tue of the absence of the inner- shell ’gravity well’ in the valence-only optimizations. Th e exponents of the decontracted primitives in fact are fairly similar. This having been established, we determined our favored ’VT Z’ and ’VQZ’ contraction patterns for each element by comparing total energies betwe en all six and four possible choices, respectively, among the four outermost primitive s. If we denote decontraction of a primitive by a 1 and the lack thereof by a 0, and start at the lo west exponent, then the favored (i.e., lowest-energy) quadruple-zeta contractio n pattern is found to be {0111}for Se, 8Br, Kr, Te, I, and Xe, but {1101}for Ga, Ge, In, and Sn. (For As, {1011}is marginally lower in energy than {0111}, while for Sb, a {1110}pattern for the swas combined with a{1101}pattern for the pfunctions.) For the triple-zeta contractions, the {0101}pattern prevails for As, Se, Br, Kr, Te, I, and Xe, but the {0110}pattern for Ga, Ge, In, Sn, and Sb. The ﬁnal basis sets for most elements were then obtained simp ly by adding the optimum (2d1f) exponents to the ‘triple-zeta’ contraction — leadin g to a [3s3p2d1f] contracted basis set —, and the optimum (3d2f1g) exponents to the ‘quadruple- zeta’ contraction — leading to a [4s4p3d2f1g] contracted basis set. For Te and I, because of the peculiarities of the d exponents noted above, this procedure does not yield a satis factory SDB-cc-pVTZ basis set. By obtaining CISD natural orbitals for Te and I using (3 d1f) primitives, it was revealed that the highest-exponent primitive contributed appreciably ( and similarly) to the lowest two d- type natural orbitals, but that the latter are mainly distin guished by a sign change in the lowest-exponent dprimitive. Consequently, the two innermost dprimitives were contracted based on their coeﬃcients in the lowest dtype natural orbital. The slight added cost should be well outweighed by the greater reliability. Considering the d-type ANOs in calculations with (3 d1f) primitives on Sb, Sn, and In revealed that the same procedur e might be beneﬁcial for In, but would not aﬀect Sn or Sb. Therefore, in our ﬁnal SDB -cc-pVTZ basis sets, the d functions in In, Te, and I are in fact (3 d)→[2d] segmented contractions. The ﬁnal basis sets generated are available on the Internet W orld Wide Web at the Uniform Resource Locator http://theochem.weizmann.ac.il/web/papers/SDB-cc.ht ml in both Gaussian 98 and MOLPRO format. D. Diﬀuse function exponents For anionic systems and some very polar compounds, the avail ability of (diﬀuse-function) ‘augmented’ basis sets, like the original aug-cc-pV nZ basis sets [39], is essential. We have obtained diﬀuse functions for use with our SDB-cc-pVTZ and S DB-cc-pVQZ basis sets using the following procedure: (a) one low-exponent sandpfunction, each, were added to the 9sppart of the underlying basis set and optimized simultaneous ly at the SCF level for the corresponding atomic anion; (b) successive angular moment a of the underlying basis set were introduced, and one additional low-exponent primitive add ed and optimized, at the CISD level for the corresponding atomic anion. The ﬁnal SDB-aug- cc-pVTZ and SDB-aug-cc- pVQZ basis sets are thus of [4 s4p3d2f] and [5 s5p4d3f2g] quality, respectively. III. APPLICATION TO DIATOMIC MOLECULES In order to validate our basis sets, we have carried out CCSD( T) calculations of the dissociation energy ( De), bond length ( re), harmonic frequency ( ωe) and ﬁrst-order anhar- monicity ( ωexe) of a number of third-row and fourth-row diatomic molecules selected from the compilation by Huber and Herzberg [40]. CCSD(T) energie s were computed at eleven points spaced evenly at 0.01 ˚A intervals around the experimental re, a ﬁfth-or sixth-order polynomial in rwas ﬁt, and a standard Dunham analysis [41] carried out on the resulting polynomial. For the open-shell systems and the constituent atoms, the CCSD(T) deﬁnition according to Ref. [42] was employed throughout. Since we are using ‘large’ cores, we also carried out calcula tions using core polarization potentials (CPPs). For elements of groups IV, V, and VI the pa rameters were taken from the work of Igel-Mann et al. [43], although the cutoﬀ parameters given in that reference are not optimal for the ECP nnMWB pseudopotentials. For group III elements, optimal cuto ﬀs were taken from Leininger et al. [44], while optimal cutoﬀs for th e halogens were taken from the online version of the SDB pseudopotentials [45]. (The valen ce basis set was left unchanged.) For systems that include at most third-row atoms, all-elect ron calculations could be carried out for comparison using the corresponding standar d cc-pV nZ basis sets [5,46,14]. re andωefor these species are given in Table I, while Devalues are given in Tables VII and V. For the remaining diatomics (which include at least one four th-row atom), the corresponding data are found in Tables II and VI, respectively. In comparing such data with all-electron calculations in wh ich only valence electrons are correlated, it should be kept in mind that the CPPs approxima tely account for both inner- 10shell relaxation/polarization (”static core polarizatio n”) and inner-shell correlation (”dy- namic core polarization”). Therefore, a direct comparison appears to be somewhat ‘unfair’ to the all-electron calculations; on the other hand, since t he standard cc-pV nZ basis sets are by deﬁnition of minimal basis set quality in the inner-shell orbitals, these basis sets are fairly limited in terms of ﬂexibility for static polarization. For heavier elements, it should also be kept in mind that the ECP calculations include relativistic eﬀects at least approximately, while their all-electron counterparts discussed here are e ntirely nonrelativistic. For the late third-row species, it seems to be clear that the p erformance of our SDB-cc- pVnZ basis sets is on a par with that of the all-electron basis set s. Introduction of the core polarization potentials results in a signiﬁcant improveme nt in the computed bond distance: agreement between SDB+CPP-cc-pVQZ and experimental bond l engths is particularly good for many species. This conclusion is less clear for the harmo nic frequencies, where the known tendency [47] of CCSD(T) to slightly overestimate harmonic frequencies may mask any small improvements. The computed anharmonicities (not reported in Table I) agree very well between the various methods and experiment. For the early third-row species, we noticed the at ﬁrst sight peculiar phenomenon (Table III) that, while our data with CPP are in very good agreement w ith experiment, the all- electron bond lengths are considerably too long, e.g. 0.05 ˚A in GeF. (These diﬀerences are too large to be plausibly ascribed to relativistic eﬀects accou nted for by the pseudopotentials.) The cause lies in the impossibility to make a meaningful sepa ration between ‘valence’ and (3d) orbitals in these molecules: if correlation from the (3 d) orbitals is admitted, a dramatic improvement is seen in the computed bond distances. Needles s to say, such calculations are vastly more expensive than those with the large-core pse udopotentials, and if the all- electron basis set would be expanded with the appropriate an gular correlation functions for (3d) correlation (i.e., high-exponent f and g functions), t his would further increase the cost diﬀerential. As an illustration, we will consider the GaH molecule in some what greater detail (Table IV). The all-electron calculations with standard cc-pV nZ basis sets fortuitously reproduce ωe 11very well, but overestimate the bond distance by almost 0.03 ˚A. In contrast, 4-electron ECP calculations with an 28-electron pseudopotential both und erestimate ωeand overestimate re. Admitting (3d) correlation with the cc-pV nZ basis sets leads to a dramatic shortening of re, but also to severely overestimated ωeand anharmonicity. Obviously, this basis set needs to be signiﬁcantly extended before it is suitable for (3d) co rrelation. We have generated such basis sets, denoted cc-pDVTZ and cc-pDVQZ, in the follo wing manner. All basis func- tions in the original cc-pVQZ basis set were retained, but fo ur additional dprimitives were decontracted. After this, successive layers of f,g, and ﬁnally hprimitives were optimized at the CISD level (13 electrons correlated) on top of the orig inal basis set. We found that the ﬁrst h, second g, and third f function yielded similar low erings of the atomic energy, and hence added (3f2g1h) primitives to the basis set. (Expon ents and other details can be found in the Supplementary Material.) Then we restored th e original dfunctions and progressively uncontracted primitives: while the ﬁrst add itional uncontracted dyields a very large energy lowering, the second adds a 10 millihartree amo unt comparable to that of the h functions, while lowerings decay rapidly after that. Hence the ﬁnal cc-pDVQZ basis set is of [7s6p6d5f3g1h] quality. By similar arguments, we ﬁnd that t he cc-pVTZ basis set requires addition of (2f1g) functions and decontraction of two addit ionaldprimitives, leading to a cc-pDVTZ basis set of [6s5p6d3f1g] quality. (While the fexponents in the cc-pDVQZ basis set span a continuous range, a ‘gap’ is present in the cc-pVTZ case. A similar phenomenon is seen in an earlier Ga basis set of Bauschlicher [48].) These b asis sets indeed do represent an improvement (Table IV) but the 14-electron results clearly are still deﬁcient in some respect. We considered also including (3s3p) correlation: to accomm odate this, we uncontracted two additional s and p primitives each in the cc-pDVTZ basis set, as well as (to ensure adequate coverage of angular correlation from these orbitals) one ad ditional dfunction. The resulting spectroscopic constants are in excellent agreement with ex periment, which might lead to the conclusion that (3s3p) correlation is essential for a prope r description of GaH. However, as we reduce the number of correlated electrons from 22 to 14, we see only quite minor eﬀects on the spectroscopic constants. At that stage, the addition aldfunction can be removed with 12essentially no eﬀect on the computed spectroscopic constan ts; the cc-pDVTZ+2s2p basis set is thus of [8 s7p6d3f1g] quality. Correlating valence orbitals only leads to rebeing too long andωetoo low, conﬁrming that the excellent ωewith standard cc-pV nZ basis sets is indeed the result of an error compensation. We now consider the use of small-core ECPs. The eﬀect of reopt imizing exponents was deemed minimal: instead we simply (a) carried out an ECP10MW B Hartree-Fock calcula- tion with an uncontracted cc-pVTZ or cc-pVQZ basis set; (b) d eleted all primitives with coeﬃcients ×degeneracies that are signiﬁcantly less than 10−4; repeated the SCF calculation and recontracted the basis set with the orbital coeﬃcients t hus obtained. In the cc-pVTZ basis set, we were able to delete the innermost (5s2p) primit ives; in the cc-pVQZ basis set, the innermost (6s3p) primitives could be deleted. The recon tracted basis sets (which are of [4s4p5d3f1g] and [are again given in the supplementary mate rial. (As given, these basis sets are of [4s4p5d3f1g] and [5s5p6d5f3g1h] contracted size; to this should be added the additional decontracted sandpprimitives mentioned above, leading to an SDB-cc-pDVTZ+2s 2p ba- sis set of [6 s6p5d2f1g] quality and an SDB-cc-pDVQZ basis set of [8 s8p6d5f3g1h] quality.) We indeed ﬁnd performance with these basis set-ECP combinat ions to be quite satisfactory (Table IV). (Note that for technical reasons, the SDB-cc-pD VQZ results do not include h functions.) It should also be noted that the eﬀects of (3d) correlation, w hile still important in accurate work, are signiﬁcantly smaller with the cc-pDVTZ+2s2p and c c-pDVQZ+3s3p basis sets than with their less extended counterparts. The very large core c orrelation contributions seen in such studies as Ref. [49] are thus at least in part basis set ar tefacts. Results for GeH (Table IV) follow similar trends as those for GaH, although the deviation from experiment incurred by neglecting (3d) correlation is deﬁnitely smaller. Continuing the series, our computed results for AsH, SeH, and HBr suggest no need for including (3d) correlation in these systems. We also applied the cc-pDVTZ and cc-pDVQZ basis sets to the po lar Ga and Ge com- pounds (Table III). A (sometimes notable) improvement is ma inly seen in the vibrational 13frequencies. Decontracting additional (sp) primitives in the Ga basis set was considered for GaF, and does not appear to greatly aﬀect results. This paral lels a ﬁnding noted earlier [50] for inner-shell correlation in ﬁrst-row compounds, where ﬂ exibility of the core correlation basis set appears to be more important for A–H than for A–B bon ds. For the fourth-row systems, only a comparison with experime nt is possible. Especially the SDB+CPP-pVQZ results agree very well with experiment, w hile the errors for the SDB+CPP-pVTZ basis sets are not dissimilar from those seen f or the lighter-atom sys- tems. A notable exception is constituted by a number of indiu m compounds, for which abnormally short bond distances are found. This problem has been noted previously for large-core pseudopotential calculations on heavy group II I halides [51]. We attempted a number of calculations in which Bauschlicher’s correlatio n-consistent basis set for In was used in conjunction with regular cc-pV nZ basis sets on H–Ar and SDB-cc-pV nZ on Ga–Kr and Sn–Xe. The In (4d) electrons were correlated in these cal culations. This completely re- solves the problem. Discrepancies between all-electron an d ECP28MWB basis sets on In are not inconsistent with the expected magnitude of relativist ic eﬀects on reandωe. For InBr and InI, consideration of a core polarization potential on t he halogen has eﬀects of -0.007 ˚A and -0.013 ˚A, respectively, on the bond distance, bringing them into ex cellent agreement with experiment. (Note that the +0.005 ˚A discrepancy between computed and observed re(InCl) found by Bauschlicher [34] with his largest basis set appears to be almost entirely due to (2 s2p) correlation in Cl: its inclusion reduces reby 0.005 ˚A. ) Finally, we shall consider dissociation energies. These ar e found in Tables V, VII, and VI, together with experimental data from two sources. These are the 1979 Huber and Herzberg (HH) book [40], and a more recent compilation by Kerr and Stoc ker (KS) [52] which contains data through November 1998. All computed dissociation energies are corrected for atomi c and molecular ﬁrst-order spin-orbit splitting, with the data taken from the experime ntal sources for the molecules and from Ref. [53] for the atoms. Aside from atomization energies with SDB-cc-pV nZ, SDB-aug-cc-pV nZ, and all-electron 14cc-pVnZ basis sets, the tables contain extrapolations to the inﬁni te basis limit using the expression taken from W1 theory [54]: E∞=E[V QZ]+(E[V QZ]−E[V TZ])/((4/3)3.22−1), where the exponent 3.22 is speciﬁc to the VTZ/VQZ basis set co mbination. This is in fact a damped variant of the simple A+B/l3formula of Halkier et al. [55]: the damping is required [54] because the VTZ and VQZ basis sets are still not extended enough and lead to overshooting if the A+B/l3formula is applied to them. (The latter is the extrapolation of choice for larger basis sets.) One conspicuous feature of the experimental results is just how uncertain they are for many molecules in these tables. For the late third-row syste ms, agreement between exper- iment and our extrapolated results including CPP is excelle nt for those molecules where the experimental value is precisely known. For most of the ot her systems, the computed value falls within the combined uncertainties of the experi mental values. Agreement in fact appears to be slightly better than for the all-electron calc ulations, but this is not an entirely ‘fair’ comparison since the latter include neither inner-s hell correlation nor scalar relativistic corrections, while both are included approximately in the S DB+CPP results through the core-polarization potential and the relativistic pseudop otential, respectively. For the Ga, Ge, and In compounds, experimental dissociation energies are so uncertain that a meaningful comparison is essentially impossible. Fo r those fourth-row systems where precise experimental data are available, agreement with ex periment is still quite satisfactory, albeit less good than for the third-row compounds. In partic ular, an account for higher- order spin-orbit eﬀects might be mandatory for some of the io dine compounds. Dolg [56] carried out benchmark calculations on the hydrogen halides and dihalides, and found near- exact spin-orbit contributions to De(HI) and De(I2) of 0.26 and 0.49 eV, respectively: simply considering the ﬁne structures of the constituent atoms (as done here) yields 0.315 and 0.63 eV, respectively. In other words, our calculated Devalues for HI and I 2are intrinsically too low by 0.07 and 0.14 eV, respectively. As expected, the use of (diﬀuse function) ‘augmented’ basis sets yields improved results forreandωeof highly polar molecules (e.g., GaF); for Devalues, diﬀerences of up to 0.05 15eV are seen after extrapolation, which are deﬁnitely signiﬁ cant in accurate thermochemical work. As is the general rule [57,54], the addition of diﬀuse f unctions considerably improves the success of extrapolation methods and improves agreemen t with (precise) experimental dissociation energies. Finally, we should address the question whether or not the RE CPs used here provide an approximate account for scalar relativistic eﬀects. Vis scher and coworkers studied rel- ativistic eﬀects on the hydrogen halogenides [58], dihalog enides [59], and interhalogenides [60] by means of full four-component relativistic CCSD(T) a s implemented by Visscher, Lee, and Dyall [61]. (Pisani and Clementi [62] also carried out Di rac-Fock calculations on the chalcogen hydrides — including SeH — and found an eﬀect of -0. 005˚A onre.) Since down to Br, the eﬀects are fairly small (e.g. +0.003 ˚A and –6 cm−1in BrF), a comparison be- tween all-electron and ECP results is somewhat dubious as an indicator for the recovery of relativistic eﬀects. Given however the sizable relativist ic contributions found in that work for the iodine compounds (e.g. IF: +0.012 ˚A and -23 cm−1), the level of agreement with experiment found in the present paper is somewhat hard to exp lain unless the ECPs indeed recover most of the scalar relativistic eﬀects. IV. CONCLUSIONS We have derived (fairly) compact valence basis sets of cc-pV nZ and aug-cc-pV nZ quality (n=T,Q) for the elements Ga–Kr and In–Xe, to be used in conjunct ion with large-core Stuttgart-Dresden-Bonn pseudopotentials. For the third r ow, the basis sets appear to be quite comparable to the corresponding all-electron cc-pV nZ basis sets. Agreement with experiment is quite satisfactory for compounds of the later heavy p-block elements. Highly accurate calculations on Ga and, to a lesser extent, Ge compo unds require treating the (3d) electrons explicitly: we propose (3 d)-correlation basis sets for these elements. For In compounds, inclusion of (4d) correlation is a must, as previ ously found by Bauschlicher [34]: we recommend the basis sets in that reference. Our principal objective was having extended basis sets avai lable for studies on 16organometallic compounds, including those with one or more heavy group V, VI, and VII elements. This objective appears to have been reached. ACKNOWLEDGMENTS JM is the incumbent of the Helen and Milton A. Kimmelman Caree r Development Chair. Research at the Weizmann Institute was supported by the Mine rva Foundation, Munich, Germany, and by the Tashtiyot program of the Ministry of Science (Israel). AS acknowl- edges a Minerva Postdoctoral Fellowship. The authors would like to thank Dr. Charles W. Bauschlicher Jr. (NASA Ames Research Center) for critical r eading of the manuscript prior to submission. SUPPLEMENTARY MATERIAL The SDB-cc-pVTZ, SDB-cc-pVQZ, cc-pDVTZ, and cc-pDVQZ basi s sets developed in this paper are available for download on the Internet World W ide Web at the URL http: //theochem.weizmann.ac.il/web/papers/SDB-cc.html APPENDIX: F-AND G-FUNCTION EXPONENTS FOR THE TRANSITION METALS For use in conjunction with the above SDB-cc-pVTZ basis set o n Ga-Kr and In–Xe, and the standard cc-pVTZ basis set on the ﬁrst two rows of the peri odic table, we recommend the following basis set/ECP combination for transition met als. For ﬁrst-row transition metals, the pseudopotential denot ed as ECP10MDF [63] (which has a small 10-electron core) was used in conjunction with th e [6s5p3d] contraction of an (8s7p6d) primitive set given in Ref. [63]. For second-and th ird-row transition metals, we used the ECP28MWB and ECP60MWB quasirelativistic pseudopo tentials, respectively, as given in Ref. [64], together with the [6s5p3d] contracted va lence basis sets given in the same reference. 17Two (4 f)-type functions and one (5 g)-type function were added, and their exponents op- timized at the CISD level for the lowest-lying ( s)1(d)n−1and (s)2(d)n−2states. (In addition, optimizations were carried out for the ( s)0(d)10ground state of Pd.) Proper symmetry and spin eigenfunctions were used for the Hartree-Fock referen ce, and only valence electrons were correlated. The optimum exponents for the two states consid ered (three in the case of Pd) are not very diﬀerent: we recommend their averages as the fandgexponents, which are given in Table VIII. 18REFERENCES [1] R. J. Bartlett, J. Phys. Chem. 93, 1697 (1989); P. R. Taylor,, in Lecture Notes in Quan- tum Chemistry II (ed. B. O. Roos), Lecture Notes in Chemistry 64, 125 (1994); R. J. Bartlett and J. F. Stanton, in Reviews in Computational Chemistry, Vol. V (Lipkowitz, K. B., Boyd, D. B., Eds.) VCH, New York, 1994, pp. 65–169; T. J. 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CCSD(T) and experimental spectroscopic constants (Rein˚A,ωein cm−1) for molecules containing third row atoms molecule basis Re ωe SDB CPP all e−Exp SDB CPP all e−Exp AlBr VTZ 2.3149 2.3088 2.3271 2.294807 383.0 384.4 377.3 378 .0 VQZ 2.3073 2.3014 2.3180 381.1 382.3 377.0 AVTZ 2.3164 377.9 AVQZ 2.3050 374.6 As2 VTZ 2.1265 2.1126 2.1348 2.1026 424.0 426.8 424.6 429.55 VQZ 2.1098 2.0963 2.1284 432.1 435.1 428.7 AsF3Σ−VTZ 1.7298 1.7224 1.7464 1.7360 697.0 700.3 698.1 685.78 VQZ 1.7291 1.7217 1.7446 693.6 696.9 697.1 AVTZ 1.7298 687.1 AVQZ 1.7185 700.9 AsH3Σ−VTZ 1.5320 1.5269 1.5354 1.5231a2135.3 2138.2 2151.9 2155.503a VQZ 1.5286 1.5234 1.5335 2156.0 2159.5 2161.1 AsN VTZ 1.6292 1.6227 1.6374 1.61843 1063.2 1069.7 1064.5 10 68.54 VQZ 1.6188 1.6124 1.6326 1079.1 1085.8 1071.9 AsO2Π VTZ 1.6260 1.6200 1.6378 1.6236 973.6 978.2 971.2 967.08 VQZ 1.6199 1.6138 1.6344 978.5 983.2 975.5 AsP VTZ 2.0216 2.0142 2.0276 1.999 597.7 600.2 597.3 604.02 VQZ 2.0083 2.0011 2.0194 607.8 610.3 604.1 AsS2Π VTZ 2.0393 2.0319 2.0435 2.0174 559.5 561.5 561.3 567.94 VQZ 2.0246 2.0174 2.0349 570.0 571.9 568.0 BBr VTZ 1.8964 1.8908 1.9060 1.8882 690.5 693.8 683.7 684.31 VQZ 1.8942 1.8887 1.9034 685.8 688.6 681.1 AVTZ 1.8918 1.9068 684.4 677.9 AVQZ 1.8894 1.9044 700.9 678.8 Br2 VTZ 2.3138 2.3014 2.3108 2.28105 313.6 316.4 319.3 325.321 VQZ 2.2970 2.2856 2.2983 323.8 326.1 325.9 AVTZ 2.2941 2.3127 318.1 317.3 AVQZ 2.2808 2.2986 326.9 325.1 BrCl VTZ 2.1616 2.1555 2.1627 2.136065 434.3 436.2 435.9 444 .276 VQZ 2.1491 2.1432 2.1504 442.2 443.9 443.3 AVTZ 2.1545 434.6 AVQZ 2.1420 443.5 BrF VTZ 1.7680 1.7628 1.7685 1.75894 661.0 664.0 666.5 670.7 5 VQZ 1.7611 1.7559 1.7619 672.0 674.5 677.0 AVTZ 1.7622 667.7 AVQZ 1.7547 676.5 25CSe VTZ 1.6921 1.6863 1.6966 1.67647 1019.3 1025.4 1024.1 10 35.36 VQZ 1.6834 1.6776 1.6911 1032.7 1038.8 1031.2 HBr VTZ 1.4147 1.4106 1.4203 1.414435 2656.1 2665.0 2660.0 2 648.975 VQZ 1.4155 1.4112 1.4205 2657.7 2667.2 2661.2 AVTZ 1.4166 1.4124 1.4213 2647.2 2656.2 2657.2 AVQZ 1.4163 1.4120 1.4211 2651.9 2661.5 2658.1 SeH VTZ 1.4702 1.4654 1.4731 1.46432(6)b2398.2 2406.3 2417.6 2421.715(23)b VQZ 1.4680 1.4630 1.4721 2416.6 2425.5 2425.4 NSe2Π VTZ 1.6659 1.6606 1.6671 1.6518 946.8 952.3 955.8 956.81 VQZ 1.6539 1.6487 1.6599 964.9 970.4 966.1 Se23Σ− gVTZ 2.1911 2.1783 2.1915 2.1660 381.3 384.6 384.7 385.303 VQZ 2.1752 2.1628 2.1830 388.3 391.5 389.6 SeO3Σ−VTZ 1.6476 1.6425 1.6500 1.6484 914.4 918.8 922.2 914.69 VQZ 1.6379 1.6328 1.6454 924.9 929.3 926.6 SeS3Σ−VTZ 2.0546 2.0482 2.0558 2.0367 547.7 550.2 549.5 555.56 VQZ 2.0407 2.0344 2.0458 556.2 558.7 556.6 SiSe VTZ 2.0834 2.0770 2.0882 2.058324 572.0 574.6 571.3 580 .0 VQZ 2.0713 2.0650 2.0790 578.5 581.2 576.9 AVTZ 2.0840 2.0775 2.0887 569.3 572.1 569.1 AVQZ 2.0723 2.0659 2.0794 576.1 578.9 576.1 SDB: calculations using large-core SDB pseudopotentials; CPP: ditto with core polarization potentials added.aAsH ( Reandωe): K. D. Hensel, R. A. Hughes, and J. M. Brown J. Chem. Soc. Faraday Trans. II 91, 2999 (1995).bSeH (Reandωe): R. S. Ram and P. F. Bernath, J. Mol. Spectrosc. 203, 9 (2000). 26TABLE II. CCSD(T) and experimental spectroscopic constant s (Rein˚A,ωein cm−1) for molecules containing fourth row atoms molecule basis Re ωe SDB CPP Exp SDB CPP Exp AlI VTZ 2.5591 2.5478 2.537102 321.1 323.1 316.1 VQZ 2.5521 2.5408 318.4 320.1 AVTZ 2.5519 316.3 AVQZ 2.5441 316.8 GaI VTZ 2.6240 2.5955 2.57467 212.2 213.6 216.6 VQZ 2.6150 2.5871 210.1 211.3 GeTe VTZ 2.3802 2.3596 2.340165 313.1 315.7 323.9 VQZ 2.3554 2.3358 312.4 322.6 HI VTZ 1.6147 1.6073 1.60916 2314.7 2325.7 2309.014 VQZ 1.6131 1.6054 2320.3 2332.7 AVTZ 1.6162 1.6088 2310.4 2321.4 AVQZ 1.6135 1.6058 2318.1 2330.3 I2 VTZ 2.7072 2.6831 2.6663 212.2 215.8 214.502 VQZ 2.6876 2.6655 218.4 221.3 IBr VTZ 2.5049 2.4870 2.468989 263.6 266.9 268.640 VQZ 2.4862 2.4698 271.5 274.2 AVTZ 2.4805 267.9 AVQZ 2.4652 274.6 ICl VTZ 2.3482 2.3371 2.320878 383.2 386.1 384.293 VQZ 2.3349 2.3239 388.3 391.0 AVTZ 2.3358 385.4 AVQZ 2.3225 391.6 IF VTZ 1.9296 1.9204 1.90975 611.8 616.2 610.24 VQZ 1.9158 1.9066 622.0 626.1 AVTZ 1.9196 616.8 AVQZ 1.9070 625.7 InBr VTZ 2.5377 2.5029 2.54318 229.1 231.2 221.0 VQZ 2.5080 2.4716 228.7 230.5 CWB(t) 2.5556 224.0 CWB(q) 2.5486 224.0 ACWB(t) 2.5677 2.5606 218.1 219.0 ACWB(q) 2.5540 2.5474 220.7 221.6 InCl VTZ 2.3849 2.3553 2.401169 327.1 330.0 317.4 VQZ 2.3571 2.3246 326.7 328.6 ACWB(t) [34] 2.423 309 ACWB(q) [34] 2.411 314 ACWB(5) [34] 2.406 316 all e- CWB(t) [34] 2.423 317 27all e- CWB(q) [34] 2.412 2.4068e318 316.0e all e- CWB(5) [34] 2.406 319 (a) 2.4206 319.4 InF VTZ 1.9329 1.9109 1.985396 568.8 577.0 535.35 VQZ 1.8976 1.8729 579.4 587.1 CWB(t) 1.9833 538.5 CWB(q) 1.9822 541.3 ACWB(t) 1.9910 526.6 ACWB(q) 1.9853 534.3 all e- CWB(t) 1.9833 550.7 all e- CWB(q) 1.9821 553.7 all e- ACWB(t) 1.9910 539.4 all e- ACWB(q) 1.9857 546.6 InH VTZ 1.8653 1.8517 1.8380 1482.3 1472.7 1476.04 VQZ 1.8317 1.8171 1503.5 1497.3 CWB(t) 1.8395 1462.4 CWB(q) 1.8374 1471.9 all e- CWB(t) 1.8491 1493.4 all e- CWB(q) 1.8473 1504.4 InI VTZ 2.7698 2.7273 2.75365 179.0 181.1 177.1 VQZ 2.7412 2.6983 178.9 180.6 CWB(t) 2.7740 2.7605 177.8 179.0 CWB(q) 2.7674 2.7543 177.0 178.1 Sb2 VTZ 2.5294 2.5045 2.476b266.3 268.2 269.623b VQZ 2.5056 2.4816 273.0 275.1 SbF3Σ−VTZ 1.9252 1.9138 1.9177 615.3 619.8 609.0d VQZ 1.9137 1.9023 620.0 623.9 SbH3Σ−VTZ 1.7259 1.7181 1.7107c1910.9 1914.0 1923.179c VQZ 1.7188 1.7107 1930.3 1933.6 SbP VTZ 2.2335 2.2211 2.205 495.0 497.8 500.07 VQZ 2.2183 2.2058 503.5 506.5 SnO VTZ 1.8426 1.8297 1.832505 763.3 768.7 814.6 VQZ 1.8271 1.8137 781.9 788.5 SnS VTZ 2.2382 2.2220 2.209026 466.9 468.6 487.26 VQZ 2.2214 2.2050 474.9 476.8 SnSe VTZ 2.3595 2.3379 2.325601 316.3 318.1 331.2 VQZ 2.3398 2.3181 321.0 323.2 AVTZ 2.3541 2.3319 315.5 317.9 AVQZ 2.3421 2.3202 319.2 321.6 SnTe VTZ 2.5563 2.5294 2.522814 253.5 255.4 259.5 VQZ 2.5359 2.5091 256.9 258.9 AVTZ 2.5492 2.5218 253.3 255.6 28AVQZ 2.5375 2.5104 255.6 257.8 (a) using ECP28MDF for In, (4d) electrons correlated (b) Sb 2(Re,ωe): H. Sontag and R. Weber, J. Mol. Spectrosc. 91, 72 (1982). (c) SbH ( Re,ωe): R.-D. Urban, K. Essig, and H. Jones, J. Chem. Phys. 9915911 993. (d) SbF ( ωe): D. K. W. Wang, W. E. Jones, F. Pr´ evot, and R. Colin, J. Mol. S pectrosc. 49, 377 (1974). (e) This work, using the MTavqz basis set [54] on Cl and includ ing (2s,2p) correlation in Cl. CWB(t) and CWB(q) indicate the Bauschlicher [34] cc-pVTZ an d cc-pVQZ basis sets on indium, and regular cc-pV nZ or SDB-ccpV nZ basis sets on the other atom. ACWB(t) and ACWB(q) indicate the same, but in conjunction with an augmen ted basis set on the other atom. 29TABLE III. CCSD(T) and experimental spectroscopic constan ts (Rein˚A,ωein cm−1) for diatomics involving Ga or Ge and an electronegative element molecule Re ωe SDB CPP all e−(a) all e−(b) Exp. SDB CPP all e−(a) all e−(b) Exp GaBr VTZ 2.3912 2.3681 2.4013 2.3618 2.35248 261.7 262.6 268 .6 274.9 263.0 VQZ 2.3829 2.3602 2.4007 2.3644 258.7 259.5 266.6 266.8 13e−DVTZ 2.3564 2.3668 275.5 270.9 DVQZ 2.3486 2.3656 268.9 267.7 GaCl VTZ 2.2411 2.2227 2.2554 2.2031 2.201690 353.5 353.6 38 1.3 382.4 365.3 VQZ 2.2320 2.2135 2.2572 2.2010 350.7 351.1 373.3 370.4 13e−DVTZ 2.2063 2.2070 372.9 374.7 DVQZ 2.1987 2.2044 368.4 370.1 GaF VTZ 1.7851 1.7688 1.8031 1.7756 1.774369 589.0 589.8 631 .9 652.3 622.2 VQZ 1.7753 1.7587 1.8043 1.7697 586.3 588.0 625.4 644.8 13e−DVTZ 1.7704 1.7709 638.7 644.2 DVQZ 1.7674 1.7716 636.1 637.6 ADVTZ 1.7798 1.7812 619.7 623.7 ADVQZ 1.7693 1.7741 631.2 628.9 +2s2p DVTZ 1.7711 629.1 +3s3p DVQZ 1.7714 629.3 GeF2Π VTZ 1.7655 1.7550 1.8047 1.7469 1.7452 642.6 645.3 746.2 68 8.6 665.67 VQZ 1.7577 1.7473 1.8096 1.7424 640.3 642.8 746.1 681.2 13e−DVTZ 1.7411 1.7450 683.1 677.6 13e−DVQZ 1.7407 1.7424 675.3 679.5 GeO VTZ 1.6382 1.6290 1.6491 1.6341 1.624648 925.1 930.8 987 .4 993.7 985.5 VQZ 1.6295 1.6205 1.6485 1.6276 943.2 949.1 989.3 1000.5 13e−DVTZ 1.6254 1.6293 995.4 992.3 13e−DVQZ 1.6214 1.6269 996.7 996.6 GeS VTZ 2.0461 2.0356 2.0416 2.0280 2.012086 552.2 554.1 568 .4 578.0 575.8 VQZ 2.0300 2.0200 2.0356 2.0192 560.5 562.4 572.3 578.2 13e−DVTZ 2.0207 2.0284 577.9 572.3 13e−DVQZ 2.0133 2.0190 577.5 578.2 GeSe VTZ 2.1743 2.1580 2.1713 2.1574 2.134629 384.8 387.4 39 7.5 403.7 408.7 VQZ 2.1531 2.1377 2.1658 2.1506 393.5 444.5 400.2 403.5 13e−DVTZ 2.1479 2.1583 403.2 401.4 13e−DVQZ 2.1370 2.1495 405.0 403.8 (a) all d electrons of Ga, Ge are frozen for the all electron ca lculation. (b) d-shell correlated. 30TABLE IV. Spectroscopic constants for GaH and GeH. All resul ts at the CCSD(T) level. Nonvalence Pseudopotential Basis set re ωe ωexe re ωe ωexe e−correlated ˚A cm−1cm−1 ˚A cm−1cm−1 GaH GeH Experiment 1.660149(2)a1603.9559(20)b28.4227b1.58724c1900.3820c33.5024(28) — all-electron cc-pVDZ 1.6860 1610.0 26.43 1.6083 1896.1 32 .96 — all-electron ditto full CI 1.6867 1607.0 26.55 1.6090 1891 .7 33.24 — all-electron cc-pVTZ 1.6878 1603.2 26.80 1.6039 1897.9 32 .80 — all-electron cc-pVQZ 1.6864 1604.3 27.74 1.6021 1903.1 33 .14 (3d) all-electron cc-pVTZ 1.6573 1691.8 31.15 1.5860 1956. 9 36.24 (3d) all-electron cc-pVQZ 1.6449 1698.5 40.56 1.5764 1965. 0 43.78 — ECP28MWB SDB-cc-pVTZ 1.6969 1530.9 24.27 1.6084 1844.4 30 .67 — + CPP 1.6876 1524.2 24.35 1.6014 1845.8 30.72 — ECP28MWB SDB-cc-pVQZ 1.6820 1546.3 25.09 1.5990 1862.3 31 .30 — + CPP 1.6729 1540.4 25.13 1.5921 1863.5 31.46 (3d) all-electron cc-pDVTZ 1.6582 1658.2 32.91 1.5893 1900 .4 31.65 (3d) all-electron cc-pDVQZ 1.6565 1653.6 36.20 1.5821 1950 .2 42.03 (3d) all-electron cc-pDVTZ+2d 1.6500 1659.5 33.07 (3s,3p,3d) all-electron cc-pDVTZ+2s2p1d 1.6647 1602.5 26 .56 (3d) all-electron cc-pDVTZ+2s2p1d 1.6635 1604.4 25.82 — all-electron cc-pDVTZ+2s2p1d 1.6863 1598.0 25.56 (3d) all-electron cc-pDVTZ+2s2p 1.6637 1604.0 25.69 1.588 9 1906.2 32.30 — all-electron cc-pDVTZ+2s2p 1.6863 1597.6 25.75 1.6020 18 99.3 32.64 (3d) ECP10MWB cc-pDVTZ 1.6550 1649.3 31.47 1.5783 1950.6 37 .24 (3d) ECP10MWB cc-pDVQZ 1.6446 1662.8 37.57 1.5751 1941.7 40 .79 (3d) ECP10MWB cc-pDVTZ+2s2p 1.6613 1588.8 25.09 1.5865 189 3.2 32.12 — ECP10MWB cc-pDVTZ+2s2p 1.6838 1584.4 25.12 1.5993 1887.7 32.40 (3d) ECP10MWB cc-pDVQZ+2s2p 1.6584 1607.4 27.33 (3d) ECP10MWB cc-pDVQZ+3s3p 1.6586 1605.6 26.92 1.5881 191 0.8 32.86 — ECP10MWB cc-pDVQZ+3s3p 1.6829 1592.1 27.38 1.6006 1897.4 32.40 (3s,3p,3d) ECP10MWB cc-pDVQZ+3s3p 1.6602 1601.5 26.00 1.5 898 1907.4 32.82 aM. Molski, J. Mol. Spectrosc. 182, 1 (1997). bF. Ito, T. Nakanago, H. Jones, J. Mol. Spectrosc. 164, 379 (1994). cJ. P. Towle and J. M. Brown, Mol. Phys. 78, 249 (1993). 31TABLE V. Binding energies ( Dein eV) for molecules containing third row atoms. molecule SDB CPP all e−Experiment VTZ VQZ ∞ VTZ VQZ ∞ VTZ VQZ ∞ HH [40] KS [52] AlBr 4.31 4.45 4.55 4.33 4.48 4.57 4.29 4.43 4.52 4.43 4.42 ±0.06 ditto aug-cc 4.35 4.49 4.58 As2 3.42 3.78 4.02 3.48 3.83 4.06 3.48 3.71 3.86 3.96 3.93 ±0.10 AsF3Σ−3.88 4.13 4.30 3.91 4.16 4.32 3.85 4.08 4.22 4.2 4.21 ditto aug-cc 4.12 4.26 4.37 AsH3Σ−2.63 2.72 2.79 2.63 2.73 2.79 2.64 2.71 2.76 2.76 (AsD) 2.80 ±0.03 AsN 4.48 4.85 5.09 4.54 4.90 5.14 4.47 4.72 4.88 — 5.03 ±0.02 AsO2Π 4.58 4.90 5.11 4.63 4.94 5.14 4.56 4.80 4.97 ≤4.98 4.95 ±0.08 AsP 3.90 4.24 4.46 3.94 4.27 4.49 3.90 4.16 4.33 — 4.45 AsS2Π 3.47 3.79 3.99 3.51 3.81 4.02 3.50 3.73 3.88 — 3.90 ±0.07 BBr 4.23 4.32 4.38 4.25 4.34 4.40 4.18 4.28 4.34 ≤4.49a4.07 ditto aug-cc 4.26 4.34 4.40 4.21 4.29 4.35 Br2 1.58 1.80 1.94 1.61 1.82 1.96 1.69 1.84 1.93 1.9707 idem ditto aug-cc 1.76 1.88 1.96 1.75 1.87 1.94 BrCl 1.90 2.08 2.20 1.91 2.09 2.21 1.94 2.10 2.20 2.233 2.223 ±0.003 ditto aug-cc 2.01 2.13 2.21 BrF 2.18 2.40 2.54 2.20 2.41 2.55 2.25 2.44 2.57 2.548 2.87 ±0.12 ditto aug-cc 2.41 2.50 2.56 CSe 5.72 5.92 6.06 5.76 5.97 6.06 5.73 5.89 6.00 5.98 6.08 ±0.06 HBr 3.62 3.69 3.74 3.64 3.71 3.76 3.63 3.70 3.75 3.758 idem ditto aug-cc 3.66 3.71 3.74 3.68 3.73 3.76 3.67 3.72 3.75 NSe2Π 3.23 3.51 3.70 3.27 3.55 3.73 3.29 3.51 3.66 4.0 3.80 ±0.11 Se23Σ− g 2.94 3.19 3.35 2.99 3.23 3.39 3.05 3.23 3.34 3.410 3.417 ±0.004 SeH 3.08 3.16 3.22 3.09 3.18 3.23 3.11 3.19 3.24 — 3.221 ±0.01 SeO3Σ−4.02 4.29 4.47 4.05 4.32 4.49 4.08 4.29 4.42 4.41 4.78 ±0.22 SeS3Σ−3.42 3.65 3.80 3.45 3.68 3.83 3.46 3.65 3.78 3.7 3.81 ±0.07 SiSe 5.13 5.35 5.49 5.16 5.38 5.53 5.14 5.34 5.47 5.64 5.54 ±0.13 ditto aug-cc 5.16 5.36 5.49 5.20 5.40 5.53 5.18 5.36 5.48 (a) Predissociation: Ref. [40] notes a possible potential h ump of up to 0.13 eV in the upper a1Π state. 32TABLE VI. Binding energies ( Dein eV) for molecules containing fourth row atoms. molecule SDB CPP Experiment VTZ VQZ ∞ VTZ VQZ ∞ HH [40] KS [52] AlI 3.49 3.60 3.67 3.52 3.63 3.71 3.77 3.81 ±0.02 ditto aug-cc 3.53 3.64 3.72 GaI 3.18 3.34 3.45 3.26 3.42 3.53 3.47 3.47 ±0.10 GeTe 3.75 4.08 4.30 3.84 4.16 4.37 4.24 4.09 ±0.03 HI 2.90 2.96 3.00 2.93 2.99 3.03 3.0541 idem ditto aug-cc 2.94 2.99 3.01 2.96 3.01 3.04 I2 1.07 1.25 1.37 1.13 1.30 1.42 1.54238 idem ditto aug-cc 1.15 1.36 1.50 IBr 1.37 1.57 1.70 1.42 1.61 1.74 1.817 idem ditto aug-cc 1.54 1.67 1.75 ICl 1.73 1.90 2.02 1.76 1.93 2.04 2.1531 idem ditto aug-cc 1.86 1.98 2.06 IF 2.24 2.48 2.64 2.27 2.51 2.66 2.879 ≤2.78 ditto aug-cc 2.51 2.61 2.68 InBr 3.76 4.03 4.21 3.85 4.13 4.32 3.99 4.27 ±0.22 (b) 3.73 3.85 3.93 (c) 3.75 3.86 3.93 3.77 3.88 3.95 InCl 4.32 4.60 4.79 4.41 4.70 4.89 4.44 4.52 ±0.08 (c) [34] 4.29 4.45 (e) [34] 4.32 4.43 4.46f InF 5.41 5.87 6.17 5.50 5.97 6.28 5.25 5.21 ±0.15 (b) 5.18 5.35 5.46 (c) 5.35 5.42 5.47 (d) 5.26 5.42 5.53 (e) 5.42 5.50 5.55 InH 2.45 2.58 2.66 2.44 2.58 2.66 2.48 idem (b) 2.43 2.47 2.49 (d) 2.46 2.50 2.53 InI 3.06 3.29 3.44 3.17 3.40 3.55 3.43 3.41 ±0.01 (b) 3.07 3.16 3.22 3.11 3.20 3.26 Sb2 2.51 2.88 3.13 2.60 2.97 3.21 2.995a3.07±0.07 SbF3Σ−3.77 4.12 4.36 3.82 4.17 4.40 4.4 4.5 ±0.1 ditto aug-cc 4.11 4.26 4.35 SbH3Σ−2.43 2.52 2.59 2.44 2.54 2.60 — — SbP 3.20 3.51 3.71 3.26 3.57 3.78 3.68 idem SnO 4.86 5.28 5.54 4.96 5.38 5.65 5.49 5.48 ±0.13 SnS 4.31 4.61 4.81 4.39 4.69 4.89 4.77 4.78 ±0.03 SnSe 3.83 4.12 4.31 3.92 4.22 4.41 4.20 4.13 ±0.06 ditto aug-cc 3.97 4.15 4.27 4.06 4.24 4.35 33SnTe 3.29 3.56 3.74 3.39 3.66 3.84 3.69 idem ditto aug-cc 3.40 3.58 3.70 3.50 3.68 3.79 (a) Sb 2(De): H. Sontag and R. Weber, J. Mol. Spectrosc. 91, 72 (1982). (b) using Bauschlicher [34] cc-pV nZ basis sets on In in conjunction with cc-pV nZ and SDB- cc-pVnZ basis sets on other element. (c) ditto, but using ‘augmented’ basis sets on other element . (d) as (b), but using all-electron basis set on In. (e) as (c), but using all-electron basis set on In. (f) This work, correlating (2s2p) electrons on Cl and using t he MTavqz core-correlation basis set [54] on Cl. 34TABLE VII. Binding energies ( Dein eV) for polar molecules of Ga and Ge molecule basis SDB CPP SDB- all e−alle−with Experiment cc-pDV nZ cc-pV nZ cc-pV nZ cc-pDV nZ HH [40] KS [52] (3d) corr. no no yes no yes yes GaBr VTZ 3.90 3.97 3.86 3.97 4.04 4.02 4.31 4.58 ±0.18 VQZ 4.09 4.16 4.15 4.08 4.15 4.12 ∞ 4.22 4.29 4.35 4.16 4.23 4.19 GaCl VTZ 4.46 4.52 4.60 4.53 4.63 4.60 4.92 4.96 ±0.13 VQZ 4.66 4.72 4.73 4.65 4.75 4.72 ∞ 4.79 4.84 4.81 4.73 4.83 4.80 GaF VTZ 5.67 5.74 5.83 5.74 5.87 5.85 5.98 5.95 ±0.15 VQZ 5.94 6.01 6.00 5.91 6.03 6.01 ∞ 6.12 6.18 6.12 6.02 6.14 6.11 AVTZ 5.95 6.01 AVQZ 6.08 6.09 ∞ 6.17 6.15 DVTZ+2s2p 5.80 DVQZ+3s3p 5.97 ∞ 6.09 GaH VTZ 2.69 2.69 2.81 2.77 2.86 2.82 <2.84 ≤2.80 VQZ 2.77 2.77 2.85 2.80 2.89 2.84 ∞ 2.82 2.81 2.88 2.82 2.90 2.85 GeF2Π VTZ 4.78 4.83 4.99 4.83 4.99 4.98 5.0 5.0 ±0.2 VQZ 5.07 5.12 5.17 5.02 5.20 5.18 ∞ 5.27 5.31 5.31 5.14 5.33 5.28 AVTZ 5.06 AVQZ 5.20 ∞ 5.30 GeH2Π VTZ 2.57 2.58 2.69 2.63 2.70 2.66 <3.3 ≤3.3 VQZ 2.65 2.66 2.72 2.68 2.76 2.72 ∞ 2.71 2.71 2.75 2.72 2.80 2.76 GeO VTZ 6.19 6.27 6.57 6.41 6.54 6.53 6.78 6.80 ±0.13 VQZ 6.57 6.65 6.76 6.60 6.77 6.74 ∞ 6.82 6.89 6.88 6.73 6.93 6.87 GeS VTZ 5.10 5.16 5.36 5.25 5.35 5.31 5.67 5.50 ±0.03 VQZ 5.40 5.57 5.52 5.44 5.55 5.51 ∞ 5.59 5.83 5.63 5.56 5.68 5.63 GeSe VTZ 4.45 4.53 4.71 4.64 4.73 4.70 4.98 ±0.10 4.99 ±0.02 VQZ 4.79 4.86 4.89 4.81 4.91 4.87 ∞ 5.01 5.07 5.01 4.92 5.03 4.98 35TABLE VIII. State-averaged optimum fandgexponents for the transition metals, to be used in conjunction with Stuttgart-Dresden ECPs and [6s5p3 d] contracted valence basis sets. The cc-pVTZ and SDB-cc-pVTZ basis sets are recommended for the o ther elements (s)1(d)n−1state4F5F6D7S6D5F4F3D2S N/A (s)2(d)n−2state2D3F4F5D6S5D4F3F2D1S Sc Ti V Cr Mn Fe Co Ni Cu Zn ζf1 0.180 0.285 0.425 0.640 0.795 0.871 1.019 1.182 1.315 1.498 ζf2 0.764 1.264 1.788 2.555 3.118 3.516 4.076 4.685 5.208 5.871 ζg 0.347 0.636 1.048 1.712 1.964 2.269 2.711 3.212 3.665 4.365 Y Zr Nb Mo Tc Ru Rh Pd Ag Cd ζf1 0.144 0.236 0.261 0.338 0.398 0.478 0.567 0.621 0.732 0.834 ζf2 0.546 0.883 0.970 1.223 1.430 1.666 1.989 2.203 2.537 2.853 ζg 0.249 0.547 0.536 0.744 0.918 1.057 1.236 1.385 1.587 1.795 La Hf Ta W Re Os Ir Pt Au Hg ζf1 0.120 0.163 0.210 0.256 0.327 0.347 0.395 0.443 0.498 0.545 ζf2 0.456 0.557 0.697 0.825 0.955 1.067 1.189 1.323 1.461 1.580 ζg 0.209 0.352 0.472 0.627 0.636 0.860 0.982 1.100 1.218 1.384 Exponents were averaged over ( s)1(d)n−1and (s)2(d)n−2states, except for Pd where in ad- dition the ( s)0(d)10ground state was used, and Zn/Cd/Hg for which only the ( s)2(d)n−2is involved. 36
arXiv:physics/0011031v1 [physics.flu-dyn] 15 Nov 2000Waves attractors in rotating ﬂuids: a paradigm for ill-pose d Cauchy problems M. Rieutord1,2, B. Georgeot3and L. Valdettaro4 1Observatoire Midi-Pyr´ en´ ees, 14 avenue E. Belin, F-31400 Toulouse, France 2Institut Universitaire de France 3Laboratoire de Physique Quantique, UMR 5626 du CNRS, Univer sit´ e Paul Sabatier, F-31062 Toulouse Cedex 4, France 4Dip. di Matematica, Politecnico di Milano, Piazza L. da Vinc i, 32, 20133 Milano, Italy (February 2, 2008) In the limit of low viscosity, we show that the amplitude of th e modes of oscillation of a rotating ﬂuid, namely inertial modes, concentrate along an attractor form ed by a periodic orbit of characteristics of the underlying hyperbolic Poincar´ e equation. The dynamic s of characteristics is used to elaborate a scenario for the asymptotic behaviour of the eigenmodes an d eigenspectrum in the physically relevant r´ egime of very low viscosities which are out of rea ch numerically. This problem oﬀers a canonical ill-posed Cauchy problem which has applications in other ﬁelds. PACS numbers: 47.32.-y, 05.45.-a, 02.60.Lj, 04.20.Gz Rotating ﬂuids encompass all ﬂuids whose motions are dominated by the Coriolis force. These ﬂows play an important role in astrophysics or geophysics where the large size of the bodies makes the Coriolis force a promi- nent force. Some engineering problems like the stability of artiﬁcial satellites also require the study of rotating ﬂuids because of their liquid-ﬁlled tanks [1]. This latter problem is related to the existence of waves speciﬁc to rotating ﬂuids, namely inertial waves, which easily res- onate. These waves play also an important part in the oscillation properties of large bodies like the atmosphere , the oceans, the liquid core of the Earth [2], rapidly rotat- ing stars [3] or neutron stars [4]. As such, they have been considered since the work of Poincar´ e on the stability of ﬁgures of equilibrium of rotating masses [5]. Pressure perturbations of inertial modes for inviscid ﬂuids obey the Poincar´ e equation (PE) (christened by Cartan [6]) which reads ∆ P−(2Ω/ω)−2∂2P/∂z2= 0 where Ω /vector ezis the angular velocity of the ﬂuid and ωis the frequency of the oscillation. Since ω <2Ω [7], the PE is hyper- bolic (energy propagates along characteristics) and since its solutions must meet boundary conditions, the prob- lem is ill-posed mathematically. Although some smooth solutions exist (for instance for a ﬂuid contained in a full sphere or a cylinder), one should expect singular so- lutions in the general case. These latter solutions have been made explicit only recently thanks to numerical sim- ulations which include viscosity to regularize the singu- larities and let this parameter be very small as in real systems [8,9]. In this letter we wish to present a scenario, based on analytical and numerical results, for the asymptotic be- haviour of inertial modes at small viscosities. We use the case of a spherical shell as a container, which is relevant for astrophysical or geophysical problems, but it will be clear that this case is general. We will only sketch the main results, more details can be found in [9]. While the ﬂuid mechanical problem is of much interest by it- self, it opens new perspectives in the theory of PartialDiﬀerential Equations (PDE) and also oﬀers a toy model for some (very involved) problems of General Relativity which we shall present brieﬂy. The model we use is a spherical shell whose inner radius isηRand outer radius R(η <1). The ﬂuid is assumed incompressible with a kinematic viscosity ν. We write the linearized equations of motion for small amplitude perturbations for the velocity /vector uin a frame corotating with the ﬂuid; momentum and mass conservation imply: ∂/vector u ∂t+/vector ez×/vector u=−/vector∇p+E∆/vector u, /vector∇ ·/vector u= 0 (1) when dimensionless variables are used; (2Ω)−1is the time scale and E=ν/2ΩR2the Ekman number. When Eis set to zero and /vector uis eliminated, one obtains the Poincar´ e equation. In nature E≪1 and one is tempted to use boundary layer theory and singular perturbations to solve (1). However, this is feasible only when regular so- lutions exist for E= 0; this is the case when the container is a full sphere [7] but not when the container is a spheri- cal shell. Indeed, numerical solutions of the eigenvalue problem issued from (1), where solutions of the form /vector u(/vector r)eλtare searched for (with −1≤ω=Im(λ)≤1), yield eigenmodes of the kind shown in Fig. 1. In this ﬁgure we see that the amplitude of the mode is all con- centrated along a periodic orbit of characteristics of the PE; we found this property to be quite general, after ex- tensive numerical exploration of least-damped modes of (1) [8,9], and will now explain its origin and consequences on the asymptotic spectrum of inertial modes. For this purpose we will use axisymmetric modes since the az- imuthal dependence of solutions can always be separated out because of the axial symmetry of the problem. For understanding the concentration of kinetic energy along a periodic orbit of characteristics, it is necessary to consider in some details the dynamics of these lines. Characteristics of PE are, in a meridional plane, straight lines making the angle arcsin ωwith the rotation axis. A numerical calculation of their trajectories shows that 1they generally converge towards a periodic orbit which we call, after [10], an attractor . The periodic orbit of Fig. 1 is one example of such an attractor. FIG. 1. Kinetic energy in a meridional section of a spherical shell of an inertial mode in a viscous ﬂuid. For this numerica l solution, E= 10−8, 570 spherical harmonics and 250 Cheby- shev polynomials have been used (the numerical method is described in [8]). The mode is axisymmetric and symmetric with respect to equator. η= 0.35 like in the Earth’s core. ωis the frequency of this mode and τits damping rate. Stress-free boundary conditions are used. The convergence of character - istics towards the attractor is also shown (white lines). The Lyapunov exponent (LE) of a trajectory, deﬁned by Λ = lim N→∞1 N/summationtextN n=1ln/vextendsingle/vextendsingle/vextendsingledφn+1 dφn/vextendsingle/vextendsingle/vextendsingle(φnis the latitude of the nthreﬂection point), describes how fast characteris- tics are attracted or repelled. Its computation as a func- tion of frequency shows that attractors (Λ <0) are ubiq- uitous in frequency space (see Fig. 2). Their existence shows that the dynamical system described by the char- acteristics is not hamiltonian; the “dissipation” is purel y geometrical and is due to the fact that, unlike billiards, the reﬂection on boundaries is not specular but conserves the angle with the rotation axis. In fact, the dynamics of rays is a one-to-one one-dimensional map (from the outer boundary to itself), piecewise smooth, but with a ﬁnite number (twelve) of discontinuities. This kind of map has not been studied in the literature of dynami- cal systems, perhaps because it does not produce chaos because of its invertibility. Iterations of such a map gen- erate ﬁxed points which either correspond to attractors or to some neutral periodic orbits. Indeed, if η= 0 (i.e. the sphere is full), all orbits such that ω= sin( pπ/2q) with ( p, q)∈I N2, are neutral and periodic while those such that ω= sin( rπ),rbeing irrational, are neutral ergodic (quasiperiodic). When ηis non-zero only a ﬁ- nite number of such neutral periodic orbits subsist; for instance, if η= 0.35 which is the aspect ratio of the Earth’s liquid core, q= 1,2,3,4 are the only possibili- ties. Interestingly, we face here a situation which is justthe opposite of the one described by the KAM theorem in Hamiltonian systems: when the full sphere is perturbed by the introduction of an inner sphere, all ergodic orbits are instantaneously destroyed while the longer periodic orbits survive the smaller the denominator qis. FIG. 2. LE Λ( ω) of the orbits as a function of ωfor η= 0.35. Inset: blow-up showing the LE of two coexisting attractors (full and dashed thick lines). Apart from these isolated frequencies which become rarer and rarer as ηincreases, generic trajectories are in the basin of attraction of attractors. We were able to show [9] that the number of attractors at a given fre- quency is ﬁnite. The inset of Fig. 2 shows the typical case where an attractor exists in a frequency band [ ω1, ω2] with Λ( ω1) = 0, Λ( ω2) =−∞andω2−ω1∼1/N2where Nis the length of the attractor deﬁned as its number of reﬂection points. Near ω1, Λ∼√ω−ω1and near ω2, Λ(ω)∼1 NlnN(ω−ω2). The latter implies that long at- tractors have small LE in a large fraction of [ ω1, ω2] (all these results are shown in [9]). The existence of attractors for characteristics implies that solutions of the inviscid problem ( i.e.of PE) are singular. This property can be made explicit in the sim- pliﬁed case of a 2D problem. Indeed, in this case the PE may be written ∂2P/∂u +∂u−= 0 using characteris- tics coordinates; solutions may be constructed explicitly from an arbitrary function but, as shown in [11], regular eigenmodes exist only when neutral periodic orbits exist and eigenvalues are inﬁnitely degenerate. When attrac- tors are present, the scale of variations of the pressure vanishes on the attractors while its amplitude remains constant. As velocity depends on the pressure gradient, it diverges on the attractor; this divergence is like the inverse of the distance to the attractor which makes the velocity ﬁeld not square integrable. This result seems to be valid also in 3D [9]. We therefore understand why solutions of (1) look like Fig. 1: the inviscid part of the operator focuses energy of 2the modes thanks to the action of the mapping made by characteristics while viscosity opposes to this action via diﬀusion. The resulting picture of Fig. 1 therefore comes from a balance between inviscid terms and viscous ones; let us make this more quantitative. FIG. 3. Asymptotic behavior of an eigenvalue. The dashed line is ω−ωias a function of E, while the dotted line is for the damping rate τ. The solid line represents the ‘theoretical’ law E1/2.ωi= 0.403112887 is a root of Λ( ω) = 0 when η= 0.35. For this purpose we ﬁrst observe that the patterns drawn by the kinetic energy of the mode in Fig. 1 is in fact a shear layer whose width scale with Eσandσ≃1/4. Such a scaling is observed numerically and seems generic [8,9]; it implies that the damping rate of such modes scales like E1/2as clearly shown in Fig. 3. Now we may consider a wave packet travelling around an attractor in a slightly viscous ﬂuid. The above mentioned balance, when applied to both the width and the amplitude of the packet, leads to a relation between the LE and the Ekman number such as Λ ∼E1−3σwithσ <1/3 for an eigenmode of the viscous problem. We see that the con- straint σ <1/3 is met by actual shear layers. It therefore turns out that frequencies of eigenmodes of the viscous problem are such that Λ →0 when E→0 which means that they will gather around the roots of the equation Λ(ω) = 0. The above result shows the importance of the scal- ing veriﬁed by shear layers. A boundary layer analysis reveals that these shear layers are in fact nested layers which consist of an inner σ= 1/3-layer surrounded by a thicker layer. The inner 1/3-layer can be fully explicited. Using coordinates along the shear layers ( x) and perpen- dicular to it ( y), we ﬁnd that the ϕ-component of the velocity veriﬁes∂3uϕ ∂Y3=−i∂uϕ ∂q, with Y=y/E1/3and q=x/√ 1−ω2which is also the equation veriﬁed by the stream function in a steady shear layer of a rotat- ing ﬂuid [12]. Solutions which vanish in Y=±∞are self-similar and of the form uϕ=qαHα/parenleftbig Y/q1/3/parenrightbig with Hα(t) =/integraltext∞ 0e−ipte−p3p−3α−1dp. Besides, α=−1 3is the only admissible value to ensure a coherent evolution of the width and amplitudes after reﬂection on a boundary. We are now in position to propose a scenario for the asymptotic behaviour of inertial modes when the vis- cosity vanishes. Eigenfunctions reduce to nested shearlayers concentrated along attractors while the associ- ated eigenvalues converge toward the frequency ωisuch that Λ( ωi) = 0 for the associated attractor. Further- more, we can constrain this convergence of eigenfre- quencies; indeed, since Λ ∼√ω−ωi, one ﬁnds that ω=ωi+aE2−6σ+···andτ=Re(λ) =−bE1−2σwhen E→0; Fig. 3 shows that this law agrees well with the numerical results, in the case shown, with σ= 1/4. In addition, we noticed earlier that for a ﬁnite number ofωsuch that ω= sin( pπ/2q) all orbits of characteristics are periodic; this implies that in the vicinity of these fre- quencies very long attractors with very small average LE accumulate as shown by Fig. 4; therefore, these frequen- cies will be accumulation points of the asymptotic spec- trum. Moreover, around these frequencies eigenmodes are weakly damped. On the contrary, modes whose fre- quency is in the frequency band of short attractors (like the one of Fig. 1) are more strongly damped. It there- fore turns out that the LE curve in Fig. 2 will strongly constrain the distribution of least-damped modes in the complex plane at ﬁnite viscosities: such modes will avoid the large frequency bands of short-period attractors and concentrate around frequencies where Λ( ω) = 0 espe- cially those with ω= sin( pπ/2q). This general evolution of the spectrum is well illus- trated in Fig. 5. Here, the least-damped eigenvalues have been computed for E= 10−8. We clearly see frequency bands of attractors avoided by weakly damped eigenval- ues but see the gathering of these eigenvalues around sin(π/4) and, but less conspicuously, around sin( π/6). 0.662 0.664 0.666 0.668 0.67 4λ/π00.0050.011/N FIG. 4. Inverse of the length Nof attractors with N <100 forη= 0.35, near the accumulation point π/6; each point corresponds to an attractor with Λ = 0 and therefore to a point in the asymptotic spectrum. Note the lengthening of the attractors as π/6 is approached. Here η= 0.35. To complete the picture, we need now mentioning that a few regular modes survive among all these singulari- ties; such modes are purely toroidal modes or r-modes [13] which are non-axisymmetric. They avoid the con- straint of characteristics for their velocity ﬁeld has no radial component; this property makes their characteris- tics independent of frequency (they are circles and verti- 3cal straight lines) and authorizes smooth solution at zero viscosity. The associated eigenvalues ω= 1/(m+1), m∈ I N∗seem to be the only eigenvalues of the Poincar´ e op- erator in a spherical shell. FIG. 5. Distribution of the eigenvalues associated with least-damped axisymmetric modes in the complex plane. Hatched frequency bands denoted bands occupied by simple attractors; the dotted line is for sin( π/6). The Ekman number is 10−8andη= 0.35. We used a resolution of 700 spherical harmonics and 270 radial grid points. Ending this letter, it is worth emphasizing the role of the geometrical approach allowed by the dynamics of characteristics, for describing the asymptotic propertie s of inertial modes; in the domain of very low Ekman num- bers (10−10→10−20), typical of astrophysical or geo- physical ﬂuids, these modes are out of reach numerically. The foregoing presentation shows that inertial modes display a very rich dynamical behavior which comes from the ill-posedness of the underlying inviscid problem. Here we discussed the case of the spherical shell, but our re- sults are general and can be extended to any container; this is important since natural containers are usually not perfect geometrical objects. Hence, fortunately, a curve like Fig. 2 is structurally stable (see our discussion rela- tive to the core of the Earth in [2]). We note that the relevance of attractors has also been shown experimentally in stratiﬁed ﬂuids [14]. Some con- ﬁgurations of conducting ﬂuids bathed by a magnetic ﬁeld, obeying the PE, will also display attractors [15]. These properties are in fact very general and extend to mixed-type PDE as illustrated by the case of gravito- inertial modes [16]. We think that similar results should hold for systems which are solutions of PDE of hyper- bolic or mixed type meeting boundary conditions. As an example, our results may have applications in GeneralRelativity and the problem of “closed timelike curves” (CTC), that is the problem of the existence of physical systems which permit travels backward in time. Such systems like wormholes have been studied by various au- thors [17]; they set many problems among which that of causality. Such a problem is also at the origin of the ill- posedness of the Poincar´ e problem and we showed that it leads to many kinds of singularities. We therefore see that inertial oscillations of a ﬂuid in- side a container oﬀers a paradigm which may guide our intuition for problems in other ﬁelds of physics which are also ill-posed Cauchy problems. We would like to thank Boris Dintrans and Leo Maas for very helpful discussions. Part of the calculations have been carried out on the Cray C98 of IDRIS at Orsay and on the CalMip machine of CICT in Toulouse which are gratefully acknowledged. [1] R. Manasseh, J. Fluid Mech. 243, 261 (1992). [2] M. Rieutord, Phys. Earth Plan. Int. 117, 63 (2000). [3] B. Dintrans and M. Rieutord, Astron. & Astrophys. 354, 86 (2000). [4] N. Andersson, K. D. Kokkotas, and N. Stergioulas, As- trophys. J. 516, 307 (1999). [5] H. Poincar´ e, Acta Mathematica 7, 259 (1885). [6] E. Cartan, Bull. Sci. Math. 46, 317 (1922). [7] H. P. Greenspan, The theory of rotating ﬂuids (Cam- bridge University Press, 1969). [8] M. Rieutord and L. Valdettaro, J. Fluid Mech. 341, 77 (1997). [9] M. Rieutord, B. Georgeot, and L. Valdettaro, submitted to J. Fluid Mech., physics/0007007 (2000). [10] L. Maas and F.-P. Lam, J. Fluid Mech. 300, 1 (1995). [11] D. Schaeﬀer, Studies in Applied Math. 54, 269 (1975). [12] D. Moore and P. Saﬀman, Phil. Trans. R. Soc. Lond. 264, 597 (1969). [13] M. Rieutord, submitted to Astrophys. J. , astro- ph/0003171 (2000). [14] L. Maas, D. Benielli, J. Sommeria, and F.-P. Lam, Nature 388, 557 (1997). [15] W. Malkus, J. Fluid Mech. 28, 793 (1967). [16] B. Dintrans, M. Rieutord, and L. Valdettaro, J. Fluid Mech. 398, 271 (1999). [17] J. Friedman, M. Morris, I. Novikov, F. Echeverria, G. Klinkhammer, K. Thorne and U. Yurtsever, Phys. Rev. D 42, 1915 (1990); S. Hawking, ibid.52, 5681 (1995); A. Carlini and I. Novikov, Int. J. Mod. Phys. D5, 445 (1996). 4
arXiv:physics/0011032v1 [physics.plasm-ph] 15 Nov 2000Infrared Fluorescence of Xe 2Molecules in beam–excited Xe Gas at high Pressure A. F. Borghesania,b, G. Bressic, G. Carugnod, E. Contid, and D. Iannuzzic,e,1 aIstituto Nazionale per la Fisica della Materia via F.Marzolo, 8, I-35131 Padua, Italy bDepartment of Physics, University of Padua via F.Marzolo, 8, I-35131 Padua, Italy cIstituto Nazionale di Fisica Nucleare, sez. di Pavia via A. Bassi, 6, I–27100 Pavia, Italy dIstituto Nazionale di Fisica Nucleare, sez. di Padova Via F. Marzolo, 8, I–35131 Padua, Italy eDipartimento di Fisica Nucleare e Teorica, University of Pa via via A. Bassi, 6, I–27100 Pavia, Italy Abstract We report experimental results of proton– and electron–bea m–induced near–infra- red (NIR) ﬂuorescence in high–pressure Xe gas at room temper ature. The inves- tigated wavelength band spans the range 0 .7≤λ≤1.8µm.In the previously unexplored range for λ >1.05µm we have detected a broad continuum NIR ﬂuores- cence at λ≈1.3µm that shifts towards longer wavelengths as pressure is incr eased up to 1 .5MPa .We believe that this continuum is produced in a way similar to the VUV continua of noble gas excimers and that the pressure–dep endent shift can be explained by the interaction of the outer electron of the exc imer with the gas. Key words: Xe, excimers, infrared ﬂuorescence. PACS: 33.20.Ea, 34.50.Gb 1Corresponding author. E-mail:davide.iannuzzi@pv.infn. it, Fax:+390382423241 Preprint submitted to Elsevier Preprint 2 February 20081 Introduction Excited states of rare gas atoms are produced easily by means of several techniques, including electric discharges, irradiation w ith ionizing particles, or resonance lines [1,2]. Experimental studies of deexcitati on processes allowed to shed light on the potentiality of rare gases as sensitive m edia in ionizing particle detection [3] and as media for high energy electron ic transition lasers [4]. The ability to eﬃciently convert electron kinetic ener gy to electronic exci- tations and to rapidly convey the excitation energy to lower –lying atomic and excimer levels leads excited dense rare gases to emit a consi derable fraction of the released energy in a narrow band in the vacuum–ultraviol et (VUV) range [5]. Beside the emission stemming from atomic transitions, a gre at deal of work has been devoted to the VUV continuum radiation related to transitions between excited states and the repulsive ground state of neu tral diatomic rare– gas molecules. Several studies of either particle–beam– or laser–induced VUV ﬂuorescence, including time–resolved spectroscopy, have contributed to clarify the kinetics of collisional deactivation of excited atomic levels leading to the formation of rare–gas excimers and to the population invers ion required for VUV lasing [6–19]. VUV ﬂuorescence in particle–excited emission spectra of de nse noble gases is also exploited for direct and proportional scintillatio n for high–energy par- ticle detection [3]. The emission in the VUV range is produce d by several reactions between ionized, excited, and neutral gas atoms a nd free electrons leading to excimers which decay radiating the VUV continua. At moderately high pressures ( P >100 Pa) this kind of emission is believed to dominate over all other types of radiative decay including atomic emissio n [14]. Much less attention has been devoted to possible infrared (I R) or near- infrared (NIR) emission, which might be related to transiti ons between excited states of the rare–gas dimers, because eﬃcient lasers in the se wavelength bands can be accomplished by much easier means. However, the curre nt improve- ments in photomultiplier and semiconductor detector techn ology and the fact that the NIR emission can be treated with standard optical co mponents make the NIR emission worth to be accurately investigated as a fur ther promis- ing tool to complement the VUV scintillation for the detecti on of ionizing radiation and to possibly enhance the energy resolution of d etectors [20,21]. Among rare gases, Xe is one of the most used species in detecto rs, both in the gaseous and in the liquid state. In spite of this, littl e spectroscopic information is available for NIR emission involving either higher excited ex- cimer levels or excited atomic levels, and only VUV light is u sed for detection 2purposes. Two issues have been mainly investigated in Xe. On one hand, t here are studies on the pressure dependence (up to ≈1 MPa) of atomic transition lines in the 828 −1084nmband [22,12,13]. This band has been investigated in order to study the deactivations reactions of the state–sel ectivelly photoexcited 5p5[2P3/2] 6plevels of Xe, which involve radiation and collisions with gr ound state Xe atoms. These states are of particular importance fo r understanding the dominant energy pathways in excimer lasers. In fact, the y are the primary products of dissociative recombination of molecular ions a nd decay towards states belonging the 6 smanifold [11]. Further collisions of excited atoms in the 6smanifold with ground state (1S0) Xe atoms lead to the formation of the 0+ u,(1u,0− u) (in the scheme of Hund’s coupling case c) excimer states (or1,3Σu in the notation where spin–orbit coupling is neglected) [23 ]. The decay of these states towards the repulsive molecular ground state 0+ g(1Σ+ g) gives origin to the ﬁrst and second VUV continua. On the other hand, intense transient absorption bands induc ed in Xe by short electron beam pulses have been detected in the region 1 .0−1.1µm [2]. These bands are broader than atomic absorption lines ap pearing in the same region and have been thus attributed to bound–bound tra nsitions be- tween diﬀerent excimer levels. Namely, one of the observed b ands (called ﬁrst absorption ) is believed to correspond to the vibrational structure rel ated to the transitions between the excimer states 1 u,0− u(3Σu) of the A6sconﬁgura- tion (related to the excited 5 p56satomic state) and the 2 g,1g,0+ g(3Πg) of the A6pπconﬁguration (related to the atomic 5 p56pstate) [24]. This suggests that radiative decay takes place prior to the vibrational relaxa tion of the excited molecule. In contrast to the other noble gases, Xe does not show, in the e xplored range [2], a second absorption band at longer wavelengths. This may be asso- ciated with both a bound-free transition between the bound 0+ u(1Σ+ u) excimer state of the A6sconﬁguration and the (possibly) dissociative 0+ g(1Σ+ g) state of the A7pσconﬁguration, and with a bound–bound transition between th e 0+ u(1Σ+ u) and the 1 g(1Πg) excimer states. This last absorption band should also show vibrational structure [2]. It is therefore worth investigating the possibility of NIR e mission from particle–excited Xe gas at high pressure in order to ascerta in the feasibility of new scintillation detectors based on a simpler optics tha n VUV detectors. But it is also interesting to investigate possible alternat ive routes of the neu- tralization of the rare–gas dimer ions rapidly formed in thr ee-body collisions to excimer levels diﬀerent from the lowest ones in a situatio n where the large collisional frequencies will establish thermal equilibri um among the rotational and vibrational degrees of freedom on a time scale much short er than the 3relaxation of the electronic states [25]. We have thus carried out measurements of NIR ﬂuorescence in X e gas at room temperature in the spectral region 700 −1800 nm at pressures up to ≈1.5 MPa ,corresponding to a gas density N≈5×1026m−3.The ﬂuorescence has been induced by irradiating the gas sample with pulsed be ams of either ≈5 MeV protons or ≈70 keV electrons. We have detected an intense and broad band centered about 1300 nm ,which shows unexpected properties as a function of the gas density. Here we report the ﬁrst results o btained. 2 Experimental Details The experimental technique is based on the analysis of the NI R spectrum emitted by a gas sample irradiated with an ionizing particle beam. In our ex- periment, we have measured the response of a gaseous Xe sampl e excited by either an electron– or proton beam. In the ﬁrst case, electro ns are produced by a home-made ≈70 keV electron gun described elsewhere [27]. The elec- tron bunches have a duration of ≃35 ns and contains approximately 0 .1−1 nC charge. In the proton case, a 5 MeV, 1 nA continuous beam is e xtracted by a van de Graaf accelerator (at INFN-LNL laboratories) and chopped into bunches 50 through 400 µs long. The chopper frequency is ≈100 Hz. A sim- pliﬁed schematics of the experimental setup is shown in Fig. 1. The Xe sample Particle Beam Xe gas Entrance windowLight output windowSpectrometer Gas Chamber QA SAVbiasLarge area PD QA SAVbias Small area PD CPU+PAD82 BoardTC20 Board Fig. 1. Schematics of the experimental set–up. QA and SA are t he charge and shaping ampliﬁers, respectively. PD means InGaAs photodio de. is kept at room temperature inside a cylindrical stainless- steel chamber. The chamber is previously evacuated down to about 10−5mbar and then ﬁlled with 4gas. The nominal impurity content (mainly O 2) is≈1 part per million. The ﬁlling pressure is measured by a pressure transducer (Super TJE, Sensotec). The density is calculated by means of a standard equation of s tate [28]. The particle beam enters the chamber through a suitable wind ow (20 µm thick Fe window for protons, 8 µm thick Kapton window for electrons). The emitted light exits the cell through an opposite quartz or sa pphire window. The emitted light spectrum is recorded and analyzed by a Four ier trans- form infrared spectrometer (Equinox55, Bruker Optics). Th e light exiting the interferometer is detected by a InGaAs photodiode with a sen sitive area of 75 mm2(C30723G, EG&G) and quantum eﬃciency >60 % in the 950 −1600 nm range. The scheme of the electronic read-out of the photod iode is reported in Fig. 1. A similar but smaller InGaAs photodiode (GAP3000, Germanium Power Device) collects a fraction of the NIR light at the entr ance of the spec- trometer, just in front of the interferometer, for normaliz ation purposes. Both photodiodes are kept at room temperature. The electronic board controlling the movable mirror of the i nterferome- ter (TC20, Bruker Optics) and the read-out circuits of both p hotodiodes are connected to a personal computer equipped with an high-spee d 16 bit A/D converter (PAD82, Bruker Optics). Since the particle bunch es are too fast to take a complete interferogram in one single shot, we use the s o-called step-scan technique. The mirror is moved, step by step, in the interval between two con- secutive bunches. The signals of the two InGaAs detectors ar e digitized and stored by the computer. The ﬁnal interferogram is obtained b y sorting the large area photodiode data as a function of the mirror positi on. The signal of the smaller photodiode is used to weight the interferogram w ith the integrated light intensity in order to get rid of ﬂuctuations of the part icle beam intensity during the experimental run. Moreover, several acquisitio ns are stored for each mirror position in order to improve the signal-to-noise rat io. The acquisition range is usually set between 5000 and 15000 c m−1. The resolution ranges between 50 and 100 cm−1. The interferogram-to-spectrum conversion is performed by the OPUS 3.03 system software (Br uker Optics). The spectrum is then weighted by the quantum eﬃciency of the d etector. The acquisition system has been calibrated by means of an inf rared laser- diode (PGAS1S03/S, EG&G). An Al 2O3(Ti) sample has been irradiated in this experimental setup and its well-known laser emission s pectrum has been obtained [29]. We have to ﬁnally note that the signal–to–noise ratio is not v ery high for several reasons, mainly because not many photons are emitte d for each particle bunch and the statistics is therefore quite small. Moreover , the background noise level of the InGaAs detectors is pretty large since the y cannot be cooled 5below room temperature. 3 Experimental Results Time–integrated emission spectra were obtained from 0 .1 up to ≈1.5 MPa at room temperature. The corresponding gas density range is approximately (0.3< N < 5)×1026m−3.In ﬁgure 2 we show a typical NIR emission spec- trum from high–pressure proton–beam excited Xe gas at P= 0.35 MPa. The spectra obtained with an electron–beam are similar. The mos t important fea- 0.00.20.40.60.81.0 70090011001300150017001900I (arb. un.) λ (nm)atomic 6p-6s decay lines 3Σu 3Πg excimers transitions Fig. 2. Time integrated NIR emission spectrum of 5 MeV proton –excited Xe at P= 0.35 MPa at room temperature and density N≈0.95×1026m−3.The boxes frame the regions where atomic 6 p−6sdecay lines and the excimer3Σu→3Πg absorption bands have been previously observed [13,2]. The broad peak centered at λ≈1300 nm is the new spectroscopic feature observed. ture of this spectrum is the central, largely unstructured, continuum centered about λ≈1300 nm, which was not revealed before. In the ﬁgure we have framed in boxes the regions where atomic 6 p−6sdecay lines [13] and the (possible) excimer3Σu→3Πgabsorption bands [2] were previously observed. It cannot be ruled out the possibility that some lines might b e obscured under such a broad continuum. A wavelength λ= 1300 nm corresponds to an energy diﬀerence ≈0.95 eV, i.e., of the right order of magnitude for electronic transit ions in diatomic rare– gas molecules [23]. This fact and that the emission spectrum in this range 6is a continuum suggest that the broad NIR peak might be associ ated with a transition from a bound excimer level (possibly endowed wi th vibrational structure) to a dissociative state of lower energy diﬀerent from the repulsive Xe2ground state. This hypothesis would agree also with the obse rvation that there is no emission at wavelengths shorter that the ﬁrst and second VUV continua [1]. Moreover, a semiquantitative analysis of the potential energy curves of Xe suggests that such processes could result in emi ssion in the 2000 − 3000 nm region [24]. Upon increasing the gas pressure, the NIR continuum shifts t o longer wavelengths. In ﬁgure 3 the spectra recorded at four diﬀeren t pressures are shown as a function of the inverse wavelength λ−1. 00.20.40.60.81 567891011120.32 1.65 3.17 4.31I (arb. un.) λ-1 (103 cm-1) Fig. 3. Pressure dependent red–shift of the NIR continuum. T he four spec- tra have been recorded at P= 0.12,0.62,1.19,and 1.62 MPa ,corresponding to N= 0.32,1.65,3.17and 4 .31×1026m−3,as reported in the inset. In order to give an estimate of the density–dependent red–sh ift of the spectra, we plot in ﬁgure 4 the position of the emission maxim um determined by ﬁtting a Lorentzian curve to the the observed spectra. The error bars in the ﬁgure are the statistical uncertainties of the ﬁt. The posit ion of the maximum shifts linearly with density towards the red wing. In any cas e, the maximum shift observed amounts to ≈10 %,well in excess of the experimental accuracy. A similar shift has been observed in the same density range fo r the second VUV continuum in Xe [1]. However, in that case the maximum rel ative shift of the emission peak amounts to ≈1% and is comparable with the experimental 76500700075008000 0 1 2 3 4 5λ-1 (cm-1) N (1026 m-3) Fig. 4. Density dependence of the inverse wavelength λ−1of the emission maximum. The straight line is the prediction of the model (see text). accuracy of the data. It is also known that even the atomic lin es of noble gases (for instance Kr and Xe in the wavelength range 118 −150 nm) exhibit a weak density dependent red shift, of the order of 0 .1 % in a density range much larger than the present one, which is interpreted in ter ms of density dependent local ﬁeld corrections in the classical dispersi on theory [26] . It can be clearly noted from ﬁgure 3 that also the peak width is aﬀected by pressure. Namely, the NIR continuum broadens as Pincreases. In ﬁgure 5 050010001500200025003000 0 1 2 3 4Δ (cm-1) N (1026 m-3) Fig. 5. Pressure–broadening of the NIR emission spectra as a function of the gas density N.∆ is the spectrum FWHM. The line is only a guide for the eye. the emission FWHM, ∆ ,is plotted as a function of the gas density. (The error bars are again the statistical uncertainties of the ﬁt.) At s mall densities, ∆ 8corresponds to an energy spread of ≈0.1 eV and increases up to ≈0.3 eV at the high density boundary of this experiment. This amount is too large to be due to thermal ﬂuctuations only ( ≈0.025 eV at room temperature). Moreover, the increase of ∆ is not linear with N.Therefore, a simple assumption of collisional broadening might be not completely adequate. Under this res pect the observed behavior is opposite to that of the second VUV continuum. In f act, in the latter case the FWHM decreases almost linearly by ≈15 % in the same pressure range of the present experiment. The narrowing of the VUV con tinua with increasing pressure has been explained in terms of a hypothe tic absorption in a ground state population that increases with increasing pr essure. This fact would also be responsible for the red shift of the wavelength of the emission maximum [1]. 4 Discussion Owing to the similarities and diﬀerences of the NIR emission continuum with the observed VUV continua we would like to suggest a poss ible model for understanding the present experimental results. The wavelength band of the NIR emission is centered about 130 0 nm. This value roughly corresponds to the energy diﬀerence between t he stable excimer levels 0± u,1u,2uof the A7dπconﬁguration, correlated with the Xe (1S0) + Xe∗(5p5[2P3/2]6p) dissociation limit, and the energy level of the Xe (1S0) + Xe∗(5p5[2P3/2]6s) system. According to Mulliken [24], the latter system may give origin, among others, to short–distance excimer poten tial energy curves, which are mainly dissociative in nature. Namely, there are t he mainly repulsive energy curves of the states 0+ g,(1g,0− g) of the A7pσexcimer conﬁguration and the repulsive energy curves of the excimer states 1 g,2gbelonging to the B6s conﬁguration. The latter ones intersect the potential ener gy curves of the stable A7dπexcimers. We therefore assume that excited Xe atoms belonging to the 6 pmanifold can be deactivated to lower–lying excited Xe atoms in the 6 smanifold not only through direct atomic transitions but also via the form ation of highly excited excimers Xe∗∗ 2.These, in turn, decay radiatively to the dissociative states (1 g,2g),or to the mainly dissociative states 0+ g,(1g,0− g),leading to dissociation into a ground–state Xe atom and one excited Xe a tom in the 6 s manifold. Finally, excited Xe atoms in the 6 smanifold give origin to the lower lying excimer levels responsible for the production of the w ell–known VUV continua. The NIR emission may originate from high vibratio nally excited levels directly populated, or from relaxed vibrational lev els populated through collisional processes [30]. 9The nature of the bound excimer levels of the A7dπconﬁguration is pre- dissociative owing to the intersection of their potential c urves with those of the dissociative states [23]. This should produce a further inc rease of the width of the NIR band in addition to the usual collisional broadening . Moreover, at the temperature of the experiment several rotational degrees o f freedom should be excited, too [25]. The most important feature of the observed NIR continuum is i ts large density–dependent red shift. It is well known [30] that the e lectronic structure of homonuclear excimers can be described quite accurately b y an ionic molec- ular core and an electron in a diﬀuse Rydberg orbital much lar ger in diameter than the internuclear distance [2]. Such a state can exist ev en in a high pres- sure environment provided that the Rydberg electron is weak ly scattered by the gas atoms, or, in other words, if the electron mean free pa th is much larger of the radius of the orbit of the Rydberg state. In a pretty dil ute gas, like in the present experiment, this condition is fulﬁlled because the electron mean free path is several nanometers long [31]. In order to simplify the discussion, let us therefore treat t he excimer levels with the aid of the Bohr model of a hydrogen–like atom in the sa me way as Wannier–Mott excitons are dealt with in liquids or solids [3 2]. The excimers can be considered as impurities in the gas and their highest e xcited states are given by the equation [32] En=−13.6 n2K2(1) where nis the principal quantum number and Kis the dielectric constant of the gas. The energy is expressed in eV. Eq. (1) is valid prov ided that the electron orbit is suﬃciently large as to encompass several a toms of the gas. However, if the radius of the orbit of the Rydberg electron is pretty large, the interaction of the outer electron with the atoms of the ho st gas gives origin to a density–dependent shift of the electron energy. This ph enomenon aﬀects the absorption lines of alkali vapors immersed in a buﬀer gas [33]. The energy shift depends on the ordinal number of the spectral lines of t he series and converges to a limit, V0(N) proportional to the density of the buﬀer gas. This limit has been calculated by Fermi [33] as V0(N) =2π/planckover2pi12 mNa (2) where mis the electron mass and ais the scattering length for the interaction of the slow, Rydberg electron with an atom gas. The energy level s of the highest 10excited states must be then corrected for this contribution yielding [32] En=−13.6 n2K2+2π/planckover2pi12 mNa (3) For attractive electron–atom interaction potential a <0 and V0(N) gives origin to a density–dependent red shift of the spectral line s [33]. Let us furthermore assume that the excimer decays to a lower– lying, ex- cited atomic level of Xe, whose energy levels can be approxim ated by the same Bohr equation Eq.(1). The Bohr formula does indeed give an io nization energy of 13.6 eV to be compared with the actual value of 12.1 eV for at omic Xe. The correction V0has been included only in the energy levels of the excimer because it is assumed that the orbit of the electron in the les s excited atom is smaller. On the contrary, the correction due to the gas pol arizability is accounted for in the same way. Obviously, the validity of the se approximations will depend on the agreement with the experimental data. The energy released in the transition from the excimer level to the atomic one corresponds to a wavelength λgiven by 1 λ=13.6e hcK2/parenleftBigg1 n2 f−1 n2 i/parenrightBigg +/planckover2pi1 mcNa (4) where nfandniare the principal quantum numbers of the ﬁnal and ini- tial states. The ﬁrst contribution is positive because nf< n i.The dielectric constant of Xe can be obtained by the usual Lorentz–Lorenz fo rmula but, in the density region of interest, it can be approximated, wi thin 0 .02 %,by K= 1 + Nα/ǫ 0where α= 4.45×10−40F·m2is the atomic polarizability of Xe [34]. Actually, the value of the quantum numbers niandnfare not known, the 6selectron in the Xe atom gives origin to four non–degenerate s tates 1P1,3P0,1,2,and there are also vibrationally excited states of the excim er. Therefore, we cast Eq. (4) in a form better suited for further analysis 1 λ=A(1−2Nα ǫ0) +/planckover2pi1 mcNa (5) where A={(13.6e/hc)(1/n2 f−1/n2 i)}>0 is a yet unkown constant to be determined and we have expanded 1 /K2≈1−2Nα/ǫ 0.By suitably collecting 11the terms proportional to the density we ﬁnally obtain 1 λ=A−/parenleftBigg 2Aα ǫ0+/planckover2pi1 mc\|a\|/parenrightBigg N (6) where we have exploited the fact that the electron–Xe atom sc attering length is negative, a≈ −0.309 nm [35]. Eq. (6) predicts a linear decrease of the inverse wavelength with increasing density. Moreover, it contains only one ﬁtting parameter, A,which is obtained only as the intercept of the straight line at zero density. On ceAhas been determined by the ﬁtting procedure, the slope is no longer ad justable because it only contains additional contributions given by univers al constants. The straight line in ﬁgure 4 has been drawn with slope given by −(2A(α/ǫ0) + (/planckover2pi1/mc)\|a\|) =−1.98×10−22m2 withA≈7800 cm−1,as determined from the zero–density extrapolation of the data. If a straight line is ﬁtted to the data a slope of valu e−(2.05±0.09)× 10−22m2is obtained. The agreement with the experimental slope is ex cellent. We stress once more the fact that as soon as the zero–density v alue of λ−1has been determined, there are no more free parameters left. By n eglecting either the screening of the Coulomb interaction due to polarizatio n or the density– dependent shift of the energy levels of the Rydberg electron in a large–radius orbit, the slope of the straight line would become nearly 50 % smaller than actually measured. Obviously, the nature of the excimers in the A7dπconﬁguration is predis- sociative because their potential energy curves intersect the repulsive potential energy curves of (1 g,2g) states leading to dissociation in a (1S0) Xe atom plus an excited one in the 6 smanifold. Also the (0± g,1g) excimer states of the A7pσconﬁguration are mainly repulsive, though their potential energy curves do not intersect those of the A7dπexcimer states. Therefore, there is a contin- uum of kinetic energy available to the dissociation product s and a continuum NIR emission band is produced. Nonetheless, the continuum o f kinetic energy is probably distributed around the ﬁnal states in such a way t hat an average value for it can be well approximated by Eq. (1). Moreover, th e predissociative nature of the excimer states might be responsible for the non linear increase of the NIR continuum width. This might also be the reason for t he very dif- ferent behavior of the width of the VUV continuum [1], that de creases with increasing density. In fact, for the (0+ u,(1u,0− u))→0+ gtransitions there are no intersections between potential energy curves and the high er excimer states are not predissociative. However, a quantitative description would require much more accurate potential energy curves than those actually a vailable. 12The assumptions leading to the previously described model c ould raise se- vere criticisms. In particular, the radius of the Rydberg el ectron in the excimer might not be large enough to guarantee that the energy levels are aﬀected ei- ther by the dielectric constants or by the Fermi shift V0(N),or both. However, we believe that the striking agreement of the model with the e xperimental data gives some credit to the model itself. Further measurements at higher densities in Xe are needed (and are in progress) to conﬁrm the ﬁrst data r eported here. 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arXiv:physics/0011033v1 [physics.atom-ph] 15 Nov 2000Random Scattering by Atomic Density Fluctuations in Optica l Lattices M. Blaauboer,a,bG. Kurizki,aV.M. Akulin,c aChemical Physics Department, Weizmann Institute of Scienc e, Rehovot 76100, Israel bDepartment of Interdisciplinary Studies, Faculty of Engin eering, Tel Aviv University, Tel Aviv 69978, Israel cLaboratoire Aim´ e Cotton, CNRS II, Bˆ atiment 505, Orsay Ced ex 91405 France (September 14, 2013) We investigate hitherto unexplored regimes of probe scat- tering by atoms trapped in optical lattices: weak scatterin g by eﬀectively random atomic density distributions and multip le scattering by arbitrary atomic distributions. Both regime s are predicted to exhibit a universal semicircular scattering l ine- shape for large density ﬂuctuations, which depend on temper - ature and quantum statistics. PACS numbers: 32.80.Pj, 05.40.-a, 42.50.Lc, 71.23.-k Recent advances in trapping and manipulation of cold atoms interacting with external ﬁelds have primarily been implemented thus far either in single-atom systems, such as sparse (low-density) optical lattices [1], or in Bose-Einstein condensates with macroscopic numbers of atoms [2]. Between these two limiting regimes lies the scarcely in- vestigated domain of processes involving a ﬁnite number of interacting atoms . Both the potential interest and the diﬃculties involved in studying such processes are evident in the example of optical lattices with appreciable atomic ﬁlling factors [3], in which the transition between ”insulating” (localized) and ”metallic” (superﬂuid) phases have been studied in the framework of the Bose-Hubbard model [4]. As con- ﬁrmed by the above study, the mean-ﬁeld approximation is inadequate for small numbers of interacting atoms in lattices due to the presence of large quantum ﬂuctua- tions. Still more diﬃcult is the analysis of ﬂuctuations in systems of cold atoms coupled by long-range (1 /ror 1/r3) ﬁeld-induced interactions [5]. This leads to the intruiging question: Is there a way to circumvent the formidable task of treating the full dynamics of such systems and still infer their important characteristics, e.g., their dependence on temperature, quantum statis- tics (Bose or Fermi), number of atoms and lattice pa- rameters? And: Are there universal measurable features which can be a ”signature” of the statistical ensemble (distribution function) of such systems? Here we consider the possibility of inferring such sta- tistical characteristics from the spectral features of pro be photons or particles that are scattered by the density ﬂuctuations of trapped atoms, notably in optical lat- tices, in two hitherto unexplored scenarios: (a) The probe is weakly (perturbatively) scattered by the local atomic density corresponding to the random occupancy of dif-ferent lattice sites (Fig. 1 - inset a). (b) The probe is multiply scattered by an arbitrary (possibly unknown a priori) multi-atom distribution in the lattice (Fig. 1 - in- set b). At the heart of our analysis is the idea that the Green function of the scattered photon or particle, which em- bodies the relevant spectral information, can be qualita- tively estimated without resorting to cumbersome pertur- bative calculations of the probe-multiatom interaction by replacing this interaction Hamiltonian by an equivalent random matrix. The random matrix approach, which has been successfully applied to various disordered sys- tems [6], allows the evaluation of probe spectra toall orders of scattering , expressing them by means of only the ﬁrst two moments (the mean and variance) of the random interaction, averaged over the statistical ensem- ble of the multiatom system. The highlight of our anal- ysis, based on this random matrix approach, is the pre- diction of a semicircular spectral lineshape of the probe scattering in the large-ﬂuctuation limit of trapped atomic ensembles. Thus far, the only known case of quasi- semicircular lineshapes in optical scattering has been pre - dicted [7] and experimentally veriﬁed [8] in dielectric mi- crospheres with randomly distributed internal scatterers . The Green function of the probe (P) at energy ǫ= ¯hω is given by GP(ǫ) = Tr S/bracketleftbigg1 ǫ−ˆHP−ˆVˆρS/bracketrightbigg , (1) where ˆHP,ˆVand ˆρSare, respectively, the unperturbed probe Hamiltonian, the probe-system interaction Hamiltonian and the density operator for the ensemble of the multiatom system (S). We shall assume that the following conditions hold. (i) There is no appre- ciable back-eﬀect of the probe on the multiatom system (otherwise it is no longer a probe). (ii) The state of the multiatom system does not change during the interaction time with the probe, i.e., the multiatom system remains ”frozen”, as is applicable for optical or atomic probing. This situation then cannot be described as Markovian re- laxation (exponential decay) of the probe state into the multiatom reservoir, since the correlation time of this reservoir is now much longer than that of the probe, in contrast with the basic assumption of relaxation. (iii) The probe spectrum is broadband, i.e., it encompasses many of its eigenstates. 1For an ensemble ”frozen” during the interaction time, the tracing in (1) implies statistical averaging over re- peated realizations of the multiatom system, every time the probe scattering is recorded, or taking the expec- tation value with respect to the quantum state of the system. For simplicity, let us explicitly consider elastic scattering (the extension to inelastic scattering is straightforward), for which ˆV=/summationdisplay /vectorkf/vectorkˆρ† P/vectorkˆρS/vectork+ h.c. or (2a) ˆV=/summationdisplay /vectorkf/vectorka† /vectorkˆρS/vectork+ h.c. (2b) Heref/vectorkis the scattering amplitude for momentum ex- change ¯ h/vectorkbetween the probe and the system and the /vectork-mode Fourier components of the probe (system) den- sity operators ˆ ρP/vectork(ˆρS/vectork) are deﬁned in terms of their respective creation and annihilation operators ˆ ρP/vectork=/summationtext /vector qa† /vector qa/vector q+/vectork, ˆρS/vectork=/summationtext /vector qc† /vector qc/vector q+/vectork. Equations (2a) and (2b) stand, respectively, for bilinear and linear probe-system coupling. For optical probes (2a) and (2b) correspond to Raman and single-photon scattering, respectively. For atom or neutron probes the coupling (2a) is appropriate. The Green function (1) is obtainable, to all orders in ˆV[9], by solving the set of equations for its diagonal elements G/vectork/vectork(ǫ) = [ǫ−ǫ/vectork−/summationdisplay /vectork′/angbracketleftˆV2 /vectork/vectork′/angbracketrightG/vectork′/vectork′(ǫ)]−1, (3) where ǫ/vectorkare the probe energy eigenvalues in the absence of potential ﬂuctuations and pointed brackets denote the expectation value. The spectral information contained in these G/vectork/vectorkis given by the density of states (DOS) of the probe g(ǫ) =−1 πIm/summationtext /vectorkG/vectork/vectork(ǫ). In order to extract information on the system we shall make two simplifying assumptions regarding the probe and the coupling potential (2): (i) f/vectorkis ﬂat in /vectork(the coupling is strongly localized in space) within a band exceeding the relevant band of the system, so that f/vectork≈f; (ii) the statistical distribution of the probe is also ﬂat in /vectorkand its second moment in /angbracketleftˆV2/angbracketrightis replacable by the square of its mean ﬂux (or density) ¯ n2 Pfor the bilinear coupling (2a) or by its mean ﬂux (density) ¯ nPfor the linear coupling (2b). Under these assumptions we can rewrite the squared coupling potential in (3) as /angbracketleftˆV2 /vectork/vectork′/angbracketright=/angbracketleftˆV/vectork/vectork′/angbracketright2+FPS/vectork/vectork′. (4) Here/angbracketleftˆV/angbracketrightis the mean coupling potential and FP∼ \|f\|2¯n2 PorFP∼ \|f\|2¯nPin the case of (2a) and (2b), respectively. The quantity of interest for the system is the Fourier-transformed density-density correlation o f the atomic system S/vectork/vectork′=/angbracketleftˆρ† S/vectorkˆρS/vectork′/angbracketright+ c.c. (5)Its diagonal element S/vectork/vectorkis the static structure factor S/vectork, which is the Fourier transform of the van-Hove correla- tion function /angbracketleftˆρ† S(/vector r, t= 0)ˆρS(/vector r′, t′= 0)/angbracketrightfor the spatial density ﬂuctuations of the ”frozen” atomic ensemble. The diﬃculty of having to evaluate or measure the ma- trix elements S/vectork/vectork′is avoided for a spatially random den- sity distribution of the atomic system, due to random site occupancy (Fig. 1, inset a) and short-range interac- tion with the probe (e.g., a neutron or thermal atom). The elements S/vectork/vectork′in (4) and (5) can then be replaced by the average of the structure factor over all relevant /vectork: S/vectork/vectork′→¯S=/integraldisplay d/vectorkS/vectork∼ /angbracketleftˆn2 S/angbracketright − /angbracketleftˆnS/angbracketright2, (6) where the right-hand side of ¯Sdenotes the local atomic density or number variance averaged over the ensemble. The implications of evaluating the probe DOS g(ǫ) using (3)-(6) will be examined for random ﬂuctuations about a mean scattering potential /angbracketleftVS(x)/angbracketright(correspond- ing to the mean atomic density distribution) that is 1D- periodic. The ”unperturbed” probe dispersion associated with/angbracketleftVS(x)/angbracketrightisǫ/vectork=−2Jcos(kxd)+A,Jbeing the hop- ping frequency, dthe lattice period and Athe band en- ergy oﬀset. This gives rise to the following expression for the Green function (3) G(ǫ) = ǫ−ǫ/vectork− /angbracketleftW2/angbracketright/summationdisplay /vectork′(ǫ−ǫ/vectork′−Λ(ǫ) +i∆(ǫ))−1 −1 . (7) Here/angbracketleftW2/angbracketright ≡ F P¯S, Λ(ǫ) =/angbracketleftW2/angbracketright//radicalbig (ǫ−A)2−4J2for \|ǫ−A\|>2J, ∆(ǫ) =/angbracketleftW2/angbracketright//radicalbig 4J2−(ǫ−A)2for\|ǫ−A\|< 2Jand both zero otherwise. Figure 1 shows how the probe DOS g(ǫ) changes from that of a periodic band structure corresponding to the mean potential /angbracketleftVS(x)/angbracketright to a semicircular shape as the amount of ﬂuctuations measured by /angbracketleftW2/angbracketrightincreases. In the multiple-scattering scenario, which pertains to resonantly scattered atomic probes or to intracavity optical probes (Fig. 1, inset b), semicir- cular lineshapes are obtained even when the S/vectork/vectork′cannot be claimed to be- long to a random distribution (Fig. 2, inset). In the case of strongly-interacting atoms within a lat- tice site or longe-range intersite density correlations [5 ] the distribution may be quite intricate, corresponding to sharp peaks of S/vectork/vectork′. Nevertheless, the universal spectral trends of Fig. 1 can be shown to hold in this scenario, provided /angbracketleftˆV2/angbracketright1/2g0(ǫ)≫1, g0(ǫ) denoting the ”unperturbed” probe DOS. This con- dition allows us to estimate G/vectork/vectorkin (3) to all orders in ˆV, upon replacing the state of the atomic system by a gaus- sian random ensemble [6,7]. The result is the following 2universal formula [7] for the renormalized probe energy ˜ ǫ at a given input energy ǫ ǫ= ˜ǫ+/angbracketleftW2/angbracketrightTrP/parenleftbigg1 ˜ǫ−ˆHP−i0/parenrightbigg . (8) The use of (8) leads to a semicircular lineshape similar to the one in Fig. 1, as if the potential were random. In order to illustrate the role of temperature, quantum statistics and the mean lattice potential in producing the semicircular lineshape, we proceed to evaluate /angbracketleftW2/angbracketrightfor several simple models: 1. The isolated-site limit : The tightly-bound Bose or Fermi distributions in a lattice can be estimated by tak- ing the potential of each site to be that of a harmonic well of depth V0. The isolated-site approximation holds for atoms in the lowest vibrational band, when the cou- pling energy is much smaller than the excitation energy to the next band [4],/radicalbig /angbracketleftW2/angbracketright ≪¯hων=2π¯h λ/radicalBig 2V0 m,λ being the wavelength of the laser light. −4 −2 0 2 4 ε−0.250.000.250.50gΛ∆(a)probe (b) probe FIG. 1. Density of states g(ǫ) of a probe scattered by bosonic atoms in a 1D optical lattice. Solid, dotted and dashed curves stand for ∆( ǫ) and the thick curve stands for Λ(ǫ) (dispersion), see text. All curves are numerically com- puted from G(ǫ) and correspond to average random couplings /angbracketleftW2/angbracketright= 0.4, 2 and 10 respectively. The hopping frequency J= 1, and for all curves/integraltext dǫ g(ǫ) = 1. Inset a: A probe weakly scattered by a randomly occupied lattice. Inset b: A probe multiply scattered by a regular atomic distribution. In the absence of additional external perturbations, the coupling /angbracketleftW2/angbracketrightarises because of temperature-dependent ﬂuctuations in the site-occupancy of the optical lattice, which has an approximately gaussian distribution [3]. The resulting random coupling energies averaged over all states yield /angbracketleftW2/angbracketright ∼ F P/angbracketleftbig /angbracketleftn/vectork, n/vectork′/angbracketright − /angbracketleftn/vectork/angbracketright/angbracketleftn/vectork′/angbracketright/angbracketrightbig /vectork/vectork′= FP/angbracketleft/angbracketleftni, nj/angbracketright − /angbracketleftni/angbracketright/angbracketleftnj/angbracketright/angbracketrightij≈ F P/parenleftbig /angbracketleftn2 S/angbracketright − /angbracketleftnS/angbracketright2/parenrightbig . Here n/vectork≡c† /vectorkc/vectork,iandjlabel atomic sites, and /angbracketleftnS/angbracketrightis the average number of atoms per site. The last step applies whenever ni≈nj∀i, jand the density ﬂuctuations are approximately site-independent. We have veriﬁed this by numerical simulation, considering 2 to 4 identical atoms on a 1D lattice with 6 sites and calculating the density ﬂuctuations if the probability of an occupied site is 1/10of the probability of an empty site. In all cases the max- imum relative diﬀerence between FPS/vectork/vectork′and/angbracketleftW2/angbracketrightwas less than 10 %. The kinetic contribution to /angbracketleftW2/angbracketrightdue to evaporation of atoms from the lattice is the dominant one at high temperatures, regardless of the statistics. If all the atom s are in the lowest energy band, we may adopt the rate equation used to describe the formation of electron-hole clusters in a plasma [10] and ﬁnd /angbracketleftW2/angbracketrightevap=a/angbracketleftnS/angbracketrightT2e−βV0. (9) Herea=kBmcp, with kBthe Boltzmann constant, mthe mass of the atoms and cptheir speciﬁc heat, Tdenotes the temperature, β−1≡kBTandV0is the optical lat- tice potential. The inﬂuence of evaporation becomes the dominant eﬀect for T∼25µK. Around T∼300µK these ﬂuctuations become comparable in size to the square of the optical lattice potential ( /angbracketleftW2/angbracketright ∼V2 0∼100 (neV)2) and atoms then largely escape from the lattice. At low temperatures (well below 100 µK) the density- density ﬂuctuations depend on whether the atoms in the lattice are bosons or fermions. For bosonic atoms in the lowest vibrational state we obtain [11] /angbracketleftW2/angbracketrightstat,Bose=z 1−z+/parenleftbiggz 1−z/parenrightbigg2 +d3 λ3 T∞/summationdisplay α=1zα α1/2.(10) Here we have approximated the motion of the atoms in the potential wells by a harmonic oscillation with fre- quency ωv[12],z≡e−kBT/¯hωv,ddenotes the average lat- tice spacing and λT= (2π¯h2/mk BT), the thermal wave- length, is the length scale separating quantum statistical behavior (for λT∼d) from classical Maxwell-Boltzmann behavior (for λT≪d). For fermionic atoms in an optical lattice [13] one starts with the analog of the coupling (2) for particles with spin, using creation and annihilation operators c† /vectorkσandc/vectorkσand performing an additional sum over the spin index σ, and follows the same analysis as above. One then ﬁnds /angbracketleftW2/angbracketrightstat,Fermi=z 1 +z+/parenleftbiggz 1 +z/parenrightbigg2 +2d3 λ2 TλF∞/summationdisplay α=1zα α1/2, (11) withλFthe Fermi wavelength. At high temperatures z→0 and both (10) and (11) reduce to the classi- cal Maxwell-Boltzmann result /angbracketleftW2/angbracketrightstat,clas=z. At low temperatures, fermionic ﬂuctuations approach a constant value, whereas bosonic ﬂuctuations become very large as Tdecreases below ∼1µK, marking the Bose-Einstein condensation. In Fig. 2 we have taken typical parameters for avail- able optical lattices to show how /angbracketleftW2/angbracketrightevolves as a func- tion of temperature both for bosons (Cs atoms) and for 3fermions (Li atoms). The total density-density ﬂuctua- tions consist of the sum of (9) and either (10) or (11), depending on the statistics. Note that since Li atoms are lighter than Cs atoms, their ﬂuctuations are larger. The isolated-site condition is satisfed for the entire tem- perature range displayed in Fig. 2. Since the hopping frequency J∼V0, the random coupling changes from /angbracketleftW2/angbracketright/J2∼0.1 to/angbracketleftW2/angbracketright/J2∼10, when going from T∼8µK toT∼100µK. Simultaneously the DOS then evolves from the periodic to the semicircular shape as in Fig. 1. 1 10 100 T (µK)010203040<W2> (10−18 eV2) <W2>stat,Bose<W2>evap x 1/7.8κS κ FIG. 2. Density-density ﬂuctuations /angbracketleftW2/angbracketright(in units of (neV)2) as a function of temperature T for bosonic and fermionic atoms in an optical lattice. Thin solid line - ﬂuc- tuations due to evaporation (9), thin dashed line - statisti cal ﬂuctuations (10). Thick solid line - total ﬂuctuations /angbracketleftW2/angbracketright for bosonic (Cs) atoms; thick dashed line - their counterpar t for fermionic (Li) atoms (scaled by a factor of 1/(7.8)). Pa- rameters used for Cs: V0= 5 neV, /angbracketleftnS/angbracketright= 0.1, d = 0.1 µm, cp(Cs) ∼(0.2)·103J kg−1K−1andωv(Cs) ∼4·105s−1; for Li:cp(Li) ∼(3.6)·106J kg−1K−1,λF(Li) = 6 ·10−10m andωv(Li)∼2·106s−1[12]. Inset: Static structure factor vs. κfor phonons (solid curve) and nearly-free fermions (dashed curve) in a lattice at ﬁnite T. 2. The nearly-free limit : A Bose or Fermi gas weakly modulated by the lattice potential yields S/vectork/vectork′= S/vector κ=/vectork−/vectork′=\|φ/vector κ\|2S(free) /vector κ. Here /vector κis a reciprocal lattice vec- tor,φ/vector κis the corresponding Fourier harmonic of the lat- tice potential (normalized to 1) and S(free) /vector κis the struc- ture factor for momentum transfer ¯ hκin a free Bose or Fermi gas. For a Fermi gas S(free) /vector κ=±Θ(kf−κ), the Fourier transform of pair correlations with parallel or anti-parallel spins (which determines the sign): it is the well-known step function which vanishes for κlarger than the Fermi wavevector kf. At ﬁnite temperatures this dis- tribution broadens. The replacement of the nearly-free fermionic S/vectork/vectork′by the average value (6) is then justiﬁ- able only in the multiple-scattering scenario, while in the weak-scattering scenario the lattice potential harmonicsφ/vectorkpick out well-deﬁned S/vectork−/vectork′=/vector κ(Fig. 2, inset - dashed line). 3. The phonon regime : Excitations at frequencies below the chemical potential of a Bose condensate trapped in a lattice can produce collective phonon modes [14] whose ”frozen” spectrum is characterized by S/vector κ=/summationtext /vector q[(/angbracketleftn/vector q/angbracketright+ 1)/summationtext /vectorGδ(/vector κ−/vector q−/vectorG) +/angbracketleftn/vector q/angbracketright/summationtext /vectorGδ(/vector κ+/vector q+/vectorG)], where /angbracketleftn/vector q/angbracketright is the mean number of phonons at temperature Twith wavevector /vector q, and /vectorGdenotes the reciprocal lattice vector. The phonon mode spectrum includes quasi-local modes in the case of ﬂuctuating atomic distributions. This nat- urally leads to the limit (6) and an eﬀectively random coupling (Fig. 2, inset - solid line). To conclude, we have identiﬁed novel regimes of probe scattering by atoms trapped in optical lattices in the random-density and multiple-scattering regimes. These regimes cannot be treated by the mean-ﬁeld approxi- mation, but are characterized by a universal feature of large density ﬂuctuations, namely, semicircular scatter- ing lineshapes. This is the atom-optical analog of the semicircular broadening of the DOS in disordered elec- tronic systems, which as far as we know has not yet been observed unambiguously. The observation of this atom-optical counterpart presents a nontrivial but feasi- ble challenge for experimentalists. This work was supported by US-Israel BSF , the Israel Council for Higher Education, Minerva and Arc-en-Ciel. [1] P. Verkerk et al., Phys. Rev. Lett. 68, 3861 (1992); M. Raizen et al., Phys. Today July 1997, 30 (1997); M. Ben- Dahan et al., Phys. Rev. Lett. 76, 4508 (1996). [2] A.S. Parkins and D.F. Walls, Phys. Rep. 303, 1 (1998); S. Inouye et al., Nature 392, 151 (1998); D.M. Stamper- Kurnet al., Phys. Rev. Lett. 80, 2027 (1998); B.D. An- derson and M.A. Kasevich, Science 282, 1686 (1998); M.O. Scully, Phys. Rev. Lett. 82, 3927 (1999). [3] M.T. DePue et al., Phys. Rev. Lett. 82, 2262 (1999). [4] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998). [5] D. O’Dell, S. Giovanazzi, G. Kurizki and V.M. Akulin, Phys. Rev. Lett. 84, 5687 (2000). [6] M.L. Mehta, Random Matrices , (Academic, New York, 1991); B. Velicky et al., Phys. Rev. 179, 747 (1968). [7] V.M. Akulin and G. Kurizki, Phys. Lett. A 174, 267 (1993); V.M. Akulin, Phys. Rev. A 48, 3532 (1993). [8] D. Ngo and R.G. Pinnick, J. Opt. Soc. Am. A 11, 1352 (1994). [9] P.W. Anderson, Phys. Rev. 124, 41 (1961). [10] C. Klingshirn and H. Haug, Phys. Rep. 70, 315 (1981). [11] L.E. Reichl, A modern Course in Statistical Physics , (Wi- ley, New York, 1998). [12] A. Kastberg et al., Phys. Rev. Lett. 74, 1542 (1995). [13] M.R. Andrews, Nature 398, 195 (1999). [14] D.M. Stamper-Kurn et al., Phys. Rev. Lett. 83, 2876 4(1999). 5
arXiv:physics/0011034v1 [physics.optics] 15 Nov 2000Theory of self phase-locked optical parametric oscillator s J.-J. Zondy and A. Douillet Laboratoire Primaire du Temps et des Fr´ equences Bureau National de M´ etrologie / Observatoire de Paris 61, avenue de l’Observatoire, F-75014 Paris, France. A. Tallet, E. Ressayre and M. Le Berre Laboratoire de Photophysique Mol´ eculaire Batiment 210, Universit´ e de Paris Sud, 91405, Orsay Cedex, France. (February 2, 2008) The plane-wave dynamics of 3 ω→2ω, ωsubharmonic optical parametric oscillators containing a second harmonic generator of the idler-wave ωis analyzed analytically by using the meanﬁeld approximation and numerically by taking into account the ﬁe ld propagation inside the media. The resonant χ(2)(−3ω;2ω, ω) :χ(2)(−2ω;ω,ω) cascaded second-order nonlinearities induce a mutual injection-locking of the signal and idler waves that leads t o coherent self phase-locking of the pump and subharmonic waves, freezing the phase diﬀusion noise. I n case of signal-and-idler resonant devices, largely detuned sub-threshold states occur due to a subcritical bifurcation, broadening out the self-locking frequency range to a few cavity linewidths . PACS: 42.65.-k, 42.65.Yj, 42.65.Sf, 42.65.Ky I. INTRODUCTION The synthesis of phase-coherent (sub)harmonic optical rad iation ( ω,2ω, ..., Nω ) is useful in high precision optical measurements, such as optical frequency metrology [1], [2] . In frequency metrology, a phase-coherent optical by-N divider allows to reduce the absolute measurement of an opti cal frequency Nω(e.g, couting cycles in the hundreds of THz range) to the measurement of a smaller frequency interva l ∆ = ωby use of a femtosecond laser comb generator whose radio-frequency intermode spacing is phase-locked t o a primary microwave clock [3]. The generation of a comb of phase-locked harmonic radiation can also be the starting basis for the generation of attosecond pulse train by Fourie r synthesis [4], [5], [6]. Indeed, the superposition (Fourie r synthesis [7], [8]) of a comb of Nequal amplitude optical harmonic ﬁelds with a controllable relative phase relation ship can lead to a temporal train of ultra-short pulses with a sub-optical cycle duration ( τ∼2π/Nω ) and a repetition rate equal to the fundamental harmonic ω. Phase-locked optical harmonic generators are often based on many lasers l inked via nonlinear upconversion parametric processes [1], [9], [10], [11], [12]. Phase coherence among the harmon ic waves is usually achieved by use of complex electronic phase-locking loops. The high conversion eﬃciencies of opt ical parametric oscillators (OPOs) combined with the engineering ﬂexibility oﬀered by periodically poled (PP) n onlinear materials may allow a compact implementation of such optical subharmonic generators or attosecond pulse ge nerator. The study of OPOs as by- Ndividers ( N= 2,3,4) of a pump frequency νpis motivated by their capacity to perform the phase-coherent division of a pump photon into tw o highly phase-correlated subharmonic photons. In precision measurement setups, subharmonic generation lea ds to a subsequent phase noise reduction ( ÷N2) compared to harmonic generation ( ×N2). Graham and Haken have ﬁrst demonstrated that the phase sum ϕ1+ϕ2of the idler and signal waves from an OPO follows adiabatically the phase noise ϕpof the pump laser, while their phase diﬀerence ϕ2−ϕ1undergoes a phase diﬀusion process stemming from the intera ction of both modes with the vacuum ﬂuctuations [13]. To act as pertinent phase-coherent divid ers, the idler and signal waves have thus to be phase-locked, for instance by use of an electronic servo control that force sϕ2−ϕ1to copy the phase of an external RF oscillator referenced to a primary clock. Such an electronic servo tech nique has been implemented to develop phase-coherent divide-by-2 (2:1) [14], and divide-by-3 (3:1) OPOs for high resolution optical metrology [15], [16] with correspondin g residual phase diﬀerence variance well below 1 rad2. Electronic phase-locking loops are however subject to limi ted response time bandwidth and require fast OPO cavity length actuators. As an alternative, all-optical self phas e-locking (SPL) methods are currently being investigated [ 17], 1[18] to by-pass the bandwidth limitation of electronic serv os and simplify the implementation of phase-locked divider s. These methods are based on the self injection-locking of the subharmonic waves, similarly to the injection locking process of a slave laser oscillator by a master laser which po ssesses a better spectral purity and frequency stability [7 ], [19]. In injection-locked systems, the phase of the slave os cillator, φ(t) =ωt+ϕ(t), is a perfect copy of the phase of the injecting master oscillator. Phase locking can occur on ly when the two frequencies are close enough, e.g. within a certain locking range ∆ωlock=\|ωs−ωm\|whose extent is proportional to the squared ratio/radicalbig Pm/Psof the injecting master power to the slave power. It is thus expected that the s elf-locking range in an OPO divider should depend on whether the OPO is conﬁgured as a doubly-resonant (DRO), a triply-resonant (TRO) or a pump-enhanced singly resonant oscillator (PRSRO). Strong self injection-locki ng regime occurs only when the signal and idler waves are simultaneously resonant. For divide-by-2 OPOs based on a type-I nonlinear process (th e signal and idler are identically polarized), self- injection locking occurs naturally when the OPO is tuned clo se enough to the frequency-degeneracy [20]. Recently a type-II self phase-locked 2:1 DRO was demonstrated, in whic h a small mixing of the two orthogonally polarized signal and idler waves, performed by an intracavity wave plate, ind uces the injection locking [17]. In both cases the evidence of self phase locking was provided by the high level of phase- coherence between the frequency degenerate signal and idler waves. Following the experiment in ref [17], the theor y oflinearly coupled type-II 2:1 SPL-DROs has been reported [21]. In the case of 3:1 OPO dividers, obviously suc h a linear coupling cannot be implemented, and one must use a nonlinear coupling via χ(2)(−3ω; 2ω, ω) :χ(2)(−2ω;ω, ω) cascading processes (for instance by introducing inside the OPO cavity a second nonlinear material phase-matched fo r the SHG of the idler wave). The ﬁrst implementation of such a self phase-locked (SPL) OPO was recently reported, and used a dual-grating periodically poled lithium niobate chip carrying the OPO and SHG sections in a pump-reso nant idler-resonant conﬁguration (PRSRO) [18]. An extremely small locking range of ∼500 kHz, corresponding to a fraction of the idler cavity line width, was reported due to the weak (non resonant) doubled idler power. A diﬀeren t situation should occur when both the signal and idler waves are resonant, as in a DRO or a TRO. Because of the en hanced energy ﬂow exchange between the 2 ωand ωmodes, the dynamics of such a self-phase locked OPO (SPL-OPO ) is expected to diﬀer signiﬁcantly from that of a conventional OPO. Strictly speaking the 3:1 SPL-DRO or TRO c an be regarded as a degenerate sub/second harmonic generator (2 ω⇆ω+ω) internally pumped by the signal wave of a non-degenerate OP O (3ω⇆2ω+ω). While theories of OPO devices containing additional up or down conversion o f the signal and idler waves have been investigated in the past with the aim of generating new frequencies [22], [23 ], [24], [25], or revealing the quantum noise signature of competing χ(2)nonlinearities [26], [27] such a subharmonically resonant conﬁguration of competing nonlinearities in OPOs has never been theoretically investigated in detail. A 3:1 DRO-SHG rough rate equation analysis based on photon ﬂux conservation, but neglecting the role of cavity d etunings and ﬁeld phase coherence, was previously given by Zhang et al[28] with the conclusion that self-phase locking should man ifest through an imbalance of the signal and idler intensities at exact 3:1 division. Actually our in-de pth analysis shows that such an imbalance holds only for zero detunings: When the nonlinear phase shift due to cascading i s compensated by a non-zero optimal cavity detuning, one can obtain the same output intensities as in a convention al DRO. In this paper, we provide a detailed plane-wave description of the dynamics of divide-by-3 SPL-OPOs that encom- passes all the three cavity conﬁgurations (only the main res ults are summarized in the conclusion for the PRSRO, which will be detailed elsewhere). The dynamics of the nonli nearly coupled OPO is shown to diﬀer substancially from the linear coupling case of a type-II 2:1 SPL-DRO [21]. In par ticular, the nonlinear coupling gives rise to a subcrital bifurcation for any non-zero cavity detuning while convent ional OPOs undergo subcritical bifurcations only for the case of largely pump-detuned TROs [29], [30]. As a consequen ce of the nonlinear phase shifts due to the cascading processes, SPL-DROs or TROs will be shown to display large-i ntensity stable stationary states that would correspond to non-lasing detuning domains in conventional OPOs, exten ding thus the phase-locking detuning range up to a few cavity linewidths. This paper is organized as follows. In section II we describe the basic plane-wave ring cavity model leading to the meanﬁeld solutions that will be compared with the full propa gation solutions. In section III, we ﬁrst treat in detail the DRO conﬁguration case. We then extend the theory to the ca se of a TRO in section IV. Main results on the PRSRO conﬁgurations will be also summarized in the conclusi on. Section V discusses the practical implementation of such SPL-OPOs on the basis of the theoretical ﬁndings and s ection VI concludes with some prospective studies aroused by this ﬁrst theoretical approach. 2II. BASIC MODEL AND EXACT NUMERICAL SOLUTIONS The OPO-SHG device described in the paper is schematically s ketched in Fig.1 . We consider a ring cavity containing the cascaded OPO and SHG nonlinear media, each of length L1andL2respectively and assume that the pump, signal and idler waves satisfy already the 3:2:1 frequency ratios. Though the drawing depicts the case of a dual-grating PP material, phase-matched for an ( eee) interaction, the model is valid also for any separate biref ringent material sections, or a single material phase-matched for both interactions. I n the case of birefringent phase-matching however, the OPO should be of type-II kind (e.g. e→o+eoro→e+o) and the SHG of type-I kind (resp. e+e→ooro+o→e), so that the system is described by only 3 ﬁeld variables. A. Plane-wave propagation equations We denote by z=Z/L1andz′=Z/L2the normalized propagation distances within each crystal, such that 0≤z≤1 between points ”0” and ”1” and 0 ≤z′≤1 between ”1” and ”2”. Let Ej(in m/V) be the slowly varying complex ﬁeld amplitudes where the subscripts j=p,2,1 stand respectively for pump, signal and idler ǫj(Z, t) =1 2Ej(Z, t)exp [ i(ωjt−kjZ)] +c.c. (1) and let Njbe the complex ﬁeld variable, such that \|Nj\|2is the number of photons in mode jat plane zinside the ring cavity, by Nj=/radicalBigg ǫ0cV 2ℏωjEj (2) where Vis the average volume occupied by the modes inside the resona tor. Then, the reduced ﬁeld amplitudes Aj are introduced by scaling Njwith the small signal gain coeﬃcient g1L1of the OPO crystal Aj=g1L1Nj (3) with g1=dOPO c/radicalBigg 2ℏω1ω2ωp ǫ0V n1n2np(4) In Eq.(4), dOPO=ep·[χ(2)(−3ω; 2ω, ω)/2]:e2·e1(in m/V) is the eﬀective nonlinear coeﬃcient for the OPO interaction and njare the refractive indices at frequency ωj. The reduced Maxwell equations for these ﬁeld amplitudes Aj(z,t), when propagating through the (3 ω→2ω, ω) crystal without any diﬀraction eﬀect, are dAp dz=iA1A2 (5a) dA2 dz=iApA∗ 1 (5b) dA1 dz=iApA∗ 2 (5c) using the standard slowly varying amplidude approximation and the usual change of variables z−→z, t−→t−nz/c, where nis the mean linear refractive index. In Eqs. 5 perfect phase- matching has been assumed, ∆ kOPO=kp− k2−k1= 0. The propagation equations in the SHG crystal, allowing f or a non-vanishing wavevector mismatch ∆kSHG=k2−2k1, are dAp dz′= 0 (6a) dA2 dz′=iSA2 1exp(+ i2ξz′) (6b) dA1 dz′=iSA2A∗ 1exp(−i2ξz′) (6c) 3with initial conditions Aj(z′= 0) = Aj(z= 1). The two cascaded crystals are usually phase-matched by adju sting their temperature or angle. But, exact zero mismatch for both interactions may be diﬃcult to achieve, so that a phase-mismatch parameter ξ= ∆kSHGL2/2 is introduced. The parameter Sin eqs. (6) is the ratio of the SHG to OPO small signal gains S=g2L2 g1L1(7) where the small signal gain SHG coeﬃcient (in m−1) is g2=dSHG c/radicalBigg 2ℏω2 1ω2 ǫ0V n2 1n2(8) For a type-I (eee) PP crystal the eﬀective nonlinear coeﬃcie nts satisfy dOPO≈dSHG, so that with n1≈n2≈np andωp= 3ω,ω2= 2ω,ω1=ω,the relation (7) can be approximated by S≈(L2/L1)/√ 3. For L2/L1= 1/3 for instance, one has S≈0.2. The solutions at the SHG crystal exit can be limited to few ter ms of the Mac-Laurin ﬁeld expansion serie at the cell entrance, because of the smallness of the parametric gain co eﬃcients. We shall refer to ML1 approximation, keeping only the lowest-order perturbative terms, i.e. quadratic t erms in powers of the ﬁeld amplitudes. At the exit of the SHG crystal, the ML1 axpproximation provides the solutions for the signal and idler amplitudes as functions of their value at the entrance of the OPO crystal (point ”0” in Fig.1) Ap(t, L1+L2) =Ap(t,0) +iA1(t,0)A2(t,0) (9a) A2(t, L1+L2) =A2(t,0) +iAp(t,0)A∗ 1(t,0) +iχ∗A2 1(t,0) (9b) A1(t, L1+L2) =A1(t,0) +iAp(t,0)A∗ 2(t,0) +iχA2(t,0)A∗ 1(t,0) (9c) with the nonlinear coupling constant χ χ=Sexp(−iξ)(sinξ/ξ) (10) B. Boundary conditions The reduced Maxwell equations have to be completed by the bou ndary conditions for the three waves at the entrance of the OPO crystal to derive the cavity equations. The ﬁeld Aj(t+τ,0) at location ”0” and at a time t+τ, where τ= Λ/cis the cavity roundtrip time (Λ is the total cavity optical pa th), is the ﬁeld Aj(t, L1+L2) at point ”2”of Fig.1, which propagates freely after bounces at the totally reﬂecting ( R= 1) and output coupling ( rj, tj) mirrors and eventually summed with an input ﬁeld, as it is the case for the pump ﬁeld. We deﬁne by rj(tj) the overall mirror amplitude reﬂectivities (transmissivities), such that r2 j+t2 j= 1 and deﬁne the amplitude loss coeﬃcients κj= 1−rj. Then, the boundary conditions take then the form Ap(t+τ,0) =rpexp(i∆p)Ap(t, L1+L2) +Ain (11a) A2(t+τ,0) =r2exp(i∆2)A2(t, L1+L2) (11b) A1(t+τ,0) =r1exp(i∆1)A1(t, L1+L2) (11c) where Aindenotes the input pump amplitude inside the cavity, Ain=tpAext in. Each ∆ jis the usual cavity detuning between the frequency of the waves and the corresponding col d cavity frequency, scaled to the HWHM cavity resonance width ( Only nearly resonant waves will be considered, ∆ j<<2π). The set of equations (9) and (11) forms a mapping of the ﬁeld am plitudes at location ”0”, from which meanﬁeld equations may be derived. Themeanﬁeld model was originally derived from a similar set of equations for a t wo-level cell ring cavity device when the atomic dephasing time is much greater than the round-tri p time τ, but much smaller than the photon lifetime τ/κ [31]. For an OPO, the situation is somewhat diﬀerent because the response of the crystal is assumed to be instantaneous (see Eqs. 5). Nevertheless, meanﬁeld equations can be also d erived from the ML1 equations (9) and the boundary conditions (11) if the ﬁeld amplitudes are slowly varying du ring a round-trip time ,i.e.τ(dAj/dt)<< A j(t) [32], [33]. 4In the conventional TRO case ( \|χ\|= 0 , ∆ j<<1, κj<<1),nontrivial homogeneous stationary solutions exist for the idler and the signal only if the following relation holds [29], [30] ∆= ∆1/κ1= ∆2/κ2. (12a) and the intracavity pump intensity Ip=\|Ap\|2is clamped to the threshold input intensity, Ith=κ1κ2+ ∆1∆2 (12b) whatever the input intensity \|Ain\|2might be above the threshold. In the case of the DRO, the relat ion (12a) still holds. But solutions for the signal and idler intensities ca n be determined only if the expansion (9) is continued up to the cubic terms, leading to . I1= 2κ2/bracketleftbigg/radicalBig 1 + (1+ ∆2)(Ip/Ith−1)−1/bracketrightbigg (12c) I1/I2= ∆2/∆1=κ2/κ1 (12d) cot(ϕp−ϕ1−ϕ2) =−∆ (12e) The bifurcation is supercritical for any detuning . Besides , the phase diﬀerence ϕ1−ϕ2is an undetermined quantity while the sum phase ϕ1+ϕ2depends on the pump laser phase [21]. Actually, this classic al phase indetermination is compatible with the result of the quantum ﬂuctuation theory of parametric oscillators [13]. III. SPL-DRO CASE A. Stationary solutions In the case of a DRO, only the signal and idler waves resonate w ith the cavity while the pump wave is a travelling wave ( tp= 1,rp= 0). Assuming κ1,2≪1,∆1,2≪1,Eqs.(9-11) provide the stationary solutions, Aj(t+τ,0) = Aj(t,0)≡Aj,which are also those of the meanﬁeld model, Ap=Ain (13a) (κ2−i∆2)A2=iApA∗ 1+iχ∗A2 1 (13b) (κ1−i∆1)A1=iApA∗ 2+iχA2A∗ 1 (13c) Eqs (13) can be conveniently solved by setting Aj=αjexp(iϕj), giving rise to (κ2−i∆2)α2=iαpα1exp(iµ) +i\|χ\|α2 1exp(−iη) (14a) (κ1−i∆1)α1=iαpα2exp(iµ) +i\|χ\|α1α2exp(+ iη) (14b) µ=ϕp−ϕ2−ϕ1 (14c) η=ϕ2−2ϕ1−ξ (14d) The non-trivial solutions ( Aj/negationslash= 0) can be easily handled by setting X= (\|χ\|α1)2andIp=α2 p.Then the scaled idler intensity is found to be solution of X2−2(Ip−κ1κ2+ ∆1∆2)X+\|(κ1−i∆1)(κ2+i∆2)−Ip\|2= 0 (15) Let us note that the coupling terms in Eqs.(14) allow to obtai n the non-trivial solutions at the ML1 approximation order, unlike for the conventional DRO. Three cases have to be distinguished in order to solve Eq .(15), ∆ 1,2= 0,∆1∆2/negationslash= 0 (but with ∆ 1∆2>0),and ∆1,2= 0, ∆ 2,1/negationslash= 0. For ∆ 1∆2/negationslash= 0, there are two solutions X±= (\|χ\|α1)2 ±=Ip−κ1κ2+ ∆1∆2±2/radicalBig ∆1∆2[Ip−I0] (16) for any input Ip≥I0,where I0is the input intensity at the saddle-node bifurcation 5I0=1 4/parenleftBigg κ1/radicalbigg ∆2 ∆1+κ2/radicalbigg ∆1 ∆2/parenrightBigg2 (17) while the threshold intensity, determined by dX−/dIp= 0, is Ith=I0+ ∆1∆2 (18) which is minimum for ∆ 1/κ1= ∆2/κ2and equal to κ1κ2+ ∆1∆2,like for the conventional DRO. The signal intensity I2is related to I1via the relation (α1/α2)2= ∆2/∆1 (19) deduced from Eqs.(14a-b), so that the two detunings must hav e the same sign. The solutions X+andX−are represented by the lines a and a’, respectively, in Fig. 2 that displays a subcritical bifurcation: Indeed, only the stationary solution X+is stable ( solid line a ) and extends from the saddle-node int ensity I0to the Hopf bifurcation intensity IH,above which the solution is periodic (dashed portion) . The s tationary solution X−is marginally unstable (dashed line a’), whatever the input intensity might be [See Appendix]. Let us notice that the conventional DRO detuning condition, ∆= ∆1/κ1= ∆2/κ2does no longer necessarily hold and hence the SPL-DRO can oscillate with a wider detuning ran ge. This result is similar to the result of a 2:1 degenerate SPL-DRO, in the case of a linear coupling. Howeve r, this latter case does not display any subcriticality: The linear coupling gives rise instead to two self phase-loc ked states corresponding to two distinct thresholds, for a given pumping rate [21]. The phases µandηfulﬁll the relations sinµ=−1 2αp/bracketleftBigg κ1/radicalbigg ∆2 ∆1+κ2/radicalbigg ∆1 ∆2/bracketrightBigg =−/radicalBigg I0 Ip(20a) sinη=−1 2\|χ\|α1/bracketleftBigg κ1/radicalbigg ∆2 ∆1−κ2/radicalbigg ∆1 ∆2/bracketrightBigg (20b) withαpcosµ+\|χ\|α1cosη=−/radicalbig ∆1∆2 (20c) Eqs.(20) completely determine the absolute phases of the su bharmonic waves, which are hence self-locked. These relations on the idler and signal phases, while similar to th ose in ref [21], will be shown to present substantial diﬀerences. As a consequence of the self-locking, highly ph ase-coherent subharmonic outputs are expected from the SPL-DRO. In the case ∆ 1,2= 0, the bifurcation is supercritical with X=Ip−Ith (21a) Ith=κ1κ2 (21b) The real and imaginary parts of Eqs. (14a-b) give rise to sinµ=−1 2αp/bracketleftbigg κ1α1 α2+κ2α2 α1/bracketrightbigg (22a) sinη=−1 2/radicalbig Ip−κ1κ2/bracketleftbigg κ1α1 α2−κ2α2 α1/bracketrightbigg (22b) αpcosµ+/radicalbig Ip−κ1κ2cosη= 0 (22c) Eq.(22c) implies that cos µand cos ηare either of opposite signs or simultaneously nil. The ﬁrst case corresponds to α1 α2=κ1 κ2= 1 with sin η= 0 and sin µ=−κ1,2 αp(23) The case cos µ= cos η= 0 is compatible only with κ1/negationslash=κ2, leading to 6α1 α2=/radicalbiggκ2 κ1/parenleftBig√ N∓√ N−1/parenrightBig (24a) sinη=±1 and sin µ=−1 (24b) where N=Ip/κ1κ2is the pumping rate. Eq.(24a) displays a dependence ofα1 α2on the pump amplitude, unlike eq.(19) and consequently the solutions are not continuous when ∆ 1,2→0,in the general case κ1/negationslash=κ2. For ∆ 1,2= 0,∆2,1/negationslash= 0,Eq. (15) has no real solution. Actually the exact solutions w ill be shown to display a time-dependent regime. Finally let us note that the signal and idler intensities are inversely proportional to \|χ\|2as displayed by Eq.(16). This dependence is a consequence of the ML1 approximation, E qs.(9). (The limit \|χ\| →0 is irrelevant since the meanﬁeld equations (13) are misleading for the conventiona l DRO). B. Numerical results The time evolution of the ﬁeld amplitudes is obtained by solv ing the propagation equations (5-6) with the boundary conditions (11) for a given input amplitude and small initia l signal and idler amplitudes. The solutions are obtained by numerical integration of Eqs. (5-6) using of a fourth-ord er Runge Kutta algorithm, until convergence is achieved. In most of the calculations, the cavity loss coeﬃcients are t aken constant and equal, κ1,2= 0.005,and the SHG phase mismatch ξ= ∆kL2/2 = 0, unless otherwise stated. The input amplitude, the detu nings and the coupling constant are varied. •Stationary solutions As predicted by the linear stability analysis of the meanﬁel d solutions, a single solution, corresponding to X+,occurs by numerical integration for an input amplitude above the th reshold given by Eq.(18), where the trivial solution is unstable. In a general manner, the signal and idler intensit ies satisfy the relations predicted by the mean-ﬁeld, eithe r for non-zero detunings in Eq. (19) or vanishing detunings in Eqs. (23-24). Nevertheless, for zero detunings, numerical intensities agree with the meanﬁeld intensities, only for s mall coupling parameters ( \|χ\|/lessorequalslant0.15). As already pointed out, there is no continuity for the intensities when ∆ 1,2→0 in the case κ1/negationslash=κ2. The exact stationary amplitude \|χ\|α1and its stability domain depend on the magnitude of \|χ\|as displayed in Fig. 2 by line bfor\|χ\|= 0.05 and line cfor\|χ\|= 0.2.In this latter case, the exact solution is close to the meanﬁe ld one, the stationary solution extends almost from I0in the subthreshold domain, reachable only by backward adia batic decrementation of the pump intensity, to the Hopf bifurcati on threshold intensity Inum H, slightly larger than the meanﬁeld value .As\|χ\|decreases, the domain of stability is shortened below thres hold ( Inum 0> I0),but it is enlarged above threshold, because the Hopf bifurcation threshold is shifted towards larger input intensity, Inum H> IH( For κ1,2= ∆1,2= 0.005,the meanﬁeld predicts IH= 2Ith, while the numerical values are Inum H≃2.6Ithfor\|χ\|= 0.2 and Inum H>4Ithfor\|χ\|= 0.05). Subcriticality is also evidenced in Fig.3 , when starting the numerical integration from ∆= 0 and performing an adiabatic increase of ∆,for a ﬁxed input pump intensity Ip= 4κ1κ2.The diﬀerent curves in solid line correspond to diﬀerent values of \|χ\|. They are symmetric for negative detunings, so that the tuni ng curve of the SPL-DRO appears as a widened double-sided fringe, which reminds of resonant ly phase-modulated Fabry Perot devices (the same kind of fringe is obtained from a resonator containing an electro -optic phase modulator driven by a RF oscillator whose frequency is equal to the resonator free spectral range [34] ). In the case of the conventional DRO ( \|χ\|= 0, dashed curve), lasing begins at ∆= 0 where the intensity is maximum as seen in Eq.(12c) and stop s for∆=√3; indeed, the bifurcation is supercritical so that lasing may occur only i f the input pump intensity is larger than the threshold value κ1κ2+ ∆1∆2.The case \|χ\|= 0.01 (curve a), corresponding to nascent bistability, displa ys lasing, approximately in the same range of detuning as in the \|χ\|= 0 case, with the signiﬁcant diﬀerence that there is self pha se-locking (see below on Fig.6). As \|χ\|increases further, the bifurcation becomes subcritical, t he saddle-node moves away from the threshold, approaching the meanﬁeld location, independen tly of ∆. The intensity reaches the same maximum Imaxfor any 0≤ \|χ\| ≤0.25,but at a detuning ∆max(\|χ\|),for which the pump is entirely depleted; then the intensity d ecreases to zero when increasing further the detuning. The maximum Imaxis equal to the maximum conventional DRO intensity, which occurs at ∆max= 0. In case of the parameters used in Fig.3, Eq. (12c) gives ri se to√Imax= 0.1,in agreement with the numerical result. The detuning ∆maxbecomes approximately proportional to \|χ\|for large enough coupling strength. For \|χ\| ≥0.25,the idler intensity becomes time-dependent before ∆maxis reached. In summary, the cascaded SHG nonlinearity induces a \|χ\|−dependent phase that shifts the optimum detuning from 0 to ∆max. 7The temporal response of the system subject to stepwise-lik e detuning jumps has been also studied in order to check the stability of the subthreshold states against external p erturbations that tend to modulate the detuning parameter (via the cavity length for instance). Figure 4 , associated with \|χ\|= 0.2, shows that the system recovers its steady state operating point as long as the perturbation amplitude does not exceed ±0.12∆max.Note the longer decay time, which is characteristic of a critical slowing-down phenome non, on the positive detuning step side that brings the DRO very close to the saddle-node bifurcation. We can deﬁne a self-locking domain in the (∆ 1,∆2) plane over which the DRO is self phase-locked with a well-de ﬁned phase relationship. Contour plots of the numerical solutio ns for the idler I1, the signal I2, the total subharmonic intensity I1+I2and the phase diﬀerence ηare shown in Figs 5 for\|χ\|= 0.2 and for a given pumping rate N= Ip/κ1κ2= 4 . The subharmonic intensities are scaled to the conventio nal zero-detuning DRO values as deduced from Eqs.(12c), IDRO 1,2= 2κ2,1. These contour plots are obtained by adiabatical following of the stationary solutions in order to reveal the 2D subthreshold domain. The granular sma ll regions adjacent to the ∆ 1= 0 axis correspond to a time-dependent regime. The I1andI2intensity distributions in Figs 5a-5b display oﬀ-diagonal maxima that are larger than the conventional DRO signal and idler outputs ( I1,2/IDRO 1,2= 1.6). Note also that the exact intensities I1and I2are not invariant with respect to the product ∆ 1∆2,diﬀerently from the predicted meanﬁeld intensities (16) bu t they satisfy the relation (19). The ﬁgure (5c) shows that the maximum total intensity, slightly larger ( I1+I2= 2.2 ) than the conventional DRO total intensity, I1+I2= 2, occurs in the subthreshold domain close to the saddle-no de bifurcation (see diagonal line d in Fig.3). These curves sho w that the relative phase control of the subharmonics can be achieved via the control of the relative output intensiti es, provided that an independent control of both cavity detunings is implemented (see section V). •Phase-locking The numerical phase ηrelated to the diﬀerence between the signal and idler phases agrees extremely well with the mean-ﬁeld prediction in Eqs. (20b),( 22b) or (23),(24b) for any values of the detuning and cavity loss parameter, but the numerical phase µrelated to the sum of the signal and idler phases is found to de part signiﬁcantly from the mean- ﬁeld value (20a) when ∆ 1,2/κ1,2/followsorequal1. We have checked numerically the self phase-locking eﬀect of the signal-idler output, by varing randomly the initial phase of the signal an d idler noise. This general result is illustrated in Fig.6 , for ∆ 1/κ1= ∆2/κ2where the phase diﬀerence ηfollows randomly the initial values (blank circles), when \|χ\|= 0, as expected for a conventional DRO. However, yet for a vanishin gly small value S= 0.001, the phase diﬀerence ηlocks to 0 or π(mod kπ) forξ= 0 and the sum phase locks to a constant value determined only by the pumping rate. •Time-dependent solutions For ∆ 1,2/negationslash= 0 and ∆ 1/κ1= ∆ 2/κ2,the periodic solution remains generally stable on a large ra nge of the input intensity. For instance, for \|χ\|= 0.2 and ∆ = 1 (case of line c in Fig.2) ,the periodic solution is stable until the pump parameter reaches αp/αth∼12 , well above any experimentally achievable input. The per iod of the amplitude oscillation in the vicinity of Inum His found to be T∼300τ. But the case \|∆1/κ1−∆2/κ2\| /negationslash= 0 leads to a more complex dynamics, which depends on the nonlinear coupling \|χ\|and departs from the meanﬁeld predictions . In the zero-detuning case, no time-dependent solutions are found for realistic pumping rate ( N < 25) for equal cavity loss κ1=κ2. Diﬀerently, the case κ1/negationslash=κ2displays time-dependent regimes, for \|χ\|/followsorequal0.2. In the case κ1/κ2= 2.5,\|χ\|= 0.2 shown in Fig.7, a periodic regime arises at Iin∼=2Ithand a weak chaotic pulsating behaviour is found at Iin∼=4Ith, with a pulsating signal intensity. Vanishingly small detu nings (∆ 1,2) however give rise to steady state solutions satisfying relations (16). The numerical solutions associated with ∆ 1,2= 0,∆2,1/negationslash= 0,are not stationary, as predicted by the meanﬁeld model; they are periodic with time for any input above thresh old. •Inﬂuence of SHG phase-mismatch We have studied the inﬂuence of a moderate SHG phase mismatch (ξ/negationslash= 0) on the dynamical behaviour of the system. This phase mismatch is usually controlled by the tem perature or the angular orientation of the SHG crystal. It provides a control of the strength of the nonlinear coupli ng parameter \|χ\|, via the relation (10), and oﬀsets the value to which the phase diﬀerence ηlocks (Fig.6). For instance, a small phase ξ≤1 does not signiﬁcantly change the idler amplitudes of Fig.3 but induces a slight imbalance of t he signal and idler intensity ratio compared to the ratio (19) and slightly reduces the self-locking range. The relat ionµ= 0 (mod π) is still valid and ηlocks to ξ. In some undesirable operating conditions, for instance, when the s ystem approaches a Hopf bifurcation for a given pumping rate and detuning set (see Fig.2 for S= 0.2, ξ= 0), a small amount of phase mismatch produces a shift of the p eriodic oscillation threshold towards higher pump rate. Diﬀerentl y, for ∆= 0 and the same other parameters as for curve (c) 8of Fig.2, a small phase mismatch ( ξ= 0.1) leads to a slow periodic regime ( T∼4×10+6τatαin/αth= 1.2) which is suppressed when vanishingly small detunings (always pre sent in practical devices) are introduced. From the simulations carried out, we conclude that a moderat e SHG phase-mismatch does not change signiﬁcantly the main bifurcation dynamics studied up to this point, exce pt for the ∆ 1,2= 0 case. Finally let us point out that the SPL-DRO solutions expanded up to the second-order in Eq. (9) (ML2 approxima- tion) agree very well with the exact solutions in almost the w hole range of the pump intensity and detunings below the Hopf bifurcation threshold, except for the singular cas e ∆1,2= 0 corresponding to the transition from subcriti- cality to supercriticality, where it fails to converge. Onl y the propagation model can solve this case, which requires a double-precision computation due to the slow convergence a ssociated to a critical slowing-down phenomenon. IV. SPL-TRO CASE In the case of SPL-TRO, the meanﬁeld equations (13b)-(13c) a re still valid, except that Ip=α2 pdenotes now the circulating intracavity pump intensity. The intracavity p ump amplitude obeys the boundary condition (11a), leading to [1−rpexp(i∆p)]Ap−irpexp(i∆p)A1A2=Ain (25) where the pump detuning may be arbitrarily large (∆ p≤2π) . The relation between the input pump parameter Iin=\|Ain\|2=α2 in,and the cavity pump intensity is obtained, for an arbitrary p hase of the input ﬁeld Iin= (1 + r2 p−2rpcos∆ p)α2 p+r2 pα2 1α2 2+ 2rpα1α2αp[sin(∆ p−µ) +rpsinµ] (26) where α1,α2andµcan be deduced from Eqs.(16) and (19-20). The threshold for o scillation is easily found from Eq. (26), on the basis of simple considerations. Below the th reshold, where α1,2= 0,Ipgrows linearly as a function ofIin. When it reaches the value given by Eq.(21), oscillation sta rts, which leads to the following threshold input intensity ISPL−TRO th= (1 + r2 p−2rpcos∆ p)(I0+ ∆1∆2) (27) In a practical TROs, although the pump ﬁnesse is lower than th e ﬁnesse at the subharmonic waves (typically an order of magnitude), the condition κp≪1 is still satisﬁed . If we consider a small enough pump detuni ng and the speciﬁc case ∆= ∆1/κ1= ∆2/κ2, Eq.(27) reduces to the conventional TRO threshold [30] ITRO th=\|Ain\|2 th=κ2 pκ1κ2(1+∆2)(1+∆2 p) (28) where ∆p= ∆ p/κp.Fig. 8 shows the normalized intra-cavity pump and idler bifurcati on diagrams as a function of/radicalbig Iin=/radicalBig Iin/ITRO thwhen condition (12a) holds and for ∆ p= 0. Only the details of the \|χ\|α1solutions in the vicinity of the threshold are plotted in the top frame of Fig .8 that displays the subcriticality. The numerical and meanﬁeld solutions of the idler amplitude as given by Eq. (16 ) are confounded over a much larger input intensity range. The thick solid and dashed lines for αpcorrespond to the meanﬁeld solution (26), while the thin sol id line is the exact numerical pump solution. Notice that the numerica l solution agrees very well with the meanﬁeld solution. As the input parameter is increased from threshold, Ipdecreases from the clamped value of the conventional TRO to a minimum I0(Eq.17) for an input pump intensity Imin ingiven by Imin in=/bracketleftbigg 1 + (κ1/κp)(− ∆/\|χ\|)2/bracketrightbigg2 /(1+− ∆2 ) (29) for ∆ p= 0. The meanﬁeld model hence predicts that, unlike for conve ntional TRO, the intracavity pump intensity is not clamped. For− ∆→0,Imin in→1, e.g. there is a transition to supercriticality as in the SP L-DRO case. In pump-detuned conventional TROs, subcriticality occurs only when the condition ∆p∆>1 holds [29], [30]. The experimental observation of this subcriticality requires a high pumping level because of the pump detuning dependence in Eq.(28) [35] . It is interesting to study how the subcritic ality originating from the nonlinear OPO/SHG cascading would aﬀect the intrinsic detuned TRO bistability curve. In Fig.9 we have plotted the numerically computed idler intensity versus the normalized input intensity for \|χ\|= 0,0.2,0.6 and ∆p∆= 2 . The subthreshold domain extends 9as\|χ\|increases from zero, it has also increased, when compared to the∆p= 0,\|χ\|= 0.2 case, shown in Fig. 8. Furthermore, as \|χ\|increases, the numerical threshold is smaller than the mean ﬁeld threshold given by Eq.(27). We have also veriﬁed that the numerical amplitudes and phase s agree with the meanﬁeld predictions for Iin<10 and for detuning values such that ∆ j/κj≼1. Far above threshold and for larger detunings, only the pha se diﬀerence relations remain exact, like in the SPL-DRO case. Furthermore, the detuning range ∆for self-locking is less dependent on the coupling strength than in the SPL-DRO case. The high-pump-ﬁnesse SPL-TRO device does not exhibit the ti me-dependent periodical solutions observed above threshold with the SPL-DRO as long as \|χ\| ≤0.52 (which corresponds roughly to equal OPO and SHG crystal le ngth for a PPLN) and for realistic maximum pumping rate− Iin=Iin/ISPL−TRO th≤2500, corresponding to a Watt-level pump power and typical TRO pump thresholds in the mW range. Fo r larger coupling \|χ\|= 0.6,periodic oscillations occur, with a period varying in a very complex way when the inp ut pump is increased. Nevertheless no chaotic regime is observed in the range of considered input pump. In many practical experiments, one cannot avoid a weak pump r esonance in DROs due to the multiwavelength coatings of the mirrors. It is thus interesting to investiga te how a moderate pump resonance would aﬀect the (∆ 1,∆2) self-locking domain of Figs.5. The input external pump inte nsity is kept equal to Iext in= 4κ1κ2to provide a comparison with Figs.5 ( \|χ\|= 0.2). Such a pumping level corresponds to an internal TRO pumpi ng rate− Iin/ITRO th= 8/κp= 160. In Fig. 10, the pump reﬂectivity is rp= 0.8 with ∆ p= 0. Apart from the slight enlargement of the self-locking domains, as compared to Fig.5, which is mainly attributed to the high pumping rate, the intensities are smaller than for the DRO, also the intensity distributions are strongly m odiﬁed, with oﬀ-diagonal maxima. In summary, the only improvement due to a moderate pump reson ance is the stabilization eﬀect respective to the onset of temporal dynamics. The extended self-phase lockin g range is paid back with lower output intensities. V. PRACTICAL IMPLEMENTATION OF SPL-OPOS In order to avoid spurious cavity loss, the use of a dual-grat ing quasi-phase matched periodically poled crystal is particularly well suited to the implementation of SPL-OP Os. However, the grating period should be accurately designed so as to phase-matched simultaneously both intera ctions for the same chip temperature, even though the theoretical analysis predicts a minor inﬂuence of SHG phase mismatch. From an experimental point of view, it is desirable to implem ent 3:1 SPL-OPOs using the least constraining OPO conﬁguration. Diagnosis methods to check the high phase coh erence between the subharmonics under SPL have to be implemented. In the frequency domain, this can be achieved b y monitoring the beatnote between the signal wave and the externally frequency-doubled idler wave (or between th e summed subharmonics and the pump). Because under SPL operation these two waves are frequency degenerate, the output signal wave must be preliminary frequency- shifted by a suitable RF frequency ωRFusing, e.g., an acousto-optic modulator. When the OPO opera tes within the locking range, the beatnote frequency should be ﬁxed to ωRFand its power spectral density should approach a Dirac function. Another equivalent method to check the phase cohe rence is to perform an interferometric fringe pattern measurement by overlapping the two beams on a slow detector [ 17]. In the following, other indirect methods, based on the theoretical analysis will be outlined. As a starting point of the analysis, we have assumed perfect 3 :2:1 frequency ratios for the pump, signal and idler waves. In practical devices, such a situation will be unlike ly met at once. The major diﬃculty will come from the ﬁne (continuous) tuning of the signal and idler frequencies close enough to the 2:1 degeneracy, in order to fall within the capture range of the self-phase locking. For a ﬁxed pump f requency, such a ﬁne tuning is usually performed via the temperature or angle tuning of the phase-matching. S ingly resonant devices (PRSROs) oﬀer an easy and relaxed mode-hop free frequency tuning because only one sub hamonic wave is resonant, especially in a dual-arm cavity conﬁguration to control independently the pump and s ignal resonances [36]. However from our analysis of the SPL-PRSRO with a resonant signal wave, which also displays s ubcriticality, extremely small self-locking ranges are predicted due to the weak coupling between the injecting fre quency-doubled idler and the resonant signal. While in a conventional PRSRO the signal ﬁeld is constrained to oscil late with a nil cavity detuning, the nonlinear coupling is found to allow oscillation over a small detuning range not exceeding the cavity linewidth. The increase of the locking range versus the coupling parameter (e.g. the SHG cr ystal length) is only moderate, even with \|χ\|= 1 (which would correspond to a SHG crystal ∼1.5 times longer than the OPO crystal supposing that both OPO and SHG interactions have the same nonlinearity magnitude) . Due to this limited capture range, experimentally conﬁrmed in ref [18], 3:1 SPL-PRSROs would probably require an addition al electronic servo on the cavity length to operate as a stable divider device. A convenient criterion for the asses sment of SPL in PRSROs would be the slight enhancement 10of the signal intensity when the nonlinear coupling is switc hed on, compared to the slight decrease predicted for the SPL-DRO (Fig.3 ). A possible way to realize such a switching i s to have an OPO-only grating section patterned beside the OPO/SHG dual-grating section on the periodicall y-poled wafer. From our theoretical study, the widest SPL range (a few cavit y linewidths) is obtained with the doubly resonant conﬁguration (SPL-DRO) due to the strong self-injection re gime, with eventually a weak pump resonance (SPL-TRO) to stabilize the device. Even though the amount of nonlinear coupling required can be extremely small ( \|χ\|= 0.01 ) - and such low level doubled idler can even be spontaneously g enerated via non-phase matched or higher-order quasi- phase matching in PP single-grating nonlinear OPO crystals - a coupling strength corresponding to \|χ\|= 0.1−0.2 will ensure a robust self-phase locking of the subharmonic wave. Preliminary single mode-pair operation of DRO/TROs usually require an intensity sidelock servo to control the s tability of the cavity length. This sidelock servo compares the output signal (or idler) intensity to a stable electroni c voltage reference which sets the operating (usually non-z ero) signal and idler cavity detuning values. When a linear cavit y is used, the signal and idler (plus eventually the pump) detunings cannot be independently controlled via the cavit y length. It is then probable that the oscillating mode pair will have equal normalized detunings (condition (12a)) tha t satisfy the minimum threshold. Under sidelock servo the transition from conventional to SPL states is accompanied n ecessarily by a detuning transition (see Fig.3). It is then important to set the sidelock reference voltage as close as t o the maximum fringe intensity in order that the new detuning value does not exceed the allowed subthreshold ran ge. In the case of a well resolved DRO mode pair cluster, the observation of these subthreshold states, and the assoc iated broadened mode pair fringe, should be made possible via adiabatic cavity length tuning. Such an observation wou ld be an indirect diagnosis of SPL. But it is necessary to have an independent control of the signal and idler detuning s to explore the full allowed range of (∆ 1,∆2) detunings depicted in Figs.5 and 10. A dual-cavity DRO/TRO design woul d then be appropriate. The control of these detunings allows the control of the relative phase between the pump, si gnal and idler. The output subharmonic intensity ratio I1/I2(see Eq.19) can be used as an error signal for the relative pha se control. VI. CONCLUSIONS AND OUTLOOK We have theoretically demonstrated that resonant χ(2):χ(2)nonlinear cascaded OPO/SHG processes induce a self injection-locking between the subharmonic waves of an OPO l eading to the self-phase locking of the three interacting waves, unlike in a conventional OPO for which the absolute ph ases of the signal and idler are undetermined. The theoretical treatment encompasses the meanﬁeld model as we ll as a full propagation model. The doubly and triply resonant oscillator conﬁgurations lead to the widest self p hase-locking ranges. The main conclusions are: a) the minimum threshold of oscillation of these devices are ident ical to the conventional devices; b) the nonlinear cascadin g leads to a subcritical behaviour, even in the case of a DRO or a PR-SRO, and can lead to the occurrence of self-pulsing instabilitities. This subcriticality is diﬀerent from the standard subcriticality reported in pump detuned conventi onal TROs. The nonlinear coupling removes the detuning constrai nts of conventional systems, allowing for a potentially accurate control of the relative phase between the subharmo nic waves. The range of allowable detunings over which the ﬁeld phases are locked depends on the magnitude of the non linear coupling and on the self-injection regime. While this range is smaller than the cavity linewidth for PRS ROs (weak injection regime), it spans over several cavity linewidths under strong injection-locking regime o btained in signal/idler resonant devices (DRO/TROs). The SPL-DRO/TRO give rise to a richer dynamics than the singly re sonant PRSRO for which no occurence of a Hopf bifurcation is found in the whole range of the system paramet ers. It is thus interesting to extend the theoretical model by including diﬀraction eﬀects in order to investigat e the possible occurrence of new spatio-temporal dynamics, [33], [37]. These self-phase locked OPOs will be useful tool s for applications requiring a high degree of optical phase coherence between optical harmonic waves, such as precisio n optical measurements in the mid-IR or Fourier synthesis of ultra-short optical pulses. The model developed can be easily extended to the study of div ide-by-4 SPL-OPOs based on the cascading OPO/OPO processes 4 ω⇄2ω⇄ωwhich has the potential to generate up to 8 phase-locked harm onic waves by additional up-conversion processes. Such a strong nonli nearly coupled system can be viewed as a secondary degen- erate OPO (DOPO) embedded in a primary DOPO. Our future work w ill be directed to the theoretical investigation of the stability of 4:1 OPO dividers. A classical signature o f the system, derived from the meanﬁeld analysis, is the clamping of the secondary pump 2 ωto the threshold power for the fundamental oscillation. The present study of 3:1 OPO dividers, which makes the simple st assumption of exact 3:2:1 frequency ratios, arouses another interrogation. An interesting situation not consi dered regards the behaviour of the nearly 3:1 OPO/SHG system when the frequency ratios slightly departs from the p erfect 3:2:1 division ratio by a radio-frequency quantity δ << ω , e.g. when 3 ω→2ω−δ, ω+δ. The frequency diﬀerence beween the signal wave and the doub led idler is then\|2ωi−ωs\|= 3δ. When the doubled idler frequency does not match one of the ca vity eigenmode frequencies, the 11operation of the OPO would be merely that of a conventional DR O. But if the OPO is tuned such that 3 δ=FSR s≈ c/Λ, FSR sbeing the signal free spectral range of the cavity, then the d oubled idler will be enhanced to a point where it may lead to the creation of a new mode pair with frequencies (2ω+ 2δ, ω−2δ) and so on. The parametric gain bandwidth of OPOs extending usually over several THz or seve ral tens of THz (in case of a wavelength non-critical phase matching), a multitude of self-phase locked mode pair s equally spaced by 3 δmay potentially oscillate, provided that the pumping level is suﬃciently high. Such a complex sys tem opens the prospect of building a mode-locked, dual-band OPO frequency comb generator using cascaded seco nd-order nonlinearities as the passive mode-locking mechanism. In the time domain, the output of such a system wou ld consist of a train of short optical pulses with a repetion rate set by the FSR sintermode spacing, provided that the relative phase betwee n adjacent mode pairs is preserved and group velocity dispersion is compensated. We note that a similar cw-DRO running near frequency degeneracy with thousands of mode pairs actively locked by a n intracavity electro-optic phase modulator has been recently reported, with a striking passive output stabilit y feature compared to a conventional quasi-degenerate sing le- mode pair DRO [38]. Acknowledgement 1 This work is partially supported by an INCO-Copernicus Euro pean network program (contract n◦ERBIC 15CT 98 0814 ). VII. APPENDIX I: LINEAR STABILITY ANALYSIS OF THE SPL-DRO With Aj(t)≡Aj(t,0),Eqs. (9-11) give rise to the mapping equations A2(t+τ) =r2ei∆2[A2(t) +iAp(t)A∗ 2(t) +iχ∗A2 1(t)]] (A1) A1(t+τ) =r1ei∆2[A1(t) +iAp(t)A∗ 1(t) +iχA2(t)A∗ 1(t)] (A2) Ap(t) =Ain (A3) the stationary solutions of which are deduced for large r1,2and small ∆ 1,2.(See Eqs.(19),(22)-(23)). The linear stability analysis consists in assuming small de viations δA1,2(t) from the stationary solutions A1,2, A1,2(t) =A1,2+δA1,2(t) (A4a) δA1,2(t) =/summationdisplay λδA1,2(λ)eλt(A4b) where λmay be complex. The stationary solutions A1,2are stable only if the real part of any λis negative. At the instability threshold, therefore the system may underg o a Hopf bifurcation ( λ0=±iβ),so that the intensitiy oscillates with time at angular frequency β. Eqs. (A1-A4) lead to a linearized system of four equations, t he determinant of which satisﬁes D=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΛ +α1−iχ1A2−iχ1A∗ 1−iχ1Ain,1 iχ∗ 1A∗ 2Λ +α∗ 1iA∗ in,1iχ∗ 1A1 −2iχ∗ 2A1−iAin,2Λ +α20 iA∗ in,22iχ2A10 Λ + α∗ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0 (A5) with the notations Λ =eλτ−1 (A6) and α1,2=κ1,2−i∆1,2 (A7a) χ1=ei∆1χr1, χ2=e−i∆1χr2 (A7b) Ain,1,2=r1,2ei∆1,2Ain (A7c) 12Then, the eigenvalues Λ are solutions of the quartic charact eristic equation Λ4+ Φ3Λ3+ Φ2Λ2+ Φ1Λ + Φ 0= 0 (A8) where all the coeﬃcients Φ iare real Φ3=−2(κ1+κ2) (A9a) Φ2=\|α1\|2+\|α2\|2+ 4κ1κ2−2r1r2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2cos(∆ 1−∆2)−/vextendsingle/vextendsingleχr2A2/vextendsingle/vextendsingle2 +4r1r2/vextendsingle/vextendsingleχA1/vextendsingle/vextendsingle2cos(∆ 1+ ∆2) (A9b) Φ1= 2r1r2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2[(∆1+ ∆2)sin(∆ 1+ ∆2)−(κ1+κ2)cos(∆ 1−∆2)] + 2( κ2\|α1\|2+κ1\|α2\|2) −2κ2/vextendsingle/vextendsingleχr2A2/vextendsingle/vextendsingle2+ 4/vextendsingle/vextendsingleχr1r2A1/vextendsingle/vextendsingle2[(κ1+κ2)cos(∆ 1+ ∆2) + (∆ 1+∆2)sin(∆ 1+ ∆2)] −ir1r2\|χ\|2cos∆ 2(AinA∗ 2A∗ 1−c.c.) + 3r1r2\|χ\|2sin∆ 2(AinA∗ 2A∗ 1+c.c.) (A9c) Φ0=\|α1\|2\|α2\|2−2r1r2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2[(κ1κ2+ ∆1∆2)cos(∆ 1−∆2) + (∆ 1κ2−∆2κ1)sin(∆ 1−∆2)] +4r1r2/vextendsingle/vextendsingleχA1/vextendsingle/vextendsingle2[(κ1κ2−∆1∆2)cos(∆ 1+ ∆2) + (∆ 1κ2+ ∆2κ1)sin(∆ 1+ ∆2)] −/vextendsingle/vextendsingleχr1α2A2/vextendsingle/vextendsingle2+/parenleftBig 4/vextendsingle/vextendsingleχr2A1/vextendsingle/vextendsingle2−r2 2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2/parenrightBig/parenleftBig/vextendsingle/vextendsingleχr1A1/vextendsingle/vextendsingle2−r2 1/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2/parenrightBig (A9d) . The roots for Λ and consequently the eigenvalues λhave been calculated for various detunings ∆ 1,2as a function of the input amplitude Ainfor the trivial solutions A1,2= 0 and the non trivial solutions ( A1,2)±deduced from Eqs.(16) and (20). The trivial solution is veriﬁed to be stable for any input pum p intensity below Ith(Eq.18). On the other hand the solution X−is found to be unstable with respect to any constant perturba tion for any detuning and input pump intensity: Indeed, in this case, Eq. ( A8) has a root Λ = 0 ,i.e,λ= 0,because the constant coeﬃcient Φ 0is identically nil, when replacing X−by its expression (16). Diﬀerently, the upper branch displays two sets of complex co njugate eigenvalues, ( σ1,2±iβ1,2) with negative real parts for a range of the input pump intensity, lying from I0toIHwhere the system undergoes a Hopf bifurcation (σ1= 0). For Ip> IH,the signal and idler intensities oscillate with time at angu lar frequency β1. Therefore the upper branch is stable for I0<Ip< IH. The Hopf bifurcation threshold intensity IHmay be either smaller or larger than the threshold intensity Ith, depending on detunings ∆ 1and ∆ 2. Let us introduce the parameter µHthat measures the departure between the threshold for lasing Ithand the Hopf bifurcation threshold as µH= (Ith−IH)/Ith (A10) The variation of µHis presented in Fig. 11 as a function ∆ 1= ∆ either with ∆ 1= ∆2in curve (a) or ∆ 1=1 2∆2 in curve (b). For ∆ 1= ∆ 2,the Hofp bifurcation ( σ1= 0, σ2<0) occurs above threshold only for ∆ /lessorsimilarκ.Therefore lasing stationary solutions can be reached from rest only for detun ings smaller than the cavity loss coeﬃcient. Otherwise, lasing at larger detunings can be reached, when increasing a diabatically the detunings from values smaller than κ. In the other case, ∆ 1=1 2∆2,µHis positive for any ∆ 1/lessorsimilar5κ. For higher detuning, σ2reaches zero at an input intensity much smaller than the value at which σ1= 0 crosses zero. This causes an abrupt decrease of IH,so that µHsuddenly becomes negative for ∆ 1/greaterorequalslant5κ,as shown in the curve (b) of Fig. 11 . Finally, the angular frequency βHat the Hopf bifurcation threshold, not reported here, is fou nd to vary propor- tionally to the detuning. 13Figure captions Fig.1: Schematic ring cavity model of SPL-OPOs. All intracavity lo sses are lumped into the output mirror trans- missivities tj. Fig.2: Comparison of the meanﬁeld and numerical stationary soluti ons of SPL-DRO, with κ1,2= ∆ 1,2= 0.005, as a function of the scaled pump input amplitude ( αth=√Ith). (a): meanﬁeld upper branch (The dotted portion corresponds to unstable solutions), (a’): meanﬁeld lower b ranch . (b): numerical solution for \|χ\|= 0.05. (c): numerical solutions for \|χ\|= 0.2. Fig.3: Numerical SPL-DRO self locking ranges as a function of the de tuning ∆ 1/κ1(= ∆ 2/κ2) , for Iin= 4κ1κ2, with \|χ\|= 0.01 in (a), 0 .05 in (b), 0 .1 in (c), 0 .2 in (d) and 0 .25 in (e). The dashed line is for \|χ\|= 0. The parameters are κ1,2= 0.005. Fig.4: Time response of the SPL-DRO stationary state operating at/braceleftBig \|χ\|= 0.2,∆opt= 4/bracerightBig , under step-wise linear detuning jumps ∆opt(1±β), with modulation index β= 0.109 (curve a). Curve (b) gives the idler amplitude response and curve (c) the phase diﬀerence response. The re-capture r ange of the detuning perturbation is βmax= 0.12. Fig.5: Contour plots of the stationary SPL-DRO signal (a), idler (b ), and total (c) intensities and phase diﬀerence η (d) in the 2D detunig plane, obtained from the numerical comp utation with κ1,2= 0.005,\|χ\|=S= 0.2,Iin= 4κ1κ2 . The intensities are scaled to I1,2= 2κ2,1. The pale blue domain are trivial solutions. Fig.6: Numerically computed distribution of stationary signal/i dler phase diﬀerence as deﬁned in eq.(14d) versus the initial random phase for ∆ 1,2=κ1,2= 0.005,αin/αth= 2 and S= 0.001, for a SHG phase mismatch ξ=π(i.e χ= 0) in blank circles; ξ= 0 in solid black circles and ξ= 3π/2 in black triangles. Note that each phase data point is associated to the same stationary signal-idler intensit ies and the same phase sum µ. Fig.7: Time-dependent numerical solutions for κ1= 0.005,κ2= 0.002, ∆ 1,2= 0,χ= 0.2,Iin= 1.96κ1κ2(bottom frame) and Iin= 4κ1κ2(top frame). Figs.8: Meanﬁeld and numerical SPL-TRO stationary solutions for ∆ p= 0,rp= 0.9 and the same other parameters as for Fig.2. The top frame shows details of the hysteresis lo op of the idler amplitude in the vicinity of the input pump threshold. The thin lines correspond to the numerical s olutions, interrupted by the vertical thin dashed line. The thick lines correspond to the meanﬁeld solutions. The th ick dashed line in the bottom frame is the meanﬁeld unstable branch of αp/αth(αth=√κ1κ2+ ∆1∆2), and the thinner horizontal dashed line shows the conventi onal TRO pump clamping. 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Lett. 24, 1747 (1999). 15This figure "fig1.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig2.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig3.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig4.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig5.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig6.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig7.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig8.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig9.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig10.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1This figure "fig11.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0011034v1
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arXiv:physics/0011036v1 [physics.gen-ph] 16 Nov 2000Why We Can Not Walk To and Fro in Time as Do it in Space? (Why the A rrow of Time is Exists?) L.Ya.Kobelev Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru Existence of arrow of time in our world may be easy explained i f time has multifractal nature. The interpretation of nature of time arrow is made on the base of m ultifractal theory of time and space presented at works [5]- [19]. In this paper shown possibilit y to walk to and fro in space and necessity of huge amount of energy for stopping time and changing direc tion of it in microscopic volumes. CONTENTS : 1. Introduction 2. Universe as Time and Space with Fractional Dimensions 3. Why Time has Direction Only to Future and Why Impossible to Walk in Time To and Fro? 4. Is It Possible to Change Direction of Time and How Much Ener gy It Needs? 5.How Much Energy Needs for Stopping Time and Moving it Back i n the Volume of Cubic Centimeter During One Second? 6. Why We Can Walk To and Fro in Our Space? 7. Conclusions 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION As is well known the time ﬂows in our world only in future (arrow of time of Eddington [1] and Prigogin [2]) and now nobody knows is it possible to change direction of the time ﬂow or is it impossible and why if it is pos- sible or impossible. The problems of arrow of time are very intrigues and many of physicians presented interest- ing models of it phenomenon (see for example [3], or new interesting experiments [4] . For analyzing this problem it is necessary to investigate the models of time in which the time is not simple time axes, but has complicated structure. The purpose of this paper is to investigate problem of time arrow in the multifractal model of time presented in fractal model of space and time at [5]- [19]. In this model the time and the space are real material ﬁelds with fractional dimensions and multifractal struc- ture (multyfractal sets) deﬁned on sets of their carriers of measure. In every time (or space) points the dimensions of time (or space) determine densities of Lagrangians en- ergy for all known physical ﬁelds ( or Lagrangian of new physical ﬁelds for space dimensions) in these points. This model allows understand reason of existence of the arrow of time: this reason has pure energetically nature and consists in necessity to use huge amount of energy for changing of direction of time.II. UNIVERSE AS TIME AND SPACE WITH FRACTIONAL DIMENSIONS In this paragraph we summary the main results of the fractal theory of time and space [5]- [19]. In this theory, when Universe was born ( the moment of ”big bang”) from vacuum (in this theory Universe was born from the set of carrier of measure which plays role of phys- ical vacuum) only material ﬁelds were born (or appear on surface of carrier of measure-vacuum): the time and the space ﬁelds with fractional dimensions. So our Uni- verse are real ﬁelds of time and space and not conclude something yet. Fractional dimensions of time and space are appearing in our world as physical ﬁelds (all physical ﬁelds, known and new ﬁelds that will be ﬁnd, are char- acteristics of time with fractional dimensions). When temperature of Universe (i.e. the temperature of ﬁelds of time and space) was changing, fractional dimensions was changing too and new physical ﬁelds were appear- ing in accordance with known theory of ”big bang” and broken symmetry ( depending at Universe temperature). The equations for physical ﬁelds appear as consequences of principle of fractal dimensions functional (FDF) mini- mum . These equations are Euler equations with general- ized fractional Riemann-Liouville derivatives (the gener - alization consists in propagating the Riemann-Liouville derivatives on domain of multifractal sets where frac- tional dimensions are functions of time and coordinates) . For case of integer dimensions these derivatives and equations coincide with ordinary derivatives and the the- ory in whole coincides with known physical theories. So 1only diﬀerence of the theory [5]- [19] from known theories consists in using the algorithm for propagating modern theories on the domain of the Universe with fractional dimensions of time and space and geometrization of all physical ﬁelds in frame of fractal geometry of world. In cited works many equations of modern physics were re- searched in fractal space and time. It was shown that in multifractal model all physical ﬁelds are geometriza- tioned. It diﬀers this theory positively from general rel- ativity theory (the latter is a special case of multifractal theory of time and space in special selection of measure carrier and integer dimensions (see [13]) ) where only one ﬁeld (gravitation ﬁeld) is geometrizationed. In the world of fractional dimensions there are many new special physical characteristics and peculiarities. The main of them are: a) in the world with fractal dimensions there are no con- stant physical values because fractional derivative with respect to constant value is not zero. So all physical values are changing in time and in space ( for example, electric charge of electron has changed from the time of big bang as δe=eε/twheret=t0+δtandδtis current time,t0-is the time of existence of Universe). Conse- quences of it are absence in this world any rigorous laws of preservation (they are fulﬁlled only as very good ap- proach, but not as rigorously laws); b) there are no inertial systems in Universe because there are no constant velocities and special Einstein theory must be replaced by theory of almost inertial systems (it coincide with special relativity theory on the surface of Earth till velocities v=c−δv, δv ∼100msec−1; On the surface of Earth the fractional dimensions of time dif- fers from unity on value ∼10−9. c) all systems of reference are absolute systems (time and space are non homogeneous and non isotropic). So Michelson experiments had proved independence of speed of light at moving of origin of light are only very good approach ( on the surface of Earth the changing of direc- tionvon−vin the fractal world gives change of speed of light on value ∼0.1cmsec−1); d) in the fractal world velocities of moving bodies may be equal speed of light in vacuum (there are no singular- ity atv=c) or exceed it and it is possible to propagate information with any wished velocities; e) there are no singularities in the theory because for V(r)∼ ∞ (Vis a potential of any ﬁelds) all fractional diﬀerential operators of the theory turns into fractional integrals; f) all physical (i.e. Newton ,Shr¨ odinger, Maxwell, Dirac, Einstein and so on ) equations of modern physics are ir- reversible; g) the laws of thermodynamics are consequences of mul- tifractal structure of Universe for its domain with state near thermodynamics equilibrium; h) the theory predicts existence of new ﬁelds originated by fractional dimensions of space ; i) the theory predicts existence of new physics in the do- main of superluminal velocities in vacuum for ordinary(not taxions) particles with new eﬀects that may be ex- perimentally discovered; j) in the fractal world the time has inertial characteris- tics described by mrin analogy with well known Newton massmtand equations for t(r) are exist (analogies of Newton, Dirac and so on equations) ; k) the theory have two masses: masses mtas measure of inertia for moving in the time(Newton masses) and massesmras measure of inertia for moving time in space (masses concerned with inhomogeneous time in the space); l) the presented theory is a natural generalization of all modern physical theories for domains of time and space with fractional dimensions and coincide with any of them in the case when fractional parts of dimensions are zero. This theory is not the variant of theories of quantum time and quantum space, because the multifractal ”intervals” of space and time had used in the theory (the time and the space are consist of them) are very composed multi- fractal sets and researching its structure lay in f uture. The question about irriversibility of time (the arrow of time) in the theory of fractal time and space ( [5]- [16]) was not researched. This paper has purpose to investi- gate why the time has only one direction in our world on the base of the multifractal theory of time and space [5]- [19]. III. WHY TIME HAS DIRECTION ONLY TO FUTURE AND WHY IMPOSSIBLE TO WALK IN TIME TO AND FRO? In the theory of fractal time and space the problem of existing in Universe the arrow of time may be considered (we show it below) as the problem of decreasing of the energy for the states when the time arrow has direction to future. We will show that in the domain of multifractal Universe with time dimensions less than unit and when the fractional parts of time dimensions are small addi- tions (with negative signs) to unit , the arrow of time with direction to future gives decreasing of energy for any body in this domain of Universe. So the arrow of time gives spontaneous decreasing (diminishing) of Uni- verse energy in these domains (and in Universe on the whole). For demonstrating it let us write the quantum equations for model particle with a rest mass mand momentum p= 0 for two cases: in the time space with integer di- mensions and in the time space with fractional dimen- sions i¯h∂ ∂tψ(r,t)−mc2ψ(r,t) = 0 (1) i¯hDdt 0,+ψ(r,t)−m2ψ(r,t) = 0 (2) In (2) we used generalized fractional derivative Ddt +,tde- ﬁned as (see [5]- [19]). Following these works we consider 2both time and space as the initial real material ﬁelds ex- isting in the world and generating all other physical ﬁelds by means of their fractal dimensions. Assume that every of them consists of a continuous, but not diﬀerentiable bounded set of small intervals (these intervals further treated as ”points”). Consider the set of small time in- tervalsSt(their sizes may be evaluated in rude approach as Planck sizes). Let time be deﬁned on multifractal subsets of such intervals, deﬁned on certain measure car- rierRN. Each interval of these subsets (or ”points”) is characterized by the fractional (fractal) dimension (FD) dt(r(t),t) and for diﬀerent intervals FD are diﬀerent. In this case the classical mathematical calculus or fractiona l (say, Riemann - Liouville) calculus [22] can not be applied to describe a small changes of a continuous function of physical values f(t), deﬁned on time subsets St, because the fractional exponent depends on the coordinates and time. Therefore, we have to introduce integral function- als (both left-sided and right-sided) which are suitable to describe the dynamics of functions deﬁned on multifrac- tal sets (see [5]- [7]). Actually, these functionals are sim - ple and natural generalization of the Riemann-Liouville fractional derivatives and integrals: Dd +,tf(t) =/parenleftbiggd dt/parenrightbiggn/integraldisplayt af(t′)dt′ Γ(n−d(t′))(t−t′)d(t′)−n+1(3) Dd −,tf(t) = (−1)n/parenleftbiggd dt/parenrightbiggn/integraldisplayb tf(t′)dt′ Γ(n−d(t′))(t′−t)d(t′)−n+1 (4) where Γ(x) is Euler’s gamma function, and aandbare some constants from [0 ,∞). In these deﬁnitions, as usu- ally,n={d}+ 1 , where {d}is the integer part of dif d≥0 (i.e.n−1≤d < n ) andn= 0 ford <0. If d=const, the generalized fractional derivatives (GFD) (1)-(2) coincide with the Riemann - Liouville fractional derivatives ( d≥0) or fractional integrals ( d<0). When d=n+ε(t), ε(t)→0, GFD can be represented by means of integer derivatives and integrals. For n= 1, that is, d= 1 +ε,\|ε\|<<1 it is possible to obtain: D1+ε +,tf(r(t),t)≈∂ ∂tf(r(t),t) + +a∂ ∂t[ε(r(t),t)f(r(t),t)] +ε(r(t),t)f(r(t),t) t(5) whereais aconstant and determined by choice of the rules of regularization of integrals ( [5])-( [6]), ( [11]) ( for more detailed see [11]) and the last addendum in the right hand side of (5) is very small. The selection of the rule of regularization that gives a real additives for usual derivative in (3) yield a= 0.5 ford <1 [5]. The functions under integral sign in (3)-(4) we consider as the generalized functions deﬁned on the set of the ﬁnite functions [23]. The notions of GFD, similar to (3)-(4), can also be deﬁned and for the space variables r. Thedeﬁnitions of GFD (3)-(4) needs in connections between fractal dimensions of time dt(r(t),t) and characteristics of physical ﬁelds (say, potentials Φ i(r(t),t), i= 1,2,..) or densities of Lagrangians Li) and it was deﬁned in cited works. Following [5]- [19], we deﬁne this connection by the relation dt(r(t),t) = 1 +/summationdisplay iβiLi(Φi(r(t),t)) (6) whereLiare densities of energy of physical ﬁelds, βiare dimensional constants with physical dimension of [ Li]−1 (it is worth to choose β′ iin the form β′ i=a−1βifor the sake of independence from regularization constant). The deﬁnition of time as the system of subsets and deﬁnition of the FD for dt(see ( 4)) connects the value of fractional (fractal) dimension dt(r(t),t) with each time instant t. The latter depends both on time tand coordinates r. If dt= 1 (an absence of physical ﬁelds) the set of time has topological dimension equal to unity. The multifractal model of time allows ( as was shown [9]) to consider the divergence of energy of masses moving with speed of light in the SR theory as the result of the requirement of rig- orous validity of the laws pointed out in the beginning of this paper in the presence of physical ﬁelds (in the mul- tifractal theory there are only approximate fulﬁllment of these laws). We bound consideration only the case when relationdt= 1−ε(r(t),t)),\|ε\| ≪1 are fulﬁlled. In that case the GFD may be represented (as a good approach) by ordinary derivatives and relation (5) are valid. So the equation (2) reeds i¯h∂ ∂tψ(r,t)−mc2ψ(r(t),t) + +i¯h∂ ∂t[εψ(r(t),t)] +i¯hεψ t= 0 (7) This equation describes behavior of the particle with point sizes in time and space (we remind that it is only the approach that we use and in reality minimal size of time intervals and minimal sizes of space intervals in the theory are bound, for example, by Planck sizes, thou the last are multifractal sets too) For free ( more rightly al- most free) particle choose solution for ψas a plane wave with energy depending of time ( ψ=ψ0exp−iE(t) ¯h) and for domain of time-space where by members with∂ε ∂tmay neglect ( i.e. fractional additives almost constant) recei ve ψ(t) =ψ0exp(−i ¯hE(t)t) (8) E(t) =mc2+π¯hε t−i¯hεlnt t(9) or ψ(t) =ψ0 tεexp(−i ¯h˜E(t)t) (10) where 3˜E=mc2+ε¯h t(11) The equations (10) - (11) allow to conclude that the frac- tal dimensions time leads to the two sorts of phenomena: a) decreasing of energy with time ﬂow on value ε¯ht−1; b) spread (run) of wave function on value t−ε∼(1−εlnt). Thus when time t(or current time ( t−t0)) is increasing (t0is age of Universe) both energy and wave function are decreasing. Evaluation of both decreasing values (if take into account only gravitation ﬁeld on surface of Earth and take into account that ε∼10−7,t∼t0∼1017) gives: △E= ¯hεt−2 0∼10−56evfor one second of current time . We pay attention that so little value of damping energy follows from the points model for describing the particle (the particle consists are of one ”interval” of time and of one ”interval” of space, described above). For more re- alistic model it is necessary to take into account a really sizes of particles ( partly the energy of damping will be evaluates in next paragraph). Rigorously say we made evaluation of the lose energy in the point of space where point particle presents because the fractional dimensions that are sources of the real particle mass may be so large that approach ε<< 1 is not work. IV. IS IT POSSIBLE TO CHANGE THE DIRECTION OF TIME AND HOW MUCH ENERGY IT NEEDS ? In this paragraph we research the question: may the time be turned in back direction? The energies needs for is possible to evaluate if use rude approach . The ex- ample for behavior of the free model particle has demon- strated the damping of energy in Universe with fractional time dimensions ( the case of decreasing energy as con- sequences of existence of fractional dimensions of space is analogies). Thus the question why the time has only one direction towards future has natural answer: it is because only in that direction of time the energy of par- ticles ( or any bodies consisting of particles) decrease. If somebody wants to change the direction of time it is necessary to spend energy for changing structure of real ﬁelds of time and space. On the ﬁrst look this energy is very small (see above paragraph), but its smallness is related with the case when multifractal ”intervals” of time and space ( △tand△r) were treated as ”points” with fractional (global for sets consisting the time ﬁeld) dimensions and equation (11) describes the lose energy only of such points ”intervals”. What are values of time and space ”intervals ” in our Universe? The theory of fractal time and space in her present state can not an- swer on this question. So we use some hypotheses about their values. As the rude approach we may take for its values Planck sizes: △t∼10−44secand△x∼10−33cm. Then one second consists of 1044of ”intervals” of time and one centimeter consists of 1033”intervals” of space ( we needs to remember that every of ”intervals” is multi-fractal sets with very composed characteristics which do not researched in present work (see [9]). For current time (t−t0) the relation for lose energy by the domain space with volume ∼10−42cm3during one second ( the volume of elementary particles) reads △E∼ε¯h t2 0104410−421099∼1045ev (12) If such gigantic energy will be received by time ﬁeld with space volume 10−42cm3secthe ﬂow of time during one second be stopped and if double this energy during one second the time ﬂow change its direction (i.e. t→ −t) and time will ﬂow one second in back direction). Thus the direction of time in our Universe may be changed but it needs in the gigantic amount of energy. Of course, the value of this energy depends at the evaluation of the ”in- tervals” of time and space values and if last values more than Planck intervals (for example at 105−1010) the en- ergy will be smaller but also huge. Result is: in principle the inversion of direction of time may be reached but it is impossible on the modern state of humankind tech- nology even for microscopic volumes. If values of time and space intervals needs in corrections, the evaluation of energy needs for inversion of time direction must be corrected too. V. HOW MUCH ENERGY NEEDS FOR STOPPING THE TIME AND MOVING IT BACK IN THE VOLUME OF ONE CUBIC CENTIMETER DURING ONE SECOND? In the multifractal fractal theory of time and space where time and space ﬁelds are real themselves and are real origin of all physical ﬁelds in principle (it was shown in above paragraph) there are possibilities to inverse the time ﬂow in back direction. Now we write the equation for changing with time of one Planck interval of space xp and see how the energy of it (as a part of real space ﬁeld it has energy of rest which damping with ﬂow of time) changes in time (let this Planck volume is in rest, i.e. p= 0) i¯hDdt +,txp=E0xp (13) or i¯h∂ ∂txp=E0xp−i¯hε txp (14) The solution of (14) may be represented as xp=x0exp−i ¯hE(t)t (15) where E(t) =E0+ε¯h t(16) 4Now evaluate the volume xpusing Planck interval and use earlier values of ε,t0,¯h. If connection binding every element rwith element thas the form dr2−dt2= 0 and for each element of space spending energy each element of time spend energy too, thus for the energy lose of space volume equal one cm3during one second write △E∼ε t2 0109910441012ev∼10−6810155ev∼1087ev(17) So we got the order of values of energy needs for stop- ping the time in the cm3sec. For inversion of time ﬂow in this volume needs double this value of energy. For stopping time in the volume of one elementary particle necessary multiply above value at 10−42(if size of parti- cle∼10−14cm). It gives △E∼1045ev. Nobody knows is it possible in far future to receive such energies and concentrate them in small volumes. VI. WHY WE CAN WALK TO AND FRO IN OUR SPACE Why we can not walk to and fro in time had been ex- plained in the frame of multifractal time in paragraph above on the language of energetically reasons. The pos- sibility of walking to and fro in space is conditioned by vector characteristics of fractional addendum to space derivatives in multifractal Universe. For simplicity we consider non relativistic case when particle is described by Shr¨ odinger equation in multifractal space (see [5], [6] , [16]) . Let multifractional addendum to integer space di- mensionsεiis very small ( \|εi\|<<1). Than for GFD we can right (we conserve only main addendum necessary for our purpose) D1−εr +,r∼(∂ ∂r+εr¯h r) (18) and for Shr¨ odinger equation in fractal space for free par- ticle receive i¯hD1−εt +,tψ=−¯h2 2mD1−εr −,rD1−εr +,rψ (19) Now replace fractional space derivatives by means of (13) then (14) reads (if neglect the members of order ε2and non essential scalar members in right hand part of equa- tion ) i¯hD1−εt +,tψ=−¯h2 2m△ψ+∇(εr¯h r)ψ (20) If in the (20) we replace rby−rthe sign of fractal ad- dendum from fractional space dimensions do not change its sign, so there are no energetically reasons forbidding walking to and fro in the fractal Universe. Of course, in the equation (15) omitted the members describing the lose of energy by particle reasoned by the fractional struc- ture of space. We do not evaluate the energy lose rea- soned by multifractal structure of space because in thiscase the evaluation is very diﬃcult ( the value of εris unknown). It value deﬁned by new ﬁelds (not discovered yet) that borne by fractional space dimensions (see [5], [6], [7]. VII. CONCLUSIONS The arrow of time in considered model of multifractal time and space as was seen above is consequence of en- ergetically reasons. Direction of time may be changed (thou only in principle in our epoch because of huge amount of energy that needs for it). There are three main results of this article: a) the explanation of nature of arrow of time by natural lose of energy of our Universe and by necessity for energy compensation of this lose for changing of direction of time in any domain (small or large) of space and time; b) it is point out at the huge amount of energy in every bodies and ﬁelds ( more de- tailed consideration will be in special paper ) caused by the real nature of ﬁelds of time and space ; c) the principle possibility to change directions of time and space ﬁelds in the remote future epoch. We considered the energy needs for changing direction of time, thou it is necessary to return the space in earlier state too. Some general remarks . Any fractal or multifractal sets (Universe is multifractal set) always not belong and not coincide with measure of carrier on that they are deﬁned (it include the cases when measure of carrier is multi- fractal set itself and not space RNtype ). If describe the measure carrier of our Universe in terms of ”physical vacuum”, then ”vacuum” do not belongs to our Universe. Main part of it lays out of Universe (see also [3]) . The Universe may be treated as a gigantic energetic ﬂuctua- tion (”metastable” long living ﬂuctuation) in the measure of carrier and as the ﬂuctuation it has strong binding with its ”mother” . The existence of strong binding with the vacuum (measure of carrier) consist in continual trans- ferring to vacuum the huge amount of energy that was got from vacuum in the moment of ”big bang” (or in the moment of birth in any over scenario) as was demon- strated in this paper. What future wait our Universe in the model of multifractal space and time ? Universe will spend her supply of energy that was got from vacuum in the moment of big bang. When all energy supply be spent process of Universe dying will be ﬁnished till time when new Universe borne from measure of carrier (vacuum). In this model many of universes (may be inﬁnity) may exis- tent because measure of carrier can give birth any huge amount of Universes (with their own times and spaces that can diﬀerent at our time and space by it dimensions and energetical characteristics) and die of one of them is not essential for carrier of measure in this model of multi Universes structure of ”vacuum ” not belonging to our Universe. 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Do Electromagnetic and Gravitation Fields Have Rest Masses? Preprint at http://arXiv. org/abs/physics/0006043 [20] Klimontovith Yu.L., Statistical Theory of Open Sys- tems. Vol.1 , Moscow, Yanus, 1995, 686p. (in Russian); Kluwer Academic Publishers, Dordrecht, 1995; Klimon- tovich Yu.L., Statistical Physics of Open systems. Vol.2 , Moscow, Yanus, 1999, 450p. (in Rusian). [21] Mandelbrot B., Fractal Geometry of Nature , W.H.Freeman, San Francisco, 1982 [22] Samko S.G, Kilbas A.A., Marithev I.I., Integrals and Derivatives of the Fractional Order and Their Applica- tions, (Gordon and Breach, New York, 1993). [23] I.M.Gelfand, G.E.Shilov, Generalized functions (Aca- demic Press, New York, 1964) 6
arXiv:physics/0011037v1 [physics.atom-ph] 16 Nov 2000Contribution of the screened self-energy to the Lamb shift o f quasidegenerate states ´Eric-Olivier Le Bigot∗, Paul Indelicato†, Vladimir M. Shabaev‡ Laboratoire Kastler-Brossel, Case 74, ´Ecole Normale Sup´ erieure et Universit´ e P. et M. Curie Unit´ e Mixte de Recherche du CNRS n◦C8552 4, pl. Jussieu, 75252 Paris CEDEX 05, France (February 20, 2014) Abstract Expressions for the eﬀective Quantum Electrodynamics (QED ) Hamiltonian due to self-energy screening (self-energy correction to th e electron-electron interaction) are presented. We use the method of the two-tim e Green’s func- tion, which handles quasidegenerate atomic states. From th ese expression one can evaluate energy corrections to, e.g., 1 s2p3P1and 1s2p1P1in helium and two-electron ions, to all orders in Zα. Typeset using REVT EX 1In the last ten years, experiments in the spectroscopy of hel ium [1–5] have become two orders of magnitude more precise than the best theoretic al energy level calculations available (see, e.g., Refs. [6,7] and references therein). Several experiments are now focusing on Helium and heliumlike ions 1 s2p3PJﬁne structure [8–12], with the aim of providing a new determination of the ﬁne structure constant and of check ing higher-order eﬀects in the calculations. In this case the theory is again a limiting fac tor. In this context a direct determination of all α2contributions to all order in Zαis necessary to improve reliability and accuracy of theoretical calculations ( αbeing the ﬁne structure constant, and Zthe charge of the nucleus). A diﬃculty in the study of the (1 s2p1/2)1and (1s2p3/2)1levels is that they are quaside- generate for low and middle Zions [13]; this precludes the use of the Gell-Man–Low and Sucher method [14,15] to evaluate QED energy shifts of atomi c levels. In fact, this method has two important drawbacks: it does not handle quasidegene rate energy levels, and it leads to a diﬃcult renormalization procedure when applied t o degenerate states. (The latter problem has only been tackled up to second-order in α[16,17].) We use the method of the two-time Green’s function [18–20], r igorously derived from QED (for the most detailed description of this method, see [2 1]). To the best of our knowl- edge, only the method recently proposed by Lindgren [22], cl osely modeled to multireference- state Many-body perturbation techniques, is designed to wo rk for quasidegenerate states. We evaluate the contribution of the screened self-energy di agrams/A1 /CZ/D2/C8 /B4/BE/B5 /D2/C8 /B4/BD/B5/D2 /BC/C8 /BC/B4/BE/B5 /D2 /BC/C8 /BC/B4/BD/B5/A1(1) to quasidegenerate energy levels in heliumlike ions. Our re sults can be easily extended to ions with more than two electrons along lines similar to thos e found in [23]. First approximate evaluations of the contribution of these diagrams for isolated states in two- and three-electron ions were performed in Refs. [24– 27]. Accurate calculations from the ﬁrst principles of QED were accomplished in Refs. [28–30 ] for the ground state ofheli- umlike ions and in Refs. [31,32] for the 2 sand 2p1/2states of lithiumlike ions. The other 2twoα2corrections to the electron-electron interaction have als o been calculated for isolated states in two- and three-electron ions: the vacuum-polarization screening [13,29,30,33,34], and the two-photon exchange diagrams [35–38]. In [13], the vacuum polarization screen- ing for quasidegenerate states of heliumlike ions was evalu ated as well. Some results for the direct contribution of the self-energy correction to th e Coulomb interaction are also available [24,39]. As depicted in diagrams (1), the interaction between the two electrons through photons is treated perturbatively. On the contrary, the binding to t he nucleus is included non- perturbatively in the method we use, since the corresponding coupling const ant isZα. Such a treatment is obviously mandatory for highly-charged ions . Furthermore, it allows one to compare non-perturbative (in Zα) results to (semi-)analytic expansions in Zα(see [40] for a review). We derive the eﬀective (ﬁnite-sized) matrix hamiltonian H, whose eigenvalues give the contribution of QED to a group of energy levels [23]. The diag onal entries of the hamilto- nian that we evaluate correctly reproduce previous express ions of the screened self-energy, while the new, non-diagonal entries that we derive allow one to obtain a second-order QED correction to quasidegenerate ordegenerate energy levels. Relativistic units ¯ h=c= 1 are used throughout this paper. If we havesquasidegenerate energy levels E(0) 1...s, the eﬀective hamiltonian His ans×s matrix restricted to these levels [23]. Let us introduce som e notations in order to express this hamiltonian. The second-order contribution H(2)to this hamiltonian H=H(0)+H(1)+ H(2)+...is constructed from a projection matrix Pand an energy matrix K[23]: H(2)=K(2)−1 2{P(1),K(1)} −1 2{P(2),K(0)} +3 8{[P(1)]2,K(0)}+1 4P(1)K(0)P(1), (2) where the notation {,}represents the usual anticommutator, and where the supersc ripts indicate the number of photons of the diagrams that contribu te to each term of the pertur- bative expansion P=P(0)+P(1)+...andK=K(0)+K(1)+...; thes×smatricesPand 3K, which are deﬁned as [20]: P≡1 2πi/contintegraldisplay ΓdEg(E) (3a) K≡1 2πi/contintegraldisplay ΓdEEg (E), (3b) whereg(E) is thes×smatrix restriction of the Green’s function to the sunperturbed atomic levels under consideration, and where Γ is a contour t hat encloses each of the Dirac atomic energy levels with a positive orientation [23]. We directly evaluate the hamiltonian matrix elements of Eq. (2) between states of dif- ferent energiesE(0) nandE(0) n′, and put them in a form that readily displays the limiting cas e ofidentical energies; we checked by a direct calculation of the diagonal matrix elements that they can be obtained from non-diagonal elements H(2) nn′by taking the formal limit E(0) n→E(0) n′.Allthe subsequent derivations of H(2) nn′will thus be done with E(0) n/ne}ationslash=E(0) n′. The ﬁrst diagram of (1) appears only in the second-order matr icesK(2)andP(2)in Eq. (2). As usual, we must calculate a reducible and an irreducible contribution; as can be seen in subsequent calculations, it turns out that the corre ct extension of these notions to quasidegenerate states is the following: in the ﬁrst diagra m of Eq. (1), the contribution of intermediate electrons with a Dirac energy εksuch thatεk+εn′ P′(2)coincides with one of thesenergy levels under consideration and must be separated out from the contribution of the other intermediate electron states; the ﬁrst contrib ution (called reducible ) requires a diﬀerent mathematical treatment from that of the second con tribution (called irreducible ). Thus, the irreducible contribution is obtained by summing over almost all electro n states kin the ﬁrst diagram of Eq. (1); we ﬁrst show that it is suﬃcient to remove only onestatek from the sum over states in the ﬁrst diagram of Eq. (1). We see t hat an intermediate energy εk+εn′ P′(2)can coincide with an unperturbed atomic levels E(0) 1...sonly if the electron khas the same principal quantum number as the electron n′ P′(1)on the other side of the self-energy, because otherwise the total energy εk+εn′ P′(2)would lie largely out of the range spanned by the unperturbed quasidegenerate energy levels located aro undE(0) n′=εn′ P′(1)+εn′ P′(2). There is an additional selection on the electrons kto be removed: since the total angular 4momentum, its projection, and parity are conserved by the se lf-energy operator Σ [Eq. (6) below], as can be seen by integrating over angles using st andard techniques [41], the contribution of electrons kthat do not share the same quantum numbers ( κ,m) as the electronn′ P′(1)in the ﬁrst diagram of Eq. (1) is exactly zero. We denote the individual electrons of a state nbyn1andn2, in an order which is arbitrary but that must remain ﬁxed. With these notations, our evaluat ion of the irreducible part of the ﬁrst diagram of (1) to the eﬀective hamiltonian (2) takes a simple form and reads (Dirac energies are still denoted by εk): Hscr. SE, irr. nn′ =/summationdisplay P,P′(−1)PP′/parenleftBigg/summationdisplay k/negationslash=nP(1)/an}bracketle{tnP(1)\|Σ(εnP(1))\|k/an}bracketri}ht1 εnP(1)−εk/an}bracketle{tknP(2)\|I(εnP(1)−εn′ P′(1))\|m′ 1m′ 2/an}bracketri}ht +/summationdisplay k/negationslash=n′ P′(1)/an}bracketle{tnP(1)nP(2)\|I(εnP(1)−εn′ P′(1))\|n′ P′(1)n′ P′(2)/an}bracketri}ht1 εn′ P′(1)−εk/an}bracketle{tk\|Σ(εn′ P′(1))\|n′ P′(1)/an}bracketri}ht/parenrightBigg +O[α2(E(0) n′−E(0) n)], (4) where ( −1)PP′is the signature of the permutation P◦P′(PandP′are permutations of{1,2}.), where the sum over kis over (almost) all possible intermediate Dirac states, and where the photon exchange and the self-energy of diagram s (1) are represented by the following usual operators [32]: /an}bracketle{tab\|I(ω)\|cd/an}bracketri}ht ≡e2/integraldisplay d3x1/integraldisplay d3x2[ψ† a(x1)αµψc(x1)] ×[ψ† b(x2)ανψd(x2)]Dµν(ω;x1−x2) (5) /an}bracketle{ta\|Σ(p)\|b/an}bracketri}ht ≡1 2πi/integraldisplay dω/summationdisplay k/an}bracketle{tak\|I(ω)\|kb/an}bracketri}ht εk(1−i0)−(p−ω), (6) in whicheis the charge of the electron, αµ≡(1,α) are the Dirac matrices, and where ψ denotes a Dirac spinor; the photon propagator Dis given in the Feynman gauge by Dνν′(ω;r)≡gνν′exp/parenleftBig i\|r\|√ω2−µ2+i0/parenrightBig 4π\|r\|, (7) whereµis a small photon mass that eventually tends to zero, and wher e the square root branch is chosen such as to yield a decreasing exponential fo r large real-valued energies ω. 5The last term in Eq. (4) represents a contribution of order α2which is multiplied by a factor that tends to zero as E(0) n′−E(0) n→0. It can be shown (see Ref. [21]) that such a term does not contribute to order α2and that it can therefore be omitted. We note that result (4) readily yields diagonal elements by t aking the (formal) limit E(0) n−E(0) n′→0. The hamiltonian (2) contains the contribution of many ﬁrst-order diagrams through the operatorsP(1)andK(1). We must consider here the contribution of the photon exchan ge and of the self-energy/A1/D2/C8 /B4/BE/B5 /D2/C8 /B4/BD/B5/D2 /BC/C8 /BC/B4/BE/B5 /D2 /BC/C8 /BC/B4/BD/B5/A1; (8) their contribution to Eq. (2) cancels a part of the reducible screened self-energy. We thus evaluate in the following the contribution of both diag rams of Eq. (8) to the terms −1 2{P(1),K(1)}+3 8{[P(1)]2,K(0)}+1 4P(1)K(0)P(1)of the eﬀective hamiltonian. The energy and projection matrices KandPof Eq. (3) have been calculated for the photon-exchange diagram in [13]; this allows one to evaluat e any integral due to the photon exchange that appears in the eﬀective hamiltonian (2). In order to derive the contribution of the one-electron self-energy, let us show that the evaluation of the self-energy contributions to the hami ltonian (2) boils down to the calculation of contour integrals of the form 1 2πi/contintegraldisplay ΓndEgSE nn(E) and1 2πi/contintegraldisplay ΓndEEgSE nn(E) (9) wheregSE nn(E) are diagonal elements of the self-energy Green’s function ; in other words, the contour Γ that surrounds allthe levels in Eq. (3) can be replaced by the contour that surroundsE(0) nonly, andnon-diagonal elements of the self-energy Green’s function are not relevant. The contour integrals of Eq. (9) have both been eva luated in [32], so that no further quantity is required in order to obtain the self-energy cont ribution to the hamiltonian (2). Let us prove the above statements. As mentioned before, angu lar momentum conserva- tions constrain the self-energy operator Σ to be zero betwee n states with diﬀerent angular 6quantum numbers ( κ,m); and since the atomic levels we consider have the same princ ipal quantum number (they are quasidegenerate), the self-energ y Green’s matrix is diagonal: gSE nn′(E) = 0 ifn/ne}ationslash=n′, (10) wherenandn′are the setsof quantum numbers of two of the slevels under consideration. Furthermore, the Green’s function gSE nn(E) has only onepole inside the integration con- tour Γ, namely at E=E(0) n. Therefore, integrating over the full contour Γ in the hamil tonian (2) amounts to integrate over the contour Γ nthat surrounds only E(0) n, since the Green’s function is analytic inside the contours that encircle the o ther energies. We thus see that the contribution of the self-energy to Eq. (2 ) depends only on contour integrals of the form (9), which are known analytically [32] . With the help of some published analytical formulas, we obta in the following contribution of the photon exchange (see Eqs. (27) and (28) in [13]) and of t he self-energy (see Eqs. (36) and (37) in [32]) to the eﬀective hamiltonian (2): −/summationdisplay P,P′(−1)PP′/braceleftBigg1 4/bracketleftBigg/parenleftBig /an}bracketle{tnP(1)\|Σ′(εnP(1))\|nP(1)/an}bracketri}ht+/an}bracketle{tn′ P(1)\|Σ′(εn′ P(1))\|n′ P(1)/an}bracketri}ht/parenrightBig ×/parenleftBig /an}bracketle{tnP(1)nP(2)\|I(∆1)\|n′ P′(1)n′ P′(2)/an}bracketri}ht+/an}bracketle{tnP(1)nP(2)\|I(∆2)\|n′ P′(1)n′ P′(2)/an}bracketri}ht/parenrightBig/bracketrightBigg +1 2/bracketleftBigg/parenleftBig /an}bracketle{tnP(1)\|Σ(εnP(1))\|nP(1)/an}bracketri}ht+/an}bracketle{tn′ P(1)\|Σ(εn′ P(1))\|n′ P(1)/an}bracketri}ht/parenrightBig ×1 2πi/integraldisplay dω/an}bracketle{tnP(1)nP(2)\|I(ω)\|n′ P′(1)n′ P′(2)/an}bracketri}ht/parenleftBigg1 (ω+ ∆ 1−i0)(ω−∆2−i0)+1 (ω+ ∆ 2−i0)(ω−∆1−i0)/parenrightBigg/bracketrightBigg/bracerightBigg , where Σ′represents the derivative of the self-energy operator (6) w ith respect to the energy that ﬂows in it, and where the two possible energies for the ph oton in the photon-exchange diagram are ∆ 1≡εnP(1)−εn′ P′(1)and ∆ 2≡εnP(2)−εn′ P′(2). As seen above, the reducible part of the ﬁrstdiagram of Eq. (1) represents the con- tribution of an intermediate electron k=n′ P′(1). (For the second diagram, the reducible part is similarly obtained through an intermediate electro nk=nP(1).) The evaluation of the reducible contribution follows steps similar to those u sed for the irreducible part. The contribution of diagrams (8) to the eﬀective hamiltonian H(2), which is given in Eq. (11), 7cancels a few terms of the contribution of the reducible diagram, as f or diagonal matrix elements [32]; the total reducible contribution to Eq. (2) is then found to be: Hscr. SE, red. nn′ =/summationdisplay P,P′(−1)PP′1 2/bracketleftbigg ∂p\|εnP(1)/parenleftbigg /an}bracketle{tnP(1)\|Σ(p)\|nP(1)/an}bracketri}ht/an}bracketle{tnP(1)nP(2)\|I(p−εn′ P′(1))\|n′ P′(1)n′ P′(2)/an}bracketri}ht/parenrightbigg +∂p′\|εn′ P′(1)/parenleftBig /an}bracketle{tnP(1)nP(2)\|I(εnP(1)−p′)\|n′ P′(1)n′ P′(2)/an}bracketri}ht/an}bracketle{tn′ P′(1)\|Σ(p′)\|n′ P′(1)/an}bracketri}ht/parenrightBig/bracketrightbigg +O[α2(E(0) n′−E(0) n)], (12) where∂x\|x0represents the derivative with respect to xat the point x0. For the vertex diagram [second diagram of (1)], the two-time Green’s function method yields the following contribution to (2): Hvertex nn′=/summationdisplay P,P′(−1)PP′/summationdisplay i1,i2/an}bracketle{ti1nP(2)\|I(εnP(1)−εn′ P′(1))\|i2n′ P′(2)/an}bracketri}ht ×i 2π/integraldisplay dω/an}bracketle{tnP(1)i2\|I(ω)\|i1n′ P′(1)/an}bracketri}ht [εi1(1−i0)−(εnP(1)−ω)][εi2(1−i0)−(εn′ P′(1)−ω)]+O[α2(E(0) n′−E(0) n)],(13) with the same notations as before; the sum is over all pairs of Dirac states. We thus have obtained the full contribution [Eq. (4) + Eq. (12 )+Eq. (13)] of the screened self-energy diagrams (1) to a ﬁnite-sized eﬀective hamilto nian which acts on a few atomic energy levels (in the general case: quasidegenerate, fully degenerate or isolated); the eigen- values of this hamiltonian give the QED prediction for the en ergy levels. We have also taken into account the contribution of the ﬁrst-order diagrams (8 ) to the second-order hamiltonian (2). 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arXiv:physics/0011038v1 [physics.gen-ph] 16 Nov 2000How Much Energy Have Real Fields Time and Space in Multifract al Universe? L.Ya.Kobelev Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru On the base of multifractal theory of time and space ( see [1]- [16]) in this paper shown presence in every space and time volumes of real space and time ﬁelds a h uge supply of energy . In the multifractal Universe every space volume or time interval p ossesses by huge amount of energy( ∼ 1060cm3) and we discuss the problem is it possible this new for mankin d sorts of energy to extract. CONTENTS : 1. Introduction 2. What are Energy Densities of Real Space and Time Fields in M ultifractal Universe? 3. How Much Energy Space and Time Continually Lose? 4. Where the Loses Energy Go To? Why We did not Discovered it Ti ll Now? 5. Is It Possible to Extract the Energy from Fields of Time or S pace? 6. Conclusions 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION The fractal model of space and time [1]- [16] treats the time and the space with fractional dimensions as real ﬁelds. Universe is formed only by these ﬁelds , i.e. our Universe is fractional material time and fractional mate- rial space and include not any more. As the time and the space are material ﬁelds with fractional dimensions and multifractal structure (multifractal sets) they deﬁne d on sets of their carriers of measure. In each time (or space) point ( ”points” are approach for very small in- tervals of time or space and ”intervals” are multifractal sets with global dimensions for its sets playing role of lo- cal dimensions for Universe in whole ) the dimensions of time (or space) determine densities of Lagrangians energy for all physical ﬁelds ( or new physical ﬁelds for space) in these points. Time and space are binding by relation dr2−c2dt2= 0 (this relation is only good approach, more precise relations see at [2]). As real ﬁelds time and space own huge supply of energy ( the question about its energy was considered partly at [16]) and these energies may be evaluated. The purpose of this paper is more detailed consideration ( in the mathematical formalism of multi- fractal model of time and space presented in [1]- [16]) of problem existence huge supply of energy owned by each element of time and space. The reason having energy lay in multifractal nature of time and space, i.e. multifractal nature of our Universe. Time and space formed Uni- verse and by means of their multifractal dimensions con- struct picture of all physical ﬁelds and got huge amount of energy when Universe was born. In the [16] where evaluated these energies for case if current time must be turned back. In this paper we consider continually loses of these energies by time and space, evaluate values ofloses energies and there is a discussion: may humankind be provided with part of these energies. II. WHAT ARE ENERGY DENSITIES OF TIME AND SPACE REAL FIELDS IN MULTIFRACTAL UNIVERSE? For answer on these questions it is necessary to con- struct and investigate equations describing behavior of very small ”intervals” of space and time ( in considered theory each minimal interval space or time ( i.e. multi- fractal sets) treats as a ”point” [1]- [16]). It is well known that behavior of objects with very small sizes describes by quantum laws. Equations for moving such objects are equations describing diﬀusion with imaginary coeﬃ- cient of diﬀusion. Then equations for space point xpand time point tpmay be written in the frame of multifrac- tal theory of time and space as quantum equations with generalized fractional derivatives (GFD) [1] - [16]. i¯hD1−εr +,txp=mrc2xp (1) and i¯hD1−εt +,rtp=mtc2tp (2) where mrandmtare rest masses time and space vol- umes. In (1)-(2) we used generalized fractional deriva- tiveDdt +,tandDdr +,rdeﬁned on multifractal sets (see [1]- [16]). Following these works we consider both time and space as the initial real material ﬁelds existing in the world and generating all other physical ﬁelds by means of their fractal dimensions. Assume that every of them consists of a continuous, but not diﬀerentiable bounded 1set of small intervals (these intervals further treated as ”points”). Consider the set of small time intervals St (their sizes may be evaluated in rude approach as Planck sizes). Let time be deﬁned on multifractal subsets of such intervals, deﬁned on certain measure carrier RN. Each interval of these subsets (or ”points”) is characterized by the fractional (fractal) dimension (FD) dt(r(t), t) and for diﬀerent intervals FD are diﬀerent. In this case the clas- sical mathematical calculus or fractional (say, Riemann - Liouville) calculus [19] can not be applied to describe a small changes of a continuous function of physical values f(t), deﬁned on time subsets St, because the fractional exponent depends on the coordinates and time. There- fore, we have to introduce integral functionals (both left- sided and right-sided) which are suitable to describe the dynamics of functions deﬁned on multifractal sets (see [1]- [3]). Actually, these functionals are simple and nat- ural generalization of the Riemann-Liouville fractional derivatives and integrals: Dd +,tf(t) =/parenleftbiggd dt/parenrightbiggn/integraldisplayt af(t′)dt′ Γ(n−d(t′))(t−t′)d(t′)−n+1(3) Dd −,tf(t) = (−1)n/parenleftbiggd dt/parenrightbiggn/integraldisplayb tf(t′)dt′ Γ(n−d(t′))(t′−t)d(t′)−n+1 (4) where Γ( x) is Euler’s gamma function, and aandbare some constants from [0 ,∞). In these deﬁnitions, as usu- ally,n={d}+ 1 , where {d}is the integer part of dif d≥0 (i.e. n−1≤d < n ) and n= 0 for d <0. If d=const, the generalized fractional derivatives (GFD) (1)-(2) coincide with the Riemann - Liouville fractional derivatives ( d≥0) or fractional integrals ( d <0). When d=n+ε(t), ε(t)→0, GFD can be represented by means of integer derivatives and integrals. For n= 1, that is, d= 1 + ε,\|ε\|<<1 it is possible to obtain: D1+ε +,tf(r(t), t)≈∂ ∂tf(r(t), t) + +a∂ ∂t[ε(r(t), t)f(r(t), t)] +ε(r(t), t)f(r(t), t) t(5) where ais aconstant and determined by choice of the rules of regularization of integrals ( [1]- [2], [7]) (for mo re detailed see [7]) and the last addendum in the right hand side of (5) is very small. The selection of the rule of reg- ularization that gives a real additives for usual deriva- tive in (3) yield a= 0.5 for d <1 [1]. The functions under integral sign in (3)-(4) we consider as the gener- alized functions deﬁned on the set of the ﬁnite functions [20]. The notions of GFD, similar to (3)-(4), can also be deﬁned and for the space variables r. The deﬁnitions of GFD (3)-(4) needs in connections between fractal di- mensions of time dt(r(t), t) and characteristics of physical ﬁelds (say, potentials Φ i(r(t), t), i= 1,2, ..) or densities of Lagrangians Li) and it was deﬁned in cited works. Fol- lowing [1]- [15], we deﬁne this connection by the relationdt(r(t), t) = 1 +/summationdisplay iβiLi(Φi(r(t), t)) (6) where Liare densities of energy of physical ﬁelds, βiare dimensional constants with physical dimension of [ Li]−1 (it is worth to choose β′ iin the form β′ i=a−1βifor the sake of independence from regularization constant). The deﬁnition of time as the system of subsets and deﬁnition of the FD for dt(see (6)) connects the value of fractional (fractal) dimension dt(r(t), t) with each time instant t. The latter depends both on time tand coordinates r. If dt= 1 (an absence of physical ﬁelds) the set of time has topological dimension equal to unity. The multifractal model of time allows ( as was be shown [5]) to consider the divergence of energy of masses moving with speed of light in the SR theory as the result of the requirement of rigorous validity of the laws pointed out in the beginning of this paper in the presence of physical ﬁelds (in the multifractal theory there are only approximate fulﬁllment of these laws). We bound consideration only the case when relation dt= 1−ε(r(t), t)),\|ε\| ≪1 are fulﬁlled. In that case the GFD (as was shown) may be represented (as a good approach) by ordinary derivatives and relation (1, (5)) are valid. So the equations (1) -(2) reeds ( we used for GFD approach of (5)( [16])) i¯h∂ ∂txp−mtc2xp+ +i¯h∂ ∂t[εxp] +i¯hε txp= 0 (7) ci¯h∂ ∂rtp−mrc2tp+ +ci¯h∂ ∂r[εtp] +ci¯hε rtp= 0 (8) where c-speed of light. These equations describe behav- ior of the volumes with ”point” sizes in time and space (we remind once more that it is only the approach that we use and in reality minimal size of time intervals and minimal sizes of space intervals in the theory are bound, for example, by Planck sizes, thou the last are multifrac- tal sets too) For free volumes choose solutions for xpand tpas a plane waves with energy depending of time (we consider further only case of xpand omit the members with masses) xp=x0exp−iE(t)t ¯h(9) and for domain of time-space where by members with∂ε ∂t may neglect ( i.e. fractional additives almost constant) receive (1 t=P(1 t)−iπδ(t), P-mean value of integral, δ(t) -δfunction) E(t) =−i¯hεt t+¯hεtπ t(10) We do not consider the solutions of equation (8). If admit relation x=ctas was admitted in [1]- [2] the energy of time ”volume” will be of same order that the energy of space volume. Then (10) (or the energy of time volume from the equation (8) gives if fractal dimensions deﬁned by gravitation ﬁeld ( ε∼10−7, t∼1017) 2E∼10−51(11) This energy belongs to ”point” with coordinate xp. In the considered model multifractal time and space each points is multifractal set with global dimensions dtor dr. Let us characterize the volumes of these points by Planck sizes: tp∼10−44sec,xp∼10−33cm. The density of energy in the one cm3is equal E∼1060ev (12) So each cubic centimeter of space has such huge en- ergy. This gigantic energy is determined by fractional dimensions of time in the domain of space where we live. The energy determined by fractional dimensions of space (ε(t(r), r)) may be evaluated if ﬁnd value of the fractional additives in this case. What is nature of these energies? It originates by all physical ﬁelds born by fractional di- mensions of time ( fractional dimensions gravitational, weak-electro-magnetic, strong interaction and so on and their vacuum ). We took into account only gravitational ﬁeld as example. There are vacuum states of all phys- ical ﬁelds in any point of our Universe, they give con- stant additives at fractional dimensions (stress diﬀerenc e of vacuum physical ﬁelds from vacuum (carrier of mea- sure) that born our Universe). Thus huge energies of space and time are constructed by the multifractal struc- ture of our world and are consequences of multifractal nature Universe. III. HOW MUCH ENERGIES TIME AND SPACE CONTINUALLY LOSE? The huge supply of energy are constructed by the frac- tional nature of time and space . The relation (10) (or analogies relation for time volume) demonstrated dimin- ishing of fractal structure with increasing time ﬂow and with space expending. Both time and space tends to the state where their fractal structures tends to zero and it will the end of Universe. How much energy time lose each second? The relation (10) allows to evaluate its value. We use (10) and write △E∼ −¯hετ t2 0(13) where τ=t−t0is a current time. So the losees space energy by one cm3secare △E∼1043evsec (14) This is the huge ﬂow of energy (in our case the ﬂow of gravitation ﬁeld energy, because we do not consider ﬂows of other ﬁelds). Where this energy going to?IV. WHERE THE LOSEES ENERGY GO TO? WHY WE DID NOT DISCOVERED IT TILL NOW It seems only one the answer at the question put above exist: the ﬂow of energy are born by diminishing of frac- tal structure of space and time goes directly to the carrier of measure that created our Universe. The fractal the- ory of time and space now are in study of construction thou main principle of it formulated and main predic- tion are made ( see [1]- [16]). Our Universe connected (bind) with vacuum (measure carrier) and continually re- turns it the energy that got when ”big bang” happened. So Universe must be ﬁlled of ”radiation” (gravitational, electro-magnetic and so on energy ﬂows ) that it con- tinually losing. We mention once yet that in the form of these energy ﬂows the time and the space (Universe) directly return the energy (it is the energy they had got from the carrier of measure (physical vacuum) that born them when ”big bang” had happened) back to the car- rier of measure (vacuum). Universe behaves himself as a big reserver of energy strong connected with vacuum and directly return vacuum her energy. Stress that carrier of measure in considered model of fractal world not belongs our Universe ( as forest not belongs Earth till it died). The Universe (Universe consists of the real ﬁelds of time and the space) only deﬁned on the carrier of measure as part of it and may be many such Universes deﬁned on the same carrier of measure. So time an space con- stantly return energy to carrier of measure in the form of energy ﬂows diminishing the fractal structure and de- creases gigantic energy of all physical ﬁelds. We can not see or sense these energies as we can not sense for exam- ple very low frequency of electro-magnetic ﬁeld. For ex- perimental discovers of these ﬂows (time ﬂows and space ﬂows) it seems necessary some a new devices based on a new theory of vacuum that born our Universe and more deep penetration in the nature of time and space ﬁelds. Physics needs more knowledge about the nature of carrier of measure, characteristics of interactions between time and space ﬁelds and measure carrier. These problems may be solved in future. V. IS IT POSSIBLE TO EXTRACT THE ENERGY FROM FIELDS OF TIME OR SPACE? As was shown in the above paragraph the multifractal theory of time and space the real ﬁelds of time and space are ﬁlled by energy. The one cm3only space ﬁeld is a container of the huge amount of energy because only lose of energy during one second of time consists ∼1043ev. The energy of rest mass is almost nothing as compared with this energy. This energy transfers directly in the measure carrier (see above paragraphs). Is it possible ex- tract it and use for practical purpose? It is very diﬃcult to answer on this question now because there are many non researched problems in the theory of real fractal time 3and space ﬁelds. This problem is one of the main prob- lems of the theory . If somebody asks the Neanderthal man about possibility to use for needs of his tribe the energy of water ﬂow of near river what he will answer? He could not even understand what he was asked about. Now the method of transferring of the energy of physi- cal ﬁelds ﬂows directly at the energy ﬁelds of carrier of measure unknown for mankind. What new devises must be invented for discover and using these ﬂows of energy? What new physics needs for it? Nobody can answer on these question now exactly. Nevertheless for any ener- getical ﬂow must exist some of stopping ﬂow devices that may take some of ﬂow energies. I think early or later such devices will be invented. For it the new theory of vacuum is necessary . Now such the theory of vacuum is absent. It is necessary ﬁnd out how correctly describe interac- tions between time and space ﬂows, physical ﬁelds plows and measure carrier (vacuum). If it will be done (nobody knows how much time it needs) the stopping (bar) ﬂows devices will be invented. We suppose that these prob- lems will be solved in future then the two problem are decided: a)the problem of sources of any amount of energy; b) the problem of governing by time, because the taking energy from time ﬁeld ﬂow partly stopped or partly in- verse the time ﬂow. In the beginning of the last century had appeared the known relation E=mc2. It was not simple under- stand its inﬂuence on human life. It relation allowed hu- mankind to develop new technology of atomic era. May be huge amount of energy condensed in the time and the space ﬁelds allow mankind in future era use it too. How to do it nobody knows now. May be very intensive beams of electro-magnetic ﬁeld energy received by means of beams of particles moving with speed of light ( see [4]) allow to do ”the opening” in the time ﬁeld and to re- ceive from it energy? Is it is possible to use the energy of long wave” vibrations” that ﬁlled Universe? Optimistic point of view allows to have hope that in future these new sources of energy when energy may be got directly from time and space ﬁelds ﬂows (in case that the fractal the- ory of time and space will ﬁnd experimental veriﬁcation) may be used. VI. CONCLUSIONS 1.The multifractal theory of time and space forecast existence of gigantic supply of energy in the real time and space ﬁelds (the order of these losees is 1060evsec); 2. The multifractal theory of time and space forecast existence of gigantic losees of energy by each volumes of space and time (the order of these losees is 1043evsec); 3. Discussion of possibilities to use the huge supply of time ﬁeld and space ﬁeld energies by humankind gives more optimistic then pessimistic perspective for futureof mankind. [1] Kobelev L.Ya., Fractal Theory of Time and Space ,Preprint at Dep. in VINITI 19.08.99, No.2677-B99 , (in Russian) [2] Kobelev L.Ya., Fractal Theory of Time and Space , Kon- ros,1999, p.136(in Russian) [3] Kobelev L.Ya., What Dimensions Do the Time and Space Have : Integer or Fractional? Preprint at http://arXiv.org/abs/physics/0001035 [4] Kobelev L.Ya., Can a Particle’s Velocity Exceeds the Speed of Light in the Empty Space? Preprint at http://arXiv.org/abs/gr-qc /0001042 [5] Kobelev L.Ya., Multifractality of Time and Space, Co- variant Derivatives and Gauge Invariance, Preprint at http://arXiv.org/abs/hep-th /0002005 [6] Kobelev L.Ya. Does Special Relativity Have Limits of Applicability in the Domain of Very Large Energies? Preprint at http://arXiv.org/abs/physics/0005069 [7] Kobelev L.Ya.,Generalized Riemann -Liouville Frac- tional Derivatives for Multifractal Sets,Preprint at http://arXiv.org/abs/math.CA/0002008,; [8] Kobelev L.Ya.The Theory of Fractal Time: Field Equations (the Theory of Almost Inertial Systems and Modiﬁed Lorentz Transformations), Preprint at http://arXiv.org/abs/physics/0005068 [9] Kobelev L.Ya. The Theory of Gravitation in the Space - Time with Fractal Dimensions and Modiﬁed Lorentz Transformations, Preprint at http://arXiv.org/abs/ physics/0006029 [10] Kobelev L.Ya. Physical Consequences of Moving Faster than Light in Empty space, Preprint at http://arXiv.org/abs/gr-qc /0001043 [11] Kobelev L.Ya. The Multifractal Time and Irriversibili ty in Dynamic Systems, Preprint at http://arXiv.org/abs/ physics/0002002 [12] Kobelev L.Ya. Maxwell Equation, Shr¨ odinger Equation , Dirac Equation, Einstein Equation Deﬁned on the Mul- tifractal Sets of the Time and the Space Preprint at http://arXiv.org/abs/gr-qc/0002003 [13] Kobelev L.Ya. Is it Possible to Transfer an Information with the Velocities Exceeding Speed of Light in Empty Space? Preprint at http://arXiv.org/abs/physics /0002003 [14] Kobelev L.Ya. Are the Laws of Thermodynamics Conse- quences of a Fractal Properties of Universe? Preprint at http://arXiv.org/abs/physics/0003036 [15] Kobelev L.Ya. Do Electromagnetic and Gravitation Fields Have Rest Masses? Preprint at http://arXiv. org/abs/physics/0006043 [16] Kobelev L.Ya.Why We Can Not Walk To and Fro in Time as Do it in Space? (Why the Arrow of Time is Exists?) Preprint at http://arXiv.org/abs/ [17] Klimontovith Yu.L., Statistical Theory of Open Sys- tems. Vol.1 , Moscow, Yanus, 1995, 686p. (in Russian); 4Kluwer Academic Publishers, Dordrecht, 1995; Klimon- tovich Yu.L., Statistical Physics of Open systems. Vol.2 , Moscow, Yanus, 1999, 450p. (in Rusian). [18] Mandelbrot B., Fractal Geometry of Nature , W.H.Freeman, San Francisco, 1982 [19] Samko S.G, Kilbas A.A., Marithev I.I., Integrals and Derivatives of the Fractional Order and Their Applica- tions, (Gordon and Breach, New York, 1993). [20] I.M.Gelfand, G.E.Shilov, Generalized functions (Aca- demic Press, New York, 1964) 5
arXiv:physics/0011039v1 [physics.atom-ph] 16 Nov 2000Extension of Kohn-Sham theory to excited states by means of a n oﬀ-diagonal density array Abraham Klein∗ Department of Physics, University of Pennsylvania, Philad elphia, PA 19104-6396 Reiner M. Dreizler† Instit¨ ut f¨ ur Theoretische Physik, Universit¨ at Frankfu rt, D-60054 Frankfurt (February 20, 2014) Early work extending the Kohn-Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding aver age density. We suggest and develop an alternative to this description of excited states that ut ilizes the matrix of the density operator taken between any two states of the included space. Such an ap proach provides more detailed information about the states included, for example, transi tion probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space of states that we wish to descr ibe viewed, however, as a functional of the associated array of matrix elements of the density. It ﬁnds expression in a matrix version of Kohn-Sham theory. To illustrate the formalism, we study a su itably deﬁned weak-coupling limit and derive from it an eigenvalue equation that has the form of the random phase approximation. The result can be identiﬁed with a similar equation derived dire ctly from the time-dependent Kohn-Sham equation and applied recently with considerable success to molecular excitations. We prove, within the deﬁned approximations, that the eigenvalues can be inte rpreted as true excitation energies, a result not accessible to the time-dependent Kohn-Sham sche me. 31.15.Ew, 32.15.Ne, 31.15.Pf I. INTRODUCTION Density functional theory (DFT) was designed originally as a theory of the ground-state density and energy of a many-particle system [1–5]. For an extension to include the calculation of excitation energies, several lines of thoug ht have been developed. The earliest one was based on a minimum p rinciple [6,7] for the trace of the Hamiltonian over a set of the lowest-energy eigenstates of the system. This the ory was then extended to a suitably weighted sum over the same set of eigenstates [8]. The expanded version of the Hohe nberg-Kohn theorem, in either case, is that the average energy is a unique functional of the corresponding average d ensity. Excitation energies are obtained (essentially) by taking diﬀerences between averages over almost overlappin g sets. This approach has not been developed beyond the cited work. Recently, considerable attention has been focused on the de velopment of other methods for studying excitation energies. One powerful approach is based on time-dependent density functional theory (TDDFT) [9–14]. In this approach, one studies the linear response of the time-depen dent density to a time-dependent external ﬁeld. The Fourier transform of the susceptibility (density-density correlation function), which is the essential ingredient f or the calculation of dynamic polarizabilities, has poles at the t rue eigenstates of the system. By application of TDDFT one can derive both a formally exact inhomogeneous integral equ ation for the correlation function and a related eigenvalue equation for the excitation energies. Results obtained for simple systems by the approximate solution of this equation are promising [10,14]. TDDFT has also been applied to the excitation-energy proble m in a diﬀerent way, with less ` a priori justiﬁcation than the above method, but with impressive results upon applicat ion [15–20]. In this approach, an eigenvalue equation that has the form of a random phase approximation (RPA) is derived directly from the Kohn-Sham (KS) time-dependent equation, which we call TDKST, in analogy with the procedure applied to time-dependent Hartree-Fock theory. The ∗aklein@nucth.physics.upenn.edu †dreizler@th.physik.uni-frankfurt.de 1interpretation of the eigenvalue as a true excitation energ y is taken for granted in the literature cited. One of the results of the present work is that this interpretation can b e justiﬁed for a suitably deﬁned set of excitations. Finally, we call attention to several recent studies of the e xcited state problem that involve extensions of the variationally based KS theory to individual excited states [21,22]. For these methods, as well, applications to simple systems seem promising. Improved exchange and correlation kernels necessary for all these methods and a connection with many-body perturbation theory are discussed in [23], w hereas in [24] an improved exchange-correlation potential is utilized to provide more accurate continuum KS orbitals n eeded for excited state and polarizability calculations. In this paper, we appear initially to be taking a step backwar ds by returning to a study of the trace variational principle [25–27]. Instead of considering the average ener gy as a functional of the average density, however, we argue for the introduction of a matrix array of densities, i. e., al l matrix elements of the density operator among all states of the chosen ensemble, and for an investigation of the avera ge energy as a functional of this matrix array. In Sec. II we present arguments to indicate how the Hohenberg-Kohn ( HK) analysis can be extended to this case yielding a matrix Thomas-Fermi (MTF) equation. We subsequently (Sec . III) generalize the KS analysis, deriving a matrix Kohn-Sham equation (MKS), that contains not only the expect ed ingredient, a matrix eﬀective potential, but also a matrix of Lagrange multipliers arising from number conserv ation in each state of the chosen subset; this matrix can be diagonalized, but not otherwise transformed away. By combi ning solutions of the MKS equations, we can construct the density array. As an application of this theory, we study, in Sec. IV, the MKS equations in what we term the weak-coupling limit. In this limit, we include only the ground state and exc ited states characterized (largely) as linear combination s of Slater determinants with only one excited particle compa red to the ground-state determinant (and therefore one hole). Reference to higher excited states and simple assump tions concerning their properties do eventually enter the discussion. The major consequence of this analysis is an eigenvalue equation for the aforementioned Lagrange multipliers (relative to their ground-state value) that ha s the form of the random phase approximation. This equation has the same structure as that deduced from TDKST. Assuming t hat the ground-state KS problem has been solved, the major unknown ingredient in these equations, an exchang e-correlation interaction, can be identiﬁed with the corresponding quantity utilized in TDKST, at least in the ad iabatic limit utilized in the RPA calculations. There remains the problem of the physical signiﬁcance of the eigenvalues of the RPA formalism. In the work based on TDKST, it is simply assumed that these may be identiﬁed wit h true excitation energies. In our work, they appear as Lagrange multipliers to enforce number conservation in e xcited states. In our formalism true excitation energies can be calculated, in principle, from a diﬀerence of adjacen t averages of the Hamiltonian, as in previous applications o f the trace variational principle. In Sec. V we carry out such a calculation, and show that with an extended deﬁnition of the weak coupling approximation, consonant with the tradit ional interpretation of the RPA as a boson approximation, the interpretation of the eigenvalues as excitation energi es is justiﬁed. In a concluding section, we summarize our considerations. II. HOHENBERG-KOHN ARGUMENTS The Hamiltonian is written as ˆH=ˆT+ˆV+ˆW+ˆY, (2.1) the sum of the kinetic energy, the electrostatic interactio n of the electrons with the nucleus, the Coulomb repulsion of the electrons, and an additional ﬁctitious external sour ce term that will be set to zero for actual calculations. The following considerations apply, however, to any many body H amiltonian of similar structure. The various terms have the forms ( xstands for the space-spin pair ( r,s)), in atomic units, ˆT=/integraldisplay dxˆψ†(x)(−1 2∇2)ˆψ(x) =/integraldisplay ˆψ†τˆψ, (2.2) ˆV=/integraldisplay dxˆψ†(x)ˆψ(x)v(r), (2.3) ˆW=/integraldisplay dxdx′1 \|r−r′\|ˆψ†(x)ˆψ†(x′)ˆψ(x′)ˆψ(x), (2.4) ˆY=/integraldisplay dxdx′y(x,x′)ˆη(x,x′), (2.5) ˆη=ˆψ†(x)ˆψ(x)ˆψ†(x′)ˆψ(x′). (2.6) 2ˆYis a combination of one and two body forces. For the traces of t hese operators over the ensembles introduced below, we use the same symbols without hats. In the following we shall base our arguments on the variation al principle for the trace of the Hamiltonian over the lowestMeigenstates of the system [6–8,25–27]. We consider the case where theM+ 1ststate has a higher energy than theMthstate. This is the normal, but not absolutely necessary, cri terion for choosing M. In order to achieve our goals, beyond a certain point our considerations will be heuristic rather than rigorous. Let S={\|I/an}b∇acket∇i}ht} (2.7) be the space of included states ( I= 1...M). For any operator ˆO, we deﬁne the restricted trace O(M)=M/summationdisplay I=1/an}b∇acketle{tI\|ˆO\|I/an}b∇acket∇i}ht, (2.8) where it is convenient in the further development not to divi de byM. Unless more than one value of Moccurs in the same equation, we shall otherwise drop the superscript. We then consider a set of propositions formulated in imitation of the Hohenberg-Kohn (HK) theorem [1]: (i) Every choice of a function y(x,x′) in (2.5) determines a space Sthrough the solution of the Schr¨ odinger equation. (ii)Sdetermines the correlation function η(x,x′) =/summationtext/an}b∇acketle{tI\|ˆη(x,x′)\|I/an}b∇acket∇i}ht. (iii) This relationship is single-valued and invertible. T his can be proved by an adaptation of the standard HK argument, as we now show. Suppose that S →η,S′/ne}ationslash=S →η′. (2.9) It follows that η/ne}ationslash=η′. We prove this by using the trace variational principle to es tablish two inequalities, HS[y]<H S′[y′] +/integraldisplay (y−y′)η′, (2.10) HS′[y′]<H S[y] +/integraldisplay (y′−y)η. (2.11) Here, for example, HS[y] is the ensemble average of ˆHover the set S, where it is further emphasized that this average is a functional of y. Adding (2.10) and (2.11) and assuming that η=η′, we obtain the usual contradiction HS[y] +HS′[y′]<H S′[y′] +HS[y]. (2.12) Thus Sis a single-valued functional of η. Considering Hto be a functional of η, we write the variational principle in the form δH=/integraldisplayδH δηδη= 0. (2.13) We shall not attempt, however, to implement the variational principle in this version. Instead, using completeness, we introduce the formula η(x,x′) =M/summationdisplay I=1∞/summationdisplay I′=1/an}b∇acketle{tI\|ˆψ†(x)ˆψ(x)\|I′/an}b∇acket∇i}ht ×/an}b∇acketle{tI′\|ˆψ†(x′)ˆψ(x′)\|I/an}b∇acket∇i}ht. (2.14) As long asMis ﬁnite, this is an asymmetric formula. Since our aim is to ut ilize the quantities n(x)I′I=/an}b∇acketle{tI\|ˆψ†(x)ˆψ(x)\|I′/an}b∇acket∇i}ht (2.15) as variational parameters, this asymmetry presents a probl em that can be dealt with (approximately) in two ways. In the ﬁrst method, which will be studied in this paper, we shall deﬁne the “matrix” nas a square matrix, M×M, but chooseMonly large enough to encompass a well-deﬁned small set of sta tes. (In extreme cases, this may well be only the ground state and one or a few excited states.) Neverthele ss, in (2.14) we must allow completeness to have its full sway, as a matter of both mathematical and physical rigor. In deed, for any physical situation of which we are aware, 3there will always be values of I′outside the set M, for which the matrix elements connecting these states to st atesI within the set are as numerically signiﬁcant as essential el ements belonging to the set n. We deal with this situation by assuming that the matrix elements nII′,I≤M,I′> M can be approximated as functionals of n. We call this assumption a closure approximation, whose speciﬁc form wil l depend on the physics of the speciﬁc application. In the second method, which applies, for example, to the rota tional spectrum of molecules or nuclei, we have a situation, where starting from the ground state, there is a c hain of matrix elements of the density that are signiﬁcantly (an order of magnitude or more) larger than can be found for an y other chain (without the intervention of at least one smaller matrix element). We have in mind the rotational b ands built upon the ground state. Of course there are similar structures built upon excited (vibrational) state s, but starting from the ground state, such a sequence involv es at least one smaller matrix element of the density connectin g the ground and vibrational structures. In such cases, in order to produce correct physics, the initial set Mmust be very large or, in an ideal limit, inﬁnite. To deal with the vibrational excitations moreover, we have to deal with sets of large sets. This is not as formidable as it sounds, but, in any event, will not be studied in the present work. Returning to the formal development, with the help of (2.15) , (2.14) can be rewritten (summation convention) η(x,x′) =n(x′)II′n(x)I′I =n(x′)II′n∗(x)II′. (2.16) Thus we may replace the variational principle (2.13) by the f orm δH=/integraldisplayδH δnδn. (2.17) We emphasize that our conﬁdence in the application of (2.17) , which is expressed in terms of the matrix elements of nwithin the included space, depends on the validity of the clo sure approximation. ¿From Eq. (2.17) we can derive a generalized Thomas-Fermi (TF) equation by imposing the nu mber conservation constraints. If Nis the number of electrons, we have /integraldisplay dxn(x)II′=NδII′. (2.18) Introducing a set of Lagrange multipliers µII′, we now write δH−µII′/integraldisplay δn(x)I′I= 0, (2.19) and conclude that δH δn(x)I′I=µII′, (2.20) which is the generalized TF equation for the present case. III. GENERALIZED KOHN-SHAM SCHEME n(x)II′is the limit x→x′of the oﬀ-diagonal one-body density matrix ρ(xI\|x′I′) =/an}b∇acketle{tI′\|ˆψ†(x′)ˆψ(x)\|I/an}b∇acket∇i}ht. (3.1) Sinceρis a positive deﬁnite matrix, it can be brought to diagonal fo rm, a move that generalizes the concept of natural orbitals. We thus write ρ(xI\|x′I′) =/summationdisplay JλJΦJ(xI)Φ∗ J(x′I′), (3.2) λJ≥0, (3.3) /summationdisplay I/integraldisplay dxΦ∗ J(xI)ΦJ′(xI) =δJJ′, (3.4) /integraldisplay dxρ(xI\|xI′) =NδII′. (3.5) 4Here Eqs. (3.2) and (3.3) deﬁne the eigenfunctions and eigen values of the generalized density matrix, (3.4) expresses the property that the Φ J(xI) are unit vectors in the space labeled jointly by the single- particle coordinates and the eigenvalues of the states in the set S, and (3.5) expresses number conservation. It follows from t hese equations that /summationdisplay I/integraldisplay dxρ(xI\|xI) =/summationdisplay JλJ=NM. (3.6) In imitation of ground-state KS theory, we introduce a mappi ng from the oﬀ-diagonal density to a quasi-independent- particle oﬀ-diagonal density, n(x)II′→ns(x)II′, (3.7) ns(x)II′=/summationdisplay JϕJ(xI)ϕ∗ J(xI′), (3.8) /summationdisplay I/integraldisplay dxϕ∗ J(xI)ϕJ′(xI) =δJJ′, (3.9) /integraldisplay dxns(x)II′=NδII′. (3.10) Though we use the same symbol Jto label orbitals as for the case of natural orbitals, here th e similarity stops. For the latter,Jis, in principle, an unbounded set. For the present alternat ive, the set labeled by Jis strictly a ﬁnite set as determined by the sum (cf. (3.6)), /summationdisplay J1 =NM. (3.11) We next show how the variational principle may be used to obta in equations for the orbitals ϕJso that in fact the matrices nandnsare equal. We shall utilize the variational principle in the form /summationdisplay/integraldisplayδH δϕ∗ J(xI)δϕ∗ J(xI) + c.c.= 0, (3.12) together with its complex conjugate. Setting the extra sour ce termY, deﬁned in (2.5) to zero and imitating the procedure for the ground-state theory, we decompose H=Ts+ (V+W+T−Ts), (3.13) Ts=/summationdisplay J/integraldisplay ϕ∗ JtϕJ. (3.14) Enforcing the equality of nandns, we deﬁne an eﬀective single-particle potential matrix, vs(x)II′=δ δn(x)I′I(V+W+T−Ts), (3.15) =δ δns(x)I′I(V+W+T−Ts). (3.16) The discussion of the decomposition of this matrix single-p article operator into constituent interesting parts will b e taken up in Sec. IV. With the help of Eqs. (3.13-3.16), we derive from the variati onal principle (3.12) the conditions /summationdisplay/integraldisplay δϕ∗ J(xI)[τδII′+vs(x)II′]ϕJ(xI′) + c.c.= 0. (3.17) To derive generalized single-particle equations of motion from the variational principle, we add the constraint condi - tions −/summationdisplay/integraldisplay δϕ∗ J(xI)[ǫJδII′+ν(x)II′]ϕJ(xI′) + c.c.= 0. (3.18) 5HereǫJis the Lagrange multiplier for the normalization condition contained as part of (3.9). (As usual, the or- thogonality condition need not be imposed, since it will be a utomatically satisﬁed by the solutions of the emerging equations.) The unfamiliar term containing the Lagrange mu ltiplier matrix ν(x)II′has the form of an additional potential matrix, whose purpose is to enforce the condition [28] that n=ns. We shall study this quantity further be- low. Combining Eqs. (3.17) and (3.18), we derive (together w ith it complex conjugate) the generalized single-particle equation ǫJϕJ(xI) = [tδII′+vs(x)II′−νs(x)II′]ϕJ(xI). (3.19) At this juncture it is appropriate to wonder if (3.19) can be r elated to TDKST. We cannot expect a general connection, since the latter describes the consequences of the application of a time-dependent external ﬁeld, whereas in the theory under development, the “time dependence” is a p urely internal matter expressed by an oﬀ-diagonal array of densities and eﬀective potentials. Nevertheless, a connection between the two formalisms will be made for the application studied in Sec. IV, the so-called weak-coup ling limit. We conclude the present section by showing that (cf. Eq. (2.2 0)) ν(x)II′=µII′, (3.20) up to an additive constant. It is thus a non-trivial matrix an d cannot be absorbed into the eigenvalues ǫJ. To prove (3.20), we can work backwards from the sum of (3.17) and (3.18 ) to the equation 0 =/summationdisplay/integraldisplay/bracketleftbiggδH δns(x)II′−ν(x)II′/bracketrightbigg δns(x)I′I (3.21) =/summationdisplay/integraldisplay/bracketleftbiggδH δn(x)II′−ν(x)II′/bracketrightbigg δn(x)I′I (3.22) =/summationdisplay/integraldisplay/bracketleftbiggδH δn(x)II′−µII′/bracketrightbigg δn(x)I′I. (3.23) In passing from (3.21) to (3.22), we have used the equality ns=n. In writing (3.23), we have repeated (2.19). Comparing (3.22) with (3.23), we arrive at (3.20), again up t o an additive constant. In the following sections, we shall use the summation convention consistently both for the coor dinate xand for the index I, and for the index Jmost of the time. IV. APPLICATION TO THE WEAK COUPLING LIMIT In the course of this section, we shall transform and approxi mate Eq. (3.19), leading to an eigenvalue equation that will determine oﬀ-diagonal elements of the matrix n. We shall do so in an approximation, the weak-coupling approximation, that is roughly equivalent to a linear respo nse approach. Assuming that the matrix µcan be chosen diagonal (see immediately below), the eigenvalues are the q uantities λI=µII−µ00. (4.1) The proof that the matrix µcan be chosen diagonal goes as follows: Though we trace over a set of states labeled Iand originally identiﬁed as eigenstates of the reference sy stem, the entire formalism is invariant under a unitary transformation within the included space. Such a transform ation can be chosen to diagonalize µif it isn’t already diagonal. The relation of the quantities in Eq. (4.1) to the e xcitation energies of the system is not immediately apparent, even though this identiﬁcation has been made in th e recent literature with remarkable empirical success [15–20]. We shall address this problem in Sec. V. Though the derivation of the main result of this section, the eigenvalue equation, can be carried out directly from the generalized KS equation, we present the discussion in a f orm that makes more immediate contact with the density functional form of the theory. The ﬁrst step, which is comple tely general, is to transform Eq. (3.19) into an equation for the matrix ns II′(x,x′). First rewrite Eq. (3.19), remembering Eq. (4.1), as ǫJϕJ(xI) = (hs−λ)II′(xx′)ϕJ(x′I′), (4.2) Recalling the deﬁnition ns II′(xx′) =/summationdisplay JϕJ(xI)ϕ∗ J(x′I′), (4.3) 6we can form from Eq. (4.2) and its complex conjugate two equiv alent but distinct values of the sum/summationtext JǫJϕJ(xI)ϕ∗ J(x′I′). The diﬀerence of these forms yields the generalized densi ty-matrix equation ns II′(xx′)(λI′−λI) =ns II′′(xx′′)hs I′′I′(x′′x′)−hs II′′(xx′′)ns I′′I′(x′′x′), (4.4) that will provide the starting point for our further conside rations. Before continuing on our main path we note that by introducin g time-dependent matrix elements OII′(t)≡OII′exp[−i(λI−λI′)t], (4.5) whereOtakes on the values nsandhs, Eq. (4.4), may be written in the form −id dtns(t) = [ns(t),hs(t)]. (4.6) This resembles the fundamental equation of TDKST, in densit y matrix form, except that the bold-face type reminds us that we are dealing with quantum-mechanical operators ra ther than c-numbers. This can be converted into a form of TDKST, however, by assuming the existence of a wave packet \|Ψ/an}b∇acket∇i}htthat is a linear combination of the ground state and excited states of interest, for which we can also replace the average of the products that appear in the commutator by the product of the averages. However, this derivation of T DKST is not suitable for our purposes. We therefore return to the direct study of Eq. (4.4) in the limit of interes t. In the weak coupling approximation, we conﬁne our attention to the ground state 0 and to a single excited state 1 (up to magnetic degeneracy) which belongs to a subset of the s tatesIto be characterized. It will turn out that the equations to be derived will characterize an entire subset o f the states I, i. e., the state 1 will belong to a well-deﬁned subset. We associate the ground state with the Slater determ inant of the ground-state KS scheme. The excited states of immediate interest to us will be associated with linear co mbinations of determinants of the same complete set of orbitals in which one particle in a previously occupied or bital is promoted to a previously unoccupied orbital, a so-called particle-hole (ph) excitation. Here the word ass ociation is meant to imply that these are states that have overwhelmingly larger overlap with such determinants than they have with any other determinant of KS orbitals. We may also imagine that there are states that have maximum over lap with determinants characterized by νparticle-ν hole excitations. It is convenient below to designate the sp ace of 1p-1h states as I1, as opposed to the general Iν. To reduce Eq. (4.4) to a useful and ultimately recognizable f orm, we introduce a set of assumptions concerning relative orders of magnitude of certain matrix elements, wh ose validity is obvious in the limit of vanishing two-partic le interaction (and is discussed further below) \|ns 00\|>>\|ns 0I1\|>>ns \|0I2\|>>..., (4.7) \|ns I1I1\| ≈ \|ns 00\|, (4.8) \|ns I1I′ 1\| ≈ \|ns 0I2\|ifI1/ne}ationslash=I′ 1. (4.9) We shall consider diagonal elements to be of zero order, elem ents connecting states IνtoIν+pto be ofpth order. We interrupt the formal development in order to examine the a ssumptions Eqs. (4.7-4.9). Since the density matrix elements are bilinear combinations of the generalized sing le-particle amplitudes ϕJ(xI), it is convenient to discuss the assumptions of the weak coupling approximation in terms of t he latter quantities. We assume that the indices Jcan be identiﬁed as a pair ( I,h) whereIis now any state, ground or excited, of the reference system, andhidentiﬁes one of the occupied single-particle orbitals of the KS theory. T hus each value of Jof interest to us speciﬁes a one-hole state with parentage (largely) in one of the states of the ref erence system. We introduce next the concept of hierarchy of states. Here the ground state stands by itself, and we shal l think of it roughly as a Slater determinant occupied by the lowest orbitals in an eﬀective external potential, as in the KS theory. At the ﬁrst level of the hierarchy is a set of excited states of approximately one-particle, one-h ole character, formed by linear combinations of particle-h ole excitations, At the next level are the two-particle, two-ho le excitations, etc. In Sec. V we go further and treat the excited states as boson excitations, as suggested by the for m of the eigenvalue equation that is the major result of this section. Notice that in the weak coupling picture, not o nly arenII′andvs II′matrices in the space of states of the reference system, but so also is ϕIh(I′). Considering assumption (4.8) ﬁrst, it asserts that for Ibelonging to the ﬁrst few levels of the hierarchy, if N, the number of particles is not too small, in lowest approximatio n matrix elements diagonal in Iare equal to their value forI= 0. It is easiest to see this for the density itself, since the wave functions of the excited states diﬀer from those of the ground state by at most a few particles out of N. That it follows for the other quantities is a consequence of their relation to the density, as will be seen from further st udy below. We shall consider all diagonal matrix elements to be zero order quantities. A further assumption, in terms o f this scale, is that matrix elements in which IandI′ 7belong to adjacent levels in the hierarchy are, on the averag e, of order (1 /√ N) compared to zero order quantities. For the sorting of our equations, we also need the assumption tha t matrix elements in which I,I′diﬀer by two levels or refer to two diﬀerent states of the same level are second orde r quantities, i. e., of the order of the product of ﬁrst order quantities. Of course, it has to be veriﬁed a posteriori that the solutions found are in accord with these statements . Our aim is to apply these assumptions to choose those matrix e lements of Eq. (4.4) that characterize the state 0 and the states I1. To carry out this program, we must look more closely into the structure of the eﬀective interaction vs. First we rewrite the trace of the Hamiltonian in the form H=Ts+V+Wc+Hxc, (4.10) Wc=1 2ns II′(x)1 \|x−x′\|ns I′I(x′), (4.11) which deﬁnes Hxc. It follows that vs II′(x) =δ δns I′I(x)(V+Wc+Hxc) (4.12) =v(x)δII′+vc II′(x) +vxc II′(x), (4.13) vc II′(x) =1 \|x−x′\|ns II′(x′). (4.14) The main reason for exhibiting these formulas is to recogniz e, as we shall see in more detail below, that the oﬀ-diagonal elements of hare at least linear in the corresponding oﬀ-diagonal elemen ts ofns. This is obvious from Eq. (4.14) for the Coulomb contribution and will be argued more closely lat er forvxc. Thus we may safely assume that that the matrix elements of hare the same order of magnitude as the corresponding matrix e lements of ns. Turning ﬁnally to the matrix elements of Eq. (4.4), we consid er ﬁrst the ground or 00 element. Neglecting terms of second order and higher, we ﬁnd ns 00(xx′′)hs 00(x′′x′)−hs 00(xx′′)ns 00(x′′x′) = 0. (4.15) It is consistent with our approximations to identify ns 00(in leading approximation only) with the ground state densi ty of KS theory and hs 00with the KS single-particle Hamiltonian. Equation (4.15) i s thus the KS equation in density matrix form and determines a complete set of orbitals ϕa(x), wherea=hwill refer to the orbitals occupied in the ground-state determinant and a=pthose unoccupied. Consider next the ﬁrst-order matrix element 01. Retaining o nly ﬁrst-order contributions (leading corrections are third order), we may write λ1ns 01(xx′) =ns 00(xx′′)hs 01(x′′x′) +ns 01(xx′′)hs 11(x′′x′)−hs 00(xx′′)ns 01(x′′x′)−hs 01(xx′′)ns 11(x′′x′). (4.16) As a ﬁrst step in the evaluation of this equation, we may, acco rding to Eq. (4.8), set the 11 matrix elements equal to the 00 ones. We also drop the subscripts 00 understanding the se according to the previous identiﬁcation to be the standard KS quantities. If we can exhibit hs 01as an (approximate) linear functional of ns 01, Eq. (4.16) will have the form of a linear eigenvalue problem. First we have (the matri x elements in question are local functions of x) hs 01(x) =vc 01(x) +vxc 01(x), (4.17) vc 01(x) =1 \|x−x′\|ns 01(x′). (4.18) We see that vcis, by deﬁnition, already of the desired form. We turn then to vxc. Our approach to this quantity is to revert to the study of Hxc, deﬁned in Eq. (4.10), which we consider, in line with assumptions previously made, a fun ctional ofn00≈n, ofns 01, and ofns 10, the latter two considered as small quantities. (It is also a functional of t he other oﬀ-diagonal elements, ns 01′andns 1′0, where 1′refers to any of the other states at level one of the hierachy of state s. It is simply that this dependence does not enter into the current discussion). We then expand Hxcas a functional Taylor series in these quantities, Hxc=Hxc\|0+δHxc δns 10(x)\|0ns 10(x) +δHxc δns 01(x)\|0ns 01(x) +1 2δ2Hxc δns 10(x)δns 10(x′)\|0ns 10(x)ns 10(x′) +δ2Hxc δns 10(x)δns 01(x′)\|0ns 10(x)ns 01(x′) +1 2δ2Hxc δns 01(x)δns 01(x′)\|0ns 01(x)ns 01(x′) +... . (4.19) 8Strictly, the quantity Hxc\|0 and its functional derivatives still depend on n11as well asn00. It suﬃces to ignore the diﬀerence of the two quantities in the present discussion, b ut we shall have to remember and include the diﬀerence in the arguments of Sec. V. We note further that only the ﬁrst and fourth of the terms shown explicitly in this equation are non-vanishing. Recall that Hxcis a trace and therefore invariant under a unitary transform ation in the space of statesI. Its dependence on the matrix nmust also be in the form of traces over these indices. As we can see on the example of the Coulomb interaction, this dependence is m ore general than traces of products of nat the same point, but in any event it follows that for every factor of ns 10at some spatial point, there must be a factor of ns 01, at a generally diﬀerent point. The simpliﬁcation described ab ove follows. We thus compute to ﬁrst order vxc 01(x) =δ2Hxc δns 10(x)δns 01(x′)\|0ns 01(x′) ≡f10,10(\|x−x′\|,n)ns 01(x′) ≈f(\|x−x′\|,n)ns 01(x′). (4.20) In passing from the second to the third line of this equation, i. e., in ignoring the state-dependence of f, we are making an approximation equivalent to the adiabatic approx imation widely used in TDKST. With the deﬁnition (the dependence on nbeing understood) feff(\|x−x′\|) =1 \|x−x′\|+f(\|x−x′\|), (4.21) Eq. (4.16) may be rewritten as λ1ns 01(xx′) =ns(xx′)feff(\|x′−x′′\|)ns 01(x′′) +ns 01(xx′′)hs(x′′x′) −hs(xx′′)ns 01(x′′x′)−ns(xx′)feff(\|x−x′′\|)ns 01(x′′). (4.22) . The ﬁnal task with respect to this equation is to convert it in to a standard RPA form. Toward this end we reexpress the matrices nsandns 01in terms of the KS single-particle functions, ϕa(x), satisfying the KS equation hs(xx′)ϕa(x′) =ǫaϕa(x). (4.23) First of all we have the familiar equation ns(xx′) =ϕh(x)ϕh(x′). (4.24) Next we must evaluate the sum ns 01(xx′) =ϕJ(x0)ϕ∗ J(x′1). (4.25) Here we must introduce assumptions concerning which values ofJcontribute to the required order. In the space of the eigenstates of the fully interacting system, we are conc erned with the ground state and with states that are largely phexcitations of this state. When we remove one particle (crea te a holeh), we expect to encounter states that can be characterized as either 0 hor 1h, and these are the values of Jthat we assign in the sum (4.25). If we consistently use the approximations ϕ0h(0)≈ϕ1h(1)≈ϕh, the weak-coupling value of Eq. (4.25) becomes ns 01(xx′) =ϕh(x)ϕ∗ 0h(x′1) +ϕ1h(x0)ϕ∗ h(x′). (4.26) The ﬁnal form for this quantity is achieved by expanding the ﬁ rst-order amplitudes in terms of KS modes, ϕ0h(1) =ϕpXph, (4.27) ϕ1h(0) =ϕpY∗ ph. (4.28) The restriction of the sums on the right-hand sides of these e quations is also consistent with the weak-coupling picture painted above. Strictly the amplitudes X,Yshould carry superscripts 1, identifying the eigenstate to which they refer, but we shall suppress these except when required for clarity , as in Sec. V. Finally then, ns 01(xx′) =ϕh(x)ϕ∗ p(x′)X∗ ph+ϕ∗ p(x)ϕh(x′)Y∗ ph. (4.29) Introducing Eqs. (4.24) and (4.29) into Eq. (4.22), we can pr oject out equations for X∗ phandY∗ ph. We quote the complex conjugate of these equations: 9(ǫh−ǫp+λ1)Xph= (feff)ph′hp′Xp′h′+ (feff)pp′hh′Yp′h′, (4.30) (ǫh−ǫp−λ1)Yph= (feff)hp′ph′Yp′h′+ (feff)hh′pp′Xp′h′, (4.31) (feff)abcd=ϕ∗ a(x)ϕ∗ b(x′)feff(\|x−x′\|)ϕc(x)ϕd(x′). (4.32) The equations found are of the same form as those of the random phase approximation (RPA). Solutions are to be normalized in the usual way, according to the conditions (Ap pendix B), /summationdisplay ph(\|Xph\|2− \|Yph\|2) = 1. (4.33) As is well known, two diﬀerent non-degenerate solutions of t he RPA equations are orthogonal with the same metric as in (4.33). It is important to emphasize what has been accomplished by th e calculations of this section. With the help of Eq. (4.29), for instance, we can calculate the oﬀ-diagonal m atrix elements of the density between the ground state and the ﬁrst level of excited states. This can be applied, for example to the calculation of the corresponding matrix elements of the electric dipole moment. However, just as in t he case of KS theory, where we ﬁnd single-particle energies that bear no simple relation, except for the most lo osely bound orbit, to physical energy diﬀerences, so in the present case as well the eigenvalues, which ﬁrst enter as Lagrange multipliers in the variational principle, do not appear to have a simple relation to excitation energies. We t urn next to a more detailed study of this question. V. EXCITATIONS AS ENERGY DIFFERENCES We shall discover in this section that with the help of additi onal assumptions concerning the RPA limit that are consonant with its signiﬁcance as a quasi-boson approximat ion, the eigenvalues λ1of Eqs. (4.30) and (4.31) can be identiﬁed with true excitation energies of the system. In pr inciple the energy diﬀerences can be calculated from the expression H(2)−2H(1)≡/summationdisplay I=0,1/an}b∇acketle{tI\|ˆH\|I/an}b∇acket∇i}ht −2/an}b∇acketle{t0\|ˆH\|0/an}b∇acket∇i}ht =E1−E0, (5.1) whereEIis the energy of state I. This diﬀerence will be evaluated with the aid of Eqs. (4.10) , (4.11), and the simpliﬁed version of (4.19). These equations refer in turn t oH(2)orH(1), as required. The result that we shall establish is E1−E0= (ǫp−ǫh)(\|Xph\|2− \|Yph\|2) +X∗ ph[fph′hp′Xp′h′+fpp′hh′Yp′h′] +Y∗ ph[fhp′ph′Yp′h′+fhh′pp′Xp′h′].(5.2) But the right hand side of this equation is easily seen from Eq s. (4.30) and (4.31) to equal λ1, provided that we make use of Eq. (4.33). It is simplest to evaluate the diﬀerence (5.1) ﬁrst for the in teraction terms. Consider, for instance, the Coulomb diﬀerence, Vc(2)−2Vc(1)=1 21 \|x−x′\|[ns 11(x)ns 11(x′)−ns 00(x)ns 00(x′) + 2ns 01(x)ns 10(x′)] ≈1 \|x−x′\|{[ns 11(x)−ns 00(x)]ns 00(x′) +ns 01(x)ns 10(x′)}, = [ns 11(x)−ns 00(x)]vc(x) +1 \|x−x′\|ns 01(x)ns 10(x′)], (5.3) where the simpliﬁcation is made possible by the fact that the diﬀerencens 11−ns 00, as we shall prove below, is quadratic in the RPA amplitudes. The corresponding diﬀerence involvi ng the exchange-correlation energy can be written Hxc(2)−2Hxc(1)= [ns 11(x)−ns 00(x)]vxc(x) +f(\|x−x′\|)ns 01(x)ns 10(x′)], (5.4) The ﬁrst term of this equation is the value, to the required or der, ofHxc(2)\|0−2Hxc(1)\|0. Next we see that the second terms of Eqs. (5.3) and (5.4) combi ne to give 10feff(\|x−x′\|)ns 01(x)ns 10(x′) =X∗ ph[fph′hp′Xp′h′+fpp′hh′Yp′h′] +Y∗ ph[fhp′ph′Yp′h′+fhh′pp′Xp′h′], (5.5) which has been evaluated with the help of Eq. (4.29). This is a lready seen to be the interaction terms of Eq. (5.2). The remaining terms of Eqs. (5.3) and (5.4), as well as the con tributions arising from the kinetic energy and the external potential depend on the value of ns 11(x)−ns 00(x) =ϕ∗ J(x1)ϕJ(x1)−ϕ∗ J(x0)ϕJ(x0). (5.6) To enumerate the states Jthat contribute to this diﬀerence we shall picture the state 1 as an elementary boson excitation, as is done in the standard approach to the RPA. Th e relations that follow from this assumption will lead, as we shall see, to a quantitative form of closure approximat ion that is essential to the calculation. By the notation 1×1, we shall mean a double boson excitation with the same boson , whereas by 1 ×1′we shall mean a double excitation with diﬀerent bosons. Thus for the amplitudes ϕJ(1), we consider the values J= 0h,1h,1×1h,1×1′h. The contributions from the latter two values are evaluated i n boson (closure) approximation as ϕ1×1h(1) =√ 2ϕ1(0), (5.7) ϕ1×1′h(1) =ϕ1′(0). (5.8) For the amplitude ϕJ(0), the required values are J= 0h,1h,1′h. For the diﬀerence (5.6), we thus ﬁnd ns 11−ns 00=ϕ∗ 0h(1)ϕ0h(1) +ϕ∗ 1h(0)ϕ1h(0) +ϕ∗ 1h(1)ϕ1h(1)−ϕ∗ 0h(0)ϕ0h(0). (5.9) The total contribution of the ﬁrst two terms of Eq. (5.9) to th e energy diﬀerence under study, obtained by substi- tuting Eqs. (4.27) and (4.28) and applying the result to the s um of single-particle operators that add up to the KS Hamiltonian hs, is found to be ǫp(\|Xph\|2+\|Yph\|2), one of the single-particle terms in Eq.(5.2). The evaluat ion of the remaining terms of Eq. (5.9) is carried by studying the norma lization conditions, Eq. (3.9). We calculate 1 =/summationdisplay I\|ϕ0h(I)\|2 =\|ϕ0h(0)\|2+\|ϕ0h(1)\|2+/summationdisplay I′/negationslash=I\|ϕ0h(1′)\|2, (5.10) 1 =/summationdisplay I\|ϕ1h(I)\|2 =\|ϕ1h(1)\|2+\|ϕ1h(0)\|2+\|ϕ1h(1×1)\|2+/summationdisplay 1′/negationslash=1\|ϕ1h(1×1′)\|2 ≈ \|ϕ1h(1)\|2+\|ϕ1h(0)\|2+ 2\|ϕ0h(1)\|2+/summationdisplay 1′/negationslash=1\|ϕ0h(1′)\|2, (5.11) where the last evaluation has made use of the boson approxima tion expressed by Eqs. (5.7) and (5.8). These equations are satisﬁed by the normalization changes ϕ0h(x0) =ϕh(x)[1−1 2\|Xph\|2−1 2/summationdisplay 1′/negationslash=1\|X1′ ph\|2], (5.12) ϕ1h(x1) =ϕh(x)[1− \|Xph\|2−1 2\|Yph\|2−1 2/summationdisplay 1′/negationslash=1\|X1′ ph\|2]. (5.13) Combining these results and applying them to the last two ter ms of Eq. (5.9), suitably multiplied by the sum of terms that comprise hsleads to the ﬁnal contribution −ǫh(\|Xph\|2+\|Yph\|2) to the theorem stated in Eq. (5.2). VI. CONCLUDING REMARKS In this paper, we have developed yet another formalism for th e study of excited states within a framework that generalizes the basic ideas of KS theory. The main novelty in our approach compared to other methods is that the latter work with a single density, be it the average in the gro und state, in an excited state, an ensemble average, or the average in a suitably chosen time-dependent state. On the ot her hand, we arrive by somewhat circuitous reasoning 11at a formalism involving an entire array of matrix elements o f the density operator taken among a pre-selected set of states. The application of the variational principle for th e trace of the Hamiltonian then leads to a generalized KS scheme in terms of orbitals that depend not only on the coordi natex, but also on a label Ifor one of the included states. We have examined the consequences of this formalism for the weak-coupling limit. We did this by framing a set of assumptions, including a closure approximation, in order to identify the most important amplitudes and their equations that characterize the ground state and a sim ple class of excited states that are composed of 1p-1h excitations of the ground state. In this way, we regained ﬁrst the ground-state KS theory and s econd derived an eigenvalue equation of RPA form. By approximating a state-dependent (frequency-dependent ) eﬀective interaction by a state-independent (frequency independent) eﬀective interaction, the eigenvalue equati on became identical to one that can be derived from TDKST, that has been quite successful in application, especially t o the description of excited states that are known to be of the simple type included in our assumptions. A problem of i nterpretation remains in that the derivation from TDKST contains no argument to justify that the eigenvalues c an be associated with observed excitations. The same diﬃculty applies to our derivation, in that the eigenvalues enter the formalism as Lagrange multipliers arising from th e conservation of electrons in the given state. Exploiting ou r assumptions to the fullest extent, we are able, neverthele ss, to prove a theorem that the Lagrange multipliers that enter t he scheme can be equated to real energy diﬀerences. As formulated, the reasoning described in this paper can be e xtended to improve the approximations that we have so far achieved for 1p-1h states, as well as to study more comp licated exited states, e. g., of 2p-2h character. The application to rotational spectra might also be intriguing . ACKNOWLEDGMENT One of the authors (AK) is grateful to the Humboldt Foundatio n for support of this work and to his co-author for his hospitality. APPENDIX A: RELATION OF WEAK-COUPLING LIMIT TO TIME-DEPEND ENT DENSITY FUNCTIONAL THEORY In this section, we shall connect the linearized RPA equatio ns (4.30) and (4.31) with a corresponding linearized approximation to TDDFT. We start with TDDFT in density-matr ix form idρs dt= [(τ+vs(t)),ρs], (A1) ρs(xt,x′t) =/summationdisplay hϕh(xt)ϕ∗ h(x′t), (A2) vs(xt) =δ δn(xt)(V(t) +W(t) +T(t)−Ts(t)). (A3) Hereϕ(xt) are theNinstantaneous eigenfunctions of τ+vs(t) of lowest energy, deﬁning a time-dependent Slater determinant whose kinetic energy is Ts(t), andV(t), for example, is the expectation value of ˆVin the time-dependent wave-function \|Ψ(t)/an}b∇acket∇i}ht. We are interested in the physical situation where the time-d ependence of the state vector arises not from an explicitly time-dependent external ﬁeld but from the fact that initial ly the state vector is a superposition of the ground state (predominately) and a small amplitude for one of the excited states. We thus assume that ρs(xt,x′t) =ρ0(x,x′) + [ρ1(x,x′)exp(−iλt) + c.c.], (A4) ρ1(x,x′) =/summationdisplay ph[Xphϕp(x)ϕ∗ h(x′) +Yphϕh(x)ϕ∗ p(x′)]. (A5) In (A4) and below the superscript 0 identiﬁes quantities ass ociated with the KS ground-state theory. If ρs(t) was the physical one-particle density matrix, we could understand λas a physical excitation energy, but no such claim can be made for what we are doing. What follows now is close to a standard derivation of the RPA. We insert (A4) and (A5) into (A1) and, considering the amplitudes XandYas ﬁrst order quantities, we expand to ﬁrst order. For this pu rpose, we need the expansion, 12vs(xt) =v0(x) +/integraldisplay f(x,x′)n1(x′), (A6) f(x,x′) =δv0(x) δn0(x′), (A7) n1(x) =ρ1(x,x). (A8) In Eqs. (A6) and (A7), we have already made the adiabatic appr oximation by ignoring the time dependence of f. As a consequence, the quantity called fin this appendix can be identiﬁed with the quantity feffof the text. ¿From the zero order term, we regain the KS theory for the ground state. ¿From the ﬁrst order terms proportional to exp( −iλt), for example, we ﬁnd λρ1(x,x′) = [(τ+v0),ρ1](x,x′) +/integraldisplay dx′′[δv0 δn(x′′),ρ0](x,x′)n1(x′′). (A9) Taking, in turn, the phandhpmatrix elements of (A9), we ﬁnd the familiar equations [ǫh−ǫp+λ]Xph=fph′hp′Xp′h′+fpp′hh′Yp′h′, (A10) [ǫh−ǫp−λ]Yph=fhp′ph′Yp′h′+fhh′pp′Xp′h′. (A11) APPENDIX B: RPA NORMALIZATION CONDITION We deﬁne mode operators for the ﬁeld ˆψ(x) by expanding in terms of the KS modes, ˆψ(x) =/summationdisplay aaaϕa(x), (B1) a={h,p}. From the commutation relations for particle-hole pairs, [a† hap,a† p′ah′] =δhh′δpp′−δhh′a† p′ap−δpp′ah′a† h, (B2) we obtain an approximate sum rule by taking the expectation v alue in the state \|0/an}b∇acket∇i}ht, introducing a complete set of intermediate states \|i/an}b∇acket∇i}ht, and retaining only the ﬁrst term on the right hand side (on th e justiﬁed assumption that, for instance, /an}b∇acketle{t0\|a† pa′ p\|0/an}b∇acket∇i}htis, on the average small compared to unity). With the deﬁniti ons ξi ph=/an}b∇acketle{t0\|a† hap\|i/an}b∇acket∇i}ht, (B3) ηi ph=/an}b∇acketle{t0\|a† pah\|i/an}b∇acket∇i}ht, (B4) we have /summationdisplay i[ξi phξi∗ p′h′−ηi p′h′ηi∗ ph] =δpp′δhh′. (B5) We would like to identify the quantities ξandηwith the quantities XandY, where the latter satisfy Eqs. (4.30) and (4.31). Equation (B5) would then constitute the complet eness relation for the solutions of these equations, and as is well-known, a completeness relation and orthogonalit y of solutions with the corresponding metric implies the normalization condition Eq. (4.33). Toward this end, we con sider two diﬀerent evaluations of /an}b∇acketle{t0\|ˆψ†(x)ˆψ(x)\|i/an}b∇acket∇i}ht=ni0(x). On the one hand we have in an approximate evaluation based on t he physical picture, ni0(x) =/summationdisplay abϕ∗ a(x)ϕb(x)/an}b∇acketle{t0\|a† aab\|i/an}b∇acket∇i}ht ∼=/summationdisplay ph[ϕ∗ p(x)ϕh(x)/an}b∇acketle{t0\|a† pah\|i/an}b∇acket∇i}ht] +ϕ∗ h(x)ϕp(x)/an}b∇acketle{t0\|a† hap\|i/an}b∇acket∇i}ht. (B6) On the other hand, from the generalized KS mapping ni0→ns i0and Eq. (4.29, we have ni0(x) =/summationdisplay ph[ϕ∗ p(x)ϕh(x)Yi ph+ϕ∗ h(x)ϕp(x)Xi ph]. (B7) 13The identiﬁcations ξ=Xandη=Yare consistent with these equations. We actually have, /summationdisplay ph[ϕ∗ h(x)ϕp(x)(ξi ph−Xi ph) +ϕ∗ p(x)ϕh(x)(ηi ph−Yi ph)] = 0. (B8) If the points in the single-particle functions were distinc t, the result we seek would follow trivially from orthonorma lity of these functions. If we take the modes to be complex functio ns and assume that we can cut oﬀ the expansion (B1) at a ﬁnite number of terms, then by choosing a suﬃciently larg e set of distinct values of x, we can still obtain the desired consequence from Eq. (B8). [1] P. Hohenberg and W. Kohn, Phys. Rev. 136B , 864 (1964). [2] W. Kohn and L. J. Sham, Phys. Rev. 140A , 1133 (1965). [3] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules , (Oxford U. Press, New York, 1989). [4] R. M. Dreizler and E. K. U. Gross, Density Functional Theory, An Approach to the Quantum Many- Body Problem , (Springer-Verlag, Berlin, 1990). [5] A. Holas and M. H. March, in Topics in Current Chemistry , Vol. 180, ed. R. F. 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Phy s.104, 5134 (1996). [15] R. Bauernschmitt and R. Ahlrichs, Chem. Phys. Lett. 256, 454 (1996). [16] R. Bauernschmitt, M. H¨ aser, O. Treutler, and R. Ahlric hs, Chem. Phys. Lett. 264, 573 (1997). [17] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 302, 375 (1999). [18] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 314, 291 (1999). [19] K. B. Wiberg, R. E. Stratmann, and M. J. Frisch, Chem. Phy s. Lett. 297, 60 (1998). [20] R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, J. Chem . Phys. 109, 8218 (1998). [21] A. G¨ orling, Phys. Rev. A 59, 3359 (1999). [22] M. Levy and A. Nagy, Phys. Rev. Lett. 83, 4361 (1999) [23] X. Gonze and M. Scheﬄer, Phys. Rev. Lett. 82, 4416 (1999). [24] D. J. Tozer and N. C. Handy, J. Chem. Phys. 109, 10180 (1998). [25] A. K. Kerman and A. Klein, Phys. Rev. 132, 1326 (1963). [26] G. Do Dang et al, Nucl. Phys. A114 , 501 (1968). [27] A. Klein, Phys. Rev. C30, 1680 (1984). [28] A. Klein and R. M. Dreizler, Phys. Rev. A58, 1581 (1998). 14
arXiv:physics/0011040v1 [physics.gen-ph] 17 Nov 2000The Status and Programs of the New Relativity Theory Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, GA. 30314 November, 2000 Dedicated to the memory of my parents Abstract A review of the most recent results of the New Relativity Theo ry is presented. These include a straight- forward derivation of the Black Hole Entropy-Area relation and its logarithmic corrections; the derivation of the string uncertainty relations and generalizations ; ; th e relation between the four dimensional gravitational conformal anomaly and the ﬁne structure constant; the role o f Noncommutative Geometry, Negative Prob- abilities and Cantorian-Fractal spacetime in the Young’s t wo-slit experiment. We then generalize the recent construction of the Quenched-Minisuperspace bosonic p-brane propagator in Ddimensions ( AACS [18] ) to the full multidimensional case involving all p-branes : the construction of the Multidimensional-Partic le propagator in Cliﬀord spaces ( C-spaces ) associated with a nested family of p-loop histories living in a target D-dim background spacetime . We show how the eﬀective C-space geometry is related to extrinsic curvature of ordinary spacetime. The motion of rigid particles/brane s is studied to explain the natural emergence of classical spin. The relation among C-space geometry and W, Finsler Geometry and ( Braided ) Quantum Groups is discussed. Some ﬁnal remarks about the Riemannian long distance limit of C-space geometry are made. 1. Introduction : The New Relativity Theory Before we behin the present status and prsopects of the New Re lativity Theory [1] we deem it very important to review in 1the New Extended Scale Relativity and its postulates. This i s all one needs to con- struct in 2the generalization of the Quenched-Minisuperspace bosoni cp-brane propagator in Ddimensions [18] ; i.e the Multidimensional-Particle propagator in C-spaces ( Cliﬀord ) associated with a nested family of p-loop histories living in a D-dimensional target spacetime background. In 3we discuss the derivation of the Black Hole entropy and its logarithmic corrections directly from a p-loop Harmonic oscillator in C-space ( Cliﬀord space ). In 4we review the derivation of the string uncertainty relation s and corrections thereof due to all p-brane values ranging from p= 0 to p=∞. ; In5we show how the eﬀective C-space geometry is related to extrinsic curvature of ordinary spacetime. The motion of rigid partic le/branes is studied in rela- tion to the natural emergence of classical spinfromC-space.Some ﬁnal remarks are made about the relation that the C-space geometry bears to WGeometry, Finsler Geometry and ( Braided ) Quantum Groups. I n6 the calculation of 4+ φ3(φis the Golden Mean 0 .618...) as the average dimensions of the world is discussed and the relation between the four dimensional gravitationa l conformal anomaly, Fractal Spacetime and the ﬁne structure constant is explicitly shown. The role of Nonc ommutative Geometry, Negative Probabilities and Cantorian-Fractal spacetime in the Young’s two-slit ex periment of an indivisible quantum particle. is derived in 7. In8we brieﬂy discuss Quantum Groups, and the Braided Hopf Quant um Cliﬀord Algebra associated with the Master Action Functional for the quantu m dynamics of the Master Field in C-space. Finally, we present our conclusions. Some ﬁnal remarks abou t the Riemannian long distance limit of C-space geometry are made. 1.1 Historical Background Recently we have proposed that a New Relativity principle ma y be operating in Nature which could reveal important clues to ﬁnd the origins of Mtheory [1]. We were forced to introduce this New Relativity principle, where all dimensions and signatures of spacetim e are on the same footing, to ﬁnd a fully covariant 1formulation of the p-brane Quantum Mechanical Loop Wave equations. This New Rel ativity Principle, or the principle of Polydimensional Covariance as has been cal led by Pezzaglia, has also been crucial in the derivation of Papapetrou’s equations of motion of a spinnin g particle in curved spaces that was a long standing problem which lasted almost 50 years [2]. A Cliﬀord calculus was used where all the equations were written in terms of Cliﬀord-valued multivector quanti ties; i.e one had to abandon the use of vectors and tensors and replace them by Cliﬀord-algebra valued quan tities, matrices, for example . The New Extended Scale Relativity Theory has already allow u s to derive, from ﬁrst principles, the String Uncertainty Relations, corrections thereof, and th e precise connection between the Regge trajectory behaviour of the string spectrum and the areaquantization . The full blown inﬁnite-dimensional Quantum Spacetime Generalized Uncertainty relations that include d the contributions of allp-branes, not only due to strings, was given [3]. In [4] we were able to show that ther e arenosuch things as EPR paradoxes in this New Scale Relativity. The latter is truly a Machian Re lativity theory, were relationships are the only meaningful statements one can make. Originally it was f ormulated in an abstract Categorical space, C-spaces [1]. The New Relativity sprang out from Laurent Nott ale’s Scale Relativity [5]; Mohammed El Naschie Transﬁnite Cantorian-Fractal Spacetime [6]; Garn et Ord’s original work on fractal random walks [7] ; the classic Geoﬀrey Chew’s Bootstrap hypothesis : all p-branes are made of each other ( William Pezzaglia’s principle of polydimensional covariance ); and p-adic Physics and Non-Archimedean Geometry by Siddarth, Pitkannen, Khrennikov, Freund, Volovich, Vladimorov, Zel enov and many others [8]. The goals and partial list of sucesses of the New Relativity, formulated in Cantorian-Fractal Spacetime, are : 1- Hope to provide with a truly background independent formul ation of Quantum Gravity and M theory. 2- To furnish the physical foundations underlying String and Mtheory. So far nobody has been able to answer the question : ” What isString Theory ? ”. The New Relativity may provide with a plaus ible answer. In particular we were able to derive the string uncer tainty relations directly from the wave equations of the New Relativity [10] and its p-brane generalizations. 3- Prove why we live in 3+1 dimensions. In [9] we have shown that the average dimension of the world is of the order of 4 + φ3. 4- Solve the cosmological constant problem [9] , in a similar f ashion that Nottale did in his initial work on Scale Relativity. And discuss of the existence of a dimens ional phase transition from 4 + φ3toφ3, the so called Noncommutative quasi-crystal phase. 5- Provide with a physical meaning to negative topological di mensions and to the notion of negative- information entropy, or anti-entropy as has been called by M . Conrad. . 6- Write down the Unique Quantum Master Action functional of t he world, in an abstract C-space, outside spacetime, a generalized Twistor space, that gover ns the quantum dynamics for the creation of spacetime itself; gravity and all of the other fundamental f orces in Nature. 7- Propose a solution to how chiral symmetry-breaking occurs in Nature and much, much more [10]. With this preamble let us summarize brieﬂy what are the basic postulates of the New Relativity Theory next. This will allow us to explain in full rigour why the fund amental constants in Nature, like Planck’s constant, change with ” time ”; i.e. with the Renormalizatio n Group ﬂow of the microscopic ( scaling ) arrow of time: with energy or resolutions. We will explain why ther e is an eﬀective value of the Planck constant when one approaches scales comparable with the Planck lengt h : when quantum gravitational phenomena are important. 1.2 The Postulates of the New Scale Relativity Theory 1. Physics is an experimental science. Physics is about measu rements. In order to perform a measure- ment one needs a standard of measure to compare measurements with. In Einstein’s Relativity he introduced the speed of light as the velocity standard to compare veloci ties with. From the mathematical point of view one can say : let us have a ﬁeld at x. To deﬁne the real number x, mathematically, requires knowing its value with inﬁnite precision : simply adding digits at will. To do it physically is another story. It would require a computer an inﬁnite amount of memory just to store a ll this inﬁnite amount of information. One way to avoid the problem of using real numbers is to introduce p-Adic numbes in Physics [8]. 2For these reasons, Laurent Nottale [5] introduced his origi nal Scale Relativity by postulating that the Planck scale is the universal standard of measurements. It i s the minimum, impassible distance in Nature, invariant under scale-relativistic dilatations. Exactly along similar lines, Einstein’s motion Relativity was based on taking the speed of light as the maximum, and invariant , attainable speed in Nature. The Planck scale in four dimensions is given in terms of the 3 fundamenta l constants, speed of light c, Newton’s constant in four dimensions Gand Planck’s constant ¯ h: Λ =/radicalbigg ¯hG c3= 10−33cms. (1) Notice that if ¯ h= 0,G= 0,c=∞the Planck scale would have been automatically zero. Meanin g that one would not have ”quantum mechanics ” ; the gravitatio nal force will reduce to 0 and there would not be ” Lorentz invariance ”, only Galilean symmetry. The in variant minimum scale in all dimensions, Λ will be set to unity ; in units where ¯ h=c= 1. The Planck scale is explictly dimension dependent through its dependence on the Newton constants. So is the New ton gravitational constant. For example, in D, D−1, D−2, ...., the Planck scale ( that we set to unity) is given in terms of th e Newton constants : Λ =G(1/D−2) D =G(1/D−3) D−1=......= 1. (2a) Taking logarithms ( in any base if one wishes ) on eq-(2a), one has the relationship among the Newton constants, in DandD−1 diemnsions, which does notrequire any compactiﬁcations whatsoever, as it is assumed in conventioanl string and Kaluza-Klein supergrav ities, : D−3 D−2=ln G D−1 ln G D=log G D−1 log G D. (2b) Notice that when D= 2, the Newton constant in two-dimensions is set to G2= 1 so that 1∞= 1. For example in D= 2, G2= 1→ln G 2=ln1 = 0 which is consistent with all the denominators of eq-(2b) . Variable speed of light Cosmologies are becoming very popul ar today. This does not imply that Einstein was wrong at his time. This only means, as Dirac pointed out lo ng ago, that the fundamental constants can change slowly with ” time ” : the value of the constants slo wly ﬂow with the Renormalization group, from the Ultraviolet ( small scales ) to the Infrared ( large s cales ). The speed of light when Einstein formulated his theory is the same as today. Only during the ea rly Universe there were substantial changes. The constants ﬂow with the provision that the truly fundamen tal ” relativistic ” invariant in Nature, the Planck scale, remains ﬁxed. Eq-(1) by inspection, entails t hat ¯h, c, G could ﬂow with the RG ﬂow in such a fashion that they will leave Λ invariant. This explains how , at very large energies ( Planckian ) , at very small scales ( Planck scales), we can begin to see the correct ions to the fundamental constants. We have now an understanding as to why one has an eﬀective Planck cons tant that can vary with energy once we approach scales comparable to the Planck scale. Exactly the same thing happens when we approach the speed of light : the masses begin to grow compared to their res t mass values. 2. The Principle of Poly-dimensional covariance and the Cliﬀ ord-algebra-multivector calculus. The New Relativity is a true Machian one. Relationships are t he only meaningful statements one can make. To view a single p-brane as an isolated entity is a meaningless concept. p-branes are solely deﬁned in terms of others. For this reason, we included the Chew boos trap hypothesis as a crucial ingredient, and wrote : Allp-branes are made of each-other. This is, in essence, the orig ins of the dualities in Mtheory. Pezzaglia [2] using Chew’s bootstrap hypothesis coined the term ” poly-dimensional-covariance ”. Since now we have all p-branes, of all dimensionalities from p= 0,1,2....∞, the poly-dimensional covariance is the statement that all p-branes ( dimensions ) rotate into each-other. Exactly what happens with ordinary Lorentz transformations : the axis are entangled and space a nd time are mixed. For the role of objects of negative topological dimensions see [10]. For example, i n ordinary string theory, an object of dimension p=−1 spans a p+ 1 = −1 + 1 = 0-dimensional ”worldline” : it is an instanton. Space and time have diﬀerent units. Einstein was able to conn ect them by the introduction of an invariant velocity parameter : the speed of light. If one set sc= 1 in Einstein’s theory, space and time have the same ” units ” : they are exchangeable. Since the Planck sc ale has units of length, upon setting Λ = 1, 3it means that all dimensions are are exchangeable also. Dime nsions then can be ” rotated ” into each-other. p-branes can be transformed into each-other ( duality princi ple ). Einstein’s Relativity required the use of a Lorentz four-vector to embrace space with time; energy with momentum, etc.... In the New Relativity one needs Cliﬀord-algebra-valued multivectors. The latter mu lti-vector is a mathematical object that encodes all objects of diﬀerent dimensionalities. It encodes all p-brane histories, embedded in a target spacetime background, into one single scoop. For the analog of ordinar y Lorentz transformations that rotate space into time, for example, one has polydimensional transformation s, that rotate p-branes among each-other. For this reason we will write our New Relativistic wave equat ions in a Cliﬀord-space, whose derivative elements will be quadratic and have, in addition to the ordinary quadratic derivatives with respect to ordinary point coordinates, xµ, quadratic derivatives w.r.t the holographic area-coordi nates, holographic volume-coordinates, ..... We will explain this in detail in section 2what these holographic area, volume, hypervolume....coordinates really are. 3. Einstein’s General Relativity required a Riemannian Geom etry; the New Relativity requires a Cantorian-Fractal Spacetime model developed by Mohammed E l Naschie [6]. The latter is an example of Von Neuman’s PointlessNoncommutative Geometry. The world is multi-fractal. For a detailed analyis of Noncommutative geometry, etc...see [10] and references th erein. Essentially one has that because the Planck scale is the mini mum distance in Nature; there are no such things as ” points ” in Nature. Only in Mathematics. A ” po int ” is smeared into a fuzzy ball of all possible topological dimensision. On average, the ” points ” are four-dimensional spheres, for this reason we live eﬀectively in four-dimensions : we perceive an average dimension taken over all the inﬁnite-dimensional spacetime [10]. This occurs exactly in the same fashion that we only measure the average velocity of an ensemble of molecules in a room, when we measure the temperat ure of the room, pressure, etc...Cantorian- Fractal Geometry is a ” pointless ” Noncommutative Geometry . As we zoom into what looks like a ” point” from a distance, we realize that it is a four-dimensional sph ere. As we zoom in deeper into what we thought looked like a point inside that small sphere we realize that i t is a smaller four-dimensional sphere; and so forth ad inﬁnitum. We cannot reach the Planck scale. Like a ma ssive object cannot reach the speed of light. Nature is multifractal. 4Non-Archimedean geometry and p-Adic Physics [8] Since the Planck scale is the minimum attainable scale in Nat ure, Nottale provided with the scaling analogs of Lorentz transformations : The composition of two scalings ( contractions ) cannot yield scales smaller than the Planck scale. In the same way that the additi on of two velocities cannot exceed the speed of light. This forces one to abandon the Archimedean Ge ometry for a Non-Archimedean one and the replacement of real numbers for p-Adic ones [8]. Using a p-Adic norm, which allows to deﬁne p-adic numbers based on another extension of the rational numbers , one can s how that the net resulting p-adic norm, of the composition/sum of two p-adic numbers, is less than the p-adic norm of the larger norm of the initial twop-adic numbers. Roughly speaking, the p-adic norm obeys an ultra-metricity condition. p-Adics are the natural numbers to use in the New Relativity. The latter Rela tivity is consitent with a Non-Archimedean Geometry. What we ﬁnd most important in using primes, the atoms of numbe rs, in the New Relativity is that all prime numbers must appear on equal footing. In the same way th at all dimensions did. Since dimensions change this automatically entails that one must include als o all topologies on equal footing : a “ Topological Relativity” as has been called by Finkelstein and others. p-Adic numbers have been used to label topologies. Not surprisingly, this ﬁts very well within this new framewo rk. It is also well known to the experts that there is a very deep connection between Quantum groups and p-Adic numbers when the deformation parameter q= 1/p. The Real number limit p=∞is equivalent to the classical group limit q= 0. Quantum Symmetric spaces “ interpolate “ between the Real numbers and the p-Adic ones. To sum up, the world is not scale invariant. Dimensions are in the eye of the beholder [1 ,2]. They are resolution-dependent [5,6]. As we probe into smaller re gions, larger energies, with the ”microscope” of the RG ﬂow, more dimensions are accesible to us. Things loo k diﬀerent to two observers living in two diﬀerent scale-frames of references. The most famous examp le : it is meaningless to compare the vacuum energies in two completely diﬀerent scale-frames of refere nces : the Planck scale and the Hubble scale. This is the fundamental reason why the ” cosmological” constant d iﬀers by 60 orders of magnitude : the so-called cosmological constant ” problem ”. This inconsistency, was elegantly solved by Nottale [5] and recently by 4us [9] within the context of Renormalization Group techniqu es and self-organized non-equlibrium critical phenomena in Cosmology, intially emphasized by Smolin and K auﬀmann [19]. Based on this cursory introduction to the basic principles o f the New Relativity, we can understand now why the Planck constant ¯ h, for example, is not a true constant, but it varies with energ y, resolution. Once, of course, we approach the Planck scale. At ordinary energie s, ¯his a constant. This is why these eﬀects have not been detected in ordinary experiments. The energy i s extremely low in comparison with the Planck energy of 1019GeV. . 2. Quenched-Minisuperspace Bosonic p-brane Propagator and its C-space generalization Recently AACS [18] we were able to write down the Quenched-Minisuperspace Bosonic p-brane prop- agator by borrowing the Minisuperspace approximation from Cosmology, and the “ quenching “ procedure fromQCD. This new approximation provided an exact description of both the collective mode deformation of the brane and the center of mass dynamics in the target spac etime. Earlier work based on loopspaces allowed [30] to write down the Schwinger-Fock proper time fo rmulation of wiggled p-branes. Imagine one has a family of p-branes moving in a target spacetime background, where the v alues of prange from p= 0, a point history; p= 1 a closed string history; p= 2 a closed membrane history. A membrane of topology of a sphere, for example : a 2-loop; and so forth. Until one saturates the spacetime with the spacetime ﬁlling p-brane : p+ 1 = D. The family of p-brane degrees of freedom are encoded in term of hyper-matrix coordinates [1]. . Generalized ” hyper-matrix coordinates ” transformations in the New Relativity reshuﬄe, for example, a loop history into a membrane history; a membrane history in to a into a 5-brane history; a 5-brane history into a 9-brane history and so forth; in particular it can tran sform a p-brane history into suitable combinations of other p-brane histories as building blocks. This is the bootstrap i dea taken from the point particle case to to the p-branes case : each brane is made out of all the others. ” Loren tz” transformations in C-spaces involve hypermatrix changes of ” coordinates ” [1] . The naiv e Lorentz transformations do not apply in the world of Planck scale physics. Only at large scales the Ri emannian continuum is recaptured . For a discussion of the more fundamental Finsler Geometries impl ementing the minimum scale ( maximal proper acceleration ) in String Theory see [13]. There was a one-to-one correspondence between the nested hi erarchy of point, loop, 2-loop,3-loop,...... p- loop histories encoded in terms of hypermatrices [1] and wav e equations written in terms of Cliﬀord-algebra valued multivector quantities.[2] This permitted us to rec ast the QM wave equations associated with the hierarchy of nested p-loop histories, embedded in a target spacetime of Ddimensions , where the values ofprange from : p= 0,1,2,3......D−1, as a single QMlinefunctional wave equation whose lines live in a Noncommutative Cliﬀord manifold of 2Ddimensions. p=D−1 is the the maximum value of pthat saturates the embedding spacetime dimension. An action linefunctional, associated with the interacting QFT of lines in Noncommutative Cliﬀord manifolds C-spaces, was launched forward in [1]. The QFT program of such interacting ﬁeld theory of C-lines in Noncommutative spaces, a generalized Twistor theory, is currently under investigation [18]. One will have cubic interactions associated with the product and coprodu ct of a Braided-Hopf-Quantum-Cliﬀord algebra [15]. The product represents the annihilation of two C-lines into a third one. The coproduct represents the creation of two lines from one line . The quartic interaction s correspond to the braiding of two-lines into another two-lines : scattering. One has here more complicat ed statistics than the ordinary bose/fermions one : it is a braided one !. The two-point vertex corresponds t o a pairing of the algebra representing the composition of two lines into a 0-line. The kinetic terms are the extensions of Witten-Zwiebach open/closed string ﬁeld theory, based on the Batalin-Vilkovisky Quantu m Master action [16,17] . The closed-string ﬁeld theory action required the used of Operads and Gerstenhaber algebras [17]. Such QFT in Noncommutative spaces is a very complex one due to the Ultra-Violet/Infrare d entanglement. Due to string duality, there is a maximum scale dual to the minimum Planck scale. In [9] we prov ided with integral expressions to determine the maximum scale that is dual to the Planck scale. The line functional wave equation in the Cliﬀord manifold, C-space, for the simplest “ linear “ case is : 5/integraldisplay dΣ (δ2 δX(Σ)δX(Σ)+E2)Ψ[X(Σ)] = 0 . (3) where Σ is an invariant evolution parameter of lDdimensions generalizing the notion of the invariant proper time in Special Relativity linked to a massive point particl e line ( path ) history : (dΣ)2= (dΩp+1)2+ Λ2p(dxµdxµ) + Λ2(p−1)(dσµνdσµν) + Λ2(p−2)(dσµνρdσµνρ) +....... (4) Λ is the Planck scale in Ddimensions : Λ = G1 D−2 Dwhere GDis Newton’s constant in Ddimensions. X(Σ) is a Cliﬀord-algebra valued ” line ” living in the Cliﬀord man ifold ( C-space) : X= Ω p+1+ Λpxµγµ+ Λp−1σµνγµγν+ Λp−2σµνργµγνγρ+......... (5a) The multivector Xencodes in one single stroke the point history represented b y the ordinary xµcoor- dinates and the holographic projections of the nested famil y of1-loop,2-loop,3-loop... p-loop histories onto the embedding coordinate spacetime planes given respectiv ely by : σµν, σµνρ......σ µ1µ2...µp+1 (5b) The scalar Ω p+1is the invariant proper p+ 1 = D-volume associated with the motion of the ( maximal dimension ) p-loop across the D=p+ 1-dim target spacetime. It naturally couples to the unit ma trix of the Cliﬀord algebra. There was a coincidence condition [1] that required to equat e the values of the center of mass coordinates xµ, for all the p-loops, with the values of the xµcoordinates of the point particle path history. This was due to the fact that upon setting Λ = 0 all the p-loop histories collapse to a point history. The latter hist ory is the baseline where one constructs the whole hierarchy. Th is also required a proportionality relationship : τ∼A Λ∼V Λ2∼.......∼Ωp+1 Λp. (6) τ, A, V.... Ωp+1represent the invariant proper time, proper area, proper vo lume,... proper p+ 1-dim volume swept by the point, loop, 2-loop, 3-loop,..... p-loop histories across their motion through the embedding spacetime, respectively. E=Tis a quantity of dimension ( mass)p+1, the maximal p-brane tension ( p=D−1) . AC-line in C-space is nothing but a Cliﬀord algebraic extension of Penro se’s twistors. From a distance, the line looks like a point history : a one-dimensional world line. Upon closer inspection, upon zooming in, we realize that it corresponds to the center-of-mass motion of a closed string history, a 1-loop. And that the line turns into a two-dimensional surface : the lateral a rea-swept by the closed-string. Upon a further inspection, as we zoom in deeper, we realize that the closed- string history is really a closed-membrane history; and so forth and so forth. Dimensions are resolution and ener gy dependent. All these p-brane histories that have a common center-of-mass coordinate , are encoded in ter ms of the Cliﬀord-algebra-valued C-lines , a generalized twistor. The holographic, or shadow-projecti ons, of the areas, volumes, hypervolumes,....onto the respective coordinate planes are nothing but the hologr aphic coordinates of the C-lines. The wave functional Ψ is in general a Cliﬀord-valued, hyperc omplex number. In particular it could be a complex, quaternionic or octonionic valued quantity. At t he moment we shall not dwell on the very subtle complications and battles associated with the quaternioni c/octonionic extensions of Quantum Mechanics [14] based on Division algebras and simply take the wave function to be a complex number. The line functional wave equation for lines living in the Cliﬀord manifold ( C-spaces) are diﬃcult to solve in general. To obtain the Bekenstein-Hawking Black-Hole Entropy-Area relation s, and corrections thereof, one needs to simplify them. The most simple expression ( all modes are frozen except the z ero modes ) is to write the simpliﬁed wave equation for the family of free ( non-interacting ) p-loops in D-dimensions , in units ¯ h=c= 1 : { −1 2Λp−1[∂2 ∂xµ∂xµ+ Λ2∂2 ∂σµν∂σµν+ Λ4 ∂2 ∂σµνρ∂σµνρ+......]}Ψ =TΨ[xµ, σµν, σµνρ, .....]. (7) 6where Tis the tension associated with the maximal spacetime ﬁlling p-brane : p+ 1 = D. It has units of energy per unit p-Volume ; i.e ( mass)p+1. Following the result for the ordinary point-particle propa gator K(xb, xa;τb−τa) obeying : −1 2m∂2 ∂x2aK(xb, xa;τb−τa) =i∂ ∂τaK(xb, xa;τb−τa). (8) whose solution is : K(xb, xa;τb−τa) = (m 2π(τb−τa))1 2exp[1/2im(xb−xa)2 τb−τa]. (9) One can notice the role of the quantities m( mass ) and τ( proper time ) in the expression for the kernel. One can generalize this result to C-spaces by ﬁnding the analog of a C-space invariant; i.e a polydimensional invariant parameter of dimensions of length λcosntructed out of the two C-space invariants : The C-space analog of proper time interval Σ given in eq-(4) and the analo g of “ mass “ : the quantity mp+1obeying the on shell condition for the polydimensional Cliﬀord-algebr a valued momentum : P2= (1 Λ)2p(pµpµ) + (1 Λ)2p−2(pµνpµν) + (1 Λ)2p−4(pµνρpµνρ)........... + (µo)2= (mp+1)2. (10) where the canonical conjugate variable to the worldvolume Ω p+1of the maximal spacetime ﬁlling p-brane ( p+1 = D) is nothing but the cosmological constant µ oof dimensions ( mass)p+1. Because the cosmological “ constant “ µois itself a component of the Cliﬀord-algebra valued polymom entum Pthis means that the cosmological “ constant “ is not aC-space invariant by deﬁnition ! Its value can rotate under polydimensional rotations ! [3, 9 ] The natural C-space invariant quantity Lof dimensions of ( length )p+1allows to deﬁne a natural length scaleλ, which is just nothing but the analog of the Fock−Schwinger −Feynman formulation based on the auxiliary parameter λ[18, 30] : L2≡Σp+1 mp+1⇒λ=L(1/p+1)= (Σp+1 mp+1)(1/2p+2). (11) that involves the analog of mass : mp+1and the analog of proper time Σ p+1for the Multidimensional-Particle. These quantities are genuine invariants under automorphis ms of the Cliﬀord algebra ; i.e polydimensional rotations or automorphisms of the Cliﬀord algebra basis “ ve ctors “ γµ[2] in such a way that the Cliﬀord valued multivector Xin eq-(5a) rotates as follows : γµ→Γ−1γµΓ. X→MX (12) where Mis a 2D×2Dmatrix. One must arranged the 2Dcomponents of the Cliﬀord algebra valued multivector X: Ωp+1;xµ, σµν, σµνρ...into a column matrix of 2Dentries. The polydimensional rotations willreshuffle the components/dimensions in such a way that a membrane hist ory rotates into a ﬁve-brane history; a nine-brane rotates into an eleven-brane history and so forth, or even mixtures of all brane histories . This is the p-brane Bootstrapping generalization of Chew’s particle bo otstrap ideas extended to dimensions : Pezzaglia’s polydimensional covariance principle. Hence, these polydimesnional rotations are the extensions of ordinary Lorentz transformations leaving theC-space interval invariant : (d˜Σ)2= (d˜Ωp+1)2+ Λ2p(d˜xµd˜xµ) + Λ2p−2(d˜σµνd˜σµν) +.................. = (dΣ)2= (dΩp+1)2+ Λ2p(dxµdxµ) + Λ2p−2(dσµνdσµν) +................... (13) The double covering of ordinary Lorentz transformations, i n spinorial notation, can be seen as SL(2, C) rotations/Mobius transformations using 2 ×2 complex valued matrix whose entries are respectively ( a, b, c, d ) : z→az+b cz+d. ad−bc= 1. 7This is attained by assigning to each four vector xµ= (x0, x1, x2, x3) the 2 ×2 matrix Xusing the three Pauli spin matrices σiand unit matrix I: xµ= (x0, x1, x2, x3)↔X=x0I+x1σ1+x2σ2+x3σ3⇒A−1XA=˜X= ˜x0I+ ˜x1σ1+ ˜x2σ2+ ˜x3σ3.(14) where Ais anSL(2, C) matrix. In this fashion one obtains the Lorentz transforma tions of the four vector ˜xµpreserving the Lorentz norm x2 0−x2 1−x2 2−x2 3of the four-vector xµ. ; i.e it is just given by taking the trace Tr(X2) =Tr(˜X2) due to its cyclic property. Based on the one-to-one correspondence bewteen hypermatrices and Cliﬀord algebra valued multivec- torsX( 5a ) one can extend the Lorentz transformations, using Paul i spin matrices , as polydimensional rotations or automorphisms of the Cliﬀord algebra , that can be recast as automorphisms of hypermatrices , similarily to what occurs with ordinary Lorentz transforma tions expressed in terms of SL(2, C) transforma- tions of the 2 ×2 matrix Xgiven by eq-(14) . The analog of Null lines ( photons ) in C-space naturally correspond to tensionless p -branes : mp+1= 0. Instead of null lines one has null tubes. For example, in D= 4 the degree of the Cliﬀord algebra is 24= 16 , meaning that one has 16 independent 4×4 matrices : I, γµ, γµ∧γν, ...spanning the Cliﬀord algebra basis. These 16 4 ×4 matrices can be arranged as 4cubic hypermatrices 4×4×4 :Y0, Y1, Y2, Y3where a similar type of transformation as eqs-(12, 14 ) via automorphisms of the Cliﬀord algebra will map these 4 cubic hypermatrices into a new set ˜Y0,˜Y1,˜Y2,˜Y3. The generalization of Mobius transformations for Rn=R2Dcan be obtained via the Vahlen matrices and the Vahlen group [25] . Whatever prescription one wishes to t ake, the idea is essentially to rotate the Cliﬀord algebra multivector X, with 2Dindependent components, by right multiplication with a 2D×2Dmatrix :˜X=MXleaving invariant the C-space interval or norm of the XCliﬀord algebra valued multivector. The matrix Mshould obey the analog of the orthogonal/unitary matrix MT=M−1andM+=M−1 respectively. Having discussed polydimensional rotations, autmorphism s of the Cliﬀord algebra that leave the C-space inteval invariant, the invariant parameter λwill allow us to extend the point particle kernel to the C-space case by counting the number of degrees of fredom associated w ith the collective excitations : there are D degrees of freedom associated with the center of mass motion . There are D(D−1)/2 degrees of freedom associated with the holographic area excitations σµν. There are D(D−1)(D−2)/6 degrees of freedom associated with the holographic area excitations σµνρ. and so forth. The kernel in C-space can then be recast in terms of the analog of the Fock-Schwinger-Feynman paramter λ[18, 30] and factorizes as follows : K(Xa, Xb; Σb−Σa) =Kxµ(xµ a, xν b; Σb−Σa)Kσµν(σµν a, σµν b; Σb−Σa)Kσµνρ(σµνρ a, σµνρ b; Σb−Σa)......(15) where Kxµ= (1 λ2)D 2exp[1/2i(xµ b−xµ a)2 λ2]. (16) Kσµν= (1 λ4)D(D−1) 2.2exp[1/2i(σµν b−σµν a)2 λ4]. (17) Kσµνρ= (1 λ6)D(D−1)(D−2) 2.2.3exp[1/2i(σµνρ b−σµνρ a)2 λ6]. (18) ............... TheC-space Kernel/Propagator K(Xa, Xb; Σb−Σa) is what allows to evaluate the wavefunction at two separate locations Xa, Xbin a time span of Σ b−Σa: Ψ(xµ b, σµν b, σµνρ b.....) =/integraldisplay dxµ adσµν adσµνρ a.....dΩp+1,aK(Xa, Xb; Σb−Σa) Ψ(xµ a, σµν a, σµνρ a.....).(19) Notice that the temporal evolution is recast in terms of the a nalog of the Fock-Schwinger-Feynman paramter λ[18, 30] which depends on Σ p+1. For further details we refer to [18,26, 30 ]. 83 . Explicit Derivation of the Black Hole Entropy-Area Relat ions from the New Relativity Theory Having gone through a brief tour of the the postulates of the N ew Relativity, we shall present in this section the simple steps to derive the Black-Hole Entropy-A rea linear relation and its logarithmic ( and higher order ) corrections. leaving all the details to refer ences [20,21]. We only need to discuss in detail the generalized wave equations ( for the p-loop oscillator ) in a Cliﬀord-manifold that encode the dyn amics such family of p-brane or p-loop harmonic oscillators . The most simple expression ( all modes are frozen except the z ero modes ) is to write the simpliﬁed wave equation for the p-loop harmonic oscillator , in units ¯ h=c= 1 : { −1 2Λp−1[∂2 ∂xµ∂xµ+ Λ2∂2 ∂σµν∂σµν+ Λ4 ∂2 ∂σµνρ∂σµνρ+......]+ mp+1 2L2[Λ2pxµ2+ Λ2p−2σ2 µν+......Ω2 p+1]}Ψ =TΨ[xµ, σµν, σµνρ, .....]. (20) The solutions of the latter p-loop oscillator wave equations where given in [21,22 ] in te rms of the dimensionless ( rescaled ) variables : ˜xµ=Λpxµ L.˜σµν=Λp−1σµν L.˜Ωp+1=Ωp+1 L. (21) where the analog of oscillator amplitude is Lgiven by : L2=Λp+1 mp+1⇒Λp+1< L <1 mp+1. (22) where mp+1is the analog of the Compton momentum. Ψ∼exp[−(˜x2 µ+ ˜σ2 µν+....)]Hni(˜xµ)Hnjk(˜σµν)Hnjkl(˜σµνρ)..... (23) we are expressing as usual the ground state as the Gaussian an d the excited ones by the product of the Hermite polynomials. The excitations of the p-loop oscillator are collective ones given by the set of center of mass excitations; holographic area, holographic volume , .... excitations : N={ni;nij;nijk, ......}. (24) The Tension is quantized as follows [ 21, 22 ] : TN= (N+1 22D)mp+1 (12) the degree of the Cliﬀord algebra in Ddimesnions ( “ number of bits “ ) is 2D. The first collective excited state corresponds to setting all the quantum numbers equal t o 1 in full compliance with the principle of dimensional democracy ( p-brane democracy ) or poly-dimensional covariance : alldimensions must appear on the same footing : {ni= 1;nij= 1;nijk= 1, ......} ⇒N1= 2D(25) The degeneracy of the the Nth=kstate is given as a function of D, N : dg(D, N k) =Γ(2D+Nk) Γ(Nk+ 1)Γ(2D). N k=k2D. k= 1,2,3.... (26) The degeneracy of the ﬁrst collective excited state is N1= 2Dand is naturally given by simply setting N= 2Din the above equation : dg(N= 2D) =Γ(2N) Γ(N+ 1)Γ( N)(27) 9The Entropy is deﬁned as the natural logarithm of the degener acy . Hence taking the logarithm and using Stirling’s asymptotic expansion of the logarithms of the Gamma functions yields for the ﬁrst collective excited state the following Entropy : Entropy =S= 2Nln(2)−1 2ln(N)−1 2ln(4π)−O(1/N). (28) Before invoking Shannon’s information entropy by setting t he number of holographic p-loop bits to coincide precisely with the ratio of the d−2-dimensional area associated with a black hole horizon in d- dimensions ( d/ne}ationslash=D) of radius Rand the area of radius Λ in the following manner : N∼A/G, one needs to justify this assumption. Most importantly is to answer th e following questions : Where does Einstein’s gravity come from ? How is it obtained a s the long distance eﬀective theory from the ‘” gas “ of highly-excited p-loop oscillators quanta associated with New Relativty in C-spaces ( Cliﬀord manifolds ) ? •At the moment we cannot answer such diﬃcult questions ; howev er we recall that the low energy limit of string theory ( the eﬀective string action after int egrating out the massive string modes ) reproduces Einstein-Hilbert action with the ordinary scalar curvatur e term plus a series of higher powers of the curvature. Einstein’s gravity is recovered as the low energy limit from strings propagating in curved backgrounds. •Fujikawa [23] has given strong reasons why Shannon’s Inform ation Entropy is related to the Quantum Statistical ( Thermodynamical) Entropy. In particular in u nderstanding the meaning of Temperature. •Li and Yoneya [24] , among many others, have derived the Beken tein-Hawking entropy-area linear relation by taking the logarithm of the degeneracy of the hig hly excited massive ( super ) string states in d-dimensions. •The New Relativity principle advocates that dimensions are in the eye of the beholder [1,2] . In one reference frame an observer sees a gas of p-loop oscillators in D-dimensions in the ﬁrst collective excited state of N= 2D. In another frame of reference another observer sees only st rings in ddimensions ( d/ne}ationslash=D) in a very highly excited state n. If we identify Shannon’s information entropy as the number o f bits N= 2Dof the p-loop oscillator which is just the degree of the Cliﬀord algebra in Ddimensions , Shannon’s information entropy can them be re-expressed in terms of the number of bits as follows : N=SShannon =log2(2N) =N= 2D⇒Number of states =N= 2N= 22D. (29) Notice the double exponents deﬁning the number of states. The Black-Hole Horizon litera rily is an information horizon as well ! Now we are ready to make our only assumption. We will identify the number of p-loop bits N= 2D inDdimensions to coincide precisely with the number of areabits contained by a Black-Hole horizon ( in ddimensions ) of a given Area in units of the Planck scale. Name ly it is the ratio of the areas of radius R and Λ that gives the vlaue of the number of geometrical bits : N= 2D=Ad−2(R) Ad−2(Λ)∼Ad−2 Λd−2. G d= Λd−2(30) where we just wrote down the value of the Newton constant Gdind-dimensions as Λd−2. The number of transverse dimensions to the radial rand temporal coordinates tof a spherically-symmetric black hole is d−2. So the Horizon area refers to a ( d−2)-dimensional one. This is allwe need to obtain precisely the Black-Hole entropy-area rel ation in the literature including the logarithmic corrections [22] , up to numerical coeﬃcien ts , directly from eq-(28) : S∼[2ln(2)](A/G)−1 2ln(A/G)−1 2ln(4π)−O(1/(A/G)) +....... (31) To conclude : We have obtained in a very straightforward fash ion not only the Bekenstein-Hawking entropy-area linear relation for a Black Hole in any dimensi ons but also the logarithmic corrections plus 10higher order corrections [ 21, 22 ] . All of these corrections appear with a minus sign and the entropy-area relation satisﬁes the second law of black hole thermodynami cs : If A 3> A1+A2⇒S(A3)−S(A1)−S(A2)>0. (32) If two black holes of areas A1, A2merge to give another black hole of area A3> A1+A2then the resulting entropy cannot decrease. For an upper bound on the values of A3see [ 21 ]. In Planck units we obtained : N1N2> N3> N1+N2⇒A1A2 G2>A3 G>A1+A2 G. (33) For further details of all the technicalities behind this co nstruction we refer to [21, 22 ] . Perhaps the most salient feature is the intricate realtion b etween d, D, R, Λ in all these relations. Based on the deﬁnition of number of geometrical bits as the ratio of two areas, one for radius Rand the other for radius Λ one can immediately infer : N=Ad−2(R) Ad−2(Λ)= 2D= (R Λ)d−2. (34) Taking the logarithms on both sides we get the desired relati onship among D, d, R, Λ : D(ln2) = ( d−2)ln(R Λ)⇒R= Λ→d=∞!. (35) This is a remarkable conclusion. When R= Λ , for D≥2 ( dimensions where Cliﬀord algebras are deﬁned) one recovers autmatically Nottale’s scale relativ istic results that when one reaches the impassible Planck scale the fractal dimension of spacetime blow up. We c an see also why Ncannot be equal to unity. IfN1=N2= 1 this violates the second law of black hole thermodynamics since 1 + 1 = 2 >1.1. One must have at least N = 2 in order that 2 + 2 = 4 = 2 .2 and the relation (33) is not violated. 4. The Generalized Spacetime Uncertainty Relations 4.1. The String Uncertainty Relations follow from the New Re lativity We have studied the p-loop harmonic oscillator using the C-space wave equations. The free case admits plane wave type solutions : Ψ =ei(kµxµ+kµνσµν+kµνρσµνρ+....). (36) Inserting this plane wave type of solution into the wave equa tion fro the free p-loop case yields the generalized dispersion relation : ¯h2(k2+1 2Λ2(kµν)(kµν) +1 3!Λ4(kµνρ)(kµνρ) +.....) =Λ2pm2 p+1 ¯h2p. (37) This is just the extension of : p2= ¯h2k2. p2=m2. (38) On dimensional analysis and using the principle of polydime nsional covariance one can infer that : k2≡kµkµ.(kµν)(kµν) =βk4.(kµνρ)(kµνρ) =βk6...... (39) where βis a proportionality coeﬃcient; i.e all dimensions are weig hted by the same value in compliance with polydimensional covariance. Using this relation and inserting into the square root of the dispersion relation one obtaines an effective value of the Planck constant : ¯heff= ¯h(1 +1 4βΛ2k2+O(Λ4k4) +....). (40) 11Recurring to the well known relation ( due to the Schwartz ine quality and that \|z\| ≥ \|Im z\|) : ∆x∆p≥1 2\|<[ˆx,ˆp >\|=¯heff 2. (41) After a little algebra and using the relations : ¯hk=p. < p2>≥(∆p)2. < p4>≥(∆p)4.... (42) one arrives at : ∆x∆p≥1 2¯h+1 2βΛ2 4¯h(∆p)2+O[(∆p)4] +... (43) Finally one derives the string uncertainty relation by keep ing the ﬁrst two leading terms : ∆x≥1 2¯h ∆p+1 2βΛ2 4¯h(∆p). (44) Eq-(44) has for minimum value of ∆ xof the order of the Planck length Λ corroborating once more th at distances below the Planck scale have no physical meaning. 4.2 The Full Blown Generalized Uncertainty Relations The stringy uncertainty relations are notthe most fundamental ones. The contribution to an infinite number of p-branes where p= 0,1,2,3, ....∞to the eﬀective Planck constant is : ¯h2 eff= ¯h2∞/summationdisplay r=1(kΛ)2(r−1) r!=ez2−1 z2. z≡kΛ. k=\|\|/vectork\|\|=\|\|kµkµ\|\|1/2(45) Following the same procedure as above one recovers the fullblown uncertainty relations for Quantum Spacetime due to allextended objects from p= 0 all the way to p=∞: ∆x≥√ 2 Λe(∆z)2/4 (∆z)2/radicalbigg sinh[(∆z)2 2].∆z= (∆k)Λ. (46) which yields a minimum distance of ∆ x∼1.2426 Λ; i.e compatible with Nottale’s Scale relativity post ulate that the Planck scale ( resolution ) is unattainable in Natur e. It takes an infinite amount of energy to resolve such scales ; i.e reaching inﬁnite dimensions in the process. The eﬀective Planck constant is : ¯heff= ¯h√ 2ez2/4 z/radicalbigg sinh[(z)2 2]. z≡kΛ. (47) Which means that the effective momentum-squared of point particle moving in the bakground geometry ofC-space is ( we do not include numerical factors for convenien ce ) : p2 eff(p) =k2(¯heff[k])2∼¯h2k2+ ¯h2βk4Λ2+..=∼(¯h2k2) + (¯h2k2)(βΛ2k2) +... (48) Therefore the eﬀective effective squared-momentum due to the total contribution of all p-branes is : p2 eff=p2e(pΛ/¯h)2/2 [(pΛ/¯h)2/2]sinh[(pΛ/¯h)2 2]. p= ¯hk. (49) We would like to point out a common and widely spread misconception about the modiﬁed uncertainty relations. Oned can perform a noncanonical transformation from ( x, p) to a new pair of ( noncanonical ) variables x′, p′such as : Given [x, p] =i¯h x→x′. p→p′.[x′, p′] =i¯heff(p′) = [x, p′] =i¯h∂p′ ∂p. (50) 12Hence to ﬁrst order corrections the relationship between pandp′after setting the βparameter to unity is : p=p(p′) =/integraldisplaydp′ heff(p′)∼/integraldisplayp′ 0dp′ [1 + (p′Λ)2/4¯h2+...]⇒p=¯h Λtan−1[p′Λ ¯h]. (51) Inverting the last relation yields : p′=p′(p) =¯h Λtan[pΛ ¯h]. (52) We deem very important to emphasize that the new momentum p′of the ( noncanonical ) pair of new variables x′, p′( the commmutator of x, pisnotpreserved ) must not be confused with the effective momentum peff(p) given by (48) . Eqs-(48-51 ) indicate clearly that p′/ne}ationslash=peff! Upon setting the Planck momentum to be pPlanck = ¯h/Λ in eq-(52) one can see that in a sense one may compactify the momentum simply by having the momentum values bounded as : 0 <(p/pPlanck )<2π. And this justiﬁes naturally the introduction of an ultraviolet cuttoﬀ . Based on the Ultravi olet/Infrared entanglement in Noncommutative Geometry one can also postulate an infared cuttoﬀ, in additi on to an ultaviolet cuttoﬀ, consistent with Nottale’s impassible upper length scale [5]. A sort of strin gyT-duality analog. This idea allow us to postulate a phase transition for the universe, from the meta stable vacuum whose average dimension is close to 4 + φ3, to the Noncommutative quasi-crystal phase of dualdimension φ3= [1/(4 +φ3)] [9] . This phase transition from the metastable vacuum to the ﬁnal quasi-cry stal phase of average dimensions equal to φ3is completed once the size of the universe has reached the upper impassible dualscale to the Planck scale in the fashion we shall indicate next. A rough estimate of such u pper scale Lwas given by the geometric mean relation between the Planck scale and the Hubble radius RH∼1060Λ : R2 H= ΛL.(RH Λ) = (L RH)⇒Λ< R H<L (53) This sort of “ renormalization group “ argument was given by N ottale [5] and recently by us [9] which provides a very plausible and elegant resolution to the cosm ological constant problem. The Universe self tunes itself along the renormalization group ﬂow given by th e scaling temporal evolution ( size ) as follows : Evac(Λ) Evac(RH)= (RH Λ)2∼10120. (54) To ﬁnalize this section we wish to add that we have arrived at s imilar results as Majid [12] : at scales close to the Planck scale we do nothave the standard Lorentz invariance but a Quantum Group def ormation as indicated by Majid and others [12]. We will discuss the rol e of quantum groups, Braided Hopf Quantum Algebras, Braided QFT next in the construction of the Master Action Functional . Especially, the importance it has in order to recover, in the long distance limit , ordina ry Einstein-Riemannian Geometry from the more fundamental C-space Geometry. 5 Rigid Branes, Spin, Extrinsic Curvature and the Eﬀective C-space Geometry In this section we will provide with a geometrical meaning to the eﬀective momentum peffappearing in eq-(48). To the author, this is probably one of the most fasci nating results from the New Relativity . It is the emergence of an eﬀective background geometry linked to a spi nning particle and the actions with an explicit extrinsic curvature ( rigidity ) terms, directly from C-space. We shall argue why this eﬀective Geometry may in fact be connected to WGeometry and Finsler Geometries [13] . WGeometry is the geometry related to higher conformal spin theories, from spin 1 all the way to ∞. The subject of extended conformal ﬁeld theories and Wstrings is very vast that we just refer to [34, 36] for referen ces. Pavsic [30] long ago has shown that the classical equations o f motion for a rigid p -brane in a curved background can be derived from a Lagrangian which contains extrinsic curvature terms . The world line for a rigid point particle admits for equations of motion the Pap apetrous’ s equations ( also studied by Pezzaglia using Cliﬀord algebras ) for a spinning particle ( not a geode sic ) . Though our rigid particle is pointlike it 13has eﬀectively spindue to the extrinsic curvature term in the action which forces the particle to move in ahelical path. Pavsic was also able to show that the action for a rigid partic le can be obtained via a “ Kaluza-Klein “ reduction of an openstring wound up around a compact direction. The particle equ ations of motion follow from the truncated string equations of motion. If the string is spacelike, and i s compactiﬁed along a spacelike direction, the derived rigid particle action has only tachyonic non-trivial solutions. This could be relevant to the current tachyon condensation studies in Mtheory ( Sen and Ghoskal [8] ) and a plausible explanation of the apparent superluminal group velocities found in the rec ent experiments. It is clear from the generalized dispersion relation in C-space, given by eq-(37), that one may encounter eﬀective gr oup velocities faster than light. This is not surprising since after all we are working i nC-space and notin the ordinary spacetime of Special Relativity. If the string is timelike then the der ived rigid particle has subluminal ( bradyonic) non-trivial solutions corresponding to a helical ( or circu lar ) motion. This helical motion is precisely the one associated with the values of the eﬀective peffgiven by eq-(48), as we intend to show next. Pavsic considered the action for rigid pbranes in a target curved Dspacetime: S=/integraldisplay dp+1σ/radicalbig \|γ\|(T−µgµνHµHµ). γ AB=∂AXµ∂BXνgµν. (55a) where γABis the induced p-brane metric as a result of the embedding of the p-brane into spacetime. γis the determinant of γAB.Tis the p-brane tension; µis the rigidity parameter ( like a “ friction “ term that opposes bending ) ; Xµ(σA) are the embedding coordinates of the p-brane whose worldvolume coordinates areσAwithA= 1,2,3....p+ 1< D.Hµis a vector related to the extrinsic curvature : Hµ AB=DADBXµ(σA) + Γµ αβ∂AXα(σA)∂BXβ(σA)≡ D ADBXµ(σA). Hµ≡Hµ ABγAB. (55b) The Lagrangian is of second order and the equations of motion contain quartic derivatives whe nHµ/ne}ationslash= 0. When Hµ= 0 the equations of motion coincide with the harmonic equati ons of motion associated with the minimal embedding surface in a curved background : Hµ=DADBXµ= 0. In the special case of a rigid particle the fourth order equation associated with its worldline is nothing but Papapetrou’s equation for a spinning particle ! : 1√γdpµ dτ+1√γΓµ αβpβdxα dτ+1√γRν ναβSναdxα dτ= 0. HµdXν dτ=Sµν. (56) This has a straighforward explanation based on C-space geometry. The geodesic Cliﬀord-algebra valued lines in C-space are not geodesics in ordinary spacetime ! Secondly, to diﬀerentiat e with respect to τisnot the same than to diﬀerentiate with respect to the Σ p+1( which is the true C-space generalization of proper time ). The eﬀective particle motion was shown to be helical and was due to the classical spin, “ induced” by the extrinsic curvature terms. The classical spin or intr insic angular momentum ( associated with the second order action ) is given by : Sµν=πµdxν dτ−πνdxµ dτ. (57) where πµis the second order momentum conjugate to the variable ˙Xµ. Whereas the ﬁrst order momentum pµis the conjugate to the variable Xµand gives the standard orbital angular momentum Lµν=pµxν−pνxµ. Jµν=Lµν+Sµν. Thesecond order action of Pavsic, involving extrinsic curvature rigi dity terms, contains for basic variables Xµand (dXµ/dτ) ( and their assocated ﬁrst and second order canonical momen ta ) . This occurs also in the more fundamental Finsler Geometries ( Jet Bundles ) wher e the metric is both a function of the position and velocities. Finsler Geometries have both a maximum spee d and maximum four acceleration ( maximum value of tidal forces ). The maximum four acceleration is a=c2/Λ. If the Planck scale is seto to zero, Finsler Geometry collapses to Riemannian one. Finsler Geometry [13 ] has natural connection to Wgeometry ( see 14the references of C. Hull in [34]) . And the latter is related t o theextrinsic geometry of embedded surfaces in CPNspaces as Sotkov, and then Gervias and Matsuo have shown, [36 ] . The realtion to the Moyal-Fedosov Deformation Quantization was shown by the author following the work of C. Hull , see [34]. All this has a natural interpretation in C-space ( Pezzaglia [2] has also discussed the relation betwe en spin, Cliﬀord algebras and polydimensional covariance ). O ne has an ordinary center of mass of motion associated with the p-loop histories. The ﬁrst order corrections are the nothing but the arearemnants of the extended objects or p-loop holographic coordinates : The holographic area proye ctions σµνhave for their Fourier conjugates , the two-vector kµν. By keeping only the leading term in the expansion , the eﬀecti ve momentum ( 48 ) will have two pieces, one from the center of mass motion, and another from the holog raphic components kµν. The latter are the Fourier dual to the holographic Area coordinates σµνor Spin tensor/Spin-two-vector Sµν. The temporal change of the holographic area σµν/Sµνis nothing but the kµνas we intend to show. The eﬀective momentum due to the center of mass and spinning m otion found in [30] was : pβ= [m+µHµHµ] [1√γdXβ dτ]−1√γdXα dτ[pα1√γdXβ dτ−pβ1√γdXα dτ]. (58) To make this connection more explicit with the eﬀective pefffound in eq-( 48 ) we need ﬁrstly to study the special solutions which correspond to the case when Hµ/ne}ationslash= 0 but with HµHµ= 0 ( null-like acceleration ); i.e these solutions precisely correspond to the helical mot ion in ﬂat spacetime. Notice that in general ﬂatness does not necessarily imply torsionless. Flat Superspace Su pergravity has Torsion . The torsion is associated tospin. The connection in superspace has the ordinary Levi-Civita piece but an extra piiece due to the fermion bilinear terms ( torsion ) . In this special case the r igid point particle follows a helical world line ( as a result of the spin ) with center of mass momentum given by thetranslational motion along the axis of the helix and the angular rotation frecuency is precisely related to the rigidity parameter ω2= (m/2µ). The vector Hµsatisﬁed Hµ= (d2Xµ/dτ2) ( an acceleration ) and it had a nullnorm HµHµ= 0. The rate of change of the Spin two-vector ( Areal coordinates ) is : DSαβ Dτ=−[pα1√γdXβ dτ−pβ1√γdXα dτ]. (59a) Which clearly has the form of a two-vector momentum : pαβ=m2Λ2kαβ↔m[pα1√γdXβ dτ−pβ1√γdXα dτ]. λ= (Σp+1 mp+1)(1/2p+2)↔1 m(59b) where λis the the analog of the Fock-Schwinger-Feynman evolution p arameter ( a length scale ) induced in eq-(11). In the proper time gauge γ= 1, when HµHµ= 0, the rigidity constant µ decouples from eq-(58 ) ( not from the theory ) : pβ=mdXβ dτ−dXα dτ[pαdXβ τ−pβdXα dτ]. (60) one can compare then : pβ=mdXβ dτ−dXα dτ[pαdXβ dτ−pβdXα dτ]↔peff= (¯hkβ) + (¯hkαkαβΛ2). (61) Hence : mdXβ dτ↔¯hkβ. mdXα dτ[pα mdXβ dτ−pβ mdXα dτ]↔¯hkαkαβΛ2. m↔1 λ. (62) Which means then : m[pαdXβ dτ−pβdXα dτ]↔m2kαβΛ2=pαβ. (63) 15We have not ﬁnish yet. We still need to be more precise. If we wa nt to match exactly the quantity : ¯h2k2βΛ2k2appearing in the second term of the expression for p2 effin eq-(48) with the square of the last term of eq-(61) : (¯ hkαkαβ)(¯hkαkαβ) wemust have a proportionality factor ( m2λ2 h) between them as follows. Using the condition that kαβkαβ=βk4and the relation : kαkβkαβ= 0 due to the antisymmetry of the two-vector kαβone arrives : (¯h2k2)(βΛ2k2) = (m2λ2 h)(¯h2k2)(kαβkαβ)(Λ4) = (m2λ2 h)(¯h2k2)(βk4)Λ4⇒(m2)(λ2 h)(k2)(Λ2) = 1.(64) This last equation (64) in conjunction with the mass-shell c ondition forces a relationship among Λ , λh, m as follows : Given m2=k2⇒1 m2= (λh)(Λ)⇒Λ<1 m< λh. (65a) Andonceagain we recover a geometric mean relation among 3 scales : Λ; λh; (1/m) as we did for the p-loop harmonic oscillator earlier on in the derivation of the Blac k-Hole Area-Entropy relation and its logarithmic corrections. The physical meaning of the geometric mean relation is the following : The rigid particle follows a helical world line in such a fashion [30] that the circular motion mov es along a nullsurface in such a way the the net spacetime interval spanned b y its motion is precisely equal to the spacetime interval spanned by the center-of-mass mot ion ( see [30] for details ). If one deﬁnes the temporal-step interval parameter to complete one full revo lution to be λhelix≡λhthen the geometric mean relation is clear. The temporal-step paameter, or period, t o complete one full revolution is greater or equal to the Compton wavelength of the particle ( in units pf c= 1). This immediately allows us to evaluate the value of the rigidity parameter µfrom the frecuency relation [30] : ω2=m 2µ= (2π λh)2⇒µ=mλ2 h 2(2π)2.(65b) From eq-(65a) one can deduce the expression for the rigidity parameter that has dimensions of length : µ=1 2(2π)2m3Λ2=1 m1 2(2π)21 mΛ2 . (65c) If one sets the value of mto be of the same order of the Planck mass Mp∼1/Λ one gets a rigidity value of : µ=1 2(2π)2m3Λ2= Λ1 2(2π)2. (65d) which will give us a natural estimate for the value of the rigi dity parameter in Nature. It would be interesting to explore other values for µand see if there are any Astrophysical signals [30] of its exi stence. The most important conclusion of this section is that a geode sic in C-space does notcorrespond to a geodesic in ordinary spacetime. The ﬁrst leading correcti ons of the eﬀective C-space Geometry furnish naturally the Papapetrou’s equations for a rigid particle . ; i.e a classical spin is induced by the extrinsic curvature rigidity terms and, consequentlty, the worldlin e is a helical ( circular in some limiting case ) motion in ordinary spacetime; i.e the rigid particle does notfollow a geodesic, the four acceleration is not zero, although it has null norm . Roughly speaking, the free “ multidimensional- particle “ in C-space, to the ﬁrst leading approximation, corresponds to a spinning p article ( helical motion ). We took the Planck scale Λ to be the natural length scale of thi s problem because of its relation to the C-space geometry. There are also the two other natural length scales λh( the helical motion period ) and 1/m( the Compton wavelength ) and we were able to derive the geometric mean relationship among these 3 scales. The latter relation allowed us to estimate the value of the rigidity constant µexactly in terms of m,Λ. The Compton wavelength of the rigid particle falls in betwee n the Planck scale and the λhscale ( period ) which emerged from the full C-space propagator. Pezzaglia [2] took another diﬀerent val ue for the natural scale related to the properties of the electron. For interes ting ideas pertaining the role of Grassmanian time in QM see [39] and for superluminal travel through extra dime snions see [40]. 6. The Four Dimensional Conformal Anomaly, Fractal Spaceti me and the Fine Structure Constant 16In this section we will brieﬂy summarize the most relevant fe atures of Fractal spacetime to cosmological applications. In essence, the universe began as a process of non-equilibrium self-organized critical phenomena. Following the spirit of Cantorian-Fractal spacetime [6] in [9] we computed the eﬀective ( time dependent ) average dimension of the world by taking the statistical ave rage of an inﬁnity family of p-loops , of bubbles whose dimension ranged from D=−2 toD=∞. The gamma function was derived as the dimensional ensemble distribution. The average dimension ( observed to day ) relative to the zeropoint dimension Do=−2 was : < D−Do>=< D′>=/integraltext∞ 0dD′D′√πD′ [Γ(D′+2 2)]−1 /integraltext∞ 0)dD′√πD′[Γ(D′+2 2)]−1∼6.236 ⇒< D > = [6.236...+ (−2)] = 4 .236 = 4 + φ3. (66) where φis the Golden Mean : (√ 5−1)/2 = 0.618.... The average dimension was of the same magnitude as the average dimension of the transﬁnite Cantorian-Fractal spacetime E(∞)developed by M. S. El Naschie [6]. The average dimension <E(∞)>= 4 + φ3coincides precisely with the Hausdorf dimension of the setdimE(4)= 4 + φ3= 1/φ3= (1 + φ)3that is packed densely onto a smooth manifold of four topolog ical dimensions. Using the bijection formula [6] : dimE(n)= (1 φ)n−1⇒dimE(−2)=φ3=1 4 +φ3=1 dimE(4). (67) Hence the set E(−2)( Hausdorﬀ dimension equal to φ3and densely embedded into a smooth space of topological dimension −2 ) is the dimension dualto the set E(4)( Hausdorﬀ dimension equal to 4 + φ3and densely embedded into a smooth space of topological dimensi on 4 ). . The backbone set E(0)is packed densely onto a point , topological dimension 0, and its Hausdorﬀ dimension equals the Golden Mean φ. The backbone set is a random Cantor set whose dimension is φ < ln(2)/ln(3) with probability oneaccording to the celebrated Maudlin-Williams theorem. The transﬁnite Cantorian-Fractal spacetime model of El Naschie is the randomly constructed geometric space that ﬁts very naturally within the Random Process Physics ( Self Referent ial Noise as a model of reality ) program of Cahill and Klinger [29] , based on Godel and Chaitin’s work , a nd consistent with the most recent work on Quantum Information Theory and Quantum Algorithmic Random Processes in Nature [31] . In addition, it adopts von Neumann’s Noncommutative Pointless Geometry at its very core. The authors [32] more than two years ago computed the intrins ic Hausdorﬀ dimension of spacetime at the infrared ﬁxed point of the quantum conformal factor in 4 DGravity. The fractal dimension was determined by the coeﬃcient Q2of the Gauss-Bonnet topological term associated with the fo ur dimensional conformal anomaly ( trace anomaly ) and was computed to be greater than four. We were able to show that one can relate the value of the Hausdorﬀ dimension compu ted by [32] to the universal dimensional ﬂuctuation of spacetime given by φ3/2 = 0 .11856 ...[38]. Based on the infared scaling limit and using recent Renormalization group arguments by El Naschie [6] we conjectured that the unknown coeﬃcient Q2 associated with the four dimensional conformal anomaly may be equal to the inverse ﬁne structure constant of values ranging from 137 .036 to 137 .641 [38] . The idea is based on Eddington’s old belief that the inverse ﬁne structure constnat cod play the role of an internal ( electron’s dimension ). The Hausdorﬀ dimension computed in terms of the conformal an omaly coeﬃcient Q2by [32] was : dH= 41 +/radicalBig 1 +8 Q2 1 +/radicalBig 1−8 Q2≥4. (68) Inverting this relation allows one to express directly the a nomaly coeﬃcient Q2in terms of dH. By re-writing : dH= 4 + ǫ= 4(1 + ǫ/4) = 4(1 + φ3/8) = 4 δ⇒Q2= 2(δ2+ 1)2 δ(δ2−1)= 137 .6414382326 . (69) 17Thus, if one sets 4 + ǫto be 4 plus the universal dimensional ﬂuctuation ǫ=φ3/2 one obtains for the anomaly coeﬃcient Q2a value very close to the experimental value of 137 .036. The value of 137 .641..will be then associated with the infrared contribution to the inverse ﬁne structure constant due to th e quantum ﬂuctuations of the conformal mode of the metric ! 7. Noncommutative Geometry, Negative Probabilities and Ca ntorian Fractal Spacetime Recently [38] we provided a straightforward explanation of the Young’s double-slit experiment of a QM particle based on the Noncommutative Geometric nature of th e transﬁnite spacetime E(∞)and Negative probabilities. Since spacetime is in essence a randomly constructed space and the path of the QM parti- cle is fractal then any question about the exact spatial location of a micro scopic point is fundamentally undecidable due to the inherent uncertainty and fuzziness of the geometr ical structure of such space [6]. Since von Neumann’s Noncommutative Geometry is a pointless one a “ point “ in E(∞)can in fact occupy two diﬀerent locations at the same time; i.e unions and intersec tions are indistinguishable. The same argument follows from Pitkannen’s construction of p-Adic Fractals : there is a fundamentally indeterminism in t hese spaces. The Topology of E(∞)is in fact a p-Adic topology : every point is the center of a disk becuase ev ery point can occupy many places at once. The Young’s double-slit experiment admitted a straighforw ard interpretation by simply assigning to the particle a fractal zig-zag motion around the two slits A, B: a Peano-Hilbert curve. The negative probability came very natural due to the opposite orientations of the two Peano-Hilbert curves around slit A, B. The probability was related to the inverse fractal dimesnions of the sets around slit A, B. The respective probabilty assignments were respectively : p(A) =φ. p(B) =−φ2. p(A∧B) =−p(B∧A) =−φ3. p(A\|B) =φ. p(B\|A) =φ2. (70) in such a way that the total sum of probabilities yields unity exactly : [φ−φ2−φ3] + [φ+φ2] = 0 + 1 = 1 . (71) Since φ+φ2=φ(1 +φ) = 1. φ−φ2=φ(1−φ) =φφ2=φ3. φ=1 1 +φ. (72) 8. Quantum Groups, Braided Hopf Quantum Cliﬀord Algebras an d the Master Action Functional It was argued by the author in [1] that the master action funct ional for the nested family of p-loop histories could be given by a Braided QFT given by the action : S=/integraldisplay [DX(Σ)]1 2δΨ[(X(Σ)] δX(Σ)∗δΨ[(X(Σ)] δX(Σ)+ (mp+1)2Ψ[(X(Σ)]∗Ψ[(X(Σ)]+ g3 3!Ψ[(X(Σ)]∗Ψ[(X(Σ)]∗Ψ[(X(Σ)] + +g4 4!Ψ[(X(Σ)]∗Ψ[(X(Σ)]∗Ψ[(X(Σ)]∗Ψ[(X(Σ)].(73) Where the action functional is invariant under a Braided Hop f Quantum Cliﬀord algebra associated with the Quantum-Cliﬀord-algebra valued master ﬁeld Ψ[( X(Σ)]. The X(Σ) are the Cliﬀord algebra valued hyperlines ( higher dimensional version of twistors for exa mple or hypercomplex numbers ) in C-space parametrized by the C-space extension of the “ proper time “ Σ as shown in eq-(4) . As mentioned earlier, the ﬁrst term corresponds to the quadr atic kinetic term. The second ones to the analog of mass-squared terms for a scalar ﬁled theory ; the tr iple interaction vertex corresponds to both a product ( two hyperlines join to give a third one ) and co-prod uct ( one hyperline breaks into two ) of the Quantum algebra. The quartic terms correspond to the braide d scattering of the four lines. These are the only terms allowed consitent with the Noncommutative Braid ed Hopf Quantum Cliﬀord algebra [15] encoded by the star product ∗. This star product is reminiscent of the Noncommutative star product associated with the BRST and Batalin-Vilkoviski formulation of string ﬁeld theory [16, 17] where anticommuting variables are introduced. 18The string ﬁeld Ψ[ Xµ, c] is now a functional of the string world−sheet coordinates Xµ(σa) and of the ghost coordinates c(σa) with a= 1,2. Compare this with the Master ﬁeld which is a Cliﬀord-line f unctional. The C-space Cliﬀord-alegebra valued ( matrix ) multivector X(Σp+1) that solely depends on one parameter. Similar deformed star products have been obtained by Vasili ev in his construction of higher conformal spin algebras from the deformation of the Anti de Sitter Alge bra. These higher spin theories have been essential to construct Higher Spin Supergravity Theories w hich are conjectured to be the eﬀective ﬁeld theory limit of MTheory compactiﬁcations on S7×AdS4[38]. This Braided QFT [33] is highly nontrivial. To derive the wav e equations used for the p-loop harmonic oscillator we had to make several assumptions. The ﬁrst one i s to freeze or quench the higher order modes of the functionals so one can approximate the dynamics by writi ngordinary diﬀerential equations. Secondly one must assume a flat C -space metric. and thirdly we had to set the cubic and quartic couplings to zero . See [20, 21]. In general one has a nonlinear oscillator which has far richer properties than the linear c ase. Planck scale Hopf algebra, co-gravity ..... Future Prospects Riemannian Geometry as the large distance limit of C-space Geometry Mahid has proposed a Planck scale Hopf algebra, a sort of grav ity/co-gravity dulaity : meaning a curved phase space ( perfectly consistent with the Finsler and WGeometry ) to formulate the quantum group algebraic properties of a plausible Planck scale quan tum geometry that should reproduce ordinary classical Riemannian Geometry in the long distance limit. I n particular he derived the analogous ( not identical ) commutations relations to our commutation rela tions eq-(41) : [x, p] =i¯heff=i¯h[1−e−x/L]. (74) where Lis a suitable length scale. The quantum ﬂat space limit is att ained when L= 0. The classical limit is attained when ¯ h= 0. The most interesting aspect of majid PLanck scale Hopf Al gebra construction is that one can simulate the dynamics of a particle with a position dependent momentum p′given by : p′=po[1−e−x/L].⇒[x′, p′] = [x, po[1−e−x/L] =i¯heff. (75) In such a fashion that it mimics the motion of a free falling particle into a black hole whose v elocity mesaured by an asymptotic observer at inﬁnity is : v(r) =v∞(1−1 1 +r/L+...) +O(¯h). (76) As the particle approaches the origin it apperas to move more slowly relative to the asymptotic observer and will take an inﬁnite amount of time ( relative to the observer ) to reach the origin. This is excatly similar ( up to a factor of 1 /2 ) to the formula for the radial infalling particle velocity in the vecinity of a Black Hole of mass Mand Schwarschild radius R: v(r) =v∞(1−1 1 +r/2L). L=GM=1 2(2GM) =1 2RSchwarzchild . c= 1 (77 a) Another way to recapture the Riemannian ( Black Hole ) Geomet ry in the large distance limit x=r >> Lis to consider using a nonlinear p-loop oscillator. Then to linearize its solutions Ψ[ X] and evaluate the expectation values of the multicomponets of the C-space Geometry in the large ecitation level nlimit and show that : <Ψ\|Gµν[X]\|Ψ>=<Ψ\|Gµν[xµ;σµν;σµνρ, ...]\|Ψ>→=gµν(Schwarzschild ) +O(r/Λ)..... (77b) <Ψ\|Gµνρτ[X]\|Ψ>=<Ψ\|Gµνρτ[xµ;σµν;σµνρ, ...]\|Ψ>→= 0, etc..... (79c) Based on what we have shown in this work it seems undoubtledly that the New Relativity is marching forward. 19Acknowledgements We thank A. Granik, C. Handy, E.Spallucci, S.Ansoldi, E. Goz zi, T. Smith, G. Bekkum, D. Finkelstein, A. Schoeller, J. Boedo, S. Duplij, M. Pavsic, E. Guendelman, G. Kalberman, L. Nottale , M.S. El Naschie , D. Chakalov, M. Pitkannen, W. Pezzaglia, S. Paul King, L. Ba querom J. Mahecha, J. Giraldo for many discussions. Special thanks go to B. G. 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Granik : “ P-loop Oscillator on Cliﬀord mani folds and Black Hole Entropy “ physics/0008222 v2. 22-P. Majumdar : “Quantum Aspects of Black Hole Entropy “ hep -th/0009008. 23-K. Fujikawa “ Shannon’s Statistical Entropy and the H-Theorem in Quantum Statistical Mechanics “ cond-mat/0005496 24- M. Li, T. Yoneya : Journal of Chaos, Solitons and Fractals “10(2-3 ) ( 1999 ) page 429. 25. P. Lounesto : “ Cliﬀord Algebras and Applications “ Lectu re Notes in Mathematics 26. C. Castro, M. Pavsic : “ The Multidimensional-Particle P ropagator from the New Relativity “ CTSPS-preprint ( Atlanta ) and IJS-Ljubljana preprint ( 200 0 ) . To appear. 27-B.G. Sidharth : “ Fractal Statistics “ physics/0009083 B.G. Sidharth : “ Quantum Superstrings and the Quantized Fra ctal Spacetime “ physics/0010026 W. da Cruz : “ Fractal Statistics, Fractal Index and Fractons : “ hep-th/0008222” 28- H. Maris : Jour. Low Temperature Physics vol 120 ( 2000 ) 173. 29-Cahill , Klinger : “ Self Referential Noise as “ 30- M. Pavsic : Class. Quant. Gravity 7(1990 ) L-187. M. Pavsic : Phys. Letts B 205 ( 1998) 231. Phys. Letts B 221 ( 1999) 264. M. Pavsic : Nuc. Phys. B 57 ( Proc. Suppl) ( 1997) 265-268 M. Pavsic : Riv. Il Nuovo Cimento 110 A ( 4 ) ( 1997) 369. 31- G. Segre : “ Quantum Algorithmic Randomness “ Ph. D Thesis , quant-ph/0009009 v3 . 32-I. Antoniadis, P. Mazur , E. Mottola : “ Fractal Geometry o f Quantum Spacetime at Large Scales “ hep-th/9808070 : 33- R. Oeckl : “ Braided Quantum Field Theory “ hep-th/990622 5 v2. 34- C. Castro : “ WGeometry from Fedosov Deformation Quantization “ Jour. Geo m. and Physics 33 ( 2000 ) 173. 35- C. Castro, J. Mahecha : “ Comments on the Riemann Conjectu re and Index Theory on Cantorian- Fractal spacetime “ hep-th/0009014 36- P. Bouwknegt, K. Schouetens : Phys. Reports 223(1993) 183-270 37- M . Vasiliev : “ Higher Spin gauge Theories, Star products and AdS “ hep-th/9910096. E. Sezgin, P. Sundell : Higher Spin N= 8 Supergravity in AdS4“ hep-th/9910096. 38. C. 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page 1 of 2The generation of gravitational waves A. Loinger * Dipartimento di Fisica, Università di Milano, Via Celoria, 16 - 20133 Milano, Italy Abstract. – A proof that the generation of gravitational waves is physically impossible. PACS Code: 04.30 - Gravitational waves: theory. Many papers have been written on the gravitational waves, see e.g. the ample bibliography at the end of the review article by Schutz1[1]. Now, I have recently demonstrated the non-existence of a physical “mechanism” for the production of gravitational waves2[2]. In the present Letter I give a simpler proof of this thesis. As is known, we can always choose a Gaussian (“synchronous”, in Landau’s terminology) system of space-time coordinates [3]. In such a reference frame the four- dimensional interval has the following form: βα αβ− = xxtxxxgtcs dd),,,( d d321 22 2 , )3,2,1,(=βα , (1) and the time lines coincide with the geodesics. Let us consider several point masses which interact via gravitational forces only. Of course, their motions can be quite complicated, but we are sure that any particle follows a geodesic line. Now, a geodesic motion is a free motion, it is the perfect physical analogue of a rectilinear and uniform motion of a pointlike electric charge in the customary Maxwell- Lorentz electrodynamics. No electromagnetic wave is generated by this motion of the charge − and no gravitational wave is generated by the geodesic motion of a point mass. If we add some internal actions − as hydrodynamical pressures, etc. − the result does not change: indeed, in this case the world lines of the particles do not coincide with geodesic lines,page 2 of 2but any acceleration of the new motion of a given point mass can be reproduced by a geodesic motion of the same particle in a suitable, “fixed”, purely gravitational field. A confirmation of the above argument is represented by the following remark. In the well-known Einstein-Infeld-Hoffmann method for deriving the motions of pointlike singularities of the gravitational field from Einstein equations, there is the possibility − at each stage of approximation − to pass to a reference frame for which our system of particles is non-radiating. And this is not an ad hoc trick: on the contrary, the mentioned possibility is rooted in the basic structure of general relativity. ( Fock was of a different opinion, because he attributed a physically privileged role to the harmonic frames [4]. However, Fock’s conviction − as emphasized by Einstein − is repugnant to the intrinsic nature of general relativity). The conclusion is obvious: the undulatory solutions of Einstein field equations do not describe physical phenomena. 3 * e-mail: angelo.loinger@mi.infn.it 1 B. F. Schutz, Class. Quantum Gravity 16, p. A131, (1999). 2 A. Loinger, Nuovo Cimento 115B, p. 679, (2000). 3 D. Hilbert, Mathem. Annalen 92, p. 1, (1924); also in Gesammelte Abhandlungen (J. Springer, Berlin, 1931), Dritter Band , p. 258. 4 V. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, Oxford, etc., 1964), Second Revised Edition, p. 398.
A Historical Perspective on the Topology and Physics of Hyperspace Ian T. Durham 2 di••••men••••sion n 1 a … (2) : one of a group of properties whose number is necessary and sufficient to determine uniquely each element of a system of usu. mathematical entities (as an aggregate of points in real or abstract space) <the surface of a sphere has two ~s>; also : a parameter or coordinate variable assigned to such a property <the three ~s of momentum> … 1 Introduction Throughout history, the human mind has sought to understand its surroundings. One of the most fundamental aspects of our universal surroundings is the array of spatial and temporal dimensions within which we exist. Humanity has slowly and discontinuously managed to unfold eleven (or twelve) of these dimensions over the last 2550 years or so. In this paper the historical development of the mathematics and physics behind the discovery of these dimensions is examined from the earliest records of Greek geometers and scientists starting in the sixth century BCE through to the most recent developments in theoretical physics. Historical glimpses of the people who have helped to shape these developments are given as a basis for the mathematical processes that build up to the overall worldview. It is wise to note that there were many parallel developments, notably during ancient times, in the Middle East and Far East. For the sake of brevity they have not been included here, however, the reader is encouraged to explore these areas in greater depth. In particular, the legacy of Euclid and Klein are developed in depth working from the Euclidean concept that the topology of the universe is inherently flat and moving into Klein’s first use of a curved dimension in modifying Kaluza’s initial work on five dimensions. This paper relies heavily on secondary sources as it is merely meant to be an introduction to the topic. Early Developments The earliest developments in Greek mathematics are attributed to Pythagoras and his followers. Pythagoras was born around 570 BCE on the island of Samos off the Ionian coast. He supposedly left the island around 540 BCE out of disenchantment with the ruling Polycrates and fled to Croton. Croton was a Greek settlement on the southeastern coast of Italy on the lower Adriatic. Once in Croton he attracted a group of followers who have since been known as the Pythagoreans, a mysterious group who have been revered and copied, reportedly, by druids, masons, and secret societies over the centuries. Their teachings and, as a result, those of Pythagoras himself were kept secret. Most knowledge of their teachings was not revealed until nearly a century later in the writings of Philolaus. Thus the teachings are far from a direct account of Pythagorean thought. Recent tradition even indicates Pythagoras may have learned much of his teachings from other sources, possibly on a series of travels he had undertaken. It is said these travels were in the East, though there is a persistent legend that quotes him as saying: 1 F.C. Mish Ed., Webster’s Ninth New Collegiate Dictionary , Merriam-Webster, 1991. 3 All I know I learned from a Druid.2 The earliest written evidence of Greek mathematics is in fact Euclid’s Elements . This work dates from the fourth century BCE though much of it is thought to be the work of earlier mathematicians, including the Pythagoreans. Eudemus the Peripatetic attributes to them the theorem that describes the sum of the interior angles in a triangle being equal to the sum of two right angles. His description of the Pythagoreans’ proof of the theorem is as follows: Let ABC be a triangle, and let the line DE be drawn through A parallel to BC. Now since BC and DE are parallel, and the alternate angles are equal, the angle DAB is equal to the angle ABC and the angle EAC is equal to the angle ACB . Let the angle BAC be added to them both. Then the angles DAB, BAC, and CAE (that is to say, the angles DAB and BAE, i.e., two right angles) are equal to the three angles of the triangle ABC . Hence the three angles of the triangle are equal to two right angles. 3 According to this proof, the Pythagoreans were already familiar with the concept of parallel lines as well as two-dimensional objects. Certainly, humans were aware of two dimensions from simple sensory perceptions, however, this proof is one of the earliest pieces of evidence that indicates an understanding of the mathematical nuances lying behind the physical reality. Of course, the more famous mathematical construct attributed to the Pythagoreans is the famous Pythagorean theorem that describes the sum of the squares on the shortest two sides of a right triangle as being equal to the square of the hypotenuse, or longest side. The Pythagoreans’ use of one-dimensional lines laid down to describe a two-dimensional feature was one of the earliest examples of extending a single dimension in a way that creates an extra or “higher” orthogonal dimension. In fact, if a single straight line is 2 A legendary quote often found in popular accounts and products relating to the druids and druidry. 3 Proclus in Euclid I , qtd. in J.M. Robinson, An Introduction to Early Greek Philosophy , Houghton Mifflin, 1968. 4 considered to be one-dimensional, and a second straight line, non-parallel to the first, is laid down and connected at any point to the first, a two-dimensional space is automatically created where a minimum of two position coordinates must be specified in order to describe a single point in that space. Realizing that such abstract spaces as Riemannian geometry had yet to be developed, this can be considered one of the first mathematical realizations of multi-dimensional space. In contrast, other Greek geometers used a method that consisted of applying a method of “area mathematics” to decipher algebraic problems. Since they did not possess a form of algebra similar to our own, this method of area application allowed them to solve second-degree equations and formed the basis of Euclid’s work on irrationals. Euclid’s Elements Euclid was born around 325 BC. Not much is known of his life other than the fact that he lived and worked in Alexandria, Egypt. From various accounts including those of Proclus, Euclid compiled and refined the work of many of his predecessors in his famous anthology, The Elements . The exact nature of his personal contribution and ideas is sketchy. However, Proclus wrote that Euclid also brought “to irrefutable demonstration the things which had been only loosely proved by his predecessors.” 4 Euclid’s Elements begins by defining certain terms vital to the understanding of geometrical space. It is important to make note of a few of these definitions in our study of advancing dimensions. First, Euclid defines a point as “that which has no part.” 5 He also defines a line as a “breadthless point”6 asserting the Pythagorean use of lines as one dimensional objects. In addition, the definition of a point is of particular importance as it is one of the earliest assertions that zero-dimensional objects can be represented mathematically. As we will see, this plays an important role later in the definition of string theory as the original concept of point-particles based on the Euclidean assumption of an indivisible point is amended and the actual definition of “point-like” is no longer quite the same thing. Euclid defines a surface as “that which has length and breadth only.” 7 It is interesting to note that Euclid differentiated between this simple definition of a surface and that of a plane surface which he defined as “a surface which lies evenly with the straight lines on itself.” 8 What is interesting to note is that the definition of height (as a separate concept from length and breadth) is not given until Book VI and is given completely independently of length and breadth as “the perpendicular drawn from the vertex to the base.” 9 4 Euclid in Elements I , D. E. Joyce Ed., Clark University Mathematics Department Web, 1998. 5 Ibid. 6 Ibid. 7 Ibid. 8 Ibid. 9 Euclid in Elements VI , D. E. Joyce Ed. 5 Euclid’s first indication that a third spatial dimension exists specifically and mathematically in relation to the first two spatial dimensions, comes in Book XI. However in Book VII he states that “when three numbers having multiplied one another make some number, the number so produced be called solid , and its sides are the numbers which have multiplied one another.” 10 The definition in Book XI is the more familiar one. Here a solid is defined as “that which has length, breadth, and depth.”11 Further, a surface is defined as the side of a solid. Thus Euclid took the Pythagorean concept a step further (at least in ‘print’) and created a third dimension orthogonal to both the previous two. Relating this to my assertion that by simply creating two non-parallel, yet connected lines, a two-dimensional space is immediately created, to ensure the existence of the third dimension in an additional space, three lines must be connected (only once each) and be completely non-parallel, and one must be non-coplanar, to each other to ensure three spatial dimensions. The easiest way to do both of these extensions is to make the lines themselves completely orthogonal, thus creating a visual aide in perceiving three orthogonal dimensions much as we create x, y, and z axes when plotting a point in three-dimensions (it should be noted, the dimensions are always orthogonal, but to create them, the lines only need to be non-parallel, with one non-coplanar, and connected once – a moment or two of thought should confirm this). It is interesting to note that nowhere does Euclid define addition and subtraction. He assumes that these basic functions are known and understood. However, multiplication is specifically defined. Euclid also represents numbers solely in the context of a line while it was apparent that the Pythagoreans represented numbers as figures. 12 Thus, by the end of the third century BCE, the three spatial dimensions as we can perceive them, were mathematically known and rigorously defined. Euclidean geometry then remained the only accepted description of the spatial universe until well into the Renaissance in the 17 th century CE. There is one aspect of Euclid’s work that bedeviled mathematicians and physicists for nearly two millennia. This is frequently referred to as Euclid’s fifth postulate . The fifth postulate states “that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.” 13 This, it turns out, is the essence of Euclidean geometry. It assumes a non-curved space in all dimensions. Every other postulate of Euclidean geometry can hold true on certain surfaces except the fifth postulate. Mathematicians struggled to prove or disprove or disassociate this principle until well into the 19 th century when it seems Gauss was the first to accept the possibility of non-Euclidean geometry and, in conjunction with Bolyai and Lobachevski, showed through the development of curved geometrical spaces, that 10 Euclid in Elements VII , D. E. Joyce Ed. 11 Euclid in Elements XI, D. E. Joyce Ed. 12 D. E. Joyce, Guide to Euclid’s Elements VII , Clark University Mathematics Department Web, 1998. 13 Euclid in Elements I , D. E. Joyce Ed. 6 Euclid’s fifth postulate was indeed independent.14 This opened the door to a vastly new way of representing dimensions. We will see that this has an important impact on the discovery of the dimensions beyond the fourth (time) and third (spatial). The difficulty in proving this postulate also secured Euclid’s legacy for two millennia as dimensions were seen as completely flat. It should be noted that Farkas Bolyai, a geometer in his own right and a lifelong friend of Gauss, attempted, in vain, to stop his son Janos from contemplating this problem, but, luckily, failed in his attempt. 15 A Brief Foray into Phase Space The dimensions described here are limited to actual physical dimensions. However, the use of a mathematical tool called phase space has utilized the concept of a dimension as a direction orthogonal to other defined directions in a unique and handy way. Phase space is defined as the number of dimensions that can be utilized to represent the state of a particular system at a given time and is equal to the number of degrees of freedom of the that system. Frequently momentum is the quantity represented in addition to our customary space and time, although, in the first known representation of its kind, in 1698 by Varignon, velocity was instead used in place of momentum 16. The important distinction of phase space is that it is a useful mathematical tool but does not describe dimensions in the same way used here. It merely allows for an easier representation of the state of a particular system at a given time. This is particularly useful in the representation of quantum states where momentum (through the uncertainty principle) plays an important role in the description of the given state. This actually does end up playing an important role later on as we will see that string theory relies heavily on the principles outlined by the uncertainty principle and draws on the fundamentals of quantum mechanics. However, strictly speaking, it is not the type of dimension we are interested in. We will pay close attention only to those dimensions that are physical actualities and allow the physical transfer of energy in one or two directions within that dimension (finding a clear cut definition of a dimension in physics is not an easy task, as we will see, particularly in the context of temporal dimensions) and that can be represented in length units alone (this includes temporal dimensions). Time as a Dimension The use of time as a dimension in mathematical plots dates from well before the 19 th century. It was in the latter portion of this century, however, when the use of time as a true dimension, able to be represented by a length , and consistent with the previous uses of spatial dimensions, was brought to bear. Actually, for the first representation of time as an independent orthogonal coordinate in a four-dimensional space-time, we must look to the early 20 th century. In fact, Einstein used a purely algebraic form of math to describe special relativity in 1905. It was not until 1908 that time was included as a 14 S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity , John Wiley & Sons, 1972. 15 http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Bolyai_Farkas.html 16 J. Stachel, Einstein: A Man for the Millennium? , lecture to the Spring 2000 New England Section meeting of the American Physical Society and American Association of Physics Teachers. 7 coordinate in a four-dimensional space-time by Minkowski.17 T h i s g a v e r i s e t o t h e geometrical representation of relativity and led directly to the development of general relativity in 1914-16. It is imperative to remember, however, that until we reach Klein’s formation of the fifth dimension, we are still in Euclidean space-time which means all dimensions are still flat. At first, it is not imperative to represent time in spatial units to visualize the logical extension of space to space-time. However, in order to do any meaningful mathematics, it would be desirable to develop some way in which to represent time with the same unit measurement as space. Building on Einstein’s postulate that the speed of light is invariant and universal, it can be represented as a unitless number. The most logical choice of number in this case is 1. Therefore, if the speed of light is c then c = 1. More specifically, the speed of light is defined as: In order to ensure that this number is unitless, we must represent time spatially. So, in SI units, applied to relativity, time is measured in meters. This representation is often referred to as natural units . This allows us to construct yet another useful tool in relativity: the space-time diagram. In this diagram, a spatial coordinate (usually x in two dimensions) is plotted as one axis with time as the other, both being represented in natural units as meters: In this diagram, the angles A and B are both equal to 45 °. This equality represents the invariability of the speed of light. The time and space axes are not always at right angles in this diagram but they cannot ‘pass through’ the world-line of light which represents a barrier. The slope of this line is dt/dx = 1/v. Given an event (any single point on this 17 B. F. Schutz, A First Course in General Relativity , Cambridge University Press, 1990. 8 diagram), the interval between any two events separated by coordinate increments ( ∆t, ∆x, ∆y, ∆z) is given as: 2 2 2 2 2)()()()( z y x t s ∆+∆+∆+∆−=∆ (1). This interval is invariant and forms the basis for building a space-time metric. The Minkowski metric which we use in the still flat space-time of special relativity is defined as:    − = 1000010000100001 αβη (2). So in brief review, by 1908, we have seen the recognition of four dimensions beginning with Pythagorean representations of two-dimensional objects and properties, working through Euclid’s definitions surrounding three-dimensional objects, and finally reaching, over two millennia later, the representation by Minkowski and Einstein of time as the fourth dimension. It is presumably safe to say that the four dimensions of space and time as presented here, represent the limit of human sensory perception. This limitation is most likely the reason that over two thousand years passed between the establishment of the initial three spatial dimensions and the addition of the next spatial dimension. The perception of time as a dimension could quite possibly be viewed as a concept behind its time. It certainly contains a sophisticated level of mathematics, but is consistent enough in its linearity that it could be considered a mere fluke that it had not been perceived as a dimension by even the Greeks. Evidence points to its having appeared on the same plot as spatial coordinates two-hundred years prior to Minkowski’s use of time as a coordinate. However, ignoring this fact, the limit of human sensory perception can be considered the greatest barrier that needed to be overcome in order to even conceptualize higher dimensions. Einstein’s relativity opened the door for this, a lifelong dream of Riemann, but it was an obscure mathematician named Theodor Kaluza who first mathematically developed the idea (Nordstrom was also successful in this effort, but has not been the beneficiary of a ‘named’ theory in this area). The Fifth Dimension We would be remiss, however, if we do not mention the fact that two rather colourful scientists publicly proposed the fifth dimension (they referred to it as the fourth dimension as Minkowski’s work had yet to be published) less than thirty years prior. 9 In 1877, a bizarre and sensational trial took place in London. The then renowned psychic Henry Slade sat accused of fraud for supposedly deceiving his clients who were some of England’s elite. Quite possibly the most bizarre part of the trial was the fact that several prominent physicists of the time, including some future Nobel Prize winners, came to Slade’s defense by supporting the notion of a ‘fourth dimension’ (spatially speaking – for our sake, it is the fifth dimension). One of these physicists was Johann Zollner a professor of physics and astronomy at the University of Leipzig. Zollner enlisted the aide of famous physicists William Crookes, Wilhelm Weber, J.J. Thompson, Lord Rayleigh, and others in an attempt to prove Slade’s innocence. Slade was ultimately convicted, at no surprise to us (for we now know the amount of energy required to manipulate the ‘fourth dimension’), but Zollner was so convinced of the existence of the ‘fourth dimension’ and its ability to be manipulated that he published articles in both scientific and pseudo-scientific journals in defense of it. 18 In the very same year as the Slade trial, a mathematician named Charles Howard Hinton graduated from Oxford. Being the son of famous ear surgeon and renowned bigamist James Hinton, his personal life began as anything but dull. He eventually became a bigamist himself, having taken the widow of George Boole (of Boolean algebra fame) as his first wife, and Maude Weldon as his second. Despite his arrest, his first wife, Mary, declined to press charges and they both fled to the United States. It was here that Hinton eventually found his way from Princeton to the US Naval Observatory and finally, to the place where another great physicist of the time was ‘born:’ the patent office (though not the same patent office, of course). Hinton was known as the man who could ‘see’ the fourth dimension and spent his life laboring to develop ingenious visual descriptions of the fourth dimension. These descriptions eventually became known as hypercubes and unraveled hypercubes became known as tesseracts, a term coined by Hinton himself. 19 But, despite all this laboring on the part of physicists and mathematicians during the late 19 th century, it was not until after Einstein published his seminal work on general relativity that Theodor Kaluza was able to become to the first to mathematically describe the fifth (or fourth spatial) dimension. Theodor Kaluza was born in 1885 in Ratibor, Germany, now known as Raciborz, Poland, eight years after the famous Slade trial. He was a professor at Königsberg when in 1919 he sent Einstein a paper he had been working on that unified Einstein’s relativity with Maxwell’s theory of light. The very means for unifying these two theories was the addition of a fifth dimension. What separated Kaluza’s work from that of Riemann, Zollner, and Hinton, was that Kaluza was proposing a true field theory. He simply wrote down Einstein’s field equations in five dimensions. He then showed that the new five-dimensional equations contained Einstein’s four-dimensional relativity theory plus an additional piece. It turned out the additional piece was exactly Maxwell’s theory of light. In Kaluza’s original theory, all the fields involved were independent of the fifth dimension. By starting with pure gravity written in five dimensions, though independent 18 M. Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension , Anchor Books, 1994. 19 Ibid. 10 of the fifth dimension, the field breaks down to four dimensions which ultimately leaves a metric, a Maxwell field, and a scalar. Kaluza’s major restriction on the fifth dimension was that it was cylindrical in form thus forcing it not to appear in the physics of the problem (i.e. it was a convenient mathematical device, but held little real meaning). Oskar Klein (born in 1894), then professor at the University of Michigan, refined Kaluza’s ideas in 1926. These combined works are now known as Kaluza-Klein theory. Klein did not assume total independence of the fifth dimension. Returning to Euclid’s fifth postulate, we recall that Gauss, Bolyai, and Lobachevski were the first to prove the independence of Euclid’s fifth postulate. This opened the door to curved geometries. Kaluza had initially employed a cylindrical shape (topology) for his fifth dimension, but it was still Euclidean in geometry. Klein utilized the new idea of non-Euclidean geometry and postulated that Kaluza’s fifth dimension was actually curved in geometry and microscopic in size. In fact, Klein assumed this dimension would have the topology of a circle with the radius on the order of the Planck length (we will see the exact radius becomes important in string theory). We can then write the topology for all five dimensions as B 4 x S1 where the fifth coordinate, y, is periodic – 0 ≤ my ≤ 2π - and m is the inverse radius of the circle. The periodicity of the extra dimension allows us to make a Fourier expansion in this coordinate. The first order terms of this expansion correspond to the reduction initially introduced by Kaluza. Working with the convention adopted by Derix and van der Schaar 20 we will define hatted quantities as being five-dimensional and unhatted quantities as four-dimensional. Five dimensional indices will run as: µˆ = 0,1,2,3,5 and the four-dimensional indices will run as: µ = 0,1,2,3 ( xµˆ = ( xµ, y)). Kaluza wrote the five-dimensional metric as follows, with a 4+1 split: gˆ νµˆˆ=     − −− − φ σφ φ νµ νµ µν AA AA g (3) This allows the four-dimensional fields to have the proper transformation characteristics in four dimensions. As developed by Derix and van der Schaar, we must first consider an infinitesimal coordinate transformation in five dimensions: x µˆ→xµˆ + εξµˆ(xµ) where the transformation is independent of the fifth coordinate. Given this coordinate transformation, we can transform the five dimensional metric in the following way: ρµνµ ˆˆ ˆˆ ˆ ˆ g g=∂()() ()νµρρ ρ µνρρ ν ξξ ξˆˆ ˆˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ g g ∂+∂+∂ (4). 20 M. Derix and J. P. van der Schaar, Stringy Black Holes , Master’s Thesis, University of Groningen, 1998. 11 We can then derive the transformation properties of the four-dimensional vector µA as follows: ()()µ µ µ φφ A A g ∂−∂−=∂5ˆ ()()5ˆ 5ˆ ˆ ˆµρρ ρ µρ ξξ g g ∂+∂= () () () ( ) φ ξφξξφξφµρρ µρρ µρ µρ D D A ∂−∂−∂−∂−=5 therefore ()()5ξ ξξµ µρρ ρ µρ µ ∂+∂+∂=∂ A A A (3). The last term in this equation is a U(1) gauge term and µA has right transformation properties in four dimensions. The invariance of general coordinates in five dimensions and the independence of the fifth dimension (still held by Klein despite his topology change from flat to curved) results in gauge symmetry of the four-dimensional vector. The gauge symmetries become more complicated in four dimensions and are a result of more complicated compactifications, which is an important part of string theory and Calabi-Yau spaces (as we will later see). The four-dimensional metric and scalar also have the correct transformations: ()() ()µνρρ ρ µρνρ νµρ µν ξξ ξ g g g g ∂+∂+∂=∂ and φξφρρ∂=∂ . Here, Derix and van der Schaar have set φ−=55ˆg which keeps the scalar field positive while also keeping the fifth coordinate space-like. Keeping in mind the fact that ν µνρ ρµˆ ˆˆˆ ˆˆˆˆ ∂=gg the inverse metric can be written as:     +−−−=2 1ˆˆˆA AA gg φνµ µν νµ (4). 12 To develop Kaluza’s idea, we begin with pure gravity, meaning a source-free space-time in five dimensions. The action integral for this system is given by Derix and van der Schaar 21 as: Rgxd S ˆˆ5 )5(∫−= (5). The constant in front of the integrals in equation 6 can be inserted here, but was left out by Derix and van der Schaar. See Overduin and Wesson for a more in depth discussion of this. 22 Compare this to the action integral given by Visser23 (for comparison, see those given by Weinberg24 and Misner, Thorne, and Wheeler25) from which we can derive general relativity in four dimensions: ∫∫ ∫ ΩΩ ∂Ω+ − −= xdg xdg KGcxdgRGcS4 3 33 43 8 16Lπ π (6). The four dimensional action integral is far more complicated than the five dimensional one. This is one of the most important aspects of multi-dimensional physics. In this way, physicists have used the addition of extra dimensions to simplify complex mathematical problems, the most important example being string theory. Returning to the five dimensional model, the determinant of the metric can be reduced to: ()()φφµν νµ g g g g −= −= = det ˆdetˆˆˆ (7). Derix and van der Schaar present the result of the derivation of the Ricci curvature scalar in five dimensions as: ()φφ φ1 21ˆ2 2−∂+=RR )( )(41A FA Fµν µνφφ+ (8) where µννµ µν A A F ∂−∂= . Putting this back into equation 5 and assuming that integration over the fifth coordinate is 1 ( 15=dx ), the action becomes: ()φφφφ1 212 24 )4(+ ∂−−−=∫R g xd S φ −2)(41AFφ (9). Both terms involving derivatives of φ can be written as total derivatives thus not contributing to the action and simplifying equation 9 to: 21 Ibid. 22 J. M. Overduin and P. S. Wesson, Kaluza-Klein Gravity, Phys. Rep. 283, 303, 1997. 23 M. Visser, Lorentzian Wormholes: From Einstein to Hawking , AIP Press/Springer, 1996. 24 S. Weinberg, 1972. 25 C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation , W.H. Freeman & Company, 1973. 13 ∫ −−−=2 2/1 4 )4()(41AF R g xd S φ φ (10). Derix and van der Schaar take this a few steps further by deriving a form involving an Einstein term and additional exotic matter terms. They first do this by performing a conformal rescaling of the metric: µν µν µν φg g g21 =′→ Non-trivially, the Ricci scalar transforms to the following four-dimensional form: ()        ∇−   ∇∇+′=2 221 41 1 23φ φφφφρ ρ R R (11). Finally, they transformed the other terms in the action as follows: 2 2F F′=φ , g g ′−=−−1φ , φ φφ log3=′→ . Finally, the four-dimensional action can be written in the conventional form (Derix and van der Schaar dropped the primes): ∫ −∂∂+−−=− µν µνφ µ µφφ FF e R g xd S3 4 41 21 (12). A detailed construction of the action for gravitational fields is contained in Chapter 12 of Weinberg. 26 In this way, we see that by adding a curled-up fifth dimension, Klein, building on Kaluza’s initial work, succeeded in unifying electromagnetism and gravity. Apparently, the scalar in the action was considered a bit of an embarrassment in the 1920’s, but in recent years Kaluza-Klein theory has experienced a revival as the expanding notions of string theory have created a need for defining actions in higher dimensions. It is interesting to note that not only did Kaluza and Klein succeed in unifying electromagnetism and gravity, but matter and geometry as well, as the photon appeared in four dimensions as a manifestation of empty five-dimensional space-time. 27 26 S. Weinberg, 1972. 27 J. M. Overduin and P.S. Wesson, 1997. 14 The Fifth Dimension Revisited In recent years, the concept of five-dimensional gravity has been revisited by a consortium of researchers led by Paul Wesson at the University of Waterloo in Ontario, Canada. The consortium also includes members of Stanford University’s Gravity Probe-B program. 28 The major difference is that in the consortium’s research, the fifth dimension is not compactified. The result of this is that new terms enter into the physics, even at low energies. In standard four dimensional space-time, these terms appear as matter and energy. By moving them to the right-hand side of the four dimensional equations they provide an induced energy-momentum tensor. According to Wesson, they have shown that, in fact, no five dimensional energy-momentum tensor is required. The results can include new forms of matter ultimately uniting gravity with its source, as well as with other fields. An interesting point that Overduin and Wesson have shown is that should φ be constant and the electromagnetic potential be set to zero, 0 =µA , the result is a Brans-Dicke-type scalar field theory. The resulting metric can be written as:    =200ˆ φαβggAB (13). Combining this with the field equations and Kaluza’s assumptions, the action integral becomes: ∫− −= φπRg xdGS4 161 (14). Compare this with equation 12. Neglecting the constant in front of the integral (as Derix and van der Schaar have done), and making the assumptions we have made with regard to the potential, A, and φ, we see that equation 14 is a direct result of equation 12 (we show this to merely bridge the methods of Derix & van der Schaar and Overduin & Wesson – and we should also note that µνF ∝ µA which allows us to drop the last term in equation 12)29. Overduin and Wesson show, through a Kaluza-Klein ansatz metric, that for the metric to satisfy Einstein’s equations in 4+ d dimensions, the Killing vectors must be independent of the extra coordinate, which means that the compact manifold is flat. Ultimately, they show that µνgˆ must also be flat.30 So we flip-flop from “Kleinian” assumptions to Euclidean. Conventional compactification models require either that the extra dimensions be under a state of constant curvature or must include other modifications 28 See http://astro.uwaterloo.ca/~wesson for more information on the consortium. 29 C.W. Misner, K.S. Thorne, and J.A. Wheeler, 1973. 30 J.M. Overduin and P.S. Wesson, 1997. 15 such as torsion or higher-derivative terms. This is where Overduin and Wesson begin laying the groundwork for a non-compactified five-dimensional theory. The groundwork for their development is the dependence of physical quantities on the fifth coordinate. This is something we have not seen as yet in our development of the fifth dimension. This dependence is precisely what produces the electromagnetic radiation as well as a general form of matter from geometry via the higher-dimensional field equations. 31 One of the interesting outcomes of this research is that it ultimately becomes a more easily testable theory. The compactified dimensions of Klein are not necessarily lengthlike in nature, but the new non-compactified dimensions, with the cylindrical condition removed, can be represented as lengthlike. Another point of interest is that previous authors have maintained Klein’s mechanism of harmonic expansion which means the compact manifold must have finite volume. With non-compactified dimensions, no such requirement exists. Overduin and Wesson thus write the metric as:    = 200)ˆ(εφαβggAB (15). The ε term is introduced to allow for a timelike as well as a spacelike signature for the fifth dimension requiring only that 12=ε . Please note that there is a difference between timelike and temporal here. Having a timelike signature does not mean the dimension is necessarily temporal (and thus non-causal). Time has rarely been considered in compactified dimensions due to a variety of problems that arise from its inclusion. However, in non-compactified theories, some of these problems vanish. The components of the Ricci tensor can then be represented as: ()     ∂∂−∂∂+∂−∂∂+∂∇−=2 2ˆ4 4 4 4 44 4 2αβ γδγδ βδ αγγδ αβαβ αβ αβ αβφφ φε φφ g g gg g g ggR R ()() 2 2 2 4ˆ4 4 4 4 44 44 444 4βγβγ α γαββγ γαβγ β αβ γ αβγβγ αg g g g g gg g g gggR∂∂−∂∂+∂∂+∂∂−∂∂ = () 4 4 24 4 4 βγαβγ δεβγαδεβγ βγαβγg g g g gg g g ∂∂+∂∂+∂∂− εφ−=44ˆR() 4 2 2 24 4 4 4 4 4 4 4 αδ γβγδαβ αβαβ αβαβ αβαβ φφφg g gg g g g g g g ∂∂−∂∂+∂∂−∂∂− (16) 31 Ibid. 16 Assuming that no higher-dimensional matter exists (that’s a tricky line we’re going to avoid crossing), the four-dimensional Ricci tensor becomes: ()()    ∂∂−∂∂+∂∂−∂∂−∂∇=2 24 4 4 4 4 44 4 2αβ γδγδ βδ αγγδ αβαβ αβ αβφφ φε φφ g g gg g g ggR (17) Overduin and Wesson then write the second of equations 16 in the form of a conservation law: 0=∇ β αβP (18) where they have defined a new four-tensor as: ()γεγεβ α γαβγ β α δ g g g g gP4 4 44ˆ21∂−∂ ≡ (19) Finally, the third of equations 16 takes the form of a scalar wave equation for φ : εφ() φφφαβαβ αβαβ αβαβ 2 2 44 4 4 4 4 4 g g g g g g ∂∂+∂∂−∂∂−= (20) Equations 17 through 20 form the basis for the non-compactified five dimensional Kaluza-Klein theory developed by Overduin and Wesson. 32 The physical meaning of the components of these equations as well as their application to cosmology and astrophysics are discussed in depth in their paper. Their results can be compared to those derived by Derix and van der Schaar. The non-compactified equations are more complicated, and this has been an overriding motivation for compactification in the past, but the physical significance of the non-compactified equations is interesting to note (again, see Overduin and Wesson 33 as well as other reports from the consortium34). Classical String Theory In order to more fully understand the next dimensional jump, it is necessary to digress for a moment into explaining some of the underlying methods of classical string theory, which primarily deals with, at least here, bosonic strings. To fully understand string theory, it is necessary to understand quantum field theory as modern string theory is simply a theory of quantum gravity. In addition, the mathematics of string theory can get phenomenally complex. In the interest of brevity, a few choice topics and equations will be presented in an attempt to give the flavor of string theory as a basis for moving into higher dimensional analysis. 32 Ibid. 33 Ibid. 34 http://astro.uwaterloo.ca/~wesson. 17 String theory is a relatively recent phenomenon. The first hint at its existence came in 1968 when Gabriele Veneziano at CERN in Geneva needed a solution to a vexing problem he was working on at the time. Surprisingly, he found he was able to use a little-used purely mathematical tool developed by Leonard Euler nearly 200 years before called the Euler beta-function. The solution however lacked some sense of physical meaning or justification – i.e. no one was certain as to why it worked. However, two years later, the concept of strings was developed in the works of Yoichiro Nambu of the University of Chicago, Holger Nielsen of the Niels Bohr Institute, and Leonard Susskind of Stanford University and the physical meaning to Veneziano’s problem was introduced. Nambu, Neilsen, and Susskind postulated that zero-dimensional point-particles were actually one-dimensional vibrating strings (thus instantaneously adding an additional dimension to the mix, at least in theory). They showed that the nuclear interactions of particles modeled as these strings were exactly described using the Euler beta-function. 35 The resonances of the vibrating strings determines the masses of the point particles we observe in nature. Unfortunately, string theory sat dormant for over a decade due to inconsistencies and problems in some of the predictions. In 1984, John Schwarz of Cal Tech and Michael Green of Queen Mary College launched what is now known as “the first superstring revolution.” During this three year period, from 1984 through 1986, more than one-thousand research papers were published on the subject. 36 Even in its initial form, superstring theory was able to unite the four forces in nature as well as matter. As a note, the “super” in superstring comes from the incorporation of supersymmetry into the theory (which has profound implications on the length scale of the actual strings as we will see in coming sections). For now, let us delve into a bit of the basics of classical string theory. As we stated earlier, the concept of a string in its most basic form is that of a zero-dimensional point-particle magnified to such an extent that it is actually a one-dimensional vibrating string. Initially we will consider this string to be a closed loop (there are other string theories that will be discussed shortly that include non-closed loops). Just as a point particle draws out a worldline as it travels in space-time, a string sweeps out a world-sheet – one-dimension higher than a worldline. In this way, construction of a space-time diagram becomes more complicated. As such, h= c = 1 are not natural units for strings (mass has the unit of inverse length). Additional introduced quantities include a new coupling constant in the form of a string tension, T, which has the units of ()2−length when h= c = 1 which then introduces a characteristic length squared, 2L. Conversion to ordinary units defines this length as T c L π/h= . Being ultimately a theory of quantum gravity, this length must be on the order of the Planck length, 3/cG Lp h= , which is the only length that can be constructed from G, h, and c. 35 B. Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory , Norton, 1999. 36 Ibid. 18 Brian Hatfield has a detailed discussion of the units and scales of strings in Chapter 21 of his book.37 Strings also happen to be Wilson loops which are closed strings (of glu) of infinitesimal width. This model for strings goes back to its original development in the 1970’s when it was used to describe properties in hadron physics. The strings were the gluons holding the bound quark states together to form hadrons. This led to work on developing QCD and Yang-Mills as string theories of Wilson loops. If all point-particles are indeed strings, there would presumably be a string interaction here between the quarks and the gluons, both actually being strings. The oddly fascinating part of string theory is that when strings interact they simply produce another string which is, geometrically, though not necessarily topologically, the same as the strings that formed it. In this way we have no way of telling if a particular string was formed as a result of interactions simply by looking at it. The benefit of this is that standard perturbation theory is trivial since it really only would describe the topology of the new string. In addition, we are unable to detect exactly where a sting interaction has occurred as it will look different in two Lorentz frames. 38 For given string theories there exists a maximum allowable space-time dimension beyond which the theory ultimately breaks down. The critical dimension is determined by the number of local supersymmetries on the string’s world-sheet. If absolutely no supersymmetry is present, the critical dimension is D = 26. This is the maximum number of existing dimensions proposed by any string theory. It was in fashion as a possible solution for several years but, as we will see, Witten’s “second superstring revolution” in 1995 may have doused that option. For 1 supersymmetry, the critical dimension is D = 10. This was the most widely accepted theory until Witten’s revolution in 1995 and still forms the basis of D = 11 theory. It is also the theory that we will be focusing on it our brief glimpse at classical string theory. The action for a relativistic particle of rest mass m is given as: ∫−=µ µτ xxdm S && (21) or, in a form without the nasty square root: ()∫+ −=2)()(1 21m xx d S τλτλτµµ&& (22). In this case τ is not necessarily the proper time but is instead a parametrization of the particle’s world-line. Equation 22 is the Lagrange multiplier version of equation 21. Hatfield presents a more detailed description of the derivation of equation 22 from 37 B. Hatfield, Quantum Field Theory of Point Particles and Strings , Addison-Wesley, 1992. 38 Ibid. 19 equation 21.39 In comparison, the string action that corresponds to equation 21 (the point-particle action) is called the Nambu-Goto action. It is defined as: () ( ) () ∫′−′⋅ −=2 2 2 νµ µµτσ x x xx ddT S & & (23) where the dotted coordinates are the customary τ derivatives and the primed coordinates are the σ derivatives. Similarly, the equivalent string description for equation 22 is the Polykov action defined as: ∫∂∂− −=µµτσ x x gg ddTSb aab 2 (24). Compare the results of the action integrals of these topological defects to equations 12 and 14, the action integrals for the five-dimensional systems of Kaluza-Klein theory. (Keep in mind the actions here are describing paths of particles and strings while the Kaluza-Klein actions describe a field). A cursory comparison shows that one of the major introductions to the string action is the string tension, T. However, we should note that by once again expanding from a zero-dimensional point-particle up one dimension to a one-dimensional string we get an equation (24) that is suspiciously similar to equation 14, the Overduin and Wesson action integral. Supergravity and Superstrings: 10 or 11 dimensions? In expanding beyond five dimensions, we actually explode into more than double that. This is based on the critical dimension we mentioned in the previous section. String theory got “stuck in the mud” for many years, in particular in the early 1990’s, within a maze of infinities and other odd problems including the lack of a consensus on the value of the critical dimension. In 1995, however, Edward Witten of the Institute for Advanced Study in Princeton, New Jersey, presented a seminal lecture at the Strings 1995 conference at USC that launched the “second superstring revolution.” The meat of his lecture showed that a minimum of 11 dimensions was required for a Kaluza-Klein theory to unify all of the forces in the standard model of particle physics (namely it contained the gauge groups of the strong and electroweak interactions). Prior to this, 11 was precisely the same number of dimensions determined by Nahm to be a maximum for consistency with, none other than, the graviton (with a maximum of spin 2)! 40 So, it seems, the unification of the four forces of nature necessarily required 11 dimensions! In fact, there were even more conditions that were discovered to apply that fixed the critical dimension at 11. In addition, the four dimensions of the visible world split out perfectly from the total 11 leaving 7 compactified or non-physical dimensions in its wake. The supergravity concept was developed to add the extra matter fields to the equations. The easiest way to do this was to make the theory supersymmetric which means every 39 Ibid. 40 J.M. Overduin and P.S. Wesson, 1997. 20 boson has some, as yet undiscovered, fermionic superpartner. This very development poses an enormous barrier to the experimental verification of string theory, however. Obtaining the energy levels necessary to produce these supermassive superpartners in a laboratory is well beyond our current reach. This isn’t the only problem with the D = 11 supergravity theory. One major problem that turned up is that the compact manifolds did not produce quarks and leptons. Several other issues involving chirality and a rather large cosmological constant also arise. The real breakthrough came with the development of two separate ten-dimensional supergravity models that were able to solve the anomaly problems while also maintaining the uniqueness that the eleven dimensional theory held (the minimum/maximum critical dimension issues). These two theories were based on the groups SO(32) and 8 8E E×. The extra terms that needed to be added corresponded to those that appeared naturally in low-energy superstring theory. The first sign of trouble with these two theories is that they predicted five separate string theories between them. However, Witten has proposed an entirely new theory called M-theory (M for membrane) that unites the five complete string theories along with supergravity under one umbrella: Type I (the bosonic string theory we looked at in the previous section), Type IIA, Type IIB, Heterotic-O ( 8 8O O×), Heterotic-E (8 8E E×), and D = 11 supergravity. Details of this unification depend on the introduction of a new concept into the fray: that of duality. Duality and M-Theory Duality was really the essence of Witten’s lecture at Strings 1995. The idea behind duality is that a singular physical system can be described by two seemingly separate theories. More to the point, it’s like looking at a house from the front and then from the back. Initially there might be no indication that you’re actually looking at one-in-the-same house when, in fact, further research eventually proves it is indeed one house. One fantastically interesting application of duality in physics is that when shrinking down to the scale of the Planck length while looking at a circular dimension of radius R, once we pass through the Planck length we find that the physics described by the system with radius 1/ R is precisely the same as that described by the radius R. Therefore, essentially, the universe at sub-Planck scales on the order of say something as absurd as 8010− m is exactly the same as the universe at 1/8010− m (which is huge). So we see that the Planck length mirrors us back outward if we try to continue to probe to smaller lengths, all thanks to the introduction of duality. This means that there is an exact lower limit to the size of compactified dimensions – the Planck length (radial in this example). Another interesting artifact of duality is the fact that the exact shape of the compactified dimensions (taking the form of a Calabi-Yau space as we will see) is not necessarily important. Two completely different shapes can produce the exact same physics. Witten used this idea to develop M-Theory, proposing that the different superstring theories as well as D = 11 supergravity were all portions of the same theory that appeared different simply on the surface but, thanks to duality, described exactly the same physics. To couple the various theories to each other and to M-Theory as a whole, dualities have been 21 employed to show that Type I and Heterotic-O are coupled, while Heterotic-O is also coupled to Heterotic-E which is coupled to M-Theory’s core, which is coupled to Type-IIA which is coupled to Type-IIB which is finally coupled to itself. More work is being performed in this area in an effort to unite supergravity and also to more clearly develop the exact form of M-Theory. Of particular interest to us in regard to this paper is what all this has to say about the extra dimensions it offers us. Traversable Dimensions and F-Theory In superstring theory the extra 7 dimensions are compactified into a complex set of shapes that is dictated by the equations of the theory. It turns out the geometrical shapes dictated by string theory satisfied a previously known set of geometrical spaces known as Calabi-Yau spaces (after Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University). The mathematics of Calabi-Yau shapes is quite complex and a visual representation of 7 spatial dimensions on a sheet of paper is quite complicated (though, see Greene 41 page 207 for a reasonable approximation) so we will not delve deeper into them here. As we stated in the previous section, duality allows for a veritable zoo of Calabi-Yau shapes that ultimately describe the exact same physics. The physics described by Calabi-Yau spaces in string theory is actually indirectly experimentally testable. As we stated earlier, the resonances of the vibrating strings determines the masses of the elementary particles in physics. The strings are free to vibrate in virtually any direction in the spatially extended dimensions and can also vibrate within the compactified dimensions. However, when vibrating in the compactified dimensions, the precise nature of the Calabi-Yau space describing the higher dimensions constrains the motion of the vibrating string. So in understanding the precise Calabi-Yau spatial geometry of a particular manifold, additional constraints can be placed on the strings making it theoretically easier to determine the precise physical nature of the string – e.g. the mass and charge of the particle it describes. Physicists consider this to be one of the most far-reaching and profoundly insightful facts of string theory. Additional work by Andrew Strominger and others allowed for the slight modification of this theory to solve the problem of collapsing dimensions. In this theory, a one-dimensional string is called a one-brane and can completely surround a one-dimensional piece of space. If this one-dimensional string is blown up like an inner-tube or a tire it becomes two-dimensional and is called a two-brane. A two-brane can completely surround a two-dimensional piece of space. One can easily see where this is heading. The idea is that by surrounding the extra spatial dimensions with a brane (a multi-dimensional string) the cataclysmic effects of collapse can be blocked. 42 Based on these concepts, the compactified dimensions are traversable by strings, but not by anything larger. So technically to us the extra dimensions are not traversable. Objects on the order of a point-particle (as we see them) and larger can only traverse the four non-compactified dimensions. The nature of these dimensions is not as well known as we think. The precise nature of the Euclidean spatial dimensions appears locally to be 41 B. Greene, 1999. 42 Ibid. 22 flat, though general relativity has shown that the manifold of these three dimensions along with time can be bent and warped in the presence of gravity. On a larger scale the precise shape of the universe has a profound effect on the ultimate shape of the Euclidean dimensions. Technically speaking the most interesting case would be a closed universe in which the Euclidean dimensions eventually bent back on themselves (legitimate physicists have recently suggested this could be possible 43). Unfortunately for those who find this notion romantic, a recent paper by P. De Bernardis, et. al. in the journal Nature based on balloon research in Antarctica has shown that the universe is indeed flat (Euclidean - at least for now). The fifth dimension as described by the original Kaluza-Klein theory would not be traversable except possibly by vibrating strings. However, the new non-compactified fifth dimension as proposed by Wesson’s consortium might allow for a fifth fully traversable dimension. Whether this dimension is truly Euclidean as well based on the recent observations of De Bernardis, et. al. would need to be probed. Standard traversable Euclidean dimensions have two degrees of physical freedom. To our knowledge, temporal dimensions do not. Based on standard causal-based physics, the only time dimension that we are aware of has a single degree of freedom – forward. Science fiction writers and some physicists have speculated that time travel is possible. An extensive base of scientific research has been performed on wormholes, some of which suggests the possibility of time travel, though the research has taken this to be a useful mathematical tool rather than speculating on its actual physical existence (see Visser for a detailed look at current wormhole research and the mathematical foundations for theoretical time travel 44). M.J. Duff of the University of Michigan began his compilation on string theory with the following quote from Mother Goose: “Nature requires five, Custom allows seven, Idleness takes nine, And wickedness eleven.” The appropriateness of this quote became apparent when Cumrun Vafa of Harvard in February of 1996 first suggested F-Theory (building on work by a number of others – for a very interesting and detailed overview of F-Theory, David R. Morrison at Duke, a leading string theorist and F-Theorist, has archived six lectures on RealVideo on his website 45). In this theory, the 11 dimensions of M-Theory are extended to 12 in order to solve a few select problems inherent in M-Theory. The interesting thing is that this additional dimension is temporal . Immediately, the mere philosophical implications are staggering if there is physical fact lying behind the mathematics. But, for now, let’s remain with the idea that there is a single degree of freedom in all possible temporal dimensions combined and simply say that F-Theory is a convenient mathematical tool. Creation and Conclusion One final point in discussing this dizzying array of dimensions is to briefly mention how they formed. At some point a few fractions of a second after the Big Bang, spontaneous symmetry breaking occurred causing the non-compactified dimensions to expand while 43 Ibid. 44 M. Visser, 1996. 45 See http:// www.cgtp.duke.edu/~drm/ftheory/ to access the RealVideo lectures. 23 the compactified ones curled up into a Calabi-Yau “ball.” This was a result of the presence of a tremendous amount of tension that, when released during the symmetry breaking, caused the dimensions to “snap into position.” Whether these dimensions will be united at some time in the tremendously distant future is of course unknown. But what we have learned over the last two millennia (a short time, in perspective) is tremendous. We have slowly developed, dimension by dimension, a world of multiple dimensions, some seen, some unseen. The implications of the physics and topology of these dimensions are far reaching and years of research are still ahead of us. Ultimately, the reward should be well worth the hunt but “only time will tell…” 24 Bibliography Print Resources - Technical Baulieu, Laurent, Di Francesco, Philippe, Douglas, Michael, Kazakov, Vladimir, Picco, Marco, and Windey, Paul (eds.), Strings, Branes, and Dualities , Kluwer Academic Publishers, Dordrecht, the Netherlands, 1997. Davies, A.T., and Sutherland, D.G. (eds.), Superstrings and Supergravity , Scottish Universities Summer School in Physics, Edinburgh, Scotland, United Kingdom, 1986. Derix, M. and van der Schaar, J.P., Stringy Black Holes , Master’s Thesis, University of Groningen, 1998. Duff, M.J. (ed.), The World in Eleven Dimensions , Institute of Physics Publishing, Bristol, United Kingdom, 1999. Hatfield, Brian, Quantum Field Theory of Point Particles and Strings , Addison-Wesley, Reading, Massachusetts, 1992. Ludvigsen, Malcolm, General Relativity: A Geometric Approach , Cambridge University Press, Cambridge, United Kingdom, 1999. Misner, Charles W., Thorne, Kip S., and Wheeler, John Archibald, Gravitation , W.H. Freeman and Company, New York, New York, 1973. Overduin, J.M. and Wesson, P.S., Kaluza-Klein Gravity, Phys. Rep. 283, 303, 1997 Peacock, John A., Cosmological Physics , Cambridge University Press, Cambridge, United Kingdom, 1999. Peebles, P.J.E., Principles of Physical Cosmology , Princeton University Press, Princeton, New Jersey, 1993. Peskin, Michael E., and Schroeder, Daniel V., An Introduction to Quantum Field Theory , Perseus Books, Reading, Massachusetts, 1995. Robinson, John Mansley, An Introduction to Early Greek Philosophy , Houghton Mifflin Company, Boston, Massachusetts, 1968. Schutz, Bernard F., A First Course in General Relativity , Cambridge University Press, Cambridge, United Kingdom, 1990. Vilenkin, A., and Shellard, E.P.S., Cosmic Strings and Other Topological Defects , Cambridge University Press, Cambridge, United Kingdom, 1994. 25 Visser, Matt, Lorentzian Wormholes: From Einstein to Hawking , AIP Press/Springer- Verlag, New York, New York, 1996. Weinberg, Steven, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity , John Wiley & Sons, New York, New York, 1972. Print Resources – General Greene, Brian, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory , W.W. Norton & Company, New York, New York, 1999. Kaku, Michio, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10 th Dimension , Anchor Books/Doubleday, New York, New York, 1994. Internet Resources Joyce, D.E. (ed.), Euclid’s Elements , http://aleph0.clarku.edu/~djoyce/java/elements/elements.html , 1998. O’Connor, J.J. and Robertson, E.F., The MacTutor History of Mathematics Archive , http://www-history.mcs.st-andrews.ac.uk/history/ , 2000. Wesson, P.S., 5-D Space-Time-Matter Consortium, http://astro.uwaterloo.ca/~wesson , 1998. Miscellaneous Resources Stachel, John, Einstein: A Man for the Millennium? , lecture given to the 2000 Spring meeting of the New England Section of the American Physical Society and the American Association of Physics Teachers. Cover painting of Oskar Klein by unknown artist appearing on website for the Oskar Klein Memorial Lectures, Royal Swedish Academy of Sciences, http://vanosf.physto.se/klein/lectures.html.
arXiv:physics/0011043v1 [physics.chem-ph] 18 Nov 2000New access to very weak interactions in molecules P.L. Chapovsky∗ Institute of Automation and Electrometry, Russian Academy of Sciences, 630090 Novosibirsk, Russia (February 2, 2008) Abstract It is predicted that nuclear spin conversion in molecules ca n be eﬃciently con- trolled by strong laser radiation resonant to rovibrationa l molecular transition. The phenomenon can be used for substantial enrichment of spi n isomers, or for detection of very weak (10 −100 Hz) interactions in molecules. Typeset using REVT EX ∗E-mail: chapovsky@iae.nsk.su 1I. INTRODUCTION Many symmetrical molecules exist in Nature only in the form o f nuclear spin isomers [1]. These isomers diﬀer by symmetry of nuclear spin wave fun ction and, consequently, by symmetry of molecular spatial wave function. Relaxation be tween diﬀerent spin states (spin conversion) is extremely slow process if it is compared with other gas kinetic rates, e.g., vibrational relaxation. It makes nuclear spin isomers uniq ue objects with many potential applications. It was shown in a recent paper [2] that external radiation can inﬂuence conversion of spin isomers both through level populations and through optical ly induced coherences. Purpose of the paper [2] was to investigate main features of the coheren t control of isomer enrichment and conversion. In order to achieve the goal, the process was considered in a simplest arrangement in which microwave radiation excited molecula r rotational transition. In the present paper we will consider the process in more comp licated arrangement which promises signiﬁcantly better control of spin conversion. O ne outcome of this high eﬃciency is that the coherent control in new arrangement can be used fo r detection of very weak (10−100 Hz) interactions in molecules. II. EQUATION OF CHANGE First, we give qualitative picture of the process. Let us ass ume that a test molecule has two nuclear spin states, ortho and para, and that there is a laser radiation resonant to the rovibrational transition, m−n, in the ortho subspace, Fig. 1. The low state nis not mixed with para states, but the upper state mis mixed by the intramolecular perturbation ˆVwith the para state k. In addition, there is the ortho-para level pair, m′−k′, in the ground vibrational state mixed by another intramolecular perturbation ˆV′. We assume that collisions of the test molecules cannot alter their spin sta te. This arrangement corresponds to the general formulation of quantum relaxation in which one has collisionally isolated subspaces of states mixed together by internal perturbatio n [3]. Suppose that the molecule is placed initially in the ortho subspace of the ground vibra tional state. Due to collisions the test molecule will undergo fast rotational relaxation inside the ortho subspace. This will proceed until molecule jumps to the state m′, which is directly mixed with the para 2state k′, or to the state nwhich is mixed by combined action of the external ﬁeld and intramolecular perturbation ˆVwith the para state k. Admixture of a para state implies that the next collision can move the molecule to another para states and thus localize it inside the para subspace. It is clear that resonant radiatio n can signiﬁcantly modify the spin conversion. One knows from the literature on resonant inter action of strong laser radiation with matter (see, e.g., [4,5]) that radiation can change pop ulation of states, split states and create coherences in the system. All these can aﬀect the mixi ng of ortho and para states and thus the spin conversion process. Quantitative description of the problem can be performed wi th the help of kinetic equa- tion for density matrix, ˆ ρ. The molecular Hamiltonian consists of four terms, ˆH=ˆH0+ ¯hˆG+ ¯hˆV+ ¯hˆV′. (1) The main part, ˆH0, has the eigen ortho and para states shown in Fig. 1. ¯ hˆGdescribes the molecular interactions with the external radiation, ˆG=−(E0ˆd/¯h) cosωLt, (2) where E0andωLare the amplitude and frequency of the electromagnetic wave ;ˆdis the operator of the molecular electric dipole moment. We have ne glected molecular motion in the operator ˆGin order to simplify the theory. In the representation of the eigen states of ˆH0kinetic equation reads, dˆρ/dt= (dˆρ/dt)coll−i[ˆG+ˆV+ˆV′,ˆρ], (3) where ( dˆρ/dt)collis the collision integral. Further, collisions in our syste m will be described by the model standard in the theory of molecular interaction with laser radiation. The oﬀ-diagonal elements of ( dˆρ/dt)collwill be assumed to have only decay terms, (dρj,j′/dt)coll=−Γρjj′;j/negationslash=j′. (4) Here, jandj′indicate rovibrational states of the molecule. These state s are assumed to have no degeneracy. The decoherence rates were taken equal f or all oﬀ-diagonal elements of collision integral. Diagonal terms of the collision integr al will be described in the framework of the strong collision model. 3Our goal is to determine time dependence of the total concent ration of molecules in one spin state. For example, for the total concentration of orth o molecules, ρo, one can get from Eq. (3) the following equation of change, dρo/dt= 2Re i(ρmkVkm+ρm′k′V′ k′m′). (5) In fact, this result is valid for any model of collision integ ral, as long as collisions do not alter the molecular spin state which implies,/summationtext j(dρjj/dt)coll= 0, if j∈ortho, or j∈para. One has to make a few simpliﬁcations in order to ﬁnd the oﬀ-dia gonal density matrix elements, ρmkandρm′k′. We assume ˆVandˆV′being small and consider zero and ﬁrst order terms of the density matrix, ˆρ= ˆρ(0)+ ˆρ(1). (6) We start with zero order perturbation theory. ρ(0)is determined by the kinetic equation, dˆρ(0)/dt= (dˆρ(0)/dt)coll−i[ˆG,ˆρ(0)]. (7) In zero order perturbation theory, para molecules are at equ ilibrium, ρ(0) p(g, j) = (n−ρ(0) o)w(j);ρ(0) p(e, j) = 0, (8) where nis the total concentration of the test molecules; w(j) is the Boltzmann distribution over rotational states. We will assume the same function w(j) for each of four vibrational states. We have neglected in Eq. (8) vibrational resonance e xchange between ortho and para molecules which would populate the upper vibrational s tate of para molecules. It was taken into account also that ortho-para exchange is on many o rders of magnitude slower than vibrational and rotational relaxations. Equations for stationary level populations of ortho molecu les are obtained from Eqs. (4) and (7). Thus one has, (νV+νR)ρ(0) o(e, j) =νRw(j)ρ(0) o(e) +ρ(0) op δjm; νRρ(0) o(g, j) =νRw(j)ρ(0) o(g) +νVw(j)ρ(0) o(e)−ρ(0) op δjn; ρ(0) op=2ΓG2 Γ2+ Ω2/bracketleftBig ρ(0) o(g, n)−ρ(0) o(e, m)/bracketrightBig , (9) where ρ(0) o(e) and ρ(0) o(g) are the concentrations of ortho molecules in excited and gr ound vibrational states; νVandνRare the rates of vibrational and rotational relaxations; pis 4the probability of optical excitation of ortho molecules. I ntroduction of diﬀerent relaxation rates for diﬀerent degrees of freedom makes the model of stro ng collisions more accurate. Eqs. (9) correspond to rotational wave approximation. Matr ix element of ˆGis given by Gmn=−Ge−iΩt;G≡E0dmn/2¯h, (10) where Ω = ωL−ωmnis the radiation frequency detuning from the absorption lin e center, ωmn; the line over symbol indicate a time-independent factor. R abi frequency, G, is assumed to be real. Solution of Eqs. (9) has no diﬃculty. Concentration of ortho molecules in excited vibra- tional state, ρ(0)(e), and in the state mwhich one needs for further calculations read, ρ(0) o(e) =ρ(0) op νV;ρ(0) o(e, m) =ρ(0) op νm;p=2ΓG2w(n) Γ2 B+ Ω2; ν−1 m=w(m)ν−1 V+ (1−w(m))(νV+νR)−1. (11) Here, Γ Bis the homogeneous linewidth of the absorption spectrum pro ﬁle, Γ2 B= Γ2+ 2ΓτG2;τ=ν−1 m+ν−1 n;ν−1 n=w(n)ν−1 V+ (1−w(n))ν−1 R.νmandνnare the eﬀective population decay rates of the corresponding states. In a sim ilar way, one can calculate from Eq. (7) the oﬀ-diagonal density matrix element, which ampli tude is equal to, ρ(0) o(m\|n) =iGρ(0) ow(n)Γ +iΩ Γ2 B+ Ω2. (12) In zero order perturbation theory, one neglects perturbati onsˆVandˆV′. It implies that there are no coherences between ortho and para states, ρ(0) mk= 0;ρ(0) m′k′= 0. Consequently, one has, dρo/dt= 2Re i(ρ(1) mkVkm+ρ(1) m′k′V′ k′m′), (13) instead of Eq. (5). Note, that the spin conversion appears in the second order approximation. The ﬁrst order correction to the density matrix, ρ(1), is determined by the equation, dˆρ(1)/dt= (dˆρ(1)/dt)coll−i[ˆG,ˆρ(1)]−i[ˆV+ˆV′,ˆρ(0)]. (14) Forρ(1) m′k′one has from this equation, ρ(1) m′k′=−iV′ m′k′ Γ +iω′[ρ(0) p(g, k′)−ρ(0) o(g, m′)], (15) where ω′≡ωm′k′.ρ(1) mkcan be obtained from equations which are deduced from Eq. (14 ), 5dρ(1) mk/dt+ Γρ(1) mk+iGmnρ(1) nk=iVmkρ(0) o(m); dρ(1) nk/dt+ Γρ(1) nk+iGnmρ(1) mk=iVmkρ(0) o(n\|m). (16) Substitutions, Vmk=V eiωt,(ω≡ωmk);ρ(1) mk=ρ(1) mkeiωt;ρ(1) nk=ρ(1) nkei(ωL−ωknt), transform Eqs. (16) to algebraic equations from which one ﬁnds ρ(1) mk. Using ρ(1) mkandρ(1) m′k′from Eq. (15) one has an equation of change (13) in the form, dρo dt=2Γ\|V′\|2 Γ2+ω′2/bracketleftBig ρ(0) p(g, k′)−ρ(0) o(g, m′)/bracketrightBig − 2\|V\|2Re[Γ +i(Ω +ω)]ρ(0) o(m) +iGρ(0) o(n\|m) (Γ +iω)[Γ +i(Ω +ω)] +G2. (17) III. ENRICHMENT AND CONVERSION The denominator of the second term in the right-hand side of E q. (17) is convenient to present as, (Γ + iω1)(Γ + iω2), where ω1,2=ω+Ω 2±/radicalBigg/parenleftbiggΩ 2/parenrightbigg2 +G2. (18) New parameters, ω1andω2, can be interpreted as the gaps between the two components, \|m1>and\|m2>, of the ortho state \|m >, split by the optical ﬁeld, and the para state \|k >. The ortho state, \|m2>, crosses the para state, \|k >, at Ω = −ω(1−G2/ω2), see Fig. 2. The splitting of states by resonant laser radiation is well- known phenomenon in nonlinear spectroscopy [4,5]. Using Eq. (17) one can present equation of change in the ﬁnal f orm, dρo/dt=nγ′ op−ρoγ;γ≡γ′ op+γ′ po−γ′ n+γn+γcoh. (19) In writing this equation we have neglected in the right-hand side of Eq. (17) small diﬀerence between ρ(0) oand the total concentration of ortho molecules, ρo. In Eq. (19) the following partial conversion rates have been introduced. The ﬁeld ind ependent rates, γ′ op=2Γ\|V′\|2 Γ2+ω′2w(k′);γ′ po=2Γ\|V′\|2 Γ2+ω′2w(m′). (20) The rate γfree≡γ′ op+γ′ podetermines the equilibration rate in the system without an e xternal ﬁeld. The ﬁeld dependent term, 6γ′ n=γ′ pop/νV, (21) appears because of depletion of the ground vibrational stat e of ortho molecules by optical excitation, Index ninγ′ ncomes from “noncoherent”, i.e., induced by level populatio ns. Another term of similar “noncoherent” origin appears due to the level population, ρ(0) o(m), in Eq. (17), γn= 2\|V\|2p νmReΓ +i(Ω +ω) (Γ +iω1)(Γ + iω2). (22) And ﬁnally the “coherent” term, γcoh, originated from ρ(0) o(n\|m), in Eq. (17), γcoh= 2\|V\|2p 2ΓReΓ−iΩ (Γ +iω1)(Γ + iω2). (23) Solution to Eq. (19) can be presented as, ρo=ρo+ (ρo(0)−ρo) exp( −γt);ρo=nγ′ op/γ. (24) Hereγis the equilibration rate in the system in the presence of ext ernal ﬁeld; ρois the stationary concentration of ortho molecules. Without an ex ternal radiation (at the instant t= 0), the equilibrium concentration of para molecules is equ al to, ρp(0) = n−ρo(0) = nγ′ po/γfree, (25) if the Boltzmann factors are assumed to be equal, w(k′) =w(m′). This implies γ′ op=γ′ po (see Eq. (20)), the laser ﬁeld produces a stationary enrichm ent of para molecules, β≡ρp ρp(0)−1 = 1 −2γ′ op γ. (26) One can see from this equation that external ﬁeld changes con centration of para isomers if γ/negationslash=γfree. We assume in further analysis the following parameters, ω= 100 MHz, ω′= 130 MHz, V′ m′k′= 5 kHz, Γ = 2 ·108s−1/Torr and the Boltzmann factors of the states m′,k′,m, andkall equal 10−2. This set of parameters gives the ﬁeld free conversion rate, γfree= 10−2s−1/Torr, which coincides with the conversion rate in13CH3F. Nuclear spin conversion in these molecules is governed by quantum relaxation (see th e review [6]). The rotational and vibrational relaxation rates will be taken equal, νR= 0.1Γ and νV= 0.01Γ, respectively. First, we consider relatively low optical ﬁelds, thus small G. In this case one has two peaks in enrichment at frequencies Ω ≃ −ωand Ω = 0, see Fig. 3. The data shown in this 7ﬁgure correspond to Vmk= 3 kHz, and Γ = 2 MHz. The peak at Ω = 0 appears because the excitation probability, p, has maximum at this frequency. Amplitude of this peak is determined mainly by the rate γn. AsGgrows, the amplitude of the peak 2 reaches the value γn/γfree∼(V ω′/2V′ω)2which constitutes ≃15%. The peak at Ω = 0 in isomer enrichment was predicted in [7] by considering only the leve l population eﬀects. Peak at Ω ≃ −ωappears because the ortho state \|m2>crosses the para state \|k >at this frequency of the external ﬁeld (see Fig. 2). This peak is determined mainly by γcoh. When Gincreases its amplitude grows to much bigger values than the amplitude of the peak at Ω = 0. At resonant frequency Ω ≃ −ωthe rate γcohis enhanced by large factor ( ω/Γ)2. This explains much larger enrichment at Ω ≃ −ω. Note, that large enrichment occurs only if the excitation probability at this frequency, p(−ω), is not very low. The data shown in Fig. 4 correspond to strong optical ﬁeld, G= 50 MHz, and three values of Vmk. Γ was taken equal 2 MHz. One can see, that strong optical ﬁeld is able to convert almost all molecules to the para state if Vmk≃Vm′k′. Thus relatively weak (3 kHz) coupling in upper state is able to produce macroscopi c eﬀect, viz., almost complete enrichment of spin isomers. It is of fundamental importance , that even for much weaker coupling in upper state, enrichment is still signiﬁcant. Fo r example, if the perturbation in upper state, Vmk= 30 Hz, one has the enrichment, β≃1%. Enrichment at this level can easily be measured. It is important that the enrichment peak at Ω≃ −ωis narrow (the width ≃Γ) and thus can be distinguished from much wider structures ( the width ≃ΓB) induced by population eﬀects. Equilibration rate in the system is given by γ, see Eq. (24). It is convenient to characterize the conversion rate in relative units, γrel=γ/γfree−1, (27) Conversion rate, like enrichment, has two peaks in its frequ ency dependence at low G. If Rabi frequency, G, is large and the ortho-para couplings in upper and low state s have the same order of magnitude, conversion can be signiﬁcantly enh anced (Fig. 4, upper panel). Again, this enhancement appears because of the crossing of o rtho and para states in upper vibrational state by external ﬁeld. 8IV. DISCUSSION The phenomenon considered in the paper is based on the level s plitting produced by resonant electromagnetic radiation. Sometimes, this spli tting is called in optics the dynamic Stark eﬀect. The essence of the eﬀect can be understood as fol lows. Mixing of ortho and para states depends on magnitude of the perturbation ˆVbut also on the ortho-para level gap. Optical ﬁeld splits the molecular state and thus change the gaps between the ortho and para states. Conversion rate is signiﬁcantly enhanced when the ortho and para states cross. Similar enhancement occurs when ortho and para states are cr ossed by ordinary Stark eﬀect in an external DC electric ﬁeld [8]. In the same way, one can understand high sensitivity of the ph enomenon to weak ortho- para couplings in excited state. Equilibrium concentratio ns of ortho and para molecules are achieved when one has the ortho-to-para ﬂux in excited state equal to the back ﬂux in the ground vibrational state. Back ﬂux is slow because it is dete rmined by the non-degenerate ortho-para level pair. On the other hand, the ﬂux in excited s tate can be signiﬁcantly enhanced by proper choice of radiation parameters which all ows to cross the ortho and para states in upper vibrational state. High eﬃciency of the proposed enrichment method can be used t o detect weak pertur- bations in excited vibrational state. First of all, it can be the hyperﬁne perturbations of the same origin and similar magnitude as the perturbations i n the ground vibrational state. We have seen that hyperﬁne coupling of the order of ∼103Hz is able to convert almost all molecules in one spin state. One can also detect much weak er interactions in molecules. In this case one should select the ortho-para level pair in ex cited vibrational state which is not mixed by “ordinary” hyperﬁne interactions in order to av oid the weak interaction to be hidden by stronger, ordinary hyperﬁne interactions. An int eresting case is the crossings of states having opposite parity. Mixing of such ortho and para states can be performed only by spin-dependent, parity-odd interactions which are not o bserved in molecules yet. V. CONCLUSIONS We have performed analysis of the spin isomer enrichment and conversion governed by molecular rovibrational excitation. This analysis was don e using a few simpliﬁcations. We 9have neglected the Doppler broadening of the absorbing tran sition, degeneracy of molecular states, and resonant vibrational exchange between excited and unexcited molecules. These simpliﬁcations are not crucial for the existence of the phen omenon. More detailed analysis will be done elsewhere. We have shown that coherent control of nuclear spin conversi on in molecules can be eﬃciently performed by strong radiation resonant to rovibr ational molecular transition. A possible applications of this phenomenon is the enrichment of molecular spin isomers. An- other application is the detection of very weak (10 −100 Hz) interactions in molecules, which can be, e.g., parity-odd interactions. ACKNOWLEDGMENTS This work was made possible by ﬁnancial support from the Russ ian Foundation for Basic Research (RFBR), grant No. 98-03-33124a. 10REFERENCES [1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1981). [2] P. L. Chapovsky, http://arXiv.org/abs/physics/00110 12 . [3] P. L. Chapovsky, Physica A (Amsterdam) 233, 441 (1996). [4] S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear resonances in atom and molecular spectra (Nauka, Siberian Branch, Novosibirsk, Russia, 1979), p. 31 0. [5] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wi- ley, New-York, 1992). [6] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Chem .50, 315 (1999). [7] L. V. Il’ichov, L. J. F. Hermans, A. M. Shalagin, and P. L. C hapovsky, Chem. Phys. Lett.297, 439 (1998). [8] B. Nagels, N. Calas, D. A. Roozemond, L. J. F. Hermans, and P. L. Chapovsky, Phys. Rev. Lett. 77, 4732 (1996). 11ortho param' k'm k V'm'k'Vmk noptical excitation FIG. 1. Level scheme. Horizontal lines indicate the ortho-para mix ing in the ground and excited vibrational states. There are rotational relaxati on inside each vibrational state (rate νR) and vibrational relaxation from upper states (rate νV). 12-100 0100200 -1000100200300 \|k>\|m> \|m2>\|m1> Ortho-para gaps (MHz) Frequency, Ω (MHz) FIG. 2. Gaps between the ortho states \|m1>and\|m2>and para state \|k >. Rabi frequency was taken equal G= 50MHz and the radiation free ortho-para gap, ω= 130 MHz. 13-200 -100 0 100 2000510152025 peak 2peak 1 Enrichment, β (%) Frequency, Ω (MHz) G=1 G=2 FIG. 3. Frequency dependence of the enrichment of para molecules, β, atG= 1and 2 MHz. 14020406080100 V = 3 kHz /G01/G02/G03/G04/G05/G06/G07/G08/G02/G09/G0A/G0B β /G0B/G0C/G0D/G0E 020406080 V = 3 kHz Conversion rate, γrel 01020304050 V = 300 Hz Enrichment, β (%) 0.00.20.40.60.8 V = 300 Hz Conversion rate, γrel -200-100 0100200-2.6-2.4-2.2-2.0-1.8-1.6-1.4 V = 30 Hz Enrichment, β (%) Frequency, Ω (MHz)-200-100 0100200-2.4-2.2-2.0-1.8-1.6-1.4 V = 30 Hz Conversion rate, γrel(%) Frequency, Ω (MHz) FIG. 4. Enrichment of para molecules, β, and conversion rate, γrel, for three values of Vmk. 15
arXiv:physics/0011044v1 [physics.atom-ph] 18 Nov 2000version 1.0 Logarithmic two-loop corrections to the Lamb shift in hydro gen Krzysztof Pachucki∗ Institute of Theoretical Physics, Warsaw University, Ho˙ z a 69, 00-681 Warsaw, Poland Abstract Higher order ( α/π)2(Z α)6logarithmic corrections to the hydrogen Lamb shift are calculated. The results obtained show the two-loo p contribution has a very peculiar behavior, and signiﬁcantly alter the theore tical predictions for low lying S-states. PACS numbers 31.30 Jv, 12.20 Ds, 06.20 Jr, 32.10 Fn Typeset using REVT EX ∗E-mail address: krp@fuw.edu.pl 1The calculation of the two-loop contribution to the Lamb shi ft in hydrogen is one of the most challenging projects in bound state QED [1,2]. Sinc e direct numerical calculations with the use of Dirac-Coulomb propagators have not yet been c ompleted, one has to rely on theZ αexpansion: ∆E=m/parenleftbiggα π/parenrightbigg2 (Z α)4/braceleftbigg B40+ (Z α)B50 +(Z α)2/bracketleftbigg ln3(Z α)−2B63+ ln2(Z α)−2B62+ ln(Z α)−2B61+B60/bracketrightbigg +. . ./bracerightbigg . (1) The leading order correction B40can be obtained from the slope of the electron form-factor F1andF2atq2= 0. It is known analytically and its numerical value is quite small (for S-states includic vacuum polarization) B40= 0.538941 . (2) The calculation of the next order correction B50was completed only a few years ago inde- pendently by two groups in [3] and [4]. The value was surprisi ngly large B50=−21.5561(31) . (3) Moreover, this correction led to strong disagreement in He+Lamb shift with the most precise experimental value in [5], while for hydrogen Lamb s hift, led to agreement with the Mainz value for the proton charge radius [6]. This large v alue of B50compared to B40 indicates a very slow convergence or even might suggest an no nperturbative behavior of the two-loop contribution. Indeed, the direct numerical ca lculations of one diagram, the loop-by-loop electron self–energy by Mallampali and Sapir stein in [7], shows that the value of this correction at Z= 1 is of diﬀerent sign and magnitude, that the one based on ﬁrs t two terms of analytic expansion. Moreover, this numerical c alculation was in disagreement with the analytical value of B63in [8], while it was argued in [8], that this correction comes only from this one diagram in the covariant gauge. A yea r later another group [9] calculated numerically this one diagram and found agreemen t with the analytic expansion including ln3(Z α)−2term. While this may suggest that the ﬁrst numerical calcula tions were incorrect, a very recent, third numerical result by Yer okhin in [10] conﬁrmed the ﬁrst one [7]. So, this situation with the two-loop contribution i s very unclear. Moreover, the analytic value ln3(Z α)−2term corresponding to all diagrams, was conﬁrmed independe ntly by several groups, so this situation is even more confusing. It was argued, by Yerokhin in [10], that the ln3(Z α)−2term for this one loop-by-loop diagram is diﬀerent from the total value of B63, and in fact found an additional contribution. However, the value for this term, coming from all diagrams might be correct, becaus e other diagrams may contain compensating terms. The goal of this work is to shed some ligh t into higher order two-loop corrections and calculate all logarithmic terms: B63, B62, and B61. We ﬁnd that indeed the two-loop contribution has a very peculiar behavior, as the h igher order term B61dominates and reverses the sign for the overall logarithmic contribut ion. In next sections we present some details of this calculation. First, a simple example is worked out to demonstrate the method, then we pass to the most diﬃcult two-photon-loop dia grams and complete with remaining diagrams containing an electron loop. Conclusio ns with prospects of calculation ofB60summarize this work. 2I. SIMPLE EXAMPLE The example to demonstrate the calculational method is the a symptotic expansion of P(ω)≡ /an}b∇acketle{tφ\|p1 E−(H+ω)p\|φ/an}b∇acket∇i}ht (4) =−1 ω+2 ω2−4√ 2 ω5 2+4−12 ln(2) + 4 ln( ω) ω3+. . . (5) around large ωfor ground state of the hydrogen atom. More precisely, we con centrate on theω−3term. For simplicity, we put here m= 1, α= 1. From one side P(ω) is known analytically [11] P(ω) =−384τ5 (1 +τ)8(2−τ)2F1(4,2−τ,3−τ, ζ), (6) where ζ=/parenleftbigg1−τ 1 +τ/parenrightbigg2 , τ=1/radicalBig 2 (ω+ 1/2), (7) so one could get this coeﬃcient from here. However, our ﬁnal g oal is to calculate the two- loop contribution, for which no analytic formula has been de rived so far. Therefore, we use a diﬀerent approach, based on the eﬀective Hamiltonian. Fir st, we regularize the Coulomb interaction by the following replacement: V(r) =−1 r→ −1 r(1−e−λ r). (8) With the regularized potential ( P→PR) one can expand PRin (H−E)/ωwhich leads to PR=−1 ω3/an}b∇acketle{tφ\|p(H−E)2p\|φ/an}b∇acket∇i}ht=−1 ω3/an}b∇acketle{tφ\|V′(r)2\|φ/an}b∇acket∇i}ht, (9) /an}b∇acketle{tφ\|V′(r)2\|φ/an}b∇acket∇i}ht= 2λ+ 8 ln(3) −8 ln(λ)−2, (10) where the last expectation value is taken from [12]. The rema ining part, which was left out by this replacement, is obtained from the subtracted forwar d scattering amplitude. Two photon exchange is P2=/integraldisplayd3p (2π)364π/bracketleftbiggp p4(−1) p2/2 +ωp p4−p p4λ2 p2+λ2(−1) p2/2 +ωλ2 p2+λ2p p4/bracketrightbigg =2λ ω3,(11) where we keep only the ω−3term. The three photon exchange requires more subtractions . One Coulomb exchange between photon vertices gives P3A P3A=/integraldisplayd3p (2π)3/integraldisplayd3p′ (2π)364π/bracketleftbiggp′ p′4(−1) p′2/2 +ω(−4π) q2(−1) p2/2 +ωp p4 −p′ p′4λ2 p′2+λ2(−1) p′2/2 +ω(−4π) q2λ2 λ2+q2(−1) p2/2 +ωλ2 p2+λ2p p4/bracketrightbigg =4 lnω−8 lnλ−8 ln 3 + 20 ln 2 ω3. (12) 3Coulomb exchanges out of photon vertices gives P3B P3B=−2048π2/integraldisplayd3p (2π)3/integraldisplayd3p′ (2π)3/parenleftbigg1 p′41 q21 p2+ 2ω1 p4 −1 p′4λ2 λ2+p′21 q2λ2 λ2+q21 p2+ 2ωλ2 λ2+p21 p4/parenrightbigg =2−32 ln(2) + 16 ln(3) ω3. (13) There is an implicit subtraction at p′= 0 for removal of small p′divergence. It corresponds to subtraction of lower order contributions. Additionally , only the ω−3term is selected. The sum P=PR+P2+P3A+P3B=4−12 ln(2) + 4 ln( ω) ω3(14) is independent of λin the limit of large λand agrees with that from the expansion of analytic formula in Eq. (5). The advantage of this method is the direct application to the two-loop Lamb shift. II. TWO-LOOP LAMB SHIFT The calculations of two-loop Lamb shift in the order of α2(Z α)6is more complicated due to the presence of powers of ln( Z α). It reﬂects the fact that several energy and momentum regions contribute. For these calculations we introduce a n umber of cutoﬀ parameters to separate diﬀerent regions and calculate them independentl y. In Fig. 1 the integration region of two photon energies ω1andω2is split with the help of ǫ1, ǫ2, ǫ′ 1, ǫ′ 2. Additionally λ’splits’ the integration over electron momenta. The splitt ing itself, does not help too much. The key trick is the assumption that after expansion in Z αone goes to the limits ǫ2→0, ǫ1→0, ǫ′ 2→0, ǫ′ 1→0, λ→ ∞, in the order as written. The two-loop contribution is split accordingly ∆E=EL+EM+EF+EH, (15) and calculated separately, each term in the most convenient gauge. In the following sec- tions we calculate all logs. The constant term B60is left unevaluated, however we lay the groundwork for its calculation. III. CONTRIBUTION EL Diagrams in the Coulomb gauge in NRQED are presented on Fig. 2 . We calculate them ﬁrst, for photon energies inside a rectangular box 0 < ω 1< ǫ1,0< ω 2< ǫ2, ǫ2<< ǫ 1, and after combine to the region ELas shown in Fig. 1. The expression derived from nonrelativistic QED for all these diagrams is: 4EL=/parenleftbigg2α 3π m2/parenrightbigg2/integraldisplayǫ1 0dω1ω1/integraldisplayǫ2 0dω2ω2 /braceleftbigg /an}b∇acketle{tφ\|pi 1 E−(H+ω1)pj 1 E−(H+ω1+ω2)pi 1 E−(H+ω2)pj\|φ/an}b∇acket∇i}ht +1 2/an}b∇acketle{tφ\|pi 1 E−(H+ω1)pj 1 E−(H+ω1+ω2)pj 1 E−(H+ω1)pi\|φ/an}b∇acket∇i}ht +1 2/an}b∇acketle{tφ\|pi 1 E−(H+ω2)pj 1 E−(H+ω1+ω2)pj 1 E−(H+ω2)pi\|φ/an}b∇acket∇i}ht +/an}b∇acketle{tφ\|pi 1 E−(H+ω1)pi1 (E−H)′pj 1 E−(H+ω2)pj\|φ/an}b∇acket∇i}ht −1 2/an}b∇acketle{tφ\|pi 1 E−(H+ω1)pi\|φ/an}b∇acket∇i}ht /an}b∇acketle{tφ\|pj 1 [E−(H+ω2)]2pj\|φ/an}b∇acket∇i}ht −1 2/an}b∇acketle{tφ\|pi 1 E−(H+ω2)pi\|φ/an}b∇acket∇i}ht /an}b∇acketle{tφ\|pj 1 [E−(H+ω1)]2pj\|φ/an}b∇acket∇i}ht +m/an}b∇acketle{tφ\|pi 1 E−(H+ω1)1 E−(H+ω1)pi\|φ/an}b∇acket∇i}ht −m ω1+ω2/an}b∇acketle{tφ\|pi 1 E−(H+ω2)pi\|φ/an}b∇acket∇i}ht −m ω1+ω2/an}b∇acketle{tφ\|pi 1 E−(H+ω1)pi\|φ/an}b∇acket∇i}ht/bracerightbigg . (16) It is a two-loop analog of Bethe logs. We have not found a way to calculate its matrix elements analytically in a compact form, therefore we proce ed in a diﬀerent way. One ﬁnds, thatELas in Eq. (16) depends on αonly through ǫ1andǫ2: EL=EL/parenleftBigǫ1 α2,ǫ2 α2/parenrightBig . (17) To ﬁnd the logarithmic dependence, we diﬀerentiate ELoverǫ1andǫ2which with the help ofǫ2<< ǫ 1leads to much simpler expression. The ﬁrst derivative leads to ǫ1∂EL ∂ǫ1=/parenleftbigg2α 3π m2/parenrightbigg2/integraldisplayǫ2 0dω2ω2δπ δ3(r)/an}b∇acketle{tφ\|pi 1 E−(H+ω2)pi\|φ/an}b∇acket∇i}ht, (18) where δπ δ3(r)denotes ﬁrst order corrections to φ, H, E due to π δ3(r) operator. This integral was considered and calculated in the context of hyperﬁne spl itting in hydrogen-like systems [13], since the Fermi spin-spin interaction is also proport ional to δ3(r). The result from that paper which is extended here to any value of principal quantu m number is: 2α 3π m2δπ δ3(r)/integraldisplayǫ 0dω ω/an}b∇acketle{tφ\|pi 1 E−(H+ω2)pi\|φ/an}b∇acket∇i}ht=α πα2F(n) n3, (19) F(n) =−2 3ln2¯ǫ+ ln ¯ǫ/bracketleftbigg 2 (1−2 ln(2)) +8 3/parenleftbigg3 4+1 4n2−1 n−ln(n) + Ψ( n) +C/parenrightbigg/bracketrightbigg +N(n), (20) where Nhas been calculated only for n= 1 N≡N(1) = 17 .8299093 , (21) 5and Ψ = Γ′/Γ with Euler Γ function and Euler Cconstant Ψ(1) = 0; Ψ( n) = 1 +1 2+1 3+. . .+1 n−1−C (22) We have introduced here a notation ¯ ǫ=ǫ/α2, which is to be used throughout this work. The result for n= 1 with E=m(α/π)2α6is: ǫ1∂EL ∂ǫ1=E2 3/bracketleftbigg −2 3ln2(¯ǫ2) + 2 (1 −2 ln 2) ln(¯ ǫ2) +N/bracketrightbigg . (23) The second derivative, over ǫ2, is little more diﬃcult to calculate: ǫ2∂EL ∂ǫ2=/parenleftbigg2α 3π/parenrightbigg2/integraldisplayǫ1 0dω1ω1ǫ2 2/braceleftBig . . ./bracerightBig =/parenleftbigg2α 3π/parenrightbigg2/parenleftbigg/integraldisplayǫ′ 1 0+/integraldisplayǫ1 ǫ′ 1/parenrightbigg dω1ω1ǫ2 2/braceleftBig . . ./bracerightBig =A+B . (24) One splits it into two parts, with the assumption ǫ′ 1<< ǫ 2. The ﬁrst term Ahas the same form as that in Eq. (23) with ǫ2replaced by ǫ′ 1. The second term Bis in turn split into two parts B=BL+BH, where BLis calculated with the regularized Coulomb potential, as in Eq. (8). One can expand here in the ratio ( H−E)/ωwhich leads to the expression: BL=E 9ln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg /braceleftbigg /an}b∇acketle{tφ\|4π δ3 λ(r)1 (E−H)′4π δ3 λ(r)\|φ/an}b∇acket∇i}ht+1 2/an}b∇acketle{tφ\|∇24π δ3 λ(r)\|φ/an}b∇acket∇i}ht/bracerightbigg . (25) Both terms in above braces have already been calculated in th e context of positronium energy levels in [12] /an}b∇acketle{tφ\|4π δ3 λ(r)1 (E−H)′4π δ3 λ(r)\|φ/an}b∇acket∇i}ht=−8 n3/bracketleftbiggλ 2+ 2 lnλ 3+ 8 ln3 4−3 2+2 n +2(ln( n)−Ψ(n)−C)/bracketrightbigg , (26) /an}b∇acketle{tφ\|∇24π δ3 λ(r)\|φ/an}b∇acket∇i}ht=−8 n3/bracketleftbigg −1 n2+λ−4 + 6 ln3 4/bracketrightbigg , (27) withn= 1 in our case. BHis the diﬀerence between BandBL. In this diﬀerence only large electron momenta contribute, therefore it could be obtaine d in the scattering amplitude approximation, in the same way as P2andP3in a simple example in the previous section. The result is BH=E4 9/bracketleftbigg 8 + 5π2−ln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg + 2λln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg −50 ln(2) ln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg + 18 ln(3) ln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg +ln/parenleftbigg¯ǫ′ 1 ¯ǫ2/parenrightbigg2 + 4 ln/parenleftbigg¯ǫ1 ¯ǫ′ 1/parenrightbigg ln/parenleftbiggλ√¯ǫ2/parenrightbigg/bracketrightbigg . (28) The complete Bterm is B=E4 9[8 + 5 π2+ 3 ln(¯ ǫ1)−6 ln(2) ln(¯ ǫ1)−2 ln(¯ǫ1) ln(¯ǫ2) + ln(¯ ǫ2)2−3 ln(¯ǫ′ 1) +6 ln(2) ln(¯ ǫ′ 1) + ln(¯ ǫ′ 1)2]. (29) 6We can now go back to Eq. (24) for the second derivative of ELwhich is a sum of AandB ǫ2∂EL ∂ǫ2=E4 9[8 +3N 2+ 5π2+ 3 ln(¯ ǫ1)−6 ln(2) ln(¯ ǫ1)−2 ln(¯ǫ1) ln(¯ǫ2) + ln(¯ ǫ2)2].(30) The expression for ELwhich matches both derivatives is: EL(¯ǫ1,¯ǫ2) =E/bracketleftbigg2Nln(¯ǫ1) 3+32 ln(¯ ǫ2) 9+2Nln(¯ǫ2) 3+20π2ln(¯ǫ2) 9+4 ln(¯ǫ1) ln(¯ǫ2) 3 −8 ln(2) ln(¯ ǫ1) ln(¯ǫ2) 3−4 ln(¯ǫ1) ln(¯ǫ2)2 9+4 ln(¯ǫ2)3 27/bracketrightbigg . (31) The constant term (no logs) is not included here. ELas shown in Fig. 1 is integrated over the region which is a combination of three rectangles: EL=EL/parenleftbiggǫ′ 1 α2,ǫ2 α2/parenrightbigg +EL/parenleftbiggǫ′ 2 α2,ǫ1 α2/parenrightbigg − EL/parenleftbiggǫ1 α2,ǫ2 α2/parenrightbigg . (32) IV. CONTRIBUTION EM In the one-loop case, contribution to energy, coming from ph oton energies k0> ǫis δE=/an}b∇acketle{tφ\|V\|φ/an}b∇acket∇i}ht, (33) V(ǫ) =α2δ3(r)/bracketleftbigg10 9−4 3ln(2ǫ)/bracketrightbigg . (34) EMis aVcorrection to the Bethe log: EM=2α 3πδV(ǫ1)/integraldisplayǫ2 0dω ω/an}b∇acketle{tφ\|pi 1 E−(H+ω)pi\|φ/an}b∇acket∇i}ht. (35) It has the same form as Eq. (23), so after symmetrization ǫ1↔ǫ2it is: EM=E 2/parenleftbigg10 9−4 3ln(2ǫ′ 1)/parenrightbigg /bracketleftbigg −2 3ln2ǫ2 α2+ 2 (1 −2 ln 2) lnǫ2 α2+N/bracketrightbigg + (ǫ1↔ǫ2).(36) V. CONTRIBUTION EF EFis the two-loop contribution with regularized Coulomb inte raction and with both photon energies limited from below by ǫ. It is a sum of three terms EF=E1 F+E2 F+E3 F, (37) deﬁned and calculated as follows. E1 Fis a second order correction coming from V(ǫ1) and V(ǫ2) with Vdeﬁned in (34), here additionally with λ-regularization E1 F=/an}b∇acketle{tφ\|V(ǫ1)1 (E−H)′V(ǫ2)\|φ/an}b∇acket∇i}ht. (38) 7The corresponding matrix element is given in Eq. (26), so E1 Fbecomes E1 F=E 16/parenleftbigg10 9−4 3ln(2ǫ1)/parenrightbigg /parenleftbigg10 9−4 3ln(2ǫ2)/parenrightbigg /parenleftbigg −4λ−16 lnλ 3−4/parenrightbigg . (39) One needs only ln λterm, since others do not give ln α.E2 Fis the contribution from electron formfactors F′ 1andF2atq2= 0 on relativistic (Dirac) wave function. We know it from the one-loop case that for vacuum-polarization A61=A40/2. The same holds for two-loop contribution, thus we have E2 F=Elnα−2B40 2. (40) Diagrams with closed fermion loop are automatically includ ed in the above formula. Other contributions coming from these diagrams are calculated in Section VII. E3 Fis the contribution from F′′ 1andF′ 2calculated with nonrelativistic wave functions. It leads to the matrix element /an}b∇acketle{tφ\|∇2δ3(r)λ\|φ/an}b∇acket∇i}htwhich does not lead to ln λ. Hence, it does not contribute to ln α. VI. CONTRIBUTION EH EHis the contribution obtained from the two–loop three–photo n exchange forward scat- tering amplitude. It requires subtractions of terms, contr ibuting to Lamb shift at lower orders. After subtractions it is ﬁnite and depends on ǫ1, ǫ2and Λ = λ α. When combined withELandEF, the dependence on ǫ1, ǫ2and Λ should cancel out. Having this in mind, the ln αcontribution could be obtained by the replacement λ→1/αinE1 Fin Eq. (39). However, the constant term B60requires complete calculation of EH, which we think is the most diﬃcult of the contributions. VII. DIAGRAMS WITH CLOSED FERMION LOOP There is a small logarithmic contribution coming from diagr ams with a closed fermion loop. They are partially included in E2 F. Two other contributions E1 V P, E2 V Pare the follow- ing. The second order correction coming from the one-loop va cuum polarization is E1 V P=E/parenleftbigg −4 15/parenrightbigg2 /an}b∇acketle{tφ\|δ3 λ(r)1 (E−H)δ3 λ(r)\|φ/an}b∇acket∇i}ht → E/parenleftbigg4 15/parenrightbigg2 lnα . (41) The second contribution E2 V Pis electron self–energy in the Coulomb potential including vacuum polarization correction. It is calculated in the sim ilar way, as previous corrections. One splits it into three parts E2 V P=CL+CM+CH. (42) CLis a v.p. correction V=−(4/15)δ3(r) to the Bethe log: 8CL=2α 3πδV/integraldisplayǫ 0dω ω/an}b∇acketle{tφ\|pi 1 E−(H+ω)pi\|φ/an}b∇acket∇i}ht (43) =E/parenleftbigg −4 15/parenrightbigg /parenleftbigg −2 3ln2ǫ α2+ 2 (1 −2 ln 2) lnǫ α2+N/parenrightbigg . (44) CMis a second order correction coming from self–energy and v.p . CM= 2/parenleftbiggα π/parenrightbigg2/parenleftbigg10 9−4 3ln 2ǫ/parenrightbigg /parenleftbigg −4 15/parenrightbigg /an}b∇acketle{tφ\|δ3 λ(r)1 (E−H)δ3 λ(r)\|φ/an}b∇acket∇i}ht (45) →2E/parenleftbigg10 9−4 3ln 2ǫ/parenrightbigg /parenleftbigg −4 15/parenrightbigg lnα . (46) CHis given by the scattering amplitude. Since we calculate onl y the logarithmic part, instead of calculating BHwe replaced ln λby−lnαin the equation above. The logarithmic part of electron self–energy in the Coulomb potential including va cuum polarization correction is E2 V P=E4 15/bracketleftbigg2 3ln2α−2+ 4/parenleftbigg2 9+ ln 2/parenrightbigg lnα−2/bracketrightbigg . (47) This completes the treatment of two-loop logarithmic corre ction VIII. SUMMARY The sum of all logarithmic terms in Eqs. (32,36,37,41,47) is B63=−8 27=−0.296296 , (48) B62=104 135−16 ln2 9=−0.461891 , (49) B61=39751 10800+4N 3+55π2 27−616 ln 2 135+3π2ln 2 4+40 ln22 9−9ζ(3) 8(50) = 50.309654 . First of all the result for B61is surprisingly large, and reverses the sign of the overall logarithmic contribution. B63agrees with the result obtained ﬁrst in [8]. However, as it wa s pointed out by Yerokhin [10], the loop-by-loop diagram is th e source of additional terms, which were not accounted for in the calculation in [8]. An add itional result of this work is the state dependence of Bcoeﬃcients which is obtained from n-dependence of matrix elements in Eqs. (20,26,27) B62(n) =B62+16 9/parenleftbigg3 4+1 4n2−1 n−ln(n) + Ψ( n) +C/parenrightbigg , (51) B61(n) =B61+4 3(N(n)−N) +/parenleftbigg304 135−32 9ln(2)/parenrightbigg /parenleftbigg3 4+1 4n2−1 n−ln(n) + Ψ( n) +C/parenrightbigg .(52) n-dependence of B62agrees with the former result in [14] (apart from the misprin t in the overall sign there). B61depends on N-coeﬃcient, the Dirac delta correction to Bethe logs, 9which has not been calculated yet for other states than 1S, th erefore its complete state de- pendence is unknown. However, one may expect to a good approx imation Nis independent ofn, as it is for Bethe logs. Because of the large value of B61theoretical predictions for hydrogen Lamb shift are going to be changed. The total logarithmic contribution is 16.9 kH z for the 1S state, compared to the previous one, based only on B63-28.4 kHz. Theoretical predictions for Lamb shift in hydrogen with proton radius rp= 0.862(12) fm from [15], using recent updates: analytical calculations of the three-loop contribution by Melnikov an d Ritbergen in [16] and direct numerical calculation of one-loop self-energy by Jentschu raet al. in [17] are (see details in the appendix) EL(1S)th= 8 172 816(10)(32) kHz , (53) EL(2S−2P1/2)th= 1 057 842(1)(4) kHz , (54) where we assumed for B60= 0±100, which gives the ﬁrst uncertainty. For P-states we neglect B-terms completely. The second uncertainty comes from the pr oton charge radius. Since it dominates the theoretical error, we emphasize the import ance of the muonic-hydrogen measurement, from which rpcould be precisely obtained. Current theoretical predicti ons agrees well with the most precise experimental values: EL(1S)exp= 8 172 837(22) kHz [18 ,19], (55) EL(2S−2P1/2)exp= 1 057 845(9) kHz [20] , (56) EL(2S−2P1/2)exp= 1 057 842(12) kHz [21] . (57) (58) Due to large uncertainty and ambiguities with the proton cha rge radius, one may regard the Lamb measurement as a determination of rp. In this way, from 1S Lamb shift, one obtains: rp= 0.869(12) fm . (59) Logarithmic two-loop corrections signiﬁcantly alter theo retical predictions for the Lamb shift in the single ionized helium as well. The current theoretica l value is EL(2S−2P1/2)th= 14 041 .57(8) MHz . (60) It does not agree with both: the experimental value from [22] and the recent update in [23] respectively: EL(2S−2P1/2)exp= 14 042 .52(16) MHz , (61) EL(2S−2P1/2)exp= 14 041 .13(17) MHz . (62) One may wonder about B60and further higher order terms, keeping in mind the large val ue ofB61. There are two possible and complementary undergoing proje cts: direct calculation of this term or numerical calculation of complete two-loop d iagrams with Dirac-Coulomb propagators. While the second would be the best way, the nume rical accuracy might be limited at small Z, such as Z= 1. In the direct calculation of B60one has to consider three points: two-loop Bethe logs with ǫcut-oﬀs, two-loop scattering amplitude with the photon massµ, and the transition terms between ǫandµ. This project seems to be achievable using the methods developed for B50, positronium decay rate and the one applied here. 10ACKNOWLEDGMENTS I gratefully acknowledge interesting discussions and help ful comments from Jonathan Sapirstein. I wish to thank M. Eides for inspiration. This wo rk was supported by Polish Comittee for Scientiﬁc Research under Contract No. 2P03B 05 7 18. APPENDIX A: FORMULAS FOR CALCULATIONS OF LAMB SHIFT In the calculation of hydrogen and helium Lamb shift we use th e following physical constants: R= 10973731 .568516(84) m−1, c= 299792458 m s−1, α−1= 137 .03599958(50) , mp me= 1836 .1526675(39) , mα me= 7294 .299508(16) , rp= 0.862(12) fm , rα= 1.673(1) fm . (A1) In general, Lamb shift in light hydrogen like systems is a sum of nonrecoil, recoil and the proton structure contributions. In the nonrecoil limit, kn own terms are: EL=mα(Z α)4 π n3/parenleftbiggµ m/parenrightbigg3 {A40+A41L+ (Z α)A50+ (Z α)2[A62L2+A61L+A60(Z α)] +α π[B40+ (Z α)B50+ (Z α)2(B63L3+B62L2+B61L+B60(Z α))] +/parenleftbiggα π/parenrightbigg2 C40}, (A2) where µis a reduced mass, m=me, and L= ln[m/(µ(Z α)2)]. Most of these coeﬃcients could be ﬁnd in any review, such as [1] or [2]. The recent resul t is the direct numerical calculations of one-loop self-energy, what give for hydrog en (Z= 1) A60(1S, α) =−30.29024 +/bracketleftbigg −0.6187 +/parenleftbigg19 45−π2 27/parenrightbigg/bracketrightbigg , A60(2S, α) =−31.18515 +/bracketleftbigg −0.8089 +/parenleftbigg19 45−π2 27/parenrightbigg/bracketrightbigg , A60(2P1/2, α) =−0.9735−0.0640, (A3) and for He+(Z= 2) A60(2S,2α) =−30.64466 +/bracketleftbigg −0.7961 +/parenleftbigg19 45−π2 27/parenrightbigg/bracketrightbigg , A60(2P1/2,2α) =−0.94940 −0.0638, (A4) 11where the second term is the vacuum polarization [24]. Anoth er recent result is analytical calculation of three-loop contribution in [16]. Together w ith the previously known vacuum polarization and anomalous magnetic moment it amounts C40= 0.417508 . (A5) In this work we calculate all logarithmic two-loop correcti ons for S-states. However, for P-state only B62is known. For this reason in the theoretical predictions for hydrogen and helium we totally neglect higher order two loop corrections , butB40forPstates. We neglect also dependence of Nin Eq. (20) on principal quantum number n, since Nhas not yet been calculated for n/ne}ationslash= 1. Recoil corrections, not included in Eq. (A2) sum to δE=µ3 m M(Z α)5 π n3/braceleftbigg1 3δl0ln(Z α)−2−8 3lnk0(n, l) +14 3δl0/bracketleftbigg ln/parenleftbigg2 n/parenrightbigg + Ψ(n) +C+1 2n+ 1/bracketrightbigg −1 9δl0−2 M2−m2δl0/bracketleftbigg M2ln/parenleftbiggm µ/parenrightbigg −m2ln/parenleftbiggM µ/parenrightbigg/bracketrightbigg −7 31−δl0 l(l+ 1) (2 l+ 1)/bracerightbigg −α(Z α)5 n3m2 Mδl0[1.364 49(2)] +(Z α)6 n3m2 MD60, (A6) where D60(nS1/2) = 4 ln(2) −7 2, D60(l≥1) =/bracketleftbigg 3−l(l+ 1) n2/bracketrightbigg2 (4l2−1)(2l+ 3). (A7) The ﬁnite charge distribution of the nucleus and its self-en ergy give corrections: δE=2 3n3(Z α)4µ3r2δl0+4 3π n3µ3 M2(Z2α) (Z α)4/bracketleftBigg ln/parenleftbiggM µ(Z α)2/parenrightbigg δl0−lnk0(n, l)/bracketrightBigg .(A8) In the theoretical predictions, presented in this paper we h ave neglected higher order proton structure corrections and higher order recoil corrections , which at present are negligible. 12REFERENCES [1] J.R. Sapirstein and D.R. Yennie, in Quantum Electrodynamics , edited by T. Kinoshita (World Scientiﬁc, Singapore, 1990). [2] M.I. Eides, H. Grotch, and V.A. Shelyuto, Phys. Rep. in print . [3] K. Pachucki, Phys. Rev. Lett. 72, 3154 (1994). [4] M. Eides and V. Sheluto, Phys. Rev. A 52, 954 (1995). [5] A. van Wijngaarden, J. Kwela, and G.W.F. Drake, Phys. Rev . A43, 3325 (1991). [6] K. Pachucki et al., J. Phys. B 29, 177 (1996) [7] S. Mallampali and J. Sapirstein, Phys. Rev. Lett. 80, 5297 (1998). [8] S.G. Karshenboim, Zh. Eksp. Teor. Fiz. 103, 1105 (1993). 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Mohr and B.N. Taylor, Rev. Mod. Phys. 72, 351 (2000). 13FIGURES εε ε1 12 '' EE LM FE+Hω ω2 1ε2EL EME FIG. 1. division of integration region into 4 parts, dependi ng on the value of both photon frequencies, ǫ2<< ǫ 1/A2 /BX FIG. 2. Two–loop diagrams in the Coulomb gauge in NRQED 14
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/CS/D3/D1/CP/CX/D2/D7/BN/B9 /CC /D6/CP/D2/D7/CX/CT/D2 /D8 /D7/CT /D3/D2/CS/CP/D6/DD /D7/D8/D6/D9 /D8/D9/D6/CT /CX/D7 /CU/D3/D6/D1/CT/CS /CQ /CT/CU/D3/D6/CT/D8/CT/D6/D8/CX/CP/D6/DD /D7/D8/D6/D9 /D8/D9/D6/CT/BN/BD/B9 /CC /D6/CP/D2/D7/CX/CT/D2 /D8 /CP /D9/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CZ/CX/D2/CT/D8/CX /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT/D7/BN/B9 /BX/DC/CX/D7/D8/CT/D2 /CT /D3/CU /CU/D3/D0/CS/CX/D2/CV /D4/CP/D8/CW /DB /CP /DD/D7/BN/B9 /C8 /D3/D7/D7/CX/CQ/D0/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /D2/D3/D2/B9/D2/CP/D8/CX/DA /CT /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT/D7/CU/D6/D3/D1 /D2/D3/D2/B9/D2/CP/D8/CX/DA /CT /D3/D0/D0/CX/D7/CX/D3/D2/D7/BN/B9 /CB/D3/D0/DA /CT/D2 /D8 /DA/CX/D7 /D3/D7/CX/D8 /DD /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/CU /CU/D3/D0/CS/CX/D2/CV /D6/CP/D8/CT/D7/BN/B9 /BY /D3/D0/CS/CX/D2/CV /D6/CP/D8/CT/D7 /CP/D2/CS /CU/CP /DA /D3/D6/CT/CS /D4/CP/D8/CW /DB /CP /DD/D7 /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2/D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/BN/B9 /CC/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /D1/D3 /DA /CT /CS/CX/AR/D9/D7/CX/DA /CT/D0/DD /D9/D2/CS/CT/D6 /D8/CW/CT /CX/D2/B9/AT/D9/CT/D2 /CT /D3/CU /CX/D2 /D8/CT/D6/D2/CP/D0 /CP/D2/CS /D6/CP/D2/CS/D3/D1 /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3/D6 /CT/D7/B8 /CP/D2/CS/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/B9/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D3/D0/D0/CX/D7/CX/D3/D2/D7 /D3 /D9/D6/BA /CC/CW/CT /CS/DD/B9/D2/CP/D1/CX /D7 /D3/CU /CU/D3/D0/CS/CX/D2/CV /CX/D7 /D7/CX/D1 /D9/D0/CP/D8/CT/CS /CQ /DD /CP /D7/CT/D8 /D3/CU /CS/CX/AR/D9/B9/D7/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /CS/CT/D7 /D6/CX/CQ /CT /D8/CW/CT /D1/D3/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CX/B9 /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CX/D2 /CP/D5/D9/CT/D3/D9/D7 /D7/D3/D0/D9/D8/CX/D3/D2/B8 /CP/D2/CS /CQ /DD /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /D4/D6/D3 /DA/CX/CS/CT /CU/D3/D6 /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /D3/D0/D0/CX/B9/D7/CX/D3/D2 /CP/D2/CS /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/CP/D0/CT/D7 /CT/D2 /CT/BA /CC/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2/B9 /D3/D0/D0/CX/D7/CX/D3/D2/CS/DD/D2/CP/D1/CX /D7 /CX/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CP/D7 /CP /D2/CT/D8 /DB /D3/D6/CZ /D3/CU /D7/D8/CT/D4/D7/B8 /CT/CP /CW /D3/D2 /D8/CP/CX/D2/CX/D2/CV /CP /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D4/CP/CX/D6 /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2/B8 /CX/D2 /DB/CW/CX /CW/D8/CW/CT /D6/CP/D8/CT /D3/CU /D3/CP/D0/CT/D7 /CT/D2 /CT /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /D4/CW /DD/D7/CX /CP/D0 /D4/D6/D3/D4/B9/CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/BA /CC/CW/CT /D6/CP/D8/CT/D7 /CP/D2 /CQ /CT /CP/D2/CP/D0/DD/D8/B9/CX /CP/D0/D0/DD /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D4/CW /DD/D7/CX /CP/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA/C4/CT/D8 /D9/D7 /D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /CP/D2/CP/D0/DD/D8/CX /CP/D0 /D1/D3 /CS/CT/D0/D8/D3 /CP/D0 /D9/D0/CP/D8/CT /D8/CW/CT /CU/D3/D0/CS/CX/D2/CV /D6/CP/D8/CT /D3/CU /D8 /DB /D3 /D3/D2/D2/CT /D8/CT/CS /D1/CX/B9 /D6/D3 /CS/D3/D1/CP/CX/D2/D7/B8 /DB/CW/CX /CW /CX/D7 /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/CP/D6/DD /D7/D8/CT/D4 /CX/D2 /D8/CW/CT/CS/CX/AR/D9/D7/CX/D3/D2/B9 /D3/D0/D0/CX/D7/CX/D3/D2 /D1/D3 /CS/CT/D0/BA /BV/D3/D2/D7/CX/CS/CT/D6 /D8 /DB /D3 /D3/D2/D2/CT /D8/CT/CS /D1/CX/B9 /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /BT /CP/D2/CS /BU /D8/CW/CP/D8 /D3/CP/D0/CT/D7 /CT /CX/D2 /D8/D3 /BT/BU/BA A+B−> AB /B4/BD/B5/CC/CW/CT /CS/DD/D2/CP/D1/CX /CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CX/D7/D1/D3 /CS/CT/D0/CT/CS /CQ /DD /CP /CS/CX/AR/D9/D7/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/BA /CB/CX/D2 /CT /DB /CT /CW/CP /DA /CT /CP/D7/DD/D7/D8/CT/D1 /D3/CU /D8 /DB /D3 /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/B8 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D3/D9/B9/D4/D0/CT/CS /CJ/BF℄/BA /CC/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /D1/D3/D8/CX/D3/D2 /CS/CX/AR/D9/D7/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7/CJ/BF℄/BM ∂ ∂t/parenleftbiggρ1 ρ2/parenrightbigg =D∇2/parenleftbiggρ1 ρ2/parenrightbigg +/parenleftbigg−λ1 λ1λ2 −λ2/parenrightbigg /parenleftbiggρ1 ρ2/parenrightbigg/B4/BE/B5/DB/CW/CT/D6/CT ρ /CX/D7 /CP /BE /CT/D0/CT/D1/CT/D2 /D8 /DA /CT /D8/D3/D6/B8 ρ1 /CQ /CT/CX/D2/CV /CP /D4/D6/D3/CQ/CP/B9/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD /CU/D3/D6 /CQ /D3/D8/CW /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CU/D3/D0/CS/CT/CS/B8 /CP/D2/CSρ2/D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/D7/CX/D8 /DD /CU/D3/D6 /CP/D0/D0 /D3/D8/CW/CT/D6 /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8/CX/CT/D7/BA /BW/CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /CS/CX/AR/D9/D7/CX/D3/D2 /D3/D2/D7/D8/CP/D2 /D8/B8 /CP/D2/CS /D8/CW/CT /D6/CP/D8/CT /D3/D2/B9/D7/D8/CP/D2 /D8/D7 /CP/D6/CTλ1 /B9 /CU/D6/D3/D1 /CQ /D3/D8/CW /CU/D3/D0/CS/CT/CS /D7/D8/CP/D8/CT /D8/D3 /CP/D0/D0 /D3/D8/CW/CT/D6/D7/B8 /CP/D2/CSλ2 /D8/CW/CT /D6/CP/D8/CT /CU/D3/D6 /D8/CW/CT /D6/CT/DA /CT/D6/D7/CT /D4/D6/D3 /CT/D7/D7/BA /BX/D5/D9/CP/D8/CX/D3/D2 /BE /D3/D9/D4/D0/CT/D7 /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/B9/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D6/CT/D0/CP/D8/CX/DA /CT /CS/CX/AR/D9/D7/CX/D3/D2/DB/CX/D8/CW /D8/CW/CT /D8 /DB /D3/B9/D7/D8/CP/D8/CT /CU/D3/D0/CS/CX/D2/CV/B9/D9/D2/CU/D3/D0/CS/CX/D2/CV /D4/D6/D3 /CT/D7/D7 /CP/D6/D6/CX/CT/CS/D3/D9/D8 /CX/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /CQ /DD /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/BA /CC/CW/CT /D3/D2/D2/CT /D8/B9/CX/D2/CV /CW/CP/CX/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D1 /D0/CX/D1/CX/D8/D7 /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2 /D7/D4/CP /CT /CU/D3/D6/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /B9 /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D6/CT/D0/CP/D8/CX/DA /CT /D1/D3/D8/CX/D3/D2/BA /BT/D2 /CX/CS/CT/B9/CP/D0/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D1/CP/CS/CT /D8/CW/CP/D8 /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CP/D6/CT /D7/D4/CW/CT/D6/CT/D7 /D3/D2/B9/D2/CT /D8/CT/CS /CQ /DD /CP /D4 /D3/D0/DD/D4 /CT/D4/D8/CX/CS/CT /CW/CP/CX/D2 /D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/D3 /CQ /CT /CP /AT/CT/DC/B9/CX/CQ/D0/CT /CU/CT/CP/D8/D9/D6/CT/D0/CT/D7/D7 /D7/D8/D6/CX/D2/CV/BA /CC/CW/CT /D3/D0/D0/CX/D7/CX/D3/D2 /CP/D2/CS /D3/CP/D0/CT/D7 /CT/D2 /CT/D3/CU /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CP/D6/CT /CV/D3 /DA /CT/D6/D2/CT/CS /CQ /DD /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CU/D3/D6 /BX/D5/D9/CP/D8/CX/D3/D2 /BE/BA /CC/CW/CT /CX/D2/D2/CT/D6 /CQ /D3/D9/D2/CS/CP/D6/DD /CX/D7 /D8/CW/CT /D0/D3/D7/CT/D7/D8 /CP/D4/D4/D6/D3/CP /CW /D7/D4/CW/CT/D6/CX /CP/D0 /D7/CW/CT/D0/D0/B8 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /DA /CP/D2 /CS/CT/D6/CF /CP/CP/D0/D7 /CT/D2 /DA /CT/D0/D3/D4 /CT/D7 /D3/CU /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/BA /CC/CW/CT /D0/D3/D7/CT/D7/D8/CP/D4/D4/D6/D3/CP /CW /CS/CX/D7/D8/CP/D2 /CT /D3/CU /D8 /DB /D3 /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CX/D7 /D8/CW/CT /D7/D9/D1 /D3/CU/D8/CW/CT/CX/D6 /D6/CP/CS/CX/CXRmin. /CC/CW/CT /D3/D8/CW/CT/D6 /D3/D2/D7/D8/D6/CP/CX/D2 /D8 /D3/D2 /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2/D7/D4/CP /CT /CX/D7 /D8/CW/CT /D1/CP/DC/CX/D1/CP/D0 /D6/CP/CS/CX/CP/D0 /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2 Rmax /CQ /CT/D8 /DB /CT/CT/D2/D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7/B8 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT/D7/D8/D6/CX/D2/CV /CQ /CT/D8 /DB /CT/CT/D2 /BT /CP/D2/CS /BU/BA /CB/D3 /DB /CT /CW/CP /DA /CT/BM Rmin=RA+RB /B4/BF/B5 Rmax=RA+RB+ /D7/CW/D3/D6/D8/CT/D7/D8 /CX/D2 /D8/CT/D6/DA /CT/D2/CX/D2/CV /CW/CP/CX/D2 /D0/CT/D2/CV/CW /D8/CC/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/D2 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/D2/B9/D7/CX/D8 /DD /CP/D6/CT /D7/D4 /CT /CX/AS/CT/CS /CP/D7/BM ∂ρ1,2 ∂r\|Rmax= 0 /B4/BG/B5/DB/CW/CX /CW /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CP/D2/D2/D3/D8 /CV/CT/D8/CU/D9/D6/D8/CW/CT/D6 /CP /DB /CP /DD /CU/D6/D3/D1 /D3/D2/CT /CP/D2/D3/D8/CW/CT/D6 /D8/CW/CP/D2 Rmax /CP/D2/CS /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/BM ∂ρ2 ∂r\|Rmin= 0 /B4/BH/B5/D1/CT/CP/D2/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D9/D2/CU/D3/D0/CS/CT/CS /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /CP/D2 /D2/D3/D8/CV/CT/D8 /D0/D3/D7/CT/D6 /D8/D3 /D3/D2/CT 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n/parenrightbigg xn= 2n/B4/BD/BF/B5/DB/CW/CT/D6/CT x= 1 /B8 /D7/D3 /CU/D3/D6 /D3/D2/CT /D4/CP/CX/D6 /D7/D8/CP/D8/CT/D7 /DB /CT /CW/CP /DA /CT/parenleftbign 1/parenrightbig/CS/CX/AR/CT/D6/CT/D2 /D8 /D7/D8/CP/D8/CT/D7/B8 /CU/D3/D6 /D8 /DB /D3 /D4/CP/CX/D6 /D7/D8/CP/D8/CT/D7 /DB /CT /CW/CP /DA /CT/parenleftbign 2/parenrightbig/CS/CX/AR/CT/D6/CT/D2 /D8 /D7/D8/CP/D8/CT/D7/B8 /CP/D2/CS /D7/D3 /D3/D2/BA /BT /D7 /CW/CT/D1/CP/D8/CX /DB /CP /DD /D3/CU/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CX/D2/CV /D8/CW/CT /CQ/CX/D2/D3/D1/CX/CP/D0 /D3 /CTꜶ /CX/CT/D2 /D8/D7 /CX/D7 /D8/CW/CT /CU/CP/D1/CX/D0/CX/CP/D6/C8 /CP/D7 /CP/D0/D7 /D8/D6/CX/CP/D2/CV/D0/CT /DB/CW/CT/D6/CT /D8/CW/CT /CQ/CX/D2/D3/D1/CX/CP/D0 /D3 /CTꜶ /CX/CT/D2 /D8/D7 /CX/D2 /D8/CW/CT/D2/CT/DC/D8 /D6/D3 /DB /CP/D6/CT /D7/CX/D1/D4/D0/DD /D6/CT/D0/CP/D8/CT/CS /CQ /DD /CP/CS/CS/CX/D8/CX/D3/D2/B8 /D8/D3 /D8/CW/CT /DA /CP/D0/B9/D9/CT/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D6/D3 /DB/BA /CC/CW/CX/D7 /D1/D3/D8/CX/DA /CP/D8/CT/D7 /D9/D7/CX/D2/CV 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/DD/D7 /B4/BD/BL/BK/BD/B5/BJ/CC /CP/CQ/D0/CT /C1/BA /BX/DC/CP/D1/D4/D0/CT /D3/CU /CP /D8/CW/D6/CT/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D4 /D6/D3/D8/CT/CX/D2 /CS/CP/D8/CP/BA /B6/C5/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /CA/CP/CS/CX/D9/D7 /B4/FH/B5 /BW/CX/D7/D8/CP/D2 /CT /B4/FH/B5 /BT /CT/D7/D7 /CP /D6/CT/CP /B4/FH/BE/B5/BT /D6/BT /BTA/BU /D6/BU /BTB/BV /D6/BV /BTC/BT/BU /BT/BU /BTAB/BT /BV /BTAC/BU/BV /BU/BV /BTBC/BT/BU/BV /BTABC/B6/C6/D9/D1/CT/D6/CX /CP/D0 /DA/CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT/D7/CT /D4 /D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CP /D6/CT /D8/CW/CT /CX/D2/D4/D9/D8 /D4/CP /D6/CP/D1/CT/D8/CT/D6/D7 /CX/D2 /CS/CX/AR/D9/D7/CX/D3/D2/B9 /D3/D0/D0/CX/D7/CX/D3/D2 /D1/D3 /CS/CT/D0 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7/BA/CC /CP/CQ/D0/CT /C1 /C1/BA /C8 /D3/D7/D7/CX/CQ/D0/CT /D7/D8/CP/D8/CT/D7 /CU/D3 /D6 /CP /D8/CW/D6/CT/CT/B9/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D4 /D6/D3/D8/CT/CX/D2/BA /B6/CD/D2/CU/D3/D0/CS/CT/CS /C7/D2/CT/B9/C8 /CP/CX/D6 /CC /DB /D3/B9/C8 /CP/CX/D6 /BY /D3/D0/CS/CT/CS/AZ /BU/BV /BT /BV /BT/BU /AZ /BU/BV /BT /BV /BT/BU /AZ /BU/BV /BT /BV /BT/BU /AZ /BU/BV /BT /BV /BT/BU/BE /BC /BC /BD /BG /BC /BD /BD/BD /BC /BC /BC /BF /BC /BD /BC /BI /BD /BC /BD /BK /BD /BD /BD/BH /BD /BC /BC /BJ /BD /BD /BC/B6/BT /D7 /CW/CT/D1/CP/D8/CX /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /CZ/CX/D2/CT/D8/CX /CX/D2/D8/CT/D6/D1/CT/CS/CX/CP/D8/CT /D7/D8/CP/D8/CT/D7/BA /BW/CT /CX/D1/CP/D0 /D2/D9/D1/CQ /CT/D6/D7 /CT/D2/D9/D1/CT/D6/CP/D8/CT /D8/CW/CT /D7/D8/CP/D8/CT/D7/B8 /DB/CW/CX/D0/CT /D8/CW/CT /CQ/CX/D2/CP /D6/DD /CS/CX/CV/CX/D8 /BD /CX/D2 /D8/CW/CT/CQ/CX/D2/CP /D6/DD /D2/D9/D1/CQ /CT/D6/D7 /D6/CT/D4 /D6/CT/D7/CT/D2/D8/D7 /CP /D3/CP/D0/CT/D7 /CT/CS /D4/CP/CX/D6/BA /CC/CW/CT /CS/CT /CX/D1/CP/D0 /D2/D9/D1/CQ /CT/D6 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /CP/CS/CS/CX/D2/CV /D3/D2/CT /D8/D3 /D8/CW/CT /CS/CT /CX/D1/CP/D0 /DA/CP/D0/D9/CT /D3/CU /D8/CW/CT /CQ/CX/D2/CP /D6/DD /D2/D9/D1/CQ /CT/D6/BA/BK/CC /CP/CQ/D0/CT /C1 /C1 /C1/BA /CC /D6/CP/D2/D7/CX/D8/CX/D3/D2 /CB/D8/CP/D8/CT/D7/B8 /BU/D3/D2/CS/D7 /CP/D2/CS /C8 /CP /D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/D6/CT/CT/B9/D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2 /D4 /D6/D3/D8/CT/CX/D2/BA /B6/CC /D6/CP/D2/D7/CX/D8/CX/D3/D2 /C1/D2/CX/D8/CX/CP/D0 /CB/D8/CP/D8/CT /BU/D3/D2/CS /BY /D3 /D6/D1/CT/CS /BY/CX/D2/CP/D0 /CB/D8/CP/D8/CT Rmin Rmax/BD/B9> /BE /BT/B9/BU/B9/BV /BT/BU /BT/BU/B9/BV /D6/BT/B7/D6/BU /D6/BT/B7 /BT/BU/B7/D6/BU/BD/B9> /BF /BT/B9/BU/B9/BV /BT /BV /BU/B9/BT /BV /D6/BT/B7/D6/BV /D6/BT/B7 /BT/BU/B7/BE/D6/BU/B7 /BU/BV/B7/D6/BV/BD/B9> /BH /BT/B9/BU/B9/BV /BU/BV /BT/B9/BU/BV /D6/BU/B7/D6/BV /D6/BU/B7 /BU/BV/B7/D6/BV/BE/B9> /BG /BT/BU/B9/BV /BT /BV /BT/BU/BV1 /D6/BT/BU/B7/D6/BV /D6/BT/BU/B7 /BU/BV/B7/D6/BV/BE/B9> /BI /BT/BU/B9/BV /BU/BV /BT/BU/BV2 /D6/BT/BU/B7/D6/BV /D6/BT/BU/B7 /BU/BV/B7/D6/BV/BF/B9> /BG /BU/B9/BT /BV /BT/BU /BT/BU/BV1 /D6/BU/B7/D6/BT /BV /D6/BU/B7 /BU/BV/B7/D6/BT /BV/BF/B9> /BJ /BU/B9/BT /BV /BU/BV /BT/BU/BV3 /D6/BU/B7/D6/BT /BV /D6/BU/B7 /BU/BV/B7/D6/BT /BV/BH/B9> /BI /BT/B9/BU/BV /BT/BU /BT/BU/BV2 /D6/BT/B7/D6/BU/BV /D6/BT/B7 /BT/BU/B7/D6/BU/BV/BH/B9> /BJ /BT/B9/BU/BV /BT /BV /BT/BU/BV3 /D6/BT/B7/D6/BU/BV /D6/BT/B7 /BT/BU/B7/D6/BU/BV/BG/B9> /BK /BT/BU/BV1 /BU/BV /BT/BU/BV4 /D6/BU/B7/D6/BV π /D6/BT/BU/BV/BI/B9> /BK /BT/BU/BV2 /BT /BV /BT/BU/BV4 /D6/BT/B7/D6/BV π /D6/BT/BU/BV/BJ/B9> /BK /BT/BU/BV3 /BT/BU /BT/BU/BV4 /D6/BT/B7/D6/BU π /D6/BT/BU/BV/B6/BU/CP/D7/CT/CS /D3/D2 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0 /D4/CP /D6/CP/D1/CT/D8/CT/D6/D7/B8 /D8/CW/CT /CS/CP/D8/CP /CX/D2 /D8/CW/CX/D7 /D8/CP/CQ/D0/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2/B9 /D3/D0/D0/CX/D7/CX/D3/D2/D1/D3 /CS/CT/D0 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT /CS/CP/D8/CP /CX/D7 /D8/CW/CT/D2 /D9/D7/CT/CS /D8/D3 /CP/D0 /D9/D0/CP/D8/CT /D8/CW/CT /D4 /D6/D3/CQ/CP/CQ/CX/D0/CX/D8/CX/CT/D7 /D3/CU /D8/CW/CT /CZ/CX/D2/CT/D8/CX /D7/D8/CP/D8/CT/D7/CP/D2/CS /D8/CW/CT /CU/D3/D0/CS/CX/D2/CV /CP/D2/CS /D9/D2/CU/D3/D0/CS/CX/D2/CV /D6/CP/D8/CT/D7 /CQ /DD /D7/D3/D0/DA/CX/D2/CV /D8/CW/CT /CS/CX/AR/D9/D7/CX/D3/D2 /CT/D5/D9/CP/D8/CX/D3/D2/BA/CC /CP/CQ/D0/CT /C1/CE/BA /BV/D3/D1/CQ/CX/D2/CP/D8/D3 /D6/CX/CP/D0 /CS/CT/D4 /CT/D2/CS/CT/D2 /CT /D3/D2 /D8/CW/CT /D2/D9/D1/CQ /CT/D6 /D3/CU /D4/CP/CX/D6/CX/D2/CV/D7/BA /B6/AZ /D3/CU /D4/CP/CX/D6/CX/D2/CV/D7 /AZ /D3/CU /D7/D8/CP/D8/CT/D7 /AZ /D3/CU /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2/D8 /AZ /D3/CU /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/D7/D2 /BEn/D4/CP/D8/CW/DB /CP /DD/D7 /D2/AX /D2/BEn−1/BD /BE /BD /BD/BE /BG /BE /BG/BF /BK /BI /BD/BE/BG /BD/BI /BE/BG /BF/BE/BH /BF/BE /BD/BE/BC /BK/BC/BI /BI/BG /BJ/BE/BC /BD/BL/BE/BJ /BD/BE/BK /BH/BC/BG/BC /BG/BG/BK/BK /BE/BH/BI /BG/BC/BF/BE/BC /BD/BC/BE/BG/BL /BH/BD/BE /BF/BI/BE/BK/BK/BC /BE/BF/BC/BG/B6/BV/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D0/CP /D6/CV/CT /D6/D2/D9/D1/CQ /CT/D6 /D3/CU /D4/CP/CX/D6/CX/D2/CV/D7 /D2/B8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /D3/CU /D8/CW/CT /D4 /D6/D3/D8/CT/CX/D2/B8 /D8/CW/CT /D2/D9/D1/CQ /CT/D6/D3/CU /D7/D8/CP/D8/CT/D7/B8 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/D7/B8 /CP/D2/CS /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2/D8 /D4/CP/D8/CW/DB /CP /DD/D7 /CX/D2 /D6/CT/CP/D7/CT/D7 /D5/D9/CX /CZ/D0/DD /B8 /D8/CW/D9/D7 /D6/CT/CP/D8/CX/D2/CV /D3/D1/CQ/CX/D2/CP/D8/D3 /D6/CX/CP/D0 /D3/D1/D4/D0/CT/DC/CX/D8 /DD /CX/D2 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7/BA/BL/CC /CP/CQ/D0/CT /CE/BA /CB/CX/D1/D4/D0/CT /CT/D0/D0/D9/D0/CP /D6 /CP/D9/D8/D3/D1/CP/D8/D3/D2 /D4/CP/D8/D8/CT/D6/D2/BC /BC /BC /BC /BC /BC /BC /BD /BC /BC /BC /BC /BC /BC /BC/BC /BC /BC /BC /BC /BC /BD /BC /BD /BC /BC /BC /BC /BC /BC/BC /BC /BC /BC /BC /BD /BC /BC /BC /BD /BC /BC /BC /BC /BC/BC /BC /BC /BC /BD /BC /BD /BC /BD /BC /BD /BC /BC /BC /BC/BC /BC /BC /BD /BC /BC /BC /BC /BC /BC /BC /BD /BC /BC /BC/BC /BC /BD /BC /BD /BC /BC /BC /BC /BC /BD /BC /BD /BC /BC/BC /BD /BC /BC /BC /BD /BC /BC /BC /BD /BC /BC /BC /BD /BC/BD /BC /BD /BC /BD /BC /BD /BC /BD /BC /BD /BC /BD /BC /BD/CC /CP/CQ/D0/CT /CE/C1/BA /BT /CS/CY/CP /CT/D2 /DD /D1/CP/D8/D6/CX/DC /CU/D3 /D6 /C6/BP/BD/BD /DB/CX/D8/CW /D2/D3 /D4/CP/CX/D6/CX/D2/CV/D7/BT/C5/B4/BD/B8/BM/B5 /BC /BD /BC /BC /BC /BC /BC /BC /BC /BC /BC/BT/C5/B4/BE/B8/BM/B5 /BD /BC /BD /BC /BC /BC /BC /BC /BC /BC /BC/BT/C5/B4/BF/B8/BM/B5 /BC /BD /BC /BD /BC /BC /BC /BC /BC /BC /BC/BT/C5/B4/BG/B8/BM/B5 /BC /BC /BD /BC /BD /BC /BC /BC /BC /BC /BC/BT/C5/B4/BH/B8/BM/B5 /BC /BC /BC /BD /BC /BD /BC /BC /BC /BC /BC/BT/C5/B4/BI/B8/BM/B5 /BC /BC /BC /BC /BD /BC /BD /BC /BC /BC /BC/BT/C5/B4/BJ/B8/BM/B5 /BC /BC /BC /BC /BC /BD /BC /BD /BC /BC /BC/BT/C5/B4/BK/B8/BM/B5 /BC /BC /BC /BC /BC /BC /BD /BC /BD /BC /BC/BT/C5/B4/BL/B8/BM/B5 /BC /BC /BC /BC /BC /BC /BC /BD /BC /BD /BC/BT/C5/B4/BD/BC/B8/BM/B5 /BC /BC /BC /BC /BC /BC /BC /BC /BD /BC /BD/BT/C5/B4/BD/BD/B8/BM/B5 /BC /BC /BC /BC /BC /BC /BC /BC /BC /BD /BC/CC /CP/CQ/D0/CT /CE/C1 /C1/BA /BT /CS/CY/CP /CT/D2 /DD /D1/CP/D8/D6/CX/DC /CU/D3 /D6 /C6/BP/BD/BD /DB/CX/D8/CW /D1/CX /D6/D3 /CS/D3/D1/CP/CX/D2/D7 /BF/B9/BL /D4/CP/CX/D6/CX/D2/CV/BT/C5/B4/BD/B8/BM/B5 /BC /BD /BC /BC /BC /BC /BC /BC /BC /BC /BC/BT/C5/B4/BE/B8/BM/B5 /BD /BC /BD /BC /BC /BC /BC /BC /BD /BC /BC/BT/C5/B4/BF/B8/BM/B5 /BC /BD /BC /BD /BC /BC /BC /BD /BC /BD /BC/BT/C5/B4/BG/B8/BM/B5 /BC /BC /BD /BC /BD /BC /BC /BC /BD /BC /BC/BT/C5/B4/BH/B8/BM/B5 /BC /BC /BC /BD /BC /BD /BC /BC /BC /BC /BC/BT/C5/B4/BI/B8/BM/B5 /BC /BC /BC /BC /BD /BC /BD /BC /BC /BC /BC/BT/C5/B4/BJ/B8/BM/B5 /BC /BC /BC /BC /BC /BD /BC /BD /BC /BC /BC/BT/C5/B4/BK/B8/BM/B5 /BC /BC /BD /BC /BC /BC /BD /BC /BD /BC /BC/BT/C5/B4/BL/B8/BM/B5 /BC /BD /BC /BD /BC /BC /BC /BD /BC /BD /BC/BT/C5/B4/BD/BC/B8/BM/B5 /BC /BC /BD /BC /BC /BC /BC /BC /BD /BC /BD/BT/C5/B4/BD/BD/B8/BM/B5 /BC /BC /BC /BC /BC /BC /BC /BC /BC /BD /BC/BD/BC/CC /CP/CQ/D0/CT /CE/C1 /C1 /C1/BA /BT /CS/CY/CP /CT/D2 /DD /D1/CP/D8/D6/CX/DC /D4/CP/D8/D8/CT/D6/D2 /D9/D7/CT/CS /D8/D3 /AS/D2/CS /CA/D1/CP/DC /BA/C1/D2/CX/D8/CX/CP/D0 /BD/BT/C5/B4/BD/B8/BM/B5 /BC /BD /BC /BC /BC /BC /BC /BC /BC /BC /BC/BT/C52/B4/BE/B8/BM/B5 /BD /BC /BD /BC /BC /BC /BC /BC /BD /BC /BC/BT/C53/B4/BF/B8/BM/B5 /BC /BF /BC /BE /BC /BC /BC /BE /BC /BE /BC/BT/C54/B4/BG/B8/BM/B5 /BF /BC /BL /BC /BE /BC /BE /BC /BL /BC /BE/BT/C55/B4/BH/B8/BM/B5 /BC /BE/BD /BC /BE/BC /BC /BG /BC /BE /BC /BC /BE /BC /BC/BT/C56/B4/BI/B8/BM/B5 /BE/BD /BC /BK/BD /BC /BE/BG /BC /BE/BG /BC /BK/BD /BC /BE /BC/BT/C57/B4/BJ/B8/BM/B5 /BC /BD/BK/BF /BC /BD/BK/BI /BC /BG/BK /BC /BD/BK/BI /BC /BD/BK/BE /BC/BT/C58/B4/BK/B8/BM/B5 /BD/BK/BF /BC /BJ/BF/BJ /BC /BE/BF/BG /BC /BE/BF/BG /BC /BJ/BF/BJ /BC /BD/BK/BE/BT/C59/B4/BL/B8/BM/B5 /BC /BD/BI/BH/BJ /BC /BD/BJ/BC/BK /BC /BG/BI/BK /BC /BD/BJ/BC/BK /BC /BD/BI/BH/BI /BC/BT/C510/B4/BD/BC/B8/BM/B5 /BD/BI/BH/BJ /BC /BI/BJ/BE/BL /BC /BE/BD/BJ/BI /BC /BE/BD/BJ/BI /BC /BI/BJ/BE/BL /BC /BD/BI/BH/BI/BT/C511/B4/BD/BD/B8/BM/B5 /BC /BD/BH/BD/BD/BH /BC /BD/BH/BI/BF/BG /BC /BG/BF/BH/BE /BC /BD/BH/BI/BF/BG /BC /BD/BH/BD/BD/BG /BC/BD/BD/BY/CX/CV/D9/D6/CT /BE/BA 52 4 6 78 1 3 A schematic view of the states and transitions for 3 microdom ains with n =3./CC/CW/CT/D6/CT /CP /D6/CT /BK 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arXiv:physics/0011046v1 [physics.comp-ph] 20 Nov 2000MZ-TH/00–32 Juli 2000 Approximate 3-Dimensional Electrical Impedance Imaging C. Lehmann∗, K. Schilcher Institut f¨ ur Physik, Johannes-Gutenberg-Universit¨ at, Staudinger Weg 7, D-55099 Mainz, Germany Abstract We discuss a new approach to three-dimensional electrical i mpedance imaging based on a reduction of the information to be demande d from a reconstruction algorithm. Images are obtained from a sing le mea- surement by suitably simplifying the geometry of the measur ing cham- ber and by restricting the nature of the object to be imaged an d the information required from the image. In particular we seek t o es- tablish the existence or non-existence of a single object (o r a small number of objects) in a homogeneous background and the locat ion of the former in the ( x,y)-plane deﬁned by the measuring electrodes . Given in addition the conductivity of the object rough estim ates of its position along the z-axis may be obtained. The approach may have practical applications. 1 Introduction The aim of electrical impedance tomography (EIT) is to recon struct the conductivity distribution σ(x) in the interior of an object Ω ⊂R3from electrical measurements on the boundary ∂Ω . For this purpose a number of diﬀerent current distributions are applied to the surfac e of the object via electrodes and the the resulting potentials on the surfa ce are recorded. Applications can be envisaged both in medicine and industry [1]. Conservation of the current j(x) and Maxwell’s equations in the quasi- static limit lead to the following diﬀerential equation for the potential Φ( x): ∇ ·[σ(x)∇Φ(x)] = 0 . (1) 1In the following we take as the object a rectangular box and in vestigate whether statements on the conductivity distribution can be made if the sur- face potential can only be measured on one side of the box. Suc h a model relates to typical situation in geological and medical imag ing. The general inverse conductivity problem for the box requir es current- and potential-measurements for a large number (in principl e inﬁnite) of ap- plied current conﬁgurations on the surface of the box. For th e reconstruction of the conductivity distribution in this and related proble ms the boundary conditions must be known precisely and all calculations of p otentials be per- formed with high accuracy. All these conditions are diﬃcult to be achieved in practice, which explains the comparative lack of success of the impedance method in medical applications. In many cases, speciﬁcally breast cancer screening, it is actually not absolutely necessary to have a complete image of the region. If we restrict the reconstruction to a shadow o n a plane and require only rough information on size and location of the ca ncerous region, the reconstruction can be done analytically using a single measurement. This problem has also been discussed from diﬀerent points of view [2], [3]. 2 Description of the problem We are interested in the conductivity distribution σ(x) inside a rectangular box with sides a,b,c, as pictured in ﬁgure (1 ). ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏ ✲✲✲ ✲✲✲ ✲✲✲ ✲✲✲ ✚✙✛✘
arXiv:physics/0011047v1 [physics.chem-ph] 20 Nov 2000Analytical Cell Potentials for Clathrate-Hydrates from In version of Langmuir Constant Versus Temperature Curves Martin Z. Bazant Department of Mathematics, Massachusetts Institute of Tec hnology, Cambridge, MA 02139 Bernhardt L. Trout Department of Chemical Engineering, Massachusetts Instit ute of Technology, Cambridge, MA 02139 (November 20, 2000) Experimental Langmuir constants for clathrate-hydrates, which are inferred from phase equilibrium data using an equation of state and the statisti cal thermodynamical model of van der Waals and Platteeuw (vdWP), contain a wealth of infor mation about intermolecular forces. In the simplest (Lennard-Jones and Devonshire) app roximation of a spherical cell, the Langmuir constant is related to a conﬁgurational integr al involving a spherically averaged cell potential. All previous attempts to interpret experim ental data based on this formalism have involved numerical ﬁtting with ad hoc functional forms, such as the popular Kihara potential. In addition to lacking analytical insight, howe ver, such empirical ﬁts often produce parameters contradicting those obtained by other means. As a more appealing alternative, the spherical-cell vdWP formula for the Langmuir constant vers us temperature can be viewed as a nonlinear integral equation for the cell potential. In thi s article, a variety of exact analytical solutions are derived, and the mathematical theory is used t o interpret experimental data for ethane and cyclopropane clathrate-hydrates. I. INTRODUCTION Clathrate-hydrates exist throughout nature and are potent ially very useful technological materials [1]. For example, existing methane hydrates are believed to hold muc h more energy than any fossil fuel in use today. Carbon dioxide hydrates are being considered as eﬀective ma terials for the sequestration and/or storage of CO2. In spite of their great importance, however, the theory of cl athrate-hydrate phase behavior is not very well developed, still relying for the most part on the ad hoc empirical ﬁtting of experimental data. Since being introduced in 1959, the statistical thermodynamical model of van der Waals and Platteeuw (vdWP) has been used almost exclusively to model the phase behavior of clathrate-hydrates, usually together with a spherical cell (SC) model for the interaction potential bet ween the enclathrated or “guest” molecule and the cage of the clathrate-hydrate. The SC model was also introdu ced by vdWP and was inspired by an analogous approximation made by Lennard-Jones and Devonshire in the c ase of liquids [2,3]. In the general formulation of vdWP [2], the chemical potenti al diﬀerence between an empty, unstable hydrate structure with no guest molecules, labeled MT, and the stabl e hydrate, labeled H, is related to the so-called Langmuir hydrate constant CJiand the fugacity of the guest molecule ˆfJ ∆µMT−H=kT/summationdisplay iνiln(1 +/summationdisplay JCJiˆfJ) (1) whereidesignates the type of cage, νithe number of cages of type iper water molecule and Jthe type of guest molecule. In practice, experimental phase equilibri a data is used to determine ∆ µMT−H. The connection with intermolecular forces within vdWP theo ry is made by expressing the Langmuir hydrate constant as the conﬁgurational integral ZJidivided by kT, which is written explicitly as an integral over the volumeV CJi(T) =1 8π2kT/integraldisplay Ve−Φ(r,θ,φ,α,ξ,γ )/kTr2sinθsinξdrdθdφdαdξdγ (2) where Φ(r,θ,φ,α,ξ,γ ) is the general six dimensional form of the interaction pote ntial between the guest molecule at spherical coordinates ( r,θ,φ) oriented with Euler angles ( α,ξ,γ ) with respect to all of the water molecules in the clathrate-hydrate. In the SC approximatio n, which is made without any careful mathematical 1justiﬁcation, the intermolecular potential Φ is replaced b y a spherically averaged “cell potential” w(r), which reduces the Langmuir constant formula (2) to a single, radia l integration CJi(T) =4π kT/integraldisplayR 0e−w(r)/kTr2dr (3) where the cutoﬀ distance Ris normally and arbitrarily taken as the radius of the cage, a lthough its exact value does not usually matter because typical temperatures are so low that the high energy portion of the cage r≈R makes a negligible contribution. Although the SC approxima tion may appear to be a drastic simpliﬁcation, it is very useful for theoretical studies of intermolecular fo rces based on Langmuir constant measurements. Before this work, the functional form of the cell potential w(r) has always been obtained by ﬁrst choosing a model interaction potential between the guest molecule in a cage and each nearest neighbor water molecule essentially ad hoc , and then performing the spherical average from (2) to (3) an alytically. The most common potential form in use today is the Kihara potential, which is simply a shifted Lennard-Jones potential with a hard-core. Using the Kihara potential and spherically ave raging the interaction energy, typically over the ﬁrst-shell only, yields the following functional form for w(r): w(r) = 2zǫ[σ12 R11r(δ10+a Rδ11)−σ6 R5r(δ4+a Rδ5)] (4) where δN=1 N[(1−r R−a R)−N−(1 +r R−a R)−N] (5) andzis the coordination number, Ris the radius of the cage, and σ,ǫ, andaare the Kihara parameters. As a result of the averaging process leading from (2) to (3), t he functional form of w(r) is fairly complicated, and the parameters ǫandσare generally determined by ﬁtting monovarient equilibriu m temperature-pressure data numerically [1,4]. There are several serious drawbacks to this ubiquitous nume rical ﬁtting procedure, which suggest that the Kihara parameters lack any physical signiﬁcance: ( i) The Kihara parameters are not unique, and many diﬀerent sets can ﬁt the experimental data well; ( ii) the Kihara parameters found by ﬁtting Langmuir curves do not match those of found by ﬁtting other experimental data , such as the second virial coeﬃcient or the gas viscosity [1]; and ( iii) comparisons of Langmuir constants found via the SC approxi mation (3) and via explicit multi-dimensional quadrature (2) show that the two can diﬀe r by over 12 orders of magnitude [5,6] (which results from the exponentially strong sensitivity of the La ngmuir constant to changes in the cell potential). These problems call into question the validity of using the K ihara potential as the basis for the empirical ﬁtting, and even the use of the SC approximation itself. It would clearly be preferable to extract more reliable info rmation about the interatomic forces in clathrate- hydrates directly from experimental data without any ad hoc assumptions about their functional form. Such an approach may be possible in the case of clathrate-hydrate s which contain a single type of guest molecule occupying only one type of cage. In this case, each of the sums in Eq. (1) contains only one term, and by using an equation of state to compute the fugacity ˆfJ, the Langmuir constant CJican be determined directly from experimental phase equilibria data. Typical data sets obtained in this manner are shown in Fig. 1 for Structure I ethane and cyclopropane clathrate-hydrates [7 ]. Because the full potential Φ in (2) is multi-dimensional (wh ile the Langmuir constant only depends on a single parameter T), the general vdWP theory is too complex to pose a well-deﬁne d inverse problem for the interatomic forces. The SC approximation, on the other h and, introduces a very convenient theoretical construct, the spherically averaged potential w(r), which has the same dimensionality as the Langmuir curve CJi(T) of a single type of guest molecule occupying a single type of cage. (Since we consider only this case, we drop the subscripts Jihereafter.) Although one can question the accuracy of the SC approximation, its simplicity at least allows precise connections to be made be tween the Langmuir curve and the cell potential. Such analytical insights, which are necessary for an evalua tion of the SC approximation on physical and mathematical grounds, cannot be obtained from ad hoc numerical ﬁtting schemes. As an appealing alternative to empirical ﬁtting, therefore , in this article we view Eq. (3) as an integral equation to be solved analytically for w(r), given a particular Langmuir curve C(T). Lettingβ= 1/kT, we rewrite (3) as C(β) = 4πβ/integraldisplay∞ 0e−βw(r)r2dr, (6) 2where we have also set the upper limit of integration to R=∞, which introduces negligible errors due to the very low temperatures (large β) accessible in experiments. (This will be justiﬁed a posteriori below with a precise deﬁnition of “low” temperatures.) In our analytical approach, some straight-forward ﬁtting o f the raw experimental data is needed to construct the function C(β), but after that, the “inversion” process leading to w(r) is exact. For example, typical sets of experimental data are well described by a van’t Hoﬀ temper ature dependence C(β) =Coemβ(7) as shown in Fig. 2 for ethane and cyclopropane clathrate hydr ates [7], and the constant mis generally positive. (Note that the exponential dependence which we call ”van’t H oﬀ dependence” in this paper can be expected based on quite general thermodynamic considerations [8].) Aided by the analysis, the quality and functional form of the se ﬁts are discussed below in section VI. In order to allow for deviations from the dominant van’t Hoﬀ beh avior, however, in this article we consider the more general form C(β) =βF(β)emβ(8) wheremis a constant deﬁned by m= lim β→∞logC(β)/β (9) whenever this limit exists and is ﬁnite, i.e. when the prefac torF(β) in Eq. (8) is dominated by the exponential term at low temperatures. We exclude the possibility of hype rexponential behavior at low temperatures, logC/β→ ∞, which is not physically meaningful, as explained below. Th e set of possible prefactors includes power-laws, F(β) =β−µ, as well as various rational functions. Although we are not aware of any previous attempts to derive i nteratomic potentials for chemical systems by solving an inverse problem, the basic idea of “exact inver sion” has recently been pursued in solid state physics, albeit based on a very diﬀerent mathematical forma lism. This approach was pioneered by Carlsson, Gelatt and Ehrenreich in 1980 for the case of interatomic pai r potentials for crystalline metals [9,10]. These authors had the following insight: Assuming that the total c ohesive energy E(x) of a crystal with nearest neighbor distance xcan be expressed as a lattice sum over all pairs of atoms ( i,j) E(x) =/summationdisplay ijφ(xsij) (10) wheresijare normalized atomic separation distances, then a unique p air potential φ(r) can be derived which exactly reproduces the cohesive energy curve E(x). (Lattice sums also appear in some clathrate-hydrates models [18], but to our knowledge they have never been used as the basis for an inversion procedure.) The mathematical theory for the inversion of cohesive energy cu rves has been developed considerably in recent years and applied to wide variety of solids [11–15]. For exam ple, the successful extension of the inversion formalism to semiconductors has required solving a nonline ar generalization of Eq. (10) representing many- body angle-dependent interactions [14,15]. These theoret ical advances have recently led to improvements in the modeling of silicon, beyond what had previously been obt ained by empirical ﬁtting alone [16,17]. Inspired by such developments in solid state physics, in this work we s eek analogous insights into clathrate-hydrate interatomic forces from the exact inversion of Langmuir con stant versus temperature curves. The article is organized as follows. In section II we discuss various necessary and suﬃcient conditions for the existence of physically reasonable solutions, and we al so derive the asymptotics of the Langmuir curve at low temperature from the behavior of the cell potential at near its minimum. In section III, we perform the analysis in the general case, and in section IV, we discus s the speciﬁc case of van’t Hoﬀ dependence (7), which leads to a cubic solution as well as various unphysical solutions involving cusps. In section V, we derive analytical solutions for several diﬀerent temperature dep endences, which reveal the signiﬁcance of possible deviations from van’t Hoﬀ behavior for the form of the potent ialw(r). The theoretical curves are compared with the experimental data in section VI, and the physical co nclusions of the analysis are summarized in section VII. Relevant mathematical theorems are proved in t he Appendix. II. GENERAL ANALYSIS OF THE INVERSE PROBLEM 3A. Necessary Conditions for the Existence of Solutions On physical grounds, it expected that the cell potential w(r) is continuous (at least piecewise) and has a ﬁnite minimum at ro≥0 somewhere inside the clathrate cage, w(r)≥w(ro) =wo. We also allow the possibility that w(r) is inﬁnite for certain values of r(e.g. outside a “hard-wall radius”) by simply omitted such values from the integration in Eq. (6). As proved in the A ppendix, these simple physical requirements suﬃce to imply the asymptotic relation (9), where m=−wo. They also place important constraints on the prefactorF(β) deﬁned in (8), which must be (i) analytic in the half plane Re β >c and (ii) real, positive and non-increasing for β >c on the real axis wherec≥0 is a real number. (Note that we view the inverse temperature βas a complex variable, for reasons soon to become clear.) Moreover, if the set S={r≥0\|wo<w(r)<∞} (11) has nonzero measure, then F(β) is strictly decreasing on the positive real axis. It is stra ightforward to generalize these rigorous results to the multi-dimensiona l integral of vdWP theory, Eq. (2), without making the spherical cell approximation, as described in the Appen dix, but hereafter we discuss only the spherically averaged integral equation, Eq. (6), because it makes possi ble an exact inversion. In this section, we give simple arguments to explain the resu lts proved in Theorem 1 of the Appendix. First, we consider the illustrative example of a constant cell pote ntial with a hard wall at r=rhw>0, w(r) =/braceleftbigg woif 0≤r<r hw ∞ifr>r hw(12) which satisﬁes the assumptions stated above. The integral ( 6) is easily performed in this case to yield C(β) =4 3πr3 hwβe−woβ(13) which implies m=−wo, the well depth, and F(β) =4 3πr3 hw, the volume of negative energy. Consistent with the general results above, F(β) is constant in this case since S=∅. For continuous potentials w(r), however, the prefactor F(β) must be strictly decreasing because S∝\e}atio\slash=∅. The integral equation (6) can be simpliﬁed by a change of vari ables from radius to volume. In terms of a shifted cell potential versus volume u(x) =w(r)−wowherex=4π 3r3(14) the integral equation is reduced to the form C(β) =βF(β)e−w(ro)β(15) where F(β) =/integraldisplay∞ 0e−βu(x)dx. (16) Sinceu(x)≥0 by construction, the function F(β) is clearly non-increasing. In the Appendix, it is proved that if the potential varies continuously near its minimum ( in a very general sense), then F(β) does not decay exponentially. Since F(β) also positive and bounded above, we conclude lim β→∞logC(β)/β=−wo (17) which implies m=−wo. Therefore, the slope of the Langmuir curve on a “van’t Hoﬀ plot” (logCversus β= 1/kT)in the low temperature limit is equal to (minus) the minimum e nergy of the cell potential . Since it is generally observed that mis positive, as in the case of ethane and cyclopropane clathr ate-hydrates shown in 4Figs. 1–2, the cell potential must be attractive, wo=−m<0, which simply indicates that the total internal energy is lowered by the introduction of guest molecules int o clathrate-hydrates. The fact that F(β) must be nondecreasing has important consequences for the e xistence of solutions which are piecewise continuous and bounded below. For example, co nsider the class of Langmuir curves of the form C(β) =βνemβ(18) which is useful in ﬁtting experimental data (see below). We h ave already addressed the borderline case, ν= 0, in which a discontinuous hard-wall solution is possible. Al though it is not obvious a priori , there are no solutions to the inverse problem if ν >0, since in that case F(β) =βν−1is increasing. On the other hand, if ν >0, then well behaved continuous solutions are possible, bec auseF(β) is strictly decreasing. B. Low Temperature Asymptotics of the Langmuir Curve From Eq. (17), we see that the minimum energy wo=w(ro) determines the leading order asymptotics ofC(β) in the low temperature limit. More generally, one would exp ect thatC(β) at low temperatures is completely determined by shape of the cell potential at low e nergies, close to its minimum. Using standard methods for the asymptotic expansion of Laplace integrals [ 19], it is straightforward to provide a mathematical basis for this intuition. For simplicity, here we consider t he usual case of a parabolic minimum w(r) =wo+1 2k(r−ro)2+O((r−ro)3) (19) for some constants k>0 andro≥0, although below we will derive many exact solutions with no n-parabolic minima. Due to the factor of r2appearing in the integrand in Eq. (6), the two cases of a non-c entral or central minimum,ro>0 andro= 0, respectively, must be treated separately. Physically, this qualitative diﬀerence between central and non-central-wells is due to the spheric al averaging process going from Eq. (2) to Eq. (6): A central well in w(r) corresponds to a unique local minimum of the multidimensio nal potential Φ, but a non-central-well in w(r) corresponds to a nonlocal minimum of Φ which is smeared acro ss a sphere of radius ro. Beginning with non-central-well case, r>0, we have the following asymptotics as Re β→ ∞: C(β)∼4πr2 oβ/integraldisplayro+ǫ ro−ǫe−β(wo+1 2k(r−ro)2)dr ∼4πr2 o/parenleftbigg2β k/parenrightbigg1/2 e−βwo/integraldisplay∞ −∞e−t2dt = 4πr2 o/parenleftbigg2πβ k/parenrightbigg1/2 e−βwo(20) which is the usual leading order term in the expansion of a Lap lace integral [19]. Therefore, the experimental signature of a non-central-well is a Langmuir constant whic h behaves at low temperatures like C(β)∼Coemββ1/2asβ→ ∞. (21) Comparing (20) and (21), we can identify the well depth wo=−m, consistent with the general arguments above, but it is impossible to determine independently the l ocation ofroand the curvature kof the minimum. Instead, any roandksatisfying 4 πr2 o/radicalbig 2π/k=Cowould exactly reproduce the same large- βasymptotics of the Langmuir curve (as would a completely diﬀerent centra l-well solution described in section V). This degeneracy of non-central-well solutions revealed in the l ow temperature asymptotics is actually characteristic of all non-central-well solutions, as explained below. In the central-well case, ro= 0, the asymptotics must be carried out more carefully becau se the leading term derived in (20) vanishes: C(β)∼4πβ/integraldisplayǫ 0e−β(wo+1 2kr2)r2dr 5∼4π/parenleftbigg2 k3β/parenrightbigg1/2 e−βwo/integraldisplay∞ 0t1/2e−tdt =/parenleftbigg2π k/parenrightbigg3/2e−βwo β1/2. (22) The experimental signature of a parabolic central well in th e Langmuir curve, C(β)∼Coemββ−1/2asβ→ ∞, (23) is qualitatively diﬀerent from (21), which provides an unam biguous way to separate the two cases using low temperature measurements. Moreover, unlike the non-ce ntral-well case, the curvature k= 2πC−2/3 oof a parabolic central minimum is uniquely determined by the lo w temperature asymptotics of the Langmuir curve. Consistent with asymptotic results, we shall see in s ection III that central-well solutions to the inverse problem are unique, while non-central-well solutions are n ot. C. Suﬃcient Conditions for the Existence of Solutions The primary diﬃculty in solving Eq. (6) lies in its being a non linear integral equation of the “ﬁrst kind” for which no general theory of the existence and uniqueness of so lutions exists [20,21]. In the linear case, however, there is a special class of ﬁrst-kind equations which can be s olved using Laplace, Fourier or Mellin transforms, namely integral equations of the additive or multiplicativ e convolution type [22,23] Ψ(x) =/integraldisplay∞ −∞K(x−y)ψ(y)dy (24) or Ψ(x) =/integraldisplay∞ 0K(xy)ψ(y)dy, (25) respectively, where ψ(x) is the unknown function and Ψ( x) is given. Integral equations of the form (25) often arise in statistical mechanics [24]. For example, the parti tion function Q(β) for the canonical ensemble is simply the Laplace transform of the energy density of states n(E) Q(β) =/integraldisplay∞ 0e−βEn(E)dE, (26) which means that the n(E) is the inverse Laplace transform of Q(β). In this way, the density of states in the canonical ensemble can also be obtained from derivatives of the partition function such as the speciﬁc heat. The formalism based on inverse Laplace transforms can also b een extended in various ways to solve related multiplicative convolution equations arising in Bose-Ein stein [11,25–27] and Fermi-Dirac statistics [28]. Although our nonlinear, ﬁrst-kind equation (6) is not of the convolution type because the unknown function w(r) appears in the exponent, it does somewhat resemble a Laplac e transform. This connection is more obvious in the alternative formulation (16) relating F(β) andu(x), which is equivalent to the original equation (6) according to the analysis above. In the next section, it is sh own that physically reasonable solutions exist ifF(β) has an inverse Laplace transform f(y) which is positive, nondecreasing and non-constant for y >0. In light of the necessary condition that F(β) be analytic the right half plane Re β > c , the deﬁning contour integral for f(y) f(y) =1 2πi/integraldisplayc′+i∞ c′−i∞eβyF(β)dβ (27) must converge for any c′>c. By closing the contour in the left half plane, it can be shown that a suﬃcient (but not necessary) condition to ensure the assumed propert ies off(y) is thatF(β) decay in the left half plane (lim ρ→∞\|F(ρeiθ)\|= 0 forπ/2≤θ≤3π/2) and have isolated singularities only on the negative real axis or at the origin with positive real residues. The particular exam ples ofF(β) considered in section V satisfy these conditions, but the weaker assumptions above regarding f(y) suﬃce for the following general derivation. 6III. ANALYTICAL SOLUTIONS FOR ARBITRARY LANGMUIR CURVES A. The Unique Central-Well Solution It is tempting to change variables y=u(x) in the integral (16) to reduce it to a Laplace transform, but care must be taken since u−1(y) may not be single-valued. This leads us to treat solutions w hich are monotonic separately from those from those which are not, an important distinction foreshadowed by the asymptotic analysis above. As a natural ﬁrst case, we seek diﬀerentiabl e solutions u(x) which are strictly increasing without bound ( u(∞) =∞) from a central minimum ( u(0) = 0). Such “central-well solutions” correspond to cell potentials w(r) which are strictly increasing from a ﬁnite minimum w(0) =woat the center of the cage. We proceed by considering the inverse cell potential v(y) =u−1(y) with units of volume as a function of energy, which is single-valued and strictly increasing with v(0) = 0, as shown in Fig. 3(a). With the substitution y=u(x), Eq. (16) is reduced to Laplace’s integral equation [23] fo r the unknown function v′(y) F(β) =/integraldisplay∞ 0e−βydy u′(u−1(y))=/integraldisplay∞ 0e−βyv′(y)dy. (28) Upon taking inverse Laplace transforms, we arrive at a diﬀer ential equation for v(y) v′(y) =f(y) (29) whose unique solution is v(y) =/integraldisplayy 0f(y)dy (30) using the boundary condition v(0) = 0. According to (29), the continuity of f(y) fory >0 (which is not assumed) would the guarantee diﬀerentiability of v(y) fory>0, and hence of u(x) forx>0. Equivalently, we can also simplify (28) with an integration by parts F(β) =β/integraldisplay∞ 0e−βyv(y)dy. (31) Therefore, the inverse cell potential is given by v(y) =g(y) (32) whereg(y) is the inverse Laplace transform of the function G(β) =F(β) β=C(β)eβwo β2. (33) The cell potential u(x) is determined implicitly by the algebraic equation g(u) =x. (34) Returning to the original variables, we have a general expre ssion forw(r) in the central-well case w(r) =wo+g−1/parenleftbigg4 3πr3/parenrightbigg . (35) This equation uniquely determines the central-well potent ial that exactly reproduces any admissible Langmuir curve. 7B. Non-Central-Well Solutions The simplest kind of non-central-well solution is the centr al-well (35) shifted by a “hard-core” radius rhc>0 w(r) =/braceleftbigg∞ if 0≤r<r hc wo+g−1/bracketleftbig4 3π(r3−r3 hc)/bracketrightbig ifr≥rhc(36) which exempliﬁes a peculiar general property of our integra l equation: An arbitrary hard core can be added to any solution. Note that, if u(x) is any solution of the rescaled equation (16), then so is ˜u(x) =/braceleftbigg ∞ if 0≤x<x hc u(x−xhc) ifx≥xhc(37) for any hard-core volume xhc≥0. The proof is simple: /integraldisplay∞ 0e−β˜u(x)dx=/integraldisplay∞ xhce−βu(x−xhc)dx=/integraldisplay∞ 0e−βu(x)dx=F(β). (38) Physically, a hard core for the cell potential could represe nt the presence of a second guest molecule (in a spherically symmetric model) in the same clathrate-hydrat e cage. Alternatively, a hard-core could represent a water molecule (again in a spherically symmetric model) at the node of several adjacent clathrate cages, in which case the cell potential actually describes the “super -cage” surrounding the central water molecule. The arbitrary hard-core just described only hints at the vas t multiplicity of non-monotonic solutions to the integral equation (16), which is a common characteristic of ﬁrst-kind equations [21]. Next, we consider the general case of a non-central-well, shown in Fig. 3(b), whic h includes (36) as a special case. To be precise, we seek continuous solutions u(x) on an interval x1< x < x 2composed of a non-increasing function u−(x) and a nondecreasing function u−(x) which are piecewise diﬀerentiable and non-negative. We al so allow for a possible hard-core in the central region x<x 1as well as a “hard wall” beyond the clathrate cage boundary x>x 2. The general form of such a non-central-well solution is u(x) =  ∞ if 0<x<x 1 u−(x) ifx1<x≤xo u+(x) ifxo≤x<x 2 ∞ ifx<x 2(39) whereu−(xo) =u+(xo). We do not assume u′ +(xo) =u′ −(xo), which would imply diﬀerentiability at the minimumu′(xo) = 0, although we do not rule out this case either. Instead, we allow for the mathematical possibility of a discontinuous ﬁrst derivative at xo, i.e. a “cusp” at the well position, at least for the moment. As before, it is convenient to express the solution (39) in te rms of two diﬀerentiable functions v−(y) =u−1 −(y) andv+(y) =u−1 +(y) which describe the multi-valued inverse cell potential. N ote thatv−(∞) =x1,v+(∞) =x2 andv−(0) =v+(0) =xo. In terms of the inverse cell potentials, the integral equat ion (16) takes the form F(β) =/integraldisplay∞ 0e−βu(x)dx =/integraldisplayxo x1e−βu−(x)dx+/integraldisplayx2 xoe−βu+(x)dx =/integraldisplay∞ 0e−βy/bracketleftbig v′ +(y)−v′ −(y)/bracketrightbig dy (40) which implies v′ +(y)−v′ −(y) =f(y). (41) In this case, the continuity of f(y) would only guarantee the diﬀerentiability of the diﬀerenc ev+(y)−v−(y), but not of the individual functions v+(y) andv−(y). Integrating (40) by parts before taking the inverse transform yields a general expression for the solution v+(y)−v−(y) =g(y) (42) 8where again g(y) is the inverse Laplace transform of F(β)/β. Unfortunately, we have two unknown functions and only one equation, so the set of non-central-well soluti ons is inﬁnite. The scaled Langmuir curve F(β) uniquely determines only v+(y)−v−(y), the volume diﬀerence as a function of energy between the two branches of the scaled cell potenti alu(x), but not the branches v+(y) andv−(y) themselves. An inﬁnite variety of non-central-well soluti ons, which exactly reproduce the same Langmuir curve as the central well solution, can be easily generated by choo sing any non-increasing, non-negative, piecewise diﬀerentiable function v−(y) such that the function v+(y) deﬁned by (42) is nondecreasing. Even the position of the well v−(0) =xocan be chosen arbitrarily. For example, one such family of solutions with a central “sof t-core” (x1= 0) is given by u(x) =/braceleftbigg u−(x) if 0 ≤x≤xo u+(x) ifx≥xo(43) where v−(y) =u−1 −(y) =/braceleftbigg xo−aybif 0≤y≤yc 0 if y≥yc(44) and v+(y) =u−1 +(y) =v−(y) +g(y), (45) for anya,b > 0 andxo≥0. (In the limit a→0, we recover the unique central-well solution.) Note that yc=u(0) = (xo/a)1/bis the height of the central maximum of the potential. These s olutions, all derived from a single Langmuir curve, exist whenever g(y) increases quickly enough that v+(y) is nondecreasing, which is guaranteed if g′(y)≥abyb−1for 0<y<y c. Another family of non-central-well solutions with a soft-c ore can be constructed with the choice v−(y) =/braceleftbigg xo−λg(y) if 0 ≤y≤yc 0 if y≥yc(46) for any 0< λ < 1 andxo≥0, whereyc=g−1(xo/λ). In this case, the cell potential is easily expressed in terms ofg−1(x) as u(x) =  g−1/parenleftbigxo−x λ/parenrightbig if 0≤x≤xo g−1/parenleftBig xo−x 1−λ/parenrightBig ifxo≤x≤xc g−1(x) ifx≥xc(47) wherexc=v+(yc) =xo/λ. This class of solutions exists whenever g(y) is nondecreasing (or f(y)≥0). If yc=∞, then there is a hard core u(0) =∞. Otherwise, if there is a soft core u(0) =yc<∞, then there is typically a cusp (discontinuous derivative) at xc, as explained below. As demonstrated by the preceding examples, it is simple to ge nerate an enormous variety of non-central-well solutions, with an arbitrarily shaped soft or hard core, and an arbitrary position of the minimum. In spite of the multiplicity of non-central-well solutions, however, our analysis of the inverse problem at least determines v+(y)−v−(y) uniquely from any experimental Langmuir curve. This important analytical constraint is not satisﬁed by empirical ﬁtting procedures. C. Soft Cores and Outer Cusps Non-central-well solutions with a soft-core satisfy v−(y) = 0 fory≥yc>0, as in the examples above. In such cases, v+(y) =g(y) fory≥ycregardless of whether or not there is an outer hard wall, whic h implies thatu(x) =g−1(x) forx≥xc, wherexc=g−1(yc). Iff(y) is continuous for y >0, then, unless u−(x) has an “inverted cusp” at the origin ( v′ −(yc) = 0 andu′ −(0+) =−∞), any non-central-well solution u(x) with a soft-core must have a cusp at x=xc, as in the examples above. This “outer cusp” in u(x) could only be avoided iff(y) itself has a cusp at ycwhich would allow v+(y) to be continuous. However, an inverted cusp in u(x) at the origin does not necessarily imply an cusp in w(r) at the origin due the transformation x=4 3πr3. For example, if v−(y)∼(yc−y)3/2asy→yc, oru(x)∼yc−x2/3asx→0, thenw(r) would have a physically 9reasonable, parabolic soft core w(r)∼wo+yc−(4π/3)2/3r2asr→0. Nevertheless, even in such cases, if f(y) were continuous for all y>0, then both u(x) andw(r) would have unphysical second-derivative discontinuitie s atx=xcrelated to the soft core. In general, a continuously diﬀeren tiable, non-central-well solution with a central soft core could only arise if f(y) were discontinuous at some yc>0, and such discontinuities are generally not present. D. Cusps at a Non-Central Minimum As mentioned above, the behavior of the cell potential near i ts minimum (whether central or not) is deter- mined by the behavior of the Langmuir curve at low temperatur e, or equivalently, at large inverse temperature, β= 1/T. The Laplace transform formalism makes this connection tra nsparent and mathematically rigorous. The asymptotic behavior of G(β) as Reβ→ ∞ is related to the asymptotics of the inverse transform g(y) as y→0, which in turn governs the local shape of the energy minimum through Eq. (32) for a central well or Eq. (42) for a non-central well. The leading order asymptoti cs has already been computed above for parabolic minima, but the general solutions above show how various non -local properties of the potential are related to ﬁnite temperature features of the Langmuir curve. Here, we c omment on a subtle diﬀerence in diﬀerentiability between central and non-central-wells, related to the smal l-ybehavior of f(y). For typical sets of experimental data, including the van’t H oﬀ form (7), the prefactor F(β) has a bounded inverse Laplace transform in the neighborhood of the origin lim y→0f(y) =f(0)<∞. (48) This generally implies the existence of a cusp at a non-centr al minimum of u(x), which is signiﬁed by a nonzero right and/or left derivative. When u(x) is diﬀerentiable at its minimum, it satisﬁes u′ −(x− o) =u′ +(x+ o) = 0. In the central-well case xo= 0, the existence of a cusp in u(x) follows from (29) u′(0+) = 1/v′(0+) = 1/f(0)>0, (49) but this does not imply a cusp in the unscaled potential w(r) as long as f(0)>0 because in that case w(r)−wo=u(4πr3/3)∼(4π/3f(0))r3asr→0+. (50) In the non-central-well case, however, the bounded inverse transform (48) implies a cusp at the minimum because,v′ +(0)−v′ −(0) =f(0)<∞from (42) along with v′ +(0)≥0 andv′ −(0)≤0 implies that v′ +(0)<∞ and/orv′ −(0)>−∞which in turn implies u′ +(0)>0 and/oru′ −(0)<0. Unlike the central-well case, however, a cusp inu(x) at the non-central minimum xo>0 implies a cusp at the corresponding non-central minimum ofw(r). Therefore, we conclude that whenever (48) holds, the only physically reasonable solution is the central-well solution (35). E. Asymptotics at High Energy and Temperature The high energy behavior of the cell potential is related to ( but not completely determined by) the high temperature asymptotics of the Langmuir hydrate constant, through the function g(y). For example, the cell potential would have a hard wall at x2<∞, if and only if g(y) were unbounded lim y→∞g(y) =∞. (51) Since the empirical modeling of Langmuir curves using Kihar a potentials assumes an outer hard-core, Eq. (51) could be used to test the suitability of using the Kihara pote ntial form, although experimental data is often not available at suﬃciently high temperatures to make a full y adequate comparison (see below). Whenever (51) holds, the non-central-well solutions u(x) are also universally asymptotic to the central-well solut ion u(x)∼g−1(x) (52) at large volumes x→ ∞. This follows from (42) and the fact that v−(y) is bounded, which implies v+(y)∼ g(y). The exact inversions performed in section V provide furth er insight into the relationship between small βasymptotics of the Langmuir constant and high energy behavi or of the cell potential. 10IV. LANGMUIR CURVES WITH VAN’T HOFF TEMPERATURE DEPENDENCE Experimental Langmuir hydrate-constant curves C(β) are well ﬁt by an ideal van’t Hoﬀ temperature de- pendence (7), demonstrated by straight lines on Arrhenius l og-linear plots logC=mβ+ logCo (53) as shown in Figs. 1 and 2 for ethane ( Co= 4.733×10−7atm−1,m= 9.4236 kcal/mol) and cyclopropane (Co= 1.9041×10−7atm−1,m= 10.5939 kcal/mol) clathrate-hydrates [7]. This data is analyz ed carefully in section VI, where alternative functional forms are conside red. In the ideal van’t Hoﬀ case, we have F(β) = Co/βandG(β) =Co/β2. The inverse Laplace transforms of these functions are simp lyf(y) =CoH(y) and g(y) =CoyH(y), respectively, where H(y) is the Heaviside step function. We begin by discussing the unique central-well solution, wh ich is illustrated by the solid line in Fig. 4 for the case of ethane. The central-well solution is linear in vo lumeu(x) =g(x) =CoyH(y), and cubic in radius w(r) =4πr3 3Co−m. (54) A curious feature of this exact solution is that it has a vanis hing “elastic constant”, w′′(0) = 0, a somewhat unphysical property which we address again in section VI. The simple form of (54) makes it very appealing as a means of in terpreting experimental data with van’t Hoﬀ temperature dependence. We have already noted that the s lope of a van’t Hoﬀ (Fig. 2) plot of the Langmuir constant is equal to the well depth m=−wo, but now we see that the y-intercept log Cois related to the well-size, e.g. measured by the volume of negative ene rgymCo. This volume corresponds to a spherical radius of rs=/parenleftbigg3mCo 4π/parenrightbigg1/3 (55) which is 0.4180˚A for ethane and 0 .3208˚A for cyclopropane. There are inﬁnitely many non-central-well solutions repro ducing van’t Hoﬀ temperature dependence, but each of them has unphysical cusps (discontinuous derivativ es). There will always be a cusp at the minimum of the potential, since f(y) satisﬁes the general condition (48). For example, the cent ral-well solution can be shifted by an arbitrary hard-core radius ro≥0 w(r) =/braceleftBigg ∞ if 0≤r<r o 4π(r3−r3 o) 3Co−mifr≥ro(56) In the case of a soft core, there must be a second cusp in the out er branch of the potential at the same energy as the inner core due to the continuity of f(y), as explained above. This is illustrated by the following piecewise cubic family of soft-core solutions of the genera l form (47): w(r) =/braceleftBigg 8π\|r3 o−r3\| 3Co−mif 0≤r≤21/3ro 4πr3 3Co−m ifro≥21/3ro(57) which are shown in Fig. 4 in the case of ethane guest molecules . An inﬁnite variety of other piecewise diﬀerentiable solutions exactly reproducing van’t Hoﬀ dep endence of the Langmuir curve could easily be generated, as described above, but each would have unphysic al cusps. Previous studies involving ad hoc ﬁtting of Kihara potentials have reported non-central-wel ls [1], but these empirical ﬁts may be only approximating various exact, cusp -like, non-central-well solutions, such as those described above. Moreover, given that the central-well sol ution (54) can perfectly reproduce the experimental data, it is clear that the results obtained by ﬁtting Kihara p otentials to Langmuir curves are simply artifacts of the ad hoc functional form, without any physical signiﬁcance. Kihara ﬁts also assume a hard wall at the boundary of the clathrate cage (by construction), whereas a ll of the exact analytical solutions (both central and non-central-wells) have the asymptotic dependence w(r)∼4πr3 3Co(58) 11asr→ ∞ according to (52). Any deviation from the cubic shape at larg e radii, such as a hard wall, would be indicated by a deviation from van’t Hoﬀ behavior at high te mperatures, but such data would be diﬃcult to attain in experiments (see below). The preceding analysis shows that the only physical informa tion contained in a Langmuir curve with van’t Hoﬀ temperature dependence is the depth woand the eﬀective radius rsof the spherically averaged cell potential, which takes the unique form (54) in the central-w ell case. In hindsight, the simple two-parameter form of the potential is not surprising since a van’t Hoﬀ depe ndence is described by only two parameters, m andCo. It is clearly inappropriate to ﬁt more complicated ad hoc functional forms, such as Eq. (4) derived from the Kihara potential, since they contain extraneous ﬁt ting parameters and do not reproduce the precise shape of any exact solution. V. ANALYSIS OF POSSIBLE DEVIATIONS FROM VAN’T HOFF BEHAVIOR A. Dimensionless Formulation The general analysis above makes it possible to predict anal ytically the signiﬁcance of possible deviations from van’t Hoﬀ temperature dependence, which could be prese nt in the experimental data (see below). We have already discussed the experimental signatures of vari ous low and high energy features of the cell potential in the asymptotics of the Langmuir curve. In this section, we derive exact solutions for Langmuir curves of the form (8) where F(β) is a rational function. Such cases correspond to logarithm ic corrections of linear behavior on a van’t Hoﬀ plot of the Langmuir curve, which are s mall enough over the accessible temperature range to be of experimental relevance, in spite of the domina nt van’t Hoﬀ behavior seen in the data. Fitting to the dominant van’t Hoﬀ behavior (53) introduces n atural scales for energy, m, and pressure, C−1 o, so it is convenient and enlightening to introduce dimension less variables. With the deﬁnitions ˜β=mβ, ˜C(˜β) =C(˜β/m)/Co,and ˜F(˜β) =F(˜β/m)/mC o, (59) the Langmuir curve can be expressed in the dimensionless for m ˜C(˜β) =˜β˜F(˜β)e˜β. (60) For consistency with these deﬁnitions, the other energy-re lated functions in the analysis are nondimensionalized as follows ˜G(˜β) =G(˜β/m)/m2Co,˜y=y/m, ˜f(˜y) =f(m˜y)/Co,˜g(˜y) =g(m˜y)/mC o, (61) where ˜f(˜y) and ˜g(˜y) are the inverse Laplace transforms of ˜F(˜β) and ˜G(˜β), respectively. The natural scales for energy and pressure also imply natural scales for volume, mCo, and distance, rs, as described in the previous section, which motivates the following deﬁnitions of the di mensionless cell potential versus volume ˜x=x/mC o,˜u(˜x) =u(mCo˜x)/m (62) and radius ˜r=r/rs,˜w(˜r) =w(rs˜r)/m. (63) Note that ˜x= ˜r3. With these deﬁnitions, the central-well solution takes th e simple form, ˜u(˜x) = ˜g−1(˜x) (64) in terms of the dimensionless volume, or ˜w(˜r) =−1 + ˜g−1(˜r3) (65) in terms of the dimensionless radius. We now consider variou s prefactors ˜F(˜β) which encode valuable infor- mation about the energy landscape in various regions of the c lathrate cage. 12B. The Interior of the Clathrate Cage 1. Power-Law Prefactors The simplest possible correction to van’t Hoﬀ behavior invo lves a power-law prefactor ˜F(˜β) =˜β−µfor anyµ>0, (66) which corresponds to a logarithmic correction on a van’t Hoﬀ plot of the Langmuir constant log˜C=˜β+ (1−µ)log(˜β) (67) as shown in Fig. 5(a). In this case, we have ˜f(˜y) = ˜yµ−1H(˜y)/Γ(µ) and ˜g(˜y) = ˜yµH(˜y)/Γ(µ+ 1), (68) where Γ(z) is the gamma function. In general, power-law prefactors at low temperatures signify an energy minimum with a simple polynomial shape. 2. The Central-Well Solution The unique central-well solution is also a simple power law ˜u(˜x) = [Γ(µ+ 1)˜x]1/µ(69) or equivalently ˜w(˜r) =−1 + Γ(µ+ 1)1/µ˜r3/µ. (70) The cubic van’t Hoﬀ behavior is recovered in the case µ= 1, as is the (asymptotic) parabolic behavior from (19) and (23) in the case µ= 3/2. Because ˜ w(˜r)+1∝˜r3/µ, a power-law correction to van’t Hoﬀ behavior with a positive exponent ( µ<1) corresponds one which is “wider” than a cubic, while a nega tive exponent ( µ>1) corresponds to a potential which is “more narrow” than a cubi c, as shown in Fig. 5(b). On physical grounds, the smooth polynomial behavior described by (70) is always t o be expected near the minimum energy of the cell potential. Therefore, the power-law correction to van ’t Hoﬀ behavior (67) has greatest relevance for low temperature measurements in the range ˜β≫1, from which it determines interatomic forces in the interi or of the clathrate cage at low energies \|˜w(˜r)\| ≪1. 3. Non-Central-Well Solutions As described above, there are inﬁnitely many non-central-w ell solutions. One family of solutions of the form (47) withλ= 1/2 is given by ˜w(˜r) + 1 =/braceleftBigg/bracketleftbig 2Γ(µ+ 1)\|˜r3−˜r3 o\|/bracketrightbig1/µif 0≤˜r≤21/3˜ro/bracketleftbig Γ(µ+ 1)˜r3/bracketrightbig1/µif ˜r≥21/3˜ro(71) where ˜ro=ro/rsis arbitrary, as shown in Fig. 5(c) for the case ˜ ro= 0.65. These solutions are unphysical since they all have cusps at ˜ r= 21/3˜ronear the outer wall of the cage. However, they can still have r easonable behavior near the minimum at ˜ rofor certain values of µ, which could have experimental relevance for low temperature measurements. Near the minimum, the exact solu tions (71) have the asymptotic form ˜w(˜r)∼ −1 +/bracketleftbig 6Γ(µ+ 1)˜r2 o\|˜r−˜ro\|/bracketrightbig1/µas ˜r→˜ro, (72) which is cusp-like for µ >1/2, but diﬀerentiable for 0 < µ≤1/2. For example, the non-central-well has a parabolic shape in the case µ= 1/2, which agrees with the asymptotic analysis in Eqs. (19)–(2 1) when the units are restored, and it has a cubic shape when µ= 1/3. On the other hand, in the central-well case µ= 3/2 and µ= 1 correspond to parabolic and cubic minima, respectively. Therefore, this example nicely illustrates the diﬀerence between the low-energy asymptotics of central an d non-central-wells described above in section II, which would be useful in interpreting any experimental Lang muir constant data showing deviations from van’t Hoﬀ behavior. 13C. The Outer Wall of the Clathrate Cage 1. Rational Function Prefactors The behavior of the Langmuir curve in the high temperature re gion˜β=O(1) is directly linked to properties of the outer wall of the clathrate cage, described by the cell potential at high energies ˜ w(˜r) + 1 =O(1). Although this region of the Langmuir curve does not appear to be accessible in experiments (see below), in this section we derive exact solutions possessing diﬀerent kinds of outer walls, whose faint signature might someday be observed in experiments at moderate temperature s. In order to isolate possible eﬀects of the outer wall, we consider Langmuir curves which are exactly as ymptotic to the usual van’t Hoﬀ behavior at low temperatures with small logarithmic corrections (on a v an’t Hoﬀ plot) at moderate temperatures. These constraints suggest choosing rational functions for ˜F(˜β) such that ˜F(˜β)∼1/˜βasβ→ ∞. 2. Central Wells with Hard Walls We begin by considering a “shifted power-law” prefactor ˜F(˜β) = 1/(˜β+α),for anyα>0 (73) which corresponds to a shifted logarithmic deviation from v an’t Hoﬀ behavior, log˜C=˜β−log(1 +α/˜β). (74) As shown in Fig. 6(a), this suppresses the Langmuir constant at high temperatures, which intuitively should be connected with an enhancement of the strength of the outer wall compared to the cubic van’t Hoﬀ solution. Taking inverse Laplace transforms we have ˜f(˜y) =e−α˜yH(˜y) and ˜g(˜y) = (1 −e−α˜y)H(˜y)/α, (75) and indeed, since ˜ g(˜y) is bounded, all solutions must have a hard wall regardless o f whether or not the well is central, as described above. For example, the unique centra l-well solution is ˜w(˜r) =−1−log(1−α˜r3)/αfor 0≤˜r<α−1/3(76) which has an outer hard wall at ˜ r=α−1/3, as shown in Fig. 6(b). The solution is also asymptotic to the cubic van’t Hoﬀ solution at small radii ˜ r≪α−1/3. Therefore, in the limit α→0, the radius of the outer hard wall diverges, and the solution reduces to the cubic shape as the deviation from van’t Hoﬀ behavior is moved to increasingly large temperatures. Since empirical ﬁttin g with Kihara potential forms arbitrarily assumes an outer hard wall, this example provides analytical insight i nto the nature of the approximation at moderate to high temperatures, where the Langmuir constant should be su ppressed according to (74). 3. Central Wells with Soft Walls Next we consider the opposite case of a Langmuir constant whi ch is enhanced at high temperatures compared to van’t Hoﬀ behavior, which intuitively should indicate th e presence of a “soft wall”, rising much less steeply than a cubic function. An convenient choice is ˜F(˜β) =˜β/(˜β2−γ2),for anyγ>0. (77) which is analytic except for poles at β=±γon the real axis. Although this function diverges at β=γdue to the overly soft outer wall, the corresponding Langmuir cu rve log˜C=˜β−log/bracketleftBig 1−(γ/˜β)2/bracketrightBig (78) shown in Fig. 6(a) could have experimental relevance at mode rate temperatures ˜β≫γ, ifγwere suﬃciently small. In this case, we have 14˜f(˜y) = cosh(γ˜y)H(˜y) and ˜g(˜y) = sinh(γ˜y)H(˜y)/γ (79) which yields the central-well solution ˜w(˜r) =−1 + sinh−1(γ˜r3)/γ. (80) As shown in Fig. 6(b), this function follows the van’t Hoﬀ cub ic at small radii ˜ r≪γ−1/3but “softens” to a logarithmic dependence for large radii ˜ r≫γ−1/3. VI. INTERPRETATION OF EXPERIMENTAL DATA We begin by ﬁtting Langmuir constant curves, computed from e xperimental phase equilibria data, an equation of state, and a heat capacity model [7] for ethane an d cyclopropane clathrate-hydrates to the van’t Hoﬀ equation logC=mβ+b (81) using least-squares linear regression. This leads to rathe r accurate results, as indicated by the small uncer- tainties in the parameters displayed in Table I (63% conﬁden ce intervals corresponding to much less than one percent error). The high quality of the regression of log Conβis further indicated by correlation coeﬃcients very close to unity, 0 .99650 and 0 .99998 for the ethane and cyclopropane data, respectively. U sing the ﬁtted values formandCo=eb, the data for the two clathrate-hydrates can be combined int o a single plot in terms of the dimensionless variables ˜Cand˜β, as shown in in Fig. 7, which further demonstrates the common linear dependence. Converting the experimental data to dimensionless variabl es also reveals that the measurements correspond to extremely “low temperatures”. This is indicated by large values of ˜β=m/kT in the range of 16 to 24, which imply thatkTis less than 6% of the well depth m. As such, physical intuition tells us that the experiments c an probe the cell potential only very close to its minimum. This intuition is ﬁrmly supported by the asymptotic analysis above, which (converted to dimensionless variabl es) links the asymptotics of the Langmuir constant for˜β≫1 to that of the cell potential for \|˜r−˜ro\| ≪1. In this light, it is clear that any features of the cell potential other than the local shape of its minimum, which ar e determined by empirical ﬁtting, e.g. using Eq. (4) based on the Kihara potential, are simply artifacts o f anad hoc functional form, devoid of any physical signiﬁcance. Since the shape of the potential very close to its minimum sho uld always be well approximated by a polynomial (the leading term in its Taylor expansion), the a nalysis above implies that only simple power-law prefactors to van’t Hoﬀ behavior should be considered in ﬁtt ing low temperature data. Therefore, we reﬁt the experimental data, allowing for a logarithmic correction, logC=mβ+b+νlog(β) (82) as in Eq. (67). The results are shown in Table I, and the best-ﬁ t functions are displayed in dimensionless form in Fig. 7. In the case of ethane, the best-ﬁt value of µ= 1−νcorresponds to a roughly linear central-well solution ˜w∝˜r0.9or a cusp-like non-central-well solution ˜ w∝ \|˜t−˜ro\|0.3. Although these solutions are not physically reasonable, perhaps the qualitative increase i nµcompared with ideal van’t Hoﬀ behavior ( µ= 1) is indicative of a parabolic central well ( µ= 3/2). In the case of cyclopropane, we have µ=−1.4±0.9, which violates the general condition µ≥0 needed for the existence of solutions to the inverse proble m. If this ﬁt were deemed reliable, the the basic postulate of vdWP theory , Eq. (2), would be directly contradicted, with or without the spherical cell approximation (see the Append ix). It is perhaps more likely that the trend of decreasing µ <1 could indicate a non-central parabolic minimum in the sphe rically averaged cell potential (µ= 1/2). Although it appears there may be systematic deviations from ideal van’t Hoﬀ behavior in the experimental data for ethane and cyclopropane, ν∝\e}atio\slash= 0 orµ∝\e}atio\slash= 1, the results are statistically ambiguous. For both types of guest molecules, adding the third degree of freedom νsubstantially degrades the accuracy of the two linear parameters mandb, with errors increased by several hundred percent. Moreove r, the uncertainty in νis comparable to its best-ﬁt value. Therefore, it seems that we cannot trust the results with ν∝\e}atio\slash= 0, and, by the principle of Occam’s razor, we are left with the more parsimo nious two-parameter ﬁt to van’t Hoﬀ behavior, which after all is quite good, and its associated simple cubi c, central-well solution. 15On the other hand, there are diﬀerent two-parameter ﬁts, mot ivated by the inversion theory, which can describe the experimental data equally well, but which are s omewhat more appealing than the cubic solution in that they possess a non-vanishing elastic constant (seco nd spatial derivative of the energy). For example, the ﬁts can be done using (82) with the parameter νﬁxed at either 1 /2 or−1/2, corresponding to either a non-central or central, parabolic minimum, respectively . The results shown in Table I reveal that these physically signiﬁcant changes in the functional form have l ittle eﬀect on the van’t Hoﬀ parameters mand b= logCo. The diﬃculty with the experimental data as a starting point f or inversion is its limited range in ˜βof roughly one decade, which makes it nearly impossible to detect corre ctions proportional to log βrelated to diﬀerent polynomial shapes of the minimum. It would be very useful to e xtend the range of the data, using the analytical predictions to interpret the results. In general, it is noto riously diﬃcult to determine power-law prefactors multiplying a dominant exponential dependence, but at leas t the present analysis provides important guidance regarding the appropriate ﬁtting functions, which could no t be obtained by ad hoc numerical ﬁtting. Moreover, the clear physical meaning of the dominant van’t Hoﬀ paramet ers elucidated by the analysis also makes them much more suitable to describe experimental data than the ar tiﬁcial Kihara potential parameters. VII. CONCLUSION In this article, we have shown that pairwise intermolecular potentials (within the spherical shell approxi- mation Lennard-Jones and Devonshire) can be determined ana lytically from Langmuir constant versus tem- perature curves for clathrate-hydrates which contain only a single type of guest molecule occupying a single type of cage, e.g. ethane and cyclopropane clathrate-hydra tes. The availability of this analytical approach obviates the need for empirical ﬁtting procedures for such h ydrates [1,4]. Moreover, this approach also allows a systematic analysis of the functional forms which are typi cally assumed ad hoc . For example, the ubiqui- tous ﬁtting procedure involving Kihara potentials actuall y approximates unphysical exact solutions possessing cusps. Therefore, such empirical approaches cannot be expe cted to have predictive power beyond the data sets used in parameter ﬁtting. Further conclusions of our an alysis are listed below: •The experimental Langmuir constant data for ethane and cycl opropane clathrate-hydrates is very well ﬁt by an ideal van’t Hoﬀ dependence, which corresponds to a cu bic central well. However, the data is also equally consistent with both central and non-central ( spherically averaged) parabolic wells because the range of temperatures is insuﬃcient to distinguish betw een these cases. •Experimental data tends to be taken at very “low” temperatur es,kT≪m, which means that only the region of the potential very close to the minimum \|r−ro\| ≪rsis probed. Therefore, only simple polynomial functions are to be expected, and ﬁtting to more c omplicated functional forms, such as the Kihara potential, has little physical signiﬁcance. •Physically reasonable solutions to the inverse problem, wh ich are piecewise continuous and bounded below, exist only if the Langmuir curve has a low-temperatur e van’t Hoﬀ dependence, lim β→∞logC/β= m, with a prefactor function F(β) =C(β)e−mβ/βwhich is (i) analytic in some half plane Re β >c and (ii) positive, real and non-increasing on the real axis. In g eneral, the minimum of the potential is given bywo=−m. •For any admissible Langmuir curve, the unique central-well potential is given by Eq. (54). •The central-well solution for ideal van’t Hoﬀ temperature d ependenceC(β) =Coemβis a simple cubic given by Eq. (54). The attractive region of the potential has depthm, volumemCo, and radius rs= (3mCo/4π)1/3. Each non-central-well solution for van’t Hoﬀ dependence h as two unphysical cusps, one at the minimum. •There also exist inﬁnitely many non-central-well solution s of the general form (39), constrained only to satisfy Eq. (42). Several classes of such solutions with a ce ntral “soft core” (a ﬁnite maximum at the center of the cage) are described explicitly in Eqs. (43)–(4 7). •Each one of the multitude of non-central-well solutions wit h a soft-core typically possesses unphysical cusps (slope discontinuities), while the unique central-w ell solution is a well-behaved analytic function. 16•The experimental signature of a parabolic, non-central-we ll is a Langmuir curve that behaves like C(β)∼Coemββ1/2at low temperatures ( β→ ∞ ), while a parabolic central well corresponds to C(β)∼Coemββ−1/2. •If there is a pure power-law prefactor multiplying van’t Hoﬀ behaviorC(β) =Co(mβ)1−µemβwith µ >0, the central-well solution is also a power-law (70). For ce rtain values of the prefactor exponent 0<µ≤1/2, there are also non-central-well solutions with diﬀerent iable minima such as (71), although such solutions still possess cusps at higher energies. •Rational function prefactors multiplying van’t Hoﬀ behavi or, such as (74) or (78), are associated with non-cubic behavior at the outer wall of the cage, such as a “ha rd wall” (76) or a “soft wall” (80), respectively. These analytical results shed much more light on the physica l meaning of experimental Langmuir hydrate constant data than does the standard approach of numerical ﬁ tting. The method of “exact inversion” developed here may also be useful in analyzing other statistical therm odynamical problems. In practical applications to clathrate-hydrates, the full power of our analysis could be exploited by measuring Langmuir hydrate constants over a broader range of temperatures than has previously bee n done. ACKNOWLEDGMENTS We would like to thank Z. Cao for help with the experimental ﬁg ures, J. W. Tester for comments on the manuscript, and H. Cheng for useful discussions. This work w as supported in part by the Idaho National Engineering and Environmental Laboratory. APPENDIX: MATHEMATICAL THEOREMS The ﬁrst theorem provides necessary conditions on the Langm uirC(β) so that the cell potential w(r) is bounded below and continuous. It also interprets the slope o f a van’t Hoﬀ plot of the Langmuir curve in the low temperature limit as the well depth, under very general c onditions. As pointed out in the main text, it is convenient to view the inverse temperature βas a complex variable. Theorem 1 Letw(r)be real and continuous (except at possibly a ﬁnite number of d iscontinuities) for r≥0 with a ﬁnite minimum, w(r)≥wo=w(ro)>−∞for somero≥0, and suppose that the integral C(β) = 4πβ/integraldisplay∞ 0e−βw(r)r2dr (83) converges for some β=con the real axis. Then C(β) =βF(β)e−woβ(84) where the complex function F(β)is (i) real, positive and non-increasing on the real axis for β >c and (ii) analytic in the half plane Re β >c. If, in addition, the set Sǫ={r≥0\|wo< w(r)< w o+ǫ}has nonzero measure for some ǫ=ǫo>0, then F(β)is strictly decreasing on the positive real axis (for β > c). Moreover, if Sǫhas ﬁnite, nonzero measure for every 0<ǫ<ǫ o, then lim β→∞logC(β)/β=−wo (85) where the limit is taken on the real axis. 17Proof: Deﬁne a shifted cell potential versus volume, u/parenleftbig4π 3r3/parenrightbig =w(r)−wo. Substituting u(x) forw(r) reduces Eq. (83) to Eq. (84), where F(β) =/integraldisplay∞ 0e−βu(x)dx. (86) Sinceu(x)≥0 is real, the function F(β) is real and positive for all real βfor which the integral converges. Moreover, for any complex βandβ′with Reβ >Reβ′>c, we have the bound \|F(β)\| ≤/integraldisplay∞ 0e−Reβ·u(x)dx≤/integraldisplay∞ 0e−Reβ′·u(x)dx≤F(c)<∞ (87) which establishes that the deﬁning integral (86) converges in the right half plane Re β≥cand is non-increasing on the real axis, thus completing the proof of (i). Next letw(r) be larger than its minimum value (but ﬁnite), w(ro)< w(r)<∞, on a setS∞of nonzero measure, so that 0 <u(x)<∞for the corresponding set of volumes. Then for every β >β′>con the real axis we have /integraldisplay S∞e−βu(x)dx</integraldisplay S∞e−β′u(x)dx. (88) On the complement Sc ∞= (0,∞)\S∞, eitheru(x) = 0 oru(x) =∞, which implies /integraldisplay Sc∞e−βu(x)dx=/integraldisplay Sc∞e−β′u(x)dx. (89) From Eqs. (88)–(89) we conclude that F(β) is strictly decreasing on the real axis. Next we establish the low-temperature limit (85). Given 0 <ǫ<ǫ o, we have the following lower bound for anyβ >c on the real axis: eǫβF(β) =/integraldisplay∞ 0e−β[u(x)−ǫ]dx≥eǫβ/2/integraldisplay Sǫ/2dx+/integraldisplay Sc ǫ/2e−β[u(x)−ǫ]dx≥eǫβ/2/integraldisplay Sǫ/2dx. (90) Combining this with the upper bound, F(β)≤F(c)<∞, we obtain e−ǫ/2Mǫ≤F(β)≤F(c) (91) whereMǫ=/integraltext Sǫ/2dxis a ﬁnite, nonzero constant (because Sǫ/2is assumed to have ﬁnite, nonzero measure). Substituting Eq. (84) in Eq. (91), we arrive at logβ+ logMǫ−ǫβ/2−βwo≤logC(β)≤logβ+ logF(c)−βwo (92) which yields −wo−ǫ/2≤lim β→0logC(β)/β≤ −wo. (93) The desired result is obtained in the limit ǫ→0. Finally, we establish the analyticity of F(β) in the open half plane Re β >c by showing that its derivative exists and is given explicitly by F′(β) =−/integraldisplay∞ 0e−βu(x)u(x)dx. (94) This requires justifying the passing a derivative inside th e integral (16), which we have just shown to converge for Reβ≥c. Using a classical theorem of analysis [20], it suﬃces to sho w that the integral in (94) converges uniformly for Re β > c +ǫfor everyǫ >0 because the integrand is a continuous function of βandx. (The possibility of a ﬁnite number of discontinuities in u(x) is easily handled by expressing (94) as ﬁnite sum of integrals with continuous integrands.) It is a simple calcu lus exercise to show that te−t<1/e, and hence te−(c+ǫ)t≤e−ct eǫ(95) 18for all realt≥0. This allows us to derive a bound on the “tail” of the integra l (94): \|/integraldisplay∞ Xe−βu(x)u(x)dx\| ≤/integraldisplay∞ Xe−Reβ·u(x)u(x)dx≤/integraldisplay∞ Xe−(c+ǫ)u(x)u(x)dx<1 eǫ/integraldisplay∞ Xe−cu(x)dx (96) which is independent of β. This uniform bound vanishes in the limit X→ ∞ because it is proportional to the tail of the convergent integral deﬁning F(β), which completes the proof. ✷ The proof of Theorem 1 does not depend in any way on the dimensi onality of the integral and thus can be trivially extended to the general multi-dimensional case o f vdWP theory without the spherical cell approxi- mation. Theorem 2 LetΦ(r,θ,φ,α,ξ,γ )≥Φ(ro,θo,φo,αo,ξo,γo) =wobe real and continuous, and suppose that the integral C(β) =β 8π2/integraldisplay Ve−βΦ(r,θ,φ,α,ξ,γ )r2sinθsinξdrdθdφdαdξdγ (97) converges for some β=c(real). Then all the conclusions of Theorem 1 hold. The six-dimensional integral (97) of Theorem 2 does not pres ent a well-posed inverse problem for the intermolecular potential Φ. However, the spherically aver aged integral equation (83) of Theorem 1 can be solved for the cell potential w(r) for a broad class of Langmuir curves C(β) speciﬁed in the following theorem. 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Chen and G.-Y. Li, Phys. Lett. A 149, 357 (1990). [26] B. D. Hughes, N. E. Frankel and B. W. Ninham, Phys. Rev. A 42, 3643 (1990). [27] B. W. Ninham, B. D. Hughes, N. E. Frankel and M. L. Glasser , Physica A 186, 441 (1992). [28] N.-X. Chen and G.-B. Ren, Phys. Lett. A 160, 319 (1991). TABLE I. Linear regressions of the experimental Langmuir co nstant data [7] for ethane and cyclopropane clathrate-hydrates on the form log C=mβ+b+νlog(β). Errors reﬂect 63% conﬁdence intervals. The rows where ν= 0,1/2,or−1/2 correspond to two-parameter ﬁts with νheld constant. Guest Molecule m b ν Ethane 9.422±0.043 -14.561 ±0.081 0 9.180±0.044 -14.419 ±0.082 1/2 9.664±0.043 -14.703 ±0.080 -1/2 10.52±0.85 -15.2 ±0.50 -2.3 ±1.8 Cyclopropane 10.594 ±0.012 -15.474 ±0.022 0 10.335 ±0.011 -15.302 ±0.021 1/2 10.853 ±0.012 -15.646 ±0.024 -1/2 9.36±0.47 -14.66 ±0.31 2.37 ±0.90 203.4 3.5 3.6 3.7 3.8 3.9 x 10−320406080100120140160180200 1/T(1/K)Langmuir Constant (atm−1) (a) 3.43.53.63.73.83.944.14.2 x 10−3200400600800100012001400160018002000 1/T(1/K)Langmuir Constant (atm−1) (c) 3.43.63.844.24.44.64.85 x 10−310002000300040005000600070008000900010000 1/T(1/K)Langmuir Constant (atm−1) (b) FIG. 1. Exponential ﬁts of Langmuir constants over the measu red temperature range plotted with linear axes for (a)-(b) cyclopropane and (c) et hane clathrate hydrates. An en- largement of the high temperature data for cyclopropane is s hown in (a). The experimental data is taken from Ref. [7]. 213.5 4 4.5 5 x 10−3100101102103104 1/T(1/K)Langmuir Constant (atm−1)(a) 3.4 3.6 3.8 4 4.2 x 10−3101102103 1/T(1/K)Langmuir Constant (atm−1)(b) FIG. 2. Exponential dependence with inverse temperature of experimental Langmuir curves from Fig. 1 plotted with log-linear axes for (a) ethan e and (b) cyclopropane clathrate hydrates. Straight lines indicate pure van’t Hoﬀ behavior. 220 0y x(a)g(y)v(y) u(x) ycv (y)- v (y)+ xo x xu (x) u (x)+ - 0 0y x1 2 g(y)(b) xc FIG. 3. (a) Sketch of a central-well solution, where xis the scaled volume of interaction andu(x) is the spherically averaged cell potential with inverse v(y) =u−1(y). (b) Sketch of a non-central-well solution composed of a non-increasin g function u−(x) and a nonde- creasing function u+(x) joined at a minimum of zero at xo, along with a possible hard core atx1and hard wall at x2. The two branches v−(y) and v+(y) of the multi-valued inverse cell potential v(y) are also shown, along with other variables deﬁned in the tex t. 23-10-50510 0 0.1 0.2 0.3 0.4 0.5 (A)r(kcal/mol) ow FIG. 4. Analytical cell potentials for the ethane clathrate -hydrate which exactly repro- duce the experimental data in Fig. 2(a). The unique central- well solution (54) is indicated by a solid line, while a family of non-central-well solution s with soft cores (57) is also shown as dashed lines with cusp-like minima at ro= 0.2,0.3,0.4˚A. Each of these solutions also has a cusp at r= 21/3ro, where the energy is the same as the central maximum, and beyond this distance joins the central-well solution. 24110 0 0.5 1 1.5 2 2.5 βC∼ ∼(a) µ µ µ= 1/2 = 1 = 3/2van't Hoff -1-0.500.511.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4µ µ µ= 1/2 = 1 = 3/2(b) rw~ ~van't Hoff µ µ µ= 1/2 = 1 = 3/2(c) rw~ ~-1-0.500.511.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4van't Hoff FIG. 5. Exact inversion of Langmuir curves with power-law co rrections to van’t Hoﬀ behavior in terms of dimensionless variables, as in Eq. (67) . (a) Plots of ˜C=C/C oversus ˜β=m/kT for the cases µ= 1/2,1,3/2. (b) The corresponding (unique) central-well potentials plotted as ˜ w=w/m versus ˜ r=r/(3mCo/4π)1/3. (c) Examples of soft-core non-central-well solutions of the form (71) with an arbitra rily chosen minimum at ˜ r= 0.65, which all have cusps at ˜ r= 21/3(0.65)≈0.819. 25α = 2, γ = 0 α = 0, γ = 2α = 0, γ = 1α = 1, γ = 0 α = γ = 0 βC∼ ∼(a) 0.1110100 0 0.5 1 1.5 2 2.5 3 3.5 4van't Hoff α = 0, γ = 2α = 0, γ = 1α = 2, γ = 0 α = 1, γ = 0 α = γ = 0 ∼ ∼-1-0.500.511.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4(b)van't Hoff rw FIG. 6. Exact inversion of Langmuir curves of the form ˜C=e˜β/(1 +α/˜β−(γ/˜β)2) in terms of the dimensionless variables deﬁned in Fig. 5. (a) La ngmuir curves in this class of functions have anomalous high temperature (small ˜β) behavior but are asymptotic to the van’t Hoﬀ curve ( α=γ= 0). (b) The corresponding central-well solutions depart f rom the cubic van’t Hoﬀ curve at large radius and energy, indicat ing diﬀerent properties at the boundary of the clathrate cage: “hard walls”, if α >0 and γ= 0, or “soft walls”, if α= 0 andγ >0. 26β∼161718192021222324 16 17 18 19 20 21 22 23 24Clog van't Hoff ethane data ethane fit cyclopropane data cyclopropane fit∼ FIG. 7. Experimental Langmuir hydrate constant data for eth ane and cyclopropane from Fig. 2 plotted in terms of the dimensionless variables ˜C=C/C oversus ˜β=m/kT , where Coandmare obtained by each data set ﬁtting to log C=m/kT +logCo. Ideal van’t Hoﬀ behavior ˜C= exp( ˜β) is shown as a solid line. Fits including power law correctio ns of the form log C=m/kT + log Co+νlog(m/kT ) are also shown as the dotted and dashed lines (which are very close to the van’t Hoﬀ line). 27
arXiv:physics/0011048v1 [physics.flu-dyn] 20 Nov 2000The Case for 2-D Turbulence in Antarctic Data Mayer Humi Department of Mathematical Sciences Worcester Polytechnic Institute 100 Institute Road Worcester, MA 0l609 August 4, 2013Abstract In this paper we examine the data that was collected at Haley S tation in Antarctica on June 22, 1987. Using a test devised by Dewan [9] we interpre t the ﬂow as one which represents two-dimensional turbulence. We also construct a model to interpret the spectrum of this data which is almost independent of the wave number for a range of frequencies. 21 Introduction Two dimensional turbulence has been the subject of intense t heoretical research [1, 2] and simulation experiments [3]. The reason for this interest st ems from the fundamental diﬀer- ences between 3-d isotropic and 2-d turbulence. To begin wit h, vortex stretching is absent in 2-d as a direct consequence of Navier-Stokes equations. F urthermore in 3-d the energy cascade is from the large eddies to small one but this process reverses itself in 2-d and leads to the formation of large scale coherent eddies. Another diﬀ erence between two and three dimensional turbulence exists in the inertial range of the s pectrum. Kraichnan showed [4] that in 2-d in addition to Kolmogorov inertial range there is (due to ensotrophy conservation in zero viscosity) another scaling law in the form E(k) =cη2/3k−3 where ηis ensotrophy dissipation rate. While many simulations [5, 6] conﬁrm these theoretical pred ictions the actual observation and detection of 2-d turbulence as a natural phenomena remai ns (as far as we know) an open questions. One of the objectives of this paper is to weigh in the pros and c ons for 2-d turbulence in the Antarctic data that was obtained by the British observ ation post as Haley Station in Antarctica on June 22, 1987 (for further description of this data see [7, 8]). The importance of these measurements stem from the fact that the ﬂow ﬁeld u= (u, v, w ) and the temperatures were measured simultaneously at three diﬀerent heights viz . 5m, 16m and 32m. These simultaneous readings enable us to apply a test devised by E. Dewan [9] for the detection of 2-d turbulence. According to this test 2-d turbulence is cha racterized by small values for the coherence [20] between the time series which represent the v arious meteorological variables at diﬀerent heights. From another point of view the Antarctic data represent a sta bly stratiﬁed medium. (According to mission records the temperature gradient wit h height can reach up to 1 K/m). Under these circumstances Bolgiano [10, 11] and others [9] s peculated about the existence of “buoyancy range turbulence” (BRT) which should lead to a ﬂat tening of the spectra in parts of the inertial range. In this paper we shall estimate the pow er spectrum for the data using 3the usual Fourier transform and by the method of maximum entr opy (brieﬂy the reason for this duplicatcy is due to the existence of “discontinuit ies” in the data). Both of these estimates show a spectral range in which the spectrum is almo st ﬂat and thus support the theoretical arguments that were advanced for the existence of BRT. The plan of the paper is as follows: In section 2 we describe th e method used to ﬁlter out the mean ﬂow and waves from the data and the tests that were applied to verify that the residuals actually represent turbulence. In section 3 w e apply the coherence test for 2-d turbulence and discuss its consequences. In section 4 we present a model for the power spectrum of the data and its implications. We end up in sectio n 5 with some conclusions. 2 Data Detrending The statistical approach to turbulence splits the ﬂow varia bles˜u,˜T(where ˜Tis the temper- ature) into a sum ˜u=u+u′+u,˜T=T+T′+t where u, Trepresent the mean (large scale) ﬂow, u′, T′represent waves and u, t“turbulent residuals” [12] To eﬀect such a decomposition in our data we used the Karahuna n-Loeve (K-L) decom- position algorithm (or PCA) which was used by many researche rs (for a review see [13]). Here we shall give only a brief overview of this algorithm wit hin our context. Let be given a time series X(of length N) of some geophysical variable. We ﬁrst deter- mine a time delay ∆ for which the points in the series are decor related. Using ∆ we create ncopies of the original series X(k), X(d+ ∆), . . . , X (k+ (n−1)∆). (To create these one uses either periodicity or choose to con sider shorter time-series). Then one computes the auto-covariance matrix R= (Rij) Rij=N/summationdisplay k=1X(k+i∆)X(k+j∆). (2.1) 4Letλ0> λ1, . . . , > λ n−1be the eigenvalues of Rwith their corresponding eigenvectors φi= (φi 0, . . . , φi n−1), i= 0, . . . , n −1. The original time series Tcan be reconstructed then as X(j) =n−1/summationdisplay k=0ak(j)φk 0 (2.2) where ak(j) =1 nn−1/summationdisplay i=0X(j+i∆)φk i. (2.3) The essence of the K-L decomposition is based on the recognit ion that if a large spectral gap exists after the ﬁrst m1eigenvalues of Rthen one can reconstruct the mean ﬂow (or the large component ( of the data by using only the ﬁrst m1eigenfunctions in (2.2). A recent reﬁnement of this procedure due to Ghil et al [13] is th at the data corresponding to eigenvalues between m1+ 1 and up to the point m2where they start to form a “continuum” represent waves. The location of m2can be ascertained further by applying the tests devised by Axford [14] and Dewan [9] (see below). Thus the original data can be decomposed into mean ﬂow, waves and residuals (i.e. data corresponding to eigenvalues m2+ 1, . . . , n −1 which we wish to interpret at least partly as turbulent residuals). For the data under consideration we carried out this decompo sition using a delay ∆ of 1024 points (approximately 51 sec.) for all the geophysical variables. In table 1 we present the values of m1, m2that were used in this decomposition for the ﬂow variables at diﬀerent heights. (In all cases n= 64). The residuals of the time series which are reconstructed as Xr(j) =n−1/summationdisplay k=m2+1ak(j)φk 0 (2.4) contain (obviously) the measurement errors in the data. How ever to ascertain that they should be interpreted primarily as representing turbulenc e we utilize the tests devised by Axford [14] and Dewan [9]. According to these tests turbulen ce data (at the same location) 5is characterized by low coherence between u, v, w and a phase close to zero or πbetween w andt. (A phase close to π/2 is characteristic of waves). Figs. 1,2,3 show samples of th e coherence between the residuals of u, v, w at diﬀerent heights. They demonstrate that for most frequencies the coherence is less than 0.1. Fig. 4 gives a scatter plot of the phase between wandtat height 5m. This ﬁgure is less deﬁnitive as there are still q uite a few points in the wave sector/parenleftbiggπ 4,3π 4/parenrightbigg . However out of the 200 points in this plot 125 are in the “turbulence sector”. These tests show that to a large extent the residuals that wer e obtained from the K-L decomposition represent actual turbulence. 3 Tests for 2-d turbulence In today literature [15] a spectral slope of −3 in part of the inertial range is considered to be a strong indicator for 2-d turbulence. However as noted al ready by Lily [5] “geophysical consideration” might modify this slope. Since the spectral plots for the ﬂow under consider- ation (for sample see ﬁgs. 8,9,10) do not exhibit this depend ency (except for wat 16m in the low frequencies) we must resort to other tests to bolster the claim that the ﬂow described by this data corresponds to 2-d turbulence. To this end we utilize a test devised by Dewan [9]. According t o this test inviscid two dimensional turbulence is characterized by the fact that th e temporal statistical coherency [20] between the time series representing the ﬂow variables at diﬀerent altitudes is zero. With viscosity taken into account some vertical separation of th e order of (10m for air) is needed for the coherency to become small. (Strong coherency with va lues close to one indicates a strong linear relationship between the two time series [20] ). Some typical plots for the coherency in the data is presented in ﬁgs. (5,6,7). In these plots the coherency for wbetween the diﬀerent heights is plotted for diﬀerent wave nu mbers. We observe that for most sampled frequencies the coherency i s well below 0.1 and according to Dewan [9] “these values constitute evidence for 2-d turbu lence and against other types of ﬂuctuations”. 64 The spectrum Two dimensional ﬂow of incompressible and inviscid ﬂuid con serve both the energy Eand the enstrophy Ω. Fir viscous ﬂuid these quantities decay acc ording to −ǫ=∂E ∂t=−2νΩ,−ǫω=∂Ω ∂t=−ν\|∇ω\|2(4.1) The energy spectrum is determined therefore by both paramet ersǫ, ǫωwhich leads to the deﬁnition of a length scale Lω=/parenleftbiggǫ ǫω/parenrightbigg1/2 (4.2) From dimensional considerations one concludes then that [1 6] the energy spectrum in the inertial range must have the form E(k) =f(kLω)ǫ2/3k−5/3(4.3) where fis a function of the dimensionless variable kLω. If at one end of the inertial range only ǫis essential (and the eﬀect of ǫωis negligible) then f∼=constant and the energy spectrum obey Kolmogorov 5 /3 power law. If on the other end of this range ǫis not essential then f must have the form f∼=(kLω)−4/3(4.4) and consequently E(k) =Cǫ2/3 ωk−3(4.5) (where Cis a constant). For stratiﬁed medium Obukov [17] introduced the temperatur e inhomogeneity dissipation rate ǫT= 2χ/integraldisplay∞ 0k2ET(k)dk (4.6) where ETis the temperature spectra and χis the heat conductivity of the medium. He further postulated that the turbulent component of Tis dependent on this parameter. 7For the (stratiﬁed) Antarctic medium we would like to enlarg e the domain of this pos- tulate to include the velocity components of the ﬂow. This en ables us to introduce the buoyancy (length) scale [16, 18] LB= (αg)−3/2ǫ5/4ǫ−3/4 T (4.7) where ( αg) is the buoyancy parameter. The existence of this second len gth scale for stratiﬁed two dimensional ﬂow lead us to replace (4.3) by E(k) =f(kLω, kLB)ǫ2/3k−5/3(4.8) However since stratiﬁcation and enstrophy conservation ar e independent of each other we infer that fmust have the form f∼=(kLω)r(kLB)s. (4.9) It follows then that the spectral dependence on kis given by E(k)∼kr+s−5/3. (4.10) We conclude therefore that various combinations of r, sare possible and this will lead to diﬀerent spectral dependencies on k. Thus if E(k)∼k−q and the dissipation ǫis negligible we must have then r+s= 5/3−q,r 2+5 4s+2 3= 0 which yields r=33−15q 9, s=15q−18 9. From the spectral plots for the data under consideration we s ee that (approximately) E(k)∼k0 for a large segment of the inertial range which is characteri stic of the “buoyancy range turbulence” as predicted by Bolgiano [10, 11]. 8It is interesting to note in this context that Kriachnen [19] already observed that the “energy spectrum of the ﬂow depends on the details of the nonl inear interaction embodied in the equations that govern the ﬂow and can not be deduced sol ely from the symmetries, invariances and dimensionality of the equations”. Finally we would like to observe that the data under consider ation contains some dis- continuities. These can change completely the asymptotic b ehavior of the spectrum. To demonstrate this assume that the data is described by D(x) =CH(x−x0) +g(x) (4.11) where g(x) is a smooth function whose Fourier transform (FT) decays ex ponentially and H(x) is the Heaviside function H(x) =  1x≥0 0x <0. Diﬀerentiating (4.11) we have D′(x) =Cδ(x−x0) +g′(x) (4.12) and the FT of (4.12) is ˜D′(k) =C+ ˜g′(k) (4.13) The FT of Dis obtained then by dividing (4.13) by kwhich shows clearly that the asymptotic behavior of ˜D(k) is proportional to k−1. We conclude then that a proper ﬁlter for the removal of these d iscontinuities from the data is needed in order to obtain the true spectrum of the turb ulent residuals. Such a ﬁltering algorithm is given by the K−Ldecomposition which was described in Sec. 2. 5 Conclusion Using the coherency test advanced by Dewan we are able to char acterize the ﬂow under consideration as one that has the characteristics of 2-d tur bulence. One stumbling block for 9this interpretation is the absence of −3 slope in part of the inertial range. To explain this we introduced a model that takes into account the stratiﬁcatio n of this ﬂow. This model shows that when buoyancy eﬀects are taken into account diﬀerent sl opes of E(k) are possible. Thus we believe that we introduced evidence for the interpretati on of this spectra as one belonging to BTR. Acknowledgment The author is deeply indebted to Dr. J. Rees and the British An tarctic Survey Team, Cambridge, UK for access to the antarctic data and to Dr. J. Re es and O. Cote for bringing to his attention the peculiar spectrum of this data. References [1] V. M. Canuto, M.S. Dubovikov and D.J. Wielaard - A dynamic al model for turbulence vs Two-dimensional Turbulence, Phys. Fluids, 9p. 2141-2147 (1997). [2] M.E. Maltrud and G.K. Vallis - Energy spectra and chohere nt structure in forced two-dimenmsional and beta plane turublence, J. Fluid Mech. 228p. 321-342 (1991). [3] Wendal Horton and A. Hasegawa - Quasi two-dimensional dy namics of plasmas and ﬂuids, Chaos, 4, p. 227-251 (1994). [4] R. Kraichnan - Phys. Fluids, 10, p. 1417 (1967). [5] D. K. Lily - Numerical Simulation of two-dimensional tur bulence, Phys. Fluid Supp. 2, II-233 (1969). [6] G.K. Batchelor - Computation of the energy spectrum in ho mogeneous two- dimensional turbulence, Phys. Fluid Supp. 2, II-240 (1969) . [7] J. C. King, S. D. Mobbs, J.M. Rees, P.S. Anderson and A.D. C ulf. The stable Antarc- tic boundary layer experiment at Haley Station, Weather, 44, p. 398-405 (1989). 10[8] N.R. Edwards and S.D. Mobbs - Observation of isolated wav e-turbulence interactions in the stable atmospheric boundary layer, Q.J.R. Meteorol. Soc.,123, p. 561-584 (1997). [9] E.M. Dewan - On the nature of atmospheric waves and turbul ence, Radio Sci., 20, p. 1301-1307 (1985). [10] R. Bolgiano, Jr. - Turbulent spectra in a stably stratiﬁ ed atmosphere, J. Geo Res. 64, p. 2226-2229 (1959). [11] R. Bolgiano, Jr. - Structure of turbulence in stratiﬁed media, J. Geo Res. 67, p. 3015-3023 (1962). [12] F. Einaudi and J.J. Finnigan - Wave turbulence dynamics in the stably stratiﬁed boundary layer, J. Atmos. Sci., 50, p. 1841-1864 (1993). [13] C. Penland, M. Ghil and K.M. Weickmann - Adaptive ﬁlteri ng and maximum entropy spectra with applications to changes in atmospheric angula r momentum, J. Geo. Res. 96p. 22659-22671 (1991). [14] D.N. Axford - Spectral analysis of an aircraft observat ion of gravity waves, Q.J. Roy. Met. Soc., 97, p. 313-321 (1971). [15] U. Frisch - Turbulence, Cambridge Univ. Press. (1995). [16] A.S. Monin and R.V. Ozmidov - Turbulence in the ocean, D. Reidal Pub. Co. (1985). [17] A.M. Obukhov - Structure of temperature ﬁeld in turbule nt ﬂow, Izv. Ale. Nauk SSSR, Ser. Geoﬁz 13p. 58-69 (1949). [18] A.M. Obukhov - On stratiﬁed ﬂuid dynamics, Dokledy AK, N auk SSSR, 145p. 1239-1242 (1962). [19] R. Kraichnan - On Kolmogorov inertial-range theories, J. Fluid Mech. 62, p. 305-330 (1974). 11[20] W.N. Venables and B.D. Ripley - Modern applied statisti cs with S-plus, Springer- Verlag (1996). 12m1 m2 u at 5m 2 42 v at 5m 2 26 w at 5m 2 30 T at 5m 4 26 u at 16m 2 42 v at 16m 2 40 w at 16m 3 37 T at 16m 2 41 u at 32m 4 48 v at 32m 1 40 w at 32m 4 51 T at 32m 2 42 Table 1 13
arXiv:physics/0011049v1 [physics.data-an] 21 Nov 2000A Variational Formulation of Optimal Nonlinear Estimation Gregory L. Eyink∗ CCS-3 MS-B256 Los Alamos National Laboratory Los Alamos, NM 87545 Abstract We propose a variational method to solve all three estimatio n problems for nonlinear stochastic dynamical systems: prediction, ﬁltering, and s moothing. Our new approach is based upon a proper choice of cost function, termed the eﬀective action . We show that this functional of time-histories is the unique statistically w ell-founded cost function to determine most probable histories within empirical ensembles. The en semble dispersion about the sample mean history can also be obtained from the Hessian of t he cost function. We show that the eﬀective action can be calculated by a variational p rescription, which generalizes the “sweep method” used in optimal linear estimation. An iterat ive numerical scheme results which converges globally to the variational estimator. Thi s scheme involves integrating forward in time a “perturbed” Fokker-Planck equation, very closely related to the Kushner- Stratonovich equation for optimal ﬁltering, and an adjoint equation backward in time, similarly related to the Pardoux-Kushner equation for opti mal smoothing. The variational estimator enjoys a somewhat weaker property, which we call “ mean optimality”. However, the variational scheme has the principal advantage—crucia l for practical applications—that it admits a wide variety of ﬁnite-dimensional moment-closu re approximations. The moment approximations are derived reductively from the Euler-Lag range variational formulation and preserve the good structural properties of the optimal esti mator. ∗Permanent address: Department of Mathematics, University of Arizona, Tucson, AZ 85721 10 Introduction The three classical problems of stochastic estimation are p rediction, ﬁltering, and smoothing of time series; e.g. see [1]. These correspond to estimating the future, present, and past states, respectively, based upon current available information. I n more detail, the nonlinear estima- tion problem may be described as follows: assume as known som e nonlinear (Ito) stochastic diﬀerential equation for a time-series X(t): dX=f(X,t)dt+ (2D)1/2(X,t)dW(t). (0.1) Herefis a (drift) dynamical vector, Dis a nonnegative diﬀusion matrix, and W(t) is a vector Wiener process. Suppose also that some imperfect observati onsr(t) are taken of a function Z(X(t),t) of the basic process, including some measurement errors ρ(t) with covariance R(t): r(t) =Z(X(t),t) +ρ(t). (0.2) It will generally be assumed that the distribution of the mea surement errors is known as well. For example, the errors may be assumed to be proportional to a white noise: ρ(t) =R1/2(t)η(t). Then the problem is, given the data R(tf) ={r(t) :t < t f}up to ﬁnal measurement time tf, to obtain the best estimate of X(t) at times t > t f, t=tfandt < t f. The optimal ﬁltering problem in the above general setting ha s been exactly solved by Stratonovich [2] and Kushner [3, 4] within a Bayesian formul ation. Those authors have shown that the conditional probability density P(x,t\|R(t)), given the current data R(t), obeys a stochastic partial diﬀerential equations, nowadays calle d the Kushner-Stratonovich equation . Explicitly, denoting P∗(x,t) =P(x,t\|R(t)),the KS equation is of the form ∂tP∗(x,t) =ˆL(t)P∗(x,t) +h⊤(t)[Z(x,t)− /an}bracketle{tZ(t)/an}bracketri}ht∗t]P∗(x,t). (0.3) where ˆL(t) =−∇x·[f(x,t)(·)] +∇x·[D(x,t)·∇x(·)] (0.4) 2is the standard Fokker-Planck linear operator, and h(t) =R−1(t)[r(t)− /an}bracketle{tZ(t)/an}bracketri}ht∗t] (0.5) is a random forcing term constructed from the particular rea lization of the observation r(t) obtained in a given sample run of the system. Note that /an}bracketle{t·/an}bracketri}ht∗t=E(·\|R(t)) denotes conditional average, i.e. the average with respect to the distribution P(x,t\|R(t)) itself. Hence, the KS equation is nonlinear. The integration of this equation for ward in time, with sequential input of the fresh observations r(t) as they become available, solves, in principle, the ﬁlteri ng problem. The prediction problem is then solved in theory, by integrat ing the standard Fokker-Planck equation with P(x,tf\|R(tf)) as initial data to obtain P(x,t\|R(tf)) for t > t f. The optimal smoothing problem has also been solved, in princ iple, by Kushner [5] and Pardoux [6]. They have shown that P(x,t\|R(tf)) for t < t fcan be written as P(x,t\|R(tf)) =A∗(x,t)P∗(x,t), (0.6) where P∗(x,t) is as above and A∗(x,t) solves the adjoint equation ∂tA∗(x,t) +ˆL∗(t)A∗(x,t) +h⊤(t)[Z(x,t)− /an}bracketle{tZ(t)/an}bracketri}ht∗t]A∗(x,t) = 0. (0.7) This equation, with the random forcing h(t), must be interpreted as a “backward stochastic equation”. It is solved subject to the ﬁnal condition A(x,tf) = 1. In certain cases, these estimators reduce exactly to solvin g a ﬁnite number of ODE’s. For example, in the linear case, where f(x,t) =A(t)x,D(x,t) =D(t), and Z(x,t) =B(t)x,the KS optimal ﬁlter reduces exactly to the ﬁnite-dimensional K alman-Bucy optimal linear ﬁlter [7]. This reduction occurs in the linear case because the con ditional PDF is known rigorously to be multivariate Gaussian, uniquely speciﬁed by its mean a nd covariance. The conditional mean E[X(t)\|R(t)] coincides with the Kalman-Bucy ﬁlter estimate ξ(t) of the current state X(t),which is determined by the solution of a stochastic ODE with s equential input of the observations. The covariance matrix C(t) =E[X(t)X⊤(t)\|R(t)]−ξ(t)ξ⊤(t) is obtained as well from a linear Ricatti equation integrated forward in time. 3The Pardoux-Kushner smoother is also ﬁnite-dimensional fo r linear systems. In fact, it coincides there with an alternative variational formulation of the linear estimation problem. The latter can be motivated most naively from the idea of leas t-square-error estimation. That is, one may introduce a weighted square-error functional fo r the dynamics, ΓX[x] =1 4/integraldisplaytf tidt[˙x−A(t)x]⊤D−1(t)[˙x−A(t)x], (0.8) which measures the “cost” for a history x(t) to depart from the solution of the linear, deter- ministic dynamics ˙x=A(t)x. The integral is weighted by the “error covariance” D(t) which arises from the random noise. A similar cost function may be i ntroduced for the observation error of the data: ΓR[ρ] =1 2/integraldisplaytf tidtρ(t)⊤R−1(t)ρ(t). (0.9) In that case, the solution to the estimation problem may be ob tained by minimizing with respect toxthe combined cost function ΓX,R[x,r] := Γ X[x] + Γ R[r−Bx] (0.10) when the set of observations {r(t) :ti< t < t f}is input into the second term. The minimizer x∗=x∗[r] is then the optimal history , which solves simultaneously all three estimation problem s. It may be shown that x∗(t) =ξ(t) fort≥tf, so that the variational estimator coincides with the Kalman-Bucy ﬁlter and predictor. Furthermore, it may be shown that, for t < t f, the variational estimator is given by the Ansatz x∗(t) =ξ(t) +C(t)α(t). (0.11) Here, α(t) is the solution of a linear adjoint equation integrated backward in time with the ﬁnal condition α(tf) =0and thus vanishes identically for t≥tf. However, it makes a contribution fort < t fto the smoother, proportional to C(t).This adjoint algorithm to calculate the mini- mizer is called the “sweep method” [8], and it gives the same r esultx∗(t) =E[X(t)\|R(tf)] as calculated by the Pardoux-Kushner equation. Hence, it prov ides a ﬁnite-dimensional represen- tation of the optimal smoother for linear systems. 4In general, however, the optimal estimators are inﬁnite-di mensional, i.e. they require the solution of (stochastic) PDE’s. For many of the spatially-e xtended, continuum systems of greatest interest in geophysics and in engineering, this is , in fact, a functional PDE. Even discretization for numerical solution results in a (stocha stic) PDE on a phase space of dimension literally a billion or more. It has therefore been clear sinc e their original formulation that, for such spatially-extended or distributed systems, the exact calculation of the optimal nonlinear estimator will be numerically unfeasible. Kushner himself wrote an early paper [9], in which he stressed this point and set up a formalism for approximating the optimal ﬁlter. As he observed there, the problem is formally the same as the “closure probl em” in turbulence theory. The approximation scheme he proposed was also the same as that tr aditionally adopted in turbulence theory: namely, a moment closure of the full KS equation. Such a scheme results in a set of equations with a number of variables comparable to that in the starting equation (0.1), which may still be large but tractable. Constructing ﬁnite- dimensional approximate estimators continues to be a pressing research problem up to the present day, e.g. see [10]. Indeed, general approximation schemes for the full estimation problem (pre diction, ﬁltering and smoothing) that are at once computationally practicable and faithful t o the optimal solution remain to be developed. The fact that the problem of estimation for exten ded systems is formally equivalent to the turbulence problem—a notoriously diﬃcult one—sugge sts that the solution here, too, will be nontrivial. Not only must the formal properties of th e optimal estimator be retained by any approximation, but also the physical properties of the u nderlying dynamical system must be suﬃciently represented. The problem of approximating th e optimal estimator is not, in our opinion, just one of mathematics but also of physics. The aim of this paper is to formulate a new approach to the prob lem of optimal nonlinear estimation, based upon a variational formulation. The crux of the method is to identify an action functional, analogous to (0.8), which is statistica lly justiﬁed to use as a cost function for estimation of nonlinear dynamics. This is the quantity w hich we have called the eﬀective action in previous works [11]-[13]. This functional of state histo ries is uniquely characterized as 5that which selects the most probable value under arbitrary c onditions on the empirical sample averages. In fact, the cost function (0.8) appropriate for l inear systems has been motivated only rather crudely but it has a more fundamental probabilis tic justiﬁcation. It was apparently ﬁrst observed by the chemist Lars Onsager that the dynamical cost function (0.8) is the unique functional whose minimum determines the statistically mos t probable time-history of a linear dynamics of form (0.1), subject to an arbitrary sets of const raints. In the statistical physics literature, the functional (0.8) is known as the Onsager-Machlup action [14]. The cost function (0.9) for the current observation error can be similarly sho wn to give the most probable error in the case of a Gaussian white-noise distribution. The combin ed cost function (0.10)—under the assumption that dynamical noise and observation error are i ndependent random functions— then indeed gives by minimization the most probable time-hi story subject to currently available information. Thus, the minimizer x∗[r] is the unequivocal optimal estimator in the linear case. Previous attempts to develop variational methods for optim al nonlinear estimation have not paid suﬃcient attention to the statistical requirements on the cost function. For example, a functional has often been employed similar to (0.8) for the l inear case, ΓX[x] =1 4/integraldisplaytf tidt[˙x−f(x,t)]⊤D−1(t)[˙x−f(x,t)], (0.12) naively based upon least-square-error philosophy. Howeve r, the use of this cost function has no statistical justiﬁcation, except in the weak-noise limit D→0. In that case, (0.12) is known as the “nonlinear Onsager-Machlup action” and it is proved to g ive the leading-order asymptotics of probabilities of time histories for small noise [15, 16]. However, except for the weak-noise limit or for linear dynamics, the Onsager-Machlup action ha s no probabilistic signiﬁcance. Only the eﬀective action—and no other cost function, such as (0.1 2) above—will even have as its minimum the correct mean value. The eﬀective action thus pla ys the role of a “ﬂuctuation potential” in the theory of empirical ensemble averages con structed from independent samples, analogous to the Onsager-Machlup action for weak-noise or f or linear systems. In fact, the eﬀective action is known to coincide with the Onsager-Machl up action for weak-noise or for 6linear systems [15]. Thus, the optimal estimator proposed i n this work coincides with the standard ones for those special cases. Unlike (0.8), the eﬀective action cannot generally be writt en as an explicit function of the state histories. However, it has been shown in [11]-[13] that it may be calculated by a constrained variational method. The Euler-Lagrange equat ions that result are a pair of forward and backward equations, very similar to the Kushner-Strato novich-Pardoux (KSP) equations. Despite this, the variational estimator is not quite equiva lent to the optimal estimator which follows from the KSP equations. It possesses instead a prope rty that we call mean-optimality . Although somewhat weaker than the optimality enjoyed by KSP , mean-optimality distinguishes it from other “suboptimal” estimators which have in fact no o ptimality whatsoever. However, the main advance of the variational approach is in the proble m of constructing ﬁnite-dimensional approximations. Because the variational estimator is base d upon an Euler-Lagrange variational principle, it is very easy to develop consistent approximat ions by a Rayleigh-Ritz scheme. In this method, parameterized trial functions are selected to represent the solutions of the forward- backward equations. Inserted into the variational functio nal and varying over parameters, one obtains approximations to the exact forward-backward equa tions and thereby to the eﬀective action. A straightforward use of this scheme in fact leads to a moment-closure approximation for the forward ﬁltering equation, much like that originall y proposed by Kushner [9]. However, now also backward equations are obtained for the smoothing p roblem. Necessary consistency properties with the forward equations are guaranteed by the fact that these arise together as the Euler-Lagrange system of an approximate action functio nal. This paper is organized as follows: in Part I, we present our v ariational formulation of optimal estimation. We ﬁrst explain the unique statistical signiﬁcance of the eﬀective action, which makes it appropriate for variational estimation. We r eview there also the deﬁnition and properties of the eﬀective action, including the notion s of joint and conditional eﬀective actions. The optimality property will be established for th e variational estimator and compared with that of the KSP estimator. We next discuss how to calcula te the eﬀective action based 7upon its variational characterization. An iterative numer ical scheme is outlined to numerically calculate the variational estimator, which reduces to solv ing KSP-type equations. Some matters important for practical applications will ﬁnally be discus sed: the case when measurements are taken, not continuously, but at a discrete set of times, and t he evaluation of the ensemble dispersion around the most probable value of the sample mean . In Part II the very important issue is addressed of constructing ﬁnite-dimensional appr oximations to the variational estimator, crucial for application of the methods to spatially-extend ed (or distributed) systems with many degrees of freedom. A Rayleigh-Ritz moment-closure scheme is developed, based upon the ﬁnite- dimensional reduction of the nonequilibrium action. The us e of this approximation scheme for solution of practical estimation problems is ﬁnally discus sed. I Variational Formulation of Optimal Estimation I.1. Ensemble Theory of Estimation There are intrinsic limits to our ability to estimate, which can be understood most simply from anensemble point of view. If the stochastic dynamics (0.1) is run many ti mes with diﬀerent realizations of the noise or, even in the deterministic case D(t)≡0, if the initial data are selected randomly from some starting distribution P(0), then the solutions will be generally quite distinct. Thus, in Ndiﬀerent trials there will be Ndiﬀerent outcomes X1(t),...,XN(t). It is therefore not obviously very meaningful to give a singl e value x∗(t) as an estimate of the state given some partial information R(unless, of course, that information included the exact initial data or realization of the random noise!) It is true t hat the average over samples will converge to the mean in the ensemble conditioned on the avail able information: lim N→∞1 NN/summationdisplay n=1Xn(t) =E[X(t)\|R]. (I.1) However, the individual sample points will show a scatter, p ossibly quite large, about this mean 8value. A useful measure of this scatter is the covariance mat rix CR(t) :=E[δX(t)δX⊤(t)\|R] (I.2) in the ensemble conditioned on R, where δX(t) :=X(t)−E[X(t)\|R]. In particular, Tr CR(t) := E[/bardblδX(t)/bardbl2\|R] gives the mean square radius σ2 R(t) of scatter of the sample points around the mean. The ensemble mean has the one virtue that it minimizes this rms radius of scatter. In other words, if one took δX(t) :=X(t)−x∗(t) for any other non-random estimator x∗(t)/ne}ationslash= E[X(t)\|R], one would increase σ2 R(t). This is an elementary fact of probability theory: for any random variable, the expectation value is the unique determ inistic estimator for which the mean- square error is a minimum. This important property of the mea n value as a predictor—that it minimizes rms forecast error—has been emphasized before by Leith in the ﬁeld of climatology [21]. Of course, the above considerations show that one shou ld have not only an estimate of the state of a system, but also an estimate of the reliability or certainty of that state. The covariance matrix CR(t) is a good such measure. Any state within a few standard devia tions σR(t) of the mean must be regarded as having a good degree of probab ility to occur. Such considerations are precisely those which justify the s tandard Bayesian approach of Kushner-Stratonovich-Pardoux. Granted the limitations i mplied above, one cannot do better than to give the probability density P(x,t\|R) of the state variable conditioned on the available information. The minimal requirement on a variational appr oach to estimation is thus that it should give at least the mean and covariance of such conditio ned ensembles. This has motivated us to consider a cost function, the “eﬀective action”, which has a proper foundation in the theory of empirical ensembles. A brief review of its deﬁnition and b asic properties is here required. I.2. Basic Theory of the Eﬀective Action The quantity which we have termed the eﬀective action [11]-[ 13] has appeared, in various guises and by various names, in quantum ﬁeld theory, in theor y of stochastic processes, and in dynamical systems theory. We shall here just brieﬂy recapit ulate its deﬁnition and properties. One interpretation of the eﬀective action is as a generating functional for multi-time cor- 9relations. This is the way in which the functional is general ly introduced in ﬁeld theory [17]. Consider any vector-valued random process Z(t). Then, the cumulant generating functional WZ[h] is deﬁned as WZ[h] = log /an}bracketle{texp/parenleftbigg/integraldisplaytf tidth⊤(t)Z(t)/parenrightbigg /an}bracketri}ht. (I.3) Thenth-order multi-time cumulants of Z(t) are obtained from WZ[h] by functional diﬀerenti- ation with respect to the “test history” h(t): Ci1···in(t1,...,t n) =δnWZ[h] δhi1(t1)···δhin(tn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=0. (I.4) It is not hard to check from its deﬁnition (I.3) that WZ[h] is a convex functional of h. The Legendre dual of this functional is deﬁned to be the eﬀective action of Z(t): ΓZ[z] = max h{<h,z>−WZ[h]}, (I.5) with<h,z>:=/integraltextdth⊤(t)z(t). It is a generating functional of so-called irreducible correlation functions ofZ(t): Γi1···in(t1,...,t n) =δnΓZ[z] δzi1(t1)···δzin(tn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=z. (I.6) The functional derivatives here are evaluated at the mean hi storyz(t) :=/an}bracketle{tZ(t)/an}bracketri}ht. It is not hard to check from the deﬁnition (I.5) that Γ Z[z] is a convex, nonnegative functional with a unique global minimum (equal to zero) at the mean history z=z. The eﬀective action has another important interpretation a s the rate function in the theory of large deviations of empirical ensemble averages for time -series. See [18] for the original, so- called Cram´ er theory of single real variables and [19] for the extension to genera l vector spaces. This theory involves the empirical or sample mean: ZN(t) :=1 NN/summationdisplay n=1Zn(t), (I.7) whereZn(t) forn= 1,...,N are independent, identically distributed realizations of the random process Z(t). The law of large numbers states that, in the limit of number of samples Ngoing to inﬁnity, ZN(t)→z(t). However, for ﬁnite N,ZN(t) is itself a random process with some 10probability of achieving a ﬂuctuation value z(t) diﬀerent from the ensemble mean z(t). The basic result of the Cram´ er theory is that this probability d ecreases exponentially in the limit asN→ ∞: P/parenleftBig ZN(t)≈z(t) :ti< t < t f/parenrightBig ∼exp (−N·ΓZ[z]). (I.8) Thus, Γ Z[z] forz/ne}ationslash=zgives the rate of decay of the probability to observe ZN≈z. Since ΓZ[z] = 0 only for z=z, the probability to observe the empirical N-sample mean ZNequal to anything other than the ensemble mean zmust go to zero as N→ ∞. Thus, the large deviation result (I.8) includes, and generalizes, the usual law of lar ge numbers. Furthermore, the ensemble mean is now also seen to be characterized by a variational pri nciple of least eﬀective action . That is, the most probable value of the sample mean for large N, orz=z, is just that which minimizes the eﬀective action Γ Z[z]. It is worth emphasizing that the eﬀective action is the unique function possessing all of these properties. This is a conse quence of a general theorem on uniqueness of rate functions for large deviations [20]. There is one other general theorem of large deviation theory which will prove important to us in the sequel. This is the so-called Contraction Principle . Suppose that WN(t) is a random process which is deﬁned as a continuous functional Wof an empirical mean ZN(t): WN(t) :=W[t;ZN]. (I.9) Then, WN(t) also satisﬁes a large deviations principle with rate funct ion given by ˜ΓW[w] = min {z:W[z]=w}ΓZ[z]. (I.10) See [20]. When the functional Wis linear, then obviously ˜ΓW[w] = Γ W[w]. Thejoint eﬀective action ΓX,Y[x,y] of a pair of random time series X(t) andY(t) can be deﬁned most simply as the eﬀective action Γ Z[z] of the composite vector Z(t) := (X(t),Y(t)). Of course, this notion may be extended to a joint eﬀective act ion Γ X1···Xn[x1,...,xn] of an arbitrary number nof variables Xi(t), i= 1,...,n. It is a simple application of the Contraction Principle to see that elimination of one of the variables is a ccomplished by minimizing over its 11possible values. For example, ΓX[x] = minyΓX,Y[x,y] (I.11) recovers the eﬀective action Γ X[x] ofX(t) alone. The joint eﬀective action of a pair of independent time series X(t) andY(t) is easily shown from the deﬁnition to be given just by the sum: ΓX,Y[x,y] = Γ X[x] + Γ Y[y]. (I.12) In general, for dependent time series, one may deﬁne the noti on of a conditional eﬀective action by means of ΓX\|Y[x\|y] := Γ X,Y[x,y]−ΓY[y]. (I.13) Thus, when X(t) andY(t) are independent, Γ X\|Y[x\|y] = Γ X[x]. The term “conditional action” is justiﬁed by the relation to large- Nasymptotics of conditional probabilities for empirical averages: P/parenleftBig XN≈x\|YN=y/parenrightBig ∼exp/parenleftBig −N·ΓX\|Y[x\|y]/parenrightBig . (I.14) Using the deﬁnition of the conditional probability, (I.14) is a simple consequence of the ba- sic large deviation estimate (I.8). The conditional action is also a generating functional for irreducible multi-time correlation functions in the condi tioned ensemble, in the limit N→ ∞. I.3. Proposal for the Variational Estimator These considerations motivate our following proposal: we propose to take as optimal estima- tor x ∗[r]the minimizer of the conditional eﬀective action ΓX\|R[x\|r]of the history xgiven the current observation history {r(t) :t∈[ti,tf]}. From our discussion of the eﬀective action in the previous section, we can infer the crucial property of th is estimator: it is the mean value within the subensemble in which the empirical N-sample average takes on the value r, that is, the sub-ensemble in which rN(t) =r(t), t∈[ti,tf]. (I.15) In fact, as discussed above, the conditional eﬀective actio n is a variational functional whose minimum coincides with the subensemble mean history for an a rbitrary set of constraints on 12the empirical sample average. Thus, the estimator x∗[r] is exactly of the form E[X(t)\|R] discussed above, with Rspeciﬁed by (I.15). It is also possible to obtain from the con ditional eﬀective action the error covariance C∗[t\|r], essentially by evaluating its Hessian matrix (see Section I.6). This is precisely the ensemble dispersion tha t would be observed via the spread of sample histories in an ensemble forecasting scheme [21], if the ensemble considered were the one speciﬁed by the condition (I.15) for the limit of large N. Our proposal is clearly similar in spirit to the Bayesian for mulation of Kushner-Stratonovich- Pardoux. However, they are distinct. The diﬀerence can best be understood by considering the problem from an experimental point of view. Suppose that a ve ry large number Nof samples of the system (1) are run, drawing initial conditions random ly from the same distribution P0. Then the conditional distribution P(x,t\|R(tf)) considered by KSP corresponds to the very small sub-ensemble in which that particular realizati onr(non-random) of the observation history occurred. It obviously diﬃcult to prepare such sub- ensembles, since one must wait patiently for the particular observation rto spontaneously occur and, each time it does, add it as a member to the sub-ensemble. This makes it very diﬃcult to directly test the predictions of the KSP-equations. More to the point, it is very diﬃcult to carry out an ensemble or Monte Carlo approach to calculate directly the conditional average. The variational estimation method proposed above corresponds to a diﬀerent—and somewh at larger—sub-ensemble. As noted above, it corresponds to considering the sub-ensembl e speciﬁed by the condition {rN(t) = r(t), t∈[ti,tf]}. It is clear that the sub-ensemble described by the KSP-equa tions is, also, a subset of the new, larger one. In fact, if it is true as in the KS P sub-ensemble that rn=rin every realization, n= 1,...,N then it is a fortiori true that rN=r. However, there will clearly be many members of the new ensemble in which rN=rbut for which not every term rnof theN-sample average is equal to r. Thus, the new sub-ensemble is clearly much larger than that considered by Kushner-Stratonovich-Pardoux, but, st ill, too small a subset of the whole ensemble to be reproducible by direct methods. Despite the diﬃculty of directly testing the KSP-equations , they are truly the optimal 13method for the ﬁltering and smoothing problems. While it is d iﬃcult to prepare the conditional ensemble corresponding to a given observation r, there is no diﬃculty in preparing one member of such an ensemble. After all, just running the system once a nd collecting one observation rprovides one realization in which that particular observat ion occurs! In fact, it is exactly this type of situation which occurs in practical prediction problems, such a meteorology. One has no control over which particular weather pattern will be observed up to today, but, given the one that has occurred, one would like to predict tomorrow ’s weather. The best predictor will be that corresponding to the conditional ensemble in wh ich all of the available information is used. The variational predictor we have proposed corresp onds to a larger sub-ensemble, which means that somewhat less detailed information about t he system is used in making the prediction. Therefore, the variational predictor is optim al, but in a somewhat weaker sense than the KSP one. The variational estimator x∗[r]is optimal given the data just on empirical sample averages, which is somewhat less than the information one ac tually possesses. We shall refer to the weaker optimality property possessed by the variatio nal estimator as mean-optimality . In the language of statistical physics, the variational est imator could be termed a “mean-ﬁeld approximation” to the optimal one, since it exploits condit ions deﬁned only through the sample mean. How diﬀerent as predictors are the variational and KSP optimal estimators will depend upon how much variation occurs with the larger sub-ensemble . If all the terms rn≈rin the sample average whenever rN=rfor a given r, then there will be little diﬀerence between the two subensembles. This may be expected to occur whenever the re are “preferred paths” in the dynamical evolution. In the linear case, the variational method proposed above co incides with the standard one described in the Introduction. To see this is true it is enoug h to point out the well-known fact that, for a linear dynamics, the eﬀective action Γ X[x] coincides with the Onsager-Machlup action [15]. Hence, in the linear case, the variational esti mator coincides with the optimal estimator of Kushner-Stratonovich-Pardoux, so far as the p roblems of prediction and ﬁltering are concerned. This will not be true in general for nonlinear systems. As we shall see in the 14next subsection, there is nevertheless a close formal conne ction between the Bayesian approach of Kushner-Stratonovich-Pardoux and the variational appr oach. I.4 Calculation of Eﬀective Action & Variational Estimator It remains to consider how the eﬀective action and its minimi zer, the variational estimator, may actually be calculated. It was shown in [11, 13] that the eﬀec tive action may be obtained from a constrained variational formulation. We shall here brieﬂ y review the results of those works and then explain how to obtain the minimizing history x∗[r] itself. Consider any Markov times series X(t) andZ(t) :=Z(X(t),t) given by a continuous func- tionZ(x,t). In this general context there is a useful variational char acterization of the eﬀective action Γ Z[z]. To explain this result, we must introduce a few notations. Because the process X(t) is Markov, its distribution P(x,t) at time tis governed by the forward Kolmogorov equation ∂tP(x,t) =ˆL(t)P(x,t), (I.16) withˆL(t) the instantaneous Markov generator. The diﬀusion process governed by the stochastic equation (0.1) is a particular example, for which the genera tor is the Fokker-Planck operator deﬁned in (0.4). Observables, or random variables, A(x,t) evolve under the corresponding backward Kolmogorov equation ∂tA(x,t) =−ˆL∗(t)A(x,t), (I.17) in which ˆL∗(t) is the adjoint operator of ˆL(t) with respect to the canonical bilinear form on L∞×L1, i.e.<A,P>:=/integraltextdxA(x)P(x). The backward and forward Kolmogorov equations may be simultaneously obtained as Euler-Lagrange equation s for stationarity of the action functional Γ[A,P] :=/integraldisplaytf tidt <A(t),(∂t−ˆL(t))P(t)> (I.18) when varied over P ∈L1with initial condition P(ti) =P0andA ∈L∞with ﬁnal condition A(tf)≡1. 15For the above situation, the eﬀective action of Z(t) :=Z(X(t),t) has been shown [11, 13] to be obtained by a constrained variation of the action Γ[ A,P]. In fact, ΓZ[z] = st.pt.A,PΓ[A,P] (I.19) when varied over the same classes as above, but subject to con straints of ﬁxed overlap <A(t),P(t)>= 1 (I.20) and ﬁxed expectation <A(t),ˆZ(t)P(t)>=z(t) (I.21) for all t∈[ti,tf]. Note that ˆZ(t) is used to denote the operator (in both L1andL∞) of multiplication by Z(x,t). The Euler-Lagrange equations for this constrained varia tion may be obtained by incorporating the expectation constraint (I.2 1) with a Lagrange multiplier h(t). The overlap constraint could also be imposed with a Lagrange multiplier w(t). However, it turns out to be preferable to impose it through the deﬁnition s A(t) = 1 + [ B(t)− /an}bracketle{tB(t)/an}bracketri}htt] := 1 + C(t), (I.22) with the ﬁnal conditions B(tf) =C(tf)≡0. Note that /an}bracketle{tB(t)/an}bracketri}htt:=<B(t),P(t)>is the expecta- tion with respect to the distribution P(t). Hence, the overlap constraint (I.20) is satisﬁed when B(t) is varied independently of P(t). Like A(t), the variable C(t) is not independent of P(t), but must satisfy the orthogonality condition <C(t),P(t)>= 0. We shall mostly make use here of the original variable A(t) rather than C(t), but the latter will play an important role in our formulation of moment-closures in Part II. Although obtained by varying over B(t),P(t), the Euler-Lagrange equations are most use- fully written instead in terms of the original variables A(t),P(t): ∂tP(t) =ˆL(t)P(t) +h⊤(t)[Z(t)− /an}bracketle{tZ(t)/an}bracketri}htt]P(t) (I.23) 16and ∂tA(t) +ˆL∗(t)A(t) +h⊤(t)[Z(t)− /an}bracketle{tZ(t)/an}bracketri}htt]A(t) = 0. (I.24) The calculation via B(t) has allowed the Lagrange multiplier to be evaluated explic itly, as w(t) =h⊤(t)/an}bracketle{tZ(t)/an}bracketri}htt.The eﬀective action Γ Z[z] evaluated at a speciﬁc history z(t) is now obtained from the solutions of (I.23),(I.24) by substituti ng them back into the action functional Γ[A,P] in (I.18), when the “control ﬁeld” h(t) is chosen so that (I.21) reproduces the considered history z(t). It is not accidental that the same notation h(t) was chosen above as for the argument of the cumulant generating functional WZ[h]. In fact, it can be shown that also WZ[h] =/integraldisplaytf tidth⊤(t)/an}bracketle{tZ(t)/an}bracketri}htt, (I.25) using just the solution P(t) of the forward equation (I.23) for the control history h(t) which appears as the argument of WZ. For more details, see [11, 13]. It should not have escaped the attention of the reader that the forward equation (I.23) is very similar to the Kushner- Stratonovich equation (0.3) for the conditional distribut ionP∗(t) =P(t\|R(t)) and that the backward equation (I.24) is likewise similar to the Kushner -Pardoux equation (0.7) for A∗(t) = P(t\|R(tf))/P(t\|R(t)). This observation will be developed below. (See also Appe ndix 1.) Having completed our review of established results, we now c onsider how to calculate the variational estimator. It is helpful to observe that the min imizer x∗[r] of Γ X\|R[x\|r] over x withrﬁxed is the same as of Γ X,R[x,r],the joint action of xandr. For simplicity, the observation errors will be assumed to be white-noise in time and independent of the dynamical noise. Another important simplifying assumption we shall m ake here is that the function of the process which is observed is linear: Z(x,t) =B(t)x. (I.26) We postpone to later the consideration of the general case, w hich is somewhat more complicated but no diﬀerent in principle. By our assumptions, the joint a ction is given as ΓX,R[x,r] = Γ X[x] +1 2/integraldisplaytf tidt[r(t)−B(t)x(t)]⊤R−1(t)[r(t)−B(t)x(t)]. (I.27) 17The second term is Γ R[ρ] given in (0.9). We abbreviate Γ ∗[x] := Γ X,R[x,r] and its functional derivative as k∗[t;x] =δΓ∗ δx(t)[x]. It is an easy calculation, using the expression (I.27), to show that k∗[t;x] =k[t;x] +B⊤(t)R−1(t)[B(t)x(t)−r(t)], (I.28) withk[t;x] :=δΓX δx(t)[x]. Observe that we are using here the notation k(t) for the control associated to X(t), whereas we reserve h(t) for the control ﬁeld associated to Z(t). What makes ﬁnding the minimizer x∗[r] less trivial is the fact that Γ X[x] andk[t;x] are not calculable directly, but only as the result of another optimization pro blem, like that in Eq.(I.5): Γ X[x] = maxk{<x,k>−WX[k]}.Thus, the problem to be solved is really of minimax type: Γ∗[x∗[r]] = minxmax k{ΓR[r−Bx]+<x,k>−WX[k]}. (I.29) Numerical schemes to obtain the minimizer x∗[r] must thus address this minimax problem. The simplest approach conceptually is to reformulate it as a double minimization, i.e. Γ∗[x∗[r]] = minx/braceleftbigg ΓR[r−Bx]−min k{WX[k]−<x,k>}/bracerightbigg . (I.30) In this case, each of the minimizations may be carried out in n ested fashion, via any of the common iterative methods. For example, a conjugate gradient (CG) algorithm applied to the outer problem will produce a sequence x(n)converging as n→ ∞ to, at least, a local minimum x∗of Γ∗[x]. We mention conjugate gradient only as an example of an iter ative scheme to ﬁnd the minimum of a convex function, which requires as its in put at each step the gradient k(n) ∗(t) =δΓ∗ δx(t)[x(n)]. Any such scheme requiring the gradient might be used inste ad. From (I.28) such algorithms require knowing k[x(n)]. Conveniently, this is exactly what is obtained from the solution of the inner problem, since k[x(n)] is the unique minimizer k(n)of the convex functional W(n)[k] :=WX[k]−<k,x(n)> .This inner minimization problem may also be attacked by a CG-type method, noting that the gradient is δW(n) δk(t)[k] =x[t;k]−x(n)(t). (I.31) 18This gradient is now directly calculable via formula (I.21) above for a given k(t). Each evaluation ofx[k] by (I.21) requires one forward and one backward integratio n over the time interval [ ti,tf]. A CG-type method applied to W(n)[k] will then produce a sequence k(n,m)which converges to k(n)=k[x(n)] asm→ ∞. This inner minimization thus provides the gradient k(n)required for thenth CG step of the ﬁrst minimization. To initiate the algorith m, one must specify x(0)and k(0,0). For this purpose, one may, for example, set A ≡1 as a ﬁrst approximation in (I.21). This gives x(0)(t) =/an}bracketle{tX(t)/an}bracketri}htt (I.32) and, from the equation k∗[t;x(0)] =0, the ﬁrst guess k(0,0)(t) =B⊤(t)R−1(t)[r(t)−B(t)x(t)] (I.33) If (I.33) is substituted into the forward equation (I.23), t he latter may be integrated with sequential input of the observations r(t). Thence, both x(0)andk(0,0)are determined. At each successive stage one may take k(n+1,0)=k(n)to ﬁnd the gradient k(n+1)for the ( n+ 1)st CG step. This entire procedure can be regarded as a nonlinear generalization of the “sweep method” [8] used to ﬁnd the minimizer of the Onsager-Machup a ction (0.8). While this method has the advantage of conceptual simplicit y, it suﬀers numerically from loss of precision and computational ineﬃciency. It is well- known in numerical optimization that minimizers are in general obtained to only half the prec ision of the minimum values them- selves. As it is the outside minimizer which is of direct inte rest here, the double minimization algorithm requires working in a precision quadruple to that desired for the optimizing history. Furthermore, the nested algorithm requires the square of the number of iterations as for a single minimization. It is thus advantageous to reformulate the mi nimax problem in terms of a single numerical minimization. This can be easily accomplished by rewriting it as Γ∗[x∗[r]] = min k{ΓR[r−Bx[k])]+<x[k],k>−WX[k]}. (I.34) (We thank M. Anitescu for this observation.) Note again that x[t;k] is given directly by (I.21) via one integration each of the forward and backward Kolmogo rov equations over the time 19interval [ ti,tf]. The result of this single minimization is a control ﬁeld k∗[r], which then yields the desired optimal history x∗[r] asx[k∗[r]]. The only disadvantage of this formulation is that the gradient of the functional in brackets in (I.34), GX[k,r] := Γ R[r−Bx[k]]+<x[k],k>−WX[k], (I.35) is δGX δk(t)[k,r] =/integraldisplaytf tidt′δx δk(t)[t′;k]/bracketleftBig k(t′) +R−1(t′)/parenleftbigB(t′)x[t′;k]−r(t′)/parenrightbig/bracketrightBig . (I.36) This expression involves δx δk(t)[t′;k] =δ2WX δk(t)δk(t′)[k], (I.37) the Hessian of the dual functional WX[k]. Thus, this 2nd-derivative must be evaluated and stored for use. The storage issue is nontrivial for spatiall y-extended or distributed systems, because the Hessian then involves a number of elements of the order of the spacetime grid squared. However, these problems can be overcome. First, th ere are eﬃcient direct and adjoint algorithms for calculating higher-order derivatives, suc h as Hessians, in addition to those for ﬁrst derivatives. For example, see [23], Chapter 7, and also [13]. Second, it is not really the Hessian itself which must be stored but only its matrix produ cts with certain vectors, those in (I.36). Hence, storage requirements can be reduced in int elligent schemes to vectors of the same order as required for the double minimization algorith m. We give further details of such algorithms elsewhere, which we regard as the most promising numerical implementations of our estimation method. Whichever of these iterative optimization methods is emplo yed, Γ ∗[x] is a convex functional, and the iterates will therefore converge to the global minim izerx∗[r].Observe that the zeroth- order of the double iteration scheme coincides formally wit h the KSP equations (0.3)-(0.7). In fact, it is then easy to see that equation (I.33) for k(t) at zeroth-order reduces to k(t) =B⊤(t)h(t), (I.38) 20withh(t) given precisely by (0.5). Substituting this value, the for ward-backward equations in our iterative scheme reduce in form to the KSP equations (0 .3),(0.7). In general, there is no reason to believe (except for linear dynamics), that the v ariational ﬁlter and KS ﬁlter will coincide. However, one may hope that the variational estima tor, acting as a ﬁlter, is not too far from the optimal KS ﬁlter. The formal coincidence of these tw o in the case of linear observations at the start of the iterative construction provides possibl y a convenient algorithmic approach to assess the diﬀerences. We emphasize, however, the word “f ormal” in this context, because the variational equations (I.23), (I.24), while appearing in form identical to the KSP equations (0.3),(0.7), have a quite diﬀerent mathematical interpret ation. Whereas the control ﬁeld h(t) in the variational equations is non-random, the KSP equations are stochastic PDE’s. In particular, the numerical discretization schemes appropriate to the tw o mathematical interpretations are quite diﬀerent and lead to quantitatively distinct results . This will be discussed in more detail below for the case of discrete-time measurements. When the measured function Z(x,t) is nonlinear in x, then our approach must be slightly generalized. In this case, we consider the joint action Γ X,Z,R[x,z,r], whose minimum over x,z withrﬁxed yields the optimum state estimate x∗[r] and also the optimum value of the measured variable z∗[r]. The advantage to considering this joint action is that it i s simply expressed in terms of the eﬀective action Γ R[ρ] of the observation error, which is still assumed independe nt but not necessarily Gaussian. Indeed, a simple calculation in this case gives ΓX,Z,R[x,z,r] = Γ X,Z[x,z] + Γ R[r−z]. (I.39) In contrast, the joint action Γ X,R[x,r] does not have such a simple expression, but instead must be calculated via the Contraction Principle as Γ X,R[x,r] = min zΓX,Z,R[x,z,r].In the case of a linear observed variable, Z(x,t) =B(t)x, the joint action Γ X,Z[x,z] is found to be ΓX,Z[x,z] =  ΓX[x] ifz=Bx +∞ otherwise(I.40) Hence Γ X,R[x,r] = Γ X[x] + Γ R[r−Bx] and the estimation strategy we have proposed for a 21nonlinear measurement function reduces to the earlier one i n the linear case. The minimization of Γ X,Z,R[x,z,r] overx,zmay be done in two steps, which can be carried out independently. These are, ﬁrst, to minimize ΓZ,R[z,r] = Γ Z[z] + Γ R[r−z]. (I.41) over all zat ﬁxed r, and, second, to minimize Γ X,Z[x,z] over all xwithzﬁxed. From the solutions of these two problems, z∗[r] andx∗[z], respectively, the ﬁnal variational estimator of the state of the system is then the obtained as the compositio nx∗[r] =x∗[z∗[r]]. The equivalence of this two-step formulation with the direct one is an applic ation of the Contraction Principle. Clearly, minimizing Γ X,R[x,r] over all xcan be achieved by minimizing ﬁrst Γ X,Z,R[x,z,r] over all xwithzﬁxed, and then by minimizing over all z. The minimization over xyields the joint eﬀective action of zandr, since Γ Z,R[z,r] = min xΓX,Z,R[x,z,r] by the Contraction Principle. The minimum is achieved here for some x∗[z], the optimal state history xfor a given z-history. There is no dependence upon r. To see this, observe that the minimization may be directly carried out in equation (I.39), with the result tha t ΓZ,R[z,r] is given by (I.41). The Contraction Principle has been employed again to infer Γ Z[z] = min xΓX,Z[x,z]. It is from this minimization that x∗[z] is determined, which therefore cannot involve r. All of the dependence upon measurements is now isolated in (I.41), whose minimiza tion over zyieldsz∗[r]. This ﬁrst minimization of Γ Z,R[z,r] overzis a problem of the same type as for the case of linear measurement functions discussed in the text. As ther e, a CG-type method applied to Γ∗[z] := Γ Z,R[z,r] may be employed to calculate z∗[r],based upon the Legendre dual relations h[t;z] =δΓZ δz(t)[z],z[t;h] =δWZ δh(t)[h]. (I.42) Any of the algorithms discussed in the text may be employed. F or example, in the double minimization scheme, the gradient for the outer minimizati on, h∗[t;z(n)] :=δΓ∗ δz(t)[z(n)] =h[t;z(n)] +R−1(t)[z(n)(t)−r(t)], (I.43) 22would be obtained from an inner one. The iteration could be in itiated by z(0)(t) =/an}bracketle{tZ(t)/an}bracketri}htt (I.44) and h(0,0)(t) =R−1(t)[r(t)− /an}bracketle{tZ(t)/an}bracketri}htt]. (I.45) Just as before—but now quite in general—the zeroth-order co ntrolh(0,0)(t), when substituted into the forward-backward equations (I.23), (I.24) recove rs formally the KSP equations. The second minimization of Γ X,Z[x,z] overxis similar. Note that ΓX,Z[x,z] = max k,h{<k,x>+<h,z>−WX,Z[k,h]}. (I.46) Hence, the problem ΓX,Z[x∗[z],z] = minxmax k,h{<k,x>+<h,z>−WX,Z[k,h]} (I.47) is again of minimax type. A doubly iterative scheme would the refore carry out the maximization overk,hat ﬁxed x,zto obtain not only Γ X,Z[x,z] but also the gradients k[t;x,z] =δΓX,Z δx(t)[x,z],h[t;x,z] =δΓX,Z δz(t)[x,z] (I.48) that are used in the next minimization over x(at ﬁxed z). Alternatively, one may solve this problem as before via a single minimization over k,hof a functional GX,Z[k,h] :=<k,x[k,h]>+<h,z[k,h]>−WX,Z[k,h] (I.49) but with the diﬀerence that this minimization is now subject to a nonlinear constraint that z[t;k,h] =z(t), t∈[ti,tf]. (I.50) This may be addressed using algorithms from nonlinear progr amming or stochastic/ quasi- random methods. 23I.5. Estimation with Discrete-Time Data So far, we have considered the case where the measurements em ployed in our estimation are taken continuously in time. However, this can only be an idea lization of a situation where the data are obtained at a discrete series of times. In many pr actical examples, the instants of measurement will be so widely separated that the idealiza tion of continuous acquisition is far from valid. It is thus a very practical concern to address the issue of state estimation of continuous in time, stochastic dynamical systems such as (0 .1) based upon discrete-time data. In addition, we shall ﬁnd that some fundamental new concepts are required that are important in other contexts. For example, the calculation of ensemble dispersions at an instant of time will turn out to be closely related to the problem of estimati on with discrete-time data. The only change in the statement of the problem in the Introdu ction is that now the mea- surements are of the form rk=Z(x(tk),tk) +ρk, k= 1,...,n (I.51) where ρkrepresents a measurement error with covariance Rk. If the measurement error is taken to be an independent Gaussian at each time tk, then the cost function for the observations is ΓR[ρ] =1 2n/summationdisplay k=1ρ⊤ kR−1 kρk, (I.52) where the sum includes all of the observation times t1,...,t nup to the present time. The combined cost function Γ ∗[z] := Γ Z,R[z,r] for the estimation is then, analogous to (I.41), Γ∗[z] = Γ Z[z] +1 2n/summationdisplay k=1[rk−z(tk)]⊤R−1 k[rk−z(tk)]. (I.53) For simplicity, we shall only consider here the problem of es timating the optimal z-history. As discussed in the previous section, there remains the prob lem of estimating the optimal state or x-history, given the z-history. This can be handled in the same way as discussed there. Alternatively, we might formulate the problem as a di rect estimation of x. The changes necessary to our discussion below should be obvious to the re ader. If we seek the minimizer of 24(I.53), we must satisfy 0=δΓ∗ δz(t)[z] =h[t;z] +n/summationdisplay k=1R−1 k[z(tk)−rk]δ(t−tk). (I.54) Thus, we see that h[t;z∗] for the optimal z∗[r] must be a sum of delta functions at the obser- vation times. This suggests that we consider only the estima tion of zat the observation times. In fact, we will see that this suﬃces. The cost function H∗(z1,...,zn) :=HZ,R(z1,...,zn;r1,...rn) for estimating zk:=z(tk), k= 1,...,n is obtained in the following way. First, we deﬁne a cumulant generating function FZ(λ1,...,λn) := log /an}bracketle{texp[n/summationdisplay k=1λ⊤ kZ(tk)]/an}bracketri}ht. (I.55) This is entirely analogous to the cumulant generating funct ionalWZ[h] deﬁned in subsection 2.1. In fact, they are equal with h(t) =n/summationdisplay k=1λkδ(t−tk). (I.56) The Legendre transform of FZis the dynamical part of the cost function: HZ(z1,...,zn) = max λ1,...,λn/braceleftBiggn/summationdisplay k=1z⊤ kλk−FZ(λ1,...,λn)/bracerightBigg . (I.57) This quantity is called the multitime (relative) entropy . It may also be obtained via the Con- traction Principle directly from the eﬀective action throu gh a constrained minimization: HZ(z1,...,zn) = min {z:z(tk)=zk,k=1,...,n}ΓZ[z]. (I.58) The combined cost function is then H∗(z1,...,zn) =HZ(z1,...,zn) +1 2n/summationdisplay k=1[rk−zk]⊤R−1 k[rk−zk]. (I.59) Its minimization yields the optimal values of z(t1),...,z(tn). The condition for the minimum is λk=R−1 k[rk−zk], (I.60) which can already be inferred from (I.54),(I.56). We may reg ard (I.60) as a nonlinear equation for either the λ’s or the z’s. 25To calculate numerically the cost function HZ(z1,...,zn) we see that we must integrate the forward and backward equations (I.23),(I.24) with a con trol ﬁeld h(t) consisting of delta- function spikes, as in (I.56). It is easiest to formulate thi s integration in terms of suitable jump conditions at the observation times. That is, we may integrate the ordin ary forward and backward Kolmogorov equations (I.16),(I.17) with h=0between the observation times but make discrete jumps at those times. We shall show that the pro per jump conditions are simply P(x,tk+) =eλ⊤ kZ(x,tk) W(tk−)P(x,tk−), (I.61) and A(x,tk−) =eλ⊤ kZ(x,tk) W(tk−)A(x,tk+). (I.62) Here we deﬁned W(tk−) :=/integraldisplay dxeλ⊤ kZ(x,tk)P(x,tk−) =/an}bracketle{teλ⊤ kZ(tk)/an}bracketri}httk−, (I.63) so that division by that factor guarantees proper normaliza tion of the results after the jump. We prove now the validity of these jump conditions. For the ﬁr st, it is useful to make a reformulation of the forward equation (I.23). The same solu tion found for that equation may be obtained by solving instead ∂tQ(t) =ˆL(t)Q(t) +h⊤(t)Z(t)· Q(t) (I.64) and then renormalizing subsequently P(x,t) :=Q(x,t) N(t)(I.65) with N(t) :=/integraldisplay dxQ(x,t). (I.66) It is not hard, by diﬀerentiating (I.65) with respect to time , to show that P(t) so-deﬁned satisﬁes (I.23). This is actually a standard device to solve the Kushner-Stratonovich equation. 26In that context, the analogue of equation (I.64) is called th eZakai equation [26]. With the delta-function control ﬁeld, we obtain ∂tlnQ(t) =ˆL(t)Q(t) Q(t)+n/summationdisplay k=1λkZ(tk)δ(t−tk). (I.67) We then integrate in time over the range ( tk−ǫ,tk+ǫ) and take the limit as ǫ→0. The ﬁrst term on the righthand side is continuous and does not contrib ute. From the delta function contribution we easily obtain Q(x,tk+) Q(x,tk−)=eλ⊤ kZ(x,tk). (I.68) Renormalizing Q(x,tk+), we recover (I.61), as claimed. The second jump condition can be similarly obtained. The bac kward equation (I.24) must likewise be rewritten so that the source terms stand alone be fore integration. In fact, by integrating (I.64) over x, one ﬁnds that d dtlnN(t) =h⊤(t)/an}bracketle{tZ(t)/an}bracketri}htt. (I.69) Using this result, one easily derives from (I.24) that ∂tln/parenleftbiggA(t) N(t)/parenrightbigg +ˆL∗(t)A(t) A(t)+h⊤(t)Z(t) = 0. (I.70) Let us now consider the case where h(t) is given by (I.56), as a sum of delta-functions. Inte- grating (I.70) over the range ( tk−ǫ,tk+ǫ) and taking the limit as ǫ→0, then yields A(x,tk+)/N(tk+) A(x,tk−)/N(tk−)=e−λ⊤ kZ(x,tk). (I.71) Of course, N(t) itself experiences a jump across the observation time tk, changing as we have seen by the ratio N(tk+) N(tk−)=/an}bracketle{teλ⊤ kZ(tk)/an}bracketri}httk−=W(tk−). (I.72) This is a direct consequence of (I.68). From (I.71) and (I.72 ), the second jump condition immediately follows. 27Using the jump conditions (I.61) and (I.62) to replace the co ntrolled forward and backward equations (I.23),(I.24), the calculation of the cost funct ion proceeds as follows. Integrating (I.69) over the time-interval [ ti,tf] and comparing with (I.25), we see that FZ(λ1,...,λn) = log N(tf). (I.73) By writing N(tf) =/producttextn k=1N(tk+) N(tk−)(where N(t1−) = 1 was used), we can decompose this into a sum of contributions for each time tk FZ(λ1,...,λn) =n/summationdisplay k=1(∆F)k(λ1,...,λk) (I.74) with (∆F)k(λ1,...,λk) := log /an}bracketle{teλ⊤ kZ(tk)/an}bracketri}httk−= logW(tk−). (I.75) Whereas the dependence upon λkis explicit, note that the dependence upon the remaining variables λ1,...,λk−1is only implicit through P(x,tk−). Having determined FZ(λ1,...,λn), the entropy HZ(z1,...,zn) can then be obtained by the Legendre transform formula (I.5 7). In that formula zk:=z(tk), k= 1,...,n, with z(t) given for all times tby z(t) =/integraldisplay dxZ(x,t)A(x,t)P(x,t). (I.76) It is worth emphasizing that the history z(t) is a continuous function of time. This will be true even though the solutions P(x,t),A(x,t) have jump discontinuities at the observation times t=tk, k= 1,...,n. In fact, it is easy to see by a direct diﬀerentiation that dz dt(t) =<{∂tˆZ+ [ˆL∗,ˆZ]}A(t),P(t)> . (I.77) In particular, all of the delta-function sources cancel fro m this equation. Hence, z(t) is contin- uous but will generally have a time-derivative with jump-di scontinuities. The rest of the estimation protocols outlined in sections I. 4 are the same. For example, the double minimization algorithm may be carried out using the L egendre dual pair HZ(z1,...,zn), FZ(λ1,...,λn). The iteration may be initiated by taking z(0) k=/an}bracketle{tZ(tk)/an}bracketri}httk+ (I.78) 28and λ(0,0) k=R−1 k[rk− /an}bracketle{tZ(tk)/an}bracketri}httk−]. (I.79) The ﬁnal result will be an optimal estimated history z∗(t) given by (I.76), where P∗(t),A∗(t) therein are the solutions of the forward and backward equati ons for the λ∗ k,z∗ kobtained as convergents of the minimization algorithm. It is worthwhile to compare this procedure for calculating t he variational estimator with discrete data to that for calculating the optimal KSP estima tor in the same circumstances. It is shown in Appendix 1 that the optimal estimator may be obt ained as well by integrat- ing the forward and backward Kolmogorov equations (I.16),( I.17) for P∗(t),A∗(t) between the observation times and by making discrete jumps at those time s. The proper jump conditions are P∗(x,tk+) =1 W∗(tk−)exp/bracketleftbigg λ⊤ kZ(x,tk)−1 2δZ⊤(x,tk−)R−1 kδZ(x,tk−)/bracketrightbigg P∗(x,tk−),(I.80) and A∗(x,tk−) =1 W∗(tk−)exp/bracketleftbigg λ⊤ kZ(x,tk)−1 2δZ⊤(x,tk−)R−1 kδZ(x,tk−)/bracketrightbigg A∗(x,tk+).(I.81) In these equations λk=R−1 k[rk− /an}bracketle{tZ(tk)/an}bracketri}httk−], (I.82) the same as the zeroeth-order (I.79) above, δZ(x,tk−) :=Z(x,tk)− /an}bracketle{tZ(tk)/an}bracketri}httk−, (I.83) and W∗(tk−) :=/integraldisplay dxexp/bracketleftbigg λ⊤ kZ(x,tk)−1 2δZ⊤(x,tk−)R−1 kδZ(x,tk−)/bracketrightbigg P(x,tk−) (I.84) is the factor to keep the density P∗(x,t) normalized. The solutions of these equations after a single forward and backward integration then yield the conditiona l probability density, via the formula P(x,t\|R(tf)) =A∗(x,t)P∗(x,t). See Appendix 1. 29It is now clear that the zeroeth-order variational estimato r, calculated after one forward- backward sweep initialized with (I.78),(I.79), does notcoincide with the optimal KSP estimator, for the case of discrete-time measurements. The main diﬀere nce, one can easily see, lies in the extra term in the exponent quadratic in δZ(x,tk−). This term has the eﬀect of preventing the estimate after the jump from being too far from the prior esti mate/an}bracketle{tZ(tk)/an}bracketri}httk−. The absence of this term in the zeroeth-order variational equations hel ps to make clear in what sense it is a “mean-ﬁeld” approximation of the optimal estimator, obta ined by neglect of “ﬂuctuations”. In fact, if we estimate not the variable Z(t) itself, but rather its sample mean ZN(t), then heuristically λremains unchanged but δZN(t) =O(N−1/2). Hence, the quadratic term in the exponent is O(1/N) and may be neglected. Of course, it must be realized that we a re here comparing the optimal KSP estimator with only a zeroeth -order approximation to the variational estimator, not to the variational estimator it self for the converged values of λ∗ k, k= 1,...,n. We hope that, in general, the value of the variational estim ator, calculated with the “mean-ﬁeld” jump rules (I.61),(I.62) for the minimizing va luesλ∗ k, will not be too far from the optimal KSP estimator given by the jump rules (I.80),(I.81) . The diﬀerence between the zeroeth-order variational estim ator and the optimal KSP esti- mator which we have illustrated above for the case of discret e-time measurements, of course also holds in the case of continuous-time measurements. In t hat case it is a consequences of the diﬀerence in mathematical interpretation of the zeroet h-order variational equations and the KSP stochastic equations, despite their formal identity. O f course, at this point one might question the utility of the variational formulation compar ed with the straightforward Bayesian approach based upon the KSP equations. Whether in the discre te- or continuous-time formu- lations, it is essentially just as diﬃcult to solve the KSP eq uations as to make one “sweep” in the iterative solution of the variational problem. However , the latter requires in most cases a large number of “sweeps” and furthermore provides a subopti mal estimate compared with the KSP approach! The advantage of the variational approach wil l become apparent in Part II, when we consider making ﬁnite-dimensional approximations . 30I.6. Calculation of the Ensemble Dispersion As discussed in section I.1, one would like to have not just th e mean state x∗(t) but also the covariance matrix C∗(t) in the conditioned ensemble at each time t. As we shall show now, the covariance may be readily calculated from the cost funct ion itself. For simplicity, we shall conﬁne our discussion to the calculation of the covariance o f the measured variable Z(t). The changes required for the determination of the state covaria nce will be obvious. Let us discuss ﬁrst the case with discrete-time data. The ent ropy function H∗(z1,...,zn) := HZ,R(z1,...,zn;r1,...,rn) that we introduced in (I.59) of the last subsection has also an in- terpretation as a generating function for (irreducible) mu ltitime correlations in the ensemble conditioned on rN(tj) =rj, j= 1,...,n. Thus, one may calculate the 2-time irreducible corre- lator Γ∗(tk,tj) =∂2H∗ ∂zk∂zj(z1,...,zn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=z. (I.85) This irreducible correlator is related to the multitime cov ariance matrix by matrix inversion: C∗(tk,tj) = [Γ∗(tk,tj)]−1. (I.86) The same quantity could also be obtained from F∗(λ∗ 1,...,λ∗ n) :=FZ,R(λ∗ 1,...,λ∗ n;r1,...,rn), the Legendre dual of H∗(z1,...,zn), as C∗(tk,tj) =∂2F∗ ∂λ∗ k∂λ∗ j(λ∗ 1,...,λ∗ n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ∗=0. (I.87) From this one may obtain the single-time covariance at any of the times tkby considering the diagonal C∗(tk) =C∗(tk,tk). Without loss of generality, one may include any time of int erest as one of the “measurement times” by simply taking the corres ponding value of its observation error as inﬁnite, or R−1 k=0. While this procedure gives the correct result, it is not so pr actical because the quantity of interest, the diagonal C∗(tk), is obtained only through the intermediary of the full 2-ti me covariance C∗(tk,tj). A more useful approach is based upon the single-time gener ating function 31obtained from the Contraction Principle H∗(z;tk) := min ˜z:˜zk=zH∗(˜z1,..,˜zn). (I.88) This is just the (conditional) relative entropy at time tk. We have chosen here to make the time-dependence explicit in the instantaneous entropy. On e can calculate the Hessian of this function Γ∗(tk) =∂2H∗ ∂z∂z(z;tk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=zk. (I.89) and then obtain C∗(tk) = [Γ∗(tk)]−1. (I.90) To employ this method, one must carry out the minimization in (I.88). This leads as before to the condition ∂H∗ ∂zj(˜z1,...,˜zn) := λ∗ j(˜z1,...,˜zn) =λj(˜z1,...,˜zn) +R−1 j(˜zj−rj) =0, (I.91) forj/ne}ationslash=kwith˜zk=zﬁxed. This minimization problem can be solved computationa lly with the same methods used to ﬁnd the global minimum, e.g. the doub le CG-type algorithm, but now with ˜zk=zheld invariant and H∗minimized only over the remaining variables ˜zj, j/ne}ationslash=k. The result will be the constrained minimizers z∗ j(z;tk) that, substituted into λ∗ j(˜z1,...,˜zn) with ˜zk=z, give0for all j/ne}ationslash=k. However, λ∗(z;tk) :=λ∗ k(˜z1,...,˜zn)\|˜zk=z;˜zj=z∗ j(z;tk), j/ne}ationslash=k (I.92) will not be zero. In fact, it is not hard to see that λ∗(z;tk) =∂H∗ ∂z(z;tk). (I.93) Then, from (I.89), Γ∗(tk) =∂λ∗ ∂z(z;tk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=zk. (I.94) 32This gives rise to a simple, practical algorithm to calculat e the covariance, by means of a ﬁnite-diﬀerence approximation for some small δ Γ∗ αβ(tk)≈λ∗ α(z+β;tk)−λ∗ α(z−β;tk) 2δ =λα(z+β;tk)−λα(z−β;tk) 2δ+ [R−1 k]αβ, (I.95) with z±β:=zk±δ·ˆeβ, (I.96) andˆeβa unit vector in the β-direction. We have set λ(z;tk) =λk(˜z1,...,˜zn)\|˜zk=z;˜zj=z∗ j(z;tk), j/ne}ationslash=k. This approximation requires the calculation of λ∗(z;tk) for the two new values z=z±βdisplaced slightly from zk. This can be accomplished using (I.92) and the double minimi zation algorithm. Suitable guesses to initiate the minimization would be ˜z(0) j=  z±βj=k zjj/ne}ationslash=k(I.97) and λ(0,0) j=R−1 j[rj−˜z(0) j], j= 1,...,n. (I.98) Of course, ˜zk=z±βis held ﬁxed in the iteration. This procedure must be followe d to calculate the covariance at each time tkof interest. For each scalar variable, calculating its vari ance by this method is roughly twice the work as calculating the opti mal estimate itself over the whole interval of time. However, this statement is misleadingly p essimistic. In fact, the initial points considered, ˜zk=z±β,˜zj=zj, j/ne}ationslash=kare very close to the optimal history, which is assumed known. Hence, only small changes will occur in the ˜zj’s,O(δ) corrections to the zj’s, and the minimization algorithm should converge quite quickly. The contribution of the various small changes can be read oﬀ from (I.95). The direct contribution f rom the change in ˜zktoz±βis [C(tk)]−1+R−1 k, where C(tk) is the covariance in the unconditioned ensemble. The addit ional contributions from the small changes in the ˜zj, j/ne}ationslash=kwill be similar, but will decay according to the distance of tjfromtkin time. The rate of decay will be determined by some internal relaxation or memory time of the system. 33If the number of variables whose variance is required is larg e, then even the matrix inversion in (I.90) is diﬃcult and should be avoided. This can be accomp lished by following an alternative procedure, based upon implementing the constraint ˜zk=zby a Lagrange multiplier. In this case, (I.88) is replaced by an unconstrained minimization H∗(z;tk) := min ˜z1,...,˜zn˜H(˜z1,..,˜zn;˜λ), (I.99) where ˜H(˜z1,..,˜zn;˜λ) :=H∗(˜z1,..,˜zn) +˜λ⊤(z−˜zk), (I.100) and the Lagrange multiplier ˜λis chosen subsequently to impose the constraint ˜zk=z. The condition for the minimum over all the variables ˜zj, j= 1,...n,is ∂˜H ∂˜zj(˜z1,...,˜zn) := λ∗ j(˜z1,...,˜zn)−˜λδjk =0. (I.101) Thus, we see that the minimizing ˜zj(˜λ;tk)’s in (I.99) are nothing more than ˜zj(˜λ;tk) =z∗ j(λ∗ 1,...,λ∗ n)/vextendsingle/vextendsingle/vextendsingleλ∗ k=˜λ;λ∗ j=0, j/ne}ationslash=k(I.102) where z∗ j(λ∗ 1,...,λ∗ n) :=∂F∗ ∂λ∗ j(λ∗ 1,...,λ∗ n). (I.103) Then, using (I.87), one obtains C∗(tk) =∂z∗ k ∂λ∗ k(λ∗ 1,...,λ∗ n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ∗ j=0, j=1,...,n =∂˜zk ∂˜λ(˜λ;tk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜λ=0. (I.104) Incidentally, it is clear also from (I.102) that the value of the Lagrange multiplier to achieve the constraint ˜zk=zis just ˜λ=λ∗(z;tk) as given in (I.92),(I.93). The important point for our considerations here is that formula (I.104) involves no matrix inversion. Thus, formula (I.104) becomes the basis of an alternative pr ocedure to numerically compute the covariance. In this procedure, one minimizes ˜H(˜z1,..,˜zn;λ±β) for the values λ±β=±δ·ˆeβ (I.105) 34to obtain ˜zj(λ±β;tk), j= 1,...,n. Then one may approximate C∗ αβ(tk)≈˜zkα(λ+β;tk)−˜zkα(λ−β;tk) 2δ. (I.106) The minimization to obtain the ˜zj(λ±β;tk) may be carried out with similar methods as before, e.g. the double CG-type algorithm initiated with the guesse s ˜z(0) j=zj, j= 1,...,n (I.107) and λ(0,0) j=R−1 j[rj−zj] +λ±βδjk, j= 1,...,n. (I.108) Our discussion above carries over straightforwardly to the case of continuous-time data acqui- sition. The entropy at any time t0is, by the Contraction Principle, given as HZ,R(z,r;t0) = min z:z(t0)=zΓZ,R[z,r] (I.109) with Γ Z,Ras in (I.27). Alternatively, one has HZ,R(z,r;t0) = minz˜ΓZ,R[z,r;˜λ], (I.110) with ˜ΓZ,R[z,r;˜λ] := Γ Z,R[z,r] +˜λ⊤[z−z(t0)]. (I.111) Either of the approaches outlined above may be used to ﬁnd C(t0;r). For example, in the second method C(t0;r) =∂˜z ∂˜λ[t0;r,˜λ]/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜λ=0(I.112) where ˜z[t;r,˜λ] is the solution of the minimization condition 0=δ˜ΓZ,R δz(t)[z,r;˜λ] =h[t;z,r]−˜λδ(t−t0). (I.113) This is solved with the double CG-type algorithm, using jump conditions (I.61),(I.62) at t0. 35II Moment-Approximation of the Optimal Estimator II.1. The Rayleigh-Ritz Method Until now all of our theoretical work has been exact and witho ut any approximation, other than that involved in conditioning on sample averages, the “mean -ﬁeld” approximation discussed in the section I.3. However, it is clear that additional approx imations are required to achieve a computationally tractable estimation scheme for spatiall y-extended or distributed systems. As discussed in the Introduction, the exact calculation of the optimal nonlinear estimator by the KSP equations is already known to be numerically unfeasible in such situations. Furthermore, computation of the exact variational estimator is just as im practical as the computation of the exact KSP optimal estimator for a system with a large number o f degrees-of-freedom. It will not be possible for almost any system of real, practical inte rest. The existence of a variational principle does not ameliorate the basic computational diﬃc ulty imposed by the enormously many variables. Just as for the KS ﬁlter, moment-closure app ears to be the only tractable numerical approach to an approximate solution. The advanta ge of the variational formulation is that it permits ﬁnite-dimensional approximations to be con structed by a Rayleigh-Ritz method which preserves the main structural properties of the exact estimator, discussed previously. We shall brieﬂy discuss these features here, referring to prev ious works [11, 13] for many details. The Rayleigh-Ritz approximation to the cost function is obt ained by means of the charac- terization of that functional through the constrained vari ation in (I.19). Rather than varying over all A ∈L∞,P ∈L1, one varies only over ﬁnitely parametrized trial functions . The trial functions are constructed from the usual elements of a momen t-closure: a set of moment func- tions Mi(x,t), i= 1,...,R and a PDF Ansatz P(x,t;µ), which is conveniently parametrized by the mean values which it attributes to the moment-functio ns,µ:=/integraltextdxP(x,t;µ)M(x,t). The left trial function may be taken to be A(t) = 1 + [ B(t)− /an}bracketle{tB(t)/an}bracketri}htt] with B(x,t;α) :=R/summationdisplay i=1αiMi(x,t). (II.1) Following the discussion in section 2.3, we have chosen the l eft trial state in the form (I.22), 36to incorporate automatically the overlap constraint (I.20 ). The histories α(t),µ(t) are the parameters to be varied over. Substituting the trial forms, one obtains the reduced action Γ[α,µ] =/integraldisplaytf tidtα⊤(t)[˙µ(t)−V(µ(t),t)] (II.2) with V(µ,t) :=/an}bracketle{t(∂t+ˆL∗)M(t)/an}bracketri}htµ(t). (II.3) Of course, /an}bracketle{t·/an}bracketri}htµ(t)denotes average with respect to the PDF Ansatz . An unconstrained variation of (II.2) recovers the standard moment-closure equation: ˙µ=V(µ,t). For the calculation of the action, however, there is the additional expectation co nstraint (I.21). In terms of the trial functions, it becomes z(t) =ζ(µ(t),t) +CZ(µ(t),t)α(t). (II.4) Here, ζ(µ,t) :=/an}bracketle{tZ(t)/an}bracketri}htµ (II.5) is the Z-expectation within the PDF Ansatz and CZ(µ,t) :=/an}bracketle{tZ(t)M⊤(t)/an}bracketri}htµ−ζ(µ,t)µ⊤(II.6) is the corresponding ZM-covariance matrix. It is remarkable that ζ(µ,t),CZ(µ,t) are the only inputs of the PDF Ansatz actually required for the calculation. When the constraint (II.4) is incorporated into the action f unctional (II.2) by means of a Lagrange multiplier h(t), the resulting Euler-Lagrange equations are ˙µ=V(µ,t) +C⊤ Z(µ,t)h(t) :=VZ(µ,h,t). (II.7) and ˙α+/parenleftbigg∂VZ ∂µ/parenrightbigg⊤ (µ,h,t)α+/parenleftbigg∂ζ ∂µ/parenrightbigg⊤ (µ,t)h(t) =0. (II.8) These are solved subject to an initial condition µ(ti) =µ0and a ﬁnal condition α(tf) =0. When the solutions of the integrations are substituted into (II.2), there results a Rayleigh-Ritz 37approximation ˜ΓZ[z] to the eﬀective action of Z(t). The value z(t) of the argument is that given by the constraint equation (II.4) for the given value of the c ontrol ﬁeld h(t). A corresponding approximation of the cumulant generating functional is giv en by ˜WZ[h] =/integraldisplaytf tidth⊤(t)ζ(µ(t),t) (II.9) in which µ(t) is the solution of just the forward equation (II.7) for the c ontrol history h(t). It is a very attractive feature of the above approximation sc heme that the resulting func- tionals ˜ΓZ[z],˜WZ[h] remain formal Legendre transforms of each other. That is, ˜WZ[h] +˜ΓZ[z] =<h,z> (II.10) and h[t;z] =δ˜ΓZ δz(t)[z],z[t;h] =δ˜WZ δh(t)[h]. (II.11) This fact makes it possible to carry over directly all of the m inimax algorithms discussed in section I.4 for determination of optimal histories using th e exact cost function to the Rayleigh- Ritz approximate one. Incidentally, the form of the constra int (II.4) makes it more apparent that this approach generalizes the “sweep method” employed in the case of linear dynamics [8]. The iterative constructions that were discussed for the exa ct optimal estimator can be followed also to calculate the moment-closure approximati on. For example, consider the two- step method. As the ﬁrst step, one can calculate the approxim ate optimal ˜z∗[r] given r, by minimizing ˜ΓZ,R[z,r] =˜ΓZ[z] +1 2/integraldisplaytf tidt[r(t)−z(t)]⊤R−1(t)[r(t)−z(t)]. (II.12) overzwithrﬁxed. This can be accomplished, for example, with a double CG method as before, taking now z(0)(t) =ζ(µ,t) (II.13) h(0,0)(t) =R−1(t)[r(t)−ζ(µ,t)] (II.14) as the zeroth-order inputs. It is clear that (II.14), substi tuted into the approximate forward equation (II.7), is formally equivalent to a moment-closur e of the KS-equation. (Although it 38must be emphasized once more that, in the case of the KS ﬁlter, the closure equation analogous to (II.7) must be regarded as a stochastic diﬀerential equat ion.) The second step is to calculate the approximate optimal ˜x∗[z] by minimizing ˜ΓX,Z[x,z] over xwithzﬁxed. Of course, the Rayleigh-Ritz approximation ˜ΓX,Z[x,z] is calculated by the analogous equations as (II.7),(II.8) : ˙µ=V(µ,t) +C⊤ X(µ,t)k(t) +C⊤ Z(µ,t)h(t) :=VX,Z(µ,k,h,t). (II.15) and ˙α+/parenleftbigg∂VX,Z ∂µ/parenrightbigg⊤ (µ,k,h,t)α+/parenleftbigg∂ξ ∂µ/parenrightbigg⊤ (µ,t)k(t) +/parenleftbigg∂ζ ∂µ/parenrightbigg⊤ (µ,t)h(t) =0, (II.16) where ξ(µ,t),CX(µ,t) are the closure X-mean and XM-covariance, respectively. The ﬁnal approximate estimator is then the composition ˜x∗[r] =˜x∗[˜z∗[r]]. However, there is a potential diﬃculty in applying the minim ization algorithms: the Rayleigh- Ritz approximations to the cost functions need not be convex at all! Lack of convexity would correspond to a failure of realizability of the predicted mu lti-time correlations [11]. As a conse- quence of this failure, there might exist local minima in add ition to the global one or, possibly, no minimum at all, local or global. In the former case, a CG alg orithm could be trapped in a local minimum, and, in the latter, it would not converge at al l. Thus, for numerical purposes, it is exceedingly desirable to maintain convexity. It was sh own in [12] that convexity will be maintained —at least for an expansion of the action to quadra tic order in small departures from the minimum—whenever the relative entropy is a Lyapunov sta bility function for the closure dynamics. It is possible to construct closures for nonlinea r stochastic dynamics which guaran- tee the validity of such an H-theorem [27], using methods previously developed for Bolt zmann kinetic equations in transport theory [28]. One example of t he general scheme are closures based upon an exponential PDF Ansatz . Such closures have the property that the relative entropy satisﬁes an H-theorem and thus (local) convexity of the Rayleigh-Ritz ap proximations is guaranteed. This is discussed further in [27] and in secti on II.3 below. 39II.2. Discrete-Time Data and Ensemble Dispersion We have seen in section I.5. that the estimation problem base d upon discrete-time data has, in the exact formulation, a simple solution in terms of certa in jump conditions. The situation is worse for closure approximations. In fact, we shall see be low that, for general moment- closures, the approximate “smoother” with discrete-time d ata may not even be continuous at the observation times! This is an important failing since th ese same methods are also involved in the calculation of instantaneous ensemble dispersions, as we have seen in section I.6. Let us illustrate the nature of the problem for a general mome nt-closure. If one diﬀerentiates the expression for z(t) in (II.4) with respect to time, using the variational equat ions (II.7), (II.8), simple computations give a result of the form dza dt(t) =dζa dt(µ,t) +αj/bracketleftBigg dCZ aj dt(µ,t)−CZ ai(µ,t)∂Vj ∂µi(µ,t)/bracketrightBigg +hb(t)/bracketleftbigg CZ bi(µ,t)∂ζa ∂µi(µ,t)−CZ ai(µ,t)∂ζb ∂µi(µ,t)/bracketrightbigg +hb(t)/bracketleftBigg CZ bi(µ,t)∂CZ aj ∂µi(µ,t)−CZ ai(µ,t)∂CZ bj ∂µi(µ,t)/bracketrightBigg αj. (II.17) For any function of µ,twe setd dt:=∂ ∂t+V(µ,t)·∇µ.Equation (II.17) should be compared with the exact result in (I.77). In contrast to the cancellation o f the explicit h(t) terms found there, such terms remain in the second and third lines above. Only in the case where a single scalar variable z(t) is considered and thus a=b= 1 is there an obvious cancellation in the last two terms. This means that, in general, the delta-functions wil l not cancel if one considers a control ﬁeld of the form h(t) =/summationtext kλkδ(t−tk), as appropriate for discrete-time data, and z(t) itself will have jump-discontinuities at the measurement times tk. Of course, one take observations, not instantaneously, but instead averaged over a small inte rval of time τ. The delta functions are then replaced by approximate delta’s δτ(t−tk) with time-window τ. However, the problem will reappear when τis taken very small, for then z(t) will change sharply at times tk. A related problem has to do with the formulation of proper jum p conditions in the same circumstances. Let us even assume that z(t) is a single scalar. Then, the forward equation for 40the moment variable µbecomes ˙µi=Vi(µ,t) +h(t)CZ i(µ,t). (II.18) Ifh(t) is a sum of delta-functions, then one cannot integrate the e quation to obtain the jumps inµiat the observation times. The diﬃculty is that CZ i(µ,t) will then also have jump- discontinuities at those times and it is impermissable to in tegrate a delta-function against a discontinuous function. The obvious strategy is ﬁrst to di vide both sides by CZ i(µ,t) and only afterward integrate across the jump. There is still a pr oblem however. The other variables besides µiin the integrand also make jumps and it is therefore ambiguou s which value should appear as the integration range shrinks to zero. Thus, the st rategy only works in the case where there is also a single scalar moment variable µ(t). In that case, we can integrate and obtain a jump condition in the form of an “area rule”: /integraldisplayµ+ k µ− kdµ CZ(µ,tk)=λk. (II.19) We have set µ± k=µ(tk±). The backward jump condition for the adjoint variable αthen follows most easily from the continuity of z(t) noted above. With notations as above, α± k=α(tk±) and so forth, we have ζ+ k+CZ+ kα+ k=ζ− k+CZ− kα− k. (II.20) The jumps in ζ,CZare known, because these are assumed continuous functions o fµ,tand the jump in µis known from (II.19). Solving for the backward jump gives α− k=(ζ+ k−ζ− k) +CZ+ kα+ k CZ− k. (II.21) Hence, only in the case of a single scalar moment function and observation variable is it obvious how to formulate jump conditions, in the case of a general mom ent-closure. It still remains in that case to formulate the algorithm to ca lculate the cost function itself. The proper deﬁnition turns out to be FZ(λ1,...,λ n) :=n/summationdisplay k=1(∆F)k(λ1,...,λ k) (II.22) 41where the increment at time tkis given by a second area rule: /integraldisplayµ+ k µ− kζ(µ,tk) CZ(µ,tk)dµ= (∆F)k. (II.23) There is a simple heuristic motivation for both this rule and the previous one. In fact, the basic approximation is to replace the exponentially-modiﬁed PDF by a PDF from the Ansatz with an adjusted moment. That is, 1 W(λk;µ− k,tk)eλkZ(x,tk)P(x,tk;µ− k)≈P(x,t;µ+ k), (II.24) with the normalization factor W(λk;µ− k,tk) :=/integraldisplay dxeλkZ(x,tk)P(x,tk;µ− k). (II.25) TheM-moment of (II.24) is µ+ k(λk;µ− k,tk) :=1 W(λk;µ− k,tk)/integraldisplay dxM(x,tk)eλkZ(x,tk)P(x,tk;µ− k) ≈µ+ k. (II.26) Diﬀerentiating once and using again (II.24) thus gives ∂µ+ k ∂λk(λk;µ− k,tk) =CZ(µ+ k,tk). (II.27) The ﬁrst area rule (II.19) is just an integral form of this lat ter relation (II.27). Likewise, if we deﬁne (∆F)k(λk;µ− k,tk) := log W(λk;µ− k,tk), (II.28) then we see by applying (II.24) twice again that ∂(∆F)k ∂λk(λk;µ− k,tk) =ζ(µ+ k,tk). (II.29) The second area rule (II.23) is likewise the integral form of (II.29). Note from (II.28) that all of the dependence of (∆ F)kupon λ1,...,λ k−1is through µ− k, analogous to (I.75). The cost function HZ(z1,...,z n) is ﬁnally deﬁned as the Legendre dual of FZ(λ1,...,λ n) given by (II.22). 42The jump conditions (II.19),(II.21) may be used very much as the exact ones (I.61),(I.62) for the purposes of estimation with discrete-time data and of en semble variance calculation. Only the Rayleigh-Ritz approximations of the cost functions nee d be substituted for the exact ones in the algorithms described earlier. In calculating the Leg endre dual HZ(z1,...,z n) the adjoint equation is used to evaluate zk=ζ± k+CZ± kα± k. Of course, one should check that this gives the same result as direct diﬀerentiation zk=∂FZ ∂λk. This is true but we shall not give the proof here, because we prove a very similar result in the Appendix 2. The p roof is based upon the easily established relations ∂µ+ k ∂µ− k=CZ+ CZ−,∂(∆F)k ∂µ− k=ζ+ k−ζ− k CZ−. (II.30) Of course, the adjoint equation need not be employed at all, b ut it is a convenient way of evaluating the required derivative. Substantial simpliﬁcations in the jump conditions occur in the important special case where Z=M. In that case ζ(µ,t) =µ, CZ(µ,t) =C(µ,t). We can then deﬁne a function λ(µ,t) :=/integraldisplayµ µ(t)d¯µ C(¯µ,t)(II.31) where µ(t) is the solution of the unperturbed moment equation. If µ(λ,t) is the inverse function, then we can also deﬁne F(λ,t) :=/integraldisplayλ 0µ(¯λ,t)d¯λ. (II.32) It follows by our deﬁnitions that F′(λ,t) =µ, F′′(λ,t) =C. (II.33) In terms of the function λ(µ,t) the ﬁrst area rule (II.19) becomes λ(µ+ k,tk)−λ(µ− k,tk) =λk. (II.34) Also, using (II.33) we note that (∆F)k=/integraldisplayµ+ k µ− kµ dµ/parenleftBig dµ dλ/parenrightBig 43=/integraldisplayλ+ k λ− kµ(λ)dλ =F(λ+ k,tk)−F(λ− k,tk). (II.35) Hence, the “area rules” are replaced by equations involving discontinuities of explicit functions, always assuming, of course, that integrals deﬁning the func tions in (II.31),(II.32) may be evalu- ated. The key to this simpliﬁcation was the relations in (II. 33), which imply that Fis a convex “potential” generating the ﬁrst and second moments of the PD FAnsatz . Such a potential will always exist for functions of one variable, but not in genera l for multivariate functions. II.3. Exponential PDF Closures We have seen above that, for a general closure, there is a sati sfactory treatment of estimation with discrete-time data onlyfor the case where there is both a single measured variable Z(t) and a single closure variable M(t). Obviously, this is an extreme limitation. However, it may be possible to circumvent this severe restriction within sp ecial classes of closures. In fact, as we show now, closures constructed with an exponential PDF Ansatz have better properties. We shall see that they guarantee continuity of optimal estimat ors. Furthermore, they provide very simple “jump-conditions” for estimation with discrete-ti me data. Exponential PDF closures are one example of the general clas s considered in [27]. Hence, we shall only make a quick summary of the properties required here and refer the interested reader to the paper [27] for more details. Most concretely, t he class of closures we consider are those built from a PDF Ansatz of the exponential form: P(x,t;λ) =exp(λ⊤M(x,t)) N(λ,t)P∗(x,t) (II.36) with N(λ,t) :=/integraldisplay dxexp(λ⊤M(x,t))P∗(x,t). (II.37) HereP∗(x,t) is a reference PDF . To guarantee some of the good properties of the closure discussed in [27], the reference PDF must be a solution (or ap proximate solution) of the Fokker- Planck equation. However, for the properties discussed her e,P∗(t) may be an arbitrary PDF. 44The exponential family in (II.36) is parameterized by the “p otential” variables λ, rather than by the moments µof the closure variables M(x,t). However, there are simple relationships between these quantities. We may deﬁne F(λ,t) := log N(λ,t), (II.38) which is a cumulant-generating function for the variables M(t) in the PDF Ansatz . Likewise, its Legendre transform H(µ,t) := max λ/braceleftBig µ⊤λ−F(λ,t)/bracerightBig (II.39) is a generating function for irreducible correlation funct ions of M(t). It is the relative entropy for the PDF Ansatz in (II.36) with respect to the reference PDF P∗(t). Under some conditions discussed in [27], it satisﬁes an H-theorem for the closure d ynamics constructed with the Ansatz . However, the role of F,Has generating functions will be more important for us here. T hus, µ=∂F ∂λand conversely λ=∂H ∂µ. It is a consequence of the former that∂µ ∂λis the covariance matrix CofMand that∂C ∂λis the 3rd-order cumulant. These relationships will prove t o be important in the following. We shall now show that, for the exponential PDF closures, the history z(t) is continuous even for h(t) consisting of delta-function spikes, when the variables Z(t)are among the closure variables M(t)themselves . This last condition places some restriction, but a fairly m odest and natural one. Without any loss of generality, we can consider Z(t) to consist of the entire set of closure variables M(t). As before, some of our previous formulas then simplify con siderably. For example, ζ(µ,t) =µandCZ(µ,t) =C(µ,t), the usual MM-covariance matrix. Then (II.4) is replaced by m(t) =µ(t) +C(µ(t),t)α(t). (II.40) The time-derivative of the latter, given in general in (II.1 7), also simpliﬁes. In fact, the term in the bracket in the second line becomes Cba(µ,t)−Cab(µ,t) = 0 (II.41) 45which vanishes by the symmetry of the covariance matrix. The term in the bracket in the third line of (II.17) becomes Cajb(µ,t)−Cbja(µ,t) = 0 (II.42) where Cabc(µ,t) is the 3rd-order cumulant of M(t). Indeed, ∂Caj ∂µi=∂Caj ∂λk∂λk ∂µi=CajkΓki. (II.43) Since the irreducible 2nd correlator is the inverse covaria nce matrix, Γ=C−1, the expression in (II.42) follows from the corresponding expression in (II.1 7). However, it is obvious that (II.42) vanishes, by the symmetry of the 3rd-order cumulant. Puttin g together all of these results, we have dm dt(t) =V(µ,t) +/bracketleftBigg d dt/parenleftbigg∂µ ∂λ/parenrightbigg −/parenleftbigg∂V ∂λ/parenrightbigg⊤/bracketrightBigg α. (II.44) This should be compared with the exact expression (I.77). Ju st as there, we see that the terms directly involving h(t) all cancel. Hence, m(t) remains continuous even with h(t) containing delta-function spikes. We shall ﬁnally show that the exponential PDF closures also p ermit the formulation of simple jump conditions at the times tkwhere the delta functions occur. This should not be too surprising, when one considers that the exact jump conditio ns in (I.61),(I.62) consist simply of suitable exponential modiﬁcations of the solutions of the f oward, backward equations. To derive the jump conditions in the closure, we use a strategy motivat ed by that in section II.2. In fact, observe by C=∂µ ∂λand the chain rule that ˙µ=C(λ)˙λ.Thus, if one deﬁnes W(λ) :=Γ(µ)V(µ) andγ:=C(λ)α,then in terms of the new variables γ,λ, the nonequilibrium action, including the constraint term with the Lagrange multiplier, becomes Γ[γ,λ] =/integraldisplaytf tidt/braceleftBig γ⊤[˙λ−W(λ)] +h⊤(t)[m(t)−µ(λ)−γ]/bracerightBig . (II.45) The Euler-Lagrange equations in terms of these variables be come ˙λ=W(λ,t) +h(t), (II.46) 46˙γ+/parenleftbigg∂W ∂λ/parenrightbigg⊤ (λ,t)γ+C(λ,t)h(t) =0, (II.47) and the constraint equation m(t) =µ(λ,t) +γ. (II.48) In the ﬁrst equation (II.46) we may integrate across the spik eλkδ(t−tk) inh(t) to obtain λ(µ+ k,tk)−λ(µ− k,tk) =λk. (II.49) These are the forward jump conditions . As should not be unexpected, the potential λ(µ,t) is simply incremented by λkat the spike. A similar result can be obtained by integrating the backward closure equation (II.47) across the spike. Howeve r, it is simpler to use the continuity ofm(t) at the jump, which was established above. Then from (II.48) one immediately derives γ− k= (µ+ k−µ− k) +γ+ k. (II.50) These are the backward jump conditions . The multi-time cumulant-generating function F(λ1,...,λn) can be obtained from (I.73) with the observation that N(tf) =/producttextn k=1N(λ+ k,tk) N(λ− k,tk)and thus (I.74) holds with (∆F)k(λ1,...,λk) = F(λ+ k,tk)−F(λ− k,tk) =F(λ− k+λk,tk)−F(λ− k,tk), (II.51) generalizing (II.35). Then the multi-time entropy H(m1,...,mn) is obtained by the Legendre transform H(m1,...,mn) =n/summationdisplay k=1m⊤ kλk−F(λ1,...,λk). (II.52) withmkgiven by (II.40), mk=m(tk), for t=tk, k= 1,...,n. Of course, it must be shown that mk=∂F ∂λk(II.53) for all k= 1,...,n in order for (II.52) to be valid. Cf. equation (I.57). The pro of is somewhat technical, so it is given in the Appendix 2. 47While the previous approximation has a rather elegant and tr actable formulation, there is nevertheless also an unpleasant asymmetry between forwa rd and backward time directions. Thus, information propagates forward in time via the nonlin ear closure equation (II.46), but information propagates backward in time via the equation (I I.47) which is linear in the adjoint variable γ. Ultimately, this asymmetry is due to our employment of a non linear (exponen- tial)Ansatz (II.36) for the PDF, while the solution of the backward equat ion is taken to be of the linear form (II.1). However, there is nothing in the Ra yleigh-Ritz method which re- quires the use of the linear Ansatz (II.1) for the left trial state. In fact, that expression has other unpleasant features. The exact backward Euler-Lagra nge equation (I.24) is known to be positivity-preserving, so that the solution A(x,t) starting from ﬁnal data A(t)≡1 must be everywhere nonnegative. However, the linear Ansatz A(x,t) = 1 +/summationtextR i=1αi(t)[Mi(x,t)−µi(t)] may easily become negative, if the adjoint variables αbecome large enough in magnitude. It is therefore desirable to consider more general Ans¨ atze for the left trial state than the linear one. Within the context of exponential PDF closures a particular ly symmetric and attractive choice is to make the double exponential Ansatz : P(x,t) = exp/bracketleftBig β⊤M(x,t)−F(β,t)/bracketrightBig P∗(x,t) (II.54) for the right trial state and A(x,t) = exp/bracketleftBig α⊤M(x,t)−(∆αF)(β,t)/bracketrightBig (II.55) for the left trial state. Here (∆ αF)(β,t) :=F(α+β,t)−F(β,t) so that the normalization constraint <A(t),P(t)>= 1 is automatically satisﬁed. It is then clear that, for smal lα, (II.55) coincides with the linear Ansatz . (Note that (∆ αF)(β,t) =α⊤µ(β,t)+O(α2).) However, this newAnsatz is globally nonnegative and symmetric in form to the exponen tial for the right trial state. An even more attractive feature of this double e xponential Ansatz is that, within it, the Rayleigh-Ritz eﬀective action of the closure variab lesMthemselves may be calculated 48analytically in closed form. The result is: Γ[m] =1 4/integraldisplaytf tidt[˙m(t)−V(m,t)]⊤Q−1(m,t)[˙m(t)−V(m,t)], (II.56) where Qij(m,t) :=/an}bracketle{t(∇xMi)⊤D(∇xMj)/an}bracketri}htλ(m,t). (II.57) This eﬀective action has precisely the Onsager-Machlup for m. The statement generalizes a previous result in [12], for general closures, that the Rayl eigh-Ritz eﬀective action has the Onsager-Machlup form to quadratic order. Let us just brieﬂy sketch the derivation, which will be given in detail elsewhere [29], along with a complete disc ussion of its remarkable properties. It is a straightforward calculation to show that (∂t+ˆL∗)A(x,t) =/braceleftBig ˙α⊤M(x,t) +α⊤˙M(x,t) +∇x(α⊤M)·D·∇x(α⊤M)−∆α˙F(β,t)/bracerightBig A(x,t). (II.58) In that case <(∂t+ˆL∗)A(t),P(t)>=˙α⊤µ(λ,t) +α⊤V(λ,t) +α⊤Q(λ,t)α−∆α˙F(β,t),(II.59) where λ:=α+β.However, it is easy to see that the second constraint on the me an values becomes in these variables m(t) =<A(t),M(t)P(t)>=µ(λ,t).Thus, holding the history m(t) ﬁxed is equivalent to holding λ(t) ﬁxed. We cannot vary independently over α(t) and β(t), but one is determined from the other via the relation λ(t) =α(t) +β(t). Since, for ﬁxed m(t), (II.59) implies that Γ[α,β] =/integraldisplaytf tidt/braceleftBig α⊤[˙m−V(m,t)]−α⊤Q(m,t)α/bracerightBig , (II.60) maximizing over αyields (II.56). Although the Onsager-Machlup form (II.56) is most interest ing for theory, practical estima- tion is easier with the latter expression (II.60). Includin g the cost function for the observations, the total action to be minimized is Γ∗[α,m] =/integraldisplaytf tidt/braceleftBig α⊤[˙m−V(m,t)]−α⊤Q(m,t)α/bracerightBig +1 2n/summationdisplay k=1[m(tk)−rk]⊤R−1 k[m(tk)−rk]. (II.61) 49The Euler-Lagrange equations of this problem are ˙m=V(m,t) + 2Q(m,t)α, (II.62) ˙α+/parenleftbigg∂V ∂m/parenrightbigg⊤ α+∂ ∂m/parenleftBig α⊤Qα/parenrightBig =n/summationdisplay k=1R−1 k[m(tk)−rk]δ(t−tk). (II.63) Solving these equations with boundary values m(ti) =m0andα(tf) =0can give directly the optimal history, without the need of applying any explic it minimization algorithm. It is transparent in this formulation that the optimal history m∗(t) is continuous at the observation times, because the ﬁrst equation (II.52) contains no delta- functions in time. Only the adjoint variables α(t) suﬀer jumps at the measurement times t=tk. The same circle of ideas may be applied to constructing closu res of the KSP equations for the optimal history itself, rather than just the variationa l approximation. In fact, assume that the closure variables Mconsist of the measured variables Zand their tensor products Z⊗Z, M:= (Z,Z⊗Z), with mean values given by m= (ζ,Σ) for a double exponential Ansatz. Let the exponential parameters in the left trial state then b e denoted as ( α,A) and those in the right trial state as ( β,B). Because the states evolve by the (unperturbed) forward an d backward Kolmogorov equations between measurements, the E uler-Lagrange equations within the closure are of the same form as those in (II.62),(II.63). As there, there are no jumps at measurement times in the equations for m= (ζ,Σ).On the other hand, there are simple jump conditions for the adjoint variables ( α,A), which may be read oﬀ directly from (A.3),(A.9): α− k=α+ k+R−1 krk, (II.64) A− k=A+ k−1 2R−1 k, (II.65) fork= 1,...,n. Further details, including the formulation for continuous -time observation, will be given elsewhere. We only note here that there is a price to b e paid for constructing a closure of the KSP equation: the necessity of including among the clo sure variables the squares of the observed variables in addition to those variables themselv es. 50III Conclusion This paper is intended to serve as a primer and technical refe rence for the application of the proposed variational estimation scheme to concrete proble ms. We have discussed the meaning of the variational estimator within ensemble theory and empha sized its character as a “mean-ﬁeld approximation” to the optimal estimator. Neither the varia tional method nor the optimal KSP method can be directly applied in practice to complex, high- dimensional systems. An action functional can be used to construct Rayleigh-Ritz or moment -closure approximations of both the variational and KSP estimators, but the variational sch eme has the advantage of requiring simpler, lower order closures. We have discussed a number of special closure schemes, based in particular upon exponential Ans¨ atze , that preserve good properties of the exact estimators. We have discussed also the numerical implementation of the var iational estimation scheme, both exactly and within a Rayleigh-Ritz approximation, both to o btain the estimator itself and also to approximate the variance or ensemble dispersion. Most of the algorithms discussed here have already been implemented in [30] and in our forthcoming work [31]. In addition to providing a practical estimation scheme, we h ope that the variational frame- work will provide also some additional physical insight int o the complex stochastic systems to which it is applied. It exploits a thermodynamic formalism f or far from equilibrium systems and provides a motivation to understand better the concepts of action and entropy in concrete physical systems, e.g. atmospheres, oceans, ecosystems, l iving organisms, etc. Acknowledgements. The author wishes to thank F. Alexander, M. Anitescu, C. E. Le ith, C. D. Levermore and J. Restrepo for valuable conversations and suggestions which contributed to this work. He thanks the Isaac Newton Institute for its hospi tality during his stay there for the 1999 Turbulence Programme, when part of this work was done. T his paper was prepared as Los Alamos report LA-UR00-5264 and supported by the DOE gran t LDRD - ER 2000047. 51A Appendices Appendix 1: Optimal Estimation with Discrete-Time Data We give here a simple derivation of the Kushner-Stratonovic h-Pardoux equations for estimation with data taken at a discrete set of times tk, k= 1,...,n. The problem set-up is the same as described in Section I.5. We deﬁne P∗(x,t) :=P(x,t\|r1,...,rk) fortk+1> t≥tk, so that P∗(t) is right-continuous in time. It is then clear that between me asurement times, P∗(t) evolves by the forward Kolmogorov equation (I.16). At measurement tim es, P∗(x,tk+) =P∗(x,tk− \|Z(tk) +ρk=rk) (A.1) fork= 1,...,n. Thus, by Bayes’ rule, P∗(x,tk+) =P∗(Z(tk) +ρk=rk\|x,tk−)P∗(x,tk−)/integraltextdyP∗(Z(tk) +ρk=rk\|y,tk−)P∗(y,tk−). (A.2) By our assumptions, ρkis a normal random variable of mean 0and covariance Rk,independent of the process X(t). Hence, if Z,ρares-dimensional P∗(Z(tk) +ρk=rk\|x,tk−) =1/radicalbig (2π)sDetRkexp/bracketleftbigg −1 2(Z(x,tk)−rk)⊤R−1 k(Z(x,tk)−rk)/bracketrightbigg . (A.3) The term1√ (2π)sDetRkexp/bracketleftBig −1 2r⊤ kR−1 krk/bracketrightBig may be cancelled between numerator and denominator in (A.2). Hence we obtain ﬁnally the forward “jump condition ” P∗(x,tk+) =exp/bracketleftBig r⊤ kR−1 kZ(x,tk)−1 2Z⊤(x,tk)R−1 kZ(x,tk)/bracketrightBig W(r1,...,rk)P∗(x,tk−) (A.4) with the normalization factor W(r1,...,rk) :=/integraldisplay dyexp/bracketleftbigg r⊤ kR−1 kZ(y,tk)−1 2Z⊤(y,tk)R−1 kZ(y,tk)/bracketrightbigg P∗(y,tk−).(A.5) Next, we deﬁne fortk−1< t≤tk, A∗(x,t) :=P(x,t\|r1,...,rn) P∗(x,t)=P(x,t\|r1,...,rn) P(x,t\|r1,...,rk−1)(A.6) 52and for t > t n. A∗(x,t) := 1 (A.7) Writing this deﬁnition as A∗(x,t) :=P(x,t\|r1,...,rn) P(x,t\|r1,...,rk)·P(x,t\|r1,...,rk) P(x,t\|r1,...,rk−1)(A.8) and using the already derived condition (A.4), we obtain for t→tk−that A∗(x,tk−) =A∗(x,tk+)exp/bracketleftBig r⊤ kR−1 kZ(x,tk)−1 2Z⊤(x,tk)R−1 kZ(x,tk)/bracketrightBig W(r1,...,rk). (A.9) This is the backward “jump condition”. It remains only to show that A∗(x,t) deﬁned via (A.6) satisﬁes the backward Kolmogorov equation (I.17) between measurements. We apply again Bayes ’ rule, in the form P(x,t\|r1,...,rn) =P(rk,...,rn\|x,t;r1,...,rk−1)P(x,t\|r1,...,rk−1) P(rk,...,rn\|r1,...,rk−1). (A.10) However, by the Markov property, P(rk,...,rn\|x,t;r1,...,rk−1) = P(rk,...,rn\|x,t) =/integraldisplay dykP(rk,...,rn\|yk,tk)P(yk,tk\|x,t) (A.11) when tk−1< t≤tk. Putting together (A.6),(A.10),(A.11), we conclude that A∗(x,t) =/integraldisplay dykP(rk,...,rn\|yk,tk) P(rk,...,rn\|r1,...,rk−1)P(yk,tk\|x,t). (A.12) Since the transition probability satisﬁes the backward equ ation in the variables x,t (∂t+ˆL∗)P(yk,tk\|x,t) = 0, (A.13) it then immediately follows from the integral representati on (A.12) that A∗(t) satisﬁes (I.17) fortk−1< t < t k, k= 1,...,n. It is not hard to show that the jump conditions above, (A.4),( A.9), are equivalent to those given in the text, (I.80),(I.81). One simply multiplies the numerators and denominators in (A.4),(A.9) by the factor exp/bracketleftBig −1 2/an}bracketle{tZ(tk)/an}bracketri}ht⊤ tk−R−1 k/an}bracketle{tZ(tk)/an}bracketri}httk−/bracketrightBig and rearranges terms in the expo- nents by completing the square. 53This is an appropriate place to discuss the close formal rese mblance of the KSP jump condi- tions, (I.80),(I.81), to the jump conditions, (I.61),(I.6 2), employed in calculating the multitime entropy HZ(or, more correctly, its Legendre dual FZ.) In fact, the exponential PDF Ansatz (II.36) has also an interpretation as a conditional PDF. The conditioning is now upon the event that the empirical average ZN=zin the limit as N→ ∞: lim N→∞P∗(x,t\|ZN(t) =z) =exp(λ⊤Z(x,t)) N(λ,t)P∗(x,t) (A.14) withλ=λ(z,t). More precisely, the result is that lim N→∞P⊗N ∗(x1,..,xN,t\|ZN(t) =z) = /producttextN i=1exp(λ⊤Z(xi,t)) N(λ,t)P∗(xi,t).Here the product measure P⊗N ∗(x1,..,xN;t) =/producttextN i=1P∗(xi,t) is taken, to correspond to an ensemble of Nindependently prepared samples. Convergence to the new product measure holds for any ﬁnite-dimensional mar ginals (i.e. for i∈S,any ﬁnite set, as N→ ∞). Statistical physicists will recognize this as an equival ence of ensembles result, in which the “microcanonical ensemble” corresponding to th e condition ZN(t) =zbecomes equivalent in the thermodynamic limit to the “canonical ens emble” with potential λ(z,t). As a consequence of this, we may interpret the solution P∗(x,t) of the forward equation, with the jump conditions (I.61) at measurement times less th ant,as P∗(x,t) =P(x,t\|z1,...,zk), t k+1> t≥tk, (A.15) where the righthand side is shorthand for the PDF conditione d upon the event ZN(t1) =z1,..., ZN(tk) =zkin the limit N→ ∞. Likewise, A∗(x,t)P∗(x,t) =P(x,t\|z1,...,zn), (A.16) where the conditioning is now upon ZN(ti) =zifor the full set of times ti, i= 1,...,n, both those before and after the time t. The proof of this assertion is exactly the same as for the corresponding results proved earlier in this appendix rega rding PDF’s conditioned upon obser- vations r1,...,rn. This relation to conditional PDF’s helps to explain the clo se similarity to the KSP formalism. Note, however, that this is the wrong set of co nditions to use for estimation, as it is upon ZNitself and not upon (empirical means of) observations rN=ZN+ρN. 54Appendix 2: Adjoint Calculation of a Derivative For completeness, we shall give a direct proof of (II.53) her e. We ﬁrst observe by (I.74) and causality that ∂F ∂λk=n/summationdisplay l=k∂(∆F)l ∂λk. (A.17) Then we note from (II.51) that ∂(∆F)k ∂λk=µ+ k(A.18) while ∂(∆F)k ∂λ− k=µ+ k−µ− k. (A.19) Thus, by (A.17),(A.18) and the chain rule ∂F ∂λk=µ+ k+n/summationdisplay l=k+1∂(∆F)l ∂µ− l∂µ− l ∂λk. (A.20) Furthermore, by (A.19) and the chain rule, ∂(∆F)l ∂µ− l=∂(∆F)l ∂λ− l∂λ− l ∂µ− l = [µ+ l−µ− l]⊤Γ− l. (A.21) Therefore, it only remains in (A.20) to evaluate∂µ− l ∂λkforl > k. The Jacobian matrix for arbitrary times t/ne}ationslash=tl, l > k satisﬁes the linearized equation ∂t∂µ(t) ∂λk=A(t)∂µ(t) ∂λk, (A.22) where A(t) :=∂V ∂µ(µ(t),t). The initial condition ∂µ(t) ∂λk/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=tk+=C+ k(A.23) is provided by the formula µ+ k=µ(λ− k+λk) whence∂µ+ k ∂λk=C+ k.At the measurement times t=tl, l > k there is an additional multiplicative factor, which follow s from the Jacobian ∂µ+ l ∂µ− l=∂µ+ l ∂λ− l∂λ− l ∂µ− l=C+ lΓ− l. (A.24) 55The solution for tl> t > t l−1, l > k , is ∂µ(t) ∂λk= T exp/bracketleftBigg/integraldisplayt tl−1A(s)ds/bracketrightBigg  l−1/productdisplay j=k+1C+ jΓ− j·T exp/bracketleftBigg/integraldisplaytj tj−1A(s)ds/bracketrightBigg  C+ k. (A.25) Here T exp denotes the time-ordered exponential with matric es at increasing times to the left, and likewise Π is the time-ordered product in the same sense. Thus, the ﬁnal result ∂µ− l ∂λk= T exp/bracketleftBigg/integraldisplaytl tl−1A(s)ds/bracketrightBigg  l−1/productdisplay j=k+1C+ jΓ− j·T exp/bracketleftBigg/integraldisplaytj tj−1A(s)ds/bracketrightBigg  C+ k(A.26) follows upon setting t=tl−. This may be compared with the solution of the adjoint equatio n ∂tα(t) +A∗(t)α(t) =0 (A.27) integrated backward in time for t/ne}ationslash=tland subject to the jump conditions (II.50) at t=tl, l= 1,...,n. The explicit solution for tk< t < t k+1is α(t) =n/summationdisplay l=k+1T exp/bracketleftbigg/integraldisplaytk+1 tA∗(s)ds/bracketrightbigg/braceleftBigg/productdisplayk+2 j=lΓ− j−1C+ j−1·Texp/bracketleftBigg/integraldisplaytj tj−1A∗(s)ds/bracketrightBigg/bracerightBigg Γ− l[µ+ l−µ− l]. (A.28) NowTexp denotes anti-time-ordered exponential with matrices at decreasing times to the left, andΠ is the anti-time-ordered product. Setting t=tk+ and regrouping terms gives α+ k=n/summationdisplay l=k+1/braceleftBigg/productdisplayk+1 j=l−1T exp/bracketleftBigg/integraldisplaytj−1 tjA∗(s)ds/bracketrightBigg ·Γ− jC+ j/bracerightBigg Texp/bracketleftBigg/integraldisplaytl tl−1A∗(s)ds/bracketrightBigg Γ− l[µ+ l−µ− l]. (A.29) Finally, substituting (A.21),(A.26) into (A.20), and usin g (A.29), gives ∂F ∂λk=µ+ k+/parenleftBig α+ k/parenrightBig⊤C+ k=mk, (A.30) which is exactly the result required. 56References [1] A. Gelb, ed., Applied Optimal Estimation . (MIT Press, Cambridge, MA, 1974). [2] R. L. 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arXiv:physics/0011050v1 [physics.atom-ph] 21 Nov 2000Angular momentum spatial distribution symmetry breaking i n Rb by an external magnetic ﬁeld Janis Alnis, Marcis Auzinsh∗ Department of Physics, University of Latvia, 19 Rainis blvd ., Riga LV-1586, Latvia (July 23, 2013) Typeset using REVT EX ∗Corresponding author, Fax +371-7820113, e-mail mauzins@l atnet.lv 1Abstract Excited state angular momentum alignment – orientation con version for atoms with hyperﬁne structure in presence of an external mag netic ﬁeld is investigated. Transversal orientation in these condition s is reported for the ﬁrst time. This phenomenon occurs under Paschen Back condit ions at in- termediate magnetic ﬁeld strength. Weak radiation from a li nearly polarized diode laser is used to excite Rb atoms in a cell. The laser beam is polarized at an angle of π/4 with respect to the external magnetic ﬁeld direction. Grou nd state hyperﬁne levels of the 5 S1/2state are resolved using laser-induced ﬂuo- rescence spectroscopy under conditions for which all excit ed 5P3/2state hyper- ﬁne components are excited simultaneously. Circularly pol arized ﬂuorescence is observed to be emitted in the direction perpendicular to b oth to the direc- tion of the magnetic ﬁeld Band direction of the light polarization E. The obtained circularity is shown to be in quantitative agreeme nt with theoretical predictions. I. INTRODUCTION In the absence of external forces an ensemble of unpolarized atoms can only be aligned [1] by linearly polarized light. The fact that the atoms are only aligned implies that, although the magnetic sublevels of diﬀerent \|mJ\|are populated unequally, magnetic sublevels of + mJ and−mJare equally populated. For this reason, atoms excited by lin early polarized laser radiation are not expected to produce circularly polarized ﬂuorescence. In the presence of external forces, however, excitation by linear polarized l ight can produce an orientated pop- ulation of atoms (with diﬀerent + mJand−mJpopulations.) This eﬀect, called alignment – orientation conversion, was predicted and experimentall y observed in the late sixties in the anisotropic collisions of initially aligned atoms [2–5 ]. Later an electric ﬁeld was also shown to induce alignment – orientation conversion [6]. Ele ctric-ﬁeld-induced alignment – 2orientation conversion has since been studied in great deta il [7]. Contrary to the case of an electric ﬁeld, linear perturbatio n by a magnetic ﬁeld is not able to orient an initially aligned angular momentum distri bution. This inability to induce alignment – orientation conversion is a result of the reﬂect ion symmetry of axial vector ﬁelds. This symmetry can be broken if, in addition to the line ar Zeeman eﬀect, there exists nonlinear dependencies of the magnetic sublevel ene rgies on the ﬁeld intensity and the magnetic quantum numbers mJ. Such nonlinear perturbations can have a variety of causes including predissociation [8–11] and hyperﬁne inte raction. Alignment – orientation conversion as a result of hyperﬁne interaction in a magnetic ﬁeld in context of nuclear spin I= 1/2 was studied by J. Lehmann for the case of optically pumped ca dmium in a magnetic ﬁeld [12,13]. W. Baylis described the same eﬀect in sodium [1 4]. The ﬁrst experiment to detect directly a net circular polarization of ﬂuorescence from an initially aligned excited state in an external magnetic ﬁeld was reported by M. Krainsk a – Miszczak [15]. In this work the optical pumping of85Rb by a π-polarized D 2line was studied. This eﬀect was also examined by X. Han and G. Schinn in sodium atoms [16]. They des cribe this alignment - orientation conversion process as resulting from hyperﬁn e-F-level mixing in an external magnetic ﬁeld and the interference of diﬀerent excitation – decay pathways in such mixed levels. In all above cases, a joint action of the magnetic ﬁeld and hyp erﬁne interaction creates diﬀerent population of magnetic sublevels + mFand−mFof hyperﬁne levels F. This means thatlongitudinal orientation of atoms along the direction of an external magn etic ﬁeld is cre- ated. Recently it was predicted that joint action of a magnet ic ﬁeld and hyperﬁne interaction from an initially aligned ensemble would create transverse orientation of angular momentum of atoms or molecules [17]. Transverse orientation implies orientation in a direction perpen- dicular to the external magnetic ﬁeld B. In this particular case magnetic sublevels + mF and−mFare equally populated, but orientation is a result of cohere nce between pairs of wave functions of magnetic sublevels mFwith ∆ mF= 1. Creation of transverse orientation is achieved if the excitation light polarization vector is n either parallel nor perpendicular 3to the external magnetic ﬁeld direction with the largest eﬀe ct occurring for the case of a light polarization – magnetic ﬁeld angle of π/4. In a previous paper [17], parameters of the NaK molecule were used for numerical simulations of orie ntation and ﬂuorescence cir- cularity signals. We found that transverse orientation onl y occurred when the rotational angular momentum Jis small enough to be comparable with the nuclear spin I. For levels with larger angular momentum quantum number, the magnitude of created orientation was found to decrease rapidly. Previously transverse alignment – orientation conversion was studied in detail for the case of an external electric ﬁeld [7]. In this case the conver sion occurs with or without hyperﬁne interaction. In this paper we report the ﬁrst exper imental observation to our knowledge of alignment – orientation conversion that creat es net transverse orientation of atoms with hyperﬁne structure in an external magnetic ﬁeld. As we will show, this eﬀect is interesting not only as a new way to create orientated atoms, but also can be used to increase the accuracy with which constants related to the hyperﬁne in teraction can be determined. II. THEORETICAL DESCRIPTION The general scheme how transverse orientation of angular mo mentum is created from an aligned ensemble of atoms is the following: Initial alignme nt, for example by absorption of a linearly polarized light, is created at some non-zero acute angle with respect to the direction of an external-ﬁeld (in this case a B-ﬁeld.) The optimum angle is π/4, but the eﬀect will take place at any angle that diﬀers form 0 and π/2. The perturbing ﬁeld together with the hyperﬁne interaction causes unequally spaced magnetic sublevel splittings. Under these conditions, angular momenta orientation at the direction p erpendicular to the direction of the external ﬁeld is created [22]. A semiclassical interpre tation of this eﬀect in terms of angular momentum precession in an external ﬁeld can be found in a previous publication [7]. In this vectorial model, alignment – orientation conve rsion is the result of a diﬀerent precession rate for diﬀerent orientations of angular momen tum with respect to the external 4ﬁeld. In what follows we explain this transverse-orientati on in terms of a accurate quantum mechanical model. In the present study we exploit laser excitation of pure isot opes of Rb atoms from their ground state 5 S1/2to the ﬁrst excited state 5 P3/2(resonance D 2line) (see inset Figure 5.) The two most common naturally occurring isotopes of Rubidiu m are85Rb(72.15 %, nuclear spinI= 5/2) and87Rb(27.85%, nuclear spin I= 3/2.) As a result of hyperﬁne interac- tions, the ground-state level of85Rb is split into components with total angular momentum quantum numbers Fi= 2 and Fi= 3 and the ground-state level of87Rb is split into compo- nents with total angular momentum quantum numbers Fi= 1 and 2. The ground-state-level splittings for85Rb and87Rb are approximately 3 GHz and 6 GHz respectively. In contras t, the four excited state hyperﬁne components are separated by only several hundred MHz (see Figures 1 and 3). Excited-state-hyperﬁne structure in absorption is not res olved due to Doppler broadening and laser-frequency jittering. To make an accurate signal m odeling assuming broad line excitation, laser frequency is modulated by a few hundred MH z superimposing a 10 kHz sine wave on laser current. This allows accurate modeling to be done assuming that the excitation radiation is broad enough to excite all hyperﬁne components of the excited state without frequency selection, yet narrow enough to complete ly resolve the two ground-state components. Magnetic ﬁeld caused mixing takes place between sublevels w ith diﬀerent total angular momentum Fe, but with identical magnetic quantum numbers mF. Only levels of identi- calmFmix because, as far as the magnetic ﬁeld possesses axial symm etry, the magnetic quantum number mFremains a good quantum number. However, levels of diﬀerent Femix because an intermediate strength magnetic ﬁeld partially d ecouples the electronic angular momentum Jeand nuclear spin I. As a consequence Feceases to be a good quantum num- ber. The fact the mFremains a good quantum number whereas Fedoes not is important to the interpretation of the data. A convenient way to describe excited state atoms is by means o f a quantum density 5matrixklfmm′[18]. Upper indices characterize atomic states in a magneti c ﬁeld. In the weak ﬁeld limit these states correspond to hyperﬁne levels Fe. Lower indices characterize magnetic quantum numbers. We consider an atom possessing th e hyperﬁne structure which is placed in an external magnetic ﬁeld. We further assume tha t this atom absorbs laser light polarized in the direction characterized by light electric ﬁeld vector Eexc. In this situation the density matrix that characterizes coherence between ma gnetic sublevels with quantum numbers mandm′is given as [21] klfmm′=/tildewideΓp Γ +ikl∆ωmm′/summationdisplay jµ/angbracketleftγkm\|/hatwideE∗ exc·/hatwideD\|ηjµ/angbracketright /angbracketleftγlm′\|/hatwideE∗ exc·/hatwideD\|ηjµ/angbracketright∗. (1) Here/tildewideΓpis a reduced absorption rate, Γ is the excited state relaxati on rate andkl∆ωmm′= (γkEm−γlEm′)//planckover2pi1is the energy splitting of magnetic sublevels mandm′belonging to the excited state levels kandl. Magnetic quantum numbers of the ground state level ηjare denoted by µand magnetic quantum numbers of the excited state level γkbymandm′. In an external magnetic ﬁeld, ground- and excited- state lev elsηjandγkare not charac- terized by a total angular momentum quantum numbers FiandFe, but are instead mixtures of these states: \|γkm/angbracketright=Fe=Je+I/summationdisplay Fe=Je−IC(e) kFe\|Fe, m/angbracketright,\|ηjµ/angbracketright=Fi=Ji+I/summationdisplay Fi=Ji−IC(i) jFi\|Fi, µ/angbracketright. (2) The wave-function-expansion coeﬃcients C(e) kFe, C(i) jFirepresent the mixing of ﬁeld free hy- perﬁne state wave functions by the magnetic ﬁeld. These expa nsion coeﬃcients along with the magnetic sublevel energy splittingskl∆ωmm′can be obtained by a standard procedure of diagonalization of a Hamilton matrix that contains both t he diagonal hyperﬁne elements and the oﬀ-diagonal magnetic ﬁeld interaction elements (se e for example [17].) There are several methods how to tell whether or not a particu lar atomic state described by a density matrix (1) possesses orientation. One possibil ity is to expand this matrix over the irreducible tensorial operators. Then those expan sion coeﬃcients can directly be attributed to the alignment and orientation of the atomic en semble [18–20]. Alternatively, 6one may calculate directly the ﬂuorescence circularity rat e in spontaneous transitions from a particular excited state of an atom: C=I(Eright)−I(Eleft) I(Eright) +I(Eleft)(3) Observed circularity of the ﬂuorescence in a speciﬁc direct ion can diﬀer from zero only for the case that the ensemble of atoms possesses overall orientati on in this direction [18]. I(Eright) andI(Eleft) are intensities of two ﬂuorescence components with opposi te circularity. We choose to calculate this expected circularity rate because it is the experimental measure used to register the appearance of orientation in an ensemble of a toms (see for example [22]). We consider the case that spontaneous emission is detected w ithout hyperﬁne-state reso- lution. The intensity of the ﬂuorescence with deﬁnite polar ization characterized by a vector Efin a spontaneous transition from an excited state Jecharacterized by a set γkof levels in an external ﬁeld to the ground state Jfcharacterized by a set ηjof levels can be calculated according to a previous work [21] as I(Ef) =I0/summationdisplay mm′µ/summationdisplay klj/angbracketleftγkm\|/hatwideE∗ f·/hatwideD\|ηjµ/angbracketright /angbracketleftγlm′\|/hatwideE∗ f·/hatwideD\|ηjµ/angbracketright∗klfmm′. (4) To ﬁnd the circularity rate C,one needs to not only determine the matrix elements appearin g in (1) and (4), but also the hyperﬁne level splitting and magn etic sublevel mixing coeﬃcients. In Figure 1 the hyperﬁne energy level splitting of the ﬁrst ex cited state 5 P3/2for85Rb is presented. In these calculations the following published [ 23] hyperﬁne splitting constants and magnetic moment for the rubidium atom in its ﬁrst excited state are used: a= 25.009 MHz, b= 25.83 MHz, gJ=−1.3362, gI= 0.000293. In Figure 1 level crossing positions for magnetic sublevels with ∆ mFe= 2 are indicated by circles and crossings with ∆ mFe= 1 by squares. At values of magnetic ﬁeld strength for which coherently excited magnetic sublevels undergo a l evel crossing,kl∆ωmm′= 0 the prefactor appearing in Equation (1) becomes large. This leads to resonance behavior of the observed signal. For case of excitation with linearly polarized light, the intensity of the resonance depends upon the angle between polarizatio n direction of the laser light 7and external magnetic ﬁeld direction. If the angle between t hese directions is 0, diﬀerent magnetic sublevels are diﬀerently populated but no coheren ce is created in the ensemble. If the angle is π/2 coherence is created between magnetic sublevels with ∆ mFe= 2. If the angle diﬀers form 0 and π/2 then magnetic sublevels with ∆ mFe= 1 and 2 [22] are excited coherently. This ∆ mFe= 1 coherence is required for transverse orientation. We now consider the ﬂuorescence circularity enhancement du e to ∆ mFe= 1 level crossing for the case that the linear polarization and external ﬁeld m eet at an angle of π/4 (inset Figure 2.) The circularity Cis calculated assuming an excited state relaxation rate [24 ] Γ = 3.8×107s−1and observation along an axis normal to the plain containing the external ﬁeldBand the polarization vector Eexc. The smooth lines of Figure 2 give the expected signals for both resolved absorption lines. Both signals ar e maximum at an approximate magnetic ﬁeld strength of 10 G. For both absorption lines we c alculate a total ﬂuorescence circularity with unresolved hyperﬁne components in a trans ition back to the ground state 5S1/2. The resonance peak is more pronounced for the Fi= 2− →Feabsorption transition than for the Fi= 3− →Fetransition. Because a 10 G ﬁeld is weak enough not to cause substantial hyp erﬁne level mixing (i.e., the magnetic sublevel splitting in the magnetic ﬁeld still is small in comparison with hyperﬁne splitting,) the increase in orientation for Fi= 2 absorption can be understood using the relative transition probability WFi− →Fegiven by Sobelman [20]: WFi−→Fe= (2Fi+ 1)(2 Fe+ 1)(2 Ji+ 1)(2 Je+ 1)  JiFiI FeJe1    LiJiS JeLe1  2 . (5) HereJi, JeandLi, Leare quantum numbers of total and orbital electronic angular mo- mentum of the initial and ﬁnal atomic state and Sis the electronic spin of the atomic state. Quantities in curled brackets are 6 −jsymbols. This expression predicts that the Fi= 2− →Fe= 2 absorption contributes 39% of the total allowed ( ∆ F= 0,±1) absorption fromFi= 2.In contrast, the Fi= 3− →Fe= 2 absorption contributes only 8% of the total allowed (∆ F= 0,±1) absorption from Fi= 3. At the same time the Fe= 2 state is 8the state for which the magnetic sublevels undergo a level cr ossing in the vicinity of a 10 G magnetic ﬁeld. Thus the absorption from the Fi= 2 state leads to a greater degree of transverse orientation. Similar level splitting diagrams (Figure 3) and expected ci rcularity signals (Figure 4) are calculated also for rubidium isotope87Rb. In this case the following atomic constants are used: I= 3/2,a= 84.845 MHz, b= 12.52 MHz, gJ=−1.3362, gI=−0.000995 [23]. III. EXPERIMENTAL In our experiment we use isotopically enriched rubidium (99 % of85Rb) contained in a glass cell at room temperature to keep atomic vapor concentr ation low and avoid reabsorp- tion. The 5 s2S1/2to 5p2P3/2transition at 780 .2 nm is excited using a temperature- and current- stabilized single-mode diode laser (Sony SLD114V S). Absorption signal is mea- sured using a photodiode. As the laser frequency is swept usi ng a ramped current drive, two absorption peaks with half-width of about 600 MHz separa ted by ∼3 GHz appear due to the85Rb ground state hyperﬁne structure. The excited-state hype rﬁne structure is not resolved under the Doppler proﬁle and introduced laser-fre quency jittering. The laser line width without jittering is about 60 MHz. To avoid optical pum ping and other nonlinear eﬀects, neutral density ﬁlters are used to reduce the laser i ntensity until absorption lines at 60 G broaden by less than 10%. During the level crossing and circularity measurements, th e laser wavelength is stabilized on one of the two absorption peaks. Fluorescence is monitore d on an axis normal to the electric vector Eexcand external magnetic ﬁeld B. A two-lens system is used to image the ﬂuorescence on a photodetector containing a 3 ×3 mm photodiode (Hamamatsu S1223- 01) and a transimpedance ampliﬁer. A rotating ( f= 240 Hz) sheet polarizer is inserted between the lenses. The photodetector signal is fed to a lock -in ampliﬁer (Femto LIA-MV- 150) that measures the intensity diﬀerence of two orthogona l linearly polarized ﬂuorescence components. A magnetic ﬁeld of up to 65 G is produced by passin g current through a pair 9of Helmholz coils 20 cm in diameter. The uncertainty of the ma gnetic ﬁeld is estimated to ±0.3 G. The sweep time is 5 s and 256 sweeps are averaged on an IBM co mpatible computer with a National Instruments data acquisition card. A lock-i n time constant of 10 ms is used. Several adjustments are made to record symmetrical level cr ossing signals while sweeping the magnetic ﬁeld in opposite directions. First, a linear po larizer is placed in a laser beam before the rubidium cell to ﬁne adjust the laser polarizatio n. Second, the lock-in phase is adjusted and, third, the Earth magnetic ﬁeld components are compensated with additional Helmholz coils. To detect circularly polarized light the gain electronics a re ﬁrst adjusted so that the linear polarization signals are symmetrical in opposite magnetic ﬁeld directions. A λ/4 wave plate is then placed before the polarizer so that right- and left- h anded circularly polarized light components are converted to opposite polarizations. It is c hecked that circularity signal at B= 0 is zero. During the circularity measurements the magneti c ﬁeld is swept alternatively in one and another direction and both traces are averaged. Th e experimentally recorded signal actually is I(Eright)−I(Eleft) and not the ratio ( I(Eright)−I(Eleft))/(I(Eright) + I(Eleft)). Numerical simulations reveal that these two signals hav e almost the same shapes, the relative diﬀerence is less than 3%. Experimentally reco rded signals are scaled vertically to ﬁt the calculated ones. Figures 2 and 4 compare experimentally obtained circularit y to the theoretical ones. After the scaling factor to the experimental signal is appli ed (no other adjustable parameters are used) an excellent agreement between theoretical predi ctions and experimental signals can be observed. For both isotopes circularity signals with amplitude of several percents are measured. In case of the measurements with87Rb another cell was used, that contained isotopically enriched87Rb (99%). For this isotope the signal starting from Fi= 1 exhibits stronger resonance circularity than the one starting from the Fi= 2 ground state. The reason for this is the same as already discussed in a Section II for the85Rb isotope. Only in this case the excited state hyperﬁne component Fe= 1 undergoes level crossings with ∆ mFe=±1. 10This resonance intensity ratio for two measured signals reﬂ ects the general situation that transitions with ∆ F= 0 are more intense than transitions with ∆ F=±1. IV. COMPARISON OF CIRCULARITY MEASUREMENTS TO OTHER LEVEL-CROSSING MEASUREMENTS In previous studies, atoms are excited by a linearly polariz ed light with Eexcvector perpendicular to an external magnetic ﬁeld. The ﬂuorescenc e emitted along the magnetic ﬁeld is then detected. Fluorescence linear polarization as a function of magnetic ﬁeld is measured. Here we repeat this experiment for the case of85Rb (see inset of Figure 5.) Two signals are numerically simulated and experimentally reco rded, the ﬁrst one when absorption occurs on the transitions ( Fi= 2→Fe) and a second one for a ( Fi= 3→Fe) absorption transition. In both cases the conditions are maintained so t hat the excited state hyperﬁne levels are not resolved. For the ﬁrst absorption transition in the absence of the magnetic ﬁeld electric dipole transitions are allowed only to the lev elsFe= 1,2,and 3. For the second absorption transition in absence of the external ﬁeld hyper ﬁne components with Fe= 2,3, and 4 can be excited. In presence of the magnetic ﬁeld selection rules change subs tantially. As it was mentioned before, Feis no longer a good quantum number. Each hyperﬁne level in the presence of external ﬁeld is mixed together with others. As far as mFeremains a good quantum number in the presence of the external ﬁeld, only components with th e same mFeare mixed. This implies that for the present example of85Rb, magnetic sublevels with mFe= 4 and −4 at any ﬁeld value are unmixed because only Fe= 4 contains such sublevels and there is no counterpart for these states to be mixed with. In case of mFe= 3 and −3 only two magnetic sublevels originating from Fe= 3 and 4 are mixed together, etc. This means that magnetic sublevels mFe= 0,±1 in external ﬁeld are composed from Fe= 1,2,3,4 components, mFe=±2, from Fe= 2,3,4 components mFe=±3, from Fe= 3,4 components butmFe=±4, contain only one component Fe= 4. 11In Figure 1 we can see several ∆ mFe= 2 magnetic sublevel crossings. The ﬁrst crossing takes place at zero magnetic ﬁeld when all magnetic sublevel s belonging to the same hyperﬁne level have the same energy. This crossing is the zero ﬁeld lev el crossing. Because all magnetic sublevels belonging to the same hyperﬁne state cro ss at zero ﬁeld, the zero ﬁeld level crossing leads to the largest resonance amplitude. Th en subsequent crossings take place at approximately 2 .4,4.2,8.2,24,44,52 and 74 G magnetic ﬁeld strength. A resonant peak occurs on both linear polarization signals for almost e very one of these level crossings, although with diﬀering amplitudes (see Figure 5.) There is o ne exception. The strong resonance peak at 52 G that is present in Fi= 3→Fesignal is missing in Fi= 2→Fe signal. This seeming inconsistency can be easily explained . This resonance appears when the magnetic sublevels mFe=4=−4 and mFe=3=−2 are crossing. But as it was mentioned due to dipole transition selection rules the mFe=4=−4 level can not be excited from Fi= 2 and this restriction can not be removed by external ﬁeld because mFe=4=−4 remains unmixed at any ﬁeld strength. In the same Figure 5 along with the theoretically simulated s ignal the experimentally registered signal is depicted as well. The only adjustable p arameter in this comparison is a scaling factor for the overall intensity of the experimenta lly detected signal. The observed signals agrees very well with level crossing signal registe red by several groups before us [24,25]. However, in these previous studies the ﬁrst deriva tive from the intensity was mea- sured and so we were able to compare only the exact positions o f resonances. These coincide perfectly. In our case we are able to calculate not only the po sitions of resonances, but also the shape, width and relative amplitudes of the resonance pe aks. V. CONCLUSIONS In this study we report for the ﬁrst time the appearance of tra nsverse orientation in atoms with hyperﬁne structure after excitation by linearly polarized light in the presence of an external magnetic ﬁeld. We have also presented a theory th at is in quantitative agreement 12with our data. The use of transverse alignment-orientation as a probe of level crossings is compared to previous measurements The case of measured circ ularity has an advantage over the more conventional measurement of the degree of line ar polarization: For the case of alignment – orientation conversion, there is no signal in the absence of the external ﬁeld. This implies that we do not have the ﬁrst trivial resonance po sition at zero ﬁeld value which is always present in traditional geometry (Hanle eﬀect [26] .) This allows measurements of ﬁrst level crossing positions that are very close to the zero ﬁeld resonance. In traditional methods these resonances are hidden under the zero ﬁeld peak . For example, from the inset of Figure 1 we can see that there must exist several resonance s of ∆ mFe= 2 crossings around 3 and 6 G. However these resonances are hidden in a traditiona l level crossing signal and can not be observed (see Figure 5.) At the same time ∆ mFe= 1 crossings that appear even at smaller ﬁeld values 2 ,4 and 8 G in alignment – orientation conversion signals, alth ough not fully resolved are clearly visible. The possibility to d etect these resonances can improve the precision of atomic hyperﬁne splitting constants. VI. ACKNOWLEDGMENTS This work could not be completed without valuable comments a nd advice of Dr. Habil. Phys. Maris Tamanis from Institute of Atomic Physics and Spe ctroscopy, University of Latvia and continuous support of Prof. Sune Svanberg from De partment of Physics at Lund Institute of Technology. We are thankful to Prof. Neil S hafer-Ray from University of Oklahoma for careful reading of the manuscript and valuable comments. Financial support from Swedish Institute Visby program is greatly acknowledg ed. 13REFERENCES [1] In this work the terms alignment, orientation, and polar ization take on their technical meaning. Alignment refers to a distribution of angular mome ntum direction with respect to an axis, but not a preferred direction. Orientation refer s to a distribution of angular momentum direction that favors a given direction. [2] M. Lombardi, C. R. Acad. Sci., Ser. B, 265B , 191 (1967) [3] V.N. Rebane, Opt. Spectrosc. USSR, 51, 163 (1968) [4] M. Lombardi, in Beam Foil Spectroscopy , edited by I.A. Sellin and D.J. Pegg, vol. 2, (Plenum Press, New York, 1976), p. 731 [5] T. Manabe, T. Yabuzaki, T. Ogawa, Phys. Rev. Lett., 46, 637 (1981) [6] M. Lombardi, J. Phys. (Paris), 30, 631 (1969) [7] M. Auzinsh, Can. J. Phys. 75, 853 (1997); M. Auzinsh, R. Ferber, Phys. Rev. Lett. 69, 3463 (1992); M. Auzinsh, R. Ferber, and A.V. Stolyarov. J. C hem. Phys. 101, 5559 (1994). [8] J. Vigue, M. Broyer, J.-C. Lehmann, J. Phys. B, 7, L158, (1974) [9] J. Vigue, M. Broyer, J.-C. Lehmann, J. Chem. Phys. 62, 4941 (1975) [10] I. Klintsare, M. Tamanis, A.V. Stolyarov, M. Auzinsh an d R. Ferber, J. Chem. Phys. 99, 5748 (1993) [11] M. Auzinsh, A.V. Stolyarov, M. Tamanis, R. Ferber, J. Ch em. Phys., 105, 37 (1996) [12] J.C. Lehmann, Journal de Physique, 25, 809 (1964) [13] J.C. Lehmann, Phys. Rev. 178, 153 (1969) [14] W.E. Baylis, Phys. Lett., 26A, 414 (1968) [15] M. Krainska – Miszczak, J. Phys. B: Atom. Molec. Phys., 12, 555 (1979) 14[16] X.L. Han, G.W. Schinn, Phys. Rev. A, 43, 266 (1991) [17] J. Alnis, M. Auzinsh, European Physical Journal D, 11, 91 (2000) [18] M. Auzinsh, R. Ferber, Optical Polarization of Molecules , (Cambridge University Press, Cambridge UK, 1995) [19] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Mo- mentum , (World Scientiﬁc, Singapore, 1988). [20] I.I. Sobelman, Atomic Spectra and Radiative Transitions , (Springer–Verlag, Berlin, 1979) [21] M. Tamanis, M. Auzinsh, I. Klincare, O. Nikolayeva, A. V . Stolyarov, R. Ferber, J. Chem. Phys., 106, 2195 (1997) [22] M. Auzinsh, R. Ferber, J. Chem. Phys. 99, 5742 (1993) [23] E. Arimondo, M. Inguscio, P. Violino, Rev. Mod. Phys, 49, 31 (1977) [24] G. Belin, S. Svanberg, Physica Scripta 4, 269 (1971) [25] R.W. Schmieder, A. Lurio, W. Happer and A. Khadjavi, Phy s. Rev. A2, 1217 (1970) [26] G. Moruzzi, F. Strumia, Hanle Eﬀect and Level-Crossing Spectroscopy , (Plenum Press, New York, London, 1991). 15FIGURES FIG. 1. Hyperﬁne structure energy-level diagram of85Rb 5p2P3/2in an external magnetic ﬁeld. Symbols /squareindicate ∆ mFe= 1 level crossings, /circlecopyrt— ∆mFe= 2 level crossings. FIG. 2. Numerically simulated (smooth line) and experiment ally measured (signal with noise) level crossing signals in ﬂuorescence circularity for85Rb in conditions of alignment – orientation conversion and production of transversal orientation. FIG. 3. Hyperﬁne structure energy-level diagram of87Rb 5p2P3/2state in an external magnetic ﬁeld. Symbols /squareindicate ∆ mFe= 1 level crossings, and /circlecopyrt— ∆mFe= 2 level crossings. FIG. 4. Numerically simulated (smooth line) and experiment ally measured (signal with noise) level crossing signals in ﬂuorescence circularity for87Rb in conditions of alignment – orientation conversion and production of transversal orientation. FIG. 5. Numerically simulated (smooth line) and experiment ally measured (signal with noise) level crossing signals in linearly polarized ﬂuorescence f or85Rb. 16/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13 /G19/G13 /G1A/G13 /G1B/G13 /G1C/G13 /G14/G13/G13/G10/G14/G18/G13/G10/G14/G13/G13/G10/G18/G13/G13/G18/G13/G14/G13/G13/G14/G18/G13 /G03 /G2C /G03/G20/G03/G18/G12/G15 /G29/G4C/G4A/G58/G55/G48/G03/G14 /G2D/G11/G24/G4F/G51/G4C/G56/G03/G44/G51/G47/G03/G30/G11/G24/G58/G5D/G4C/G51/G56/G4B/G48/G50/G29 /G10/G15/G10/G14 /G13 /G14 /G15/G13/G10/G15 /G10/G16 /G10/G17 /G10/G15/G10/G14 /G10/G16 /G10/G14/G03 /G03/G1B/G18/G35/G45/G03/G03/G2B/G29/G36/G03/G48/G51/G48/G55 /G4A /G5C/G03/G03 /G28/G12/G4B /G03/G03/G0B/G30/G2B/G5D/G0C /G30/G44 /G4A /G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G03 /G25 /G03/G03/G0B/G2A/G0C/G13 /G15 /G17 /G19 /G1B /G14/G13/G10 /G1B/G19/G10 /G1B/G18/G10 /G1B/G17/G10 /G1B/G16/G10 /G1B/G15/G10 /G1B/G14/G10 /G1B/G13/G10 /G1A/G1C/G03 /G03 /G03/G03/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13 /G19/G13 /G1A/G13 /G1B/G13 /G1C/G13 /G14/G13/G13/G10/G13/G11/G13/G17/G10/G13/G11/G13/G16/G10/G13/G11/G13/G15/G10/G13/G11/G13/G14/G13/G11/G13/G13 /G03/G03 /G2C /G03/G20/G03/G18/G12/G15 /G29/G4C/G4A/G58/G55/G48/G03/G15 /G2D/G11/G24/G4F/G51/G4C/G56/G03/G44/G51/G47/G03/G30/G11/G24/G58/G5D/G4C/G51/G56/G4B/G17/G18/G13/G48/G5B/G46/G11 /G52 /G45/G56/G11/G5D /G5C /G5B/G28/G48/G5B/G46 /G2C/G55/G4C/G4A/G4B/G57/G2C/G4F/G48/G49/G57/G25/G29/G4C/G03/G20/G03/G16/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C /G29/G4C/G03/G20/G03/G15/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C/G03 /G03/G1B/G18/G35/G45/G03/G26/G4C/G55/G46/G58/G4F/G44/G55/G03/G53 /G52 /G4F/G44/G55/G4C/G5D/G44/G57/G4C /G52 /G51/G03/G55/G44/G57/G48/G03/G03 /G26 /G03/G20/G03/G0B/G03 /G2C/G55/G4C/G4A/G4B/G57/G03/G10/G03 /G2C/G4F/G48/G49/G57/G03/G0C/G12/G0B/G03 /G2C/G55/G4C/G4A/G4B/G57/G03/G0E/G03 /G2C/G4F/G48/G49/G57/G03/G0C /G30/G44 /G4A /G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G03 /G25 /G03/G03/G0B/G2A/G0C/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13 /G19/G13 /G1A/G13 /G1B/G13 /G1C/G13 /G14/G13/G13 /G14/G14/G13 /G14/G15/G13 /G14/G16/G13/G10/G17/G13/G13/G10/G16/G13/G13/G10/G15/G13/G13/G10/G14/G13/G13/G13/G14/G13/G13/G15/G13/G13/G16/G13/G13 /G29/G4C/G4A/G58/G55/G48/G03/G16 /G2D/G11/G24/G4F/G51/G4C/G56/G03/G44/G51/G47/G03/G30/G11/G24/G58/G5D/G4C/G51/G56/G4B/G03/G03 /G2C /G03/G20/G03/G16/G12/G15 /G48/G50/G29 /G13/G10/G15 /G10/G16 /G10/G15/G10/G14 /G13 /G14/G03 /G03/G1B/G1A/G35/G45/G2B/G29/G36/G03/G48/G51/G48/G55 /G4A /G5C/G03/G03 /G28/G12/G4B /G03/G03/G0B/G30/G2B/G5D/G0C /G30/G44 /G4A /G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G03 /G25 /G03/G03/G0B/G2A/G0C/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13 /G19/G13 /G1A/G13 /G1B/G13 /G1C/G13 /G14/G13/G13 /G14/G14/G13 /G14/G15/G13 /G14/G16/G13/G10/G13/G11/G13/G15/G10/G13/G11/G13/G14/G13/G11/G13/G13 /G29/G4C/G4A/G58/G55/G48/G03/G17 /G2D/G11/G24/G4F/G51/G4C/G56/G03/G44/G51/G47/G03/G30/G11/G24/G58/G5D/G4C/G51/G56/G4B/G29/G4C/G03/G20/G03/G14/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C/G03 /G2C /G03/G20/G03/G16/G12/G15/G29/G4C/G03/G20/G03/G15/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C/G03 /G03/G1B/G1A/G35/G45/G26/G4C/G55/G46/G58/G4F/G44/G55/G03/G53 /G52 /G4F/G44/G55/G4C/G5D/G44/G57/G4C /G52 /G51/G03/G55/G44/G57/G48/G03/G03/G03/G26/G03/G20/G03/G0B/G03 /G2C/G55/G4C/G4A/G4B/G57/G03 /G10/G03 /G2C/G4F/G48/G49/G57/G03/G0C/G03/G12/G03/G0B/G03 /G2C/G55/G4C/G4A/G4B/G57/G03/G0E/G03 /G2C/G4F/G48/G49/G57/G03/G0C /G30/G44 /G4A /G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G03 /G25 /G03/G03/G0B/G2A/G0C/G13 /G14/G13 /G15/G13 /G16/G13 /G17/G13 /G18/G13 /G19/G13 /G1A/G13 /G1B/G13 /G1C/G13 /G14/G13/G13/G13/G11/G13/G13/G13/G11/G13/G18/G13/G11/G14/G13/G13/G11/G14/G18/G13/G11/G15/G13 /G29/G4C/G4A/G58/G55/G48/G03/G18 /G2D/G11/G24/G4F/G51/G4C/G56/G03/G44/G51/G47/G03/G30/G11/G24/G58/G5D/G4C/G51/G56/G4B/G29/G4C/G03/G20/G03/G15/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C/G03 /G2C /G03/G20/G03/G18/G12/G15 /G18/G33/G16/G12/G15 /G18/G36/G14/G12/G15 /G29/G4C/G29/G48/G17/G16/G15/G14 /G16 /G15 /G48/G5B/G46/G11/G52 /G45/G56/G11 /G5D /G5C /G5B /G28/G48/G5B/G46ΙΙΙΙ⊥ /G25/G29/G4C/G03/G20/G03/G16/G03→ /G03 /G29/G48/G03→ /G03 /G29/G4C/G03 /G03/G1B/G18/G35/G45/G03/G03/G2F/G4C/G51/G48/G44/G55/G03/G53 /G52 /G4F/G44/G55/G4C/G5D/G44/G57/G4C /G52 /G51/G03/G55/G44/G57/G48/G03/G03/G03Ρ = ( ΙΙΙ − Ι⊥ ) / ( ΙΙΙ + Ι⊥ ) /G30/G44 /G4A /G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G03 /G25 /G03/G03/G0B/G2A/G0C
arXiv:physics/0011051v1 [physics.atom-ph] 21 Nov 2000Reversed Dark Resonance in Rb Atom Excited by a Diode Laser Janis Alnis, Marcis Auzinsh∗ Department of Physics, University of Latvia, 19 Rainis boul evard, Riga, LV-1586, Latvia (October 31, 2013) Typeset using REVT EX ∗Corresponding author, Fax +371-7820113, e-mail mauzins@l atnet.lv 1Abstract Origin of recently discovered reversed (opposite sign) dar k resonances was explained theoretically and veriﬁed experimentally. It is shown that the reason for these resonances is a speciﬁc optical pumping of ground s tate level in a transition when ground state angular momentum is smaller th an the excited state momentum. I. INTRODUCTION Coherent population trapping was discovered in the interac tion of sodium atoms with a laser ﬁeld in 1976. [1]. Due to this eﬀect a substantial part of population, because of destructive quantum interference between diﬀerent excita tion pathways, is trapped in a coherent superposition of ground state sublevels – dark sta tes. With a coherent population trapping are associated dark resonances when due to this eﬀe ct absorption and as a result ﬂuorescence from atoms decreases, but intensity of the tran smitted light increases when part of the atomic population is trapped in dark states. If in addi tion to the optical excitation an external magnetic ﬁeld is applied, it can destroy coherence between ground state sublevels and return trapped population into absorbing states and, as a result, increase absorption and ﬂuorescence, but decrease a transmitted light. A review of a pplications of dark resonances was published some years ago by Arimondo [2]. Coherence in an atomic ground state attracted substantial attention in connection with lasing without inversion [3], magnetometry [4] and laser cooling [5]. As a result dark resonances recent ly are studied in detail, including open systems [6] and systems with losses [7]. In course of the se studies a new and unexpected phenomenon was observed by authors of [8]. In this study D 2line of85Rb atoms was excited by a diode laser. Radiation was tuned to the absorption from o ptically resolved ground state hyperﬁne level Fg= 3 originating from atomic 5S 1/2state. The ﬁnal state of the transition was Rb 5P 3/2excited state. Hyperﬁne components of this level was not res olved and all allowed in a dipole transition excited state hyperﬁn e levels with quantum numbers 2Fe= 2,3,4 were excited. In contrary to the usual dark resonance signa l when in the absence of the magnetic ﬁeld one can observe increased transmittanc e and decreased ﬂuorescence intensity authors observed opposite eﬀect — decreased tran smittance which increases with magnetic ﬁeld applied and an increased ﬂuorescence intensi ty which decreased when ﬁeld was applied. Authors of the paper [8] write that the physical reason of this eﬀect remains still unclear . They suppose that one of the reasons for the peak in the ﬂuore scence could be the non-coupled states on the Zeeman sublevels of the exci ted hyperﬁne levels. These non-coupled states, as it is supposed in [8], inhibit the sti mulated emission induced by the laser ﬁeld. The decrease of stimulated emission leads to an i ncrease of the ﬂuorescence. In this letter we oﬀer, in our opinion, very simple and strait forward explanation of the origin of these ”reversed” dark resonances and perform expe rimental and numerical studies of them. II. REVERSED RESONANCE In a simple qualitative explanation traditional dark reson ances can be connected with a well known optical pumping phenomenon. Let us assume that we excite atomic transition Fg= 2−→Fe= 1 with a linearly polarized light. Direction of the zaxis is chosen along the light electric ﬁeld vector E. As a result πabsorption takes place and transitions occur between ground and excited state magnetic sublevels w ith ∆M=Mg−Me= 0, where Mg, Meare magnetic quantum numbers of the ground and excited state s respectively, see Fig. 1. According to this scheme absorption does not take place from ground state magnetic sublevels with quantum number Mg=±2, because for these states there are no corresponding excited state magnetic sublevel with the sam e magnetic quantum number value. In the spontaneous decay dipole transitions from optically populated excited state mag- netic sublevels Me=±1 to the nonabsorptive ground state sublevels Mg=±2 are allowed. As a result, if relaxation in the ground state is slow, in a ste ady state conditions substantial 3part of the population will be optically pumped to the ground state sublevels with quan- tum numbers Mg=±2 and will be trapped there. As a result traditional decrease of the absorption and ﬂuorescence and increase of the transmittan ce will be observed, because the population of absorbing ground sate magnetic sublevels wil l be reduced. If we now apply an external magnetic ﬁeld in a direction perpe ndicular to the zaxis, ﬁeld will mix ground state sublevels eﬀectively and will return t rapped population into the states from which absorption takes place. As a result absorption an d ﬂuorescence will increase. This is a qualitative explanation of the usual dark resonanc e. A similar reasoning can be exploited to explain the ”reverse d” resonance observed in [8] and in this paper. Let us assume that we excite with πradiation atomic transition Fg= 1−→Fe= 2. In this case there are no ground state sublevels not invol ved in the absorption that can trap atomic population. The actual rela tive transition rates proportional to the squared respective Clebsch – Gordan coeﬃcients in thi s system of sublevels are shown in Fig. 2. As one can see ground state magnetic sublevel Mg= 0 is the most absorbing - with highest relative absorption rate. At the same time just to th is sublevel intensively with high rates decay all three excited state magnetic sublevels popu lated by the light. One can expect that in a conditions of a steady state excitation, as a result of interplay of absorption and decay rates, population of the intensively absorbing groun d state magnetic sublevel Mg= 0 will be increased and, as a result, one can expect increased a bsorption and ﬂuorescence from this atom and decreased transmittance of the resonant laser light. If an external magnetic ﬁeld is applied perpendicularly to zaxis it will mix ground state magnetic sublevels and redistribute population between th e ground state magnetic sublevels. As a result population of intensively absorbing magnetic su blevel Mg= 0 will be decreased. At the same time population of less absorbing magnetic suble velsMg=±1 will be increased This means that the total absorption and ﬂuorescence will be decreased and transmittance will be increased. This means that reversed dark resonance w ill be observed. To prove this qualitative consideration let us solve balanc e equations for the magnetic sublevel stationary population nMgin the scheme shown in Fig. 2. In a steady state 4conditions for the ground state magnetic sublevels we will o btain n−1=6(6Γ + 5Γ p) 51Γ + 100Γ png, n0=9(9Γ + 10Γ p) 51Γ + 100Γ png, (1) n+1=6(6Γ + 5Γ p) 51Γ + 100Γ png, where Γ is excited state relaxation rate, Γ pabsorption rate, and ngground state magnetic sublevel population in absence of the radiation. In a condit ion when absorption is slow Γp≪Γ — weak absorption, we have n−1≈36 51ng≈0.706ng, n0≈81 51ng≈1.59ng, (2) n+1≈36 51ng≈0.706ng. If we now keep in mind absorption rates from diﬀerent magneti c sublevels of the ground state, see Fig. 2, and calculate the overall absorption from such st ate and compare it with the absorption from the equally populated magnetic sublevels ( when magnetic ﬁeld is applied) than we see an increase in the absorption rate by a factor 18 /17≈1.059 or by approximately 5.9%. The same calculation can be performed for the transitions Fg= 2−→Fe= 3 and Fg= 3−→Fe= 4. For these schemes in a similar way we will obtain even larg er increase of the absorption due to this speciﬁc optical pumping. The in crease will be by a factor 540/461≈1.17 and 4004 /3217≈1.24 respectively. This means that the described eﬀect increases with increase of the quantum numbers of involved l evels. Of course presented description is only qualitative, but in our opinion gives a good idea what is happening when reversed dark resonances are observe d. To have a quantitative description of the phenomenon one must solve equations for t he density matrix for an open system with losses. We will not do this in present paper. Inst ead a simple analysis will be carried out. 5An analysis of the probabilities of optical transitions ori ginating from Fg= 3 between excited hyperﬁne levels of the Rb atom show that levels Fe= 2,3,4 are populated in the ratio (5 /18≈0.278) : (35 /36≈0.972) : (9 /4 = 2.25). It means that a hyperﬁne transition leading to the reversed dark resonances discussed above is m ost strongly excited. For this scheme let us calculate a signal shape using a full density ma trix approach. We solved a rate equations for a density matrix, see [9], Chapter 5, for Fg= 3←→Fe= 4 transition. A broad line approximation was used. It means that we assume t hat in a magnetic ﬁeld all magnetic sublevels are in equally good resonance with radia tion. Secondly, we assumed that at a magnetic ﬁeld strength used in the experiment (resonanc e width is less than 100 mG) hyperﬁne levels experience linear Zeeman eﬀect. No substan tial hyperﬁne level mixing at applied ﬁeld strength takes place. Direct Rb atom magnetic s ublevel splitting in a magnetic ﬁeld calculations and measurements prove that these assump tion are valid, see for example [10]. For signal simulation the following rate constants we re used. Excited state relaxation rate Γ = 3 .8×107s−1[11], absorption rate Γ p= 3×106s−1, ground state relaxation rateγ= 2×105s−1(mainly due to collisions with the walls of the cell and ﬂy-th rough the excitation laser beam) . Lande factors gg=−0.3336, ge=−0.5013 were calculated in a standard way from the atomic and nuclear data available i n [12]. We suppose that a magnetic ﬁeld is applied along zaxis. Laser light excites Fg= 3−→Fe= 4 transition and is linearly polarized along yaxis. Intensity of the ﬂuorescence with the same polarizati on is calculated and the intensity of the transmitted beam is also calculated as a function of the magnetic ﬁeld. The results are presented in a Fig. 3. They dem onstrate well pronounced reverse resonances and are in a very good qualitative agreem ent with the measurements obtained in [8], see Fig. 5. there. Width of these resonances can be varied by changing ground state Lande factor value, ground state relaxation ra te and absorption rate. At week absorption Γ p≪Γ the dark resonance width is determined by a condition when ground state Larmor frequency is equal to the ground state re laxation rate. For Fg= 3 state of85Rb atom at low concentration it can be as narrow as 20 −30 mG. This is a width that was actually observed in [8]. 6Obtained signals in some sense are the same as the ground stat e Hanle eﬀect measured in atoms as well as in molecules in great extent, see. for exam ple [9,13]. In case of molecules also reversed structure in ground state Hanle eﬀect was obse rved. In case of molecules, when optical pumping takes place in an open cycle and a total g round state population is substantially reduced, this structure can be attributed to a high order coherence created between ground state magnetic sublevels [14,15] III. EXPERIMENTAL We performed measurements of these reversed resonances als o in our laboratory. In our experiment we use isotopically enriched rubidium (99 % of85Rb) that is contained in a glass cell at room temperature to keep atomic vapor concentration low and avoid reabsorption. Transition 5 s2S1/2to 5p2P3/2at 780 .2 nm is excited using both temperature- and current- stabilized single-mode diode laser (Sony SLD114VS) with be am diameter 7 mm. Absorption signal (transmitted light) is monitored by a photodiode. As the laser is swept applying a ramp on a drive current, two absorption peaks with half-widt h of about 600 MHz separated by∼3 GHz appear due to the85Rb ground state hyperﬁne structure. The excited state hyperﬁne structure is not resolved under Doppler proﬁle. During the resonance measurements, the laser wavelength is stabilized on absorption peak originating from ground sate hyperﬁne level Fg= 3. Helmholz coils are used to sweep the magnetic ﬁeld over zero G auss region, the Earth magnetic ﬁeld components are compensated. Signal detected in a transmitted light is averaged over 64 cy cles and the result is presented in a Fig. 4, data points and Lorenz ﬁtting, curve 1. On the same ﬁgure a calculated signal, curve 2, is presented. Model for this calculation is the same as in case on Fig. 3. Amplitude of the experimental and calculated signals are in arbitrary units and in ﬁgure are scaled in a way to make them easy to compare. For theoretical curve th e parameters are chosen to have values that maximally reproduce our experimental co nditions. The ground state 7relaxation rate γ=vp/r0= 0.07µs−1was chosen as a reciprocal time of thermal motion of Rb atoms at room temperature with most probable velocity vp= 0.24 mm/ µsthrough the laser beam of radius r0= 3.5 mm [9]. Absorption rate was chosen to be Γ p= 1.5µs−1. Other parameters are as in Fig. 3. Agreement between calcula ted and measured signal is remarkable. As far as our model does not account for transiti ons to other hyperﬁne levels and so does not reproduce experiment in full, we do not attemp t to ﬁt experimental points with theoretical curve, nevertheless achieved agreement f ully convinces us that the proposed explanation of the reversed dark resonances is correct. To o btain quantitative description of this signal in future one must take into account in the numeri cal model all other hyperﬁne levels involved in the process. IV. ACKNOWLEDGMENT One of us MA is thankful to Prof. Neil Shafer-Ray for fruitful discussions. Financial support from Swedish Institute Visby program is greatly ack nowledged. 8REFERENCES [1] G. Alzetta, A. Gozzini, L. Moi, and G. Orrioli, Novo Cimen to B36, (1976) 5 [2] E. Arimondo, Progr. Opt. 35(1996) 257 [3] M. Scully, S-Y Zhu, A. Gavrielides, Phys. Rev. Lett. 62(1989) 2813 [4] M. Scully, M. Fleischhauer, Phys. Rev. Lett. 69(1992) 1360 [5] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. K ohen-Tannoudji, Phys. Rev. Lett.61(1988) 826 [6] Ferrucio Renzoni, Albrecht Lindner, and Ennio Arimondo , Phys. Rev. A 60, (1999) 450 [7] F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, Phy s. Rev. A 55(1997) 3710 [8] Y. Dancheva, G. Alzetta, S. Cartalava, M. Taslakov, Ch. A ndreeva, Opt. Comm. 178 (2000) 103 [9] M. Auzinsh, R. Ferber, Optical Polarization of Molecules , Cambridge University Press, Cambridge UK, 1995, 305 [10] J. Alnis, M. Auzinsh, Phys. Rev. A, submitted [11] G. Belin, S. Svanberg, Physica Scripta 4, 269 (1971) [12] E. Arimondo, M. Inguscio, P. Violino, Rev. Mod. Phys, 49, 31 (1977) [13] M.P. Auzinsh, R.S. Ferber, Phys. Rev. A, 43, 2374 (1991) [14] M.P. Auzinsh, R.S. Ferber, Sov. Phys. Usp. 33, 833 (1990) [15] M.P. Auzinsh, R.S. Ferber, Opt. Spectrosc. (USSR), 55,674 (1983) 9FIGURES FIG. 1. Allowed dipole transition scheme for ground state op tical pumping in case of πabsorp- tion for Fg= 2→Fe= 1. FIG. 2. Transition scheme and rate constants for πabsorption in case of Fg= 1→Fe= 2. FIG. 3. Calculated intensity of ﬂuorescence and transmitte d light for reverse dark resonance forFg= 3→Fe= 4. FIG. 4. Measured (points and Lorenz ﬁtting — curve 1) and calc ulated (curve 2) reversed dark resonance in85Rb. 10□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ Figure□1□ J□Alnis□and□M.□Auzinsh□Me□=-1□Me□=□0□Me□=□1□ Mg□=-1□Mg□=□0□Mg□=□1□Mg□=□2□ Mg□=-2□□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ Figure□2□ J.□Alnis□and□M.□Auzinsh□Mg□=-1□Mg□=□0□Mg□=□1□Me□=-2□Me□=-1□Me□=□0□Me□=□1□Me□=□2□103p
1The paper was accepted at the 32nd EGAS Conference (European Group Atomic Spectroscopy), Vilnius, July 4-7, 2000 as a poster presentation DEPENDENCE OF THE ENERGY OF MOLECULES ON INTERATOMIC DISTANCE AT LARGE DISTANCES I. A. Stepanov Latvian University, Rainis bulv. 19, Riga, LV-1586, Latvia Abstract Earlier it has been supposed that energy of molecules depends on interatomic distance according to the curve 1, Fig. 1. However, dissociation of molecules (for example, Te 2→2Te) often is a chemical reaction. According to chemical kinetics, chemical reactions overcome a potential barrier. This barrier is absent at the curve 1. It is a very strong argument against the curve 1. It has been shown that the molecule energy dependence on interatomic distance can behave at large distances not so but like the curve 2, Fig. 1. Earlier it has been supposed that quantum chemical methods give a wrong result at big distances if the wave function does not turn to zero. In this paper it has been shown that it must not turn to zero.21. Introduction According to the traditional point of view, the energy of molecules depends on interatomic distance according to the curve 1, Fig. 1 (the energy of independent atoms is supposed to be zero). This dependence has the following disadvantages. Dissociation of molecules (for example, F 2→2F) often is a chemical reaction. According to chemical kinetics, chemical reactions overcome a potential barrier. This barrier is absent at the curve 1. It is a very strong argument against the curve 1. Rupture is a transition from the less stable state to the more stable state, from the state with bigger energy to the state with lower energy. According to non-equilibrium thermodynamics, the system being deflected greatly from equilibrium, looses steadiness, and the system changes to a qualitatively new steady state (with lower energy) [1]. In [2-4] it has been shown that molecules failure during stretching of solids happens like this: at strong stretching of interatomic bonds, molecules loose stability and turn to a qualitatively new steady state with lower energy: ruptured molecules. One can assume that dissociation of a single molecule also happens like this. According to the curve 1, energy must not be released during separation of interatomic bond. However, it has been found experimentally that during stretching of solids, chemical bond ruptures lead to microheating of substance to a few hundred grades [5]. 2. Methods3The dependence of the energy of diatomic molecule on the distance between the atoms must be the following one: curve 2, Fig. 1. During stretching of molecule, its energy becomes larger than that of independent atoms and in the point A the molecule becomes greatly unstable and turns to the dissociated state. There are 2 possibilities: from the point A the curve tends smoothly to zero, or transition to the dissociated state happens by a jump. With this the energy of elastic stretching is released. Pay attention that E(R) at R →∞ does not turn to zero. The energy E(R) at R →∞ has the sense of the energy of fictitious molecule being stretched to the infinitive distance. Dependence E(R) near the bottom of the potential well is found experimentally, behavior of E(R) at big distances is an invention of physicists. It has been supposed that E(R) at R→∞ turns to zero. It is not obvious. Molecule must not obligatory fail if E(R)=0. In [6] the H2+ ion has been solved exactly taking into account that the energies of electron - nuclei interaction are E1=1/2Kr12(1) E2=1/2Kr22(2) where K is the coefficient of proportionality, ri is the distance between nucleus and the electron, and the energy of nuclei interaction is E3=λ/R2, λ>0 (3) where R is the internuclear distance. According to this calculation, the energy of the ground state is E(R)=3/2((2K/m)1/2)+KR2/4+λ/R2(4) where m is the hydrogen atom nucleus mass. The attraction force in such ion is greater than that in the real one, and the repulsion force is less. It means that the binding energy4of such ion is bigger and dE/dR for it is bigger than that for the real ion. Let's build the following H2+ ion model: near the bottom of the potential well energy is described by (4), at bigger distances it is described by the same equation but K and λ depend on R. In this model E(R) behaves like the curve 1 > Fig. 1. By fitting of K(R) and λ(R) in (4) one can ascertain that the depth of the potential well is the same as that with the real H2+. In such model dependence E(R) is stiffer and must reach zero by smaller R than that of the real ion. It is a contradiction: the ion with larger binding energy fails earlier than that with the smaller one. Therefore, the initial supposition that bond failure happens at E ≈0, is not true. The rupture of chemical bond begins in the real ion at R 0, E(R0)>0, but the bond rupture in the model ion begins at RM>R0, E(RM)>E(R0). 3. Results and Discussion Earlier it was supposed that quantum chemical methods give a wrong result at big distances if the wave function does not turn to zero. It is necessary to make the conclusion that the wave function must not turn to zero. This result explains the paradox: experimental dissociation energies usually are much bigger than theoretical ones [7, 8]. In astrophysics the barriers at the curve 2, Fig. 1 are found experimentally for some molecules. They are barriers of chemical reaction [9]. References51. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Moscow, Mir, 1973, (Russian transl.). 2. I. A. Stepanov, "The scale effect is a consequence of the cellular structure of solid bodies. Thermofluctuation nature of spread in the values of strength," Fiziko- Khimicheskaya Mekhanika Materialov, V. 31, No 4, p. 36-42 (1995) [English Translation: I. A. Stepanov Materials Science, V. 31, No 4, p. 441-447 (1995)]. 3. I. A. Stepanov, "The mathematical law of scatter of strength of solids and some composite materials," J. Macromol. Sci. Phys., V. B36, No 1, p. 117-124 (1997). 4. I. A. Stepanov, "Mathematical dependence of the dispersion of solid states and certain composite materials strength," Zhur. Tekh. Fiz., V. 66, No 2, p. 185-189 (1996). [English Translation: I. A. Stepanov, Tech. Phys. (1996) 5. V. R. Regel, A. I. Slutsker, and E. E. Tomashevskij, Kinetical Nature of the Strength of Solids (Nauka, Moscow, 1974). 6. B. Crandall, R. Bettega, and R. Whitnell, "A class of exactly soluble three-body problems," J. Chem. Phys., V. 83, No 2, p. 698-702 (1985). 7. H. Schaefer III, J. Russel Thomas, Y Yamaguchi, B. De Leeuw, and G. Vacek, Modern Electronic Structure Theory (Ed. David R Yarkony, World Scientific, Part 1, 1995), pp. 3-54. 8. V. Minkin, B. Simkin, and R. Minjaev, The Theory of Molecules (Vysshaja Shkola, Moscow, 1979), p. 107-110. 9. D. A. Varshalovich, private communication.6A RE 01 2 Figure 1 Dependence of the energy of molecule on the distance between atoms: 1- traditional theory, 2 - according to this paper . Point A is the point where the bond begins to fail.
arXiv:physics/0011053v1 [physics.comp-ph] 21 Nov 2000Faster Evaluation of Multidimensional Integrals A. Papageorgiou J.F. Traub Department of Computer Science Columbia University New York, NY 10027 June 1997 Abstract In a recent paper Keister proposed two quadrature rules as al ternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasi-Monte Carlo method with gene ralized Faure points is very eﬀective for a variety of high dimensional integrals occuring in mathematical ﬁnance. In this paper we report test results of this method on Keister’s examples of dimension 9 and 25, and also for examples of dimension 60, 80 a nd 100. For the 25 dimensional integral we achieved accuracy of 10−2with less than 500 points while the two methods tested by Keister used more than 220,000 points. In all of our tests, for nsample points we obtained an empirical convergence rate pro portional ton−1rather than the n−1/2of Monte Carlo. 1 Introduction Keister [1] points out that multi-dimensional integrals ar ise frequently in many branches of physics. He rules out product rules of one-dimensional me thods because the number of integrand evaluations required grows exponentially in the number of dimensions. He observes that although Monte Carlo (MC) methods are desirable in high dimension, a large number, n, of integrand evaluations can be required since the expecte d error decreases as n−1/2. This motivates Keister to seek non-product rules for a certa in class of integrands deﬁned below. He proposes two quadrature rules, one by Mc Namee and S tenger (MS) [2], and a second due to Genz and Patterson (GP) [3],[4], which he tests on a speciﬁc example of his class of integrands. In this paper we report test results on Keister’s example usi ng quasi-Monte Carlo (QMC) methods. QMC methods evaluate the integrand at determinist ic points in contrast to MC 1methods which evaluate the integrand at random points. The d eterministic points belong to low discrepancy sequences which, roughly speaking, are uniformly spread as we will see in the next section. Niederreiter [5] is an authoritative monogra ph on low discrepancy sequences, their properties, and their applications to multi-dimensi onal integration. The Koksma-Hlawka inequality (see the next section for a pre cise statement) states that low discrepancy sequences yield a worst case error for multi variate integration bounded by a multiple of (log n)d/n, where nis the number of evaluations and dis the dimension of the integrand. A similar bound on the average error is implie d by Wo´ zniakowski’s theorem [6]. The proof of this theorem is based on concepts and result s from information-based complexity [7]. Fordﬁxed and nlarge, the error (log n)d/nbeats the MC error n−1/2. But for nﬁxed anddlarge, the (log n)d/nfactor looks ominous. Therefore, it was believed that QMC methods should not be used for high-dimensional problems; d= 12 was considered high [8, p. 204]. Traub and a then Ph.D. student, Paskov, decided t o test the eﬃcacy of QMC methods for the valuation of ﬁnancial derivatives. Softwar e construction and testing of QMC methods for ﬁnancial applications was began in Fall 1992. Th e ﬁrst tests were run on a very diﬃcult ﬁnancial derivative in 360 dimensions, which requi red 105ﬂoating point operations per evaluation. Surprisingly, QMC methods consistently be at MC methods. The ﬁrst published announcement was in January 1994 [9]. Det ails appeared in [10], [11], [12]. Tests by other researchers [13], [14] lead to similar c onclusions for the high-dimensional problems of mathematical ﬁnance. These results are empirical. A number of hypotheses have bee n advanced to explain the observed results. One of these is that, due to the discounted value of money, the ﬁnancial problems are highly non-isotropic with some dimensions far more important than others. Perhaps the QMC methods take advantage of this. A generally a ccepted explanation is not yet available. Since Keister’s test integral is isotropic it provides an ex ample which is very diﬀerent than the examples from mathematical ﬁnance. To our surprise the QMC method beat both MC and two other methods tested by Keister by very convincing margins. The problems in [1] require the computation of a weighted mul ti-dimensional integral /integraldisplay Rdf(x)ρ(x)dx, (1) where dis the dimension of the problem, f:Rd→Ris asmooth function, and the weight ρ(x),x∈Rdsatisﬁes ρ(x) =d/productdisplay j=1η(xj), (2) withη(−xj) =η(xj),xj∈R. Thus, the weight is symmetric with respect to permutations and changes of sign of the variables. The example in [1] (see a lso [15]) is /integraldisplay Rdcos(/bardblx/bardbl)e−\|\|x\|\|2dx, (3) 2where /bardbl · /bardbldenotes the Euclidean norm in Rd. The integral in (3) can be reduced, via a change of variable, t o a one-dimensional integral which can be analytically integrated. As we will see, the QMC method takes advantage of the dependence on the norm automatically and provides a numerical solution with error similar to a one-dimensional integral. The QMC method that we test in this paper uses points from the g eneralized Faure sequence, which was constructed by Tezuka [16]. We will refe r to it as QMC-GF. This sequence has been very successful in solving problems of mat hematical ﬁnance [12], [14]. The performance of QMC-GF on the integral (3) is most impress ive. For example, for the 25-dimensional integral it achieves error 10−2using less than 500 points, far superior to all the other methods. Its error over the range we tested, whi ch was up to 106points, was c·n−1, with c <110,d= 9,25,80,60,100. That may be compared with the MC method whose error was proportional to n−1/2. We summarize the remainder of this paper. In the next section we provide a brief introduction to low discrepancy sequences. Test results ar e given in the third section. A summary of our results and future research concludes the pap er. 2 Low Discrepancy Sequences Discrepancy is a measure of deviation from uniformity of a se quence of real numbers. In particular, the discrepancy of npoints x1, . . ., x n∈[0,1]d,d≥1, is deﬁned by D(d) n= sup E/vextendsingle/vextendsingle/vextendsingle/vextendsingleA(E;n) n−λ(E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (4) where the supremum is taken over all the subsets of [0 ,1]dof the form E= [0, t1)×···× [0, td), 0≤tj≤1, 1≤j≤d,λdenotes the Lebesgue measure, and A(E;n) denotes the number of the xjthat are contained in E. A detailed analysis of low discrepancy sequences can be found in [5] and in the references therein. A sequence x1, x2, . . .of points in [0 ,1]dis a low discrepancy sequence iﬀ D(d) n≤c(d)(logn)d n,∀n >1, (5) where the constant c(d) depends only on the dimension d. Neiderreiter, see [5], gives a general method for constructing ( t, d)-sequences, t≥0, which are low discrepancy sequences. The discrepancy of the ﬁrst npoints in a ( t, d)-sequence is given by D(d) n≤c(t, d, b)(logn)d n+O/parenleftbigg(logn)d−1 n/parenrightbigg , where b≥2 is an integer parameter, upon which the sequence depends, a ndc(t, d, b)≈ bt/d!·(b/2 logb)d. Hence, the value t= 0 is desirable. 3The generalized Faure sequence [16] is a (0 , d) sequence and is obtained as follows. For a prime number b≥dandn= 0,1, . . ., consider the base brepresentation of n, i.e., n=∞/summationdisplay i=0ai(n)bi, where ai(n)∈[0, b) are integers, i= 0,1, . . .. The j-th coordinate of the point xnis then given by x(j) n=∞/summationdisplay k=0x(j) nkb−k−1,1≤j≤d, where x(j) nk=∞/summationdisplay s=0c(j) ksas(n). The matrix C(j)= (c(j) ks) is called the generator matrix of the sequence and is given b y C(j)=A(j)Pj−1, where A(j)is a nonsingular lower triangular matrix and Pj−1denotes the j−1 power of the Pascal matrix, 1 ≤j≤d. We conclude this section by stating the Koksma-Hlawka inequ ality which establishes the relationship between low discrepancy sequences and multiv ariate integration, see [5]. If f is a real function, deﬁned on [0 ,1]d, of bounded variation, V(f), in the sense of Hardy and Krause, then for any sequence x1, . . ., x n∈[0,1)dwe have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay [0,1]df(x)dx−1 nn/summationdisplay i=1f(xi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤V(f)D(d) n. 3 Methods and Test Results We transform the integral (3) to one over the cube [0 ,1]d. We have Id(cos) =/integraldisplay Rdcos(/bardblx/bardbl)e−/bardblx/bardbl2dx= 2−d/2/integraldisplay Rdcos(/bardbly/bardbl/√ 2)e−/bardbly/bardbl2/2dy (6) =πd/2/integraldisplay Rdcos(/bardbly/bardbl/√ 2)e−/bardbly/bardbl2/2 (2π)d/2dy =πd/2/integraldisplay [0,1]dcos /radicaltp/radicalvertex/radicalvertex/radicalbtd/summationdisplay j=1(φ−1)2(tj)/2 dt, where φis the cummulative normal distribution function with mean 0 and variance 1, φ(u) =1√ 2π/integraldisplayu −∞e−s2/2ds, u∈[−∞,∞]. 4We obtain the ndeterministic sample points xi= (xi1,...,x id)∈Rd,i= 1, . . ., n , by setting xij=φ−1(tij), where ti= (ti1, . . ., t id)∈[0,1]d,i= 1, . . ., n , arenconsecutive terms of a low discrepancy sequence. Our method, Id,n, is deﬁned by: Id,n(cos) =πd/2 nn/summationdisplay i=1cos /radicaltp/radicalvertex/radicalvertex/radicalbtd/summationdisplay j=1(φ−1)2(ti,j)/2 . (7) Our test problem could be reduced to a one-dimensional integ ral. We did not do this because we wanted test how QMC methods perform on d-dimensional integrals. As we will see, the empirical rate of convergence of QMC-GF is n−1which suggests that this method takes advantage of the dependence on the norm automat ically without a dimension reducing transformation. A method corresponding to (7) can be derived for the more general integration problem (1) with weight function satisfying (2 ). We report test results. We used the generalized Faure1low discrepancy sequence [16] to derive the sample points for the QMC method. We remind the r eader that we call this the QMC-GF method. We compared this method to the McNamee-St enger (MS) and Genz- Patterson (GP) [2], [3], [4] methods. We also tested using a M onte Carlo method of the form (7), i.e., using randomly generated points ti,j. Hence, we use the same change of variable for QMC-GF and MC. The value, Id(cos), of the integral (6) is, see [1], I9(cos) = −71.633234291 and I25(cos) = −1.356914 ·106. We used Mathematica to compute I60(cos) = 4 .89052986 ·1014,I80(cos) = 6.78878724 ·1019andI100(cos) = 4 .57024396 ·1024. We measure the accuracy of an approxi- mation by computing its relative error (fractional deviati on). We observe the least number of sample points required by an method to achieve and maintain a relative error below a speciﬁed level, e.g. 10−3, until the end of the simulation. We introduced this more con ser- vative way of assessing the performance of a method in [12]. T hus, we study the error of an method throughout a simulation. We believe that this has a dvantages over performance reports that are based only on values at the end of a simulatio n. We summarize our ﬁndings and then provide some details. •The QMC-GF method outperforms the MS and GP methods for d= 25. •The MS and GP methods are sensitive to the dimension. They per form quite well ford= 9 and very poorly for d= 25. For example, for d= 25 and for accuracy of the order 10−2these methods use some 220 ,000 points while the QMC-GF method uses less than 500 points. Therefore, they should only be use d when the dimension is relatively low. •The QMC-GF method performs well for d= 9, 25, 60, 80 and 100. •The relative error of the QMC-GF method is bounded by cd·n−1, cd<110, n≤106, d= 9,25,60,80,100. (8) 1The generalized Faure and the Sobol’ low discrepancy sequen ces are included in FINDER, a Columbia University software system, and are available to researche rs upon request by writing the authors. 5Note that this is an empirical conclusion. We write cdto suggest that, in principle, this constant depends on dalthough we did not see a strong dependence in our tests. •The QMC-GF method achieves relative error 10−2using about 500 points. •The relative error of the MC method is bounded by β·n−1/2as predicted by the theory. First we consider the case d= 9. The performance of the QMC-GF, and the GP methods is comparable for accuracy less than 10−4. The relative error of the MS method ﬂuctuates about the value 10−4for sample sizes between 36 ,967 and 96 ,745 points, see [1, Table I], and is slower than the QMC-GF method since it requires at least fo ur times as many function evaluations. (For this level of accuracy the MC method requi res more than 106points). Ford= 25 the results are striking. The MS and GP methods require ab out 220 ,000 points for accuracy of order 10−2while the QMC-GF method requires less than 500 points. Table I is from [1, Table II] and exhibits the performance of t he MS and GP methods. Method Number of Points Relative Error GP and MS 1,251 2.00 GP 19,751 0.40 MS 20,901 0.75 GP 227,001 0.06 MS 244,101 0.07 Table I. Comparison of MS and GP methods, d=25 Table II summarizes the performance of the QMC-GF method. Method Number of Points Relative Error QMC-GF 500 10−2 QMC-GF 1,200 10−3 QMC-GF 14,500 5·10−4 QMC-GF 214,000 5·10−5 Table II. The Quasi-Monte Carlo method, d=25 As we mentioned above, we are using a very conservative crite rion when we report relative error. It takes about 219 ,000, 490 ,000, and many more than 106points for the MC method to reach accuracies of 10−3, 5·10−4, and 5 ·10−5, respectively. Figure 1 exhibits the relative error of the QMC-GF method for d= 25. The horizontal axis shows the sample size n, while the vertical axis shows the relative error. The horiz ontal lines depict the accuracy. Figure 2 shows the convergence rate of the QMC-GF method. We p lot the logarithm of the relative error as a function of the logarithm of the sampl e size for d= 25 and obtain the linear convergence summarized in (8). Recently, Keister [17] obtained good results using a public domain version of the Sobol’ low discrepancy sequence. Keister [1] did not perform tests for d >25. We tested the QMC-GF method for d= 60, 80 and 100 and we found that its performance is comparable to t hat of the lower values of 6d. We did not ﬁnd evidence suggesting that its performance suﬀ ers as the dimension grows. This is shown in the empirical error equation (8) and is furth er demonstrated in Figure 3, which shows the convergence of QMC-GF for d= 100. In particular, in Figure 3 we plot the logarithm of the relative error as a function of the logarith m of the sample size. 4 Summary and Future Research We have shown that the QMC-GF method beats MC methods and the M S and GP meth- ods by a wide margin for Keister’s 25-dimensional example. W e have also shown that its good performance is maintained when the dimension takes muc h higher values. Other high dimensional problems motivated by applications to physics should be tested. Extensive testing on a variety of high-dimensional integra ls which occur in mathematical ﬁnance also ﬁnd QMC methods consistently beating the MC meth od. Preliminary results from our tests on high-dimensional integrals arising from s everal very diﬀerent applications again point to the superiority of QMC over MC. The results are empirical. There is currently no theory whic h explains why, for a variety of applications, QMC methods are much better than one would e xpect from the Koksma- Hlawka inequality or from Wo´ zniakowski’s theorem. Findin g the theoretical justiﬁcation for the superiority of QMC methods for certain classes of int egrands is a most important direction of future research. Acknowledgments We thank Bradley Keister for his comments on a draft of this pa per. We are grateful to Richard Palmer for directing us to Bradley Keister’s paper, and to Henryk Wo´ zniakowski for his comments on the manuscript. References [1] Keister, B.D., Multidimensional Quadrature Algorithm s,Computers in Physics , 10:20, 119–122, 1996. [2] Mc Namee, J., and Stenger, F., Construction of Fully Symm etric Numerical Integration Formulas, Numer. Math. , 10, 327–344, 1967. [3] Genz, A., A Lagrange Extrapolation Algorithm for Sequen ces of Approximations to Multiple Integrals, SIAM J. Sci. Stat. Comput. , 3, 160–172, 1982. [4] Patterson, T.N.L., The Optimum Addition of Points to Qua drature Formulae, Mathe- matics of Computation , 22, 847–856, 1968. 7[5] Niederreiter, H., Random Number Generation and Quasi-M onte Carlo Methods, CBMS- NSF Regional Conference Series in Applied Math. No. 63, SIAM , 1992. [6] Wo´ zniakowski, H., Average case complexity of multivar iate integration, Bulletin of the American Mathematical Society , 24, 185–194, 1991. [7] Traub, J.F., Wasilkowski, G.W., and Wo´ zniakowski, H., Information-Based Complexity, Academic Press , New York, 1988. [8] Bratley, P., Fox, B.L., and Niederreiter, H., Implement ation and Tests of Low- Discrepancy Sequences, ACM Trans. on Modeling and Computer Simulation , 2:3, 195– 213, 1992. [9] Traub, J.F. and Wo´ zniakowski, H., Breaking Intractabi lity,Scientiﬁc American , 270, 102–107, 1994. [10] Paskov, S.H. and Traub, J.F., Faster Valuation of Finan cial Derivatives, The Journal of Portfolio Management , 113–120, Fall 1995. [11] Paskov, S.H., New Methodologies for Valuing Derivativ es, in Mathematics of Deriva- tive Securities, S. Pliska and M. Dempster eds., Isaac Newto n Institute, Cambridge University Press , Cambridge, UK, 1997. [12] Papageorgiou, A., and Traub, J.F., Beating Monte Carlo ,Risk, 9:6, 63–65, 1996. [13] Joy, C., Boyle, P.P., and Tan, K.S., Quasi-Monte Carlo M ethods in Numerical Finance, working paper, University of Waterloo, Waterloo , Ontario, Canada N2L 3G1, 1995. [14] Ninomiya, S., and Tezuka, S., Toward real-time pricing of complex ﬁnancial derivatives, Applied Mathematical Finance , 3, 1–20, 1996. [15] Capstick, S., and Keister, B.D., Multidimensional qua drature algorithms at higher de- gree and/or dimension, Journal of Computational Physics , 123, 267–273, 1996. [16] Tezuka, S., Uniform Random Numbers: Theory and Practic e,Kluwer Academic Pub- lishers , Boston, 1995. [17] Keister, B.D., Private communication, 1997. 800.00050.0010.00150.0020.0025 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 Figure 1. QMC-GF, relative error as a function of the sample s ize, d=25 1e-081e-071e-061e-050.00010.0010.010.1 1000 10000 100000 1e+06 Figure 2. QMC-GF, log(relative error) as a function of log(s ample size), d=25 91e-081e-071e-061e-050.00010.0010.010.1 1000 10000 100000 1e+06 Figure 3. QMC-GF, log(relative error) as a function of log(s ample size), d=100 10
arXiv:physics/0011054v1 [physics.atom-ph] 22 Nov 2000Pickoﬀ and spin-conversion quenchings of ortho-positroni um in oxygen N. Shinohara,1,∗T. Chang,2N. Suzuki,3and T. Hyodo1 1Institute of Physics, College of Arts and Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 2Institute of High Energy Physics, Academia Sinica, P.O. Box 2732, Beijing 100080, PR China 3The Institute of Physical and Chemical Research (RIKEN), Hi rosawa 2-1, Wako, Saitama 351-0198, Japan (Dated: December 22, 2013) The quenching processes of the thermalized ortho-positron ium(o-Ps) on an oxygen molecule have been studied by the positron annihilation age-momentu m correlation techinique(AMOC). The Doppler broadening spectrum of the 511 keV γ-rays from the 2 γannihilation of o-Ps in O 2has been measured as a function of the o-Ps age. The rate of the quenchi ng, consisting of the pickoﬀ and the spin-conversion, is estimated from the positron lifetime s pectrum. The ratio of the pickoﬀ quench- ing rate to the spin-conversion rate is deduced from the Dopp ler broadening of the 511 keV γ-rays from the annihilation of the o-Ps. The pickoﬀ parameter1Zeﬀ, the eﬀective number of the electrons per molecule which contribute to the pickoﬀ quenching, for O 2is determined to be 0 .6±0.4. The cross-section for the elastic spin-conversion quenching i s determined to be (1 .16±0.01)×10−19cm2. PACS numbers: 36.10.Dr, 78.70.Bj, 34.50.-s I. INTRODUCTION Recently the annihilation of the low energy positrons on collision with various gas molecules has been stud- ied systematically by Surko and his collaborators [1, 2]. They have measured the annihilation rate, λ+, of the thermalized positron and estimated the positron annihi- lation parameter, Zeﬀ, deﬁned as λ+=πr2 0cnZeﬀ, (1) where r0is the classical electron radius, cis the speed of light, and nis the number density of the molecules. TheZeﬀhas been revealed to be very sensitive to small changes in the molecular structure and increasing para- metrically from 1 to 107[2]. A parameter similar to Zeﬀis deﬁned for the case of the positronium(Ps) annihilation in gases. It is related to the pickoﬀ quenching of the ortho-positronium(o-Ps) and called1Zeﬀ. This parameter represents the eﬀective number of the electrons per molecule in a spin singlet state relative to the positron in the o-Ps [3, 4, 5]. The pickoﬀ quenching rate is expressed as λpickoﬀ = 4πr2 0cn1Zeﬀ. (2) The values of1Zeﬀfor various gases reported so far are collected in Ref. [5]. They lie, except for O 2, between 0.1 and 1.3, very small in contrast to Zeﬀ. In a paramagnetic gas such as O 2, o-Ps can be also quenched by spin-conversion, i.e., the conversion of o-Ps into para-positronium(p-Ps) followed by its prompt self- annihilation [6, 7, 8]. There exists two kinds of Ps spin- conversion processes in O 2. One excites the O 2molecule to the excited state a1∆g(or b1Σ+ g), and thus may be ∗Electronic address: shino@rsaixsv.icepp.s.u-tokyo.ac. jpcalled inelastic conversion. This process is active when the energy of the Ps is larger than 0 .977eV (or 1 .62eV). The other leaves the O 2molecule in the ground state X3Σ− gand may be called elastic conversion. The latter process has no threshold and thus active for the thermal- ized o-Ps. These two spin-conversions have quite diﬀerent cross-sections [7, 8]; the cross-section for the former is o n the order of 10−16cm2, and that for the latter is on the order of 10−19cm2[8]. In Ref. [5], the quenching rate of o-Ps in O 2including the eﬀect of the spin-conversion is expressed in terms of 1Zeﬀas 44±3. The proper1Zeﬀdeﬁned by Eq. (2) is not known. In order to measure this, the pickoﬀ quenching has to be separated from the spin-conversion. In the present work, we obtain the1Zeﬀand the spin-conversion cross-section for O 2by positron anni- hilation age-momentum correlation technique(AMOC). The AMOC consists in a correlated measurement of the positron lifetime and the energy of the annihilation γ- rays. The Doppler broadening spectrum of the 511 keV γ-rays from the 2 γannihilation of the o-Ps in O 2has been measured as a function of the time that the o-Ps atoms have spent from their formation to annihilation. The 2 γannihilation in the time range in which only o-Ps exists results either from the pickoﬀ quenching or from the self-annihilation of the p-Ps created through the spin-conversion. The1Zeﬀand the spin-conversion cross- section are determined by separating the pickoﬀ quench- ing from the spin-conversion. Preliminary results with a diﬀerent setup were re- ported in Ref. [9]. II. EXPERIMENTAL The experimental setup is shown schematically in Fig. 1. A 0.5 µCi22Na positron source was sandwiched between two sheets of 100 µm thick plastic scintillator.2 PMT Light guide SiO aerogel2 Ge□detector 2□cm Pb□sheet Plastic Scintillator 22NaAluminized Mylar FIG. 1: Schematic diagram of the experimental setup. The source-scintillator assembly was placed between two pieces of 1 cm thick SiO 2aerogel, which was used as Ps formation medium [10]. The macroscopic density and the averaged grain diameter of the SiO 2aerogel are 0.1 g/cm3and about 5 nm, respectively. Most of the positrons from the source pass through the scintillators and give scintillation lights. The lights pa ss through the transparent SiO 2aerogel and are directed to a photomultiplier tube(PMT) by a light guide. A γ-ray emitted when a positron annihilates was detected by a high purity Ge detector. The output of the Ge detector was fed to a timing-ﬁlter-ampliﬁer(TFA). The time inter- val between the anode signal of the PMT and the output of the TFA was converted into an peak amplitude of an output pulse by a time-to-amplitude converter(TAC) and used for the lifetime spectroscopy. The other output of the Ge detector was processed by a shaping ampliﬁer and the peak hight was recorded for the energy information. The observed energy region was limited in the neighbor- hood of the 511 keV peak by a biased ampliﬁer. All the data were stored in the list mode. A 2 mm thick Pb sheet was placed in front of the Ge detector to prevent low energy scattered γ-rays from si- multaneously hitting the Ge detector. The time resolution was 4.8 ns(FWHM), and the energy resolution was 1.12 keV(FWHM) at 512 keV(106Ru). The chamber was ﬁlled with O 2of purity 99.9995% to 1.05 atm after evacuating it with a turbo molecular pump, and then isolated from the rest of the gas handling system. The measurement lasted for 16 hours. During the measurement, the room temperature was controlledto be 25 .5±0.5◦C. For data analysis, a measurement without a gas was also made for 16 hours. III. RESULTS AND DISCUSSION 0 50 100 150100101102103104105106107 203040 time(ns)counts Fitted /c108(/c109s-1):□time□spectrum :□fitted /c108 (a)(b) (c) FIG. 2: The time spectrum for the energy range from 500 keV to 516 keV. The closed circles show the time interval distrib u- tion between a positron emission and the eventual annihila- tion. The closed triangles show the ﬁttd λas a function of the start point of the ﬁt. The energy spectra of the 511keV γ-rays in the time windows (a), (b), and (c) are shown in Fig. 4. The closed circles in Fig. 2 show the positron lifetime spectrum for the energy range from 500 keV to 516 keV. The prompt peak is followed by the slow decay curve and subsequently the ﬂat background. The prompt peak arises from the positron annihilations without Ps forma- tion as well as the self-annihilations of p-Ps. The slow decay part originates from the o-Ps annihilations. The total annihilation rate of the o-Ps, λoPs, is obtained from the slow decay part. It is the sum of the self-annihilation rate,λ3γ, and the annihilation rates for three diﬀerent collisional quenching modes; λoPs=λ3γ+λSiO2+λox+λspin, (3) where λSiO2is the rate of the pickoﬀ quenching on the grain surface, λoxis that on the O 2molecule, and λspin is the rate of the spin-conversion on the O 2molecule. The latter two collisional quenching rates are propo- tional to the number density of the O 2molecules, n, λq=nσqvPs, (4) where σqis the quenching cross-section and vPsis the speed of the Ps. The speed of the o-Ps relative to the target molecule has been replaced by vPs, neglecting the relatively small velocity of the molecules. The initial kinetic energy of the Ps depends on whether it is formed inside a SiO 2grain or on a grain surface [11]. The Ps formed and thermalized inside a SiO 2grain es- capes into the free space between the grains with the3 kinetic energy of 1 eV determined by the negative work function for the Ps. On the other hand, the Ps formed on the surfaces of the grains is emmitted with the ki- netic energy of 3 eV. It penetrates shallowly back into another grain and thermalizes, and reemitted with the kinetic energy of 1 eV. Once the Ps energy reaches 1 eV, it does not enter another grain any more. Then the energy loss of the Ps becomes much slower [12, 13, 14]. This leads to the time dependence of the collisional quenching rates, resulting in time dependence of the total annihilation rate λoPs and subtle and complicated shape of the o-Ps time spec- trum. The time spectrum cannot be represented as a simple exponential decay before the o-Ps is thermalized. Denoting the survival probability of the o-Ps up to time t byD(t) = exp( −/integraltextt 0dt′λoPs(t′)), the time spectrum is given by P(t) =N0(−ǫd dtD(t)) +C =N0[ǫ3γλ3γ+ǫ2γ(λSiO2(t) +λox(t) +λspin(t))]D(t) +C , (5) where ǫ3γandǫ2γare the absolute detection eﬃciencies for the γ-rays from the 3 γannihilations and the 2 γan- nihilations, respectively, and N0andCare constants. In order to determine the total annihilation rate λoPsfor the themalized o-Ps, we ﬁt the lifetime spec- trum(Fig. 2) to a function A exp( −λt) + B, where A and B are constants, and the parameter λis obtained as the start time of the ﬁt t∗is stepped out. The results are plotted by triangles as a function of the start time of the ﬁt in Fig.2. The ﬁtted λgradually decreases, indicating the o-Ps slowing down process, and then becomes ﬂat, indicating that the o-Ps is thermalized. Once the o-Ps is thermalized, the ﬁtted λrepresents the total annihilation rate of the o-Ps averaged over the Maxwell-Boltzmann velocity distribution at the measuring temperature. The λoPsfor the thermalized o-Ps is determined from the value in the region where the ﬁtted values are statisti- cally consistent. We choose t∗= 71ns to yield λoPs=λ3γ+λ∗ SiO2+λ∗ ox+λ∗ spin = 32.5±0.2µs−1, (6) where the quenching rates for the thermalized o-Ps are represented with the superscript∗. The value for λ3γ+λ∗ SiO2is determined similarly from the measure- ment without a gas, to be λoPs=λ3γ+λ∗ SiO2 = 7.41±0.04µs−1. (7) By using λ3γ= 7.040±0.003µs−1[15], the values for λ∗ SiO2andλ∗ ox+λ∗ spinare determined to be, λ∗ SiO2= 0.37±0.04µs−1(8) λ∗ ox+λ∗ spin= 25.1±0.2µs−1. (9)508 509 510 511 512 513 514100101102103104 energy(keV)counts71□ns□-□150□ns FIG. 3: The energy spectrum of the annihilation γ-rays(closed circles) for 1 atm of O 2with the time range from 71 ns to 150 ns. The background has been subtracted. The open circles show the o-Ps self-annihilation component. Figure 3 shows an example of the time-selected energy spectrum for 1 atm of O 2measured by the AMOC. The intensity and the shape of the background spectrum have been estimated from the energy spectrum in the time range from 400 ns to 700 ns and subtracted. Figure 4 shows the Doppler broadening spectra in the time ranges indicated. The spectra have been corrected for the 3 γannihilation component, which is estimated using theoretical 3 γspectrum convoluted with the energy resolution curve, i.e., the shape of the 512 keV peak from 106Ru and normalized to the counts in the energy range from 485 keV to 500 keV, as shown in Fig. 3. The spectra in Fig.4 are ﬁtted to two gaussian func- tions(solid curves). The dashed curves show the broad component. The width of this component was ﬁxed to that of the pickoﬀ component for SiO 2aerogel only, be- cause the change in the width due to the presence of O 2 molecules was not appreciable[8]. The spectra in Fig. 4(a) and (b) include the compo- nents which contributes to the prompt peak; i.e., those from the annihilations of the non-Ps positrons and p- Ps. Hence they are not appropriate to the analysis. The spectrum in Fig. 4(c) represents the 2 γannihilations of the thermalized o-Ps. The pickoﬀ quenching gives the extremely low intensity broad component representing the momentum distribution of the electrons bound in O 2 molecules and those on the SiO 2surfaces. The narrow component results from the spin-conversion quenching and represents the center-of-mass momentum distribu- tion of the p-Ps at the moment of the annihilation after the conversion from o-Ps. A log scale is used for the vertical axis in Fig. 5 to blow up the pickoﬀ quenching component. The ratio of the intensity of the broad component to4 0.51[×105] counts0□ns□-□7.4□ns 0.51[×104] counts7.4□ns□-□14.8□ns 508 509 510 511 512 513 5140.511.52[×103] energy(keV)counts71□ns□-□150□ns(c)(a) (b) FIG. 4: Examples of the time-selected energy spectra of the 511keV γ-rays for 1 atm of O 2. that of the narrow component is Ibroad Inarrow= (3.3±1.3)×10−2. (10) An alternative analysis with the width of the broad com- ponent as a free parameter gives a similar result. The intensity of the 2 γquenching component in the time- selected energy spectrum, Iq(q representing SiO 2, ox, spin), for the time range from t∗to 150 ns is given by Iq=/integraldisplay150ns t∗N0ǫ2γλq(t)D(t)dt =λ∗ q/integraldisplay150ns t∗N0ǫ2γD(t)dt . (11) 508 509 510 511 512 513 514100101102103 energy(keV)counts71□ns□-□150□nsHence we have the relation: Ibroad Inarrow=ISiO2+Iox Ispin=λ∗ SiO2+λ∗ ox λ∗ spin. (12) Combining the results (8) ∼(10), and (12), the quench- ing rates for O 2are obtained as, pickoﬀ quenching rate : λ∗ ox= 0.4±0.3µs−1(13) spin conversion rate : λ∗ spin= 24.7±0.2µs−1.(14) From the relation(2), we conclude that 1Zeﬀ= 0.6±0.4. This value is on the order of magnitude as the other gases [5]. The spin-conversion cross-section σspinis σspin= (1.16±0.01)×10−19cm2. This cross-section is for the elastic conversion process [8 ]. In conclusion, we have studied the 2 γannihilation of the thermalized o-Ps in oxygen by AMOC. The1Zeﬀand the elastic spin-conversion cross-section of the thermal- ized o-Ps are estimated by separating the pickoﬀ quench- ing from the spin-conversion. The1Zeﬀfor O 2is revealed to be on the order of magnitude as the other gases. Acknowledgement We would like to acknowledge Dr. Y. Nagashima and Dr. H. Saito for valuable discussions.5 [1] K. Iwata, R.G. Greaves, T.J. Murphy, M.D. Tinkle, and C.M. Surko, Phys. Rev. A51, 473 (1995). [2] K. Iwata, G.F. Gribakin, R.G. Greaves, C. Kurz, and C.M. Surko, Phys. Rev. A61, 022719 (2000). [3] P.A. Fraser, Adv. Mol. Phys. 4, 63 (1968). [4] T.C. Griﬃth and G.R. Heyland, Phys. Rep. 39, 169 (1978). [5] M. Charlton, Rep. Prog. Phys. 48, 737 (1985). [6] R.A. Ferrell, Phys. Rev. 110, 1355 (1958). [7] M. Deutsch and S. Berko, in Alpha-, Beta- and Gamma- ray Spectroscopy , edited by K. Siegbahn, (North-Holland, Amsterdam, 1965), p.1583. [8] M. Kakimoto et al., J. Phys. B20, L107 (1987); M. Kaki- moto, T. Hyodo, and T. Chang, J. Phys. B23, 589 (1990). [9] T. Chang, G. Yang, and T. Hyodo, in Positron Annihi-lation, edited by Zs. Kajcsos and Cs. Szeles, Materials Science Forum 105-110, 1509 (1992). [10] T. Chang, Y. Wang, C. Chang, and S. Wang, in Positron Annihilation , edited by P.G. Coleman, S.C. Sharma, and L.M. Diana, (North-Holland, Amsterdam, 1982), p. 696. [11] Y. Nagashima et al., Phys. Rev. B 58, 12676 (1998). [12] T. Chang, M. Xu, and X. Zeng, Phys. Lett. A126 , 189 (1987). [13] Y. Nagashima et al., Phys. Rev. A 52, 258 (1995). [14] Y. Nagashima, T. Hyodo, K. Fujiwara, and A. Ichimura, J. Phys. B31, 329 (1998). [15] S. Asai, S. Orito, and N. Shinohara, Phys. Lett. B357 , 475 (1995); S. Asai, Ph.D. thesis, University of Tokyo (1994).
arXiv:physics/0011055 22 Nov 2000 1 Explicitly correlated trial wave functions in Quantum Monte Carlo calculations of excited states of Be and Be- Luca Bertinia, Massimo Mellaa†, Dario Bressaninib‡, and Gabriele Morosib§ a) Dipartimento di Chimica Fisica ed Elettrochimica, Universita` di Milano, via Golgi 19, 20133 Milano, Italy. b) Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita’ dell’Insubria, via Lucini 3, 22100 Como, Italy E-mail: bert@csrsrc.mi.cnr.it †E-mail: Massimo.Mella@unimi.it ‡E-mail: Dario.Bressanini@uninsubria.it §E-mail: Gabriele.Morosi@uninsubria.it2Abstract We present a new form of explicitly correlated wave function whose parameters are mainly linear, to circumvent the problem of the optimization of a large number of non-linear parameters usually encountered with basis sets of explicitly correlated wave functions. With this trial wave function we succeeded in minimizing the energy instead of the variance of the local energy, as is more common in quantum Monte Carlo methods. We applied this wave function to the calculation of the energies of Be 3P (1s22p2) and Be- 4So (1s22p3) by variational and diffusion Monte Carlo methods. The results compare favorably with those obtained by different types of explicitly correlated trial wave functions already described in the literature. The energies obtained are improved with respect to the best variational ones found in literature, and within one standard deviation from the estimated non-relativistic limits.3Introduction The description of the electron correlation plays a central role in highly accurate quantum chemistry calculations. Mean-field methods give a qualitative description for many atomic and molecular systems, but in order to get quantitative results the instantaneous correlation between electrons must be taken into account. The most common way to include correlation is, starting from the Hartree-Fock picture, to approximate the exact wave function using MC-SCF or CI expansions. Unfortunately methods based on the orbital approximation converge very slowly to the non-relativistic limit. The reason is that these wave functions include the interelectronic distances only in an implicit form. Furthermore this implicit dependence is quadratic instead of linear, so the cusp conditions [1] of the exact wave functions are reproduced only for infinite expansions. A very efficient and effective approach to accurately describe the local behavior of the wave function when two electrons collide is the explicit inclusion of the interelectronic distances into an approximate wave function. Hylleraas [2], Pekeris [3], James and Coolidge [4], and Kolos and Wolniewicz [5-7] showed how to obtain very accurate results for two electron systems by including the interelectronic distance into the wave function. An alternative possibility is the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates [8, 9]. Unfortunately it is not easy to generalize these methods to many-electron systems since the resulting integrals are extremely difficult to evaluate analytically. Beyond four electron systems, with at most two nuclei, the analytical approach becomes almost unfeasible [10, 11]. Instead of computing the integrals analytically, one could resort to a numerical method. The variational Monte Carlo (VMC) method [12, 13] is a very powerful numerical technique that estimates the energy, and all the desired properties, of a given trial wave function without any need to analytically compute the matrix elements. For this reason it poses no restriction on the functional form of the trial wave function, requiring only the evaluation of the wave function value, its gradient and its Laplacian, and these are easily computed. Using the VMC algorithm, essentially a stochastic numerical integration scheme, the expectation value of the energy for any form of the trial wave function can be estimated by averaging the local energy )(/)(Hˆ T T RRΨΨ over an ensemble of configurations distributed as 2 TΨ, sampled during a random walk in the configuration space using Metropolis [14] or Langevin algorithms [15]. The fluctuations of the local energy depend on the quality of the function ΨT, and they are zero if the4exact wave function is used (zero variance principle). VMC can also be used to optimize the trial wave function ΨT, and we refer the reader to the literature for the technical details. A popular and effective approach to building compact explicitly correlated wave functions is to multiply a determinantal wave function by a correlation factor, the most commonly used being a Jastrow factor [16]. The inclusion of the Jastrow factor does not allow the analytical evaluation of the integrals, so the use of VMC is mandatory. However, departing from the usual determinantal wave function form can be very fruitful [11], allowing an accurate and, at the same time, compact description of atomic and molecular systems. Very few terms are needed to reach a good accuracy, in comparison to more common wave function forms. The recovery of the remaining correlation energy can be done using the diffusion Monte Carlo (DMC) method. Since this method is already well described in the literature, we refer the reader to the available reviews [12, 13]. We only recall here that in this method the exact, but unknown, wave function is interpreted as a probability density. In the fixed-node (FN) approximation [17] the nodal surfaces of the trial wave function ΨT are used to partition the space and within each region the wave function can be safely interpreted as a probability density. It can be shown that the FN-DMC energies are an upper bound to the exact ground state energy. This paper is part of an ongoing project in our laboratory to develop accurate and compact wave functions for few-electron systems. In our previous works [11, 18, 19] we used linear expansions of explicitly correlated wave functions for calculations on the ground state of few-electron systems. In all cases good VMC energies were obtained, both in infinite nuclear mass approximation calculations and non- adiabatic calculations. In particular we used a linear expansion of explicitly correlated exponential functions to develop accurate wave functions for two test systems: the beryllium atom and the lithium hydride molecule in their ground state. Here we present a new form of explicitly correlated wave function and we use VMC to extend the application of correlated trial wave function to excited states and five electron systems. Furthermore we use DMC to approximate the exact energies and compare them with the estimated non-relativistic limits. We choose the Be 3P (1s22p2) and Be- 4So (1s22p3) states which are involved in Beryllium electron affinity determination. We compare VMC and DMC energies and variances of the energy and examine the nodal properties of the trial wave functions comparing FN-DMC results with the best variational calculations and the non-relativistic limits estimated by Chung and coworkers [20, 21].5Explicitly Correlated functional form For an N electron atomic system we write an explicitly correlated trial wave function [11] as { }N MSSgfA,)(ˆ Θ =Ψ φr (1) In this equation Aˆ is the antisymmetrizer operator, φ is a function of all the electron- nucleus distances and g is a function of all the electron-electron distances called correlation factor. Both functions include variational parameters. ΘSMN S, is an eigenfunction of the spin operators 2Sˆ and zSˆ of the correct spin multiplicity. The functions φ and g, being dependent only on interparticle distances, are rotationally invariant. This means that their product can describe only S states, with zero angular momentum. To describe higher angular momentum states, it is necessary to include a function f(r) with the correct rotational symmetry. f(r) is a function of the Cartesian electronic coordinates (x,y,z), but might include also the electron-nucleus distances [11]. This Ψ function might be generalized including products of the interparticle distances, that is Ψ is the two-body term of a many-body expansion of the wave function. It is possible to further generalize the wave function by taking linear combinations of such terms. To assure a high quality wave function it is particularly important that the function Ψ satisfy the cusp conditions [1], representing the behavior of the exact wave function at the coalescence of two particles. It is also important to take into account the asymptotic conditions [22], which represent the behavior when one of the particles go to infinity. The first type of functional form we examined is generated assuming a Pade’ factor [ ])1()(exp2cr brar + + for the electron-nucleus part φ and a Jastrow factor [ ])r'c1(r'aexp + for the interelectronic part g.   Θ     +    ++=Ψ ∑ ∑ <e sn MS ji ijij iii rcra icrbrarfA,2 '1'exp1exp)(ˆr /G03 /G0B/G15/G0C In the following this wave function will be called Pade'-Jastrow. The Pade’ factor is a good choice for the electron-nucleus part, because it is the best compromise between flexibility and small number of parameters. In fact this function goes as are for 0r→and ()rcbefor ∞→r. So with different exponents it can accommodate both the coalescence at the nucleus and the decay for large r. It is also important to point out that this factor can accurately describe both 1s and 2s orbitals as we have shown in our previous work [11]. The main problem with linear expansions of explicitly correlated trial wave functions is the huge number of non-linear parameters to optimize. In our previous work [11] for more6sophisticated factors like Pade’ or Jastrow we succeeded in optimizing trial wave functions including a maximum of two terms. To overcome this problem we choose a second type of functional form, similar to the first one:   Θ     +    ++=Ψ ∑ ∑ <e sn MS ji ijij i iii rcra crbrarGfA,2 '1'exp1exp)()(ˆ rr (3) We limit the expansion to a single term and so we have few non-linear parameters to optimize. However to add extra flexibility to the wave function we introduce a pre-exponential factor G(r) written as a sum of powers of interparticle distances weighted by linear parameters: .... )( ∑∑∑ ∑∑∑ ∑∑∑∑ + + + + = > < sm jn i ijs ro ijn i iijr p qn ij jiqn i ipss rr q prrg rrg rg rg Gr /G03 /G0B/G17/G0C In the following we will call Eq. 3 pre-exponential wave function (prex). Even if this kind of wave function allows us to reduce the effort for the optimization of the parameters, we recall that the CPU time needed to evaluate explicitly correlated trial wave functions is very large and proportional to the number of permutations generated by the antisymmetrizer. No matter the form we choose for explicitly correlated wave functions, they are limited to few electron systems. In this paper we also compare Pade'-Jastrow and pre-exponential wave functions with a more standard form, widely used in QMC calculations [23, 24], that is the product of a determinantal function times the Schmidt-Moskowitz [25] correlation factor (SM): ∑ ∑    =Ψ <↓↑ l jiijjiij lll rrrU DetDetC ),( exp,(5) ↑Det /G03and /G03↓Det /G03are the determinants for α and /G03β /G03electrons. The function U for atoms is given by ∑ + Δ= ko ijn im jn jm ikkk ijkkk kk r)rrrr(c)n,m( U (6) where ck are trial parameters and r = ar/(1+br). The determinants are generated from ab initio calculations, in general SCF or MCSCF calculations, for a given basis set. Then the correlation factor is added, and its variational parameters optimized using VMC calculations.7Optimization of the trial wave functions Our previous work [11] showed that departing from the usual determinantal wave function form can be very fruitful, allowing to write very compact and at the same time very accurate wave functions. However it is computationally much more demanding and for this reason special care must be given to the design of an efficient way of generating and optimizing the trial wave function. These steps must be implemented in the most effective, fast and efficient way. The standard way to optimize a trial wave function using VMC is to minimize the variance of the local energy using a fixed sample of walkers; a method proposed by Frost [26] and Conroy [27] and described in detail by Umrigar, Wilson, and Wilkins [28] and by Mushinsky and Nightingale [29]. This has been proved to be numerically much more stable than the energy minimization. For our trial wave functions we have found very effective the minimization of the variance of the energy 22 2HˆHˆ)Hˆ( −=σ (7) or, even better, of the second moment with respect to an arbitrary parameter ER, µ(ER): ()()()2 R22 R R EHˆHˆ EHˆ)E( −+σ=−=µ (8) where the parameter ER can be set equal to the exact energy of the system E0. Both)H(σ and µ(E0) go to zero as 0 TΨ→Ψ , where 0Ψ is the exact eigenfunction: their values for a given trial wave function ΨT can be used to evaluate the quality of the trial wave function ΨT. We used µ(ER) as cost function for the optimization of both Pade’-Jastrow and SCF-MS wave functions. As to Eq. 2, the optimization of the first term of the expansion is usually performed starting from a trial wave function with a reasonable electron-electron Jastrow factor, and with the electron-nucleus functions coming from some standard Slater orbital basis set, or from small basis sets optimized at the SCF level. As we showed in our previous work [11], it is possible to build a trial wave function as a linear expansion of n terms by adding an extra term to an optimized n-1 term wave function. This procedure worked well for two and three electron systems with simple exponential basis sets, but not for more sophisticated Pade’ and Jastrow basis sets. For these reasons in this work we optimized only one term Pade’-Jastrow functions. Let us now consider the case of the pre-exponential trial wave function. The function in Eq. 3 can be written as a linear combination: ∑ =Φ=Ψ 1lllg (9)8where the term lΦ is given by [])(........ pΦ =Φm ijp ilrr (10) )(pΦ is a single term Pade’-Jastrow function whose parameters p are optimized minimizing µ(ER) in a preliminary step, and then are fixed during the optimization of the linear parameters. As to the linear parameters, we succeeded in minimizing the energy instead of the variance of the local energy or the second moment µ(ER), so we could choose the best linear parameters according to the observable we are interested in. The standard linear variational methods requires the solution of the secular problem, and so the calculation of the matrix elements ∫∫ ΦΦ=ΦΦ= RR d SdH H ji ijji ijˆ (11) These integrals are evaluated during a VMC simulation. We show the main features of this pre-exponential trial wave function using as benchmarks the Be and Li ground states. In table 1 we compare the energy for Be 1S0 ground state obtained by the one term Pade’- Jastrow function with the value calculated by the function obtained adding a pre-exponential including all the electron-nucleus and electron-electron distances and their products, a total of 66 terms. The two values evidence a large gain of correlation energy for the addition of the pre- exponential factor and further improvement is obtained adding 20 more terms, that is the third and fourth powers of the electron-nucleus and electron-electron distances, to the linear expansion. So the pre-exponential factor adds flexibility to the wave function in a very efficient way. To examine the relative efficiency of linear and non linear parameters in adding flexibility to the wave function, for the Li ground state (see table 2) we compare the energy of a 28-term pre-exponential function with the result of a 8-term expansion of explicitly correlated exponential functions, a simplified form of the Pade'-Jastrow wave function in which both the factor φ and the correlation factor g are in the form exp(cr). These two trial wave functions give a similar gain of correlation energy, but in the pre-exponential case there are 12 non-linear and 28 linear parameters, while the 8 term expansion of correlated exponential functions includes 48 non-linear and 8 linear parameters. In spite of the smaller number of parameters the pre-exponential function gives a better result, and its optimization process was much easier and faster. We optimized the linear parameters also by minimization of µ(ER). The calculated energy at VMC level is worse than the one obtained by minimization of the energy: this is obviously related to the different minima of the energy and µ(ER).9Results and discussion For Be 3P (1s22p2) and Be− 4So (1s22p3) we computed SCF and CASSCF trial wave functions using GAMESS with the Slater orbital basis sets reported in table 3. Each orbital was fitted with 6 Gaussian functions. We optimized the Schmidt-Moskowitz correlation factor of the SCF-SM functions and the non-linear parameters of the Pade’-Jastrow and pre-exponential functions minimizing the variance of the local energy. Beside the VMC energies, we report the variance of the local energy, given by Eq. 7, estimated using VMC. DMC energies were obtained by a linear fit of the energy at three time steps (τ=5, 3, and 1 mhartree-1) and extrapolation to τ=0 mhartree-1. We compare our results with the best variational energies obtained by Chung and coworkers [20, 21], who used linear expansions of Slater orbitals in the L-S-coupling scheme. Be 3P (1s22p2) The calculations for the excited state Be 3P (1s22p2) were carried out with the following three trial wave functions: 1) a single determinantal function times a nine term Schmidt-Moskowitz correlation factor; 2) a one term Pade’-Jastrow function; 3) a pre-exponential function with 33 terms. The spin eigenfunction used for the Pade’-Jastrow and pre-exponential function is ) (4 1,1 βααααβαα− =Θ . The pre-factor f(r) that defines the state symmetry is 3443)( yxyxrf −= (12) The results are reported in table 4. As to the determinantal wave function, we used only a single determinant, as the energy lowers by only 0.003 hartree on going from the SCF to a CASSCF function for two electrons in an active space of two p shells (15 configurations, the highest weight of the first double excitation being equal to 0.04). The explicit inclusion of the interelectronic distances in the wave function by the SM factor results in a large improvement of the quality of the wave function, as shown from the lowering of the energy (0.043 hartree) and the variance of the energy (almost four times smaller) on going from SCF to SCF-SM/VMC.10A further improvement of the trial wave function at VMC level is found using more sophisticated functional forms [11], like the Pade’-Jastrow and the pre-exponential ones. For the pre-exponential function we used a 33 term expansion, including all ri, ri2, and the products ririj. In particular in the case of the pre-exponential function we were able to optimize the linear parameters of G(r) minimizing the energy, not the variance of the local energy, obtaining an energy 0.9 mhartree higher than the best variational one. We also notice that this wave function is very compact with 45 (12 non-linear and 33 linear) variational parameters on the whole. At DMC level already the SCF-SM wave function gives a lower energy than the best variational value, and 0.2 mhartree higher than the estimated non-relativistic limit (NRL). It means that the nodal surfaces of this function are fairly good, at variance with the SCF-SM trial wave function for the Be ground state, whose energy is 11 mhartree higher than the NRL [30]. This large nodal error is due to the strong contribution of the first double excitation in improving the quality of the nodal surfaces, because of the quasi-degeneracy of the 2s and 2p orbital. DMC energies for Pade'-Jastrow and pre-exponential functions have the estimated NRL within one standard deviation, that is the nodal surfaces of these wave functions are correct and better than the SCF ones. Be- 4So (1s2 2p3) The calculations for the excited state Be- 4So were carried out with these four trial wave functions: 1) a single determinantal function times a nine term Moskowitz-Schmidt correlation factor; 2) a multideterminantal function times a nine term Moskowitz-Schmidt correlation factor; 3) a one term Pade’-Jastrow function; 4) a pre-exponential function G(r) with 61 terms. The CASSCF wave function for three electrons in an active space of two p shells includes 20 configurations. The first two highest weights, relative to the first double and single excited configurations, are equal to 0.125 and 0.03 and indicate a more marked multiconfigurational character of the wave function. As we have seen for Be 3P state, the gain in energy and the lowering of the variance of the energy between SCF and SCF-SM are very large, while between SCF-SM and CASSCF-SM they are an order of magnitude less. The spin eigenfunction used for the Pade’-Jastrow and pre-exponential function is ) (5 2/3,2/3 βαααα αβααα− =Θ . The pre-factor f(r) that defines the state symmetry is f rxyzxyzxyzxyzxyzxyz ()= + + − − −345534453354543435 (13)11The results are reported is table 5. For Be- using the Pade’ and pre-exponential functions we obtained better energies and variances of the energy than the SCF-SM and CASSCF-SM ones, as already seen in the Be 3P case. In particular, with the pre-exponential function with 61 linear parameters G(r) (all ri, ri2 and the products ririj), the VMC energy is 0.9 mhartree higher than the best variational energy. From DMC simulations we see clearly that the nodal surfaces don’t change on going from the SCF to the CASSCF trial wave function and in both cases we have around 0.4 mhartree of nodal error. Pade'-Jastrow and pre-exponential functions have better nodal surfaces and their DMC energies have the estimated NRL within one standard deviation. Conclusions We have used explicitly correlated functional forms to improve the quality of the trial wave functions usually adopted to calculate the energy of a system. For the two excited states of Be and Be- we obtained better non-relativistic energies with very compact trial wave functions compared to the best variational results. Using a suitable pre-exponential factor we were able to improve the flexibility of the trial wave function without including too many non-linear parameters: this kind of trial wave function allowed us to minimize directly the energy instead of the variance of the local energy. As to the computational time, the optimization of the Be- five electron trial wave function and the VMC calculation required around a week on a modern PC .It is not possible to compare this CPU time with calculations by correlated Gaussians as at present they are limited to four electron systems. Our DMC energies are in good agreement with the estimated NRL obtained by Chung and coworkers [20, 21]. From our best values for Be 3P (-14.39547(5) hartree) and Be− 4So (-14.40620(6) hartree) we compute an electron affinity of 0.01073(8) hartree = 292(2) meV, within two standard deviations from the experimental value 295.49(25) meV [31] and the theoretical value 295.0(7) calculated by Hsu and Chung [20]. A significant comparison would require the reduction of the calculated standard deviation by one order of magnitude. Acknowledgements CPU time for this work has been partially granted by the Centro CNR per lo studio delle relazioni tra struttura e reattività chimica, Milano.1213Table 1. VMC results for Be ground state. VMC Energy (hartree) % Correlation energy one term Pade’-Jastrow -14.6528(2) 84.57 66 term prex -14.6633(3) 95.70 86 term prex -14.6651(2) 97.60 HF limit -14.57302 NR limit -14.6673514Table 2. VMC results for Li ground state. VMC Energy (hartree) % Correlation energy 8 term exp. -7.4775(2) 98.29 28 term prex -7.47770(8) 99.20 HF limit -7.43274 NR limit -7.4780615Table 3. Basis sets for SCF and CASSCF calculations SystemBe 3P Be- 4So 1s 5.7 5 .7 1s 4.2 4 .2 2s 4.3 4 .3 2s 2.4 2 .4 2p 1 .65 1 .65 2p 0 .76 0 .76 2p 0 .37616Table 4. Be 3P (1s22p2) energies and VMC variances of the energy Method Energy (hartree)σVMC(H) SCF -14.3340 1 .68(2) SCF-SM/VMC -14.3769(2) 0.48(1) Pade’/VMC -14.3930(1) 0.27(1) 33 term prex /VMC-14.3942(1) 0.22(1) Best variational-14.3951086 SCF-SM/DMC -14.39521(5) Pade’/DMC -14.39541(7) 33 term prex /DMC-14.39547(5) Estimated LNR -14.395440417Table 5: Be 4So (1s22p3) energies and VMC variances of the energy Method Energy (hartree) σVMC(H) SCF -14.326976 1 .68(1) CASSCF -14.334010 1 .68(1) SCF-SM/VMC -14.3769(2) 0.48(1) CASSCF-/VMC -14.3836(1) 0.48(1) Pade’/VMC -14.4031(2) 0.29(1) 61 term prex/VMC-14.4051(2) 0.21(1) SCF-SM/DMC -14.40594(8) CASSCF-SM/DMC-14.40597(7) Best variational -14.4060320 Pade’/DMC -14.40620(4) 61 term prex /DMC-14.40620(6) Estimated LNR -14.406282(26)18References [1]Kato T 1957 Commun. Pure Appl. Math. 10, 151. [2]Hylleraas E A 1929 Z. Phys. 54, 347. [3]Pekeris C L 1958 Phys. Rev. 112, 1649. [4]James H M and Coolidge A S 1933 J. Chem. Phys. 1, 825. 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[18]Bressanini D, Mella M and Morosi G 1995 Chem. Phys. Lett. 240, 566. [19]Bressanini D, Mella M and Morosi G 1997 Chem. Phys. Lett. 272, 370. [20]Hsu J-J and Chung K T 1995 Phys. Rev. A 52, R898. [21]Zhu X W and Chung K T 1995 Phys. Scr. 52, 654. [22]Patil S H, Tang K T and Toennies J P 1999 J. Chem. Phys. 111, 7278. [23]Grossman J C, Mitas L and Raghavachari K 1995 Phys. Rev. Lett. 75, 3870. [24]Filippi C and Umrigar C J 1996 J. Chem. Phys. 105, 213. [25]Schmidt K E and Moskowitz J W 1990 J. Chem. Phys. 93, 4172. [26]Frost A A 1964 J. Chem. Phys. 41, 478. [27]Conroy H 1964 J. Chem. Phys. 41, 1327. [28]Umrigar C J, Wilson K G and Wilkins J W 1988 Phys. Rev. Lett. 60, 1719.19[29]Mushinski A and Nightingale M P 1994 J. Chem. Phys. 101, 8831. [30]Umrigar C J, Nightingale M P and Runge K J 1993 J. Chem. Phys. 99, 2865. [31]Kristensen P, Petrunin V V, Andersen H H and Andersen T 1995 Phys. Rev. A 52, R2508.
arXiv:physics/0011058v1 [physics.chem-ph] 23 Nov 2000Generalized Heitler–London Theory for H 3: A Comparison of the Surface Integral Method with Perturbation Theory Tanja I. Sachse Max-Planck-Institut f¨ ur Str¨ omungsforschung, Bunsenst r.10, D-37073 G¨ ottingen, Germany Ulrich Kleinekath¨ ofer Institut f¨ ur Physik, Technische Universit¨ at, D-09107 Ch emnitz, Germany (July 21, 2013) Abstract The generalized Heitler–London (GHL) theory provides a str aightforward way to express the potential energy surface of H 3in terms of Coulomb and exchange energies which can be calculated either by perturb ation theory or us- ing the surface integral method (SIM). By applying the Rayle igh–Schr¨ odinger perturbation theory, GHL theory for the quartet spin state o f H3is shown to yield results equivalent to the symmetrized Rayleigh–Sc hr¨ odinger version of symmetry adapted perturbation theory (SAPT). This equiv alence allows a comparison with the corresponding results obtained by the surface integral method. The surface integral result calculated with a produ ct of atomic wave functions is found to have certain advantages over the pertu rbation approach. 1I. INTRODUCTION The generalized Heitler–London (GHL) theory provides a use ful framework to calculate the potential energy surfaces for polyatomic systems [1–4] . Since the potential energy is expressed in terms of Coulomb and exchange energies it is pos sible to systematically separate out many–body eﬀects in every single term contributing to th e potential energy. In this paper some aspects of the three–body exchange eﬀects occurring in H3are examined in more detail. Axilrod, Teller and Muto [5] were the ﬁrst to suggest a formul a describing the leading long range three–body dispersion term for three sphericall y symmetric atoms. Since then the non–additive eﬀects have been intensively studied and s everal review articles have been published [6–8]. In the GHL approach the potentials can be de composed into Coulomb and exchange energies , whereas in symmetry adapted perturbation theory (SAPT) th ese interactions are expressed in terms of Coulomb and exchange integrals in the manner ﬁrst introduced by Heitler and London. Recently, SAPT was formul ated for the interactions of trimers [9] and has been applied to numerical calculations u p to third order for the quartet spin state of H 3[10] and for the helium–trimer [11] up to third order. Other t hree–body calculations for H 3are based on Heitler–London type calculations [12] and pert urbation calculations making use of Uns¨ old approximations [13]. In the former the splitting into Coulomb and exchange part is as pointed out by the author hims elf not completely rigorous. In a previous paper [3] analytical results were reported for the doublet as well as for the quartet spin state for the H 3system based on the GHL theory. Two kinds of exchange energies appear: cyclic exchange energies, where all three electrons are involved, and two– body exchange energies in the presence of the respective thi rd atom. The cyclic exchange energy of three hydrogen and three helium atoms [14] was calc ulated using the surface integral method (SIM) which was previously applied to two at oms [1, 2, 4, 15–17]. In a forthcoming paper [18] it will be demonstrated that all exch ange energies occurring in the H3–system can be calculated either by the surface integral met hod or by using perturbation theory, and the corresponding results for the implicit thre e–body eﬀect on the two–body exchange energies will be derived and compared. For H 2it was previously shown that SAPT and GHL are equivalent [19] . The purpose of this paper is to compare the surface integral method calcula tions of the three–body eﬀects in the exchange energies based on an atomic product wave functi on with the results of ﬁrst to third order of SAPT which are only available for the quartet s pin state of H 3[10]. In order to perform this comparison it is necessary to ﬁrst prove that the SAPT and GHL theory expressions for the energy of the quartet state are equivale nt. The results reveal that with the zeroth order wave function the surface integral result c ontains parts of the second order SAPT result and is therefore more eﬃcient. In Sections II and III the basic ideas of the GHL theory and pol arization approximation are described. In Section IV the equivalence of the GHL and th e symmetrized Rayleigh– Schr¨ odinger (SRS) theories is demonstrated order by order . The latter is designated a weak symmetry forcing SAPT. Section V reviews the surface integr al method (SIM). Thereafter in Section VI the advantages of SIM over the perturbation app roach will be demonstrated by comparing the numerical results of perturbation theory a nd SIM. 2II. GENERALIZED HEITLER–LONDON THEORY FOR H 3 The application of generalized Heitler–London theory to H 3was previously discussed in Ref. [3]. The generalized Heitler–London equation is given by ˆHF =/summationdisplay gǫgˆT(g)F (1) whereFis the localized, i.e. non–symmetrized wave function, ˆT(g) designates a permutation operator for the electron coordinates, and ǫgstands for the Coulomb ( g=I) and exchange energies (g/negationslash=I). Applying results from the theory of the symmetric group, t he energy eigenvalues of the Hamiltonian can be derived. For the H 3–system, the result for the two doublet states is 1/2EGHL=ǫI−ǫ123±/radicalBigg 1 2[(ǫ12−ǫ23)2+ (ǫ23−ǫ13)2+ (ǫ13−ǫ12)2] (2) and for the quartet state 3/2EGHL=ǫI−ǫ12−ǫ23−ǫ13+ 2ǫ123. (3) The remainder of this paper will be concerned only with the qu artet state. III. POLARIZATION APPROXIMATION AND GENERALIZED HEITLER–LONDON (GHL) THEORY The Born–Oppenheimer non–relativistic Hamiltonian of the three–body system is given by ˆH=ˆH0+ˆV (4) using ˆH0=ˆH0 A+ˆH0 B+ˆH0 C (5) ˆV=ˆVAB+ˆVBC+ˆVAC (6) where ˆH0 A,ˆH0 Band ˆH0 Care the Hamiltonians of three free hydrogen atoms and ˆVAB,ˆVBC and ˆVACdescribe the interaction between atoms AandB,BandC, as well as AandC, respectively. The polarization approximation [20] is base d on the equation ˆHF =EpF (7) where the polarization wave function Fand the polarization energy Epcan be written as perturbation series F=/summationdisplay φn, (8) Ep=/summationdisplay ǫn. (9) 3The zeroth order polarization wave function φ0is the eigenfunction of the free Hamilto- nian ˆH0and thus is a product of three free hydrogen wave functions. S tarting from the GHL equation with Fchosen as the polarization wave function, Eq. (1) together w ith the Hamiltonian Eq. (4) can be written as (ˆH0+ˆV)\|/summationdisplay nφn/angb∇acket∇ight=/summationdisplay gǫgˆT(g)\|N/summationdisplay n=0φn/angb∇acket∇ight. (10) Forming scalar products with ˆT(g)φ0for each group element g (ˆT(g)φ0,(ˆH0+ˆV)/summationdisplay n=0φn) =/summationdisplay g′ǫg′(ˆT(g)φ0,/summationdisplay n=0ˆT(g′)φn) (11) a system of linear equations can be derived for the Coulomb en ergyǫIas well as for the exchange energies ǫg(g/negationslash=I) in terms of Coulomb integrals J, exchange integrals Kg, and overlap integrals Sg: E0+J≈ǫI+/summationtext g′/negationslash=gǫg′Sg′−1:g=I E0Sg+Kg≈ǫg+/summationtext g′/negationslash=gǫg′Sg′−1g:g/negationslash=I. (12) The following notation for the nth order overlap, Coulomb and exchange integrals was used: Sg:=M/summationdisplay n=0Sn g (13) J:=M/summationdisplay n=0Jn(14) Kg:=M/summationdisplay n=0Kn g=M/summationdisplay n=1Kn g, (15) where Sn g:= (ˆT(g)φ0, φn) (16) Jn:= (φ0,ˆV φn−1) (17) J0=E0 (18) Kn g:= (φ0,ˆVˆT(g−1)φn−1). (19) The equalities Sn g−1=Sn gandKn g−1=Kn ghold. In Ref. [18] it will be shown how the Coulomb and exchange energies can be expressed in terms of Coulomb, e xchange and overlap integrals and how the order–by–order contributions to the Coulomb and exchange energies can be found. The convergence properties of the polarization theory have been extensively discussed for the case of two hydrogen atoms [21]. For low orders it was s hown that the perturbation series rapidly converges to the Coulomb energy [19, 21–23] t hough this is not the limit for the inﬁnite order expansion. It is assumed that the behavior of this perturbation theory for a system of two atoms also roughly holds in the case of three at oms [9, 10]. Since here we are only interested in low orders, especially the ﬁrst, this expected behavior justiﬁes approx- imating the localized wave function via the polarization ap proximation for three hydrogen atoms as well. 4IV. EQUIVALENCE OF THE GHL AND SRS THEORY FOR QUARTET H 3 In this section the order–by–order equivalence of the compl ete energy expressions ob- tained by using either the GHL or the SRS theory will be demons trated. Both the GHL and SRS theories start with the Hamiltonian Eq. (4) and a zeroth o rder wave function which is a product of three free hydrogen atom wave functions. To demo nstrate the equivalence of the ﬁrst order expressions the ﬁrst order SRS term will be exp ressed in terms of Coulomb and exchange energies. In Eq. (12) of Ref. [10] this term is gi ven by 3/2E1 SRS=N0/bracketleftBigg <ψ0\|ˆV(1−ˆT(12)−ˆT(23)−ˆT(13) + ˆT(123) + ˆT(132)) \|ψ0>/bracketrightBigg ,(20) which can be expressed with Eqs. (16) to (19) as 3/2E1 SRS=N0/bracketleftBigg J1−K1 12−K1 23−K1 13+K1 123+K1 132/bracketrightBigg , (21) where N0= 1−S0 12−S0 23−S0 13+S0 123+S0 132. (22) With Eq. (12) it is possible to express the ﬁrst order contrib utions as J1=ǫ1 I+ǫ1 12S0 12+ǫ1 23S0 23+ǫ1 13S0 13+ǫ1 123S0 123+ǫ1 132S0 123 (23) K1 12=ǫ1 12+ǫ1 IS0 12+ǫ1 23S0 123+ǫ1 13S0 123+ǫ1 123S0 23+ǫ1 132S0 13 (24) K1 23=ǫ1 23+ǫ1 IS0 23+ǫ1 12S0 123+ǫ1 13S0 123+ǫ1 123S0 13+ǫ1 132S0 12 (25) K1 13=ǫ1 13+ǫ1 IS0 13+ǫ1 12S0 123+ǫ1 23S0 123+ǫ1 123S0 12+ǫ1 132S0 23 (26) K1 123=ǫ1 123+ǫ1 IS0 123+ǫ1 12S0 23+ǫ1 23S0 13+ǫ1 13S0 12+ǫ1 132S0 123 (27) K1 132=ǫ1 132+ǫ1 IS0 123+ǫ1 12S0 13+ǫ1 23S0 12+ǫ1 13S0 23+ǫ1 123S0 123 (28) On inserting into Eq. (21) many terms cancel and Eq. (21) is eq uivalent to the ﬁrst order contribution to Eq. (3) 3/2E1 SRS=N0/bracketleftBigg J1−K1 12−K1 23−K1 13+K1 123+K1 132/bracketrightBigg =ǫ1 I−ǫ1 12−ǫ1 23−ǫ1 13+ǫ1 123+ǫ1 132=3/2E1 GHL. (29) The rest of the proof will be done by complete induction. The c laim of the induction is the equivalence of the GHL and SRS energy expressions up to nth order. From Eq. (12) of [10] the general nth–order expression for the interaction energy in SRS theor y is found to be 3/2En SRS=N0/bracketleftBigg <ψ0\|ˆV(1−ˆT(12)−ˆT(23)−ˆT(13) + ˆT(123) + ˆT(132)) \|ψ(n−1) pol> −n−1/summationdisplay k=13/2Ek SRS<ψ0\|(1−ˆT(12)−ˆT(23)−ˆT(13) + ˆT(123) + ˆT(132)) \|ψ(n−k) pol>/bracketrightBigg =N0/bracketleftBigg Jn−Kn 12−Kn 23−Kn 13+Kn 123+Kn 132 −n−1/summationdisplay k=13/2Ek SRS(−Sn−k 12−Sn−k 23−Sn−k 13+Sn−k 123+Sn−k 132)/bracketrightBigg (30) 5whereN0is given by Eq. (22). Thus it is necessary to prove that 3/2En GHL=ǫn I−ǫn 12−ǫn 23−ǫn 13+ǫn 123+ǫn 132 (31) =3/2En SRS. (32) To perform a proof by induction it is necessary to show that al so the (n+1)st order terms of both theories are equal. To do so, the ( n+1)st order of GHL theory is expressed in terms of the quantities occurring in SRS theory. This can be achiev ed by inserting the solutions of the set of linear equations Eq. (12) into the complete GHL e nergy for the H 3–quartet state [24] 3/2EGHL=ǫI−ǫ12−ǫ23−ǫ13+ǫ123+ǫ132 (33) ≈M/summationdisplay n=03/2En GHL =M/summationdisplay n=0/bracketleftBig ǫn I−ǫn 12−ǫn 23−ǫn 13+ǫn 123+ǫn 132/bracketrightBig =E0+/bracketleftBig J−K12−K23−K13+K123+K132/bracketrightBig /bracketleftBig 1−S12−S23−S13+S123+S132/bracketrightBig−1(34) whereJ,Kg, andSghave been deﬁned in Eqs. (13) to (15). To ﬁnd the expression fo r the (n+ 1)st order contribution to the energy of the quartet state, the left hand side is ﬁrst multiplied by the denominator /parenleftBigM/summationdisplay n=03/2En GHL/parenrightBig /bracketleftBig 1−M/summationdisplay n=0(Sn 12+Sn 23+Sn 13) +M/summationdisplay n=0(Sn 123+Sn 132)/bracketrightBig =E0/bracketleftBig 1−M/summationdisplay n=0(Sn 12+Sn 23+Sn 13) +M/summationdisplay n=0(Sn 123+Sn 132)/bracketrightBig +M/summationdisplay n=0[Jn−Kn 12−Kn 23−Kn 13+Kn 123+Kn 132]. (35) Collecting terms of ( n+ 1)st order leads to 3/2En+1 GHL(1−S0 12−S0 23−S0 13+S0 123+S0 132) =Jn+1−Kn+1 12−Kn+1 23−Kn+1 13+Kn+1 123+Kn+1 132 +E0(−Sn+1 12−Sn+1 23−Sn+1 13+Sn+1 123+Sn+1 132) −n/summationdisplay k=03/2Ek GHL(−Sn+1−k 12 −Sn+1−k 23 −Sn+1−k 13 +Sn+1−k 123 +Sn+1−k 132 ) (36) with the result that 3/2En+1 GHL=N0/bracketleftBigg Jn+1−Kn+1 12−Kn+1 23−Kn+1 13+Kn+1 123+Kn+1 132 −n/summationdisplay k=1EGHL,k 3/2(−Sn+1−k 12 −Sn+1−k 23 −Sn+1−k 13 +Sn+1−k 123 +Sn+1−k 132 )/bracketrightBigg . (37) Using the claim of the proof, which stated that for all orders up to thenth the GHL term is equal to the SRS–term, EGHL,k 3/2in the last line can be replaced by3/2E(n+1) SRS for all orders 1,...,n . Thus Eq. (37) can be transformed into 63/2En+1 GHL=N0/bracketleftBigg Jn+1−Kn+1 12−Kn+1 23−Kn+1 13+Kn+1 123+Kn+1 132 −n/summationdisplay k=13/2Ek SRS(−Sn+1−k 12 −Sn+1−k 23 −Sn+1−k 13 +Sn+1−k 123 +Sn+1−k 132 )/bracketrightBigg (38) =3/2En+1 SRS (39) and the equality also holds for the ( n+ 1)st order. Thus the contributions to the energy of the H 3–quartet state in the SRS and GHL theories are equal order by o rder. One advantage of the GHL theory is that it permits the calcula tion of the exchange energies by other methods, such as the surface integral meth od. In Ref. [10], the non– additive energy terms of the quartet spin state of H 3have been calculated up to third order. The ﬁrst order terms can be split into a polarization and an ex change part. Since the ﬁrst order polarization energy is pairwise additive, the only no n–additive term in ﬁrst order is contained in the exchange term which in Eqs. (23) and (55) of R ef. [9] is given by E1 exch(3,3) =<ψ0\|ˆVAB/parenleftBigˆT(23) + ˆT(13) + ˆT(123) + ˆT(132) −S0 23−S0 13−S0 123−S0 132/parenrightBig \|ψ0> +<ψ0\|ˆVAB/parenleftBigˆT(12) + ˆT(13) + ˆT(123) + ˆT(132) −S0 12−S0 13−S0 123−S0 132/parenrightBig \|ψ0> +<ψ0\|ˆVAB/parenleftBigˆT(12) + ˆT(23) + ˆT(123) + ˆT(132) −S0 12−S0 23−S0 123−S0 132/parenrightBig \|ψ0>, (40) which can be expressed in terms of exchange energies as E1 exch(3,3) =ǫ1 123(1−S0 123)−/bracketleftBig ǫ1 12(1 +S0 12)−ǫH2,1 12(1 +S0 12)/bracketrightBig −/bracketleftBig ǫ1 23(1 +S0 23)−ǫH2,1 23(1 +S0 23)/bracketrightBig −/bracketleftBig ǫ1 13(1 +S0 13)−ǫH2,1 13(1 +S0 13)/bracketrightBig . (41) This term is also obtained if the pure two–body contribution s are subtracted from Eq. (29). V. SURFACE INTEGRAL METHOD (SIM) FOR THE CALCULATION OF EXCHANGE ENERGIES As shown in Refs. [14] and [18] all exchange energies occurri ng in the GHL–description of the H 3system, i.e. the two–body as well as the cyclic exchange ener gies, can be calculated by the surface integral method (SIM). The exchange energy ǫg0associated with the arbitrary group element g0/negationslash=Iis given accordingly by εg0=/bracketleftBigg/integraldisplay Vdv/bracketleftBig F2−(ˆT(g0)F)2/bracketrightBig/bracketrightBigg−1/bracketleftBigg1 2/integraldisplay Σ/braceleftBig F/vector∇9/bracketleftBigˆT(g0)F/bracketrightBig −/bracketleftBigˆT(g0)F/bracketrightBig/vector∇9F/bracerightBig ·d/vector s/bracketrightBigg −/summationdisplay g/negationslash=I,g0εg/integraldisplay Vdv/bracketleftBig F(ˆT(g0g)F)−(ˆT(g0)F)(ˆT(g)F)/bracketrightBig/bracketrightBigg . (42) In order to compare numerical results for three–body exchan ge eﬀects with the published SAPT results for H 3[10], an expression for the non–additive exchange energy ha s to be ob- tained using SIM. The non–additive exchange energy basical ly contains the cyclic exchange 7energy and the implicit three–body eﬀects on the two–body ex change energies. As already pointed out in Ref. [14] it can be shown that for a choice of the partial volume Vsuch thatFis localized inside, all quantities occurring in the sum of E q. (42) go to zero with at least a factor of e−Rfaster than the surface integral itself if all internuclear distances are larger or equal to R. This holds for all exchange energies. In a diﬀerent paper [1 8] it will be shown how to ﬁnd the implicit three–body eﬀect from th e complete surface integral expression for the two–body exchange energies. For product wave functions as used here the pure two–body part is given by the ﬁrst line of formula Eq. (42), i.e. surface integral (SI) over denominator. The implicit three–body eﬀect is con tained in the second line of Eq. (42), i.e. the products of partial overlap integrals with ex change energies. Following the same scheme used in the Appendix of Ref. [14], these terms can be shown to asymptotically go to zero as e−5Rwhich is faster by a factor of e−3Rthan the surface integral (SI) itself. Using these results a GHL non–additive exchange energy for t he quartet state of H 3 can be deﬁned by simply subtracting the pure two–body contri bution from the two–body exchange energies in the GHL result for the quartet state Eq. (3) (3/2EGHL)exch= 2ǫ123−/bracketleftBig ǫ12−ǫH2 12/bracketrightBig −/bracketleftBig ǫ23−ǫH2 23/bracketrightBig −/bracketleftBig ǫ13−ǫH2 13/bracketrightBig (43) which can be calculated either by SIM or perturbation theory . The ﬁrst order contribution to this non–additive term (3/2E1 GHL)exch= 2ǫ1 123−/bracketleftBig ǫ1 12−ǫH2,1 12/bracketrightBig −/bracketleftBig ǫ1 23−ǫH2,1 23/bracketrightBig −/bracketleftBig ǫ1 13−ǫH2,1 13/bracketrightBig (44) diﬀers from the respective SRS–term Eq. (41) only by overlap integrals that are negligible compared to one. A comparison of the numerical results of the ﬁrst order non–a dditive exchange energy Eq. (41) of SRS theory and the GHL term [Eq. (44)] calculated b y SIM using the zeroth order product wave function F= 1/π3/2exp(−r1A−r2B−r3C) is given in Tables I and II and will be discussed in the next Section. In summary, the complete three–body exchange eﬀect in H 3, which consists of the cyclic exchange energy and the eﬀect of the presence of the third ato m on the two–body exchange energies, can asymptotically be approximated by the surfac e integral for the cyclic exchange energy. VI. RESULTS In Tables I and II as well as Figures 1 and 2 the numerical resul ts for the ﬁrst order non–additive exchange energy of SRS theory are compared wit h three diﬀerent SIM–terms: (i) the non–additive exchange energy of GHL theory Eq. (43), (ii) the cyclic exchange energy (complete SIM expression Eq. (42) with overlaps), (iii) the surface integral (SI) of the cyclic exchange energy only (without overlaps). All these quantit ies have been calculated using the zeroth order localized wave function F= 1/π3/2exp(−r1A−r2B−r3C). Since the exchange energies calculated by SIM cannot be given a deﬁnite perturb ative order (due to the fact that only part of the complete space is used in the calculation) th e quantity (i) is not expected to yield the same numerical results as the ﬁrst order non–add itive exchange energy of SRS theory. But since the same zeroth order product wave functio n was used to calculate both 8terms it is expected that both quantities exhibit a similar o verall behavior in the range of parameters studied. In Table I results for equilateral triangular geometry of th e nuclei ranging between R= 4 andR= 10 atomic units are listed. Generally, all terms calculate d by SIM have smaller absolute values than the ﬁrst order perturbative ones. At R= 4 a.u., the absolute value of the complete SIM term Eq. (43) is 27 % below the SRS result Eq. ( 41), the cyclic exchange energy is 38 % smaller, and only the surface integral of the cy clic exchange energy is 25 % greater in absolute value. At R= 10 a.u., however, all three quantities calculated by SIM are no longer distinguishable and are only 6 % below the SRS re sult. In Table II the results for isosceles triangles with equal si des of length of 6 a.u. and with anglesγBvarying between 30◦and 180◦are shown. All quantities except for the surface integral without overlaps exhibit a change of sign in the reg ion around 120◦and 150◦. At 30◦, (i) the absolute value of the SIM term Eq. (43) is 31 % smaller than the SRS result, (ii) the cyclic exchange energy is 41 % smaller, and again (ii i) the surface integral of the cyclic exchange energy only is 13 % greater in absolute value . At 180◦on the other hand, only the value for the surface integral has the wrong sign, wh ile both the other terms have become indistinguishable and are now 35 % greater in absolut e value than the SRS term. The diﬀerences between the numerical results for the quanti ties compared in Tables I and II are, as already pointed out, not due to numerical problems but due to the fact that the quantities are diﬀerent by deﬁnition. From the Tables it appears that for triangular geometries of the nuclei and internuclear distancesR≥4 a.u. the ﬁrst order non–additive exchange energy for the qu artet state of H3can be quite well approximated by the surface integral of the cyclic exchange energy. This was stated in Ref. [14] and has now been explained by the f act that all the SIM approximations (see section V and in Ref. [14]) hold in this r egion. In Tables III and IV as well as Figures 1 and 2 higher orders of S RS theory are also taken into account and compared with the complete GHL non–ad ditive exchange energy Eq. (43) in order to show that SIM goes beyond the ﬁrst order of SRS theory. For equilateral triangular geometries of the nuclei and internuclear dista nces larger than 6 a.u. the results of GHL theory lie between the ﬁrst order SRS term and the sum of the ﬁrst and second order terms, approaching the ﬁrst order term for increasing distances. At 6 a.u. GHL is very close to the ﬁrst plus second order of SRS, and even at 4 a. u. GHL is only 17 % below the total sum up to third order of SRS theory. For isosceles structures of the nuclei with equal internucl ear distances of 6 a.u. the advantage of SIM over the ﬁrst order SRS theory is even more ap parent. Starting at 60◦, the GHL result is closer to the ﬁrst plus second order than to t he ﬁrst order SRS term. The change of sign occurs for the ﬁrst order between 120◦and 150◦whereas for all other terms already between 90◦and 120◦. The diﬀerences of the GHL to the ﬁrst plus second order SRS term range from 0.4% at 60◦to 33% at 120◦and 10% at 180◦. At 30◦the GHL result is again only 16% smaller than the SRS term with the third orde r term included. The advantage of SIM over the perturbative approach is that t he surface integral SI is easily calculated numerically, and including the partial o verlap terms provides part of the second order SRS contributions. 9VII. CONCLUSIONS This paper demonstrates how the perturbation series consis ting of Coulomb, exchange and overlap integrals can be used to express the Coulomb and e xchange energies occurring in GHL theory. Combining the perturbation series with the GH L theory yields an energy expression for the quartet spin state equivalent to that of s ymmetrized Rayleigh–Schr¨ odinger perturbation theory given in [10]. It is possible to evaluate the exchange energies using the su rface integral method (SIM). The SIM has the advantage that it derives from a clear physica l picture for the exchange process in terms of the electrons continuously trading plac es. For the cyclic exchange energies this method has already been described in detail in Ref. [14] , and for the implicit three–body eﬀect on the two–body exchange energies it will be shown in Re f. [18]. The long range behavior of the three–body terms entering the two–body exchange ener- gies and of the partial overlap integrals — multiplied by two –body exchange energies in the expression for the cyclic exchange energy in Eq. (42) — indic ate that for large internuclear separations the surface integral for the cyclic exchange en ergy is suﬃcient to describe the non–additive contribution to the exchange part of the quart et spin state. The numerical results in Tables I and II conﬁrm this conclusion. VIII. ACKNOWLEDGEMENTS We thank K. T. Tang and J. P. Toennies for helpful discussions . U. K. gratefully ac- knowledges ﬁnancial support from the DFG. 10REFERENCES [1] K.T. Tang, J.P. Toennies and C. L. Yiu, Int. Rev. Phys. Che m.17, 363 (1998). [2] S. H. Patil and K. T. Tang, Asymptotic Methods in Quantum Mechanics: Applications to Atoms, Molecules and Nuclei (Springer, Berlin, 2000). [3] U. Kleinekath¨ ofer, K.T. Tang, J.P. Toennies, and C.L. Y iu, J. Chem. Phys. 111, 3377 (1999). [4] U. Kleinekath¨ ofer, Chem. Phys. Lett 324, 403 (2000). [5] B. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299 (1943); Y. Muto, Proc. Phys. Soc. Jpn.17, 629 (1943). [6] M. J. Elrod and R. J. Saykally, Chem. Rev. 94, 1975 (1994). [7] W. J. Meath and M. Koulis, J. Mol. Struct. (Theochem.) 226, 1 (1991). [8] W. J. Meath and R. A. Aziz, Mol. Phys. 52, 225 (1984). [9] R. Moszynski, P. E. S. Wormer, B. Jeziorski, and A. van der Avoird, J. Chem. Phys. 103, 8058 (1995). [10] T. Korona, R. Moszynski, and B. Jeziorski, J. Chem. Phys .105, 8178 (1996). [11] V. F. Lotrich and K. Szalewicz, J. Chem. Phys. 112, 112 (2000). [12] R. J. Wheatley, Mol. Phys. 84, 899 (1995). [13] Z. C. Zhang, A. R. Allnatt, J. D. Talman, and W. J. Meath, M ol. Phys. 81, 1425 (1994). [14] U. Kleinekath¨ ofer, T. I. Sachse, K. T. Tang, J. P. Toenn ies, and C. L. Yiu, J. Chem. Phys.113, 948 (2000) [15] K. T. Tang, J. P. Toennies, and C. L. Yiu, J. Chem. Phys. 94, 7266 (1991). [16] K. T. Tang, J. P. Toennies and C. L. Yiu, J. Chem. Phys. 99, 377 (1993). [17] U. Kleinekath¨ ofer, K. T. Tang, J. P. Toennies, and C. L. Yiu, J. Chem. Phys. 107, 9502, (1997). [18] T. I. Sachse, K. T. Tang and J. P. Toennies, in preparatio n. [19] T. Cwiok, B. Jeziorski, W. Ko/suppress los, R. Moszynski, J. Rych lewski und K.Szalewicz, Chem. Phys. Lett. 195, 67 (1992). [20] J.O. Hirschfelder, Chem. Phys. Lett. 1, 325 (1967). [21] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev .94, 1887 (1994). [22] G. Chalasinski, B. Jeziorski, and K. Szalewicz, Int. J. Quantum Chem. 11, 247 (1977). [23] K. T. Tang, J. P. Toennies, and C. L. Yiu, Chem. Phys. Lett .162, 170 (1989). [24] The explicit expressions will be given in a forthcoming paper [18]. 11TABLES E1 exch[Eh] R[a0] SRS Eq. (41) GHL Eq. (43) 2 ǫ123(SIM) 2 SI 4 −3.83·10−3−2.79·10−3−2.39·10−3−4.21·10−3 5 — −4.31·10−4−4.16·10−4−5.26·10−4 6 −5.90·10−5−5.19·10−5−5.15·10−5−5.70·10−5 7 −5.88·10−6−5.32·10−6−5.31·10−6−5.55·10−6 8 −5.33·10−7−4.89·10−7−4.89·10−7−4.98·10−7 10 −3.6·10−9−3.4·10−9−3.4·10−9−3.4·10−9 TABLE I. Comparison of the numerical results for the ﬁrst ord er non–additive exchange energy of SRS–theory (SRS 1Eq. (41)) with a similar but still diﬀerent quantity derived from GHL theory Eq. (43), with the cyclic exchange calculated by SIM (2 ǫ123(SIM)) including overlaps, and with the surface integral SI of the cyclic exchange energy withou t overlaps (2 SI). The nuclei form equilateral triangles with sides of lengths R. E1 exch[Eh],RAB=RBC= 6 a.u. γB[degrees] SRS Eq. (41) GHL Eq. (43) 2 ǫ123(SIM) 2 SI 30 −3.75·10−4−2.60·10−4−2.23·10−4−4.25·10−4 60 −5.90·10−5−5.19·10−5−5.15·10−5−5.70·10−5 90 −7.40·10−6−6.05·10−6−6.03·10−6−7.95·10−6 120 −3.42·10−72.61·10−72.60·10−7−1.62·10−6 150 8.84·10−71.31·10−61.30·10−6−5.83·10−7 180 1.10·10−61.48·10−61.48·10−6−4.10·10−7 TABLE II. Comparison of the numerical results of SRS–theory with the same quantities as in Table I. The nuclei form isosceles triangles with two sides o f lengths RAB=RBC= 6 a.u., γBis the angle included. Eexch[Eh] R[a0] SRS1Eq. (41) SRS 2 SRS3 GHL Eq. (43) 4 −3.83·10−3−3.60·10−3−3.34·10−3−2.79·10−3 6 −5.90·10−5−5.21·10−5−5.03·10−5−5.19·10−5 7 −5.88·10−6−4.77·10−6−4.62·10−6−5.32·10−6 8 −5.33·10−7−3.71·10−7−3.57·10−7−4.89·10−7 10 −3.6·10−9−0.7·10−9−0.7·10−9−3.4·10−9 TABLE III. Comparison of the numerical results for the non–a dditive exchange energy in GHL theory (GHL Eq. (43)) with the ﬁrst order non–additive excha nge energy of SRS–theory (SRS 1 Eq. (41)), with the SRS non–additive exchange energy up to se cond order (SRS 2) [10] , and with up to third order SRS 3[10] . The nuclei form equilateral triangles with sides of le ngths R. 12Eexch[Eh],RAB=RBC= 6 a.u. γB[degrees] SRS1Eq. (41) SRS 2 SRS3 GHL Eq. (43) 30 −3.75·10−4−3.33·10−4−3.08·10−4−2.60·10−4 60 −5.90·10−5−5.21·10−5−5.03·10−5−5.19·10−5 90 −7.40·10−6−5.67·10−6−4.98·10−6−6.05·10−6 120 −3.42·10−73.88·10−79.02·10−72.61·10−7 150 8.84·10−71.43·10−61.88·10−61.31·10−6 180 1.10·10−61.63·10−62.07·10−61.48·10−6 TABLE IV. Comparison of the numerical results of GHL–theory with the same quantities as in Table III. The nuclei form isosceles triangles with two si des of lengths RAB=RBC= 6 a.u., γB is the angle included. 13FIGURES FIG. 1. Comparison of diﬀerent orders of the non–additive ex change energy in SRS theory with the GHL result (ﬁlled triangles) calculated with SIM from Eq . (43) for equilateral triangles. The ﬁrst order SRS contribution is denoted by circles, and with a ll terms up to second order by open triangles. The stars show twice the surface integral of the c yclic exchange energy. 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 45678910/0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0/0/0/0 /1/1/1/1/1/1R A/0/0/0/0/0/0 /1/1/1/1/1/1 C[a.u.]RRBNAexchE- internuclear distance R [a.u.] 14FIG. 2. Comparison of diﬀerent orders of the non–additive ex change energy in SRS theory with the GHL result (ﬁlled triangles) calculated with SIM fr om Eq. (43) for isosceles triangles with RAB=RBC= 6 a.u. as a function of the included angle γB. The ﬁrst order SRS contribution is denoted by circles, and wi th all terms up to second order by open triangles. The stars show twice the surface integral of the cyclic exchange energy only. Note the change in the energy axis from linear to logarithmic scal e. 10-3 10-4 10-5 10-6 0.5 x 10-6 0 -0.5 x 10-6 -10-6 -1.5 x 10-6 -2 x 10-6 -2.5 x 10-6 4080120160Bϑ AB C /0/0/0/0/0/0 /1/1/1/1/1/1 R[a.u.] AB= = 6 a.u.BCR/0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0/0/0/0 /1/1/1/1/1/1- EexchNA ϑBangle [degrees] 15
arXiv:physics/0011059v1 [physics.plasm-ph] 24 Nov 2000On a correspondence between classical and quantum particle systems Klaus Morawetz Max-Planck-Institute for the Physics of Complex Systems, N oethnitzer Str. 38, 01187 Dresden, Germany An exact correspondence is established between a N-body classical interacting system and a N−1- body quantum system with respect to the partition function. The resulting quantum-potential is a N−1-body one. Inversely the Kelbg potential is reproduced whi ch describes quantum systems at a quasi-classical level. The found correspondence between c lassical and quantum systems allows also to approximate dense classical many body systems by lower or der quantum perturbation theory replacing Planck’s constant properly by temperature and de nsity dependent expressions. As an example the dynamical behaviour of an one - component plasma is well reproduced concerning the formation of correlation energy after a disturbance utilis ing solely the analytical quantum - Born result for dense degenerated Fermi systems. As a practical g uide the quantum - Bruckner parameter rshas been replaced by the classical plasma parameter Γ as rs≈0.3Γ3/2. Several hints in recent literature conjecture that there seem to exist a correspondence between quantum systems and higher dimensional classical systems. The authors of [1] argue that a higher dimensional classical non-Abelian gauge theory leads to a lower dimensional quantum ﬁeld theory in the sense of chaotic quantisation. The corre- spondence has been achieved by equating the tempera- ture characterising chaotization of the higher dimensiona l system with ¯ hof the lower dimensional system by ¯h=aT. (1) Recalling imaginary time evolution as a method to cal- culate correlated systems in equilibrium such correspon- dence seems suggestible. We will ﬁnd a similar relation (20) but only as a best ﬁt of quantum - Born calculations to dense interacting classical systems. In condensed matter physics it is a commonly used trick to map a two - dimensional classical spin system onto a one - dimensional quantum system [2]. This sug- gests that there might exist a general relation between classical and higher dimensional quantum systems. We will show that a classical many body system can be equally described by a quantum system with one particle less in the system but with the price of complicated non- local potential. This can be considered analogously to the Bohm interpretation of quantum mechanics [3] where the Schroedinger equation is rewritten in a Hamilton-Jacobi equation but with a nonlocal quantum potential. Another hint towards a correspondence between classi- cal and quantum systems was found recently in [4] where it was achieved to deﬁne a Lyapunov exponent in quan- tum mechanics by employing the marginal distribution which is a representation of Wigner function in a higher dimensional space. Since the Lyapunov exponent is es- sentially a concept borrowed from classical physics this ﬁnding points also in the direction that there exists a correspondence between quantum systems and higher di- mensional classical systems.On the opposite side there are systematic derivations of constructing eﬀective classical potentials such that th e many body quantum system is described by the classical system. An example is the Kelbg potential for Coulomb systems [5–8] VKelbg 12(r) =e1e2 r/parenleftBig 1−e−r2/l2+√πr lerfc/parenleftBigr l/parenrightBig/parenrightBig (2) withl2= ¯h2/µTand 1/µ= 1/m1+ 1/m2describing the two-particle quantum Slater sum correctly by a classical system. Improvements and systematic applications can be found in [9–11]. Here in this paper it should be shown that a classical N-particle system can be mapped exactly on a quantum N−1-particle system with respect to the partition func- tion. Though the resulting eﬀective N−1 body quantum potential is highly complex it can lead to practical ap- plications for approximating strongly correlated classic al systems. In the thermodynamical limit it means that the dense classical system can be described alternatively by a quantum system with properly chosen potential. This ﬁnding suggests that the quantum calculation in lowest order perturbation might be suitable to derive good approximations for the dense classical system. This is also motivated by an intuitive picture. Assume we have a dense interacting classical plasma system. Then the correlations will restrict the possible phase space for travelling of one particle considerably like in dense Fermi systems at low temperatures where the Pauli exclusion principle restrict the phase space for scattering. There- fore we might be able to describe a dense interacting classical system by a perturbative quantum calculation when properly replacing ¯ hby density and temperature expressions. Indeed we will demonstrate in a one - com- ponent plasma system that even the time evolution and dynamics of a very strongly correlated classical system can be properly approximated by quantum - Born cal- culations replacing the quantum parameters by proper 1classical ones. Let us now start to derive the equivalence between clas- sical and quantum systems by rewriting the classical N- particle partition function. The conﬁguration integral reads QN(β) =/integraldisplay dx1...dx NN/productdisplay i<j(1 +fij) (3) where we used Meyer’s graphs fij= exp( −βu(xi−xj))−1 with the interaction potential of the classical particles uand the inverse temperature β. It is now of advantage to consider the modiﬁed conﬁguration integral ˜QN(β) =QN(2β) =/integraldisplay dx1...dx Ndx′ 1...dx′ Nδ(x1−x′ 1)...δ(xN−1−x′ N−1) × (1 +f12)(1 +f13)(1 + f14)...(1 +f1N) × (1 +f21′)(1 + f23)(1 +f24)...(1 +f2N) × ...... ×(1 +fN1′)(1 + fN2′)(1 +fN3′)...(1 +fNN−1′)(4) where we have completed a quadratic schema in fij′. Now we assume a complete set of N−1 particle wave functions Ψ nN−1such that δ(x1−x′ 1)...δ(xN−1−x′ N−1) =/summationdisplay i1..iN−1Ψ∗ i1..iN−1(x′ 1...x′ N−1)Ψi1..iN−1(x1...xN−1) (5) with some ”quantum numbers” {i}characterising the state. Further we propose the following eigenvalue prob- lem deﬁning the wave function /integraldisplay dx1N/productdisplay j=2(1 +f1j)Ψi1..iN−1(x1...xN−1) =Ve−ε{i}Ψi2..iN−1i1(x2...xN) (6) with the system volume V. This allows to calculate the conﬁgurational integral (4) exactly by successively inte- grating x1...xN ˜QN(β) =/summationdisplay i1..iN−1/integraldisplay dx1...dx Ndx′ 1...dx′ N−1Ψ∗ i1..iN−1(x′ 1...x′ N−1) ×Ψi1..iN−1(x1...xN−1) × (1 +f12)(1 +f13)(1 + f14)...(1 +f1N) × (1 +f21′)(1 + f23)(1 +f24)...(1 +f2N) × ...... ×(1 +fN1′)(1 + fN2′)(1 +fN3′)...(1 +fNN−1′) =/summationdisplay i1..iN−1/integraldisplay dx′ 1...dx′ N−1Ψ∗ i1..iN−1(x′ 1...x′ N−1) ×VNe−Nε{i}Ψi1..iN−1(x′ 1...x′ N−1) =VN/summationdisplay i1..iN−1e−Nε{i}. (7)This establishes already the complete proof that we can map a classical N-body system on a N−1-body quantum system since (6) is the eigenvalue problem of a N−1-body Schroedinger equation /parenleftBigg ¯hω{i}−N−1/summationdisplay ip2 i 2mi−U/parenrightBigg Ψi1...iN−1= 0 (8) where ¯ hω{i}−/summationtext ip2 i 2mi= ΩVe−ε{i}with some character- istic energy density, e.q. Ω = ¯ hω0/V. The equivalent quantum potential reads now < x1i1...xN−1iN−1\|U\|x′ 1i′ 1...x′ N−1i′ N−1> = eβ[u12(x′ 1−x1)+...+u1N(x′ 1−xN−1)] ×Ωδ(x1−x′ 2)...δ(xN−2−x′ N−1)δi′ 1,i1...δi′ N−1,iN−1(9) where we rewrote Meyer’s cluster graphs in terms of the classical potential u. The resulting equivalent quantum potential (9) is a N−1-body nonlocal potential with respect to the coordinates but depends on Nstrength function parameter (e.g. charges). Therefore we have casted a classical N-body problem into a nonlocal quan- tumN−1 body problem. One could easily give also a symmetrised or anti-symmetrised form of the potential using symmetries of the wave function and permuting coordinates of (9) respectively. We do not need it here since we will restrict to applications neglecting exchange correlations further on. While the above correspondence holds for any particle number and might be useful to ﬁnd solvable models for classical three - body problems, we will consider in the following many - body systems. First let us invert the problem and search for an eﬀective classical potential ap- proximating quantum systems. This should us lead to the known Kelbg-potential (2). For this purpose we assume a quantum system described in lowest approximation by a Slater determinant or a complete factorisation of the many - body wave function into single wave function Ψi1...iN(x1...xN) =φi1...φiN. We neglect for simplicity exchange correlations in the following. The correspond- ing eigenvalue equation for φitself one can obtain from (6) or (8) by multiplying with Ψ∗ i2..iN−1(x2...xN−1) and integrate over x2...xN−1. To see the generic structure more clearly we better calculate the correlation energy by multiplying (6) or (8) by Ψ∗ i2..iN−1i1(x2...xN) and in- tegrating over x2...xN. This provides also the eigenvalue ǫ{i}and leads easily to approximations for the partition function (3). To demonstrate this we choose the lowest order approximation taking identical plane waves for φ. Than the pressure can be obtained from the partition function QNvia (7) P=T∂ ∂VlnQN=T/parenleftbiggN V−N(N−2) V2/integraldisplay dr/parenleftBig e−βu(r)/2−1/parenrightBig/parenrightbigg (10) 2where Vis the volume of the system. We recognise the standard second virial coeﬃcient for small potentials while for higher order potential the factor 1 /2 appears in the exponent instead as a pre-factor indicating a diﬀerent partial summation of diagrams due to the schema behind (7) and (8). To go beyond the plane wave approximation we multi- ply (6) by Ψ∗ i2..iN−1i1(x2...xN) and the kinetic part of the statistical operator before integrating over x2...xN. This means we create an integral over the N−1 particle den- sity operator and the potential (9) which together repre- sents the correlation energy. This expression is a succes- sive convolution between the cluster graphs fijand the relative two - particle correlation function ρi1i2(x1−x2). The resulting mean correlation energy density reads U V=/summationdisplay {i}/integraldisplaydy1... dyN−1 VN−1ρi1i2(y1)ρi2i3(y2)...ρiN−1i1(yN−1) ×(1−f12(y1))(1−f13(y1+y2))...(1−f1N(y1+...+yN−1)) ≈/summationdisplay {i}/integraldisplaydy1... dyN−1 VN−1ρi1i2(y1)... ρiN−1i1(yN−1) ×u12(y1)u13(y1+y2)... u1N(y1+...+yN−1) +... (11) in dimensionless units where all other cluster expansion terms lead either to lower mean ﬁeld or disconnected terms. While these terms can be calculated as well we re- strict to the highest order convolutions in the correlation energy (11) which have now the structure of mean corre- lation energy U/V=/summationtext i1i2/integraltextdx Vρi1i2(x)Veﬀ 12with a classical eﬀective potential Veﬀ 12 Veﬀ 2(r)∝/summationdisplay 3/integraldisplaydx1 Vρ12(x1)u12(x1)u23(x1+r) (12) Veﬀ 3(r)∝/summationdisplay 34/integraldisplaydx1dx2 V2ρ12(x1)u12(x1)u13(x1+x2) ×ρ23(x2)u34(x1+x2+r) ... (13) where the two-particle, three-particle etc. approximatio n can be given. In equilibrium the nondegenerate correla- tion function reads [ l2= ¯h2/µT=λ2/2π] ρi1i2(x1−x2)=/integraldisplaydp (2π¯h)3eipr/¯hλ3e−βp2 2µ= e−r2/l2.(14) Using the Coulomb potential u∝1/rwe obtain from the two-particle approximation (12) just the Kelbg po- tential (2). The three - particle approximation (13) can be calculated as well and reads [ x=r/l] Veﬀ 3∼1 x erf2/parenleftbiggx√ 2/parenrightbigg +23/2x√π∞/integraldisplay xdz ze−z2/2erf/parenleftbiggz√ 2/parenrightbigg . (15)With the schema (13) one can easily integrate higher or- der approximations as successive convolutions. Also in principle the degenerate case could be calculated using Fermi-Dirac distributions in (14). But one should then consider also the neglected exchange correlations during factorisation of Ψ as well. Let us summarise that the known eﬀective classical potential describing a quantum system in binary approximation has been recovered by identifying the eﬀective two-particle interaction within the correlation energy. We want now to proceed to a phenomenological level in that the above correspondence between quantum and classical systems motivates to ﬁnd good approximations for the dynamics of classical many - body systems by employing quantum - Born approximations and replacing ¯hproperly. Let us consider an one-component plasma system which is characterised by two values. The classical cou- pling is described by the plasma parameter Γ =e2 dTas a ratio of the length where Coulomb energy becomes larger than kinetic energy,e2 T, to the interparticle distance or Wigner size radius d= (3 4πn)1/3. Ideal plasmas are found for Γ<<1 while around Γ = 1 non-ideal eﬀects become important. A second parameter which controls the quan- tum features is the Bruckner parameter as the ratio of the Wigner size radius to the Bohr radius aB= ¯h2/me2. Quantum eﬀects will play a role if rs≤1. We will con- sider the situation that the interaction of such system is switched on at initial time. Then the correlations are formed by the system which is seen in an increase of temperature accompanied by the build up of negative correlation energy. This theoretical experiment has been investigated numerically by [12] for classical plasmas wit h diﬀerent plasma parameter Γ. In [13,14] we have calcu- lated the formation of such correlations by using quan- tum kinetic equations in Born approximation. The time dependence of kinetic energy was found at short times to be Ecorr=−/summationdisplay ab/integraldisplaydkdpdq (2π¯h)9V2 D1−cos/braceleftbig1 ¯ht∆E/bracerightbig ∆E ×f′ af′ b(1−fa)(1−fb) (16) where fare the initial distributions and ∆ E=k2 2ma+ p2 2mb−(k−q)2 2ma−(p+q)2 2mb. The statical screened Coulomb in- teraction is VD(q) = 4πe2¯h2/(q2+¯h2κ2) with the inverse screening length expressed by density nand temperature Tasκ2= 4πe2n/Torκ2= 6πe2n/ǫffor the high or low temperature limit. For both cases dynamical as well as statical screening it was possible to integrate analyt- ically the time dependent correlation energy (16). This has allowed to describe the time dependence of simula- tions in the weak coupling limit Γ <1 appropriately [13]. For stronger coupling Γ ≥1 the Born approximation fails since the exact correlation energy of simulation is lower 3than the ﬁrst order (Born) result κe2/2T=/radicalbig 3/2Γ3/2. Moreover there appear typical oscillations as seen in ﬁg- ure 1. Now we will employ the ideas developed above and will use the quantum Born approximations in the strongly degenerated case to describe the classical strongly corre- lated system. For strongly degenerated plasmas the time dependence of correlation energy was possible to inte- grate as well with the result [14] expressed here in terms of plasma parameter Γ and quantum Bruckner parameter rsas ET corr(t)−E0 corr(t) nT=1 (36π4)1/6r3 s Γ/parenleftbiggsinyτ yτ−1/parenrightbigg ×/parenleftbigg1 blarctan(1 bl) +1 b2 l+b4 l/parenrightbigg (17) withbl= ¯hκ/2pf=√ Γ/(48π2)1/6,yτ= 4ǫft/¯h= (2)4/3π5/335/6τ/√rswhere the time is scaled in plasma periods τ= 2πt/ω p. Now we try to ﬁt this quantum result to the simulation using the Bruckner parameter as free parameter. For the available simulations between 1≤Γ≤10 we obtain a best ﬁt rﬁt s=c/radicalbigg 3 8Γ3/2c≈0.5. (18) The quality of this ﬁt is illustrated in ﬁgure 1 which is throughout the range 1 ≤Γ≤10. This is quite astonish- ing since not only the correct classical correlation energy [15] is described but also the correct time dependence i.e. dynamics. 0 1 2 3 4 5 6 7 t [2π/ωp]−7−5−3−1Ecorr/(nT)Simulation Γ=10 Quantum Born rB ~ Γ3/2 FIG. 1. The time evolution of a classical one-component plasma after sudden switching of interaction [12] compared to the quantum Born result when the Bruckner parameter is replaced according to (18). The long time equilibrium value is remarkably well reproduced by the quantum - Born result(17) . Let us try to understand what this phenomenological ﬁnding means. Deﬁning the thermal De Broglie wave length as λ2= ¯h2/4mTwe can rewrite (18) as λ d≈κ−1 λ(19) which means that there is a geometrical relation between the thermal De Broglie wave length λ, the interparticledistance dand the screening length κ−1. Using the de- generated screening length we rewrite (18) ¯hﬁt≈/radicalbiggm nT e/parenleftbigg283 π/parenrightbigg1/6 (20) which shows remarkably the structure of the result (1) in literature. But here it is only a best ﬁt to reproduce dense classical results by simpler quantum Born approx- imations and bears no fundamental importance. We summarise that it is indeed possible to ﬁnd a good approximation of classical dense interacting systems by quantum Born calculations replacing the quantum pa- rameter properly by classical ones. For equilibrium we have shown that there exist an exact relation between a N-body classical system and a N−1-body quantum sys- tem. This has allowed to recover the quantum Kelbg po- tential easily. We conjecture that a similar relation like in equilibrium between classical N-body and quantum N−1 body system might exist also for nonequilibrium. As practical consequence we suggest to describe dense interacting classical many body systems by the simpler perturbative quantum calculation in degenerate limit re- placing properly ¯ hby typical classical parameters of the system. I would like to thank S. G. Chung for numerous dis- cussions and valuable hints. [1] T. S. Bir´ o, S. G. Matinyan, and B. M¨ uller, (2000), hep- th/0010134. [2] S. G. Chung, Phys. Rev. B 60, 11761 (1999). [3] D. Bohm and B. J. Hiley, Foundations of Physics 14, 255 (1984). [4] V. I. Man’ko and R. V. Mendes, Physica D 45, 330 (2000). [5] G. Kelbg, Ann. Physik 13, 354 (1964). [6] G. Kelbg, Ann. Physik 14, 394 (1964). [7] W. Ebeling, H. Hoﬀmann, and G. Kelbg, Beitr¨ age aus der Plasmaphysik 7, 233 (1967). [8] D. Kremp and W. D. Kraeft, Ann. Physik 20, 340 (1968). [9] W. D. Kraeft, D. Kremp, W. Ebeling, and G. R¨ opke, Quantum Statistics of Charged Particle Systems (Akademie Verlag, Berlin, 1986). [10] W. D. Kraeft and D. Kremp, Zeit. f. Physik 208, 475 (1968). [11] J. Ortner, I. Valuev, and W. Ebeling, Contrib. Plasma Phys. . [12] G. Zwicknagel, Contrib. Plasma Phys. 39, 155 (1999). [13] K. Morawetz, V. ˇSpiˇ cka, and P. Lipavsk´ y, Phys. Lett. A 246, 311 (1998). [14] K. Morawetz and H. K¨ ohler, Eur. Phys. J. A 4, 291 (1999). [15] S. Ichimaru, Statistical Plasma Physics (Addison-Wesley Publishing company,, Massachusetts, 1994), p. 57. 4
arXiv:physics/0011060v1 [physics.comp-ph] 24 Nov 2000Complex Scaling of the Faddeev Equations1,2 E. A. Kolganovaa,b, A. K. Motovilovb, Y. K. Hoa aIAMS, Academia Sinica, P.O.Box 23-166, Taipei, Taiwan, ROC bJINR, 141980 Dubna, Moscow Region, Russia Abstract In this work we compare two diﬀerent approaches to calculati on of the three-body resonances on the basis of Faddeev diﬀerential equations. T he ﬁrst one is the com- plex scaling approach. The second method is based on an immed iate calculation of resonances as zeros of the three-body scattering matrix con tinued to the physical sheet. Key words: three-body systems, complex scaling, resonances 1 Introduction The complex scaling method [1,2] invented in early 70-s rema ins one of the most eﬀective approaches to calculation of resonances in fe w-body systems. This method is applicable to an N-body problem in the case where inter- particle interaction potentials are analytic functions of coordinates. The com- plex scaling gives a possibility to rotate the continuous sp ectrum of an N-body Hamiltonian in such a way that certain sectors of unphysical sheets neighbor- ing the physical one turn into a part of the physical sheet for the resulting non-selfadjoint operator. Resonances appear to be complex eigenvalues of this operator [1,2] while the binding energies stay ﬁxed during t he scaling trans- formations. Therefore, when searching for the resonances w ithin the complex scaling approach one may apply the methods which are usually employed to locate the binding energies. Some reviews of the literature on the complex scaling and its many applications can be found, in particula r, in [4–6]. Here 1Contribution to Proceedings of the International Conferen ce “Modern Trends in Computational Physics”, July 2000, Dubna, Russia 2This work was supported by Academia Sinica, National Scienc e Council of R.O.C., and Russian Foundation for Basic Research Preprint submitted to Elsevier Preprint November 24, 2000we only mention that there is a rigorous mathematical proof [ 3] that for a rather wide class of interaction potentials the resonances given by the com- plex scaling method coincide with the “true scattering reso nances”, i.e. the poles of the analytically continued scattering matrix in th e unphysical sheets. Along with the complex scaling, various diﬀerent methods ar e also used for calculations of the resonances. Among the methods develope d to calculate directly the scattering-matrix resonances we, ﬁrst, menti on the approach based on the momentum space Faddeev integral equations [7,8] (see , e.g., Ref. [9] and references cited therein). In this approach one numericall y solves the equations continued into an unphysical sheet and, thus, the three-bod y resonances arise as the poles of the continued T-matrix. Another approach to c alculation of the scattering-matrix resonances is based on the explicit repr esentations [10,11] for the analytically continued T- and S-matrices in terms of the physical sheet. From these representations one infers that the three-body r esonances can be found as zeros of certain truncations of the scattering matr ix only taken in the physical sheet. Such an approach can be employed even in t he coordinate space [11,12]. To the best of our knowledge there are no published works appl ying the com- plex scaling to the Faddeev equations. Therefore, we consid er the present investigation as a ﬁrst attempt undertaken in this directio n. However, the pur- pose of our work is rather two-fold. On the one hand, we make th e complex scaling of the Faddeev diﬀerential equations. On the other h and we compare the complex scaling method with the scattering-matrix appr oach suggested in [11,12]. We do this making use of both the approaches to exa mine reso- nances in a model system of three bosons having the nucleon ma sses and in the three-nucleon ( nnp) system itself. 2 Formalism First, we recall that, after the scaling transformation, th e three-body Schr¨ odinger operator reads as follows [1–3] H(ϑ) =−e−2ϑ∆X+3/summationdisplay α=1Vα(eϑxα) (1) whereϑ= iθis the scaling parameter with θ∈R. By ∆ Xwe understand the six–dimensional Laplacian in X≡(xα,yα) where xα,yαare the standard Jacobi variables, α= 1,2,3. Notation Vαis used for the two-body potentials which are assumed to depend on xα=\|xα\|but not on /hatwidexα=xα/xα. The corresponding scaled Faddeev equations which we solve r ead 2[−e−2ϑ∆X+vα(eϑxα)−z]Φ(α)(z;X) +Vα(eϑxα)/summationdisplay β/negationslash=αΦ(β)(z;X) =fα(X), α= 1,2,3. (2) Heref= (f1,f2,f3) is an arbitrary three-component vector with components fαbelonging to the three-body Hilbert space L2(R6). The partial-wave version of the equations (2) for a system of three identical bosons at the zero total angular momentum L= 0 reads e−2iθH(l) 0Φl(z;x,y)−zΦl(z;x,y) +V(eiθx)Ψl(z;x,y) =f(l)(x,y) (3) wherex>0,y>0 andH(l) 0denotes the partial-wave kinetic energy operator, H(l) 0=−∂2 ∂x2−∂2 ∂y2+l(l+ 1)/parenleftBigg1 x2+1 y2/parenrightBigg , l = 0,2,4,..., while Ψ lstands for the partial-wave component of the total wave func tion, Ψl(z;x,y) = Φ l(z;x,y) +/summationdisplay l′+1/integraldisplay −1dηh ll′(x,y,η ) Φl′(z;x′,y′). (4) Here,x′=/radicalBig 1 4x2+3 4y2−√ 3 2xyηandy′=/radicalBig 3 4x2+1 4y2+√ 3 2xyη. Explicit expression for the geometric function hll′(x,y,η ) can be found, e.g., in [8]. The partial-wave equations (3) are supplied with the bounda ry conditions Φl(z;x,y)\|x=0= 0 and Φ l(z;x,y)\|y=0= 0. (5) For compactly supported inhomogeneous terms f(l)(x,y) the partial-wave Fad- deev component Φ l(z;x,y) also satisﬁes the asymptotic condition Φl(z;x,y) =δl0ψd(eiθx) exp(i√z−ǫdeiθy)/bracketleftBig a0(z) +o/parenleftBig y−1/2/parenrightBig/bracketrightBig +exp(i√zeiθρ)√ρ/bracketleftBig Al(z;y/x) +o/parenleftBig ρ−1/2/parenrightBig/bracketrightBig ,(6) For simplicity it is assumed in this formula that the two-bos on subsystem has only one bound state with the energy ǫd, andψd(x) represents its wave function. The values of a 0andAl(y/x) are the main asymptotical coeﬃcients eﬀectively describing the contributions to Φ lfrom the elastic (2 + 1 →2 + 1) and breakup (2 + 1 →1 + 1 + 1) channels, respectively. Hereafter, by√ζ, ζ∈C, we understand the main (arithmetic) branch of the function ζ1/2. 3In the scaling method a resonance is looked for as the energy zwhich produces a pole to the quadratic form Q(θ,z) =/angbracketleftBig [HF(θ)−z]−1f,f/angbracketrightBig whereHF(θ) is the non-selfadjoint operator resulting from the comple x-scaling transformation of the Faddeev operator. The latter operato r is just the opera- tor constituted by the l.h.s. parts of Eqs. (2). The resonanc e energies should not, of course, depend on the scaling parameter θand on the choice of the termsf(l)(x,y). In the scattering-matrix approach we solve the same partial -wave Faddeev equations (3) with the same boundary conditions (5) and (6) b ut forθ= 0 and f(l)(x,y) =−V(x)+1/integraldisplay −1dηh l0(x,y,η )ψd(x′) sin(√z−ǫdy′). The resonances are looked for as zeroes of the truncated scat tering-matrix (see [12] for details) s 0(z) = 1+2ia 0(z), where the (1+1 →1+1) elastic scattering amplitude a 0(z) for complex energies zin the physical sheet is extracted from the asymptotics (6). For numerical solution of the boundary-value problem (3–6) we employ its ﬁnite-diﬀerence approximation in the hyperradius-hypera ngle coordinates. A detail description of the ﬁnite-diﬀerence algorithm used c an be found in Ref. [13]. 3 Results In the table we present our results obtained for a complex-sc aling resonance in the model three-body system which consists of identical b osons having the nucleon mass. To describe interaction between them we emplo y a Gauss-type potential of Ref. [12] V(x) =V0exp[−µ0x2] +Vbexp[−µb(x−xb)2] withV0=−55MeV,µ0= 0.2fm−2,xb= 5fm,µb= 0.01fm−2andVb= 1.5. The ﬁgures in the table correspond to the roots of the inver se function [Q(θ,z)]−1forL= 0 andl= 0 only taken into account. In the present calculation we have taken up to 400 knots in both hyperradius and hyper- angle variables while for the cut-oﬀ hyperradius we take 40f m. One observes from the table that the position of the resonance depends ver y weakly on the scaling parameter θwhich conﬁrms a good numerical quality of our re- sults. We compare the resonance values of the table to the res onance value 4Fig. 1. Trajectory of the resonance zresin the model system of three bosons with the nucleon masses. Values of the barrier Vbin MeV are given near the points marked on the curve. zres=−5.952−0.403 iMeV obtained for the same three-boson system with exactly the same potentials but in the completely diﬀerent s cattering-matrix approach of Ref. [12]. We see that, indeed, both the complex s caling and the scattering matrix approaches give the same result. θzres(MeV) θzres(MeV) 0.25 −5.9525−0.4034 i 0.50 −5.9526−0.4032 i 0.30 −5.9526−0.4033 i 0.60 −5.9526−0.4033 i 0.40 −5.9526−0.4032 i 0.70 −5.9526−0.4034 i We also watched the trajectory of the above resonance when th e barrier ampli- tudeVbvaried (see. Fig. 1). While the complex scaling method was ap plicable it gave practically the same positions for the resonance. Fo r the barrier ampli- tudesVbsmaller than 1.0 only the scattering-matrix approach allow s to locate the resonance (which ﬁnally, for Vb<0.85, turns into a virtual level). As to thennpsystem in the S–state where we employed the MTI–III [14] potential model, both the methods applied give no resonance s on the two- body unphysical sheet (see [12]). Moreover, we have found no resonances in the part of the three-body sheet accessible via the complex s caling method. Thus, at least in the framework of the MTI–III model we can not conﬁrm the experimental result of Ref. [16] in which the point −1.5±0.3−i(0.3±0.15)MeV was interpreted as a resonance corresponding to an exited st ate of the triton 3H. The triton virtual state can be only calculated within the sc attering-matrix method but not in the scaling approach. Our present improved scattering- matrix result for the triton virtual state is −2.690MeV (i.e. the virtual level lies 0.47MeV below the two-body threshold). This result has been obtained with the MTI-III potential on a grid having 1000 knots in both hyperradial 5and hyperradial variables and with the value of cut-oﬀ hyper radius equal to 120fm. Notice that some values for the virtual-state energy obtained by dif- ferent authors can be found in [9] and all of these values are a bout 0.5MeV below the two-body threshold. References [1] E. Balslev, J. M. Combes, Commun. Math. Phys., 22(1971), 280. [2] M. Reed, B. Simon, Methods of modern mathematical physics. IV: Analysis of operators , Academic Press, N.Y., 1978. [3] G. A. Hagedorn, Comm. Math. Phys. 65(1979), 81. [4] Y. K. Ho, Phys. Rep. 99(1983), 3; Chin. J. Phys. 35(1997), 97. [5] B. R. Junker, Adv. Atom. Mol. Phys. 18(1982), 208. [6] W. P. Reinhard, Ann. Rev. Phys. Chem. 33(1982), 223. [7] L. D. Faddeev, Mathematical aspects of the three–body problem in quantum mechanics , Israel Program for Scientiﬁc Translations, Jerusalem, 19 65. [8] L. D. Faddeev, S. P. Merkuriev, Quantum scattering theory for several particle systems, Kluwer Academic Publishers, Dorderecht, 1993. [9] K. M¨ oller, Yu. V. Orlov, Fiz. Elem. Chast. At. Yadra. 20(1989), 1341 (Russian). [10] A. K. Motovilov, Theor. Math. Phys. 95(1993), 692. [11] A. K. Motovilov, Math. Nachr. 187(1997), 147. [12] E. A. Kolganova, A. K. Motovilov, Phys. Atom. Nucl. 60(1997), 235. [13] E. A. Kolganova, A. K. Motovilov, S. A. Soﬁanos, J. Phys. B.31(1998), 1279. [14] R.A.Malﬂiet, J.A.Tjon, Nucl. Phys. A 127(1969), 161. [15] Yu. V. Orlov, V. V. Turovtsev, JETP 86(1984), 1600 (Russian). [16] D. V. Alexandrov et. al. , JETP Lett. 59(1994), 320 (Russian). 6
arXiv:physics/0011061v1 [physics.bio-ph] 24 Nov 2000Predicting Optimal Lengths of Random Knots February 2, 2008 Akos Dobay1, Pierre-Edouard Sottas1,2, Jacques Dubochet1and Andrzej Stasiak1 1Laboratory of Ultrastructural Analysis, University of Lau sanne, 1015 Lausanne, Switzerland 2Center for Neuromimetic Systems, Swiss Federal Institute o f Technology, EPFL-DI, 1015 Lausanne, Switzerland Abstract In thermally ﬂuctuating long linear polymeric chain in solu tion, the ends come from time to time into a direct contact or a close vic inity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show their highest occurrence for shorter random walks than more compl ex knots. However up to now there were no rules that could be used to pred ict the optimal length of a random walk, i.e. the length for which a gi ven knot reaches its highest occurrence. Using numerical simulatio ns, we show here that a power law accurately describes the relation between t he optimal lengths of random walks leading to the formation of diﬀerent knots and the previously characterized lengths of ideal knots of the c orresponding type. keywords: knots, polymers, scaling laws, DNA, random walks. A random walk can frequently lead to the formation of knots an d it was proven that as the walk becomes very long the probability of f orming nontrivial knots upon closure of such a walk tends to one [1, 2]. Many diﬀe rent simula- tion approaches were used to study random knotting [3, 4, 5, 6 , 7]. Probably the most fundamental one is by simulation of ideal random cha ins where each segment of the chain is of the same length and has no thickness [4, 8]. In ideal random chains the neighboring segments are not correlated w ith each other and thus show the average deﬂection angle of 90◦. Ideal random chain behavior is interesting from physical point of view as it reﬂects statis tical behavior of long 1polymer chains in so-called melt phase and in θsolvents where excluded volume eﬀect vanishes [8]. Highly diluted polymer chains in θsolvents are unlikely to interact with each other and therefore upon circularizatio n will form mainly knots rather than links. In thermally ﬂuctuating long linea r polymers the ends of the same chain can come from time to time into a close vicini ty of each other. This can lead to a cyclization of the polymer whereby the end c losure frequently traps a nontrivial knot on the chain. By studying knotting in simulated ideal random chains we thus can gain insight into knotting of real p olymer chains in θsolvents and in the dense melt phase frequently used for the p reparation of such synthetic polymeric materials like fabrics, paints or adhesives [9]. However, ideal chains do not reﬂect the behavior of real polymer chain s in good solvent. Intramolecular interactions cannot be neglected in these c onditions, but can be well approximated by introducing an eﬀective diameter. Whe n such a constraint is introduced into simulated chains one can also model knott ing of polymers in good solvents like for example knotting of DNA molecules in t ypical reaction buﬀers used for biochemical experiments [4]. Our simulatio ns can be adjusted to both situations and we shall present here results for rand om chains with and without an eﬀective diameter. Several earlier studies of random knotting showed that simp ler knots reach a maximum of their occurrence for shorter length of random wal ks than this re- quired for the formation of more complex knots [5, 6, 10]. In c onsidering the equilibrium ensemble of closed walks, these studies showed that the relative fre- quency of occurrence of each type of knot ﬁrst increases with the length of the chain, then passes through a maximum and ﬁnally decreases ex ponentially at very long chains. However, these earlier studies did not att empt to establish a relation between the type of a knot and the optimal length of a random walk leading to the maximal occurrence of this knot. If we conside r a thermally ﬂuc- tuating polymer with ends that can stick to each other with th e energy much smaller than kT, then from time to time these ends will stay in contact for a short period and at this moment the polymer will form a trivia l or nontrivial knot. In this study, we characterize statistical ensembles of ﬂuctuating linear polymers in order to ﬁnd speciﬁc lengths (expressed in numbe r of statistical segments) at which a given type of knot or rather a virtual kno t reaches its highest occurrence. Recently we have characterized ideal geometric conﬁgurati ons of knots corre- sponding to the shortest trajectories of ﬂexible cylindric al tube with uniform diameter to form a given knot [11]. The ratio of the length to d iameter of the tube forming ideal conﬁguration of a given knot is a topol ogical invariant and we call it here the length of ideal knots. Ideal knots turn ed out to be good predictors of statistical behavior of random knots. So for example the writhe of ideal conﬁguration of a given knot was equal to the a verage writhe of thermally ﬂuctuating polymer forming a given random knot [1 1]. We showed also that electrophoretic migrations of various types of kn otted DNA molecules 2of the same molecular weight or their expected sedimentatio n constants were practically proportional to the length of the correspondin g ideal knots [12, 13]. Therefore we decided here to check whether the length of idea l knots is related to the length of ideal random chains for which diﬀerent knots reach their high- est occurrence. To this aim we used the following simulation procedure. 2 ·109 random walks of 170 segments were started and each time the gr owing end ap- proached the starting end to a distance smaller than the leng th of one segment the conﬁguration was saved upon which the walk was continued for the remain- ing number of steps. Each vector (segment) of the chain was ra ndomly chosen from uniformly distributed vectors pointing from the cente r to the surface of the unit sphere. Thus some of the random walks showed one or more a pproaches of the growing and starting ends and we collected 2 ·109random walks for every number of segments between 5 and 170. Each saved conﬁguratio n with nearby ends was then closed with a connecting segment and the type of the formed knot was determined by the calculation of its Alexander polynomi al [7, 14, 15, 16]. For random linear walks to eﬃciently form diﬀerent knots a co mpromise has to be met between the length optimizing their close approach an d the length which is suﬃcient to form a knot of a given type. The present analysi s diﬀers from earlier studies [4, 5, 10] where the statistics was based onl y on equilibrium knot- ting of closed walks. In our case, we consider the formation o f knots through the approach of the terminal segments of linear chains. Therefo re not only closed chains, but also linear chains are taken into account in our s tatistics. Figure 1 shows the occurrence proﬁles of diﬀerent knots with up to six cross- ings as a function of the length of random walk which leads to t he formation of these knots. It is visible that trefoil knots show their hi ghest occurrence for 25±1 segments while 4 1knots form most frequently for 42 ±1 segments. The formation of more complicated knots happens much less frequ ently than this of simpler knots, therefore in the insert in Figure 1 a change of scale is applied to better visualize the occurrence of more complicated knots. We observed that the obtained probabilities values for diﬀerent knots can be well ﬁtted with the function Pk(N) =a(N−N0)bexp(−Nc d) (1) where for each knot a,banddare free parameters, cis an empirical constant equal to 0 .18,N0is the minimal number of segments required to form a given type of knot [17] without the closing segment and Nis the number of segments in the walk. Our ﬁtting function was adapted from Katritch et al.2000 [18] but modiﬁed to take into account the probability of cyclizat ion. Table 1 lists the positions of maximal occurrence for the analyzed types o f knots. In order to concentrate on the position of the maximum for diﬀerent kn ots and not on their actual probability values we decided to present proba bility proﬁles for each knot upon normalizing them by assigning a value 1 to the respe ctive maximum of probabilities. 3Figure 1: Probability of forming a given knot amoung all rand om walks of a given size is plotted as a function of the number of segments i n the walk. Note the change of the scale between the main panel and the insert. Diagrams of the corresponding knots are drawn to visualize the diﬀerences b etween analyzed types of knots. The notations accompanying the drawn diagra ms correspond to those in standard tables of knots [21], where the main numb er indicates the minimal number of crossings possible for this knot type and t he index indicates the tabular position amongst the knots with the same minimal crossings number. Formed knot types were recognised by computing their Alexan der polynomial. Since Alexander polynomial does not distinguish between le ft-handed and right- handed knots of the same type, we have to group them together a nd therefore the drawn diagrams of the knots do not show the handedness. Th is polynomial has sometimes the same value for diﬀerent knots like for exam ple knot 6 1and 9 46 [22]. However within groups of knots with the same Alexander polynomial more complicated knots have such a low occurrence that their eﬀec t on the position of the maximum of the simplest knot within the group can be neg lected. 4Figure 2 presents normalized probability proﬁles for the an alyzed knots. It is visible that diﬀerent knots show now quite similar type of pr oﬁles (e.g. knot 51and 5 2) whereby the diﬀerences in the position of maximum between k nots with diﬀerent minimal number of crossings can be easily perc eived. It may be surprising that we observed here such a short optimal length for analyzed knots while earlier studies showed that several hundred segments are needed to ob- serve maximum occurrence of a given knot among closed walks o f a given size [5, 10, 19]. This is simply due to the fact that our system take s into account the probability of cyclization. Figure 2: Normalized probability proﬁles for the analyzed k nots. In Figure 3 we show the relation between the optimal length of random knots and the length of the corresponding ideal knots. This relati on is well approxi- mated by a power law function. Upon ﬁtting the free parameter s of this function in the simulation data obtained for the knots with up to 7 cros sings, we decided to check if by knowing the length of ideal conﬁgurations of mo re complicated knots we can predict positions of the maximum of occurrence f or the corre- sponding random knots. As the statistics of random knotting gets poor for knots with increasing crossing number we limited veriﬁcati ons of our predic- 5tions to these knots with eight crossings which at their maxi ma of occurrence were represented more than 500 times out of 2 ·109random walks with a given number of segments. Analysis of our simulation data (Figure 3) positively veri- ﬁed our predictions for optimal sizes of random walks leadin g to the formation of these knots. Figure 3: Relation between the length of the ideal geometric representations of knots [23] and positions of maximal occurrence for the corre sponding random knots. The lower curve: the optimal length of random knots wi th an eﬀective di- ameter set to zero. The simulation data for the knots with up t o seven crossings were ﬁtted with a power law function and the best ﬁt curve was e xtrapolated. Data points for eight crossing knots for which we obtained go od statistics co- incide with the extrapolated curve. The upper curve: data po ints of maximal occurrence of knots for random chains with an eﬀective diame ter set to 0.05 of the segment length. In both cases a power law function adequa tely describes the relation between the optimal length of random knots and t he length of ideal knots of a given type. Best ﬁt parameters for both cases are in dicated. As already mentioned, ideal random chains have no thickness and this causes that they reﬂect the behavior of polymers in the melt phase wh ere thin polymers have practically no exclusion volume [7, 8]. However when po lymers are sus- pended in a good solvent, like DNA in aqueous solution, the ex clusion volume of polymers becomes not negligible and this strongly decrea ses the probability of forming knots [7]. It was observed that the higher the eﬀec tive diameter of 6the polymer the lower the probability of forming knots by ran dom cyclization [4, 7, 19]. We decided therefore to investigate whether a pow er law relation between the length of ideal knots and the optimal length of ra ndomly knotted chains also holds for chains with an exclusion volume. To thi s aim from our original set of 2 ·109ideal random walks for every segment length from 5 to 100 we selected the walks which never showed a closer approach be tween any pair of non neighboring segments than the considered eﬀective di ameter (terminal segments of the chain are considered as neighboring ones). S ubsequently we analyzed all conﬁgurations with approached ends for the typ es of formed knots and calculated the probabilities of various knots among all random chains which fulﬁlled the criteria of a given eﬀective diameter. We obser ved that as the ef- fective diameter grows the probability of forming various k nots decreases and positions of the maximum move toward longer chains. Figure 3 (dashed line) shows the relation between the length of ideal knots and the o ptimal length of corresponding random knots formed by chains with the eﬀecti ve diameter being set to 0.05 of the segment length. The eﬀective diameter 0.05 corresponds to this of diluted solutions of DNA molecules in about 100 mM NaC l [4]. In the case of DNA each segment in the random chain corresponds to 30 0 base pair long region [20]. It is visible that the data can be again appr oximated by a power law function. Fact that lengths of ideal knots shows a c orrelation with the optimal sizes of corresponding random knots formed by ch ains with a given eﬀective diameter provides another example that ideal knot s are good predictors of physical behavior of real knots [11]. Post factum it might seem to be obvious that knots requiring h igher length of the rope to tie them should require higher length of a random w alk to reach their highest occurrence. However until recently the minim al length of the rope to tie a given knot was not known. In addition the relation bet ween the optimal length of random walk producing a given knot and the length of ideal knot was not yet proposed in the literature. On the other hand a simple expectation would dictate that the shorter the length of ideal knot the hi gher the prob- ability of its formation. So for example trivial knots are mo re frequent than trefoils and these are more frequent than 4 1knots. However this does not hold for 5 1and 5 2knots. Ideal knot 5 1is slightly shorter than ideal 5 2knot (which is consistent with the optimal size of random walks leading t o the formation of corresponding knots), but 5 2knot formation by random walks is circa twice more frequent than formation of 5 1knot. Therefore the values of random knots probabilities (in contrast to the positions of the maxima) a re not related by a relatively simple growing function to the values of lengths of the corresponding ideal knots. What can be the possible applications resulting from the det ermination of the optimal size of knots? For chemical cyclization of polymer c hains we can use linear polymer of a speciﬁc length and thus promote formatio n of a given type of knot. Materials with interesting properties could be for med by this way. 7Table 1: Optimal sizes Osof random walks (in number of segments) leading to the formation of corresponding knots, the length/diamet er ratio LDvalues of ideal conﬁgurations of these knots Kn[23] and the values of the parameters in the ﬁts of the observed probabilities (see Fig. 1). The pre sented data are limited to knots with up to 7 crossings since obtained by us, s tatistics for more complex knots is less good. KnOs LD a b d N 0 3125±1 16.33 (1.84 ±0.01)×10−11.57±0.01 0.165 ±0.001 5 4142±1 20.99 (0.45 ±0.01)×10−12.24±0.01 0.134 ±0.001 6 5154±2 23.55 (1.28 ±0.02)×10−22.65±0.01 0.121 ±0.001 7 5256±2 24.68 (2.31 ±0.04)×10−22.77±0.01 0.118 ±0.001 7 6174±2 28.30 (0.78 ±0.03)×10−23.75±0.02 0.095 ±0.001 7 6275±2 28.47 (0.74 ±0.03)×10−23.67±0.02 0.096 ±0.001 7 6376±2 28.88 (0.39 ±0.02)×10−23.69±0.02 0.097 ±0.001 7 7189±3 30.70 (4.09 ±0.47)×10−73.95±0.06 0.093 ±0.001 8 7292±3 32.41 (1.72 ±0.16)×10−34.33±0.05 0.086 ±0.001 8 7392±3 31.90 (9.43 ±0.85)×10−44.03±0.05 0.092 ±0.001 8 7497±3 32.53 (5.55 ±0.67)×10−44.25±0.06 0.087 ±0.001 8 7597±3 32.57 (1.32 ±0.09)×10−34.24±0.04 0.089 ±0.001 8 7698±3 32.82 (1.71 ±0.14)×10−34.36±0.04 0.086 ±0.001 8 7795±3 32.76 (8.82 ±0.96)×10−44.31±0.06 0.087 ±0.001 8 Acknowledgment. We thank Alexander Vologodskii and Vsevolod Katritch for making available their routine for the calculation of Alexa nder polynomial. We also thank Piotr Pieranski for numerous discussions. This w ork was supported by the Swiss National Science Foundation (31-58841.99 and 3 152-061636.00). References [1] Sumners, D.W. and Whittington, S.G: Knots in self avoidi ng walks, J. Phys. A: Math. Gen. 21(1988), 1689-1694 [2] Matthews, R.: Knotted rope: a topological example of Mur phy’s law, Math- ematics Today 33(1997), 82-84. [3] Frank-Kamenetskii, M.D., Lukashin, A.V. and Vologodsk ii, A.V.: Statisti- cal mechanics and topology of polymer chains, Nature 258(1975), 398-402. [4] Rybenkov, V.V., Cozzarelli, N.R. and Vologodskii, A.V. : The probability of DNA knotting and the eﬀective diameter of the DNA double he lix,Proc. Nat. Acad. Sci. U.S.A. 90(1993), 5307-5311. 8[5] Deguchi, T. and Tsurusaki, K.: A statistical study of ran dom knotting using the Vassiliev invariants, J. Knot Theo. Ram. 3(1994), 321-353. [6] Janse van Rensburg, E.J., Orlandini, E., Sumners, D.W., Tesi, M.C. and Whittington, S.G.: J. Knot Theo. Ram. 6, (1997), 31-44. [7] Koniaris, K. and Muthukumar, M.: Knottedness in ring pol ymers, Phys. Rev. Lett. 66(1991), 2211-2214. [8] de Gennes, P.G.: Scaling Concepts in Polymer Physics , Cornell University Press, Ithaca, New York, 1979. [9] Alper, J. and Nelson, G.L.: Polymeric Materials: Chemistry for the Future , American Chemical Society, Washington, DC, 1989. [10] Orlandini, E., Janse van Rensburg, E.J., Tesi, M.C. and Whittington, S.G.: Entropic exponents of knotted lattice polygons, In: Topology and Geome- try in Polymer Science , edited by Whittington, S.G. and Sumners D.W., Springer, New York, 1998, pp. 9-21. [11] Katritch, V., Bednar, J., Michoud, D., Scharein, R.G., Dubochet, J. and Stasiak, A.: Geometry and physics of knots, Nature (London) 384(1996), 142-145. [12] Stasiak, A., Katritch, V., Bednar, J., Michoud, D. and D ubochet, J.: Elec- trophoretic mobility of DNA knots, Nature (London) 384(1996), 122. [13] Vologodskii, A., Crisona, N., Laurie, B., Pieranski, P ., Katritch, V., Dubo- chet, J. and Stasiak, A.: Sedimentation and electrophoreti c migration of DNA knots and catenanes, J. Mol. Biol. 278(1998), 1-3. [14] Alexander, J.W.: Topological invariants of knots and l inks,Trans. Amer. Math. Soc. 30(1928), 275-306. [15] Frank-Kamenetskii, M.D. and Vologodskii, A.V.: Topol ogical aspects of polymer physics: theory and its biophysical applications, Sov. Phys. Usp. 24(1981), 679-696. [16] Adams, C. C.: The Knot Book , Freeman, W.H. and Company, New York, 1994. [17] Calvo, J.A. and Millett, K.C.: Minimal edge piecewise l inear knots, In: Ideal Knots , edited by Stasiak, A., Katritch, V. and Kauﬀman, L.H. Edi- tors, World Scientiﬁc Publishing Co, Singapore, 1998, pp. 1 07-128. [18] Katritch, V., Olson, W.K., Vologodskii, A., Dubochet, J. and Stasiak, A.: Tightness of random knotting, Phys. Rev. E 61(2000), 5545-5549. 9[19] Deguchi, T. and Tsurusaki, K., In: Lectures at Knots 96 , edited by Suzuki, S., World Scientiﬁc Publishing Co, Singapore, 1997, p. 95. [20] Vologodskii, A.V.: Topology and physics of circular DNA . Physical ap- proaches to DNA, CRC Press, Boca Raton, 1992. [21] Rolfsen, D.: Knots and links , Publish or Perish Press, Berkeley, CA., 1976. [22] de la Harpe, P.: Introduction to knot and link polynomia ls, In: Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechani cs, edited by Amann, Cederbaum, A.L. and Gans, W. Editors, Kluwer Acade mic Publishers, Dordrecht, 1988, pp. 233-263. [23] Stasiak, A., Dubochet, J., Katritch, V. and Pieranski, P.: Ideal knots and their relation to the physics of real knots, In: Ideal Knots , edited by Stasiak, A., Katritch, V. and Kauﬀman, L.H. Editors, World Scientiﬁc Publishing Co, Singapore, 1998, pp. 1-19. 10
1The Mass Discrepancies That Are Implied By Galactic Rotation Curves [ i.e. Dynamically Inferred Mass Minus Observed Mass ] Are Not Correctly Resolved by the Proposed Hypotheses: [That Galaxies Contain a Large Amount of "Dark Matter"] [or of a Modification of Newtonian Dynamics, MOND]; Rather, The Discrepancies Are Apparent, Not Real, and Are a Further Effect or Aspect of the Exponential Decay of the Overall Universe. Roger Ellman Abstract In general, galaxies are rotating systems, a balance of gravitational attraction [G·M·m/R2] and centripetal force [ m·V2/R] maintaining the structure. When the central mass is far greater than the orbiting masses the dynamics, referred to as Keplerian, aresuch that the orbital velocities are inversely proportional to the square root of the radialdistance from the central mass [ V =(G·M/R)½], as for example in our solar system. A curve or plot of velocity vs. path radius is termed a Rotation Curve. For galaxies that we view as the thin disk not the spiral or globular spread in space, we see one end moving toward us relative to the center and the other end movingaway. The rotational velocities are measured by observing the variations in redshiftalong the galactic diameter represented by the disk. Galactic rotation curves so obtainedare not of the expected Keplerian form, an inverse square root; rather, [beyond the radiusof the dense galactic central core] the curves are essentially flat. In a solid sphere, where the density is uniform throughout, all parts move at rotational velocities directly proportional to radius. Since the observed flat galacticrotation curves fall between the Keplerian inverse square root of radius and the solid'sdirect proportion to radius, it is inferred that matter that we have not observed must bedispersed throughout the galaxy, a halo of "dark matter" that causes the rotation to takethe form that the rotation curve exhibits -- thus the "dark matter" hypothesis. 1 The mass discrepancy phenomenon only appears where the acceleration is V2/R < ∼ 10-8 cm/sec2. Modeling related to that gives an alternative hypothesis: Modification of Newtonian Dynamics or MOND, that gravity and or inertia behave in amodified manner when g or a < 10-8 cm/sec2. No justification has been developed other than that the hypothesis correlates with the mass discrepancies.2,3 An alternative explanation is presented -- the general exponential decay of the overall universe, which has been analyzed and developed in several papers.4,5,6,7,8 The universal decay accounts for the mass discrepancies because the effect of the decay is tomake the rotation curves appear to deviate from the form expected in a Keplerian outergalactic disk although the actual rotational behavior does not. Roger Ellman, The-Origin Foundation, Inc. 320 Gemma Circle, Santa Rosa, CA 95404, USA RogerEllman@The-Origin.org2The Mass Discrepancies That Are Implied By Galactic Rotation Curves [ i.e. Dynamically Inferred Mass Minus Observed Mass ] Are Not Correctly Resolved by the Proposed Hypotheses: [That Galaxies Contain a Large Amount of "Dark Matter"] [or of a Modification of Newtonian Dynamics, MOND]; Rather, The Discrepancies Are Apparent, Not Real, and Are a Further Effect or Aspect of the Exponential Decay of the Overall Universe. Roger Ellman Background Of The Problem [While unnecessary for astronomers and astrophysicists this review is included for the benefit of other scientists, who may not be familiar with the details of thedevelopment, details which are essential to understanding the issues.] In general, galaxies are rotating systems, a balance of gravitational attraction [ G·M·m/R2] and centripetal force [ m·V2/R] maintaining the structure. A curve or plot of rotational velocity vs. path radius is termed a Rotation Curve. When the central mass is far greater than the orbiting masses the dynamics are such that the orbital velocities are inversely proportional to the square root of the radialdistance from the center mass [ V =(G·M/R)½], as for example in our solar system and as illustrated in Figure 1, below. Such rotational dynamics and rotation curves are referredto as Keplerian. Figure 1 - A Keplerian Rotation Curve In the case of a solid sphere, where the density is uniform throughout, all parts move at rotational velocities directly proportional to radius as illustrated in Figure 2,below. 3Figure 2 - The Rotation Curve of a Solid Sphere The form of galaxies as we are able to directly observe them is that of a fairly spherical dense central core and a transition from that to the much more extensive flatdisk which has a far smaller density of more widely dispersed stars. The portion ofgalactic rotation curves that pertains to the dense central core of the galaxy would beexpected to exhibit approximately the same velocity - proportional - to - radius form asillustrated for a solid sphere in Figure 2. Likewise, the more dispersed flat disk, minor inmass compared to the dense central core, would be expected to exhibit the Keplerianform of Figure 1. Consequently, the expected form of galactic rotation curves would be that of the above Figures 1 and 2 combined plus a smooth transition between the two regions, asillustrated in Figure 3, below. Figure 3 - The Expected Form of Galactic Rotation Curves Based on Their Observable Matter For galaxies that present themselves in an edge view of the thin disk not as their spiral or globular spread in space, it is possible to measure the rotational velocities andobtain a rotation curve. We see one end of the presented flat disk moving toward usrelative to the center and the other end moving away. The rotational velocities are 4measured along the galactic diameter represented by our view of the disk by observing the variations in redshift, the variations being a Doppler effect. Galactic rotation curves so obtained turn out not to exhibit the expected Keplerian form, an inverse square root of radius. Rather, actual galactic rotation curvesderived from redshift observations along the disk exhibit a flat form, that is they exhibitrotational velocity independent of radius. The overall curve, after the portion pertainingto the dense central core of the galaxy, is a transition to a flat curve in the regioncorresponding to the spread-out galactic disk as in Figure 4, below. Figure 4 - A Typical Actual Galactic Rotation Curve as Obtained From Galactic Disk Measurements of Redshifts Because the form of the flat portion of galactic rotation curves lies between the case of a dominant central mass, as in the Keplerian inverse square root of radius form[Figure 1], and the case of a uniformly dense mass, with its direct proportion to radiusform [Figure 2], it has been inferred that matter that we have not observed must bedispersed throughout the galaxy as a halo of "dark matter" that causes the rotation to takethe form that the rotation curve exhibits -- thus the "dark matter" hypothesis. 1 This mass discrepancy phenomenon only appears where the acceleration is V2/R < ∼ 10-8 cm/sec2. Modeling related to that gives an alternative hypothesis: Modification of Newtonian Dynamics or MOND, that gravity and or inertia behave in amodified manner when ag or ai < 10-8 cm/sec2. No justification or cause for that behavior has been developed other than that as a hypothesis it gives results that correlatewith the mass discrepancies. 2,3 Flat rotation curves are observed not only in individual galaxies, but in galaxy clusters and in even larger groups of galactic clusters, all of which rotate under the same G·M·m/R2 = m·V2/R. law. As a consequence it has been hypothesized that there is a substantial amount of "dark matter" throughout the universe to the largest scale. But, there is an explanation of these data alternative to that of "dark matter" or MOND, an explanation that avoids the several problems that have been encountered inexploring and developing the "dark matter" galactic halos and MOND 1,2, and which correlates with other recent discoveries.6,7 That is the general exponential decay of the overall universe, which has been analyzed and developed in several papers. 4,5,6,7,8 Exponential decay is found throughout nature so that overall decay of the universe is notunreasonable. 5The universal decay accounts for essentially all of the rotational dynamics mass discrepancies [there may well be, nevertheless, some modicum of non-luminous matter ingalaxies and throughout the universe]. The General Universal Decay The general exponential decay of the overall universe is derived and developed in The Origin and Its Meaning . 8 The decay is of the same form as the myriad exponential decays found throughout nature because all such decays are aspects of the generalsolution to the 2 nd order linear differential equation with constant coefficients. The universal decay is decay of the quantities that we refer to as the fundamental constants, c, h, q, G, and so forth, essentially a decay of the fundamental substance of material reality. The values of these fundamental constants are the same everywhere in the universe at any instant of time. The decay means that they are everywhere uniformly andconsistently exponentially decaying with time. That the laws of physics and theirfundamental constants must be the same everywhere in the universe [Einstein's"invariance"] includes within it the decay processes acting consistently everywhere. These fundamental constants interact through the various physical laws of nature and, therefore, the decay of each constant must be consistent with the decays of all of theothers. Analysis of all of the implications of that requirement shows that the decay is ofthe length dimensional component of those constants. That is, from among thefundamental dimensional components length [L], mass [M] and time [T], it is length [L] that is in decay. That develops as follows. The decay being an exponential function the independent variable of which is time, t, as in for example equation 1, it cannot be the time dimensional component, [T], that is decaying. (1) -t/τ c(t) = c0·ε Furthermore, mass and time, the inverse of frequency, are closely interrelated as in equation 2, (2) h·f = E = m·c2 so that if mass, [M], were to decay it would imply that frequency decays and that time, [T], the inverse of frequency, inversely decays, which the independent variable cannot do. That leaves only length, [L], to be the dependent variable in the decay. Equation 2 also illustrates another point. Planck's constant, h, appears in the equation with an exponent of 1 whereas the speed of light, c, appears with an exponent of 2. For the two decays, that of h and that of c, to be consistent their time constants must be different. Planck's constant, h, must decay twice as rapidly as the speed of light, c; its time constant, τh, must be half that of light, τc. That is, (3) [ -t/τ ]1 [ -t/τ ]2 [ ε h ] = [ε c ] for consistency of the decays, ∴∴∴∴ τh = 0.5 · τc The time constant of the general exponential decay of the overall universe is derived and calculated in The Origin and Its Meaning.8 The value for c, the "fundamental" value as compared to that for, for example, h = 1/2 of that for c, is (4) τc = 3.57532·1017 sec = about 11.3 billion years "c" dimensions are L1/T6The values for other constant's decays are the appropriate multiple or sub- multiple of the value for c. For example: (5) τh = 1/2·τc = 1.78766·1017 sec = about 5.65 billion years "h" dimensions are M· L2/T τG = 1/3·τc = 1.19177·1017 sec = about 3.77 billion years "G" dimensions are L3/M·T2 Occurrences of the Effect of the Decay The first definitive experimental observation of this decay was in the tracking of the Pioneer 10 and 11 satellites. The observations were reported in 1998 in Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration. 7 and were further analyzed in 1999 in The Apparent Anomalous, Weak, Long-Range Acceleration of Pioneer 10 and 11.10 These reported that a weak long-range acceleration towards the Sun has been observed in the Pioneer 10 and 11satellites for which no satisfactory explanation had been obtained in spite of diligentefforts by a number of parties, for which reason it was described as "anomalous". The interpretation of the anomalous acceleration as being a direct effect of the universal decay was presented in Exponential Decay of the Overall Universe is the Cause of "The Apparent Anomalous, Weak, Long-Range Acceleration of [the spacecraft]Pioneer 10 and 11" . 6 Decay in the gravitational acceleration, aG, acting on the satellites and due to the Sun means that aG was greater in the past, which means that the satellites were slowed more in the past than we now would expect in terms of the current value of aG. That effect is the "anomalous acceleration" toward the Sun. The time constant for this decay, τa,G, is as given in equation 6. (6) τa,G = τc = 3.57532·1017 sec = about 11.3 billion years "a" dimensions are L1/T2 For that the corresponding [that is the decay-related] acceleration toward the Sun is 8.38505·10-8 cm/sec2 (the observed value was reported as 8.5·10-8 cm/sec2 including other secondary effects) and the anomalous frequency drift, stated as clock acceleration, is 2.79695·10-18 sec/sec2 (the observed value was 2.8·10-18 sec/sec2). While this was the first definitive, experimental observation of the decay, every redshift measurement is a partial such observation. That is, the decay in the speed oflight, c, means that the light from far distant sources, which we now observe a long time after it was emitted, was emitted at a larger value of c than the value we know now. That greater speed means that the wavelengths all are longer, are redshifted as weperceive them. Furthermore, the more distant the source the earlier its light was emitted and the less decayed is the light's speed. That means that the greater the [decay-caused] redshiftthe more distant the source is. That relationship is non-linear as is the exponential decayfunction and unlike the Hubble model linear relationship. However, the sources of such light are, nevertheless, moving away from us so that there is also some Doppler effect. The redshifts that we observe are a combination ofDoppler and decay effects. Likewise, recent discrepancies in the determination of the distance of some far distant galaxies is a currently unresolved problem that has resulted in some ratherextreme hypotheses. In the Hubble model of the universe, the distance to far distant7sources is determined from the redshift, from which the speed of regression, v, [the redshift being deemed due to its Doppler effect] and then the distance v ⋅ H [where H is called the Hubble constant] are determined. Recently it has become possible to determine the distance to Type Ia supernovae by other independent means.11,12 The intrinsic brightness [luminosity] of such supernovae is related to the pattern [light curve] of their flare up and back down, aprocess taking weeks. By comparing the intrinsic brightness, determined from thatpattern, to the observed brightness the distance can be determined from the inverse squarelaw. Those new distance determinations exceed the Hubble distance by 10 - 15% . The explanation others propose is that an "antigravity effect" is accelerating the universe'expansion, which had hitherto been thought to be slowing down because of gravitation.That has led to their proposing reinstatement of Einstein's "cosmological constant", aterm in his equations introduced to account for gravitation not promptly collapsing theuniverse and which he disavowed upon Hubble's discovery of the expansion of theuniverse. And that has further led to their proposing some form of the Ancients' fifthessence, quintessence [the first four being earth, air, fire and water], to account for the"antigravity effect". Any "antigravity effect", regardless of its cause, would have the effect of counteracting ordinary gravitation. Inasmuch as one of the major current problems incosmology is to identify more gravitation to account for the cosmos' large scale structureand galaxies' centripetal force, any "antigravity effect" to act as the cause of accelerationwould not appear to fit with the rest of the cosmological situation. An alternative explanation is the general exponential decay of the overall universe, which accounts for the greater distances and the necessary cosmic energywithout the challenge to theory and to reasonableness that acceleration, its unknowncause, and a cosmological constant involve. This is fully developed in The Explanation of the New Astronomical Distance Data That Has Resulted From Measurements of TypeIa Supernovae Lies Not In a "Cosmological Constant" and Accelerating Expansion;Rather, It Is Another Aspect / Effect of the General Exponential Decay of the OverallUniverse. 7 The analysis of the universal decay in The Origin and Its Meaning.8 addresses the problem of determining what part of the observed redshifts is due to the Dopplereffect and what part to decay. The results are that the Doppler-caused part of theredshifts could not be more than 10% of the total redshift and is more likely on the order of only 1% or less. The remainder of the observed amounts of redshift, 90-99% of them, are due to the universal decay of the speed of light. The reasons for this are asfollows. At the Big Bang the material of the universe was thrust rapidly outward in all directions. Since then the mutual gravitational attraction of all of that material has beenslowing it all down. The amount of the gravitational slowing is inversely proportional tothe square of the distant between the mutually attracting bodies. Starting at a very largespeed the distance of separation increased rapidly, meaning that the slowing was rapidlyreduced. Therefore, most of the slowing, most of the speed loss, had to occur early afterthe "Big Bang". A very large part of the slowing must have taken place by the time the earliest galaxies formed, about 2½ to 3 billion years after the "Big Bang". Even if the initial speeds, immediately after the Big Bang, of the material of those earliest galaxieswere almost the speed of light, c, their speeds 2½ to 3 billion years later could not have been more than 1/10 as much, c/10, and more likely were on the order of c/100, or8less. Thus most of the observed amounts of redshift, 90-99% is due to the universal decay of the speed of light. Application of the Decay to Rotation Curves The above-cited behavior of the Pioneer 10 and 11 satellites presents one of the effects of the universal decay -- an always present, essentially constant, independent ofseparation distance increment of gravitational attraction in addition to the amount ofgravitational attraction expected in Newtonian terms. This comes about because decay ofgravitation means that earlier gravitation was greater. The analysis in Exponential Decay of the Overall Universe is the Cause of "The Apparent Anomalous, Weak, Long-RangeAcceleration of [the spacecraft] Pioneer 10 and 11" 6 develops that the magnitude of the increment, ∆aG, is dependent on the speed of light and its decay time constant as follows. -t/τc Where, from equation 1, the speed of light decay is c(t) = c0·ε , then (6) d[c(t)] c [where c is the local ∆aG = = contemporary value] dt τc The contemporary local value of c is 2.9979…·1010 cm/sec for our solar system. With the time constant, τc = 3.575…·1017 sec from equation 4, our solar system contemporary local increment of constant gravitational acceleration is (7) 2.997…·1010 ∆aG = = 8.385 …·10-8 cm/sec2 [Sol local] 3.575…·1017 For galaxies distant up to on the order of 500 million light years this value increases to on the order of 8.7·10-8 cm/sec2 the higher values due to the slightly lesser amount of decay that has taken place that many years earlier than thepresent at the very long term time constant involved, on the order of 11.3 billion years. For the present analysis, consistent with the limited accuracy of rotation curves, the intermediate value of 8.5·10-8 cm/sec2 will be used, for all galaxies in that distance range. For any specific galaxy the more precise value of ∆aG can be calculated by obtaining that galaxy's less decayed value of c from equation 1 and then ∆aG from equation 6. That acceleration, ∆aG, acting alone as a gravitational acceleration maintaining a mass in orbit, would produce a rotation curve as in Figure 5, below. Figure 5 - The Rotation Curve of the Decay-Produced Increment of Gravitational Acceleration Acting Alone That rotation curve of ∆aG is of the correct form to convert a galactic rotation curve exhibiting a Keplerian form [as in Figure 3] to a flat one [as in Figure 4], as illustrated inFigure 6, below, by superimposing the curves. 9Figure 6 - The Decay-Produced Increment of Gravitational Acceleration Acting Alone Superimposed on the Expected and Actual Rotation Curves [Fig. 3 & 4] Of course, the rotational velocities corresponding to the components of the total acceleration cannot properly be added. Rather, the accelerations must be summed andthe resulting rotational velocities then obtained as follows, (8) Total Acceleration = "natural acceleration" + ∆aG V2 G·M  =  + ∆aG R R2 G·M  ½ V =  + R·∆aG   R  which produces the observed actual flat portion of the rotation curve in the region corresponding to where the "expected" form is Keplerian and the "natural" accelerationsare G·M/R2 << ∆aG. This behavior largely corresponds to the MOND behavior which was obtained essentially by fitting behavior to observation in spite of lacking a physical justificationfor that action. That is, the effect of the universal decay is also to explain why theMOND hypothesis has had whatever success it has experienced -- MOND is a simulationof reality but for incorrect, rather missing, physical reasons. Conclusion The universal decay avoids the considerable difficulties with the hypothesized "dark matter" that is assumed to exist on the basis of rotation curves, and it provides therequisite physical justification behind, and completely replaces, MOND. It does so withphysical justification based upon completely independent observations -- the Pioneer 10and 11 data. McGaugh 3 has shown that if MOND is assumed the resulting universe is one in which structure grows rapidly and to large scales, and the universe is made up ofbaryons. Those same results apply more securely to the universe of the universal decay. Actions Needed to Complete the Verification of the Universal Decay The universal decay can be verified and further investigated by conducting two experiments set forth in The Origin and Its Meaning 8; the measurement of the value of 10each of the two fundamental constants, c and h, directly as they are in the light from far distant astronomical sources. The measurements must be of the actual light emitted longago from a far distant astronomical source, not local, just emitted, light. The measurements must directly measure the constant sought; they cannot be a measurement of other quantities with the calculation of the fundamental constant using laws of physics relating the quantities. For example, in the usual determinations of thevalues of the various fundamental constants Planck's constant is not directly measured.Rather its value is inferred from other measurements [e.g. the Rydberg constant] andcalculated via other formulations [e.g. the fine structure constant]. Such indirectprocedures may not give correct results in the present experiments. The expected results of the experiments are given in Figure 5, below, which gives the multiples of our contemporary value of the constants c and h that are expected to be found in light that was emitted at various times in the past. Figure 7 Measuring The Speed of Light, c Modern measurements of the speed of light are done by measuring certain frequencies and wavelengths that are measurable with very great precision, c being the product of a frequency and its related wavelength. To measure the speed of ancient lightfrom far distant sources the product of frequency and wavelength is useless. We alreadyknow that the wavelength is significantly different from that in our local light, thedifference being the redshift. If that redshift were entirely due to universal decay then thefrequency-wavelength product would give the correct speed, but at least some of theredshift is due to the Doppler effect [on the order of 1 - 10%]. The data of interest is a comparison of the c in ancient light with that in contemporary light. That can be determined by an interferometer type measurement suchas those of Michaelson / Pease and Pearson using the Foucault method. In thoserevolving mirrors or a toothed wheel were used to break a monochromatic [singlefrequency] light beam into segments. The beam was then split into two beams whichwere directed over two different paths of known length and then recombined. If the speedof travel over the two paths were the same then the recombination would produce aperfect overlap of the waves, but if it were different the difference would show in theresulting interference wave pattern. To compare far distant ancient light against contemporary local light the interference must be generated between a single frequency of the ancient light [as 11selected by a spectroscope, one of the lines of the distant source's line spectrum being selected] and a beam of local light [the same frequency line as in the ancient light spectrum being spectroscopically selected], no beam splitting being involved. Asindicated in the sample data above, the speed difference of the two light beams will belarge and the resulting interference pattern will be accordingly. Measuring Planck's Constant, h. Planck's Constant, h, can be directly measured using the photoelectric effect. Figure 6, below, illustrates the photoelectric effect and its relationship to Planck'sconstant. While the accuracy using the photoelectric effect is not nearly as good as thatprovided by other less direct means, the method is quite sufficiently accurate for theaccuracies involved for the present purposes. The lines in the figure [which are straightlines] can be plotted from as little as two data points for any one substance [of courseaccuracy improves with a greater number of data points and interpolation among them]. Figure 8 Each data point is obtained by shining light [in the present situation the light must be from a far distant astronomical source] of a single frequency [as selected by aspectroscope, one of the lines of the source's line spectrum being selected] on aphotosensitive surface that emits photoelectrons [the selected line must be of a frequencygreater than the cut-off frequency, e.g. f 1 or f2, for the particular photosensitive substance being used]. Normally in the use of the photoelectric effect the objective is to readily collect a current of photoelectrons so that the collection anode is set at a positive electricalpotential relative to the photoelectron source, the photosensitive surface on which thelight is shined. [Of course, the entire structure must be in a vacuum for the photoelectronsto be free to travel without the interference of a relatively dense gas.] In the present experiment the collection anode is set negative relative to the photoelectron source, that negative potential being adjustable. Then the negative potentialis made progressively less negative until the first, initial photoelectron current is detected.That potential is the energy of the most energetic photoelectron produced by theparticular frequency of the light being used [the photoelectrons emitted at lesser energieshaving been freed from the photosensitive surface with the same high energy but havinglost some within the material before becoming free]. The data point is the energy and thefrequency. As indicated in the figure, Planck's constant is the slope of the resulting line(s), which develops as follows. The energy of a photon of light is given by (16) E = h·f where: E is the energy, h is Planck's constant, and f is the frequency of the particular photon. 12The initial energy datum is the electric retarding potential and must be converted to the units of Planck's constant times frequency as required for the E of equation 16. That done, then the slope of the line in the figure is (17)Energy/frequency = h·f/f = h, Planck's constant. This measurement performed on light from distant astronomical sources will result in values for Planck's constant quite noticeably larger than our domestic value, thedifference being the decay that has taken place since the time the sample light wasoriginally emitted at its distant source. References [N.B. The first three references themselves further reference many of the large and significant number of papers on their topics.] [1] J.A. Sellwood and A. Kosowsky, Does Dark Matter Exist ? , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, astro-ph/0009074. [2] Mordehai Milgrom, The Modified Dynamics -- Status Review , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, astro-ph/9810302. [3] Stacy McGaugh, MOND in the Early Universe , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, astro-ph/9812328. [4] R. Ellman, A Conjecture Concerning Red Shifts , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, physics/9808051. [5] R. Ellman, Further Analysis of the Universal Decay Suggested In "A Conjecture Concerning Red Shifts" , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, physics/9809029. [6] R. Ellman, Exponential Decay of the Overall Universe is the Cause of "The Apparent Anomalous, Weak, Long-Range Acceleration of [the spacecraft] Pioneer10 and 11" , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, physics/9906031. [7] R. Ellman, TheExplanation of the New Astronomical Distance Data That Has Resulted From Measurements of Type Ia Supernovae Lies Not In a "CosmologicalConstant" and Accelerating Expansion; Rather, It Is Another Aspect / Effect of theGeneral Exponential Decay of the Overall Universe , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, physics/0007058. [8] This paper is based on development in R. Ellman, The Origin and Its Meaning , The- Origin Foundation, Inc., http://www.The-Origin.org, 1997, in which thedevelopment is more extensive and the collateral issues are developed. [It may bedownloaded from http://www.The-Origin.org/download.htm]. [9] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an ApparentAnomalous, Weak, Long-Range Acceleration , Phys. Rev. Lett. 81, 2858 (1998), Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, gr-qc/9808081. [10] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, The Apparent Anomalous, Weak, Long-Range Acceleration of Pioneer 10 and 11 , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, gr- qc/9903024.13[11] A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, R.L. Gilliland, C.J. Hogan, S. Jha, R.P. Kirshner, B. Leibundgut, M.M. Phillips, D.Reiss, B.P. Schmidt, R.A. Schommer, R.C. Smith, J. Spyromilio, C. Stubbs, N.B.Suntzeff, J. Tonry, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, astro-ph/9805201. [12] S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro, S. Deustua, S. Fabbro, A. Goobar, D.E. Groom, I.M. Hook, A.G. Kim, M.Y. Kim, J.C.Lee, N.J. Nunes, R. Pain, C.R. Pennypacker, R. Quimby, C. Lidman, R.S. Ellis, M.Irwin, R.G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B.J. Boyle, A.V.Filppenko, T. Matheson, A.S. Fruchter, N. Panagia, H.J.M. Newberg, W.J. Couch,Measurements of Ω and Λ From 42 High-Redshift Supernovae , Los Alamos National Laboratory Eprint Archive at http://xxx.lanl.gov, astro-ph/9812133.
arXiv:physics/0011063v1 [physics.atm-clus] 27 Nov 2000Thermodynamics of Na 8and Na 20clusters studied with ab initio electronic structure methods Abhijat Vichare and D. G. Kanhere1 Department of Physics, University of Pune, Ganeshkhind, Pu ne 411 007, INDIA S. A. Blundell2 D´ epartement de Recherche Fondamentale sur la Mati` ere Con dens´ ee, CEA Grenoble 17, rue des Martyrs, F-38054 Grenoble CEDEX 9, France Abstract We study the thermodynamics of Na 8and Na 20clusters using multiple-histogram methods and an ab initio treatment of the valence electrons within density functional theory. We consider th e inﬂuence of various electron kinetic-energy functionals and pseudo potentials on the canonical ionic speciﬁc heats. The results for all model s we con- sider show qualitative similarities, but also signiﬁcant t emperature shifts from model to model of peaks and other features in the s peciﬁc- heat curves. The use of phenomenological pseudopotentials shifts the melting peak substantially ( ∼50–100 K) when compared to ab ini- tioresults. It is argued that the choice of a good pseudopotenti al and use of better electronic kinetic-energy functionals ha s the poten- tial for performing large time scale and large sized thermod ynamical simulations on clusters. 1 Introduction The physics of ﬁnite-sized systems such as clusters continu es to invoke consid- erable interest both in theory and experiment. A particular ly intriguing and poorly understood phenomenon is the melting behavior of suc h ﬁnite-sized systems. Only recently have Haberland et al.[1] succeeded in measuring the 1(amv, kanhere)@unipune.ernet.in 2sblundell@cea.fr 1heat capacity of free (i.e. unsupported) Na+ nclusters, with nranging from 70 to 200 atoms. Interestingly, they ﬁnd a nonmonotonic behavi or of the melt- ing temperature as a function of cluster size, with pronounc ed maxima at n= 57 and 142. These sizes correspond neither to closed-shell Mackay icosa- hedra ( n= 55 and 147) nor to closed shells of valence electrons ( n= 59 and 139), but are intermediate between the two. This clearly ind icates that both geometric and electronic shell eﬀects contribute to the mel ting phenomenon in a rather subtle manner. Prior to this measurement, there have been a few experimenta l studies on melting of clusters. Martin et al.[2] reported measurements on the melting temperature of Na clusters for the sizes of the order of thous ands of atoms and their results indicated that the melting temperatures i ncreased with size, but had not reached the experimental bulk value. Peters et al.[3] have noted the existence of surface melting on supported Pb nanopartic les using X-Ray diﬀraction. Clearly, the melting behavior at small sizes is cluster spec iﬁc and dependent on the nature of the electronic structure and geometry. Furt her, the transi- tion found by Haberland et al.[1] is not sharp and has a broadened peak in the speciﬁc heat with a width of approximately 40 K. The expec ted mono- tonic increase of melting point has been seen only for very la rge clusters con- taining upwards of several thousand atoms. On the theoretic al side, much insight into the ﬁnite-temperature properties has been gai ned via molecu- lar dynamics (MD) and Monte-Carlo (MC) numerical simulatio ns. Most of these simulations have been carried out using classical emp irical two-body potential functions, [4] mostly of Lennard-Jones (LJ) or Mo rse type. These studies have revealed that small clusters exhibit a melting transition over a broad temperature range, unlike bulk systems, and have broa d heat-capacity curves. In addition, they also exhibit a variety of other phe nomena such as isomerization (including surface isomerization) and surf ace melting, which are generically referred to as “premelting” phenomena. Som e clusters also exhibit coexistence of liquid-like and solid-like phases w ithin the melting temperature range. MD and MC simulations have also been reported using classica l embedded- atom potentials, such as the single-moment approximation ( SMA), [5, 6] which contain approximations to the N-body forces found in metallic sys- 2tems like Na clusters. Calvo and Spiegelmann [7, 8] performe d extensive simulations on Na clusters with from 8 to 147 atoms using the S MA poten- tial of Li et al. [6] with a view to probe the melting phenomena of small Na clusters. They ﬁnd that premelting phenomena dominate th e melting process at small cluster sizes ( n <75), while the larger sizes exhibit a pref- erence for a single-process melting. They also observe that the nature of the ground state is critical to the thermodynamics of the cluste r. However, as they clearly point out, their simulations do not incorporat e the electronic structure eﬀects directly. An alternative approach for metallic clusters is that of Pot eauet al. [9, 10] who developed a tight-binding Hamiltonian to incorporate q uantal eﬀects approximately. They use a H¨ uckel-type Hamiltonian and MC t o sample the phase space for small Na clusters with 4, 8, and 20 atoms. Calv o and Spiegel- mann [8] have performed more extensive calculations for siz es up to 147 atoms with the same potential. However, during the last decade dev elopments in ab initio methods have opened up practical possibilities of performi ng accurate simulations by combining density functional theory (DFT) w ith classical MD or MC. The most accurate form of DFT is the Kohn-Sham (KS) form ulation. Although these methods have been used to investigate the str uctural prop- erties with remarkable success, relatively few applicatio ns of such ab initio methods have been to the simulation of melting. Jellinek et al.[11] have, however, combined a hybrid Hartree-Fock/DFT method with MC sampling to study the thermodynamics of Li 8. Although it is most desirable to have a full quantum mechanic al treatment of electrons, as in the KS method, such simulations turn out t o be expensive. It is also to be noted that typical simulation times used in pu rely empirical potential MD are of the order of a few 100 ps or more per energy p oint. Considering that the most relevant sizes for the experiment are in excess of 50 atoms, the full ab initio simulation may turn out to be practically too expensive.[12] Hence approximate methods leading to pr actical and fast algorithms have been developed. One such technique is densi ty-based (DB) molecular dynamics, where the electronic kinetic energy is approximated as a functional of density only. For example, Vichare and Kanher e [13] performed ab initio simulations on an Al 13cluster to investigate its melting behavior. The DB method has also been used by Aguado et al.[14, 15] to study the melting of Na clusters ranging in size from 8 to 142 atoms. The ir simulations 3are of constant total energy type using empirical pseudopot entials with up to 50 ps of observation time per energy point for small cluste rs (8 and 20 atoms), and up to 18 ps per energy point for larger ones. Anoth er approach is that of Blaise et al.[16] who carried out DB simulations for Na clusters up to size 274, but using soft, phenomenological pseudopotent ials rather than ab initio pseudopotentials. In addition to permitting signiﬁcantly longer observation times, these soft pseudopotentials were shown to reproduce well properties such as the volume and surface energies, ionizat ion energies, and the frequency of collective ionic monopole and quadrupole o scillations. These above-mentioned studies on Na clusters bring out a num ber of issues which need further investigation. Clearly, it is desirable to have both long simulation times and a full quantum mechanical treatment of electrons. Full KS being expensive, however, the attractive propositions o f DB or soft pseu- dopotentials as practical alternatives for the simulation of such systems need to be assessed as to their quality. This is particularly impo rtant because of some discrepancies seen in the above studies. For example, i n the case of Na8, Calvo and Spiegelmann, [7, 8] using an SMA potential, ﬁnd th at the canonical speciﬁc heat is broad in nature and shows a ﬂattene d peak between about 110 K to 250 K. For the same cluster, the tight-binding p otential [8, 9] leads to a less broad peak, with a width of about 70 K and peakin g at 160 K. However, the microcanonical speciﬁc heat obtained by Agu adoet al.[14] for Na 8in a constant-energy DB study shows a peak at a much lower valu e of 110 K and is sharp with a width of less than 30 K. This is in qua litative disagreement with the SMA and tight-binding results. Further, there is a diﬀerence in the way the data has been anal yzed by these workers. Aguado et al.[14, 15] have used the traditional trajectory-based analysis, which uses the caloric curve supplemented by Lind emann type cri- teria for identifying the transition. Since the transition is never sharp, such an analysis may not lead to an unambiguous determination of t he melting temperature. In addition, the observation times of Aguado et al.are signif- icantly less than those used by Calvo and Spiegelmann [7, 8] i n their SMA or tight-binding work. In fact, it is desirable to calculate appropriate ther- modynamic indicators such as the ionic entropy and the speci ﬁc heat. In Refs. [7, 8, 9], the authors have used the multiple histogram (MH) technique [17, 18] to extract the entropy and the speciﬁc heat from the s imulation data, as we do here. 4In the present work, we therefore examine the melting of Na 8and Na 20clus- ters with a view to resolving these issues. We have carried ou t the follow- ing simulations on Na 8: a full KS (orbital-based) simulation using ab initio pseudopotentials; a DB simulation, where the electronic ki netic energy is ap- proximated, but with identical pseudopotential and time sc ales; and both KS and DB simulations with soft pseudopotentials. The same sim ulations have been carried out for Na 20, with the exception of the full KS simulation with ab initio pseudopotentials. In all the cases we have calculated the en tropy and the canonical speciﬁc heat via the MH method, as well as th e tradi- tional indicators like the RMS bond length ﬂuctuation and me an squared displacements. In the next section, we brieﬂy describe the formalism, analy sis methods, and numerical details. In Section 3, we present our results and d iscuss them in the light of earlier studies. Finally, our conclusions are p resented in Section 4. 2 Method Following the usual procedure in DFT,[19] we write the total energy of a system of Nastationary Na+ions with coordinates R≡ {Ri}andNevalence electrons as a functional of the electron density ρ≡ρ(r) Epot[ρ, R] =T[ρ] +Eext[ρ, R] +EH[ρ] +Exc[ρ] +Eii[R], (1) where T[ρ] andEH[ρ] are the kinetic and Hartree energy, respectively, of the valence electrons, Eext[ρ, R] is the interaction energy of the valence electrons with the ions, evaluated using the pseudopotential formali sm,Exc[ρ] is the electron exchange-correlation energy in the local density approximation (us- ing the parametrization of Perdew and Zunger[20]), and Eii[R] is the ion-ion interaction energy. In the standard KS approach, T[ρ] is expressed as a sum of expectation values, over each KS orbital, of the electron kinetic-energy operator −(1/2)∇2. In contrast, in the DB approach T[ρ] is expressed as a functional of ρonly, without introducing orbitals, leading to a faster tho ugh in practice less accurate calculational scheme. For each ap proach we use either ab initio (AI) pseudopotentials or soft, phenomenological (SP) pseu d- potentials. We consider two forms for T[ρ] in the DB approach: in our DB-AI 5approach we use a functional form proposed for clusters,[21 , 22] while in our DB-SP approach we take T[ρ] as a sum of the Thomas-Fermi energy and a scaled Weizs¨ acker term. [16] Theab initio pseudopotentials used in the KS-AI and DB-AI approaches are those proposed by Bachelet, Hamann and Schl¨ uter.[23] The s oft, phenomeno- logical pseudopotential used in the KS-SP and DB-SP approac hes is given by [16] Vsoft(r) =  −1 r, r > r c −1 6rc/bracketleftbigg 7−/parenleftBig r rc/parenrightBig6/bracketrightbigg , r≤rc,(2) for a single Na+ion at the origin, where rc= 3.55a0in the DB-SP approach andrc= 3.7a0in the KS-SP approach. The choice of rcfor the DB-SP approach follows from a ﬁt to volume and surface energies,[1 6] while for the KS-SP approach the choice of rcensures close agreement with ionization energies and dissociation energies given by an ab initio pseudopotential, for small clusters in the size range n= 3 to 8. Use of the phenomenological pseudopotential permits a larger grid step size or equivale ntly a smaller plane- wave energy cut-oﬀ, thus leading to a faster solution in eith er the KS or DB formalisms. The Car-Parinello (CP) algorithm [24] was used in the DB-AI and DB-SP schemes, while the damping scheme proposed by Joan opoulous et al. [25] was used to minimise the electronic degrees of freedom i n KS-AI. The trajectories collected were analyzed using traditiona l indicators of melt- ing like the rms bond-length ﬂuctuation, deﬁned as δrms=2 Na(Na−1)/summationdisplay i<j/parenleftBig /an}b∇acketle{tr2 ij/an}b∇acket∇i}htt− /an}b∇acketle{trij/an}b∇acket∇i}ht2 t/parenrightBig1 2 /an}b∇acketle{trij/an}b∇acket∇i}htt, (3) where rijis the distance between ions iandj, and /an}b∇acketle{t. . ./an}b∇acket∇i}httdenotes a time average. According to the Lindemann criterion, a system may no longer be considered to be solid if δrmsis greater than 0.1 to 0.15. Short time averages over the trajectory data, e.g. over data points correspondi ng to 1 ps, 2 ps, 5 ps, etc., were evaluated to obtain the dependence of δrmson the duration of the time average. Another indicator we have used is the mean s quare ionic displacement, deﬁned as /an}b∇acketle{tr2(t)/an}b∇acket∇i}ht=1 Nntnt/summationdisplay m=1Na/summationdisplay i=1[ri(t0m+t)−ri(t0m)]2. (4) 6We have set the total number of time-steps ntused in the time average to nt=nT/2, where nTis the total simulation time (usually about 50 ps). A more complete thermodynamic analysis of the simulations i s possible us- ing the multiple histogram method (MH),[17, 18] and all simu lations were analysed using this method. It requires the conﬁgurational energy, which corresponds here to the classical potential energy Epot[ρ, R] of Eq. (1), over various points in the ionic phase space accessed by the syste m along the tra- jectory. This is used to evaluate the classical ionic densit y of states Ω( E), and thereby the ionic entropy S(E) = ln Ω( E), as well as the partition function via a least-squares ﬁtting procedure. The sampled values of the conﬁgura- tional energy are ﬁtted to the theoretical probability dist ribution and the ﬁtted coeﬃcients are then used to evaluate the various therm odynamic func- tions. We consider in particular the canonical speciﬁc heat , deﬁned as usual by C=∂U ∂T, (5) where U=/an}b∇acketle{tEpot+Ekin/an}b∇acket∇i}htTis the average total internal energy in a canonical ensemble at temperature T. We here exclude the contribution of the center- of-mass motion to the ion kinetic energy Ekin, so that from the equipartition theorem /an}b∇acketle{tEkin/an}b∇acket∇i}htT=3 2(Na−1)kBT . (6) The canonical probability distribution for observing a tot al energy Eat tem- perature Tis given by the usual Gibbs distribution p(E, T) =1 Z(T)Ω(E) exp/parenleftbigg −E kBT/parenrightbigg , (7) with Ω( E) the classical density of states extracted from the MH ﬁt, an d Z(T) the normalizing canonical partition function. Note that a lthough here we shall discuss results in the canonical ensemble, once Ω( E) is known, one may also evaluate properties in the microcanonical ensembl e, such as the microcanonical temperature T(E) 1 T(E)=∂ ∂ElnΩ(E). (8) Simulated annealing was used to obtain the ground-state ion ic structures from a randomly chosen initial conﬁguration for each cluste r. For Na 8the 7ground-state geometry is found to have a dodecahedral D2dsymmetry in both the KS and DB formalisms and for both the AI and SP pseudopoten tials, in agreement with the structure found by R¨ othlisberger and Andreoni [26] in a KS approach. For Na 20in the DB formalism, we ﬁnd a ground state consisting of a double icosahedron with a single cap on its wa ist, which is the second of the two structures found in Ref. [26]. In the KS f ormalism, the ground state for Na 20is a double icosahedron missing one end cap and with two caps on the waist, in agreement with Ref. [26]. Our DB stru ctures agree with those found by Aguado et. al.[14] We have considered two approaches to the statistical sampli ng of the ionic phase space, required as input to the MH analysis. In each app roach the clusters are eﬀectively heated slowly from the ground-stat e structure at 0 K to a liquid-like state at upwards of 250 K. The ﬁrst approach inv olves a canonical sampling of the phase space and was used with the AI pseudopot entials in both the KS and DB schemes. Successive simulation temperatu res of 60 K, 80 K, 100 K, 125 K, 150 K, 175 K, 200 K, 225 K, and 250 K were chosen . Each temperature was maintained within ±10 K using velocity scaling,[30] except for the 60 K and 80 K temperatures, where the temperatures wer e maintained within ±5 K. The total observation time for both KS and DB is about 57.5 ps per temperature point. The initial condition at each temper ature was taken as the ﬁnal state of the previous temperature, and the initia l 1.25 ps of simulation time were used to raise the previous temperature . The next 5 ps were then discarded to allow for thermalization of the syste m at the new temperature. The analysis was performed on the data corresp onding to the last∼50 ps. The simulations for the clusters Na 8and Na 20were performed within a cubical supercell of edge 40 a.u. [27] or more. All Fo urier space evaluations were carried out on a mesh of 64 ×64×64 for DB and 48 ×48×48 for KS with a cutoﬀ of about 21 Ry. The conﬁguration energy ran ge was divided into bins whose width was chosen to give at least abou t 30 points for the lowest temperature distribution. About 500 bins wer e typically used to cover the entire conﬁguration energy range. The canonica l speciﬁc heats obtained using the MH analysis were then plotted as a multipl e of their valueC0at 0 K given by C0= (3Na−9/2)kB, which is the zero-temperature classical limit of the rotational plus vibrational speciﬁc heats. Our second approach consists of a microcanonical sampling o f the phase space, and was used with the SP pseudopotential in both the KS and DB 8schemes. Constant total energy simulations were performed at closely spaced values of the total energy, such as to give good overlap of suc cessive his- tograms of the potential energy Epot. The simulations were performed in order of increasing total energy, with the initial conditio n at one energy ob- tained by scaling the velocities of the ﬁnal state of the prev ious energy, and 20–30 energy points were used to scan the required energy ran ge. Each en- ergy point consisted of from 50–100 ps of observation time, o f which 5–10 ps were discarded for equilibration. Several scans of the enti re energy range were made in this way, giving totalsimulation times of about 15 ns for (Na 8, DB-SP), 5 ns for (Na 8, KS-SP), 6 ns for (Na 20, DB-SP), and 3 ns for (Na 20, KS-SP). The microcanonical sampling requires a modiﬁed MH a nalysis.[28] Note that DB-SP results have been reported elsewhere,[29] a nd are repro- duced here for purposes of comparison. The dominant error in our speciﬁc-heat curves is statistica l, due to the ﬁnite duration of the sampling of the phase space. By adding extra d ata points, or complete additional scans of the whole temperature range , to the MH analysis, we ﬁnd the speciﬁc-heat curves to be stable to abou t 10% or better, and the positions of peaks to be stable to about ±20 K or better. We take this as an informal estimate of the statistical error. Howev er, in dynamical simulations such as these, it may be that some processes of im portance (e.g. isomerizations) occur on a physical time scale rather longe r than we have considered, so that we have imperfect ergodicity; all we can say is that our curves do appear to be rather stable on the time scales that we have con- sidered. We are currently considering recent Monte-Carlo s ampling methods such as the parallel tempering method,[31] which are design ed to overcome the problem of long time scales and improve ergodicity. 3 Results We begin the discussion by considering some of the conventio nal trajectory- based indicators of isomerization and melting. In Fig. 1, we show the rms bond-length ﬂuctuations δrmsof Na 8in the KS-AI model as a function of simulation time, for diﬀerent temperatures in the range 60 K to 250 K. The ﬁgure makes it clear that for temperatures up to about T= 200 K, 25 ps 9are suﬃcient to converge the value of δrms, while for higher temperatures of the order of 250 K or more, even longer simulation times may be required. Similar behaviour is seen in Fig. 2 for the 20-atom cluster si mulated within the DB-AI model. In Fig. 3, we show δrmsaveraged over 37.5 ps and over 5 ps, as a function of temperature. Note that the 5 ps curve nev er crosses the Lindemann criterion of 0.1, while the 37.5 ps curve crosses t he Lindemann criterion of 0.1 around 190 K. The behavior of δrmsin DB-AI over identical simulation times is very similar. The mean square ionic displacement /an}b∇acketle{tr2(t)/an}b∇acket∇i}ht(4) has also often been used as an indicator of isomerization or of a solid-like to liquid-l ike transition. In Figs. 4 and 5 we show /an}b∇acketle{tr2(t)/an}b∇acket∇i}hton diﬀerent time scales of 1 ps and 25 ps, respectively, for Na 8in the KS-AI model. One observes that /an}b∇acketle{tr2(t)/an}b∇acket∇i}htat low temperatures T <100 K reaches a horizontal plateau for t >∼0.25 ps, indica- tive of a solid-like behavior in which atoms vibrate around ﬁ xed points with an amplitude squared that increases in rough proportion to t he temperature. On the other hand, the rising curve for T≥250 K suggests a liquid-like behavior with diﬀusion throughout the entire volume of the c luster. The curve for T≥250 K would eventually reach a plateau with a /an}b∇acketle{tr2(t)/an}b∇acket∇i}htvalue characteristic of the square of the linear dimension of the c luster, but even att= 25 ps this plateau has not yet been attained. Somewhere betw een these two limiting temperatures is a region of isomerizatio n processes with a character intermediate, in some sense, between solid and li quid. The MH analysis may be used to probe further the thermodynami cs of the cluster in any particular model. One here extracts the io nic entropy S(E) = ln Ω( E), which is a functional of the potential-energy surface (1) . As expected, all entropy curves show a monotonic increase, the curve for KS-AI, shown in Fig. 6, being typical. The canonical speciﬁc heats ( 5) for Na 8ob- tained via the MH technique for all four models are shown in Fi gs. 7–10. In general, all the Na 8speciﬁc-heat curves show broad peaks with widths over 100 K. The initial rise is around 70 K for both DB models. In the KS models, the initial rise of the main peak for the SP pseudopotential i s at a higher temperature than for the AI pseudopotential, namely, at 200 K compared to 150 K. However, the KS-SP model has a shoulder feature around 80 K not visible in the KS-AI results. Turning to Na 20in Figs. 11–13, we ﬁnd main peaks that are less broad than for Na 8, with a width generally somewhat less than 100 K. In the DB models, the main peak for the AI pseudopot ential is 10at a higher temperature than for the SP pseudopotential, nam ely, at about 250 K compared to about 150 K. If on the other hand we compare th e KS- SP model with the DB-SP model, we ﬁnd that both main peaks occu r at about the same temperature. However, the KS-SP model has a “p remelting” feature around 80 K that is more distinct than for DB-SP model . It is diﬃcult to draw simple, general conclusions from these observations concerning the eﬀect of the KS approach versus the DB approac h, or the eﬀect of AI versus SP pseudopotentials. For example, for Na 8in the KS model, the SP pseudopotential gives a main peak at higher a te mperature than for the AI pseudopotential (if one ignores the small pre melting feature in the former), while for Na 20in the DB model, it is the AI pseudopotential that gives a peak at the higher temperature. For these small c luster sizes, the precise form of the speciﬁc-heat curves can evidently be very sensitive to the model used. One observes a similarly large variation i n speciﬁc-heat curves between the SMA and TB models reported in Ref. [8]. Evi dently, the important features of the potential-energy landscape can b e rather sensitive to the model employed. Some insight into the model-dependence of the potential-en ergy surface may be gained from the energetic ordering of a selection of possi ble isomers of Na8. We consider the dodecahedron D2d(the ground state in all DB and KS models of this work, as well as in the SMA model [7]), the cappe d pentagonal bipyramid Cs(the ground state for LJ 8), and the stellated tetrahedron Td. In the DB models and the SMA model, the Csstructure forms a relatively low-lying excited isomer at 0.03–0.05 eV above the D2dground state, while in our KS models and in the KS approach of Ref. [26], the Csstructure is unstable and collapses to D2dupon relaxation. On the other hand, the Td structure forms a higher-lying isomer at around 0.09–0.12 e V in the present DB and KS models, in the KS approach of Ref. [26], and in the SMA model, while in the TB model [8] and in an all-electron conﬁguration -interaction approach,[32] the Tdstructure is the ground state. This illustrates how even for Na 8the ordering of isomers given by ab initio calculations is uncertain. We note that, while the heights of the barriers separating is omers are a more important determining factor than the simple energy di ﬀerences, the existence of the low-lying Csisomer in the DB models, but not in the KS models, is consistent with the lower-temperature shoulder of the Na 8speciﬁc heat curve in the DB models. 11Our speciﬁc-heat curves are in general qualitatively quite similar to those for the SMA potential:[7, 8] Na 8in the SMA model has a broad peak, and Na 20a rather narrower peak with a small premelting feature on the l ow-temperature side. On the other hand, there are some diﬀerences with the TB speciﬁc- heat curves.[8] For instance, the speciﬁc heat of Na 8has a somewhat narrower peak in the TB model than in the DB, KS, or SMA models. However, as noted above, the ground-state structure of Na 8has a Tdsymmetry in the TB model, but a D2dsymmetry in the DB, KS, and SMA models. Finally, Aguado et. al., [14] using a model quite similar to the present DB-AI model , givemicrocanonical speciﬁc heats for Na 8and Na 20, derived from a trajectory- based analysis, that appear to disagree qualitatively with the present DB and KS results (and with the SMA results.[7, 8]) Their curve for N a8has a single narrow peak with a width less than 30 K located around 110 K, wh ile their curve for Na 20has two distinct peaks, each with a width less than 30 K and of similar height, located at about 110 K and 170 K. The precise r eason for the diﬀerences between their results and the present results is unclear at present and requires further investigation. A re-evaluation of our own results in the microcanonical ensemble shows that the change of ensemble i s insuﬃcient to explain these diﬀerences, and we note that our DB ground-sta te geometries agree with theirs. Given the similarity between their DB mod el and ours, the discrepancies may be due simply to methodological diﬀer ences: in the present work we have derived the speciﬁc-heat curves from a M H analysis, and have used longer sampling runs, checking that the speciﬁ c-heat curves are reasonably stable against the addition of further data. 4 Conclusions We have investigated the thermodynamics and melting of the s mall clusters Na8and Na 20using interionic potentials derived from several DFT model s for the valence electrons. The data have been analyzed using a MH analy- sis, which is an eﬃcient and reliable way of probing the melti ng transition. Of the various DFT models, the most accurate one considered h ere should be the KS-AI model. The other models involve substituting le ss accurate electron kinetic-energy functionals T[ρ] (the DB approaches), or else soft, phenomenological pseudopotentials (the SP approaches) in place of ab initio 12pseudopotentials, in each case with a view to accelerating t he calculation and permitting better statistics. While there are qualitative similarities between the curves obtained from the various models, we also observe substantial shifts in temperatures of the main peaks and other features o f the curves from model to model. Concerning the choice of pseudopotential, while the SP pseu dopotential is known to predict ground-state geometries and certain other properties well, as mentioned previously, it does not necessarily follow tha t energetic bar- riers and other important features of the potential-energy surface are well described. Given the substantial diﬀerences in the speciﬁc -heat curves ob- tained from the SP and AI pseudopotentials, it would therefo re seem wise to prefer AI pseudopotentials. Note, however, that the soft pseudopoten- tial used here is highly phenomenological: it lacks entirel y a repulsive core, and deviates from the asymptotic value −1/rfor r<∼3.7a0, which is well outside the physical core of the Na+ion, r<∼2.0a0. It may be possible to construct a better soft pseudopotential that minimizes the diﬀerence with the speciﬁc-heat curve obtained from an AI pseudopotential and yet still yields a signiﬁcantly cheaper calculation, thus permittin g a very useful gain in statistics in thermodynamic simulations. We are current ly investigating such possibilities. As to the DB approach versus the full KS approach, we note that one impor- tant approximation in the DB approach in its present form is i ts diﬃculty in accounting for quantum shell eﬀects accurately. The two DB f orms for T[ρ] considered here yield energies that vary smoothly with clus ter size Naccord- ing to a liquid-drop formula (as shown explicitly in Ref. [16 ] for the extended Thomas-Fermi functional), without showing the ﬂuctuation s in energy asso- ciated with quantum shell closures of the valence electrons . Further, while ground-state geometries and other properties of closed-sh ell systems can be predicted rather successfully by the DB approach [16, 22], i t has trouble reproducing Jahn-Teller distortions in open-shell system s. Now, according to the Hohenberg-Kohn theorems, it should in principle be po ssible to ﬁnd a DB functional T[ρ] that fully incorporates such quantum shell eﬀects in ﬁnite systems. It appears that if the DB methods are to yield r eliable, quan- titative information, then better electronic kinetic-ene rgy functionals along these lines are required. Fortunately, signiﬁcant progres s is being made in this direction, and a number of researchers have already pro posed DB kinetic- 13energy functionals incorporating electronic shell eﬀects approximately. There is a special reason for paying careful attention to ele ctronic shell eﬀects in studies of melting that was mentioned earlier, related to the experiments of Haberland et al. [1]. The fact that prominent maxima in the melting point occur for sizes that are intermediate between geometric she ll closures and electronic shell closures suggests that there is an importa nt interplay between geometric eﬀects and quantum shell eﬀects. To understand th is phenomenon more closely, we are currently considering Monte-Carlo sam pling methods combined with various approximate KS schemes, with a view to extending the KS calculations presented in this work to larger sizes, w ithin the range of the Haberland et al. data. Acknowledgements We gratefully acknowledge the support of the Indo-French Ce nter for the Promotion of Advanced Research (New Delhi) / Centre Franco- Indien pour la Promotion de la Recherche Avanc´ ee under contract 1901-1 . One of us (AMV) acknowledges the hospitality of the CEA, Grenoble, Fr ance. AMV is grateful to CSIR, New Delhi, India for their research fellow ship. Thanks are also due to Matt Freigo and Stephen Johnson for an excellent F FT library. References [1] M. Schmidt, R. Kusche, W. Kronm¨ uller, B. von Issendorﬀ, and H. Haberland, Phys. Rev. B 79, 99 (1997); M. Schmidt et al., Nature 393, 238 (1998). [2] T. P. Martin, Phys. Rep. 273, 199 (1996). [3] K. F. Peters, J. B. Cohen, and Y. W. Chung, Phys. Rev. B 57, 13430 (1998). [4] D. J. Wells and R. S. Berry, Phys. Rev. Lett. 73, 2875 (1994); T. L. Beck, D. M. Leitner, and R. S. Berry, J. Chem. Phys. 89, 1681 (1993); 14T. L. Beck and R. S. Berry, ibid.88, 3910 (1993); H. L. Davis, J. Jellinek, and R. S. Berry, ibid.86, 6456 (1987); J. Jellinek, T. L. Beck, and R. S. Berry, J. Chem. Phys. 84(5), 2783 (1986). [5] N. Ju and A. Bulgac, Phys. Rev. B 48, 2721 (1993). [6] Y. Li, E. Blaisten-Barojas, and D. A. Papaconstantopoul os, Phys. Rev. B57, 15519 (1998). [7] F. Calvo and F. Speigelmann, Phys. Rev. Lett. 82, 2270 (1999). [8] F. Calvo and F. Spiegelmann, J. Chem. Phys. 112, 2888 (2000). [9] R. Poteau, F. Speigelmann, and P. Labastie, Z. Phys. D 30, 57 (1994). [10] R. Poteau and F. Spiegelmann, Phys. Rev. B 45, 1878 (1992). [11] J. Jellinek, S. Srinivas, and P. Fantucci, Chem. Phys. L ett.288, 705 (1998). [12] Standard KS scales at least as O(N2). [13] A. Vichare and D. G. Kanhere, J. Phys.: Condens. Matter 10, 3309 (1998). [14] A. Aguado, J. M. L´ opez, J. A. Alonso, and M. J. Stott, J. C hem. Phys. 111, 6026 (1999). [15] A. Aguado, J. M. L´ opez, J. A. Alonso, and M. J. Stott (to b e published). [16] P. Blaise, S. A. Blundell, and C. Guet, Phys. Rev. B 55, 15856 (1997). [17] A. M. Ferrenberg and R. H. Swendson, Phys. Rev. Lett. 61, 2635 (1988). [18] P. Labastie and R. L. Whetton, Phys. Rev. Lett. 65, 1567 (1990). [19] M. C. Payne et al., Rev. Mod. Phys. 64, 1045 (1992). [20] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [21] S. K. Ghosh and L. C. Balbas, J. Chem. Phys. 83, 5778 (1985). 15[22] D. Nehete, V. Shah, and D. G. Kanhere, Phys. Rev. B 53, 2126 (1996); V. Shah, D Nehete, and D. G. Kanhere, J. Phys.: Condens. Matte r6, 10773 (1994); V. Shah and D. G. Kanhere, J. Phys.: Condens. Ma tter 8, L253 (1996); V. Shah, D. G. Kanhere, C. Majumder, and G. P. Da s, J. Phys.: Condens. Matter 9, 2165 (1997). [23] G. B. Bachelet, D. R. Hamann, and M. Schl¨ uter, Phys. Rev . B26, 4199 (1982). [24] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 685 (1985). [25] D. C. Payne, and J. D. Joannopoulos, Phys. Rev. Lett. 56, 2656 (1986). [26] U. R¨ othlisberger and W. Andreoni, J. Chem. Phys. 94, 8129 (1991). [27] Atomic units are used in the paper, unless otherwise exp licitly stated. [28] F. Calvo and P. Labastie, Chem. Phys. Lett. 247, 395 (1995). [29] P. Blaise and S. A. Blundell, submitted to Phys. Rev. (20 00). [30] S. Nos´ e, Prog. Theor. Phys. Suppl. 103, 1 (1991). [31] K. Hukushima and K. Nemoto, J. Phys. Soc. Japan 65, 1604 (1995); U. H. E. Hansmann, Chem. Phys. Lett. 281, 140 (1997). [32] V. Bonaˇ ciˇ c-Kouteck´ y, P. Fantucci, and J. Kouteck´ y , Phys. Rev. B 37, 4369 (1988). Figures 160.020.040.060.080.10.120.140.160.180.2 0510152025303540δ Time (ps) 60 K125 K200 K250 K Figure 1: The rms bond-length ﬂuctuation of Na 8simulated using the KS-AI model as a function of time for various temperatures. Note th at the tendency to converge is faster at low temperatures. 1700.020.040.060.080.10.120.140.160.18 05101520253035δ Time (ps)75 K250 K 175 K Figure 2: The rms bond-length ﬂuctuation of Na 20simulated using the DB- AI model as a function of time for various temperatures. Note that the tendency to converge is faster at low temperatures. 180.020.040.060.080.10.120.140.160.180.2 406080100120140160180200220240260δ Temperature (K) 5 ps37.5 ps Figure 3: The rms bond-length ﬂuctuation of Na 8simulated using the KS-AI model as a function of temperature over 5 ps and 37.5 ps. 00.511.522.53 0.2 0.4 0.6 0.8 1〈r2(t)〉 Time (ps)80 K150 K200 K250 K Figure 4: Mean square ionic displacements of Na 8simulated using the KS-AI model at 1 ps time scale. 1905101520253035 0 510 15 20 25 30〈r2(t)〉 Time (ps)60 K150 K250 K Figure 5: Mean square ionic displacements of Na 8simulated using Kohn- Sham at 25 ps time scale. 0510152025 0.0000.0020.0040.0060.0080.0100.0120.014Entropy (au) Configurational Energy (au) Figure 6: Ionic entropy of Na 8for the KS-AI model extracted by the multiple histogram method. 2011.051.11.151.21.251.31.351.4 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 7: Canonical speciﬁc heat of Na 8simulated using KS-AI. 11.021.041.061.081.11.121.141.16 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 8: Canonical speciﬁc heat of Na 8simulated using DB-AI. 2111.021.041.061.081.11.121.141.16 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 9: Canonical speciﬁc heat of Na 8simulated using KS-SP. 11.051.11.151.21.251.3 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 10: Canonical speciﬁc heat of Na 8simulated using DB-SP. 2211.051.11.151.21.251.3 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 11: Canonical speciﬁc heat of Na 20simulated using KS-SP. 11.11.21.31.41.51.6 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 12: Canonical speciﬁc heat of Na 20simulated using DB-SP. 2311.051.11.151.21.25 50 100 150 200 250 300Cv/C0 Temperature (K) Figure 13: Canonical speciﬁc heat of Na 20simulated using DB-AI. 24
arXiv:physics/0011064v1 [physics.space-ph] 28 Nov 2000Particle propagation upstream of CIR shocks M. Savopulos Information Dept, Education and Technological Institute o f Thessaloniki, Greece J. J. Quenby and M.K. Joshi Physics Department, Imperial College, London, SW7 2BZ, UK M. Fraenz Queen Mary and Westﬁeld College, Astronomy Unit, Mile End Rd ., London E1 4NS, UK. Abstract The ﬁrst high solar latitude pass of the Ulysses spacecraft r evealed the pres- ence of MeV particle increases up to latitudes above 60 degre es, well outside the CIR belt but associated in time with the regular passage of these plas ma interfaces at more equa- torial latitudes. The particle increases have been explain ed variously as due to diﬀusion from a ﬁeld line connection with a CIR at a greater distance, p erpendicular diﬀusion in latitude, propagation from the inner Corona or an accelerat ion process on a connecting ﬁeld line. Numerical solutions to the 1-D propagation equat ion upstream of a CIR shock for reasonable diﬀusion mean free paths are shown here to lim it the source of the increases to within about 2 AU of the CIR’s, asssuming the CIR’s either t rap or accelerate ener- getic particles and they diﬀuse from the nearby CIR interfac e. The problem of allowing propagation from the CIR’s is eased if the CIRs are located ac cording to the predictions for the current sheet with a solar source surface at 2 .5Rs, rather than at 3 .25Rs. Ener- getic electron observations showing delays with respect to the CIR-associated ions suggest possibly an acceleration in the inner corona, rather than in the CIR-associated shock. A more likely explanation, however, is the ability of electro ns to take a longer a longer route, from beyond the point of observation, than for that of the ion s and yet arrive within a few days of CIR closest approach time. 1.Introduction In the course of the ﬁrst Ulysses southern latitude polar pas s it was found that whereas the interaction regions between high speed and low speed sol ar wind driving CIR’s were limited to solar latitudes between 13◦and 40◦, associated energetic particle increases ap- peared up to 64◦in the case of 0.5-1.0 MeV protons and to 75◦in the case of 50 keV electrons. Despite the well known longitudinal spreading o f such particle increases in equatorial regions (O’Gallagher and Simpson, 1966 ), this o bservation constituted a ma- jor surprise to those working on interplanetary energetic p article transport. It clearly provides crucial information on the relative importance of various competing plasma and energetic particle transport processes (Fisk, 1997). The p articles may have originated in the CIR’s due to interplanetary diﬀusive shock acceleratio n (Palmer and Gosling, 1978), or they may have been mainly accelerated close to the sun and c arried out in a trapping region (Lim et al., 1996). In either case, cross-ﬁeld diﬀusi on from the CIR to Ulysses (Kota and Jokipii , 1995) or sunward diﬀusion along a ﬁeld lin e connecting to an ex- panded CIR beyond the Ulysses orbit (Keppler et al., 1995, La nzerotti et al., 1995) or sun-ward diﬀusion along a ﬁeld line constrained to move in la titude under the combined inﬂuence of diﬀerential solar rotation and diﬀering wind sp eeds which eventually meet the streamer belt ( Fisk, 1996) are all possible modes of prop agation for the high latitude energetic particles. A third alternative is that particles , originally accelerated at the sun, diﬀuse in the inner corona before they move out along connect ing ﬁeld lines to Ulysses ( Quenby et al., 1996). In the following we describe data rele vant to the high latitudeincreases and discuss possible models. An additional complication in the observational data lies i n the delayed onset of the CIR- related energetic electron increases, relative to ions, ab ove the low latitude streamer belt, Simnett and Roelof (1995). 2.Data The helium data to be employed has already been presented by K eppler et al., (1995) and was obtained from the Ulysses EPAC experiment. 4 identical, three-element semiconduc- tor telescopes mounted at diﬀerent angles to the spin axis yi elded 8 sector information over 80% of the solid angle The geometry factor to measure 0.3 to 1.5 MeV protons and 0.4 to 6 MeV/N heavy ions was 0.08 cm2ster. We conﬁne ourselves to the use of 0.4-1.0 MeV/N omnidirectio nal helium data obtained on the ﬁrst Ulysses southern solar pass. Figure 1 is reproduc ed from Keppler et al., (1995) in order to clarify subsequent discussion. The regul ar series of energetic alpha particle increases, 26 days apart are shown plotted against time, spacecraft-sun distance and heliolatitude. In general, the greater part of these par ticle increases lie between well-identiﬁed forward-reverse shock pairs in the low lati tude ’streamer belt’ but cease to be enclosed by signiﬁcant plasma discontinuities at high la titudes (Phillips et al, 1995). 40◦is taken as the transition point between ’low’ and ’high’ in o ur subsequent anal- ysis. A mean spectrum for energetic He ions within a CIR is est ablished from data obtained close to the reverse shocks identiﬁed between lati tudes -32◦and-38◦. We will also use data at higher latitudes, again coinciding with the regular particle increases which seem CIR-associated, at the distance (AU) and souther n latitude (degrees) pairs; (4.2,41),(4.0,44),(3.8,47),(3.6,52),(3.4,56),(3.2,6 0). Electron and ion data are obtained from the Ulysses HI-Scale experiment, (Lanzerotti et al., 1992). We use the archived data from the thin, single sol id-state detector system with active anti-coincidence and thin foil separation of electr ons in the energy range 30 keV to 300 keV and ions in the energy range 300 keV to 5 MeV. The telesc opes involved have solid angles of 0.48 cm−2sr. 3.Shock Expansion Model Since it will turn out that the models for the high latitude in creases most stringently con- strained by the above data are those involving su-ward diﬀus ion from a CIR source located well beyond Ulysses, we will centre our numerical analysis o n one of these, namely the ex- panded CIR beyond the spacecraft orbit. In order to determin e the likely range over which diﬀusion takes place, it is thus necessary to establish plau sible estimates of the expansion history of a CIR as it moves beyond Ulysses. Gosling et al., (1 993) pointed out that the reverse shock moved poleward. Figure 2 is a cartoon repre senting a meridian plane cross-section of an interaction region in the southern hemi sphere with the equator-ward forward shock expansion and the pol-ward reverse shock expa nsion. Ulysses is located at a latitude such that it does not directly see the CIR interact ion region but lies on a radial line from the sun which subsequently intersects the expande d CIR. A simple estimate of the expansion is obtained following Quenby et al., (1995), w ho assumed a radial speed of 500 km/s and found the pole-ward or latitude expansion spe ed arising from the excess pressure within the CIR to be the Alfven speed of 150 km/s. In t his calculation, it is further assumed that the extent of the streamer belt remains constant in it’s upper lati- tude boundary at a given radial distance throughout the sout hern latitude pass. Starting from the observed extent of the streamer belt as reaching 36◦S at 4.4 AU, the CIR is then found to lie on the same radial vector as Ulysses at the follow ing pairs of distances andlatitudes; (5.8,41),(6.9,44),(8.2,47),(10.8,52), (13. 6,56),(17.1,60). This calculation of the CIR expansion is to be regarded as the maximum possible and in deed, the simulations by Pizzo, (1991) suggest a reduction in the rate of latitude inc rease beyond ∼10 AU. It is not easily established how the intensity increase insi de the CIR varies with posi- tion. We have tried correlations with radial distance, Ulys ses latitude and the maximum extent of the neutral sheet at the time. There is only a poor co rrelation with the later two parameters but while the radial increase yields the domi nant correlation, we cannot from our data and epoch extrapolate beyond 5 AU and so we take a favourable case and assume that the intensity increase stays constant. Thus we take the mean intensity found previously as applying both as the source intensity in side the CIR for the (4.2,41) observation and to the subsequent points where Ulysses is co nnected on a solar radius vector to the CIR which has propagated beyond and above this p osition. Infact, if the Van Hollebeke et al., (1978) data on the equatorial plane rad ial dependence of the peak CIR particle intensity applies, the source ﬂux within the CI R will diminish beyond 4-5 AU. All the intensities corresponding to the above pairs of p ositions are assumed to refer to observations upstream (or sun-ward) of the expanded reve rse shock of the CIR. These intensities, normalised to the intensity inside the CIR (sh ock and observation at 5.8 AU) are plotted in ﬁgures 3 and 4, together with the estimated ups tream distances on the Quenby et al., (1996) model. If the connection between the CIR from beyond Ulysses to the s pacecraft is via a ﬁeld line moving in latitude according to the Fisk, (1996) model, the r adial distance to be moved is typically 10 AU for a line seen at 70◦in the inner heliosphere. Thus calculations on the ﬂux expected upstream of shock on the expanded CIR model are a pplicable to geometries based upon the Fisk, (1996) model provided in each case, the d ominant mode of particle diﬀusion is radial. 4.Semi-quantitative consideration of the possible propagat ion The ability of MeV particles to propagate ∼10 AU to be seen with the observed intensity and to be reasonably in phase with the established low latitu de solar rotation periodicity are critically dependent on the radial or perpendicular diﬀ usion mean free path adopted. It is diﬃcult to better the ’Palmer Consensus’ value of 0.1 AU (Palmer, 1982) as applying in a wide rigidity and distance range for λrr. However a detailed study of the time vari- ability in the magnetic ﬂuctuation spectum on an hour to hour basis and of the anisotropy injection proﬁles of long lived particle events by Wanner an d Wibberenz, (1991) found λ/bardblvarying between 0.01 and 1.0 AU. Reames (1999) maintains tha t both prompt and gradual SEP events are consistent with λ/bardbl∼1 AU. The ’Palmer’ value is in accord with recent realistic, magnetometer based computations of the p arallel mean free path (Drolias et al., 1997) over the Ulysses orbit which yield λ/bardbl∼0.1 AU provided we are at ∼1 AU. There is, however, already a puzzle beyond ∼5 AU in adopting this relatively low value ofλ/bardblwhere λrr=λ/bardblcos2χwhere χ∼70◦orλrr∼0.01 AU , unless λ⊥∼λ/bardbl. Using the expression for the diﬀusion time, τ, to reach peak ﬂux over distance, L, with velocity, v, τ∼(3/4)L2/λrrv, we ﬁnd it takes 94 days to diﬀuse 10 AU, 23 days to diﬀuse 5 AU and 90 hours to diﬀuse 2 AU. These numbers suggest that unl ess the ’source’ of the high latitude increases is within a few AU of Ulysses, the 27-day periodicity in the enhancements will be diﬃcult to explain. 5.Quantitative radial back diﬀusion model In this section, we numerically solve the Fokker-Planck tra nsport equation for the case of spherically symmetric propagation back towards the sun fro m the reverse shock of a CIR.The Fokker-Planck with only radial coordinate dependence a nd assuming a steady state is1 r2∂ ∂r(r2k∂f ∂r)−(V−Vs)∂f ∂r+2 3V v r∂f ∂v= 0 (1) Here k is the radial diﬀusion coeﬃcient where k=λrrv/3 , f the distribution function, V the wind speed, Vsthe shock speed, r the solar radial distance and all quantiti es measured in a solar reference frame. The equation expresses the facts that while the diﬀusive and adiabatic deceleration divergence terms depend on distanc e from the Sun, the convective motion must be considered relative to the shock. The boundar y condition derived from continuity of streaming at the CIR reverse shock is −k(∂f ∂r)1+v 3(∂f ∂v)1(Vs−V1) =v 3(∂f ∂v)2(Vs−V2) (2) where 1 and 2 refer to upstream and downstream conditions. Us ing the inverse compres- sion ratio β,Vs−V2=β(Vs−V1). The boundary condition equates upstream diﬀusive and convective ﬂows with downstream convection with negligibl e diﬀusion. Following Savop- ulos et al., (1995), equations 1 and 2 are combined and numeri cally solved by inverting a quindiagonal matrix allowing k=k◦vαrǫ.k◦will be quoted with v measured in units of 108m s−1and r in units of 1011m. We show in ﬁgures 3 and 4 as continuous lines solutions to equa tions 1 and 2 for two relations for k. In ﬁgure 3, α=ǫ= 1 and λrr=0.008AU at 1 MeV/N and 1AU. In ﬁgure 4,α= 1.6, ǫ= 0.0 and the k◦value of 5.1435 in the above units corresponds to λrr= 0.44 AU at 1MeV/N independent of distance. The top curve is the ﬁt t o the He spectrum inside the CIR while the others correspond to the computed up stream spectra for the assumed shock-Ulysses distances on the model of section 3 fo r the latitudinal spread of a CIR. These distances can similarly be employed in the model o f the latitudinal wandering of an interplanetary ﬁeld line. The ﬁgures show that a variet y of mean free path models can ﬁt the data (as demonstrated by Savopulos and Quenby, 199 8) although independent propagation evidence clearly favours an intermediate mean free path value. However, both ﬁgures conﬁrm that there is little prospect of ﬁtting th e intensity fall-oﬀ apparently observed upstream of the reverse shock except within about 1 or 2 AU of the CIR, if either back diﬀusion model is adopted. The fall in intensity due to convective sweeping and adiabat ic expansion over ∼5AU also calls into question models requiring coronal plus interpla netary propagation. Some signiﬁcant reduction in the requirement on the distanc e travelled in latitude is gained by following Sanderson et al.,(1999) who employ the W ilcox Solar Observatory curent sheet locations based on the 2 .5Rssource surface model, rather than that with the source surface at 3 .25Rs, which is often used. Both models assume potential ﬁelds and a radial source surface ﬁeld, but the former adds a sharply pea ked polar ﬁeld to the surface line-of-sight data while the latter assumes a purely radial surface ﬁeld. Both models are reasonably, but not entirely successful in predicting to th e IMF (eg Hoeksema,1995). Us- ing the 2 .5Rsmodel the curent sheet is found to have passed within about 5◦of Ulysses up to about day 60, 1994, although there still remains a gap op ening to ∼30◦at the last ion increase seen around day 160, 1994, so the problem is only possibly partly solved with the model with the lower source surface. The active interact ion region thought to be op- erative at these times is not considered to extend much south of the maximum southward extent of the neutral sheet at the Ulysses radius.6.Interpretation Of ’Delayed Electrons’ Figures 5a and 5b illustrate the delayed electron arrival relative to the ions above the streamer bel t. Figure 5a runs from day 183 to day 273, 1993 and shows the electron channels 30-50 keV(E1 ), 50-90 keV(E2), 90-165 keV(E3), 165-300 keV(E4) and the ion channels 300-550keV (F P4), 550-keV-1MeV (FP6) and 1-5MeV (FP6). The electron and ion increases up to day 225 , corresponding to a South latitude of about 37◦, tend to peak at the same time, although the electron in- crease often extends to later times or further upstream from the reverse shock, behaviour conﬁrmed by similar plots at lower latitudes. By day 261, cor responding to a latitude of about 41◦, the electron increase is delayed 2 days in the lowest energy channel at least. Figure 5b for the time period day 274 to 365, 1993 shows a delay of about 2 days devel- oping at all energies shown, again a number conﬁrmed by early 1994 data. In seeking to understand the electron data delay we note the c onclusion of Palmer (1982) that the parallel electron mean free path between 10 keV and 1 0 MeV lies in the interval 0.1-1 AU with a likely value ∼0.1U AU. Scatter-free events with λ∼1 AU are judged to be relatively rare. Assuming perpendicular diﬀusion over t he shortest possible mean free path, let us attempt an interpretation of the extra electron time delay around day 343, 1994, when Ulysses was at ∼3.9 AU at latitude ∼46◦south. According to Sanderson et al,(1999), the current sheet was at ∼40◦when it passed closest to Ulysses. Hence the distance over which perpendicular diﬀusion is to take place from the CIR to Ulysses is ∼0.4 AU. A way of estimating the MeV proton propagation time is to take a λ⊥∼0.1λ/bardbl. This is within an acceptable range and is favourable to expla ining the electron data by the same propagation mode. With λ/bardbl∼0.1AU, λ⊥=0.01AU. This yields a 1.7 day de- lay. Suppose a total diﬀusion time across the ﬁeld above the C IR of 1.7 day (suggested by the proton diﬀusion data) plus the 2 days extra electron de lay at 230 keV. Then we require an electron λ⊥= 0.0003−0.0004 AU in the range 40-230 keV. There is thus no consistency with the ’Palmer consensus’ value of the cros s ﬁeld k⊥/β∼1021cm2s−1 orλ⊥≃7.10−3AU. Neither is the theoretical ﬁeld line wandering estimate (Forman et al.,1974) of k⊥/β≃4×1020cm2s−1satisﬁed, especially the velocity independence of this quantity. Computations by Kota and Jokippi, (1998), suggest that the 1 00keV electrons, in contrast to the MeV ions, can infact arrive from a CIR position further out than Ulysses. These authors employ a tilted dipole model of the heliospheric ﬁel d, with purely radial ﬂow, that incorporates the forward and reverse shocks at the interfac es between the slow and fast streams. The 3-D test particle transport equation is solved , including already accelerated electron and ion spectra injected at the shock fronts. The si mulations reveal a broader time peak in the high latitude arrival with several days dela y with respect to the ions, suggesting propagation predominantly from further out. Using our ’semi-quantitative approach’ as before, we estim ate a three day delay for 100 keV electrons with λ/bardbl=0.1 AU would enable a distance of 1.8 AU to be traversed along a ﬁeld line whereas this distance increases to 5.7 AU if λ/bardbl=1.0 AU. Since these relativistic electrons are in a propagation parameter parameter regime w here diﬀusion dominates convection in the transport equation (1), the Fisk and Lee (1 980) approximation to the remaining, diﬀusion and adiabatic deceleration terms shou ld yield the radial intensity dependence upstream of the shock. This solution gives f=F(v)(r/rs)2β/(1−β)provided k=k◦vrwithr◦as shock location. Hence with β= 1/4, f∝r2/3and therefore the intensity reduction over 2-6 AU is not drastic. This fact can also be inferred from ﬁg 4 which in the high velocity regime and for λ/bardbl=0.44 AU corresponds to the diﬀusiondominated solution regime. Note that the theoretical curve s approach each other at the highest velocities, denoting a low spatial gradient in the p arameter space where the radial dependence is expected to be particle velocity and species i ndependent according to the Fisk and Lee (1980) approximation. Clearly with an appropri ate choice of the k⊥/k/bardbl ratio, one can arrange for the cross ﬁeld diﬀusion electron i ntensity to be less than the along the ﬁeld diﬀusion electron intensity. Note this 1-D di scussion of an essentially 3D situation depends on k⊥<< k /bardbl. An alternative approach to the electron delay is to assume mo st acceleration takes place close to the sun within the streamer belt conﬁnes or with part icle release mainly situ- ated within this region. It is unlikely that electrons appea r at higher latitudes relative to protons because of a basic asymmetry in the output of recon nection event aceleration because such events are orientated at random. In the IMF at th e epoch of observation, the gradient and curvature drifts will produce a drift veloc ity in a direction such that the electrons appear later, but the drift speed is only is only ∼3×105cm/s. Hence this is unlikely to be a cause of the measured delay as insuﬃcient lat itude separation occurs. A possibility is that at ∼100 keV or ∼0.3MV, λ/bardbl∼1 AU. Then electrons accelerated within a few Rspropagate easily to Ulysses, outside the CIR belt, after cor onal diﬀusion while protons ∼0.4 MeV or ∼30 MV are much attenuated in intensity in the coronal diﬀusio n region or even in the IMF. This idea presupposes that favoura ble acceleration conditions involving reconnection or shock propagation across closed magnetic loops occur mainly within the streamer belt. The observation by Van Hollebeke e t al, (1978), that within CIR’s the proton intensity increases more than 100% from 0.4 AU to 1 AU simply states that some interplanetary acceleration occurs within the CI R but does not invalidate the idea of an important coronal component to acceleration. A co ronal diﬀusion model ex- plains the reduction in appearence rate with increased lati tude for electrons noticed by Lanzerotti et al, (1995). CIR particle abundances are mainl y similar to solar wind abun- dances (Keppler, 1998), rather than ﬂare abundances. Howev er, the likely source of the solar wind is small-scale reconnection events, so there can be no basic incompatibility in the source abundances. Large-scale ﬂares are clearly pot entially diﬀerent in plasma constitution. 7.Conclusions With accepted values of the radial mean free path in the few AU , few MeV/N region, the models for high solar latitude CIR associated particle i ncreases which involve diﬀu- sion over 5-10 AU, either from behind Ulysses or directly fro m the Sun, have diﬃculty in ﬁtting the relatively high intensities seen up to 60◦when spherically symmetric solutions of the Fokker-Planck equation are employed. The problem is e nhanced by the fact that favourable assumptions were made for the intensity of the so urce particles within CIR’s beyond Ulysses and for the height of the CIR expansion in lati tude. There is an additional problem in the long time required for such diﬀusion to take pl ace rendering the observed phase coherence of the particle increases diﬃcult to unders tand. An alternative is to consider a perpendicular diﬀusion mean free path not much sm aller than the parallel λ, allowing the CIR associated ﬂuxes to propagate only 1-2 AU to locations directly above the shock in latitude. Use of the neutral sheet model with sol ar source at 2 .5Rsreduces the amount of latitude propagtion required although it rema ins signiﬁcant. The relatively large value of the perpendicular diﬀusion co eﬃcient required in the above analysis ﬁts with the relatively small latitudinal gradien ts in low energy cosmic ray in- tensity seen by Ulysses which appear signiﬁcantly less than expected on full drift modelswithK⊥= 0.05K/bardbl(see Drolias et al., 1997 and references therein). 100 keV electron increases observed above the streamer belt are not easily explained on the purely perpendicular diﬀusion transport model with rea sonable parameters. Instead, it is possible that a coronal source is the basic accelerator and the scattering mean free path outside CIR’s at sub MV rigidities is of the order of 1 AU. Most ion transport would then be eﬀected by trapping within the CIR’s while most elect ron transport would be ﬁrst via the corona and then in the quiet IMF. However, back diﬀusi on from the expanding CIR beyond Ulysses is a likely explanation for the delayed an d extended electron arrival as fewer special assumptions arise in this last model. 8.Acknowledgements Helpful discusions with Bern Blake, Erhard Keppler, Bob For syth, Trevor Sanderson and Bernd Heber are acknowledged. PPARC visitors’ support for M . Savopulos was gratefully received. 9.References Drolias,B.,Garaud,P.,Quenby,J.J. and Smith,E.J.:1997, Proc.25th. ICRC,1,229. Fisk,L.A.:1996, JGR,101,A7,15,547. Fisk,L.A.:1997, Proc.25th. ICRC,8,27. Fisk,L.A and Lee,M.A.:1980, Astrophys. J. 237,620. Forman,M.A.,Jokippi,J.R. and Owens.:A.J.,1974, Astroph ys.J.,192,535. Gosling,J.T.,Baker,D.N.,Bame,S.J.,Feldman,W.C.,Zwic kel.R.D.and Smith,E.J.:1993, Geo- phys. Res. Lett.,20, 2789. Hoeksema, J.T.: 1995, Space Sci., Rev.,72/1-2,137. Keppler,E.,Franz,M.,Korth,A.,Reuss, M.K.,Blake,J.B., Seidel,R,Quenby,J.J. and Witte,M.:1995,Science,268,1013. 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Figure Captions Figure 1 Overview of the ﬁrst Ulysses south high latitude pas s showing the solar wind, magnetic ﬁeld strength and energetic helium ﬂux as a functio n of heliolatitude. Figure 2 Representation of a heliospheric meridian plane sh owing the oscillating neutral current sheet, the developing forward and reverse shocks an d the radial line through the Ulysses spacecraft. Figure 3 Observed high latitude He intensities as a function of particle velocity measured in MeV/N units. Positions of observation and modelled posit ions where the expanding CIR’s reach the radial line from the sun through Ulysses beyo nd the spacecraft are indi- cated. Computational solutions corresponding to these ups tream of the shock positions are shown as solid lines. The diﬀusion parameters employed a re stated at the bottom. Figure 4 Similar to ﬁgure 3 for alternative diﬀusion paramet ers. Figure 5 Ulysses electron CIR observations above the stream er belt, 1993, days 183-273. Comparison ion channels are also shown. Figure 6 Ulysses electron CIR observations above the stream er belt 1993, days 274 to 365. Comparison ion channels are also show.HED: magnetic field strength EPAC: 0.4 - 1 MeV/n heliumSWOOPS : solar wind speed(a) (b) (c)400600800Speed [km/s] 0246B [nT] 10-510-410-310-210-1110count rate [1/s] 92180 92280 93014 93114 93214 93314 94049 94149 94249 5 4.5 4 3.5 32.5 -20 -25 -30 -35-40 -50 -60-70r [AU] q [o]5AU F RF R U10AU HCS/G19/G17/G19 /G19/G17/G24 /G20/G17/G19 /G20/G17/G24 /G21/G17/G19/G16/G23/G16/G21/G19/G21/G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G21/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G19/G36/G56/G15/G3/G53/G86/G32/G25/G17/G28/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G27/G36/G56/G15/G3/G53/G86/G32/G27/G17/G21/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G25/G36/G56/G15/G3/G53/G86/G32/G20/G19/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G23/G36/G56/G15/G3/G53/G86/G32/G20/G22/G17/G25/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G21/G36/G56/G15/G3/G53/G86/G32/G20/G26/G17/G20/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G21/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G19/G36/G56/G15/G3/G53/G86/G32/G25/G17/G28/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G27/G36/G56/G15/G3/G53/G86/G32/G27/G17/G21/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G25/G36/G56/G15/G3/G53/G86/G32/G20/G19/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G23/G36/G56/G15/G3/G53/G86/G32/G20/G22/G17/G25/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G21/G36/G56/G15/G3/G53/G86/G32/G20/G26/G17/G20/G36/G56/G47/G82/G74/G3/G82/G73/G3/G45/G87/G15/G3/G45/G87/G3/G81/G82/G85/G80/G68/G79/G76/G93/G72/G71/G3/G87/G82/G3/G20/G3/G68/G87/G3/G89/G32/G19/G17/G24/G26/G24/G48/G72/G57/G18/G81/G88/G70/G15/G3/G53/G86/G32/G24/G17/G27/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27 /G57/G72/G79/G82/G70/G76/G87/G92/G3/G76/G81/G3/G48/G72/G57/G18/G81/G88/G70 /G38/G82/G80/G83/G68/G85/G76/G86/G82/G81/G3/G39/G76/G68/G74/G85/G68/G80/G3/G73/G82/G85/G3/G78/G19/G32/G20/G17/G19/G71/G16/G20/G15/G3/G46/G19/G32/G24/G17/G20/G23/G22/G24/G15/G3/G3/G78/G32/G78/G19/G13/G11/G89/G13/G13/G20/G17/G25/G12/G19/G17/G19 /G19/G17/G24 /G20/G17/G19 /G20/G17/G24 /G21/G17/G19/G16/G25/G16/G23/G16/G21/G19/G21/G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G21/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G19/G36/G56/G15/G3/G53/G86/G32/G25/G17/G28/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G27/G36/G56/G15/G3/G53/G86/G32/G27/G17/G21/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G25/G36/G56/G15/G3/G53/G86/G32/G20/G19/G17/G27/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G23/G36/G56/G15/G3/G53/G86/G32/G20/G22/G17/G25/G36/G56 /G3/G49/G88/G80/G72/G85/G76/G70/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G21/G36/G56/G15/G3/G53/G86/G32/G20/G26/G17/G20/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G21/G36/G56/G15/G3/G53/G86/G32/G24/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G23/G17/G19/G36/G56/G15/G3/G53/G86/G32/G25/G17/G28/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G27/G36/G56/G15/G3/G53/G86/G32/G27/G17/G21/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G25/G36/G56/G15/G3/G53/G86/G32/G20/G19/G17/G27/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G23/G36/G56/G15/G3/G53/G86/G32/G20/G22/G17/G25/G36/G56 /G3/G82/G69/G86/G72/G85/G89/G68/G87/G76/G82/G81/G68/G79/G15/G3/G53/G82/G69/G86/G32/G22/G17/G21/G36/G56/G15/G3/G53/G86/G32/G20/G26/G17/G20/G36/G56/G47/G82/G74/G3/G82/G73/G3/G45/G87/G15/G3/G45/G87/G3/G81/G82/G85/G80/G68/G79/G76/G93/G72/G71/G3/G87/G82/G3/G20/G3/G68/G87/G3/G89/G32/G19/G17/G24/G26/G24/G48/G72/G57/G18/G81/G88/G70/G15/G3/G53/G86/G32/G24/G17/G27/G15/G3/G53/G82/G69/G86/G32/G24/G17/G27 /G57/G72/G79/G82/G70/G76/G87/G92/G3/G76/G81/G3/G48/G72/G57/G18/G81/G88/G70 /G38/G82/G80/G83/G68/G85/G76/G86/G82/G81/G3/G39/G76/G68/G74/G85/G68/G80/G3/G73/G82/G85/G3/G78/G19/G32/G21/G17/G24/G24/G71/G16/G22/G15/G3/G78/G32/G78/G19/G13/G11/G89/G13/G13/G20/G12/G13/G11/G85/G13/G13/G20/G121993: Days 183 to 273 : ALL 01101001000 183 186 189 192 195 198 201 204 207 210 213 216 219 222 225 228 231 234 237 240 243 246 249 252 255 258 261 264 267 270 273 Days 183 to 273Electrons + (Ions (FP5-6-7) : Counts per DayE1 =40-65 keV E2=60-107 keV E3=107-170 keV E4=170-280 keV FP5=540-765 keV FP6=765-1223 keV FP7=1.223-4.942 MeV1993: Days 274 to 365 : ALL 110100 274277280283286289292295298301304307310313316319322325328331334337340343346349352355358361364 Days 274 to 365Electrons + (Ions: FP5-6-7) : Counts per DayE1=40-65 keV E2=60-107 keV E3=107-170 keV E4=170-280 keV FP5=540-765 keV FP6=765-1223 keV FP7=1.223-4.942 MeV
arXiv:physics/0011065v1 [physics.geo-ph] 28 Nov 2000A natural phenomenon that may pose a severe aircraft hazard? Thomas Gold Professor Emeritus, Cornell University, Ithaca NY 14853; E mail: tg21@cornell.edu There have been many serious aircraft accidents in recent ye ars that have not had a satisfactory explanation despite exhaustive researches, and that have certain features in common. Those features include apparently a situation of ex treme urgency and danger, so that there was no time for the ﬂight crew to communicate detai ls to the ﬂight controllers; in some cases there were circumstances that seemed quite une xpected and perplexing to the ﬂight crew, suggesting an urgent need to override the usu al automated control systems and manually put the plane into a steep dive. In several cases this was followed by actions to avoid excessive speed that would threaten the structural integrity of the aircraft. Several accidents have another feature in common: they occurred alo ng the edge of the North- Eastern American continental shelf. These include, among o thers, TW 800 on July 17, 1996, Swissair 111 on September 2, 1998, Egypt Air 990 on Octo ber 1999, and also the crash of J. F. Kennedy Jr. The case of the EgyptAir crash has re cently come under public debate again as some new information has become known, and th e explanation tentatively oﬀered by the National Transportation Safety Board (NTSB), suggesting a suicide attempt by a co-pilot, has come under strong attack by Egyptian autho rities, and does not ﬁt with the new information. In view of the statistically quite improbable occurrence of these accidents, it seems prudent now to widen the search to causes that have so far not b een included among possible aircraft hazards, and that have possibly a relatio nship to geographical features. Among such, the massive emission of gases from the seaﬂoor (o r land surface) seems to us most worthy of attention. Massive sudden eruptions of gases have occurred in many loca tions, bursting up through the ground both from ocean ﬂoors and from dry land. Th ey often occur repet- itively in the same area, and on land create what is known as ”m ud volcanoes”. The amounts of expelled material accumulated in some mud volcan oes in the last million years are as large as 10 or 20 billion tons, and the estimates of the a mounts of gas responsible are several times larger than that. The erupting gases are us ually dominated by methane. Since methane is lighter than air, it races upwards at high sp eed. Many cases are known where the gas spontaneously ignited, and ﬂames to a height of 6,000 ft have been pho- tographed from Baku, in the active mud volcano area on the Wes t shore of the Caspian Sea. Much higher brief ﬂashes have been reported, up to 30,00 0 ft but these were too brief to be photographed. Massive ﬂammable gas eruptions at or nea r times of earthquakes (before, during or after) are reported in historical and in r ecent times from many parts of the world. Similar eruptions are indicated on the sea ﬂoors, where larg e areas are densely covered with ”pockmarks”, quite characteristic circular features in the ocean mud, with diameters of between 10 and 200 meters. These features were ﬁrst detect ed in the North Sea by Dr. Martin Hovland, of Statoil, (the Norwegian oil company) , overlying known gas and oil ﬁelds. Similar ﬁelds have since been detected in many par ts of the world by sonar,again often showing a relation to underlying hydrocarbon ﬁe lds, and also there showing features of repetition of outbursts, with methane again the major component. Both in mud volcanoes and in pockmark ﬁelds the emitted quantities o f gas in any single event may well amount to some millions of tons. Another set of observations has now to be added: it is the occu rrence of ”mystery clouds” in the air. Satellite photography over a ten year per iod revealed more than two hundred clouds that rose up at a high speed from a small area of land or sea, forming an expanding funnel. Temperature observations showed a muc h lower temperature in the funnel cloud than in the outside air at the same height, and th is implied that the rising gas must be one that is intrinsically much lighter than air. O nly methane and hydrogen are candidates, and both are combustible. The largest such c loud on record was seen and reported by several airline pilots ﬂying between Tokyo and A laska, North-East of Japan, on April 9, 1984. They described it as a mushroom cloud that re ached up to 50,000 ft, attaining a diameter of more than 200 miles. Evidence of massive gas emissions have recently been report ed by the Woods Hole Oceanographic Institute, who conducted a sonar survey of th e mid-Atlantic US continental shelf edge. Along a major fault line they found many and very l arge pockmarks, similar to those described by Dr. Hovland, indicating that sudden al most explosive gas eruptions had taken place there. Also recent reports from the Province of Quebec, of frequent and large displays of lights in the sky, clearly related to the sw arm of earthquakes between November 1988 and end of January 1989 in the region of Sanguen ay and Quebec City, leave little doubt that massive gas eruptions occurred ther e, with some ﬂames reaching high into the sky. Altogether 46 such sightings were recorde d in that period, some but not all coincident with earthquake shocks. Earthquake-relate d lights have been well known and reported since antiquity, and indeed one very large even t involving gas ﬂames was reported in 1663, not far from the Sanguenay region, close to the St.Lawrence River. I had investigated in 1982 a ”near disaster” of a British Airw ays 747 plane ﬂying at 37,000 ft over a volcanic region of Java. All four engines sto pped shortly after it had entered a visible but tenuous volcanic cloud. After gliding down to 1 5,000 ft without power, and there apparently leaving the cloud, all engines could be sta rted again immediately. The same sequence of events was experienced two weeks later by an Air Singapore 747 plane over a nearby region, and many years later by a KLM ﬂight over t he Aleutian Islands. A gas lighter than air, and hence combustible, must have been r esponsible in all three cases, to have carried small volcanic dust grains to these altitude s, and its combustion may have been responsible for the engine failures that were so sharpl y limited to the ﬂight within the cloud, probably due to the fuel-rich and oxygen poor mixt ures of the gas adding to the airplane fuel. Gas eruptions of volcanoes are known of ei ther kind: eruptions of a ground-hugging heavy gas identiﬁed as carbon dioxide, but a lso eruptions of a light and ﬂammable gas, probably methane, whose density is a little mo re than half that of air. With three large planes having come so close to disaster, but yet able to give a precise account of the events, one has to take the threat of gas emissi on seriously. The belief that such emissions can come only from volcanoes has been voi ced, but is clearly wrong in view of the facts already mentioned. What threats would mass ive gas emissions pose for aircraft?One eﬀect I have already described: the possibility of induc ing failure of all engines. But several other aircraft hazards have also to be considere d. One is due to the great upward speed the light gases would have, greatly in excess of the vertical speeds in ordinary atmospheric turbulence, and structural damage to the plane or serious injuries to persons may result from the ensuing violent vertical movement. The i gnition and explosion of a large mass of gas external to the plane may be initiated by the engine exhausts and may be violently destructive, yet the recovered airplane skin w ould not show the shrapnel holes that would be the usual signs of explosions. Other consequences of gas emissions are the dangerous and mi sleading indications that the ﬂight instruments would provide. Air speed indicat ors and air pressure altimeters would give quite false and ﬂuctuating readings. The autopil ots, programmed for air, may have totally erroneous responses in the light gas, as indeed may the pilots themselves, who would be perplexed by a situation they had never encountered or contemplated before. A further hazard is that clouds of low density gas may not supp ort a plane, even at a ﬂying speed that would be amply high enough in air. This woul d cause a stall of the aircraft, or be preceded by automatic stall-warning that re quires the pilot to turn the nose down into a dive, and then confront the danger of excessive sp eed. Then there are the various ﬁre hazards resulting from combus tible air-gas mixtures, especially in some conﬁned spaces in the airplane where ﬂame s could be supported, even if the same gas-air mixture would readily be extinguished in th e external high speed airﬂow. That danger may be highest in cable ducts where damage could d estroy the airplane control system. The North-Eastern coastline or edge of the continental shel f of the US and Canada, is the northward continuation of the line whose investigati on I have already mentioned. This extension also has a history of earthquakes and gas emis sion from sand beaches and water surfaces beyond the shoreline. Such emissions had not ceased around the times of the aircraft disasters. A large number of reports were phone d in to police and emergency services in New Brunswick and Nova Scotia on October 27, abou t three days before the Egypt Air crash, stating that at 9:30 p.m. a large ﬁreball had been seen streaking across the night sky. The details reported did not correspond to a me teorite, but included reports of ﬂames and events much slower than those caused by meteors. A peak in the number of reports recorded prior to an event must be taken seriously, i f the number greatly exceeds the number on other days, as was the case here. There were simi lar reports also before and after the TW 800 crash. There was also a report from Swiss A ir 111 of a strange smell about three minutes before the declaration of emergency. Th is is particularly suggestive of gas eﬀects, as a similar report was made in one of the near ac cidents over Java, where gas certainly was involved. We may then wish to investigate whether some features of airc raft disasters along this region, the four disasters mentioned and several others tha t have also occurred along this corridor, could have an explanation in terms of the list of ha zards I have mentioned, or others that have not yet been considered, that could be attri buted to gas eruptions. Mr. Jack Reed retired from the Sandia National Laboratory, a n expert in sound propagation, has noted that the ”loud” boom heard by many eye witnesses at the time of the TW 800 crash on a 25 mile stretch of Long Island, nearest point to the plane 15miles away, was far too loud to have been caused by the propose d explosion of the empty central fuel tank. In his view a one ton bomb of TNT would have b een the least required to make such a sound at that distance. Nor would such an explosio n have caused the various external luminous phenomena that have been reported by many . Also it is doubtful that an explosion of such a small amount of fuel vapor could have ha d the power to tear oﬀ the entire front section of the fuselage. The absence of shrapne l holes in the recovered skin of TW 800 was taken to exclude a bomb explosion inside or outside the plane. However, a massive external gas explosion would produce no shrapnel. The facts newly announced about the EgyptAir disaster make c lear that a deliberate dive had seemed imperative to the pilot then at the controls, and that a dangerous over- speed situation had then arisen. After a brief recovery to le vel ﬂight, again a dive seemed imperative, and the overspeed may then have destroyed the pl ane. There are many steps that can be taken to ﬁnd whether the seque nce of disasters along this heavily traveled corridor may be due to gas emissi ons. As an immediate step I urge the continuation of the sonar search for pockmarks on t he ocean ﬂoor along this coastline in the regions of the four disasters mentioned and others that occurred near this geographical line, since this will have a good chance of show ing whether these accidents were indeed over locations at which strong gas outbursts had occurred. A routing change may then be indicated as the ﬁrst step to avoid further disast ers.
arXiv:physics/0011066v1 [physics.gen-ph] 28 Nov 2000Consistent Equation of Classical Gravitation to Quantum Limit and Beyond Shantilal G. Goradia Physics Department University of Notre Dame Notre Dame, IN-46556 (USA) Email: Shantlal.Goradia.1@nd.edu 1 Introduction General Relativity makes a distinction between mass and spa ce. Mass tells space how to curve and space tells mass how to move. Newtonian gravi ty equation makes a distinction between them by having its numerator as mass eﬀ ect and its denom- inator as inverse square law space eﬀect at macroscopic appr oximation. At micro- scopic distances it makes sense to substitute surface-to-s urface distance between two nucleons for center-to-center distance between them to account for the mass space distinction, keeping in mind the smallest distance be tween coupled nucleons is Planck length. Any distance less than Planck makes no sens e in the classical world. When we calculate the force between two nucleons of on e femtometer diam- eter each, separated by a surface-to-surface distance of Pl anck length, we get the force that matches well known nuclear force i.e. 1040times the value of the force of gravitation “g” calculated by assuming the Newtonian cente r-to-center distance of 1 femtometer. What we get is what is described as the nuclear f orce in scientiﬁc lit- erature. This leads to the question: Is the nuclear force (we ll recognized secondary eﬀect of color force) high intensity gravitation? 2 Analysis Consider the following equations (1)and(2). The notation dnin Equation (2)is the diameter of the nucleons in question. Newtonian FN=Gm1xm2/D2(1) Proposed FP=Gm1xm2/(D−dn)2(2) The notation Din the denominator in Newton’s equation (1)of the gravitational force denotes the separating distance between the centers o f mass of the particles in question. The validity of equation (1)has been veriﬁed for distances as low as a few centimeters. Its validity is not veriﬁed when the sub atomic separating distance between nuclei of atoms is a few femtometers ( fm). Newton’s equation is an approximation that explains macroscopic observation s. If Newton meant his equation to hold true at microscopic distances, he would hav e explained the binding 1energy. Newtonian physics implies point masses and action b etween points. A point has no mass. I am asserting that, instead, the classically de terministic Newtonian gravity originates at the surfaces of nucleons, not at a cent ral point within the nucleons. I am not addressing electrons and coulomb forces i n this paper. This paper is dedicated to the investigation of the nuclear force alone at this stage. The deviation from inverse square logic resulting from our p roposal is insigniﬁcant. The consequences of mass and space microscopic distinction are enormous. The proposed correction is the injection of dn. Equation (2)is good for all distances greater than Planck length (10−35meters = 10−20fm). I require the Planck length as a lower bound so as to include the dominant ﬁrst order quant um eﬀect in this classical model. The relative strength of the proposed equa tion is the ratio obtained by dividing equation (2)by equation (1), which is The ratio FP/FN=D2/(D−dn)2. (3) 3 Strength of Gravity at Short Range per Eqn. (2) When Dis very large compared to dn,D2is almost equal to ( D−dn)2. The diam- eter of atoms is hundreds of millions of times greater than th e diameter of nucleons located at the center of atoms. The force of gravitation calc ulated by these two equations between the nucleons of two adjoining atoms is pra ctically the same, be- cause the ratio D2/(D−dn)2is almost equal to one. When Dis small compared to dn, the force of attraction calculated by equation (2)will be signiﬁcantly greater than that calculated by equation (1). The following results bring home the con- cept. If we call the force of gravitation calculated by equat ion(1)“g”, the force of gravitation calculated by equation (2)would be higher by the ratio D2/(D−dn)2. When Dexceeds dn, by Planck length (10−20fm), one obtains the ratio: D2/(D−dn)2= (dn+ Planck length)2/(Planck length)2(All lenghts in femtometers) = (1 + 10−20)2/(10−20)2(dn= 1 femtometer) = 1040. At surface to surface separations of 1, 2, 3, 4 and 10 femtomet ers, the calculated nuclear forces rapidly diminish to 4.0, 2.1, 1.77, 1.56 and 1 .23 times the gravita- tional forces respectively and match Newtonian gravitatio n at 1000 femtometers as tabulated below. My calculations meet the observed bound ary values. Nucleon deformation is neglected. 2Separating Distance Nuclear Force / Gravity One Planck length, 10−20fm 1040 1 Femtometer 4.0 2 Femtometers 2.1 3 Femtometers 1.77 4 Femtometers 1.56 5 Femtometers 1.44 6 Femtometers 1.36 7 Femtometers 1.31 8 Femtometers 1.26 9 Femtometers 1.23 10 Femtometers 1.21 15 Femtometers 1.15 20 Femtometers 1.11 25 Femtometers 1.09 50 Femtometers 1.04 100 Femtometers 1.02 1000 Femtometers 1.00 34 Analogy Considering a hollow metal sphere containing smaller metal balls rumbling inside the sphere, a probe inside the sphere or close to the outside surf ace would detect non- central high intensity, indeterministic noise with intens ity increasing with distance from the center. For a distant listener, the sound would be of deterministic nature originating from the center of the sphere with its intensity decreasing with distance. Indeterministic high intensity noise at a short range is det erministic low intensity sound at large distances. What this analogy brings home is th at the color force is potentially the high intensity, non-central, classical ly indeterministic interaction. The gravity is potentially the low intensity, macroscopica lly central, Newtonian deterministic manifestation of the same fundamental inter action. I am taking the liberty to use the prevailing view that the nuclear force is t he secondary eﬀect of the color force to reach the following conclusion. This view does not need to be reestablished. 5 Conjectures (A) We do not have quantum gravity: gravity is potentially no t a separate funda- mental interaction of Nature. (B) Rutherford’s scattering experiments showed nuclear fo rces as far as 10 fem- tometers [2]: not that they do not exist beyond that range. At higher distances they are too weak to detect. (C) There is no proof of a central force detected inside the nu cleons. (D) Despite its theoretical justiﬁcation, Yukawa potentia l does not predict the observations. (E) There is no feature of nuclear force that distinguishes i t from gravitation. (F) Einstein attempted to explain nuclear force in terms of g ravity [1]. (G) The Standard Model does not incorporate gravity. 6 Conclusions Newtonian gravitation is potentially a deterministic mani festation of the classically indeterminate color forces addressed in QCD. The prevailin g view is that the nu- clear force is the secondary eﬀect of the color force. The pro posed theory connects the nuclear force with gravitation in one common equation. T he combined contri- butions of these two clearly imply that gravitation is the Ne wtonian deterministic manifestation of the classically indeterminate color forc es. 7 Acknowledgements. I am grateful to Professor Fridolin Weber (University of Not re Dame) for his com- ments following the presentation of the concepts. 4References [1] The conceptual foundation of quantum ﬁeld theory, edite d by Tian Yu Cao of Boston University. (QC174.45.A1C646 1999) Page 85, “Doe s quantum ﬁeld theory need a foundation?” See discussion referring to Nobel Prize Winner Sheldon Glas how. [2] Introductory Nuclear Theory by L. R. B. Elton, D. Sc., F. I nst. P, Professor of Physics, Battersea College of Technology, Second Editio n, 1966. Section 1.7, Nuclear Forces [3] http://ar.Xiv.org/abs/math-ph/0009025 , By S. G. Goradia, 9/15/00 5
arXiv:physics/0011067 v2 18 Dec 2000 1"Ideal-Chain Collapse" in Biopolymers Richard M. Neumann The Energy Institute The Pennsylvania State University University Park, PA 16802 (12/19/00) Abstract A conceptual difficulty in the Hooke's-law description of ideal Gaussian polymer-chain elasticity is sometimes apparent in analyses of experimental data or in physical models designedto simulate the behavior of biopolymers. The problem, the tendency of a chain to collapse in theabsence of external forces, is examined in the following examples: DNA-stretching experiments,gel electrophoresis, and protein folding. We demonstrate that the application of a statistical-mechanically derived repulsive force, acting between the chain ends, whose magnitude isproportional to the absolute temperature and inversely proportional to the scalar end separationremoves this difficulty. Introduction For nearly 60 years, Hooke's law has provided a convenient means of describing the entropic elasticity of an ideal Gaussian polymer chain. [1] Frequently, the force law for a single chain is given by fh = -(3kBT/nl2)r, (1) where fh is the average attractive force acting between the chain ends separated by a scalar distance r. kB is the Boltzmann constant, T the absolute temperature, n the number of chain links, and l the link length. It is apparent that in the absence of an external stretching force, such an equation, if interpreted as a macroscopic equation of state, predicts the collapse of the chainbecause a retractive force is present for r > 0. In fact, this interpretation has prompted one popular reference [2] to state "Therefore, we cannot bring in anything like the relative2deformation Δr/r which appears in the usual form of Hooke's law". This statement appears in the context of defining an elastic modulus to be the ratio of the stress to strain, where the strain, bydefinition, would require division by zero; zero being the magnitude of r associated with the absence of an external stretching force. Although this paradox is frequently ignored, someauthors have attempted to deal with it by focussing solely on real chains and relying on excludedvolume to prevent chain collapse. [3] Others have invoked a pair of "Maxwell demons", one ateach end of the chain, pulling on it with a force equal to the retractive force. [4] Still others havesuggested that the origin of the problem lies in the microscopic nature of the system, resulting ina nonequivalence among ensembles derived by holding either the length, force, or displacementconstant. [5,6] Here, in the spirit of the latter view, we show that the following thermodynamic relationship, which expresses the average repulsive force acting between any pair of particles as a function oftheir separation-distance r, fr = 3kBT/r, (2) provides a means of avoiding the paradoxical behavior noted. [7] In analyzing DNA-stretchingdata, Eq. 2 used in conjunction with Eq. 1 predicts a chain end separation, at zero stretchingforce, equal to the root-mean-square value, ro (ro = n1/2l), in agreement with experiment [8], rather than the zero result predicted by Eq. 1 alone, used by the authors [8] to interpret theirresults. In the lakes-straits model for polyelectrolyte migration in gel electrophoresis, the chain-collapse problem is circumvented by the authors through the derivation of a repulsive forceproportional to T/n. [9] We demonstrate that in the absence of an external field, such a force will move chain segments from a pore having few segments to an adjoining pore having many, inapparent violation of the Second law. We suggest that Eq. 2 provides a statistical-thermodynamic remedy for this problem. In a recent model of protein folding based on springsand beads [10], the excluded volume of the beads is used by the authors to prevent the collapseof the protein into a point singularity. Their approach leads to an equilibrium bead separationthat is not dependent on the temperature. We demonstrate, using Eq. 2, that the classicaldependence of the end separation of a spring on the temperature [11] is recovered and thatexcluded volume is not required to prevent the collapse of a spring.3Theory Whereas there are a number of ways to derive Eq. 2 [7], a particularly simple and instructive approach is to examine a Brownian particle located inside a spherical volume of radius r. The particle occupies a volume element equal to 4 πr3/3, and the particle concentration, c, is proportional to r-3. A chemical potential may be defined by µ = kBT ln(c), permitting the derivation of Eq. 2 using fr = -(∂µ/∂r)T. fr may be viewed as the average repulsive force acting between two particles separated by a distance r, one particle being located at the center of a spherical-coordinate system, r = 0. Equations 1 and 2 may be combined to yield a force equation for an ideal Gaussian chain for which the force vanishes when r = ro, rather than when r = 0: f = 3kBT(1/r – r/ro2) = m(1/h – h), (3) where h = r/ro, and m = 3kBT/ro. Unlike Eq. 1, which does not permit the definition of an elastic modulus in the usual sense, Eq. 3 yields an elastic modulus equal to 2 m. Results and Discussion DNA-Stretching Experiment Figure 1 shows the extension of a DNA molecule, visualized with fluorescence microscopy, as a function of force derived from the flowing solvent surrounding it. [8] The figure depictsonly the qualitative features of the experiment in the weak-stretching region. The molecule washeld stationary against the flow by means of a microsphere attached to one end; the microsphere,in turn, was secured by means of optical trapping. The measure of extension used by the authorsis the average maximum visual elongation in the direction of flow, which for our purposes maybe approximated by r. The straight line depicts an analysis based on Eq. 1 used by the authors. The discrepancy between the data and theory is obvious, most notably the fact that the extensionmeasured at zero flow rate is not zero. Equation 3 is consistent with the experimental data in thatat zero external force, r approximates the random-coil value, h = 1; with increasing force, Hooke's law (Eq. 1) is recovered because fr becomes insignificant for large values of r.4Lakes-Straits Model in Gel Electrophoresis Figure 2 shows a model for a gel where the pores are represented by "lak es", and the straits are the narrow regions connecting the lakes, where there is just sufficient room through which anidealized DNA molecule can migrate. The chain is sufficiently long so that its various sectionsreside in a series of lakes. In describing the elastic behavior of the migrating chain, the authors[9] derive a force equation for an arbitrary section of chain using the distribution function for anideal Gaussian chain in a given lake of size r (the strait-separation r coincides with the end-to- end separation of the chain section), P(r,n) ≈ n-3/2exp(-3r2/2nl2). The entropy is calculated in the usual manner via the Boltzmann expression, S = k ln(P). Here, the lake size is regarded as fixed, the number of links, n, present in the pore fluctuates, and S is differentiated with respect to n (rather than r) to obtain the force equation, fn = (3kBT/2nl)(1 - r2/nl2). (4) As in Eq. 3, a positive value for fn indicates a repulsive force; a negative value indicates an attractive force. It is readily apparent that the force vanishes for nl2 = r2; i.e., when the number of chain links is equal to the value for a random coil having an end separation equal to r. Thus, it would appear that the paradox of chain collapse has been avoided. It is instructive, nevertheless,to consider the following example where two adjacent lakes ( i and j), connected by a strait, are prepared so that nj > ni > (r/l)2. Equation 4 predicts that, in the absence of an applied field, segments will migrate from lake i to lake j, in apparent violation of the Second law. The reason for this behavior is subtle and based on the implicit use of a microcanonical-ensemble in derivingEq. 4; the reader should consult the references for additional background. [9] We simply suggestthat for calculations or simulations of electrophoretic behavior based on the lakes-straits model,Eq. 3 rather than Eq. 4 should be used to calculate the entropic-elasticity contribution to theequation of motion.Gaussian Model of Protein Folding Here a protein is modeled by beads and springs where in all the interactions between any given pair of beads are governed by a single quadratic potential. [10] The beads represent monomersthat may be polar or hydrophobic, and the springs represent the covalent and noncovalent5attractive forces acting between the beads. The model is Gaussian because of the use of aquadratic potential that mimics the Hookean behavior of an ideal Gaussian polymer-chainnetwork. The authors note that, as in the classical theory of rubber elasticity, the springs in theirprotein model are prone to collapse [3,10], regardless of the temperature. Thus, they include arepulsive potential whose magnitude is adjusted to provide the various stable conformations ofthe protein being modeled. Because the calculations for an entire protein require the use of amatrix method to describe the folding (relaxation) process and the stable states and because weare concerned solely with the behavior of a given pair of beads and their connecting spring(where the spring energy is of order kBT) in a heat bath , we shall consider the simplest system described by the model: two beads attached to each other by a spring – a Hookean dumbbell. Following the authors' approach for the multi-bead system, we shall solve the Langevin equation for the dumbbell, - ζdr/dt + f(t) - ∂U/∂r = 0, (5) where r is the vector separation between the first bead and the origin of a coordinate system located on the second bead, ζ is the friction factor for a bead, f(t) is the random-force vector, and U is a potential, here equal to ar2/2. With the use of a coarse-grained time scale to permit f(t) to average to zero, the relaxation expression resulting from the solution of Eq. 5 is r(t) = roexp(-t/τ). Note that the relaxation time τ is independent of temperature; τ = ζ/a. Thus, at long times (equilibrium) and in the absence of excluded volume, r = r = 0. However, in the classical treatment of Brownian motion by Chandrasekhar [11], the equilibrium separation is given by, <r2> = 3kBT/a. (6) This result indicates that it is inappropriate to average out f(t), which is the only term in Eq. 5 that reflects the thermal energy in the spring-bead system through the relationship < f(t1)f(t2)> = (2ζkBT)δ(t1– t2) [11]. Equation 2 provides an alternate, and very much simpler derivation of Eq. 6; the repulsive force 3kBT/r is simply equated to the spring-force ar to yield r2 = 3kBT/a. Thus, in the Gaussian model of protein folding, the expressions for the equilibrium average separation between pairs of6beads must include terms proportional to kBT/a in addition to the contributions from whatever excluded volume is present. The authors claim to present a "thermodynamic" (and presumablystatistical mechanical) approach to protein folding but, in fact, have obtained a mechanical,nonthermal result because of their neglect of f(t) in solving the Langevin equation. In other words, the temperature does not appear in their results. Conclusion Equation 2 is essentially a manifestation of the ideal gas law for a microscopic system consisting of one particle. Equation 3 was anticipated by Flory [12] in his derivation of themolecular expansion factor, α; however, most of his work on rubber-like elasticity is based on Hooke's law (Eq. 1). Using Eq. 2, we have presented realistic solutions to three differentproblems, all associated with a polymer chain (or spring) in a heat bath. It is obvious that the conventional Hooke's-law approach cannot provide an adequate description for polymer-stretching experiments in the weak-force regime where the chain'selongation is measured as the scalar end-to-end separation. Equation 3 provides a description in qualitative agreement with experiment in that it yields the random-coil end separation at zeroapplied force; in the intermediate-force regime, it reduces to the Hookean force law. Whereas the lakes-straits model force equation (Eq. 4) describes an ideal chain that does not collapse in the absence of an external field, it does predict spontaneous segmental flow from aregion of low density to one of high density. The present approach using Eq. 3 avoids both thereverse-flow and the chain-collapse problems. Finally, the Gaussian model of protein folding, despite the claim of its authors that it is a thermodynamic model, does not incorporate the effect of thermal motion. This occurs becausethe random-force vector in the Langevin equation is averaged out, resulting in dynamicalsolutions and equilibrium-state configurations characteristic of a mechanical, nonthermal system.Applying Eq. 2 to a simple bead-spring dumbbell in a heat bath, we obtain the classical result forthe equilibrium size of a harmonic oscillator and suggest that the protein-folding model requiresterms proportional to kBT/a in its expressions for the equilibrium separations of its beads.7References 1. H. M. James and E. Guth, J. Chem. Phys. 11, 455 (1943). 2. A. Y. Grosberg and A. R. Khokhlov, Giant Molecules, Here There and Everywhere (Academic Press, New York, 1997) p. 99; Statistical Physics of Macromolecules (AIP Press, New York, 1994) p. 55. 3. L. R. Treloar, The Physics of Rubber Elasticity (Oxford, London, 1958) p. 78. 4. M. Doi a nd S. F. Edwards, J. Chem. Soc., Faraday Trans. II 74, 1802 (1978). 5. R. M. Neumann, Phys. Rev. A 31, 3516 (1985); ibid. 34, 3486 (1986). 6. D. H. Berman and J. H. Weiner, J. Chem. Phys. 83, 1311 (1985); J. Polymer Sci., Polymer Phys. Ed., 24, 389 (1986). 7. R. M. Neumann, J. Chem. Phys. 66, 870 (1977); Am. J. Phys. 48, 354 (1980). 8. T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, Science 268, 83 (1995). 9. B. H. Zimm, J. Chem. Phys. 94, 2187 (1991); D. Loomans, I. M. Sokolov, and A. Blumen, Macromol. Theory Simul. 4, 145 (1995). 10. B. Erman and K. Dill, J. Chem. Phys. 112, 1050 (2000). 11. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 12. P. J. Flory, Principles of Polymer Chemistry (Cornell, Ithaca, 1971) p. 600.8100 10 1 .1.1110100 Fluid velocity (um/s)Extension (um) Figure 1 (•) depict extension measurements as a function of the fluid velocity for an individual, fluorescently labeled DNA molecule using data in the weak/moderate-stretching regime adaptedfrom Fig. 1 of Ref. 8. The initial extension here is based on an estimate for ro, as the authors provided no value for this quantity. The solid line (—), also borrowed from Ref. 8, illustratesHookean behavior "predicted" from a dumbbell model based on Eq. 1. In the experiment, aseries of "maximum visual extensions in the direction of flow" for a single fluctuating molecule(at a fixed solvent flow rate in the x direction) was measured. The individual measurements, taken at approximately 1-second intervals, were averaged to obtain the reported extension, whichcorresponds to <\| x\|> rather than < r>. Because <\| x\|> → <r> with increasing force, a distinction between these two measures of extension is not necessary for the present purpose.9 ← r → Figure 2. Two adjoining lakes are shown, each of size r. The polymer chain is depicted as the "scribble" line passing through both; the strait is the point of contact between the two circles.
arXiv:physics/0011068v1 [physics.comp-ph] 29 Nov 2000Fast and stable method for simulating quantum electron dyna mics Naoki Watanabe, Masaru Tsukada Department of Physics,Graduate School of Science, Univers ity of Tokyo 7-3-1 Hongo, 113-0033 Bunkyo-ku, Tokyo, Japan (Published from Physical Review E. 62, 2914, (2000).) A fast and stable method is formulated to compute the time evo lution of a wavefunction by numerically solving the time-dependent Schr¨ odinger equa tion. This method is a real space/real time evolution method implemented by several computational tec hniques such as Suzuki’s exponential product, Cayley’s form, the ﬁnite diﬀerential method and an operator named adhesive operator. This method conserves the norm of the wavefunction, manages periodic conditions and adaptive mesh reﬁnement technique, and is suitable for vector- and pa rallel-type supercomputers. Applying this method to some simple electron dynamics, we conﬁrmed th e eﬃciency and accuracy of the method for simulating fast time-dependent quantum phenome na. 02.70.-c,03.67.Lx,73.23,42.65.-k I. INTRODUCTION There are many computational method of solving the TD-Schr¨ odinger equation numerically. Conventionally, a wavefunction has been represented as a linear com- bination of plane waves or atomic orbitals. However, these representations entail high computational cost to calculate the matrix elements for these bases. The plane wave bases set is not suitable for localized orbitals, and the atomic orbital bases set is not suitable for spreading waves. Moreover, they are not suitable for paralleliza- tion, since the calculation of matrix elements requires massive data transmission among processors. To overcome those problems, some numerical methods adopted real space representation [1–4]. In those meth- ods, a wavefunction is descritized by grid points in real space, and with them some dynamic electron phenomena were simulated successfully [6–8]. Among these real space methods, a method called Cayley’s form or Crank-Nicholson scheme is known to be especially useful for one-dimensional closed systems because this method conserves the norm of the wave- function exactly and the simulation is rather stable and accurate even in a long time slice. These characteris- tics are very attractive for simulations over a long time span. Unfortunately, this method is not suitable for two- or three-dimensional systems. This problem is fatal for physically meaningful systems. Though there are many other computational methods that can manage two- or three-dimensional systems, these methods also have dis- advantages. In the present work, we have overcome the problems associated with Cayley’s form and have formulated a new computational method which is more eﬃcient, more adaptable and more attractive than any other ordinary methods. In our method, all computations are performed in real space so there is no need of using Fourier transform. The time evolution operator in our method is exactly unitary by using Cayley’s form and Suzuki’s exponential productso that the norm of the wavefunction is conserved during the time evolution. Stability and accuracy are improved by Cayley’s form so we can use a longer time slice than those of the other methods. Cayley’s form is a kind of implicit methods, this is the key to the stability, but im- plicit methods are not suitable for periodic conditions and parallelization. We have avoided these problems by introducing an operator named adhesive operator. This adhesive operator is also useful for adaptive mesh reﬁne- ment technique. Our method inherits many advantages from many ordi- nary methods, and yet more improved in many aspects. With these advantages, this method will be useful for simulating large-scale and long-term quantum electron dynamics from ﬁrst principles. In section II, we formulate the new method step by step. In section III, we apply it to some simulations of electron dynamics and demonstrate its eﬃciency. In sec- tion IV, we draw some conclusions. II. FORMULATION In this section, we formulate the new method step by step from the simplest case to complicated cases. Throughout this paper, we use the atomic units ¯ h= 1,m= 1,e= 1. A. One-dimensional closed free system For the ﬁrst step, we consider a one-dimensional closed system where an electron moves freely but never leaks out of the system. The TD-Schr¨ odinger equation of this system is simply given as i∂ψ(x,t) ∂t=−∂2 x 2ψ(x,t). (1) 1The solution of Eq. (1) is analytically given by an ex- ponential operator as ψ(x,t+ ∆t) = exp/bracketleftBig i∆t∂2 x 2/bracketrightBig ψ(x,t), (2) where ∆tis a small time slice. By using Eq. (2) repeat- edly, the time evolution of the wavefunction is obtained. An approximation is utilized to make a concrete form of the exponential operator. We have to be careful not to destroy the unitarity of the time evolution operator, oth- erwise the wavefunction rapidly diverges. We adopted Cayley’s form because it is unconditionally stable and accurate enough. Cayley’s form is a fractional approxi- mation of the exponential operator given by exp/bracketleftBig i∆t∂2 x 2/bracketrightBig ≃1 + i∆t∂2 x/4 1−i∆t∂2x/4. (3) It is second-order accurate in time. By substituting Eq. (3) for Eq. (2) and moving the denominator onto the left-hand side, the following basic equation is obtained: /bracketleftBig 1−i∆t 2∂2 x 2/bracketrightBig ψ(x,t+ ∆t) =/bracketleftBig 1 + i∆t 2∂2 x 2/bracketrightBig ψ(x,t).(4) This is identical with the well-known Crank-Nicholson scheme. The wavefunction is descritized by grid points in real space as ψi(t) =ψ(xi,t) ;xi=i∆x, i = 0,· · ·,N−1 (5) where ∆xis the span of the grid points. We approximate the spatial diﬀerential operator by the ﬁnite diﬀerence method (FDM). Then Eq. (4) becomes a simultaneous linear equation for the vector quantity ψi(t+ ∆t). For example, in a system with six grid points, Eq. (4) is ap- proximated in the following way:  A−1 0 0 −1A−1 0 0−1A−1 0 0 −1A  ψ1(t+ ∆t) ψ2(t+ ∆t) ψ3(t+ ∆t) ψ4(t+ ∆t)  = B1 0 0 1B1 0 0 1B1 0 0 1B  ψ1(t) ψ2(t) ψ3(t) ψ4(t) (6) In the above, A≡ −4i∆x2 ∆t+ 2, B≡ −4i∆x2 ∆t−2 (7) andψ0andψ5are ﬁxed at zero due to the boundary condition. It is easy to solve this simultaneous linear equation be- cause the matrix appearing on the left-hand side is easilydecomposed into the LU form as  u−1 10 0 0 −1u−1 20 0 0−1u−1 30 0 0 −1u−1 4  1−u10 0 0 1 −u20 0 0 1 −u3 0 0 0 1  × ψ1(t+ ∆t) ψ2(t+ ∆t) ψ3(t+ ∆t) ψ4(t+ ∆t) = b1(t) b2(t) b3(t) b4(t) (8) Herebianduiare auxiliary vectors deﬁned as below bi(t)≡ψi−1(t) +Bψi(t) +ψi+1(t), (9) ui≡1/(A−ui−1), u 0≡0 (10) The auxiliary vector uiis determined in advance, and it is treated as a constant vector in Eq. (10). 26 Nﬂoat- ing operations are heeded to solve Eq. (10); here Nis the number of the grid points in the system, about twice that of the Euler method. Unlike the Euler method, it ex- actly conserves the norm because the matrices in Eq. (6) are unitary. Moreover, the expected energy is conserved because the time evolution operator commutes with the Hamiltonian in this case. B. Three-dimensional closed free system It is easy to extend this technique to a three- dimensional system. The formal solution of the TD- Schr¨ odinger equation in a three-dimensional system is given by an exponential of the sum of three second dif- ferential operators as ψ(r,t+ ∆t) = exp/bracketleftBig i∆t/parenleftBig∂2 x 2+∂2 y 2+∂2 z 2/parenrightBig/bracketrightBig ψ(r,t).(11) These diﬀerential operators in Eq. (11) are commutable among each other, so the exponential operator is exactly decomposed into a product of three exponential opera- tors: ψ(r,t+ ∆t) = exp/bracketleftBig i∆t∂2 x 2/bracketrightBig exp/bracketleftBig i∆t∂2 y 2/bracketrightBig ×exp/bracketleftBig i∆t∂2 z 2/bracketrightBig ψ(r,t).(12) Each exponential operator is approximated by Cayley’s form as ψ(r,t+ ∆t) =1 + i∆t∂2 x/4 1−i∆t∂2x/4·1 + i∆t∂2 y/4 1−i∆t∂2y/4· ×1 + i∆t∂2 z/4 1−i∆t∂2z/4ψ(r,t).(13) 78Nﬂoating operations are required to compute Eq. (13); where Nis the total number of grid points in the system. The norm and energy are conserved exactly. 2By the way, a conventional method, Peaceman- Rachfold method [1,8], utilizes similar approximation ap- pearing on Eq. (13), which is a kind of the alternating direction implicit method (ADI method). However, by using exponential product, we have found that there is no need of ADI. This fact makes the programming code simpler and it runs faster. C. Static potential Next we consider a system subjected to a static exter- nal scalar ﬁeld V(r). The TD-Schr¨ odinger equation and its formal solution in this system are as follows: i∂ψ(r,t) ∂t=/bracketleftBig −△ 2+V(r)/bracketrightBig ψ(r,t). (14) ψ(r,t+ ∆t) = exp/bracketleftBig i∆t△ 2−i∆tV(r)/bracketrightBig ψ(r,t).(15) To cooperate with the potential in the framework of the formula described in the previous subsections, we have to separate the potential operator from the kinetic operator using Suzuki’s exponential product theory [9,10] as ψ(r,t+ ∆t) = exp/bracketleftBig −i∆t 2V/bracketrightBig exp/bracketleftBig i∆t△ 2/bracketrightBig ×exp/bracketleftBig −i∆t 2V/bracketrightBig ψ(r,t).(16) This decomposition is correct up to the second-order of ∆t. The exponential of the potential is computed by just changing the phase of the wavefunction at each grid point. The exponential of the Laplacian is computed in the way described in the previous subsections. Each op- erator is exactly unitary, so the norm is conserved exactly. But due to the separation of the incommutable operators, the energy is not conserved exactly. Yet it oscillates near around its initial values and it never drifts monotonously. This algorithm is quite suitable for vector-type super- computers because all operations are independent by grid points, by rows, or by columns. The outline of this pro- cedure for a two-dimensional system is schematically de- scribed by Fig. 1. V V y x K K FIG. 1. The procedure for a two-dimensional closed sys- tem with a static potential. Here Vshows the operation of the exponential of the potential, which changes the phase of the wavefunction at each grid point. KxandKyshow the operation of Cayley’s form along the x-axis and the y-axis re - spectively. They are computed independently by grid points , by rows, or by columns.The decomposition (16) is a second-order one. Higher- order decompositions are derived using Suzuki’s fractal decomposition [9–11,13]. For instance, a fourth-order fractal decomposition S4(∆t) is given by S4(∆t) =S2(s∆t)S2(s∆t)S2((1−4s)∆t) ×S2(s∆t)S2(s∆t) (17) where S2(∆t)≡exp/bracketleftBig −i∆t 2V/bracketrightBig exp/bracketleftBig i∆t△ 2/bracketrightBig exp/bracketleftBig −i∆t 2V/bracketrightBig s≡1/(4−3√ 4). (18) D. Dynamic potential To discuss high-speed electron dynamics caused by a time-dependent external ﬁeld V(r,t), we should take ac- count of the evolution of the potential itself in the TD- Schr¨ odinger equation given as i∂ψ(r,t) ∂t=H(t)ψ(r,t) ;H(t) =−△ 2+V(r,t).(19) The analytic solution of Eq. (19) is given by a Dyson’s time ordering operator Pas ψ(r,t+ ∆t) =Pexp/bracketleftBigg i/integraldisplayt+∆t tdt′/braceleftBig△ 2−V(r,t′)/bracerightBig/bracketrightBigg ψ(r,t). (20) The theory of the decomposition of an exponential with time ordering was derived by Suzuki [12]. The result is rather simple. For instance, the second-order decompo- sition is simply given by ψ(r,t+ ∆t)≃exp/bracketleftBig −i∆t 2V(r,t+∆t 2)/bracketrightBig exp/bracketleftBig i∆t△ 2/bracketrightBig ×exp/bracketleftBig −i∆t 2V(r,t+∆t 2)/bracketrightBig ψ(r,t) (21) and the fourth-order fractal decomposition is given by ψ(r,t+ ∆t) =S2(s∆t;t+ (1−s)∆t) ×S2(s∆t;t+ (1−2s)∆t) ×S2((1−4s)∆t;t+ 2s∆t) ×S2(s∆t;t+s∆t) ×S2(s∆t;t)ψ(r,t), (22) S2(∆t;t)≡exp/bracketleftBig −i∆t 2V(r,t+∆t 2)/bracketrightBig exp/bracketleftBig i∆t△ 2/bracketrightBig ×exp/bracketleftBig −i∆t 2V(r,t+∆t 2)/bracketrightBig .(23) These operators are also unitary. These procedures are quite similar to those of the static potential except that we take the dynamic potential at the speciﬁed time. 3E. Periodic system In a crystal or periodic system, the wavefunctions must obey a periodic condition: ψ(r+R,t) =ψ(r,t)exp [iφ], φ ≡k·R,(24) where kis the Bloch wave number and Ris the unit vector of the lattice. The matrix form equation corre- sponding to Eq. (6) in this system takes the following form:  A−1 0e+iφ −1A−1 0 0−1A−1 e−iφ0−1A  ψ1(t+ ∆t) ψ2(t+ ∆t) ψ3(t+ ∆t) ψ4(t+ ∆t)  = B1 0e−iφ 1B1 0 0 1B1 e+iφ0 1B  ψ1(t) ψ2(t) ψ3(t) ψ4(t) (25) These matrices have extra elements, so the equation can no longer be solve eﬃciently. We propose a trick to avoid this problem. We repre- sent the second spatial diﬀerential operator ∂2 xas a sum of two operators: ∂2 x=∂2 xtd+∂2 xad. (26) Multiplying by ∆ x2, the above representation reads in the matrix form:  −2 1 0 e−iφ 1−2 1 0 0 1 −2 1 e+iφ0 1 −2  = −1 1 0 0 1−2 1 0 0 1 −2 1 0 0 1 −1 + −1 0 0e−iφ 0 0 0 0 0 0 0 0 e+iφ0 0 −1 .(27) The ﬁrst matrix on the right-hand side, which corre- sponds to∂2 xtd, is tri-diagonal, and the second one, which corresponds to ∂2 xad, is its remainder, and it has a quite simple form. The exponential of the second diﬀerential operator is decomposed by these terms: exp/bracketleftBigi∆t 2∂2 x/bracketrightBig = exp/bracketleftBigi∆t 4∂2 xad/bracketrightBig exp/bracketleftBigi∆t 2∂2 xtd/bracketrightBig exp/bracketleftBigi∆t 4∂2 xad/bracketrightBig . (28) The exponential of ∂2 xadis exactly calculated by the fol- lowing formula: exp/bracketleftBig iC/parenleftBig−1e−iφ e+iφ−1/parenrightBig/bracketrightBig =I+1−e−2iC 2/parenleftBig−1e−iφ e+iφ−1/parenrightBig . (29)This operation is exactly unitary and easy to compute. The exponential of ∂2 xtdis computed in the ordinary way. Thus the norm is conserved. We named ∂2 adan “adhesive operator” because this operator plays the role of an adhesion to connect both edges of the system. The outline of the procedure for a two-dimensional periodic system is schematically described by Fig. 2. y VV xK Y-adhesive Y-adhesiveX-adhesive X-adhesive K FIG. 2. The procedure for a two-dimensional periodic sys- tem. Here KxandKyshow the operations of Cayley’s form, and they operate as if this system is not periodic. X-adhesiv e and Y-adhesive mean the operations of the exponential of the adhesive operators along the x-axis and the y-axis, respec- tively. The operation of the adhesive operator needs only th e values at the edges of the system. F. Parallelization The adhesive operator plays another important role. It makes Cayley’s form suitable for parallelization. We use the adhesive operator to represent the second ﬁnite diﬀerence matrix in the following way:  −2 1 0 0 1−2 1 0 0 1 −2 1 0 0 1 −2  = −2 1 0 0 1−1 0 0 0 0 −1 1 0 0 1 −2 + 0 0 0 0 0−1 1 0 0 1 −1 0 0 0 0 0 .(30) The interior of the ﬁrst matrix on the right-hand side is separated into two blocks, which means this system is separated into two physically independent areas. The second matrix, which is the adhesive operator, connects the two areas. A large system is separated into many small areas, and each area is managed by a single pro- cessor. Since the exponential of a block diagonal ma- trix is also a block diagonal matrix, each block is com- puted by a single processor independently. Data trans- mission is needed only to compute the adhesive oper- ator. The amount of data transmission is quite small, nearly negligible. The outline of the procedure for a two- dimensional closed system on two processors is schemat- ically described by Fig. 3. 4x V y V AdhesiveK Adhesive K FIG. 3. The procedure for a two-dimensional closed sys- tem on two processors. Adhesive shows the operation of the exponential of the adhesive operator for parallel computin g. The operation of the adhesive operator needs only the values at the edges of the areas, so the data transmission between the processors is quite small. G. Adaptive mesh reﬁnement It is necessary for real space computation to be equipped with an adaptive mesh reﬁnement to reduce the computational cost or to improve the accuracy in some important regions. We improved the adhesive operator to manage a connection of between two regions whose mesh sizes are diﬀerent, as illustrated in Fig. 4. xx∆x ∆x ∆2x∆ 1 23 45 62 FIG. 4. An example of adaptive mesh reﬁnement. The element in the left area is twice as large as that in the right area. The adhesive operator connects these areas. The second diﬀerential operator ∂2 xshould be Hermite, but in this case the condition required for the matrix rep- resentation ( ∂2 x)ijis given by (∂2 x)ij∆x2 i= (∂2 x)ji∆x2 j; for all i,j. (31) Considering this condition, an approximation of the second diﬀerential operator is given as ∂2 x=1 ∆x2-1/2 1/4 1 1/4 -1/2 1/8 1/8 2 1/2 -3/2 1 3 1/2 -3/2 1 4 1 -2 5 1 -2 6(32) The indices attached to this matrix indicate the corre- sponding mesh indices described in Fig. 4. This matrix is also divided into a block-diagonal one and an adhesive operator as ∂2 xbd=1 ∆x2-1/2 1/4 1 1/4 -1/4 2 -1 1 3 -1 1 4 1 -2 5 1 -2 6(33)∂2 xad=1 ∆x21 -1/4 1/8 1/8 2 1/2 -1/2 3 1/2 -1/2 4 5 6(34) The exponential of the adhesive operator is calculated using the following formula: exp i∆t 4∆x2 −1/4 1/8 1/8 1/2−1/2 0 1/2 0 −1/2  = I+ 2c1−c1 −c1 −4c12c1+c22c1−c2 −4c12c1−c22c1+c2 ,(35) where c1≡1 6exp/bracketleftBig −3i√ 2∆t 8∆x2/bracketrightBig −1 6, (36) c2≡1 6exp/bracketleftBig −2i√ 2∆t 8∆x2/bracketrightBig −1 6. (37) In this way, it is found that the adhesive operator is important to simulate a larger or a more complicated system by the present method. III. APPLICATION In this section, we show some applications of our nu- merical method. Though these applications treat simple physical systems, they are suﬃcient for verifying the re- liability and eﬃciency of the method. Throughout this section, we use the atomic units (a.u.). A. Comparison with conventional methods As far as we know, the conventional methods of solv- ing the TD-Schr¨ odinger equation are classiﬁed into three categories: 1) the multistep method [3], 2) the method developed by De Raedt [2] and 3) the method equipped with Cayley’s form [5]. In this section, we make brief comparisons between Cayley’s form and other conventional methods by sim- ply simulating a Gaussian wave packet moving in a one- dimensional free system as illustrated in Fig. 5. x2W xopo 5FIG. 5. The model system for comparison with con- ventional methods. 256 computational grid points are allo- cated in the physical length 8 .0a.u. A Gaussian wave packet is placed in the system, whose initial average location xoand momentum poare set at xo= 2.0a.u. and po= 12.0a.u., re- spectively. The TD-Schr¨ odinger equation of this system is simply given by i∂ψ(x,t) ∂t=−∂2 x 2ψ(x,t). (38) The wavefunction at the initial state is set as a Gaus- sian: ψ(x,t= 0) =1 4√ 2πW2exp/bracketleftBig −\|x−xo\|2 4W2+ ipox/bracketrightBig ,(39) whereW= 0.25a.u., xo= 2.0a.u., po= 12.0a.u. The evolution of this Gaussian is analytically derived as ψ(x,t) =1 4/radicalbig 2πW2+ (π/2)(t/W)2 ×exp/bracketleftBig −(x−xo−pot)2 4W2+ (t/W)2+ ipox/bracketrightBig .(40) Therefore, the average location of the Gaussian /angbracketleftx(t)/angbracketrightis derived as if it is a classical particle: /angbracketleftx(t)/angbracketright=/angbracketleftx(t= 0)/angbracketright+pot . (41) This characteristic is useful to check the accuracy of the simulation. We use the second-order version of the multistep method and the De Raedt’s method in order to compare with Cayley’s form since Cayley’s form is second-order accurate in space and time. The second-order multistep method we used in this system is given by ψ(t+ ∆t) =ψ(t−∆t) + i2∆t∂2 x 2ψ(t),(42) where∂2 xis approximated by a ﬁnite diﬀerence matrix as ∂2 x≃1 ∆x2 −2 1 0 0 0 0 1−2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 . (43) Extra memories are needed for the wavefunction at the previous time step ψ(t−∆t). Though the time evolution of this method is not unitary, the norm of the wavefunc- tion is conserved with good accuracy on the condition that ∆t/∆x2≤0.5. This method needs only 10 Nﬂoat- ing operations per time step, which is the fastest method in conditionally stable methods.Meanwhile, the second-order De Raedt’s method is given by ψ(t+ ∆t) = exp/bracketleftBigi∆t 2∂2 xa 2/bracketrightBig exp/bracketleftBig i∆t∂2 xb 2/bracketrightBig exp/bracketleftBigi∆t 2∂2 xa 2/bracketrightBig ψ(t) (44) where∂2 xaand∂2 xbare the parts of the second diﬀeren- tial operator and are approximated by ﬁnite diﬀerence matrices as below: ∂2 xa≃1 ∆x2 −1 1 0 0 0 0 1−1 0 0 0 0 0 0 −1 1 0 0 0 0 1 −1 0 0 0 0 0 0 −1 1 0 0 0 0 1 −1 ,(45) ∂2 xb≃1 ∆x2 −1 0 0 0 0 0 0−1 1 0 0 0 0 1 −1 0 0 0 0 0 0 −1 1 0 0 0 0 1 −1 0 0 0 0 0 0 −1 .(46) The exponentials of those matrices are exactly calcu- lated using the following formula: exp/bracketleftBig iC/parenleftbigg −1 1 1−1/parenrightbigg/bracketrightBig =I+1−e−2iC 2/parenleftbigg −1 1 1−1/parenrightbigg .(47) The time evolution of this method is exactly unitary, and the norm is exactly conserved unconditionally. However, it seems that the accuracy tends to break down on the condition that ∆ t/∆x2>1.0. This method needs 18 N ﬂoating operations per time step, which is the fastest method in unconditionally norm-conserving methods. Cayley’s form with the ﬁnite diﬀerence method is given by ψ(t+ ∆t) =1 + i∆t/4∂2 x 1−i∆t/4∂2xψ(t), (48) where the spatial diﬀerential operator is approximated by the ordinary way in Eq. (43). The time evolution of this method is exactly unitary, and the norm is exactly conserved unconditionally. More- over, this method maintains good accuracy even under the condition that ∆ t/∆x2>1.0. This method needs 26Nﬂoating operations per time step, which is the fastest method in unconditionally stable methods. We have simulated the motion of the Gaussian by those methods. First we show a comparison of Cayley’s form with the conventional methods in the framework of the FDM. Figure 6 shows the time evolution of the error in the energy, which is evaluated by the ﬁnite diﬀerence method as described below ǫ(t) =E(t)−E(t= 0) (49) E(t) =−1 2∆xReN−1/summationdisplay i=0ψ∗ i(t)/parenleftbig ψi−1(t)−2ψi(t) +ψi+1(t)/parenrightbig . (50) 6The initial energy is evaluated as 73 .03a.u., though it is theoretically expected to be 74a.u. The ratio ∆ t/∆x2is set at 0.5 to meet the stable condition required for the multistep method. The energies violently oscillate in the results of the multistep method and De Raedt’s method, as a result of the fact that these time evolution operators do not commute with the Hamiltonian. These energies seem to converge after the wave packet is delocalized in a uniform way over the system. Meanwhile, the energy is conserved exactly in the result of Cayley’s form because Cayley’s form commutes with the spatial second diﬀerential oper- ator which is the Hamiltonian itself in this system. Figure 7 shows the relation of the time slice ∆ tto the error in the average momentum of the Gaussian, which is evaluated by the ﬁnite diﬀerence method as described below: ǫ(∆t/∆x2) =/angbracketleftx(t=T)/angbracketright − /angbracketleftx(t= 0)/angbracketright T− /angbracketleftp(t= 0)/angbracketright, (51) /angbracketleftx(t)/angbracketright= ∆xN−1/summationdisplay i=0xi\|ψi(t)\|2(52) /angbracketleftp(t)/angbracketright=1 2ImN−1/summationdisplay i=0ψi(t)∗/parenleftbig ψi+1(t)−ψi−1(t)/parenrightbig ,(53) whereTis a time span set at 0 .4a.u. The initial momen- tum/angbracketleftp(t= 0)/angbracketrightis calculated as 11 .7a.u., which is diﬀerent from the theoretical value po= 12.0a.u. due to the ﬁnite diﬀerence method. -0.0500.050.10.150.20.25 00.02 0.04 0.06 0.08 0.1Error of the energy [a.u.] Time [a.u.]Multistep De Raedt Cayley FIG. 6. Time variances in the energies computed by the three methods. The time slice is set at ∆ t= 1/2048a.u. and the spatial slice is set at ∆ x= 1/32a.u. so that the ratio ∆t/∆x2is equal to 0 .5. The energies violently oscillates in the result of the multistep method and De Raedt’s method. Meanwhile, the energy is conserved exactly in the result of Cayley’s form.-0.100.10.20.30.40.50.60.70.80.9 0.0625 0.125 0.25 0.5 1 2Error of the mean velocity [a.u.] ∆ ∆t / x2Multistep deRaedt Cayley FIG. 7. Errors in the average momentum computed by the three methods in several time slices. The multistep method cannot be performed when ∆ t/∆x2>0.5. The error of De Raedt’s method is too large when ∆ t/∆x2>1. The error of Cayley’s form is rather small. The spatial slice is s et at ∆x= 1/32a.u. In the multistep method, the computation cannot be performed due to a ﬂoating exception, if the ratio ∆t/∆x2exceeds 0.5. In De Raedt’s method, the error be- comes too large to plot in this graph if the ratio ∆ t/∆x2 exceeds 1.0. Meanwhile, in Cayley’s form, the error is not so large even if the ratio ∆ t/∆x2exceeds 1.0. In this way, Cayley’s form is found rather stable. Therefore, we can use a longer time slice than those of the other methods. And this Cayley’s form becomes suitable for three-dimensional systems, potentials, periodic cond i- tions, adaptive mesh reﬁnement, and parallelizations by our improvements in this paper. B. Test of the adhesive operator To verify the reliability and eﬃciency of the adhesive operator for periodic condition and parallelization, we have simulated the motion of a Gaussian wave packet in a two-dimensional free system. As illustrated in Fig. 8, this system has periodic conditions along both the x-axis and the y-axis, and it is divided into nine areas, each of them is managed by a single processing element; the ad- hesive operator connects them. The initial wavefunction is set as a Gaussian given as ψ(r,t= 0) =1√ 2πW2exp/bracketleftBig −\|r−ro\|2 4W2+ ipo·r/bracketrightBig ,(54) where rois set as the center of this system and po= (1a.u.,1a.u.),W= 1a.u. The energy of this Gaussian is theoretically derived as 1 .0625a.u. 7PE6 PE9 PE2PE1 PE3 PE3 PE1PE5 PE4 PE7PE1 PE4PE7 PE9PE9 PE2PE8 PE6PE7 PE1PE9 PE3 PE3 PE7 PE8 FIG. 8. The model system for the test of the adhesive operator for periodic conditions and parallelization. Thi s sys- tem is periodically connected and is divided into nine areas . Each area is managed by a single processing element. 32 ×32 computational grid points are allocated in each area whose physical size is set at 8 .0a.u.×8.0a.u. The time slice is set at ∆t= 1/16a.u. Figure 9 shows snapshots of the time evolution of the Gaussian, which is observed to go through these areas smoothly. Figure 10 shows the evolution of the energy, which is observed to oscillate around its initial value. FIG. 9. Evolution of the density. The Gaussian is ob- served to go through these areas smoothly. 1.05301.05351.05401.05451.05501.05551.0560 051015202530Energy [a.u.] Time [a.u.]FIG. 10. Time variance in the energy. The initial energy is theoretically derived as 1 .0625a.u., but it is evaluated as 1.0553a.u. by the FDM. The energy oscillates near its initial value but never drifts monotonously. Second, we allocate 64 ×64 grid points only in the cen- tral area as illustrated in Fig. 11. We utilize the adhesive operator for the adaptive mesh reﬁnement. Figure 12 shows the snapshots, with the Gaussian going through these areas smoothly. Figure 13 shows the evolution of the energy, which is observed to oscillate near its initial value. In this way, the reliability of the adhesive operator is proved. PE6 PE9 PE2PE1 PE3 PE3 PE1PE5 PE4 PE7PE1 PE4PE7 PE9PE9 PE2PE8 PE6PE7 PE1PE9 PE3 PE3 PE7 PE8 FIG. 11. The model system for the test of the adhesive op- erator for the adaptive mesh reﬁnement. This system is also periodically connected and is divided into nine areas. Each area is managed by a single processing element. The size of each area is set at 8 .0a.u.×8.0a.u. 32×32 computational grid points are allocated in each areas except the central ar ea. The central area has 64 ×64 computational grid points, which makes it twice as ﬁne as those of the other areas. The time slice is set at ∆ t= 1/16a.u. FIG. 12. Evolution of the density. The Gaussian is ob- served to go through these areas smoothly. 81.0541.0551.0561.0571.0581.0591.0601.0611.0621.063 051015202530Energy [a.u.] Time [a.u.] FIG. 13. Time variance in the energy. The initial energy is theoretically derived as 1 .0625a.u., but it is evaluated as 1.0591a.u. by the FDM. The energy oscillates near its initial value but it never drifts monotonously. C. Excitation of a hydrogen As the last application of the present method, we demonstrate its validity and eﬃciency in describing the process of photon-induced electron excitation in a hy- drogen atom in a strong laser ﬁeld. The laser is treated as a classically oscillating electric force polarized in th e z-direction: Ez=Eosinωt. (55) The spatial variation of the electric ﬁeld of the light is neglected, because the electron system is much smaller than the order of the wave length. Then the interaction term of the Hamiltonian is approximated as Hint=−eEzz . (56) In other words, we only take into account the electro- dipole interaction of the electron with the light, and ne- glect the electro-quadrapole, the magnetic-dipole, and other higher interactions. The amplitude Eois set at 1/64a.u. = 0.80V/˚A, which is as strong as a usual pulse laser. The angular frequency ωis set at 0.3125a.u. = 8 .5eV, less than the transition en- ergy between 1S and 2P. Ordinarily, such low energetic electric force has no eﬀect on the electronic excitation. But with such a strong amplitude, various nonlinear op- tical eﬀects are caused by the electron dynamics. We allocate 1283grid points in a 323a.u.3cubic closed system. The hydrogen nucleus is located at the center of the system, and the nucleus potential is constructed by solving the Poisson equation in the discretized space to avoid the singularity of the nucleus potential. The 1S- orbital is assumed as the initial state of the wavefunction. Then we turn on the electric ﬁeld and start the simula- tion. The time slice is set at 0 .0785a.u. = 2 .0×10−3fs so as to follow the rapid variation of the wavefunctionand the electric force. We follow the evolution for 32k iteration. Figure 14 shows the time variance in the polarization of the electron. The oscillation of the polarization generate s another electric ﬁeld, which corresponds to a non-linearly scattered light from the atom. By Fourier-transforming the polarization along the time axis, we obtained the spectrum of the scattered light shown in Fig. 15. -0.2-0.100.10.2 0 5 10 15 20 25 30Polarization [angstrom] Time [fs] FIG. 14. Time variance in the polarization of the electron. 0246810 0 5 10 15 20 25 30Intensity [arbitrary units] Photon energy [eV] FIG. 15. Spectrum of the scattered light generated by the oscillation of the electron. Several sharp peaks are found, which are interpreted as follows: The peak at 8 .5eV comes from Rayleigh scatter- ing, whose frequency is identical with the injected light: ω. The peak at 10 .2eV comes from Lyman αemission, which is generated by the electron transition from the 2P- orbital to the 1S-orbital: ωLα. On the other hand, the peak at 12.1eV comes from Lyman βemission, which is generated by the electron transition from the 3P-orbital to the 1S-orbital: ωLβ. The peak at 6 .8eV comes from hy- per Raman scattering, whose frequency is identical with 2ω−ωLα. Moreover the peak at 25 .5eV comes from the third harmonic generation, whose frequency is identical with 3ω. 9The simulation is also performed for a diﬀerent laser frequency; the injecting photon energy ωis set at 10.2eV, which is the same as the transition energy between 1S and 2P. In this case the electron starting from a 1S orbital is expected to excite to a 2Pz orbital. Figure 16 shows the snapshots of the density during the simulation time span. FIG. 16. Evolution of the density of the electron in the hy- drogen atom. The density starting from a 1S orbital oscillat es with time and becomes a 2Pz orbital. Figure 17 and Fig. 18 show the polarization and the spectrum, respectively. Three peaks are found, at 9 .9eV, 10.2eV, and 10 .5eV. These peaks are derived from the theory of the Dressed atom or the AC stark eﬀect as below: ω−eEo/angbracketleft2Pz\|z\|1S/angbracketright, ω, ω +eEo/angbracketleft2Pz\|z\|1S/angbracketright.(57) -0.4-0.3-0.2-0.100.10.20.30.4 0 5 10 15 20 25 30Polarization [angstrom] Time [fs] FIG. 17. Time variance in the polarization of the electron. 05101520 0 5 10 15 20 25 30Intensity [arbitrary units] Photon energy [eV] FIG. 18. Spectrum of the scattered light generated by the oscillation of the electron.One could obtain such behavior analytically by using perturbation theory; however, with the present method, we could directly calculate them without perturbation theory and without information on the excited states of the system. IV. CONCLUSION We have formulated a new method for solving the time-dependent Schr¨ odinger equation numerically in real space. We have found that by using Cayley’s form and Suzuki’s fractal decomposition, the simulation can be fast, stable, accurate, and suitable for vector-type su- percomputers. We have proposed the adhesive operator to make Cayley’s form suitable for periodic systems and parallelization and adaptive mesh reﬁnement. These techniques will also be useful for the time- dependent Kohn Sham equation, which is our future work. V. ACKNOWLEDGMENTS We are indebted to Takahiro Kuga for his suggestions concerning non-linear optics. Calculations were done us- ing the SR8000 supercomputer system at the Computer Centre, University of Tokyo. [1] R. Varga, Matrix Iterative Analysis (Prentice-Hall, En- glewood Cliﬀs, NJ, 1962), p.273. [2] H. De Raedt and K. Michielsen, Computers in Physics, 8, 600 (1994). [3] T. Iitaka, Phys. Rev. E 49, 4684 (1994). [4] H. Natori and T Munehisa, J. Phys. Soc. Japan 66, 351 (1997). [5]Numerical Recipes in C , chapter 19, section 2, W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flan- nery, (Cambridge University Press, 1996). [6] H. De Raedt and K. Michielsen, Phys. Rev. B. 50, 631 (1994) [7] T. Iitaka, S. Nomura, H. Hirayama, X. Zhao, Y. Aoyagi and T. Sugano, Phys. Rev. E 56, 1222 (1997). [8] H. Kono, A. Kita, Y. Ohtsuki and Y. Fujimura, J. Com- put. Phys. (USA), 130, 148 (1997). [9] M. Suzuki, Phys. Lett. A 146, 319 (1990). [10] M. Suzuki, J. Math. Phys. 32, 400 (1991). [11] K. Umeno and M. Suzuki, Phys. Lett. A 181, 387 (1993). [12] M. Suzuki, Proc. Japan Acad. 69Ser. B, 161 (1993). [13] M. Suzuki and K. Umeno, Vol. 76 of Springer Proceedings in Physics, (Computer Simulation Studies in Condensed- Matter Physics VI, editied by D. P. Landau, K. K. Mon, H. B. Sch¨ uttler, Springer, Berlin, 1993), p. 74. [14] M. Suzuki, Phys. Lett. A 201, 425 (1995). 10
arXiv:physics/0011069v1 [physics.comp-ph] 29 Nov 2000Finite element approach for simulating quantum electron dy namics in a magnetic ﬁeld Naoki Watanabe, Masaru Tsukada Department of Physics,Graduate School of Science, Univers ity of Tokyo 7-3-1 Hongo, 113-0033 Bunkyo-ku, Tokyo, Japan (Published from Journal of Physical Society of Japan, 69, No.9, 2962, (2000).) A fast and stable numerical method is formulated to compute t he time evolution of a wave function in a magnetic ﬁeld by solving the time-dependent Schr¨ oding er equation. This computational method is based on the ﬁnite element method in real space to improved accuracy without any increase of computational cost. This method is also based on Suzuki’s ex ponential product theory to aﬀord an eﬃcient way to manage the TD-Schr¨ odinger equation with a ve ctor potential. Applying this method to some simple electron dynamics, we have conﬁrmed its eﬃcie ncy and accuracy. 02.70.-c,03.67.Lx,73.23,42.65.-k I. INTRODUCTION Conventionally, wave functions have been represented as a linear combination of plane waves or atomic orbitals in the calculations of the electronic states or their time evolution. However, these representations entail high computational cost to calculate the matrix elements for these bases. The plane wave bases set is not suitable for localized orbitals, and the atomic orbital bases set is not suitable for spreading waves. To overcome those problems, some numerical methods adopted real-space representation to solve the time de- pendent Schr¨ odinger equation [1–4]. In those methods, a wavefunction is descritized by grid points in real space and the spatial diﬀerential operator is approximated by the ﬁnite diﬀerence method (FDM). With those meth- ods, some dynamic electron phenomena were simulated successfully [7–9]. In the previous work [11], we have formulated a new computational method for the TD-Schr¨ odinger equation by using some computational techniques such as, the FDM, Suzuki’s exponential product theory [12–17], Cay- ley’s form [6] and Adhesive operator. This method af- forded high-stability and low computational cost. In the ﬁeld of engineering, for example, numerical anal- ysis of ﬂuid dynamics or of strength of macroscopic con- structions, the ﬁnite element method (FEM) has been widely and traditionally used for approximating the ap- propriate partial diﬀerential equations. Recently, the FEM has been found useful for the time-independent Schr¨ odinger equation of electrons in solid or liquid mate- rials [10]. In this paper, we have utilized the FEM for solving the TD-Schr¨ odinger equation as an extension of the pre- vious work [11]. By using Cayley’s form and the FEM, this method aﬀords high-accuracy without any increase of computational cost. Moreover, we have formulated a new eﬃcient method which manages the time evolution of a wave function in a vector potential or in a magnetic ﬁeld. These techniques are especially useful for simu- lating dynamics of electrons in a variety of meso-scopic systems.II. FORMULATION In this section, we formulate a new method derived by the FEM and a new scheme to manage a vector poten- tial eﬃciently. Throughout this paper, we often use the atomic unit ¯ h= 1,m= 1,e= 1. A. FEM for the TD-Schr¨ odinger equation First, we utilize the FEM for the time evolution of a wave function in a one-dimensional closed system de- scribed by the following TD-Schr¨ odinger equation: i¯h∂ψ(x,t) ∂t=−¯h2 2m∂2 ∂x2ψ(x,t). (1) The FEM starts by smoothing the wavefunction around a grid point. We smoothed ψ(x) around a grid pointxiby eq. (2), as illustrated in Fig. 1: ψ(x,t) =ψi(t) +ψ′ i(t)(x−xi) +1 2ψ′′ i(t)(x−xi)2,(2) where ψi(t)≡ψ(xi,t), ψ′ i(t)≡ψi+1(t)−ψi−1(t) 2∆x, ψ′′ i(t)≡ψi+1(t)−2ψi(t) +ψi−1(t) ∆x2.(3) iφi φ φi+1i-1 xi+1 xi-1 x FIG. 1. The FEM starts by smoothing the wavefunction around a grid point. The wavefunction is supplemented by a quadratic equation. 1By substituting eq. (3) for eq. (2), ψ(x,t) is expressed as ψ(x,t) =uai(x)ψi−1(t) +uoi(x)ψi(t) +ubi(x)ψi+1(t), (4) whereuai(x), uoi(x) andubi(x) are the base functions deﬁned below: uai(x) =(x−xi)2 2∆x2−(x−xi) 2∆x, uoi(x) = 1 −(x−xi)2 ∆x2, ubi(x) =(x−xi)2 2∆x2+(x−xi) 2∆x.(5) Substituting eq. (4) for eq. (1) and multiplying both side of the equation by the base function uoi(x) and in- tegrating by xin the range [ xi−1, xi+1] as i¯h/integraldisplayxi+1 xi−1dxuoi(x)/bracketleftBig uoi(x)˙ψi(t) +uai(x)˙ψi+1(t) +ubi(x)˙ψi−1(t)/bracketrightBig =−¯h2 2m/integraldisplayxi+1 xi−1dxuoi(x)/bracketleftBig ∂2 xuoi(x)ψi(t) +∂2 xuai(x)ψi+1(t) +∂2 xubi(x)ψi−1(t)/bracketrightBig ,(6) the following formula is obtained after some algebra: i¯h1 10/bracketleftbig˙ψi−1(t) + 8˙ψi(t) +˙ψi+1(t)/bracketrightbig =−¯h2 2m∆x2/bracketleftbig ψi−1(t)−2ψi(t) +ψi+1(t)/bracketrightbig .(7) To simplify the expression, it is useful to deﬁne a vector and two matrices as below: ψ(t)≡(ψ0(t),... ,ψ N−1(t))T, (8) S≡1 10 8 1 0 0 1......0 0......1 0 0 1 8 , D≡ −2 1 0 0 1......0 0......1 0 0 1 −2 .(9) Clearly SandDsatisfy the following equation: S=I+1 10D. (10)Using these notations, eq. (7) is expressed simply as i¯hS∂ψ(t) ∂t=−¯h2 2m∆x2Dψ(t). (11) Equation (11) is the ﬁnite element equation for this case. It has been thought that the existence of the matrix Sis troublesome since the inverse of this matrix is re- quired to obtain the time derivative of the wave function, namely, ∂ψ(t) ∂t=i¯h 2m∆x2S−1Dψ(t). (12) However, we have found that this diﬀerential equation is easily solved by using an approximation called Cayley’s form. The formal solution of eq. (12) is given by, ψ(t+ ∆t) = exp/bracketleftBigi¯h 2m∆t ∆x2S−1D/bracketrightBig ψ(t).(13) The exponential operator is approximated by Cayley’s form: ψ(t+ ∆t)≃I+i¯h 4m∆t ∆x2S−1D I−i¯h 4m∆t ∆x2S−1Dψ(t). (14) Multiplying both the numerator and the denominator of the righthand side by the matrix Sand using the relation (6), the required formula is obtained: ψ(t+ ∆t) =I+i¯h 4meﬀ∆t ∆x2D I−i¯h 4m∗ eﬀ∆t ∆x2Dψ(t), (15) wheremeﬀis an “eﬀective mass” of an electron deﬁned as ¯h meﬀ≡¯h m−i2 5∆x2 ∆t. (16) In this way, the solution of the partial diﬀerential equa- tion, eq. (1) is computed by eq. (15) with the concept of the FEM. It is quite a remarkable result that formula eq. (15) is almost the same as the formula derived by the FDM [11]. In this time evolution, the norm of the wave function is exactly conserved since the time evolution op- erator appearing in eq. (1) is strictly unitary. Moreover, accuracy is dramatically improved without any increase in the computational cost, as demonstrated in the next section. It is easy to extend this idea for two-dimensional systems, since the time evolution operator in a two- dimensional system is decomposed into a product of the time evolution operators in one-dimensional systems [11]. The approximated solution utilizing the FEM is given by ψ(r,t+ ∆t) =I+i¯h 4meﬀ∆t ∆x2Dx I−i¯h 4m∗ eﬀ∆t ∆x2Dx·I+i¯h 4meﬀ∆t ∆y2Dy I−i¯h 4m∗ eﬀ∆t ∆y2Dyψ(r,t), (17) 2whereDxandDyare the ﬁnite diﬀerence matrices along thexandyaxes respectively, and their appearances are the same as Ddeﬁned in eq. (9). B. Evolution in a magnetic ﬁeld Though there are many interesting phenomena in a magnetic ﬁeld, there has been no eﬃcient methods that numerically manage the dynamics in a magnetic ﬁeld as far as we know. We have improved our method to aﬀord an eﬃcient way to solve the TD-Schr¨ odinger equation with a vector potential given as below i¯h∂ψ(r,t) ∂t=−¯h2 2m/parenleftBig ∇ −ie ¯hA/parenrightBig2 ψ(r,t).(18) In this subsection, we present the method for only the case of a two-dimensional system lying on the xyplane subjected to a uniform external magnetic ﬁeld along the z axis. We do not mention the case of a non-uniform mag- netic ﬁeld speciﬁcally, but the extension of the method is straightforward. We adopt the following vector potential Afor this magnetic ﬁeld: A= (−By,0,0)T. (19) The TD-Schr¨ odinger equation of this system is given by i¯h∂ψ(r,t) ∂t=/bracketleftBig −¯h2 2m/parenleftBig ∂x−ie ¯hBy/parenrightBig2 −¯h2 2m∂2 y/bracketrightBig ψ(r,t). (20) The strict, analytical solution is also given by an expo- nential operator: ψ(r,t+ ∆t) = exp/bracketleftBigi¯h 2m∆t/parenleftBig ∂x−ie ¯hBy/parenrightBig2 +i¯h 2m∆t∂2 y/bracketrightBig ψ(r,t). (21) Note the following identity: exp/bracketleftbigg ∆ti¯h 2m/parenleftBig ∂x−ie ¯hBy/parenrightBig2/bracketrightbigg = exp/parenleftBig +ie ¯hBxy/parenrightBig ×exp/bracketleftbigg ∆ti¯h 2m∂2 x/bracketrightbigg exp/parenleftBig −ie ¯hBxy/parenrightBig . (22) Equation (21) is approximated by the following second- order exponential product: ψ(r,t+ ∆t) = exp/bracketleftbigg∆t 2i¯h 2m∂2 y/bracketrightbigg exp/parenleftBig +ie ¯hBxy/parenrightBig ×exp/bracketleftbigg ∆ti¯h 2m∂2 x/bracketrightbigg exp/parenleftBig −ie ¯hBxy/parenrightBig ×exp/bracketleftbigg∆t 2i¯h 2m∂2 y/bracketrightbigg ψ(r,t) +O(∆t3).(23)Moreover, we have found that the hybrid decomposi- tion [17] is rather easy in this case. Note the following identity: /bracketleftbig ∂2 y,/bracketleftbig (∂x−iay)2,∂2 y/bracketrightbig/bracketrightbig =−8a2∂2 y. (24) Then, equation (21) is approximated by the following fourth-order hybrid exponential product: ψ(r,t+ ∆t) = exp/bracketleftBig ∆ti¯h 2m/parenleftBig1 6−e2B2∆t2 72m2c2/parenrightBig ∂2 y/bracketrightBig ×exp/parenleftBig +ie ¯hBxy/parenrightBig exp/bracketleftBig∆t 2i¯h 2m∂2 x/bracketrightBig exp/parenleftBig −ie ¯hBxy/parenrightBig ×exp/bracketleftBig2∆t 3i¯h 2m∂2 y/bracketrightBig ×exp/parenleftBig +ie ¯hBxy/parenrightBig exp/bracketleftBig∆t 2i¯h 2m∂2 x/bracketrightBig exp/parenleftBig −ie ¯hBxy/parenrightBig ×exp/bracketleftBig ∆ti¯h 2m/parenleftBig1 6−e2B2∆t2 72m2c2/parenrightBig ∂2 y/bracketrightBig ψ(r,t) +O(∆t5). (25) The exponential of the magnetic ﬁeld just changes the phase of the wave function, so it is very easy to compute. Therefore, this method is adaptable to systems subjected to a magnetic ﬁeld. The outline of the procedure for a two-dimensional system subjected to a magnetic ﬁeld is schematically described by Fig. 2. x y +Bxy -Bxy y K K K FIG. 2. The procedure for a two-dimensional system sub- jected to a magnetic ﬁeld. Here Bxymeans the operation of the exponential of the magnetic ﬁeld. In this way, the phase of the wavefunction is turned forward before the operation o f Cayley’s form along the x-axis and is turned backward after Cayley’s form. III. APPLICATIONS A. Comparison between FDM and FEM In this subsection, we brieﬂy compare Cayley’s form and other conventional methods by simply simulating a Gaussian wave packet moving in a one-dimensional free system as illustrated in Fig. 3. x2W xopo 3FIG. 3. The model system for comparison with the con- ventional methods. 256 computational grid points are allo- cated in the physical length 8 .0a.u. A Gaussian wave packet is placed in the system, whose initial average location xoand momentum poare set as xo= 2.0a.u. and po= 12.0a.u., re- spectively. The TD-Schr¨ odinger equation of this system is simply given by i∂ψ(x,t) ∂t=−∂2 x 2ψ(x,t). (26) The wavefunction at the initial state is set as a Gaussian: ψ(x,t= 0) =1 4√ 2πW2exp/bracketleftBig −\|x−xo\|2 4W2+ ipox/bracketrightBig ,(27) whereW= 0.25a.u.xo= 2.0a.u.po= 12.0a.u. The evo- lution of this Gaussian is analytically derived as ψ(x,t) =1 4/radicalbig 2πW2+ (π/2)(t/W)2 ×exp/bracketleftBig −(x−xo−pot)2 4W2+ (t/W)2+ ipox/bracketrightBig . (28) Therefore, the average location of the Gaussian ∝angb∇acketleftx(t)∝angb∇acket∇ightis derived as if it is a classical particle: ∝angb∇acketleftx(t)∝angb∇acket∇ight=∝angb∇acketleftx(t= 0)∝angb∇acket∇ight+pot . (29) This characteristic is useful to check the accuracy of the simulation. Cayley’s form with the FDM is given by ψ(t+ ∆t) =1 + i∆t/4∂2 x 1−i∆t/4∂2xψ(t), (30) where∂2 xis approximated by a ﬁnite diﬀerence matrix as ∂2 x≃1 ∆x2 −2 1 0 0 0 0 1−2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 . (31) Meanwhile, Cayley’s form with the FEM is given by ψ(t+ ∆t) =meﬀ+ i∆t/4∂2 x m∗ eﬀ−i∆t/4∂2xψ(t), (32) where the spatial diﬀerential operator is approximated in the ordinary way and meﬀis the eﬀective mass: 1 meﬀ≡1 m−i2 5∆x2 ∆t, (33) ∂2 xis approximated by eq. (31).We have simulated the motion of the Gaussian by those methods. Figure 4 shows the error in the average momen- tum. The errors are evaluated in the following way: ǫ(∆t/∆x2) =∝angb∇acketleftx(t=T)∝angb∇acket∇ight −xo T−po, (34) ∝angb∇acketleftx(t)∝angb∇acket∇ight= ∆xN−1/summationdisplay i=0xi\|ψi(t)\|2in FDM. (35) ∝angb∇acketleftx(t)∝angb∇acket∇ight=∆x 30ReN−1/summationdisplay i=0xiψ∗ i/parenleftbig 24ψi+ 4ψi+1+ 4ψi−1 −ψi+2−ψi−2/parenrightbig in FEM. (36) -0.6-0.5-0.4-0.3-0.2-0.10.00.1 0.0625 0.125 0.25 0.5 124Error in the average momentum ∆ ∆t / x2Cayley with FDMCayley with FEM FIG. 4. The errors in the average momentum computed by Cayley’s form with the FDM and Cayley’s form with the FEM. The error of the FEM is smaller than that of the FDM. The spatial slice is set as ∆ x= 1/32a.u. It is found that the accuracy is dramatically improved by using the FEM. It is remarkable that in spite of the improvement of accuracy, the computational cost does not increase at all. B. Cyclotron motion We demonstrate the cyclotron motion in the frame- work of quantum mechanics. We have simulated the mo- tion of a Gaussian wave packet in a uniform magnetic force as illustrated in Fig. 5. y xB 4FIG. 5. The model system for the cyclotron motion. This system is subjected to a static magnetic force perpendicula rly. and it is surrounded by inﬁnitely high potentials. 64 ×64 computational grid points are allocated in the physical len gth 8a.u.×8a.u. The strength of the static magnetic force Bis set as 2a.u. A Gaussian is placed as the initial state of the wavefunction, whose average location and momentum are set as (6a.u. ,4a.u.) and (0a.u. ,4a.u.), respectively. The time slice is set as ∆ t= 1/64a.u. The initial wavefunction ψ(r,t= 0) is set as the fol- lowing Gaussian: ψ(r,t= 0) =1√ 2πW2exp/bracketleftBig −\|r−ro\|2 4W2/bracketrightBig exp/bracketleftBigieB ¯h(x−L/2)y/bracketrightBig , (37) where rois set asxo= 6a.u.,yo= 4a.u. and Wis set as 0.5a.u. The initial density ρ(r,t= 0) and the initial current density j(r,t= 0) derived from this wave function are as follows: ρ(r,t= 0) =1 2πW2exp/bracketleftBig −\|r−ro\|2 2W2/bracketrightBig , (38) j(r,t= 0) =e2Bρ(r) mc(0,x−L/2,0)T.(39) We adopt a gauge of the vector potential Aas A= (−By,0,0)T. (40) In classical mechanics, the average momentum of this Gaussian at the initial state is evaluated as po=m e\|∝angb∇acketleftj∝angb∇acket∇ight\|=eB c\|xo−L/2\|. (41) This means the classical cyclotron radius is \|xo−L/2\|. Some snapshots of the simulation time span are illus- trated in Fig. 6. The average location of the wave packet is observed to circle around as plotted in Fig. 7. t=0 t=3/8 t=6/8 t=9/8 t=12/8 t=15/8 t=18/8 t=21/8 FIG. 6. The evolution of the density and the current vec- tor. The Gaussian is observed to circle around. 012345678 012345678y xStart FIG. 7. The orbit of the average location of the wave packet. The radius of this circular trace is estimated as 2 .0a.u. The initial average location and momentum of this Gaussian are set as (6a.u. ,4a.u.) and (0a.u. ,4a.u.), respectively. This trace is not a perfect circle but a swirl due to the reﬂection by the closed walls around the system. A more perfect circular trace is observed by enlarging the system or shortening the cyclotron radius to reduce the eﬀect of the reﬂection. Figure 8 shows the result of another simulation. 5012345678 012345678y xStart FIG. 8. The another orbit of the average location of the wave packet. The radius of this circular trace is estimated as 1.0a.u. The initial average location and momentum of this Gaussian are set as (5a.u. ,4a.u.) and (0a.u. ,2a.u.), respec- tively. These results aﬀord good agreement with the result by classical mechanics. C. Aharonov-Bohm eﬀect We demonstrate Aharonov-Bohm eﬀect by simulating an electron dynamics on a system as illustrated in Fig. 9. Wall Free spacea magnetic fluxWall with Space witha vector potential y=ldD Wx yLLL FIG. 9. The model system for the Aharonov-Bohm eﬀect. The shape of this system is rectangular. A double-slit lies at the center. A magnetic ﬂux Φ goes through a wall lying between the slits. 64 ×128 computational grid points are allocated in the physical size 8a.u. ×16a.u. The initial wave- function is set as a plane wave kin front of the double-slit. The time slice is set as ∆ t= 1/64a.u. The vector potential is constructed as follows: A(x,y) = (0, Ay(x),0)T;Ay(x) =−/integraldisplayx −L/2dx′B(x′,y). (42) ThusAy(x) has a ﬁnite value only inside the right slit: Ay(x) =/braceleftbigg −B(D−d) : inside the upper slit. 0 : in other area.,(43)wheredandDmean the width of the slits and the span of the slits respectively. Thus D−dis the length of the wall where a magnetic ﬂux goes through. In an analogy to semi-classical photon interference, the electron interference pattern I(x) in this AB system is approximately described by the following form: I(x)∝/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ℓ kdxsin/bracketleftBigkd 2ℓx/bracketrightBig cos/bracketleftBigkD 2ℓx−eΦ 2¯h/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 .(44) In the above, ℓis aycoordinate where the pattern is evaluated. Figure 10 shows the result of this simulation for the case of no magnetic ﬂux, Φ = 0. These data were taken soon after the pattern appeared in order to prevent the pattern from extra interference due to the reﬂected waves from side walls. The interference pattern basically agrees with the semi-classical one derived from eq. (44). -400 -200 0 200 400I(x) x FIG. 10. The interference pattern observed in the back of the double-slit and at the line y=ℓ=L/4 in a case of no magnetic ﬂux, Φ = 0. The solid line indicates the numerical result; the dashed line indicates the semi-classical one de rived from eq. (44). Further, the results for the case of magnetic ﬂux Φ =h/2eand Φ =h/eare shown in Figs. 11 and 12, respectively. The patterns are observed to shift to the right-hand side, and these behaviors also agree with the semi-classical one. However, the patterns are diﬀerent from the the semi-classical one in their details. This is of course due to the quantum eﬀect. 6-400 -200 0 200 400I(x) x FIG. 11. The interference pattern observed in the back of the double-slit and at the line y=ℓ=L/4 in a case of Φ = h/2e. The solid line indicates the numerical result; the dashed line indicates the semi-classical one derived fr om eq. (44). -400 -200 0 200 400I(x) x FIG. 12. The interference pattern observed in the back of the double-slit and at the line y=ℓ=L/4 in a case of Φ = h/e. The solid line indicates the numerical result; the dashed line indicates the semi-classical one derived fr om eq. (44).IV. CONCLUSION We have improved the computational method for the time-dependent Schr¨ odinger equation by utilizing the ﬁ- nite element method and by formulating a new scheme for a magnetic ﬁeld. We have found that by using the FEM, the accuracy of the simulation is dramatically im- proved without any increase in the computational cost. We have also found that the new scheme is quite eﬃcient for simulating systems in a magnetic ﬁeld. 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arXiv:physics/0011070v1 [physics.atm-clus] 29 Nov 2000Time dependent energy absorption of atomic clusters from an intense laser pulse Christian Siedschlag and Jan M. Rost Max-Planck-Institute for the Physics of Complex Systems, N ¨ othnitzer Str. 38, D-01187 Dresden, Germany (November 2000) For the energy absorption of atomic clusters as a function of the laser pulse duration we ﬁnd a similar behavior as it has been observed for metallic cluste rs [K¨ oller et al., Phys. Rev. Lett. 82, 3783 (1999)]. In both situations there exists an optimum ra diusRoof the cluster for energy absorption. In the metallic case the existence of Rohas been interpreted as a consequence of the collective oscillation of a delocalized electron cloud in r esonance with the laser frequency. Here, we give evidence that in the atomic cluster the origin of Rois very diﬀerent. Based on ﬁeld assisted tunneling it can be related to the phenomenon of enhanced ion ization as it occurs in small molecules. The dependence of Roon the laser frequency turns out to be the key quantity to dist inguish the processes. PACS numbers: 36.40 -c, 33.80 Rv, 36.40 Gk Exposed to an intense laser pulse a cluster absorbs a considerable amount of energy which is released sub- sequently through fragmentation into fast electrons [1], multicharged ions [2] and radiation in the x-ray regime [3]. These eﬀects depend on the seize of the cluster, i.e., the number of constituent atoms and the pulse duration, somewhat less on the kind of atoms and the wavelength of the light. In a recent experiment K¨ oller et al. measured the in- tensity dependent energy absorption of a cluster consist- ing of some ten platinum atoms as a function of laser pulse duration [4]. This has been done by keeping the energy content (ﬂuence) of the laser pulse constant and varying the pulse duration as well as the peak intensity accordingly. Interestingly, the absorbed energy decreases within- creasing laser intensity, after having reached a maximum. We have found the same behavior in calculations for atomic clusters containing a similar number of atoms. However, as we will see below, the mechanism responsi- ble for the phenomenon is quite diﬀerent from the one described in [4] for the metallic clusters. We will show that a sensitive indicator for the underly- ing mechanism is the existence of an optimal mean inter- nuclear distance Rofor energy absorption which changes with the laser frequency ωfor a cluster of delocalized electrons (a metal cluster) while it is independent of ωin the case of an atomic cluster. In both cases the existence ofRois also the origin of the peculiarity that the energy absorption can decrease with increasing peak intensity of the laser, as mentioned above. In order to discriminate between diﬀerent mechanisms we had to choose an approach which is capable of han- dling, at least in principle, both situations: atomic clus- ters with localized electrons and delocalized electrons as they are typical for metallic behavior. Furthermore,the numerical treatment had to be fast enough to follow an appreciable number of particles (ions and electrons). Clearly, this cannot be done fully quantum mechanically, for the time being. Our approach is a combination of the ones described in [6–9], i.e., essentially based on classic al equations of motion for all ionized charged particles un- der full mutual Coulomb interactions. As in [7] we have used Coulomb soft-core potentials Ve(r) = (r2+a2)−1/2. (1) We will see later, that the choice of aallows us to describe an atomic cluster with localized electrons ( aa∼1a.u.) or to simulate a metallic cluster with delocalized electrons (am≫aa). The initial ionization of an electron bound to an atom or ion is described with an analytically known rate [5], dependent on the instant (static) electric ﬁeld at the po- sition of the atom/ion to be ionized. The ﬁeld is created by all surrounding charges (ions and electrons) and the laser. In contrast to [7] we do not include additional elec- tron impact ionization. Its eﬀect is small (see [9]), more- over, its implementation based on empirical cross sec- tions, such as the Lotz formula, bares the danger that the contribution of electrons to ionization is counted twice: through ﬁeld ionization and through additional impact ionization. The actual computation goes as follows: After a relax- ation to an equilibrium under Lennard-Jones potentials the atomic conﬁguration is exposed to the laser pulse. We compute a probability for ionization for each atom (later ion) from the rate in an time interval ∆ t. Is it larger than a generated random number 0 ≤s≤1, the atom is considered as ’ionized’ [12] and turns into an ion, and a new electron is created outside the instant po- tential barrier with zero kinetic energy. The ionization rate for the ion is adjusted to the corresponding higher 1binding energy and the procedure is repeated, of course, simultaneously for all atoms/ions. Newton’s equations are solved for the time evolution of all charged particles interacting through mutual Coulomb soft-core potentials Eq. (1) with a2= 2a.u.. In the following we will discuss the energy transfer to a Ne 16cluster from a laser pulse with sin2-envelope and an optical frequency of ω= 0.055 a.u.. If we record the energy gain after the pulse as a function of pulse duration T, we obtain Fig. 1. Since for constant ﬂuence under a variation of T, the peak intensity behaves as I∝1/T, one recognizes the increasing energy absorption for decreas- ing intensity. Only for very short pulses (high intensities ) the trend is reversed indicating that in this regime the pure atomic response dominates cooperate cluster eﬀects. An analogous behavior of the energy absorption, includ- ing the rise for very short pulses, is found in calculations for excitation of an Na 9cluster [11], and in the exper- iment on platinum clusters (exempliﬁed by the depen- dence of the charge states for ejected ions as a function of the pulse duration, see Fig. 2 in [4]). 1415161718 01500 3000 4500absorbed energy (keV) pulse duration T (a.u.) FIG. 1. Absorbed energy as a function of laser pulse du- ration Tfor constant ﬂuence such that a peak intensity of 1015W/cm2is reached with a pulse of 20 cycles ( ω= 0.055 a.u.). Note that an atomic time unit is 0.0242 fs. The line is to guide the eye. The authors of [4] provided an appealing interpretation in terms of a plasma model for the delocalized electron density of the platinum cluster: The eigenfrequency Ω of the electron cloud depends on its density, which, in turn, is a function of the cluster radius, i.e. Ω = Ω( R(t)). When the cluster expands due to the net positive charge after initial ionization, the electron density decreases a nd so does the plasma frequency Ω which will eventually match the laser frequency ω. Then, energy absorption becomes resonant and is greatly enhanced. The maxi- mum in the absorption as a function of pulse duration is now essentially a matching problem: The best condition is a coincidence of the peak intensity with the time when Ω(R(t)) =ω. If the laser pulse is too short, the resonance condition is reached when the pulse is already over. Onthe other hand, if the pulse is too long, the cluster has expanded beyond Rowhen the peak intensity is reached. We deﬁne as a characteristic length scale for the cluster R(t) =/parenleftBiggN/summationdisplay iR2 i/N/parenrightBigg1/2 , (2) the mean over all individual internuclear distances Ri. Equivalently, we will speak of the cluster radius which is directly proportional to Rfor a ﬁxed number of atoms in the cluster [12]. Typically, R(t) increases adiabatically slowly compared to the electronic and optical time scales. This allows us to gain more insight into the dynamics by considering the energy absorption of the cluster for dif- ferent but ﬁxedmean internuclear separations R. Figure 2a demonstrates that the energy absorption for ﬁxed R peaks at a critical Roindependent of the laser frequency. This is a key observation which has several conse- quences: Firstly, the existence of Rofor an atomic cluster explains the shape of the energy absorption in Fig. 1 with a maximum due to the monotonic increase of Ras a func- tion of time. Large energy absorption occurs for a pulse duration Tsuch that peak intensity is reached at T/2, when the cluster has the optimal seize R(T/2)≈Ro. Secondly, the mechanism which leads to the existence of Romust be diﬀerent from the one proposed for a metal cluster in [4], since a resonant absorption with Ω( Ro) =ω points to optimal cluster radii Rowhich change with the frequency ω. Rather, the mechanism we have identiﬁed for these relatively small atomic clusters is akin to a be- havior in small molecules which has been described under the name enhanced ionization [13] or CREI (charge reso- nant enhanced ionization) [14]. In short, the idea is that in a diatomic molecule the electron, localized on one atom most of the time due to the oscillatory light ﬁeld, can eas- ier tunnel through the barrier formed by the attractive Coulomb potential and the electric ﬁeld since this bar- rier is lowered by the additional electric ﬁeld generated by the neighboring (positively charged) nucleus. An optimal internuclear distance Rofor this ﬁeld as- sisted tunneling exists since in the united atom limit R= 0 there is only one well (and deeper binding) while in the separated atom limit R→ ∞ the additional ﬁeld simply goes to zero. The signature of this mechanism is the existence of Roand its independence of the laser frequency ω, precisely as seen in Fig. 2a. Hence, the mechanism of ﬁeld assisted tunneling is indeed also op- erative for our cluster where many surrounding charged ions form a strong ﬁeld for the speciﬁc atom or ion to be ionized in the cluster. Having established the origin of the peculiar behavior of energy absorption in an atomic cluster as a function of laser pulse duration, we may subject our modelling of cluster dynamics to an ultimate test by comparing exact quantum results for the simplest system H+ 2to predic- tions from our approach. This is done in one dimension 20 5 10 15 20 25 30absorbed energy ( arb. units) R (a.u.)(a) 0 1 2 3 4 5 6absorbed energy ( arb. units) R/Re(b) 0 5 10 15 20 2500.10.20.30.40.5ionization probability R (a.u.)(c) FIG. 2. Energy absorption from an intense laser pulse ( T= 55 fs) in diﬀerent situations: (a) for Ne 16as a function of ﬁxed mean interatomic distance Rat two diﬀerent laser frequencies, ω= 0.055 a.u. (solid), ω= 0.11 a.u. (dashed) and with peak intensity I= 1015W/cm2, (b) as in (a) but for the 16 atom metallic cluster model as a fu nction of the initial mean ion distance Reand for I= 3.51×1012W/cm2, see text, (c) for H+ 2as a function of ﬁxed internuclear distance with the one dimensional quantum result (solid) and the present tunneli ng approach (dashed) at a peak intensity of I= 5.6×1013W/cm2. (where the internuclear axis is aligned along the electric ﬁeld of the laser) in Fig. 2c. Although we model the bound electron being attached to one proton and calcu- late its tunneling rate subject to the laser ﬁeld and the ﬁeld generated by the second proton, the actual ioniza- tion probability is in surprisingly good agreement with the exact quantum result, particularly compared to a purely classical over barrier model whose ionization yield is too small to be visible in Fig. 2c. Note that Rofor the cluster (Fig. 2a) is even roughly equal to RoinH+ 2 (Fig. 2c). Designed for an interaction of several ions with many electrons and an intense laser pulse the fairly accurate description of H+ 2is an unexpected conﬁrmation of the modelling. However, it raises also the question if the mechanism we have identiﬁed for energy absorption in clusters being akin to that for molecules in intense laser ﬁelds is merely a consequence of the modelling which seems to be ideally suited to describe tunneling related phenomena. To doublecheck that our result is independent of the modelling and also, to clarify further the diﬀerent mech- anism which seems to be responsible for the (similar) energy absorption and existence of a critical mean dis- tance Roin metal clusters we have simulated within our approach the behavior of delocalized electrons as they occur in a metal cluster. This has been achieved by arti- ﬁcially softening the potential Eq. (1) with a2= 30 a.u.. As a consequence the cluster ions at equilibrium distance of each other form one structureless well for the ”collec- tive ” binding of the electrons. Comparing the excitation spectrum of the electrons, one sees for the original situ- ation of atomic clusters with localized electrons a single peak which corresponds to the local excitation (Fig. 3a) while for the delocalized electrons with a2= 30 one sees two peaks (Fig. 3b), the lower and wider one correspondsto the softened local excitation of the local binding, the higher peak is the new feature of collective excitation which is believed to be responsible for the mechanism of resonant energy absorption as described above. If this is true, we would expect in our model for delocalized electrons a dependence of the optimal cluster radius for energy absorption on the laser frequency. This is indeed the case, as one can see in Fig. 2b. As expected for a de- creasing electron density with growing cluster radius, and corresponding decreasing eigenfrequency Ω of the elec- tron cloud, we ﬁnd that Rois smaller for the higher laser frequency. This conﬁrms the existence of a diﬀerent mechanism which leads to a critical cluster radius in a situation of delocalized electrons, in accordance with what has been found by very diﬀerent modelling of the valence electrons in sodium clusters [15,16]. It also demonstrates that our 0.5 0.6 0.7σ(ω) (arb. units) ω (a.u.)(a) 0 0.13 0.26 ω (a.u.)(b) FIG. 3. Excitation spectrum for the electrons in the clus- ter, (a) localized electrons in Ne 16, (b) delocalized electrons in the metallic cluster model. 3formulation of intense ﬁeld dynamics of clusters is ca- pable of describing both, atomic clusters with localized electrons and, at least qualitatively, the situation of del o- calized electrons as they occur in metal clusters. Hence, the result reassures that the mechanism of ﬁeld enhanced ionization by surrounding charged particles is not an ar- tifact of the theoretical description. To summarize, we have found that the energy absorp- tion in small atomic clusters depends strongly on the laser pulse duration, similarly as in metallic clusters and large (N≈106) atomic clusters. However, the mechanism is very diﬀerent. While in metallic cluster [4] as well as in large atomic clusters [17] a similar plasmon resonance mechanism prominently involving delocalized electrons has been advocated to explain the observations, we ﬁnd that in small atomic clusters ﬁeld assisted tunneling is re- sponsible. By making use of the adiabaticity of the ionic motion compared to the electronic motion we could show that a critical cluster radius exists for maximum energy absorption which is independent of the laser frequency ω. This behavior is akin to the one known from diatomic molecules as ”enhanced ionization ” [13,14] and can be attributed to the same physical eﬀect of ﬁeld assisted tunneling ionization. As a sideﬀect, we have shown that our approach also describes the ionization of the smallest molecule, H+ 2, in a strong laser pulse rather well. Furthermore, we have simulated the behavior of delo- calized electrons within the same theoretical approach. Thereby, we could conﬁrm that for delocalized electrons enhanced energy absorption can be attributed to a plas- mon type resonance. It occurs when the eigenfrequency of the delocalized electron density and the laser frequency agree, as suggested by K¨ oller etal to interpret their ex- periment [4]. However, we could only clearly identify this type of resonance behavior if exclusively the valence elec- trons are involved in the ionization dynamics, i.e., if the laser intensity is suﬃciently weak (in our case 3 .51×1012 W/cm2). Once electrons from the ionic cores are ionized, the local character of the electron binding starts to domi- nate. Moreover, ﬁeld ionization triggered by surrounding charges takes over the ionization caused by the collective electron cloud and the laser ﬁeld. Since in the experi- ment [4] the peak intensity of the laser was rather large (more than 1015W/cm2) and highly charged ions have been detected (which probably were even higher charged through the initial ionization before recombination took place), it is possible that the actual mechanism for the energy absorption pattern as a function of pulse duration is closer to that of ﬁeld assisted tunneling as in atomic clusters than to the plasmon resonance enhanced ioniza- tion of metallic valence electrons. As we have shown, the two mechanisms diﬀer by their dependence on the laser frequency. Hence, it would be desirable to repeat the experiment of [4] at a higher laser frequency, the best choice being an ωhigh enough that the resonance condi- tion cannot be fullﬁlled. If the energy absorption patternstill shows a pronounced maximum, one could exclude the plasmon induced absorption mechanism and rather would have to conclude that the ﬁeld assisted tunneling scenario is a universal mechanism for intense laser ﬁeld ionization in molecules and clusters of moderate seize. It is a pleasure to thank K.H. Meiwes-de Broer, P. Corkum, R. Schmidt, and P.-G. Reinhard for fruitful dis- cussions. We also acknowledge O. Frank’s input at the initial stages of this work which has been supported by the DFG through the Gerhard Hess-program. [1] Y. L. Shao et al, Phys. Rev. Lett. 77, 3(1996). [2] T. Ditmire et al, Nature (London) 386, 54 (1997). [3] A. Mc Pherson et al, Nature (London) 370, 631 (1994). [4] L. K¨ oller, M. Schumacher, J. K¨ ohn, S. Teuber, J. Tiggesb¨ aumker, and K. H. Meiwes-Broer, Phys. Rev. Lett.82, 3783 (1999). [5] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965) [6] C. Rose-Petruck et al, Phys. Rev. A 55, 1182 (1977). [7] T. Ditmire, Phys. Rev. A 57, R4094 (1998). [8] I. Last and J. Jortner, Phys. Rev. A 62, 013201 (2000). [9] K. Ishikawa, and T. Blenski, Phys. Rev. A 62, 063204 (2000). [10] In a simple model of dense spheres of atoms, the acu- tal cluster radius Ris directly proportional to Rwith R=N1/3R/2, see, e.g., Kreibig and Vollmer, Optical properties of Metal Clusters (Springer, 1995) . [11] private communication, R. Schmidt, see also U. Saal- mann and R. Schmidt, Phys. Rev. Lett. 80, 3213 (1998). [12] ’Ionization’ means here that the electron is no longer bound to its mother atom, however, it may be still bound to the entire cluster. See also the distinction of ’outer’ an d ’inner’ ionization in [6]. [13] T. Seidemann, M. Yu. Ivanov, and P. B. Corkum , Phys. Rev. Lett. 75, 2819 (1995). [14] T. Zuo and A. D. Bandrauk, Phys. Rev. A 52, R2511 (1995). [15] P. G. Reinhard et al, Physics Reports 337, 493-579 (2000) [16] E. Suraud and P. G. Reinhard, Phys. Rev. Lett. 85, 2296 (2000) [17] J. Zweiback, T. Ditmire, and M. D. Perry, Phys. Rev. A 59, R3166 (1999). 4
arXiv:physics/0011071v1 [physics.bio-ph] 29 Nov 2000Compound Poisson Statistics and Models of Clustering of Rad iation Induced DNA Double Strand Breaks. E.Gudowska-Nowak ,1,2 M.Kr¨amer ,2G.Kraft2and G. Taucher-Scholz ,2 1M. Smoluchowski Institute of Physics, Jagellonian Univers ity, ul. Reymonta 4, 30059 Krak´ ow, Poland 2Biophysik, Gesellschaft f¨ ur Schwerionenforschung, Plan ckstr. 1 , 64291 Darmstadt, Germany According to the experimental evidence damage induced by densely ionizing radiation in mammalian cells is distribut ed along the DNA molecule in the form of clusters. The most critical constituent of DNA damage are double-strand break s (DSBs) which are formed when the breaks occur in both DNA strands and are directly opposite or separated by only a few base pairs. The paper discusses a model of clustered DSB formation viewed in terms of compound Poisson process along with the predictive assay of the formalism in application to experimental data. PACS numbers: 87.10.+e, 05.40.+j I. INTRODUCTION In living cells subjected to ionizing radiation many chemical reactions are induced leading to various bio- logical eﬀects such as mutations, cell lethality or neo- plastic transformation [1,2]. The most important tar- get for radiation induced chemical transformation where these changes can be critical for cell survival is DNA dis- tributed within the cell’s nucleus. Nuclear DNA is orga- nized in a hierarchy of structures which comprise the cel- lular chromatin. The latter is composed of DNA, histones and other structural proteins as well as polyamines. Or- ganization of DNA within the chromatin varies with the cell type and changes as the cell progresses through the cell cycle. Ionizing radiation produces variety of damage to DNA including base alterations and single- and double strand breaks (DSBs) in the sugar-phosphate backbone of the molecule [1,3]. Single strand breaks (SSBs) are eﬃciently repaired with high ﬁdelity and probably con- tribute very little to the loss of function of living cells. On the other hand, DSBs are believed to be the critical lesions produced in chromosomes by radiation; interac- tion between DSBs can lead to cell killing, mutation or carcinogenesis. The purpose of theoretical modeling of radiation action [4]– [7] is to describe qualitatively and quantitatively the results of radiobiological eﬀects at th e molecular, chromosomal and cellular level. The basic consideration in such an approach must be then descrip- tive analysis of breaks in DNA caused by charged particle tracks and by the chemical species produced. Production of DSBs in intracellular DNA can be stud- ied by use of the pulsed ﬁeld gel electrophoresis (PFGE) [8] in which the gel electrophoresis is applied to elutehigh molecular weight DNA fragments from whole cel- lular DNA embedded in an organic gel (agarose). Two main approaches of this technique are usually applied. One is the measurement of the fraction of DNA leaving the well in PFGE, i.e. the amount of DNA smaller than a certain cutoﬀ size deﬁned by the electrophoretic condi- tions. This method has proven to be very sensitive, allow- ing reproducible measurements at relatively low doses. The second approach is to describe fragment-size distri- butions obtained after irradiation as a function of dose, taking advantage of the property of PFGE to separate DNA molecules based on how quickly they reorient in a switching (pulsed) electrical ﬁeld. The major goal of the experiments is to quantify number of induced DSBs based on changes in the amount of DNA or the average fragment size in response to dose. In both cases data obtained are related to average number of DSBs. To an- alyze the data, the formalism describing random depolar- ization of polymers of ﬁnite size is usually adopted [9,10] giving very well ﬁts to experimental results with X-ray induced DNA fragmentation. In contrast to the ﬁndings for sparsely ionizing irradiation (X and γrays) charac- terized by low average energy deposition per unit track length (linear energy transfer, LET ≈1 keV/ µm), the densely ionizing (high LET) particle track is spatially lo- calized [2,11]. In eﬀect, multiplicity of ionizations with in the track of heavy ions can produce clusters of DSBs on packed chromatin [13]. The formation of clusters de- pends on chromatin geometry in the cell and radiation track structure. DSBs multiplicity and location on chromosomes may determine the distribution of DNA fragments detected in PFGE experiments. Modeling DNA fragment-size- distributions provides then a tool which allows to elu- cidate experimentally observed frequencies of fragments. Even without detailed information on the geometry of chromatin, models of radiation action on DNA can serve with some predictive information concerning measured DNA fragment-size-distribution. The purpose of the present paper is to discuss a model which can be used in analysis of DNA fragment-size- distribution after heavy ion irradiation. The background of the model is the Pois- son statistics of radiation events which lead to formation of clusters of DNA damage. The formation of breaks to DNA can be then described as the generalized or com- pound Poisson process for which the overall statistics of damage is an outcome of the random sum of random 1variables (Section 2). Biologically relevant distributio ns are further derived and used (Section 3) in description of fragment size distribution in DNA after irradiation with heavy ions. Practical use of the formalism is discussed by ﬁtting the distributions to experimental data. II. RANDOM SUMS OF RANDOM VARIABLES AND COMPOUND POISSON DISTRIBUTIONS Consider [14,15] a sum SNofNindependent random variables X SN=N/summationdisplay i=1Xi (2.1) where Nis a random variable with a probability gener- ating function g(s) g(s) =∞/summationdisplay i=0gisi(2.2) andXiarei.i.d. variables (independent and sampled from the same distribution) whose generating function f(s) is f(s) =∞/summationdisplay j=1fjsj(2.3) By use of the Bayes rule of conditional probabilities the probability that SNtakes value jcan be then written as P(SN=j)≡hj=∞/summationdisplay n=0P(SN=j\|N=n)P(N=n) (2.4) For ﬁxed value of nand by using the statistical inde- pendence of Xi’s, the sum SNhas a probability gener- ating function F(s) being a direct product of f(s),i.e. F(s) =f(s)n=/summationtext∞ j=0Fjsjfrom which it follows that P(SN=j\|N=n) =Fj. The formula (2.4) leads then to the compound probability generating function of SN given by h(s) =∞/summationdisplay j=0hjsj= =∞/summationdisplay j=0∞/summationdisplay n=0Fjgnsj= =∞/summationdisplay n=0gnf(s)n≡g{f(s)} (2.5) Conditional expectations rules can be used to determine moments of a random sum. Given E[N] =ν,E[Xi] =µ, V ar[N] =τ2andV ar[Xi] =σ2, the ﬁrst and the second moment of the random sum SNareE[SN] =µν, V ar [SN] =νσ2+µ2τ2(2.6) The above compound distribution is describing “clus- tered statistics” of events grouped in a number Nof clusters which itself has a distribution. As such, it is sometimes described in literature [16] as “mixture of dis- tributions”. Out of many interesting biological applica- tions of compound distributions [17]- [20], a special class constitute Poisson point processes which can be also an- alyzed in terms of random sums with Poisson distributed random events N. It can be shown that a mixture of Poisson distributions resulting from using any unimodal continuous function f(λ) is a unimodal discrete distribu- tion. It is not so, however, in case of unimodal discrete mixing. In particular, mixtures of Poisson-Poisson or Poisson-binomial, known in literature as Neyman distri- butions [21] can exhibit strongly multinomial character. By virtue of the above formalism and by using the for- mulae (2.5) , the generating function of the compound Poisson-Poisson distribution is: g= exp( −λ(1−f(s))) (2.7) where the random variables Xiare distributed according to a Poisson law f(s) = exp( −µ+µs) (2.8) and the total SNis a random variable with a compound Poisson-Poisson ( Neyman type A ) distribution: P(SN=x)≡P(x;µ, λ) =∞/summationdisplay N=0(Nµ)xe−Nµ x!λNe−λ N!(2.9) for which the mean and variance are given by E[x] =µλ, V ar [x] =λµ(1 +µ) (2.10) The resulting distribution can be interpreted as a mix- ture of Poisson distribution with parameter Nµwhere N(number of clusters) is itself Poisson distributed with parameter λ. Figures 1,2 present function (2.9) for two various sets of parameters λ, µ. The compound Poisson distribution (CPD) has a wide application in ecology, nuclear chain reactions and que- ing theory [4,19–21]. It is sometimes known as the distri- bution of a “branching process” and as such has been also used to describe radiobiological eﬀects of densely ionizing radiation in cells [17,22–24]. When a single heavy ion crosses a cell nucleus, it may produce DNA strand breaks and chromatin scissions wherever the ion- izing track structure overlaps chromatin structure. The multiple yield of such lesions depends on the radial distri- bution of deposited energy and on the microdistribution of DNA in the cell nucleus. The latter and the geome- try of DNA coiling in the cell nucleus determine number of crossings, the “primary” incidents leading to DSBs production. By assuming for a given cell line, a “typi- cal” average number nof possible crossings per particle 2traversal, the distribution of the number of chromatin breaks ican be modelled by a binomial law: P(i\|n) =/parenleftbiggn i/parenrightbigg piq(n−i)(2.11) where pis a probability that a chromatin break occurs at each particle crossing (and qis the probability that it does not). The overall probability that ilesions will be observed after mindependent particles traversed the nucleus is given by [4] P(i\|σ, F, n ) =∞/summationdisplay m=1(nm)!piq(nm−i)(σF)me−σF i!(nm−i)!m!(2.12) which is a compound Neyman type B distribution ob- tained as a random Poisson sum of binomially distributed i.i.dvariables. In the above presentation the average number of particles crossing the cell nucleus λis pro- portional to the absorbed energy (dose) and given by a product λ=σFof particle ﬂuence Fand nuclear cross section σ. 02004006008001000 x05101520253035Probabilitydensity FIG. 1. Simulated probability density function for the Ney- man-type A distribution (2.9) with λ= 6, µ= 100 for N= 10000 points. Note the ﬁnite value at x= 0 corre- sponding to P(0;µ, λ). Aggregation of observed cellular damage potentially leads to the phenomenon of “overdispersion”– that is, the variance of the aggregate may be larger than Poisson variance yielding “relative variance” V ar rel= V ar[SN]/E[SN] larger than 1. Assuming thus the Pois- son statistics of radiative events, for any distribution of lesions per particle traversal, the condition for overdis- persion can be easily rephrased in terms of (2.6) V ar[Xi]/E[Xi] +E[Xi]<1 (2.13) If no repair process is involved in diminishing number of initially produced lesions, the surviving fraction of cell s can be estimated from formula eq.(2.12) as a zero class of the initial distribution, i.e.the proportion of cells with no breaksPN(0\|σ, F, n ) =∞/summationdisplay m=1(nm)!qnm(σF)me−σF (nm)!m!= = exp[ −σF(1−qn)] (2.14) which diﬀers by a factor (1 −qn) in the exponent from the surviving fraction for a Poisson distribution: PP(0\|σ, F, n ) = exp[ −σF] = exp[ −E[i]] (2.15) 02004006008001000 x020406080Probabilitydensity FIG. 2. Simulated probability density function for the Ney- man-type A distribution (2.9) with λ= 100 , µ= 6 for N= 10000 points III. DNA FRAGMENTS DISTRIBUTION GENERATED BY IRRADIATION: STATISTICAL MODEL. DNA double stranded molecules in a size range from a few tenths of kilobase pairs to several megabase pairs can be evaluated by the PFGE technique. Randomly dis- tributed DSBs are detected as smears of DNA fragments. The DNA mobility mass distribution may be transformed into a fragment length distribution using a calibration curve. It is obtained by relating migration distance of DNA within the gel to molecular length with the aid of size markers loaded on the same gel [25]. To interpret the experimental material one needs to relate percentage of fragments in deﬁned size ranges to number of induced DSBs. For that purpose several models have been de- rived, mainly based on the description of random depo- larization of polymers of ﬁnite size [9,10,26]. Although the models give satisfactory prediction of size-frequency distribution of fragments after sparsely ionizing radiati on (i.efor X-rays and γ), they generally fail to describe the data after densely ionizing radiation [13,25]. The exper- iments with heavy ions demonstrate that after exposure to densely ionizing particles gives rise to substantially overdispersed distribution of DNA fragments which indi- cates the occurrence of clusters of damage. The following analysis presents a model which takes into account forma- tion of aggregates of lesions after heavy ion irradiation. Fragment distribution in PFGE studies is measured by 3use of ﬂuorescence technique or radioactive labeling with the result being the intensity distribution. The generated signal is proportional to the relative intensity distribut ion of DNA fragments and can be expressed as I(x) =xD(x) (3.1) with D(x) =∞/summationdisplay j=0D(x\|j)P(j;µ, λ) (3.2) where D(x\|j) stands for the density of fragments of length xprovided jDSBs occur on the chromosome of sizeS. Frequency distribution of the number of DSBs is assumed here in the form of CPD (2.9) with parameters µ andλrepresenting average number of breaks produced by a single particle traversal and average number of particle traversals, respectively. The “broken-stick” distributi on [27,26] for jbreaks on a chromosome of size Syields a density of fragments of size x: D(x\|j) =δ(x−S) + 2j1 S(1−x S)j−1+ +j(j−1)1 S(1−x S)j−1(3.3) where the ﬁrst two terms describe contributions from the edge fragments of the chromosome and the third term de- scribes contribution from the internal fragments of length x < S . The ﬁrst term applies to the situation when j= 0; the edge contribution can be understood by ob- serving that the ﬁrst and the j+ 1 fragment have the same probability of being size x. Direct summation in formula (3.2) leads to DN(x) = exp( −λ(1−e−µ))δ(x−S) + +2λµ Sexp(−µx S+λ(e−µx S−1)) + +e−λ(1−x S)µ2λ S(1 +λe−µx S)exp(−µx S+λe−µx S) (3.4) for Neyman distribution of number of breaks jand to DP(x) = Λ exp( −Λx S)(2 + Λ −Λx S) (3.5) for a Poisson distribution with parameter Λ. Integration ofI(x) (eq.(3.1)) from 0 to some average (marker) size X∗and division by Syields the relative fraction of DNA content. For λ >> 1 and µ << 1, the Neyman-type A distribution converges to a simple Poisson. In such a case, simpliﬁed expression (3.4) leads to results known in literature as “Bl¨ ocher formalism” [9,10,26] which de- scribes well the DNA content in probes irradiated with X– and γ–rays.100 200 300 400 500Dose 00.20.40.60.81 x00.250.50.751 DNAcontent 100 200 300 400Dose FIG. 3. Distribution of DNA content (integrated eq.(3.1)) as a function of the dose and fragment size for S= 245 Mbp, µ = 5. The fragments length is in Mbp units. Figure 3 presents predicted dose-response curves for the model. The amount of DNA content is shown in function of dose and fragment size. In calculation, the parameter S= 245 mega base pairs has been used which is the mean chromosome size for Chinese hamster cells, the cell line for which experimental data are displayed in Figure 4. The increase in multiplicity of DSBs produced per one traversal of a particle leads to pronounced increase in production of shorter fragments which is illustrated in the shift of the peak intensity towards smaller xvalues. FIG. 4. Fraction of DNA content observed experimentally within the range of sizes 0.1-1.0 Mbp. Data show higher prob- ability of producing short fragments after irradiation wit h particles than for sparsely ionizing radiation at comparat ive dose. Lines represent the best ﬁt to eq.(3.1) by use of DN(x) function for heavy ions (Au: λ= 3×10−3, µ= 6×102; C: λ= 6×10−3, µ= 6×102) and DP(x) for X-rays (Λ = 0 .85). 4IV. SPATIAL CLUSTERING OF BREAKS AND NON-POISSON STATISTICS. Clustering of breakage events can be viewed as the pro- cess leading to non-exponential “spacing” between sub- sequent events, similar to the standard analysis of level repulsion in spectra of polyatomic molecules and com- plex nuclei. For a random sequence, the probability that a DSB will be in the inﬁnitesimal interval (X+x, X+x+dx) (4.1) proportional to dxis independent of whether or not there is a break at X. This result can be easily changed by using the concept of breaks “repulsion’. Given a break atX, letP(x)dxbe the probability that the next break (x≥0) be found in the interval ( X+x, X+x+dx). We then have for the nearest-neighbour spacing distribution of breaks the following formula: P(x)dx=Prob(1∈dx\|0∈x)Prob(0∈x) (4.2) where Prob(n∈dx\|m∈x) is the conditional probability that the inﬁnitesimal interval of length dxcontains n breaks wheras that of length xcontains mof those. The ﬁrst term on the right-hand side of the above equation isdxtimes a function of xwhich we denote by r(x), depending explicitly on the choices 1 and 0 of the discrete variables nandm. The second term is given by the probability that the spacing is larger than x: /integraldisplay∞ xP(y)dy (4.3) Accordingly, one obtains P(x) =r(x)/integraldisplay∞ xP(y)dy, (4.4) whose solution can be easily found to be P(x) =Cr(x)exp(−/integraldisplayx r(y)dy) (4.5) where Cis a constant. The Poisson law, which reﬂects lack of correlation between breaks, follows if one takes r(x) =λ, where λ−1is the mean spacing between DSBs. If choosing on the other hand r(x) =λxλ−1(4.6) i.e.by assuming clustering of points (DSBs) along a line, one ends up with the Weibull density . The constants C andλcan then be determined from appropriate condi- tions, e.g. /integraldisplay P(x)dx= 1, (4.7) and/integraldisplay xP(x)dx=λ−1(4.8) One then ﬁnds that P(x) =λe−λx(4.9) for the Poisson distribution and P(x) =λxλ−1exp(−xλ) (4.10) for the Weibull analogue. Note that the above density can be derived as a generalization of the law eq.(4.9): the Weibull density can be obtained as the density of random variable y=x1/λwithxbeing an exponential random variable. For λ≥1, the Weibull distribution is unimodal with a maximum at point xm= (1−λ−1)λ−1. In this one easily recognizes for λ= 2 the spacing dis- tribution of the Wigner law . The latter displays “re- pulsion” of spacing, since P(0) = 0, in contrast to the Poisson case which gives maximum at x= 0. Fractional exponent λ <1 describes, on the other hand, enhanced frequency of short spacings which, in fact, matches bet- ter experimental data for heavy ions( cf.Figure 4). The above analysis brings also similarities with random walks [29,30] where symmetry breaking transition manifests it- self as a change in the spectral spacing statistics of decay rates. In such cases, the statistics of events of interest de - viates, as a counting process, from the regularity of Pois- son process, for which the subsequent event arrivals are spaced with a constant mean λ−1. The clustered statis- tics of breakage can be thus viewed as a (fractal) random walk or a cumulative distribution of a random sum of random variables eq.(2.1). The problem of characteriz- ing the limit distribution for such cases with underlying “broad” distributions g(x) ofXihas been studied ex- tensively in mathematical literature [32] and has been solved with classiﬁcation of the possible limit distribu- tions provided that requirement of “stability” is fulﬁlled under convolution. Following the deﬁnition, the distri- bution g(x) is stable, if for any Nthere exist constants cN>0 and dNsuch that SNhas the same density as the variable y=cNXi+dN. The stability condition can be rephrased in terms of the canonical representation given by a form of the characteristic function ( i.e.the Fourier transform g(k)) of stable distributions [33,32] lng(k) =iγk−C\|k\|λ[1−iωβsign (k)] (4.11) where γis real, 0 ≤λ≤2,ωis real and \|ω\| ≤tan(πλ/2)\|. The cases relevant for biological modelling are covered by 1≤λ≤2 (stable distributions have no variance if λ <2 and no mean if λ <1). In particular, positiv- ity of steps in the random walk modelled by eq.(2.1) al- lows for g(k) = exp[ −C\|k\|λ] which gives asymptotically g(x)≈x−λ−1. Probability distribution that x≥zsat- isﬁes then f(z)≈z−λforz→ ∞. The resulting distri- bution is “self-similar” in the sense that rescaling zto Azandf(z) toA−λf(z) does not change the power law 5distribution. In other words, the number of realizations larger than AzisA−λtimes the number of realizations larger than z. The power-law probability distribution function describes then the same proportion of shorter and larger fragments whatever size is discussed within the power law range. For λ= 1/2, C= 1, ω= 1 the form of L` evy-Smirnov law is recovered g(x) = (2 π)−1/2x−3/2e−1 2x (4.12) The probability density eq.(4.12) has a simple interpre- tation as the limiting law of return times to the origin for a one-dimensional symmetrical random walk and as such has been also used to describe the fragment size distribu- tion of a one dimensional polymer [31,34]. In the prob- lems related to polymer fragmentation induced by irradi- ation, the approach based on a random walk with ﬂuctu- ating number of steps (or, equivalently, on a point proce- ses model with a clustered statistics of waiting times) is a legitimate one as it can comprise the natural random- ness of primary events ( i.e.particle hits of biological target) and secondary induction of multiple (clustered) lesions. Further investigations in this ﬁeld should lead to better understanding of possible emergence of power-law distributions of larger fragments on kbp and Mbp scales. V. CONCLUSIONS An existing substantial evidence demonstrates that ex- posure to densely ionizing charged particles gives rise to overdispersed distribution of chromatin breaks and DNA fragments which is indicative of clustered damage occur- ing in irradiated cells. The clustering process can be expressed for any particular class of events such as ion- izations or radical species formation and is a consequence of energy localization in the radiation track. Chromoso- mal aberrations expressed in irradiated cells are formed in process of misrejoining of fragments which result from production of double-strand breaks in DNA. The loca- tion of double-strand breaks along chromosomes deter- mines DNA fragment-size distribution which can be ob- served experimentally. The task of stochastic modeling is then to relate parameters of such distributions to rele- vant quantities describing number of induced DSBs. Ap- plication of the formalism of clustered breakage oﬀers thus a tool in evaluation of the radiation respone of DNA fragment-size distribution and assessment of radiation in - duced biological damage. Acknowledgements . E.G-N acknowledges partial support by KBN grant 2PO3 98 14 and by KBN–British Council collaboration grant C51.REFERENCES [1] E.L. 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L¨ obrich, P. Cooper and B. Rydberg, Int. J. Ra- diat. Biol. 70(1996) 493; H.C. Newman, K.M. Prise, M. Folkard and B.D. Michael, ibid71(1997) 347; E. H¨ oglund, E. Blomquist, J. Carlsson and B. Sternl¨ ow, ibid 76(2000) 539. [14] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry , (North Holland, Amsterdam, 1981). [15] A. Papoulis, Probability, Random Variables and Stochas- tic Processes , (McGraw-Hill, Tokyo, 1981). [16] M. Kendall and A. Stuart, The Advanced Theory of Statistics , Charles Griﬃn & Co., London, 1977. [17] N. Goel and N. Richter-Dyn, Stochastic Processes in Bi- ology, (Academic Press, New York, 1974). [18] T. Maruyama, Mathematical Modeling in Genetics , (Springer Verlag, Berlin, 1981). [19] A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications , (Dover Publications, New York, 1988). [20] S. Karlin and H. Taylor, First Course in Stochastic Pro- cesses , (Academic Press, New York, 1976). [21] J. Neyman, Am. Math. 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arXiv:physics/0011072v1 [physics.atom-ph] 29 Nov 2000Ultracold collisions of metastable helium atoms P.J. Leo Atomic Physics Division, National Institute of Standards a nd Technology, Gaithersburg, Maryland 20899 V. Venturi∗and I.B. Whittingham School of Mathematical and Physical Sciences, James Cook Un iversity, Townsville 4811, Australia J. F. Babb Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 (February 21, 2014) Abstract We report scattering lengths for the1Σ+ g,3Σ+ uand5Σ+ gadiabatic molecular potentials relevant to collisions of two metastable 23Shelium atoms as a function of the uncertainty in these potentials. These scat tering lengths can be used to calculate elastic cross sections and inelastic ra tes of experimental interest at temperatures where the Wigner threshold approx imation is valid. Importantly, we show that the scattering length of the5Σ+ gpotential, crucial to the experimental attainment of Bose-Einstein condensat ion in a gas of spin- polarized metastable atoms, can be obtained from a measurem ent of the total elastic cross section in an unpolarized gas. PACS numbers: 03.75.Fi, 32.10.Hq, 32.80.Dz, 32.80Pj, 33.1 5.Pw, 34.20Cf, 34.50.-s Typeset using REVT EX ∗Present address: Department of Computing Science, Glasgow University, Glasgow G12 8QQ, UK 1I. INTRODUCTION Metastable helium has been the subject of many experimental investigations at cold and ultracold temperatures [1–26] and there have also been seve ral theoretical studies [27–32]. Much of this interest has been stimulated by the prospect of o btaining a Bose-Einstein condensate with spin-polarized metastable helium 23Satoms [2,14,30,31]. We denote a He(23S) atom by the symbol He∗. The Penning ionization (PI) and associative ionization (AI) processes, He∗+ He∗→/braceleftBigg He + He++ e−(PI) He+ 2+ e−(AI)(1) have high threshold rates in an unpolarized gas and limit the achievable density of trapped atoms. However, these autoionization processes are spin-f orbidden and suppressed [31,32] from the spin-polarized state and only via the weak spin-dip ole interaction can such processes can occur. Consequently, a suﬃcient number of spin-polariz ed metastable atoms should remain trapped. In addition, the scattering length associa ted with the quintet potential, which controls the collision dynamics of spin-polarized me tastable helium atoms, is predicted to be large and positive, a necessary requirement for a stabl e Bose-Einstein condensate. Although some theoretical studies [29–31] have estimated t he scattering length associated with the quintet potential to be large and positive, no detai led study of the scattering lengths for metastable helium has been previously undertaken. This present investigation not only calculates the possible ranges of values for the scattering lengths directly associated with the molecular potentials, but also reports experimentally obs ervable scattering lengths, elastic cross sections and inelastic rate constants for collisions of metastable helium atoms in the presence of a magnetic ﬁeld. For this theoretical investigation we have chosen to simula te the Penning and associative ionization processes that occur at small internuclear sepa rations from the singlet and triplet molecular states by a complex optical potential. The comple x interaction potentials then have the form2S+1V(R)−1 2i2S+1Γ(R), where Ris the internuclear separation of the two atoms,2S+1V(R) is the usual adiabatic molecular potential for the molecul ar state2S+1Σ+ g,u with total spin S, and2S+1Γ(R) is the corresponding total autoionization width represen t- ing ﬂux loss due to the ionization processes. Since the Penni ng and associative ionization processes are spin-forbidden from the quintet state,5Γ(R) = 0. The adiabatic molecular potentials required in this invest igation for the1Σ+ g,3Σ+ uand 5Σ+ gmolecular states were constructed using data from various s ources. The long-range interaction potential was described by a multipole expansi on of the form −C6/R6−C8/R8− C10/R10using the most accurate dispersion coeﬃcients available fo r the He(23S)–He(23S) interaction [33]. The short-range1Σ+ gand3Σ+ umolecular potentials and their corresponding autoionization widths for Penning and associative ionizat ion were obtained from M¨ uller et al.[35], while the short-range5Σ+ gmolecular potential was taken from St¨ arck and Meyer [34]. The5Σ+ gpotential was reported with an uncertainty of 0.5% in the rep ulsive part of the potential and 1% in the attractive part of the potential. The molecular potentials for metastable helium were constr ucted by ﬁtting the three short-range potentials smoothly onto the long-range dispe rsion interaction. The uncertain- ties in the short-range potentials, the procedure used to co nnect these to the long-range 2potential, and the form used for the autoionization widths l ead to uncertainties in the scattering lengths for the1Σ+ g,3Σ+ uand5Σ+ gpotentials and subsequently to the ultracold scattering properties of metastable helium atoms. To deter mine the extent of these uncer- tainties we vary the short-range potentials by ±2.5% for ﬁve diﬀerent interaction potentials which use diﬀerent methods to connect the long-range and sho rt-range potentials or have diﬀerent forms of the autoionization widths. We have chosen to vary the short-range poten- tial by more than their stated uncertainty to ensure that we o btain conservative estimates for the range of scattering lengths. The ﬁrst of these potentials, labeled (A), uses the short-ra nge5Σ+ gpotential ﬁtted smoothly onto the long-range potential at R≈20a0. The1Σ+ gand3Σ+ umolecular po- tentials of Ref. [35] are used for R <11.5a0and for larger R, where the electronic structure calculations become inaccurate, we replace the potentials by5V(R)−Vexch(R). The ex- change term has the form Vexch(R) =A2S+1Rγexp(−βR), where γ= 4.91249, β= 1.183933, A1= 6.3245×10−3andA3= 4.6317×10−3. The autoionization widths2S+1ΓM(R) of Ref. [35] were used to represent the Penning and associative ioni zation processes. Potential (B) is identical to (A) except that the short-rang e5Σ+ gform is ﬁtted to the long-range potential at R≈35a0. Potential (C) is identical to (A) except that the ex- change term has the form Vexch(R) =A2S+1exp(−βR) with β= 0.704921, A1= 4.29808 andA3= 3.14764. Potentials (D) and (E) are identical to (A) but employ diﬀerent forms for the autoionization widths. The autoionization width Γ GMS(R) = 0.3 exp(−R/1.086), given by Garrison et al. [36], is used in (D). This autoionization width has a steeper exponential form which doesn’t dampen at small internuclear separation s like1ΓM(R) or3ΓM(R). Po- tential (E) uses another alternative form of the autoioniza tion widths which was arbitrarily constructed to assess the sensitivity of the calculated res ults to the form of Γ( R) and is given by: Γ(R) =/braceleftBigg ΓGMS(R) + (R−6.5)2e−0.75R,forR≤6.5 ΓGMS(R), forR >6.5.(2) All the molecular potentials considered have the same long- range form since the uncertainties in the long-range multipole potential were found to have a ne gligible eﬀect on the scattering lengths. The potentials (A) to (E) with unmodiﬁed short-ran ge forms possess the same number of bound states, calculated to be 28 for1Σ+ g, 27 for3Σ+ uand 15 for5Σ+ g. II. SCATTERING LENGTHS ASSOCIATED WITH THE MOLECULAR POTENTIALS The scattering lengths for the1Σ+ g,3Σ+ uand5Σ+ gmolecular potentials were obtained by solving a single channel radial Schr¨ odinger equation of th e form, /braceleftbiggd2 dR2−l(l+ 1) R2−/bracketleftBig 2S+1V(R) −1 2i2S+1Γ(R)/bracketrightBig +k2/bracerightbigg uS,l(k, R) = 0 (3) in the limit where k→0. Here k=/radicalBig 2µE/¯h2,µis the reduced mass of the atomic system, Eis the total energy of the system and lis the relative rotational angular momentum. 3As a result of the complex interaction potential, the scatte ring equation (3) and the wave functions uS,l(k, R) are complex. Solution of this equation allowing for the non -unitarity of the Hamiltonian, and subsequent ﬁtting to free-ﬁeld bounda ry conditions provides a complex K-matrix and corresponding non-unitary S-matrix ( SS), as described previously [31]. The complex phase shift ηS, deﬁned by SS= exp( i2ηS), can then be used to calculate the complex scattering lengths a2S+1=are 2S+1−iaim 2S+1for each molecular state2S+1Σ+ g,u: are 2S+1=−1 2ktan−1/parenleftBiggSim S Sre S/parenrightBigg aim 2S+1=ln/parenleftBig SSS† S/parenrightBig 4k, (4) where the scattering lengths are deﬁned by ηS=−k a2S+1and the superscripts ‘re’ and ‘im’ denote real and imaginary components, respectively. The scattering lengths for the three molecular states were c alculated as a function of the percentage variation in the corresponding short-range molecular potential for the ﬁve potential cases (A) to (E) and are displayed in Fig. 1. For the a5scattering length only the results for potential (A) are plotted because the5Σ+ gpotentials are identical for potential cases (A), (C), (D) and (E) and the results obtained with pote ntial (B) diﬀered by less than 5%. The a5scattering length has no imaginary component since the Penn ing process is spin-forbidden from the S= 2 molecular state. Of particular interest is the resonance in a5at a percentage variation of ≈1.875 where the short-range potential is made suﬃciently shallow that a bound state is removed from the5Σ+ gpotential. For percentage variations >1.875 it is found that a5is negative in contradiction to recent experimental eviden ce that a5is large and positive [37]. With potentials (A) and (B) the sc attering lengths a1anda3 were nearly identical and are denoted by a single solid curve . The scattering lengths associated with the molecular poten tials are not observable ex- perimentally, with the exception of a5, which is approximately equal to the scattering length for the spin-polarized state. However, these scattering le ngths provide unique parameteri- zation of the1Σ+ g,3Σ+ uand5Σ+ gpotentials, from which the threshold scattering propertie s of metastable helium atoms can be obtained. Of more practica l interest are the scattering lengths for collisions between atoms in given atomic states in the presence of a magnetic ﬁeld. III. COLLISIONS IN THE PRESENCE OF A MAGNETIC FIELD To study collisions in the presence of a magnetic ﬁeld a full m ultichannel scattering calculation must be undertaken. The details of such a quantu m-mechanical multichannel scattering model for metastable helium is described elsewh ere [31]. In brief, we perform the present calculations for atoms in initial atomic states αandβ, including both sandd-waves, and calculate the full non-unitary S-matrix which has eleme ntsSα,β,l;α′,β′,l′. Here we let α andβdenote the atomic states ( s, m s), where msis the space-ﬁxed projection of the spin s for an individual atom. For collision energies up to 100 µK the contributions of entrance pandd-waves are negligible (note that due to symmetrization p-waves only contribute in collisions between 4atoms in diﬀerent atomic states), so that only the s-wave entrance channel [ α, β], l= 0 needs to be considered. The elastic cross section σel α,βand inelastic rate Kinel α,βare then given by [38] σel α,β=π k2\|1−Sα,β,l=0;α,β,l=0\|2 Kinel α,β=vπ k2/parenleftBig 1− \|Sα,β,l=0;α,β,l=0\|2/parenrightBig , (5) where vis the relative atomic velocity. In the Wigner threshold reg ion (ka << 1) one can deﬁne the scattering lengths using ηα,β=−kaα,βand obtain expressions for the observable scattering lengths aα,βby replacing SSwith the matrix element Sα,β,l=0;α,β,l=0in Eq. (4). The elastic cross sections and inelastic rates can then be ob tained using σel α,β= 4π/bracketleftBig (are α,β)2+ (aim α,β)2/bracketrightBig Kinel α,β= 4π aim α,β/k. (6) The inelastic rate Kinel α,βincludes both contributions from the ﬂux loss due to Penning ion- ization and that due to the atoms exiting in diﬀerent atomic s tates. Since we calculate the full S-matrix, the contributions of these two processes can be easily separated. We note that for (1,1) + (1 ,1),(1,1) + (1 ,0),(1,−1) + (1 ,−1) or (1 ,−1) + (1 ,0) collisions, where the total projection of the spin ( M) is non-zero, the collision is dominated by the5Σ+ gpotential and inelastic processes can only occur via the weak relativi stic spin-dipole interaction. The scattering lengths for these states are then almost identic al toa5but with a small imaginary component. The properties of (1 ,1)+(1 ,1) collisions were investigated in detail in a previous paper [31]. The inelastic rates for (1 ,0) + (1 ,0) and (1 ,1) + (1 ,−1) collisions, from which ionization can occur directly via strong exchange forces, a re much larger and dominate the total inelastic rate for an unpolarized gas. The (1 ,0) + (1 ,0) and (1 ,1) + (1 ,−1) inelastic rates contain two diﬀerent contributions. The ﬁrst is due to exothermic inelastic processes which incl udes the Penning rate KP α,β and the much smaller collision rate for exothermic ﬁne-stru cture changing collisions Kex α,β. The second is the rate for degenerate ﬁne-structure changin g collisions Kdeg α,β. For example, in ultracold (1 ,0) + (1 ,0) collisions the entrance channel [(1 ,0) + (1 ,0)], l= 0 can decay exothermically to the three channels [(1 ,−1)+(1 ,−1)], l= 2; [(1 ,0)+(1 ,−1)], l= 2 and the Penning channel, and to the two degenerate channels [(1 ,1) + (1 ,−1)], l= 0 and [(1 ,1) + (1,−1)], l= 2. The ﬂux loss to the degenerate d-wave exit channels or exothermic d-wave exit channels (ie KP) only occurs via weak spin-dipole forces and is at least 3 ord ers of magnitude smaller than that lost to the Penning channel or to degenerat el= 0 exit channels that occurs through strong exchange forces. Importantly, exothermic a nd degenerate inelastic processes exhibit diﬀerent threshold properties. Exothermic inelas tic rates tend to a constant in the Wigner threshold region whereas degenerate inelastic rate s fall oﬀ as 1 /ksince, as for elastic processes, the incident and ﬁnal wave number are identical. To represent these separate threshold behaviors in the inelastic rates, we write aim α,β=aimex α,β+k aimdeg α,β. The slope and intercept of ln( Sα,β,l=0;α,β,l=0S† α,β,l=0;α,β,l=0)/4kvsk, forkin the Wigner threshold region, then gives the degenerate and exothermic scattering length saimdeg α,βandaimex α,β, respectively. We have calculated these imaginary and the real scattering l engths for all possible col- lision processes in spin-polarized metastable helium for t he ﬁve diﬀerent potentials under 5investigation. From these calculated scattering lengths o ne can use Eq. (6) to calculate the partial rates or the total rates in an unpolarized gas at t emperatures where the Wigner threshold approximation is valid. The scattering lengths a re calculated assuming a magnetic ﬁeld of 10 Gauss, however we ﬁnd only a weak dependence on the m agnetic ﬁeld and results for ﬁelds in the range 0 to 20 Gauss diﬀer by less than 1%. The sc attering lengths can be used to calculate the rates and cross sections up to typicall y≈100µK, except where the scattering lengths become >1000a0. Scattering lengths for (1 ,0) + (1 ,0) and (1 ,1) + (1 ,−1) collisions (with sandd-waves) depend on both the5Σ+ gand1Σ+ gpotentials and so their scattering lengths are a function of both the percentage variation of the short-range5Σ+ gand1Σ+ gpotentials for potential cases (A) to (E). However, we ﬁnd that for a given percentage variat ion of the5Σ+ gpotential the uncertainty in the scattering lengths induced by varying th e short-range1Σ+ gpotential by ±2.5% is similar to that calculated by ﬁxing the percentage vari ation in the1Σ+ gpotential to zero and using the ﬁve diﬀerent potential cases (A) to (E). In all instances the percentage variation in the5Σ+ gpotential has the largest eﬀect on the scattering lengths an d resulting rates. The (1 ,1) + (1 ,1),(1,1) + (1 ,0),(1,−1) + (1 ,−1) or (1 ,−1) + (1 ,0) interactions depend only weakly on the singlet potential via the weak rela tivistic spin-dipole interaction and we ﬁnd that varying the short-range1Σ+ gpotential for these collisions produces negligible changes in the scattering length. Therefore we only report s cattering lengths as a function of the percentage variation in the5Σ+ gpotential for potential cases (A) to (E), with the understanding that similar uncertainties result in the (1 ,0) + (1 ,0) and (1 ,1) + (1 ,−1) scattering lengths by varying the short-range singlet pote ntial. Figures 2, 3 and 4 show the real and imaginary scattering leng ths for (1 ,1)+(1 ,1),(1,0)+ (1,0) and (1 ,1) + (1 ,−1) collisions. The real scattering lengths all possess a res onance in the region where a bound state is removed from the5Σ+ gpotential and a5goes through ±∞. Similar plots exist for (1 ,1) + (1 ,0),(1,−1) + (1 ,−1) and (1 ,−1) + (1 ,0) collisions but are almost identical to that shown in Fig. 2 for (1 ,1) + (1 ,1) since all are dominated by the 5Σ+ gpotential. The underlying5Σ+ gpotentials are identical for potential cases (A), (C), (D) and (E) and we found that are (1,1),(1,1)calculated with these potentials diﬀered from those obtained using potential (B) by less than 2%. These small diﬀ erences are not observable on the scale used in Fig. 2 and so for clarity a single solid curve is used to represent are (1,1),(1,1) for the ﬁve potential cases. Similarly for aimex (1,1),(1,1)the results were identical except for cases (D) and (E) where diﬀerent forms of the autoionisation width s were used, and so we show only results for (A), (D) and (E) potential cases. We note tha t imaginary scattering lengths for collisions where the total spin projection is non-zero p ossess no degenerate component and the exothermic contributions are negligible when compa red to those for (1 ,0) + (1 ,0) and (1 ,1) + (1 ,−1) collisions where Penning ionization can occur via exchan ge forces. For (1 ,0)+ (1 ,0) and (1 ,1)+ (1 ,−1) collisions arein Fig. 3 were almost identical for the ﬁve potential cases and are represented by a single solid cur ve for (1 ,0)+(1 ,0) and a dashed curve for (1 ,1)+(1 ,−1). In Fig. 4 we show aimexandaimdegfor these collisions. The scattering lengths aimex, which measure KP α,β+Kex α,β, are independent of the percentage variation in the5Σ+ gpotential and thus a5, except very near the a5resonance where the contribution fromKex α,βis no longer negligible and a small increase in aimex α,βis observable. Therefore, the measurement of the ionization signal from trapped metas table helium atoms does not provide information about a5, the parameter which is required to make predictions of the 6formation or properties of a Bose condensate of spin-polari zed metastable helium atoms. If Kex α,βis neglected then a simple examination of the Hamiltonian sh ows that 2 KP (1,0),(1,0)= KP (1,1),(1,−1). We have veriﬁed that this relation is valid to better than 1% and so in Fig. 4 we plot results for aimex (1,0),(1,0)for the ﬁve potential cases with the understanding that 2 aimex (1,0),(1,0)= aimex (1,1),(1,−1). The curves labeled aimdegin Fig. 4 provide the degenerate temperature-dependent in- elastic rates for either (1 ,0) + (1 ,0)→(1,1) + (1 ,−1) or (1 ,1) + (1 ,−1)→(1,0) + (1 ,0). These equal, exchange-dominated rates strongly mix the (1 ,1), (1,0) and (1 ,−1) atoms and are equal to, or larger than, KPat temperatures greater than 500 µK or when the quintet potential is near resonance. Of the potentials tested only t hose with very diﬀerent exchange terms provided signiﬁcantly diﬀerent results and conseque ntlyaimdegfor potentials (A), (B), (D) and (E) were nearly identical. For convenience only aimdeg (1,0),(1,0)for potentials (A) and (C) have been plotted in Fig.4. The elastic cross section depends on the real and imaginary s cattering lengths and its measurement in a spin-polarized or unpolarized gas may prov ide useful information on a5. In Figs. 5–8 we provide the total elastic cross sections and P enning ionization rates for (1,1) + (1 ,1) collisions and for an unpolarized gas calculated using Po tential (A). Also shown are unthermalized results for 1 µK and 500 µK calculated from the scattering lengths using Eq.(6). In general the results obtained using Eq.(6) f or temperatures up to 100 µK are identical to the thermalized results whereas at higher t emperatures, outside the Wigner regime, the use of scattering lengths is inappropriate and t hermalization is required. The rate equations ∂nα ∂t=Kinel α,αn2 α ∂nα ∂t=Kinel α,βnαnβ (7) deﬁne our unthermalized partial rates for, respectively, i dentical and non-identical atom collisions, where nαis the number of colliding atoms in state αand the superscript ‘inel’ denotes ‘P’, ‘ex’ or ‘deg’. The total thermalized Penning ra tes and cross sections for an unpolarized gas are obtained assuming an equal population o f thes= 1 ﬁne structure levels so that nα=n/3 and hence ∂n ∂t=1 9/summationdisplay α,β/angbracketleftKP α,β/angbracketrightn2 ∂n ∂t=1 9/summationdisplay α,β/angbracketleftvσinel α,β/angbracketright /angbracketleftv/angbracketrightn2, (8) where /angbracketleft.../angbracketrightdenotes the thermal average. In this case the assumption tha t the ﬁne structure levels are evenly populated in an unpolarized gas is well jus tiﬁed on collisional grounds. At temperatures above 500 µK the degenerate rates Kimdegevenly mix (1 ,1), (1,0) and (1 ,−1) atoms. At lower temperatures the Penning rates KP (1,0),(1,0)andKP (1,1),(1,−1), which dominate the exothermic inelastic rates, deplete the three diﬀerent atomic populations nαequally since 2KP (1,0),(1,0)=KP (1,1),(1,−1)and the collision of (1 ,0) + (1 ,0) results in the loss of two (1 ,0) atoms. Here we have neglected the small contribution from sp in-dipole processes, that is 7Kex (1,1),(1,−1), and assume that any initial asymmetry in the populations nαdue to preparation of the atoms in a light ﬁeld for instance is small or has become small once the measurement of the collisional rate in the absence of light is performed. The thermalized results were calculated by averaging over a Maxwell-Boltzmann distribu tion of atomic velocity using 71 velocity nodes which correspond to collision energies in th e range 0 .01µK to 10 ,000µK. Since the results are for the case (A) potentials, with the pe rcentage variation in the singlet potential set to zero, we estimate from the uncertainties in the scattering lengths that the errors in the elastic cross sections and total inelastic rat es are of the order of 10% and 40% respectively. The Penning rates possess a larger uncertain ty to account for the percentage variation of the1Σ+ gpotential whereas the unpolarized elastic rates, which are dominated by the real scattering lengths belonging to collisions with M= 2 or 1, are controlled only by5Σ+ g. In an unpolarized gas the p-waves can contribute in (1 ,1) + (1 ,−1), (1,1) + (1 ,0) and (1,−1) + (1 ,0) collisions. These contributions to the total thermalize d Penning rates were found to be negligible at 1 µK. However the p-wave contributions increased the total Penning rate (compared to that obtained using only s-waves) by approximately 7% at 500 µK and 12% at 1 mK. The p-waves modiﬁed the total elastic cross sections by less than 1% at all temperatures. For (1 ,1)+(1 ,1) collisions (and similarly for (1 ,1)+(1 ,0),(1,−1)+(1 ,−1) or (1 ,−1)+ (1,0)) we observe a resonance in the inelastic rates at a percent age variation of +1.875 due to the resonant enhancement of the exothermic rates. We ﬁ nd that KP α,β> Kex α,β, indicating that most but not all of the ﬂux leaving the [(1 ,1) + (1 ,1)], l= 0 entrance channel is subsequently lost through ionization. These rat es are much smaller than those from the (1 ,0)+(1 ,0) and (1 ,1)+(1 ,−1) collisions and the total contribution to KP α,βfrom (1,1)+(1 ,1),(1,1)+(1 ,0),(1,−1)+(1 ,−1) and (1 ,−1)+(1 ,0) collisions is only observable in Fig. 8 as a small peak at +1.875 in the unpolarized ionizati on rate. The total elastic cross sections of an unpolarized or a polar ized gas show strong depen- dences on the form of the5Σ+ gpotential and provide possible measures of a5. IV. CONCLUSIONS The scattering lengths associated with the three molecular potentials relevant to colli- sions of metastable helium atoms have been reported. The unc ertainties in the molecular potentials and autoionization widths have been considered and probable ranges of values given for the scattering lengths for each molecular state. S cattering lengths for collisions involving the various atomic states have also been calculat ed and related to the elastic cross sections and inelastic collision rates for temperatures in the Wigner threshold region, with the aim of providing a correspondence with experimentally m easurable quantities. In par- ticular, it has been shown that measurement of the total elas tic cross section in a polarized or unpolarized gas should provide a means of experimentally determining the a5scattering length, which is of importance in the attainment of a Bose-Ei nstein condensate in a gas of spin-polarized metastable helium atoms. In Fig. 8 we compare the total Penning rates calculated here t o those from experiment. Not shown are the theoretical uncertainties of ≈40% which arise from uncertainties in the molecular potentials and in the form of the autoionization w idths. The total elastic cross 8sections and Penning rates are consistent with those report ed in Ref. [32] where slightly diﬀerent molecular potentials and autoionization widths w ere used. The experimental results possess uncertainties on the order of 50% which are not shown in Fig. 8. The experimental results correspond to the case of zero magnetic ﬁeld whereas the theoretical predictions are made for B= 10 G. However the scattering lengths were found to vary by le ss than 1% over the range 0–20 G which is negligible when compared to t hese uncertainties that arise from the form of the autoionization width. The compari son between theoretical and experimental data is satisfactory given these uncertainti es, however the experimental results are consistently higher than the theoretical predictions. Finally, using the scattering lengths reported in this inve stigation, one can estimate the scattering lengths for the other isotopes of helium by mass s caling the vibrational defect. This is related to the scattering length by [39] a2S+1=−∂ν ∂κ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle κ→0/bracketleftbigg cot(π t−2) + cot( νS(0))/bracketrightbigg , (9) where νS(0) is the vibrational defect and tis deﬁned by the leading term −Ct/Rtin the long- range potential ( t= 6 for He). The term∂ν ∂κ\|κ→0is an asymptotic property which depends only on the long-range potential and can be approximated by 0 .956×0.5(2µC6)0.25≈35 for He [40]. To mass scale the vibrational defect we ﬁrst calc ulateνS(0) for4He for a given potential. Since the trigonometric function is periodic, t his only gives the fractional part of the vibrational defect and one must include the multiple of nπwhere nis the number of bound states supported by that potential, ie νS(0)→nπ+νS(0). This vibrational defect can then be scaled using ( µx/µ4)0.5×νS(0), to determine the vibrational defect for isotope x. Here µxandµ4are the reduced masses ofxHe and4He respectively. ACKNOWLEDGMENTS VV acknowledges partial support from the Engineering and Ph ysical Sciences Research Council. The Institute for Theoretical Atomic and Molecula r Physics is supported by a grant from the NSF to the Harvard College Observatory and the Smithsonian Astrophysical Observatory. 9REFERENCES [1] A. Aspect et al., Phys. Rev. Lett. 61, 826 (1988). [2] H. Metcalf, J. Opt. Soc. Am. B 6, 2206 (1989). [3] A. Aspect et al., J. Opt. Soc. Am. B 6, 2112 (1989). [4] A. Aspect et al., Chem. Phys. 145, 307 (1990). [5] C. Westbrook et al., inTENICOLS ’91, papers presented at Tenth International Con- ference on Laser Spectroscopy, France, June 1991 , edited by E. G. M. Ducloy and G. Camy (World Scientiﬁc, Singapore, 1991), pp. 48–9. [6] N. Morita and M. Kumakura, Jpn. J. App. Phys. 30, L1678 (1991). [7] M. Kumakura and N. Morita, Jpn. J. App. Phys. 31, L276 (1992). [8] F. Bardou et al., Europhys. Lett. 20, 681 (1992). [9] J. Lawall et al., Phys. Rev. Lett. 73, 1915 (1994). [10] J. 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Nowak et al., Appl. Phys. B: Lasers Opt. 70, 1 (2000). [24] A. Browaeys et al., Euro. Phys. J. D 8, 199 (2000). [25] N. Herschbach, P. J. J. Tol, W. Hogervorst, and W. Vassen , Phys. Rev. A 61, 050702(R) (2000). [26] N. Herschbach et al., Phys. Rev. Lett. 84, 1874 (2000). [27] P. S. Julienne and F. H. Mies, J. Opt. Soc. Am. B 6, 2257 (1989). [28] C. J. Williams and P. S. Julienne, Theory of Penning Ioni zation of Metastable He, 1992, presented at the Symposium on Cold Atom Collisions, Ca mbridge, MA, 1992 (unpublished). [29] G. V. Shlyapnikov, J. T. M. Walraven, U. M. Rahmanov, and M. W. Reynolds, Phys. Rev. Lett. 73, 3247 (1994). [30] P. O. Fedichev, M. W. Reynolds, U. M. Rahmanov, and G. V. S hlyapnikov, Phys. Rev. A53, 1447 (1996). [31] V. Venturi, I. B. Whittingham, P. J. Leo, and G. Peach, Ph ys. Rev. A 60, 4635 (1999). [32] V. Venturi and I. B. Whittingham, Phys. Rev. A 61, 060703(R) (2000). [33] Z.-C. Yan and J. F. Babb, Phys. Rev. A 58, 1247 (1998). [34] J. St¨ arck and W. Meyer, Chem. Phys. Lett. 225, 229 (1994). 10[35] M. W. M¨ uller et al., Z. Phys. D. - Atoms, Molecules and Clusters 21, 89 (1991). [36] B. J. Garrison, W. H. Miller and H. F. Schaefer, J. Comput . Phys. 59, 3193 (1973). [37] W. Vassen (2000) (Private Communication) [38] L. D. Landau and E. M. Lifshitz , Quantum Mechanics , Vol 3, 3rd ed. (Pergamon Press, Oxford, 1977). [39] F. H. Mies et al., J. Res. Natl. Inst. Stand. Technol. 101, 521 (1996). [40] G. F. Gribakin and V.V. Flambaum, Phys. Rev. A 48, 546 (1998). 11FIGURES FIG. 1. Real and imaginary components of the scattering leng thsa1,a3anda5plotted against variation in the short-range potential. For a5the ﬁve potential cases produced similar results and are encompassed by the solid curve with a dashed line to denot e the position of the resonance. Fora1anda3, potentials (A) and (B) produced identical results denoted by (—), potential (C) by (· · ·), potential (D) by (- - -) and potential (E) by ( − · −). 12FIG. 2. Complex scattering lengths for (1 ,1) + (1 ,1) collisions. The solid line includes the real components of the scattering lengths obtained from all ﬁve p otential cases. The dashed and dotted lines give the imaginary components of the scattering lengt hs. Results for potentials (A), (B) and (C) are given by ( · · ·), potential (D) by (- - -) and potential (E) by ( − · −). 13FIG. 3. Real components of the scattering lengths for (1 ,0)+(1 ,0) and (1 ,−1)+(1 ,1) collisions. The solid line represents the results for (1 ,0)+(1 ,0) collisions for all ﬁve potential cases, the dashed line includes results for (1 ,−1) + (1 ,1) collisions for all ﬁve potential cases. 14FIG. 4. Imaginary components of the scattering lengths for ( 1,0) + (1 ,0) and (1 ,−1) + (1 ,1) collisions. The near horizontal lines are for (1 ,0) + (1 ,0) collisions with results for potentials (A) and (B) encompassed by the solid curve, potential (C) by ( ···), potential (D) by (- - -) and potential (E) by ( − · −). Note that 2 aimex (1,0),(1,0)=aimex (1,−1),(1,1).aimdegfor (1,0) + (1 ,0) and (1 ,−1) + (1 ,1) collisions are equal and results for potentials (A),(B),(D ) and (E) are encompassed by the solid curve and those for potential (C) by the dotted curve. 15FIG. 5. Thermalized elastic cross section for (1 ,1) + (1 ,1) collisions with potential (A) at various temperatures denoted by (—) for 1 µK, (- - -) for 500 µK, (− · −) for 1000 µK. Results for 1µK and 500 µK calculated using the scattering lengths are denoted by ✷and/circlecopyrtrespectively. 16FIG. 6. Thermalized and unthermalized inelastic rates for ( 1,1) + (1 ,1) collisions for potential (A) with curves and symbols labeled using the same scheme as i n Fig. 5. Thick lines labeled KP denote the Penning rate KPand the thinner lines labeled KP+Kexgive the total inelastic rate. 17FIG. 7. Thermalized elastic cross sections for an unpolariz ed gas with curves labeled as per Fig.5. 18FIG. 8. Thermalized Penning rates for an unpolarized gas wit h curves labeled as per Fig.5. The theoretical predictions possess an error of ≈40% and the experimental results have uncertainties on the order of 50%. Experimental results are denoted by △for [20], /circlecopyrtfor [17], ✷for [21] and ⋄ for [26]. 19
arXiv:physics/0011073v1 [physics.gen-ph] 30 Nov 2000Electric Charge as a Vector Quantity Gerald L. Fitzpatrick PRI Research and Development Corp. 12517 131 Ct. N. E. Kirkland, WA 98034 425–820–1905 glﬁtzpatrick@yahoo.com Abstract Starting with the premise that the electric charge associat ed with fun- damental fermions (quarks and leptons) can, under certain c ircum- stances, be appropriately represented as a real internal 2-vector, the mathematical “machinery” implicit in the associated inter nal 2-space is shown to apply to allfundamental fermions. In particular, it is shown that ﬂavor eigenstates ,ﬂavor doublets andfamilies of funda- mental fermions can all be represented in the 2-space, and th at such things as internal colors,family replication , and the observed number (three) of families, are more-or-less implicit in the new 2- space de- scription. Moreover, the model predicts that, unlike the ca se in the standard model, particles such as the u,candtquarks are character- ized by signiﬁcant internal (topological and other) diﬀere nces. Similar diﬀerences may help explain recent observations of (nearly ) maximal νµ−ντmixing. 1 Introduction If quantum eﬀects are ignored it is completely appropriate t o treat elec- tric charge in external spacetime as a scalar quantity. In particular, in 4- dimensional spacetime, the electric charge carried by an is olated particle is appropriately treated as a Lorentz invariant 4-scalar [1]. However, when 1quantum eﬀects are taken into account it is not at all clear th at the internal description of electric charge should be limited to scalar q uantities. The pur- pose of this paper will be to argue that in the case of fundamen tal fermions (quarks and leptons) there are good reasons for treating ele ctric charge as a vector quantity associated with a new (abstract) internal 2-space [2]. The motivation for taking such an unusual step is that it will be s hown to lead to anextension of the standard model description of quarks and leptons. Before embarking on this exploration, it is appropriate to b rieﬂy review the conventional description of ﬂavor doublets of fundamen tal fermions. Cer- tain aspects of the conventional description via the SU(2) isospin formalism will suggest the new description. 1.1 The conventional description of ﬂavor doublets According to the standard model of particle physics, all lef t-handed quarks and leptons (right-handed antiquarks and antileptons) are members of SU(2) weak isospin doublets [3]. To be speciﬁc let us limit the present discussion to the ﬁrst-family quark states \|u/an}bracketri}htand\|d/an}bracketri}ht, which constitute an SU(2) weak isospin doublet (also called a ﬂavor doublet). These states are properly to be thought of as being two diﬀerent (isospin) states of a single quark ﬁeld [4]. Using the conventional isospin language, there exists an is ospin (vector) operator τ=1 2σ, where σ= (σ1,σ2,σ3) is a vector form whose components are the familiar Pauli matrices σ1,σ2andσ3. All physically observable states carrying (weak) isospin (e.g., \|u/an}bracketri}htand\|d/an}bracketri}ht) must be simultaneous eigenstates of the square of the total isospin vector τ2=τ2 1+τ2 2+τ2 3, and the third- component of isospin τ3, i.e., they take the form \|τ2, τ3/an}bracketri}ht. The states \|u/an}bracketri}htand \|d/an}bracketri}htare said to span a two-dimensional Hilbert space, and to cons titute a 2-dimensional representation D(1/2)ofSU(2). In particular, given that the states \|u/an}bracketri}htand\|d/an}bracketri}htare eigenstates of τ3=1 2σ3, (1) where σ3=/parenleftbigg 1 0 0−1/parenrightbigg , (2) we have the eigenvalue equations (use τ3\|τ2, τ3/an}bracketri}ht=τ3\|τ2, τ3/an}bracketri}htorτ3\|3 4,±1 2/an}bracketri}ht= 2±1 2\|3 4,±1 2/an}bracketri}htand/or the column-vector forms \|u/an}bracketri}ht={1,0}and\|d/an}bracketri}ht={0,1}) τ3\|u/an}bracketri}ht= +1 2\|u/an}bracketri}ht τ3\|d/an}bracketri}ht=−1 2\|d/an}bracketri}ht/bracerightBigg . (3) Now the electric charge of the quark ﬁeld in either of the states \|u/an}bracketri}htor\|d/an}bracketri}ht is given by operating on these states with the operator for el ectric charge, which may be expressed in this particular case as QQQ=τ3+1 6. (4) Then we have for the electric charges of these twopossible states of the quark ﬁeld QQQ\|u/an}bracketri}ht=q1\|u/an}bracketri}ht (5) and QQQ\|d/an}bracketri}ht=q2\|d/an}bracketri}ht, (6) where q1= +2 3andq2=−1 3(7) are the speciﬁc electric charges in question. While the foregoing description is certainly correct as far as it goes it is, nevertheless, unnecessarily restrictive. In particul ar, we will show in the next section that the foregoing properties of ﬂavor doublet s are suggestive of a (complementary) extension of this conventional descript ion, which treats electric charge as a real vector quantity in a new internal 2-dimensional linear vector space. 1.2 Electric charge as a vector quantity It is implicit in the description of SU(2) isospin doublets that if one knows the state \|u/an}bracketri}htone can (must) infer the existence of a second state \|d/an}bracketri}ht, and vice versa. The 2 ×2 matrix form of the charge operator QQQ, and the isospin operator τ3makes this abundantly clear. 3If one has a state with τ3= +1 2, and one knows that one is dealing with anSU(2) isospin doublet ﬁeld, then one must also have a state with τ3=−1 2. As a consequence of these simple facts, one can have simultaneous knowledge of the two electric charges q1andq2associated with the two states \|u/an}bracketri}htand \|d/an}bracketri}ht, respectively, given either one of the states \|u/an}bracketri}htor\|d/an}bracketri}ht. In a certain sense then, there could be a physical meaning to a geomet- ric object deﬁned by the ordered pair of observable (real num bers)electric charges associated with the states \|u/an}bracketri}htand\|d/an}bracketri}ht, namely, a real2-vector or “charge vector” Q={q1, q2}. (8) However, it should be stressed that if the 2-vector Qwere to be “carried” by, or “associated” with, each of the states \|u/an}bracketri}htand\|d/an}bracketri}htthere would be deﬁnite nontrivial physical consequences. Clearly, the assignment of Qto\|u/an}bracketri}htand\|d/an}bracketri}htwould mean that these states carryadditional information (besides isospin) in the quantum sense [5]. In particular, because the 2-vector Qhas to “live” in some abstract internal 2-space, all of the mathematical “machinery” associated wi th this 2-space would have to be taken into account (e.g., the 2-space metric and various in- ternal transformations in the 2-space) when describing fun damental fermions. In short, this new description would promise a far richer internal structure than that implied by the description of ﬂavor doublets using SU(2) alone. This implied richness constitutes nothing less than a possi bleextension of the standard model [3, 6], and encourages us to seriously con sider the idea of representing electric charge (of fundamental fermions) by an internal 2- vector. In the next section we will show how this idea leads naturally to, among other things, an explanation for ﬂavor doublets in other fam ilies, i.e., to an explanation for family replication . 2 Consequences of Treating Electric Charge as a 2-Vector In this section we derive a number of consequences of applyin g the 2-space mathematical machinery to fundamental fermions. Many of th e results pre- sented here were arrived at in earlier works by a somewhat diﬀ erent route. 4In particular, in [7] we began by generalizing the scalar fermion number fto a 2×2 real, generally non-Hermitian matrix F(i.e.,f→F), and in [8] we arrived at this same matrix Fby analytically continuing the fermion number operator F(op)→F. In the present paper, by contrast, we begin by generalizing electric charge (call it e) from a scalar to a 2- vector Q(i.e.,e→Q) or “charge vector.” Only later do we arrive at the matrix Fdescribed above. The interested reader is encouraged to consult the indicated references fo r further details regarding these earlier works. 2.1 The 2-space metric If the 2-vector Q={q1, q2}is assigned to eachof the matter states \|u/an}bracketri}htand \|d/an}bracketri}ht, then there must exist a vector (call it Qc) that is assigned to eachof the corresponding anti-matter states \|u/an}bracketri}htand\|d/an}bracketri}ht, respectively. Recalling the deﬁnition of the charge vector Qgiven in Sec. 1.2 (see Eq. 8), the (anti) charge vector Qcmust be formed in some way from the ordered pair of real numbers −q1and−q2corresponding, respectively, to the electric charges of the \|u/an}bracketri}htand\|d/an}bracketri}htantimatter states. Now, assuming that the scalar product of QandQc, namely, Q·Qcshould vanish ( Q andQcshould be orthogonal) so as to insure that matter and antimat ter states can be distinguished in the 2-space, it follows that no matter what the metric is, as long as it is realandﬂat,QandQcmust also be linearly independent . To see this, notice ﬁrst that if QandQcarenotlinearly independent, then they are, by deﬁnition, necessarily, linearly dependent , in which case Qc=−Q={−q1,−q2}. Assuming that the 2-space metric gisrealandﬂat,gcan be represented by the 2 ×2 matrix g= (g11= 1, g22=s, g12=g21= 0) or g=/parenleftbigg1 0 0s/parenrightbigg , (9) where \|s\|= 1. Then, given A·B=2/summationtext i,j=1gijaibj, where A= (a1, a2) is a row- vector and B={b1, b2}is a (conformable) column-vector, the scalar product ofQandQc=−Q(Note that q1is never equal to q2because these charges correspond to diﬀerent eigenstates of the electric charge operator QQQ; see Eqs. 55–7) is, from (8) and (9), Q·Qc= (q1, q2)/parenleftbigg−q1 −q2/parenrightbigg =−q2 1−sq2 2, (10) which is generally nonzero for any \|s\|= 1, i.e., s=±1 (also see Footnote 9). Therefore, if QandQcare to be orthogonal (Q·Qc= 0), the fact that (10) is generally nonzero ensures that QandQcmust be linearly independent . In this case, given Q={q1, q2}, it must be true that Qc={−q2,−q1}, (11) which is notproportional to Q, as required to ensure linear independence . Finally, given the general form for the metric (9), and the li near indepen- dence (and orthogonality) of QandQc, one has the following result for s Q·Qc= (q1, q2)/parenleftbigg−q2 −q1/parenrightbigg =−q1q2−sq1q2= 0, (12) if, and only if, s=−1. Therefore, the 2-space metric is given by g=/parenleftbigg 1 0 0−1/parenrightbigg , (13) and we see that the requisite 2-space is, necessarily, “Lore ntzian” or non- Euclidean [10]. 2.1.1 Scalar products of 2-vectors Using the metric given in (13), and the general formula for A·Bgiven in Sec. 2.1, we immediately have the scalar product of two diﬀerent real 2-vectors (a, b){e, f}=ae−bf. (14) Similarly, the square of a real 2-vector is given by (a, b){a, b}=a2−b2. (15) Here we remind the reader that ( ,) is a rowvector while {,}is a (conformable) column vector. Clearly, the scalar products in (14) and (15) transform like charge-conjugation-reversing or C-reversing (2-scalar) charges . 6For example, using (8), (11) and (15) we immediately have the square of the 2-vectors QandQc, namely, Q2=Q·Q=q2 1−q2 2 (16) and (Qc)2=Qc·Qc=q2 2−q2 1. (17) Therefore, Q2=−(Qc)2, (18) which means that Q2and (Qc)2each transform like C-reversing 2-scalar charges [9]. It happens that these particular charges can be identiﬁed wi th the baryon - orlepton -number carried by quarks or leptons, respectively (see Ref. 7, p. 72). We will see in a later section that when charge vectors su ch asQ(orQc) areresolved in the 2-space into pairs of linearly independent vectors (e .g., Q=U+V), not only are the components ofQ,UandV,C-reversing charges , but also given Q2=U2+ 2U·V+V2, (19) U2, 2U·VandV2are, like Q2,C-reversing charges. The foregoing collection of 2-scalar charges will be used to deﬁne and describe ﬂavor eigenstates ,ﬂavor doublets , and eventually families of fundamental fermions. 2.1.2 A conjectured “duality” Given the number of ﬂavors of quarks and leptons, and an appro priate (renormalizable) Lagrangian, the so-called “accidental s ymmetries” of the Lagrangian [11] are known to “explain” the separate conserv ation of vari- ous (global) ﬂavor-deﬁning (Lorentz 4-scalar) “charges” [ e.g., lepton num- ber, baryon number, strangeness, charm, the third-compone nt of (strong or global) isospin, truth, beauty, electron-, muon-, and tau- numbers]. Now, as demonstrated in detail in Ref. 7, pp. 67–71, given certain re alC-reversing scalars—components of various vectors and matrices deﬁned on the inter- nal non-Euclidean 2-space, and various scalar products of 2 -vectors—it is possible to describe the ﬂavor eigenstates of fundamental f ermions [12]. 7In principle, what one does is to identify the mutually-comm utingC- reversing 2-space “charges” (call them Ci) or charge-like quantum numbers associated with a particular ﬂavor, and then write the corre sponding simul- taneous ﬂavor-eigenstate as \|C1, C2, C3, . . . , C n/an}bracketri}ht. (20) HereC1, C2, C3, . . . , C n, are said to be the “good” charge-like quantum num- bers (charges) associated with a particular ﬂavor [13]. Now it also happens that these observable real numbers can be identiﬁed with quantum numbers such as electric charge, lepton number, baryon number, strangenes s, charm , the third-component of (strong or global) isospin ,truth andbeauty (see Ref. 7, p. 72). In short, these 2-space charges look very much like those associated with the aforementioned “accidental symmetries” of the Lag rangian ! The foregoing properties of the non-Euclidean charge-like sca lars, leads naturally to the following “duality” conjecture: The global (ﬂavor-deﬁning )charges associated with the “accidental sym- metries” of the Lagrangian describing strong and electrowe ak interactions, and the global (ﬂavor-deﬁning )charges associated with the non-Euclidean 2-space, are (essentially )one and the same charges. 2.2 Charge conjugation in the 2-space Given that there are numerous C-reversing scalars in the 2-space (see Sec. 2.1.1), there must exist a 2 ×2 matrix, call it X, that serves to transform thesescalars , various 2- vectors such as QorQc, and various 2 ×2matrices , to their corresponding C-reversed (2-space) counterparts. In particular, a matrix Xshould exist such that (use Eqs. 8 and 11) XQ=Qc(21) and XQc=Q. (22) From (21) and (22) it follows that Xmust equal its multiplicative inverse (X=X−1), and thus X2=I2, (23) 8where I2is the 2 ×2 identity matrix. WriteXin the general form ( Xis real) X=/parenleftbigga b c d/parenrightbigg , (24) and consider the situation where one of the two charges (say q2) associated withQ={q1, q2}iszero, and the other charge ( q1) isnonzero (this is actually the case for leptons). In this particular case, we have (use Eqs. 8, 11, 21 and 24) /parenleftbigga b c d/parenrightbigg /parenleftbiggq1 0/parenrightbigg =/parenleftbigg0 −q1/parenrightbigg , (25) which means that aq1= 0 (26) and cq1=−q1. (27) And therefore, since q1/ne}ationslash= 0, it must be true from (26) and (27) that a= 0 andc=−1. SinceXQc=Qit must also be true that /parenleftbigg0b −1d/parenrightbigg /parenleftbigg0 −q1/parenrightbigg =/parenleftbiggq1 0/parenrightbigg , (28) which means that −bq1=q1 (29) and −dq1= 0. (30) Finally, since q1/ne}ationslash= 0 it must be true from (29) and (30) that b=−1 and d= 0. Collecting the foregoing matrix elements, we have X=/parenleftbigg 0−1 −1 0/parenrightbigg , (31) 9or X=−σX, (32) where σXorσ1is one of the familiar Pauli matrices. In general, the matrix X=−σXshould apply (in the 2-space) to 2- scalars , 2-vectors , 2×2matrices , and to both quarks andleptons . 2.2.1 Transformation of the metric and other matrices under X Any 2 ×2 matrix M, appropriate to the 2-space description of fundamental fermions, should transform under X=−σXto itsC-reversed counterpart Mcaccording to the similarity transformation X M X−1=Mc, (33) or because X=X−1=−σX, equivalently as (−σX)M(−σX) =Mc. (34) For example, the metric g(see Eq. 13) is found to be C-reversing since (−σX)g(−σX) =−g. (35) A matrix that is C-invariant (e.g., the matrix X) would, necessarily, have the form N=/parenleftbigg a b b a/parenrightbigg , (36) where it is clear that (−σX)N(−σX) =N. (37) Now let us apply the foregoing similarity transformation to the matrix F, which represents the generalized fermion number in this 2-space [see Ref. 7, pp. 4–12]. 102.3 The generalized fermion number F (v) There are other 2 ×2 matrices besides Xthat act on the 2-vectors Qand Qc. Let us deﬁne a matrix Fwhose eigenvalues fare the fermion numbers fm= +1for matter, and fa=−1for antimatter, and whose eigenvectors are the electric-charge vectors QandQc,respectively . Then FQ=fmQ (38) and FQc=faQc. (39) Clearly, (38) and (39) require Fto be equal to its multiplicative inverse, i.e., F=F−1orF2=I2. Given that the eigenvalues of Farefmandfa, the diagonal form for F is simply (see Ref. 7, p. 4) Fdiag=/parenleftbiggfm0 0fa/parenrightbigg . (40) And, from (31) and (33) the C-reversed counterpart of Fdiagis, necessarily, given by the similarity transformation (−σX)Fdiag(−σX) =−Fdiag. (41) From the minus sign on the right hand side of (41) we see that th e scalar fermion numbers fmandfaproperly change signs (are C-reversing “charges”) under −σX. Consider next a more general (nondiagonal) matrix F. Because the trace, determinant andsquare ofFareinvariants , one has trF=trFdiag=fm+fa= 0 (42) detF=detFdiag=fm·fa=−1 (43) and F2=F2 diag=I2. (44) And, because Fis traceless it must have the general form F=/parenleftbigg a b c−a/parenrightbigg . (45) 11Finally, because Fshould transform in the same way as FdiagunderX= −σXwe have the similarity transform (−σX)F(−σX) =−F. (46) Using Fas expressed by (45) and employing (46), one ﬁnds that c=−b. Hence, the most general form of Fis, necessarily, F=/parenleftbigga b −b−a/parenrightbigg , (47) where a2−b2= 1 since detF=−1. Making the following substitutions in (47), namely, a= cosh v (48) and b=±sinhv, (49) where −∞ ≤ v≤+∞is areal(dimensionless) parameter in the range indicated, ForF(v) ﬁnally assumes the general form F(v) =/parenleftbiggcoshv,±sinhv ∓sinhv,−coshv/parenrightbigg . (50) Here,F2(v) =I2for any v,F(v) satisﬁes the boundary condition F(0) = Fdiag, and the non-Euclidean 2-scalars Q2and (Qc)2are left invariant by the transformation F(v). 2.4 Distinguishing quarks and leptons Choosing the upper signs in (50), the matrix F(v) becomes F(v) =/parenleftbiggcoshvsinhv −sinhv−coshv/parenrightbigg , (51) where vis also chosen to be a positive real number (see Ref. 7, p. 50 an d 54). As described in Ref. 7, pp. 52–55, the parameter vdistinguishes between quarks andleptons . In particular, the parameter vis found to be quantized and obeys the “quantum condition”: v= lnMc, (52) 12where Mccounts both the number of fundamental fermions in a strongly- bound composite fermion, and the strong-color multiplicity . That is, Mc= 3 for quarks (strong-color triplets) and Mc= 1 for leptons (strong-color singlets). Thus we have found that a connection exists betwe en the 2-space description of quarks and leptons, and their associated strong colors! 2.4.1 Quark and lepton electric charges and BandL It has been shown (see Sec. 2.3, Eqs. 38 and 39; Ref. 7, pp. 52–5 5, and Ref. 8) that the quark and lepton electric charges are the “up”-“d own” compo- nents of the eigenvectors of the matrix F(v) speciﬁed by (51) and (52). In particular, the quark charges are given by ( Mc= 3) q1(f) =(M2 c−1) 2Mc(Mc−f)= +2 3forf= +1 and +1 3forf=−1,(53) q2(f) = q1(f)−1, (54) where the baryon number for quarks is B=q2 1(f)−q2 2(f) =±1 3forf=±1. Similarly, the lepton electric charges are given by ( Mc= 1) q′ 1(f) =−(M2 c−1) 2Mc(Mc−f)=−1 forf= +1 and 0 for f=−1,(55) q′ 2(f) = q′ 1(f) + 1, (56) where the lepton number for leptons is L= [q′ 1(f)]2−[q′ 2(f)]2=±1 for f=±1. In summary, the new 2-space description, and F(v), is found to provide an explanation for the quark-lepton “dichotomy” of fundament al fermions in ad- dition to the matter-antimatter, and “up”-“ down” type ﬂavo r-dichotomies. 2.5 Representing ﬂavor doublets in the 2-space Consider again the eigenvectors QofF(v) for fundamental fermions. Since the space on which F(v) “acts” is two-dimensional, an observable vector Q can be “resolved” into two (no more or less) observable , linearly-independent vectors, call them UandV, asQ=U+V[14]. Now, because these three vectors ( Q,U, andV) aresimultaneous observables, it makes sense to speak of this “triad” of vectors as being a well deﬁned geometric ob ject, namely, a “vector triad.” 13Recognizing that the components of Q,UandVareC-reversing charge- likeobservables [13] we can write these observable “charge” vectors as Q={q1, q2} (57) U={u1, u2} (58) V={v1, v2}, (59) where q1,q2,u1,u2,v1andv2are the various observable “charges” (e.g., q1andq2are the electric charges). Given Q=U+V, the non-Euclidean metric (13), and Eqs. (57) through (59), we ﬁnd the associate dobservable quadratic-“charges” [13] Q2=U2+ 2U•V+V2(60) 2U•V= 2(u1v1−u2v2) (61) U2=u2 1−u2 2 (62) V2=v2 1−v2 2. (63) Finally, using the foregoing collection of (global) ﬂavor- deﬁning charges, we can express the twoquantum states (simultaneous ﬂavor-eigenstates) associ- ated with a single vector-triad in the form of “ket” vectors as follows (Ref. 7, pp. 16–18) \|q1, u1, v1,Q2,U2,2U•V,V2/an}bracketri}ht \|q2, u2, v2,Q2,U2,2U•V,V2/an}bracketri}ht/bracerightBigg .(64) Here, the state \|q1, u1, v1,Q2,U2,2U•V,V2/an}bracketri}htrepresents the “up”-type ﬂavor- eigenstate, and \|q2, u2, v2,Q2,U2,2U•V,V2/an}bracketri}htrepresents the corresponding “down”- type ﬂavor-eigenstate in a ﬂavor doublet of fundame ntal fermions [12, 15]. 2.6 Family replication and the number of families In Ref. 7, pp. 39–49 and pp. 59–65, it is shown that ﬂavor doubl ets (hence families) are replicated and that there are only three families of quarks and leptons. We refer the reader to [7] for a full and detailed acc ount. Here we simply outline how this situation comes about. Once again, by the deﬁnition of a linear-vector 2-space, a 2- vector such as Qcan always be resolved into a pair (no more, or less) of linear ly-independent 14vectors UandVasQ=U+V(see Sec. 2.5). And, since Qrepresents a ﬂavor doublet, so should UandVrepresent thissame ﬂavor doublet. But, if this is so, diﬀerent vector-resolutions of Q(i.e., diﬀerent vector-triads ) should correspond to diﬀerent ﬂavor-doublets having the sa meQ. In other words, ﬂavor doublets should be replicated . SinceQcan be resolved (mathematically) in an inﬁnite number of way s, we might suppose that there are an inﬁnite number of ﬂavor doublets, and hence, families. But, because of various “quantum constrai nts,” it is possible to show that Qcan be resolved in only three physically acceptable ways for Q-vectors associated with either quarks or leptons . In other words, there can be only six quark ﬂavors and six lepton ﬂavors, which leads to the (ex post facto)“prediction” of three quark-lepton families. 2.7 New internal diﬀerences and neutrino mixing It is important to understand that the new 2-space descripti on of fundamen- tal fermions (quarks and leptons) provides a distinction be tween these par- ticles that goes beyond diﬀerences that can be explained by m ass diﬀerences alone. For example, in the standard model the only diﬀerence between the u, candtquarks is that they have diﬀerent masses. Otherwise, these p articles experience identical strong and electroweak interactions . Moreover, as de- scribed in Section 2.1.2, the separate conservation of quan tum numbers such as “charm” and “truth” can be attributed to certain unavoida ble “accidental symmetries” associated with the (renormalizable) Lagrang ian describing the (strong) interactions of these particles [11]. Taken at face value, these accidental symmetries would seem to imply that there are no internal “wheels and gears” that would dist inguish a u quark from a cquark, for example. But, if the string theories are correct, these particles would be associated with diﬀerent “handles ” on the compact- iﬁed space (see Ref. 16, Vol. 2, p. 408), and so would be diﬀere nt in this additional sense. Likewise, in the present non-Euclidean 2-space desc ription, topological diﬀerences in addition to a variety of (global) 2-scalars, which are onlyindirectly related to the accidental symmetries of the Lagrangian, ser ve to provide further distinctions between matter particles. A possible experimental signal of such “internal” diﬀerenc es is to be found in the recent observations at the Super Kamiokande of bi-max imal neutrino mixing [17]. Models which begin by positing a neutrino mass- matrix and associated mixing-parameters, such as the three-generati on model proposed 15by Georgi and Glashow [18], do an acceptable job of describin g the observa- tions. However, bi-maximal mixing may have a deeper explana tion in terms of internal topological diﬀerences (in the non-Euclidean 2 -space) between νe, andνµorντneutrinos. With respect to the internal transformation F(v), the topology of the non-Euclidean “vector triad” (see Sec. 2.6, Ref. 7, p. 57, Re f. 20, and the qualifying remarks in Footnote 19) representing the νe(νµorντ), is found to be that of a cylinder (M¨ obius strip). And, assuming that a change in topology during neutrino mixing is suppressed by energy “ba rriers,” or other topological “barriers” (e.g., one cannot continuously def orm a doughnut into a sphere), while neutrino mixing without topology-change i s (relatively) en- hanced, one can readily explain the experimental observati on of (nearly) maximal νµ−ντneutrino mixing—at least maximal νµ−ντmixing over long distances, where the proposed topological inﬂuences a re expected to be cumulative [19, 20]. If this qualitative explanation is bas ically correct, then it follows that the neutrino mass-matrix and associated mix ing-parameters needed to explain bi-maximal neutrino mixing, would be the result, at least in part, of these deeper (internal) topological diﬀerences between neutrinos, and not their cause. 3 Summary and Conclusions By insisting that the electric charge associated with SU(2) ﬂavor doublets, (i.e., weak isospin doublets) of fundamental fermions (qua rks and leptons) can be treated as a real, internal 2-vector , considerations of self consistency dictate that all of the mathematical “machinery” associate d with the lin- ear vector 2-space (e.g., the metric, and various transform ations) must be brought to bear when describing fundamental fermions. Whil e this math- ematical machinery is very simple, its application to funda mental fermions immediately leads to a (modest) extension of the standard mo del description of quarks and leptons, and to a number of predictions. The model predicts, among other things, that unlike the situ ation in the standard model [3, 6], particles such as the u,candtquarks are charac- terized by signiﬁcant internal (topological and other) diﬀerences. Similar diﬀerences may help explain recent observations of (nearly ) maximal νµ−ντ mixing [17–20]. Moreover, while we began this paper with the introduction of an SU(2) ﬂavor doublet of “quarks” \|u/an}bracketri}ht,\|d/an}bracketri}htand their antiparticle coun- 16terparts (see Sec. 1.0), very little else was assumed about quarks ,leptons , internal (strong )colors orfamily replication . And yet, the subsequent 2- space description correctly predicts that all of these thin gs, and more, are properties of fundamental fermions. For example, to mainta in compatibil- ity with the standard model, we were forced to (tentatively) conclude that certain global “charges” associated with the “accidental s ymmetries” of the Lagrangian describing strong and electroweak interaction s, and various 2- space charges, are (essentially) one and the same charges (s ee the “duality” conjecture in Sec. 2.1.2). Finally, family replication eme rged here as little more than the number of diﬀerent physically acceptable ways (three physi- cally acceptable ways are predicted) the vector Qfor quarks and leptons can beresolved in the 2-space (see Sec. 2.6 and Ref. 7, pp. 39–49) into pairs o f linearly independent “basis” vectors UandV(i.e.,Q=U+V). In closing it should be pointed out that popular extensions o f the standard model such as the so-called realistic (free fermionic) thre e-generation string models [21, 22], also provide an “explanation” for family replication and the number of families. Does this mean that in spite of very signiﬁcant a nd obvious diﬀerences there are, nevertheless, “deep” connec tions between the proposed 2-space description and string theories? I thank R. Zannelli for pointing out a problem with the existi ng 2-space description of muons (see Tables II and IV, and p. 57 in Ref. 7) . This problem, together with my proposed “remedy,” are brieﬂy described in [19]. References and Footnotes [1] J. D. Jackson, Classical Electrodynamics , John Wiley & Sons, Inc., New York, 1962, p. 377; Paul Lorrain and Dale Corson, Electromag- netic Fields and Waves , (Second Edition), W. H. Freeman and Co., San Francisco, 1970, pp. 228–229. [2] Of course, in external 4-D spacetime, observed scalars such as the electric charge of a quark or a lepton are taken to correspond to a projection of areal2-vector on one of the two internal andorthogonal coordinate “axes” of such a 2-space. [3] K. Huang, Quarks, Leptons and Gauge Fields , World Scientiﬁc Publish- ing Co., Singapore (1982), pp. 6–8. 17[4] W. Greiner and B. M¨ uller, Quantum Mechanics (Symmetries ), Springer- Verlag, Berlin, 1989, pp. 95–96; W. Heisenberg, Zeitschrif t f¨ ur Physik 77, 1 (1932); K. Huang, Quarks, Leptons and Gauge Fields , World Sci- entiﬁc, Singapore, 1982, pp. 12–14; A. W. Joshi, Elements of Group Theory for Physics (Third Edition), Wiley Eastern Limited, New Delhi, 1982, pp. 147–148; O. Nachtmann, Elementary Particle Physics (Con- cepts and Phenomena ), Springer-Verlag, Berlin, 1989, p. 185. [5] If the 2-vector Q={q1, q2}characterizes the SU(2)-doublet of states \|u/an}bracketri}ht and\|d/an}bracketri}ht, this same 2-vector must also characterize the states \|u/an}bracketri}htand\|d/an}bracketri}ht, individually . That is, each of the states \|u/an}bracketri}htand\|d/an}bracketri}htmay be said to “carry” the vector Q. It is important to understand that while the assignment of an electric-charge vector Qto each of the twoquark states \|u/an}bracketri}htand\|d/an}bracketri}ht adds information in the quantum sense, it does so in a way that doesnot violate quantum mechanics or special relativity. An analog ous situation involves the isospin vector τ(see Sec. 1.1 in the main text). Note that the vector Qis analogous to the vector τin the sense that a single vector τ, likeQ, may be assigned to eachof the quark states \|u/an}bracketri}htand \|d/an}bracketri}ht. Moreover, τ2andτ3are Lorentz 4-scalars that serve to (partially) deﬁne the states \|u/an}bracketri}htand\|d/an}bracketri}htvia the states \|τ2, τ3=1 2/an}bracketri}htand\|τ2, τ3=−1 2/an}bracketri}ht, respectively. Similarly, the states \|u/an}bracketri}htand\|d/an}bracketri}htare (partially) deﬁned by the Lorentz 4-scalars Q2,q1andq2via the states \|Q2, q1/an}bracketri}htand\|Q2, q2/an}bracketri}ht, respectively. Here, it happens that Q2provides additional information (i.e., the baryon number B=Q2) on the states \|u/an}bracketri}htand\|d/an}bracketri}ht. [6] C. Quigg, Gauge Theories of the Strong, Weak and Electromagnetic In- teractions , The Benjamin/Cummings Publishing Co., Reading, Mass., 1983. It is well known that the standard model adds “richness ” to the description of SU(2) ﬂavor-doublet states such as \|u/an}bracketri}htand\|d/an}bracketri}htby incor- porating additional symmetries such as SU(3) color. We assume here that the added “richness” associated with the new 2-space de scription, not only incorporates many features of the standard model, b ut also leads to features that lie outside the domain of the standard model, i.e., to features that eﬀectively extend the standard model. [7] Gerald L. Fitzpatrick, The Family Problem-New Internal Algebraic and Geometric Regularities , Nova Scientiﬁc Press, Issaquah, WA (1997). Ad- ditional information: http://physicsweb.org/TIPTOP/ or http://www.amazon.com/exec/obidos/ISBN=0965569500. I n spite of 18the many successes of the standard model of particle physics , the ob- served proliferation of matter-ﬁelds, in the form of “repli cated” gen- erations or families, is a major unsolved problem. This book proposes a new organizing principle for fundamental fermions, i.e., a minimalis- tic “extension” of the standard model based, in part, on the C ayley- Hamilton theorem for matrices. In particular, to introduce (internal) global degrees of freedom that are capable of distinguishin g all observed ﬂavors, the Cayley-Hamilton theorem is used to generalize t he familiar standard-model concept of scalar fermion-numbers f(i.e.,fm= +1 for all fermions and fa=−1 for all antifermions). This theorem states that every (square )matrix satisﬁes its characteristic equation . Hence, if fm andfaare taken to be the eigenvalues of some real matrix F(v)—a “gen- eralized fermion number”—it follows from this theorem that bothfand F(v) are square-roots of unity. Assuming further that the compo nents of both F(v) and its eigenvectors are global charge-like quantum obser v- ables, and that F(v) “acts” on a (real) vector 2-space, both the form ofF(v) and the 2-space metric are determined. The 2-space is found to have a “Lorentzian” or non-Euclidean metric, and various associated 2-scalars are found to serve as global ﬂavor-deﬁning “charg es,” which can be identiﬁed with charges such as strangeness, charm, ba ryon and lepton numbers etc.. Hence, these global charges can be used to describe individual ﬂavors (i.e., ﬂavor eigenstates), ﬂavor double ts and families. Moreover, because of the aforementioned non-Euclidean con straints, and certain standard-model constraints, these global charges are eﬀectively- “quantized” in such a way that families are replicated. Fina lly, because these same constraints dictate that there are only a limited number of values these charges can assume, it is found that families al ways come in “threes.” [8] G. L. Fitzpatrick, “Continuation of the Fermion Number O perator and the Puzzle of Families,” in the LANL physics e-Print arch ive [physics/0007038]. [9] Note that if Qc=−Q, then ( Qc)2= (−Q)2no matter what the metric is (i.e., s=±1). That is, if Qc=−Q, (Qc)2andQ2are the same for matter and antimatter, so they do nottransform like charges , and consequently do notdistinguish between matter and antimatter. 19[10] The term “non-Euclidean geometry” is usually reserved for the geometry ofcurved spaces. However, to avoid confusing the ﬂat“Lorentzian” 2-D geometry (and discrete transformations therein) with the ﬂatLorentzian 4-D geometry of spacetime (and continuous transformations therein), we prefer to use a term other than “Lorentzian” to describe the 2 -space. In particular, because this 2-space is notEuclidean, we choose to break with tradition, and refer to this ﬂatspace as being “not-Euclidean,” or more correctly, non-Euclidean . [11] S. Weinberg, The Quantum Theory of Fields, Vol. I, Foundations , Cam- bridge University Press, New York, NY (1995), pp. 529–531; The Quan- tum Theory of Fields, Vol. II, Modern Applications , Cambridge Univer- sity Press, New York, NY (1996), p. 155. [12] When weak interactions are “turned oﬀ” ﬂavor eigenstat es and mass eigenstates are one and the same. For the most part, when we sp eak here of ﬂavor eigenstates, we are referring to the situation where ﬂavor eigenstates and mass eigenstates are the same. [13] It is important to point out that the various C-reversing 2-scalars as- sociated with the internal 2-space description are taken to be Lorentz 4-scalars in an external spacetime setting. That is, because q1andq2are Lorentz 4-scalars in 4-D spacetime, the components of the 2- vectors U andV, and the scalar products Q2,U2,V2,U·Vare also Lorentz 4- scalars. This connection strengthens the idea that these nu mbers deﬁne ﬂavors. [14] When we say that the vectors Q,UandVareobservables , we mean that their associated component-“charges” are mutually-c ommuting si- multaneous observables. Hence, all of these charge-like co mponents can be known in principle, at the same time, meaning that the vectors Q,U andVcan be known simultaneously. Thus the vector “triad” ( Q,U,V) is a well deﬁned geometric object. [15] In Section 2.4 in the main text, it is shown that strong colors are more-or- less implicit in the 2-space description of quarks and lepto ns. However, the detailed connections between the 2-space and SU(3) color, have yet to be worked out. And, because each ﬂavor doublet of quarks or lep- tons is also a weak isospin doublet, there are other weak colors (i.e., 20weak isospin and weak hypercharge) that must be taken into account. Strictly speaking, then, besides the speciﬁcation of globa l charges, the overall quantum state of a fundamental fermion would, neces sarily, in- volve a speciﬁcation of the spin state, the energy-momentum state and so on, together with a speciﬁcation of the particular mix of l ocal color (gauge)-charges R,W,B,GandYcarried by each fundamental fermion. This color-mix would be determined, in turn, by something li ke a com- plementary, local SU(5) color-dependent gauge description. [16] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vols. 1 and 2, Cambridge University Press, 1987. [17] T. Kajita, for the Super-Kamiokande, Kamiokande Colla boration, in the LANL physics e-Print archive [hep–ex/9810001]. [18] H. Georgi and S. L. Glashow, “Neutrinos on Earth and in th e Heavens,” in the LANL physics e-Print archive [hep–ph/9808293]. [19] With the possible exception of one family, all quarks and leptons within a given family are found to exhibit the same topology with res pect to F. This is certainly the case for the ﬁrstandthirdfamilies. And, because of these facts, together with the requirement of quark-lept on “univer- sality,” at least within any given family, we naturally expe ct this to be the case for the second family as well. It happens that the second-family candsquarks exhibit M¨ obius topology with respect to F. However, strictly speaking, the second-family leptons, namely, the muon and its associated neutrino (and associated antiparticles) exhib it the requisite M¨ obius topology onlyif the components of the associated V-vectors are notexactly zero (see p. 57, and Tables II and IV in Ref. 7). Or, to put it another way, the associated U-vectors must fall slightly inside the physical lepton quadrant (quadrant II) of the 2-space (s ee Ref. 7, Chapter 3 and Sec. 4.2.2). Therefore, if we wish to maintain a kind of quark-lepton “universality” within families, we must assu me that some physical mechanism exists, which ensures that the V-vector components associated with muons, though predicted to be very small, rarely if ever actually equal zero. This assumption might be justiﬁed in a m ore fun- damental treatment where (hypothetical) quantum ﬂuctuations (of the V-vector components) are taken into account. Such a proposal is en- couraged by the fact that ( small)ﬂuctuations of this kind would have 21no eﬀect on the topology of other quarks and leptons , simply because their associated V-vector components are signiﬁcantly diﬀerent from zero to begin with (see Tables II and IV in Ref. 7). [20] G. L. Fitzpatrick, “Topological Constraints on Long-D istance Neu- trino Mixtures,” in the LANL physics e-Print archive [physi cs/0007039]. Equation (11) in this paper describes a hypothetical situat ion in which topology-maintaining (topology-changing) inﬂuences in t hree-ﬂavor neu- trino mixtures are dominant (nonexistent). Because the mat rix element a[b= (1−a)/2] in equation (11) assumes its largest [smallest] possible value, namely unity [zero], the matrix element a[b= (1−a)/2] is taken to be a puremeasure of topology-maintaining [topology-changing] inﬂ u- ences in general. Hence, the product f(a) =ab=a(1−a)/2 is taken to be a puremeasure of the balance between topology-maintaining and topology-changing inﬂuences. This balance should occur at the maxi- mum of f(a), namely, when a=1 2. Given this “equilibrium” value of the matrix element a, one easily establishes that the (predicted) matrix describing long-distance three-ﬂavor neutrino (equilibr ium) mixtures is given by M= 1/8 4 2 2 2 3 3 2 3 3 . [21] A. E. Faraggi, “Towards the Classiﬁcation of the Realis tic Free Fermionic Models,” in the LANL physics e-Print archive [hep - th/9708112]. [22] G. B. Cleaver, A. E. Faraggi, D. V. Nanopoulos and T. ter V eldhuis, “Towards String Predictions,” in the LANL physics e-Print a rchive [hep- ph/0002292]. 22
arXiv:physics/0011074v1 [physics.ao-ph] 30 Nov 2000Modelling Horizontal and Vertical Concentration Proﬁles o f Ozone and Oxides of Nitrogen within High-Latitude Urban Are as James P. Nicholson and Keith J. Weston Department of Meteorology, University of Edinburgh, Edinb urgh, EH9 3JZ, U.K. David Fowler Centre for Ecology and Hydrology, Bush Estate, Midlothian, EH26 0QB, U.K. February 20, 2014 Abstract Urban ozone concentrations are determined by the balance be tween ozone destruction, chemical production and supply through advection and turbu lent down-mixing from higher levels. At high latitudes, low levels of solar insolation an d high horizontal advection speeds reduce photochemical production and the spatial ozone conc entration patterns are largely determined by the reaction of ozone with nitric oxide and dry deposition to the surface. A Lagrangian column model has been developed to simulate the mean (monthly and an- nual) three-dimensional structure in ozone and nitrogen ox ides (NOx) concentrations in the boundary-layer within and immediately around an urban area s. The short time-scale pho- tochemical processes of ozone and NOx, as well as emissions and deposition to the ground, are simulated. The model has a horizontal resolution of 1x1km and high resol ution in the vertical. It has been applied over a 100x100km domain containing the city of Edinburgh (at latitude 56◦N) to simulate the city-scale processes of pollutants. Resu lts are presented, using aver- aged wind-ﬂow frequencies and appropriate stability condi tions, to show the extent of the depletion of ozone by city emmisions. The long-term average spatial patterns in the surface ozone and NOx concentrations over the model domain are reproduced quanti tatively. The model shows the average surface ozone concentrations in the urban area to be lower than 1Nicholson et. al. 2 the surrounding rural areas by typically 50% and that the are as experiencing a 20% ozone depletion are generally restricted to within the urban area . The depletion of the ozone concentration to less than 50% of the rural surface values ex tends only 20m vertically above the urban area. A series of monitoring sites for ozone, nitri c oxide and nitrogen dioxide on a north-south transect through the city - from an urban, thro ugh a semi-rural, to a remote rural location - allows the comparison of modelled with obse rved data for the mean diurnal cycle of ozone concentrations. In the city-centre, the cycl e is well reproduced, but the ozone concentration is consistently overestimated. Key-word index: Tropospheric ozone, Lagrangian column model, urban, nitro gen oxides, verti- cal exchange. 1 Introduction Tropospheric ozone is a photochemical oxidant formed large ly by photochemical reactions. In the troposphere ozone acts as a greenhouse gas (Fishman et al . 1979), (Chalita et al. 1996) and is toxic to plants, reducing crop yields (Hewitt et al. 1990) , and to humans as a respiratory irritant (WHO 1987), as well as damaging both natural and man -made materials, such as stone, brick-work and rubber, (PORG 1993). Quantifying the dose and exposure of the human population, vegetation and materials to ozone is required t o assess the scale of ozone impacts and to develop control strategies. A network of 17 rural ozone monitoring stations across the UK provides broad-scale regional spatial patterns in tropospheric ozone concentrations in r ural areas (PORG 1993). Peak ozone concentrations increase from north to south across the UK as the south has higher concen- trations of the primary pollutants necessary for ozone prod uction (from greater emissions and proximity to continental sources) as well as greater freque ncy of meteorological conditions suit-Nicholson et. al. 3 able for ozone production. Mean ozone concentrations also i ncrease with altitude (PORG 1997) and are higher in a 5-10km coastal strip (Entwistle et al. 199 7). The mean annual-average back- ground concentration of ozone for the UK is approximately 50 µg m−3, though there is a wide variation for individual episodes about this value, rangin g up to about 400 µg m−3(PORG 1993). There are clear diurnal and annual cycles in ozone con centrations in the UK, with a mid-afternoon peak and nocturnal minimum and a spring maxim um and autumn minimum. The diurnal cycle illustrates the fundamental importance o f vertical mixing (Garland and Der- went 1979). Ozone concentrations in urban areas are of particular inter est and importance as the population is largely urban-based and ozone concentrations show great er spatial and temporal variations in urban areas. Ozone concentrations are smaller in urban area s than they are in surrounding rural areas due to the reaction of ozone with nitric oxide, emitted from combustion sources, forming nitrogen dioxide. Where air is allowed to stagnate over an ur ban area, the eﬀects of strong insolation and accumulating ozone precursors can produce v ery high ozone concentrations. An example of this is the photochemical smog that aﬀects the Los Angeles area of California, where a combination of meteorology, local topography and ve ry high pollutant emission levels produce dangerously high ozone concentrations on many days of the year (Lents and Kelly 1993). However, in much of the UK and other high-latitude are as where insolation levels are lower and wind-speeds are larger (maintaining a steady adve ction of air through the urban air-shed), there is a diﬀerent spatial distribution of conc entrations with annual mean ozone concentrations generally smaller in urban areas. There are many urban sites at which ozone is monitored in the U K (PORG 1997). Attempts to map ozone concentrations using both rural and urban monit oring are complicated by the interaction of local chemistry with the larger-scale meteo rological factors determining ozone concentrations. Recent studies on higher resolution ozone mapping have used an urbanisationNicholson et. al. 4 index (PORG 1997), but these have not been able to account for the movement of ozone and ozone-depleted air into and out of the urban areas. Various studies have examined the characteristics of ozone and nitrogen oxide concentrations around urban areas (Ball and Bernard 1978). A large number of studies have focussed on the two-dimensional structure of the ‘urban plume’ of photo chemical-ozone downwind of a city, with diﬀerent cities around the world being studied (C leveland et al. 1976), (White et al. 1976), (Varey et al. 1988), (Lin et al. 1996) and (Silibel lo et al. 1998). There have been few studies of the three-dimensional mean structure of ozon e concentrations in urban areas, especially at high latitudes, where ozone production is of s econdary importance in describing the spatial distributions around cities. These studies are largely interpretations of observational data (Angle and Sandhu 1989) and (Leahey and Hansen 1990). A boundary-layer Lagrangian column model has been develope d to simulate the mean three- dimensional structure in ozone and nitrogen oxide concentr ations in the boundary-layer within and immediately around high-latitude urban areas at a spati al scale of 1x1km. (The model is not appropriate for use with “real time” trajectories.) T he model simulates the eﬀects of ozone depletion at the surface as a consequence of the reac tion with emitted nitric oxide and dry deposition to the surface. This has been used to follo w a range of one-dimensional trajectories over a distance of 105m and a travel time in the order of 104s, through a simulated city under a variety of meteorological and pollutant emissi on regimes representing seasonal and diurnal extremes. An assessment of the extent of ozone de struction occurring, the rate of recovery of surface ozone concentrations downwind of the ci ty and the inﬂuence of meteorological parameters on the ozone concentration has been provided usi ng the model. The model has been applied over a 100x100km domain containin g a simulation of the emission ﬁeld over the city of Edinburgh. Edinburgh was used as a gener ic, high-latitude city for mod-Nicholson et. al. 5 elling purposes. A land-use array has been created as input t o the model with spatially- and temporally-variable emission and deposition values. 2 Ozone Destruction by Nitric Oxide High latitude cities are, in general, well-ventillated, so that the timescale for air traversing the city is small compared to that for ozone generation. Under th ese conditions, the three reactions that are fundamental in the determination of ozone ( O3) concentrations in urban areas (Wayne 1991) are: O+O2+M→O3+M k 1 (1) NO2+hν→NO+O J 2 (2) NO+O3→NO2+O2 k3 (3) where k1andk3are reaction rate constants and J2is the photolysis rate for nitrogen dioxide (NO2). In the UK, the main source of nitrogen oxides is the combustio n of fossil fuels, ie.the burning of coal, oil and gas in power stations and the combustion of pe trol and diesel by road traﬃc (Salway et al. 1997). The largest source in large urban areas is the exhaust from road vehicles. The emissions are mainly NOwith ratios of NO:NO2in excess of about 3:1 (Selles et al. 1996). A large amount of this NOis rapidly oxidised by ozone (Reaction 3). The diurnal cycle of urban ozone concentrations show a maxim um concentration at night fromNicholson et. al. 6 0100 to 0500 hours when traﬃc density, and hence NOemissions, are lowest (PORG 1997). Minimum concentrations occur when the NOconcentration peaks during the morning and evening rush hours. Wind-speed also has an eﬀect on urban ozo ne concentrations with ozone- rich air advected into the city from the surrounding country side in windy condition. At the same time, NOemitted in the city is diluted in the well-mixed air and so Rea ction (3) is less dominant - reducing both O3destruction and NO2production (Oke 1992). 3 Experimental Method 3.1 Description of the Model Used The TERN model (Transport over Europe of Reduced Nitrogen) w as developed in the early 1990s (ApSimon et al. 1994) to examine the release, transpor t and chemistry of ammonia. It is a Lagrangian column model with detailed vertical resolutio n and diurnally varying emissions, deposition and turbulent mixing. It was further developed t o examine ammonia and ammonium transport and deposition over the UK (Singles et al. 1998). I t can be used to obtain vertical concentration proﬁles of pollutants emitted or generated a t or near the surface and transported over long distances on a single trajectory. This Lagrangian column model was considered suitable as a ba sis for a model to be used in this study of urban ozone concentrations because it allows detai led vertical mixing to be considered and thus is able to represent the marked variations with heig ht of the concentration proﬁles. An Eulerian model with a comparable run-time would not give t he required vertical resolution (ApSimon et al. 1994). The TERN model has been considerably m odiﬁed so as to be more suitable to modelling ozone concentrations in a city but the basic processes of calculating the extent of vertical mixing and the exchange between the verti cal layers have remained the same.Nicholson et. al. 7 The detailed vertical resolution was used to provide a sophi sticated treatment of the vertical diﬀusion of the pollutant species. The column of air that is a dvected over the surface (in the absence of vertical shear in the column) is the lowest 2.5km o f the troposphere and is divided into 33 layers. There is much ﬁner resolution near the surfac e (the lowest ﬁve layers are of depth 1, 2, 2, 4 and 15m), allowing more detail where the highest con centration gradients occur. The neglect of lateral dispersion, both by shear and turbulence , is not a serious omission for two reasons: ﬁrstly, the model is principally used to determine mean ﬁelds, for which climatological wind data are used, covering the whole 360 degrees; and secon dly, the relatively small spatial scale of both the application of the model (and the resulting ﬁelds), means that changes due to lateral mixing are small compared to those due to vertical mi xing. Parameterisation of the boundary layer includes the eﬀects on the mixing layer depth and the vertical diﬀusivity of the meteorological variables - inso lation, cloud cover and wind-speed. The mixing layer depth is calculated on a diurnally varying patt ern depending on the time of day and the prevailing meteorological conditions. Carson’s model (Carson 1973) is used to calculate the development during the day, with a mechanical mixing factor proportional to the geostrophic wind-speed. At night, the depth is determined from a combina tion of the Pasquill category and the wind-speed (Pasquill 1961). The vertical mixing of gases, is determined using the diﬀusi on equation: δC δt=δ δz/parenleftbigg KzδC δz/parenrightbigg (4) The coeﬃcient of mixing, Kz, is deﬁned as a function of height for diﬀerent stability con ditions. In the model, Kzis assumed to linearly increase with height to a value of Kmaxat height zm and to be constant above this height up to the top of the mixed l ayer. During the day, zmis ﬁxed at 200m and Kmaxis dependent on the larger of two terms, one representing mec hanicalNicholson et. al. 8 mixing and the second on convective mixing. The ﬁrst of these terms is dependent on stability and wind speed and the second on surface heat ﬂux. At night, bo thKmaxandzmare dependent on wind speed and cloud cover (ApSimon et al. 1994). TERN’s treatment of mechanical turbulence takes no account of the surface roughness. How- ever, in the city the increased roughness eﬀect of the buildi ngs would increase the turbulence of the air compared to the countryside. Assuming a logarithm ic wind proﬁle over the lowest 200m (with no zero-plane displacement) and surface roughne sses of 0.05m for rural and 1.0m for urban areas (Stull 1997), the ratio of urban to rural fric tion velocities was calculated to be 1.5. This factor is used as a scaling factor for Kzto simulate enhanced mixing over urban areas. A diurnally-varying dry deposition velocity ( VD), is included for nitrogen dioxide and ozone (Hargreaves et al. 1992). The chemistry employed in the model is very simple, with ozon e destruction and production by Reactions (1-3) included but ozone production through hy drocarbon degradation ignored. Photodissociation (Reaction 2) is dependent on radiation l evels which are parameterised in terms of time of day, time of year and climatological cloud co ver. The time-step used can be varied over a wide range of values bu t the chemistry involved requires time-steps of a few seconds to be treated accurately. The tim e-step used for all model runs was 1.5 seconds (or 40min−1). The chemistry would actually have allowed for a longer tim e-step than this, which would have been desirable in keeping the run -time down, but the diﬀerential equations used to calculate vertical diﬀusion became unsta ble at low wind-speeds with longer time-steps.Nicholson et. al. 9 3.2 Edinburgh Data A large part of this study has been performed on a model of the c ity of Edinburgh: a city of 450,000 inhabitants on the east coast of Great Britain at a la titude of 56◦. The city is set on a coastal plain between the Firth of Forth to the north and a ra nge of 600m hills to the south. It is a centre of ﬁnance and service industries and as such doe s not have a large number of factories in the city centre or nearby. It does however have a rapidly expanding road vehicle ﬂeet that generates NOx. It is also a compact city, approximately 10km in diameter, w ith little suburban sprawl or outlying towns. The nearest large urban a rea is Glasgow some 70km to the west. The omission of these distant sources is discussed in s ection 7. 3.3 Nitrogen Oxide Emissions TheNOxemissions in the model comprises 1x1km data of Edinburgh emi ssions for a 12x10km grid in the centre of the 100x100km domain. These data are par t of the 1 km21995 National Atmospheric Emissions Inventory for 1995 (Salway et al. 199 7). Other emission sources - surrounding towns, large roads and point sources - were igno red as only the eﬀects of Edinburgh emissions are being examined. All areas outside the city wer e assumed to have a low, background NOxemission. Traﬃc count data from Edinburgh were used as a proxy for the di urnal variations in NOx emissions. Approximately 85% of all NOxemissions were estimated to come from road traﬃc in city centres (Lindqvist et al. 1982), the rest was assumed to be emitted by background sources and were treated as constant throughout the 24 hours . The traﬃc count data used were from a detailed city-centre traﬃc analysis, held over 1 6 hours (0600-2200hrs) at 40 sites in June 1997, and annual average traﬃc counts from seven main arterial routes into and out of the city, also for 1997. As both the city centre and surroun ding arterial roads had similarNicholson et. al. 10 diurnal variation, the average of them was used in the model f orNOxemission variations - as seen in the composite plot in Figure 1. Rural background leve ls ofNOxemissions ( ie.outside the 10x12km city centre) were spatially constant and varied temporarily in the same way as the city centre emissions. NOxemissions are put into the lowest two layers of the model and aNO:NO2ratio of 3:1 is assumed. 3.4 Wind Data The wind data used in these modelling studies were obtained f rom Turnhouse Airport, situated 8km west of the city centre, and are averages for the period 19 70-1991 (ISMCS 1995). These are 10m winds and are used rather than the local geostrophic w ind as they are available in a detailed statistical form and yet are still representative of airﬂow across the region. The data include the speed and frequency (without calm conditions) o f the wind from diﬀerent directions for midday and midnight in June and December (see Tables 1 and 2). DIRECTION 0000hrs 1200hrs Frequency (%) Speed (m s−1) Frequency(%) Speed (m s−1) 000◦−044◦6.7 2.3 9.4 3.7 045◦−089◦19.1 3.5 28.0 4.9 090◦−134◦9.6 2.7 7.2 3.6 135◦−179◦2.5 2.0 2.1 2.5 180◦−224◦6.0 3.0 6.6 5.1 225◦−269◦39.4 3.8 20.7 6.0 270◦−314◦13.4 2.6 21.7 5.2 315◦−359◦1.7 1.7 4.4 3.3 100.0 3.0 100.0 5.0 Table 1: Edinburgh wind-rose data (Turnhouse, 1971-1991) J une 0000 and 1200hrs 3.5 Observational Data Observations of ozone and NOxconcentrations were used from three monitoring sites - one urban: Princes Street, in the city centre, and two rural: Bus h and Auchencorth Moss, to theNicholson et. al. 11 DIRECTION 0000hrs 1200hrs Frequency (%) Speed (m s−1) Frequency(%) Speed (m s−1) 000◦−044◦3.8 4.7 3.1 3.7 045◦−089◦6.4 3.8 5.9 4.4 090◦−134◦9.4 5.0 10.3 5.7 135◦−179◦2.3 3.9 1.9 4.7 180◦−224◦8.2 7.4 10.0 5.3 225◦−269◦49.1 6.0 43.5 6.2 270◦−314◦18.4 3.9 21.3 4.4 315◦−359◦2.6 3.4 4.0 3.1 100.0 4.7 100.0 5.3 Table 2: Edinburgh wind-rose data (Turnhouse, 1971-1991) D ecember 0000 and 1200hrs south of the city. Details of these sites are contained in Tab le 3. The inlet heights for monitoring at the three stations are diﬀerent. Bush and Princes Street m onitoring heights are at 4m and 5m respectively (in the third level of the model) while Auche ncorth is monitored at 3m (at the top of the second level). The output from all model runs are fo r level 3 in the column - except for those that are looking at the variations in concentratio ns with height as vertical proﬁles or sections. Station Grid Type Site Species Measurement Reference Description Analysed Height Princes Street NT 254 738 Urban Urban Parkland, O3,NOx, 4m 35m from major road. NO Bush Estate NT 245 635 Rural Site surrounded by O3,NOx, 5m open and forested land. NO Auchencorth NT 221 562 Rural Moorland. Low local O3,NOx, 3m agricultural activity. NO Table 3: Location and description of the monitoring sites in and around Edinburgh In all three cases, the nitrogen oxide concentration, [ NOx], and the nitric oxide concentration, [NO], are measured directly, and then the nitrogen dioxide conc entration is calculated by subtraction.Nicholson et. al. 12 4 One-Dimensional Trajectories Through a City 4.1 Method Straight-line trajectories, featuring rural and urban con ditions, were run over 100km to examine the eﬀect on ozone concentration and the O3/NO/NO2equilibrium of diﬀerent atmospheric conditions. Starting with constant vertical concentratio ns at the upwind boundary, an initial fetch of 45km over background (rural) landscape allowed the O3/NO/NO2system to reach a dynamic equilibrium and a vertical proﬁle to form. The colu mn then passed over 10km of simulated urban area with a lower deposition velocity fro m that over the rural area and (spatially constant) NOxemissions included. Finally, the trajectory continued for another 45km downwind of the city, again over background rural lands cape, to allow the O3/NO/NO2 system to move towards attaining a new balance. The amount to which the ozone concentration recovered was compared with initial upwind concentrations . The output from the model is the ozone concentration in the third vertical level (5m) of the c olumn. 4.2 Model Input data The one dimensional model runs were performed under four diﬀ erent ‘extreme’ atmospheric conditions - summer/winter and day/night - which have large diﬀerences in their values of inso- lation, wind-speed and atmospheric stability and thus show the range of ozone concentrations in seasonal and diurnal cycles. As many factors as possible w ere held constant between the diﬀerent conditions so as to show which of the many input para meters had the largest inﬂu- ence on the ozone results. Only the wind-speed, temperature (which has a negligible eﬀect on ozone concentrations with small variations) and solar in solation values used were diﬀerent between daytime and nighttime runs in the same season. The ‘s ummer’ and ‘winter’ and ‘day’Nicholson et. al. 13 and ‘night’ conditions used in the modelling studies were ba sed on four sets of meteorological parameters outlined in Table 4. Summer Day Summer Night Winter Day Winter Night Date 15/6/97 15/6/97 15/12/97 15/12/97 Time 12:00hrs 00:00hrs 12:00hrs 00:00hrs Wind-Speed 5.0m s−13.0m s−15.3m s−14.7m s−1 Temperature 15◦C 10◦C 5◦C 0◦C Cloud Cover 4oktas 4oktas 4oktas 4oktas NO/NO 2Ratio 75:25 75:25 75:25 75:25 Initial [ NO] 1.0 µg m−31.0µg m−31.0µg m−31.0µg m−3 Initial [ NO2] 5.5 µg m−35.5µg m−39.5µg m−39.5µg m−3 Initial [ O3] 90.0 µg m−390.0µg m−375.0µg m−375.0µg m−3 Table 4: Input parameters for the model for the simulated sea sonal and diurnal trajectories In each case, the initial concentration values was chosen so that, when run over background country, the model concentrations were close to the observe d average concentrations for those conditions at Auchencorth Moss - the remote rural site. The wind-speeds used were those shown in the bottom rows in Ta bles 1 and 2, ie.the mean of the midday and midnight values so as to make the daytime and night time model runs comparable. The deposition velocity of ozone and NO2(NOis not deposited at a signiﬁcant rate) for each season was calculated by multiplying a seasonal average by a variable factor. At nighttime this factor had a value of 0.4 while in the daytime it varied accord ing to the zenith angle of the sun with a greater amplitude in summer than winter. The values of the deposition velocity used in the model are contained in Table 5 (Brook et al. 1999). Summer Day Summer Night Winter Day Winter Night VDO3(Rural) 12.0mm s−16.0mm s−14.8mm s−12.4mm s−1 VDO3(Urban) 6.0mm s−13.0mm s−12.4mm s−11.2mm s−1 VDNO2(Rural) 3.0mm s−11.5mm s−11.2mm s−10.6mm s−1 VDNO2(Urban) 1.5mm s−10.75mm s−10.6mm s−10.3mm s−1 Table 5: Annual and diurnal variations in the ozone and NOxdeposition velocities to rural and urban areasNicholson et. al. 14 4.3 Results From One-Dimensional Trajectories Figure 2 shows the concentrations for summer day and summer n ight trajectories - midday and midnight of June 15th respectively. At midday in summer, the concentration of ozone drops from an upwind value of 70 µg m−3to a minimum of 42 µg m−3within the city boundaries. However, owing to the rapid vertical mixing, once downwind o f the city the ozone concentra- tion quickly recovers again towards 60 µg m−3- the maximum reached before deposition and chemistry start to reduce the ozone again. This is an overall loss of 10 µg m−3as the air column crossed the city or about 15% of the upwind concentration. Du e to the large amount of NOx emitted - the new dynamic equilibrium of the NO/NO 2/O3system has larger concentrations of both NOandNO2. At night the boundary layer is shallow and stable which caus es both NOandNO2to accumulate, exceeding 100 µg m−3by the downwind edge of the city. The ozone within the city is completely destroyed and the NOpresent close to the surface keeps the ozone concentration at zero for almost 10km downwind of the c ity. As there is no photolysis at night and thus no reforming of NO, once the NOemitted by the city has been used up, the ozone concentration increases again due to the mixing do wn of ozone-rich air from aloft. The stability of the lower atmosphere ensures that this incr ease is relatively slow: by 30km downwind the ozone concentration has reached 25 µg m−3, approximately 50% of the value on the upwind edge of the city. Figure 3 shows two plots equivalent to Figure 2, but for the mi d-December instead of June. From Tables 1 and 2 it can be seen that the average wind-speed i s greater in winter than in summer, meaning mechanical turbulence is greater. In Dec ember the incident radiation is reduced, leading to less convective turbulence (and hence l ess vertical mixing), and lower rates of photolysis. There are also lower reaction rates and depos ition velocities, but the emission rates of NOxfrom the city are the same. It can be seen that in the daytime, t he greaterNicholson et. al. 15 wind-speed doesn’t overcome the lower convective turbulen ce levels and leads to very high NO andNO2concentrations in the city and hence less ozone present than in June. Almost all of the 55 µg m−3of ozone upwind of the city is destroyed while NOconcentrations peak at almost 300 µg m−3. The reduced vertical mixing due to greater stability of the boundary layer in the winter can be seen by the time it takes for the O3/NO/NO2system to return towards a new dynamic equilibrium. At 45km downwind, both ozone and NO2are still increasing as the destruction of all the NOin the lower layers of the column is slow. The increased wind- speed, and thus mechanical turbulence, at midnight in December ove r the June values, together with lower emissions, keeps the ozone concentration slightly ab ove zero and leads to a more rapid and complete downwind recovery. As the NOconcentration decreases, the ozone concentration reaches 35 µg m−3by 45km downwind - a 60% recovery on the upwind concentration . Figure 4 shows the vertical proﬁles of ozone concentration o ver the lowest 150m of the boundary layer (ﬁrst 10 levels in the model), taken at three positions along the 100km trajectory. The proﬁle labelled ‘Upwind’ is taken at the upwind boundary of t he city ie.after 45km of rural emissions and deposition. The ‘City’ proﬁle was taken in the very middle of the city (50km into the trajectory). It can be seen that in the middle of the city o zone has been depleted from all levels of the column, but especially from the lowest layers w here the concentration is 20 µg m−3 less than in the upwind proﬁle. The ‘Downwind’ proﬁle was fro m 10km downwind of the city. This proﬁle shows a smaller vertical concentration gradien t throughout the column than in the city, indicating that the ozone concentration has begun to r ecover near the foot of the column with the reduction of emissions, but that the ozone depletio n has spread vertically upwards to deplete the higher levels. The proﬁle is almost parallel to t he upwind proﬁle with the depletion being around 12-15 µg m−3throughout the column. Figure 5 shows how the ozone concentrations under diﬀerent s tability conditions depends on wind-speed. The ozone concentrations were measured in the t hird level of the model at 10kmNicholson et. al. 16 upwind (35km into the trajectory) and 10km downwind (65km) u nder a range of wind-speeds varying from 0.25 - 10m s−1. For the summer, both upwind and downwind, this diagram show s the expected pattern - fairly steady concentrations (indep endent of wind-speed) until the wind drops below 2m s−1at which the concentration drops quickly (PORG 1997). In Dec ember, however, the model shows the concentration being much more d ependent on wind-speed over a larger range ie.mechanical turbulence is more important in winter and so the whole mixing is dependent on wind-speed and hence so is the concentration. 5 Two-Dimensional Maps of Ozone Concentrations Within and Around Edinburgh 5.1 Method In order to simulate the two-dimensional ﬁeld of ozone conce ntrations around Edinburgh, the one-dimensional trajectory version of the model has been mo diﬁed to run over a 100x100km grid, with Edinburgh located at the centre. This has been ach ieved by combining the patterns from a series of straight-line trajectories, from a variety of angles, so that a picture is drawn up of the ozone concentration averaged over all directions. Th e directions of the trajectories are separated from each other by 15◦, giving 24 directions, each having its own frequency weight ing and wind-speed, taken from the wind-rose data from section 2 .4. The model is run at a 1x1km grid-scale. The ozone concentration recorded at the end of e ach time-step is added to the total of the grid-square currently occupied. When the whole domai n has been covered from each of the 24 angles used, the average concentration of each grid-s quare is calculated by dividing the sum of the concentrations for that square by the number of tim e-steps ﬁnishing within it. The initial NOxandO3concentration proﬁles used at the start of each trajectory a re calculatedNicholson et. al. 17 from the 100km summer/winter trajectories from section 3. T he concentrations at 20km of the trajectory over rural land (with no emissions) were used . This allowed the chemistry to reach a steady-state and vertical concentration proﬁles th at are appropriate for the atmospheric conditions to be created. 5.2 Annual Mean Ozone Plot The annual mean ozone concentration map shows the eﬀect to wh ich titration of ozone by nitric oxide has a pronounced eﬀect throughout the year. It w as generated by averaging each pixel output from twelve diﬀerent model runs: midday and mid night for October, December and February (using the ‘winter’ conditions - ie.wind-rose data and initial concentrations) and April, June and August (using ‘summer’ conditions) and can b e seen in Figure 6. The output for the annual average ozone concentration at Bus h and Auchencorth are in excellent agreement with the annual average observed ozone concentra tion for the two sites in 1995-97 - see Table 6. It should be remembered, however, that Auchenco rth values were used to determine the upwind boundary conditions. SITE MODELLED OBSERVED Princes Street 16 µg m−331µg m−3 Bush 54 µg m−353µg m−3 Auchencorth Moss 55 µg m−354µg m−3 Table 6: Annual average ozone concentrations at the three mo nitoring sites - both modelled and observed results However, at Princes Street the agreement is not nearly so goo d with the modelled value only about half the observed average; but the model value is the av erage concentration over the whole 1x1km grid-square that the monitoring station is in. In real ity the concentration varies greatly across the grid square and the observations may be taken at a s ite where the concentration is not representative of the whole area. This is in contrast to t he Auchencorth site, where theNicholson et. al. 18 uniformity of the landscape and the lack of local sources mak es it representative of a large area. It might though be expected that the observations, par ticularly at Princes Street, would be smaller than the modelled results as they are taken within a few metres of the road itself. The position of the monitoring site though is slightly unusu al in that it is positioned within a garden that is at a lower level than the road. The model treats the entire domain as it it were ﬂat and is unable to recreate the eﬀects of such topographic d etail. However, with the height of the monitoring inlet being 4m above the surrounding groun d, it is almost on the same level as the adjacent road and thus should still be sampling air tha t is depleted in ozone. It appears that the model is under-estimating ozone concentrations wi thin urban areas. The spatial extent of ozone depletion greater than 5 µg m−3from the background (Auchencorth) value of 55 µg m−3is fairly limited on an annual-average basis. The 50 µg m−3isopleth extends only a few kilometres outside the city boundary - reaching a m aximum distance of approximately 10km downwind of the dominant wind direction (see Tables 1 an d 2), to the north-east. Areas with concentrations less than 45 µg m−3(a 20% depletion) are mainly conﬁned within the city boundary and, due to the prevailing wind from the southwest, out over the Firth of Forth to the north-east. Only an area of approximately 8x8km, just oﬀset from the city centre is depleted by more than 50%. 6 Modelling Vertical and Temporal Variations in Edinburgh’ s Ozone Concentrations 6.1 Vertical Section of Ozone Concentrations Through Edinb urgh Figure 7 is the 30x30km plot of ozone concentrations for midd ay in December. The city centre ozone concentration is less than 5 µg m−3with the Bush and Auchencorth concentrations beingNicholson et. al. 19 46 and 48 µg m−3respectively. Depletion greater than 20% ( ie.less than 35 µg m−3) is again mainly conﬁned to the city and areas to the north and east, out over the Firth of Forth. Figure 8 is a north-south vertical section (0-150m) through the 1200hrs December ozone con- centration ﬁeld - passing through Princes Street. Above 100 m there is little detail of the city below that can be seen at all (except for some jagged peaks tha t are artefacts of the plotting program used). It can be seen that the section is skewed with l ower concentration spread out towards the north - this can also be seen in the two-dimension al plot (Figure 7). Concentrations of less than 25 µg m−3,ie.50% depletion, are restricted to below 20m. At nighttime, th is would be even lower - just a few metres above the ground. 6.2 Diurnal Variation of Ozone Concentrations Diurnal plots of ozone concentrations at Princes Street and Bush for December have been made by calculating the hourly ozone concentrations for 24 h ours. These are plotted in Figure 9 and are compared with observational data obtained from the two monitoring stations (hourly averages) from 3 years (1995-97). Any day with missing data f or any of the species were ignored; however overall data capture was greater than 90%. As can be seen from the diurnal variation plot, the values at B ush generated by the model are closer to the observed average values (as expected due to the deﬁnition of the upwind perimeter concentrations) than those at Princes Street but the patter n of the diurnal variation produced by the model at Princes Street more closely resembles the obs ervations, despite a systematic under-estimation of 10-20 µg m−3. At Princes Street, the only feature of the modelled variatio n that doesn’t correspond well with the observed values is in the middle of the day when there is a p eak in the concentration from the minima during the dawn and dusk rush hours. This is due to t he increased mixing in theNicholson et. al. 20 day in the model from the increased insolation. The reason th at the modelled values are on average 10 µg m−3less than those observed may be due to one or more of several re asons: •The emission used in the model may not be appropriate for the l ocality. •Vertical mixing may be underestimated. No account is taken o f the heat island eﬀect. •It is likely that the complex mixing processes taking place w ithin the street canyons are not adequately represented. •Mixing depends on stability, which is determined only by ext ernal conditions (cloud cover and wind speed) at the time. There is no ”memory” of preceedin g conditions. The observed values at Bush are lower than those produced by t he model and show that this site is also inﬂuenced by the Edinburgh rush hour. There are t wo roads within 500m of the site that carry substantial traﬃc into and out of the city in t he morning and evening and the corresponding dips in the ozone concentration at 0900 and 17 00hours can be attributed to this. There appears to be a discontinuity in the modelled data in th e very middle of the day, where the concentration dips instead of rising to a small peak at mi dday as happens at Princes Street. This may be due to the ozone deposition velocity during the da y over rural areas reaching such a value that depletion of ozone at the surface exceeds the mix ing down of ozone rich air from aloft. The small deposition velocity and greater mechanica l turbulence over the city ensure that this does not occur at the Prices Street site. 7 Conclusions The Lagrangian column model has been used in this study to pro duce a series of one- and two- dimensional maps of surface ozone concentrations under a va riety of meteorological conditions.Nicholson et. al. 21 One-dimensional straight-line trajectories under averag e summertime boundary-layer conditions show a depletion of surface ozone over an urban area of 40% in d aytime with a downwind recovery to approximately 80% of upwind rural values at 20km and 100% depletion with an eventual recovery to 50% downwind at night. Similar traject ories run in winter, with lower insolation levels and greater mean wind-speeds show 90% of t he surface ozone is depleted in both daytime and nighttime conditions. At nighttime, the do wnwind recovery to 60% of the upwind concentration is faster than the daytime recovery. Vertical proﬁles upwind, within and downwind of the city sho w how the concentration gradient in the lowest 150m increases as the column passes over the cit y and then decreases again downwind as the loss of ozone is evenly spread through the lay ers. The upwind and downwind proﬁles have very similar shapes - at midday in June the diﬀer ence between the two proﬁles is approximately 12 µg m−3throughout the proﬁle. A vertical section through the simul ated city for December shows that 50% depletion is restricted to the lo west 20m of the vertical column and that at 100m altitude, the eﬀects of the city on ozone conc entrations are barely discernible. Two-dimensional maps of annual mean surface ozone concentr ations show that the depletion of ozone by a city is restricted to very close to the urban area . A depletion of 20% of the background ozone in a simulated Edinburgh only extends beyo nd the city boundaries downwind of the prevailing wind direction while 50% depletion is comp letely conﬁned within the city boundaries. At this stage we can revisit the omission of ozone generation from the model formulation. Typical gradients of ozone concentration around the city bo undary are 20-30 µg m−3per 5km (see Fig. 6). Observations of ozone generation rates are typ ically at least an order of magnitude less than this (e.g. (Weston et al. 1989)), so that patterns o f concentration would not be signiﬁcantly aﬀected by its inclusion.Nicholson et. al. 22 The diurnal cycle of ozone concentrations at urban and rural sites has been compared with monitoring site data. The model recreates the shape of the di urnal cycle well - but consistently underestimates the city-centre surface concentrations. T his could be due to underestimat- ing atmospheric turbulence, overestimating emissions or t he fact that the single point of the monitoring station does not accurately represent the full 1 x1km grid-square. The model also overestimates the concentrations at the rural site, which i s more dependent on the emissions from local roads than used in the model. 8 Acknowledgements The authors acknowledge Dr Rod Singles and Dr Helen ApSimon e t al. for the use of the TERN model and their expert advice. Ms Mhairi Coyle, Dr Chris Flec hard (C.E.H., Edinburgh), Dr Justin Goodwin (NETCen) and Edinburgh City Council Roads Department are thanked for supplying data. J. Nicholson was in receipt of a Natural E nvironment Research Council studentship.Nicholson et. al. 23 References Angle, R. and Sandhu, H.: 1989, Urban and rural ozone concent rations in Alberta, Canada., Atmospheric Environment 23, 215–221. 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European Se- ries 23 , World Health Organisation.Vertical Ozone Concentration Profiles - Summer Day 20 30 40 50 60 70 80 Ozone Concentration020406080100120140Altitude (m) UpwindCityDownwindarXiv:physics/0011074v1 [physics.ao-ph] 30 Nov 2000Captions for Figures Figure 1 Hourly traﬃc count data for Edinburgh city-centre, 1997 - Us ed as a proxy for NOx emissions Figure 2 Surface (5m) O3,NOandNO2concentrations for a 100km straight line trajectory featuring a 10km diameter city with constant NOx emissions - Summer Day (top) and Summer Night (bottom) Figure 3 Surface (5m) O3,NOandNO2concentrations for a 100km straight line trajectory featuring a 10km diameter city with constant NOx emissions - Winter Day (top) and Winter Night (below) Figure 4 Vertical proﬁles of ozone for summer daytime taken upwind, w ithin and downwind of a modelled city Figure 5 Ozone concentrations (5m) for summer and winter midday traj ectories taken upwind and downwind of the city under a range of diﬀerent wind-speed s Figure 6 30x30km annual mean, simulated, surface ozone concentrati on for Edinburgh Figure 7 30x30km simulated, surface ozone concentration for Edinbu rgh - December, 1200hr Figure 8 North-south vertical section (0-150m) of ozone concentrat ion from a 30km trajectory through Princes Street for 1200hrs - December Figure 9 Diurnal variation of hourly-averaged ozone concentration for Princes Street and Bush, modelled and observed for DecemberDiurnal Hourly Ozone Concentrations - Modelled and Observed 0 4 8 12 16 20 24 Hour of Day0204060Ozone Concentration ( µgm-3) Princes Street (Obs)Bush (Obs) Princes Street (Mod)Bush (Mod)NO/NO2/O3 Concentrations - Summer Day 0 20 40 60 80 100 Distance - km020406080100[O3](µgm-3) O3CITY 050100150200250 [NOx](µgm-3) NONO2 NO/NO2/O3 Concentrations - Summer Night 0 20 40 60 80 100 Distance - km020406080100[O3](µgm-3)CITY 050100150200250 [NOx](µgm-3) O3NONO2arXiv:physics/0011074v1 [physics.ao-ph] 30 Nov 2000Table Headings Table 1 Edinburgh wind-rose data (Turnhouse, 1971-1990) June 0000 and 1200hrs. Table 2 Edinburgh wind-rose data (Turnhouse, 1971-1990) December 0000 and 1200hrs. Table 3 Location and description of the monitoring sites in and arou nd Edinburgh. Table 4 Input parameters for the model for the simulated seasonal an d diurnal trajectories. Table 5 Annual and diurnal variations in the ozone and NOx depositio n velocities to rural and urban areas. Table 6 Annual average ozone concentrations at the three monitorin g sites: both modelled and observed results.Ozone Concentration Dependence on Wind-Speed 0 2 4 6 8 10 Wind-Speed (m/s)020406080Ozone Concentration ( µgm-3) Dec 1200 - Upwind Dec 1200 - DownwindJune 1200 - Upwind June 1200 - DownwindNO/NO2/O3 Concentrations - Winter Day 0 20 40 60 80 100 Distance - km020406080100[O3] (µgm-3)CITY 0100200300 [NOx] (µgm-3) O3NO NO2 NO/NO2/O3 Concentrations - Winter Night 0 20 40 60 80 100 Distance - km020406080100[O3] (µgm-3)CITY 0100200300 [NOx] (µgm-3) O3NONO2
arXiv:physics/0011075v1 [physics.acc-ph] 30 Nov 2000CLNS 00/1706 Novel Method of Measuring Electron Positron Colliding Beam Parameters D. Cinabro, K. Korbiak Department of Physics and Astronomy, Wayne State Univerist y, Detroit, MI 48202, USA R. Ehrlich, S. Henderson, N. Mistry Laboratory of Nuclear Studies, Cornell Univeristy, Ithaca, NY 14853, USA (29 November 2000) Abstract Through the simultaneous measurement of the transverse siz e as a function of longi- tudinal position, and the longitudinal distribution of lum inosity, we are able to measure theβ∗ y(vertical envelope function at the collision point), verti cal emittance, and bunch length of colliding beams at the Cornell Electron-positron Storage Ring (CESR). This measurement is possible due to the signiﬁcant “hourglass” e ﬀect at CESR and the excel- lent tracking resolution of the CLEO detector. PACS numbers: 29.27.Fh, 41.75.Ht, 29.40 Submitted to Nuclear Instruments and Methods in Physics Res earch Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 1One of the diﬃcult problems in colliding beam physics is the m easurement of beam param- eters at the collision point. In this letter, we present a met hod of making such a measurement using precision measurements of the luminous region with e+e−→µ+µ−events detected by a general purpose high energy physics experiment. Key to thi s measurement is the detailed geometry of the highly focused colliding beams, which leads to the “hourglass” eﬀect.[1] Tightly focused beams have a “waist” at the focal point of the ﬁnal quadrupoles and their size grows away from this waist. The transverse beam size is g iven by σ(z) =/radicalBig ǫβ(z) (1) where β(z) is the amplitude or beta function, which depends on the long itudinal position, z, of the beam and the emittance, ǫ, which is independent of z. Near a waist, β(z) can be written as β(z) =β∗+(z−z0beta)2 β∗(2) where β∗is the value of the beta function at the waist and z0betais the longitudinal position of the waist. Thus the beam in the longitudinal and transvers e dimensions forms an hourglass shape with a minimum size at z0beta. The hourglass eﬀect arises from the beam being in Gaussian sh aped bunches with length σz. If the bunch length is long compared with the dimension of th e waist, then little of the beam is colliding where the beam is narrowest. In this case, t he longitudinal distribution of luminosity depends not only on σz, but also on the horizontal and vertical value of the beta function at the interaction point, β∗ xandβ∗ y, respectively. In addition if either β∗is smaller thanσz, luminosity does not improve as much as naively expected by m aking β∗smaller. The luminous region is deﬁned by the overlap integral of two b eams. Thus we expect the vertical width of the luminous region as a function of the lon gitudinal position to be given by σy(z) =/radicaltp/radicalvertex/radicalvertex/radicalbtǫy 2/parenleftBigg β∗y+(z−z0beta)2 β∗y/parenrightBigg , (3) and similarly for the horizontal width. It is assumed that th e emittances and β∗’s are the same for the two beams. The beam parameters for the Cornell El ectron-positron Storage Ring (CESR) for the data discussed in this paper are given in Table 1. All the parameters in Table 1 are given at zero bunch current. They are all expected to depe nd on the bunch current. We have previously observed that β∗ xis reduced by roughly a factor of two in colliding beam condit ions due to the dynamic beta eﬀect.[2] Likewise, β∗ yis expected to be reduced by about 25% due to beam-beam focusing. The vertical emittance depends on the b eam-beam tuneshift parameter; for operation in the saturated tuneshift regime, the vertic al beam size increases linearly with bunch current. Additionally, there are streak camera obser vations which show an increase in σzas the bunch current increases.[3] The method described in t his paper is aimed at measuring some of these dynamic eﬀects by direct observation of the lum inous region. Note that these measurements are taken over a long time, abou t four months of CESR and CLEO running, and at many diﬀerent machine conditions. T hus we expect only rough 2Table 1: CESR beam parameters at zero bunch current during th e time this measurement was made. Parameter Value ( µm) β∗ x 1.1996×106 ǫx 0.21 β∗ y 17900 ǫy 0.0010 σz 18100 agreement with the parameters given in Table 1, but we should be sensitive to the dynamic eﬀects discussed above. We expect to observe a larger ǫyandσz, and a smaller β∗ yandβ∗ xthan given in Table 1. Figure 1 shows σyas given by Equation 3 using the beam parameters given in Tabl e 1. Also shown is the expected longitudinal distribution of luminos ity. This is given by[1] dL dz=L0exp/parenleftBig−(z−z0bunch )2 σ2z/parenrightBig (1 +(z−z0beta)2 β∗2x)1/2(1 +(z−z0beta)2 β∗2y)1/2, (4) where z0bunch is the longitudinal position of the bunch-bunch collision. Note the longitudinal distribution of luminosity is expected to signiﬁcantly dep end on β∗ y, but the β∗ xdependence is negligible. This is due to the large size of β∗ xas compared to σz. Thus we expect a negligible hourglass eﬀect in the horizontal size of the luminous regio n as a function of longitudinal position. This is also why we consider only one value for z0betawhich could, in principle, be diﬀerent for the horizontal and vertical beta functions. Our goal is to measure the beam parameters, β∗ y,ǫy, and incidentally σz. We do this with a simultaneous ﬁt to the measured vertical width of the lumin ous region versus longitudinal position, and the longitudinal distribution of luminosity . The vertical width depends on ǫy, the longitudinal distribution on σzand they both depend on β∗ y. CESR has been described in detail elsewhere. [4] All the data used in this measurement are taken at an e+e−collision energy of 10.58 GeV, and with bunch currents in the range of 2.5 to 7.0 mA over a four month period in late 1998 and early 1999. T he CLEO detector has also been described in detail elsewhere.[5] All of the data used i n this measurement are taken in the CLEO II.V conﬁguration which includes a silicon strip verte x detector which is crucial to the measurement of the luminous region. This consists of three l ayers of silicon wafers arrayed in an octagonal geometry around the interaction point. The ﬁrs t measurement layer is at a radius of 2.3 cm and the wafers are read out on both sides by strips whi ch are perpendicular to each other. The readout strips have a pitch of about 100 µm and with charge sharing the detector has an intrinsic per point resolution of better than 20 µm in both the transverse plane and the longitudinal direction To obtain a resolution of order 10 µm on the luminous region, we selected e+e−→µ+µ− events. These are easily selected in CLEO as events with two a nd only two tracks each with 3Vertical Width Luminous Region15 II10 005 50000 25000 25000 50000 Longitudinal Position ( m)Vertical Width of Luminous Region ( m)3541100-002 Figure 1: Expected vertical width of the luminous region as a function of longitudinal position. Also shown, in arbitrary units, is the expected longitudina l distribution of luminosity. 4y-axis x-axis2 3 1<3> <2><3> <1>3541100-001 Figure 2: An ensemble of stiﬀ tracks passing through the box a llows for a precision measure- ment of the luminous region. For example, the track labeled 1 only gives a useful measure of the vertical position of the luminous region, as indicated. Tra ck 1 crosses the the entire horizontal extent of the box and its average horizontal position is simp ly the center of the box, rather than the center of the luminous region. Similarly track 2 onl y measures the horizontal position, while track 3 measures both the horizontal and vertical posi tions of the luminous region. momentum near the beam energy and a small energy deposit in th e electromagnetic calorimeter. We chose tracks with 20 or more hits in the main drift chamber, and at least two silicon vertex detector hits in the transverse and longitudinal views. We r equire that the tracks have opposite charge and that those used for the measurement of the luminou s region have at least three silicon vertex detector hits in one of the two views. We implement a method, called the “box technique,” to obtain measurements of the beam parameters and the resolution. Figure 2 shows how this techn ique is implemented. First, the size and location of the luminous region are obtained from ru n average data using hadronic events.[2] A three-dimensional box is then centered about t he measured center of the luminous region with sides ten times the measured widths of the lumino us region. The average position of a track passing through the box is found. From an ensemble o f such positions the size and 5shape of the luminous region is measured. Tracks that are parallel to an axis of interest are the most us eful for measuring the luminous region. Tracks that are perpendicular to an axis cross the fu ll length of the box in that direction and give no information about the luminous region. We select appropriate tracks by cutting on the direction cosines. Essentially, these cuts are deter mined by the size of the luminous region, which is roughly 10 µm vertically, 300 µm horizontally, and 10000 µm longitudinally. Thus a tight cut of \|py/p\| ≡ \|cosθy\|<0.1 is needed to measure the vertical luminous region, a looser cut of \|cosθx\|<0.3 for horizontal, and \|cosθz\|<0.7 for longitudinal. For tracks with large \|cosθz\|, the resolution degrades, and the \|cosθz\|cut of 0 .7 is also used on tracks to make vertical and horizontal measures. Because of these dir ection cosine cuts, a single track can measure, at most, two dimensions. We tested this method using over 100,000 e+e−→µ+µ−simulated events. To measure the change in the vertical size of the luminous region as expe cted from the hourglass eﬀect, a constant vertical resolution is necessary. Thus we made se lections in the simulated data to test the stability of the resolution and found signiﬁcant de pendences only on tracks with large values of \|cosθz\|, which are eliminated by the direction cosine cut discussed above, and on the \|cosθy\|of the tracks. The 0 .1 cut on the \|cosθy\|is a compromise between a smaller cut value with improved resolution, and a larger value with increased statistics. Thus the resolution on the vertical luminous region is expected to be 26 .4±0.4±1.5µm with the ﬁrst error due to the statistics of the simulation sample and the second due to the sharp dependence on the \|cosθy\|cut. This vertical resolution is small enough and there are s uﬃcient data to obtain a statistically useful sample to measure the size of the lumi nous region at large longitudinal positions. These can be compared with similar measures domi nated by the resolution taken at small longitudinal positions. In the analysis, we extrac t the resolution from the data itself, thus there is no dependence on the prediction from the simula tion. Figure 3 shows the vertical width of the luminous region as a f unction of its longitudinal position. This plot clearly shows the vertical distributio n growing away from the center. This is evidence of the hourglass eﬀect. We have also repeated thi s procedure for the horizontal width. We observe a horizontal width of 296 µm and see no signiﬁcant hourglass eﬀect. These observation both agree with our expectation. When measuring the longitudinal distribution of luminosit y as a function of the longitudinal position, we see a sharp enhancement in the distribution for small values of the longitudinal position. This enhancement is caused by the existence of a no n-sensitive region in the center of the detector which greatly diminishes the chance for trac ks passing through this region to be accepted for analysis. This geometric eﬀect is accuratel y modeled in our simulation, and we use it to extract a longitudinal position dependent correct ion for the longitudinal distribution of luminosity. Applying this eﬃciency eliminates large sys tematic eﬀects in our extraction of β∗ y. The longitudinal distribution of luminosity both before a nd after the eﬃciency correction is shown in Figure 4. To extract the beam parameters, we ﬁt Figure 3 to Equation 3, i ncluding resolution smear- ing. Unfortunately, such a ﬁt does not give a useful measurem ent of any of the beam parameters, and has nearly 100% correlations among β∗ y,ǫy, and the resolution. We take advantage of the dependence of the longitudinal distribution of luminosity onβ∗ yas given in Equation 4, and 63541100-004 30 0102040 25000 50000 25000 50000 0II Longitudinal Position ( m)Measuredy ( m) Figure 3: The vertical width of the luminous region as a funct ion of longitudinal position. The line shows the ﬁt discussed in the text. 7101 0 25000104 101102103102103104 0 50000 25000 50000 0II Longitudinal Position ( m)Tracks / 250 ( m)3541100-003 Figure 4: The longitudinal distribution of luminosity. The top plot shows the raw distribution. Note the slight enhancement near zero caused by detector geo metry as discussed in the text. The bottom plot shows the eﬃciency corrected distribution w ith the ﬁt discussed in the text superimposed. 8Table 2: The results of the simultaneous ﬁt to the data distri butions for the vertical width of the luminous region as function of longitudinal position an d the longitudinal distribution. Only statistical errors are shown. Parameter Fitted Value ( µm) β∗ y 15699±138 ǫy 0.0060 + 0 .0047−0.0042 σz 19288±38 resolution 25.8±1.7 z0beta −2885±101 z0bunch 852.8±34.7 perform a simultaneous maximum likelihood ﬁt to Figures 3 an d 4 to Equations 3 and 4. This simultaneous ﬁt gives us an additional constraint on β∗ ywhich breaks the correlations among the parameters. In this ﬁt we ﬁx the value of β∗ xto 417500 µm based on our observation of the horizontal width of the luminous region and the expected ǫxgiven in Table 1. The results are not sensitive to the exact value of β∗ xused in the ﬁt and change negligibly if β∗ xis left to ﬂoat. If β∗ xis left to ﬂoat the ﬁt returns a value consistent with 417500 µm but with large errors of ±500000 µm. The resolution of the box technique on the longitudinal posi tion of the event production point is better than 40 µm. This is negligible in comparison with the longitudinal si ze of the luminous region, which is over one centimeter. In fact, the b ox technique can be used to make a very high precision measurement of the bunch length, σz. As discussed in Reference [3] the bunch length in CESR is seen to depend on the bunch current and the bunches are asymmetric with their heads being narrower than their tails. Due to the c ollision of the two bunches this single bunch asymmetry is washed out in the longitudinal dis tribution of the luminous region. The luminosity distribution would only be distorted away fr om a Gaussian shape if the single bunch asymmetry were an order of magnitude larger than the ob served ∼5%. If we allow an asymmetry in the luminosity distribution, we observe none w ith a±1% accuracy. This also conﬁrms our need for an eﬃciency correction which takes out a 1.5% asymmetry in the raw data. We have tested this simultaneous ﬁt procedure with the simul ation, and expect that our ﬁt is able to measure the input beam parameters and the resoluti on. We also derived expectations on the errors and correlations the ﬁt should return based on o ur data statistics. Table 2 shows the results of the ﬁt to the data. From the ﬁt we ob serve some large corre- lations. These are between β∗ yandσz, between ǫyand the resolution, and between z0betaand z0bunch These correlations are -88%, -75%, and -85% respectively. T hey are of the size predicted by our tests on simulated data. All other correlations are sm aller than 40% in magnitude.. These are in good agreement with the errors expected from the simulation study, the CESR beam parameters of Table 1, and streak camera observations. [3] Note that we obtain very accurate measures of β∗ yandσz, along with a resolution from the data consistent with our expectation of 26 .4±1.6µm from the simulation, but only a 1.4 standard deviation meas ure 9ofǫy. The results follow our expectations from the dynamic eﬀect s caused by the non-zero bunch current. The value for β∗ yis lower than the zero bunch current value, but not as small as the lowest recorded. The value for ǫyis not measured well enough to make a meaningful test, but it is certainly consistent with an increase. The σzis increased by 6.6% which is consistent with the streak camera observations.[3] The diﬀerence betw eenz0betaandz0bunch,−3740± 130µm, is consistent with known strength diﬀerences between the ﬁnal focus quadrupoles and alignment tolerances between ﬁnal focus elements and RF cav ities which respectively determine the longitudinal position of the beta waist and the center of the bunch collision. In an attempt to improve the measure of ǫy, we repeat the data ﬁt with the resolution ﬁxed to 26.4µm, as predicted by the simulation. This ﬁt does give a slightl y improved measurement ofǫyof 0.0049±0.0028µm with the other parameters changing negligibly. When we var y the ﬁxed resolution by ±1.6µm as indicated by the simulation studies, this introduces an error of ±0.0028µm on ǫy. The combined error of ±0.0040µm on ǫy= 0.0049µm is consistent with, but not a substantial improvement over the results of Table 2 . We prefer to quote results for the ﬁt where the resolution is left ﬂoating. We varied the standard ﬁt to test its robustness. We excluded positive z, negative z, small z, and large zdata from the ﬁt. A χ2ﬁt is used rather than a likelihood ﬁt. The only parameters that show signiﬁcant disagreement with the standard result areβ∗ yandσz. Other facets of the analysis are varied and the procedure is repeated to estimat e other systematic eﬀects. Cuts on the direction cosines are varied, ±0.01 on cos θyand±0.1 on cos θz, we relax the three silicon vertex detector hits in one view to a looser two hit pe r view requirement, we vary the procedure for applying the eﬃciency for the luminosity a s a function on the longitudinal position, and use the simulation eﬃciency without errors as an estimate of the eﬀects of our limited simulation statistics. Some of the measured vertic al widths are not consistent with their longitudinal neighbors as can be seen in Figure 3. This indic ates a systematic error of about two microns in the extraction of the widths with the box metho d. Including this error has a negligible impact on the results of the ﬁt, increasing the st atistical errors by less than 10%. For all these variations, the change in the central values of the beam parameters from the standard procedure is taken as the systematic eﬀect. The combined eﬀe cts of these variations result in a systematic error of ±460µm on β∗ y,±0.0019µm on ǫyand±200µm on σz. 10In conclusion, we use a new box technique to measure the size o f the luminous region of CESR at the CLEO interaction region. This new method takes ad vantage of the hit resolution in the CLEO II.V silicon vertex detector and the well underst ood CLEO charged particle tracking system in e+e−→µ+µ−events to precisely measure the size of the luminous region. The technique has a resolution of 25 .5±2.0µm which we extract from a ﬁt to the data. The excellent resolution of the box technique, combined with th e large size of the CLEO II.V data set, allows us to make a clear observation of the hourglass eﬀ ect, the increase in the size of the luminous region away from the focal point. The technique lea ds to measurement of the CESR beam parameters: β∗ y= (15700 ±140±460)µm, (5) ǫy= (0.0060±0.0045±0.0019) µm, (6) σz= (19290 ±40±200)µm. (7) where the ﬁrst error is statistical and the second is systema tic, in a simultaneous ﬁt of the vertical width of the luminous region as a function of the lon gitudinal position and of the lon- gitudinal distribution of the luminosity. Note that these m easurements imply that the vertical width of the CESR beam at the CLEO collision point is 6 .9±2.8µm. These measurement are in good agreement with the expectations of the theoretical C ESR lattice taking into account the dynamic eﬀects caused by the non-zero bunch current. This te chnique provides a non-invasive way to measure beam parameters at the collision point, but it requires the comparatively rare e+e−→µ+µ−events and a well understood detector tracking system. References [1] D. Rice private communication; S. Milton, Cornell Colli ding Beam Note 89-1 (unpub- lished); M. A. Furman, Proceedings Particle Accelerator Conference 1991 , 422. [2] D. Cinabro, et alPhysical Review E57, 1193 (1998). [3] R. Holtzapple, et al, Phys. Rev. ST Accel. Beams 3, 034401. [4] S.B. Peck and D.L. Rubin, ”CESR Performance and Upgrade S tatus,” Proceedings Particle Accelerator Conference 1999 , 285. [5] Y. Kubota, et al. (CLEO Collaboration), Nuclear Instruments and Methods, A820 , 66 (1990); T.S. Hill, Nuclear Instruments and Methods in Physi cs Research, A418 , 32, (1998). 11
arXiv:physics/0011076v1 [physics.flu-dyn] 30 Nov 2000The Numerical Solution of Nekrasov’s Equation in the Boundary Layer near the Crest, for Waves near the Maximum Height by J.G. Byatt-Smith Department of Mathematics and Statistics University of Edinburgh. Key Words :Integral equations, water waves. Abstract : Nekrasov’s integral equation describing water waves of pe rmanent form, determines the angle φ(s) that the wave surface makes with the horizontal. The independent variable sis a suitably scaled velocity potential, evaluated at the free surface, with the origin corresponding to the crest of the wave. For all waves, except for amplitudes near the maximum, φ(s) satisﬁes the inequality \|φ(s)\|<π/6. It has been shown numerically and analytically, that as the w ave amplitude approaches its maximum, the maximum of \|φ(s)\|can exceed π/6 by about 1% near the crest. Numerical evidence suggested that this oc curs in a small boundary layer near the crest where \|φ(s)\|rises rapidly from \|φ(0)\|= 0 and oscillates about π/6, the number of oscillations increasing as the maximum amplitude is approached. McLeod derived, from Nekrasov’s equation, the following in tegral equation φ(s) =1 3π∞/integraldisplay 0sinφ(t) 1 +/integraltextt 0sinφ(τ)dτlog/vextendsingle/vextendsingle/vextendsingle/vextendsingles−t s+t/vextendsingle/vextendsingle/vextendsingle/vextendsingledt forφ(s) in the boundary layer, whose width tends to zero as the maxim um wave is approached. He also conjectured that the asymptotic form ofφ(s) ass→ ∞ satisﬁes φ(s) =π 6/braceleftbig 1 +As−1sin (βlogs+c) +o(s−1)/bracerightbig , whereA,βandcare constants with β≈0·71 satisfying the equation √ 3βtanh1 2πβ= 1. We solve McLeod’s boundary layer equation numerically and v erify the above asymptotic form. 11 Introduction This paper considers the numerical solution of the equation φ(s) =1 3π∞/integraldisplay 0sinφ(t) 1 +/integraltextt 0sinφ(τ)dτlog/vextendsingle/vextendsingle/vextendsingle/vextendsingles+t s−t/vextendsingle/vextendsingle/vextendsingle/vextendsingledt (1.1a) =−1 3π∞/integraldisplay 0k(t,s){ψ(t)−ψ(s)}dt, (1.1b) where ψ(t) = log/parenleftbigg 1 +/integraldisplayt 0sinφ(τ)dτ/parenrightbigg andk(t,s) =2s s2−t2. (1.2) This equation was derived by McLeod [1] to describe the bound ary layer behav- ior of the solution , for large µ, near the origin of the equation φµ(s) =1 3π/integraldisplayπ 0sinφµ(t) µ−1+/integraltextt 0sinφµ(τ)dτlog/vextendsingle/vextendsingle/vextendsingle/vextendsingleF(s+t) F(s−t)/vextendsingle/vextendsingle/vextendsingle/vextendsingledt, (1.3) whereF(t) = sn (Kt/π) and sn denotes the Jacobian elliptic function with quarter periods KandiK′. Equation (1.3) was ﬁrst formulated by Nekrasov [2] to describe waves of constant periodic form moving with cons tant speed on the surface of a non-viscous ﬂuid that is either of inﬁnite depth or on a horizontal bottom, when the ﬂow is taken to be irrotational. The wave is a ssumed to be symmetric about its crest and the equation is derived by conf ormally mapping the the region of the ﬂow under one wavelength onto the unit di sc cut along the negative real axis. The generic point on the circumferen ce of the disc is eis, with−π < s < π , ands= 0 corresponds to the crest. As the circumference is described in a clockwise direction from −πtoπthe horizontal coordinate decreases by one wavelength. Then the function φµis the angle that the wave surface makes with the horizontal. With this choice of coord inateφµ(s) is periodic with period 2 π. For more details, see Nekrasov [2],[3] and [4] or Milne- Thompson [5]. The wave is assumed to be symmetric about its cr est. Thus φµ(s) is an odd 2 πperiodic function of swithφµ(0) = 0. The solution is unique provided the additional assumption, that the wave ha s only one peak and one trough per period, is made. This is φµ(s)>0, s∈(0,π) withφµ(0) =φµ(π) = 0. (1.4) The constants KandiK′, the quarter periods of sn, are related to the depth h and wavelength, λ, by the relation K′/K=h/λ. (1.5) 2Ash→ ∞ we haveK→1 2π(K′→ ∞) andF(t)→sin1 2tso that (1.3) is also applicable for inﬁnite depth. Equation (1.1) is derive d by writing ˆ s=sµ and writing φµ(sµ) =ˆφ(ˆs) and letting µ→ ∞ with ˆsﬁxed. Then ˆφ(s) satisﬁes (1.1). The boundary layer behavior of the solution of (1.3) w as established numerically by Chandler and Graham [6], who were able to obta in a solution with a maximum value of φµ(s)≏30·3787...◦and to detect a small number of oscillations about φµ= 30◦forµ= 1018. The numerical diﬃculty posed by the boundary layer behavior of the solu- tions of (1.3) for large µis over come, by Chandler and Graham [6], by using a non uniform mesh for the discretisation of (1.3). This consi sts of three regions: one to cope with the rapid variation of φµ(s) in the boundary layer, whose thickness is of order µ−1,near the origin; a second to deal with the slower vari- ation away from the origin and a third for the transitional la yer in between. For further references on the analytical properties of the s olutions of (1.3) and related numerical results, see Chandler and Graham [6] and M cLeod [1]. The purpose of this paper is to solve (1.1) numerically and sh ow that the solutionφ(s) oscillates about φ(s) =π/6 and obeys the formal asymptotic result of McLeod [1] that can be written in the form φ(s) =π 6/braceleftBigg 1 +∞/summationdisplay n=0An snsin(nβlogs+cn)/bracerightBigg ass→ ∞, (1.6) whereAnandCnare constants and β= 0·71...is the root of √ 3βtanh/parenleftbigg1 2πβ/parenrightbigg = 1. (1.7) Equation (1.1) represents the solution in the boundary laye r and can thus be solved with a uniform mesh size. However (1.1) has an additio nal complication compared with (1.3) in that the range of integration is inﬁni te and the decay of the solution to its asymptotic limit is algebraic. This fact means that we require careful consideration in order to obtain an accurate numeri cal representation of the integral in (1.1). 32 The Numerical Method Following Chandler and Graham [6] we solve the integral equa tion in the form (1.1b). This formulation is better, for numerical purposes , because the integra- tion by parts that is used to convert (1.1a) to (1.1b), remove s the logarithmic singularity, at t=s,which occurs in the kernel of (1.1a). Although the corre- sponding kernel of (1.1b) has a pole, the singularity of the i ntegrand is removable since the multiple ψ(t)−ψ(s), has a simple zero at t=s. Thus we write φ(s) =1 3π/integraldisplay∞ 0K(t,s)dt, (2.1) where K(t,s) =−2s(ψ(t)−ψ(s)) s2−t2t/ne}ationslash=s (2.2a) =ψ′(t)≡sinφ(t) 1 +/integraltextt 0sinφ(τ)dτt=s, (2.2b) the value in (2.2b) being the limit of the right hand side of (2 .2a) as \|t−s\| →0. We aim to set up a numerical approximation to the integral in t erms of a discrete number of values φ(si), wheresi=ih, 0≤i≤2N, withNan integer, for suitable choices of handNand a continuous set of values φ(s) for s/greaterorequa≏s≏ant2Nh. Any values of φ(s) fors<0 required by the numerical approximation are determined by the fact that φ(s) is an odd function of s. The numerical representation of the integral requires two approaches. Th e ﬁrst is a ﬁnite diﬀerence formulation of the integral over a predetermined ﬁnite range using the discrete values of φand the second is an estimation of the remainder using an appropriate asymptotic estimate of the values of φ(s) fors/greaterorequa≏s≏ant2Nh. The details of the asymptotic form of φ(s) ass→ ∞ that is used will be discussed later. So we choose an appropriate end point 2 TwhereTis given by T=Nhand we can approximate the integral I1(s,φ) =/integraltext2T 0K(t,s)dtusing Simpson’s Rule, since the integrand is analytic. The choice of the end point 2 Tis some what arbitrary. Eventually, see below, we will want to consider I1(s,φ) for values ofs≤T. We choose an end point mT, withm= 2 in this case, so that the singularity of k(t,s) att=sis far from the end point. The reason for this is that the remainder integral, again see below, requires a d iﬀerent evaluation and it is advantageous to make sure that the singularity of k(s,t) is not close to the range of tin the remainder integral. This will become clearer when the evaluation of the remainder integral is discussed later. Assuming that for large s, φ(s) is known in the form of an asymptotic ex- pansion then truncation of this series, expansion of the int egrand and a term by term integration of the integrand will give a suitable ana lytical estimate EI2(s,φ) for the integral I2(s,φ) =/integraltext∞ 2TK(t,s)dt. Then we deﬁne the numer- 4ical representation of the integral in (2.1) as NI(s,φ) =NI1(s,φ) +EI2(s,φ). (2.3) An alternative approach, assuming that the asymptotic form ofφ(s),s > T , has been chosen, is to transform the inﬁnite range of the rema inder integral into a ﬁnite range, which can then be approximated numerically. F or this purpose it is more convenient to revert to the integral in the form (1. 1a) so we write I2(s) = log/parenleftbigg2T+s 2T−s/parenrightbigg (ψ(2T)−ψ(s)) +∞/integraldisplay 2Tk3(s,t)dt, (2.4) where k3(s) =sinφ(t) 1 +/integraltextt 0sinφ(τ)dτlog/parenleftbiggt+s t−s/parenrightbigg (2.5) Ifφ(t)→π/6 +O(t−1) and/integraltext∞ 0(φ(t)−π/6)dtis bounded, it is easily estab- lished that k3(s,t) = 2st−2+o/parenleftbig t−2/parenrightbig ast→ ∞.Thus the integral of k3, in (2.4) is convergent at inﬁnity and the substitution t= 2T/utransforms it to/integraltext1 0k4(s,u)duwithk4(s,0) =s/T.This integral can now be approximated using Simpson’s rule with a suitably chosen step length. This appr oximation can be used instead of EI2(s,φ) in (2.3). Simpson’s rule gives an approximation which is of order h4, but this rule requires an interval which consists of an even number of step lengths. However the integrand contains the function ψ(t) which involves the determination of/integraltextt 0sinφ(τ)dτat valuest=ti=ih. To obtain a numerical approximation to this which is the same order as Simpson’s rule for this integral we use an appropriate modiﬁed trapisoidal rule. We now wish to solve the approximation φ(s) =1 3πNI(s,φ). (2.6) To do this we deﬁne an approximation φN(si) to the solution φ(s) at the discrete valuessi=ih,0≤i≤N. Using the same asymptotic form at the solution as that used to deﬁne φ(s) fors/greaterorequa≏s≏ant2Nhwe deﬁne the remaining discrete values of φN(si),N+ 1≤i<2Nh, required for the evaluation of NI1(s) at the points s=si, 0≤i≤N. ThusφN(si) satisﬁes the equations φN(si) =1 3πNI(si,φN(sj)),0≤i≤N. (2.7) This gives, in a similar fashion to Chandler and Graham [6], a fully discrete non-linear system for the unknowns {φN(si),i= 0..N}. This system is solved by the iterative method φm N(si) =NI/parenleftBig si,φ(m−1) N(sj)/parenrightBig , i= 0..N, (2.8) 5starting from a suitable initial approximation φ(0) N(si). Chandler and Graham [6] were able to prove that, when the quadrature method used t o approximate their integrals was the trapisoidal rule, convergence was g uaranteed, although for computational purposes they opted for a more accurate sc heme for computa- tional purposes. Their proof cannot be extended to the numer ical approximation used here even if the quadrature method is the trapisoidal ru le because of the inﬁnite range of integration. However we ﬁnd that, as in the c ases looked at by Chandler and Graham [6], the convergence rule is very quick. 63 The necessity of rescaling We see from the deﬁnition of K(t,s),(2.2a,b), and the fact that φ(0) is zero, thatNI(0,φ) = 0 provided the initial guess φ(0) N(0) = 0. Then (2.7) gives φm N(0) = 0 for all m >0. Thus eﬀectively we can work with the Nvariables {φN(si),i= 1..N}and corresponding Nequations from (2.7). One of the aims is to verify the asymptotic result (1.6). Initially we do not assume this and report here that for a variety of sensible choices of the asym ptotic form of φ(s) we get rapid convergence to the solution of (2.8). Provided Tis suﬃciently large we can then numerically verify that (1.6) is the correc t asymptotic result, using the computed values of φ(s) fors≤T. Having veriﬁed this numerically to get the best accuracy we use (1.6) and ﬁnd that as well as pro viding a more accurate numerical solution the convergence rate is also im proved. The larger Tis, the less necessary it is to have a large number of terms fro m (1.6) and in practice we use φ(s) =π 6/parenleftbigg 1 +A ssin (βlogs+c)/parenrightbigg , s>T. (3.1) Table 1 shows the comparison of the location and the values of φ(s) at successive maximum and minimum values of φand the comparison between this method at that of Chandler and Graham [6]. Before discussing this comp arison we use the values ofsat the successive turning points to illustrate the need for r escaling the variable s. It will become clear that the computations done to obtain ta ble 1 could not be achieved by the method outlined in paragraph 1. We see that the s coordinate of each successive turning point increases by a factor of about 81, which is approximately the value of eπ/β. This is compatible with the set of turning points obtained from (3.1). The last turning poin t in 0< s < T is located ats= 2×1011. Typically we used h= 1/20 as a sensible choice of h compatible with having a large enough Tto capture the asymptotic behavior of the solutions. However with this choice of hit is not feasible to take T= 2×1011 as this would involve 4 ×1012grid points. Typically using the scheme outlined in paragraph 1 we chose T= 100 and this does not even get to the ﬁrst minimum ofφ(s). However we learn from this initial attempt at a numerical s olution that beyonds= 100,6\|φ(s)−π/6\|/π<10−2and varies very slowly. Thus for large swe do not need to take such a small step length. For the numerical scheme we have used, we require a constant s teplength so we make a simple change of independent variable. We wish to make no eﬀective change at the origin but an exponential change at in ﬁnity so we use the transformation s=ey−1.Then witht=ez−1 andθ(y) =φ(s(y)),(1.1) becomes θ(y) =−1 3π/integraldisplay∞ 02 (ey−1) (ey−ez)(ey+ez−2)log/parenleftBigg 1 +/integraltextz 0sinθ(ζ) 1 +/integraltexty 0sinθ(ζ)/parenrightBigg dy, (3.3) We are then able to reduce the step length, h, and still take T=eyT−1 to be large. Typically we take h= 1/100 andyT= 30 giving T= 1.0×1013. This requires 3000 unknowns φ(yi) whereyi=ih, i = 1..3000. 7After the rescaling, the numerical scheme is essentially th e same as that given in section 2 and is not repeated. However near y=yT, 6\|θ(y)−π/6\|/πis now of order 10−13so the form of θ(y) eﬀectively given by (3.1) will be accurate to 10−26, that isO/parenleftbig T−2/parenrightbig . 84 The Numerical Results and Conclusions All the numerical results given here are those produced by th e numerical scheme outlined in Section 2 and 3 using the rescaled problem. Table 1 shows the comparison of the successive maxima and minima of φ(s) compared with those computed for the full problem by Chandler and Graham [6]. The position of these maxima and minima for the Chandler and Graham [6] com putation, has been calculated by scaling their coordinate, s, byµcompatible with the boundary layer scaling used to derive (1.1) from (1.3). Thus s=sB−S=sCeG× µ. The number of decimal places given in table 1 for this numeri cal computation are as accurate as the numerical calculation will allow. The re are three forms of error: the ﬁrst comes from the order of the numerical appro ximation to the solution which is O/parenleftbig h4/parenrightbig which gives rise to errors of order 10−8; the second is due to machine accuracy which gives rise to an error of about 1 0−14to 10−16; thirdly there is the error that arises when predicting the po sition and size of the maxima and minima of a function, from discrete data at giv en grid points, assuming that the data is accurate. The ﬁgures quoted in tabl e 1 do not take into account the ﬁrst of two of these sources of error. The comparison with the computations of Chandler and Graham [6] is very good. The value at the ﬁrst maximum is the same to eight signiﬁ cant ﬁgures and the position the same to six signiﬁcant ﬁgures. The calcu lation of the value at the maximum always being more accurate that its posi tions. The values at the ﬁrst minimum are in similar agreement although Chandler and Graham [6] only quote the position to four signiﬁcant ﬁgures and the value at the minimum is only 4 ×10−3below 30◦so relatively the numbers do not appear to be in such good agreement as the value at the ﬁrst max imum. The ﬁrst noticeable divergence of the two computations appears at th e second minimum where the estimates of the positions diﬀer by about 4% althou gh the values at this minimum are in good agreement given that they are both of order 10−7 below 30◦.However the next maximum of Chandler and Graham [6] lies belo w 30◦and it is apparent that at this value of sthe eﬀects of the outer solution, that is the decrease from the maximum on a slower scale, are ju st beginning to show. Presumably at this value of µthe oscillations in the Chandler and Graham [6] begin to cease at or around this value of s. We wish to show that the solution behaves like (1.6) for large s. So for comparison we write Θ ( x) =φ(s), wherex=β πlogsso that we expect Θ (x)∼π 6/braceleftbigg 1 +A ssinπ(x−x0) +.../bracerightbigg asx→+∞ (4.1) or Ψ (x)≡/parenleftbigg6 πΘ (x)−1/parenrightbigg s∼Asinπ(x−x0) +... , (4.2) Compared with the transformation (3.1) which has y= 0 whens= 0 we have x→ −∞ ass→0. This makes/parenleftbig6 πΘ (x)−1/parenrightbig s→0 asx→ −∞ and introduces 9a minimum of the function Ψ ( x) before the ﬁrst maximum. The values of x=xiat the minima, maxima and the zeros of Ψ ( x) and the value of Ψ ( x) at the turning points are shown in table 2. If (4.2) were to be e xact then the diﬀerencexi−xi−1−1/2≡∆xiwould be zero and the magnitude of the value of Ψ (x) at the turning points would be constant and equal to A. Included in this table are the computed values of ∆ xi. From the table we see that a good ﬁt is obtained by choosing Aandx0so that Ψ (x) and (4.2) agree at the second maximum and fourth zero this gi ves A= 1·2364860386 ...andτ0= 0·72422... . (4.3) A plot of the asymptotic expression (4.2) with these values o fAandx0and the comparison with Ψ ( x) is given in ﬁgure 1. The two graphs are indistinguishable from each other over a surprisingly large range of values of x, from before the ﬁrst zero to beyond the sixth zero. The graphs start to diverg e after this point. This is due to the fact that the exact solution of φ(s)−π/6,or equivalently Ψ(x)/s,is so small in this range that round oﬀ error starts to become i mportant and eventually dominates the solution. This is more apparen t in ﬁgures 2 and 3 which plot the diﬀerence between Ψ ( x) and its asymptotic value. Figure 2 shows this diﬀerence multiplied by 100 in the range of values of xwhere the diﬀerence is less then one, while ﬁgure 3 shows 1000 times the diﬀerence . In both ﬁgures we see that the diﬀerence increases rapidly after x≏4. It is particularly visible in ﬁgure 3 that this rapid rise has two diﬀerent components: a systematic rise due to truncation error of the numerical scheme, which is of o rder 108and a random error on the scale of about 10−14, due to machine accuracy. The last plot, ﬁgure 4, shows the diﬀerence between Ψ ( x) and its asymp- totic value multiplied by s. This clearly shows that the dominant feature is one of a periodic function of period 1, compatible with a term proportional to s−2sin2π(x−x1) that appears in (1.6). To conclude we have presented a numerical scheme for the solu tion of (1.1), written in the form (3.3) which allows a suﬃciently accurate numerical solution over a range 0 /≏essorequa≏s≏ants/≏essorequa≏s≏ant1013, that we can verify the predicted asymptotic form (1.6). The numerical calculation is limited by the two facto rs, truncation error and machine accuracy. The numerical solutions can be made mo re accurate by a higher order integration scheme but the range of integra tion is limited because the diﬀerence between the solution and π/6 becomes the same order of magnitude as the machine accuracy. 10References 1. J.B. McLeod, The Stokes and Krasovskii Conjectures for th e wave of greatest height. Stud. App. Math. 98: 311-333 (1997) 2. A.I. Nekrasov, Izv. Ivanovo-Vosnosonk. Politehn Inst. 3: 52-65 1921; 6:155- 71 (1922) 3. A.I. Nekrasov, Izv. Ivanovo-Vosnosonk. Politehn Inst. 6:155-71 (1922) 4. A.I. Nekrasov, The exact theory of steady state waves on th e surface of a heavy liquid. Technical Summary Report No 813. Mathematic al Research center, University of Wisconsin, 1967 [D.V. Thampuran, tra nslator:C.W. Cryer, editor] 5. L.M. Milne-Thompson, Theoretical Hydrodynamics , Macmillan, London, 1968. 6. G.A. Chandler and I.G. Graham, The Computation of water wa ves modelled by Nekrasov’s Equation. SIAM J. Numer. Anal. 30: 1041-1065 (1993). 11Figure Captions Table 1. Positions of the turning points, stand the corresponding values, φ(st) and comparison with those obtained by Chandler and Graham. Table 2. The positions, xiof the zeros and the turning points of s(φ(s)−π/6) as a function of x=βlogsand the corresponding values at the turning points. ∆xiis the diﬀerence xi−xi−1−1 2 Figure 1. Comparison Ψ( x)≡(6Θ(x)/π−1)swith Asin(π(x−x0)) as a function ofx=βlogs/π. Figure 2. Diﬀerence between the solution and its Asymptotic form 100(Ψ( x)− Asin(π(x−x0))) as a function of x=βlogs/π. Figure 3. Diﬀerence between the solution and its Asymptotic form 10000(Ψ( x)− Asin(π(x−x0))) as a function of x=βlogs/π. Figure 4. Diﬀerence between the solution and its Asymptotic form Ψ 1(x)≡ s((6Θ(x)/π−1)s−Asin(π(x−x0))) as a function of x=βlogs/π. 12arXiv:physics/0011076v1 [physics.flu-dyn] 30 Nov 20006 5 4x3 2 1 -11.0 0.5 0 -0.5 -1.0 Figure 1. Comparison Ψ( x)≡(6Θ(x)/π−1)swith Asin(π(x−x0)) as a function of x=βlogs/π. 155 4 3x2 10.2 0 -0.2 -0.4 -0.6 -0.8 Figure 2. Diﬀerence between the solution and its Asymptotic form 100(Ψ(x)−Asin(π(x−x0))) as a function of x=βlogs/π. 164.5 4x3.5 3 2.5 21 0.5 0 -0.5 -1 Figure 3. Diﬀerence between the solution and its Asymptotic form 10000(Ψ(x)−Asin(π(x−x0))) as a function of x=βlogs/π. 173 2 x 1 -11.2 1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Figure 4. Diﬀerence between the solution and its Asymptotic form Ψ1(x)≡s((6Θ(x)/π−1)s−Asin(π(x−x0))) as a function of x=βlogs/π. 18arXiv:physics/0011076v1 [physics.flu-dyn] 30 Nov 2000TABLE 1: Positions of the turning points, stand the corresponding values, φ(st) and comparison with those obtained by Chandler and Graham. stThis paper stC. & G. 6φ(st)/π−1 φ(st)o−30oφ(st)o−30oC. & G. 5.706256 ×1015.7062493 ×1011.26234416 ×10−23.787032480 ×10−13.787032466 ×10−1 4.683476245 ×1034.683×103-1.5345108772 ×10−4-4.6035326316 ×10−3-4.60353 ×10−3 3.80716.261 ×1053.807×1051.88776874 ×10−65.66330622 ×10−55.6631 ×10−5 3.09513 ×1073.21×107-2.322037 ×10−8-6.966111 ×10−7-7.4218 ×10−7 2.51266 ×1092.416×1082.8545 ×10−28.5635 ×10−9-3.6722 ×10−7 2.058×1011-3.96×10−12-1.188×10−10TABLE 2: The positions, xiof the zeros and the turning points of s(φ(s)−π/6) as a function of x=βlogsand the corresponding values at the turning points. ∆ xiis the diﬀerence xi−xi−1−1 2 Nature of point xi ∆xis(6φ(s)/π−1) 1st.Minimum 0.285868285199 -1.263432342282 1st.Zero 0.72529360368 -.06057468 1st.Maximum 1.22279723767 -.00249637 1.236931699148 2nd.Zero 1.724207968438 .00141073 2nd.Minimum 2.224237124 .00002916 -1.23648103488 3rd.Zero 2.7242200786 -.00001705 2nd.Maximum 3.224219455 -.00000062 1.23648608360 4th.Zero 3.724219486 .00000003 3rd.Minimum 4.2242314 .00001191 -1.23650251 5th.Zero 4.72425830 .00002690 3rd.Maximum 5.222 -.00225830 1.234586 6th.Zero 5.716482352 -.00551765
arXiv:physics/0012001v1 [physics.atom-ph] 1 Dec 2000The Two-Time Green’s Function and Screened Self–Energy for Two-Electron Quasi-Degenerate States ´Eric-Olivier Le Bigot1, Paul Indelicato1, and Vladimir M. Shabaev2 1Laboratoire Kastler-Brossel, Case 74, ´ENS et Universit´ e P. et M. Curie Unit´ e Mixte de Recherche du CNRS n◦C8552 4, pl. Jussieu, 75252 Paris CEDEX 05, France 2Department of Physics, St. Petersburg State University Oulianovskaya 1, Petrodvorets, St. Petersburg 198904, Rus sia Abstract. Precise predictions of atomic energy levels require the use of QED, espe- cially in highly-charged ions, where the inner electrons ha ve relativistic velocities. We present an overview of the two-time Green’s function method ; this method allows one to calculate level shifts in two-electron highly-charged i ons by including in principle all QED eﬀects, for any set of states (degenerate, quasi-deg enerate or isolated). We present an evaluation of the contribution of the screened se lf-energy to a ﬁnite-sized eﬀective hamiltonian that yields the energy levels through diagonalization. 1 Experiments and Theory Experimental measurements of atomic energy levels provide more and more stringent tests of theoretical models; thus, the experimen tal accuracy of many measurements is better than the precision of theoretical ca lculations: in hydro- gen[1,2], in helium [3,4], and in lithium -like uranium [5] and bismuth [6]. The current status of many precision tests of Quantum-Electrod ynamics in hydrogen and helium can be found in this edition. Furthermore, highly-charged ions possess electrons that m ove with a velocity which is close to the speed of light. The theoretical study of such systems must therefore take into account relativity ; moreover, a perturbative treatment of the binding to the nucleus (with coupling constant Zα) fails in this situation [7]. Perturbative expansions in Zα, however, are useful in diﬀerent situations (see [8] for a review, and articles in this edition [9,10,11,12]). 2 Theoretical Methods for Highly-Charged Ions There are only a few number of methods that can be used in order to predict energy levels for highly-charged ions within the framework of Bound-State Quan- tum Electrodynamics [13]: the adiabatic S-matrix formalism of Gell-Mann, Low and Sucher [14], the evolution operator method [15,16], the two-time Green’s function method [17] and an interesting method recently pro posed by Lindgren (based on Relativistic Many-Body Perturbation Theory merg ed with QED) [18].2 ´Eric-Olivier Le Bigot et al. All these methods are based on a study of the some evolution op erator or propa- gator; the two extreme times of the propagation can be both inﬁnite (Gell-Mann– Low–Sucher), onecan be ﬁnite and the other inﬁnite (Lindgren), and both can be ﬁnite (Shabaev). But among these methods, only twocan in principle be used in order to apply perturbation theory to quasi-degenerate levels (e.g., the3P1and1P1levels in helium-like ions): the two-time Green’s function method an d Lindgren’s method (which is still under development). Both work by constructi ng a ﬁnite-sized eﬀective hamiltonian whose eigenvalues give the energy levels [19]. The two-time Green’s function method has the advantage of be ing applicable to many atomic physics problems, such as the recombination o f an electron with an ion [20], the shape of spectral lines [21] and the eﬀect of n uclear recoil on atomic energy levels [22,23]. 2.1 Overview of the Two-Time Green’s Function Method We give in this section a short outline of the two-time Green’ s function method. The basic object of this method [24] represents the probabil ity amplitude for N fermions to go from one position to the other, as shown in Fig. 1. Time t Time t' Probability amplitude? Fig.1. The 2-particle Green’s function is the amplitude for going f rom one state of two particles to another state The corresponding mathematical object is a usual N-particle correlation function between twotimes: SN Fα1...αN α′ 1...α′ N(x1,...,xN,t;x′ 1,...,x′ N,t′) (1) ≡ /an}bracketle{tΩ\|TˆΨα1(x1,t)· · ·ˆΨαN(xN,t) ×ˆΨα′ N(x′ N,t′)· · ·ˆΨα′ 1(x′ 1,t′)\|Ω/an}bracketri}ht, (2) where \|Ω/an}bracketri}htis the vacuum of the fullBound-State QED Hamiltonian ˆH, and where the quantum ﬁeld ˆΨis deﬁned as the usual canonical electron–positron ﬁeld evolving under the total hamiltonian in the Heisenberg picture [13]. A remark can be made here about Lorentz invariance : the above correlation function (or propagator) displays only two times , which are associated to many diﬀerent positions . A Lorentz transform of the space–time positions involvedAtomic Energy Levels with QED 3 therefore yields many diﬀerent individual times (one for ea ch position); thus, the object (1) must be deﬁned in a speciﬁc reference frame. And this reference frame is chosen as nothing more than the Galilean reference f rame associated to the nucleus, which is physically privileged. Fundamental Property of the Green’s Function TheN-particle Green’s function is a function of energy simply deﬁned through a Fourier transform of Eq. (1): GN(x1,...,xN;x′ 1,...,x′ N;E∈R) ≡1 i/integraldisplay d∆teiE∆tSN F(x1,...,xN,∆t;x′ 1,...,x′ N,t′= 0). (3) This function is interesting because it contains the energy levels predicted by Bound-State QED: one can show [24] that GN(x1,...,xN;x′ 1,...,x′ N;E∈R) (4) =/summationdisplay Eigenstates \|n/angbracketrightofˆH with charge −N\|e\|/an}bracketle{tΩ\|ˆψ(x1)· · ·ˆψ(xN)\|n/an}bracketri}ht/an}bracketle{tn\|ˆψ(x′ N)· · ·ˆψ(x′ 1)\|Ω/an}bracketri}ht E−(En−i0) + (−1)N2+1/summationdisplay Eigenstates \|n/angbracketrightofˆH with charge + N\|e\|/an}bracketle{tΩ\|ˆψ(x′ N)· · ·ˆψ(x′ 1)\|n/an}bracketri}ht/an}bracketle{tn\|ˆψ(x1)· · ·ˆψ(xN)\|Ω/an}bracketri}ht E−(−En+i0), where \|Ω/an}bracketri}htis the vacuum of the total hamiltonian ˆH;ˆψis the usual second- quantized Dirac ﬁeld in the Schr¨ odinger representation an dEnis the energy of the eigenstate nofˆH. The poles in Ewith a positive real part are exactly the energies of the states with charge −N\|e\|, which are physically the atomic eigenstates of an ion with Norbiting electrons (The charge of the nucleus is not counted in the total charge.), as shown graphically in Fig. 2 . Such a result is similar to the so-called K¨ all´ en–Lehmann representation [25]. In order to obtain the energy levels contained in (4), we must resort on a per- turbative calculation of the correlation function (1), whi ch belongs to standard textbook knowledge [26]. The position of the poles of (4) must then be mathe- matically found. It is possible to construct an eﬀective, ﬁn ite-size hamiltonian which acts on the atomic state that one is interested in; the e igenvalues of this hamiltonian then give the Bound-State QED evaluation of the energy levels [19]. This hamiltonian is obtained through contour integrations . 2.2 Second-Order Calculations The current state-of-the-art in non-perturbative calcula tions (inZα) of atomic energy levels within Bound-State QED consists in the theore tical evaluation4 ´Eric-Olivier Le Bigot et al. 1s21s, 2s\|gN=2(E)\|Poles at the energy levels of a 2-electron ion Energy Fig.2. The 2-particle Green’s function contains information abou t the atomic energy levels of a 2-electron atom or ion of the contribution of diagrams with two photons (i.e. of order α2, since the electron–photon coupling constant is e). For instance, for ions with two electrons, the screening of one electron by the other is described by the six diagrams of Fig. 3. Fig.3. The contributions of order α2to the electron-electron interaction However, most of the calculations of contributions of order α2were, until very recently, restricted to the very speciﬁc case of the gro und-state (see [27] for references). The extension to the calculation of the ene rgy levels of quasi- degenerate states represents one of the current trends of th e research in the domain of non-perturbative (in Zα) calculations with QED. We have calculated the contribution of the screened self-en ergy (ﬁrst and fourth diagrams of Fig. 3) to some isolated levels in [27,28, 29,30]. When energy levels are quasi-degenerate (e.g., the3P1and1P1levels in helium-like ions), the two-time Green’s function method allows one to evaluate the matrix elements of the eﬀective hamiltonian between diﬀerent states; for the ﬁ rst diagram of Fig. 3, we obtain the following contribution to this hamiltonian (T he two electrons on the left are denoted by n1andn2, and the two on the right by n′ 1andn′ 2, andAtomic Energy Levels with QED 5 other notations follow.): /summationdisplay P,P′(−1)PP′/braceleftBigg −/parenleftBig/summationdisplay k/negationslash=n′ P′(1)/an}bracketle{tnP(1)nP(2)\|Sr k(εnP(1),εn′ P′(1))\|n′ P′(1)n′ P′(2)/an}bracketri}ht +/summationdisplay k/negationslash=nP(1)/an}bracketle{tnP(1)nP(2)\|Sl k(εnP(1),εn′ P′(1))\|n′ P′(1)n′ P′(2)/an}bracketri}ht/parenrightBig (5) +1 2/bracketleftbigg ∂p\|εnP(1)/parenleftBig /an}bracketle{tnP(1)\|Σ(p)\|nP(1)/an}bracketri}ht ×/an}bracketle{tnP(1)nP(2)\|I(p−εn′ P′(1))\|n′ P′(1)n′ P′(2)/an}bracketri}ht/parenrightBig +∂p′\|εn′ P′(1)/parenleftBig /an}bracketle{tnP(1)nP(2)\|I(εnP(1)−p′)\|n′ P′(1)n′ P′(2)/an}bracketri}ht ×/an}bracketle{tn′ P′(1)\|Σ(p′)\|n′ P′(1)/an}bracketri}ht/parenrightBig/bracketrightbigg/bracerightBigg +O[α2(E(0) n′−E(0) n)], where we made use of standard notations [27]: εkis the energy of the Dirac state k, (−1)PP′is the signature of the permutation P◦P′of the indices {1,2},Σ represents the self-energy, and Irepresents the photon-exchange: /an}bracketle{tab\|I(ω)\|cd/an}bracketri}ht ≡e2/integraldisplay d3x2[ψ† a(x1)αµψc(x1)] (6) ×[ψ† b(x2)ανψd(x2)]Dµν(ω;x1−x2) /an}bracketle{ta\|Σ(p)\|b/an}bracketri}ht ≡1 2πi/integraldisplay dω/summationdisplay k/an}bracketle{tak\|I(ω)\|kb/an}bracketri}ht εk(1−i0)−(p−ω), (7) wherea,b,c anddlabel Dirac states, and eis the charge of the electron; αµare the Dirac matrices, and ψdenotes a Dirac spinor; the photon propagator Dis given in the Feynman gauge by: Dνν′(ω;r)≡gνν′exp/parenleftBig i\|r\|/radicalbig ω2−µ2+i0/parenrightBig 4π\|r\|, (8) whereµis a small photon mass that eventually tends to zero, and wher e the square root branch is chosen such as to yield a decreasing exp onential for large real-valued energies ω. In Eq. (5), ∂x\|x0is the partial derivative with respect to xat the point x0, and the skeletons of the screened self-energy diagrams with a self-energy on the left and on the right are deﬁned as: /an}bracketle{tnP(1)nP(2)\|Sr k(p,p′)\|n′ P(1)n′ P(2)/an}bracketri}ht ≡ /an}bracketle{tnP(1)nP(2)\|I(p−p′)\|kn′ P′(2)/an}bracketri}ht1 εk(1−i0)−p′/an}bracketle{tk\|Σ(p′)\|n′ P′(1)/an}bracketri}ht, /an}bracketle{tnP(1)nP(2)\|Sr k(p,p′)\|n′ P(1)n′ P(2)/an}bracketri}ht ≡6 ´Eric-Olivier Le Bigot et al. /an}bracketle{tnP(1)\|Σ(p)\|k/an}bracketri}ht1 εk(1−i0)−p/an}bracketle{tknP(2)\|I(p−p′)\|n′ P(1)n′ P(2)/an}bracketri}ht. The terms of order α2(E(0) n′−E(0) n) are not included in the above expression because they do not contribute to the level shift of order α2in which we are interested. (They contribute to higher orders, as can be see n in the particular case of two levels [31, p. 27].) This expression is only formal and must be renormalized [27] ; angular in- tegrations can then be done and numerical computations can b e performed in order to yield the Bound-State QED evaluation of the energy s hifts. For the contribution of the ﬁrst diagram of Fig. 3 to have any p hysical mean- ing, it is necessary to calculate it together with the vertex correction (fourth diagram of Fig. 3). We have obtained the following contribut ion to the eﬀective hamiltonian for the vertex correction: /summationdisplay P,P′(−1)PP′/summationdisplay i1,i2/an}bracketle{ti1nP(2)\|I(εnP(1)−εn′ P′(1))\|i2n′ P′(2)/an}bracketri}ht ×i 2π/integraldisplay dω/an}bracketle{tnP(1)i2\|I(ω)\|i1n′ P′(1)/an}bracketri}ht [εi1(1−i0)−(εnP(1)−ω)][εi2(1−i0)−(εn′ P′(1)−ω)] +O[α2(E(0) n′−E(0) n)] where (n1,n2) and (n′ 1,n′ 2) still represent the electrons of the two states that deﬁne the hamiltonian matrix element given here, and where the sum over i1 andi2is over all Dirac states. 3 Conclusion and Outlook We have presented a quick overview of the current status of th eoretical predic- tions of energy levels in highly-charged ions with Bound-St ate Quantum Elec- trodynamics. We have given a short description of the two-ti me Green’s function method, which permits the calculation of an eﬀective hamilt onian that can in principle include all QED eﬀects in energy shifts. 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arXiv:physics/0012002v1 [physics.atom-ph] 1 Dec 2000Recoil-Induced-Resonances in Nonlinear, Ground-State, Pump-Probe Spectroscopy C. P. Search and P. R. Berman Physics Department, University of Michigan, Ann Arbor, MI, 48109-1120 (September 25, 2012) Typeset using REVT EX 1Abstract A theory of pump-probe spectroscopy is developed in which op tical ﬁelds drive two-photon Raman transitions between ground states o f an ensemble of three-level Λ atoms. Eﬀects related to the recoil the atom s undergo as a result of their interactions with the ﬁelds are fully accoun ted for in this theory. The linear absorption coeﬃcient of a weak probe ﬁeld in the pr esence of two pump ﬁelds of arbitrary strength is calculated. For subreco il cooled atoms, the spectrum consists of eight absorption lines and eight em ission lines. In the limit that χ1≪χ2, where χ1andχ2are the Rabi frequencies of the two pump ﬁelds, one recovers the absorption spectrum for a probe ﬁeld interacting with an eﬀective two-level atom in the presence of a single pump ﬁe ld. However when χ1/greaterorsimilarχ2, new interference eﬀects arise that allow one to selectivel y turn on and oﬀ some of these recoil induced resonances. I. INTRODUCTION In recent years, there has been interest in special features that can be attributed to the recoil that atoms undergo on the absorption, emission, o r scattering of radiation. One example of such phenomena is recoil induced resonances (RIR ) [1-8], which occur when a weak probe and strong pump ﬁeld interact with an ensemble of t wo level atoms having transition frequency ω. We ﬁrst recall some relevant results, neglecting eﬀects re lated to atomic recoil. In this limit, the sum of the ground state and e xcited state population for each velocity subclass of atoms is conserved. Let Ω 1and Ω pbe the frequencies of the pump and probe ﬁelds, respectively, ∆ = Ω 1−ωthe pump ﬁeld detuning from atomic resonance, χ1the Rabi frequency of the pump ﬁeld, and δ= Ω p−Ω1the probe-pump detuning. In the limit that \|∆\| ≫γe, whereγeis the excited state decay rate, the probe spectrum consists of an absorption peak at δ=−/radicalbig ∆2+ 4χ2 1and an emission peak at δ=/radicalbig ∆2+ 4χ2 1. There is also a non-secular dispersive-like structure in the prob e absorption spectrum centered at 2δ= 0.The absorption and emission peaks can be easily interpreted in terms of dressed states of the atom and pump ﬁeld [9]. The interpretation of the dispe rsive-like structure centered atδ= 0 is somewhat more elusive (see [10]). For the moment, we foc us on the secular terms and neglect the non-secular terms. The absorption spectrum can undergo a dramatic change when e ﬀects related to the recoil atoms undergo as a result of absorbing or emitting radiation are included [3]. In the case of subrecoil cooled atoms for which eﬀects of Doppler broadeni ng may be ignored, the secular absorption and emission lines of the probe spectrum are each replaced by absorption-emission doublets. In addition, there appear two secular absorption-emission doublets centered at δ= 0.An example of this spectrum is shown in Fig. 1. The secular spe ctrum can be given a simple interpretation in terms of dressed states of t he atom and pump ﬁeld, taking into account atomic recoil [3]. For the eight lines to be full y resolvable, it is necessary that 4ωk/greaterorsimilarγewhereωk=/planckover2pi1k2/2mis the recoil energy associated with the emission or absorpt ion of a photon from either the pump or probe ﬁeld (the wave vector s for the pump and probe have the same magnitude but opposite sign). If ωk≪γe/4, the absorption-emission doublet consisting of the lines A1andB1in Fig. 1 collapses into a single absorption line, the lines A2andB2collapse into a single emission line, and absorption and emi ssion linesB+and B−cancel one another. However, the lines A+andA−are resolved provided [3], ωk>1 4γe/parenleftigg/radicalbig ∆2+ 4χ2 1−∆ 2/radicalbig ∆2+ 4χ2 1/parenrightigg , (1) where it has been assumed that the ground state decay rate is z ero. In the weak ﬁeld limit, χ1≪ \|∆\|, this reduces to the condition ωk>1 4γe/parenleftbigχ1 ∆/parenrightbig2which can be satisﬁed for large enough detuning. In the strong ﬁeld limit, χ1≫ \|∆\|, one has instead ωk>1 8γe.If Eq. (1) is not satsiﬁed, A+andA−reduce to a dispersive-like structure at δ= 0.However, in the limit thatωk→0, linesA+andA−become degenerate at δ= 0 and cancel each other. One thereby recovers the secular probe spectrum in which recoil is neglected. The condition ωk/greaterorsimilarγe/4 is violated for most optically allowed transitions. As an e xam- ple, for the sodium 32S1/2→32P3/2transition,ωk= 1.6×105s−1andγe= 6.3×107s−1so 3thatωk/γe= 0.0025 [11]. Consequently, to fully resolve the recoil induce d changes in the probe spectrum would require the use of electric dipole forb idden transitions. If one instead considers two-photon Raman transitions between ground sta tes of a three-level Λ-type atom, the widths of the spectral lines are determined by some eﬀect ive ground state decay rate such as the inverse of the time the atom spends interacting wi th the pump and probe ﬁelds. As a result, the recoil induced structure in the probe spectr um is fully resolvable. In this paper we present a scheme for observing RIR in pump-pr obe spectroscopy of an atomic vapor using Raman transitions with two pump ﬁelds and a weak probe. This scheme is based on the model developed in [12] but for which all eﬀect s associated with atomic recoil were ignored. We consider the case of subrecoil coole d atoms in which the magnitude of the two-photon recoil momentum is greater than the width o f the atomic momentum distribution, i.e. Mu≪/planckover2pi1q,whereuis the most probable speed of the atoms and qis the magnitude of the diﬀerence between the probe and pump wave ve ctors. An alternative form for this condition is kBT≪/planckover2pi1ωqwhereT=Mu2/2kBis an eﬀective temperature for the atoms and /planckover2pi1ωqnow represents the recoil energy associated with a two-phot on transition. As in [12], the probe spectrum displays interference eﬀects . The interference allows one to selectively turn on and oﬀ the individual lines in the abso rption-emission doublets at δ∼0 by controlling the ratio of the pump ﬁeld Rabi frequencies an d the sign of their detunings relative to the excited state transition. In Sect. II, a model is developed for the interaction of the at oms with the pump ﬁelds and dressed states of the atom plus pump ﬁelds are deﬁned. In S ect. III, the interaction with the probe ﬁeld is introduced. Section IV contains the calcul ation of the linear absorption coeﬃcient of the probe as well as examples of the probe spectr um for various experimental parameters. Moreover, in this section it is shown that the va rious spectral components can be obtained by inspection from the dressed states in the secu lar limit. Finally, in Sect. V we discuss the physical origin of the interference as well as possible applications. 4II. EQUATIONS OF MOTION The conﬁguration of the atom and ﬁelds is indicated schemati cally in Fig.2. The pump ﬁelds 1 and 2 are denoted by the coupling constants g1andg2, respectively. The pump ﬁeld 1 couples only state \|1>and excited state \|e>, while pump 2 couples only state \|2> and\|e>. The interaction of the probe ﬁeld, Ep, which couples only state \|1>and\|e>is discussed in the next section. In this section, we consider o nly the interaction of the pump ﬁelds with the atoms. The ground state levels \|1>and\|2>are pumped incoherently at rates Λ 1(p) and Λ 2(p), respectively, and the population of both states decay with r ateγ. The eﬀect of collisional dephasing is ignored so the coherences between \|1>and\|2>also decay with rate γ. If\|1> and\|2>represent stable ground states of the atom, then the pumping rates and decay rate γconstitute a simple model for the ﬁnite interaction time of t he atoms with the ﬁelds due to the velocity of the atoms transverse to the propagation di rection of the ﬁelds. Owing to spontaneous emission, the population in the excited state d ecays at a rate γe≫γ. In contrast to [12] but as in [3], we use a quantized descripti on of the pump ﬁelds. This is not necessary, but the calculation is physically more tra nsparent using quantized pump ﬁelds. The total Hamiltonian for the system is H=Ha+Hr+H1+H2; (2a) Ha=/summationdisplay p/bracketleftbigg (/planckover2pi1ωe+p2 2M)\|e,p><e,p\|+ (/planckover2pi1ω1+p2 2M)\|1,p><1,p\|+ (/planckover2pi1ω2+p2 2M)\|2,p><2,p\|/bracketrightbigg ; (2b) Hr=/planckover2pi1Ω1a† 1a1+/planckover2pi1Ω2a† 2a2; (2c) H1=/planckover2pi1/bracketleftig g1\|e>< 1\|a1eik1·R+g∗ 1\|1><e\|a† 1e−ik1·R/bracketrightig ; (2d) H2=/planckover2pi1/bracketleftig g2\|e>< 2\|a2eik2·R+g∗ 2\|2><e\|a† 2e−ik2·R/bracketrightig ; (2e) whereHais the Hamiltonian for the atom in which the center of mass mom entum phas been quantized using periodic boundary conditions in a volu meV(assuming that the atoms are free and not subject to some trapping potential). The qua ntityHris the free ﬁeld Hamiltonian for the two pump ﬁelds and H1andH2represent the interaction of pump 51 and pump 2, respectively, with the atom, in the rotating-wa ve approximation. Bare states, \|j,p;n1,n2>,are deﬁned to be eigenstates of Ha+Hr, wherej=e,1,2 labels the internal state of the atom and n1andn2are the number of photons in pump ﬁelds 1 and 2, respectively. The matrix elements of the operator eik·Rin the basis of momentum eigenstates are <p\|eik·R\|p′>=<p\|p′+/planckover2pi1k>=δp,p′+/planckover2pi1k. (3) This allows one to rewrite the interaction terms as, H1=/planckover2pi1/summationdisplay p/bracketleftig g1\|e,p+/planckover2pi1k1><1,p\|a1+g∗ 1\|1,p><e,p+/planckover2pi1k1\|a† 1/bracketrightig (4a) H2=/planckover2pi1/summationdisplay p/bracketleftig g2\|e,p+/planckover2pi1k2><2,p\|a2+g∗ 2\|2,p><e,p+/planckover2pi1k2\|a† 2/bracketrightig (4b) The Hamiltonian, H, results in an inﬁnite ladder of closed three-state manifol ds labelled by (p,n1,n2) that contain the states, \|1,p;n1,n2>,\|2,p+/planckover2pi1(k1−k2);n1−1,n2+ 1>,and \|e,p+/planckover2pi1k1;n1−1,n2>.The Hamiltonian for a single manifold can be written as the 3 ×3 matrix, H(p,n1,n2) =/planckover2pi1(Ω1n1+ Ω2n2)I+ /planckover2pi1 (ωe−Ω1) +ωp+/planckover2pi1k1 g2√n2+ 1 g1√n1 g∗ 2√n2+ 1 (ω2−Ω1+ Ω2) +ωp+/planckover2pi1(k1−k2)0 g∗ 1√n1 0 ω1+ωp (5) where Iis the identity matrix and /planckover2pi1ωp=p2 2M. For the moment, the term proportional to Iin Eq.(5) can be ignored since it is only important when consi dering transitions between manifolds. By transforming to an interaction representati on for the state amplitudes where \|Ψ>=ae,p+/planckover2pi1k1e−i(ωe−Ω1+ωp+/planckover2pi1k1)t\|e,p+/planckover2pi1k1;n1−1,n2> +a2,p+/planckover2pi1(k1−k2)e−i(ω2−Ω1+Ω2+ωp+/planckover2pi1(k1−k2))t\|2,p+/planckover2pi1(k1−k2);n1−1,n2+ 1> +a1,pe−i(ω1+ωp)t\|1,p;n1,n2>, (6) one ﬁnds that the state amplitudes evolve as (including spon taneous emission from the excited state), 6i˙ae,p+/planckover2pi1k1=χ2e−i∆2ta2,p+/planckover2pi1(k1−k2)+χ1e−i∆1ta1,p−i(γe/2)ae,p+/planckover2pi1k1; (7a) i˙a2,p+/planckover2pi1(k1−k2)=χ∗ 2ei∆2tae,p+/planckover2pi1k1; (7b) i˙a1,p=χ∗ 1ei∆1tae,p+/planckover2pi1k1. (7c) For notational convenience we have supressed the index in th e amplitudes which la- bels the number of photons in the pump ﬁelds so that, for examp le,a2,p+/planckover2pi1(k1−k2)≡ a2,p+/planckover2pi1(k1−k2),n1−1,n2+1. We have also made the deﬁnitions χ2=g2√n2+ 1,χ1=g1√n1, and ∆2= Ω2−(ωe−ω2) +ω/planckover2pi1k2−1 M(p+/planckover2pi1k1)·k2, (8a) ∆1= Ω1−(ωe−ω1)−ω/planckover2pi1k1−1 Mp·k1. (8b) One can integrate Eq.(7a) formally to obtain, ae,p+/planckover2pi1k1(t) =ae,p+/planckover2pi1k1(−∞)e−γet/2 −ie−γet/2/integraldisplayt −∞/parenleftig χ2e(−i∆2+γe/2)t′a2,p+/planckover2pi1(k1−k2)(t′) +χ1e(−i∆1+γe/2)t′a1,p(t′)/parenrightig dt′.(9) When \|∆1\|,\|∆2\| ≫γeand the probability amplitudes a2,p+/planckover2pi1(k1−k2)(t) anda1,p(t) vary slowly on the time-scales of ∆−1 2and ∆−1 1,respectively, Eq. (9) may be approximated as, ae,p+/planckover2pi1k1(t)≈χ2 ∆2e−i∆2ta2,p+/planckover2pi1(k1−k2)(t) +χ1 ∆1e−i∆1ta1,p(t), (10) where the initial condition ae,p+/planckover2pi1k1(−∞) = 0 has been used. The excited state adiabatically follows the ground states in the limit of large detuning and t his allows one to eliminate the excited state amplitude by substituting Eq. (10) into Eqs.( 7b-7c) to obtain, i˙a2,p+/planckover2pi1(k1−k2)=\|χ2\|2 ∆2a2,p+/planckover2pi1(k1−k2)+χ∗ 2χ1 ∆1e−i/tildewideδ0ta1,p; (11a) i˙a1,p=χ∗ 1χ2 ∆2ei/tildewideδ0ta2,p+/planckover2pi1(k1−k2)+\|χ1\|2 ∆1a1,p; (11b) where /tildewideδ0= ∆ 1−∆2. (12) 7The diagonal light shifts may be eliminated by a redeﬁnition of the ground state energies, ω2+\|χ2\|2 ∆2→ω2andω1+\|χ1\|2 ∆1→ω1.By making the approximationχ∗ 1χ2 ∆2≈χ∗ 1χ2 ∆1=χ∗ 1χ2 ∆≡ G=\|G\|eiφdone has an eﬀective two level system described by, i˙a2,p+/planckover2pi1k12=G∗e−i/tildewideδ0ta1,p; (13a) i˙a1,p=Gei/tildewideδ0ta2,p+/planckover2pi1k12; (13b) where the shorthand notation kij=ki−kjhas been introduced. Equation (13a-13b) could have been derived directly from an eﬀective two-photon Hamil- tonianH′given by H′=Hr+H′ a+H12; (14a) H′ a=/summationdisplay p/bracketleftbigg (/planckover2pi1ω1+p2 2M)\|1,p><1,p\|+ (/planckover2pi1ω2+p2 2M)\|2,p><2,p\|/bracketrightbigg ; (14b) H12=/summationdisplay p/bracketleftbigg /planckover2pi1g1g∗ 2 ∆\|2,p+/planckover2pi1k12><1,p\|a† 2a1+/planckover2pi1g2g∗ 1 ∆\|1,p><2,p+/planckover2pi1k12\|a† 1a2/bracketrightbigg .(14c) The Hamiltonian H′results in an inﬁnite ladder of two state manifolds ( p,n1,n2) involving the states \|1,p;n1,n2>and\|2,p+/planckover2pi1k12;n1−1,n2+1>. The Hamiltonian for the manifold (p,n1,n2) is H′(p,n1,n2) =ε(p,n1,n2)I+/planckover2pi1 −/tildewideδ(p)/2G∗ G/tildewideδ(p)/2  (15) where ε(p,n1,n2) =/planckover2pi1(Ω1n1+ Ω2n2) +/planckover2pi1 2(ωp+ωp+/planckover2pi1k12+ω1+ω2+ Ω2−Ω1) (16) and/tildewideδ(p) =/tildewideδ0. The dressed states are deﬁned to be the eigenstates of the mat rix in Eq.(15) with energies given byε(p,n1,n2)±/planckover2pi1ω(0) AB 2and with amplitudes cs A0andcs B0that obey equations of motion i˙cs A0=/parenleftigg /planckover2pi1−1ε(p,n1,n2)−ω(0) AB 2/parenrightigg cs A0; (17a) i˙cs B0=/parenleftigg /planckover2pi1−1ε(p,n1,n2) +ω(0) AB 2/parenrightigg cs B0. (17b) 8The dressed state kets are [19] are  \|A0> \|B0> =T∗ 0 \|2,p+/planckover2pi1k12;n1−1,n2+ 1> \|1,p;n1,n2>  (18) where T0= eiφd/2cosθ0−e−iφd/2sinθ0 eiφd/2sinθ0e−iφd/2cosθ0  (19) andφd= argG.The frequency separation and rotation angle of the dressed s tates are deﬁned by, ω(0) AB=/radicalig 4\|G\|2+/tildewideδ2 0=/radicaligg 4\|G\|2+/parenleftbigg δ12−ω/planckover2pi1k12−p·k12 M/parenrightbigg2 ; (20a) cosθ0=/bracketleftigg 1 2/parenleftigg 1 +/tildewideδ0 ω(0) AB/parenrightigg/bracketrightigg1/2 ; (20b) where δ12= (Ω 1−Ω2) + (ω1−ω2). (21) The value of θ0is restricted to the range 0 ≤θ0≤π/4 for/tildewideδ0>0 andπ/4≤θ0≤π/2 for /tildewideδ0<0.Forθ0∼0 (/tildewideδ0>0 and/vextendsingle/vextendsingle/vextendsingle/tildewideδ0/vextendsingle/vextendsingle/vextendsingle/\|G\| ≫1),\|A0>∼ \|2,p+/planckover2pi1k12;n1−1,n2+ 1>while for θ0∼π/2 (/tildewideδ0<0 and/vextendsingle/vextendsingle/vextendsingle/tildewideδ0/vextendsingle/vextendsingle/vextendsingle/\|G\| ≫1),\|B0>∼ \|2,p+/planckover2pi1k12;n1−1,n2+ 1>.The 0 subscripts refer to the manifold ( p,n1,n2). In the following section it will be necessary to extend the deﬁnition of the dressed states to other manifolds in order t o include the eﬀects of the probe ﬁeld. III. PROBE FIELD & DENSITY MATRIX EQUATIONS The eﬀect of the probe ﬁeld is to induce transitions between s tates in diﬀerent manifolds. In contrast to the pump ﬁelds, the probe is treated as a classi cal ﬁeld, E(R,t) =1 2/hatwideǫEpei(kp·R−Ωpt)+c.c (22) 9where/hatwideǫis a unit polarization vector. We consider the case when the d etuning of the probe ﬁeld is suﬃciently large that the population in the excited state due to absorption from the probe is negligible and the excited stat e, in the absence of the pump ﬁelds, adiabatically follows state 1. This correspond s to the condition \|∆p\|= /vextendsingle/vextendsingle/vextendsingleΩp−(ωe−ω1) +ω/planckover2pi1kp−(p+/planckover2pi1k1)·kp M/vextendsingle/vextendsingle/vextendsingle≫γe.Under this condition, all transitions involving the probe occur via two-photon transitions in which one of the ph otons comes from the probe and the other photon comes from one of the pump ﬁelds. Starting fr om the ( p,n1,n2) manifold, absorption of the probe ﬁeld involves the transitions \|1,p;n1,n2>→ \|1,p+/planckover2pi1kp1;n1+1,n2> or\|1,p;n1,n2>→ \|2,p+/planckover2pi1(k12+kp1);n1,n2+1>, where the second photon is emitted into either the pump 1 or pump 2 modes, respectively. Similarly, g ain in the probe corresponds to the transitions \|1,p;n1,n2>→ \|1,p−/planckover2pi1kp1;n1−1,n2>or\|2,p+/planckover2pi1k12;n1−1,n2+1>→ \|1,p−/planckover2pi1kp1;n1−1,n2>. The probe ﬁeld interaction therefore involves transition s from the original ( p,n1,n2) manifold to the ( p+/planckover2pi1kp1,n1+ 1,n2) manifold in the case of probe absorption and to the ( p−/planckover2pi1kp1,n1−1,n2) manifold for probe gain. This is illustrated in Fig. 3(a). The probe term can be included into Eqs. (7a-7c) by including on the right hand side of Eq.(7a) a term χpe−i∆pta1,p−/planckover2pi1kp1and on the right hand side of Eq. (7c) a term χ∗ pei(∆p−/planckover2pi1kp1·kp/M)tae,p+/planckover2pi1kpwhereχp=−1 2/planckover2pi1de1Epis the probe Rabi frequency and de1= <e\|d·/hatwideǫ\|1>is a dipole moment matrix element. This is no longer a closed s ystem of three equations. If, however, one considers a weak probe ﬁeld, the n the lowest order transitions inχpdominate and these are precisely the transitions described in the previous paragraph. By proceeding in a manner identical to Sect. II, one can adiab atically eliminate the excited state from the equations of motion. Equations (11a- 11b) now becomes i˙a2,p+/planckover2pi1k12=G∗e−i/tildewideδ0ta1,p+G∗ 2e−i(δp2+ω/planckover2pi1kp1−ω/planckover2pi1k12−p·kp2/M)ta1,p−/planckover2pi1kp1; (23a) i˙a1,p=Gei/tildewideδ0ta2,p+/planckover2pi1k12+G∗ 1e−i(δp1+ω/planckover2pi1kp1−p·kp1/M)ta1,p−/planckover2pi1kp1 +G1ei(δp1−ω/planckover2pi1kp1−p·kp1/M)ta1,p+/planckover2pi1kp1+G2ei(δp2−ω/planckover2pi1kp2−p·kp2/M)ta2,p+/planckover2pi1kp2; (23b) where 10G1=χ∗ pχ1 ∆, (24a) G2=χ∗ pχ2 ∆, (24b) δp1= Ω p−Ω1, (24c) δp2= (Ω p−Ω2) + (ω1−ω2), (24d) and the approximation ∆ 1,∆2,∆p≈∆ has been used. It has also been assumed that n1≫1 so thatχ1=g1√n1≈g1√n1±1. One sees that with respect to the eﬀective two-state manifolds, there are two ”two-photon probe” ﬁeld s with Rabi frequencies G1and G2corresponding to the two ways in which the probe ﬁeld can comb ine with a pump ﬁeld in making a two-photon transition. The light shift associat ed with the probe ﬁeld,χ∗ pχp ∆, is negligible for a weak probe ﬁeld. In the presence of a strong two-photon pump ﬁeld, the dressed states deﬁned in Eq. (18) provide a useful basis in which to view transitions induced b y the probe. It is straight- forward to generalize the dressed states deﬁned in the previ ous section to include the two manifolds coupled to the initial manifold by the probe. The 0 , 1, and 2 manifolds re- fer to ( p,n1,n2) ={\|1,p;n1,n2>,\|2,p+/planckover2pi1k12;n1−1,n2+ 1>}, (p+/planckover2pi1kp1,n1+ 1,n2) = {\|1,p+/planckover2pi1kp1;n1+ 1,n2>,\|2,p+/planckover2pi1(k12+kp1);n1,n2+ 1>},and (p−/planckover2pi1kp1,n1−1,n2) = {\|1,p−/planckover2pi1kp1;n1−1,n2>,\|2,p+/planckover2pi1(k12−kp1);n1−2,n2+ 1>}, respectively. The ampli- tudes for the six dressed states in an interaction represent ation are deﬁned as, c= cA1 cB1 cA0 cB0 cA2 cB2 = T1eiσz/tildewideδ1t/20 0 0 T0eiσz/tildewideδ0t/20 0 0 T2eiσz/tildewideδ2t/2  a2,p+/planckover2pi1k12+/planckover2pi1kp1 a1,p+/planckover2pi1kp1 a2,p+/planckover2pi1k12 a1,p a2,p+/planckover2pi1k12−/planckover2pi1kp1 a1,p−/planckover2pi1kp1 ; (25a) Ti= eiφd/2cosθi−e−iφd/2sinθi eiφd/2sinθie−iφd/2cosθi ; (25b) 11and the detunings appearing in the three manifolds are given by /tildewideδ0=/tildewideδ(p) =δ12−ω/planckover2pi1k12−p·k12 M; (26a) /tildewideδ1=/tildewideδ(p+/planckover2pi1kp1) =δ12−ω/planckover2pi1k12−(p+/planckover2pi1kp1)·k12 M; (26b) /tildewideδ2=/tildewideδ(p−/planckover2pi1kp1) =δ12−ω/planckover2pi1k12−(p−/planckover2pi1kp1)·k12 M. (26c) The dressed state rotation angles and frequencies are cosθi=/bracketleftigg 1 2/parenleftigg 1 +/tildewideδi ωi AB/parenrightigg/bracketrightigg1/2 ; (27a) ωi AB=/radicalig 4\|G\|2+/tildewideδi2. (27b) The equations of motion for the dressed state amplitudes in t he absence of the probe are i/planckover2pi1˙ c=Hoc; (28) Ho=/planckover2pi1×diag/parenleftigg −ω(1) AB 2,ω(1) AB 2,−ω(0) AB 2,ω(0) AB 2,−ω(2) AB 2,ω(2) AB 2/parenrightigg . (29) Notice that in Eq. (29), the energies for the three manifolds are all centered about the same energy.This is due to the fact that the cI’s are amplitudes in an interaction representation with respect to the energy of the center of the manifolds. The state vector, \|Ψ>,is given by \|Ψ>=e−iε(p,n1,n2)t//planckover2pi1/parenleftbig cA0\|A0>+cB0\|B0>+e−iω10t(cA1\|A1>+cB1\|B1>) +e−iω20t(cA2\|A2>+cB2\|B2>)/parenrightbig ; =cs A0\|A0>+cs B0\|B0>+cs A1\|A1>+cs B1\|B1>+cs A2\|A2>+cs B2\|B2>. (30) where /planckover2pi1ω10=ε(p+/planckover2pi1kp1,n1+ 1,n2)−ε(p,n1,n2) =/planckover2pi1/parenleftbigg Ω1+ω/planckover2pi1kp1+kp1·p M+/planckover2pi1kp1·k12 2M/parenrightbigg ; (31a) /planckover2pi1ω20=ε(p−/planckover2pi1kp1,n1−1,n2)−ε(p,n1,n2) =/planckover2pi1/parenleftbigg −Ω1+ω/planckover2pi1kp1−kp1·p M−/planckover2pi1kp1·k12 2M/parenrightbigg .(31b) The Schr¨ odinger representation amplitudes, cs I,are the amplitudes for the eigenvectors of the block diagonal Hamiltonian, 12H102=diag(H′(p+/planckover2pi1kp1,n1+ 1,n2),H′(p,n1,n2),H′(p−/planckover2pi1kp1,n1−1,n2)).(32) where theH′are given by Eq. (15). Consequently, in the Schr¨ odinger rep resentation, the centers of the 1 and 2 manifolds are displaced from the 0 manif old by an amount /planckover2pi1ω10and /planckover2pi1ω20, respectively. Figure 3(b) illustrates the energy levels f or the six dressed states in the Schr¨ odinger representation. All calculations in the rema inder of the paper will be done using thecI’s in the interaction representation. The interaction with the probe ﬁeld given by Eqs.(23a-23b) m ay now be transformed to the dressed states basis. In this manner, we obtain an equati on of the form i/planckover2pi1˙ c= (Ho+VId)c. (33) The interaction matrix in the dressed basis, VId, is given by VId= 0V100 V† 100V† 20 0V200 (34) where V10andV20are 2×2 matrices, V10= <A 1\|VI\|A0>eiω10t<A1\|VI\|B0>eiω10t <B 1\|VI\|A0>eiω10t<B 1\|VI\|B0>eiω10t ; (35a) V20= <A 2\|VI\|A0>eiω20t<A2\|VI\|B0>eiω20t <B 2\|VI\|A0>eiω20t<B 2\|VI\|B0>eiω20t . (35b) The elements of VIdare linear in χp. The elements of V10are all proportional to e−iω(1)t whereω(1)= Ω p−ω10. The elements of V20are all proportional to eiω(2)twhereω(2)= Ωp+ω20. Each nonvanishing matrix element represents a transition a mplitude between dressed states in diﬀerent manifolds. For example, <A 1\|VI\|A0>is the amplitude for the transition \|A0>→ \|A1>involving probe absorption and pump 1 or pump 2 emission. Thi s transition is illustrated in Fig. 3(c). The probe couples only to the \|1,p;n1,n2>part of \|A0>which 13has an amplitude −eiφd/2sinθ0. The absorption of the probe is followed by emission into pump 2 leading to the \|2,p+/planckover2pi1(k12+kp1);n1,n2+ 1>component of \|A1>which has an amplitude of e−iφd/2cosθ1or emission into pump 1 leading to the \|1,p+/planckover2pi1kp1;n1+ 1,n2> component with amplitude −eiφd/2sinθ1. Consequently, one ﬁnds <A 1\|VI\|A0>=/planckover2pi1(G∗ 2eiφdcosθ1−G∗ 1sinθ1)(−sinθ0)e−iΩpt. Other matrix elements are calculated in a similar manner. Fo r probe gain, pump ﬁelds 1 and 2 couple to the \|1,p;n1,n2>and\|2,p+/planckover2pi1k12;n1−1,n2+1>components of the dressed states in the 0 manifold, respectively, while the probe ﬁeld couple s to the \|1,p−/planckover2pi1kp1;n1−1,n2> component of the dressed states in the 2 manifold. Explicit e xpressions for the matrix elements are given in the appendix. It turns out that the matrix elements and dressed state energ ies can be used to obtain directly the absorption or gain proﬁles in the secular limit . However in the more general case it is necessary to solve for the various density matrix eleme nts. The dressed state density matrix satisﬁes an evolution equation, /parenleftbiggd dt+γ/parenrightbigg ρd=1 i/planckover2pi1[Ho+VId,ρd] +Λd; (36) where ρd=cc†= ρA1A1ρA1B1ρA1A0ρA1BoρA1A2ρA1B2 ρB1A1ρB1B1ρB1A0ρB1BoρB1A2ρB1B2 ρA0A1ρA0B1ρA0A0ρA0BoρA0A2ρA0B2 ρB0A1ρB0B1ρB0A0ρB0BoρB0A2ρB0B2 ρA2A1ρA2B1ρA2A0ρA2BoρA2A2ρA2B2 ρB2A1ρB2B1ρB2A0ρB2BoρB2A2ρB2B2 . (37) It is assumed that all the elements of the density matrix deca y at the same rate, γ. This assumption is certainly valid for an atomic beam that is perp endicular to the applied ﬁelds, since the atom-ﬁeld interaction does not alter the transit t ime of the atoms through the beam. 14The matrix Λdrepresents the incoherent pumping of states \|1>and\|2>.For a subrecoil cooled vapor, the states in the 1 and 2 manifolds are initiall y unoccupied when the atoms start to interact with the ﬁelds. Consequently, only the 0 ma nifold is pumped and Λdhas the block diagonal form Λd=diag(0,Λ0,0); (38a) Λ0= ΛAΛAB ΛABΛB  (38b) = Λ2(p+/planckover2pi1k12) cos2θ0+ Λ1(p) sin2θ01 2(Λ2(p+/planckover2pi1k12)−Λ1(p)) sin 2θ0 1 2(Λ2(p+/planckover2pi1k12)−Λ1(p)) sin 2θ0Λ1(p) cos2θ0+ Λ2(p+/planckover2pi1k12) sin2θ0 ; (38c) The oﬀ-diagonal terms, Λ AB, give rise to non-secular terms in the solution of Eq. (36) wh ich are on the order of γ/ωi AB≪1 in size compared to the terms proportional to Λ Aand Λ B. IV. ABSORPTION COEFFICIENT In this section the absorption coeﬃcient for the probe ﬁeld i s calculated. Before embark- ing on a rigorous calculation of the absorption coeﬃcient, i t is helpful to present a simpliﬁed approach that provides an intuitive understanding of the li ne shapes in the secular limit. A. Secular Limit In the secular limit, one may set Λ AB= 0. The absorption coeﬃcient, α, is proportional to the rate at which energy is absorbed from ( α >0) or emitted into ( α <0) the probe ﬁeld. For a transition induced by the probe between the initi al stateIand ﬁnal state J, the absorption is determined by the number of atoms excited i nto stateJ.The absorption coeﬃcient will be proportional to N/planckover2pi1ΩpγρJJ/\|Ep\|2whereρJJis the steady state population produced in state Jdue to the I→Jtransition and Nis the atom density. It is easy to see from Eq. (36) that ρJJis equal to \|< J\|VI\|I >\|2(ΛI/γ) times a Lorentzian of width γcentered at the transition frequency between states IandJ.The transition frequencies 15may be directly read oﬀ of Fig. 3(b). Consequently, one ﬁnds t hat the secular absorption coeﬃcient has the form α∝γ2 \|χp\|2/summationdisplay I={Ao,Bo} /summationdisplay J={A1,B1}\|<J\|VI\|I >\|2(ΛI/γ) (Ωp−∆IJ)2+γ2−/summationdisplay J={A2,B2}\|<J\|VI\|I >\|2(ΛI/γ) (Ωp−∆IJ)2+γ2  (39) where ∆ IJare the transition frequencies obtained from Fig. 3(b). Mat rix elements are obtained as in the previous section. For example, if one considers the absorption line for the \|A0>→ \|A1>transition illus- trated in Fig. 3(c), one ﬁnds that \|<A1\|VI\|A0>\|2=/planckover2pi12sin2θ0(−\|G2\|cosθ1+ψ\|G1\|sinθ1)2, ∆A0A1=ω10−1 2ω(1) AB+1 2ω(0) AB,and αA0→A1∝\|χ2\|2 \|∆\|2(−cosθ1+ψ\|χ1\| \|χ2\|sinθ1)2sin2θ0(ΛA/γ)γ2 /parenleftig δp1−ω/planckover2pi1kp1−kp1·p M−/planckover2pi1kp1·k12 2M−1 2/parenleftig ω(0) AB−ω(1) AB/parenrightig/parenrightig2 +γ2. (40) Here,ψ= ∆/\|∆\|is the sign of the detuning. One sees that αA0→A1is the same as the ﬁrst absorption line of/parenleftig α α0/parenrightig secgiven in the next subsection. B. Evaluation of absorption coeﬃcient The absorption coeﬃcient, α, and index change, ∆ n, arise from the imaginary and real parts of the macroscopic polarization in the Maxwell-Bloch equations for the probe ﬁeld. They are given by the expressions α=α0Im/parenleftbiggγρ′ 1e χ∗pV/parenrightbigg ; (41a) ∆n=−α0k−1 pRe/parenleftbiggγρ′ 1e χ∗pV/parenrightbigg ; (41b) α0=kpNd2 1e 2/planckover2pi1ǫ0γ. (41c) whereVis the volume and ρ′ 1e[14] is the part of the bare state density matrix element ρ1e(R,t) which is proportional to e−i(kp·R−Ωpt). 16Before proceeding, we note that in this subsection all summa tions over momentum states have been converted to an integration over a continuum of sta tes via the standard substi- tution/summationtext p→V (2π/planckover2pi1)3/integraltext d3p. The coeﬃcient ρ′ 1eis related to the coherence in position space, ρ1e(R,t), and the momentum space density matrix elements, ρ1e(p;p′) =ρ′ 1e(p;p′)eiΩpt,by ρ1e(R,t) =ρ′ 1ee−i(kp·R−Ωpt)+other terms ; (42) =1 (2π/planckover2pi1)3/integraldisplay /integraldisplay d3pd3p′ρ′ 1e(p;p′)eiΩptei(p−p′)·R//planckover2pi1. (43) The coherence, ρ′ 1e(p;p′)eiΩpt, has been written in the Schr¨ odinger representation and is obtained from the density matrix for the atom plus pump ﬁelds by tracing over the number of photons in the pump ﬁelds, ρ′ 1e(p;p′) =e−iΩpt/summationdisplay n1,n2e−i(ω1−ωe)tρI 1e(p,n1,n2;p′,n1,n2). (44) where ρI ij(p,n1,n2;p′,n′ 1,n′ 2) =e−i(ωp−ωp′)tai,p,n1,n2a∗ j,p′,n′ 1,n′ 2(45) so thatρI 1e(p,n1,n2;p′,n′ 1,n′ 2) is in the interaction representation with respect to the in ternal energy levels and pump ﬁelds. One may derive a diﬀerential eq uation forρ′ 1e(p;p′) starting from the original Hamiltonian in Eq. (2a) including the prob e ﬁeld, ˙ρ′ 1e(p;p′) =∂ ∂t/parenleftigg e−iΩpt/summationdisplay n1,n2e−i(ω1−ωe)tρI 1e(p,n1,n2;p′,n1,n2)/parenrightigg . (46) The equation of motion for ρI 1e(p,n1,n2;p′,n′ 1,n′ 2) is, ˙ρI 1e(p,n1,n2;p′,n′ 1,n′ 2) = [−i(ωp−ωp′)−(γ+γe)/2]ρI 1e(p,n1,n2;p′,n′ 1,n′ 2) −iχ∗ 1ei(Ω1−ωe+ω1)t/bracketleftbig ρI ee(p+/planckover2pi1k1,n1−1,n2;p′,n′ 1,n′ 2) −ρI 11(p,n1,n2;p′−/planckover2pi1k1,n′ 1+ 1,n′ 2)/bracketrightbig −iχ∗ pei(Ωp−ωe+ω1)t/bracketleftbig ρI ee(p+/planckover2pi1kp,n1,n2;p′,n′ 1,n′ 2) −ρI 11(p,n1,n2;p′−/planckover2pi1kp,n′ 1,n′ 2)/bracketrightbig +iχ∗ 2ei(Ω2+ω2−ωe)tρI 12(p,n1,n2;p′−/planckover2pi1k2,n′ 1,n′ 2+ 1). (47) 17By carrying out the trace in Eq. (44), one obtains terms such a s, ρI 11(p;p′−/planckover2pi1k1) =/summationdisplay n1,n2ρI 11(p,n1,n2;p′−/planckover2pi1k1,n1+ 1,n2), so that the equation of motion for ρ′ 1e(p;p′) has a form which is identical to that which would have been obtained using classical pump ﬁelds, ˙ρ′ 1e(p;p′) =−[i(Ωp+ (ω1−ωe) + (ωp−ωp′)) + (γ+γe)/2]ρ′ 1e(p;p′) −iχ∗ p/parenleftbig ρI ee(p+/planckover2pi1kp;p′/parenrightbig −ρI 11(p;p′−/planckover2pi1kp)) −iχ∗ 1e−iδp1t/parenleftbig ρI ee(p+/planckover2pi1k1;p′/parenrightbig −ρI 11(p;p′−/planckover2pi1k1)) +iχ∗ 2e−iδp2tρI 12(p;p′−/planckover2pi1k2). (48) In terms of the perturbation series solution ρI 11(p;p′) =ρ(0) 11(p;p′) +ρ+ 11(p;p′)eiδp1t+ρ− 11(p;p′)e−iδp1t; (49a) ρI 12(p;p′) = (ρ(0) 12(p;p′) +ρ+ 12(p;p′)eiδp1t+ρ− 12(p;p′)e−iδp1t)eiδ12t; (49b) whereρ(0) jj′(p;p′) are independent of χpandρ± jj′(p;p′) arelinear inχp, the steady state solution for large detuning is ρ′ 1e(p;p′)≈1 ∆/parenleftig χ∗ 2ρ+ 12(p;p′−/planckover2pi1k2) +χ∗ 1ρ+ 11(p;p′−/planckover2pi1k1) +χ∗ pρ(0) 11(p;p′−/planckover2pi1kp)/parenrightig ; (50) and ∆ = Ω p+ (ω1−ωe).By making a change of variables, p−p′→p−p′−/planckover2pi1kp, one gets, ρ1e(R,t) =e−i(kp·R−Ωpt) (2π/planckover2pi1)3/integraldisplay /integraldisplay d3pd3p′ei(p−p′)·R//planckover2pi11 ∆ ×{χ∗ 2/parenleftbig ρ+ 12(p;p′+/planckover2pi1kp2) +ρ+ 12(p−/planckover2pi1kp1;p′+/planckover2pi1k12)/parenrightbig + χ∗ 1/parenleftbig ρ+ 11(p;p′+/planckover2pi1kp1) +ρ+ 11(p−/planckover2pi1kp1;p′)/parenrightbig +χ∗ pρ(0) 11(p;p′)} (51) One must now solve for ρI 11(p;p′) andρI 12(p;p′) using equations analogous to Eq. (48) and then extract ρ+ 12(p;p′) andρ+ 11(p;p′) from these solutions using Eqs. (49a-49b). The general structure of the solution is linked to the incoheren t pumping of levels 1 and 2. For a subrecoil cooled vapor, the pumping rate density for densi ty matrix elements ρij(p;p′) is assumed to be 18Λij(p,p′) = Λ′ iV−1(2π/planckover2pi1)3δ(p)δ(p−p′)δij (52) where Λ′ iV−1has the dimensions of ( volume ×time)−1and can be interpreted as the pumping rate to state 1 or 2 in position space. With this form of pumpin g,ρI ij(p;p′),must be proportional to δ(p−p′−/planckover2pi1k′) where k′is some algebraic combination of the pump and probe ﬁeld propagation vectors. To obtain ρ′ 1efrom Eq. (51) using Eq. (42), one must keep only those terms in the integrand of Eq. (51) proportional to δ(p−p′).In this limit, the ρ(0) 11(p;p) term in Eq. (51) makes no contribution to the absorption sin ce it is real and will be ignored from this point on. The incoherent pumping of states 1 and 2 populate two diﬀerent manifolds. The pumping of state 1 populates the ( p= 0,n1,n2) manifold while the pumping of state 2 populates the (p=−/planckover2pi1k12,n1,n2) manifold since this manifold involves state 2 with zero mom entum. Thus in viewing absorption or emission, two distinct initial state manifolds must be included, leading to the possibility of sixteen rather than eight comp onents of the spectrum. For the time being we set Λ 2= 0 but will remove this restriction in the next section. The terms appearing in Eq. (51) may be expressed in terms of th e dressed state density matrix elements. The matrix elements in VIdwhich couple the 0 and 1 manifolds all have the time dependence e±iω(1)twhile those which couple the 0 and 2 manifolds have a e±iω(2)t time dependence (see appendix). Consequently, if one is int erested only in terms linear in χp, one can expand the dressed state density matrix to ﬁrst order in the VIdmatrix elements, ρII′=ρ(0) II′+ρ+ II′eiω(1)t+ρ− II′e−iω(1)t; (53a) ρJJ′=ρ(0) JJ′+ρ+ JJ′eiω(2)t+ρ− JJ′e−iω(2)t; (53b) whereI,I′={A0,B0,A1,B1}andJ,J′={A0,B0,A2,B2}. Theρ+ ij(p;p′) needed in Eq.(51) can be expressed in terms of dressed state density matrix ele ments using Eqs. (25a,37) as ρ+ 11(p;p+/planckover2pi1kp1) = sinθ0sinθ1ρ+ A0A1−sinθ0cosθ1ρ+ A0B1 −cosθ0sinθ1ρ+ B0A1+ cosθ0cosθ1ρ+ B0B1; (54a) ρ+ 11(p−/planckover2pi1kp1;p) = sinθ2sinθ0ρ+ A2A0−sinθ2cosθ0ρ+ A2B0 19−cosθ2sinθ0ρ+ B2A0+ cosθ2cosθ0ρ+ B2B0; (54b) ρ+ 12(p;p+/planckover2pi1kp2) =eiφd/parenleftbig −sinθ0cosθ1ρ+ A0A1−sinθ0sinθ1ρ+ A0B1 + cosθ0cosθ1ρ+ B0A1+ cosθ0sinθ1ρ+ B0B1/parenrightbig ; (54c) ρ+ 12(p−/planckover2pi1kp1;p+/planckover2pi1k12) =eiφd/parenleftbig −sinθ2cosθ0ρ+ A2A0−sinθ2sinθ0ρ+ A2B0 + cosθ2cosθ0ρ+ B2A0+ cosθ2sinθ0ρ+ B2B0/parenrightbig . (54d) Substituting Eqs. (54a-54d) into Eq. (51) and using the solu tions forρ+ II′andρ+ JJ′obtained from Eqs. (53a-53b) and the solution of Eq. (36), one ﬁnds aft er some manipulation the ﬁnal expression for the absorption coeﬃcient, /parenleftbiggα α0/parenrightbigg =/parenleftbiggα α0/parenrightbigg sec+/parenleftbiggα α0/parenrightbigg ns; (55a) /parenleftbiggα α0/parenrightbigg sec=\|G\| \|∆\|/bracketleftigg/parenleftbigg ψηsinθ1−1 ηcosθ1/parenrightbigg2/parenleftbiggΛ′ A γsin2θ0L1(∆′) +Λ′ B γcos2θ0L2(∆′)/parenrightbigg +/parenleftbigg ψηcosθ1+1 ηsinθ1/parenrightbigg2/parenleftbiggΛ′ A γsin2θ0L3(∆′) +Λ′ B γcos2θ0L4(∆′)/parenrightbigg −/parenleftbigg ψηsinθ0−1 ηcosθ0/parenrightbigg2Λ′ A γ/parenleftbig sin2θ2L5(∆′) + cos2θ2L6(∆′)/parenrightbig −/parenleftbigg ψηcosθ0+1 ηsinθ0/parenrightbigg2Λ′ B γ/parenleftbig sin2θ2L7(∆′) + cos2θ2L8(∆′)/parenrightbig/bracketrightigg ; (55b) /parenleftbiggα α0/parenrightbigg ns=\|G\| \|∆\|Λ′ AB γsin 2θ0 2/parenleftig (ω(0) AB)2+γ2/parenrightig/bracketleftigg −/parenleftbigg ψηsinθ1−1 ηcosθ1/parenrightbigg2 (Γ1(∆′) + Γ 2(∆′)) −/parenleftbigg ψηcosθ1+1 ηsinθ1/parenrightbigg2 (Γ3(∆′) + Γ 4(∆′)) +/parenleftbig η2−η−2−2ψcot2θ0/parenrightbig/parenleftbig sin2θ2(Γ5(∆′) + Γ 7(∆′)) + cos2θ2(Γ6(∆′) + Γ 8(∆′))/parenrightbig/bracketrightbig ; (55c) L1(∆′) =γ2 /parenleftig ∆′−ω/planckover2pi1kp1−1 2(ω(0) AB−ω(1) AB)/parenrightig2 +γ2; (55d) L2(∆′) =γ2 /parenleftig ∆′−ω/planckover2pi1kp1+1 2(ω(0) AB+ω(1) AB)/parenrightig2 +γ2; (55e) L3(∆′) =γ2 /parenleftig ∆′−ω/planckover2pi1kp1−1 2(ω(0) AB+ω(1) AB)/parenrightig2 +γ2; (55f) 20L4(∆′) =γ2 /parenleftig ∆′−ω/planckover2pi1kp1+1 2(ω(0) AB−ω(1) AB)/parenrightig2 +γ2; (55g) L5(∆′) =γ2 /parenleftig ∆′+ω/planckover2pi1kp1+1 2(ω(0) AB−ω(2) AB)/parenrightig2 +γ2; (55h) L6(∆′) =γ2 /parenleftig ∆′+ω/planckover2pi1kp1+1 2(ω(0) AB+ω(2) AB)/parenrightig2 +γ2; (55i) L7(∆′) =γ2 /parenleftig ∆′+ω/planckover2pi1kp1−1 2(ω(0) AB+ω(2) AB)/parenrightig2 +γ2; (55j) L8(∆′) =γ2 /parenleftig ∆′+ω/planckover2pi1kp1−1 2(ω(0) AB−ω(2) AB)/parenrightig2 +γ2; (55k) Γ1(∆′) =/bracketleftbigg/parenleftbigg ∆′−ω/planckover2pi1kp1−1 2(ω(0) AB−ω(1) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L1(∆′); (55l) Γ2(∆′) =/bracketleftbigg −/parenleftbigg ∆′−ω/planckover2pi1kp1+1 2(ω(0) AB+ω(1) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L2(∆′); (55m) Γ3(∆′) =/bracketleftbigg/parenleftbigg ∆′−ω/planckover2pi1kp1−1 2(ω(0) AB+ω(1) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L3(∆′); (55n) Γ4(∆′) =/bracketleftbigg −/parenleftbigg ∆′−ω/planckover2pi1kp1+1 2(ω(0) AB−ω(1) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L4(∆′); (55o) Γ5(∆′) =/bracketleftbigg −/parenleftbigg ∆′+ω/planckover2pi1kp1+1 2(ω(0) AB−ω(2) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L5(∆′); (55p) Γ7(∆′) =/bracketleftbigg/parenleftbigg ∆′+ω/planckover2pi1kp1−1 2(ω(0) AB+ω(2) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L7(∆′); (55q) Γ6(∆′) =/bracketleftbigg −/parenleftbigg ∆′+ω/planckover2pi1kp1+1 2(ω(0) AB+ω(2) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L6(∆′); (55r) Γ8(∆′) =/bracketleftbigg/parenleftbigg ∆′+ω/planckover2pi1kp1−1 2(ω(0) AB−ω(2) AB)/parenrightbigg ω(0) AB+γ2/bracketrightbigg L8(∆′); (55s) whereψ= ∆/\|∆\|is the sign of the detuning for each of the three ﬁelds, η=/radicalbig \|χ1\|/\|χ2\|, and ∆′=δp1−/planckover2pi1kp1·k12 2M. The pumping terms Λ′ A,B,AB are obtained from Eq. (38b) with the substitution Λ 1(p)→Λ′ 1and Λ 2(p+/planckover2pi1k12) = 0, Λ′ A= Λ′ 1sin2θ0; Λ′ B= Λ′ 1cos2θ0; Λ′ AB=−1 2Λ′ 1sin 2θ0; (56) In addition, phas been set equal to zero in the /tildewideδi. The ﬁrst part of the absorption coeﬃcient,/parenleftig α α0/parenrightig sec, is the absorption coeﬃcient in the secular approximation while/parenleftig α α0/parenrightig nsis the non-secular part of the absorption coeﬃcient. 21The reason the absorption coeﬃcient is expressible as a sum o f the secular term plus a non- secular term is linked to the fact that the secular approxima tion consists solely of neglecting the oﬀ-diagonal components of Λ0. Since the ﬁrst order solutions, ρ+ II′andρ+ JJ′, are linear in the pumping terms,/parenleftig α α0/parenrightig seccontain terms proportional to Λ Aand Λ Bwhile/parenleftig α α0/parenrightig nsis proportional to Λ AB. This simpliﬁcation would not occur for a more complex decay scheme for states \|1>and\|2>since the decay would couple density matrix elements in a ﬁel d dependent manner (see [13]). The ﬁrst four resonances in/parenleftig α α0/parenrightig seccorrespond to probe absorption while the last four correspond to probe gain. The secular spectrum is shown in Fi g. 4 and has the same general structure as that of Fig. 1 with an absorption doublet and an e mission doublet at ∆′∼0. One important diﬀerence is that for γe≫γthe HWHM line widths for the lines A1, A2, B1,andB2in Fig. 1 is γe/2. The lines widths for A±andB±is alsoγe/2 in the strong ﬁeld limit. However, the line widths in Fig. 4 is γforallthe lines. Consequently, the lines in Fig. 4 are narrower by a factor of 2 γ/γe≪1. The most signiﬁcant feature of/parenleftig α α0/parenrightig secis that the line strengths involve factors such as /parenleftig ψηsinθ1−1 ηcosθ1/parenrightig2 which allow one to manipulate the strength of the lines by con trolling the sign of the ﬁeld detuning and the ratio of the pump ﬁeld amp litudes. These factors are an indication of interference between the two ”two-photon p robe” ﬁelds which can both lead to absorption or gain in Ep. This will be discussed in more detail in the next section. Because the two lines in the doublets have diﬀerent strength s, one can adjust ψandηto turn oﬀ one of the lines. For example, the absorption doublet, L1andL4,consists of the lines at ∆′=ω/planckover2pi1kp1+1 2(ω(0) AB−ω(1) AB) andω/planckover2pi1kp1−1 2(ω(0) AB−ω(1) AB) with strengths ∼/parenleftig ψηsinθ1−1 ηcosθ1/parenrightig2 and∼/parenleftig ψηcosθ1+1 ηsinθ1/parenrightig2 , respectively. Choosing ψ= +1 andη2= cotθ1turns ”oﬀ” the ﬁrst line while ψ=−1 andη2= tanθ1turns ”oﬀ” the second line. This is shown in Fig. 5. When \|G\|andδ12are much larger than any of the recoil terms, θ0≈θ1≈θ2and the emission lines are also turned ”oﬀ”. Consequently, by ch oosingψ= +1 andη2= cotθ1 to turn oﬀ the L1andL2absorption lines, the L5andL6emission lines are also turned oﬀ. 22The nonsecular term,/parenleftig α α0/parenrightig ns, consist of dispersion-like structures centered at the sam e locations as the resonances in/parenleftig α α0/parenrightig sec.In the secular limit,/parenleftig α α0/parenrightig ns≪/parenleftig α α0/parenrightig secand/parenleftig α α0/parenrightig ns can usually be ignored. Notice that if one chooses ψandηsuch that a pair of absorption lines in/parenleftig α α0/parenrightig secvanish, then the corresponding terms in/parenleftig α α0/parenrightig nsalso vanish so that/parenleftig α α0/parenrightig is identically zero. However, this will not be true for the gain terms in/parenleftig α α0/parenrightig secsince the cor- responding terms in/parenleftig α α0/parenrightig nshave a diﬀerent interference coeﬃcient, ( η2−η−2−2ψcot 2θ0). Figure 6 shows a plot of the non-secular absorption coeﬃcien t for the same parameters as Fig. 4. In this plot, the non-secular terms are ∼1000 times smaller than the secular terms. A particularly interesting case occurs when k1≈k2so thatω(0) AB=ω(1) AB=ω(2) ABandθ0= θ1=θ2. This would correspond to a two-photon pump ﬁeld which impar ts no momentum to the atoms so that the recoil splitting in the absorption spec trum can be attributed solely to the recoil due to the two two-photon probe ﬁelds, kp2=kp1.In this case, the line L1(∆′) is degenerate with L4(∆′) andL5(∆′) is degenerate with L8(∆′).Consequently, the spectrum consists of three absorption-emission doublets centered a t ∆′= 0,+ω(0) AB,−ω(0) AB.Moreover, the lines within each doublet are split by 2 ω/planckover2pi1kp1which is independent of the strength or detuning of the pump ﬁelds. When the eﬀects of atomic recoil are neglected by setting all recoil momenta to zero in /parenleftig α α0/parenrightig , one obtains the same absorption spectrum given in [12]. The absorption coeﬃcient given here can also be shown to be consistent with the results given in [3] in the limit in whichη≪1 whileGremains constant. V. DISCUSSION Pumping to state 2 at a rate Λ 2introduces a new set of manifolds. However, with the exception of doubling the number of absorption and emission lines in the probe spectrum, the new manifolds do not result in any qualitatively new feat ures. To understand this, ﬁrst note that pumping to \|2,p;n1,n2>in the absence of the probe ﬁeld results in the closed three state manifold ¯0 involving the states {\|1,p−/planckover2pi1k12;n1+1,n2−1>,\|e,p+/planckover2pi1k2;n1,n2− 231>;\|2,p;n1,n2>}which can be rewritten as {\|1,¯ p; ¯n1,¯n2>,\|e,¯ p+/planckover2pi1k1; ¯n1−1,¯n2> ,\|2,¯ p+/planckover2pi1k12; ¯n1−1,¯n2+ 1>}where ¯ p=p−/planckover2pi1k12, ¯n1=n1+ 1, and ¯n2=n2−1. The presence of the probe ﬁeld couples the ¯0 manifold to the ¯1 and ¯2 manifolds which are identical to the 1 and 2 manifolds with the substitution p→¯ p,n1→¯n1, andn2→¯n2. However, the absolute number of photons in the pump ﬁelds is unimportant, only the diﬀerence in the photon number between manifolds is relevant. Consequently, all of the results derived in this paper may be immediately used to derive the absorption coeﬃcient for the transitions between the ¯0 manifold and the ¯1 and ¯2 mani- folds through the substitution, p→p−/planckover2pi1k12. This is straightforward since pappears only inω(1), ω(2),Λ0and/tildewideδi. The detunings for the dressed state manifolds are, /tildewideδ¯ı=/tildewideδi+ 2ω/planckover2pi1k12. (57) The detuning parameter ∆′now becomes ¯∆′=δp1+/planckover2pi1kp1·k12 2Mand ﬁnally the pumping terms become Λ¯0= Λ2(p) cos2θ0+ Λ1(p−/planckover2pi1k12) sin2θ01 2(Λ2(p)−Λ1(p−/planckover2pi1k12)) sin 2θ0 1 2(Λ2(p)−Λ1(p−/planckover2pi1k12)) sin 2θ0Λ1(p−/planckover2pi1k12) cos2θ0+ Λ2(p) sin2θ0 ; (58) so that, for subrecoil atoms, when Λ 2(p)/ne}ationslash= 0, Λ 1(p−/planckover2pi1k12) = 0.Therefore the dressed state pumping terms appearing in/parenleftig α αo/parenrightig are, Λ′ A= Λ′ 2cos2θ0; Λ′ B= Λ′ 2sin2θ0; Λ′ AB=1 2Λ′ 2sin 2θ0; (59) The absorption coeﬃcient contains sixteen lines in all. How ever the eight new spectral components display the same properties as the original eigh t but are displaced by an amount ∼¯∆′−∆′=/planckover2pi1kp1·k12 M.Figure 7 shows the absorption spectrum with all sixteen comp onents when Λ′ 2= Λ′ 1. In the absence of recoil, the secular absorption coeﬃcient vanishes when Λ′ 2= Λ′ 1since the dressed states have equal population (see [12]). In Ref. [12] interference eﬀects similar to those in Eq. (55b ) were found. It was not obvious that the interference would persist for Fock states of the pump ﬁelds. Nor was it obvious that the interference would survive quantization o f the center of mass motion, since 24diﬀerent ﬁelds impart diﬀerent momenta to the atoms. The int erference is not lost since the ﬁnal states for processes involving the probe ﬁeld and pump ﬁ eld 1 are the same as those involving the probe ﬁeld and pump ﬁeld 2. The existence of total destructive interference for partic ular values of ηandψin Eq. (55b) can also be interpreted in terms of dark states. Consider a Λ t ype atom with an excited state\|c>and ground states \|a>and\|b>in the presence of some interaction represented by an operator /hatwideVwhich couples each of the ground states to the excited state b ut does not couple the two ground states to each other. In this case, a dar k state corresponds to an initial superposition of the two ground states, \|Ψdark>= sinθ\|a>+eiφcosθ\|b>, (60) such that the transition probability to the excited state is identically zero for all times. This corresponds to the condition <c\|/hatwideV\|Ψdark>≡0. (61) In the context of the system described here, the coherence be tween the ground states within the 0, 1, and 2 manifolds needed to produce a dark state is provided by the dressing with the pump ﬁelds, i.e. the dressed states \|Ai>and\|Bi>are superpositions of the bare states \|1>and\|2>of the same form as Eq. (60). Since the probe only couples to st ate \|1>, the transitions leading to probe absorption (gain) involv e matrix elements between state\|1>in the 0 (2) manifold and the dressed states in the 1 (0) manifold. For certain values of the dressed state angles and ψ, it is possible to satisfy the condition for a dark state given in Eq. (61). First, consider probe absorption which involves transitio ns from \|1,p;n1,n2>to either \|A1>or\|B1>. The square of the matrix elements of VIcorresponding to these transitions are \|<A 1\|VI\|1,p;n1,n2>\|2∝(\|G2\|cosθ1−ψ\|G1\|sinθ1)2=\|G2\|2(cosθ1−ψη2sinθ1)2; (62a) \|<B 1\|VI\|1,p;n1,n2>\|2∝(ψ\|G1\|cosθ1+\|G2\|sinθ1)2=\|G2\|2(ψη2cosθ1+ sinθ1)2; (62b) 25The expressions on the right are precisely the absorption in terference terms in Eq. (55b). Therefore, for certain values of θ1relative to ψandη,\|A1>or\|B1>correspond to ”inverted dark states”. These are called ”inverted dark sta tes” since this is an eﬀective V type system with excited states \|1,p+/planckover2pi1kp1;n1+ 1,n2>and\|2,p+/planckover2pi1(k12+kp1);n1,n2+ 1>. In a similar manner, probe emission involves transitions b etween \|A0>or\|B0> and\|1,p−/planckover2pi1kp1;n1−1,n2>.This describes an eﬀective Λ system. The transition matrix elements are \|<1,p−/planckover2pi1kp1;n1−1,n2\|VI\|A0>\|2∝\|G2\|2(cosθ0−ψη2sinθ0)2; (63a) \|<1,p−/planckover2pi1kp1;n1−1,n2\|VI\|B0>\|2∝\|G2\|2(ψη2cosθ0+ sinθ0)2. (63b) These correspond to the coeﬃcients of the emission lines in E q. (55b) so that for the appropriate value of θ0,\|A0>or\|B0>correspond to dark states. Bose-Einstein condensates would represent an interesting application of the ideas pre- sented here. Bragg spectroscopy has recently been demonstr ated in condensates [15] [16] as well as the stimulated generation of matter waves in a conden sate by Rayleigh scattering [17]. In a condensate, the strength of the absorption and emi ssion lines should be enhanced since the scattering of atoms out of the condensate state and into a recoil side-mode is a stimulated process with a rate proportional to the number of atoms in the side-mode [18]. In addition, the RIR spectrum of a weakly interacting Bose cond ensate should yield information about the spectrum of elementary excitations in a condensat e. This would have important implications, since it would provide a new means for determi ning the excitation spectrum which determines the thermodynamic and hydrodynamic prope rties of a condensate. VI. APPENDIX A - MATRIX ELEMENTS OF V ID In this appendix we provide the matrix elements of VIdgiven in Eq. (34) and deﬁned to be VId=TVIT†; (64) 26whereTis given by the matrix in Eq. (25a) and VIare the matrix elements in the interaction representation involving G1andG2in Eqs. (23a-23b). The Rabi frequencies χp,χ1, andχ2 are written as χp=\|χp\|eiφ;χ1=\|χ1\|eiφ1;χ2=\|χ2\|eiφ2; (65) so thatφd=φ2−φ1+π 2(1−ψ) sinceG=\|G\|eiφd. The matrix elements are: <A 1\|VI\|A0>eiω10t=/planckover2pi1ei(φ−φ1)e−iω(1)tsinθ0(−\|G2\|cosθ1+ψ\|G1\|sinθ1); (66a) <A 1\|VI\|B0>eiω10t=/planckover2pi1ei(φ−φ1)e−iω(1)tcosθ0(\|G2\|cosθ1−ψ\|G1\|sinθ1); (66b) <B 1\|VI\|A0>eiω10t=/planckover2pi1ei(φ−φ1)e−iω(1)tsinθ0(−ψ\|G1\|cosθ1− \|G2\|sinθ1); (66c) <B 1\|VI\|B0>eiω10t=/planckover2pi1ei(φ−φ1)e−iω(1)tcosθ0(ψ\|G1\|cosθ1+\|G2\|sinθ1); (66d) <A 0\|VI\|A2>e−iω20t=/planckover2pi1ei(φ−φ1)e−iω(2)tsinθ2(−\|G2\|cosθ0+ψ\|G1\|sinθ0); (66e) <A 0\|VI\|B2>e−iω20t=/planckover2pi1ei(φ−φ1)e−iω(2)tcosθ2(\|G2\|cosθ0−ψ\|G1\|sinθ0); (66f) <B 0\|VI\|A2>e−iω20t=/planckover2pi1ei(φ−φ1)e−iω(2)tsinθ2(−ψ\|G1\|cosθ0− \|G2\|sinθ0); (66g) <B 0\|VI\|B2>e−iω20t=/planckover2pi1ei(φ−φ1)e−iω(2)tcosθ2(ψ\|G1\|cosθ0+\|G2\|sinθ0). (66h) The other elements follow from the Hermiticity of VId. The frequencies ω(1)andω(2)are given by ω(1)= Ω p−ω10=δp1−ω/planckover2pi1kp1−/planckover2pi1kp1·k12 2M−kp1·p M; (67a) ω(2)= Ω p+ω20=δp1+ω/planckover2pi1kp1−/planckover2pi1kp1·k12 2M−kp1·p M; (67b) VII. ACKNOWLEDGMENTS C. P. S. and P. R. B. are pleased to acknowledge helpful discus sions with B. Dubetsky. This research is supported by the National Science Foundati on under Grant No. PHY- 9800981 and by the U. S. Army Research Oﬃce under Grant No. DAA G55-97-0113 and No. DAAD19-00-1-0412. 27REFERENCES [1] J. Guo, P.R. Berman, B. Dubetsky, and G. Grynberg, Phys. R ev. A46, 1426 (1992). [2] J. Guo and P.R. Berman, Phys. Rev. A 47, 4128 (1993). [3] P.R. Berman, B. Dubetsky, and J.Guo, Phys. Rev. A 51, 3947 (1995). [4] B. Dubetsky and P.R. Berman, Phys. Rev. A 52, R2519 (1995). [5] J.Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, Ph ys. Rev. Lett. 72, 3017 (1994). [6] S. Guibal, C. Triche, L. Guidoni, P. Verkerk, and G. Grynb erg, Opt. Commun. 131, 61 (1996). [7] D.R. Meacher, D. Boiron, H. Metcalf, C. Saloman, and G. Gr ynberg, Phys. Rev. A 50, R1992 (1994). [8] M. Kozuma, Y. Imai, N. Nakagawa, and M. Ohtsu, Phys. Rev. A 52, R3421 (1995); M. Kozuma, N. Nakagawa, W. Jhe, and M. Ohtsu, Phys. Rev. Lett. 76, 2428 (1996). [9] C. Cohen-Tannoudji and S. Reynaud, J. Phys. B 10, 345 (1977). [10] G. Grynberg and C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993). [11] H.J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag, New York, 1999). [12] P.R. Berman and B. Dubetsky, lanl e-print: physics/000 4046. [13] P.R. Berman, Phys. Rev. A. 53, 2627 (1996). [14] The units of ρ1e(R,t) are 1/Vand this explains the appearance of Vin Eq. (42a-b). [15] D.M. Stamper-Kurn et al., Phys. Rev. Lett. 83, 2876 (1999). [16] J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999). 28[17] S. Inouye et al., Science 285, 571 (1999). [18] M. G. Moore and P. Meystre, Phys. Rev. Let. 83, 5202 (1999). [19] The dressed states deﬁned here diﬀer by the interchange of\|1>and\|2>with respect to the dressed states deﬁned in [12]. 29Figure 1. Probe absorption coeﬃcient in arbitrary units ver sus probe-pump detuning with the inclusion of recoil. In units of the excited state de cay rateγe,the pump detuning is ∆ = 50, pump ﬁeld Rabi frequency is χ1= 75, and the recoil frequency is ωk= 5. The decay rate for the ground state is taken to be zero. See Ref. [3 ]. Figure 2. Schematic diagram of atom-ﬁeld system. Figure 3. (a) Transitions between the 0 manifold and the 1 and 2 manifolds leading to probe gain or absorption in the bare state picture. The state s in the 1 and 2 manifolds are displaced from the states in the 0 manifold by an amount ∼/planckover2pi1Ω1.The Rabi frequencies shown are those which couple the states from the 0 manifold to the 1 and 2 manifolds. (b) Energy levels in the dressed state basis. The center of the 1 m anifold has an energy /planckover2pi1ω10 above the center of the 0 manifold and similarly, the center o f the 2 manifold is /planckover2pi1ω20below the 0 manifold.(c) Illustration of the coupling of the pump a nd probe ﬁelds to the dressed states for the \|A0>→ \|A1>transition. The Rabi frequencies shown are those that coupl e the bare state \|1>component of \|A0>to the bare state components of \|A1>. Figure 4. Plot of/parenleftig α α0/parenrightig secfor Λ 2= 0, Λ 1/γ= 1, ψ=−1 andη= 2 showing all eight absorption and emission lines. The detuning is δ12/γ= 300 and the two-photon pump Rabi frequency is \|G\|/γ= 250.The recoil energies are ω/planckover2pi1k12/γ= 40,ω/planckover2pi1kp1/γ= 60, and /planckover2pi1kp1·k12/Mγ= 80. Note that γ≪γewhereγeis the excited state decay rate used in Fig. 1. Figure 5. Plot of/parenleftig α α0/parenrightig secshowing destructive interference. Parameters are the same as Fig. 4 except for ψandη.The solid line corresponds to ψ=−1 andη=√tanθ1= 0.8383 while the dotted line corresponds to ψ= +1 andη=√cotθ1= 1.1928. Figure 6. Plot of/parenleftig α α0/parenrightig nsfor the same parameters as Fig. (4). The non-secular absorpt ion coeﬃcient has dispersionlike structures at the same locati on is the line centers of/parenleftig α α0/parenrightig sec. The amplitudes of these non-secular terms is typically ∼1000 times smaller than the secular line strengths, consistent with γ/ω(0) AB= 0.00177. Figure 7. Plot showing sixteen lines in the/parenleftig α α0/parenrightig secprobe spectrum for Λ 2/γ= Λ1/γ= 1, ψ= +1 andη= 0.1. The detuning is δ12/γ= 500 and the two-photon pump Rabi frequency 30is\|G\|/γ= 750.The recoil energies are ω/planckover2pi1k12/γ= 50,ω/planckover2pi1kp1/γ= 75, and /planckover2pi1kp1·k12/Mγ=−50. 31-200 -100 0 100 200 δ/γe Probe Absorption (arbitrary units)A1 B1A2B2A+ A-B+ B-e 21g2 g1 Ep/c103e /c76/c49 /c76/c50/c103/c103/c43/c87/c49 /c45/c87/c49\|1, ;n□,n□>pi 1 2 \|2, + ;n□-1,n□+1>pi 1 2/c209k12 \|2, - ;n□-2,n□+1>pi 1 2/c209 /c209k kp1 12+\|1, ;n□+1,n□>pi 1 2+/c209kp1 \|1, - ;n□-1,n□>pi 1 2/c209kp1\|2, + ;n□,n□+1>pi 1 2/c209kp2 G2G1 G2G1(a) 0 21 \|B□>0 \|A□>0\|B□>1 \|A□>1 \|B□>2 \|A□>2/c119(0□□) AB/2 -/c119(0□□) AB/2/c119(1□□) AB/2 -/c119(1□□) AB/2 /c119(2□□) AB/2 -/c119(2□□) AB/2/c1191□□0 /c1192□□0(b) A e nn enni id d 0 02 1 2 02 12 2 11 1 = + −+−−cos ,;,sin ,;,/ / θθ φφpk p12/G21A e nn e nni pi pd d 1 12 12 12 11 2 2 1 1 1 = + +− + +−cos ,;,sin,;,/ / θθ φφpk pk2/G21 /G21 Geitp 2−ΩGeitp 1−Ω(c)-1000 -500 0 500 1000 ∆'/γ-1.0-0.50.00.51.0α/αοL1 L5L3 L7 L8L2 L6L4 -1000 -500 0 500 1000 ∆'/γ-1.0-0.50.00.51.0α/αοL1L4 L2 L3 L7L5 L8L6 -1000 -500 0 500 1000 ∆'/γ-6-3036(α/αο)x104 -1800 -1600 -1400-30-1501530 α/αο -200 0 200 δp1/γ-15.0-7.50.07.515.0 1400 1600 1800-7.5-3.80.03.87.5
arXiv:physics/0012003v1 [physics.bio-ph] 1 Dec 2000Evolutionary conservation of the folding nucleus Leonid Mirny and Eugene Shakhnovich Nov 21, 2000 Running title: Conservation of folding nucleus Submitted to Journal of Molecular Biology Harvard University, Department of Chemistry and Chemical Biology 12 Oxford Street, Cambridge MA 02138 E-mail: leonid@origami.harvard.edu, eugene@belok.harv ard.edu http://paradox.harvard.edu/ ∼leonid 1Abstract In this Communication we present statistical analysis of co nservation proﬁles in families of homologous sequences for nine proteins whose folding nuc leus was determined by protein engineering methods. We show that in all but one protein (AcP ) folding nucleus residues are signiﬁcantly more conserved than the rest of the protein . Two aspects of our study are especially important: 1) grouping of amino acids into class es according to their physical- chemical properties and 2) proper normalization of amino ac id probabilities that reﬂects the fact that evolutionary pressure to conserve some amino a cid types may itself aﬀect concentration of various amino acid types in protein famili es. Neglect of any of those two factors may make physical and biological “signals” from con servation proﬁles disappear. Introduction It is now widely accepted that folding of small single-domai n proteins follows “nucleation- condensation” mechanism (Abkevich et al., 1994; Itzhaki et al., 1995; Fersht, 1997; Shakhnovich, 1997; Guo & Thirumalai, 1995; Pande et al., 1998) whereby relatively small fragment of pro- tein structure is formed in the transition state between unf olded and folded states. Residues belonging to this fragment constitute speciﬁc folding nucl eus (SFN). Considerable experimental (Itzhaki et al., 1995; Main et al., 1999; Martinez et al., 1998; Chiti et al., 1999) and theoretical (Abkevich et al., 1994; Klimov & Thirumalai, 1998; Li et al., 2000; Dokholyan et al., 2000) eﬀort has been devoted to identiﬁcation of folding nuclei in real proteins and various models as well as factors that determine its location in structure and in sequence. One of the most intriguing aspect of nucleation-condensati on mechanism of protein fold- ing is its relation to protein evolution. Indeed residues co nstituting folding nucleus can be metaphorically considered “accelerator pedals” of foldin g (Mirny et al., 1998a) since mutations in those positions aﬀect folding rate to a much greater exten t than elsewhere in a protein. One can conclude that if there is evolutionary control of fol ding rate it should have resulted in additional pressure applied on folding nucleus residues, a nd such pressure can be manifested in noticeable additional conservation of nucleus residues. This idea was ﬁrst proposed in (Shakhnovich et al., 1996) where it was applied to prediction of nucleus residues from protein structure. Many sequences were designed to ﬁt the structure of Chymotripsin Inhibitor 2 (CI2) with low energy. Position s conserved among the designed sequences were identiﬁed as a putative nucleus. This way blindpredictions of folding nucleus in CI2 were made that were veriﬁed in independent experiment s (Itzhaki et al., 1995). In related studies papers Ptitsyn studied conservatism in d istant yet related by sequence homology members of Cytochrome C (Ptitsyn, 1998) and myoglo bin (Ptitsyn & Ting, 1999) families. In both cases he found conserved clusters of resid ues without an obvious functional role which he suggested to belong to folding nucleus of those prot eins. Michnick and Shakhnovich (Michnick & Shakhnovich, 1998) carried out an analysis of co nservation in natural and designed sequences for families of three structurally related prote ins - ubiquitin, raf and ferredoxin and predicted possible folding nucleus for those proteins. 2Neverteheless the notion of folding nucleus conservation h as drawn some controvercy in the lietrature. While earlier papers (Shakhnovich et al., 1996; Michnick & Shakhnovich, 1998; Ptitsyn, 1998; Ptitsyn & Ting, 1999) suggested conservatio n of folding nucleus in some proteins, a more recent paper by Plaxco and coauthors (Plaxco et al., 2000) argued to the opposite. These authors looked at conservatism proﬁle in several protein fa milies for which protein engineering analysis of folding transition states has been carried out, and did not observe correlation between conservation and experimentally measured φ-values. This made them conclude that there is no evolutionary pressure to control the folding rates. In this work we study evolutionary conservation of the foldi ng nucleus for several homologous proteins. Conservation of the folding nucleus is systemati cally compared with the conservation in the rest of the protein sequence. In contrast to previous s tudies, we perform rigorous statis- tical test to assess signiﬁcance of higher conservation in t he folding nucleus. The main result of this study is that for all studied proteins, except AcP, fo lding nucleus is signiﬁcantly more conserved than the rest of the protein. We explain the diﬀere nce between our thorough statis- tical analysis and that of Plaxco et al (Plaxco et al., 2000) by pointing out to some technical shortcomings in the earlier work (Plaxco et al., 2000). Results and Discussion To study evolutionary conservation of the folding nucleus w e turn to nine proteins for which nucleus has been experimentally identiﬁed from protein eng ineering analysis: CI2, FKBP12, ACBP, CheY, Tenascin, CD2.d1, U1A, AcP and ADA2h. For each of them we obtain a multiple sequence alignment from HSSP database (Dodge et al., 1998) (or PFAM (Bateman et al., 2000) database if HSSP contains too few sequences). We compute var iability at position lof the alignment as s(l) =−6/summationdisplay i=1pi(l) logpi(l) (1) where pi(l) is the frequency of residues from class iin position l. We use six classes of residues to reﬂect physical-chemical properties of amino acids and t heir natural pattern of substitutions: aliphatic [A V L I M C], aromatic [F W Y H], polar [S T N Q], basic [ K R], acidic [D E], and special (reﬂecting their special conformational properti es) [G P]. As a result of this classiﬁcation mutations within a class are ignored (e.g. V→L), while mutations that change the class are taken into account. Figure 1 presents variability proﬁle fo r studied proteins with nucleation positions marked by ﬁlled circles. Importantly, we deﬁned t he folding nucleus as it was identiﬁed Fig.1 by the original experimental groups (Table 1). Figure 2 clearly shows that nucleus residues are almost alwa ys among the most conserved Fig.2 ones for all studied proteins. It also shows that nucleus res idues are not the only conserved ones: many other residues (predominantly in the cores of the proteins) are also conserved. In order to evaluate statistical signiﬁcance of nucleus con servation we compare evolutionary conservation of the folding nucleus with the conservation o f all residues in the protein using the following statistical test. We start from the null hypot hesis H0 that nucleus residues are 3nomore conserved than the whole protein sequence. To test this hypothesis we compute median variability of the nucleus residues (med[ snuc]) and compare it with the distribution of medians variability of the same number of residues random ly chosen in the same protein (f(med[srand])). The distribution f(med[srand]) is obtained by choosing 105random sets of nresidues ( nis the number of residues in the nucleus). Then the fraction o f instances with med[srand]<med[snuc] gives the probability P0of accepting H0. In other words, P0is the probability that observed lower variability of the folding nucleus is obtained by chance. Hence, P0≤αindicates statistically signiﬁcant strong evolutionary c onservation of the folding nucleus. Below we use conﬁdence level α= 2%. Table 2 presents computed P0values. The main result of this work is that in all proteins, except AcP, residues in the folding nucleus are signiﬁcantl y more conserved than the rest of the protein. Next we study how obtained results depend on the way amino aci ds are grouped into classes (see Table 2). When classiﬁcation scheme from (Bran den & Tooze, 1998) (BT) is used, still all proteins except AcP exhibit signiﬁcant cons ervation of the folding nucleus. This clearly demonstrates that observed conservation of the fol ding nucleus is not a consequence of a particular choice of the classiﬁcation scheme. However, when amino acids are notgrouped into classes, nucleus exhibits signiﬁcant conser- vation only in four out of nine proteins. Taken together thes e results indicate that substitutions in the folding nucleus may occur, but they are limited to resi dues that belong to the same class (i.e. have similar physical-chemical properties (Thompso n & Goldstein, 1996)). To study what physical-chemical properties are conserved i n the folding nucleus we used various classiﬁcation schemes. Starting from all 20 amino a cids, we grouped some of them into classes and repeated the analysis, including the stati stical tests (see Table 2). The goal is to ﬁnd a minimal classiﬁcation (i.e. grouping the minimal number of amino acids together) that provides statistically signiﬁcant conservation of th e folding nucleus. Our results show that classiﬁcation where only I, L, and V are grouped in one class w hile all other amino acids each represent their own class satisﬁes this requirement (see Ta ble 2).This classiﬁcation provides signiﬁcant conservation of the nucleus for all proteins exc ept AcP with α= 5%, and for all proteins except AcP and FKBP12 with α= 2%. This result demonstrates that I⇀↽L⇀↽V are the most common substitutions in the nucleus (and in the p rotein core in general (Henikoﬀ & Henikoﬀ, 1992; Benner et al., 1994)). These substitutions are tolerated in the nucleus a s they do not change much neither stability of the native fold n or the folding rate. Analysis of available experimental data (L.Li unpublished) shows th at changes in stability upon I⇀↽ L⇀↽Vmutations are in average /angbracketleft∆∆GN−D/angbracketright= 1.0±0.4kCal mol−1for the native state and /angbracketleft∆∆G‡−D/angbracketright= 0.2±0.3 kCal mol−1for the transition state. Note that grouping of residues into classes to assess conser vation is similar to the use of substitution matrices in sequence alignment techniques. T he underlying idea for both methods is to take into account natural physical-chemical similari ty between amino acids and their substitution patterns. Plaxco et all used all 20 types of ami no acids and failed to identify strong conservation of the folding nucleus (Plaxco et al., 2000). Similarly, a method that relies on simple sequence identity cannot detect distant ho mology. However distant homology 4between sequences can be detected using proper substitutio n matrices (Abagyan & Batalov, 1997; Brenner et al., 1998). The use of substitution matrices is physically mean ingful since they weight, e.g., I−Vmatch higher then I−D, while a method that relays on percentage of sequence identity weights I−VandI−Dequally. Likewise, our amino acid classiﬁcation scheme does not count I→Vas a mutation, while it certainly considers substitutions l ike I→Das mutations to be counted. Although, on average, nucleus is more conserved, than the re st of the protein, not all nucleating residues are strongly conserved. For example, i n CheY two out of ten nucleation residues are not conserved. In ADA2h two out of ﬁve and in tena scin one out of four residues are not conserved. Some nucleus residues may be less conserved b ecause they belong to “extended nucleus” (Mirny & Shakhnovich, 1999) or because of limitati on of our residues classiﬁcation scheme that puts aromatic and aliphatic residues into two di ﬀerent groups, while aromatic- aliphatic substitutions may occur in the core of some protei ns (i.e. tenascin, ADA2h) usually as a result of correlated mutations that are not treated proper ly in this approach (but are taken into account in the conservation-of-conservation approach (Mi rny & Shakhnovich, 1999)). Another interesting observation is that the only protein that exhib its no preferential conservation of the folding nucleus is AcP, which is the slowest folding protein among all studied two-state folding proteins ( kH2O f= 0.23s−1). Perhaps, this protein did not undergo evolutionary selec tion for faster folding and hence its folding nucleus is under no addi tional pressure to be conserved. Note that, as expected, several other residues in studied pr oteins are as conserved as the nucleating ones. (see Fig.2) Those are the residues of the ac tive site, core hydrophobic residues responsible for stabilization of the native structure and o thers. This suggests that although folding nucleus is conserved it can not be uniquely identiﬁe d just by analysis of a single protein family as a pattern of conservation is dominated by residues conserved for protein stability and function (see (Mirny & EI, )). Thus a consistent analysis sho uld discriminate between residues that are conserved for functional reasons, for stability re asons and for kinetic reasons (folding nucleus), like it was done in a more detailed conservation-o f-conservation analysis in (Mirny & Shakhnovich, 1999). Why do results of our analysis diﬀer from those of Plaxco et al (Plaxco et al., 2000)? First, we took into account physical-chemical properties of amino acids and their natural substitution patterns to group amino acids into classes. As we showed, sub stitutions of large aliphatic residues (I,L,V) are frequent in folding nuclei and this con fused previous analysis that did not apply any amino acid classiﬁcation scheme. While Plaxco et a l claimed in their paper (Plaxco et al., 2000) (without providing a supporting evidence) that grou ping of amino acids into classes did not change their conclusions, our analysis shows that pr oper classiﬁcation of amino acids is crucial for detecting conservation in the folding nucleus. Second, Plaxco et al used a diﬀerent method to compute sequen ce variability: s2(l) =−/summationdisplay ipi(l) log[pi(l)/p0 i] (2) This equation diﬀers from eq.(1), used in this study, in norm alization by p0 i- the “background” frequency of residue type iin all proteins. Although the diﬀerence may seem technical, equations 5(1) and (2) are based on two diﬀerent models of evolution. We a rgue that while equation 2 may be adequate for DNA sequence analysis (Stormo, 1998) it i s not appropriate for analysis of protein evolution. Equation 2 implicitly assumes that amino acid composition p0 iis ﬁxed a priori in each protein. Hence equation (2) tends to underestimate conserv ation of “frequent” amino acids (L,A,S etc), while overestimating conservation of less fre quent amino acids (W,C,H etc). In contrast, equation (1) assumes that conservation requirem ent itself aﬀects the composition, i.e. higher conservation of an amino acid leads to its higher freq uency in proteins. To illustrate this point consider a toy protein that consist s of two types of residues: hy- drophobic H and polar P. Assume that 70% of amino acids in this proteins are in the core and 30% are in the loops. Also assume that in the toy world sele ction for stability requires a 100% conservation of H amino acids in the core, while loops ar e under no evolutionary pressure and H and P are equally probable in the loops. Then p0 H= 1·0.70 + 0 .5·0.3 = 0 .85 and p0 P= 0.5·0.3 = 0.15. At conserved core positions s2(core) = −1 log 1 /0.85≈ −0.16, while in the loops s2(loops) = −0.5 log 0 .5/0.85−0.5 log 0 .5/0.15≈ −0.34. Hence, the use of equation (2 leads to a counterintuitive and apparently wrong result s2(core) > s2(loops), i.e. that loops are more conserved than 100% conserved core! Clearly this re sult shows inadequacy of equation (2) as applied to protein evolution with unconstrained comp osition. Similarly, application of equation 2 to real proteins leads to unreasonably low conser vation of the hydrophobic core as compared to exposed loops (data not shown). A possible way to compensate for variations in amino acid com position of proteins is to deﬁne the sequence entropy as in (Schneider, 1999): s(l) =−/summationdisplay ipi(l) logpi(l) +/summationdisplay ip0 ilogp0 i (3) where the second term gives the “background” variability du e to amino acid composition. This term however does not depend on land hence does not change the relative variability. Interestingly, the use of equation (2) by Plaxco et al (Plaxc oet al., 2000) gave rise to a surprising result that active sites in proteins are general ly no more conserved than the rest of the protein (see Fig.2 of (Plaxco et al., 2000)). Conservation of known active sites was used as a control in (Plaxco et al., 2000) for their method of analysis based on equation 2 which it apparently failed. Finally, Plaxco et al did not study conservation of the foldi ng nucleus. Instead, they focused on the residues that featured high φ-values in protein engineering experiments and compared them with low φ-value residues. As we explained above residues in the foldi ng nucleus do not necessarily exhibit high φ-values, and many low φ-value residues are conserved in evolution as they contribute to stabilization of the native structure. C omparison with low φ-value residues instead of comparison with the whole protein also confused previous analysis since most of φ- values have been measured for amino acids located in the the c ore of a protein and hence these amino acids are on average more conserved. Here, in contrast , we used the folding nucleus as it was identiﬁed for each protein by the original experimental group and compared its conservation with the conservation of all amino acids in the protein. 6In summary, we showed that folding nucleus is indeed conserv ed in most of the proteins whose folding transition states are known from protein engi neering analysis. That does not mean that folding nucleus residues are the the only conserved ones in any family of homologous proteins. That also may not mean that folding nucleus is moreconserved than other residues in the protein core, as nucleus is equally important for protei n stability and for fast folding. Our result show that the folding nucleus is more conserved than t he rest of the protein. As stated earlier it is diﬃcult to uniquely identify folding nucleus b y looking at a conservation proﬁle in just one family of homologous sequences. Nevertheless cons ervation of folding nucleus found in this paper and in other works (Mirny & Shakhnovich, 1999; Li et al., 2000) points out to an exciting possibility that folding rates may be of biologica l signiﬁcance. Biological signiﬁcance of this fact needs to be assessed in future studies. 7References Abagyan, R. & Batalov, S. (1997). Do aligned sequences share the same fold? J Mol Biol , 273:355–68. Abkevich, V., Gutin, A., & Shakhnovich, E. (1994). Speciﬁc n ucleus as the transition state for protein folding: Evidence from the lattice model. Biochemistry , 33:10026–10036. Bateman, A., Birney, E., Durbin, R., Eddy, S., Howe, K., & Son nhammer, E. (2000). The pfam protein families database. Nucleic Acids Res , 28:263–6. 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Universally conserved r esidues in protein folds. reading evolutionary signals about protein function, stability an d folding kinetics. J.Mol.Biol. , 291:177–196. Pande, V., Grosberg, A., Rokshar, D., & Tanaka, T. (1998). Pa thways for protein folding: is a “new view” needed? Curr Opin Struct Biology , 8:68–79. Plaxco, K., Larson, S., Ruczinski, I., Riddle, D., Buchwitz , B., Davidson, A., & Baker, D. (2000). Evolutionary conservation in protein folding kine tics.J. Mol.Biol. , 298:303–312. Ptitsyn, O. (1998). Protein folding and protein evolution: Common folding nucleus in diﬀerent subfamilies of c-type cytochromes? J.Mol.Biol , 278:655–666. 9Ptitsyn, O. & Ting, K. (1999). Non-functional conserved res idues in globins and their possible role as a folding nucleus. J.Mol.Biol , 291:671–682. Schneider, T. (1999). Measuring molecular information [le tter].J Theor Biol , 201:87–92. Shakhnovich, E. (1997). Theoretical studies of protein-fo lding thermodynamics and kinetics. Curr. Opin. Struct. Biol. , 7:29–40. Shakhnovich, E., Abkevich, V., & Ptitsyn, O. (1996). Conser ved residues and the mechanism of protein folding. Nature , 379:96–98. Stormo, G. (1998). Information content and free energy in dn a–protein interactions. J Theor Biol, 195:135–7. Ternstrom, T., Mayor, U., Akke, M., & Oliveberg, M. (1999). F rom snap-shot to movie: phi- value analysis of protein folding transition states taken o ne step further. Proc Natl Acad Sci USA , 96:14854–14859. Thompson, M. & Goldstein, R. (1996). Constructing amino aci d residue substitution classes maximally indicative. Proteins , 25:28–37. Vilegas, V., Martinez, J., Avilez, F., & Serrano, L. (1998). Structure of the transition state in the folding process of human procarboxypeptidase a2 acti vation domain. J Mol Biol. , 283:1027–1036. 10Figure Captions Fig.1 Variability proﬁles (sequence entropy) for nine diﬀerent p roteins computed using MS residue classes. Circles indicate positions at which φ-values have been experimentally measured. Residues forming the folding nucleus are shown by ﬁlled circ les. Fig.2 Nine studied proteins with C βatoms colored according to the degree of their con- servation (evaluated in Fig.1): from blue (high conservati on) to light-blue, green, yellow and red (no conservation). Folding nucleus residues are shown b y twice as large spheres. Notice conserved (blue) cores of the proteins and non-conserved (y ellow and red) surfaces. Also notice several conserved non-nucleus residues in the protein core . 11Protein PDB Folding Nucleus Reference CI2 2ci2I A35 L68 I76 (Itzhaki et al., 1995) Tenascin 1ten I821 Y837 I860 V871 (Hamill et al., 2000) CD2.d1 1hnf L19 I21 I33 A45 V83 L94 W35 (Lorch et al., 1999) CheY 3chy D12 D13 D57 V10 V11 V33 A36 D38 A42 V54 (Lopez-Hernandez & Serrano, 1996) ADA2h 1aye I15 L26 F67 V54 I23 (Vilegas et al., 1998) AcP 1aps, 2acy Y11 P54 F94 (Chiti et al., 1999) U1A 1urn I43 V45 L30 F34 I40 I14 L17 L26 (Ternstrom et al., 1999) ACBP 1aca F5 A9 V12 L15 Y73 I74 V77 L80 (Kragelund et al., 1999) FKBP12 1fkj V2 V4 V24 V63 I76 I101 (Main et al., 1999) Table 1: Folding nuclei as identiﬁed by the authors MS BT no grouping [I,L,V], [W,F,Y] [I,L,V] [I,L,V] [I,L,V] [R,K] [D,E] [W,F,Y] [W,F] Nclass 6 5 20 14 16 17 18 CI2 0.0041 0.01 0.0382 0.007 0.002 0.004 0.0044 FKBP12 0.0187 0.02 0.1585 0.044 0.047 0.053 0.0363 ACBP <10−5<10−50.0216 0.022 0.008 0.0080 0.0067 CheY <10−5<10−50.0011 0.0040 0.0050 0.0020 0.0022 Ten 0.008 0.018 0.2477 0.0260 0.0220 0.0130 0.0197 CD2.d1 <10−5<10−5<10−5<10−3<10−3<10−3<10−5 U1A 0.0009 0.001 0.0029 <10−3<10−3<10−30.0002 AcP 0.089 0.086 0.0126 0.025 0.021 0.009 0.0136 Table 2: Probability P0of nucleus being as conserved as the whole protein (see text f or details) computed for all nine proteins and seven diﬀerent classiﬁca tion schemes. MS as in (Mirny et al., 1998b; Mirny & Shakhnovich, 1999), BT as in (Branden & Tooze, 1998): hydrophobic [A V F P M I L],polar [S T Y H C N Q W],basic [R K],acidic [D E],gly [G]), Nclass- number of groups in each classiﬁcation0 50 100012CI2 050100150012FKBP 0 50 100012ACBP 050100150012CheY 800 850 900012TenVariability0 100 200012CD2 0 50 100012U1A 0 50 100012AcP 0 50 100012ADA2h Residue Figure 1:AcP U1ACD2 TenCheY ACBPFKBP CI2 ADA2h Figure 2:
arXiv:physics/0012004v1 [physics.atom-ph] 4 Dec 2000Collapsing dynamics of attractive Bose-Einstein condensa tes L. Berg´ e1and J. Juul Rasmussen2 1Commissariat ` a l’Energie Atomique, CEA/DAM - Ile de France , B.P. 12, 91680 Bruy` eres-le-Chˆ atel, France. 2Risø National Laboratory, Optics and Fluids Dynamics Depar tment, P.O. Box 49, 4000 Roskilde, Denmark. (February 2, 2008) The self-similar collapse of 3D and quasi-2D atom condensat es with negative scattering length is examined. 3D condensates are shown to blow up following the s cenario of weak collapse : The in- ner core of the condensate diverges with an almost zero parti cle number, while its tail distribution spreads out to large distances with a constant density proﬁl e. For this case, the 3-body recombina- tion arrests the collapse, but it weakly dissipates the atom s. The conﬁning trap then reforms the condensate at later times. In contrast, 2D condensates unde rgo astrong collapse : The atoms stay mainly located at center and recombination sequentially ab sorbs a signiﬁcant amount of particles. PACS numbers : 03.75.Fi, 32.80.Pj, 42.65.Jx. A few years ago, Bose-Einstein condensates (BECs) were discovered in trapped clouds of alkali atoms [1,2]. Among those,7Li atoms are known to be characterized by attractive interactions with a negative scattering length , a0<0 [2]. This promotes a collapse-type instability, which yields a singular increase in the BEC wavefunc- tion. It occurs when the number of particles, ˜N, exceeds a threshold value, ˜Nc∼l0/\|a0\|, wherel0= (/planckover2pi1/mω)1/2 is the amplitude of zero point oscillations in the conﬁn- ing trap of frequency ωandmis the atom mass. BECs formed in7Li gas develop several sequences of collapse [3]. From the initial cooling, the condensate is ﬁrst fed by the thermal cloud of uncondensed atoms. Then, once the number of atoms is above ˜Nc, the condensate sharply concentrates with an increasing density. As the density rises, inelastic collisions as 3-body molecular recombina - tion increase, which arrests the collapse. The thermal cloud is next believed to re-ﬁll in the condensate, that reaches ˜N > ˜Ncand then collapses again. The cycle of collapses repeats many times until the gas comes to equilibrium. This scenario is supported by numerical in- tegrations of the Gross-Pitaevskii (G.-P.) equation [4,5] : i∂tψ=−∆rψ+r2ψ− \|ψ\|2ψ−iη\|ψ\|mψ+iγψ, (1) wherer= ˜r/l0,t=˜tω/2,ψ=˜ψ(8πl2 0\|a0\|)1/2are the dimensionless time, coordinates and wavefunction of the BEC (tilde refers to physical quantities). Here, the loss/gain mechanisms are described by the last two terms of Eq. (1), where η≪1 andγ≪1 are the co- eﬃcients related to the local recombinational decrease of the condensate density and to the ﬂux of particles from the nonequilibrium thermal cloud to the conden- sate, respectively. At leading order, we consider 3-body recombinations with m= 4. The normalized number of particles,N=/integraltext \|ψ\|2d/vector r, is related to its physical coun- terpart as ˜N=N(l0/8π\|a0\|)≃86.7N. It was recently shown that stationary condensates deﬁned by ψs(r,t) = χ(r,µ)e−iµtwith chemical potential µare stable when- ever they satisfy dNeq/dµ< 0, whereNeq(µ) =/integraltext \|χ\|2d/vector r[6]. Along the curve Neq(µ), 3D condensates then re- quiresNeq≤Nc= 14.45 for being stable. In physical units, this threshold yields the precise critical number for stability: ˜Nc= 1252 atoms, for the7Li parameters a0=−1.45 nm and l0= 3.16µm used in [2,3,5]. Numerical simulations [4] of Eq. (1) revealed that, near the collapse moment, the condensate expands with a density proﬁle exhibiting a low-amplitude, almost ﬂat plateau, \|ψ\|2r2→const, from r >1. This plateau-like behavior was again numerically found in [5], where the free collapse was described as a ”black hole”. Follow- ing this scenario, the increase in the BEC density is fu- eled by particles drawn from throughout the outer region (r>1), while the density outside form imploding ripples. However, it was suggested that the asymptotic plateau in r2\|ψ\|2may not be constant, but instead diverges in time. Thus, the singular dynamics of 3D collapsing BECs is still questionable and a self-consistent model for BEC collapse is actually missing. Describing the structure of such collapses is of utmost importance, in order to under- stand the inﬂuence of recombination, trap conﬁnement and re-feeding over several collapse events. In this letter, we examine the self-similar nature of col- lapsing BECs. For 3D isotropic condensates the number of particles is analytically shown to vanish near center and, outside, a constant plateau in the density proﬁle, r2\|ψ\|2, is actually formed. From this dynamics, 3-body recombination arrests the collapse by removing a limited number of particles from the condensate. We also brieﬂy investigate 2D condensates. As justiﬁed in [7], quasi- 2D BECs can be produced from 3D atom clouds with a density frozen on a Gaussian shape ∝e−˜z2/l2 0, when the particles are tightly conﬁned along the longitudinal axis. In what follows, 2D and 3D condensates are con- sidered as isotropic, radially-symmetric objects, since t he self-compression induced by the collapse dynamics make them have comparable sizes along each direction. Let us ﬁrst discuss the inertial regime of collapse, for which we set γ=η= 0 in Eq. (1). Collapse oc- 1curs in the sense that the mean-square width of attrac- tive BECs, ∝an}b∇acketle{tr2∝an}b∇acket∇i}ht=N−1/integraltext r2\|ψ\|2d/vector r, tends to zero in ﬁ- nite time (see, e.g., Pitaevskii [8]). From the inequality N≤ ∝an}b∇acketle{tr2∝an}b∇acket∇i}ht/integraltext \|∇ψ\|2d/vector r[9], the vanishing of ∝an}b∇acketle{tr2∝an}b∇acket∇i}htleads to the blow-up of the gradient norm, which in turn implies the divergence of the integral/integraltext \|ψ\|4d/vector rin the conserved Hamiltonian of Eq. (1): H=/integraldisplay \|∇ψ\|2d/vector r−1 2/integraldisplay \|ψ\|4d/vector r+N∝an}b∇acketle{tr2∝an}b∇acket∇i}ht. (2) By virtue of the mean-value theorem \|ψ\|4≤ max r\|ψ\|2/integraltext \|ψ\|2d/vector r, the maximum amplitude of the wave- function also blows up in ﬁnite time. It should be em- phasized that the blow-up generally occurs before ∝an}b∇acketle{tr2∝an}b∇acket∇i}ht reaches zero [9]. To examine the shape of collapsing con- densates, we introduce the self-similarlike substitution : ψ(/vector r,t) =a−α(t)φ(/vectorξ,τ)eiλτ−iβξ2/4, (3) where/vectorξ=/vector r/a(t),τ(t)≡/integraltextt 0du/a2(u) andβ=−a˙a(dot means diﬀerentiation with respect to time). Here, the parameterλ∼ −µmust be positive in the absence of the trap for making φlocalized. The function a(t) represents the BEC scale length that vanishes as collapse develops, andφ(/vectorξ,τ) is a regular function with amplitude of order unity. Asa(t)→0, it is assumed that φconverges to an exactly self-similar form φ(/vectorξ), which no longer depends explicitly on time, i.e., ∂τφ→0. In this limit, the right balance between the two integrals in Eq. (2) requires α= 1, in order to assure the ﬁniteness of H. The particle numberN=/integraltext \|ψ\|2d/vector rthen reads N=aD−2(t)/integraltext \|φ\|2d/vectorξ and the dynamics drastically changes following the space dimension number D. Setting thus α= 1 and plugging Eq. (3) into (1) transforms the G.-P. equation into i∂τφ+ ∆ ξφ+\|φ\|2φ+ǫ[ξ2−ξ2 T]φ−a4ξ2φ+ iηa2−m\|φ\|mφ−iγa2φ= 0, (4) where ∆ ξ=ξ1−D∂ξξD−1∂ξ,ǫ≡ −1 4a3¨a=1 4(β2+βτ) and ξ2 T≡ǫ−1[λ+iβ(D/2−1)] is viewed as a turning point. After an initial stage during which the trap gathers the particles at center, the wavefunction diverges freely on short time scales ∆ ˜t≪ω−1. It becomes hyperlocalized atr∼0 witha(t)→0, so that the eﬀect of trapping can be neglected. Near the collapse point, refeeding from the surrounding cloud is also ineﬃcient. Moreover, 3- body recombination does not act, as long as the BEC radius satisﬁes ηa2−m\|φ\|m≪ \|φ\|2. In this regime, Eq. (4) thus reduces to the self-similarly transformed nonlin- ear Schr¨ odinger (NLS) equation, whose properties, accu- rately veriﬁed in [9–11], are recalled below. In the self-similar limit a(t)→0,∂τφ→0, the time- dependent function ǫconverges to β2/4, which, for self- consistency, converges to a constant, ǫ0. (ii) The so- lutionφin Eq. (4) can be decomposed into a corecontribution φcextending in the range ξ < ξ T, i.e., r < r T≡a(t)\|ξT\|, and a tailφTdeﬁned in the com- plementary spatial domain ξ > ξ T, i.e.,r > r T. (iii) The self-similar assumption ∂τφ→0 holds provided that ξ < ξ max≡A\|ξT\|/a(t), whereA= const ≫1. Self- similar solutions are thus limited by the boundary radius rmax=A\|ξT\| ≫1. Knowing this, the solution φde- composes as φ=φc\|ξ<ξ T+φT\|ξT<ξ<ξ max. In the long distance domain, the nonlinearity vanishes, so that φT can be determined from the linear version of Eq. (4) by means of WKB methods [9]. As a result, the wavefunc- tionψreads near the collapse point ( ǫ=β2/4) as ψ(/vector r,t) =eiλ/integraltextt 0du a2(u) a(t)× (5) {φc(r a,ǫ)e−iβr2/4a2\|0≤r<r T+C(β) (r/a)1+iλ/β0\|rT<r<r max}, \|C(β)\|2≃2φ2(0) β\|ξT\|D−2ef(λ/β), (6) f(λ β) =−πλ β+(D 2−1)(1+2 ln2) −λ βarctan[β λ(D 2−1)]. The scaling law a(t) in the inertial range of collapse is then identiﬁed through the continuity equation for ψ: /integraldisplayrmax 0∂t\|ψ\|2rD−1dr=−2rD−1\|ψ\|2∂rarg(ψ)\|rmax −2η/integraldisplayrmax 0\|ψ\|m+2rD−1dr+ 2γ/integraldisplayrmax 0\|ψ\|2rD−1dr,(7) where arg[ψ(r= 0)] = 0. By applying the solution (5), (6) to Eq. (7) and Taylor-expanding φcaround ǫ=β2/4 =ǫ0, the contraction scale a(t) is indeed readily determined for η=γ= 0 from the dynamical system: c1βτ≃ −c2 βef(λ/β)+D−2, (8) wherec1∝Re/integraltext φ∗ 0∂ǫφ\|ǫ0d/vectorξandc2>0 are constants. We specify the structure of the collapse for 3D and quasi-2D BECs separately: 1 -Three-dimensional condensates : Let us ﬁrst de- scribe the inertial range of collapse. For D= 3, Eq. (8) shows that βrapidly attains a ﬁxed point β0>0 corresponding to the self-similar state βτ= 0. The scaling law a(t)∼(tc−t)1/2follows, where tcde- notes the collapse moment. The characteristics of a 3D collapse is that the number of particles is not pre- served self-similarly in the whole spatial domain, since N=a(t)N{φ}withN{φ}=/integraltext \|φ\|2d/vectorξ. By virtue of Eq. (5), we see that, near the collapse instant, Nbehaves as 2N≃Ncore(t) +Ntail(t) =a(t)[N{φc}(t) +N{φT}(t)], where N{φc}= 4π/integraldisplayξT 0\|φc\|2ξ2dξ≃O(1), (9) N{φT}= 4π\|C(β)\|2/integraldisplayξmax ξTdξ≃4π\|C(β)\|2rmax/a(t), so that almost all Nlies in the tail, Ntail(t) = 4π\|C(β0)\|2rmax, asa(t)→0. Therefore, a 3D collapse takes place at the center of the trap where the wave- function \|ψ\|=\|φ\|/a(t) diverges. However, it is accom- panied by an expulsion of particles towards the large- distance domain r≫rT(t). Thus, after the onset of collapse, the singularity develops, not by taking parti- cles from outside as in the ”black hole” scenario pro- posed in [5], but by ejecting particles outward the core domain . This is a weak collapse , as originally deﬁned in Ref. [12] for the free NLS equation. The solution ψ blows up at center, near which the number of particles becomes zero, i.e., Ncore(t) = 4π/integraltextrT(t) 0\|ψcore\|2r2dr→0, asrT(t)∼a(t) vanishes. Accordingly, the tail of the BEC wavefunction in the outer domain extends in space with the time-independent density: r2\|ψ\|2→ \|C(β0)\|2 = const, deduced from Eq. (5). This constant de- pends on the values of β0and\|ξT\|. For the scaling law a(t) =a0(tc−t)1/2witha0set equal to the unity without loss of generality, β0= 1/2 and numerical integrations of the 3D NLS equation with no trap [11] emphasize the valuesφ0≡φ(0) = 1.39 andλ= 0.545, which yield \|ξT\| ∼3.1 and \|C(β0)\|2≃0.17∼0.2. A tail contribu- tionr2\|ψ\|2/\|ψ(t= 0)\|2about 0.2 in magnitude for r≥1 seems compatible with the density proﬁles computed in Refs. [4,5] in the vicinity of the collapse moment. The formation of stationary plateau-like density proﬁles was numerically conﬁrmed in [10,11] by means of very ﬁne numerical schemes that solved the rescaled equation (4) with high accuracy and could access huge growths in \|ψ\|2. In contrast, the same plateau was claimed to diverge in time by only reﬁning the spatial grid in [5]. We suspect that numerical computations in [5] suﬀered serious lack of resolution, which prevented the authors from concluding correctly on the constancy of r2\|ψ\|2at large distances. We now discuss the dissipative/gain regime, that in- volves collisional losses and feeding by the thermal cloud. The inﬂuence of these two eﬀects on the con- densate particle number is described by the last two terms in the continuity equation (7). By introduc- ing the self-similar shape (5), these terms are found to read−2ηa1−m/integraltextξT 0\|φc\|m+2ξ2dξand 2γa/integraltextξmax ξT\|φT\|2ξ2dξ fora(t)≪1, respectively. Hence, 3-body recombina- tion mainly acts on the core part of the solution \|ψc\|= \|φc\|/a(t), whereas re-feeding by the surrounding cloud is eﬃcient in the outer region, where the main amount of particles is residing, with 2 γ/integraltextrmax 0\|ψ\|2r2dr≃2γNtail(t).Collisions begin to be active when a(t) decreases so much that ηa2−m\|φ\|m≃ \|φ\|2[Eq. (4)], i.e., this contri- bution saturates the blow-up induced by the cubic non- linearity, for m>2. By means of this relation, the num- ber of particles ∆ Nlosslost from the condensate during one collapse event is then yielded by that in the core re- gion,Ncore. Inelastic collisions are thus able to stop the collapse, but they cannot remove a lot of particles, the major part of N≃Ntailstarting to be transferred to large distances via the weak collapse. Explicitly, we ﬁnd by using Eq. (5) expressed with arbitrary mandD: ∆Nloss≃ −2D−1(φ0\|ξT\|)Dπη(D−2)/(m−2)/D. (10) ForD= 3 andm= 4, the particle loss is estimated as ∆Nloss≃4π√ηφ3 0ξ3 T/3 = 0.5, when we choose the valueη= 2.2×10−6physically justiﬁed in [5] and em- ployφ0= 1.39 andξT= 3.1. Thus, in physical units, the number of atoms removed during a single blow-up event is ∆˜Nloss= 44 atoms. This loss of particles agrees with recent numerical observations [13]. Here, approximately 300 particles were lost from a condensate of initial num- berN0= 1260 atoms, in a sequence of 5-6 individual blow-up events with peak amplitudes \|ψ\|max∼η1/2−m. As discovered by Vlasov et al.[14] and conﬁrmed in [11], an untrapped, 3D collapsing ﬁeld is damped within a very short time window, in which the ﬁeld experiences secondary blow-up events of duration ∼η2/m−2, as long as nonlinear dissipation remains active. These arise while most of the atoms form a broad disribution surrounding the collapsing core and are directed outwards. Collapse is not arrested abruptly, but it is distributed through a series of individual spikes. This ”distributed collapse” [14] resembles the ”intermittent implosions” numerically detected in [13]. Note that these intermittent implosions, which overlap as one collapse sequence over a trap period, may not be resolved in real experiments. Their integrated eﬀect should only yield a smooth decrease in ˜N. Once collapse is limited, it is necessary to under- stand the mechanism underlying reformation and peri- odic resurgences of spiky amplitudes in the condensate [3]. We emphasize that after the collapse stage most particles lie in the outer domain 1 ≤r < r max, where the atoms feel both the trap curvature and re-feeding by the thermal cloud. The latter re-injects particles into the BEC according to N(t)≃Ntaile2γt, i.e., on time scales t≃(2γ)−1ln (1 + ∆N/N 0). For reasonable values ofγ= 3×10−3[5], the physical time for re- ﬁlling the condensate with ∆ N/N 0≈0.25 is too long for justifying the spiky oscillations occurring periodi- cally at times ∼ω−1, mentioned in, e.g., [13]. We have thus to analyze the stage of trap reconﬁnement at ear- lier instants for which refeeding is negligible. To this aim, we deﬁne the centroid of the expanded structure by /vectorX(t)≡N−1/integraltext /vector r\|ψ\|2d/vector r. From algebraic manipulations of Eq. (1) with γ=η= 0, this centroid is found to obey 3the relation¨/vectorX+ 4/vectorX=/vector0. Simple phase transformation [15] keeps equation (1) unchanged when passing over to the frame moving with X(t). It is then easy to conclude that, from a reference coordinate /vectorX(t0)∝ne}ationslash=/vector0 that locates the BEC expanded in the outer domain at a given time t0, the condensate will return to the center of the trap over physical times = π/2ω∼ω−1after the collapse mo- ment. Once reconﬁned with N < N c, the condensate is close to an equilibrium, that is described by the ground state solution ψs=χ(r,µ)e−iµtof the G.-P. equation. As shown in Ref. [6] (see also [16]), ground states are stable with dNeq/dµ < 0 for rather broad condensates (0.72< µ < 3). They are unstable and collapse with dNeq/dµ > 0 for rather narrow condensates ( µ <0.72). In this range, the threshold Nc>N eqdecreases with \|µ\| asNc(µ)≤18.94/\|µ\|. It is thus highy problable that un- der the strong compression induced by the trap, the equi- librium state conﬁned at center corresponds to a narrow, unstableψsundergoing a new collapse sequence with a lower critical number of particles. This may repeat over several trap periods, until the refeeding becomes a key- player for making the resulting BEC bifurcate to a stable equilibrium reached at a weak number of atoms [3]. 2-Quasi-two-dimensional condensates : The dynamics signiﬁcantly changes when one considers quasi-2D BECs. Forη=γ= 0, the scaling law a(t) inferred from Eq. (8) withD= 2 indeed behaves with a twice-logarithmic correction: a(t)≃a0√tc−t//radicalbig ln ln [1/(tc−t)] [9–11]. Sinceβ≃π/lnτ(t)→0 asτ(t)≃ln [1/tc−t]→+∞, the exponential contribution in the tail amplitude of Eq. (6) decreases to zero, while the core boundary ξT=/radicalbig λ/ǫ increases slowly to inﬁnity when t→tc. As a result, collapse takes place with a core solution φcproviding the principal contribution in the wavefunction ψ.Nis mainly given by/integraltext \|φc\|2d/vectorξ, which relaxes to the critical valueNc=/integraltext \|χ\|2d/vectorξ≃11.7. Below this number, sta- tionary condensates are stable [6]. So, most of parti- cles stay located around the center of the trap, when collapse develops. This scenario meets the deﬁnition of the so-called strong collapse , which contrasts with the 3D weak collapse that promotes a leak of atoms to large distances. The relative variations in the particle num- ber due to collisional losses are governed by the estimate ∆Nloss≃ −2φ−2 0ηa2−m/integraltext \|φc\|m+2d/vectorξ. Asa(t) decreases, the part burnt by recombination is here estimated by ∆Nloss/N≃ −2/(1 +m/2), when we apply the evalu- ationηa2−m≈ \|φ\|2−mand model φaround a Gaussian with intensity I0:φ=√I0e−ξ2/2. Form= 4, 3-body recombination then removes about more than one-half of the condensed atoms. Although rough, this estimate does not explicitly vary with the coeﬃcient η, as already indicated by Eq. (10), and it slightly decreases with m, in agreement with the numerical observations of Ref. [11]. Collapse is thus arrested by 3-body recombinations that sequentially burn a substantial amount of atoms per col-lapse event. This agrees with the numerical results of Refs. [11,14], from which ∆ Nlossis about 0.25−0.3×Nc. In conclusion, we have shown that BECs with attrac- tive interactions collapse with a mean radius contract- ing like√tc−tat leading order, where tcdenotes the collapse moment. In 3D, the collapse is weak : the am- plitude of the wavefunction blows up at center, while particles are expelled outwards with a constant density proﬁler2\|ψ\|2→const. Recombination losses limit the collapse, but they only damp a few percents of the atoms at each blow-up. Several blow-up events develop within one collapse sequence. The condensate is then reconﬁned by the trap and can further undergo more collapse cycles, as long as a stable state has not been reached. Quasi-2D condensates are subject to a strong collapse , in the sense that the number of particles remains mostly conﬁned at center. In this case, recombination removes a signiﬁcant fraction (up to 0.5) of particles per collapse event. This work was partly supported by the Danish Natural Sciences Foundation (snf-grant 9903273). [1] M.H. Anderson et al.,Science 269, 198 (1995); K.B. Davis et al.,Phys. Rev. Lett. 75, 3969 (1995). [2] C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet , Phys. Rev. Lett. 75, 1687 (1995). C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [3] C.A. Sackett, J.M. Gerton, M. Welling, and R.G. Hulet, Phys. Rev. Lett. 82, 876 (1999). [4] Yu. Kagan, A.E. Muryshev, and G.V. Shlyapnikov, Phys. Rev. Lett. 81, 933 (1998). [5] A. Eleftheriou and K. Huang, Phys. Rev. A 61, 043601 (2000); See also M. Ueda and K. Huang, Phys. Rev. A 60, 3317 (1999). [6] L. Berg´ e, T.J. Alexander, and Yu.S. Kivshar, Phys. Rev. A62, 023607 (2000). [7] H. Gauck et al.,Phys. Rev. Lett. 81, 5298 (1998); A.I. Safonov et al.,Phys. Rev. Lett. 81, 4545 (1998). See also D.S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000) and references therein. [8] L.P. Pitaevskii, Phys. Lett. A 221, 14 (1996). [9] L. Berg´ e, Phys. Rep. 303259 (1998); J. Juul Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986). [10] D.W. McLaughlin et al.,Phys. Rev. A 34, 1200 (1986); B.J. LeMesurier et al.,Physica D 31, 78 (1988); 32, 210 (1988). [11] N.E. Kosmatov, V.F. Shvets, and V.E. Zakharov, Physica D52, 16 (1991). [12] V.E. Zakharov and E.A. Kuznetsov, Sov. Phys. JETP 64, 773 (1986). [13] H. Saito and M. Ueda, arXiv:cond-mat/0002393 (2000). [14] S.N. Vlasov, L.V. Piskunova, and V.I. Talanov, Sov. Phys. JETP 68, 1125 (1989). [15] L. Berg´ e, Phys. Plasmas 4, 1227 (1997). [16] C. Huepe, S. M´ etens, G. Dewel, P. Borckmans, and M.E. Brachet, Phys. Rev. Lett. 82, 1616 (1999). 4
1Generation□of□electromagnetic□pulses□from□plasma□channels induced□by□femtosecond□light□strings Chung-Chieh□Cheng,□E.□M.□Wright,□and□J.□V.□Moloney Arizona□Center□for□Mathematical□Sciences,□and□Optical□Sciences□Center,□University□of□Arizona, Tucson,□AZ□85721 December□04,□2000 We□present□a□model□that□elucidates□the□physics□underlying□the□generation□of□an electromagnetic□pulse□from□a□plasma□channel□resulting□from□the□ionization□of□air□by□a femtosecond□laser□pulse.□By□a□new□mechanism□analogous□to□nonlinear□optical rectification,□the□laser□pulse□induces□a□dipole□moment□in□the□plasma□which□subsequently oscillates□at□the□plasma□frequency□and□radiates□an□electromagnetic□pulse□with□a□peak frequency□within□the□far-infrared□to□microwave□region,□depending□on□the□electron density,□with□a□bandwidth□around□hundreds□of□gigahertz. PACS:□33.80.Wz□Other□multiphoton□processes 42.65.Re□Ultrafast□processes;□optical□pulse□generation□and□pulse□compression 52.40.Db□Electromagnetic□(nonlaser)□radiation□interactions□with□plasma Recent□investigations□of□the□propagation□of□intense□femtosecond□infrared□(IR)□laser pulses□in□air□show□that□the□dynamical□interaction□between□nonlinear□self-focusing, plasma□defocusing,□and□group-velocity□dispersion□can□cause□an□initial□beam□to□break□up spatially□into□several□filaments,□or□light□strings,□with□diameters□around□a□hundred microns□that□can□maintain□themselves□over□long□distances□[1,2,3,4,5].□□It□has□been observed□experimentally□that□femtosecond□light□strings□in□turn□produce□plasma channels□by□multi-photon□ionization□(MPI)□along□their□direction□of□propagation□with lengths□ranging□from□tens□of□centimeters□to□several□meters□[6,7,8].□□Observations□of□the electromagnetic□pulses□(EMPs)□from□light□string□induced□plasmas□suggest□that□these channels□attain□dipole□moments□during□the□laser□pulse□which□subsequently□oscillate□at the□plasma□frequency□and□radiate□[6].□□The□lifetime□of□the□observed□plasma□channels□is around□a□nanosecond□in□keeping□with□the□duration□of□the□observed□EMP,□though□their frequency□content□has□not□been□determined□experimentally□as□of□yet.2
arXiv:physics/0012006v1 [physics.geo-ph] 5 Dec 2000Geomagneticcontrolofthespectrumoftraveling ionosphericdisturbancesbasedondatafroma global GPSnetwork E. L. Afraimovich1,,E. A. Kosogorov1, O.S. Lesyuta1, I. I. Ushakov1, and A. F. Yakovets2 p. o. box 4026,Irkutsk,664033,Russia,fax: +7 3952462557; e-mail: afra@iszf.irk.ru, 1InstituteofSolar-Terrestrial PhysicsSD RAS 2InstituteofIonosphere, Almaty,Kazakhstan Manuscriptsubmittedto Annales Geophysicae Manuscriptversionfrom 3December 2000 Offsetrequests to: E. L. Afraimovich InstituteofSolar-Terrestrial PhysicsSD RAS Irkutsk,Russia Send proofs to: E. L. Afraimovich InstituteofSolar-Terrestrial PhysicsSD RAS Irkutsk,RussiaAbstract In this paper an attempt is made to verify the hypothesis on th e role of geomag- netic disturbances as a factor determining the intensity of traveling ionospheric dis- turbances (TIDs). To improve the statistical validity of th e data, we have used the based on the new GLOBDET technology (Afraimovich, 2000a) me thod involving a global spatial averaging ofdisturbance spectra ofthetota l electron content (TEC).To characterize the TID intensity quantitatively, we suggest that a new global index of thedegree ofdisturbance should beused, whichisequal toth emeanvalueof therms variations in TEC within the selected range of spectral peri ods (of 20–60 min in the present case). The analysis has been made for a set of 100 to 30 0 GPS stations, and for 10days withadifferent level of geomagnetic activity ( Dstfrom 0to-350 nT;the Kpindex from 3to 9). It was found that power spectra of daytime TEC variations in t he range of 20– 60 min periods under quiet conditions have a power-law form, with the slope index k= -2.5. Withanincrease ofthelevel ofmagneticdisturbance , thereisanincrease in total intensity of TIDs, with a concurrent kink of the spectr um caused by an increase in oscillation intensity in the range of 20–60 min. The TEC va riation amplitude is found to be smaller at night than during the daytime, and the s pectrum decreases in slope, whichisindicativeofadisproportionate increasei ntheamplitudeofthesmall- scale part of the spectrum. It was found that an increase in the level of geomagnetic acti vity is accompanied by an increase in total intensity of TEC; however, it correla tes not with the absolute levelof Dst,butwiththevalueofthetimederivative of Dst(amaximumcorrelation coefﬁcient reaches -0.94). The delay of the TID response of t he order of 2 hours is consistent with the view that TIDs are generated in auroral r egions, and propagate equatorward with the velocity of about 300-400 m/s. Correspondence to: E.L.Afraimovich 1Keywords Ionospheric disturbances ·Auroral ionosphere ·Equatorial ionosphere 1 Introduction One of the most important ideas of the origin of ionospheric i rregularities and their dynamics is the wave concept, according to which the experimentally obs erved irregular structure is the result ofthesuperposition ofwaveprocesses of different origins . Forthat reason, particular signiﬁcance intheoretical and experimental studies hasbeen attached t ospectral characteristics which make it possible toidentify ionospheric irregularities of differ ent scales. Theirregularitiesdiscussedinthispaperareclassedastr avelingionosphericdisturbances(TIDs), with atypical spatial size ranging from 100 to 1000 km,and at imeperiod inthe range of 20–120 min, and the literature on this subject is quite extensive. A classiﬁcation of TIDs in their sizes (in particular, their separation into large-scale LS and me dium-scale MS disturbances) is rather arbitrary, and manyauthors attribute different physical m echanisms to this classiﬁcation. LS TIDs with typical time periods of 1–2 hours and wavelength s of about 1000 km have been studied in many publications, including some thorough revi ews (Hunsucker, 1982; Hocke and Schlegel, 1996). It is commonly accepted that LS TIDs are man ifestations of acoustic-gravity waves(AGW)whosegeneratingregionsarelocatedintheauro ralzonesofthenorthernandsouth- ern hemispheres. These ideas have been conﬁrmed in recent ex periments using GPS data (Ho et al., 1998; Afraimovich et al., 2000b). There is currently a plethora of views regarding the effecti veness of geomagnetic ﬁeld distur- bances as the source of medium-scale (MS) TIDs. According to Hunsucker (1982), the auroral source plays the dominant role for electron density irregul arities with typical periods from 10 to 60 min. An enhancement of the variation intensity during geo magnetic disturbances was pointed out by F¨ orster et al. (1994) and Fatkullin et al. (1996). At the same time Waldock and Jones (1987) showed that the auro ral sources perhaps play a minor role in the generation of MS TIDs recorded at mid-latit udes. Ogawa et al. (1987) hold that MS TIDs are constantly recorded based on observations f rom the NNSS satellites, and their occurrence frequency does not increase under disturbed con ditions. A plausible mechanism for the production of MS TIDs is attributed by some authors to met eorological processes (Bertin et al., 1975; Waldock and Jones, 1987; Oliver et al., 1997). On this basis, one is led to conclude that there has been as yet no sufﬁciently convincing evi- 2denceinsupport ofthehypothesis ofthedetermining contri bution ofgeomagnetic disturbances to the production of medium–scale TIDs. This is largely caused by the inadequate number and low spatial resolution of the radio sounding facilities curren tly in use (ionosondes, incoherent scatter radars, etc.). Themaincharacteristics ofwaveprocesses arethetemporal andspatial spectra. Sincethespec- trahavenormallyapower-lawcharacter, theslopeofthespe ctrum kandthestandard deviation of intensityvariationsinthefrequencyrange Manalyzed(theamplitudescaleofthepowerspectrum) are the most informative parameters, estimates of which wer e made in almost all publications of an experimental or theoretical nature (Drobzhev et al., 197 9; Litvinov and Jakovets, 1983; Kaliev et al., 1988; Fridman, 1990; Yakovets et al., 1999). Determining the above-mentioned characteristics of distu rbances experimentally is of crucial importance for validating the interpretation of experimen tal data in terms of different physical mechanisms of the inhomogeneous structure. Furthermore, a knowledge of irregularity spectra is required for developing an empirical model of distortion s of transionospheric signals used in special–purpose radio engineering systems ofcommunicati on, location, andnavigation intheme- ter, decimeter and centimeter ranges. Published data show a large scatter in estimates of the slope kand of the amplitude scale M of temporal and spatial spectra (see Section 4). One of the re asons for this scatter might be that different measuring techniques are used, which differ grea tly in spatial and temporal resolution. However, the main reason is determined by the differing geop hysical conditions of separate mea- surements, and by the large difference in latitude, longitu de and local time when carrying out experiments. To obtain more reliable information requires carrying out s imultaneous measurements over a large area covering regions with a different local time. How ever, none of the above-mentioned methods meets such requirements. The advent and evolution of a Global Positioning System, GPS , and also the creation on its basis of widely branched networks of GPS stations (at least 7 57 sites by November of 2000, the data from which are placed on the Internet), opened up a new er a in remote ionospheric sensing (Klobuchar, 1997). High-precision measurements of the gro up and phase delay along the line of sight (LOS) between the receiver on the ground and transmi tters on the GPS system satellites covering the reception zone are made using two-frequency mu ltichannel receivers of the GPS system at almost any point of the globe and at anytime simulta neously at twocoherently coupled frequencies f1= 1575 .42MHzand f2= 1227 .60MHz. 3One of these authors (Afraimovich, 2000a) has developed a ne w technology, GLOBDET, for global detection of ionospheric disturbances of natural an d technogenic origins using data from the international network of two-frequency multichannel r eceivers of the navigation GPS system which improves substantially thesensitivity and spatial r esolution of experiment. The objective of this study is to develop, on the basis of the G LOBDET technology, a new method for estimating global characteristics ofthe TIDspe ctrum whichexcels inahigher statisti- cal reliability that is achieved by a global spatial averagi ng of the spectra. This method is used to verify the hypothesis of the determining role of geomagneti c disturbances asthe source of TIDs. Thegeometry andageneral description ofexperiments aregi veninSection 2. Section3brieﬂy describes our developed method for determining a global spe ctrum of TIDs. The method is used to analyze the data from the international GPS network for 10 days with different levels of ge- omagnetic disturbance – Section 4. Results obtained are dis cussed in Section 5 , and compared withavailable published data. 2 General description and geometry of theexperiment This study is based on using the data from the global GPSnetwo rk of receiving stations available viatheInternet. Figure1presentsthegeometryofthegloba lGPSnetworkusedinthispaperwhen analyzing the mean amplitude spectra of total electron cont ent (TEC) disturbances. For some reasons, slightly differing sets of stations were chosen fo r different events which were analyzed; however, the geometry of the experiment for all events was vi rtually identical. We do not present here the coordinates of the stations for reasons of space. Th is information may be obtained at the electronic address http://lox.ucsd.edu/cgi-bin/allCoo rds.cgi?. As is evident from Fig. 1, the set of stations selected from th e part of the global GPS network available to us, covers rather densely North America and Eur ope, and much less densely Asia. The number of stations on the Paciﬁc and Atlantic oceans is sm aller. However, such coverage of the terrestrial surface makes it possible even today to so lve the problem of a global detection of disturbances with an as yet unprecedented spatial accumu lation. This ensures a number of statistically independent series by two orders of magnitud e larger as a minimum than could be realized by recording UHF signals from geostationary satel lites (Davies, 1980; Afraimovich et al., 1994) or from ﬁrst-generation low-orbit navigation TR ANSIT satellites (Evans et al., 1983; Ogawa,1987). Thus,inthewesternhemispherethecorrespon ding numberofstationscan,already today, beas large as 500, and thenumber of beams to satellite s can beno less than 2000-3000. 4Wecarried out ananalysis of thedata for aset of from 100 to30 0GPSstations andfor 10days fromthetimeinterval1998-2000, withadifferent levelofg eomagneticdisturbance ( Dstfrom-13 to -321 nT; Kp-index from 3 to 9). Table 1 presents information about day nu mbers, the number ofthestationsusedm,andextremevaluesof DstminandKpmax. AtotalamountoftheGPSdata exceeds 5×10730-s observations. 3 Determining the mean (global) power spectrum of TEC variat ions and its parameters from GPSdata Below, we give a brief outline of our developed method for est imating the mean (global) power spectrum ofTECvariations caused byionospheric irregular ities ofdifferent scales, onthebasisof processing thedatafromtheinternational networkoftwo-f requency multichannel receivers ofthe GPS navigation system. With the purpose of improving the sta tistical reliability of the data, we usedtheglobalspatialaveragingtechniqueforspectrawit hintheframeworkofanovelGLOBDET technology (Afraimovich, 2000a). The method implies using an appropriate processing of TEC variationsthataredeterminedfromtheGPSdata,simultane ouslyfortheentiresetofGPSsatellites (asmanyas5–10satellites) ”visible” during agiventimein terval, atallstations oftheglobal GPS network used in theanalysis. The standart GPS technology provides a means for wave distur bances detecion based on phase measurements of TECat each of spaced two-frequency GPSrece ivers. Amethods of reconstruct- ing TEC variations from measurements of the ionosphere-ind uced additional increment of the group and phase delay of the satellite radio signal was detai led and validated in a series of pub- lications (Hofmann-Wellenhof et al., 1992; Afraimovich et al., 1998, 2000b). We reproduce here only the ﬁnal formula for phase measurements Io=1 40.308f2 1f2 2 f2 1−f2 2[(L1λ1−L2λ2) +const +nL], (1) where L1λ1andL2λ2are additional paths of the radio signal caused by the phase d elay in the ionosphere, (m); L1andL2represent the number of phase rotations at the frequencies f1and f2;λ1andλ2stand for the corresponding wavelengths, (m); constis the unknown initial phase ambiguity, (m); and nLare errors indetermining the phase path, (m). Phase measurements in the GPS can be made with a high degree of accuracy corresponding to the error of TEC determination of at least 1014m−2when averaged on a 30-second time interval, with some uncertainty of the initial value of TEC, however (H ofmann-Wellenhof et al., 1992). 5This makes possible detecting ionization irregularities a nd wave processes in the ionosphere over a wide range of amplitudes (up to 10−4of the diurnal TEC variation) and periods (from 24 hours to5min). Theunit ofTEC TECU,whichisequal to 1016m−2andiscommonly accepted inthe literature, will beused inthe following. Primarydataincludeseriesof”oblique”valuesofTEC Io(t),aswellasthecorrespondingseries of elevations θ(t)and azimuths α(t)along LOS to the satellite calculated using our developed CONVTEC program which converts the GPS system standard RINE X-ﬁles on the INTERNET (Gurtner, 1993). Unfortunately, formoststationsoftheglobalGPSnetwork, thedataareprovidedbytheInternet at time intervals of 30s, which bounds the TECvariation peri od below by about 1 min. A calculation of a single spectrum of TEC variations involve s using continuous series of Io(t) seriesofadurationofnolessthan2.5hours,thusenablingu stoobtainthenumberofcountsequal to 256 that is convenient for the algorithm of fast Fourier tr ansform (FFT) used in this study. To obtain a longer series of 512 counts requires a time interval no less than 5 hours long, which is impracticable because of the limitations of the geometry of experiment with the GPS satellites. Thisbounds the range of periods analyzed byus above by about 120 min. Toexcludethevariationsoftheregularionosphere, aswell astrendsintroducedbythemotionof thesatellite, weemploytheprocedure ofremovingthelinea rtrendbypreliminarily smoothingthe initial series with a selected time window of a duration of ab out 60 min. This procedure reduces greatlytheamplitudeoflow-frequency components inthera ngeofperiodsanalyzed, butthisdoes not affect the qualitative results derived from analyzing t he spectrum below. Series of the values of elevations θ(t)and azimuths α(t)of the beam to the satellite were used to determine the coordinates of subionospheric points, and to convert the ”oblique” TEC I0(t)to thecorresponding value of the”vertical” TECbyemploying t he technique reported byKlobuchar (1986) I=Io×cos/bracketleftbigg arcsin/parenleftbiggRz Rz+hmaxcosθ/parenrightbigg/bracketrightbigg , (2) where Rzis the Earth’s radius, and hmax=300 km is the height of the F2-layer maximum. All results in this study were obtained for elevations θ(t)larger than 30◦. By considering an example of the magnetically quiet and magn etically disturbed ionosphere over the Millstone Hill incoherent scatter facility - MHR (g eographical coordinates 42.61◦N, 288.5◦E), we describe brieﬂy the sequence of data processing proce dures. Fig. 2a gives an ex- ample of a typical weakly disturbed variation in ”vertical ” TECI(t)for station WES2 (satellite 6number PRN17) on July 15, 2000 for the time interval 17:00-19 :00 UT, preceding the onset of a geomagnetic disturbance neartheMHRovertheterritory wit hthecoordinates insidetherectangle 30−50◦N,270−290◦E. For this same series, Fig. 2b presents the dI(t)variations that were ﬁltered out from the I(t)series byremoving the trend with a60-min window. The logarithmic power spectrum lgS2(F)of the dI(t)series (Fig. 2b), obtained by using a standard FFT procedure, is presented in panel c). Boldface l etters and dots along the abscissa axis on panel c (as well as d, g and h) indicate the frequency ra nges of medium-scale (MS) and small-scale (SS)irregularities. Incoherent summation of the partial power spectra lgS2(F)iof different LOS was performed by theformula /angbracketleftlgS2(f)/angbracketright=n/summationdisplay i=1lgS2(f)i, (3) where iisthe number of LOS; i=1,2, ... n. Theresultderivedfromcombiningthespectra /angbracketleftlgS2(f)/angbracketrightfor16LOSof10GPSstationslocated inthe above-mentioned MHRregion isshown in Fig.2d by athic k line. Forcomparingthespectraforthequietanddisturbed days, t hethinlineinFig.2dplotsaglobal spectrum for the quiet day of July 29, 1999 (a maximum deviati on of Dst=-4 nT) obtained in a similar manner for the time interval 11:00-13:30 UT by aver aging over n = 309 LOS of 161 stations of the global network, Fig. 1, which are relatively uniformly distributed in the western and eastern hemispheres within 30−70◦N latitudes. Values of the slope kof thepower spectrum are shown at the spectra. Asaconsequence of the statistical independence of partial spectra, the signal/noise ratio, when the average spectrum is calculated, increases due to incohe rent accumulation at least by a factor of√n, where nis the number of LOS.This is conﬁrmed by acomparison of the re sulting sum of the/angbracketleftlgS2(f)/angbracketright, Fig. 2d, withthe partial spectrum lgS2(F), Fig. 2c. Itshouldbenotedthatthespectrumthatiscalculateddirec tlyfrom dI(t)variationsisadistorted spectrumofirregularitiesasaconsequenceoftheDopplers hifteffectoftheTIDangularfrequency (Afraimovich et al.,1998) Ω = Ω 0−/vectorK/vector ω, (4) where /vectorKand/vector ωare, respectively, the TID angular vector and the vector of d isplacement of the subionospheric point at the selected height in the ionosphe re caused by the motion of the GPS satellite; Ω0isthe initial value of theTID angular frequency. 7The bulk of information about time spectra of different-sca le ionospheric irregularities, includ- ingTIDs,wasobtainedthroughtransionospheric soundings usingsignalsfromgeostationarysatel- lites(Davies,1980;Afraimovichetal.,1994). Inthiscase thevelocity ωofthebeamtothesatellite at the level of the ionospheric F2region maximum is much smaller than the velocity Vof TIDs, hence it can be neglected. For low-orbit navigation satelli tes of the ﬁrst-generation TRANSIT,on thecontrary, thevelocity ωexceedssubstantially thevelocity VofTIDs;therefore, measurements are interpreted in terms of one-dimensional spatial spectr a (Evans et al., 1983). In the case of the GPS, the velocities can be identical, which will cause the sp ectral line to be shifted toward the positive or negative sides. The resulting value of the frequency can change sign if the mo dulus of frequency shift \|/vectorK/vector ω\| exceeds the value of Ω. This means that in this case the point at which LOS traverses the main maximumofionizationmovesfasterthantheTIDwave,andint heinterferometer’s frameofrefer- encethetravellingdirectionoftheequiphase lineisrever sedwithrespecttothatintheionosphere. Such a situation, however, can be of very infrequent occurre nce because the velocity ω(usually not higher than50-70 m/swhen hmax=300 km) isdistinctly lower thanthemeanvalue of theTID phase velocity. However, as partial spectra are accumulated, which corresp ond to all GPS satellites that are visible over a given time interval, this effect will lead mer ely to a relatively uniform smearing of spectrallinesbecausethesignandmagnitudeofthefrequen cyshiftaredifferentforseparateLOS. Thus an averaging over a large number of LOS makes it possible to obtain estimates of average spectra. AsisevidentfromFig.2d,thespectrumofaquietdaycorresp onds reasonablywelltoatheoret- icalpowerspectrum ofionospheric irregularities withasl opeofabout k=-2.5,anditcantherefore be used as a reference power spectrum. This result is consist ent with published estimates of TID spectrum characteristics obtained invertical- (Kalievet al.,1988), oblique-incidence (Gajlit etal., 1983) andtransionospheric radiosoundings (Afraimoviche tal.,1994). TheTECﬂuctuation scale Min the MS range and Cin the SS range does not exceed in this case the values 0.4 and 0 .007 TECU,respectively. Whencomparing theaverage spectraofTECvariations fromJu ly15,2000forthetimeinterval 17:00-19:00 UT with the spectrum from the quiet day of July 29 , 1999, one can notice an order- of-magnitude excess of the TEC disturbance level throughou t the spectrum, with the value of the slopek=-2.56 remaining the same. However, there is also a clear dis proportionate (by 1.5 order of magnitude) increase inTECvariation intensity in the MSr ange. 8Still more drastic changes of the ionospheric irregularity spectrum occurred over the same re- gion just one hour later. Figure 2e presents the time depende nce of the disturbed value of the ”vertical” TEC I(t)for station ALGO (satellite number PRN21) for July 15, 2000, for the time interval 20:00-22:30 UT. For the same series, Fig. 2f plots t hedI(t)variations that were ﬁltered outfromthe I(t)seriesbyremovingthetrendwitha60-minwindow. Asisappar ent fromFig.2a, and from the corresponding lgS2(F)spectrum, Fig. 2g, the TEC variations increased in power as a minimum by 2 orders of magnitude as against the time inter val 17:00-19:00 UT (Fig. 2b and 2c). Besides, there was an abrupt change in the spectrum slop ek=-0.85, which is indicative of a disproportionate increase in irregularity intensity in t he MS and SS parts of the spectrum. The TEC ﬂuctuation scale Min the MS range and Cin the SS range exceeds in this case the values 4.27 and 0.5TECU,respectively. The result derived from combining the /angbracketleftlgS2(f)/angbracketrightspectra for 7 LOS is shown in Fig. 2h by a thick line. The spectrum has a power-law character, yet wit h the mean slope k=-1.85, which differs markedly from the value of kfor the magnetically quiet day. The mean intensity Mof the irregularities of the medium-scale part MS increased two or ders of magnitude, and the intensity Cof the small-scale part increased immediately by 3 orders of magnitude as compared with the level of the magnetically quiet day. As the chief goal of this paper is to obtain the mean character istics of the TID intensity, in the discussion to follow we shall be using only the above-mentio ned parameters of the spectrum, k andM. 3.1 Geomagnetic control of the TIDspectrum The dependencies of the TID variation intensities M(t)presented below were obtained by calcu- lating global spectra forall dayslisted inTable1,withthe number of stations mfortimeintervals of aduration of 2.5hours with a1-hour shift, and by a subsequ ent integration of the spectral den- sity in the range of 20–60 min periods (see Fig. 2, where boldf ace dots along the abscissa axis indicate the MS interval). The integration result is the val ue ofMwhich is equal to the standard deviation of TEC variations in the speciﬁed range of periods and is measured in TECU units. Corresponding dataarepresented inFigs.3, 4and5,andstat istical estimates arelistedinTable1. 3.1.1 Characteristics of the TIDspectrum as afunction of th e universal timeUT The data from the magnetically quiet day of July 29, 2000, cha racterized by a low level of geo- magnetic activity and by a reference power spectrum (see Fig . 2d, 2h - thin line), are used here in 9comparison with characteristics of the TID spectrum during geomagnetic disturbances (line 3 in Table 1). Figure 3e (thick line) and Fig. 3f (dashes) plot, as a function of the universal time UT, theDst-variationsofthegeomagneticﬁeldandthestandarddeviat ionoftheTECvariations M(t) inthe range of 20–60 min periods for this day. As is evident from the ﬁgure, shallow, slow Dst-variations over the course of that day were accompanied by slow, small-amplitude ﬂuctuations caused b y TIDs; the mean value Mfor that daydidnotexceed0.16TECU.Similarresultswerealsoobtai ned fortheother magnetically quiet day of January 9, 2000 (Fig. 3b - thick line, and Fig. 3c - dashe s; line 4in Table1). Letusnowconsider, forthesakeofcontrast, aglobal ionosp heric response toamajormagnetic storm of April 6–7, 2000, characterized by a maximum amplitu de ofDst-variations as large as -321 nT (Fig. 3e - thin line; line 5 in Table 1). A maximum value of the Kpindex (Fig. 3d) for this storm was as high as 8. Until about 19:00 UT on April 6, the Dst-variations varied within a very narrow range, and wereclose to0nT. After that, the valu e ofDstbegan todecrease rapidly; after 19:00 UTit reaches thevalue -129 nT, and continued dec reasing right downto -321 nT. Fig. 3f plots the dependence M(t)of the standard deviation of TEC variations in the range of 20–60 min periods (thick line), and the inverted dependen ce of the time derivative d(Dst)/dt (relative units; thin line). The derivative d(Dst)/dtwas obtained from the dependence Dst(t) (Fig.3e) that wassmoothed witha7-hour timewindow. Asisev ident fromtheﬁgure, anincrease of the level of magnetic disturbance is accompanied by a grad ual increase in total intensity of TIDs; however, it correlates not with the absolute level of Dstbut with the value of the time derivative d(Dst)/dt(the correlation coefﬁcient rin this case is -0.84). A maximum amplitude Mmax=1.07TECU,shownbythearrowinFig.3f,exceedsoneorderof magnitudeasaminimum the corresponding value for the magnetically quiet day of Ju ly 29, 1999 (Fig. 3f - dashes). Similarresults werealsoobtained forother magnetic storm sfromJuly15–16, 2000(Fig.3a-c), August 26–27, and September 24–25, 1998 (Fig. 4), yet a maxim um value Mmaxin these cases did not exceed 0.67, 0.32 and 0.42 TECU,respectively (lines 6, 1and 2inTable 1). Thedelay τ(of about 2hours) of theincrease inTECintensity withrespe ct torapid changes in magneticﬁeldstrengthiseasytoexplainbytakingintocons ideration thatthegreatest contribution inaglobal averaging ofTIDspectra ismadebythemid-latitu de chainofGPSstations. Thischain is at about 2000 km from the southern boundary of the auroral s ource of TIDs which is produced during geomagnetic disturbances. TIDs that are generated o nce this source is produced travel equatorward with the velocity of order 300-400 m/s (Francis , 1973; Maeda and Handa, 1980; Hunsucker, 1982; Haykowicz and Hunsucker, 1987; Ma et al., 1 998; Hocke and Schlegel, 1996; 10Hoet al., 1998; Balthazor and Moffett, 1999; Hall et al., 199 9; Afraimovich et al., 2000b). 4 Characteristics of theTID spectrum asa function of theloc al time LT Forstudying the diurnal dependence of TIDspectrum charact eristics, wecarried out anaveraging of the spectra with due regard for the local time LT for each GP S station. In doing so, it should be taken into consideration that as a consequence of the nonu niform distribution of the stations, the contribution of the mid-latitude stations in North Amer ica and, to a lesser extent, in Europe is predominant (see Fig. 1). Figure 5plots the diurnal LT-dependencies of the slope inde xkof the power spectrum of TIDs – a), and of the standard deviation of the TEC variations M(t)in the range of 20–60 min periods – b) for the magnetically quiet day of July 29, 1999. As is evid ent from the ﬁgure, the value of the slope index k=-2.5 remains virtually unchanged over the course of that da y, as does the mean valueM(t)which does not exceed 0.15TECU. However, for the other magnetically quiet day, January 9, 20 00 (Fig. 5c, d), one can notice a conspicuous diurnal dependence of both the slope index k(t)and the TIDintensity M(t). Also, a maximum value of M(t)is as high as 0.8 TECU around noon, and a maximum slope index kis ashigh as-2.7. Thus,accordingtoourdata,thepowerspectraofthedaytime TECvariationsintherangeof20– 60 min periods under quiet conditions have a power-law form w ith the slope index k=-2.5. With the increasing level of geomagnetic disturbance, there is a n increase in total intensity of TIDs, with a concurrent kink of the spectrum caused by an increase i n ﬂuctuation intensity in the range of 20–60 min. The TEC variation amplitude is smaller at night than during the daytime, and the spectrum decreases in slope, which is indicative of a dispro portionate growth of the amplitude of the small-scale part of the spectrum. The above-mentioned characteristic features of the diurna l variation of these parameters are most pronounced during a major magnetic storm of April 6, 200 0 - Fig. 5e, f and, to a lesser extent, during the magnetic storm of July 15, 2000 - Fig.5g, h ). The peculiarities of the diurnal variation in TID intensity pointed out above are consistent with evidence acquired using signals from geostationary satell ites (Jacobson et al., 1995; Oliver et al., 1997; Afraimovich et al., 1999). The characteristics of the spectra which we have obtained ar e in reasonably good agreement with a number of reported results, despite the fact that the p ublished data show a large scatter in 11estimates of theslope k(as well asof the amplitude scale Mof temporal and spatial spectra). One-dimensional spatial spectra can be obtained through di rect measurements of variations in local electron density along the satellite path; however, p ublished data mostly refer to the equa- torial or polar regions. Thus, by investigating the equator ial F region of the ionosphere simul- taneously through in-situ measurements by satellites AE-E and radio probing using signals from the geostationary Wideband satellite at 137 and 378 MHz freq uencies, Livingston et al. (1981) found that the one-dimensional spatial spectrum in the rang e of scales of 10-100 km has a power character withthe slope index kof about -2. Rocket measurements of electron densities in the Fregion of the auroral ionosphere made concurrently withincoherent scatter radar measurements a ndradioprobing usingsignals fromthe geostationary Wideband satellite at Chatanika (Kelleyeta l., 1980) showedthat thecorresponding slope index kof the one-dimensional spatial spectrum for the range of 0.1 -200 km scales lies within -1.2-1.8. For 500 km altitude, estimates of the slope of the one-dimens ional spatial spectrum were ob- tained from in-situ measurements aboard the ”Cosmos-900” s atellite by Gdalevich et al. (1980). While for the equatorial and high-latitude ionosphere the v alues of kwereof order -1.2, thespec- trum for the mid-latitudes showed a kink; in the range of 30-1 50 km scales, k= -3-4, and it decreases to-1.0 for irregularities smaller than 30 km. An alternative possibility involves transferring the temp oral spectrum to the spatial region pro- videdthatthetravelinginhomogeneous structureis”froze n-in”. Inthiscasetheformofthespatial spectrum is identical to that of the temporal spectrum. Such an approach in processing measure- ments of the frequency Doppler shift at oblique-incidence s oundings was used, in particular, by Gajlit et al. (1983); a corresponding estimate of the index k for ionospheric irregularities withthe size from a few tens to several hundred kilometers was obtain ed by these authors in the range of -3.8-4.6. Furthermore, the index kfor the spectra of the frequency Doppler shift varied within -0.8-1.6. Similarresultsinidentical Doppler measuremen ts wereobtained byKalievetal. (1988); thekfrom the data on average spectra was-2 both during the daytim e and at night. Afraimovichetal. (1994)investigatedthespectralproper tiesofmedium-scaleTIDsonthebasis ofanalyzing power spectraofTECvariations obtained bymea suringthepolarization ofthesignal from geostationary satellite ETS-2at 136 MHz frequency nea r Irkutsk ( 52◦N,102◦E). For three seasonsof1990, temporalspectraofTECvariations, averag edover10days, wereobtained. Inthe low-frequency range(periods from100to20min)thedaytime variation spectra haveapower-law form,withtheslopeindex-2.5,whileinthehigh-frequency range(periodsof20–10min)theyare 12withthe index - 6; at night, the index is- 4throughout the ran ge of periods under consideration. 5 Discussion and conclusion Themain results of this study maybe summarized asfollows: 1. Our ﬁndings bear witness to the determining role of geomag netic disturbances in the for- mation of the spectrum of traveling ionospheric disturbanc es. This conclusion is based on substantially more extensive (than obtained earlier) stat istical material, spans periods with a different level of geomagnetic disturbance, and has a glob al character. The analysis has been made for a set of 100 to 300 GPS stations, and for 10 days wi th a different level of geomagnetic activity ( Dstfrom 0 to-350 nT; the Kpindex from 3to 9). 2. ItwasfoundthatpowerspectraofdaytimeTECvariationsi ntherangeof20–60minperiods underquietconditionshaveapower-lawform,withtheslope indexk=-2.5. Withanincrease of the level of magnetic disturbance, there is an increase in total intensity of TIDs, with a concurrent kink of the spectrum caused by an increase in osci llation intensity in the range of 20–60 min. The TEC variation amplitude is found to be small er at night than during the daytime, and the spectrum decreases in slope, which is in dicative of a disproportionate increase inthe amplitude of the small–scale part of the spec trum. 3. It was found that an increase in the level of geomagnetic ac tivity is accompanied by an increase in total intensity of TEC;however, it correlates n ot with the absolute level of Dst, but withthe value of the timederivative of Dst(amaximum correlation coefﬁcient reaches -0.94). 4. 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Waldock,J.A.,Jones,T.B., Sourseregionsofmediumscaletravellingionosphericdist urbancesobserved at mid-latitudes, J. Atmos.Terr. Phys. ,49,105,1987. Yakovets, A. F., Kaliev, M. Z., Vodyannikov, V. V., An experimental study of wave packets in travelling ionosphericdisturbances, J.Atmos.Sol.-Terr.Phys. ,61,629–639,1999. 16Fig/. /1/: Geometry of the global GPS arra y used in this pap er when analyzing thep erturbation sp ectra of the total electron con ten t /(TEC/)/. Boldface dots sho w thelo cation of the GPS stations/.17 18 19-20246810I(t), TECU 17 18 19 Time, UT-1.0-0.50.00.51.0dI(t), TECU lg F, Hz-6.0-5.0-4.0-3.0-2.0-1.00.01.0lg S2 (F), TECU 2 -4.0-3.5-3.0-2.5-2.0lg F, Hz-6.0-5.0-4.0-3.0-2.0-1.00.01.0<lg S2 (F)>, TECU 220212223-1001020 20212223 Time, UT-10.0-5.00.05.010.0 lg F, Hz-6.0-5.0-4.0-3.0-2.0-1.00.01.0 -4.0-3.5-3.0-2.5-2.0lg F, Hz-6.0-5.0-4.0-3.0-2.0-1.00.01.0a e b f c g d hWES2, PRN17July 15, 2000 MILLSTONE HILL k=-2.57 LOSALGO; PRN21 16 LOS k=-2.56 k=-1.820-22 UT 309 LOSk=-0.85MS SS MS SS July 19, 1999 20-60 2-4 T, min 20-60 2-4 T, minMS SS MS SSFig/. /2/: Time dep endencies of the /"oblique/" TEC I /( t /) of the magnetically dis/-turb ed da y of July /1/5/, /2/0/0/0 for the time in terv al /1/7/:/0/0/-/1/9/:/0/0 UT/, precedingthe onset of a geomagnetic disturbance o v er the Millstone Hill incoheren t sta/-tion within the rectangle /4/2 /: /6/1 /N/, /2/8/8 /: /5 /E /-a/)/, and dI /( t /) v ariations /ltered fromthe I /( t /) series b y remo ving the trend with a /6/0/-min windo w for station WES/2/(PRN/1/7/) /-b/)/; c/) logarithmic p o w er sp ectrum l g S /2/( F /) of the dI /( t /) series presen t/-ed in panel b/)/; d /- a v erage /(for /1/6 b eams of /1/0 GPS stations lo cated inside thisterritory/) logarithmic p o w er sp ectrum h l g S /2/( f /) i /. The same as ab o v e/, but for theonset of a geomagnetic disturbance o v er this territory for the time in terv al /2/0/:/0/0/-/2/2/:/0/0 UT/, and for station ALGO /(PRN/2/1/) /- e/, f/, g/; h /- a v erage /(for /7 b eams/)logarithmic p o w er sp ectrum h l g S /2/( f /) i /. F or the purp ose of comparing the sp ec/-tra for the quiet and disturb ed da ys/, in panels d/) and h/) a thin line sho ws thesp ectrum for the quiet da y of July /2/9/, /1/9/9/9 obtained b y a v eraging o v er n /= /3/0/9b eams of /1/6/1 global net w ork stations/, Fig/. /1/. V alues of the slop e k of the p o w ersp ectrum are giv en at the sp ectra/. Boldface letters and dots along the abscissaaxis in panels c/, d/, g and h sho w the frequency ranges of medium/-scale /(MS/) andsmall/-scale /(SS/) irregularities/.Fig/. /3/: K p /-index /- a/, geomagnetic /eld D st /-v ariations /- b/)/, and in v erted dep en/-dence of the time deriv ativ e dD st /( t /) /=dt during a ma jor magnetic storm of July/1/5/-/1/6/, /2/0/0/0 /- c /(thin line/)/. Standard deviations of TEC v ariations M /( t /) in therange of /2/0/{/6/0/-min p erio ds /- d /(thic k line/)/. The same dep endencies for a ma jormagnetic storm of April /6/-/8/, /2/0/0/0 /- d/, e/, and f/. F or comparison/, in panel b/) thethic k line plots the D st /-dep endence/, and in panel c/) the dashed line plots theM /( t /)/-dep endence for the magnetically quiet da y of Jan uary /9/, /2/0/0/0/. The samein panels e/) and f /) for the magnetically quiet da y of July /2/9/, /1/9/9/9/.Fig/. /4/: The same as in Fig/. /3/, but for the magnetic storms of August /2/6/-/2/7 /(left/)/,and Septem b er /2/4/-/2/5/, /1/9/9/8 /(righ t/)/.Fig/. /5/: L T diurnal dep endence of the slop e index k of the TID p o w er sp ectrum /-a/)/, standard deviation of TEC v ariations M /( t /) in the range of /2/0/{/6/0/-min p erio ds/-b/) of the magnetically quiet da y of July /2/9/, /1/9/9/9/. The same/, but for the w eaklydisturb ed da y of Jan uary /9/, /2/0/0/0 /- c/)/, d/)/; for the strong magnetic storm of April/6/, /2/0/0/0 /- e/) and f /)/, and for the magnetic storm of July /1/5/, /2/0/0/0 /- g/)/, h/)/.T able /1/: General information ab out exp erimen tN Data Da y m l D stmin /,nT K pmax tmax /,UT Mmax /,TECU r/1 /2/6///2/7/./0/8/./1/9/9 /8 /2/3/8///2/3/9 /9/3///8/8 /1/3/9///1/1/7 /-/1/8/8 /8 /3 /0/./3/2 /-/0/./9/3/7/2 /2/4///2/5/./0/9/./1/9/9 /8 /2/6/7///2/6/8 /9/6///8/7 /1/1/1///5/8 /-/2/3/3 /9 /4 /0/./4/2 /-/0/./8/4/0/3 /2/9/./0/7/./1/9/9/9 /2/1/0 /1/6/1 /2/9/9 /- /4/0 /3 /2 /0/./1/6 /{/4 /9/./0/1/./2/0/0/0 /0/0/9 /3/3/2 /4/4/5 /-/1/3 /- /2 /0/./2/1 /{/5 /6///7/./0/4/./2/0/0/0 /0/9/7///0/9/8 /1/7/9///1/8/0 /4/2/8///4/3/4 /-/3/2/1 /8 /2/2 /1/./0/7 /-/0/./8/4/8/6 /1/5///1/6/./0/4/./2/0/0 /0 /1/9/7///1/9/8 /3/0/9///3/0/8 /3/1/9///5/0/3 /-/2/9/5 /9 /2/2 /0/./6/7 /-/0/./8/4/6/1
arXiv:physics/0012007v1 [physics.flu-dyn] 5 Dec 2000Finite time singularities in a class of hydrodynamic models V. P. Ruban1,2∗, D. I. Podolsky1, and J. J. Rasmussen2 1L.D. Landau Institute for Theoretical Physics, 2 Kosygin st r., 117334 Moscow, Russia 2Optics and Fluid Dynamics Department, OFD-128, Risø National Laboratory, DK-4000 Roskilde, Denmark November 11, 2012 Abstract Models of inviscid incompressible ﬂuid are considered, wit h the kinetic energy (i.e., the Lagrangian functional) taking the form L ∼/integraltextkα\|vk\|2d3kin 3D Fourier represen- tation, where αis a constant, 0 < α < 1. Unlike the case α= 0 (the usual Eulerian hydrodynamics), a ﬁnite value of αresults in a ﬁnite energy for a singular frozen-in vor- tex ﬁlament. This property allows us to study the dynamics of such ﬁlaments without necessity in some regularization procedure. The linear ana lysis of small symmetrical deviations from a stationary solution is performed for a pai r of anti-parallel vortex ﬁlaments and an analog of the Crow instability is found at sma ll wave-numbers. A local approximate Hamiltonian is obtained for nonlinear lo ng-scale dynamics of this system. Self-similar solutions of the corresponding equat ions are found analytically, which describe ﬁnite time singularity formation with all le ngth scales decreasing like (t∗−t)1/(2−α), where t∗is the singularity time. PACS numbers: 47.15.Ki, 47.32.Cc 1 Introduction The question about possibility of ﬁnite time singularity sp ontaneous formation in solutions of the Euler equation for ideal incompressible ﬂuid has been discussed for a long time [1]. At present moment this fundamental problem of ﬂuid dynamics is still far from a complete solution, though some rigorous analytical results has been obtained [2]-[4]. The nature of the presumable singularity is to be clariﬁed, although many the oretical scenarios for blow-up has been suggested until now, as well as extensive numerical sim ulations have been performed to observe the singular behavior (see [5]-[13] and referenc es therein). As very probable, the self-similar regime of singularity formation may be consid ered, which was ﬁrst recognized by ∗Electronic address: ruban@itp.ac.ru 1Leray in 1934 [1]. In this regime, all length scales decrease like (t∗−t)1/2, and the velocity increases with the law ( t∗−t)−1/2. As the result, a maximum of the vorticity behaves like (t∗−t)−1. It is very important that the curvature of vortex lines in th e assumed self-similar solution should tend to inﬁnity in the vicinity of the singul ar point. This is consistent with the result of Constantin and Feﬀerman [3] who have found that the blow-up of the vorticity must be accompanied by singularity in the ﬁeld of the vortici ty direction. In this paper, we take the point of view that inﬁnite curvatur e of frozen-in vortex lines is in some sense a more fundamental characteristics of hydrody namic singularity than inﬁnite value of the vorticity maximum. To illustrate this statemen t, we consider a class of models of an incompressible inviscid ﬂuid, diﬀerent from Eulerian hydrodynamics, such that ﬁnite energy solutions with inﬁnitely thin frozen-in vortex ﬁlam ents of ﬁnite strengths are possible. Thus, we deal with a situation when the vorticity maximum is i nﬁnite from the very begin- ning, but nevertheless, this fact itself does not mean somet hing singular in the dynamics of vortex strings, while their shape is smooth and the distance between them is ﬁnite. However, the interaction between ﬁlaments may result in formation of ﬁnite time singularity for the curvature of vortex strings. To study this phenomenon analy tically is the main purpose of present work. It is a well known fact that absence in Eulerian hydrodynamic s of solutions with singular vortex ﬁlaments is manifested, in particular, as a logarith mic divergency of the corresponding expression for the energy functional of an inﬁnitely thin vo rtex ﬁlament having a ﬁnite circulation Γ and a shape R(ξ) (this is actually the Hamiltonian functional determining entirely the dynamics of the system): HΓ{R(ξ)}=Γ2 8π/contintegraldisplay /contintegraldisplay(R′(ξ1)·R′(ξ2))dξ1dξ2 \|R(ξ1)−R(ξ2)\|→ ∞. (1) More important is that the self-induced velocity of a curved string in Eulerian hydrodynam- ics is also inﬁnite. That is why we may not actually work in the framework of Eulerian hydrodynamics with such attractive for theoretical treatm ent one-dimensional objects. The situation becomes more favorable when we consider a class of regularized models, with the divergency of the energy functional eliminated. It should b e stressed here that in regularized systems the usual relation Ω= curl vbetween the vorticity and velocity ﬁelds is no more valid, and in this case Γ is not the circulation of the velocit y around the ﬁlament, but it is the circulation of the canonical momentum ﬁeld (see the ne xt section for more details). However, dynamical properties of a de-singularized system depend on the manner of regular- ization. For instance, it is possible to replace the singula r Green’s function G(\|R1−R2\|) in (1) (G(r)∼1/r) by some analytical function which has no singular points ne ar the real axis in the complex plane (for examples by Gq(r)∼tanh(qr)/ror by Gǫ(r)∼1/√ r2+ǫ2). In that case we may not expect any ﬁnite time singularity becaus e the corresponding velocity ﬁeld created by the vortex string appears to be too smooth wit h any shape of the curve, and this fact prevents drawing together of some pieces of the str ing. With such a very smooth velocity ﬁeld, a singularity formation needs an inﬁnite tim e. In this paper we consider an- other type of regularization of the Hamiltonian functional , when the Green’s function is still singular but this singularity is integrable in the contour i ntegral analogous to the expression 2(1): HΓ α{R(ξ)} ∼Γ2 2/contintegraldisplay /contintegraldisplay(R′(ξ1)·R′(ξ2))dξ1dξ2 \|R(ξ1)−R(ξ2)\|1−α, (2) with a small but ﬁnite positive constant 0 < α≪1. Ifαis not small, then we have actu- ally rather diﬀerent models than Eulerian hydrodynamics. N evertheless, such models still have many common features with usual hydrodynamics, which a re important for singular- ity formation in a process of interaction between vortex ﬁla ments: a similar hydrodynamic type structure of the Hamiltonian and a power-like behavior of the Green’s function, with negative exponent. Therefore we believe that it is useful to investigate these models, espe- cially the question about ﬁnite time singularity formation in the vortex line curvature. We hope the results of our study will shed more light on the probl em of blow-up in Eulerian hydrodynamics. This paper is organized as follows. In the Section II, we brie ﬂy review some basic prop- erties of frozen-in vorticity dynamics in a perfect ﬂuid, wi th giving necessary deﬁnitions for theoretical conceptions used in our study. In general, our a pproach is based on the Hamilto- nian formalism for frozen-in vortex lines [14]-[17]. Then, in the Section III, we perform the linear analysis of stability for a pair of symmetric anti-pa rallel vortex ﬁlaments and ﬁnd an instability at small wave numbers, analogous to the Crow ins tability [18]. In the Section IV, we postulate a local approximate Hamiltonian for the long sc ale nonlinear dynamics of the pair of ﬁlaments and present analytical self-similar solut ions of the corresponding equations. Those solutions describe ﬁnite time singularity formation , with the length scales decreasing like (t∗−t)1/(2−α), and this is the main result of present work. In the Section V, we make some concluding remarks about vortex ﬁlaments of a ﬁnite wid th, then about long scale ap- proximation for systems with the Green’s function of a gener al form, and ﬁnally about how it is possible to improve the approximation in the case of sma llα, when the unstable region is narrow in wave number space. In the Appendix A, we write in t erms of the special math- ematical functions some integral expressions needed for ca lculation of instability increment of the vortex pair. In the Appendix B, we provide details abou t the integration procedure for the system of ordinary diﬀerential equations related to the self-similar solutions. 2 Hamiltonian dynamics of vortex ﬁlaments To clarify a meaning of the suggested models (2) and to explai n the theoretical method used, let us remind some general properties of frozen-in vor ticity dynamics in a perfect ﬂuid, starting from the Lagrangian formalism [19]-[25], [14]-[1 7]. Let a Lagrangian functional L{v}specify the dynamics of some incompressible medium of unit density, with the solenoidal velocity ﬁeld v(r, t). Especially we are interested here in systems with quadratic Lagrangians taking in 3D Fourier rep resentation the form: LM{v}=1 2/integraldisplayd3k (2π)3M(k)\|vk\|2, (3) where M(k) is some given positive function of the absolute value of the wave vector k. This expression should be understood as the kinetic energy on the group of volume preserving 3mappings x(a, t), so that the velocity ﬁeld v(x, t) is deﬁned as the time derivative ˙x(a, t) taken at the point a(x, t). It is clear that the usual Eulerian hydrodynamics corresp onds to the case M(k) = 1. Due to the presence of the Noether type symmetry with respect to relabeling of La- grangian labels of ﬂuid points [21]-[25], [14]-[16], all su ch systems have an inﬁnite number of integrals of motion, which can be expressed as conservati on of the circulations Γ cof the canonical momentum ﬁeld p(r, t), p=δL δv, (4) along any frozen-in closed contour c(t), so that the generalized theorem of Kelvin is valid: Γc=/contintegraldisplay c(t)(p·dl) = const . (5) These integrals of motion correspond to the freezing-in pro perty of the canonical vorticity ﬁeldΩ(r, t), Ω≡curlp= curlδL δv. (6) After deﬁning the Hamiltonian functional H{Ω}, H{Ω}=/parenleftBig/integraldisplay/parenleftBigδL δv·v/parenrightBig dr− L/parenrightBig/vextendsingle/vextendsingle/vextendsingle v=v{Ω}, (7) the equation of motion for the vorticity takes the form Ωt= curl/bracketleftBigg curl/parenleftBiggδH δΩ/parenrightBigg ×Ω/bracketrightBigg . (8) This equation describes the transport of frozen-in vortex l ines by ﬂow having the velocity ﬁeld v= curl/parenleftBiggδH δΩ/parenrightBigg . (9) It is very important that in this process the conservation of all topological characteristics of the vorticity ﬁeld takes place [19], [26], [27]. It follows from the equations above, that the Hamiltonian HMcorresponding to the Lagrangian LMis HM{Ω}=1 2/integraldisplayd3k (2π)3\|Ωk\|2 k2M(k)=1 2/integraldisplay/integraldisplay GM(\|r1−r2\|)(Ω(r1)·Ω(r2)dr1dr2, (10) with the Green’s function GM(r) being equal to the following integral: GM(r) =/integraldisplayd3k (2π)3eikr k2M(k)(11) The frozen-in vorticity ﬁeld can be represented in topologi cally simple cases as a contin- uous distribution of vortex lines [14]-[17]: Ω(r, t) =/integraldisplay Nd2ν/contintegraldisplay δ(r−R(ν, ξ, t))∂R ∂ξdξ, (12) 4where a 2D Lagrangian coordinate ν= (ν1, ν2), which lies in some manifold N, is a label of a vortex line, while the longitudinal coordinate ξdetermines a point on the line. Such important characteristics of the system as its momentu mPand its angular mo- mentum Mcan be expressed as follows: P=/integraldisplay Nd2ν1 2/contintegraldisplay [R×Rξ]dξ, (13) M=/integraldisplay Nd2ν1 3/contintegraldisplay [R×[R×Rξ]]dξ. (14) In the limit, when the shapes R(ν, ξ, t) of vortex lines do not depend on the label ν, we have one singular vortex ﬁlament with a ﬁnite circulation Γ =/integraltext Nd2ν. In this case, the ﬂow is potential in the space around the ﬁlament: p=∇Φ, with a multi-valued scalar potential Φ(r, t). The potential ﬂow domain is passive from the dynamical vie wpoint, because there the ﬂow depends entirely on the ﬁlament shape. The dynamics o f the shape R(ξ, t) of such inﬁnitely thin vortex ﬁlament is determined in a self-consi stent manner by the variational principle with the Lagrangian LΓ M{R}[14]-[17], LΓ M= Γ/contintegraldisplay ([R′×Rt]·D(R))dξ −Γ2 2/contintegraldisplay/contintegraldisplay GM(\|R(ξ1)−R(ξ2)\|) (R′(ξ1)·R′(ξ2))dξ1dξ2, (15) where the vector function D(R) must have unit divergence [17]: divRD(R) = 1. (16) The generalization of the expression (15) to a case of severa l ﬁlaments with the circulations Γ(n)and shapes R(n)(ξ, t),n= 1..N, is straightforward: one should write a single sum over nfor the ﬁrst term and a double sum for the Hamiltonian. It is easy to see that the Hamiltonian (2) corresponds to the f unction M(k) in the form M(k)∼kα. (17) The choice of the longitudinal parameter ξis not unique, but this does not aﬀect the dy- namics of the vortex string which is an invariant geometric o bject. Sometimes it is convenient to use parameterization of the vortex line shape by a Cartesi an coordinate: R(ξ, t) = (X(ξ, t), Y(ξ, t), ξ). (18) Then the choice D= (0, Y,0) gives immediately that X(ξ, t) and Y(ξ, t) are canonically conjugated quantities. Hereafter, we will consider vortex ﬁlaments with unit circu lation for simplicity. So the symbol Γ, if appearing in some expressions below, will me an the special mathemati- cal Gamma function. Now, for some ﬁxed value of the parameter α, let us consider the symmetrical dynamics of a pair of oppositely rotating vortex ﬁlaments, with a symm etry plane y=const. Due to 5this symmetry, it is suﬃcient to consider only one of the ﬁlam ents. It follows from the above discussion that the exact expression for the Hamiltonian of this system is the following: Hα=1 2/integraldisplay/integraldisplay(1 +X′ 1X′ 2+Y′ 1Y′ 2)dξ1dξ2 /parenleftBig (ξ1−ξ2)2+(X1−X2)2+(Y1−Y2)2/parenrightBig1−α 2 +1 2/integraldisplay/integraldisplay(−1−X′ 1X′ 2+Y′ 1Y′ 2)dξ1dξ2 /parenleftBig (ξ1−ξ2)2+(X1−X2)2+(Y1+Y2+b)2/parenrightBig1−α 2, (19) where bis the mean distance between the two ﬁlaments ( bdoes not depend on time because of the conservation law for the momentum (13)), X1=X(ξ1),X′ 1=X′(ξ1) and so on. The ﬁrst term in Eq.(19) describes the non-local self-interact ion of the ﬁlament, while the second one corresponds to an interaction with the second ﬁlament. T he Hamiltonian equations of motion have the form ˙X(ξ) =δHα δY(ξ), ˙Y(ξ) =−δHα δX(ξ). (20) 3 Crow instability for a pair of vortex ﬁlaments The system with the Hamiltonian (19) possesses the exact sta tionary solution X(ξ, t) =C(α, b)t, Y (ξ, t) = 0, (21) which describes the uniform motion of straight ﬁlaments. He re the stationary velocity C(α, b) is proportional to bα−1. But this solution appears to be unstable due to an analog of t he Crow instability [18]. In this section we consider the linea r evolution of small perturbations of the stationary solution, and derive the linear growth rat e. To perform the linear analysis of small deviations of the vor tex shape from a straight line, we need the quadratic part of the Hamiltonian (19): H(2) α=1 2/integraldisplay/integraldisplay(X′ 1X′ 2+Y′ 1Y′ 2) \|ξ1−ξ2\|1−αdξ1dξ2 +1 2/integraldisplay/integraldisplay/parenleftbiggα−1 2/parenrightbigg[(X1−X2)2+ (Y1−Y2)2] \|ξ1−ξ2\|3−αdξ1dξ2 +1 2/integraldisplay/integraldisplay(Y′ 1Y′ 2−X′ 1X′ 2) /parenleftBig (ξ1−ξ2)2+b2/parenrightBig1−α 2dξ1dξ2 −1 2/integraldisplay/integraldisplay/parenleftbiggα−1 2/parenrightbigg[(X1−X2)2+ (Y1+Y2)2] /parenleftBig (ξ1−ξ2)2+b2/parenrightBig3−α 2dξ1dξ2 −1 2/integraldisplay/integraldisplay/parenleftbiggα−1 2/parenrightbigg/parenleftbiggα−3 2/parenrightbigg2b2(Y1+Y2)2 /parenleftBig (ξ1−ξ2)2+b2/parenrightBig5−α 2dξ1dξ2. (22) 6For further consideration, it is useful to rewrite it in the 1 D Fourier representation: H(2) α=1 2/integraldisplaydk 2π/parenleftBig Aα(k)XkX−k+Bα(k)YkY−k/parenrightBig . (23) Expressions for the functions Aα(k) and Bα(k) follow from the Eq.(22). So, Aα(k) can be represented as follows: Aα(k) = 2k2bα/integraldisplay+∞ 0cos(kbζ)/parenleftBigg1 ζ1−α−1 (ζ2+ 1)1−α 2/parenrightBigg dζ +2(α−1)bα−2/integraldisplay+∞ 0(1−cos(kbζ))/parenleftBigg1 ζ3−α−1 (ζ2+ 1)3−α 2/parenrightBigg dζ = 2(1−α)2bα−2/integraldisplay+∞ 0(1−cos(kbζ))/parenleftBigg1 ζ3−α−1 (ζ2+ 1)3−α 2+/parenleftbigg3−α 1−α/parenrightbigg1 (ζ2+ 1)5−α 2/parenrightBigg dζ(24) Obviously, Aα(k) is positive everywhere. Analogous calculations for the fu nction Bα(k) give: Bα(k) = 2k2bα/integraldisplay+∞ 0cos(kbζ)/parenleftBigg1 ζ1−α+1 (ζ2+ 1)1−α 2/parenrightBigg dζ +2(α−1)bα−2/integraldisplay+∞ 0(1−cos(kbζ))dζ ζ3−α +2(1−α)bα−2/integraldisplay+∞ 0(1 + cos( kbζ))/parenleftBigg1 (ζ2+ 1)3−α 2+α−3 (ζ2+ 1)5−α 2/parenrightBigg dζ = 2(1−α)2bα−2/integraldisplay+∞ 0(1−cos(kbζ))dζ ζ3−α −2(1−α)(3−α)bα−2/integraldisplay+∞ 0(1 + cos( kbζ))/parenleftBigg2 (ζ2+ 1)5−α 2−1 (ζ2+ 1)3−α 2/parenrightBigg dζ (25) In Appendix A, Aα(k) and Bα(k) are expressed through the Euler Gamma function Γ( x) and the modiﬁed Bessel functions of the second kind Kν(x). The dispersion relation between the frequency ωαof a small amplitude perturbation of the ﬁlament shape and the corresponding wave number kis simply given by the formula ω2 α(k) =Aα(k)Bα(k), (26) since the linearized equations of motion for XkandYkare ˙Xk=Bα(k)Yk, ˙Yk=−Aα(k)Xk, (27) as follows from Eq.(20). In Fig.1 we have plotted ω2 αversus kfor several values of α. It is easy to see that at small wave numbers the product Aα(k)Bα(k) is negative. Indeed, after some calculations we obtain in leading order for kb≪1: Aα(k)≈k2bα/parenleftbigg1−α α/parenrightbigg I3−α, (28) 70 0.5 1 1.5 2 k-4-20246810 w2 Figure 1: The dependences ω2 α(k) =Aα(k)Bα(k) with b= 1 for α= 0.01, 0.025, 0 .05, 0.1, 0.25, 0.5. Lines corresponding to the given values of αintersect the horizontal axis in the indicated order. Bα(k)≈ −4(1−α)2bα−2I3−α, (29) where the constant I3−αis given by the integral I3−α=+∞/integraldisplay 0dζ (ζ2+ 1)3−α 2=√πΓ/parenleftBig 1−α 2/parenrightBig 2Γ/parenleftBig 3−α 2/parenrightBig, (30) with Γ( ..) being the Gamma function. Therefore, an instability takes place at small k. Unstable domain in the wave number space corresponds to a range \|k\|b < q 0(α) where Bα(k) is negative, with the function q0(α) behaving at small values of αlikeα1/2: q0(α)≈2α1/2, α ≪1. (31) The graphics of q0(α) is shown in Fig.2. The instability increment γα(k) =/radicalBig −Aα(k)Bα(k) is proportional to the absolute value of kat very small values of kb: γ(k)≈(1−α)I3−α·2\|k\|bα−1/radicalBig (1−α)/α. (32) However, for each αthere exists a maximum value γmax(α) of the increment, which is attained atkb∼√α. Therefore the approximate expressions (29) and (32) are va lid only if \|k\|b≪√α. It is interesting to note that the following inequality take s place: γmax(α)<2bα−2(see the Fig.1, where for the case b= 1 the minimal value of the product Aα(k)Bα(k) approaches the value−4 asα→0). For large wave numbers, \|k\|b≫1, the functions Aα(k) and Bα(k) are both positive. The asymptotic approximations in that region are: Aα(k)≈Bα(k)≈2(1−α)2k2−α/integraldisplay+∞ 0(1−cosη) η3−αdη=k2−α2(1−α) cos(πα/2)Γ(α) 2−α.(33) 80 0.2 0.4 0.6 0.8 alpha01234 q0 Figure 2: The boundary of instability q0(α). Note that this expression does not contain the parameter b. For a single vortex ﬁlament it is actually the exact expression for Aα(k) and Bα(k), which is valid in the whole range of k. A general nonlinear analysis of the non-local system (19) is diﬃcult. Therefore we need some simpliﬁed model which would approximate the nonlinear dynamics, at least in the most interesting long scale unstable regime. In the next section , we suggest such an approximate model and ﬁnd a class of solutions describing the formation o f a ﬁnite time singularity. 4 Singularity in long-scale nonlinear dynamics We note that the same long-scale limit as (28-29) can be obtai ned from the local nonlinear Hamiltonian Hl{R(ξ)}= (1−α)I3−α/contintegraldisplay(2Y)α α√ X′2+Z′2dξ, (34) where the coordinate Y(ξ) is measured from the symmetry plane. This Hamiltonian appr ox- imates the exact non-local Hamiltonian of a symmetrical pai r of vortex ﬁlaments in the case when the ratio of a typical value of Yto a typical longitudinal scale Lis much smaller than q0(α): Y/L≪√α. (35) In particular, this means that the slope of the curve with res pect to the symmetry plane should be small, and also Yshould be small in comparison with the radius of the line curvature. When Y= const, X′= const, Z′= const, this expression gives the same result for uniform stationary motion as the exact Hamiltonian. With the Cartesian parameterization (18), the correspondi ng approximate local nonlinear equations of motion have the form (after appropriate time re scaling) ˙X=1 (2−α)√ 1 +X′2 Y1−α, (36) 9˙Y=1 (2−α)α/parenleftBiggYαX′ √ 1 +X′2/parenrightBigg′ (37) and they allow to obtain a simple explanation of the instabil ity. On a qualitative level of understanding, the reason for the instability is that if i nitially some pieces of the curve were more close to the symmetry plane and convex in the direct ion of motion, then at subsequent moments in time the curvature will be increasing because of smaller values of Y and corresponding larger velocity, while Ywill be decreasing due to the curvature. Thus, the feedback is positive and the system is unstable. In the ﬁn al stage of the instability development, a locally self-similar regime in the dynamics is possible, because the above equations admit the self-similar substitution X(ξ, t) = X∗−(t∗−t)βx/parenleftBig (ξ−ξ∗)(t∗−t)−β/parenrightBig , (38) Y(ξ, t) = ( t∗−t)βy/parenleftBig (ξ−ξ∗)(t∗−t)−β/parenrightBig , (39) with arbitrary constants X∗,ξ∗,t∗, and with the exponent β=1 2−α. (40) After substitution Eqs.(38-39) into Eqs.(36-37), we obtai n a pair of ordinary diﬀerential equations for the functions x(z) and y(z): x−z·dx dz=/radicalBig 1 + (dx/dz )2 y1−α, (41) y−z·dy dz=1 α·d dz yα·(dx/dz )/radicalBig 1 + (dx/dz )2 . (42) However, with this choice of the curve parameterization, th e obviously existing symmetry of the system (34) with respect to rotation in the x-zplane is hidden. For taking advantage of this symmetry, cylindrical coordinates are more appropr iate, with the angle coordinate ϕ serving as the longitudinal parameter: (X, Y, Z ) = (R(ϕ, t) cosϕ, Y(ϕ, t),−R(ϕ, t) sinϕ). (43) Instead of the equations of motion (36-37), we obtain the equ ivalent system (where a same time rescaling as in (36-37) is performed) −(2−α)R˙R=√ R2+R′2 Y1−α, (44) −(2−α)R˙Y=1 α/parenleftBiggYαR′ √ R2+R′2/parenrightBigg′ −1 αRYα √ R2+R′2. (45) Here ( ..)′=∂ϕ(..). This system follows from the Lagrangian written in cylind rical coordinates Lϕ∼/integraldisplay/parenleftBigg (2−α)R2 2˙Y−Yα α√ R2+R′2/parenrightBigg dϕ. (46) 1005101520 -40 -30 -20 -10 0 10 20 30 40x z Figure 3: Self-similar solution x(z) forC= 50, α= 0.1. The self-similar substitution R(ϕ, t) = (t∗−t)βr(ϕ), Y(ϕ, t) = (t∗−t)βy(ϕ) (47) does not change the meaning of the angle coordinate ϕ. It leads us to the following pair of equations for the functions r(ϕ) and y(ϕ): r2=√ r2+r′2 y1−α, (48) yr=1 α/parenleftBiggyαr′ √ r2+r′2/parenrightBigg′ −1 αryα √ r2+r′2. (49) We see that there is no explicit dependence on ϕin these equations. This property helps us to integrate the system. The general solution can be represe nted in the following parametric form (see the Appendix B for a detailed derivation): ϕ(p) =ϕ0+ arctan( p)−/radicaltp/radicalvertex/radicalvertex/radicalbtα(1−α) (2−α)(1 +α)·arctan p/radicaltp/radicalvertex/radicalvertex/radicalbtα(2−α) (1−α2) , (50) y(p) =C−1 2−α/parenleftBigg(1−α2) α(2−α)+p2/parenrightBigg 1 2(2−α) , (51) r(p) =C1−α 2−α/parenleftBigg(1−α2) α(2−α)+p2/parenrightBiggα−1 2(2−α)/radicalBig 1 +p2, (52) where the parameter pruns between the limits −∞< p < +∞,Candϕ0are arbitrary constants of integration. The constant Cdetermines asymptotic slope of the curve at large distances from the origin: y≈r/Cwhenr→ ∞, while the constant ϕ0reﬂects the mentioned symmetry of the system with respect to rotations in x-zplane. The condition (35) for applicability of the local approximation (34) is satisﬁed i fC√α≫1. A typical self-similar solution x(z) is shown in Fig.3. 11It is interesting to note that the total angle ∆ ϕbetween two asymptotic directions in x-zplane does not depend on the parameter Cin the long-scale local approximation used above: ∆ϕ=π 1−/radicaltp/radicalvertex/radicalvertex/radicalbtα(1−α) (2−α)(1 +α) . (53) At small values of α, this angle approaches π. Another remark about ∆ ϕis that the above expression assumes identical values at αand at 1 −α, so the value ˜ α= 1/2 results in the extremum ∆ ϕmin= 2π/3. For this case, the curve lies on the cone y=r/C. 5 Discussion We observed that in the systems (34) with 0 < α < 1, ﬁnite time singularity formation is possible in the self-similar regime. Inasmuch as the cond ition (35) for applicability of the approximate Hamiltonian (34) is satisﬁed in a range of th e parameter Crelated to the self-similar solutions (50-52), we conclude that in the sys tems (2) self-similar collapse of two symmetrical singular vortex ﬁlaments can also take plac e. The principal question is whether it is also possible for ﬁlaments having ﬁnite width. If yes, then such solutions are analogous to the assumed self-similar Leray’s solutions of the Euler equation. Though the exponent β(40) diﬀers from 1 /2, the diﬀerence is small if αis small. However, an important diﬀerence exists between inﬁnitely thin ﬁlaments and ﬁlame nts with ﬁnite width: inside the last ones, longitudinal ﬂows take place, caused by a twist of the vortex lines constituting the ﬁlament. Those ﬂows keep the width homogeneous along the ﬁla ment if a local stretching is not suﬃciently fast. This mechanism acts against singula rity formation and, probably, in some cases it can prevent a singularity at all. [It is worth mentioning here that for ﬁnite width vortex structures in the Navier Stockes frame, t he usual ”outcome” result of the Crow instability is vortex line reconnection [29].] Thu s, a more-less consistent analysis of the general situation should take into account, besides t he dynamics of a mean shape of the ﬁlament, at least the dynamics of the width and the conj ugated dynamics of the twist. Clearly, we do not need to consider α/ne}ationslash= 0 systems, when we deal with non-singular vortex ﬁlaments. It should be emphasized that an attempt to t ake account of ﬁnite width of the ﬁlament by simple using regularized Green’s function s like Gǫ(r)∼1/√ r2+ǫ2with a constant ǫ, giving correct results for long scale limit of the lineariz ed problem, fails to describe the dynamics in highly nonlinear regime. Also, we would like to note that a local approximation analog ous to (34) is possible for arbitrary Green’s function GM(r). The corresponding long scale Hamiltonian has the form HMl{R(ξ)}=/contintegraldisplay FM(Y)√ X′2+Z′2dξ, where the function FM(Y) is related to the function GM(r) in the following way: FM(Y) = 2/integraldisplay+∞ 0/parenleftbigg GM(ξ)−GM/parenleftBig/radicalBig ξ2+ (2Y)2/parenrightBig/parenrightbigg dξ. The stationary motion with a constant coordinate Y0=b/2 is unstable if the second deriva- tive of the function FMis negative at that value: F′′ M(b/2)<0. We believe that such 12systems can exhibit locally self-similar collapse, if the a symptotics of the function FM(Y) is power-like at small Y:FM∼Yα, with 0 < α < 1. The ﬁnal remark concerns the possibility to include into the approximate long scale theory eﬀects caused by the fact that the unstable range is ﬁnite in w ave number space. Especially this is important for the case of small values of α, because in that limit the condition (35) for applicability of the Hamiltonian (34) becomes too restr ictive. The idea how to improve approximation is the following. In general, the exact expre ssion for the Hamiltonian of a pair of singular ﬁlaments, after integration by parts, can b e represented as the half of the integral over a surface Σ drawn between the ﬁlaments (one hal f inasmuch as we consider only one from two symmetric strings): Hα=1 2/integraldisplay(v·p) 2dr=Γ 2/integraldisplay Σ(v·dS) 2, because the canonical momentum ﬁeld pcreated by ﬁlaments is determined by a multi- valued scalar potential Φ( r):p=∇Φ, which has the additive increment Γ =/contintegraltext(p·dl) after passing around a ﬁlament. Also the equality div v= 0 is important for derivation of the last expression. In the case of small α, we should just more carefully take account of the contribution to the surface integral from the vicinity of ﬁl aments. As the result of such consideration, we ﬁnd that for a better approximation it is s uﬃcient to replace in (34) the projection of the arc-length element by the entire arc-leng th element and, correspondingly, use the Hamiltonian Hα≪1 l{R(ξ)} ∼/contintegraldisplayYα α√ X′2+Y′2+Z′2dξ. (54) We stress once more here that this expression is valid only in the case α≪1, Y/L ≪1. Acknowledgments The work of V.P.R. was supported by RFBR (grant 00-01-00929) and by Program of Support of the Leading Scientiﬁc Schools (grant 00-15-96007). Appendix A In order to have some closed expressions for the functions A(k) and B(k) instead of the integral representations (24) and (25), let us use the follo wing mathematical relations [28]: In−α=/integraldisplay+∞ 0dζ (ζ2+ 1)n−α 2=√π 2Γ(n−1−α 2) Γ(n−α 2), (55) I(1)=/integraldisplay+∞ 0cos(kbζ)dζ ζ1−α= (bk)−αcos/parenleftbiggπα 2/parenrightbigg Γ(α), (56) I(3)=/integraldisplay+∞ 0(1−cos(kbζ))dζ ζ3−α=(kb)2I(1) (1−α)(2−α), (57) 13/integraldisplay+∞ 0cos(qζ)dζ (ζ2+ 1)ρ=√π Γ(ρ)/parenleftbiggq 2/parenrightbiggρ−1 2Kρ−1 2(q), ρ > 0, (58) where Γ( x) is the Gamma function, Kν(x) is the modiﬁed Bessel function of the second kind. The integral (58) results in the equalities J(1)=/integraldisplay+∞ 0cos(kbζ)dζ (ζ2+ 1)1−α 2=√π Γ/parenleftBig 1−α 2/parenrightBig/parenleftBiggkb 2/parenrightBigg−α 2 K−α 2(kb), (59) J(3)=/integraldisplay+∞ 0cos(kbζ)dζ (ζ2+ 1)3−α 2=√π Γ/parenleftBig 3−α 2/parenrightBig/parenleftBiggkb 2/parenrightBigg1−α 2 K1−α 2(kb), (60) J(5)=/integraldisplay+∞ 0cos(kbζ)dζ (ζ2+ 1)5−α 2=√π Γ/parenleftBig 5−α 2/parenrightBig/parenleftBiggkb 2/parenrightBigg2−α 2 K2−α 2(kb). (61) Thus, we have from (24) and (25): A(k) = 2(1 −α)2bα−2I(3)−2k2bαJ(1)+ 2(1−α)bα−2/parenleftBig I3−α−J(3)/parenrightBig , (62) B(k) = 2(1 −α)2bα−2I(3)−2(1−α)(3−α)bα−2/parenleftBig 2(J(5)+I5−α)−J(3)−I3−α/parenrightBig .(63) Appendix B In this Appendix, we explain how the solution (50-52) of the s ystem (48-49) can be obtained. Let us introduce the designations Q= (dr/dϕ )2, s =r2, (64) then consider temporary sas independent variable, and rewrite the equation (49) as fo llows: y=2 α/parenleftBigg Qd ds/parenleftBiggyα √s+Q/parenrightBigg +yα(dQ/ds −1) 2√s+Q/parenrightBigg , (65) or equivalently y=2 α/parenleftBigg αdy dsyα−1/radicalBig s+Q−d ds/parenleftBiggyαs√s+Q/parenrightBigg/parenrightBigg . (66) Substituting into this equation the relation Q=s(sy2(1−α)−1) (67) which follows from the equation (48), we have the following e quation for y(s): y=2 αdy ds/parenleftBig αs+ (1−2α)y2(α−1)/parenrightBig . (68) This ﬁrst order diﬀerential equation is linear for the inver se dependence s(y), and its general solution is s(y) =C2y2−(1−2α) α(2−α)y2(α−1), (69) 14where Cis an arbitrary constant of integration. Thus, we have the re lation between yand s=r2. To obtain another relation, between yandϕ, let us use the equation dϕ=ds 2√sQ, (70) which gives us the integral ϕ−ϕ0=/integraldisplays′(y)dy 2s(y)/radicalBig s(y)y2(1−α)−1= =/integraldisplay/parenleftBig C2y2(2−α)+(1−2α)(1−α) α(2−α)/parenrightBig dy y /parenleftBig C2y2(2−α)−(1−2α) α(2−α)/parenrightBig/radicalbigg C2y2(2−α)−(1−α2) α(2−α)= = arctan/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg C2y2(2−α)−(1−α2) α(2−α)/parenrightBigg −/radicaltp/radicalvertex/radicalvertex/radicalbtα(1−α) (2−α)(1 +α)arctan/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBiggα(2−α) (1−α2)C2y2(2−α)−1/parenrightBigg . (71) After introducing the new parameter p=/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg C2y2(2−α)−(1−α2) α(2−α)/parenrightBigg , (72) we arrive at solution of the system (48-49) in the form (50-52 ). References [1] J. Leray, Acta Math. 63, 193 (1934). [2] J.T. Beale, T. Kato and A. Majda, Commun. Math. Phys. 94, 61 (1984). [3] P. Constantin and C. 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arXiv:physics/0012008v1 [physics.atom-ph] 5 Dec 2000Coherent transport C. Henkel Institute of Physics, Potsdam University, Am Neuen Palais 10, 14469 Potsdam, Germany 29 November 2000 Abstract We discuss the transport of matter waves in low-dimensional waveg- uides. Due to scattering from uncontrollable noise ﬁelds, t he spatial co- herence gets reduced and eventually lost. We develop a descr iption of this decoherence process in terms of transport equations for the atomic Wigner function. We outline its derivation and discuss the special case of white noise where an analytical solution can be found. Introduction We discuss in this contribution the transport of atomic matt er waves in a low- dimensional waveguide. Such structures may be created clos e to solid substrates using electro-magnetic ﬁelds: the magnetic ﬁeld of a curren t-carrying wire com- bined with a homogeneous bias ﬁeld, e.g., gives rise to a line ar waveguide [1, 2, 3]. Planar waveguides may be constructed with repulsive magnet ic [4] or optical [5] ﬁelds that ‘coat’ the substrate surface. The atomic motion i s characterised by bound vibrations in the ‘transverse’ direction(s) and an es sentially free motion in the ‘longitudinal’ direction(s) along the waveguide axi s (plane), respectively. Although direct contact with the substrate is avoided by the shielding potential, the atoms feel its presence through enhanced electromagnet ic ﬁeld ﬂuctuations that ‘leak’ out of the thermal solid, typically held at room t emperature. We have shown elsewhere that these thermal near ﬁelds are character ised by a ﬂuctua- tion spectrum exceeding by orders of magnitude the usual bla ckbody radiation [6, 7, 8, 9]. The scattering of the atoms oﬀ the near ﬁeld ﬂuctu ations occurs at a rate that may be calculated using Fermi’s Golden Rule. The co nsequences of mul- tiple scattering is conveniently described by a transport e quation that combines in a self-consistent way both ballistic motion and scatteri ng. The purpose of this contribution is to outline a derivation o f this transport equation. The status of this equation is similar to that of th e quantum-optical 1master equations allowing to describe the evolution of the r educed density matrix of an atomic system, on a time scale large compared to the corr elation time of the reservoir the system is coupled to, typically the vacuum rad iation ﬁeld. In the case of transport in waveguides, we face both temporal and sp atial dynamics and therefore restrict our attention to scales large compared t o the correlation time and length of a ﬂuctuating noise potential. Our analysis use s a multiple scale expansion adapted from [10]. Similar to the quantum-optica l case, we make an expansion in the perturbing potential to second order. In th e resulting transport equation, the noise is thus characterised by its second-ord er correlation functions or, equivalently, its spectral density. In the case of white noise, the transport equation can be explicitly solved. We have shown elsewhere [ 8] that this approxi- mation holds quite well for thermal near ﬁeld ﬂuctuations. F or technical noise, it also holds when the noise spectrum is ﬂat on a frequency scale roughly set by the ‘longitudinal’ temperature of the atoms in the waveguide. T he explicit solution yields an estimate for the spatial coherence of the guided ma tter waves as a func- tion of time. The paper concludes with some remarks on the lim its of validity of the present transport theory. It cannot describe, e.g., And erson localisation in one dimension [11] because on the coarser spatial scale of th e transport equation, the scattering from the noise ﬁeld is assumed to take place lo cally; interferences between diﬀerent scattering sequences are not taken into ac count. Decoherence in ‘curved’ or ‘split’ waveguides also needs a reﬁned theory because of the cross- coupling between the transverse and longitudinal degrees o f freedom, the former being ‘frozen out’ in our framework. 1 Statistical matter wave optics The simplest model for atom transport in a low-dimensional w aveguide is based on the Schr¨ odinger equation i¯h∂tψ(x,t) =−¯h2 2m∇2ψ+V(x,t)ψ (1) The coordinate xdescribes the motion in the free waveguide directions. The transverse motion if ‘frozen out’ by assuming that the atom i s cooled to the transverse ground state. Atom-atom interactions are negle cted, too.V(x,t) is the noise potential: for a magnetic waveguide, e.g., it is gi ven by V(x,t) =/angb∇acketlefts\|µ·B(x,t)\|s/angb∇acket∇ight, (2) where \|s/angb∇acket∇ightis the trapped internal state of the atom (we neglect spin-ch anging processes), and B(x,t) is the thermal magnetic ﬁeld. The noise potential is a statistical quantity with zero mean and second-order corre lation function CV(s,τ) =/angb∇acketleftV(x+s,t+τ)V(x,t)/angb∇acket∇ight, (3) 2where the average is taken over the realisations of the noise potential. We assume a statistically homogeneous noise, the correlation functi on being independent of x andt. As a function of the separation s, thermal magnetic ﬁelds are correlated on a length scale lcgiven approximately by the distance dbetween the waveguide axis and the solid substrate [8]. This estimate is valid as long as the wavelength 2 πc/ω corresponding to the noise frequency ωis large compared to d: for micrometre- sized waveguide structures, this means frequencies below t he optical range. The relevant frequencies of the noise will be identiﬁed below an d turn out to be much smaller than this. The coherence properties of the guided matter waves are char acterised by the noise-averaged coherence function (the time dependence is suppressed for clarity) ρ(x;s) =/angb∇acketleftψ∗(x−1 2s)ψ(x+1 2s)/angb∇acket∇ight. (4) In complete analogy to quantum-optical master equations, t his coherence function may be regarded as the reduced density matrix of the atomic en semble, when the degrees of freedom of the noise are traced over. The Wigner fu nction gives a convenient representation of the coherence function: W(x,p) =/integraldisplaydDs (2π¯h)De−ip·s/¯hρ(x;s), (5) whereDis the waveguide dimension. This representation allows to m ake a link to classical kinetic theory: W(x,p) may be viewed as a quasi-probability in phase space. For example, the spatial density n(x) and the current density j(x) of the atoms are given by n(x) =/integraldisplay dDpW(x,p) (6) j(x) =/integraldisplay dDpp mW(x,p) (7) We also obtain information about the spatial coherence: the spatially averaged coherence function Γ( s,t), for example, is related to the Wigner function by Γ(s,t)≡/integraldisplay dDxρ(x;s,t) (8) =/integraldisplay dDxdDpeip·s/¯hW(x,p,t) (9) In the next section, we outline a derivation of a closed equat ion for the Wigner function in terms of the noise correlation function. 2 Transport equation Details of the derivation of the transport equation may be fo und in the ap- pendix A. We quote here only the main assumptions underlying the theory. 3(i) The noise potential is supposed to be weakso that a perturbative analysis is possible. As in quantum-optical master equations, a clos ed equation is found when the expansion is pushed to second order in the pert urbation. (ii) The scale lcover which the noise is spatially correlated is assumed to be small compared to the characteristic scale of variation of t he Wigner func- tion. This implies a separation of the dynamics on short and l arge spatial scales, the dynamics on the large scale being ‘enslaved’ by c ertain averages over the short scale. Similarly, we assume that the potentia l ﬂuctuates rapidly on the time scale for the evolution of the Wigner func tion. These assumptions correspond to the Markov approximation of quan tum optics, where the master equation is valid on a coarse-grained time s cale. The derivation of the master equation is based on a multiple s cale expansion. Functionsf(x) of the spatial coordinate are thus written in the form f(x) =f(X,ξ) (10) where Xgives the ‘slow’ variation and the dimensionless variable ξ=x/lcgives the ‘rapid’ variation on the scale of the noise correlation l engthlc. Spatial gradi- ents are thus expanded using ∇x=∇X+1 lc∇ξ (11) By construction, the ﬁrst term is much smaller than the secon d one. Finally, the Wigner function is expanded as W(x,p,t) =W0(X,p,t) +η1/2W1(X,ξ,p,t) +O(η) (12) whereη≪1 is the ratio between the correlation length lcand a ‘macroscopic’ scale on which the coordinate Xvaries. The expansion allows to prove self- consistently that the zeroth order approximation W0does not depend on the short scale ξ, and to ﬁx the exponent 1 /2 for the ﬁrst order correction. The resulting transport equation speciﬁes the evolution of the Wigner function W0. Dropping the subscript 0, it reads /parenleftBig ∂t+p m· ∇x/parenrightBig W(x,p) = (13) /integraldisplay dDp′SV(p′−p,Ep′−Ep) [W(x,p′)−W(x,p)], whereSV, thespectral density of the noise, is essentially the spatial and time Fourier transform of the noise correlation function SV(q,∆E) =1 ¯h2/integraldisplaydDsdτ (2π¯h)DCV(s,τ) e−i(q·s−∆Eτ)/¯h. (14) 4The left hand side of the transport equation gives the free ba llistic motion of the atoms in the waveguide. If an external force were applied, an additional term F·∇pwould appear. The right hand side describes the scattering f rom the noise potential.Ep=p2/2mis the de Broglie dispersion relation for matter waves. We observe that scattering processes p→p′occur at a rate given by the noise spectrum at the Bohr frequency ( Ep−Ep′)/¯h. If the potential noise is static (as would be the case for a ‘rough potential’), then its spectral density is proportional toδ(∆E), and energy is conserved. If we are interested in the scatte ring between guided momentum states, then the initial and ﬁnal energies Ep,Ep′are typically of the order of the (longitudinal) temperature kTof the ensemble. The relevant frequencies in the noise spectral density are thus comparab le tokT/¯h. 3 Results 3.1 White noise White noise is characterised by a constant spectral density , i.e., the noise spec- trumSV(q,∆E) is independent of ∆ E. Equivalently, the noise correlation is δ-correlated in time: CV(s,τ) =BV(s)δ(τ). (15) The integration over the momentum q′in (13) is now not restricted by energy con- servation, and the right hand side of the transport equation becomes a convolu- tion. One therefore obtains a simple solution using Fourier transforms. Denoting k(dimension: wavevector) and s(dim.: length) the Fourier variables conjugate toxandp, we ﬁnd the equation /parenleftBig ∂t+¯hk m· ∇s/parenrightBig˜W(k,s) =−γ(s)˜W(k,s). (16) where we have introduced the rate γ(s) =1 ¯h2(BV(0)−BV(s)). (17) Eq.(16) is easily solved using the method of characteristic s, using s−¯hkt/mas a new variable. One ﬁnds ˜W(k,s;t) = ˜Wi(k,s−¯hkt/m)× ×exp/bracketleftbigg −/integraldisplayt 0dt′γ(s−¯hkt′/m)/bracketrightbigg , (18) where ˜Wi(k,s) is the Wigner function at t= 0. We observe in particular that the spatially averaged cohere nce function (8) shows an exponential decay as time increases: Γ(s;t) = Γ i(s) exp/bracketleftBig −γ(s)t/bracketrightBig . (19) 5We can thus give a physical meaning to the quantity γ(s): it is the rate at which two points in the matter wave ﬁeld, that are separated by a dis tances, lose their mutual coherence. This rate saturates to γ=γ(∞) =BV(0)/¯h2for distances s≫ lclarge compared to the correlation length of the noise ﬁeld (t he correlation BV(s) then vanishes). This saturation has been discussed, e.g., i n [12]. As shown in [8], the rateγis equal to the total scattering rate from the noise potentia l, as obtained from Fermi’s Golden Rule. For distances smaller than lc, the decoherence rate γ(s) decreases since the two points of the matter wave ﬁeld ‘see’ essentially the same noise potential. The exact solution (19) thus implies t hat after a time of the order of the scattering time 1 /γ, the spatial coherence of the atomic ensemble has been reduced to the correlation length lc. The estimates given in [8] imply a time scale of the order of a fraction of a second for waveguide s at a micrometre distance from a (bulk) metallic substrate. Signiﬁcant impr ovements can be made using thin metallic layers or wires, nonconducting materia ls or by mounting the waveguide at a larger distance from the substrate [8]. At timescales longer than the scattering time 1 /γ, the spatial coherence length of the atoms decreases more slowly, approximately as lc/√γt[8]. This is due to a diﬀusive increase of the width of the atomic momentum distr ibution, with a diﬀusion constant of the order of D= ¯h2γ/l2 c. This constant is in agreement with a random walk in momentum space: for each scattering time 1 /γ, the atoms absorb a momentum qc= ¯h/lcfrom the noise potential. The momentum step qc follows from the fact that the noise potential is smooth on sc ales smaller than lc, its Fourier transform therefore contains momenta up to ¯ h/lc. 3.2 Fokker-Planck equation The momentum diﬀusion estimate given above can also be retri eved from the transport equation, making an expansion of the Wigner distr ibution as a function of momentum. We assume that the typical momentum transfer qcabsorbed from the noise is small compared to the scale of variation of the Wi gner distribution, and expand the latter to second order. This manipulation cas ts the transport equation into a Fokker-Planck form /parenleftBig ∂t+p m· ∇x+Fdr(p)· ∇p/parenrightBig W(x,p) = (20) /summationdisplay ijDij(p)∂2 ∂pi∂pjW(x,p), where the drift force and the diﬀusion coeﬃcient are given by Fdr(p) = −/integraldisplay dDqqSV(q,Ep+q−Ep) (21) Dij(p) =/integraldisplay dDqqiqjSV(q,Ep+q−Ep). (22) 6In the special case of white noise, the p-dependence of these quantities drops out. Also the drift force is then zero because the noise correlati on function is real and the spectrum SV(q) even in q. Sinceqcgives the width of the spectrum, the diﬀusion coeﬃcient turns out to be of order q2 cγ, as estimated before. Casting the transport equation into Fokker-Planck form, on e can easily take into account the scattering from the noise ﬁeld in (classica l) Monte Carlo sim- ulations of the atomic motion: one simply has to add a random f orce whose correlation is given by the diﬀusion coeﬃcient. We note, however, that the Fokker-Planck equation cannot ca pture the initial stage of the decoherence process, starting from a wave ﬁeld t hat is coherent over distances larger than the correlation length lc. Indeed, it may be shown (neglecting the p· ∇xterm and the drift force, assuming an isotropic diﬀusion tensor for simplicity) that (20) yields a spatially average d coherence function ΓFP(s,t) = Γ i(s) exp/bracketleftBig −Ds2t/¯h2/bracketrightBig (23) This result implies a decoherence rate proportional to s2without saturation. It is hence valid only at large times (compared to the scatterin g time 1/γ) where the exponentials in both solutions (19, 23) are essentially zero fors≥lc. 4 Concluding remarks We have given an outline of a transport theory for dilute atom ic gases trapped in low-dimensional waveguides. This theory allows to follo w the evolution of the atomic phase-space distribution (more precisely, the a tomic, noise-averaged Wigner function) when the atoms are subject to a noise potent ial with ﬂuctua- tions in space and time. The spatial coherence of the gas can b e tracked over temporal and spatial scales larger than the correlation sca le of the noise, in a man- ner similar to the master equations of quantum optics. We hav e given explicit results in the case of white noise, highlighting spatial dec oherence and momentum diﬀusion. The transport equation has to be taken with care for strong no ise poten- tials because its derivation is based on second-order pertu rbation theory. It is certainly not valid when the ‘mean free path’ ∼¯v/γ(¯vis a typical velocity of the gas) is smaller than the noise correlation length lcbecause then the Wigner distribution changes signiﬁcantly over a small spatial sca le. (In technical terms, the approximation of a local scattering kernel in (13) is no l onger appropriate.) Also, the theory cannot describe Anderson localisation in 1 D waveguides with static noise [11]. This can be seen by working out the scatter ing kernel with SV(q,∆E) =SV(q)δ(∆E): 2m/integraldisplay dp′SV(p′−p)δ(p′2−p2) [W(x,p′)−W(x,p)] 7=mSV(2p) p[W(x,−p)−W(x,p)]. (24) We ﬁnd a divergence of the scattering rate at p→0 since the spectrum SV(2p) is ﬁnite in this limit. The one-dimensional, static case the refore merits further investigation. We also mention that is has been found recent ly that Anderson localisation is destroyed when time-dependent ﬂuctuation s are superimposed on the static disorder [13, 14]. In this context, transport (or master) equations similar to our approach have been used. Acknowledgements. We thank S. A. Gardiner, S. P¨ otting, M. Wilkens, and P. Zoller for constructive discussions. Continuous suppor t from M. Wilkens is gratefully acknowledged. A Multiple scale derivation of the transport equa- tion The Schr¨ odinger equation (1) gives the following equation for the Wigner function (∂t+p· ∇x)W(x,p) = (25) −i ¯h/integraldisplaydDq (2π¯h)D˜V(q,t) eiq·x/bracketleftBig W(x,p+1 2q)−W(x,p−1 2q)/bracketrightBig where ˜V(q,t) is the spatial Fourier transform of the noise potential. Si nce this potential is assumed weak and varies on a scale given by the co rrelation length lc, we introduce the following scaling ˜V(qcu,t) =/integraldisplay dDxe−iqcu·x/¯hV(x,t) =lD cηβˆV(u,t) (26) whereqc≡¯h/lcis the typical momentum width of ˜V(q,t) anduis a dimensionless vector. The parameter ηis given by the ratio between the small scale lcand the ‘macroscopic’ scale of the position distribution, the (pos itive) exponent βremains to be determined. We assume η≪1 and make the multiple scale expansion (12) for the Wigner function. Using the expansion (11) for the spa tial gradient, we get /bracketleftbigg ∂t+p m·/parenleftbigg ∇X+1 lc∇ξ/parenrightbigg/bracketrightbigg (W0+ηαW1) = (27) −iηβ qlc/integraldisplaydDu (2π)DˆV(u,t) eiu·x/lc[W(x,p+qcu/2)−W(x,p−qcu/2)] We now take the limit η→0, lc→0 at ﬁxedqc. The most divergent term on the left hand side is the one with (1 /lc)∇ξW0. It could only be balanced 8with a term on the right hand side involving W0, but due to the small factor ηβ, this term cannot have the same order of magnitude. We must th erefore require that (1 /lc)∇ξW0vanishes individually: the zeroth order Wigner function is independent of the short scale variable ξ. The next terms on the left hand side contain ( ηα/lc)∇ξW1and∇XW0, while on the right hand side the leading order is ( ηβ/lc)W0. We look for a connection betweenW0andW1, and therefore, the left hand W1term must be more divergent than theW0term. This is the case if ηαO(1/lc)≫ O(1/X)∼ηO(1/lc). We thus conclude that α <1. Comparing powers of ηon the left and right hand side, we ﬁndα=β, since the vector uand the scaled distance ξare of order unity. Therefore we get the equation /parenleftbigg lc∂t+p m· ∇ξ/parenrightbigg W1(X,ξ,p) = (28) −i qc/integraldisplaydDu (2π)DˆV(u,t) eiu·ξ[W0(X,p+qcu/2)−W0(X,p−qcu/2)] In the exponential, only the short length scale ξ=x/lcoccurs. We thus ﬁnd that the large scale variable Xis a parameter in this equation, and get a solution via Fourier transforms with respect to ξandt. In the spirit of the Markov approximation, we take the slowly varying W0(as a function of time) out of the time integral /integraldisplay∞ −∞dteiωtˆV(u,t)W0(...,t)≈W0(...,t)ˆV[u,ω] (29) where ˆV[u,ω] denotes the double space and time Fourier transform of the p oten- tial. We note κ,ωthe conjugate variables for the spatial Fourier transform a nd ﬁnd the following solution for the ﬁrst order Wigner functio n W1(X,ξ,p) = (30) −i qc/integraldisplaydω 2π/integraldisplaydDκ (2π)Deiκ·ξ−iωtˆV[κ,ω] iκ·p/m−ilcω+ 0(W0(X,p+qcκ/2)−W0(X,p−qcκ/2)) The +0 prescription in the denominator is related to causali ty: it ensures that the poles in the complex ω-plane are moved into the lower half plane, avoiding a blow-up of W1. This result will be inserted into the next order equation tha t also links W0to W1: /parenleftbigg ∂t+p m· ∇X/parenrightbigg W0= −iη2α qclc/integraldisplaydDu (2π)DˆV(u,t) eiu·ξ[W1(X,ξ,p+qcu/2)−W1(X,ξ,u−qcu/2)] 9Note that this equation is scaled consistently if O(1/X)∼η2αO(1/lc) =η2α−1O(1/X). This determines the exponent α=1 2. The result is an equation for W0only. We take the statistical average and make the factorisation /angb∇acketleftˆV(u,t)ˆV[κ,ω]W0(X,p)/angb∇acket∇ight=/angb∇acketleftˆV(u,t)ˆV[κ,ω]/angb∇acket∇ightW0(X,p). (31) This may be justiﬁed heuristically as follows: it seems reas onable that the statis- tical average can also be performed via ‘spatial coarse grai ning’, i.e., taking an average over the small-scale ﬂuctuations of the medium. Thi s is precisely the pic- ture behind transport theory: the individual scattering ev ents are not resolved but only the behaviour of the matter wave on larger scales. Th e lowest order Wigner function W0may be taken out of the coarse grain average because it does not depend on the short scale ξby construction. Finally, we introduce the spectral density ˆS(u,ω) of the (scaled) noise poten- tial /angb∇acketleftˆV(u,t)ˆV[κ,ω]/angb∇acket∇ight= (2π)DˆS(u,ω) eiωtδ(u+κ) (32) This allows to perform the integration over κwhen (30) is inserted into (31). The result still contains a frequency integral where denomi nators of the following form appear 1 i(u/m)·(p+qcu/2)−ilcω+ 0=−iqc Ep+qcu−Ep−¯hω−i0(33) A second term contains the sign-reversed energy diﬀerence. These denominators ensure that the kinetic energy change occurring in the scatt ering is compensated by a ‘quantum’ ¯ hωfrom the noise potential. We write the denominators (33) as a δ-function plus a principal part. For the classical noise potential considered here, the power spect rumˆS(u,ω) is even in ω, so that the δ-functions combine and the principal parts drop out. We ﬁnal ly get /parenleftbigg ∂t+p m· ∇X/parenrightbigg W0= (34) η ¯h2/integraldisplaydDu (2π)DˆS(u,∆E/¯h) [W0(X,p+qcu)−W0(X,p)] where ∆E=Ep+qcu−Ep. It is easily checked that this is the transport equa- tion (13), taking into account the relation between the scal ed and non-scaled noise spectra ηl3 c ¯h2ˆSV(u,∆E/¯h) =SV(qcu,∆E/¯h) (35) that follows from (14) and (26). 10References [1] J. Schmiedmayer, Eur. Phys. J. D 4, 57 (1998). [2] D. M¨ uller, D. Z. Anderson, R. J. Grow, P. D. D. Schwindt, a nd E. A. Cornell, Phys. Rev. Lett. 83, 5194 (1999). [3] N. H. Dekker et al., Phys. Rev. Lett. 84, 1124 (2000). [4] E. A. Hinds, M. G. Boshier, and I. G. Hughes, Phys. Rev. Let t.80, 645 (1998). [5] H. Gauck, M. Hartl, D. Schneble, H. Schnitzler, T. Pfau, a nd J. Mlynek, Phys. Rev. Lett. 81, 5298 (1998). [6] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414 (1999). [7] C. Henkel, S. P¨ otting, and M. Wilkens, Appl. Phys. B 69, 379 (1999). [8] C. Henkel, K. Joulain, R. Carminati, and J.-J. Greﬀet, Op t. Commun. (2000), in press. [9] C. Henkel and S. P¨ otting, Appl. Phys. B (2000), in press ( selected papers of the Bonn 2000 DPG meeting). [10] L. Ryzhik, G. Papanicolaou, and J. B. Keller, Wave Motio n24, 327 (1996). [11] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [12] C.-C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82, 4807 (1999). [13] J. C. Flores, Phys. Rev. B 60, 30 (1999). [14] S. A. Gurvitz, Phys. Rev. Lett. 85, 812 (2000). 11
Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox. Albert Serra -Valls Departamento de Física, Facultad de Ciencias, Universidad de Los Andes, Venezuela. To the memory of my professor and late friend Salvador Velayos, who once said “The spiral disturbs me” The inversion of cause and effect in the classic description of electromagnetism, gives rise to a conceptual error which is at the bottom of many paradoxes and exceptions. At present, the curious fact that unipolar induct ion or the Faraday Disc constitutes an exception to the Faraday induction law is generally accepted. When we establish the correct cause and effect relationship a close connection appears between mechanics and electromagnetism, as does a new induction law for which paradoxes or exceptions do not occur. Difficulties in interpreting the Faraday Disc derive directly from Faraday’s Induction Law and the equation that defines and measures magnetic induction. The electromagnetic force and the unipolar torque gene rated in the Faraday Disc depend on the shape of the circuit which connects to the disc, giving rise to an “absolute -relative” duality of the emf and unipolar torque. This gives rise to different interpretations. Analogy with mechanics suggests this dualit y derives from the twofold inert and gravitational nature of the electromagnetic mass. Some paradoxical experiments in unipolar induction involving the unique geometry of the spiral are described demonstrating this duality and the inversion of cause and ef fect. The emf and torque of the Faraday Disc and the conducting spiral is due to the continuous variation of the electromagnetic angular moment of the continuous current. This experiments confirm the Lorentz Force and invalidate Faraday’s Induction Law. Th ey show how in a closed circuit emf and unipolar torque are not produced by the variation in magnetic flux, which is constant, but by two variations in the electromagnetic angular moment. The three possible ways of varying the electromagnetic angular momen t generated by the circulation of the charges gives rise to the different forms of electromagnetic induction. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 2Introduction. 1. Unipolar or acyclic induction. Possibly, there has been no simpler, more curious and polemical experiment since the beginnings of electromagnetism than Faraday’s rotating magnet and disc. For their simplicity and beauty they have always attracted the attention of the physicist. According to Poincaré “The most curious electrodynamics experiments are those where a continuous rotation t akes place, called unipolar induction experiments.”1 Einstein, in his first paper “On the electrodynamics of moving bodies,” states that: “It is known that Maxwell’s electrodynamics –as usually understood at the present time– when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena”. “Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears . Moreover, questions as to the “seat” of electrodynamic electromotive forces (unipolar machines) now have no point.”2 It would seem the Faraday disc contributed to the development of the Theory of Relativity. When studying unipolar induction bac k 1961, and finding the conducting spiral to be a universal unipolar generator I imagined that this must have been known since the beginnings of electromagnetism. In that year I had begun my Ph. D. course in Physics at Grenoble University and found to my s urprise that the conducting spiral was unknown to my professors of electromagnetism. They suggested that I choose this for a second subject for my doctoral thesis3. It turned out to be very polemical, for as is well -known, unipolar induction continues to b e the object of discussions and publications. On completing my thesis, the Board of Examiners recommended my second subject for publication; something I was only able to do years later, for in the opinion of the journal’s referee the conducting spiral was but a “mind experiment” and couldn’t possibly revolve. Only on checking the experiment (presumably), was the article accepted. This publication4 had involved considerable difficulties and scarce attention. To start with, I - the supposed discoverer - had failed to grasp the significance of the spiral. Curiously, this experiment, as straightforward and beautiful as the Faraday Disc, is just as paradoxical. Twenty -seven years after publishing my article I began my studies of unipolar induction anew with a ser ies of experiments on conducting spirals which led me to a new understanding of electromagnetic induction, the Faraday Disc and the conducting spiral itself, establishing a new analogy between mechanics and electromagnetism. In November 1998, I attempted t o publish these findings in the same journal which in 1970 had published my first article, only to have it rejected out of hand by the editor who alleged “articles announcing new theoretical results or experiments are not accepted in this journal”. Maybe h e should have added: especially if they come from an unknown third -world Physicist, for this publication continues to carry articles on Faraday’s Induction Law and the Lorentz Force 5, 6, 7, 8, all of which deal with the old question as to how and where em f is generated in the Faraday Disc. Regarding the substance of the matter, some authors are of the opinion that the revolving magnet and the Faraday Disc are exceptions to Faraday’s Induction Law or flux rule9, and assure us that unipolar induction is due to the Lorentz Force, others deny any exceptions10, and still others see exceptions to the Lorentz Force11. The difficulties in understanding the Faraday Disc derive from Faraday’s Induction Law and the equation F = il x B, which defines B and allows it to be measured. This assumes that magnetic induction B, generated by the circuit to which the segment l belongs, is negligible with regard to B. The emf and torque generated in the Faraday Disc depend on the shape of the circuit that connects the disc, givin g rise to an “absolute – relative” duality of emf and Lorentz Force, which in turn, occasions different interpretations. This duality becomes much more evident in the conducting spiral and when the symmetry of the Faraday Disc is enhanced. Some paradoxical experiments in unipolar induction which make use of the unique geometry of the spiral are described in this article. These experiments show that the paradoxes and discrepancies that arise with unipolar induction are resolved when the following analogies between mechanics and electromagnetism are established: a) Charges, in the same way as mass, have a dual nature, inert and gravitational, in each of these pairs neither element is independent of the other. b) In electromagnetic interaction among charges, both mechanical and electromagnetic angular moments are conserved. c) Electromagnetic induction is due to the variation and conservation of the angular moments of mass and charge. d) The possible ways of varying the electromagnetic angular moment of a current in a cir cuit correspond to the forms of electromagnetic induction. e) The deformation of a circuit by electromagnetic forces tends to diminish the rate of change of the Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 3electromagnetic angular moment of the current’s charges, i.e. it will tend to conserve the angular moment. The circulation of the charges of the continuous current in a Faraday Disc, as also in a conducting spiral, generates a continuous rate of change of angular electromagnetic moment and angular moment of matter, this works in the same way as an elec trodynamic turbine. Due to the coexistence and conservation of the angular moment of the electromagnetic field and of matter, in all closed circuits there are always two equal and opposite variations of the angular moment generated. In closed circuits, con stant emf is not produced by the variation in magnetic flux, which is constant, but by two variations in the electromagnetic angular moment. This means the new induction law will be ε = -dL/dt dε/dt=-dφ/dt in which L is the electromagnetic angular moment um and φ is the magnetic flux density. According to this new induction law, unipolar induction is a consequence and not an exception. The generation and variation of the angular moments of the electromagnetic field and of matter, occur through the normal c onstraint forces acting along the path of the charges in the conductors. These constraint forces are not explicit in Maxwell’s equations. However, without these forces it would not be possible to generate or measure electric or magnetic field. The conducti ng spiral allows us to see that unipolar induction is produced by a vortex of charges, confirming the Lorentz Force and invalidating Faraday’s Induction Law, furthermore it allows us to see the true origin of electromagnetic induction and its dual nature. In the conducting spiral, an inversion of cause and effect in the description of electromagnetism also becomes evident. Fig. 1.a. The Faraday disc Fig. 1.b. The Barlow wheel • The magnetic field on t he disc has rotational symmetry. • The magnetic field on the disc doesn't have rotational symmetry. • The rotation axis of the disc and the rotational symmetry axis of the magnetic field of the magnet coincide. • The magnetic field is parallel to the rotatio n axis but is not symmetric with respect to this axis. • Eddy currents are not induced on the disc. • Eddy currents are induced on the disc. • In open circuit the magnet does not brake the disc. • In open circuit the magnet brakes the disc. • The magnet is n ot the stator and can rotate together with the disc. • The magnet is the stator and cannot rotate with the disc. • The disc and the magnet don't constitute the whole generator. • The disc and the magnet constitute the whole generator. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 42. Exceptions to F araday’s Law or the “Flux Rule”. Possibly owing to the polemical nature of the subject, even the best texbooks12 provide only a very superficial discussion of the Faraday Disc, and do not mention experimental facts which are fundamental to the comprehension of unipolar induction. An example of this are the differences that exist between the Faraday Disc and the Barlow Wheel as shown in Fig. 1. The rotation of a cylindrical magnet around its axis does not induce, nor is it braked by the presence of a conducto r. . This is due to the coincidence of the rotation axis and the rotational symmetry axis of the magnetic field of the cylindrical magnet. This fact differentiates Faraday’s Disc from other types of generators In the Faraday Disc, the circuit that closes t he current of the disc, mysteriously, is not usually taken into account in the description of the experiment; “the dark side of the force” is the stator. Here it is easy to see that the field lines are closed (∇•B = 0), increasing the diameter of the disc ; the induction lines will cross the disc twice, annulling the torque and the emf, Fig. 2. Fig. 2. The distribution of the magnetic induction lines in the Faraday Disc. emf and torque are maximum when brush takes place at the circumfere nce that separates the two directions of the induction lines. The brush on the disc should be placed on the circumference where the induction lines are inverted. The induction lines B should cross the disc once only. This fact is not taken into account in the description of the Faraday disc made by R. Feynman in his objection to the flux rule in the Faraday Disc13 The experiment described is a mixture of the Faraday disc and the Barlow wheel, and it is doubtful whether it would work, Fig. 3 Fig. 3. Description of the Faraday Disc according to The Feynman Lectures on Physics Vol. II, P. 17 -2. “When the disc rotates there is an emf from v x B, but with no change in the linked flux.” In the first place, most of the lines of the magnet cu t across the disc twice, and this hinders their functioning as generator. In the second place, the magnet’s axis doesn’t coincide with the disc’s rotation axis in which the Eddy current or Foucault current are induced, as is the case with the Barlow Wheel, due to the flux change. In this experiment the idea is to describe the Faraday Disc “in which no rate of change of flux occurs”. When the rotation axis of the disc and the rotational symmetry axis of the magnet’s field don’t coincide, Eddy currents are induced in the disc. This fact is mentioned by Scorgie at the end of his article.14 Some physicists, in order to safeguard the Flux Rule or the Faraday Law in unipolar induction, take the rate of flux swept by a radius of the Faraday Disc, as a true variatio n of the linked flux. “The emf may also be calculated by using the Faraday law. The only problem is how to choose the circuit through which the changing flux is to be calculated”15. The only problem actually, resides in that constant changing flux is not de tected by a curl -meter. As is well known, the calculation of the emf induced on the Faraday Disc, applying the Lorentz force or the swept flux rule gives the same result. emf = 1/2 Β ω R2 (1) According to Faraday’s induction law ε = −dφ/dt, the generati on of this constant emf implies a constant rate of change of magnetic flux and its indefinite growth in the circuit. For this reason, some authors believe that the Faraday Disc constitutes an exception to Faraday’s Law or the flux rule. Further on, we shal l see that according to Feynman, Galili and Kaplan16, the density magnetic flux does not vary and this is a necessary condition for the generation of a constant emf. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 53. The Faraday Disc and the absolute - relative duality. In the Faraday Disc the relative ro tation between disc and magnet is irrelevant. The variation in magnetic induction in the magnet due to its rotation (Barnett Effect) is negligible, for this reason unipolar emf does not depend on relative angular velocity between disc and magnet. Rotation occurs relative to the magnetic field generated by the magnet, but the rotation of the magnet on its axis does not change the magnetic field.17 That is to say, that the magnetic field of the cylindrical magnet has rotational symmetry and does not vary when the magnet revolves round the symmetry axis. Thus, there is no conceptual difference at all between the disc and Faraday’s rotating magnet as some authors believe.18, 19 To facilitate the analysis of unipolar induction, we can give the Faraday Disc a larger rotational symmetry. The Faraday Disc and the rest of the circuit can be replaced by two hemispheres whose rotation axis coincides with the axis of the cylindrical magnet enclosed inside it. Fig. 4. Fig. 4. Cylindrical magnet containe d inside a conducting sphere, whose hemispheres can rotate independently around their axis coinciding with the axis of rotational symmetry of the magnet’s field. If we split the cylindrical magnet in two equal parts at its equator and fix one of the halv es to each hemisphere, nothing will change in the experiment, but it will be made easier to carry out. The identical hemispheres constitute the disc or rotor and the stator respectively, in the Faraday experiment. If the conducting surface of the sphere i s traveled over by a continuous current that enters, for example, from the North Pole and goes out from the South, an equal and opposite torque is generated on the hemispheres that makes them rotate in a opposite directions, when allowed to slide on the eq uator. The rotor and the stator are indistinguishable. Reciprocally, if one hemisphere is made to rotate on top of the other, an emf is generated between the poles. We may ask whether this emf is produced by the absolute rotation of one of the hemispheres in the magnetic field, as prescribed by the Lorentz Force, and completely independently of the existence of the other hemisphere; or whether the emf is produced by the relative rotation of the two hemispheres in the magnetic field. We may observe that this constitutes an absolute - relative duality which is totally equivalent and indistinguishable in a closed circuit. This type of Faraday Disc, whose rotor and stator are identical, makes this duality easier to apprehend. Rotation in the magnetic field of the two hemispheres conjointly as if it were a single sphere, does not generate any emf between the poles. This may be attributed to the absence of relative movement, or that emfs are equal and cancel each other out. When both hemispheres rotate together as a single sphere in a magnetic field, according to the theory of relativity or to the Lorentz Force, an electric field is generated which produces an emf ∫E•dl between the pole and the equator and which is equal in the two hemispheres and which cancel each o ther out. To measure this unipolar emf between the pole and the equetor we required a stationary conductor (hemisphere) and a sliding contact. This is equivalent to producing a rotation between the hemispheres. When the two hemispheres rotate with the same angular velocity but in opposite directions, according to the theory of relativity or the Lorentz Force, an equal and opposite emf is generated which doubles the emf between the poles while the relative angular velocity between the hemispheres is also dou bled. It can be seen that the two interpretations are evidently indistinguishable and constitute an “absolute –relative” duality. Are the emfs in the twin hemispheres independent of each other? We shall see further on, that the need for the conservation of the electromagnetic angular moment of the current in a closed circuit causes the emfs generated in both hemispheres or circuit parts in relative movement not to be independent. The statement “we need only consider a radial “rod” in the rotating disc”20 supposes that the emf 1/2ΒωR2 all along the rod will not alter on closing the circuit. In order to calculate unipolar emf, the rest of the circuit is quite unnecessary, however it is absolutely essential for generating and measuring the emf and constant curren t. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 6In my opinion this is the crux of the matter. When calculating the emf produced by induction in a section of open circuit by using the Lorentz Force, as one would calculate the difference in potential in an electrostatic field ∫E•dl, we forget that the electric field lines, generated by induction, are closed, analogous to those of a magnetic field. The emf is calculated in a section of an open circuit, but is generated and measured in a closed circuit. As will be seen further on, the emf depends on the shape of the conductor that closes the circuit, this occurs when it forms part of the generator, as is the case with the Faraday Disc. Emf is attributed to the rotating hemisphere and is calculated on it. Supposedly, the emf induced in the stationary hemis phere is zero. According to the Lorentz Force, the emf is located in the rotating hemisphere. This contradicts the statement underlined in the introduction “Moreover, questions as to the “seat” of electrodynamic electromotive forces (unipolar machines) now have no point”21. It is an experimental fact that when current circulates between the poles or is generated by rotating a hemisphere, two equal and opposite torques are generated. These torques are attributed to the Lorentz force. This suggests, accordin g to the symmetry involved, that in a closed circuit , emf should also be generated in both hemispheres. Which is to say, that in a closed circuit there should not be a single unipolar emf located in a hemisphere that rotates in relation to the magnetic fie ld, but that emf is located in both hemispheres, as occurs with the torques. In a closed circuit the emf generated is the result of two equal and opposite emfs which do not cancel each other out. These are located in each one of the two parts of the circui t which are in movement, each one in relation to the other. The emf generated between the centers of the discs or the poles of the hemispheres depends on the difference in angular rotation velocity of each hemisphere or what comes to the same, the relative velocity between them. This gives rise to an absolute - relative duality in unipolar induction. Even though we calculate and explain the generation of an emf in a conductor (disc) that rotates in a magnetic field, it is not possible to generate a constant e mf and current unless we complete the circuit and produce a relative movement between the parts. In the following sections, we go on to describe some experiments in unipolar induction involving spiral -shaped conductors without any magnetic field other than that generated by the spiral itself. These experiments confirm and make clear beyond any doubt, this absolute -relative duality, showing that due to the conservation of the electromagnetic angular moment of the current, unipolar emf and constant current ar e the result of two equal and opposite emfs. Furthermore, these experiments provide an insight into the inversion of cause and effect in electromagnetic induction and the Lorentz Force, because they make it possible to see the part played by the variation and conservation of the electromagnetic angular moment generated by the circulation of the charges in the circuit. All these experimental findings must be put down to the coexistence and inviolability of the conservation of the mechanical and electromagne tic angular moments. 4. The Symmetrical or Twin Faraday Disc. This experiment with the double, symmetrical Faraday Disc, was carried out by making use of two identical rare earth magnets each fixed to one of two identical iron discs. These discs have a circu lar channel made near their borders in which some brass balls are inserted allowing the iron discs to rotate round their axis. The twin discs are connected electrically through these brass balls. A conducting disc (Faraday Disc) is placed between the magne t and the iron disc. The discs are connected electrically through their rims. In this way a radial current enters via the central axis of one of the conducting discs and leaves via the center of the other one. See Fig. 5. This experiment requires two conce ntric Faraday Discs connected electrically throughout their rims. This twin Faraday Disc has a total rotational symmetry (rotor -stator) of its magnetic induction and radial currents. The point of contact on the rim is replaced by a circumferential contact. To emphasize the Angular Moment Conservation Principle, and to ascertain that we have here a complete generator (the feed wires do not form part of this motor, as was the case with the Faraday Disc), rotor and stator are identical and the components are m ounted so that both parts can rotate in relation to the laboratory. Fig. 5. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 7 Fig. 5. The twin Faraday disc. Two conducting discs replace the conducting hemispheres of the previous figure. The cylindrical magnet is cut by the equator in t wo equal parts and fixed to the discs. ID Iron Disc, IR Isolating Ring, CD Conducting Disc, M Magnet, BB Brass Balls, CA Conducting Axis, SC Sliding Contact. If we substitute a spiral for the magnet and conducting disc, we obtain a Twin Faraday Disc which works with continuous and alternate current. 5. The Conducting Spiral as an Electrodynamic Turbine or Universal Unipolar Generator. In this section, for the sake of homogeneity and symmetry, we shall ignore the magnets and only consider the charges on the c onductors and their relative movements. Let us suppose that we substitute a circular current on the equator for the magnet enclosed in the sphere, Fig. 4. It is irrelevant whether the circular loop conductor rotates with one of the hemispheres or not. The circular current round the equator produces two equal and opposite torques on the radial or meridian currents (as demanded by the Angular Momentum Conservation Principle). The radial currents can not produce any torque in relation to the axis of the circu lar current (the forces are normal to the circular loop, and the torque in relation to the axis is null. Ampere’s third experiment).22 The fact that the torque in relation to the axis of a circular loop current will always be null, makes it irrelevant whet her the circular loop current is fixed or not to the radial current, as occurs with the magnet and Faraday’s Disc. For this reason there is no conceptual difference between the disc and Faraday’s revolving magnet, as is often stated. “Faraday's disc should not be confused with the case of unipolar induction. In the latter the rotating disc is a magnet itself. This case is much complicated conceptually and never touched on in introductory physics courses.”23 Therefore, it is [possible to connect circular and radial currents in series, thus forming a single current. In this way, we obtain a G -shape circuit which represents the Faraday Disc. With this simple line of thought in mind, I discovered the spiral to be a unipolar generator some forty years ago.24 The G -shaped circuit is a most particular type of spiral. It may be asked whether any conducting spiral be a unipolar or acyclic generator similar to the Faraday Disc? In fact it will, but it also constitutes a universal generator, which means it works with both continuous and alternate current. Further on, we shall see the difference between a logarithmic spiral and a G spiral. A G–shaped loop or circuit is formed by a circular loop and a radial segment connected in series. A conducting spiral circuit is formed by a continuum of circular and radial elements. Fig. 6. Fig. 6 ( a) The “G” loop circuit formed by a circular loop and a radius. (b) In the spiral circuit, each element of the spiral can be decomposed into an element in the radius dir ection and another normal to the radius. For mechanical and electromagnetic experimental purposes, it is irrelevant whether the radial and circular Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 8currents belong or not to the same circuit. Either way, the resistance and inertia of the radial conductor (disc) are increased. It is easy to prove experimentally that a spiral –shaped conductor is a unipolar or acyclic machine, whose rotation direction doesn't depend on the current direction. Such a machine was patented.25 When I published my article on the co nducting spiral,26 I had not realized that it constituted an electromagnetic paradox which would afford a new vision of electromagnetic induction. Every conducting spiral current element has two components, one radial, belonging to the radial current of the Faraday Disc, and another normal to the radius, belonging to the circular current or magnetic field of the magnet. When the current direction in the conducting spiral is changed, this implies changing simultaneously the current and the field direction. T he torque and the emf generated in the spiral are proportional to i2. The vector product of the radial and normal components of the charges’ velocity defines a vector or vortex, which add together in the case of the spiral, producing a single perpendicular vector to the spiral plane. While a continuous spiral, whether Archimedean or logarithmic, immediately suggests, by analogy with mechanics, an electrodynamic turbine; and that the electromagnetic torque is due to continuous rate of change of the electrom agnetic angular moment of the current. Such is not the case with the G -spiral or the Faraday Disc. According to this phenomenological description, the continuous variation of the rotation radius in a vortex of charges, produces the continuous variation in the angular electromagnetic moment of the current charges. Magnetic moment µ = πr2i = ½qvr is not really a momentum, properly speaking. This is because its units are not a momentum (Newton • s = Kg • m/s). The time rate of change of the magnetic moment is not a torque either. Γ ≠ dµ /dt (2) The definition of a magnetic moment is cinematic and does not take into account the forces that have generated it. We shall see further on, that in this case what is really happening is the circulation of the electromag netic momentum. In the G -shaped circuit (the nearest equivalent to the Faraday Disc), the charges of the continuous current in the circular part (magnet) of the G loop, that produces the constant magnetic moment, on interacting with the charges of the rad ial current segment (disc), changes the electromagnetic angular moment of the current. While some charges produce an electromagnetic angular moment when rotating on an axis, others produce the rate of change of the electromagnetic angular moment, as they i nteract with those that move away from or towards the axis. The “G” loop produces a constant rate of change of electromagnetic angular moment (torque), in the same way as a continuous spiral. The torque is proportional to the magnetic moment of its circula r part and the current of its straight part. Γ ≈ µ I (3) The interaction of the charges produces a vortex or “magnetron effect”, or transformation of the electromagnetic lineal moment into angular moment. Possibly, because this effect has never been obse rved in mechanics, the gravitation forces involved being too week, a comparison with the origin of electromagnetic induction has not been established. We shall come back to this point later. Regarding the Lorentz Force, it has been stated that: “The emf i s independent of the path in the conductor since only the radial components of the path elements contribute to the integral ∫∫((v××ΒΒ))••dl”.27 This does not apply to the spiral, for as we may observe, the normal component to the radius generates a magnetic fiel d which adds to the “external” magnetic field. In the case of the spiral it is superfluous. This fact constitutes an aspect of the spiral paradox. It flows from this, that the magnetic induction B, the torque and the emf, all depend on the path of the char ges in the conductor. Consequently, the unipolar emf and the current in the Faraday Disc are not independent of the way the disc current is closed. Does not the differential expression of the Lorentz Force applied to the calculation of the emf in the Faraday Disc not make more sense than the integral expression of Faraday’s Law? We shall see the mining of both expressions from the point of view of the rate of change of the electromagnetic angular moment. Later, the geometry of the conducting spiral will allow us to distinguish absolute unipolar emf and torque (due to the intrinsic or absolute rotation of the Faraday Disc or spiral) from the relative emf and torque, which depend at the same time on the spiral and the rest of the circuit. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 9In most cases where the Lorentz Force is used, the magnetic induction generated by the circuit to which the current i belongs is negligible compared to the external magnetic induction B 6. The “absolute – relative” duality in the spiral paradox. Since the rotation direction of t he conducting spiral doesn’t depend on the current direction, this could induce us to believe that the spiral torque only depends on the direction of the winding, and that it does not depend on the external magnetic field that can be generated by the rest of the circuit that closes the spiral or by any other circuit. We can verify experimentally that this is not the case: the circuit that closes the spiral can increase, decrease, annul or change the spiral torque direction. If the spiral is closed by a radi al conductor, its torque is “absolute”, that is to say it is specific to the spiral and does not depend on the rest of the circuit, and for this reason does not increase or diminish. The spiral’s torque does not depend on the radial conductor, however, th e torque in the radial conductor depends on the spiral. By contrast, if the spiral is closed by a curved conductor or by another spiral, their torques are dependent or “relative”. The geometry of the spiral allows us to see this paradox or duality of unipo lar induction clearly. While the radial conductor neither increases nor diminishes the torque of the spiral, we cannot generate this torque without this conductor, which closes the circuit and satisfies the principle of the conservation of the angular mom ent. This absoulte -relative duality of torque and unipolar induction suggests that the elctromagnetic mass, as also the machanical mass is inert and gravitational. The word electromagnetic expreses the gravitational -inert duality of the charge. The conduct ing spiral constitutes the most simple and beautifull form of the continuous transformation of electric into mechanical energy. This happens through the continuous variation of the lineal and angular moments of the electromagnetic field and matter. 7. The con servation of angular moments in a circuit. When we state that a continuous current in a circular loop generates a constant magnetic moment, similar to the orbital magnetic moment of an atom, there is an electric field normal to the charges’ trajectory in b oth cases. In the case of the circular loop, this electric field is generated by the normal constraint forces of the conductor. For this reason, when we mention a magnetic moment, the electric field is always normal to the charges’ velocity. When a compone nt of an electric field in the direction of the charges’ velocity exists, a rate of change of the electromagnetic moment is produced. Fig.7. If we superimpose a normal electric field on a magnetic field, a Poynting’s vector circulation is generated. Fig.7. The twin Faraday Disc. Magnetic induction B perpendicular to the discs transforms the continuous radial circulation of the electromagnetic lineal momentum p, in in two constant time rates of change of the electromagnetic angular momentum dL/dt. Electric charges moving in the direction of the radial electric field on the discs, produce two time rates of change of the angular moment of the electromagnetic field and angular moment of matter, giving rise to equal and opposite torques as demanded by the pri nciple of the conservation of angular momentum. Reciprocally, the rate of change of the angular moments of matter produces the rate of change of angular electromagnetic moment or emf. The Faraday Disc and the conducting spiral constitute experimental proof of this. We shall now see how magnetic moment or electromagnetic angular moment and its rate of change is generated in the conducting spiral. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 10A conducting spiral, just as the Faraday Disc, is an open circuit. If we were to form a closed circuit placing face to face two identical spirals (rotor - stator), the normal components of their currents would be oriented in the same direction and their magnetic fields would add on to each other, however, the radial components of the current elements are oriented in opposite directions. So, while the sum of the normal components of the current that generate the magnetic moment can have any value, the sum of the radial components of the current will always be zero in all closed circuits. See Fig. 8. Fig. 8. A closed circuit formed by two spirals of equal step and opposite direction. The normal components to the radius do not cancel each other out. From this, we may deduce, as is well known, that the rate of change of the moment of the charges of a continuous current will be zero in any closed circuit. The fact that the sum of the radial components is always zero, divides the circuit in two, and establishes the equality of the torques and of the emfs . To each radial element i r, there corresponds another, at the same distance from the axis of rotation and which produces an equal and opposite torque. This is because the electromagnetic forces are normal to the conductors. The normal elements in that generate the constant magnetic moment of the current charges can increase or diminish the rate of change of angular moment (torques) according to their direction, generated by the radial components. They are equal because of their geometric origin. When in a rigid circuit made up of one or two spirals there circulates a continuous current, two continuous equal and opposite rates of change of the magnetic and mechanical moments are generated, which produce two equal and opposite torques. Reciprocally, when the Faraday Disc or the conducting spiral is made t o rotate with a constant angular velocity, (a non -rigid circuit) the normal constraint forces of the conductor propel the charges generating a constant current which produces two constant, equal and opposite rates of change of the electromagnetic angular m oment or emfs. When a torque acts on the Faraday Disc or conducting spiral, at the same time as the angular velocity ω increases, the intensity of the current and magnetic flux density also increase, which implies an increase in the time rate of change of the electromagnetic angular moment. Reciprocally, if we vary the current intensity, the magnetic flux density will also vary in the same way as the time rates of change of the angular moments of the electromagnetic field and matter. The orbital angular mom ent of a charge depends on the velocity magnitude and on the radius, L ≈ v2/r When the conduction electrons move througt a conducting spiral with constant drift speed, the current intensity and the magnetic flux density are constant. However, a constant ra te of change of the angular moment of the electrons is produced, due to the variation of the trajectory radius, which gives rise to constant electromagnetic torque of the spiral and emf. ε = −dL/dt = −Γ (4) The variation in emf and current intensity impli es a variation in magnetic flux density, the drift speed of the electrons, the electromagnetic angular moment and the spiral’s torque. dε/dt = −dφ/dt = −dΓ/dt (5) where φ is the magnetic flux density, ε, L and Γ are the emf, the electromagnetic angular mo ment and the torque respectively. The generation of constant emf and current are due to two constant time rates of change of electromagnetic angular moment of the current, which are equal and opposite respectively in each of the two parts of the circuit th at move in relation to each other. According to Faraday’s induction Law ε = - dΦ/dt, (Φ is the magnetic flux) the generation of a constant emf and current is due to a constant time rate of change in the magnetic flux; as we might incorrectly conclude from the following experiment, the generation of a constant emf through the deformation of a circuit in a uniform magnetic field (a rectangular loop in which one of the sides moves at a constant velocity) the magnetic flux enclosed in a circuit may vary, but i ts density will remain constant. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 11The generation of a constant emf necessarily implies a constant density of the magnetic flux. On producing a deformation in a circuit, two equal and opposite variations of the electromagnetic angular moment of the current can be produced; even though the density of the magnetic flux remains unchanged. In the symmetrical Faray Disc we saw that the two emfs must be opposite (in the same way as angular momenta) for otherwise they will cancel each other out. These two emfs or c onstant time rates of change of the angular moment of the electromagnetic field, are generated respectively in each of the parts in relative movement. Here the problem of finding the seat of the emf, as mentioned in the introduction, is also solved. The em fs are just as locatable as the torques. While the origen of the two emfs is in the rotating spiral (as the Lorentz Force has it), they are both the result of the interaction between the two spirals or parts of the circuit. Let us consider an example, by way of analogy, from the point of view of mechanics. If two bodies are at rest and a force acts on one of them causing it to collide with the other, we can state that the body responsible for the collision is the body that was impelled. However as a result of the collision, two equal and opposite moments are generated in the interacting bodies. Now, taking into account these ideas we are in a position to understand the Faraday Disc and electromagnetic induction. To summarize, electromagnetic induction is du e to the coexistence and conservation of the angular moment of the electromagnetic field and angular moment of matter. For this reason a closed loop in the primary part of a transformer diminishes induction in the secondary. Lenz Law and the diamagnetism o f superconductors constitute the most perfect and constant demonstration that electromagnetic induction is due to the conservation of the electromagnetic angular moment of the current charges. We shall demonstrate, taking another aspect of the spiral parad ox, that electromagnetic torque is linked to the time rate of change of the angular moments of the electromagnetic field and matter. 8. The paradox of unipolar torque in the spiral. If the circuit that closes the spiral is very short and the only magnetic fi eld is that of the spiral itself, we should expect that while the spiral’s step diminishes and the number of turns increases (assuming the current’s intensity and maximum radius are constant), its torque as well as its magnetic moment will increase. Surprisingly, this is not the case. While the resistance and heat generated in the spiral increase rapidly with the number of turns, its torque does not change perceptibly. Spirals of 19 cm. diameter were made having different steps. The copper wire used was 3 m m. diameter. In order to facilitate rotation, the center and exterior electrical contacts to the spirals were made through the medium of mercury. Spirals and mercury were enclosed in hermetically –tight, transparent Lucite boxes. Fig. 9. Fig. 9 The spiral paradox. Conducting spirals of different step values and equal radius maximum connected in series and sufficiently far apart so that the interaction of their fields would be negligible. The spiral’s torque increases very slowly with the number of turns, which is the contrary of what we would expect. CR Collecting Ring, SC Sliding Contact . In this experiment, two spirals of the same diameter (19 cm) and widely different step (3 and 100 mm), their centers 1,5 m apart, were connected in se ries. If current, and maximum radius are constant we should expect the torque of the spiral to increase with the number of turns, as the magnetic moment Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 12increases. Quite to the contrary, torque is not found to vary perceptibly with the number or turns. To the extent that the step value of the spiral diminishes, the number of turns and the magnetic moment of the current charges increase, but this implies, necessarily, that the rate of change of the angular magnetic moment of the current diminishes. The mos t logical and simplest explanation for this paradox consists in attributing the electromagnetic torque to the time rate of change of the electromagnetic angular moment of the current. Γ= dL/dt (6) We have seen how the electromagnetic torque of the spiral conductor depends on the product of its two components (radial and normal) of the charges’ velocity. Given that the speed of the charges’ drift is constant all along the spiral conductor, the angular moment changes. By contrast to what happens in the elli ptical trajectory of planets, the angular velocity changes in order to maintain the angular moment constant (Kepler’s Second Law). At the same time as the spiral step diminishes and the number of turns increases, the normal component of the charges’ veloc ity and the magnetic moment µ (or the angular moment of the magnetic field) increases, but the radial component of the velocity diminishes, as also the time rate of change of the angular moments of the electromagnetic field and matter. That is to say that with the spiral, the normal velocity of the charges increases, as the radial velocity disminishes. The increment in magnetic induction and in magnetic moment of the current’s charges, when increasing the number of turns in the spiral lowers the rate of ch ange of the turn radius of the charges and the rate of change of the moment. When the spiral step tends to zero or infinity the torque tends to zero, too. Furthermore, the sensitivity of the spiral to self -torque depends on the relative rate of change of t he angular moment ΔL/L. For the increase in the spiral’s moment implies an increase in weight, friction –torque and resistance. A multi –layered cylindrical coil is equivalent to a spiral in which the relative rate of change of moment is very small, this is due to its magnetic moment being very large, and its rate of change very small. This is the reason why the rotation of a cylindrical coil round its axis is extremely difficult to bring about. No cases have been reported. If the radial i r and normal i n components of the current, in a conducting spiral contribute to the same extent to its electromagnetic torque and this being due to the product of the components, torque will be at a maximum when components are equal. Archimedes’ spiral has a constant step r = kθ and does not comply with the above condition because the radial component diminishes as the normal component increases with the widening of the spiral. The spiral on which all points have equal normal and radial current components, i r = in will be tha t which has maximum torque. If this condition is fulfilled, a logarithmic spiral is obtained: rdθ=dr dθ=dr/r θ=lnr r=eθ (7) Where the dimensional constant k = 1 The logarithmic spiral is the shortest spiral with the maximum electromagneti c torque. This can easily be verified by measuring the torques with a torsion balance of, for example, a G -shaped circuit and logarithmic spiral of equal length. Fig. 10. Fig. 10. Two three –arm spirals of equal length but different shape (G and logarithmic). The mobile contacts at the center and ends of the spirals were made through mercury and closed by a radial current so as not to change their self - torque. The deformation of a circuit by electromagnetic forces tends to diminish t he time rate of change of the electromagnetic angular moment. This means the Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 13deformation of the circuit will tend to conserve the electromagnetic angular moment. 9. Mechanical mass and Electromagnetic mass. In clasical mechanics a distinction is made between inert and gravitational mass as two aspeccts of matter whose nature is totally diferent and which are defined and measured by their differents effects. Apparently, the charge also has a dual, inert and gravitational nature. This is shown by the fact that t he charge units in the mks system (coul), and the Gauss system (statcoul) are physically different, and are measured by their different effects. The constant 1 /4πε0 3x109 by which they are related, has dimensions. It is usual not to express the dimensions of the constant 3x109 and to take the relation between coul and statcoul as simply a change of scale, analogous to m and cm in these unit measurement systems. While in the mks system, the charge (coul) would be measured according to its inert nature, in t he Gauss system (statcoul) this would be done according to its gravitational nature. It is said that the electron has a mass, and the relation - ship between its charge and mass is e/m e = 1.76x1011 coul/Kg. If we attribute inertia of charge to the fact it has a mechanical mass: what will the attributes of the elctromagnetic mass of the charge be? “Suppose an electron is moving at a uniform velocity through space, assuming for a moment that the velocity is low compared with the speed of light. Associated wit h this moving electron, there is a momentum – even if the electron had no mass before it was charged - because of the momentum in the electromagnetic field”.28 The electromagnetic mass of an electron m elec = 2e2/3ac3 is calculated from the density of the li neal moment of the electromagnetic field g = ε0 E x B integrated in all the space (corresponding to the inert part) and to the energy of the electrostatic field U = q2 / 8πε0 (corresponding to the gravitational part), r 0 = e2 / melecc2 = 3/2a which is call ed the classic radius of an electron.29 Electromagnetic mass thus calculated is just as inert and gravitational as mechanical mass. They coexist in the same way as the duality particle -field or, mass and energy. They are the two sides of a coin. Are the ine rt and gravitational nature of the charge independent? Can we measure them separately, as as we apparently do with mechanical mass? When we say that inert mechanical mass is intrinsic, and that it does not depend on the presence of others masses, what does this mean? Can we really see and measure the inertia of a mass without the presence of another mass? It is possible to apply a force to a mass withhout the presence of another mass?. The existence of charges of opposite sign, whose inertia is very small a nd gravity very strong allows us to change the proportion very quickly, generating very intense fields. Contrariwise, the gravitational nature of mechanical mass is weak as compared to its inertia, which makes it impossible to produce a significant and qu ick change in the gravitational field (as for example, measuring the speed of propagation of the gravitational field) In the following section we shall see that electromagnetic torque by a apparently continuous current in a conducting spiral makes the doub le, gravitational and inert nature of the electromagnetic mass clearly apparent. 10. The spiral’s torque and electromagnetic mass. In section 7, I suggested the hypothesis that the radial and normal components of a current contribute equally to the electromagn etic torque of a spiral. In justifying the hypothesis, it will be seen that this is equivalent to attributing the electromagnetic torque to the time rate of change of the electromagnetic angular moment of the current’s charges in the spiral. Also, the auto - torque of the spiral does not depend on ineraction with the rest of the circuit when this consists of a radial conductor. For this to happen, the electromagnetic auto -torque generated in a logarithmic spiral must be the measure for a kind of conducting e lectron mass circulating in the spiral. Given a logarithmic spiral r = ekθ with a constant k = 1. If the speed of displacement of the electrons v d is constant, the modules of the radial and normal components will be equal and constant throughout a logarithmic spiral of constant 1. Fig. 11. According to the Biot -Savart Law, dB=µ0/4π dl x r/r3 (8) dlxr is the normal component of the current element di, that generates the angular moment or the magnetic field at the point r. The contribution to the vari ation of the spiral’s angular moment d Γ = dl/dt or torque, produced by an element of the current di, depends on the magnetic field. This magnetic field is produced by all the normal components to it, of the spiral’s and the rest of the circuit’s current e lements. Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 14 Fig. 11. The electromagnetic auto -torque generating by the conducting electrons in a logarithmic spiral is a measure of the inertia of electrons, which depends on the shape of the circuit. If the magnetic field produced by a circuit which closes a spiral is null in relation to the spiral center and the magnetic induction B along its length varies inversely proportionally to its radius, all the currents elements of a logarithmic spiral should contribute equally to its torque and make it possible to define the constant inert mass of the conducting electrons m e in a way analogous to mechanical mass. That is to said, the angular moment dl produced by a dme would be dl = r v n dme and its contribution to the electromagnetic torque enr endmvvdmvdtdr dtdld = ==Γ (9) The variation in angular moment dl of the dm e of the conducting electrons would be constant all along the logarithmic spiral. We may also ascertain that the variation is maximum in a logarithmic spiral of constant 1, as shown i n the previous section, that is to say, d n rv vv22== ∫∫= =Γ=Γ ed edMvdmvd2 22 2 (10) in which Γ is the auto -torque of the spiral, and M e is the electromagnetic mass of all the conducting electrons contained in the logarithmic spiral. When the cir cuit closing the spiral is not radial, but long and curved, the interaction of the spiral’s current elements with the normal components of the current elements in the rest of the circuit which contributes to the total angular moment (magnetic field) causin g the spiral’s torque to change. When we change the circuit shape which closes the spiral, keeping the current constant, this is equivalent, by analogy with mechanics, to varying the inertia of the conducting electrons in the spiral. When we measure the magnetic field generated by a current that is equivalent to measuring the inertia of the conducting electrons for a specific value of its interaction energy and velocity. The electromagnetic torque of the spiral will allow us to see that gravitational and inert nature of the electromagnetic mass are not independent, as happens with mechanical mass in the General Theory of Relativity in which mass depends on its velocity and its position energy. As a result of this, the conducting spiral constitutes a simple and beautiful example of the intimate relation between mechanics and electromagnetism. 11. Electromagnetic induction and the time rate of change of angular moment. The angular electromagnetic moment generated by the circulation of the charges (just as the a ngular moment of a particle) depends on its angular velocity and its rotational radius L = I · ω (11) Because of this, there are three possible ways of varying the electromagnetic angular moment of a charge. These three ways give rise to the three known forms of electromagnetic induction: a) Variation of the angular velocity magnitude: alternating currents, transformers. b) Variation of the angular velocity direction: Alternating current machines or cyclic machines. c) Variation of the rotation radius: unipolar induction, direct current machines or acyclic machines. Unipolar induction ceases to be an exception and now confirm the new induction law. 12. The normal constraint forces to the conductor and magnetic field. For the moving charges in the conductin g spiral, the only “magnetic field” present is that generated by the charges constraint forces normal to the conductor, which Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 15change the charges’ velocity and produce the electromagnetic angular moment. This suggests there exists an inversion of cause and effect when defining the magnetic force on a current. THE MAGNETIC FORCE ON A CURRENT. “Because a magnetic field exerts a sideways force on a moving charge, it should also exert a sideways force on wire carrying a current.”30 The conducting spiral proves th at the normal constraint forces on the charges, which circulating in the conductor, produce the electromagnetic angular moment and its variation. The variation of the electromagnetic angular moment induces an opposite moment in such a away as to conserve t he electromagnetic angular moment. In this way, the reaction of the normal constraint forces gets transmitted. For this reason, the direction of rotation of the spiral depends on the total moment resulting from all the charges, whether within or outside the spiral. The constraint forces of the charges, as also the centripetal force of a particle, demonstrate the inert nature of a charge. The constraint forces of the charges, which are normal to the conductors, generate the magnetic field, which in turn gene rates the normal forces to de conductors. By analogy to what happens in mechanics: the centripetal forces gives rise to the inertia of the particle, which in turn generates the centrifugal force. The magnetic field (relativistic effect) and magnetic force, is due to the inert property of the charge. The Lorentz equation F = q (E+vxB) expresses the double inert and gravitation nature of the charge. Without the constrain forces to the charges on a conductor it is also impossible to generate and/or measure an electric field. However, these forces are not explicit in Maxwell’s equations. The fact that we are able to define and calculate, for example, the angular moment of a particle without taking into account a centripetal constrain force, does not imply that we can generate or explain the angular moment of a particle without this force. As a result of this, we can’t explain the generation of the angular electromagnetic moment and of the angular moment of matter by the circulation of the charges without taking i nto account the charges constraint forces normal to the conductors. Without them it is not possible to generate a magnetic field or the “mystic circulating flow of energy”31 produced by the circulation of charges. This circulation necessarily implies an ele ctric field normal to the trajectory. This field is produced by some kind of constraint force. Apparently, Rowland’s experiment confirm this. However, the question can be asked whether a reciprocal experiment would be meaningful. Could the rotation of the compass in place of the charged disc detect a magnetic field?. 13. Angular moment variation and mass induction. The analogies between mass and charge in experiments carried out on conducting spirals suggest the existence of a very weak induction of mass or mechanical Lorentz Force. Moreover, General Relativity Theory, which shows that mass depends on velocity and on position energy, should be borne in mind. In a continuous spiral, the velocity of particles circulating in it have two components, radial and norma l respect to the spiral’s center, and their product gives rise to the variation in angular moment of the particles. In the sections 7 and 9 we saw that the logarithmic spiral of constant 1, constitutes the turbine with maximum torque per unit length, both for charge and mass. For a particle of mass moving at constant speed to experience a continuous variation in angular moment in relation to a point, its velocity must have two components, one radial and other normal in relation to this point as happens in a spiral. In the specific case of the logarithmic spiral, when traveled over by a particle with a constant speed, the magnitude of the normal and radial components of the speed are constant throughout the spiral producing a constant variation of the angular moment on the particle in relation to spiral’s center. On the other hand, a set of two charges may undergo a variation of its angular moment in relation to a point if the speed of one of them is radial and the other normal with regard to that point. This suggests that the difference is due to the strong electric force among the charges, compared to a weak gravitational interaction between the masses. Torque or variation of unipolar electromagnetic angular moment depends on the product of the radial and normal currents, which do not need to form part necessarily, of the same circuit. We could generate a small radial current and a large normal or circular current or vice versa, with two independent circuits. There is no parallel for this in classical mechanic s. The logarithmic spiral and the G spiral (Faraday Disc) as turbines symbolize the difference between mass and charge. While in the case of the G spiral, as a hydraulic turbine, the connexion between the radial part and circular part is fundamental in or der to produce the variation in angular moment in the mass at the connexion point, this is irrelevant in the electrodynamic Electromagnetic Induction and the Conservation of Momentum in the Spiral Paradox 16turbine, where the variation in angular moment of the charge occurs all along the radius. According to this analogy, the Foucault Pe ndulum is equivalent to a mechanical Faraday Disc and should describe a small oscillation perpendicular to its plane, which would form an extremely eccentric ellipse. In a future article, some experiments to detect this very weak induction in mass, using m echanical resonance, will be suggested. In spite of the small probability that these experiments be carried out successfully, I believe it is worth trying. I remember how some colleagues made fun of me when I set out to make the spiral revolve forty years ago. A preliminary summary of this study was presented at the First Venezuelan Congress of Physics, Mérida, Venezuela, December 1997. I should like to thank my friend and colleague Gianfranco Spavieri for the useful conversations on the subject of this pa per. I should also like to thank Antonio Albornoz of the faculty laboratory, for his invaluable help in building the experiment apparatus. My thanks also go to my friend Vincent Morley who helped me check the English translation of the text. This work is financed by CDCHT (Commission for Scientific, Humanistic and Technological Development), Universidad de Los Andes, Venezuela. For further information on de subject and the reasons for my article being rejected by Am. J. Phys. I include some commentaries b y the referee and editor together with my letter which remains unanswered. 17 18 19 20The Editor Albert Serra - Valls American Journal of Physics Apartado Postal 630, Merrill Science Building, Room 222 Mérida, 5101 Box 2262 Venezuela. Amherst College e-mail: serra@cantv.net Amherst, Massachusetts 01002 Mérida, December, 13 - 1999 Dear Mr. Romer, I hope you received my letter of Nov. 9th with the manuscript of the new version of my paper, which must have been in the post at the sa me time as your letter of Nov. 10th in which you inform me your rejection of the manuscript I sent on June 30th. It is a pity that my latest manuscript did not reach you in time, for I feel sure that the referee would have viewed it more favorably. Unfortu nately, There are some mistakes in the first manuscript, as at the time I still thought that the two equal and opposite variations of the electromagnetic moment generated by the charges of the current in the closed circuit of the Faraday Disc saved Faraday ’s Induction Law. However, quite on the contrary, as you may see, my experiments with the symmetrical Faraday Disc and the conducting spiral confirm (agreeing with Feynman, Galili and Kaplan) that unipolar induction does not only constitute an exception to Faraday’s Induction Law, but that this law is wrong. On carefully rereading “Electromagnetic Induction in Deformable Circuits” I realize I should not have included Scorgie among those who question the Lorentz force, as the referee quite correctly pointed out. However, the title “Only the integral form of the law of electromagnetic induction explains the dynamo” as well as the text of the brief article published in Am. J. Phys. leads one to believe that Mr. Scorgie has objections with regard to the Lorentz force. I accept that I made some mistakes and consequently had my article rejected. What I cannot accept is the following remark “The author also misses the mark when he attacks Feynman at the bottom of the page 3. His argument there seems to be basically with Figure 4 (which is copied directly from Feynman’s classic textbook)”. Nothing could be further from the truth, for not only do I consider Mr. Feynman to be among the best contemporary physicians and pedagogues but that without his pointing the way, q uite possibly I should never have written my article. I am quite sure that if the referee should read the whole of my article posted on Nov. 9th carefully, he would see this. Precisely the line of thought in this article is based on the coexistence of the angular moment of the electromagnetic field and the angular moment of matter and its conservation, I owe this to Mr. Feynman (Lectures on Physics, Chapter 27 -5) as can be seen in Figure 8 and reference 26. The reason 21for reproducing the Figure of the disc in Feynman’s book (probably not drawn by the author himself) is to show how it differs from the Barlow Wheel and from Faraday’s Disc and rotary magnet, Figure 1, in which the field rotational symmetry axis of the magnetic field of the cylindrical magnet co incides with the rotation axis. In cases where the axes do not coincide, Eddy currents are induced in the discs. Scorgie also mentions this towards the end of his article in the Eur. J. Phys . With the same aim in mind, I mention Arago’s Disc in the introdu ction. For the referee, this pedagogical and historical reference appears estrange and incomprehensible for he writes: “Extraneous material such as the single paragraph on Arago’s disc or the second paragraph on page 4 make the manuscript very hard for any reader to follow.” When I mention “the magnetic moment of the charges”, I have in mind, as the referee suggests, the magnetic moment of the currents. In my most recent manuscript of Nov. 9th I clarify the difference between the magnetic moment of a consta nt current and the electromagnetic angular moment of a current. As may be appreciated in my revised manuscript, I share Feynman, Galili and Kaplan’s opinions that “Faraday’s Law does not account for Faraday’s generator.” So I hope that for holding this opi nion I will not be accused of attacking Faraday who I admire very much. However, in science, as is well known, no one has the last word. I have made a concentrate effort to cut to a minimum diffuse argument and to be as to the point as possible. Yet, intui tion is diffuse and deceptive by nature, while being responsible for every discovery not produced by chance. I consider that the referee is not being consistent when he says, for instance, “Some controversy does surround electromagnetic induction. In recen t AJP publications, Galili & Kaplan argue that that the integral form of Faraday’s law is neither of “explanatory nor of general power”. Scorgie challenges G&K in a short note. I don’t believe G&K either. But that is not the point. The difficulty with the present manuscript is that by diffusing the present situation so badly, the author kills the interest of the reader in understanding the author’s new contributions - his experiment and the spiral paradox.” Apparently, the referee is justifying his not heav ing read my manuscript (the first reading of a new approach is not easy going) and this is shown in his not giving any opinion on the most relevant and important points based on irrefutable experimental facts, namely: a) Unipolar emf and torque are due to th e constant rate of change of the angular moment of the electromagnetic field and matter, which coexist. This constitutes a new analogy between mechanics and electromagnetism. b) The different forms of electromagnetic induction are due to the three possible wa ys of varying the electromagnetic angular moment of the current’s charges; these three ways correspond to the different and “diffuse” ways of varying the magnetic flux mentioned by Scorgie in the Eur. J. Phys. c) The spiral paradox proves that constant unipol ar torque is due to the constant rate of change of the electromagnetic angular moment of the current’s charges. 22d) The conducting spiral proves that unipolar induction is produced by a charges vortex; which is to say that the Faraday Disc constitute an electr odynamic turbine. e) The curious “absolute -relative” duality of unipolar induction, at the origin of controversies, apparently so “diffuse”, becomes clear and manifest in the conducting spiral. The fact that the referee does not mention any of these most spe cific and important points is no compatible with the statement that “The American Journal of Physics seeks to promote discussion of controversies”. The referee does not object to the content of the article or to the experiments. It would seem that his obje ctions are more to the style and perhaps for this reason he does not reject my article outright, to quote from his comments: “ I do not believe the present manuscript is publishable without major revision” I would like to take you up on this and suggest ha ving the opinion of a second referee. In 1969 -1970 an Am. J. Phys. referee twice rejected my manuscript giving as his reason that the conducting spiral was a “mind experiment” which could not really revolve. After a further examination of my article it was published in the Am. J. Phys. (A. Serra - Valls and C. Gago - Bousquet, “Conducting Spiral as an Acyclic or Unipolar Machine”, Am. J. Phys. Vol.38, N.11, pp.1273 -1276, Nov. 1970) It is quite possible that my manuscript might contains some minor error whic h could easily be put right, but it should obvious that this hypothetical possibility should not obscure the results of a long study and much experimental work, no less than that which went into my earlier article which you published and to which my new ar ticle is the continuation, which is why I think my new article should appear in the Am. J. Phys. too. You will excuse my insisting, but had I not done so on the occasion of my first article you would never have published it. Perseverance is certainly a vir tue in science. Sincerely yours, Prof. Albert Serra - Valls Dep. De Física, Fac. de Ciencias Universidad de Los Andes Manuscipt number : 10396 23 1 Henry Poincaré, La Science et L’Hypothès, Flammarion Paris (1968) p. 231. 2 A. Einstein, “On the Electrodynamics of the Moving Bodies” Annalen der Physics, 17, (1905). English translation in The Principle of Relativity. Dover, New York, 1952, pp. 37, 65. 3 A. Serra -Valls, Deuxième Thèse D’Université. Grenoble, (1966). 4 A. Serra -Valls and C. Gago -Bousquet, Conducting Spiral as an Acyclic or unipolar Machine, Am. J. Phys. Vol. 38, (11), pp. 1273 -1276, (1970). 5 Marcel Wellner, “ Reflections on v ×B” Am. J. Phys. Vol. 60, No 9, p. 777 (1992). 6 Robert H. Romer, Editorial: “Magnetic monopoles or cross products? Is physics too difficult? Am. J. Phys. 61 (12), P. 1065, 1993. 7 I. Galili and D. Kaplan, Changing approach to teaching elec tromagnetism in a conceptually oriented introductory physics course. Am. J. Phys. 65 (7), July 1997, p. 664. 8 G. C. Scogie, Only the integral form of law of electromagnetic induction explains the dynamo, Am. J. Phys. 66, (6) p.543, 1998. 9 R Feynman, R.B. Leighton and M. Sands. Feynman Lectures on Physics Vol. II, 17-2, Addison -Wesley (1967) y referencia 7. 10 Dale R. Corson, Electromagnetic Induction in Moving Systems, Am. J. Phys. 24, pp. I26 -130 (1956), p.130. 11 Ref. 8 12 R. Feynman, R.B. Leighton and M. Sands. Feynman Lectures on Physics Vol. II, Addison -Wesley (1967). 13 Ref. 9 14 G.C. Scorgie, Electromagnetic induction in deformable circuits. Eur. J. Phys. 16 (1995) 36 -41. 15 Ref. 10. 16 Ref. 7, 9 17 J. Guala Valverde and P. Mazzoni. The principle of relati vity as applied to motional electromagnetic induction. Am. J. Phys. 63 (3), 228 -229 (1995). 18 Ref. 7, I. Galili and D. Kaplan, Ref. 60, p. 667. 19 L.D. Landau and E. M. Lifshits, Theoretical Physics, Vol. VIII, The Electrodynamics of Continuous Media, Oxfor d Pergamon Press. 1963, p. 209. 20 Ref. 7, p. 664. 21 Ref. 2, p. 55. 22 Gaylord P. Harnwell, Principles of Electricity and Electromagnetism. Ampere’s law, pp. 298 -299. Mc Graw – Hill Book Company, Inc (1949). 23 Ref. 7, I. Galili and D. Kaplan, Ref. 60, p. 667 . 24 Ref. 3. 25 A. Serra -Valls, “Homopolar Locomotive Railway” United States Patent Office, no 3,616,761. 26 Ref. 4. 27 Ref. 10, p.130. 28 R. Feynman, R.B. Leighton, M. Sands. Feynman Lectures on Physics, Vol. II, 28 -2. Addison -Wesley Publishing Company (1967). 29 Ref. 28. 30 David Halliday, Robert Resnick, Kenneth S. Krane. Physics, Part Two, 4th ed., extended version. (1992) John Wiley & Sons, Inc. p 747. 31 “This mystic circulating flow of energy which at first seemed so ridiculous, is absolute necessary. There is really a momentum flow. It is needed to maintain the conservation of angular momentum in the whole world.” R. Feynman, R.B. Leighton M. Sands. Feynman Lectures on Physics Vol. II, 27 -11. Addison -Wesley (1967).
arXiv:physics/0012010v1 [physics.atom-ph] 6 Dec 2000Atomic Energy Levels with QED and Contribution of the Screened Self-Energy ´Eric-Olivier Le Bigot, Paul Indelicato Laboratoire Kastler-Brossel, ´Ecole Normale Sup´ erieure et Universit´ e P. et M. Curie Unit´ e Mixte de Recherche du CNRS n◦C8552, Case 74 4, pl. Jussieu, 75252 Paris CEDEX 05, France We present an introduction to the principles behind atomic e nergy level calculations with Quan- tum Electrodynamics (QED) and the two-time Green’s functio n method; this method allows one to calculate an eﬀective Hamiltonian that contains all QED e ﬀects and that can be used to predict QED Lamb shifts of degenerate, quasidegenerate and isolate d atomic levels. INTRODUCTION This contribution is concerned with the evaluation of atomi c energy levels with QED. Such an evaluation yields stringent tests of QED in strong electric ﬁelds , whereas g-factor experiments and calculations currently probe QED in situations where the magnetic ﬁeld can be treated perturb atively The nuclear Coulomb ﬁeld experienced by the inner levels of highly-charged ions makes the electrons reach relativistic velocities. Such simple physical systems are thus particul arly interesting for testing relativistic eﬀects in quantu m systems (for example, see Refs. [1,2] for experimental resu lts with lithiumlike ions). Theoretical predictions of ene rgy levels in such systems obviously require the use of QED. Experiments have reached an accuracy that shows that extrem ely accurate evaluations of QED eﬀects are also needed in helium . Experiments performed during the last ten years in the spec troscopy of this atom have become two orders of magnitude more precise than the current theoretic al calculations (see for instance Refs. [3,4] and reference s therein). Several experiments are now focusing on helium and heliumli ke ions, and especially their 1 s2p3PJﬁne structure [5–8] such experiments have implications in metrology, as they co uld provide a measurement the ﬁne structure constant and provide checks of theoretical higher-order eﬀects. Very pr ecise theoretical calculations of energy levels in heliuml ike ions can be also important in the investigation of parity violation [9]. Predictions of energy levels are usually more diﬃcult to obt ain for states with one or more open shells (retardation in the interaction and exchange of electrons must be include d, and there can be quasidegenerate levels). Only a few calculations of excited energy levels in heliumlike and lit hiumlike ions have been performed up to now; the ﬁrst results have been published quite recently [10–12]. In regards to QE D shifts of quasidegenerate levels, they have only been obtained this year for the ﬁrst time [11], with the help of the method that we present in this talk. THEORETICAL METHODS As is well known, relativistic electrons orbiting a nucleus are well treated with the Dirac equation , in which the nucleus can be considered as point-like or not. We thus treat the binding to the nucleus non-perturbatively by using “Bound-State QED” [13,14] (the coupling constant of the nuc leus-electron interaction is Zα, which is not small for highly-charged ions). In this formalism, however, QED eﬀec ts are taken into account by treating the electron-electron interaction perturbatively (with coupling constant α), and both the electron and photon ﬁelds are quantum ﬁelds (i.e., insecond -quantized form); the only diﬀerence with the free-ﬁeld cas e used in high-energy physics is that electronic creation and annihilation operators create and destroy ato mic states instead of free particles. A few methods allow one to extract energy levels from the Boun d-State QED Hamiltonian: the two-time Green’s function method [15–17], the method being developed by Lind gren (based on Relativistic Many-Body Perturbation Theory merged with QED) [18,19], the adiabatic S-matrix formalism of Gell-Mann, Low and Sucher [20], and the evolution operator method [21,22]. Some other methods yiel d atomic energy levels, but they include QED eﬀects onlypartly or approximately (such as the multiconﬁguration Dirac-Foc k method [23], conﬁguration interaction calcu- lations [24] and relativistic many-body perturbation theo ry [25]). However, only twomethods can in principle be employed in order to calculate en ergy levels of quasidegenerate atomic states [e.g., the (2 s2p1/2)1and the (2 s2p3/2)1levels in heliumlike ions, which are experimentally import ant]: the two-time Green’s function method and the method being el aborated by Lindgren. We present in this talk a non- technical introduction to the ﬁrst method. The two-time Gre en’s function method has also the advantage of yielding asimpler renormalization procedure than the Gell-Mann–Low–Sucher method in the case of degenerate levels [26,27]. 1THE TWO-TIME GREEN’S FUNCTION METHOD All the methods that extract atomic energy levels from the Bo und-State QED Hamiltonian study the propagation of electrons between two diﬀerent times. The methods diﬀer i n the number of inﬁnite times used: (a) in the Gell-Mann–Low–Sucher method, the atomic state un der consideration evolves from time −∞to time +∞with an adiabatic switching of the interaction; (b) in Lindg ren’s formalism [18,19], the evolution is from time −∞to time 0, which avoids problems associated with the two inﬁn ite times in the S-matrix approach of Gell-Mann– Low–Sucher; (c) in the two-time Green’s function method, th at we present here, the adiabatic switching is completely avoided by studying the propagation of electrons between two ﬁnite times . We note that adiabatic switching of the interactions is physically motivated in the study of collis ions between particles that start very far from each other, but this switching is not so easily related to the physical de scription of the orbiting electrons of an atom. The Green’s function Time t Time t' Probability amplitude? FIG. 1. The two-time Green’s function represents the probab ility amplitude for going from one position of the electrons to another position. Theeﬀective Hamiltonian derived from QED by the two-time Green’s function method has matrix elements be- tween the various degenerate and/or quasidegenerate state s under study; the eigenvalues of this Hamiltonian are the atomic energy levels predicted by QED (to a given order). Thi s eﬀective Hamiltonian is however notassociated to a Schr¨ odinger equation of motion; our Hamiltonian is equiva lent to the submatrix used in the perturbation theory of degenerate and quasidegenerate states; in this respect, th e approach of the two-time Green’s function method diﬀers from the spirit of the Bethe-Salpeter equation. The QED Hamiltonian of the method is deﬁned with the help of a G reen’s function that represents the propagation ofNelectrons between two diﬀerent (ﬁnite) times ( Nis the number of electrons of the atom or ion that we want to study); this propagation is represented in Fig. 1. Atomic energies are in the Green’s function 1s21s, 2s\|gN=2(E)\|Poles at the energy levels of a 2-electron ion Energy FIG. 2. The two-particle Green’s function as a function of energy contains all the information about the atomic energy levels of a two-electron atom or ion. 2The energy levels of an N-electron ion or atom can be recovered by studying the energy representation GN(E) of the Green’s function, i.e., by doing a Fourier transform: this function has (simple) poles at the atomic energy levels [15–17]. Such a result is similar to the K¨ all´ en-Lehmann re presentation [28]. As an example, Fig. 2 depicts the poles of the two-particle Green’s function. The two-time Green’s function method provides a way of mathematically extracting from the Green’s function the positions of the poles, i.e., the atomic energy levels [17]; the procedure handles degenerate and quasidegenerate atom ic levels without any special diﬃculty [29]. One of the basic id eas behind the pole extraction is found in the following mathematical device, which uses anycontour Γ 0that encloses the pole in order to ﬁnd its exact position: if t he function g(E) has a simple pole at E=E0, then we have from complex analysis E0=/contintegraldisplay Γ0dE E×g(E) /contintegraldisplay Γ0dE g(E); (1) the contour Γ 0is only required to encircle the pole and to be positively ori ented, as shown in Fig. 3. Since the Green’s function has simple poles at the atomic energy levels [17], E q. (1) is a way of obtaining them. Pole of g(E) Γ0Complex energyE0 FIG. 3. The exact atomic energies can be recovered through a c ontour integration of the Green’s function. When QED shifts of degenerate of quasidegenerate levels are calculated, the scalar Green’s function gof Eq. (1) is simply replaced by a ﬁnite-size matrix deﬁned on the space of levels under consideration [29]. GRAPHICAL CALCULATIONS Obviously, analytic properties of the Green’s function [27] are important in the evaluation of Eq. (1). We have developed a set of graphical techniques that allow one to obtain the Laurent series of the Green’s function GN(E) by asystematic procedure. The idea behind these techniques consists in dis playing the analytic structure of the Green’s function step by step; each step explicitly extracts onesingularity, and we proceed until we have exhausted all the singularities of the Green’s function; at this point, conto ur integrals such as Eq. (1) can be calculated quite simply. It is impossible to give here a full account of the method we us e for deriving the eﬀective, ﬁnite-size QED Hamilto- nian. However, we can mention a particular feature of our cal culational strategy: a very special “particle” appears in our algorithm; this particle is quite simple since it “disin tegrates” immediately (zero life time) and cannot move (zer o probability for going from one position to a diﬀerent one). I n mathematical terms, the coordinate-space propagator of this particle is a four-dimensional Delta function δ(4)[(/vector x, t); (/vectorx′, t′)] that we represent by a special line in Feynman diagrams. THE SCREENED SELF-ENERGY The experimental accuracy on transition energies is so high that second-order (i.e., two-photon) eﬀects must be taken into account in order to compare experiments with theory. We thus have very recently calculated the contribution of 3the self-energy screening [30] to the QED eﬀective hamilton ian; this contribution corresponds to the following physic al processes:/A1 /A1 /A1 . Our result is part of the current theoretical eﬀort develope d with the aim of matching experimental precisions. [1] Beiersdorfer, P., Osterheld, A. L., Scoﬁeld, J. H., L´ op ez-Urrutia, J. R. C., and Widmann, K., Phys. Rev. Lett. 80, 3022–3025 (1998). [2] Schweppe, J., Belkacem, A., Blumenfeld, L., Claytor, N. , Feynberg, B., Gould, H., Kostroun, V., Levy, L., Misawa, S. , Mowat, R., and Prior, M., Phys. Rev. Lett. 66, 1434–1437 (1991). [3] Drake, G. W. F. and Martin, W. C., Can. J. Phys. 76, 679–698 (1998). [4] Drake, G. W. F. and Goldman, S. P., Can. J. Phys. 77, 835–845 (2000). [5] Minardi, F., Bianchini, G., Pastor, P. C., Giusfredi, G. , Pavone, F. S., and Inguscio, M., Phys. Rev. Lett. 82, 1112–1115 (1999). [6] Storry, C. H., George, M. C., and Hessels, E. A., Phys. Rev. Lett. 84, 3274–3277 (2000). [7] Castillega, J., Livingston, D., Sanders, A., and Shiner , D.,Phys. Rev. Lett. 84, 4321–4324 (2000). [8] Myers, E. G. and Tarbutt, M. R., Phys. Rev. A 61, 010501(R) (2000). [9] Maul, M., Sch¨ afer, A., Greiner, W., and Indelicato, P., Phys. Rev. A 53, 3915–3925 (1996). [10] Artemyev, A. N., Beier, T., Plunien, G., Shabaev, V. M., Soﬀ, G., and Yerokhin, V. A., Phys. Rev. A 60(1), 45 (1999). [11] Artemyev, A. N., Beier, T., Plunien, G., Shabaev, V. M., Soﬀ, G., and Yerokhin, V. A., Phys. Rev. A 62, 022116 (2000). [12] Mohr, P. J. and Sapirstein, J., Phys. Rev. A 62, 052501 (2000). [13] Furry, W. H., Phys. Rev. A 81, 115–124 (1951). [14] Mohr, P. J., in Physics of Highly-ionized Atoms , edited by Marrus, R., Plenum, New York, 1989, pages 111–141 . [15] Shabaev, V. M. and Fokeeva, I. G., Phys. Rev. A 49, 4489–4501 (1994). [16] Shabaev, V. M., Phys. Rev. A 50(6), 4521–4534 (1994). [17] Shabaev, V. M., “Two-time Green function method in quan tum electrodynamics of high- Zfew-electron atoms”, arXiv:physics/0009018, 2000. [18] Lindgren, I., Mol. Phys. 98, 1159–1174 (2000). [19] Lindgren, I., see contribution in this edition. [20] Sucher, J., Phys. Rev. 107(5), 1448–1449 (1957). [21] Vasil’ev, A. N. and Kitanin, A. L., Theor. Math. Phys. 24(2), 786–793 (1975). [22] Zapryagaev, S. A., Manakov, N. L., and Pal’chikov, V. G. ,Theory of One- and Two-Electron Multicharged Ions , Ener- goatomizdat, Moscow, 1985, in Russian. [23] Indelicato, P. and Desclaux, J. P., Phys. Rev. A 42, 5139–5149 (1990). [24] Cheng, K. T. and Chen, M. H., Phys. Rev. A 61(4), 044503/1–4 (2000). [25] Ynnerman, A., James, J., Lindgren, I., Persson, H., and Salomonson, S., Phys. Rev. A 50, 4671–4677 (1994). [26] Braun, M. A. and Gurchumeliya, A. D., Theor. Math. Phys. 45(2), 975–982 (1980), Translated from Teoret. Mat. Fiz. 45, 199 (1980). [27] Braun, M. A., Gurchumelia, A. D., and Safronova, U. I., Relativistic Atom Theory , Nauka, Moscow, 1984, in Russian. [28] Peskin, M. E. and Schroeder, D. V., An introduction to quantum ﬁeld theory , Addison-Wesley, Reading, Massachusetts, 1995. [29] Shabaev, V. M., J. Phys. B 26, 4703–4718 (1993). [30] Le Bigot, E.-O., Indelicato, P., and Shabaev, V. M., “Co ntribution of the screened self-energy to the Lamb shift of quasidegenerate states”, arXiv:physics/0011037, 2000. 4
arXiv:physics/0012011v1 [physics.class-ph] 6 Dec 2000SPACE-TIME EXCHANGE INVARIANCE: SPECIAL RELATIVITY AS A SYMMETRY PRINCIPLE J.H.Field D´ epartement de Physique Nucl´ eaire et Corpusculaire Univ ersit´ e de Gen` eve . 24, quai Ernest-Ansermet CH-1211 Gen` eve 4. Abstract Special relativity is reformulated as a symmetry property o f space-time: Space- Time Exchange Invariance. The additional hypothesis of spa tial homogeneity is then suﬃcient to derive the Lorentz transformation without reference to the tradi- tional form of the Principle of Special Relativity. The kine matical version of the latter is shown to be a consequence of the Lorentz transforma tion. As a dynamical application, the laws of electrodynamics and magnetodynam ics are derived from those of electrostatics and magnetostatics respectively. The 4-vector nature of the electromagnetic potential plays a crucial role in the last t wo derivations. To be published in American Journal of Physics.1 Introduction Two postulates were essential for Einstein’s original axio matic derivation [1] of the Lorentz transformation (LT) : (i) the Special Relativity Principle and (ii) the hypothesis of the constancy of the velocity of light in all inertial frames (Ei nstein’s second postulate). The Special Relativity Principle, which states that: ‘The laws of physics are the same in all inertial frames’ had long been known to be respected by Newton’s laws of mechan ics at the time Einstein’s paper was written. Galileo had already stated th e principle in 1588 in his ‘Dialogues Concerning Two New Sciences’. The title of Einst ein’s paper [1] ‘On the Electrodynamics of Moving Bodies’ and the special role of li ght in his second postulate seem to link special relativity closely to classical electr odynamics. Indeed, the LT was discovered as the transformation that demonstrates that Ma xwell’s equations may be written in the same way in any inertial frame, and so manifest ly respect the Special Relativity Principle. The same close connection between sp ecial relativity and classical electrodynamics is retained in virtually all text-book tre atments of the subject, obscuring the essentially geometrical and kinematical nature of spec ial relativistic eﬀects. The latter actually transcend the dynamics of any particular physical system. It was realised, shortly after the space-time geometrical nature of the LT was pointe d out by Minkowski [2], that the domain of applicability of the LT extends beyond the classical electrodynamics considered by Einstein, and that, in fact, Einstein’s secon d postulate is not necessary for its derivation [3, 4]. There is now a vast literature devoted to derivations of the LT that do not require the second postulate [5]. In a recent paper by the present author [6], the question of th e minimum number of postulates, in addition to the Special Relativity Princi ple, necessary to derive the LT was addressed. The aim of the present paper is somewhat diﬀ erent. The Special Relativity Principle itself is re-stated in a simple mathem atical form which, as will be shown below, has both kinematical and dynamical applicatio ns. The new statement is a symmetry condition relating space and time, which, it is con jectured, is respected by the mathematical equations that decscribe all physical laws [7 ]. The symmetry condition is ﬁrst used, together with the postulate of the homogeneity of space, to derive the LT. It is then shown that the Kinematical Special Relativity Princ iple (KSRP) is a necessary consequence of the LT. The KSRP, which describes the reciprocal nature of similar space time measurements made in two diﬀerent inertial frames [8], states that: ‘Reciprocal space-time measurements of similar measuring rods and clocks at rest in two diﬀerent inertial frames S,S′by observers at rest in S′,S respectively, yield identical results’ There is no reference here to any physical law. Only space-ti me events that may con- stitute the raw material of any observation of a physical pro cess are considered. In the previous literature the KSRP (or some equivalent condition applied to a gedankenexper- iment [9]) has been been used as a necessary postulate to deri ve the LT. The symmetry condition that restates the Special Relativit y Principle is: 1(I)‘The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more gener ally, to the exchange of the spatial and temporal components of four vect ors.’ A corollary is: (II)‘Predictions of physical theories do not depend on the metri c sign conven- tion (space-like or time-like) used to deﬁne four-vector sc alar products.’ A proof of this corollary is presented in Section 4 below. As will become clear during the following discussion, the op eration of Space-Time Exchange ( STE) reveals an invariance property of pairs of physical equati ons, which are found to map into each other under STE. The examples of this discussed below are: the Lorentz transformation equations of space and time, the Max well equations describing electrostatics (Gauss’ law) and electrodynamics (Amp` ere ’s law), and those describing magnetostatics (Gauss’ law) and magnetodynamics (The Fara day-Lenz law). It will be demonstrated that each of these three pairs of equations map into each other under STE, and so are invariants of the STE operator. In the case of the LT equations, imposing STEsymmetry is suﬃcient to derive them from a general form of the space transformation equation that respects the classical limit. The expression: ‘The equations describing the laws of physi cs’ in (I) should then be understood as including both equations of each STE invariant pair. For example, the Gauss equation of electrostatics, considered as an indepen dent physical law, clearly does not respect (I). For dimensional reasons, the deﬁnition of the exchange oper ation referred to in (I) requires the time coordinate to be multiplied by a universal parameter Vwith the dimen- sions of velocity. The new time coordinate with dimension[ L]: x0≡V t (1.1) may be called the ‘causality radius’ [10] to distinguish it f rom the cartesian spatial co- ordinate xor the invariant interval s. Since space is three dimensional and time is one dimensional, there is a certain ambiguity in the deﬁnition o f the exchange operation in (I). Depending on the case under discussion, the space coord inate may be either the mag- nitude of the spatial vector x=\|/vector x\|, or a cartesian component x1,x2,x3. For any physical problem with a preferred spatial direction (which is the cas e for the LT), then, by a suit- able choice of coordinate system, the identiﬁcation x=x1,x2=x3= 0 is always possible. The exchange operation in (I) is then simply x0↔x1. Formally, the exchange operation is deﬁned by the equations: STEx0=x1(1.2) STEx1=x0(1.3) (STE)2= 1 (1.4) where STE denotes the space time exchange operator. As shown below, fo r problems where there is no preferred direction, but rather spatial sy mmetry, it may also be useful 2to deﬁne three exchange operators: x0↔xii= 1,2,3 (1.5) with associated operations STE(i) analagous to STE=STE(1) in Eqns.(1.2)-(1.4). The operations in Eqns.(1.2) to (1.5) may also be generalised to the case of an arbitary 4-vector with temporal and spatial components A0andA1respectively. To clarify the meaning of the STE operation, it is of interest to compare it with a diﬀerent operator acting on space and time coordinates that may be called ‘Space-Time Coordinate Permutation’ ( STCP ). Consider an equation of the form: f(x0, x1) = 0. (1.6) TheSTE conjugate equation is: f(x1, x0) = 0. (1.7) This equation is diﬀerent from (1.6) because x0andx1have diﬀerent physical meanings. In the STCP operation however, the values of the space and time coordinates are inter- changed, but no new equation is generated. If x0=aandx1=bin Eqn.(1.6) then the STCP operation applied to the latter yields: f(x0=b, x1=a) = 0. (1.8) This equation is identical in form to (1.6); only its paramet ers have diﬀerent values. The physical meaning of the universal parameter V, and its relation to the velocity of light, c, is discussed in the following Section, after the derivatio n of the LT. The plan of the paper is as follows. In the following Section t he LT is derived. In Section 3, the LT is used to derive the KSRP. The space time exc hange properties of 4-vectors and the related symmetries in Minkowski space are discussed in Section 4. In Section 5 the space-time exchange symmetries of Maxwell’s e quations are used to derive electrodynamics (Amp` ere’s law) and magnetodynamics (the Faraday-Lenz law) from the Gauss laws of electrostatics and magnetostatics respectiv ely. A summary is given in Section 6. 2 Derivation of the Lorentz Transformation Consider two inertial frames S,S′.S′moves along the common x, x′axis of orthogonal cartesian coordinate systems in S,S′with velocity vrelative to S. The y, y′axes are also parallel. At time t=t′= 0 the origins of SandS′coincide. In general the transformation equation between the coordinate xinSof a ﬁxed point on the Ox′axis and the coordinate x′of the same point referred to the frame S′is : x′=f(x, x0, β) (2.1) where β≡v/VandVis the universal constant introduced in Eqn.(1.1). Diﬀeren tiating Eqn.(2.1) with respect to x0, for ﬁxed x′, gives: dx′ dx0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′= 0 =dx dx0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′∂f ∂x+∂f ∂x0(2.2) 3Since dx dx0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′=1 Vdx dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′=v V=β the function fmust satisfy the partial diﬀerential equation: β∂f ∂x=−∂f ∂x0(2.3) A suﬃcient condition for fto be a solution of Eqn.(2.3) is that it is a function of x−βx0. Assuming also fis a diﬀerentiable function, it may be expanded in a Taylor se ries: x′=γ(β)(x−βx0) +∞/summationdisplay n=2an(β)(x−βx0)n(2.4) Requiring either spatial homogeneity [11, 12, 13], or that t he LT is a unique, single valued, function of its arguments [6], requires Eqn.(2.4) to be line ar, i.e. a2(β) =a3(β) =. . .= 0 so that x′=γ(β)(x−βx0) (2.5) Spatial homogeneity implies that Eqn(2.5) is invariant whe n all spatial coordinates are scaled by any constant factor K. Noting that : −β=−1 Vdx dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′=1 Vd(−x) dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x′(2.6) and choosing K=−1 gives : −x′=γ(−β)(−x+βx0) (2.7) Hence, Eqn.(2.5) is invariant provided that γ(−β) =γ(β) (2.8) i.e.γ(β) is an even function of β. Applying the space time exchange operations x↔x0,x′↔(x0)′to Eqn.(2.5) gives (x0)′=γ(β)(x0−βx) (2.9) The transformation inverse to (2.9) may, in general, be writ ten as: x0=γ(β′)((x0)′−β′x′) (2.10) The same inverse transformation may also be derived by elimi nating xbetween Eqns.(2.5) and (2.9) and re-arranging: x0=1 γ(β)(1−β2)((x0)′+βx′) (2.11) 4Eqns(2.10),(2.11) are consistent provided that : γ(β′) =1 γ(β)(1−β2)(2.12) and β′=−β (2.13) Eqns.(2.8),(2.12) and (2.13) then give [14]: γ(β) =1√1−β2(2.14) Eqns.(2.5),(2.9) with γgiven by (2.14) are the LT equations for space-time points al ong the common x, x′axis of the frames S,S′. They have been derived here solely from the symmetry condition (I) and the assumption of spatial homoge neity, without any reference to the Principle of Special Relativity. The physical meaning of the universal parameter Vbecomes clear when the kinemat- ical consequences of the LT for physical objects are worked o ut in detail. This is done, for example, in Reference [6], where it is shown that the velo city of any massive physical object approaches Vin any inertial frame in which its energy is much greater than its rest mass. The identiﬁcation of Vwith the velocity of light, c, then follows [13, 6] if it is assumed that light consists of massless (or almost massle ss) particles, the light quanta discovered by Einstein in his analysis of the photoelectric eﬀect [15]. That Vis the lim- iting velocity for the applicability of the LT equations is, however, already evident from Eqn.(2.14). If γ(β) is real then β≤1, that is v≤V. 3 Derivation of the Kinematical Special Relativity Principle The LT equations (2.5) and (2.9) and their inverses, written in terms of x, x′;t, t′are: x′=γ(x−vt) (3.1) t′=γ(t−vx V2) (3.2) x=γ(x′+vt′) (3.3) t=γ(t′+vx′ V2) (3.4) Consider now observers, at rest in the frames S,S′, equipped with identical measuring rods and clocks. The observer in S′places a rod, of length l, along the common x, x′ axis. The coordinates in S′of the ends of the rod are x′ 1, x′ 2where x′ 2−x′ 1=l. If the observer in Smeasures, at time tin his own frame, the ends of the rod to be at x1, x2 then, according to Eqn(3.1): x′ 1=γ(x1−vt) (3.5) x′ 2=γ(x2−vt) (3.6) 5Denoting by lSthe apparent length of the rod, as observed from Sat time t, Eqns.(3.5),(3.6) give lS≡x2−x1=1 γ(x′ 1−x′ 2) =l γ(3.7) Suppose that the observer in S′now makes reciprocal measurements x′ 1, x′ 2of the ends of a similar rod, at rest in S, at time t′. InSthe ends of the rod are at the points x1, x2, where l=x2−x1. Using Eqn.(3.3) x1=γ(x′ 1+vt′) (3.8) x2=γ(x′ 2+vt′) (3.9) and, corresponding to (3.7), there is the relation: lS′≡x′ 2−x′ 1=1 γ(x2−x1) =l γ(3.10) Hence, from Eqns.(3.7),(3.10) lS=lS′=l γ(3.11) so that reciprocal length measurements yield identical res ults. Consider now a clock at rest in S′atx′= 0. This clock is synchronized with a similar clock in Satt=t′= 0, when the spatial coordinate systems in SandS′coincide. Suppose that the observer at rest in Snotes the time trecorded by his own clock, when the moving clock records the time τ. At this time, the clock which is moving along the common x, x′axis with velocity vwill be situated at x=vt. With the deﬁnition τS≡t, and using Eqn.(3.2) : τ=γ(τS−vx V2) =γτS(1−v2 V2) =τS γ(3.12) If the observer at rest in S′makes a reciprocal measurement of the clock at rest in S, which is seen to be at x′=−vt′when it shows the time τ, then according to Eqn.(3.4) withτS′≡t′: τ=γ(τS′+vx′ V2) =γτS′(1−v2 V2) =τS′ γ(3.13) Eqns.(3.12),(3.13) give τS=τS′=γτ (3.14) Eqns.(3.11),(3.14) prove the Kinematical Special Relativ ity Principle as stated above. It is a necessary consequence of the LT. 4 General Space Time Exchange Symmetry Proper- ties of 4-Vectors. Symmetries of Minkowski Space The LT was derived above for space time points lying along the common x, x′axis, so thatx=\|/vector x\|. However, this restriction is not necessary. In the case tha t/vector x= (x1, x2, x3) thenxandx′in Eqn.(2.5) may be replaced by x=/vector x·/vector v/\|/vector v\|andx′=/vectorx′·/vector v/\|/vector v\|respectively, 6where the 1-axis is chosen parallel to /vector v. The proof proceeds as before with the space time exchange operation deﬁned as in Eqns.(1.2)-(1.4). The additional transformation equations : y′=y (4.1) z′=z (4.2) follow from spatial isotropy [1]. In the above derivation of the LT, application of the STE operator generates the LT of time from that of space. It is the pair of equations that is i nvariant with respect to the STE operation. Alternatively, as shown below, by a suitable cha nge of variables, equiva- lent equations may be deﬁned that are manifestly invariant u nder the STE operation. The 4-vector velocity Uand the energy-momentum 4-vector Pare deﬁned in terms of the space-time 4-vector [2]: X≡(V t;x, y, z ) = (x0;x1, x2, x3) (4.3) by the equations: U≡dX dτ(4.4) P≡mv (4.5) where mis the Newtonian mass of the physical object and τis its proper time, i.e. the time in a reference frame in which the object is at rest. Since τis a Lorentz invariant quantity, the 4-vectors U, Phave identical LT properties to X. The properties of U, P under the STE operation follow directly from Eqns.(1.2),(1.3) and the de ﬁnitions (4.4) and (4.5). Writing the energy-momentum 4-vector as: P= (E V;p,0,0) = ( p0;p1,0,0) (4.6) theSTE operations: p0↔p1, (p0)′↔(p1)′generate the LT equation for energy: (p0)′=γ(p0−βp1) (4.7) from that of momentum (p1)′=γ(p1−βp0) (4.8) orvice versa . The scalar product of two arbitary 4-vectors C,D: C·D≡C0D0−/vectorC·/vectorD (4.9) can, by choosing the x-axis parallel to /vectorCor/vectorD, always be written as: C·D=C0D0−C1D1(4.10) Deﬁning the STEexchange operation for an arbitary 4-vector in a similar way to Eqns.(1.2),(1.3) then the combined operations C0↔C1,D0↔D1yield: C·D→C1D1−C0D0=−C·D (4.11) 7The 4-vector product changes sign, and so the combined STE operation is equivalent to a change in the sign convention of the metric from space-like to time-like (or vice versa ), hence the corollary (II) in Section 1 above. The LT equations take a particularly simple form if new varia bles are deﬁned which have simple transformation properties under the STE operation. The variables are: x+=x0+x1 √ 2(4.12) x−=x0−x1 √ 2(4.13) x+,x−have, respectively, even and odd ‘ STE parity’: STEx +=x+ (4.14) STEx −=−x− (4.15) The manifestly STE invariant LT equations expressed in terms of these variable s are: x′ +=αx+ (4.16) x′ −=1 αx− (4.17) where α=/radicalBigg 1−β 1 +β(4.18) Introducing similar variables for an arbitary 4-vector: C+=C0+C1 √ 2(4.19) C−=C0−C1 √ 2(4.20) the 4-vector scalar product of CandDmay be written as: C·D=C+D−+C−D+ (4.21) In view of the LT equations (4.16),(4.17) C·Dis manifestly Lorentz invariant. The transformations (4.12),(4.13) and (4.19),(4.20) corresp ond to an anti-clockwise rotation by 45◦of the axes of the usual ctversus xplot. The x+,x−axes lie along the light cones of the x-ctplot (see Fig.1). The LT equations (4.16),(4.17) give a parametric represent ation of a hyperbola in x+,x−space. A point on the latter corresponds to a particular spac e-time point as viewed in a frame S. The point x+=x−= 0 corresponds to the space-time origin of the frame S′moving with velocity βcrelative to S. A point at the spatial origin of S′at time t′=τ will be seen by an observer in S, asβ(and hence α) varies, to lie on one of the hyperbolae H++,H−−in Fig.1: x+x−=c2τ2 2(4.22) 8ELSEWHERE FUTURE (s < 0) ( τ > 0) PAST ELSEWHERE (s > 0) (τ < 0)x- x+H++ H-+ H+- H--Q- P+x0 = ct P- Q+cτ √2cτ √2, s √2-s √2,4 2 -2 -4-4 -2 2 4 x Figure 1: Space-time points in S’ as seen by an observer in S. T he hyperbolae H++, H−−correspond to points at the origin of S’ at time t′=τ. The hyperbolae H+−,H−+ correspond to points at x′=sandt′= 0. See the text for the equations of the hyperbolae and further discussion. 9withx+, x−>0 ifτ >0 (H++) orx+, x−<0 ifτ <0 (H−−). A point along the x′axis at a distance sfrom the origin, at t′= 0 lies on the hyperbolae H+−,H−+: x+x−=−s2 2(4.23) withx+>0,x−<0 ifs >0 (H+−) orx+<0,x−>0 ifs <0 (H−+). As indicated in Fig.1 the hyperbolae (4.22) correspond to the past ( τ <0) or the future ( τ >0) of a space time point at the origin of SorS′, whereas (4.23) corresponds to the ‘elsewhere’ of the same space-time points. That is, the manifold of all space-t ime points that are causally disconnected from them. These are all familiar properties o f the Minkowski space x-ct plot. One may note, however, the simplicity of the equations (4.16),(4.17),(4.22), (4.23) containing the ‘lightcone’ variables x+, x−that have simple transformation properties under the STE operation. Another application of STE symmetry may be found in [16]. It is shown there that the apparent distortions of space-time that occur in observ ations of moving bodies or clocks are related by this symmetry. For example, the Lorent z-Fitzgerald contraction is directly related to Time Dilatation by the STE operations (1.2) and (1.3). 5 Dynamical Applications of Space Time Exchange Symmetry If a physical quantity is written in a manifestly covariant w ay, as a function of 4-vector products, it will evidently be invariant with respect to STE as the exchange operation has the eﬀect only of changing the sign convention for 4-vect or products from space-like to time-like or vice-versa . An example of such a quantity is the invariant amplitude M for an arbitary scattering process in Quantum Field Theory. In this case STE invariance is equivalent to Corollary II of Section 1 above. More interesting results can be obtained from equations whe re components of 4-vectors appear directly. It will now be shown how STEinvariance may be used to derive Amp` ere’s law and Maxwell’s ‘displacement current’ from the Gauss law of electrostatics, and the Faraday-Lenz law of magnetic induction from the the Gauss la w of magnetostatics (the absence of magnetic charges). Thus electrodynamics and mag netodynamics follow from the laws of electrostatics and magnetostatics, together wi th space time exchange symmetry invariance. It will be seen that the 4-vector character of th e electromagnetic potential plays a crucial role in these derivations. In the following, Maxwell’s equations are written in Heavis ide-Lorentz units with V= c= 1 [17]. The 4-vector potential A= (A0;/vectorA) is related to the electromagnetic ﬁeld tensor Fµνby the equation: Fµν=∂µAν−∂νAµ(5.1) where ∂µ≡(∂ ∂t;−/vector∇) = (∂0;−/vector∇) (5.2) 10The electric and magnetic ﬁeld components Ek,Bkrespectively, are given, in terms of Fµν, by the equations: Ek=Fk0(5.3) Bk=−ǫijkFij(5.4) A time-like metric is used with Ct=C0=C0,Cx=C1=−C1etc, with summation over repeated contravariant (upper) and covariant (lower) indi ces understood. Repeated greek indices are summed form 1 to 4 and roman ones from 1 to 3. The transformation properties of contravariant and covari ant 4-vectors under the STE operation are now discussed. They are derived from the gener al condition that 4-vector products change sign under the STE operation (Eqn.(4.11)). The 4-vector product (4.9) is written, in terms of contravariant and covariant 4-vecto rs, as: C·D=C0D0+C1D1 (5.5) Assuming that the contravariant 4-vector Cµtransforms according to Eqns.(1.2) (1.3), i.e. C0↔C1(5.6) the covariant 4-vector Dµmust transform as: D0↔ −D1 (5.7) in order to respect the transformation property C·D→ −C·D (5.8) of 4-vector products under STE. It remains to discuss the STEtransformation properties of ∂µand the 4-vector poten- tialAµ. In view of the property of ∂µ:∂1=−∂x=−∂/∂x (Eqn.(5.2)), which is similar to the relation C1=−Cxfor acovariant 4-vector, it is natural to choose for ∂µanSTE transformation similar to Eqn.(5.7): ∂0↔ −∂1(5.9) and hence, in order that ∂µ∂µchange sign under STE: ∂0↔∂1 (5.10) This is because it is clear that the appearence of a minus sign in the STE transfor- mation equation (5.7) is correlated to the minus sign in fron t of the spatial components of a covariant 4-vector, not whether the Lorentz index is an u pper or lower one. Thus ∂µ and∂µtransform in an ‘anomalous’ manner under STE as compared to the convention of Eqns.(5.6) and (5.7). In order that the 4-vector product ∂µAµrespect the condition (5.8), AµandAµmust then transform under STE as: A0↔ −A1(5.11) 11and A0↔A1 (5.12) respectively. That is, they transform in the same way as ∂µand∂µrespectively. Introducing the 4-vector electromagnetic current jµ≡(ρ;/vectorj), Gauss’ law of electro- statics may be written as: /vector∇ ·/vectorE=ρ=j0(5.13) or, in the manifestly covariant form: (∂µ∂µ)A0−∂0(∂µAµ) =j0(5.14) This equation is obtained by writing Eqn.(5.13) in covarian t notation using Eqns.(5.1) and (5.3) and adding to the left side the identity: ∂0(∂0A0−∂0A0) = 0 (5.15) Applying the space-time exchange operation to Eqn.(5.14), with index exchange 0 →1 (noting that ∂0,A0transform according to Eqns(5.9),(5.11), j0according to (5.6), and that the scalar products ∂µ∂µand∂µAµchange sign) yields the equation: (∂µ∂µ)A1−∂1(∂µAµ) =j1(5.16) The spatial part of the 4-vector products on the left side of E qn.(5.16) is: ∂i(∂iA1−∂1Ai) = ∂iFi1 =∂2B3−∂3B2 = (/vector∇ ×/vectorB)1(5.17) where Eqns.(5.1) and (5.4) have been used. The time part of th e 4-vector products in Eqn(5.16) yields, with Eqns.(5.1) and (5.3): ∂0(∂0A1−∂1A0) =−∂E1 ∂t(5.18) Combining Eqns(5.16)-(5.18) gives: (/vector∇ ×/vectorB)1−∂E1 ∂t=j1(5.19) Combining Eqn.(5.19) with the two similar equations derive d derived by the index ex- changes 0 →2, 0→3 in Eqn.(5.14) gives: (/vector∇ ×/vectorB)−∂/vectorE ∂t=/vectorj (5.20) This is Amp` ere’s law, together with Maxwell’s displacemen t current. The Faraday-Lenz law is now derived by applying the space-ti me exchange operation to the Gauss law of magnetostatics: /vector∇ ·/vectorB= 0 (5.21) 12Introducing Eqns.(5.4) and (5.1) into Eqn.(5.21) gives: ∂1(∂3A2−∂2A3) +∂2(∂1A3−∂3A1) +∂3(∂2A1−∂1A2) = 0 (5.22) Making the exchange 1 →0 of space-time indices in Eqn.(5.22) and noting that ∂1trans- forms according to Eqn.(5.10), whereas ∂1,A1transform as in Eqns.(5.9),(5.11) respec- tively, gives: ∂0(∂3A2−∂2A3) +∂2(−∂0A3+∂3A0) +∂3(−∂2A0−∂0A2) = 0 (5.23) Using Eqns.(5.1)-(5.4), Eqn.(5.23) may be written as: ∂B1 ∂t+∂2E3−∂3E2= 0 (5.24) or, in 3-vector notation: (/vector∇ ×/vectorE)1=−∂B1 ∂t(5.25) The space-time exchanges 2 →0, 3→0 in Eqn.(5.22) yield, in a similar manner, the 2 and 3 components of the Faraday-Lenz law: (/vector∇ ×/vectorE) =−∂/vectorB ∂t(5.26) Some comments now on the conditions for the validity of the ab ove derivations. It is essential to use the manifestly covariant form of the electr ostatic Gauss law Eqn.(5.14) and the manifestly rotationally invariant form, Eqn.(5.22 ), of the magnetostatic Gauss law. For example, the 1-axis may be chosen parallel to the ele ctric ﬁeld in Eqn.(5.13). In this case Eqn.(5.14) simpliﬁes to ∂1(∂0A1−∂1A0) =j0(5.27) Applying the space-time exchange operation 0 ↔1 to this equation yields only the Maxwell displacement current term in Eqn.(5.19). Similarl y, choosing the 1-axis parallel to/vectorBin Eqn.(5.21) simpliﬁes Eqn.(5.22) to ∂1(∂3A2−∂2A3) = 0 (5.28) The index exchange 1 →0 leads then to the equation: ∂B1 ∂t= 0 (5.29) instead of the 1-component of the Faraday-Lenz law, as in Eqn .(5.24). The choice of the STE transformation properties of contravariant and covariant 4- vectors according to Eqns.(5.6) and (5.7) is an arbitary one . Identical results are obtained if the opposite convention is used. However, ‘anomalous’ tr ansformation properties of ∂µ, ∂µandAµ,Aµ, in the sense described above, are essential. This complica tion results from the upper index on the left side of Eqn.(5.2) whereas on the ri ght side the spatial derivative is multiplied by a minus sign. This minus sign changes the STE transformation property relative to that, (5.6), of conventional contravariant 4-v ectors, that do not have a minus sign multiplying the spatial components. The upper index on the left side of Eqn.(5.2) is a consequence of the Lorentz transformation properties of th e four dimensional space-time derivative [18]. 136 Summary and Discussion In this paper the Lorentz transformation for points lying al ong the common x,x′axis of two inertial frames has been derived from only two postulate s: (i) the symmetry principle (I), and (ii) the homogeneity of space. This is the same numbe r of axioms as used in Ref.[6] where the postulates were: the Kinematical Special Relativity Postulate and the uniqueness condition. Since both spatial homogeneity and u niqueness require the LT equations to be linear, the KSRP of Ref.[6] has here, essenti ally, been replaced by the space-time symmetry condition (I). Although postulate (I) and the KRSP play equivalent roles in the derivation of the LT, they state in a very diﬀerent way the physical foundation of s pecial relativity. Postulate (I) is a mathematical statement about the structure of the equat ions of physics, whereas the KSRP makes, instead, a statement about the relation between space-time measurements performed in two diﬀerent inertial frames. It is important t o note that in neither case do the dynamical laws describing any particular physical ph enomenon enter into the derivation of the LT. Choosing postulate (I) as the fundamental principle of spec ial relativity instead of the Galilean Relativity Principle, as in the traditional appro ach, has the advantage that a clear distinction is made, from the outset, between classical and relativistic mechanics. Both the former and the latter respect the Galilean Relativity Pr inciple but with diﬀerent laws. On the other hand, only relativistic equations, such as the L T or Maxwell’s Equations, respect the symmetry condition (I). The teaching of, and hence the understanding of, special rel ativity diﬀers greatly depending on how the parameter Vis introduced. In axiomatic derivations of the LT, that do not use Einstein’s second postulate, a universal par ameter Vwith the dimensions of velocity necessarily appears at an intermediate stage of the derivation [19]. Its physical meaning, as the absolute upper limit of the observed velocit y ofanyphysical object, only becomes clear on working out the kinematical consequences o f the LT [6]. If Einstein’s second postulate is used to introduce the parameter c, as is done in the vast majority of text-book treatments of special relativity, justiﬁed by the empirical observation of the constancy of the velocity of light, the actual universality of the theory is not evident. The misleading impression may be given that special relativ ity is an aspect of classical electrodynamics, the domain of physics in which it was disco vered. Formulating special relativity according to the symmetry p rinciple (I) makes clear the space-time geometrical basis [2] of the theory. The univ ersal velocity parameter V must be introduced at the outset in order even to deﬁne the spa ce-time exchange op- eration. Unlike the Galilean Relativity Principle, the sym metry condition (I) gives a clear test of whether any physical equation is a candidate to describe a universal law of physics. Such an equation must either be invariant under spa ce-time exchange or related by the exchange operation to another equation that also repr esents a universal law. The invariant amplitudes of quantum ﬁeld theory are an example o f the former case, while the LT equations for space and time correspond to the latter. Maxwell’s equations are examples of dynamical laws that satisfy the symmetry condit ion (I). The laws of electro- statics and magnetostatics (Gauss’ law for electric and mag netic charges) are related by 14the space-time exchange symmetry to the laws of electrodyna mics (Amp` ere’s law) and magnetodynamics (the Faraday-Lenz law) respectively. The 4-vector character [20] of the electromagnetic potential is essential for these symmetry relations [21]. Acknowledgement I thank an anonymous referee for his encouragement, as well a s for many suggestions that have enabled me to much improve the clarity of the presen tation. The assistance of C.Laignel in the preparation of the ﬁgure is also gratefully acknowledged. References [1] A.Einstein,‘Zur Elektrodynamik bewegter K¨ orper’ ’ An nalen der Physik 17, 891 (1905) [2] H.Minkowski, Phys. Zeitschr. 10, 104 (1909). The group property of the LT and its equivalence to a rotation in four-dimensional space-time h ad previously been pointed out by Poincar´ e in ‘The Dynamics of the Electron’ Rend. del C irc. Mat. di Palermo 21, 129-146, 166-175 (1906). [3] W.v Ignatowsky Arch. Math. Phys. Lpz. 17, 1 (1910) and 18, 17 (1911) Phs. Z. 11, 972 (1910) and 12, 779 (1911). [4] P.Frank and H.Rothe, Annalen der Physik 34, 825 (1911) and Phys. Z. 13, 750 (1912). [5] See, for example, Ref.[18] of V.Berzi and V.Gorini, ‘Rec iprocity Principle and Lorentz Transformations’, Journ. Math. Phys. 10, 1518-1524 (1969). More recent references may be found in Ref.[6] below, and in J.R.Lucas and P.E.Hodgs on,Space Time and Electromagnetism (Oxford University Press, Oxford) 1990. [6] J.H.Field, ‘A New Kinematical Derivation of the Lorentz Transformation and the Particle Description of Light’, Helv. Phys. Acta. 70, 542-564 (1997). [7] That is, all laws applying to physical systems where the c urvature of space-time may be neglected, so that General Relativistic eﬀects are un important, and may be neglected. [8] See, for example, A.Einstein, Relativity, the Special and General Theory (Routledge, London 1994). [9] N.D.Mermin, ‘Relativity without Light’, Am. J. Phys. 52, 119-124 (1984), S.Singh, ‘Lorentz Transformations in Mermin’s Relativity without L ight’, Am. J. Phys. 54, 183-184 (1986), A.Sen, ‘How Galileo could have derived the S pecial Theory of Rela- tivity’, Am. J. Phys. 62, 157-162 (1994). [10] In J.A.Wheeler and R.P.Feynman, ‘Classical Electrody namics in Terms of Direct Interparticle Action’ Rev. Mod. Phys. 21, 425-433(1949), this quantity is called ‘co- time’. 15[11] L.J.Eisenberg, ‘Necessity of the linearity of relativ istic transformations between in- ertial systems’, Am. J. Phys. 35, 649 (1967). [12] Y.P.Terletskii, Paradoxes in the Theory of Relativity (Plenum Press, New York, 1968), P17. [13] J.M.L´ evy-Leblond, ‘One more Derivation of the Lorent z Transformation’, Am. J. Phys.44, 271-277 (1976) [14] The positive sign for γis taken in solving Eqn.(2.12). Evidently γ→1 asβ→0. [15] A.Einstein, Annalen der Physik 17, 132 (1905). [16] J.H.Field, ‘Two Novel Special Relativistic Eﬀects: Sp ace Dilatation and Time Con- traction’, Am. J. Phys. 68, 267-274 (2000). [17] See, for example, I.J.R.Aitchison and A.J.G.Hey Gauge Theories in Particle Physics (Adam Hilger 1982), Appendix C. [18] See, for example, S.Weinberg, Gravitation and Cosmology (John Wiley and sons 1972), p36. [19] See, for example, Eqn.(2.36) of Ref.[6]. [20] For a recent discussion of the physical meaning of the 3- vector magnetic potential see M.D.Semon and J.R.Taylor ‘Thoughts on the magnetic vect or potential ’ Am. J. Phys.64, 1361-1369 (1996). [21] It is often stated in the literature that the potentials φ,/vectorAare introduced only for ‘reasons of mathematical simplicity’ and ‘have no physical meaning’. See for example: F.R¨ ohrlich Classical Charged Particles (Addison-Wesley 1990), p65-66. Actually, the underlying space-time symmetries of Maxwell’s equations c an only be expressed by using the 4-vector character of Aµ. Also the minimal electromagnetic interaction in the covariant formulation of relativistic quantum mechani cs, which is the dynamical basis of Quantum Electrodynamics, requires the introducti on of a quantum ﬁeld for the photon that has a the same 4-vector nature as the electrom agnetic potential. 16
arXiv:physics/0012012v1 [physics.chem-ph] 6 Dec 2000Perturbative treatment of intercenter coupling in Redﬁeld theory Ulrich Kleinekath¨ ofer, Ivan Kondov, and Michael Schreibe r Institut f¨ ur Physik, Technische Universit¨ at, D-09107 Ch emnitz, Germany (February 2, 2008) Abstract The quantum dynamics of coupled subsystems connected to a th ermal bath is studied. In some of the earlier work the eﬀect of intercent er coupling on the dissipative part was neglected. This is equivalent to a zero th-order perturba- tive expansion of the damping term with respect to the interc enter coupling. It is shown numerically for two coupled harmonic oscillator s that this treat- ment can lead to artifacts and a completely wrong descriptio n, for example, of a charge transfer processes even for very weak intercente r coupling. Here we perform a ﬁrst-order treatment and show that these artifa cts disappear. In addition, we demonstrate that the thermodynamic equilib rium population is almost reached even for strong intercenter coupling stre ngth. PACS: 82.30.Fi, 82.20.Wt, 82.20.Xr Typeset using REVT EX 1I. INTRODUCTION Quantum dynamics of complex molecules or molecules in a diss ipative environment has attracted a lot of attention during the last years. One speci al kind of this problem is the electron transfer dynamics in or between molecules especia lly in solution [1–3]. The bath- related relaxation can be described in a variety of ways. Amo ng others these are the path integral methods [4,5], the semi-group methods [6–8], and t he reduced density matrix (RDM) theory [9]. The latter one has been especially successful in Redﬁeld’s formulation [10, 11] and is the topic of the present investigation. As usual, the m aster equation for the RDM is derived from the equation of the full system, i.e. relevant s ystem plus bath, by tracing out the bath degrees of freedom. The main limitations of Redﬁeld theory are the second-order perturbation treatment in system-bath coupling and the neg lect of memory eﬀects (Markov approximation). In addition Redﬁeld suggested the use of th e secular approximation. In this approximation it is assumed that every element of the RD M in eigenstate representation (ER) is coupled only to those elements that oscillate at the s ame frequency. In the present study we do not perform this additional approximation which could distort the correct time evolution in transfer problems [12,13]. To be rigorous in applying Redﬁeld theory, the operators des cribing the time evolution have to be expressed in ER of the relevant system as has been do ne in the original papers [10,11]. For electron transfer this was performed in part of the literature (see for example [14–17]) while in another part of the literature [18–23] dia batic (local) representations (DRs) have been used, which signiﬁcantly reduces the numerical eﬀ ort in many cases. In NMR literature [11,24,25] most people seem to use the ER while in quantum optics most people use DRs [26,27]. Only recently the ER is used in quantum optic s [28–30]. Here we focus on electron transfer systems, but the conclusions should al so be applicable to problems in other areas. While in ER the damping term is evaluated exactly, in DR the in ﬂuence of the coupling between the local subsystems on dissipation is neglected. A s a consequence the relaxation terms do not lead to the proper thermal equilibrium of the cou pled system [6,31,32]. Only the thermal equilibrium of each separate subsystem is reach ed which can be quite diﬀerent from the thermal equilibrium of the coupled system. It will b e shown here that even for a very small intercenter coupling a completely wrong asympto tic value can be obtained. Although possibly leading to the wrong thermal equilibrium the local DR has advantages. For large problems it may be diﬃcult to calculate the eigenst ates of the system. These are not necessary in the DR. There one only needs the eigenstates of the subsystems. The quantum master equation can be implemented more eﬃciently i n DR in many cases [18,33– 35]. Moreover, almost all physical and chemical properties of transfer systems are expressed in the DR. For example, to determine the transfer rate one oft en calculates the population of the diabatic states and obtains the rate from their time ev olution. To do so one has to switch back and forth between DR and ER all the time if the time evolution is determined in ER. Using the semi-group methodology and a simple model of two fe rmion sites, DR and ER have been compared already [6]. We are interested in a more complicated system, i.e. a curve-crossing problem. The fact that we use a diﬀerent rel axation mechanism should eﬀect the ﬁndings only very little. Here we not only compare D R and ER but show how the 2relaxation term in DR can be written more precisely for small intercenter coupling. The paper is organized as follows. The next section gives an i ntroduction to the Redﬁeld theory and, using the DR, presents a zeroth-order (DR0) and a ﬁrst-order (DR1) pertur- bation expansion in the intercenter coupling. In the third s ection numerical examples are shown for two coupled harmonic oscillators. The DR and ER res ults are compared to each other and also to the improved local relaxation term derived here. The last section gives a short summary. Atomic units are used unless otherwise state d. II. INTERCENTER PERTURBATION EXPANSION WITHIN THE REDFIEL D EQUATION In the RDM theory the full system is divided into a relevant sy stem part and a heat bath. Therefore the total Hamiltonian consists of three ter ms – the system part HS, the bath part HB, and the system-bath interaction HSB: H=HS+HB+HSB. (1) The RDM ρis obtained from the density matrix of the full system by trac ing out the degrees of freedom of the environment. This reduction together with a second-order perturbative treatment of HSBand the Markov approximation leads to the Redﬁeld equation [ 9–11,36]: ˙ρ=−i[HS, ρ] +Rρ=Lρ. (2) In this equation Rdenotes the Redﬁeld tensor. If one assumes bilinear system- bath coupling with system part Kand bath part Φ HSB=KΦ (3) one can take advantage of the following decomposition [36,3 7]: ˙ρ=−i[HS, ρ] +{[Λρ, K] + [K, ρΛ†]}. (4) HereKand Λ together hold the same information as the Redﬁeld tenso rR. The Λ operator can be written in the form Λ =∞/integraldisplay 0dτ/an}bracketle{tΦ(τ)Φ(0)/an}bracketri}htKI(−τ) (5) where KI(−τ) =e−iHtKeiHtis the operator Kin the interaction representation. Assuming a quantum bath consisting of harmonic oscillators the time c orrelation function of the bath operator is given as [15] C(τ) =/an}bracketle{tΦ(τ)Φ(0)/an}bracketri}ht=∞/integraldisplay 0dωJ(ω)n(ω)(eiωt+eβωe−iωt). (6) HereJ(ω) denotes the spectral density of the bath [15], n(ω) = (eβω−1)−1the Bose-Einstein distribution, and β= 1/(kBT) the inverse temperature. 3The Hamiltonian HSof the system we are interested in can be separated according to HS=H0+V (7) where H0is the sum of all uncoupled subsystem Hamiltonians H0,n H0=/summationdisplay nH0,n (8) andVthe coupling among them which is assumed to be small. Two cano nical bases can be constructed for such a Hamiltonian. One consists of eigenfu nctions of H0. It is often called a local basis because these basis functions of the diabatic p otential energy surfaces (PESs) are located at speciﬁc subsystems (centers). Latin indices such as \|n/an}bracketri}htare used below to denote these DR basis states. The other basis diagonalizes t he system Hamiltonian HS. So it consists of eigenstates of HSand is called adiabatic basis. For these ER basis functions we use Greek indices such as \|ν/an}bracketri}ht. As discussed in the introduction Redﬁeld theory is deﬁned in ER but for transfer problems DRs have some conceptual and n umerical advantages. Here we ﬁrst calculate the dissipation in the DR for small int ercenter coupling V. In this basis the matrix elements of Λ are given by /an}bracketle{tn\|Λ\|m/an}bracketri}ht=∞/integraldisplay 0dωJ(ω)n(ω)∞/integraldisplay 0dτ(eiωτ+eβωe−iωτ)/an}bracketle{tn\|KI(−τ)\|m/an}bracketri}ht. (9) To evaluate the matrix element of Kone has to use perturbation theory in Vbecause the diabatic states \|n/an}bracketri}htare not eigenstates of HSbut of H0. Some details of the determination of/an}bracketle{tn\|Λ\|m/an}bracketri}htare given in the appendix. Using the expression for the corre lation function in frequency space C(ω) = 2π[1 +n(ω)][J(ω)−J(−ω)], (10) and denoting the transition frequency between diabatic sta tes\|m/an}bracketri}htand\|n/an}bracketri}htbyωmnthe ﬁnal result can be written as /an}bracketle{tn\|Λ\|m/an}bracketri}ht=1 2C(ωmn)/an}bracketle{tn\|K\|m/an}bracketri}ht −1 2/summationdisplay j/an}bracketle{tn\|K\|j/an}bracketri}ht/an}bracketle{tj\|V\|m/an}bracketri}ht ωjm[C(ωmn)−C(ωjn)] −1 2/summationdisplay i/an}bracketle{ti\|K\|m/an}bracketri}ht/an}bracketle{tn\|V\|i/an}bracketri}ht ωni[C(ωmn)−C(ωmi)]. (11) This ﬁrst-order result DR1 can be split into a zeroth-order c ontribution DR0 independent ofVand a ﬁrst-order contribution proportional to V. Taking the DR0 term /an}bracketle{tn\|Λ\|m/an}bracketri}ht=1 2C(ωmn)/an}bracketle{tn\|K\|m/an}bracketri}ht (12) only is equivalent to a complete neglect of the inﬂuence of th e intercenter coupling Von dissipation. This assumption has been used earlier [18–23] and is sometimes called the diabatic damping approximation [38]. In this approximation only the states \|n/an}bracketri}htand\|m/an}bracketri}ht 4contribute to the matrix element /an}bracketle{tn\|Λ\|m/an}bracketri}ht. In DR1 all states contribute to each of these matrix elements. As a consequence the spectral density of th e bath is not only probed at the transitions of the uncoupled subsystems as in DR0 but at m any more frequencies. The ER result for the matrix elements of Λ can easily be deduce d from the DR result by replacing the diabatic states by adiabatic ones and setting V= 0 in Eq. (11): /an}bracketle{tν\|Λ\|µ/an}bracketri}ht=1 2C(ωµν)/an}bracketle{tν\|K\|µ/an}bracketri}ht. (13) This result is of course correct for arbitrary intercenter c oupling strength. III. ELECTRON TRANSFER IN A TWO-CENTER SYSTEM In the following we direct our attention to electron transfe r in an example system con- sisting of two charge localization centers considered to be excited electronic states. The PESs of the localization centers are assumed to be harmonic a nd are sketched in Fig. 1. For this example the Hamiltonian of the uncoupled system is give n by H0=/summationdisplay n/bracketleftbigg Un+/parenleftbigg a† nan+1 2/parenrightbigg ωn/bracketrightbigg (14) and the coupling by V=/summationdisplay m,n/summationdisplay M,N(1−δmn)vmn\|mM/an}bracketri}ht/an}bracketle{tnN\|. (15) The ﬁrst index in each vector denotes the diabatic PES while t he second labels the vibrational level. ananda† nare the boson operators for the normal modes at center nandωnare the eigenfrequencies of the oscillators. Bilinear system-bat h coupling is assumed and the system part is given by the coordinate operator q K=q=/summationdisplay m/summationdisplay MN(2ωmM)−1/2/parenleftBig a† m+am/parenrightBig \|mM/an}bracketri}ht/an}bracketle{tmN\| (16) The mass of the system is denoted by M. In the local DR the system part of the system-bath coupling re ads /an}bracketle{tmM\|K\|nN/an}bracketri}ht= (2ωmM)−1/2δmn/parenleftBig δM+1,N√ M+ 1 + δM−1,N√ M/parenrightBig . (17) In the DR0 expansion (12) the system can emit or absorb only at intra-subsystem transition frequencies ωMN. The spectral density of the bath J(ω) is eﬀectively reduced to discrete values J(ω) =/summationtext mγmδ(ω−ωm). The advantage of this approach is the scaling behavior of the CPU time with the number Nof basis functions which results from the simple structure of the Λ matrix (12). As shown numerically [34,35] it scales l ikeN2.3. This is far better than the N3scaling of the DR1 approximation (11). In DR1 the spectral de nsity is probed at many more frequencies. One needs the full frequency depen dence of J(ω) which we take to be of Ohmic form with exponential cut-oﬀ J(ω) =ηΘ(ω)ωe−ω/ω c. (18) 5Here Θ denotes the step function and ωcthe cut-oﬀ frequency. In this study all system oscillators have the same frequency ω1(see Table I) and the cut-oﬀ frequency ωcis set equal toω1. The normalization prefactor ηis determined such that the spectral densities in DR and ER coincide at ω1. Eq. (18) together with Eq. (10) yields the full correlation function. If the system Hamiltonian HSis diagonalized and the resulting ER basis is used to calculate the elements of the operators in Eq. (4), there wil l be no longer any convenient structure in Kor Λ, so that the full matrix-matrix multiplications are ine vitable. For this reason the CPU time scales as N3, where Nis the number of eigenstates of HS. There appears to be a minimal number N0below which the diagonalization of HSfails or the completeness relation for \|ν/an}bracketri}htis violated. Nevertheless, the beneﬁt of this choice is the e xact treatment of the intercenter coupling. It is straightforwa rd to obtain the matrices for ρand K(see for example Ref. [14]). An initial wave packet at center \|n/an}bracketri}htis prepared by a δ-pulse excitation from the ground state\|g/an}bracketri}htof the system ρ1M1N(t= 0) = /an}bracketle{t1M\|g0/an}bracketri}ht/an}bracketle{tg0\|1N/an}bracketri}ht. (19) The pulse is chosen such that mainly the fourth and ﬁfth vibra tional level of the ﬁrst (left) diabatic PES is populated. The motion of the initial wave pac ket along the coordinate q models the transfer between the centers. The parameters for our calculation are taken from the work of K¨ uhn et al. [20] and are shown in Table I. Temperat ure is chosen as T= 295 K and the reduced mass of the system Mis set to 20 proton masses. The RDM is propagated in time and the occupation probabilities for each localizat ion center are calculated by means of the partial trace: Pm=/summationdisplay MρmMmM . (20) For the case of propagating in ER the RDM is transformed back t o the DR in order to apply Eq. (20). In the following we compare the population dynamics in the tw o-center electron transfer system using three diﬀerent intercenter coupling strength sVand four diﬀerent conﬁgurations of the two harmonic PESs. The diabatic PESs and eigenenergie s are shown in Fig. 1. Beginning our analysis with the weak coupling case v=v12=v21= 0.1ω1it is expected that a perturbation expansion in Vyields almost exact results. This is the reasoning why the DR0 term, which is easy to implement, has been used in earl ier work [18–23]. In conﬁguration (a) the eigenenergies of the two diabatic PE Ss are in resonance. For example, the ﬁrst vibrational eigenenergy of the ﬁrst cente r equals the third vibrational eigenenergy of the second center. It is important to note tha t in this conﬁguration no vibra- tional level of the ﬁrst center is below the crossing point of the two PESs. The calculations using ER and DR0 as well as DR1 give almost identical results, see Fig. 2a. For long times DR0 deviates a tiny bit. Redﬁeld theory in ER is known to give t he correct long-time limit (up to the Lamb shift). Conﬁguration (b) diﬀers from the ﬁrst one by shifting the ﬁrs t PES up by ω1/2. As shown in Fig. 2b the ER and DR1 results again agree perfectly. On the other hand, the DR0 results are a little bit oﬀ already at early times and the e quilibrium value departs from the correct value much more than in the ﬁrst, on-resonance co nﬁguration. 6Shifting the PESs further apart than in (a) yields conﬁgurat ion (c). The energy levels are again on-resonance but this time two vibrational levels of the ﬁrst center are below the curve-crossing point, i.e. there is a barrier for low-en ergy parts of the wave packet. As shown in Fig. 2c DR1 and the ER results agree perfectly once mo re. The DR0 results are terribly oﬀ. The long-time population of the ﬁrst center whi ch should vanish for the present conﬁguration stays ﬁnite. If we increase the energy of the ﬁrst PES by ω1/2 to obtain conﬁguration (d) DR0 fails again while DR1 gives correct results in comparison to the ER , see Fig. 2d. To understand the large diﬀerence between DR0 and DR1 we have a closer look at the ﬁnal result for the matrix elements of Λ, Eqs. (11) and (12). T he DR0 contribution (12) is independent of the intercenter coupling V. The system part of the system-bath interaction Kallows only for relaxation within each center. So there is no mechanism in the dissipative part which transfers population from one center to the other . This transfer has to be done by the coherent part of the master equation. But the coherent pa rt cannot transfer components of the wave packet with energy below the crossing point of the PESs. As tunneling is mainly suppressed, those components of the wave packet cannot leav e their center anymore although the corresponding PES might be quite high in energy. This res ults in the failure of DR0 for the conﬁgurations with barrier: Parts of the wave packet get trapped in the two lowest levels of the left center. From Eq. (11) one can explain why in the on-resonance case the DR0 results are in better agreement with the correct results . In this conﬁguration some of the DR1 terms are very small and so the DR1 correction is small er. Now we discuss the medium coupling strength v= 0.5ω1(see Fig. 3). The results for conﬁgurations (a) and (b), i.e. without barrier, look quite similar. In both cases the ER and DR1 results agree very well for short and long times. At in termediate times there is a small diﬀerence. The DR0 results already deviate at short ti mes and for long times there is too much population in the left (higher) center. For conﬁg urations (c) and (d), i.e. with barrier, again the ER and DR1 results coincide for small and l ong times. DR0 is oﬀ already after rather short times and the long-time limit is again wro ng. For the strong coupling v=ω1(see Fig. 4) the behavior of the results is quite similar to the medium coupling. For conﬁgurations (a) and (b) the diﬀer ence at intermediate times is a little larger, so is the deviation of the long-time DR0 limi t. For conﬁgurations (c) and (d) with barrier there is also a discrepancy for DR1 already at sh ort times and the correct long- time limit is not reached exactly. But the disagreement is su rprisingly small for the strong coupling. Overall DR1 still looks quite reasonable while th e DR0 results are completely oﬀ. IV. SUMMARY In addition to the approximations done in Redﬁeld theory, i. e. second-order perturbation expansion in the system-bath coupling and Markov approxima tion, we have applied pertur- bation theory in the intercenter coupling. It has been shown for two coupled harmonic surfaces that the zeroth-order approximation DR0 which is e quivalent to the diabatic damp- ing approximation [38] can yield wrong population dynamics even for very small intercenter coupling. These artifacts disappear using the ﬁrst-order t heory DR1. The scaling of DR1 is like N3not as N2.3for DR0. This is of course a serious drawback of DR1. For conﬁgurations without barrier it seems to be poss ible to use DR0 for weak to 7medium intercenter coupling. This of course depends on the a ccuracy required especially for the long-time limit. In all other cases one should either use the exact ER or DR1. Although the ﬁrst-order results are not exact for medium and strong intercenter coupling these calculations have at least two advantages. First of al l, one does not need to calculate the eigenstates and energies of the full system Hamiltonian HS. For small systems like two coupled harmonic surfaces using one reaction coordinate th is calculation is of course easy. But if one wants to study larger systems like molecular wires [6, 14] and/or multi-mode models [22,23,33] this is no longer a trivial task. The secon d advantage is related to the fact that in all transfer problems one is mainly interested i n properties which are deﬁned in a local basis, e. g. the population in each subsystem in any mo ment in time. If one uses the ER one has always to transform back to the DR in order to calcul ate these properties. So for large-scale problems using a DR together with the ﬁrst-o rder perturbation in Vshould be advantageous. In a sense the present study is an extension of the investigat ion performed by Davis et al. [6]. They compared ER and DR for a two-site problem. Her e we looked at a more general multilevel system and also calculated the ﬁrst-ord er perturbation. In their model they do not have a reaction coordinate and therefore no barri er. Their ﬁndings correspond more to cases (a) and (b) in the previous section. Besides the agreement in the case of small intercenter coupling they also found good agreement in the h igh-temperature limit. Using our model this statement could not be conﬁrmed for a general c onﬁguration, although there might be conﬁgurations where it is true. In Ref. [30] the authors followed a strategy diﬀerent from th e present work. They also studied two coupled harmonic oscillators modeling two coup led microcavities, but only one cavity was coupled to the thermal bath directly. This should not eﬀect the questions studied here. With a transformation to uncoupled oscillators they e ﬀectively reduced the intercenter coupling to zero. The result [30] is then exact for arbitrary V. The disadvantage of this strategy is that it is not easy to extend to larger systems. Th e advantage of the presently developed ﬁrst-order expansion in Vis its general applicability to problems of any size. ACKNOWLEDGMENTS Useful discussions with V. May, W. Domcke, and D. Egorova are gratefully acknowledged. We thank the DFG for ﬁnancial support. APPENDIX: The purpose of this appendix is to show some more details for t he evaluation of /an}bracketle{tn\|Λ\|m/an}bracketri}ht. To calculate /an}bracketle{tn\|KI(−t)\|m/an}bracketri}ht=/summationdisplay i,j/an}bracketle{tn\|e−iHt\|i/an}bracketri}ht/an}bracketle{ti\|K\|j/an}bracketri}ht/an}bracketle{tj\|eiHt\|m/an}bracketri}ht (A1) the operator identity [39] e−i(H0+V)t=e−iH0t 1−it/integraldisplay 0dt′eit′H0V e−it′(H0+V) , (A2) 8which can easily be proven by multiplying both sides by eiH0tand diﬀerentiating with respect tot, is used iteratively. It yields /an}bracketle{tn\|e−iHt\|i/an}bracketri}ht=/an}bracketle{tn\|e−iH0t[1−it/integraldisplay 0dt′eit′H0V e−it′H0]\|i/an}bracketri}ht+O(V2) =e−iEitδni−ie−iEnt/an}bracketle{tn\|V\|i/an}bracketri}htt/integraldisplay 0dt′ei(En−Ei)t′+O(V2) =e−iEitδni−/an}bracketle{tn\|V\|i/an}bracketri}ht En−Ei(e−iEit−e−iEnt) +O(V2) (A3) assuming that En/ne}ationslash=Ei. Here and in the following we only give the general expressio ns for the matrix elements. If a singularity can appear due to coincidi ng frequencies the appropriate expression can be obtained by taking the proper limit. Thus the matrix element (A1) is given by /an}bracketle{tn\|KI(−t)\|m/an}bracketri}ht=eiωmnt/an}bracketle{tn\|K\|m/an}bracketri}ht −/summationdisplay j/an}bracketle{tn\|K\|j/an}bracketri}ht/an}bracketle{tj\|V\|m/an}bracketri}ht ωjm(eiωmnt−eiωjnt) −/summationdisplay i/an}bracketle{ti\|K\|m/an}bracketri}ht/an}bracketle{tn\|V\|i/an}bracketri}ht ωni(eiωmit−eiωmnt) +O(V2) (A4) This result is inserted into Eq. (9). One has to evaluate inte grals of the kind ∞/integraldisplay 0dte−ǫte−iωmnt=−i ω−ωnm−iǫ(A5) which contain a convergence parameter ǫ. Using the well known identity lim ǫ→01 x±iǫ=P x∓πδ(x) (A6) one gets for the ﬁrst term of the matrix element of Λ /an}bracketle{tn\|Λ\|m/an}bracketri}ht=π 1−e−βωmn[J(ωmn)−J(−ωmn)]/an}bracketle{tn\|K\|m/an}bracketri}ht+ (Lamb shift) + . . . The Lamb shift is the imaginary part of the matrix element of Λ and leads to an energy shift in the quantum master equation. This term is a small correcti on [40,41] and is neglected in Redﬁeld theory. The other terms of the matrix elements are ca lculated in the same fashion yielding /an}bracketle{tn\|Λ\|m/an}bracketri}ht=π 1−e−βωmn[J(ωmn)−J(−ωmn)]/an}bracketle{tn\|K\|m/an}bracketri}ht −/summationdisplay j/an}bracketle{tn\|K\|j/an}bracketri}ht/an}bracketle{tj\|V\|m/an}bracketri}ht ωjm/braceleftbiggπ 1−e−βωmn[J(ωmn)−J(−ωmn)] −π 1−e−βωjn[J(ωjn)−J(−ωjn)]/bracerightbigg 9−/summationdisplay i/an}bracketle{ti\|K\|m/an}bracketri}ht/an}bracketle{tn\|V\|i/an}bracketri}ht ωni/braceleftbiggπ 1−e−βωmn[J(ωmn)−J(−ωmn)] −π 1−e−βωmi[J(ωmi)−J(−ωmi)]/bracerightbigg (A7) 10REFERENCES [1] M. Bixon and J. Jortner, Adv. Chem. Phys. 106&107 , (1999), special issue on electron transfer. [2] M. Newton, Chem. Rev. 91, 767 (1991). [3] P. F. Barbara, T. J. Meyer, and M. A. Ratner, J. Phys. 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Parameters used for the ground state oscillator and the two excited state oscillators. Center \|n/an}bracketri}ht Conﬁguration Un, eV Qn,˚A ωn, eV \|g/an}bracketri}ht 0.00 0.000 0.1 \|1/an}bracketri}ht 0.25 0.125 0.1 \|2/an}bracketri}ht a 0.05 0.238 0.1 \|2/an}bracketri}ht b 0.00 0.238 0.1 \|2/an}bracketri}ht c 0.05 0.363 0.1 \|2/an}bracketri}ht d 0.00 0.363 0.1 13FIGURES a bc d FIG. 1. The four diﬀerent conﬁgurations of the two diabatic h armonic potentials \|1/an}bracketri}htand\|2/an}bracketri}htas discussed in the text. Also included in the ﬁgures are the ene rgy levels. 103104105106 Time [a.u.]00.20.40.60.81P1103104105106 00.20.40.60.81P1 103104105106107108 Time [a.u.]103104105106107108a bc d FIG. 2. Time evolution for small intercenter coupling and fo r the four diﬀerent conﬁgurations. The results in ER are shown by the solid line while the results in diabatic basis are shown by dotted (zeroth-order) and dashed (ﬁrst-order) lines. The results for ER and DR1 are indistinguishable for small intercenter coupling. Note the logarithmic time scal e. 14102103104 Time [a.u.]00.20.40.60.81P1102103104 00.20.40.60.81P1 102103104105106107 Time [a.u.]102103104105106107a bc d FIG. 3. Time evolution for medium intercenter coupling. 102103104 Time [a.u.]00.20.40.60.81P1102103104 00.20.40.60.81P1 102103104105106 Time [a.u.]102103104105106a bc d FIG. 4. Time evolution for strong intercenter coupling. 15
arXiv:physics/0012013 6 Dec 2000Are Short and Long Gamma Ray Bursts Really of Different Origin? Ernst Karl Kunst Im Spicher Garten 5 53639 Königswinter Germany e-mail: ErnstKunst@aol.com Short and long gamma ray bursts (GRBs) are of the same origin and, furthermore, correlated with their duration, as will be shown in the following. Key words: Gamma ray bursts - origin - vacuum Cerencov radiation According to a NASA press release, Jay Norris of the Goddard Space Flight Center found that the shorter GRBs, lasting less than 2 seconds (s), have different charasteristics than longer bursts [1]. Because I could not find the original paper in the literature I refer in the following to this press release, especially since the general results given there are fully sufficient to be compared with some theoretical derivations. According to Norris have short GRBs (< 2 s) significantly fewer pulses and are their lag times (the lag of lower-energy pulses behind high-energy pulses) 20 times shorter than the lags in the longer GRBs. Therefore, he proposed that the short bursts are produced in physically different objects. In the following is demonstrated that these experimental findings do exactly coincide with theory predicting a common origin of all GRBs. In [2] the outlines of this theory have been drawn, according to which GRBs and related pheno mena in the X-rays and the ultra-violet have their origin not in known or speculative astronomical objects but rather in the vacuum Cerenkov radiation caused by the superluminal propagation of extraterrestrial spaceprobes in the interstellar space (see also [3]). Cerenkov radiation in all wavebands is generated along the flight paths of the superluminal spaceprobes, whereby the photons of highest energy depend on the superluminal velocity of the probe or craft. On the grounds of the Cerenkov angle and the known duration of GRBs their distance is basically calculable. Furthermore, a correlation between duration, distance, relative number and intensity of GRBs has been shown to exist. It is demonstrated that this correlation also comprises the above stated decrease of pulses and shortening of lag times in short GRBs. According to theory the Cerenkov radiation generated by a superluminal extraterrestrial spaceprobe hurtling through the galactic space is emitted along and in a very narrow cone in the direction of its flight path. An observer in the vicinity of Earth will observe a GRB exactly then, when the spaceprobe crosses the line of sight in a very narrow angle (Fig. 1) and the generated Cerenkov radiation is bright enough to be observed. The radiation cone comprises - depending on the velocity -2 photons of all wavebands till down to the radio waveband. Fig. 1 shows (exaggerated) that the ”visible” gamma light track in the sky constitutes the side B¯C¯ of the triangle ABC, where A¯B¯ and A¯C¯ are the “lines of sight” to the points of the track wherefrom photons of lowest (B) and highest (gamma) energy (C), respectively, are received and the distance coincides with A¯C¯ of this triangle. Due to the different Cerenkov angle of the respecitve radiation will the Cerenkov point source be observed to recede backward in time along B¯C¯ from point C to point B and beyond with decreasing frequency from ever more distant points with ever lower velocity in the plane of the sky. Fig. 1 shows two tracks of vacuum Cerenkov radiation which we assume be generated by spaceprobes crossing the line of sight of the observer at point A with the same superluminal speed - implying them to be alike in all physical properties - but in different distances. It is clearly evident that the respective distance is proportional to the length of the observable Cerenkov track of gamma radiation B¯C¯ and D¯E¯, respectively, which again is proportional to the duration of the GRB. On the other hand, the tracks B¯C¯ and D¯E¯ are, independen t on their different length or duration, exactly of the same spectral composition, with the photons of the same highest energy at the points B and D and of lowest energy at the points C and E, respectively. As has extensively been shown in [2] will owing to the track geometry the photons (pulses) of highest energy generally arrive first at the observer in point A and subsequently photons (pulses) of ever lower energy. Therefore, in connection with Fig. 1 is clear that B¯C¯ and D¯E¯ exhibit exactly the same energy spectrum, but the latter on a much shorter scale so that firstly the number of photons or pulses must be proportional and secondly lag times or arrival times of photons of different energy inversely proportional to B¯C¯/D¯E¯ = D/D /G11 n/n, where D meansB¯C¯ D¯E¯B¯C¯ D¯E¯ duration and n nu mber of bursts in the respective distance or of the respective3 duration within some time. If a mean of 30 - 40 s for the longer and of 1.5 - 2 s for the short GRBs is assumed this results in /G11 20 - the result of Norris -, exactly as it does if the number of bursts with a duration of /G11 30 s (29) is divided by the number with a duration of 1.5 s (1.45) [4] (the latter number indeed ha s been calculated, because the mean burst distribution at 1.5 s is disturbed by a small event maximum (see [2]) and, therefore, may not look very convincing). Furthermore, follows from this hypothesis that the duration times of the subsequent Cerenkov radiation in the X-rays, visible light etc. (“afterglow”) of GRBs obey the same law. Therefore, in the case of very short GRBs a shortening of the duration times of this radiation in accord with the above relation is to expect, which presumably is the cause that it has not been detected yet. References [1] ftp://www.gsfc.nasa.gov/pub/PAO/Releases/2000/00-132.htm [2] Kunst, E. K.: On the Physical Cause and the Distance of Gamma Ray Bursts and Related Phenomena in the X-Rays and the Ultra-Violet, physics/0004034 [3] Kunst, E. K.: Is the Lorentz Transformation Distant-Dependent? physics/9911022 [4] Fishman, G. J. et al., Astrophys. J. Suppl. Ser. 92, 229 (1994)
arXiv:physics/0012014v1 [physics.acc-ph] 7 Dec 2000SLAC-AP-129 LCC-0043 December 2000 Dipole Mode Detuning in the Injector Linacs of the NLC∗ Karl L.F. Bane and Zenghai Li Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Abstract The injector linacs of the JLC/NLC project include the pre- linac, the e+drive linac, the e−booster, and the e+booster. The ﬁrst three will be S-band machines, the last one, an L-ban d machine. We have demonstrated that by using detuning alone in the accelerator structure design of these linacs we will h ave acceptable tolerances for emittance growth due to both inje ction jitter and structure misalignments, for both the nominal (2 .8 ns) and alternate (1.4 ns) bunch spacings. For the L-band struc- ture (a structure with 2 π/3 phase advance) we take a uniform distribution in synchronous dipole mode frequencies, with cen- tral frequency ¯f= 2.05 GHz and width ∆ δf= 3%. For the S-band case our optimized structure ( a 3 π/4 structure) has a trapezoidal dipole frequency distribution with ¯f= 3.92 GHz, ∆δf= 5.8%, and tilt parameter α=−.2. The central frequency and phase advance were chosen to put bunches early in the trai n on the zero crossing of the wake and, at the same time, keep the gradient optimized. We have shown that for random man- ufacturing errors with rms 5 µm, (equivalent to 10−4error in synchronous frequency), the injection jitter tolerances a re still acceptable. We have also shown that the structure alignment tolerances are loose, and that the cell-to-cell misalignme nt tol- erance is /greaterorsimilar40µm. Note that in this report we have considered only the eﬀects of modes in the ﬁrst dipole passband. ∗Work supported by Department of Energy contract DE–AC03–76 SF00515.Dipole Mode Detuning in the Injector Linacs of the NLC Karl L.F. Bane and Zenghai Li September 23, 2012 1 Introduction A major consideration in the design of the accelerator struc tures in the injector linacs of the JLC/NLC[1][2] is to keep the wakeﬁeld eﬀects within tolerances for both the nominal (2.8 ns) and the alternate (1 .4 ns) bunch spacings. One important wakeﬁeld eﬀect in the injector lina cs is likely to be multi-bunch beam break-up (BBU). With this eﬀect a jitter in the injection conditions of a bunch train, due to the dipole modes of the acc elerator structures, is ampliﬁed in the linac. By the end of the linac b unches in the train are driven to large amplitudes and/or the projecte d emittance of the train becomes large, both eﬀects which can hurt machine p erformance. Another important multi-bunch wakeﬁeld eﬀect that needs to be considered is static emittance growth caused by structure misalignmen ts. To minimize the multi-bunch wakeﬁeld eﬀects in the injector linacs we need to minimize the sum wake in the accelerator structures. The dipole wake amplitude of the structures—and therefore also the sum wake amplitude— scales as frequency to the -3 power. Therefore, compared to t he main (X– band) linac, the injector linac wakes tend to be smaller by a f actor 1 /64 and 1/512, respectively, for the S– and L–band linacs. We shall see , however, that—in the S-band case—this reduction, by itself, is not su ﬃcient. Two ways of reducing the sum wake further are to detune the ﬁrst pa ss-band dipole modes and to damp them. Detuning can be achieved by gra dually varying the dimensions of the cells in a structure. Weak damp ing can be achieved by letting the ﬁelds couple to manifolds running pa rallel to the structure (as is done in the main JLC/NLC linac[3]); stronge r damping by, for example, introducing lossy material in the cells of the s tructure. In the injector linacs the dipole mode frequencies are much l ower than in the main linac, and the number of dipole mode oscillations be tween bunches 2ndis much smaller (see Table 1). Therefore, signiﬁcantly redu cing the wake envelope at one bunch spacing behind the driving bunch by det uning alone becomes more diﬃcult. In addition, for a given Q, the eﬀective damping is 4 (or 8) times less eﬀective than for X-band. Table 1: Scaling of the frequency and the wake amplitude for C , S, and L bands as compared to X band. Also given are the number of dipol e mode oscillations between bunches nd, and the damping needed to reduce the wake amplitude by 1 /eat the position of the second bunch, Qd, for the nominal (2.8 ns) and the alternate (1.4 ns) bunch spacings. Scaling ∆t= 2.8 ns ∆t= 1.4 ns Band Freq. Wake nd Qdnd Qd X 1 1 42.0 132 21.0 66 C 1/2 1/8 21.0 6610.5 33 S 1/4 1/64 10.5 33 5.3 16 L 1/8 1/512 5.3 16 2.6 8 In this report our goal is to design the accelerator structur es for the injector linacs using simple detuning alone, i.e.including no damping, to take care of the long-range wakeﬁelds. We focus mostly on the S-band in- jector linacs. We begin by discussing analytical approache s to estimating the eﬀects of BBU and structure misalignments. We then discu ss wakeﬁeld compensation using detuning. We optimize structure dimens ions for struc- tures with 2 π/3 and 3 π/4 per cell phase advance, and show that the latter is preferable. And ﬁnally we obtain tolerances to wakeﬁeld e ﬀects for all the injector linacs using both analytical formulas and numeric al tracking. Note that in this report we are only concerned with the eﬀects of mo des in the ﬁrst dipole passband, which have kick factors much larger th an those in the higher passbands. The eﬀects of the higher passband modes, h owever, will need to be addressed in the future. 2 Emittance Growth 2.1 Beam Break-up (BBU) In the case of single-bunch beam break-up in a linac the ampliﬁcation of injection jitter can be characterized by a strength paramet er dependent on 3the longitudinal position within the bunch. When the streng th parameter is suﬃciently small the growth in amplitude at the end of the l inac is given by the ﬁrst power of this parameter[4]. For the multi-bunch case we can derive an analogous strength parameter, one dependent on bu nch number m. When this strength parameter is suﬃciently small we expect that again the growth in amplitude at the end of the linac is given by the ﬁ rst power of the parameter. (But even when the strength parameter is no t suﬃciently small it can be a useful parameter for characterizing the str ength of BBU.) For the multi-bunch case the strength parameter becomes (se e Appendix A) Υm=e2NLS m¯β0 2E0g(Ef/E0,ζ) [ m= 1,... ,M ], (1) withNthe single bunch population, Lthe machine length, ¯β0the initial value of the beta function averaged over a lattice cell, E0the initial energy, Efthe ﬁnal energy, and Mthe number of bunches in a train. The sum wake Smis given by Sm=m−1/summationdisplay i=1W[(m−i)∆t] [ m= 1,... ,M ], (2) withWthe transverse wakeﬁeld and ∆ tthe time interval between bunches in a train. The wakeﬁeld, in turn, is given by a sum over the dip ole modes in the accelerator structures: W(t) =Nm/summationdisplay n2knsin(2πfnt/c)exp( −πfnt/Qn), (3) withNmthe number of modes, fn,kn, and Qnare, respectively, the fre- quency, the kick factor, and the quality factor of the nthmode. The function g(x) in Eq. 1 is one depending on the energy gradient and focusing proﬁle in the linac. For acceleration assuming the beta function va ries as ¯β∼Eζ, g(x,ζ) =1 ζ/parenleftbiggxζ−1 x−1/parenrightbigg [¯β∼Eζ]. (4) If Υm, for all m, is not large, the linear approximation applies, and this parameter directly gives the (normalized) growth in amplit ude of bunch m. If Υmis not large the projected normalized emittance growth of th e bunch train becomes (assuming, for simplicity, that, in phase spa ce, the beam ellipse is initially upright): δǫ≈/bracketleftBigg 1 +/parenleftbiggy0Υrms0 σy0/parenrightbigg2/bracketrightBigg1/2 −1 [Υ msmall] , (5) 4withy0the initial bunch oﬀset, Υ rms0the rms of the strength parameter (the square root of the second moment: the average is not subt racted), andσy0the initial beam size. Note that the quantity Sm/Min the multi- bunch case takes the place of the bunch wake (the convolution of the wake with the bunch distribution) in the single bunch instabilit y problem. As jitter tolerance parameter, rt, we can take that ratio y0/σy0that yields a tolerable emittance growth, δǫt. 2.2 Misalignments If the structures in the linac are (statically) misaligned w ith respect to a straight line, the beam at the end of the linac will have an inc reased projected emittance. If we have an ensemble of misaligned linacs then, to ﬁrst order, the distribution in emittance growth at the end of these lina cs is given by an exponential distribution exp[ −δǫ//angb∇acketleftδǫ/angb∇acket∇ight]//angb∇acketleftδǫ/angb∇acket∇ight, with[5]1 /radicalbig /angb∇acketleftδǫ/angb∇acket∇ight=e2NLa(xa)rmsSrms E0/radicalbigg Naβ0 2h(Ef/E0,ζ), (6) withLathe structure length, ( xa)rmsthe rms of the structure misalignments, Srmsis the rms of the sum wake with respect to the average ,Nathe number of structures; the function his given by (again assuming ¯β∼Eζ): h(x,ζ) =/radicalBigg 1 ζx/parenleftbiggxζ−1 x−1/parenrightbigg [¯β∼Eζ]. (7) Eq. 6 is valid assuming the so-called betratron term in the eq uation of motion is small compared to the misalignment term. We can deﬁne a misalignment tolerance by xat= (xa)rms/radicalBigg δǫt /angb∇acketleftδǫ/angb∇acket∇ight, (8) withδǫtthe tolerance in emittance growth. What is the meaning of xat? For an ensemble of machines, each with a diﬀerent collection of random misalignment errors but with the same rms xat, then the distribution of ﬁnal emittances will be given by the exponential function wi th expectation value δǫt. Note that if we, for example, want to have 95% conﬁdence to achieve this emittance growth, we need to align the machine t o a tolerance level of xat/√ −ln.05≈.58xat. 1This equation is a slightly generalized form of an equation g iven in Ref. [5]. 5Besides the tolerance to structure misalignments, we are al so interested in the tolerance to cell-to-cell misalignments due to fabri cation errors. A structure is built as a collection of cups, one for each cell, that are brazed together, and there will be some error, small compared to the cell dimen- sions, in the straightness of each structure. To generate a w ake (for a beam on-axis) in a structure with cell-to-cell misalignments we use a perturbation approach that assumes that, to ﬁrst order, the mode frequenc ies remain unchanged (from those in the straight structure), and only n ew kick fac- tors are needed[6] (The method is described in more detail in Appendix B). Note that for particle tracking through structures with int ernal misalign- ments, contributions from both this (orbit independent) wa ke force and the normal (orbit dependent) wake force need to be included. Machine properties for the injector linacs used in this repo rt are given in Table 2[2]. The rf frequencies of all linacs are sub–harmo nics of the main linac frequency, 11.424 GHz. The prelinac, e+drive linac, e−booster linac all operate at S–band (2.856 GHz), and the e+booster linac at L–band (1.428 GHz). Note that ¯βy0andζare only a rough ﬁtting of the real machine β–function to the dependence ¯β∼Eζ. In Table 3 beam properties for the injector linacs, for the nominal bunch train conﬁguration ( 95 bunches spaced at ∆t= 2.8 ns), are given. For the alternate conﬁguration (190 bunche s spaced at ∆ t= 1.4 ns) Nis reduced by 1 /√ 2. Table 2: Machine properties of the injector linacs. Given ar e the initial energy E0, the ﬁnal energy Ef, the length L, the initial average beta function iny, and the approximate scaling parameter ζ, ofβwith energy ( β∼Eζ). Name Band E0[GeV] Ef[GeV] L[m] ¯βy0[m] ζ Prelinac S 1.98 10.0 558 8.6 1/2 e+Drive S .08 6.00 508 2.4 1/2 e−Booster S .08 2.00 163 3.4 1/4 e+Booster L .25 2.00 184 1.5 1 3 Wakeﬁeld Compensation For eﬀective detuning, one generally requires that the wake amplitude drop quickly, in the time interval between the ﬁrst two bunches, a nd then remain 6Table 3: Beam properties in the injector linacs under the nom inal bunch train conﬁguration (95 bunches spaced at ∆ t= 2.8 ns). Given are the bunch population N, the rms bunch length σz, the initial energy spread σδ0, and the nominal normalized emittance in y,ǫyn. Note that under the alternate bunch train conﬁguration (190 bunches spaced at ∆ t= 1.4 ns) N is reduced by 1 /√ 2. Name N[1010]σz[mm] σδ0[%] ǫyn[m] Prelinac 1.20 0.5 1. 3×10−8 e+Drive 1.45 2.5 1. 1×10−4 e−Booster 1.45 2.5 1. 1×10−4 e+Booster 1.60 9.0 3.5 6×10−2 low until the tail of the bunch train has passed. In the main (X -band) linac of the NLC, Gaussian detuning is used to generate a fast Gauss ian fall-oﬀ in the wakeﬁeld; in particular, at the position of the second bu nch the wake is reduced by roughly 2 orders of magnitude from its initial val ue. The short time behavior of the wake can be analyzed by the so-called “un coupled” model. According to this model (see, for example, Ref. [8]) W(t)≈Nc/summationdisplay n2ksnsin(2πfsnt/c) [ tsmall] , (9) where Ncis the number of cells in the structure, and fsnandksnare, re- spectively, the frequency and kick factor at the synchronou s point, for a periodic structure with dimensions of cell n. Therefore, one can predict the short time behavior of the wake without solving for the modes of the sys- tem. (In the following we will omit the unwieldly subscript s; whether the synchronous or mode parameters are meant will be evident fro m context.) For Gaussian detuning the initial fall-oﬀ of the wake is give n by W(t)≈2¯ksin(2π¯ft)exp/parenleftBig −2/bracketleftbig π¯ftσδf/bracketrightbig2/parenrightBig [tsmall] , (10) with¯kthe average kick factor, ¯fthe average (ﬁrst band) synchronous, dipole mode frequency, and σδfthe sigma parameter in the Gaussian distribution. Suppose we want a relative amplitude reduction to 0.05 at the position of the second bunch. Considering the alternate (1.4 ns) bunch s pacing, and taking ¯f= 4.012 GHz (S-band), we ﬁnd that the required σδf= 6.5%. 7To achieve a smooth Gaussian drop–oﬀ of the wake requires tha t we take at least ∼3σδf= 20% as the full–width of our frequency distribution, a number which is clearly too large. If we limit the total frequency spread to an acceptable ∆ δf= 10% and keep the parameter σδfﬁxed, our Gaussian distribution becomes similar to a uniform distribution. For the case of a uniform distributi on with full width ∆δfthe wake becomes W≈2¯k Ncsin(2π¯ft)sin(π¯ft∆δf) sin(π¯ft∆δf/Nc)[(π¯ft/Q)small] .(11) Again considering S-band with the alternate (1.4 ns) bunch s pacing, and taking ¯f= 4.012 GHz, we obtain an amplitude reduction to 0.37 at the position of the second bunch, which is still too large. If we want to substantially reduce the wake further we need to shift the av erage dipole frequency ¯f, so that the term sin(2 π¯ft) in Eq. 11 becomes small, and the wake at the second bunch is near a zero crossing. That is, ¯f∆t=n 2[nan integer] , (12) with ∆ tthe bunch spacing. With nan even integer the bunch train will be near the integer resonance, otherwise it will be near the h alf-integer res- onance. With our parameters ¯f∆t= 5.62, and Condition 12 is achieved by changing ¯fby−2% (or by a much larger amount in the positive direc- tion). However, ¯f= 4.012 GHz is the average dipole mode frequency for the somewhat optimized structure, and a change of −2% results in a net loss of 7% in accelerating gradient, and, presumably, a 7% increase in the required lengths of the S–band injector linacs. One ﬁnal possibility for reducing the wake at one bunch spacing is to introduce heavy damping. But f or this case, just to reduce the wake at one bunch spacing by 1 /e, a quality factor of 16 would be needed (see Table 1), and such a quality factor i s not easy to achieve without a signiﬁcant loss in fundamental mode shu nt impedance. The wakeﬁeld for a uniform distribution, as given by Eq. 11, n ot only gives the initial drop-oﬀ of the wake, but also the longer ter m behavior. (However, here the mode parameters, not the synchronous par ameters, are needed. Therefore, to see whether such a mode distribution c an actually be achieved the circuit equations need to be solved.) We see tha t for a uniform distribution the wakeﬁeld resurges to a maximum again, at t=Nc/(¯f∆δf). Therefore, ∆ δfmust be suﬃciently small to avoid this resurgence occurring before the end of the bunch train; i.e.it must be signiﬁcantly less than 8Nc/(M¯f∆t) (which is about 10% in our case). The envelope of Eq. 11 for ∆δf= 8%, ¯f= 4.012 GHz, and Nc= 114 is shown in Fig. 1. Another possibility for pushing the resurgence in the wake t o larger tis to use two structure types, which can eﬀectively double the n umber of modes available for detuning. This idea has been studied; it has be en rejected in that it requires extremely tight alignment tolerances betw een pairs of such structures. Figure 1: The wake envelope (normalized to a maximum of 1) for a uniform frequency distribution. Shown is Eq. 11 with all oscillatio ns removed. The average frequency ¯f= 4.012 GHz, the bandwidth ∆ δf= 8%, and the num- ber of modes Nc= 114. Note that in the injector linacs the bunch train extends to ct= 80 m. 3.1 2π/3Phase Advance Per Cell Except for the region of the initial drop-oﬀ, we need to solve for the eigen- modes of the system to know the behavior of the wake or the sum w ake for a detuned structure. To numerically obtain these modes we us e a computer program that solves the double-band circuit model describe d in Ref. [8]. We consider structures of the disk–loaded type, with constant period and with rounded irises of ﬁxed thickness. The iris radii and cavity r adii are adjusted to give the correct fundamental mode frequency and the desir ed dipole mode spectrum. Therefore, the dimensions of a particular cell mcan be speciﬁed 9by one free parameter, which we take to be the synchronous fre quency of the ﬁrst dipole mode pass band, fsm(more precisely, the synchronous frequency of the periodic structure with cell dimensions of cell m). The computer pro- gram generates 2 Nccoupled mode frequencies fnand kick factors kn, with Ncthe number of cells in a structure. It assumes the modes are tr apped at the ends of the structure. Only the modes of the ﬁrst band (app roximately the ﬁrst Ncmodes) are found accurately by the two-band model. And since , in addition, the strengths of the ﬁrst band modes are much lar ger than those of the second band (in the S-band case the synchronous mode ki ck factors are larger by a factor ∼35), we will use only the ﬁrst band modes to obtain the wake, and then the sum wake. For our S-band structures, we will consider a uniform freque ncy distri- bution, with a central frequency ¯fchosen so that at one bunch spacing, for the alternate conﬁguration (∆ t= 1.4 ns), the wake is very close to a zero crossing. The strength of interaction with the modes, given by the kick fac- torsk, will be stronger near the downstream end of the structure, w here the iris radii become smaller. To counteract this asymmetry we w ill allow the top of the frequency distribution to be slanted at an angle, a nd therefore, our distribution becomes trapezoidal in shape. We paramete rize this slant by α=λf(fhi)−λf(flo) λf(fhi) +λf(flo), (13) where λfis the synchronous frequency distribution, and floandfhirep- resent, respectively, the lowest and highest frequencies i n the distribution. Note that −1≤α≤1. With ¯f,α, and the relative width of the distribu- tion ∆ δf, we have 3 parameters that we will vary to reduce the wakeﬁeld eﬀects—speciﬁcally by minimizing on the sum wake—for both b unch train conﬁgurations. Each S-band structure operates at a fundamental mode freque ncy of 2.856 GHz, and consists of 114 cells with a cell period of 3.5 c m (where the phase advance per cell φ= 2π/3), an iris thickness of 0.584 cm, and cavity radius ∼4.2 cm. The Qof the modes due to wall losses (copper) ∼14,500. Given our implementation of the SLED-I pulse compression sy stem[9], to optimize the rf eﬃciency the average (synchronous) dipole m ode frequency needs to be 4.012 GHz. Fig. 2 shows the dispersion curves of th e ﬁrst two dipole bands for representative constant impedance, S- band structures, withavarying from 1.30 cm to 2.00 cm. The results of a ﬁnite element , Maxwell Equations solving program, OMEGA2[10], are given b y the plotting symbols. The end points of the curves are used to ﬁx the parame ters in the 10circuit program. The two-band circuit results for these con stant impedance structures are indicated by the curves in the ﬁgure. We note g ood agreement in the ﬁrst band results and not so good agreement in those of t he second band. For a detuned structure, to obtain the local circuit pa rameters, we interpolate from these representative dispersion curves. Figure 2: The dispersion curves of the ﬁrst two dipole bands o f represen- tative constant impedance S-band structures. The phase adv ance per cell is 2π/3. Results are given for iris radii of a= 1.30, 1.51, 1.63, 1.80, 1.92, and 2.00 cm. The plotting symbols give OMEGA2 results, the curve s those of the circuit model. The dashed line is the speed of light curve . We have 3 parameters to vary in our input (uncoupled) frequen cy distri- bution: the (relative) shift in average frequency from the n ominal 4.012 GHz, δ¯f, the (relative) width of the distribution ∆ δf, and the ﬂat-top tilt param- eterα. Varying these parameters we calculate Srms0,Srms, and the peak value of \|S\|,\|ˆS\|, for the coupled results, for both bunch train conﬁguration s. These parameters serve as indicators, respectively, of emi ttance growth due to BBU (injection jitter), emittance growth due to misalign ments, and the maximum beam excursion due to BBU. From our numerical simula tions we ﬁnd that a fairly optimized case consists of δ¯f=−2.4%, ∆ δf= 7.5%, and 11α=−0.20. In Fig. 3 we display, for the optimized case, the frequency di stribu- tion (a), the kick factors (b), and the envelope of the wake (c ). The dashed curves in (a) and (b) give the uncoupled (input) values. The p lotting sym- bols in (c) give \|W\|at the bunch positions for the alternate (1.4 ns) bunch train conﬁguration. From (b) we note that there are a few mode s, trapped near the beginning of the structure, which have kick factors signiﬁcantly larger than the rest. This is a consequence of the fact that, f or all cells of this structure, the dispersion curves are backward waves . From (c) we see that, due to these few strong modes, the wake envelope doe s not nearly reach the low, ﬂat bottom that it does for the idealized, unif orm frequency distribution (see Fig. 1). We note, however, that the short- range drop-oﬀ is similar to the idealized form (see Fig. 1), for about 20 m. I n addition we note that, by setting the second bunch near the zero crossi ng, many following bunches also have wakes with amplitudes signiﬁca ntly below the wake envelope. Finally, in Fig. 4 we present the sum wake for b oth bunch train conﬁgurations. For this case, for both bunch train con ﬁgurations, Srms0=Srms=.02 MV/nC/m2. Note that if we set δ¯fback to 0, then, for the 1.4 ns bunch spacing option, Srms0becomes a factor of 20 larger. 3.2 3π/4Phase Advance Per Cell If we would like to regain some of the 7% in accelerating gradi ent that we lost by shifting ¯f, we can move to a structure where the group velocity at the synchronous point is less than for the 2 π/3 structure (for the same ¯f). One solution is to go to a structure with a 3 π/4 synchronous point. Note that in such a structure the cell length is 3.94 cm and that there are 1 02 cells per structure. Note also that for the same group velocity for the fundamental mode a higher phase advance implies larger values of iris rad iusa, which will also improve the short-range wakeﬁeld tolerances. The disp ersion curves are shown in Fig. 5. Note that, in this case, our distribution wil l have f0< fπfor the cell geometries near the beginning of the structure, f0> fπfor the cell geometries near the end of the structure, while the synchron ous phase is near pi phase advance. Consequently modes touching either end of the structure will only weakly interact with the beam (see, egRef. [8]), allowing us to have a smoother impedance function, and therefore a more uniform ly suppressed wakeﬁeld envelope. This was not the case for 2 π/3 structure, where the dispersion curves for all cells have a negative slope (betwe en 0 and πphase advance) (see Fig. 2); it is also not the case for a 5 π/6 structure, where the slopes would all be positive. 12Figure 3: The optimized 2 π/3 structure: δ¯f=−2.4%, ∆ δf= 7.5%, and α=−0.20. Given are the frequency distribution λf(a), the mode kick factors k(b), and the envelope of the wake ˆW(c). The dashes in (a) and (b) give the uncoupled results; the plotting symbols in (c) g ive\|W\|at the bunch positions for the alternate (1.4 ns) bunch train conﬁg uration. Note that we display only the modes of the ﬁrst dipole band, and the wake due to these modes. 13Figure 4: The sum wake for the optimized 2 π/3 structure, for both the nominal and alternate bunch train conﬁgurations. In both ca sesSrms0= Srms=.02 MV/nC/m2. Figure 5: The dispersion curves of the ﬁrst two dipole bands o f represen- tative cells in a 3 π/4 structure. Results are given for iris radii of a= 1.33, 1.48, 1.63, 1.80, and 1.98 cm. The plotting symbols give OMEG A2 results, the curves those of the circuit model. The dashed line is the s peed of light line. 14Figure 6: For the 3 π/4 structure: dependence of 3 sum wake parameters onδ¯ffor the nominal (left frames) and alternate (right frames) b unch train conﬁgurations. The ordinate units are MV/nC/m2. The optimum, δ¯f= −2.3%, ∆ δf= 5.8%, and α=−0.20, is indicated by the plotting symbol. 15Figure 7: For the 3 π/4 structure: dependence of 3 sum wake parameters on ∆ δf, for the nominal (left frames) and alternate (right frames) bunch train conﬁgurations. The ordinate units are MV/nC/m2. The optimum, δ¯f=−2.3%, ∆ δf= 5.8%, and α=−0.20, is indicated by the plotting symbol. 16Figure 8: For the 3 π/4 structure: dependence of 3 sum wake parameters onα, for the nominal (left frames) and alternate (right frames) bunch train conﬁgurations. The ordinate units are MV/nC/m2. The optimum, δ¯f= −2.3%, ∆ δf= 5.8%, and α=−0.20, is indicated by the plotting symbol. 17Again optimizing on the sum wake, we ﬁnd that for a fairly opti mized caseδ¯f=−2.3%, ∆ δf= 5.8%, and α=−0.20. The change of the indicators \|ˆS\|,SrmsandSrms0as we deviate from this point, for both bunch train conﬁgurations, is shown in Figs. 6-8. In Fig. 6 we show the ¯fdependence. We see that, for both bunch train conﬁgurations the results a re very sensitive to¯f. In Fig. 7 we give the ∆ δfdependence. We can clearly see the eﬀect of the resurgence in the wake when ∆ δf/greaterorsimilar7%. And ﬁnally, in Fig. 10 we give theαdependence. We note that the tilt in the distribution helps p rimarily in reducing the sensitivity to BBU for the nominal (2.8 ns) bu nch train conﬁguration. In Fig. 9 we display, for the optimized 3 π/4 case, the frequency distribu- tion (a), the kick factors (b), and the envelope of the wake (c ). From (b) we note that in this case k(f) is a relatively smooth function, as was expected from our earlier discussion. From (c) we see that the wake env elope reaches a broader, ﬂatter bottom than for the 2 π/3 structure, again as we expected. Again we note that many of the earlier bunches have wakes with amplitudes signiﬁcantly below the wake envelope. Finally, in Fig. 9 we s how the sum wake for both bunch train conﬁgurations. The rms of these sum wakes are much smaller than for the 2 π/3 structure: Srms0=Srms=.004 MV/nC/m2. 3.3 Frequency Errors How sensitive are our results to manufacturing errors? We wi ll begin to explore this question by looking at the dependence of the sum wake on errors in the synchronous frequencies of the cells of the str ucture. Note that the synchronous frequency of a cell is not equally sensitive to each of the cell dimensions. Basically there are 4 dimensions: the iris radi usa, the cavity radius b, the iris thickness d, and the period length p. The synchronous frequency fsis insensitive to dandp, and for the average S-band cell we ﬁnd that δfs=−.85δbandδfs=−.15δa. Or, a −1 micron change in b results in δfs= 2×10−5; a−1 micron change in aresults in δfs= 1×10−5. As for attainable accuracy, let us assume that each synchron ous frequency can be obtained to a relative accuracy of 10−4, or to about .5 MHz. As for systematic frequency errors we note from Fig. 6 that we are es- pecially sensitive to changes in average frequency. For exa mple, to dou- bleSrms0from its minimum, requires a relative frequency change of on ly 4×10−4. If each cell frequency has an accuracy of 10−4, and there are about 100 cells, the accuracy in the centroid frequency shou ld be ∼10−5. Therefore, the eﬀect of this type of systematic error should be negligible. As for random manufacturing errors, let us distinguish two t ypes: “sys- 18Figure 9: For the optimized 3 π/4 structure, δ¯f=−2.3%, ∆ δf= 5.8%, andα=−0.20: the frequency distribution λf(a), the mode kick factors k(b), and the envelope of the wake ˆW(c). The dashes in (a) and (b) give the uncoupled results; the plotting symbols in (c) give \|W\|at the bunch positions for the alternate (1.4 ns) bunch train conﬁgurati on. Note that we display only the modes of the ﬁrst dipole band, and the wake du e to these modes. 19Figure 10: The sum wake for the optimized 3 π/4 structure, given for both the nominal and alternate bunch train conﬁgurations. F or both cases Srms0=Srms=.004 MV/nC/m2. tematic random” and “purely random” errors. By “systematic random” we mean errors, random in one structure, that are repeated in al l structures of the prelinac subsystem. “Purely random” errors are, in ad dition, ran- dom from structure to structure. In Fig. 11 we give the result ingSrms0and Srms, for both bunch train conﬁgurations, when a random error com ponent is added to the (input) synchronous frequencies of the optim al distribution. With a frequency spacing of ∼8×10−4, an rms frequency error of 10−4is a relatively small perturbation. We see that for the alternat e (1.4 ns) bunch spacing the eﬀect of such a perturbation is indeed very small , whereas for the nominal (2.8 ns) bunch spacing the eﬀect is large. The rea son is that with the 1.4 ns bunch spacing the beam sits near a half-intege r resonance, whereas for the 2.8 ns spacing it sits near the integer resona nce. (Resonant multi-bunch wakeﬁeld eﬀects are discussed in Appendix C.) N ote, however, that if we consider the case of “purely random” machining err ors, with a relative accuracy in synchronous frequencies of 10−4, and considering we haveNstruc= 140, 127, 41 structures in, respectively, the prelinac, th ee+ drive linac, and the e−booster, then, with a 1 /√Nstrucreduction in sensi- tivity, the appropriate abscissas in the ﬁgure become .8, .9 , and 1 .6×10−5. At these points, for the 2.8 ns spacing, we see that Srms0is only a factor 2±1, 2±1, 3±2 times larger than the zero error result. Finally, as for the “systematic random” errors, it is diﬃcult to judge how la rge they might be in the real structure; however, they are likely an order of magnitude less than the purely random errors, and should therefore not yiel d a sum wake much larger than that due to the purely random manufacturing errors. 20If we make a weak damping, approximate calculation, by redoi ng the calculation but now with Q= 1000, we ﬁnd no appreciable eﬀect on the resonance behavior for the ∆ t= 2.8 ns case with frequency errors. For strong damping, taking Q= 100, however, we do ﬁnd a suppression of the resonance eﬀect. Figure 11: The optimal 3 π/4 structure: the eﬀect of random frequency errors. Shown are the relative (synchronous) frequency err or,σf,errvsSrms0. The dashed curves give Srms. Each plotting symbol, with its error bars, represents 400 diﬀerent seeds. 3.4 e+Booster For the e+booster (the L–band machine) each accelerator structure co nsists of 72 cells and the synchronous phase advance is taken to be 2 π/3. The synchronous dipole mode distribution is taken to be uniform , with ¯f= 2.05 GHz and ∆ δf= 3%. We take Q= 20,000. Note that in this case ¯f∆t= 2.87; for the second bunch to sit on the zero crossing of the wake ﬁeld would require a shift in frequency of +4 .5% or −13%. For the L-band structure not much can be gained by changing the frequency sp ectrum. We do gain, however, a factor of 8 reduction in wake in going from S- to L-band, which, as we shall see, suﬃces. 214 Tolerances For our designed structures we perform the tolerance calcul ations presented in Section 2. To check the analytical estimates, tracking, u sing the computer program LIAR[11], was also performed. The numerical simula tions were simpliﬁed in that one macro-particle was used to represent e ach bunch, and the bunch train was taken to be mono–energetic. The analytic al BBU results and those of LIAR, at the end of the four injector linacs, are c ompared in Table 4. Results are given for bunch spacings of 1.4 and 2.8 ns . Under the heading “Analytical” are given the rms of the sum wake Srms, the rms of the growth factor Υ rms, the maximum (within the bunch train) of the growth factor ˆΥ, and the tolerance rtforδǫt= 10%, i.e.that ratio y0/σy0 that results in 10% emittance growth. Under the heading “Num erical” we give the LIAR results: the maximum (within the bunch train) g rowth in normalized phase space ˆξand the tolerance rtfor 10% emittance growth, both referenced to the centroid of the ﬁrst bunch in the train . Table 4: Beam break-up calculations for the two bunch spacin gs. Given are the rms of the sum wake Srms0(in units of MV/nC/m2), the rms and the peak of the strength parameter, Υ rms0andˆΥ, respectively, and the analytically obtained tolerance rtforδǫt= 10%, i.e.that ratio y0/σy0that results in 10% emittance growth. Also shown are LIAR results : the peak blow-up in normalized phase space, ˆξ, and the tolerance rt. Analytical Numerical∆t NameSrms0Υrms0ˆΥ rtˆξ rt Prelinac .004 .007 .025 70 .020 85 e+Drive .004 .017 .066 25 .047 352.8nse−Booster .004 .009 .036 50 .026 65 e+Booster .12 .119 .227 3.8 .153 5.5 Prelinac .004 .004 .019 115 .015 140 e+Drive .004 .011 .049 45 .035 601.4nse−Booster .004 .006 .027 80 .019 105 e+Booster .30 .205 .379 2.2 .257 3.0 We note from Table 4 that for the S-band machines, Υ rms0andˆΥ are both small compared to 1, and that the injection jitter toler ance for 10% emittance growth is very large. For the L-band machine, the e+booster, the tolerances are tighter but still acceptable. We note also th at the analytical 22Table 5: Eﬀect of structure misalignments for the two bunch s pacings. Given are the rms of the sum wake Srms(in units of MV/nC/m2) and the tolerance for structure misalignments for a 10% emittance growth, xat, both obtained analytically (Eqs. 6,8) and by LIAR. Analytical Numerical∆t NameSrms xat[mm]xat[mm] Prelinac .004 2.9 3.2 e+Drive .004 100. 120.2.8nse−Booster .004 140. 170. e+Booster .022 590. Prelinac .004 4.6 4.8 e+Drive .004 150. 180.1.4nse−Booster .004 210. 260. e+Booster .040 450. ˆΥ agrees well with the numerical ˆξ, as do the two versions of rt. In Table 5 we present misalignment results. Given are Srmsand the tolerance for structure misalignments, xat, both as given analytically and by LIAR. As discussed before, the meaning of xatis the rms misalignment that (for an ensemble of machines) results, on average, in a ﬁ nal emittance growth equal to a tolerance δǫt, which in this case we set to 10%. From Table 5 we see that the analytical and numerical results agre e well, and that the misalignment tolerances are all very loose. The tightes t tolerance is for the prelinac with nominal (2.8 ns) bunch spacing, where the t olerance is still an acceptable 3 mm. The eﬀect of machining errors will tighten these tolerances for the S- band machines with the nominal (2.8 ns) bunch spacing, due to the beam being near the integer resonance. If machining adds a purely random error component that is equivalent to 10−4frequency error, we saw earlier that (for the 2.8 ns bunch spacing case only) this will tighten the injection jitter tolerances by about a factor 2 ±1 for the prelinac and e+drive linac, and about a factor of 3 ±2 for the e−booster. But even with this, the toler- ances are still very loose. The misalignment tolerances are aﬀected less by machining errors. The prelinac tolerance, with 2.8 ns bunch spacing, will become ∼2±1 mm; for 95% conﬁdence in achieving δǫ= 10%, the tolerance becomes ∼1±.5 mm. 23Finally, what is the random, cell-to-cell misalignment tol erance? Per- forming the perturbation calculation described in Appendi x B, and cal- culating for 1000 diﬀerent random structures, we ﬁnd that Srms=.27± .12 MV/nC/m2for ∆t= 2.8 ns, and Srms=.032±.003 MV/nC/m2for ∆t= 1.4 ns. We again see the eﬀect of the integer resonance on the 2.8 ns option result. (To verify that this is the case, we performed one run but with the bunch spacing changed so that the beam sits near the next h alf-integer resonance (11.5); the result was that Srmsdropped by a factor of 6.) For the prelinac the cell-to-cell misalignment tolerance beco mes 40 µm for the nominal (2.8 ns) bunch conﬁguration and 600 µm for the alternate (1.4 ns) conﬁguration. 5 Conclusion We have demonstrated that by using detuning alone, the four i njector linacs can be built to suﬃciently suppress the multi-bunch wakeﬁel d eﬀects, for both the nominal (2.8 ns) and alternate (1.4 ns) bunch spacin gs. We have studied the sensitivity to multi-bunch beam break-up (BBU) and to struc- ture misalignments through analytical estimates and numer ical tracking, and shown that the tolerances to injection jitter, in the for mer case, and to structure misalignments, in the latter case, are not diﬃcul t to achieve. We have also studied the eﬀect of manufacturing errors on these tolerances, and have shown that if the errors are purely random, with an equiv alent rms fre- quency error of 10−4, then the other tolerances are still acceptable. Finally, we have shown that the cell-to-cell misalignment tolerance is/greaterorsimilar40µm. For the L–band machine—the e+Booster—we have shown that a uni- form detuning of the dipole modes, with central frequency ¯f= 2.05 GHz and a total frequency spread ∆ δf= 3%, suﬃces. For the S–band linacs— the Prelinac, the e+Drive Linac, and the e−Booster—we have shown that the 1.4 ns bunch spacing option forces us to reduce the centra l frequency by 2.3% from the nominal 4.012 GHz. Doing this we lose 7% in eﬀe ctive gradient, which, however, can be regained by increasing the phase advance per cell from 2 π/3 to 3 π/4. Our ﬁnal, optimized distribution is trapezoidal in shape with ¯f= 3.920 GHz, ∆ δf= 5.8%, and tilt parameter α=−0.2. We have demonstrated in this report that the integer resonan ce, which we cannot avoid given the two bunch train alternatives, can m ake us more sensitive to manufacturing errors. Also, we have shown that the analytical, single-bunch beam break-up theory, when slightly modiﬁed, can be useful in predicting the behavior of multi-bunch beam break-up also. Given the rather 24loose tolerances demonstrated here makes us think that the S -band machines can be replaced with C-band ones that still have reasonable t olerances, an option which may result in savings in cost, though this needs further study. Finally, we should reiterate that in this report we were conc erned with the eﬀects of the modes in the ﬁrst dipole passband only. With the wakeﬁeld of the ﬁrst band modes greatly suppressed by detuning, the eﬀ ects of the higher bands may no longer be insigniﬁcant. This problem wil l need to be addressed in the future. Acknowledgments The author thanks the regular attendees of the Tuesday JLC/N LC linac meetings at SLAC for helpful comments and discussions on thi s topic, and in particular T. Raubenheimer, our leader, and V. Dolgashev for carefully reading parts of this manuscript. References [1] NLC ZDR Design Report, SLAC Report 474, 589 (1996). [2] See the JLC/NLC Accelerator Physics at SLAC web site at: w ww- project.slac.stanford.edu/lc/local/AccelPhysics/Acc elPhysics index.htm. [3] R. M. Jones, et al, “Equivalent Circuit Analysis of the SLAC Damped Detuned Structures,” Proc. of EPAC96, Sitges, Spain, 1996, p. 1292. [4] A. Chao, “Physics of Collective Instabilities in High-E nergy Accelera- tors”, John Wiley & Sons, New York (1993). [5] K. Bane, et al, “Issues in Multi-Bunch Emittance Preservation in the NLC,” Proc. of EPAC94, London, England, 1994, p. 1114. [6] R. M. Jones, et al, “Emittance Dilution and Beam Breakup in the JLC/NLC,” Proc. of PAC99, New York, NY, 1999, p. 3474. [7] V. Dolgashev, et al, “Scattering Analysis of the NLC Accelerating Struc- ture,” Proc. of PAC99, New York, NY., 1999, p. 2822. [8] K. Bane and R. Gluckstern, Part. Accel. ,42, 123 (1994). [9] Z. Li, et al, “Parameter Optimization for the Low Frequency Linacs in the NLC,” Proc. of PAC99, New York, NY, 1999, p. 3486. 25[10] X. Zhan, “Parallel Electromagnetic Field Solvers ...,” PhD Thesis, Stanford University, 1997. [11] R. Assmann, et al, LIAR Reference Manual, SLAC/AP-103, April 1997. [12] R. Helm and G. Loew, Linear Accelerators , North Holland, Amsterdam, 1970, Chapter B.1.4. [13] E. U. Condon, J. Appl. Phys. 12, 129 (1941). [14] P. Morton and K. Neil, UCRL-18103, LBL, 1968, p. 365. [15] K.L.F. Bane, et al, in “Physics of High Energy Accelerators,” AIP Conf. Proc.127, 876 (1985). [16] K. Bane and B. Zotter, Proc. of the 11thInt. Conf. on High Energy Acellerators, CERN (Birkh¨ auser Verlag, Basel, 1980), p. 5 81. [17] D. Schulte, presentation given in an NLC Linac meeting, summer 1999. Appendix A: Analytical Formula for Weak Multi-Bunch BBU In Ref. [4] an analytical formula for single-bunch beam break-up in a smooth focusing linac, for the case without energy spread in the bea m, is derived, the so-called Chao-Richter-Yao (CRY) model for beam break- up. Suppose the beam is initially oﬀset from the accelerator axis. The be am break- up downstream is characterized by a strength parameter Υ( t,s), where t represents position within the bunch, and sposition along the linac. When Υ(t,s) is small compared to 1, the growth in betatron amplitude in t he linac is proportional to this parameter. When applied to the speci al case of a uniform longitudinal charge distribution, and a linearly g rowing wakeﬁeld, the result of the calculation becomes especially simple. In this case the growth in orbit amplitude is given as an asymptotic power ser ies in Υ( t,s), and the series can be summed to give a closed form, asymptotic solution for single-bunch BBU. The derivation of an analytic formula for multi-bunch BBU is almost a trivial modiﬁcation of the CRY formalism. We w ill here reproduce the important features of the single-bunch deriv ation of Ref. [4] (with slightly modiﬁed notation), and then show how it can be modiﬁed to obtain a result applicable to multi-bunch BBU. Note that we a re interested in estimating the eﬀect of relatively weak multi-bunch BBU, caused by the somewhat complicated wakeﬁelds of detuned structures. The more studied 26multi-bunch BBU problem, i.e.the eﬀect on a bunch train of a single strong mode, the so-called “cumulative beam break-up insta bility” (see, e.g. Ref. [12]), is a somewhat diﬀerent problem to which our resul ts are not meant to apply. Let us consider the case of single-bunch beam break-up, wher e a beam is initially oﬀset by distance y0in a linac with acceleration and smooth focusing. We assume that there is no energy spread within the beam. The equation of motion is 1 E(s)d ds/bracketleftbigg E(s)dy(t,s) ds/bracketrightbigg +y(t,s) β2(s)=e2Nt E(s)/integraldisplayt −∞dt′y(t′,s)λt(t′)W(t−t′), (A1) withy(t,s) the bunch oﬀset, a function of position within the bunch t, and position along the linac s; with Ethe beam energy, [1 /β(s)] the betatron wave number, eNtthe total bunch charge, λt(t) the longitudinal charge distribution, and W(t) the short-range dipole wakeﬁeld. Our convention is that negative values of tare toward the front of the bunch. Let us, for the moment, limit ourselves to the problem of no acceleration an dβa constant. A. Chao in Ref. [4] expands the solution to the equation of mot ion for this problem in a perturbation series y(t,s) =∞/summationdisplay n=0y(n)(t,s), (A2) with the ﬁrst term given by free betatron oscillation [ y(0)=y0cos(s/β)]. He then shows that the solution for the higher terms at position s=L, after many betatron oscillations, is given by y(n)(t,L)≈y0 n!/parenleftbiggie2NtLβ 2E/parenrightbiggn Rn(t)eiL/β, (A3) with Rn(t) =/integraldisplayt −∞dt1λ(t1)W(t−t1)/integraldisplayt1 −∞dt2λ(t2)W(t1−t2) ···/integraldisplaytn−1 −∞dtnλ(tn)W(tn−1−tn), (A4) andR0(z) = 1. An observable yis meant to be the real part of Eq. A2. The eﬀects of adiabatic acceleration, i.e.suﬃciently slow acceleration so that the energy doubling distance is large compared to the be tatron wave 27length, and βnot constant, can be added by simply replacing ( β/E) in Eq. A3 by /angb∇acketleftβ/E/angb∇acket∇ight, where angle brackets indicate averaging along the linac froms= 0 to s=L.2For example, if the lattice is such that β∼Eζ then/angb∇acketleftβ/E/angb∇acket∇ight= (β0/E0)g(Ef/E0,ζ), where subscripts “0” and “ f” signify, respectively, initial and ﬁnal parameters, and g(x,ζ) =1 ζ/parenleftbiggxζ−1 x−1/parenrightbigg [β∼Eζ]. (A5) Chao then shows that for certain simple combinations of bunc h shape and wake function shape the integrals in Eq. A4 can be perform ed analyt- ically, and the result becomes an asymptotic series in power s of a strength parameter. For example, for the case of a uniform charge dist ribution of length ℓ(with the front of the bunch at t= 0), and a wake that varies as W=W′t, the strength parameter is Υ(t,L) =e2NtLW′t2β0 2E0ℓg(Ef/E0,ζ). (A6) If Υ is small compared to 1, the growth is well approximated by Υ. If Υ is large, the sum over all terms can be performed to give a clos ed form, asymptotic expression. Formulti-bunch BBU we are mainly concerned with the interaction of the diﬀerent bunches in the train, and will ignore wakeﬁeld f orces within bunches. The derivation is nearly identical to that for the s ingle-bunch BBU. However, in the equation of motion, Eq. A1, the independ ent variable tis no longer a continuous variable, but rather ttakes on discrete values tm=m∆t, where mis a bunch index and ∆ tis the bunch spacing. Also, W now represents the long-range wakeﬁeld. Let us assume that t here are M, equally populated bunches in a train; i.e.Nt=MN, with Nthe particles per bunch. The solution is again expanded in a perturbation s eries. In the solution, Eq. A3, the Rn(t), which are smooth functions of t, are replaced by R(n) m=1 Mm−1/summationdisplay j=1W[(m−j)∆t]R(n−1) j , (A7) (with R0 j= 1), which is a function of a discrete parameter, the bunch in dex m. Note that R(1) m=Sm/M, with Smthe sum wake. 2Note that the terms y0eiL/βin Eq. A3, related to free betatron oscillation, also need to be modiﬁed in well-known ways to reﬂect the dependence of βonE. It is the other terms, however, which characterize BBU, in which we are inte rested. 28Generally the sums in Eq. A7 cannot be given in closed form, an d there- fore a closed, asymptotic expression for multi-bunch BBU ca nnot be given. We can still, however, numerically compute the individual t erms equivalent to Eq. A3 for the single bunch case. For example, the ﬁrst orde r term in amplitude growth is given by Υm=e2NLS mβ0 2E0g(Ef/E0,ζ) [ m= 1,... ,M ]. (A8) If this term is small compared to 1 for all m, then BBU is well characterized by Υ. If it is not small, though not extremely large, the next h igher terms can be computed and their contribution added. For Υ very large, t his approach may not be very useful. From our derivation we see that there is nothing that fundame ntally distinguishes our BBU solution from a single-bunch BBU solu tion. If we consider again the single-bunch calculation, for the case o f a uniform charge distribution of length ℓ, we see that we need to perform the integrations forRnin Eq. A4. If we do the integrations numerically, by dividing the integrals into discrete steps tn= (n−1)∆tand then performing quadrature by rectangular rule, we end up with Eq. A7 with M=ℓ/∆t. The solution is the same as our multi-bunch solution. What distinguishes th e multi-bunch from the single-bunch problem is that the wakeﬁeld for the mu lti-bunch case is not normally monotonic and does not vary smoothly with lon gitudinal position. For such a case it may be more diﬃcult to decide how m any terms are needed for the sum to converge. In Fig. 12 we give a numerical example: the NLC prelinac with t he op- timized 3 π/4 S-band structure, but with 10−5systematic frequency errors, with the nominal (2.8 ns) bunch spacing (see the main text). T he diamonds give the ﬁrst order (a) and the second order (b) perturbation terms. The crosses in (a) give the results of a smooth focusing simulati on program (tak- ingβ∼E1/2), where the free betatron term has been removed. We see that the agreement is very good; i.e.the ﬁrst order term is a good approximation to the simulation results. In (b) we note that the next order t erm is much smaller. For this example we ﬁnd that even if we increase the c urrent by an order of magnitude the 1st order term alone remains a good app roximation. 29Figure 12: A numerical example: the NLC prelinac with the opt imized 3 π/4 S-band structure, but with 10−5systematic frequency errors, with the nom- inal (2.8 ns) bunch spacing (see the main text). Srms=.005 MV/nC/m2. The diamonds give the ﬁrst order (a) and the second order (b) p erturbation terms. The crosses in (a) give smooth focusing simulation re sults with the free betatron term removed. Appendix B: The Wakeﬁeld Due to Cell-to-Cell Misalignments We assume a structure is composed of many cells that are misal igned trans- versely by amounts that are very small compared to the cell di mensions. For such a case we assume that the mode frequencies are the sam e as in the ideal structure, and only the mode kick factors are aﬀected. To ﬁrst order we assume that for each mode, the kick factor for the beam on-a xis in the imperfect structure is the same as for the case with the beam f ollowing the negative of the misalignment path in the error-free structu re. In Fig. 13 we sketch a portion of such a misaligned structure (top) and the model used for the kick factor calculation (bottom). The sketch is mean t to represent a 30disk-loaded structure that has been built up from a collecti on of cups. Note that the relative size of the misalignements is exaggerated from what is ex- pected, in order to more clearly show the principle. Given th is model, the method of calculation of the kick factors can be derived usin g the so-called “Condon Method”[13],[14] (see also [15]). Note that this ap plication to cell- to-cell misalignments in an accelerator structure is prese nted in Ref. [6]. The results of this perturbation method have been shown to be consistent with those using a 3-dimensional scattering matrix analysi s[7]. We will only sketch the derivation below. bunch bunch Figure 13: Sketches of part of a misaligned structure (top) a nd the model used for the kick factor calculation (bottom). Note that the relative size of the misalignments here is much exaggerated. Consider a closed cavity with perfectly conducting walls. F or such a cavity the Condon method expands the vector and scalar poten tials, in the 31Coulomb gauge, as a sum over the empty cavity modes. As functi on of position x(x,y,z) and time tthe vector potential in the cavity is given as A(x,t) =/summationdisplay λqλ(t)aλ(x), (B1) where ∇2aλ+ω2 λ c2aλ= 0 , (B2) withωλthe frequency of mode λ, andaλ×ˆ n= 0 on the metallic surface ( ˆ n is a unit vector normal to the surface). Using the Coulomb gau ge implies that∇ ·aλ= 0. The qλare given by ¨qλ+ω2 λqλ=1 2Uλ/integraldisplay VdVj·aλ, (B3) with the normalization ǫ0 2/integraldisplay VdVaλ′·aλ=Uλδλλ′, (B4) withjthe current density. Note that the integrations are perform ed over the volume of the cavity V. The scalar potential is given as Φ(x,t) =/summationdisplay λrλ(t)φλ(x), (B5) where ∇2φλ+Ω2 λ c2φλ= 0 , (B6) with Ω λthe frequencies associated with φλ, and with φλ= 0 on the metallic surface. The rλare given by rλ=1 2Tλ/integraldisplay VdVρφλ, (B7) withρthe charge distribution in the cavity. Note that one fundame ntal diﬀerence between the behavior of A(x,t) and Φ( x,t) is that when there are no charges in the cavity the vector potential can still oscil late whereas the scalar potential must be identically equal to 0. Let us consider an ultra-relativistic driving charge Qthat passes through the cavity parallel to the zaxis, and (for simplicity) a test charge following at a distance sbehind on the same path. Both enter the cavity at position 32z= 0 and leave at position z=L. The transverse wakeﬁeld at the test charge is then W(s) =1 QLx 0/integraldisplayL 0dz[c∇⊥Az− ∇⊥Φ]t=(z+s)/c =1 QLx 0/summationdisplay λ/integraldisplayL 0dz/bracketleftbigg cqλ/parenleftbiggz+s c/parenrightbigg ∇⊥aλz(z) −rλ/parenleftbiggz+s c/parenrightbigg ∇⊥φλ(z)/bracketrightbigg ,(B8) where the integrals are along the path of the particle trajec tory. The param- eterx0is a parameter for transverse oﬀset (the transverse wake is u sually given in units of V/C per longitudinal meter per transverse m eter); for a cylindrically-symmetric structure it is usually taken to b e the oﬀset, from the axis, of the driving bunch trajectory. For s > L we can drop the scalar potential term (it must be zero when there is no charge in the c avity), and the result can be written in the form[15] W(s) =/summationdisplay λc 2UλωλLx0ℑm/bracketleftBig V∗ λ∇⊥Vλeiωλs/c/bracketrightBig [s > L], (B9) with Vλ=/integraldisplayL 0dz aλz(z)eiωλz/c. (B10) Note that the arbitrary constants associated with the param eteraλin the numerator and the denominator of Eq. B9 cancel. Note also tha t—to the same arbitrary constant— \|Vλ\|2is the square of the voltage lost by the driving particle to mode λandUλis the energy stored in mode λ. Consider now the case of a cylindrically-symmetric, multi- cell acceler- ating cavity, and let us limit our concern to the eﬀect of the d ipole modes of such a structure. We will allow the charges to move on an arb itrary, zig-zag path in the x−zplane that is close to the axis, and for which the slope is everywhere small (so that ∇⊥∼∂/∂x). For dipole modes in a cylindrically-symmetric, multi-cell accelerator struct ure, it can shown that the synchronous component of aλz(the only component that, on average, is important) can be written in the form aλz=xfλ(z) (see e.g.Ref. [16]). Then Eq. B9 becomes Wx(s) =/summationdisplay λc 2UλωλLx0× (B11) ×ℑm/bracketleftbigg eiωλs/c/integraldisplayL 0dz′x(z′)fλ(z′)e−iωλz′/c/integraldisplayL 0dz fλ(z)eiωλz/c/bracketrightbigg [s > L]. 33Note that this equation can be written in the form: Wx(s) =/summationdisplay λ2k′ λsin/parenleftBigωλs c+θλ/parenrightBig [s > L], (B12) withk′ λa kind of kick factor and θλthe phase of excitation of mode λ. Note that in the special case where the particles move parall el to the axis, at oﬀset a,k′ λ=kλ=c\|Vλ\|2/(4Uλωλa2L), the normal kick factors for the structure, and θλ= 0. For this case it can be shown that Eq. B12 is valid for alls >0[15]. Finally, note that, for the general case, Eq. B12 can o bviously not be extrapolated down to s= 0, since it implies that Wx(0)/negationslash= 0, which we believe is nonphysical, implying that a particle can kick it self transversely. To obtain the proper equation valid down to s= 0 we would need to include the scalar potential term that was dropped in going from Eq. B 8 to Eq. B9. Our derivation, presented here, is technically applicable only to struc- tures for which all modes are trapped. The modes will be trapp ed at least at the ends of the structure, if the connecting beam tubes hav e suﬃciently small radii and the dipole modes do not couple to the fundamen tal mode couplers in the end cells. For detuned structures, like thos e in the injector linacs discussed in this report, most modes are trapped inte rnally within a structure, and those that do extend to the ends couple only we akly to the beam; for such structures the results here can also be applie d, even if the conditions on the beam tube radii and the fundamental mode co upler do not hold. We believe that even for the damped, detuned structure s of the main linac of the JLC/NLC, which are similar, though they have man ifolds to add weak damping to the wakeﬁeld, a result very similar to tha t presented here applies. To estimate the wakeﬁeld associated with very small, random cell-to- cell misalignments in accelerator structures we assume tha t we can use the mode eigenfrequencies and eigenvectors of the error-free s tructure. We ob- tain these from the circuit program. Then to ﬁnd the kick fact ors we replace x(z) in the ﬁrst integral in Eq. B11 by the zig-zag path represent ing the neg- ative of the cell misalignments, a path we generate using a ra ndom number generator. The normalization factor x0is set to the rms of the misalign- ments. How can we justify using this method for ﬁnding the wak e at the spacing of the bunch positions? For example, for the 3 π/4 S-band struc- ture, the alternate bunch spacing is only 42 cm whereas the wh ole structure length L= 4.46 m. Therefore, in principle, Eq. B11 is not valid until the 11thbunch spacing. We believe, however, that the scalar potenti al ﬁelds will not extend more than one or two cells behind the driving c harge (the 34cell length is 4.375 cm), and therefore this method will be a g ood approxi- mation at all bunch positions behind the driving charge. Thi s belief should be tested in the future by repeating the calculation, but now also including the contribution from scalar potential terms. In Fig. 14 we give a numerical example. Shown, for the optimiz ed 3π/4 S-band structure (see the main text), are the kick factors an d the phases of the modes as calculated by the method described in this secti on. Note that θλis not necessarily small. Figure 14: The kick factors and phases of the modes for a cell- to-cell misalignment example. The structure is the optimized 3 π/4 S-band struc- ture (see the main text). For this example, for the nominal (2 .8 ns) bunch spacing, Srms=.32 MV/nC/m2. Appendix C: Resonant Multi-Bunch Wakeﬁeld Eﬀects It is easy to understand how resonances can arise in a linac wi th bunch trains. Consider the case of the interaction of the beam with one single structure mode. The leading bunch enters the structure oﬀse t from the axis and excites the mode. If the bunch train is sitting on an integ er resonance, i.e.iff∆t=n, with fthe mode frequency, ∆ tthe bunch spacing, and nan integer, then when the 2nd bunch arrives it will excite the mo de at the same phase and also obtain a kick due to the wakeﬁeld of the ﬁrst bun ch. The mth bunch will also excite the mode in the same phase and obtain (m−1) times the kick from the wakeﬁeld that the second bunch experi enced (for 35simplicity we assume the mode Qis inﬁnity). On the half-integer resonance, i.e.when f∆t=n+.5, the mth bunch will also receive kicks from the wakeﬁeld left by the earlier bunches, but in this case the kic ks will alternate in direction, and no resonance builds up. For a transverse wa keﬁeld eﬀect, such as we are interested in, however, this simple descripti on of the resonant interaction needs to be modiﬁed slightly. For this case the w ake varies as sin(2πft), and neither the integer nor the half-integer resonance co ndition will excite any wakeﬁeld for the following bunches. In this c ase resonant growth is achieved at a slight deviation from the condition f∆t=n, as is shown below. In the following, for simplicity, we will use the “uncoupled ” model (which is described in Chapter 3 of the main text) to investigate res onant eﬀects in the sum wake for a structure with modes with a uniform frequen cy distribu- tion. The point of using the uncoupled model is that it allows us to study the eﬀect of an idealized, uniform frequency distribution. As we have seen in the main text, an ideal (input) frequency distribution beco mes distorted by the cell-to-cell coupling of an accelerator structure. As e xample we will use the parameters of a simpliﬁed version (all kick factors are e qual) of the opti- mized 3 π/4 S-band structure described in the main text; for bunch stru cture we consider the nominal bunch spacing (∆ t= 2.4 ns). The results for the real structure, with coupled modes, will be slightly diﬀere nt yet qualitatively the same. Note that we are also aware of a diﬀerent analysis of resonant multi-bunch wakeﬁeld eﬀect[17]. Consider ﬁrst the case of a structure with only one dipole mod e, with frequency f, and a kick factor that we will normalize (for simplicity) to 1/2. Suppose there are Mbunches in the bunch train. The sum wake at the mth bunch is given by S(1) m(f∆t) =m/summationdisplay i=1sin(2π[i−1]f∆t) =sin (π[m−1]f∆t) sin(πmf∆t) sin (πf∆t). (C1) As with the nominal (2.8 ns) bunch spacing in the S-band preli nacs, let us, for an example, consider M= 95 bunches and the region near the 11th harmonic. In Fig. 15 we plot f∆tvsthe sum wake for the Mth (the last) bunch, S(1) M, near the 11th integer resonance. It can be shown that, if Mis not small, the largest resonance peaks (the extrema of the cu rve) are at f∆t≈n±3 8M[Mnot small] , (C2) 36with values ±.72M. Note that at the exact integer and half-integer resonant spacings the sum wake is zero. Figure 15: The sum wake at the last bunch in a train vsbunch spacing, due to a single mode (Eq. C1); M= 95 bunches. Now let us consider a uniform distribution of mode frequenci es. For simplicity we will let all the kick factors be equal, and be no rmalized to 1 /2. The sum wake, according to the uncoupled model, becomes Sm(¯f∆t) =1 NcNc/summationdisplay n=1S(1) m/bracketleftbigg ¯f∆t/parenleftbigg 1 +(n−Nc/2) Nc∆δf/parenrightbigg/bracketrightbigg ,(C3) withNcthe number of cells (also the number of modes), ¯fthe central fre- quency, and ∆ δfthe total (relative) width of the frequency distribution. As an example, let us consider the optimized 3 π/4 S-band structure, with Nc= 102 and ∆ δf= 5.8%. The sum wake at the last (the Mth) bunch position, SM, is plotted as function of ¯f∆tin Fig. 16. Note that the uni- form frequency distribution appears to suppress the intege r resonance. The extrema of the curve (the “horns”) that are seen at ¯f∆t= 11±.32 are res- onances due to the edges of the frequency distribution, with the condition ¯f∆t≈11/(1±∆δf/2). Note, however, that the sizes of even these spikes are small compared to those of the single mode case. 37Figure 16: The sum wake at the last bunch in a train vsbunch spacing, due to a uniform distribution of mode frequencies (Eq. C3). The t otal frequency spread ∆ δf= 5.8%, and Nc= 102. Suppose we add frequency errors to our model. We can do this by , in each term in the sum of Eq. C3, multiplying the frequency by th e factor (1+δferrrn), with δferrthe rms (relative) frequency error and rna random number with rms 1. Doing this, considering a uniform distrib ution in fre- quency errors with rms δferr= 10−4, Fig. 16 becomes Fig. 17. Note that this perturbation is small compared to the frequency spacin g 5.7×10−4, so it does not really change the frequency distribution sign iﬁcantly. Never- theless, because of resonance-like behavior we can see a lar ge eﬀect on SM throughout the range between the horns of Fig. 16 (10 .68≤¯f∆t≤11.32). To model cell-to-cell misalignments, we multiply each term in the sum of Eq. C3 by the random factor rn. The results, for a uniform distribution of errors with rms 1, are shown in Fig. 18. Again resonance-like behavior is seen throughout the range between the horns of Fig. 16. We can understand these results in the following manner: Onl y when there are no errors does using a uniform frequency distribut ion suppress the resonance in the region near the integer resonance. But othe rwise, using a uniform frequency distribution basically only reduces the size of the reso- nances, at the expense of extending the range in bunch spacin gs where they can be excited. Instead of being localized in the region near the integer 38Figure 17: The sum wake at the last bunch in a train vsbunch spacing, due to a uniform distribution of frequencies, including fre quency errors. The total frequency spread ∆ δf= 5.8%, the number of modes Nc= 102, and rms relative frequency error is 10−4. resonance ( ¯f∆t≈n), resonance-like behavior can now be excited anywhere between the limits (¯f∆t)±=n 1∓∆δf/2. (C4) Note that this implies that if ∆ δf>1/(¯f∆t), then the resonance-like be- havior cannot be avoided no matter what bunch spacing (fract ional part) is chosen. For example, for the X-band linac in the NLC, where th e total width of the dipole frequency distribution (of the dominant ﬁrst b and modes) is 10%, even for the alternate (1.4 ns) bunch spacing, where the integer part of¯f∆tis 21, the resonance region cannot be avoided. 39Figure 18: The sum wake at the last bunch in a train vsbunch spacing, due to a uniform distribution of frequencies, including ran dom misalignment errors with rms 1. The total frequency spread ∆ δf= 5.8% and then number of modes Nc= 102. 40
arXiv:physics/0012015 7 Dec 2000n/G0A /G0A/G0B0(cos/G30/G08 /G08isin/G30) /G0Dn/G0D/G0A /G0A/G0B2 0/G0A /G0A/G0B0(1) (1a)The Arrow of Time Ernst Karl Kunst Im Spicher Garten 5 53639 Königswinter Germany e-mail: ErnstKunst@aol.com It is shown that (special) relativistically dilated time is the vector sum of rest time and time induced by movement in three dimensional space exee ding the rest time component and that the first vector is orthogonally directed relative to our three dimensional space. This again implies that its origin lies in the movement of three dimensional space relative to a four dimensional manifold. The theory predicts asymmetrical time dilation. Key words: Special relativity - time - vector sum - direction of rest-time vector - asymmetrical time dilation It has been shown in [1] that translational velocity is symmetrically composite and the resulting space-time geometry Gaussian. Accordingly the Lorentz factor /G0B = 0 (1 - v/c) associated with the composite velocity v is the complex number n(/G07', 1)0 0 022-1/2 in a complex ct, x-plane of space-time so that (/G07 = vdt’/(cdt) and dt’ = dt/G0B). Furthermore, in the same study has been demonstrated' 00 0 Einstein’s “relativity of simultaneity” and the “FitzGerald-Lorentz contraction” be erroneous derivations of the Lorentz transformation. Instead the correct interpretation of the latter predicts a relativistic expansion of length or volume by the factor /G0B and0 the simultaneity of events, but dilated by the same factor. The well-known rise of interaction-radii of hadrons in high energetic collisions and related experimental phenomena are due to this relativistic expansion of volume - as has been extensively shown. Therefrom directly follows m = dt’v/c = dt/G0Bv/c, where dt = dx/c, whichtx0 x00 x means that relativistic mass is solely induced by the factor dt’v/c of the expandedx0 volume V’ = dx’dy’dz’ of a moving body and, furthermore, the existence of a fundamental length /G1B = c/G2D = /G08h and a quantum of time /G2D.00 0 Because the space-like vector dt’v/c and rest time dt are of the same origin [2] thex0 xdtx/G08 /G08dt /G0C/G0C xv0 c/G0A /G0Adt /G0C/G0C x /G2D /G2D0/G08 /G08/G12/G13/G13/G13/G12/G13/G13/G13/G2D /G2D0 dtx×dt /G0C/G0C xv0 c/G0A /G0A/G2D /G2D0 /G0C/G0C . /G0Ddt /G0C/G0C x/G0D/G0A /G0Adtx 1/G09v2 0 c2, /G0D/G2D /G2D /G0C/G0C 0/G0D/G0A /G0A/G2D /G2D0 1/G09v2 0 c2.2 (2) (2a) (3)latter must beyond its character as a “time-like vector” be a genuine, directionally well defined vector (dt), too, implying relativistically dilated time to be the vector sum:x or, especially, Thus, time in the moving frame is the vector sum of the rest time in the frame of the observer and the space-like vector dt’v/c directed into any of the dimensions of threex0 dimensional space. From (1) follows that dt’v/c is based orthogonally onto the vectorx0 dt (dt’v/c /G5D dt) so that the absolute value of modulus of the vector dt’ and thexx0 x x vector /G2D /G2D', respectively, is0 From the foregoing is evident that any observer can consider time in his own rest frame as a vector sum dt + dtv/c = dt (/G2D /G2D + I/G2D /G2DI/IdtI × dtv/c = /G2D /G2D) owingxminx0 x0min0minxmin x0 0 to the movement relative to a system where (3) attains the minimum value IdtI = dtx xmin and I/G2D /G2DI =/G2D /G2D, respectively, which presumably will be the case if the latter frame is00min resting relative to the cosmic microwave background (space). It is clear that the vector dt (/G2D /G2D) is not a vector sum and remains invariantxmin0min independen t on the sense of the vector dtv/c associated with the movement of anyx0 rest frame in three dimensional space. Thus, the direction of the vector dt (/G2D /G2D)xmin0min must show into a fourth geometrical dimension outside of three dimensional space. This result already has been infered from the equivalence of rest time and dilated time [2]. Let us assume that the absolute value of modulus according to (3) is really associated/G50xdtx/G0A /G0A/G50x/G0Cdt/G0C x1/G09v2 0 c2. /G50x/G0Cdt/G0C x/G0A /G0A/G50xdtx1/G09v2 0 c2. dtxmin/G08 /G08dtxu0 c±dt /G0C/G0C xv0 c/G0A /G0A±dt /G0C/G0C x3 with a higher velocity of the moving frame relative to space and, therewith, to the observer’s rest frame so that is valid Because of the absolute symmetry of both frames according to the principle of relativity a Lorentz transformation in either frame (dt’ = dt/G0B and dt = dt’/G0B) results in:x 0 xx0 This result is in accord with the well-known flight-time experiments with airplanes [4] and muons [5], [6]. Therefore, the twin paradox of special relativity is resolved to the result that time dilation is asymmetrically dependen t on velocity relative to the microwave background (this result has been independen tly derived in [3]). In conventional special relativity the above result is achieved by an one-sided Lorentz transformation from the moving to the resting frame. But that theory predicts the same result if the observer’s rest frame is assumed to be the moving one - contrary to experiment. Although this is frequently denied is in the framework of conventional special relativity another conclusion absolutely not possible because it only admits of relative and den ies absolute motion. Thus, to be in accord with experimentally verified asymmetric time dilation the conventional interpretation in reality tacitly involves absolute motion. In the case of collision experiments a contradiction to special relativity does not arise, because the quantized inertial motion between any two frames of reference is applied to a preferred natural rest frame /G28 implying their absolute symmetry and equality0 relative to /G28 [1] - which again is moving relative to space (micro wave background).0 This will be deduced by every observer in every frame. On the other hand the different velocity of the frames under consideration relative to space and, therewith, different absolute values of modulus of dt’ and /G2D /G2D' acccording to (3) must be real. Hence, thex 0 latter vectors always result from two successive Lorentz transformations so that in the case of parallel and antiparallel motion, respectively, we receive: /G2D /G2D0min/G08 /G08/G12/G13/G13/G13/G13/G12/G13/G13/G13/G13/G2D /G2D0min dtxmin×dtxu0 c±/G12/G13/G13/G13/G12/G13/G13/G13/G2D /G2D0 dtx×dt /G0C/G0C xv0 c/G0A /G0A±/G2D /G2D /G0C/G0C 0,4 and where u means the velocity of the rest frame of the observer relative to space and v0 0 between the frames under consideration, respectively. This implies that the principle of relativity and, therewith, the group character of the Lorentz transformation remain valid, but with the restriction that of any two frames of reference a time interval dt of the frame with the higher velocity relative to space is really longer by the factor (1a) as compared with the other system. The above formulas predict that all clocks in the solar system run slower than a clock resting relative to the microwave background, owing to its motion through the latter. Furthermore, all clocks on Earth must run slower if during the year her velocity is in the same direction spatially as the general motion of the sun and faster if her velocity is in the other direction - compared with a clock resting relative to the sun. References [1] Kunst, E. K.: Is the Kinematics of Special Relativity incomplete?, physics/9909059 [2] Kunst, E. K.: On the Origin of Time, physics/9910024 [3] Kunst, E. K.: Is the Lorentz Transformation Distant-Dependent? physics/9911022 [4] Hafele, J.,Keating, R., Science 177, 166 (1972) [5] Rossi, B., Hall, D., Phys. Rev. 59, 223 (1941) [6] Farley, F. M., et al., Nature 217, 17 (1968)
arXiv:physics/0012016v1 [physics.atom-ph] 8 Dec 2000Atomic Radiative Transitions in Thermo Field Dynamics J. L. Tomazelli Departamento de F´ ısica e Qu´ ımica, Faculdade de Engenhari a Universidade Estadual Paulista, Campus da Guaratinguet´ a , Av. Dr. Ariberto Pereira da Cunha 333, 12500-000 Guaratinguet´ a, SP, Brazil. L. C. Costa Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, 01405-900, S˜ ao Paulo, Brazil. Abstract In this work we study the energy exchange between an atomic sy stem and a thermal radiation ﬁeld, using the Dalibard, Dupont-Roc and Cohen-T annoudji (DDC) construct, in- corporating temperature eﬀects to the eigenstates of the ra diation ﬁeld operator through the electromagnetic propagator of Thermo Field Dynamics in the Coulomb gauge. We also discuss the stability of the atomic system at ﬁnite temperature. 1I. Introduction Since the 70’s, it has been argued[1][2] that the physical in terpretation of radiative phenomena, in particlar the shift in atomic energy levels, rely upon diﬀ erent choices in the ordering of atomic and ﬁeld operators in the interaction Hamiltonian. Latter, Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC)[3 ] considered the interaction between a nonrelativistic atomic electron and the quantize d electromagnetic ﬁeld, showing that such arbitrariness can be removed by requiring that the corresponding variation rates must be Hermitian, if we want them to have a “physical meaning ”. They generalized this procedure to the case of a small sistem Sinteracting with a large reservoir R(which may be in thermal equilibrium). This construct allowed them to sep arate the physical processes in two cathegories, those where Rﬂuctuates and polarizes S(eﬀects of reservoir ﬂuctuations), and those where Spolarizes R(eﬀects of self-reaction or radiation reaction). In the present work we are interested in analyzing the implem entation of temperature in the context of DDC formalism, where the statistical functio ns, which are deﬁned from two- point functions of physical observables, play a fundamenta l role. These functions enable us to obtain expressions, up to second order in perturbation th eory, in terms of products of correlation functions and susceptibilities[4]. The implementation of temperature[3] can be made directly i n such statistical functions using the equipartition theorem, leading to a ﬁnite tempera ture description of the relevant phenomena. In an alternative way, we shall study the theory using Umezaw a’s formalism, known as Thermo Field Dynamics (TFD)[5]. In TFD, the quantum statist ical average of an observable in a given ensemble is identiﬁed with its expectation value i n a thermal vacuum. In this approach, the temperature is incorporeted from the beginni ng, in the eigenstate of the number operator associated to the radiation ﬁeld. Our ideia is to investigate the thermal propagator of electr omagnetic ﬁeld in the Coulomb gauge and to identify the correlation functions and suscept ibilities of DDC formalism. We compare our results in the case of energy exchange between an atomic system and a thermal reservoir, analysing their implication to atomic stabilit y. 2II. Radiation considered as a Reservoir In the Dalibard, Dupont-Roc and Cohen-Tannoudji[3] formal ism, the interaction between an atom and the free electromagnetic ﬁeld can be seen as the inte raction of a microscopic system Swith a large reservoir R, in the sense that Rhas many degrees of freedom and the correlation time between the observables of Ris small, allowing a perturbative treatment of the eﬀect due to the coupling of SandR. In this context, the Hamiltonian of the global system S+R, can be written as H=Hs+HR+V, (1) where Hsis the Hamiltonian of S,HRthe reservoir Hamiltonian and Vthe interaction Hamil- tonian, which we assume to be of the form V=−/summationtext iRiSi, where RiandSiare Hermitian observables of RandS. In the interaction representation with respect to Hs+HR, the density operator of the global system ρ(t) evolve according to d dt˜ρ(t) =1 i¯h[˜V(t),˜ρ(t)]. (2) Here, the reservoir stands for the radiation ﬁeld. In the cas e we are interested, Rrepresents the electromagnetic ﬁeld so that we may choose the observabl esRias the space components of the vector potential of the electromagnetic ﬁeld Ai,i=x, y, z . Under these circumstances, it is easy to verify that the average value of Riin a state σRof the reservoir is zero, i.e., Tr[σRR(t)] =Tr/bracketleftBig σR˜R(t)/bracketrightBig = 0, (3) since RiandAiare linear combinations of emission and absorption operato rs of radiation quanta. Expression (3) is a one-time average. Now consider a two-tim e average g(t′, t′′) =Tr/bracketleftBig σR˜R(t′)˜R(t′′)/bracketrightBig . (4) This two-point function represents an average in a state σRof a product of two observables taken at two diﬀerent times t′andt′′. In fact, such two-point function depends only on 3τ=t′−t′′because, due to the cyclic permutation property of the trace , g(t′, t′′) =TrR/bracketleftBig σR˜R(τ)˜R(0)/bracketrightBig =g(τ). (5) Assuming that σRis a stationary state, it follows that [ σR, HR] = 0. As a consequence, we can expand σRas σR=/summationdisplay µpµ\|µ/an}bracketri}ht/an}bracketle{tµ\|, (6) where \|µ/an}bracketri}htis an eigenstate of HRwith eigenvalue Eµandpµis a given statistical weight. Note that, when the whole system is in thermal equilibrium at temp erature T, we may assume that the equipartition theorem applies and take pµ=exp[Eµ/kBT] /summationtext µexp[Eµ/kBT], (7) where kBis the Boltzmann constant. Substituting (6) into (5), we obt ain g(τ) = Tr/summationdisplay µ{pµ\|µ/an}bracketri}ht/an}bracketle{tµ\|˜R(τ)˜R(0)} =/summationdisplay µpµ/an}bracketle{tµ\|˜R(τ)˜R(0)\|µ/an}bracketri}ht =/summationdisplay µ,νpµ\|Rµν\|2eiωµντ, (8) where Rµν=/an}bracketle{tµ\|R\|ν/an}bracketri}ht, ωµν=ωµ−ωνandωµ=Eµ/¯h. Equation (8) is a superposition of exponentials oscillatin g at diﬀerent Bohr frequencies ωµν ofR. Because Ris a reservoir, it has a very dense ensemble of energy levels a nd, consequently, a quasi-continuous spectrum of Bohr frequencies, so that th e exponentials in (8) interfere destructively once τbecomes large enough. The hipoteses made about Rare equivalent to assuming that Ris in a stationary state and exerts on Sa “force” ﬂuctuating about a zero average value with a short c orrelation time τc. A. The Statistical Functions The function g(τ) deﬁned in (4) is not real, even to Hermitian operators R, because, in general, ˜R(τ) and ˜R(0) do not commute. In order to separate the real and imaginar y parts of g(τ) we 4write g(τ) =1 2/an}bracketle{t{˜R(τ),˜R(0)}/an}bracketri}htR+i 2/an}bracketle{t[˜R(τ),˜R(0)/i]/an}bracketri}htR, (9) where /an}bracketle{t,/an}bracketri}htRindicates an average on the reservoir state deﬁned by σR. The ﬁrst term in (9) corresponds to the symmetric correlation function and the s econd is related to the linear susceptibilities of the reservoir. The symmetric correlat ion function of the observable R, CR(τ) =1 2/an}bracketle{t{˜R(t′),˜R(t′′)}/an}bracketri}htR, (10) is real and tends to the ordinary correlation function in the classical limit. It gives a physical description of the dynamics of the ﬂuctuations of the observ ableRin the state σR. The explicit expression for the quantum correlation functi on deﬁned by (10) is given by the real part of expression (8) for g(τ), CR(τ) =/summationdisplay µ,νpµ\|Rµν\|2cos(ωµντ). (11) In the frequence space (11) becomes ˆCR(ω) =/summationdisplay µ,νpµπ\|Rµν\|2[δ(ω+ωµν) +δ(ω−ωµν)]. (12) The other statistical function is the linear susceptibilit yχR(τ), which characterizes the reservoir response to an external perturbation, χR(τ) =i ¯hθ(τ)/an}bracketle{t[˜R(t′),˜R(t′′)]/an}bracketri}htR =2 ¯hθ(τ)Img(−τ), (13) where θ(τ) is the step function. Using (8), χR(τ) =−2 ¯h/summationdisplay µ,νpµ\|Rµν\|2θ(τ) sinωµντ . (14) In the frequence space we have ˆχR(ω) = ˆχ′ R(ω) +iˆχ′′ R(ω), (15) where ˆχ′ R(ω) =−1 ¯h/summationdisplay µ,νpµ\|Rµν\|2/bracketleftBigg P1 ωµν+ω+P1 ωµν−ω/bracketrightBigg (16) ˆχ′′ R(ω) =π ¯h/summationdisplay µ,νpµ\|Rµν\|2[δ(ωµν+ω)−δ(ωµν−ω)]. (17) 5In (16) Pdenotes the principal value. The above expression characte rize, respectively, the response in phase and in quadrature at the frequency ω. B. Atomic Transition LetSbe an atom ﬁxed at the origin 0of the coordinate system and Ran homogeneous and isotropic broadband radiation ﬁeld. The radiation density operator is, according to (6), a statistical mixture of the eigenstates \|n1. . .n k. . ./an}bracketri}htofHR, representing n1quanta in the mode 1,...,nkquanta in the mode k..., with a weight p(n1. . .n k. . .), σR=/summationdisplay {nk}p(. . .n k. . .)\|. . .n k. . ./an}bracketri}ht/an}bracketle{tn1. . . n k. . .\|. (18) The average number of quanta in the mode kis, therefore, given by /an}bracketle{tnk/an}bracketri}ht=/summationdisplay {nk}nkp(n1. . .n k. . .). (19) Since it depends only on ωk, we hereafter use the notation /an}bracketle{tn(ωk)/an}bracketri}ht. In order to simplify the problem, let us consider a model wher e an atom with a single electron, moving in a spherically symmetric potential arou nd the center ( r=0). Further, assuming that the electron is inside a volume having small di mensions compared with the wavelenght of the incident radiation, we can make use of the l ong wavelength approximatiom to all modes whose frequence is bellow a cutoﬀ ωM. In this case, the Hamiltonian of the global system is given by (1) and the interaction Hamiltonian betwe en the atom and the ﬁeld reduces to the expression1 V=−/summationdisplay i/parenleftbigge mpi/parenrightbigg Ai(0), (20) where i=x, y, z . We can now verify directly that expression (12) for ˆCR(ω) and (15) for ˆ χR(ω) have the same form, i.e., S±(ω) =/summationdisplay µ,νpµ\|Rµν\|2f±(ωµν, ω), (21) 1In the long wavelenght approximation, the A2term is associated to a correction for the electron kinetic energy. 6where f±(ωµν, ω) is a function of a given parity with regard to ωµν: + for ˆCand−for ˆχ. In another notation, S± i(ω) =/summationdisplay {n′}p(n1, . . ., n′, . . .)× ×/summationdisplay j[\|/an}bracketle{t. . ., n j, . . .\|Aij(0)\|. . ., n j+ 1, . . ./an}bracketri}ht\|2f±(−ωj, ω) + +\|/an}bracketle{t. . ., n j, . . .\|Aij(0)\|. . ., n j−1, . . ./an}bracketri}ht\|2f±(ωj.ω)] (22) where jrepresents a given mode ( k,r) and Aij(0) =/parenleftBigg¯h 2ε0L3ωj/parenrightBigg eij[aj+a† j]. (23) Evaluating the matrix element in (22), we obtain, after repl acing the sum in the modes by a sum in the polarizations and an integral in k, S± i(ω) =/integraldisplayωM 0dω′/parenleftBigg¯hω′ 6π2ε0c3/parenrightBigg [±/an}bracketle{tn(ω′) + 1/an}bracketri}ht+/an}bracketle{tn(ω′)/an}bracketri}ht]f±(ω′, ω), (24) where the angular part has been already performed. Choosing i=xin the above expression, we obtain the corresponding correl ation function and susceptibilities for the xcomponent of the ﬁeld: ˆCxx R(ω) =1 3πε0c3/integraldisplayωM 0dω′¯hω′(/an}bracketle{tn(ω′)/an}bracketri}ht+ 1/2)[δ(ω′−ω) +δ(ω′+ω)] =1 3πε0c3¯h\|ω\|/an}bracketle{tn(\|ω\|) + 1/2/an}bracketri}ht, (25) ˆχ′xx R(ω) =1 6π2ε0c3/integraldisplayωM 0dω′ω′/bracketleftbigg P1 ω′+ω+P1 ω′−ω/bracketrightbigg , (26) ˆχ′′xx R(ω) =−1 6πε0c3/integraldisplayωM 0dω′ω′[δ(ω′+ω)−δ(ω′−ω)] =1 6πε0c3ω. (27) The correlation function for the atomic variable ( epx/m) and the corresponding suscepti- bilities, when the atom is in a given state \|a/an}bracketri}ht, are obtained in an analogous way and are given by ˆCxx Aa(ω) =/summationdisplay be2 m2\|/an}bracketle{ta\|px\|b/an}bracketri}ht\|2π[δ(ωab+ω) +δ(ωab−ω)] (28) 7ˆχ′xx Aa(ω) =/summationdisplay b−e2 ¯hm2\|/an}bracketle{ta\|px\|b/an}bracketri}ht\|2/bracketleftbigg P1 ωab+ω+P1 ωab−ω/bracketrightbigg (29) ˆχ′′xx Aa(ω) =/summationdisplay be2 ¯hm2\|/an}bracketle{ta\|px\|b/an}bracketri}ht\|2π[δ(ωab+ω)−δ(ωab−ω)]. (30) In order to study phenomena at ﬁnite temperature, we may subs titute the average number of radiation quanta /an}bracketle{tn/an}bracketri}ht, which appear in (24), by the Bose-Einstein distribution fu nction[3]. This procedure is justiﬁed by the use of (7) and accounts for t he equipartition theorem for the modes of the radiation ﬁeld. III. Thermal Correlation Functions and Susceptibilities In this section we study the thermal propagator of electroma gnetic ﬁeld in the context of Thermal Field Dynamics (TFD). Our ideia is to obtain the stat istical functions CRandχR, implementing temperature in a criterious way. We start by wr iting the space components of the electromagnetic potential Ai(t) as2 Ai(t) =A(+) i(t) +A(−) i(t), (31) where A(+) i(t) and A(−) i(t) are the components with positive and negative frequence, d eﬁned, respectively, as A(+) i(t) =/summationdisplay k,rαker i(k)ar ke−iωkt, (32) A(−) i(t) =/summationdisplay k,rαker i(k)ar k†eiωkt(33) with αk=/parenleftBigg¯h 2ε0L3ωk/parenrightBigg1/2 . (34) In TFD we double the ﬁeld degrees of freedom introducing the t ilde conjugated of Ai(t) [5][6]. Using the thermal doublet notation[6][7], we obtai n Ai(t) =/parenleftBigg Ai(t) ˜Ai(t)/parenrightBigg ¯Ai(t) = (Ai(t),−˜Ai(t) ) (35) 2As in the last section, we assume that the atom is at rest at the origin of the coordinate system ( r=0) and that we are using the dipole approximation. 8where (¯) denots the transposed and Ai(t) =/summationdisplay k,rαker i(k) (ar ke−iωkt+ar k†eiωkt) =A(+) i(t) +A(−) i(t), (36) ˜Ai(t) =/summationdisplay k,rαker i(k) ( ˜ar keiωkt+ ˜ar k†e−iωkt) =˜A(+) i(t) +˜A(−) i(t). (37) By construction, both ﬁelds Aiand˜Aiare independent; the corresponding absorption and emission operators satisfy the algebra[6] [ar k, as k′†] = [ ˜ar k,˜as k†] =δk,k′δr,s. (38) At zero temperature, the vacuum state is given by the direct p roduct \|0/an}bracketri}htA⊗ \|0/an}bracketri}ht˜A˙ =\|0/an}bracketri}ht. Using (40), it follows that A(+) i\|0/an}bracketri}ht= 0,˜A(+) i\|0/an}bracketri}ht= 0. (39) In order to ﬁnd the thermal propagator associated with the st atistical functions, we must calculate the commutator [Ai(t′),¯Aj(t′′)]µν= ∆µν ij(t′−t′′) (40) where µ, ν= 1,2 and i, j=x, y, z . The anti-diagonal components of the above quantity are identically zero when we calculate their expectation value in the \|0/an}bracketri}htstate. The component µ=ν= 1 can be written as ∆11 ij(t′−t′′) = ∆11 (+) ij(t′−t′′) + ∆11 (−) ij(t′−t′′), (41) where ∆11 ij(+)(t′−t′′) ˙ = [A(+) i(t′), A(−) j(t′′) ], (42) ∆11 ij(−)(t′−t′′) ˙ = [A(−) i(t′), A(+) j(t′′) ]. (43) Now, using (38), (39) and (40), we calculate explicitly thes e comutators, ∆11 ij(+)(τ) =/summationdisplay k,rα2 ker i(k)er j(k)e−iωkτ, (44) 9∆11 ij(−)(τ) =−/summationdisplay k,rα2 ker i(k)er j(k)eiωkτ, (45) where τ=t′−t′′. From (44) and (45), we can deﬁne two functionals: ∆11 ij(ret)(τ) ˙ = θ(τ)∆11 ij(+)(τ) +θ(τ)∆11 ij(−)(τ) = ∆11 ij(ret)(+)(τ) + ∆11 ij(ret)(−)(τ), (46) and ∆11 ij(1)(τ) ˙ = ∆11 ij(+)(τ)−∆11 ij(−)(τ). (47) By taking the Fourier transform of (46) and (47) we obtain, re spectively, ∆11 ij(ret)(ω) =/summationdisplay k,rα2 ker i(k)er j(k)/bracketleftbigg/parenleftbiggi ω−ωk+iǫ/parenrightbigg −/parenleftbiggi ω+ωk+iǫ/parenrightbigg/bracketrightbigg . (48) ∆11 ij(1)(ω) =/summationdisplay k,rα2 ker i(k)er j(k)π[δ(ω+ωk) +δ(ω−ωk)]. (49) Adopting the same procedure, we can extend the above calcula tion to the component µ=ν= 2. As a result, we obtain ∆22 ij(ret)(ω) =/summationdisplay k,rα2 ker i(k)er j(k)/bracketleftbigg/parenleftbiggi ω−ωk−iǫ/parenrightbigg −/parenleftbiggi ω+ωk−iǫ/parenrightbigg/bracketrightbigg , (50) ∆22 ij(1)(ω) = −/summationdisplay k,rα2 ker i(k)er j(k)π[δ(ω+ωk) +δ(ω−ωk)]. (51) We may write expression (48) and (50) in a more compact notati on, i.e., ∆ij(ret)(ω) =/summationdisplay k,rα2 ker i(k)er j(k)/braceleftbiggi k0−ωk+iτ3ǫ−i k0+ωk+iτ3ǫ/bracerightbigg (52) and, in the same way, we write (49) and (51) as ∆ij(1)(ω) =−/summationdisplay k,rα2 ker i(k)er j(k)πτ3[δ(ω+ωk) +δ(ω−ωk)], (53) where, in the last two expressions, τ3=/parenleftBigg 1 0 0−1/parenrightBigg . (54) In TFD, it is known that the propagator at zero temperature is related to the one calculated in the thermal vacuum through a Bogoliubov transformation[ 8]. Applying this result to (52) 10and (53), we obtain, respectively, ∆µν β ij(ret)(ω) = {B−1 k(β) ∆ ij(ret)(ω)Bk(β)}µν, (55) ∆µν β ij(1)(ω) = {B−1 k(β) ∆ ij(1)(ω)Bk(β)}µν, (56) where Bk(β) is give by Bk(β) = (1 −nk)1/2/parenleftBigg 1 −fkα −fk1−α1/parenrightBigg , (57) withα= 1/2,fk= exp[ −¯hωkβ] and nk=1 f−1 k−1=1 e¯hωkβ−1, (58) (β= 1/kT, where kis the Boltzmann constant and Tthe equilibrim temperature). The µ=ν= 1 component of (55) is found to be ∆11β ij(ret)(ω) = −i/summationdisplay k,rα2 ker i(k)er j(k)/braceleftBig P1 ωk−ω+P1 ωk+ω+ +i π[δ(ωk−ω)−δ(ωk+ω) ] (1 + 2 n(ωk))/bracerightBig , (59) and, from (56), ∆11β ij(1)(ω) =/summationdisplay k,rα2 ker i(k)er j(k)π[δ(ω−ωk) +δ(ω+ωk) ] (1 + 2 n(ωk)). (60) Now, we are in position to deﬁning the thermal correlation fu nction and susceptibilities, Cβ ij(ω) ˙ = ∆11β ij(1)(ω), (61) and χβ ij(ω) ˙ =i ¯h∆11β ij(ret)(ω), (62) where χβ ij(ω) =χ′β ij(ω) +i χ′′β ij(ω), (63) χ′β ij(ω) =1 ¯h/summationdisplay k,rα2 ker i(k)er j(k)/bracketleftbigg P1 ωk−ω+P1 ωk+ω/bracketrightbigg , (64) χ′′β ij(ω) =−1 ¯h/summationdisplay k,rα2 ker i(k)er j(k)π(1 + 2 n(ωk)) [δ(ωk+ω)−δ(ωk−ω) ].(65) 11Choosing i=j=xand substituting the summation over modes by a polarization sum and an integral in k, we obtain Cβ xx(ω) =1 3πε0c3/integraldisplayωM 0dω′¯hω′(n(ω′) + 1/2) [δ(ω′−ω) +δ(ω′+ω) ] =1 3πε0c3¯h\|ω\|(n(\|ω\|) + 1/2) (66) χ′β xx(ω) =1 6πε0c3/integraldisplayωM 0dω′ω′/bracketleftbigg P1 ω′−ω+P1 ω′+ω/bracketrightbigg (67) χ′′β xx(ω) =−1 6πε0c3/integraldisplayωM 0dω′ω′(2n(ω′) + 1) [ δ(ω′+ω)−δ(ω′−ω) ] =1 3πε0c3ω(n(\|ω\|) + 1/2). (68) IV. Energy Exchange In order to draw a conclusion we must compare the expressions for the statistical functions derived in sections II and III. It is clear that only the dissi pative part of the susceptibilities, expression (28) and (67), are diﬀerent. We must remenber tha tg(τ), which is the starting point in the construction of statistical functions, is deﬁn ed in terms of freeﬁeld eigenstates, leading to expression (29), which does not depend on the aver age number of photons. It can be show[9] that such a diﬀerence does not aﬀect calculations related to phenomena like the Lamb shift and the AC Stark eﬀect. Hence, the discrepancy acq uires an important physical meaning if, for example, we are interested in the energy exch ange between SandR. To see this, we analyze the variation rate of the mean atomic energy when, initialy, the system is in a given state a, d dt/an}bracketle{tHs/an}bracketri}hta=/summationdisplay b(Eb−Ea)Γa→b. (69) In (69), Γ a→brepresents the transition rate between the levels aandbdue to the interaction with the reservoir. It is shown in reference [3] that (69) can be written in terms of the statistical functions, giving d dt/an}bracketle{tHs/an}bracketri}hta=˙Qfr+˙Qrr, (70) where ˙Qfr=/integraldisplaydω 2πωˆCR(ω)ˆχ′′ Aa(ω), (71) 12˙Qrr=−/integraldisplaydω 2πωˆχ′′ R(ω)ˆCAa(ω). (72) The last two expression have a clear meaning: (71) is associa ted with the absorption of energy by the system when it is aﬀected by reservoir ﬂuctuations and (72) is related to the damping of the atomic motion caused by the reservoir. Using expressions (27), (32) and (71) and taking into accoun t the summation over x,y andz, we decompose ˙Qfrin ˙Qfr=˙Qfr′+˙Qfv, (73) where ˙Qfr′=/summationdisplay b(Eb−Ea)Γsp ab/an}bracketle{tn(\|ωab\|)/an}bracketri}ht, (74) ˙Qfv=/summationdisplay b(Eb−Ea)Γsp ab 2, (75) and Γsp ab=e2\|/an}bracketle{ta\|p\|b/an}bracketri}ht\|2\|ωab\| 3πε0¯hm2c3, (76) the rate of spontaneous emission relative to transition bet ween the levels aandb. Just as a random classical perturbation, the ﬂuctuations of the radiation ﬁeld transfer populations from level ato a higher or lower level b. From (74), we see that the incident radiation contributes to processes with a factor proportio nal to /an}bracketle{tn(\|ωab\|)/an}bracketri}htper mode and, from (75), we see that the vacuum ﬂuctuations contribute proport ionaly to 1 /2. The quantity ˙Qrris calculated in the same way, from (29), (30) e (72). As a resu lt, we ﬁnd ˙Qrr=−1 6πε0c3/summationdisplay be2 m2\|/an}bracketle{ta\|p\|b/an}bracketri}ht\|2ω2 ab (77) =/summationdisplay b−\|Eb−Ea\|Γsp ab 2. (78) Considering Ea< E b, it follows that \|Eb−Ea\|= (Ea−Eb) and ˙Qrr+˙Qfv= 0. (79) Further, if the system is in thermal equilibrium at temperat ureT, we note that d dt/an}bracketle{tHA/an}bracketri}hta/ne}ationslash= 0 (80) 13where we have substituted (73) and (78) in (70). On the other hand, if we use expression (66) and (68) instead ( 27) and (29), we obtain, after some manipulation, ˙Qfr β=/summationdisplay b(Eb−Ea)Γsp ab[/an}bracketle{tn(\|ωab\|)/an}bracketri}ht+ 1/2] (81) and ˙Qrr β=−/summationdisplay b(Eb−Ea)Γsp ab[/an}bracketle{tn(\|ωab\|)/an}bracketri}ht+ 1/2], (82) which gives d dt/an}bracketle{tHA/an}bracketri}hta= 0. (83) Expression (83) shows that, in thermal equilibrium, the sta bility of the atomic system is preserved, as we should expect. Note that, for T= 0, this stability is still holds, since the eﬀects of radiation reaction, ˙Qrr β(resp. second term in (82)), are cancelled by the thermal vacuum ﬂuctuation, ˙Qfv β(resp. second term in (81)). V. Concluding Remarks We have discussed the issue of temperature implementation i n DDC and argued that if we naively apply the equipartition theorem, essential inform ation may be lost. This occurs be- cause we have neglected essential physical requirements. O n the other hand, in the TFD approach the temperature is introduced at an early stage, in the eigenstates of the number operator for the radiation ﬁeld, through the modiﬁcation of the vacuum state[6]. In fact, the eigenstates of the number operator do not satisfy the same dy namics of the original (free) ﬁeld and, in this case, the Fock space of asymptotic states of the electromagnetic ﬁeld must be constructed taking into account the correlations betwee n the reservoir Rand an image reservoir ˜R, simmulating the eﬀects of thermal correlations[10]. In qu antum electrodynamics we have an analogue situation, where a consistent construct ion of particle eigenstates must take into account the long range Coulomb interaction, which modiﬁes the dynamics of these particle states in the asymptotic region[11]. 14Acknowledgements. JLT acknowledges CNPq for partial support and IFT/UNESP for the hospitality. LCC is grateful to FAPESP for the ﬁnancial supp ort. References [1] I. R. Senitzky, Phys. Rev. Lett. 31(1973) 955. [2] P. W. Milonni and W. A. Smith, Phys. Rev. A 11(1975) 814. [3] J. Dalibard, J. Dupont-Roc and C. Cohen-Tannoudji, J. de Physique 43(1982) 1617, J. de Physique 45(1984) 637. [4] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, “Ato m-Photon Interactions - Basic Processes and Applications”, J. Wiley, NY (1998). [5] H. Umezawa and Y. Takahashi, Int. J. Mod. Phys. B 10(1996) 1755. [6] H. Umezawa, “Advanced Field Theory”, AIP Press, NY (1995 ). [7] H. Matsumoto, Y. Nakano and H. Umezawa, Phys. Rev. D 31(1985) 429. [8] H. Matsumoto, Fortsh. Phys. 25(1977) 1. [9] L. C. Costa, master thesis, IFT-D.007/00, IFT/UNESP, S˜ ao Paulo (unpublished). [10] A. A. Abrikosov, Physics of Atomic Nuclei, 59(1996) 352. [11] P. Kulish and L. Faddeev, Theor. Math. Phys., 4(1970) 745. 15
arXiv:physics/0012017v1 [physics.comp-ph] 8 Dec 2000Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schr¨ odinger Equation Siu A. Chin and Chia-Rong Chen Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843 Abstract We show that the method of splitting the operator eǫ(T+V)to fourth order with purely positive coeﬃcients produces excellent algori thms for solving the time-dependent Schr¨ odinger equation. These algorithms r equire knowing the potential and the gradient of the potential. One 4th order al gorithm only requires four Fast Fourier Transformations per iteration. In a one dimensional scattering problem, the 4th order error coeﬃcients of these new algorithms are roughly 500 times smaller than fourth order algorithms w ith negative coeﬃcient, such as those based on the traditional Ruth-Fore st symplectic integrator. These algorithms can produce converged result s of conventional second or fourth order algorithms using time steps 5 to 10 tim es as large. Iterating these positive coeﬃcient algorithms to 6th order also produced better converged algorithms than iterating the Ruth-Forest algor ithm to 6th order or using Yoshida’s 6th order algorithm A directly. PACS: 31.15.-p, 02.70.Hm, 03.65.-W Keywords: time-dependent schr¨ odinger equation, operato r splitting, symplec- tic integrators. I. INTRODUCTION Understanding the dynamics of quantum evolution is of funda mental importance in all ﬁelds of physics and chemistry. Basic improvement in algori thms for solving the time- dependent Schro¨ odinger equation can therefore impact man y areas of basic research. Among numerical techniques developed for solving the time-depen dent Schr¨ odinger equation [1–3] (see T. N. Truong et al. [4] for earlier references), the method of split-operator [ 2], or its higher order variant, the method of symplectic integrator [ 5–7], has the advantage of being unitary, remain applicable in higher dimensions and easily generalizable to higher order. The disadvantage is that the time step size needed for conver gence seemed to be small and many iterations are required for evolving system forward in time. In this work, We show that the method of factorizing the evolution operator to fou rth order with purely positive coeﬃcients, which have yielded a new class of gradient sympl ectic integrators for solving classical dynamical problems [8,9], also produces algorit hms capable of solving the time- dependent Schr¨ odinger equation with time steps 5 to 10 time s as large as before. 1The quantum state is evolved forward in time by the Schr¨ odin ger evolution operator eǫH= eǫ(T+V), (1) whereǫ=−i∆t, andT=−1 2/summationtext i∇2 i,V=V(ri) are the kinetic and potential energy operators respectively. (For clarity of presentation, we w ill work in atomic units such that the kinetic energy operator has this standard form.) In the s plit operator approach, the short-time evolution operator (1) is factorized to second o rder in the product form T(2)(ǫ)≡e1 2ǫVeǫTe1 2ǫV= eǫ(T+V)+ǫ3C+···, (2) where we have indicated the error term as ǫ3C. Thus T(2)(ǫ) evolves the system according to the Hamitonian H(2)=T+V+ǫ2C+· · ·which deviates from the original Hamiltonian by an error term second order in ǫ. Since the kinetic energy operator is diagonal in momentum space, the split operator approach shuﬄes the wavefunction back and forth between real and Fourier space. (See detailed discussion by Takahashi an d Ikeda [5].) Every occurrence of eǫTrequires two Fast Fourier Transforms (FFTs), one direct and one inverse. In this approach, the generalization to higher dimension is straig htforward, limited only by the expense of higher dimensional Fourier transforms. Moreove r, every factorization of the evolution operator eǫ(T+V)in the above form is unitary. One advantage of the split operator approach is that higher o rder algorithms can be constructed easily. For example, the evolution operator ca n be factorized to arbitrarily high order in the form [10–12] eǫ(T+V)=/productdisplay ieaiǫTebiǫV, (3) with coeﬃcients {ai,bi}determined by the required order of accuracy. This factoriz ation process is identical to the derivation of symplectic algori thms for solving classical dynamical problems [13]. However, Suzuki [14] has proved that, beyond second order, any factorization of the form (3) must produce some negative coeﬃcients in the s et{ai,bi}, corresponding to some steps in which the system is evolved backward in time. Wh ile this is not detrimental in solving classical or quantum mechanical problems, it is o bserved that in the classical case the resulting higher order symplectic algorithms converge only for very small ranges of ∆ t and is far from optimal [9]. As we will show below, the same is t rue for quantum algorithms. In this work, we show that insisting on factorizing the the Sc hr¨ odinger evolution operator to 4th order with purely positive time steps yielded algorithms with excellent convergent properties at large time steps. II. FOURTH ORDER OPERATOR SPLITTINGS An example of 4th order splitting with negative coeﬃcient is the Ruth-Forest [15] scheme, T(4) FR(ǫ) =T(2)(/tildewideǫ)T(2)(−s/tildewideǫ)T(2)(/tildewideǫ) (4) wheres= 21/3is chosen to cancel the ǫ3Cerror term in T(2)and/tildewideǫ=ǫ/(2−s) rescales the sum of forward-backward-forward time steps back to ǫ. This factorization scheme has 2been independently derived many times in the context of symp lectic integrators [16,17]. The above derivation was ﬁrst published by by Creutz and Gocksch [10] in 1989. Suzuki [11] and Yoshida [12] independent published the same constructions in 1990. Identical construction can be applied to generate a ( n+ 2)th order algorithm T(n+2)from a triplet products of T(n)’s, T(n+2)(ǫ) =T(n)(/tildewideǫ)T(n)(−s/tildewideǫ)T(n)(/tildewideǫ) (5) withs= 21/(n+1). The Ruth-Forest (RF) algorithm requires 6 FFTs. The altern ative algorithm with operators VandTinterchanged is also possible, but would have required 8 FFTs per iteration. Recently, Suzuki [18] and Chin [8] have derive a number of 4th order splitting schemes with only positive coeﬃcients. In order to circumvent Suzuk i’s “no positive coeﬃcient” proof, these factorizaztions require the use of an addition al operator [ V,[T,V]] =/summationtext i\|∇iV\|2, which means that these new algorithms require knowing the gr adient of the potential. The two schemes derived by both Susuki and Chin, using diﬀerent m ethods, are: T(4) A≡eǫ1 6Veǫ1 2Teǫ2 3/tildewideVeǫ1 2Teǫ1 6V, (6) with/tildewideVgiven by /tildewideV=V+1 48ǫ2[V,[T,V]], (7) and T(4) B≡eǫ1 2(1−1√ 3)Teǫ1 2¯Veǫ1√ 3Teǫ1 2¯Veǫ1 2(1−1√ 3)T, (8) with¯Vgiven by ¯V=V+1 24(2−√ 3)ǫ2[V,[T,V]]. (9) Note that scheme A, remarkably, only requires 4 FFTs. Chin’s splitting scheme C, T(4) C≡eǫ1 6Teǫ3 8Veǫ1 3Teǫ1 4/tildewideVeǫ1 3Teǫ3 8Veǫ1 6T, (10) which minimizes the appearance of Vfor the derivation of symplectic algorithms, has 4 Toperators, corresponding to 8 FFTs. This is undesirable in t he current context. It is however easy to derive an alternate 4th order scheme with onl y 3Toperators by splitting the operator product at midpoint and concatenate the ends to gether to yield T(4) D≡eǫ1 8/tildewideVeǫ1 3Teǫ3 8Veǫ1 3Teǫ3 8Veǫ1 3Teǫ1 8/tildewideV. (11) This “split and splice” operation only works on scheme C beca use this scheme was originally derived by symmetrizing the splitted product. Scheme D is ju st the other way of symmetriz- ing the same product. These two algorithms gave identical re sults in the scattering problem solved below. Obviously then, algorithm 4D is preferable wi th two fewer FFTs. 3III. ONE DIMENSIONAL SCATTERING To gauge the eﬀectiveness of these new algorithms, we test th em on a one dimensional scattering problem, where a Gaussian wave pocket ψ0(x) =1 (2πσ2)1/4exp/bracketleftBigg ik0x−(x−x0)2 4σ2/bracketrightBigg , (12) is impinged on a smooth sech-square potential. The Hamilton ian is given by H=−1 2d2 dx2+V0sech2(x) (13) This choice of the potential is dictated by the fact that its t ransmission coeﬃcient is known analytically [7] T=1 1 + cosh2(π/radicalBig 2V0−1/4)/sinh2(πk0). (14) We chooseV0= 48.2 so that when the initial energy E0=1 2k2 0is equal toV0, the transmission coeﬃcient is 0.520001, which is practically 0.52 for our pur pose. To compute the transmission coeﬃcient, we evolve the Gaussi an wave pocket initially suﬃciently far from the barrier and then integrate the trans mitted wave pocket after a time oftmax= 20, when the latter is well separated from the reﬂected wave . We use 212= 4096 grid points over a length of 600, yielding a discretization s pacing of ∆ x≈0.15. Using more grid points than this has no measurable impact on the ﬁna l results. We found that in order to reproduce the analytical transmission coeﬃcien t, it is necessary to use a very ﬂat Gaussian incident wave pocket. We therefore take σ= 20 and place the wave pocket initially at x0=−80. Fig. 1 shows the resulting transmission coeﬀcient for vario us algorithms as a function of the time step size ∆ tat an incident energy of E0=V0. Even with such a ﬂat Gaussian incident wave pocket, at the smallest time step for the best a lgorithm, the transmission coeﬃcient converges only to T0= 0.519905 . While this value is still slightly below the exact value due to a ﬁnite sized Guassian wave pocket, it is a perfec tly acceptable benchmark to compare all algorithms with identical starting conditions . The second order results (2), denoted by asterisks, can be ac curately ﬁtted by T0−0.36∆t2 for ∆t <0.1, demonstrating its quadratic convergence. The results of the Ruth-Forest scheme (4), can also be well ﬁtted by T0−74∆t4over the same range as shown, verifying it quartic convergence. However, it is clearly obvious that th e range of convergence of the RF algorithm is not substantially greater than that of the seco nd order algorithm, perhaps at most a factor of three greater. In comparison, the four 4th or der algorithms with positive splitting coeﬃcients are distinctly superior. Whereas the fourth order error coeﬃcient of the Ruth-Forest algorithm is 74, the corresponding coeﬃcients for algorithms 4A, 4B, 4C and 4D are respectively, -1.07, -0.38, 0.14 and 0.14 respective ly. Algorithm 4C and 4D yielded identical results. Algorithm 4D’s error coﬃcient is more th an 500 times smaller than that of RF, and can achieve the same accuracy by using step sizes ne arly 5 times as large. The comparison with second order results is even more favorable ; the step size can be 10-15 4times as large. To compare the computational eﬀort involve, we timed each algorithm for 160 iterations on a Pentium II 450 MH processor using a Fortra n compiler. The second order algorithm took T2= 5.33s. Relative to this time, the time required by algorithms R F, 4A, 4B, 4C, 4D are respectively, 2.98 T2, 2.22T2, 3.37T2, 3.97T2, and 3.26T2respectively, which roughly scale with the number of FFTs used in each algorithm. Algorithm 4A is specially notable in that it is roughly 1/3 faster than RF but converges at time steps nearly 10 times as large. We have used algorithm 4A at time step size ∆ t= 0.1 to compute the transmission coeﬃcient as a function of the incident energy. Over the rang e ofE0/V0= 0.8 to 1.2, where the transmission coeﬃcient goes from 0.0016 to 0.9974, the r esults are in agreement with the exact value (14) to at least three decimal places. At present, no 6th order factorization with positive coeﬃci ents are known. However, one can use the triplet construction (5) to build a 6th order a lgorithm by iterating on three 4th order algorithms. Fig. 2 shows the resulting conve rgence curves for various 6th order algorithms. The solid triangles corresponds to itera ting on the RF algorithm to 6th order (RF6). There is no visible improvement in the converge nce range. This algorithm requires 18 FFTs. The asterisks are Yoshida’s [12] 6th order algorithm A (Y6A) , which is a product of 7 second order algorithms (2) some with negative c oeﬃcients, requiring 14 FFTs. Its convergence range is about twice that of the RF6 algorith m. The hollow diamonds, hollow circles, and solid circles are 6th order results base d on algorithms 4A, 4B, and 4D respectively, and will be referred to as due to algorithms 6A , 6B, and 6D respectively. Note that algorithm 6A only requires 12 FFTs. By ﬁtting a polynomi al of orders 6 to 12 in ∆ t, we extracted the 6th order error coeﬃcients for each algorithm s. For algorithms RF6, Y6A, 6A, 6B and 6D, the error coeﬃcients are -7675, -171, -17.42, -6.8 87, and 5.819 respectively. The new, gradient algorithms are orders of magnitude better tha n previous 6th order algorithms. Algorithm 6B’s results are so ﬂat that they can be ﬁtted by a po lynomial in (∆ t)8alone. For comparsion, we have also replotted the 4th order results due algorithm 4D as a dashed line. Algorithms RF6 and Y6A are not even better than 4D. Since all t hese 6th order algorithms, with the exception of Y6A, are just the product of three 4th or der algorithms, their running time simply triple that of their respective 4th order algori thm’s running time. Y6A’s time is obviously 7 T2, which is faster than all other algorithms except 6A. IV. CONCLUSIONS In this work, we have demonstrated that 4th order split opera tor algorithms, with no neg- ative intermediate time steps, are superior to existing sec ond order or fourth order algorithms for solving the time-dependent Schr¨ odinger equation. It i s straighforward to generalize these algorithms to higher dimension by using higher dimensional FFTs. These new algorithms require calculating the gradient of the potential, but conv erge at much large time step sizes. They should be useful for rapid simulation of large quantum s ystems with relatively simple potentials. Our comparison of 6th order algorithms suggests that higher order algorithms with in- termediate negative time steps are far from optimal. Algori thms RF6 and Y6A, which uses more negative time steps, are inferior to algorithm 6A, 6B, o r 6D. This is the same con- clusion drawn recently when higher order symplectic algori thms are compared in solving the Kepler problem [9]. This will impact current interests i n implementing higher order 5symplectic algorithms to study quantum dynamics [5–7]. This work suggests that the continual search for purely posi tive coeﬃcients factorization schemes may produce better converged algorithms for solvin g both classical and quantum dynamical problems. Currently, there are no known 6th order splitting schemes with purely positive coeﬃcients. ACKNOWLEDGMENTS This work was supported, in part, by the National Science Fou ndation grants No. PHY- 9870054 to SAC. 6REFERENCES [1] A. Goldberg, H. M. Schey, and J. L. Schwartz, Am. J. Phys. 35, 177 (1967). [2] D. Feit, J. A. Fleck, Jr., and A. Steiger, J. Comput. Phys. 47, 412 ( 1982); J. Chem. Phys.78, 301 (1982). [3] Tal-Ezer and R. Kosloﬀ, J. Chem. Phys. 81, 3967 (1984); R. Kosloﬀ, J. Phys. Chem. 92, 2087 (1988); R. Kosloﬀ, Annu. Rev. Phys. Chem. 45, 145 (1994). [4] T. N. Truong, J. J. Tanner, P. Bala, J. A. McCammon, D. J. Ko uri, B. Lesyng, and D. K. Hoﬀman J. Chem. Phys. 96, 2077 (1992). [5] Kin’ya Takahashi, Kensuke Ikeda, J. Chem. Phys. 99, 8680 (1993) [6] Stephen K. Gray, David E. Manolopoulos, J. Chem. Phys. 104, 7099 (1996) [7] Kin’ya Takahashi, Kensuke Ikeda, J. Chem. Phys. 106, 4463 (1997) [8] S. A. Chin, Phys. Lett. A226 , 344 (1997). [9] S. A. Chin and D. W. Kidwell, ”Higher Order Force Gradient Symplectic Algorithms”, Phys. Rev. E, in press, physics/0006082. [10] M. Creutz and A. Gocksch, Phys. Rev. Letts. 63, 9 (1989). [11] M. Suzuki, Phys. Lett. A146 , 319 (1990); 165, 387 (1992). [12] H. Yoshida, Phys. Lett. A150 , 262 (1990). [13] H. Yoshida, Celest. Mech. 56, 27 (1993). [14] M. Suzuki, J. Math. Phys. 32, 400 (1991). [15] E. Forest and R. D. Ruth, Physica D 43, 105 (1990). [16] M. Campostrini and P. Rossi, Nucl. Phys. B329 , 753 (1990). [17] J. Candy and W. Rozmus, J. Comp. Phys. 92, 230 (1991). [18] M. Suzuki, in Computer Simulation Studies in Condensed Matter Physics VI II, edited by D. P. Landau, K. K. Mon, and H.-B. Sh¨ uttler, Springer-Verlag, Berlin, 1996. 7FIGURES t0 0.05 0.1 0.15 0.2T 0.5100.5120.5140.5160.5180.520 FIG. 1. The transmission coeﬃcient Tas a function of time step size for various split operator algorithms. The asterisks are second order results, (2). Th e solid triangles are 4th order results corresponding to the Ruth-Forest splitting scheme with neg ative coeﬃcients, (4). The hollow diamonds and circles are results of algorithm 4A, (6) and 4B, (8) respectively. The ﬁlled circles are identical results produced by algorithms 4C, (10), and 4 D, (11). The lines are ﬁtted lines to extract the leading error coeﬃcients. See text for further d etails. 8t0 0.05 0.1 0.15 0.2T 0.51960.51980.52000.52020.5204 FIG. 2. The transmission coeﬃcient Tas a function of time step size for various iterated 6th order algorithms. The sold triangle are results of a 6th orde r algorithm based on the 4th order Ruth-Forest algorithm. The asterisks corresponds to Yoshi da 6th order algorithm A. The hollow diamonds, hollow circles, and solid circles, are 6th order a lgorithm results based on iterating the 4th order algorithm 4A, 4B, and 4D respectively. See text for further details. The solid lines are ﬁtted polynomials in powers ∆ tbeginning with powers of 6 up to 12. For comparison, the dash line corresponds to the best of the 4th order results, due to a lgorithm 4D. 9
arXiv:physics/0012018v1 [physics.class-ph] 8 Dec 2000METHOD OF REPLACING THE VARIABLES FOR GENERALIZED SYMMETRY OF D’ALEMBERT EQUATION G.A. Kotel’nikov RRC Kurchatov Institute, Kurchatov Sq. 1, Moscow 123182, Ru ssia E-mail: kga@electronics.kiae.ru It is shown that by generalized understanding of symmetry th e D’Alembert equation for one component ﬁeld is invariant with respect to arbitrar y reversible coordinate transformations. Symmetries play an important role in particle physics and qu antum ﬁeld theory [1], nuclear physics [2], mathematical physics [3]. It is propos ed some receptions for ﬁnding the symmetries, for example, the method of replacing the variables [4], the Lie algorithm [3], the theoretical-algebraic approach [5] . The purpose of the present work is the generalization of the method of replacing the var iables. We start from the following Deﬁnition of symmetry. Deﬁnition 1 Let some partial diﬀerential equation ˆL′φ′(x′) = 0 be given. By symmetry of this equation with respect to the variables repl acementx′=x′(x), φ′=φ′(Φφ)we shall understand the compatibility of the engaging equat ions sys- temˆAφ′(Φφ) = 0 ,ˆLφ(x) = 0 , where ˆAφ′(Φφ) = 0 is obtained from the initial equation by replacing the variables, ˆL′=ˆL,Φ(x)is some weight function. If the equation ˆAφ′(Φφ) = 0 can be transformed into the form ˆL(Ψφ) = 0, the symmetry will be named the standard Lie symmetry, otherwise the gener alized symmetry. Elements of this Deﬁnition were used to study the Maxwell equ ations symmetries [6-8]. In the present work we shall apply Deﬁnition 1 for inve stigation of symmetries of the one-component D’Alembert equation: 2′φ′(x′) =∂2φ′ ∂x′2 1+∂2φ′ ∂x′2 2+∂2φ′ ∂x′2 3+∂2φ′ ∂x′2 4= 0. (1) Let us introduce the arbitrary reversible coordinate trans formations x′=x′(x) and the transformation of the ﬁeld variable φ′=φ(Φφ), where Φ( x) is some unknown function, as well as take into account ∂φ′/∂x′ i=/summationtext j(∂φ′/∂ξ)(∂Φφ/∂x j)(∂xj/∂x′ i), ∂2φ′/∂x′ i2=/summationtext j(∂2xj/∂x′ i2)(∂φ′/∂ξ)(∂Φφ/∂x j)+/summationtext jk(∂2Φφ/∂x j∂xk)(∂xj/∂x′ i)(∂xk/ ∂x′ i)(∂φ′/∂ξ) +/summationtext jk(∂2φ′/∂ξ2)(∂Φφ/∂x j)(∂Φφ/∂x k)(∂xj/∂x′ i)(∂xk/∂x′ i), whereξ= Φφ. After replacing the variables we ﬁnd that the equation 2′φ′= 0 transforms into itself, if the system of the engaging equations is fulﬁlled /summationdisplay i/summationdisplay j∂2xj ∂x′ i2∂φ′ ∂ξ∂Φφ ∂xj+/summationdisplay i/summationdisplay j=k/parenleftbigg∂xj ∂x′ i/parenrightbigg2∂φ′ ∂ξ∂2Φφ ∂xj2+/summationdisplay i/summationdisplay j<k/summationdisplay k2∂xj ∂x′ i∂xk ∂x′ i∂φ′ ∂ξ∂2Φφ ∂xj∂xk+ /summationdisplay i/summationdisplay j=k/parenleftbigg∂xj ∂xi/parenrightbigg2∂2φ′ ∂ξ2/parenleftbigg∂Φφ ∂xj/parenrightbigg2 +/summationdisplay i/summationdisplay j<k/summationdisplay k2∂xj ∂x′ i∂xk ∂x′ i∂2φ′ ∂ξ2∂Φφ ∂xj∂Φφ ∂xk= 0; 2φ= 0. (2) 1Herex= (x1,x2,x3,x4), x4=ict,cis the speed of light, tis the time. Let us put the solution of D’Alembert equation φinto the ﬁrst equation of the set (2). If the obtained equation has a solution, then the set (2) will be com patible. According to Deﬁnition 1 it will mean that the arbitrary reversible trans formations x′=x′(x) are the symmetry transformations of the initial equation 2′φ′= 0. Owing to presence of the expressions ( ∂Φφ/∂x j)2and (∂Φφ/∂x j)(∂Φφ/∂x k) in the ﬁrst equation from the set (2), the latter has non-linear character. Since the anal ysis of non-linear systems is diﬃcult we suppose that ∂2φ′ ∂ξ2= 0. (3) In this case the non-linear components in the set (2) turn to z ero and the system will be linear. As result we ﬁnd the ﬁeld transformation law by int egrating the equation (3) φ′= C1Φφ+ C2→φ′= Φφ. (4) Here we suppose for simplicity that the constants of integra tion are C 1= 1,C2= 0. It is this law of ﬁeld transformation that was used within the algorithm [7, 8]. It marks the position of the algorithm in the generalized varia bles replacement method. Taking into account the formulae (3) and (4), we ﬁnd the follo wing form for the system (2): ∂2φ′ ∂ξ2= 0; φ′= Φφ; /summationdisplay j2′xj∂Φφ ∂xj+/summationdisplay i/summationdisplay j/parenleftbigg∂xj ∂x′ i/parenrightbigg2∂2Φφ ∂xj2+/summationdisplay i/summationdisplay j<k/summationdisplay k2∂xj ∂x′ i∂xk ∂x′ i∂2Φφ ∂xj∂xk= 0; 2φ= 0. (5) Since here Φ( x) =φ′(x′→x)/φ(x), whereφ′(x′) andφ(x) are the solutions of D’Alembert equation, the system (5) has a common solution an d consequently is compatible. This means that the arbitrary reversible trans formations of coordinates x′=x′(x) are symmetry transformations for the one-component D’Ale mbert equa- tion if the ﬁeld is transformed with the help of weight functi on Φ(x) according to the law (4). The form of this function depends on D’Alembert e quation solutions and the law of the coordinate transformations x′=x′(x). Next we shall consider the following examples. Let the coordinate transformations belong to the Poincar´ e group P10: x′ j=Ljkxk+aj, (6) whereLjkis the matrix of the Lorentz group L6,ajare the parameters of the transla- tion groupT4. In this case we have 2′xj=/summationtext kL′ jk2′x′ k= 0,/summationtext i(∂xj/∂x′ i)(∂xk/∂x′ i) =/summationtext iL′ jiL′ ki=δjk. The last term in the second equation (5) turns to zero. The se t reduces to the form 2Φφ= 0;2φ= 0. (7) 2According to Deﬁnition1 1 this is a sign of the Lie symmetry. T he weight function belongs to the set in [8]: ΦP10(x) =φ′(x) φ(x)∈/braceleftbigg 1;1 φ(x);Pjφ(x) φ(x);Mjkφ(x) φ(x);PjPkφ(x) φ(x);PjMklφ(x) φ(x);· · ·/bracerightbigg (8) wherePj, Mjkare the generators of Poincar´ e group, j,k,l= 1,2,3,4. In the space of D’Alembert equation solutions the set deﬁnes a rule of the ch ange from a solution to solution. The weight function Φ( x) = 1 ∈ΦP10(x) determines the transformational properties of the solutions φ′=φ, which means the well-known relativistic symmetry of D’Alembert equation [9, 10]. Let the transformations of coordinates belong to the Weyl group W11: x′ j=ρLjkxk+aj, (9) whereρ=const is the parameter of the scale transformations of the g roup ∆ 1. In this case we have 2′xj=ρ′/summationtext kL′ jk2′x′ k= 0,/summationtext i(∂xj/∂x′ i)(∂xk/∂x′ i) =/summationtext iρ′2L′ jiL′ ki= ρ′2δjk=ρ−2δjk. The set (5) reduces to the set (7) and has the solution Φ W11= CΦ P10, where C=const. The weight function Φ( x) = C and the law φ′= Cφmeans the well-known Weyl symmetry of D’Alembert equation [9, 10]. Le t here C be equal ρl, wherelis the conformal dimension1of the ﬁeld φ(x). Consequently, D’Alembert equation is W11-invariant for the ﬁeld φwith arbitrary conformal dimension l. This property is essential for the Voigt [4] and Umov [12] works as will be shown just below. Let the coordinate transformations belong to the Inversion group I: x′ j=−xj x2;x2=x12+x22+x32+x42;x2x′2= 1. (10) In this case we have 2′xj= 4x′ j/x′4=−4xjx2,/summationtext i(∂xj/∂x′ i)(∂xk/∂x′ i) =ρ′2(x′)δjk= 1/x′4δjk=x4δjk. The set (5) reduces to the set: −4xj∂Φφ ∂xj+x22Φφ= 0;2φ= 0. (11) The substitution of Φ( x) =x2Ψ(x) transforms the equation (11) for Φ( x) into the equation 2Ψφ= 0 for Ψ(x). It is a sign of the Lie symmetry. The equation has the solution Ψ = 1. The result is Φ( x) =x2. Consequently, the ﬁeld transforms ac- cording to the law φ′=x2φ(x) =ρ−1(x)φ(x). This means the conformal dimension l=−1 of the ﬁeld φ(x) in the case of D’Alembert equation symmetry with respect to the Inversion group Iin agreement with [5, 10]. In a general case the weight function belongs to the set: ΦI(x) =x2Ψ(x)∈/braceleftbigg x2;x2 φ(x);x2Pjφ(x) φ(x);x2Mjkφ(x) φ(x);x2PjPkφ(x) φ(x);· · ·/bracerightbigg .(12) 1The conformal dimension is the number lcharacterizing the behavior of the ﬁeld under scale transformations x′=ρx, φ′(x′) =ρlφ(x) [11]. 3Let the coordinate transformations belong to the Special Conformal Group C4: x′ j=xj−ajx2 σ(x);σ(x) = 1−2a·x+a2x2;σσ′= 1. (13) In this case we have 2′xj= 4(aj−a2xj)σ(x),/summationtext i(∂xj/∂x′ i)(∂xk/∂x′ i) =ρ′2(x′)δjk= σ2(x)δjk. The set (5) reduces to the set: 4σ(x)(aj−a2xj)∂Φφ ∂xj+σ2(x)2Φφ= 0;2φ= 0. (14) The substitution of Φ( x) =σ(x)Ψ(x) transforms the equation (14) into the equation 2Ψφ= 0 which corresponds to the Lie symmetry. From this equation we have Ψ = 1, Φ(x) =σ(x). Therefore φ′=σ(x)φ(x) and the conformal dimension of the ﬁeld isl=−1 as above. Analogously to (12), the weight function belongs to the set: ΦC4(x) =σ(x)Ψ(x)∈/braceleftbigg σ(x);σ(x) φ(x);σ(x)Pjφ(x) φ(x);σ(x)Mjkφ(x) φ(x);· · ·/bracerightbigg . (15) From here we can see that φ(x) = 1/σ(x) is the solution of D’Alembert equation. Combination of W11,IandC4symmetries means the well-known D’Alembert equa- tion conformal C15-symmetry [5, 9, 10]. Let the coordinate transformations belong to the Galilei group G1: x′ 1=x1+iβx4;x′ 2=x2;x′ 3=x3;x′ 4=γx4;c′=γc, (16) whereβ′=−β/γ,γ′= 1/γ,β=V/c,γ= (1−2βnx+β2)1/2. In this case we have 2′xj= 0,/summationtext i(∂x1/∂x′ i)2= 1−β′2;/summationtext i(∂x2/∂x′ i)2=/summationtext i(∂x3/∂x′ i)2= 1;/summationtext i(∂x4/∂x′ i)2=γ′2;/summationtext i(∂x1/∂x′ i)(∂x2/∂x′ i) =/summationtext i(∂x1/∂x′ i)(∂x3/∂x′ i) =/summationtext i(∂x2/ ∂x′ i)(∂x3/∂x′ i) =/summationtext i(∂x2/∂x′ i)(∂x4/∂x′ i) = 0;/summationtext i(∂x1/∂x′ i)(∂x4/∂x′ i) =iβ′γ′= −iβ/γ2. After putting these expressions into the set (5) we ﬁnd [8]: 2Φφ−∂2Φφ ∂x42−/parenleftbigg i∂ ∂x4+β∂ ∂x1/parenrightbigg2Φφ γ2=/bracketleftbigg(i∂4+β∂1)2 γ2− △/bracketrightbigg Φφ= 0. (17) In accordance with Deﬁnition 1 it means that the Galilei symm etry of D’Alembert equation is the generalized symmetry (being the conditiona l one [8]). The weight function belongs to the set [7]: ΦG1(x) =φ′(x′→x) φ(x)∈/braceleftbiggφ(x′) φ(x);1 φ(x);P′ jφ(x′) φ(x);[2′,H′ 1]φ(x′) φ(x);· · ·/bracerightbigg ,(18) whereH′ 1=it′∂x′is the generator of the pure Galilei transformations. For th e plane waves the weight function Φ( x) is [6 - 8]: ΦG1(x) =φ(x′→x) φ(x)=exp/braceleftbigg −i γ/bracketleftbigg/parenleftbigg 1−γ/parenrightbigg k·x−βω/parenleftbigg nxt−x c/parenrightbigg/bracketrightbigg/bracerightbigg ,(19) 4wherek= (k,k4),k=ωn/cis the wave vector, nis the wave front guiding vector, ω is the wave frequency, k4=iω/c,k′ 1= (k1+iβk4)/γ,k′ 2=k2/γ,k′ 3=k3/γ,k′ 4=k4, k′2=k2- inv. (For comparison, in the relativistic case we have k′ 1= (k1+iβk4)/(1− β2)1/2,k′ 2=k2,k′ 3=k3,k′ 4= (k4−iβk1)/(1−β2)1/2,k′2+k′2 4=k2+k42- inv as is well-known). The results obtained above we illustrate by means of the Tabl e 1: GroupP10W11IC4 G1 WFΦ(x)1ρlx2σ(x)exp{−i[(1−γ)k·x−βω(nxt−x/c)]/γ} For the diﬀerent transformations x′=x′(x), the weight functions Φ( x) may be found in a similar way. Let us note that in the symmetry theory of D’Alembert equatio n, the conditions (5) for transforming this equation into itself combine the r equirements formulated by various authors, as can be seen in the Table 2: AthorCoordinates GroupConditions of invariance Fields Transform. Transform. Voigtx′ j=AjkxkL6X∆1A′ jiA′ ki=ρ′2δjk φ′=φ [4] Umovx′ j=xj′(x)W11∂xj ∂x′ i∂xk ∂x′ i=ρ′2δjk φ′=φ [12] 2′xj= 0 DiJoriox′ j=Ljkxk+P10L′ jiL′ ki=δjk φ′=mαφα+ [13]aj∂2φ′ ∂φα∂φβ= 0 m0;α= 1,..,n Kotel′−x′ j=x′ j(x)C4∂xj ∂x′ i∂xk ∂x′ i=ρ′2(x′)δjk φ′ α=ψDαβφβ nikov∂2φ′ α ∂ξβ∂ξγ= 0 ξα=ψφα [6−8] 2′φ′ α= 0→ ˆAφ′ α(ψφ1,...ψφ 6) = 0,2φβ= 0α,β= 1,...,6 x′ j=x′ j(x)G1∂2φ′ α ∂ξβ∂ξγ= 0 φ′ α=ψMαβφβ 2′φ′ α= 0→ ξα=ψφα ˆBφ′ α(ψφ1,...ψφ 6) = 0,2φβ= 0α,β= 1,...,6 Heremα,m0are some numbers, DαβandMαβare the 6X6 numerical matrices. According to this Table for the ﬁeld φ′=φwith conformal dimension l= 0 and the linear homogeneous coordinate transformations from th e groupL6X△1∈W11 withρ= (1−β2)1/2, the formulae were proposed by Voigt (1887) [4, 9]. In the plain waves case they correspond to the transformations of t he 4-vector k= (k,k4) and proper frequency ω0according to the law k′ 1= (k1+iβk4)/ρ(1−β2)1/2,k′ 2= 5k2/ρ,k′ 3=k3/ρ,k′ 4= (k4−iβk1)/ρ(1−β2)1/2,ω′ 0=ω0/ρ,k′x′=kx- inv. In the case of the W11-coordinate transformations belonging to the set of arbitr ary transformations x′=x′(x) the requirements for the one component ﬁeld with l= 0 were found by Umov (1910) [12]. The requirement that the sec ond derivative ∂2φ′/∂φα∂φβ= 0 with Φ = 1 be turned into zero was introduced by Di Jorio (1974). The weight function Φ /ne}ationslash= 1 and the set (5) were proposed by the author of the present work (1982, 1985, 1995) [6 - 8]. By now well-studied have been only the D’Alembert equation s ymmetries cor- responding to the linear systems of the type (7), (11), (14). These are the well- known relativistic and conformal symmetry of the equation. The investigations corresponding to the linear conditions (5) are much more sca nty and presented only in the papers [6 - 8]. The publications corresponding to the n on-linear conditions (2) are absent completely. The diﬃculties arising here are c onnected with analysis of compatibility of the set (2) containing the non-linear pa rtial diﬀerential equation. Thus it is shown that with the generalized understanding of t he symmetry ac- cording to Deﬁnition 1, D’Alembert equation for one compone nt ﬁeld is invariant with respect to any arbitrary reversible coordinate transf ormationsx′=x′(x). In particular, they contain the transformations of the confor mal and Galilei groups re- alizing the type of standard and generalized symmetry for Φ( x) =φ′(x′→x)/φ(x). The concept of partial diﬀerential equations symmetry is co nventional. References 1. N.N. Bogoliubov, D.V. Shirkov. Introduction in Theory of Quantized Fields. Moscow, Nauka, 1973. 2. Yu. Shirokov, N.P. Yudin. Nuclear Physics. Moscow, Nauka , 1972. 3. N.X. Ibragimov. Groups of Transformations in Mathematic al Physics. Moscow, Nauka, 1983. 4. W. Voigt. Nachr. K. Gesel. Wiss., G¨ ottingen, 2, 41 (1887). 5. I.A. Malkin, V.I. Man’ko. JETP Lett., 2, 230 (1965). 6. G.A. Kotel’nikov. Proc. Second Zvenigorod Seminar on Gro up Theoretical Methods in Physics, V. 1. Ed. by M.A. Markov, V.I. Man’ko, A.E . Shabad. (Chur, London, Paris, New York, 1985) 521. 7. G.A. Kotel’nikov. Proc. Third Yurmala Seminar Group Theo retical Methods in Physics., V. 1. Ed. by M.A. Markov, V.I. Man’ko, V.V. Dodon ov. (Moscow, 1986) 479; Izv.VUZov, Fizika, 5, 127 (1989). 8. G.A. Kotel’nikov. Proc. VII International Conference Sy mmetry Methods in Physics, V. 2. Ed. by A.N. Sissakian, G.S. Pogosyan. (Dubna, 1996) 358; http://arXiv.org/abs/physics/9701006 9. W. Pauli. Theory of Relativity. Moscow-Leningrad, Goste xizdat, 1947. 10. W.I. Fushchich, A.G. Nikitin. Symmetries of Quantum Mec hanics Equations. Moscow, Nauka, 1990. 11. P. Carruthers. Phys. Reports (Sec. C of Phys. Lett.), 1, 2 (1971). 12. N.A. Umov. Collected Works, Moscow-Leningrad, Gostexi zdat, 1950. 13. M. Di Jorio. Nuovo Cim., 22B, 70 (1974). 6
arXiv:physics/0012019v1 [physics.optics] 11 Dec 2000Mach-Zehnder interferometer based all optical ﬂip-ﬂop Martin T. Hill, H. de Waardt, G. D. Khoe, H. J. S. Dorren Department of Electrical Engineering, Eindhoven Universi ty of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Abstract For the ﬁrst time an all optical ﬂip-ﬂop is demonstrated base d on two coupled Mach-Zehnder interferometers which contain semiconductor optical ampliﬁers in their arm s. The ﬂip-ﬂop operation is discussed and it is demon- strated using commercially available ﬁber pigtailed devic es. Being based on Mach-Zehnder interferometers, the ﬂip-ﬂop has potential for very high speed operation. E-mail: m.t.hill@ele.tue.nl 1I. Introduction Optical bistable devices and in particular all optical ﬂip- ﬂops can have many uses in optical telecommunications and computing such as: threshold functions, 3R regeneration, d e-multiplexing and rate conversion of telecommunication data [1]. All optical ﬂip-ﬂops based on two coupled devices provide ma ny advantages such as controllable behaviour, separate set/reset inputs and identical set/reset operations, and l arge input wavelength range. Such a ﬂip-ﬂop using two couple d lasers was demonstrated and analyzed in [2]. In particular, the function of light output by a laser versus the light injected into a laser has the correct characteristics so tha t when two lasers are coupled the following occurs: The system has more than one steady state solution, and at least t wo of the steady state solutions are stable states of the system. Integrated Mach-Zehnder interferometers (MZI) incorpora ting semiconductor optical ampliﬁers (SOA) in the in- terferometer arms have recently been developed as very high speed all optical switching devices [3],[4]. The function of light output by a MZI (with a constant bias light injected) versus the light injected into the MZI, can also have the correct characteristics for forming an all optical ﬂip-ﬂop . In this paper we experimentally demonstrate for the ﬁrst time an all optical ﬂip-ﬂop based on two coupled MZI (with SOA s in their arms). II. Operating Theory The structure of the ﬂip-ﬂop is shown in Figure 1. It consists of two MZIs (MZI 1 and MZI 2) and a SOA connecting the two. The connecting SOA provides a controllable gain bet ween the two MZIs. For the moment assume that the SOA gain is one and that it can be replaced with a direct optica l connection between MZI 1 and MZI 2. Each MZI has a continuous wave (CW) bias light input Sbias. Qualitatively the ﬂip-ﬂop functions as follows: With the ﬂi p-ﬂop in state 1 light out of MZI 1 ﬂows into the SOA of MZI 2, changing gain and refractive index such that much le ss light exits from MZI 2 and ﬂows back into MZI 1. 2State 2 is the reverse case where a large amount of light ﬂows o ut of MZI 2 suppressing light ﬂowing out of MZI 1 and back into MZI 2. To switch the ﬂip-ﬂop between state 1 or 2 light can be injecte d into the MZI that dominates (that is the one injecting the most light into the other MZI), via the In 1 or In 2 ports (Figure 1). The injected light reduces the light exiting the dominate MZI, which allows the suppressed MZI to increase its light output and become the dominate MZI. The ﬂip-ﬂop can be described quantitatively as follows. Eac h MZI can be modeled with: a rate equation for the carrier number of the SOA in its arms [5],[2], an equation rel ating carrier changes to refractive index and phase changes [6], and an equation to model the recombining of the signals i n the arms at the MZI output coupler [4]. Sout1as a function of Sin1or equivalently Sout2can be found for the steady state from the MZI model for a parti cular set of operating conditions. One of these functions of Sout1versus Sout2is plotted in Figure 2. The SOA parameters used to construct the plot were from [6] (with the additional parame ters of intrinsic losses αint= 27cm−1and group velocity in the SOA νg= 8×109cm s−1). Also plotted in Figure 2 is Sout2as a function of Sout1. The points where the two curves intersect represent steady state solutions for the system of two MZIs. The point labelle d B1 represents state 1 mentioned above. Point B2 represents state 2. Both B1 and B2 can be shown to be stable sta tes of the system [2]. The point S represents a state where the same amounts of light ﬂow from MZI 1 to MZI 2 and visa v ersa, however it is not a stable state of the system [2]. III. Experiment To demonstrate the all optical ﬂip-ﬂop just described above the setup show in Figure 1 was realized using commercially available SOAs and ﬁber based couplers. The SOAs employed a s trained bulk active region and were manufactured by JDS-Uniphase. 3The central SOA was not necessary from a theoretical stand po int. However it allowed the coupling between the MZIs to be varied as was necessary to obtain strong bistable o peration. The amount of light being injected into each MZI 1 and MZI 2 was measured by a photo diodes PD 1 and PD 2, Figure 1. To toggle the ﬂip-ﬂop between states light pulses o f power 3.3 mW, wavelength 1547 nm, and duration 5 ms were regularly injected into the inputs In 1 and In 2 (Figur e 1). The CW bias light power was 1.25 mW, and had wavelength 1552 nm for MZI 1 and 1550 for MZ1 2. The MZI SOA curr ents were such that with only the CW bias light injected into them they had a gain of 10. The inputs pulses were injected every 50 ms into alternate MZ Is. The changing of state of the ﬂip-ﬂop every 50 ms can be clearly seen in Figure 3, demonstrating proper ﬂip- ﬂop operation. Also the eﬀects of the 5 ms input pulses can be seen. IV. Conclusions In this paper we have shown that it is possible to make an all op tical ﬂip-ﬂop out of two Mach-Zehnder interferometers (with non-linear elements in their arms, here SOAs). Integr ated versions of the ﬂip-ﬂop could operate at very high speeds [3], [4], as the MZIs themselves respond quickly and t hey are located close to each other. Furthermore the integrated MZIs will be stable. The use of MZIs makes the ﬂip- ﬂop presented here inherently faster than the ﬂip-ﬂop presented in [2] that is based on couplers lasers. The attrib utes of high speed and potentially wide input wavelength range would make the ﬂip-ﬂop suitable for all optical signal processing applications in high-speed telecommunication s. Other arrangements of the MZIs apart from that shown in Figur e 1 are possible. For example it is possible to remove the SOA between the MZIs and share a common coupler bet ween the MZIs. Additional couplers could be added in the MZIs for inputs or outputs. Finally the concept o f optically bistable coupled MZIs could prove useful for increasing the sensitivity of MZI based optical sensors. 4Acknowledgments This research was supported by the Netherlands Organizatio n for Scientiﬁc Research (N.W.O.) through the ”NRC Photonics” grant. 5References [1] K. Nonaka, and T. Kurokawa, ”Simultaneous time- and wave length-domain optical demultiplexing of NRZ signals by using a side-injection-light-controlled bistable lase rdiode,” Electronics Letters , 1996, Vol. 31, No. 21, pp. 1865- 1866. [2] M. T. Hill, H. de Waardt, G. D. Khoe, and H. J. S. Dorren, ”Al l optical ﬂip-ﬂop based on coupled laser diodes,” to appear in IEEE Journal of Quantum Electronics. [3] C. Joergensen, S. L. Danielsen, T. Durhuus, B. Mikkelsen , K. E. Stubkjaer, N. Vodjdani, F. Ratovelomanana, A. Enard, G. Glastre, D. Rondi, and R. Blondeau, ”Wavelength conversion by optimized monolithic integrated Mach-Zehnder interferometer,” IEEE Photonics Technology Letters , 1996, Vol. 8, No.4, pp. 521-523 [4] R. Hess, M. Caraccia-Gross, W. Vogt, E. Gamper, P. A. Bess e, M. Duelk, E. Gini, H. Melchior, B. Mikkelsen, M. Vaa, K. S. Jepsen, K. E. Stubkjaer, and S. Bouchoule, ”All- optical demultiplexing of 80 to 10 Gb/s signals with monolithic integrated high-performance Mach-Zehnde r interferometer,” IEEE Photonics Technology Letters , 1998, Vol. 10, No.1, pp. 165-167 [5] M. J. Adams, J. V. Collins, and I. D. Henning, ”Analysis of semiconductor laser optical ampliﬁers,” IEE Pro- ceedings Part J , 1985, Vol. 132, pp. 58-63 [6] R. J. Manning and D. A. O. Davies, ”Three-wavelength devi ce for all-optical signal processing,” Optics Letters , 1994, Vol. 19, pp. 889-891 6Figure Captions Figure 1: Structure of Mach-Zehnder Interferometer (MZI) b ased optical ﬂip-ﬂop. PD: photo diode, SOA: semicon- ductor optical ampliﬁer Figure 2: Steady state light output by a MZI as function of the light injected into it by the other MZI. Figure 3: Oscilloscope traces of output of ﬂip-ﬂop showing s witching between states every 50 milli-seconds. Note that the eﬀects of the 5 ms input pulses used to switch the ﬂip-ﬂop c an also be seen in the traces. 7 Figure 1: Structure of Mach-Zehnder Interferometer (MZI) based optical flip-flop. PD: photo diode, SOA: semiconductor optical amplifier60/40 couplerSOA 1 50/50 couplerSOA PD 1MZI 1 SbiasSin1Sout1 In 1 60/40 couplerSOA 2 50/50 couplerPD 2MZI 2 SbiasSin2Sout2 In 2 Figure 2: Steady state light output by a MZI as function of the light injected into it by the other MZI.S B2B1 0 2 4 6026 4 Photon Number (x105)Photon Number (x105) Sout2Sout1Sout1 (as a function of Sout2) Sout2 (as a function of Sout1) Figure 3: Oscilloscope traces of output of flip-flop showing switching between states every 50 milli-seconds. Note that the effects of the 5 ms input pulses used to switch the flip-flop can also be seen in the traces.0 50 1 00 150 20000 Time (milli-seconds)Output PD 1 (arbitrary units)Output PD 2 (arbitrary units)
arXiv:physics/0012020v1 [physics.acc-ph] 11 Dec 2000LAL/RT 00-08 October 2000 An Improved Empirical Equation for Bunch Lengthening in Electron Storage Rings J. Gao Laboratoire de L’Acc´ el´ erateur Lin´ eaire, IN2P3-CNRS et Universit´ e de Paris-Sud, BP 34, 91898 Orsay c edex, France Abstract In this paper we propose an improved empirical equation for t he bunch lengthening in electron storage rings. The comparisons are made between the analytical and experimental results, and the agreements ar e quite well. This improved equation can be equally applied to the case where a s torage ring is very resistive (such as the improved SLC damping rings) inst ead of inductive as usual. 1 Introduction From what we know about the single bunch longitudinal and tra nsverse instabil- ities [1][2], it is clear to see that the information about th e bunch lengthening, Rz=σz/σz0, with respect to the bunch current is the keyto open the locked chain of bunch lengthening, energy spread increasing and th e fast transverse insta- bility threshold current. In this paper an improved (compar ed with what we have proposed in ref. 3) empirical bunch lengthening equation is proposed as follows: R2 z= 1 +√ 2CRavRDKtot \|\|,0Ib γ3.5(Rz)ς+C(RavRIbDKtot \|\|,0)2 γ7(Rz)2ς(1) where C=576π2ǫ0 55√ 3¯hc3(2) D=Exp/parenleftBigg −/parenleftbigg10 2πarctan/parenleftbiggZr Zi/parenrightbigg/parenrightbigg2/parenrightBigg =Exp −/parenleftBigg 10 2πarctan/parenleftBigg Ktot \|\|,0 2πL/parenleftbigg3σz0 c/parenrightbigg2/parenrightBigg/parenrightBigg2  (3) σz0is the single particle ”bunch length”, Ktot \|\|,0is the bunch total longitudinal loss factor for one turn at σz=σz0,ZrandZiare the resistive and inductive part of the machine impedance, respectively, Lis the inductance of the ring for one turn, ǫ0is the permittivity in vacuum, ¯ his Planck constant, cis the velocity of light, Ib=eNec/2πRav,Neis the particle number inside the bunch, and Ravis the 1average radius of the ring. Obviously, if Zi≫Zrone has D ≈1, which is the case for the most existing storage rings. If SPEAR scaling law [8] is used (for example), ς≈1.21 (in fact each machine has its own ς), eq. 1 can be written as R2 z= 1 +√ 2CRavRDKtot \|\|,0Ib γ3.5R1.21z+C(RavRIbDKtot \|\|,0)2 γ7R2.42z(4) In fact, the third term of eqs. 1 is due to the Collective Random Excitation eﬀect revealed in ref. 1, except a new factor Dwhich is introduced in this paper to include the special case where Zihas the same order of magnitude or even less than Zr. The second term, however, is obtained intuitively as explai ned in section 3. Now we make more discussions on ZiandZr. Being aware of the possible ambiguity coming from this frequently used term in the domain of collec tive instabilities in storage rings, we deﬁne ZrandZiused in this paper as follows: Zr=Pb I2 b=Ktot \|\|,0T2 b T0(5) and Zi=2π T0L (6) where Pb=e2N2 eKtot \|\|,0/T0,Ib=eNe/Tb,Tb= 3σz0/c, and T0is the particle revolu- tion period. By using eqs. 5 and 6 one gets explicit expressio n ofDshown in eq. 3. The procedure to get the information about the bunch lengthe ning and the en- ergy spread increasing is ﬁrstly to ﬁnd Rz(Ib) by solving bunch lengthening equa- tion, i.e., eq. 1, and then calculate energy spread increasi ng,Rε(Ib) (Rε=σε/σε,0), by putting Rz(Ib) into eq. 7 [1]: R2 ε= 1 +C(RavRIbDKtot \|\|,0)2 γ7R2.42z(7) OnceRε(Ib) is found, one can use the following formula to calculate the fast single bunch transverse instability threshold current [2]: Ith b,gao=F′fsE0 e < β y,c>Ktot ⊥(σz)(8) with F′= 4Rε\|ξc,y\|νyσε0 νsE0(9) where νsandνyare synchrotron and vertical betatron oscillation tunes, r espectively, < βy,c>is the average beta function in the rf cavity region, ξc,yis the chromaticity in the vertical plane (usually positive to control the head- tail instability), Ktot ⊥(σz) is the total transverse loss factor over one turn, σε0is the natural energy spread, and E0is the particle energy. In practice, it is useful to express Ktot ⊥(σz) asKtot ⊥(σz) = Ktot ⊥,0/RΘ z, where Ktot ⊥,0is the value at the natural bunch length, and Θ is a constant 2depending on the machine concerned. As a Super-ACO scaling l aw, Θ can be taken as 2/3 [4]. Eq. 8 is therefore expressed as: Ith b,gao=F′fsE0R2/3 z e < β y,c>Ktot ⊥,0(10) The notation Ith b,gaois used with the aim of distinguishing it from the formula giv en by Zotter [5][6]. 2 Comparison with Experimental Results In this section we look at seven machines with their paramete rs shown in table 1. Machine R(m) Rav(m) INFN-A 1.15 5 ACO 1.11 3.41 SACO 1.7 11.5 KEK-PF 8.66 29.8 SPEAR 12.7 37.3 BEPC 10.345 38.2 SLC Damping Ring 2.037 5.61 Table 1: The machine parameters. The machine energy, natural bunch length and the correspond ing longitudinal loss factor are given in table 2. Machine γ σz0(cm) Ktot \|\|,0(V/pC) INFN-A 998 3.57 0.39 ACO 467 21.7 0.525 SACO 1566 2.4 3.1 KEK-PF 3523 1.1 5.4 KEK-PF 4892 1.47 3.7 SPEAR 2935 1 5.2 BEPC 2544 1 9.6 BEPC 3953 2 3.82 SLC Damping Ring 2329 0.53 12 Table 2: The machine energy and the total loss factors. Concerning the loss factors, that of INFN accumulator ring c omes from ref. 6 and the others are obtained by ﬁtting the corresponding experim ental results with the bunch lengthening equation given in ref. 1. Figs. 1 to 10 show the comparison re- sults between the analytical and the experimental [8]-[18] bunch lengthening values, 3and Fig. 11 shows the single bunch energy spread increasing. It is obvious that this improved empirical bunch lengthening equation is quit e powerful. Among the seven diﬀerent storage rings, SLC new damping ring is the uni que and the most interesting one since it is a very resistive ring [16], on the contrary, the other rings including SLC old damping ring are quite inductive. The indu ctances of the old and the new SLC damping rings are 33 nH and 6 nH, respectively [17] . By ﬁtting the bunch lengthening experimental results, one ﬁnds that the l oss factor Ktot \|\|,0equals 12 V/pC at σz0= 0.53 cm (this value is put in table 2), which agrees quite well with the experimentally measured loss factor, 15 V/pC, at th e same bunch length [18]. From Fig. 11 one can see that the single bunch energy spr ead increasing in SLC new damping ring is rather accurately predicted by eq. 7. 3 Discussion In fact eq. 1 can be obtained from the following equation by tr uncating the Taylor expansion of the right hand side of eq. 11 up to the second orde r. R2 z= exp/parenleftBigg√ 2CRavRDKtot \|\|,0Ib γ7/2Rςz/parenrightBigg (11) From the point of view of aesthetics, eq. 11 is more attractiv e (at least for the author). Even if it doesn’t work well itself, this equation i s instructive for us to establish the second term in eq. 1. 4 Conclusion In this paper we propose an improved empirical bunch lengthe ning equation and compare the analytical results with the experimental resul ts of seven diﬀerent ma- chines where SLC new damping ring is quite resistive. The agr eement between the analytical and experimental results is quite satisfactory . The factor Dintroduced in this paper should be included (one should multiply it to Ktot \|\|,0) into the corre- sponding formulae in ref. 1 also in order to be applied to the c ase where a storage ring is very resistive. 5 Acknowledgement The author thanks J. Le Duﬀ and J. Ha¨ ıssinski for their criti cal comments and interests in this subject. I have enjoyed the interesting di scussions on SLC damping rings with K. Bane, B. Podobedov, A. Chao, G. Stupakov, S. Hei fets, and some other theory club members at SLAC. 4References [1] J. Gao, “Bunch lengthening and energy spread increasing in electron storage rings”, Nucl. Instr. and Methods ,A418 (1998), p. 332. [2] J. Gao,“Theory of single bunch transverse collective in stabilities in electron storage rings”, Nucl. Instr. and Methods ,A416 (1998), p. 186. [3] J. Gao,“An empirical equation for bunch lengthening in e lectron storage ring”, Nucl. Instr. and Methods ,A432 (1999), p. 539. [4] P. Brunelle,“Etude th´ eorique et exp´ erimentale des fa isceaux dans l’anneau VUV SUPER-ACO”, th` ese, Universit´ e Paris 7, 1990. [5] B. Zotter, “Mode-coupling or “transverse turbulence” o f electron or positron bunches in the SPS and LEP”, LEP note 363 (1982). [6] B. Zotter, “Current limitations in LEP due to vacuum cham ber bellows”, LEP note 528 (1985). [7] M. Migliorati and L. Palumbo, “Wake ﬁeld energy spread an d microwave insta- bility”, Proceedings of ICFA Beam Dynamics Workshop, Frasc ati, Oct. 20-25, 1997, p. 347. [8] P.B. Wilson, R. Servranckx, A.P. Sabersky, J. Gareyte, G .E. Fischer, A.W. Chao, and M.H.R. Donald, “Bunch lengthening and relate d eﬀects in SPEAR II”, IEEE Trans. on Nucl. Sci .NS-24 (1977) p. 1211. [9] R. Boni, et al., “Bunch lengthening and impedance measur ements and analysis in DAΦNE accumulator ring”, Note: BM-1, Frascati, March 10, 1997. [10] Le groupe de l’anneau de collisions d’Orsay, “Allongem ent des paquets dans ACO”, Rapport technique 34-69, Orsay, le 14 novembre (1969) . [11] A. Nadji, et al., “Experiments with low and negative mom entum compaction factor with Super-ACO”, Proceedings of EPAC96, Barcelona ( 1996) p. 676. [12] N. Nakamura, S. Sakanaka, K. Haga, M. Izawa, and T. Katsu ra, “Collective eﬀects in single bunch mode at the photon factory storage rin g”, Proceedings of PAC91, San Francisco, CA (1991) p. 440. [13] SPEAR Group, “SPEAR II performance”, IEEE Trans. on Nucl. Sci. NS-22 (1975) p. 1366. [14] Z. Guo, et al., “Bunch lengthening study in BEPC”, Proce edings of PAC95, Dallas TX (1995) p. 2955. 5[15] K. Bane, “The calculated longitudinal impedance of the SLC damping rings”, SLAC-PUB-4618, 1988. [16] K. Bane and K. Oide, “Simulations of the longitudinal in stability in the new SLC damping rings”, SLAC-PUB-6878, 1995. [17] K. Bane, et al., “High-intensity single bunch instabil ity behaviour in the new SLC damping ring”, SLAC-PUB-6894, 1995. [18] B. Podobedov and R. Siemann, “New apparatus for precise synchronous phase shift measurements in storage rings”, SLAC-PUB-7939, 1998 . 00.511.522.53 0 10 20 30 40 50 60Rz (analytical results) Rz (experimental results)Rz Ib (mA)INFN Accumulator Ring Figure 1: Comparison between INFN accumulator ring ( R= 1.15 m and Rav= 5 m) experimental results and the analytical results at 510 Me V with σz0=3.57 cm. 00.511.522.533.54 0 5 10 15 20 25 30 35Rz (analytical results) Rz (experimental results)Rz Ib (mA)ACO Figure 2: Comparison between ACO ( R= 1.11 m and Rav= 3.41 m) experimental results and the analytical results at 238 MeV with σz0=21.7 cm. 6012345 0 20 40 60 80 100 120Rz (analytical results) Rz (experimental results)Rz Ib (mA)Super ACO Figure 3: Comparison between Super-ACO ( R= 1.7 m and Rav= 11.5 m) experi- mental results and the analytical results at 800 MeV with σz0=2.4 cm. 00.511.522.53 0 5 10 15 20 25 30 35Rz (analytical results) Rz (experimental results)Rz Ib (mA)KEK-PF (1.8 GeV) Figure 4: Comparison between KEK-PF ( R= 8.66 m and Rav= 29.8 m) experi- mental results and the analytical results at 1.8 GeV with σz0=1.47 cm. 00.511.522.53 0 10 20 30 40 50 60 70Rz (analytical results) Rz (experimental results)Rz Ib (mA)KEK-PF (2.5 GeV) Figure 5: Comparison between KEK-PF (2.5 GeV) ( R= 8.66 m and Rav= 29.8 m) experimental results and the analytical results at 2.5 Ge V with σz0=1.1 cm. 70123456 0 10 20 30 40 50Rz (analytical results) Rz (experimental results)Rz Ib (mA)SPEAR Figure 6: Comparison between SPEAR ( R= 12.7 m and Rav= 37.3 m) experi- mental results and the analytical results at 1.5 GeV with σz0=1 cm. 0246810121416 0 10 20 30 40 50 60Rz (analytical results) Rz (experimental results)Rz Ib (mA)BEPC (1.3 GeV) Figure 7: Comparison between BEPC (1.3 GeV) ( R= 10.345 m and Rav= 38.2 m)experimental results and the analytical results at 1.3 Ge V with σz0=1 cm. 0246810 0 10 20 30 40 50 60 70Rz (analytical results) Rz (experimental results)Rz Ib (mA)BEPC (2.02 GeV) Figure 8: Comparison between BEPC ( R= 10.345 m and Rav= 38.2 m) experi- mental results and the analytical results at 2.02 GeV with σz0=2 cm. 80 1 2 3 4 Ne (1010)0.511.522.53RzSLC Old Damping Ring Bunch Lengthening Figure 9: Comparison between SLC old damping ring ( R= 2.037 m and Rav= 5.61 m) experimental (circles) and analytical (line) results of bunch lengthening at 1.19 GeV with σz0=0.53 cm. 0 1 2 3 4 Ne (1010)0.50.7511.251.51.752RzSLC New Damping Ring Bunch Lengthening Figure 10: Comparison between SLC old damping ring ( R= 2.037 m and Rav= 5.61 m) experimental (circles) and analytica (line) results o f bunch lengthening at 1.19 GeV with σz0=0.53 cm. 0 1 2 3 4 Ne (10**10)0.511.52ReSLC New Damping Ring Energy Spread Increasing Figure 11: Comparison between SLC old damping ring ( R= 2.037 m and Rav= 5.61 m) experimental (circles) and analytical (line) results of energy spread increasing at 1.19 GeV with σz0=0.53 cm. 9
arXiv:physics/0012021v1 [physics.acc-ph] 11 Dec 2000LBNL-47144 December, 2000 Nonlinear QED Eﬀects in Heavy Ion Collisions Spencer R. Klein Nuclear Science Division, Lawrence Berkeley National Labo ratory Berkeley, CA, 94720, USA E-mail: SRKLEIN@LBL.GOV Peripheral collisions of relativistic heavy ions uniquely probe many aspects of QED. Examples include e+e−pair production and nuclear excitation in strong ﬁelds. Af- ter discussing these reactions, I will draw parallels betwe enγ→e+e−andγ→qq and consider partly hadronic reactions. The scattered qqpairs are a proliﬁc source of vector mesons, which demonstrate many quantum eﬀects. Th e two ions are a two-source interferometer, demonstrating interference b etween meson waves. Mul- tiple vector meson production will demonstrate superradia nce, a ﬁrst step toward a vector meson laser. Finally, I will discuss the experiment al program planned at the RHIC and LHC heavy ion colliders. Invited talk, presented at the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, October 15-20, 2000, Capri, Italy 1 Introduction Heavy ion collisions might seem like a strange topic for an ac celerator physics conference. However, many topics of interest to accelerato r physicists also occur in peripheral heavy ion collisions. In these collisio ns, the ions do not physically collide. Instead, they interact electromagnet ically at long ranges, up to hundreds of fermi. Relativistic heavy ions carry extre mely strong elec- tromagnetic ﬁelds, allowing tests of nonperturbative elec trodynamics. These ﬁelds are strong enough to allow for multiple reactions invo lving a single pair of ions, so quantum ﬂuctuations and superluminous emission can be studied. Even for single particle production, quantum interference aﬀects the vector meson spectrum. All of these topics have parallels in advanc ed accelerator design. And, some aspects of heavy ion interactions impact d irectly on ac- celerator design. This writeup will review the physics of pe ripheral heavy ion collisions, with an emphasis on principles. Mathematical a nd experimental details are left to the references. Several diﬀerent types of peripheral reactions are possibl e. The two nuclei may exchange one or more photons (Fig 1a). One or both nuclei m ay be excited 1pair,...pair,... AA c) d) Fermion pair, vector meson...Fermion pair, vector meson...A Ab) AAAA/A Aa) A Figure 1: Some peripheral reactions: (a) Mutual nuclear exc itation. (b) Two-photon in- teractions (c) Multiple (double) interaction, possible be causeZαis large. (d) Two-photon interaction with nuclear excitation. The dashed line shows how the reaction factorizes into independent two-photon (or photon-Pomeron) and nuclear ex citation reactions. This is the dominant diagram; the amplitude for excitation by the photo n in (b) is small7. into a Giant Dipole Resonance (GDR) or higher state. Or, the p hoton may interact with a single nucleon in the nucleus in an incoheren t photonuclear interaction. Two ﬁelds may interact with each other. In a two-photon inter action, each nucleus emits a photon. The two photons collide to produ ce a leptonic or hadronic ﬁnal state, as in Fig. 1b. The ﬁelds are so strong t hat ‘two- photon’ is a misnomer- the number of photons from one nucleus may be large, and, in fact, poorly deﬁned. A photon from one nucleus may int eract with the coherent meson or Pomeron ﬁelds of the other. Although th is reaction has some similarities with incoherent photonuclear interacti ons, coherence restricts the ﬁnal state kinematics, so reactions involving two coher ent ﬁelds produce kinematically similar ﬁnal states. Here, we (by deﬁnition) require that the two nuclei physical ly miss each other and do not interact hadronically. The impact paramete rb >2RA,RA being the nuclear radius. More detailed calculation will ca lculate and use the non-interaction probability as a function of b. In the nuclear rest frame, a photon, Pomeron or meson couplin g coherently to a nucleus must have p <¯hc/R A. More precisely, the coupling is governed 2by the nuclear form factor. In a collider where each nucleus i s Lorentz boosted byγ, this coupling transforms to p⊥<¯hc/R Aand photon energy k=p\|\|< γ¯hc/R A. So, two-ﬁeld interactions can occur up to a maximum energy W= 2γ¯hc/R A, with a ﬁnal state p⊥<2¯hc/R A. For photons, p⊥is actually smaller, peaked atp⊥<¯hc/b. Two-photon, photon-Pomeron/meson and double-Pomeron/me son reac- tions are all possible. Double-Pomeron/meson interaction s are limited to a narrow range of impact parameter because of the short range o f the strong force. Therefore, they will occur with a relatively low cros s section. They will also have a quite diﬀerent p⊥spectrum. The p⊥spectral diﬀerence will allow some statistical separation between two-photon and photon -Pomeron interac- tions. For most applications, the electromagnetic ﬁelds of ultra- relativistic nuclei may be treated as a ﬁeld of virtual photons, following Weizs¨ acker-Williams. The photon ﬂux from a nucleus with charge Za distancerfrom a nucleus is d3N(k,r) dkd2r=Z2αx2 π2kr2K2 1(x) (1) wherex=kr/γ¯handK1(x) is a modiﬁed Bessel function. The two-photon luminosity is the overlap of the two photon ﬁelds. The usable two-photon luminosity Lγγis this overlap, integrated over all b >2RA. This can be calculated using Lγγ(W,Y) =LAA/integraltextdk1 k1/integraltextdk2 k2 2π/integraltext∞ RAb1db1/integraltext∞ RAb2db2/integraltext2π 0dφd3N(k1,b1) dk1d2b1d3N(k2,b2) dk2d2b2Θ(b−R1−R2) (2) whereLAAis the nuclear luminosity, Θ is the step function and the impa ct parameterb=/radicalbig (b2 1+b2 2−2b1b2cos(φ))1 2. This must be evaluated numer- ically. The requirement that the nuclei not physically coll ide (Θ function) reduces the ﬂux by about 50%. The ﬁnal state energy W= 4k1k2and rapid- ityy= 1/2 ln(k1/k2) can also be found. Usually, the slight photon virtuality q2<(¯h/R A)2can be neglected. The exception is e+e−production, since q2∼(¯h/R A)2≫m2 e. Since Pomerons and mesons are short-ranged, photon-Pomero n and pho- ton/meson interactions take place inside one of the nuclei. At a given b, the photon intensity is found by integrating the photon ﬂux over the surface of the target nucleus, and normalizing by dividing by the area πR2 A. The to- tal eﬀective photon ﬂux is this intensity, integrated over a llb >2R. It is found analytically; the result is within 15% of the integrat ed ﬂux in the region 3Table 1: Beam Species, Energies, Luminosities, compared fo r RHIC (Summer, 2000), RHIC Design and LHC. RHIC is expected to reach it’s design paramet ers in 2001. Machine Species Beam Energy Max. Luminosity (per nucleon) (cm−2s−1) RHIC 2000 Gold 65 GeV 2×1025 RHIC Gold 100 GeV 2×1026 RHIC Silicon 125 GeV 4.4×1028 LHC Lead 2.76 TeV 1×1026 LHC Calcium 3.5 TeV 2×1030 b>2RA: dNγ dk=2Z2α πk/parenleftbig XK0(X)K1(X)−X2 2[K2 1(X)−K2 0(X)/parenrightbig (3) whereX= 2RAk/γ. ForX <1, the total number of photons with kmin< k<k maxis Nγ=2Z2α πln/parenleftbigkmax kmin/parenrightbig . (4) For photo-nuclear interactions, the maximum photon energy seen by one nu- cleus is strongly boosted, by Γ = 2 γ2−1, or 20,000 for RHIC and 1 .5×107 for LHC. Thus, the photon energies reach 600 GeV with gold at R HIC, and 500 TeV for lead at the LHC; with lighter nuclei, these number s are 2-3 times higher. Fixed target heavy ion accelerators can produce e+e−pairs, with and without capture; heavier states are not energetically acce ssible. These reactions have been studied at the LBL Bevalac, BNL AGS and CERN SPS. Stu dies of hadroproduction is just beginning at the Relativistic He avy Ion Collider (RHIC) at Brookhaven National Laboratory, and the Large Had ron Collider at CERN; these colliders are energetic enough to produce a va riety of ﬁnal states. The characteristics of these colliders are shown in Table 1. Peripheral collisions have recently been reviewed by Baur, Hencken and Trautmann3. 2 Nuclear Excitation and Incoherent Photonuclear Interact ions For low energy photons, nuclear excitations are typically c ollective. For exam- ple, in a Giant Dipole Resonance, the protons oscillate in on e direction and the neutrons in the other. This vector oscillation can be induce d by a single pho- ton. Higher excitations include double (or higher) Giant Di pole Resonances, 4Table 2: Cross sections for nuclear excitation5, pair production (Eq. 5), bound-free pair production5,ρ,J/ψand double- ρproduction16. The nuclear excitation and bound e−cross sections are per ion. System σ(Exc.)σ(e+e−)σ(bounde−)σ(ρ)σ(J/ψ)σ(ρρ) RHIC-Au 58 b 33 kb 45 b 590 mb 290µb720µb RHIC-Si 150 mb 41 b 1.8 mb 8.4 mb 3.6µb LHC-Pb 113 b 150 kb 102 b 5.2 b 32 mb 8.8 mb LHC-Ca 800 mb 600 b 36 mb 120 mb 390µb highernstates of a harmonic oscillator. There are also Giant Quadru pole Resonances, which require multiple photons to produce. The se states typically decay by emitting one or more neutrons which can be detected i n far-forward calorimeters. These reactions are of interest for a couple of reasons. As Ta ble 2 shows, the cross sections are substantial4. Nuclear excitation is a substantial contrib- utor to beam loss. The photon carries little momentum, so nuc lear excitation creates a beam of particles with unchanged momentum but alte red charge to mass ratio5. This beam will escape the magnetic optics and strike the beampipe at a relatively well deﬁned point downstream, loca lly heating the magnets. This heating could cause superconducting magnets to quench. Also, this beam could be extracted from the accelerator, for ﬁxed t arget use. A single photon can excite both the emitting and target nucle i, although the cross section is smaller than for single excitation. Thi s double process is signiﬁcant for a couple of reasons. It has a clean signatur e and is useful as a luminosity monitor6. Second, it can tag small bevents. To a good approximation, the nuclear excitation photon factorizes f rom the remainder of the interaction7, as is shown in Fig. 1(d). Thus the nuclear excitation can tag collisions at low b. 3 Two-Photon Interactions Two-photon interactions have been studied extensively at e+e−colliders. Pho- tons couple to charge, so two-photons couplings measure the internal charge content of mesons; qqpairs are produced, but not charge-free states like glue- balls. Hybrids ( qqg) and 4-quark states ( qqqq) are produced at intermediate rates. Thus, coupling to two-photons is a sensitive test for exotic mesons. Meson pair production rates depend on the pair energy. Near t hreshold, charged meson pairs ( π+π−) are produced, but neutral pairs ( π0π0) are not. At higher energies, the photons see the quark structure of me sons, and both 5Wgg [GeV]dL/dWgg [1030 cm-2 s-1 GeV-1]RHIC I+I CESR LEP 10-310-210-1110102103 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 2: Two-photon luminosity expected at RHIC with gold a nd iodine beams, compared with the luminosities at LEP II (√s= 180 GeV and a luminosity of 5 ×1031cm−2s−1) and at CESR (√s= 10 GeV and a luminosity of 2 .9×1032cm−2s−1). charged and neutral mesons are produced. Two-photon interactions at heavy ion colliders are of inter est because that the luminosity scales as Z4and extremely high rates are possible. Figure 2 compares the γγluminosities at RHIC, with the LEP and CESR e+e−col- liders8; forW < 1.5 GeV, RHIC can reach the highest presently available two-photon luminosities. Heavy ion colliders also probe so me unique areas, such as multiple pair production, and bound-free pair produ ction; both are probes of strong ﬁeld QED. 3.1 Lepton Pair Production Lepton pair production can test the limits of perturbative Q ED. Perturbation theory may fail because the coupling Aαis so large. Even with perturbative approaches, e+e−production introduces additional complications. The elec - tron Compton wavelength, Λ e=386 fm, is large compared to typical impact parameters. So at W∼2me, where the bulk of the cross section is, the pair production is poorly localized. The ﬁrst perturbative calculation speciﬁc to heavy ion coll isions was by Bottcher and Strayer9. They treated the ions as sources of classical (but relativistic) electromagnetic potentials that follow ﬁxe d trajectories. This approach naturally incorporated oﬀ-shell photons. This ca lculation also ac- counted for large electron Compton radius Λ e= 386 fermi, with an appropriate cutoﬀ. In the two-photon approach, Λ eshould replace the minimum impact 6parameter, RA∼7 fermi, in Eq. 2. This reduces the cross section signiﬁcantl y compared to earlier calculations. A slightly later, more reﬁned calculation by Baur and Bertul ani included Coulomb corrections, to account for the fact that the pair is produced deep in a Coulomb potential10. With this reﬁnement, the cross section is given by σ=28Z4α4¯h2 27πm2ec2/parenleftbig ln3(Γδ 2)−3 2(1 + 2f)ln2(Γδ 2)/parenrightbig (5) wheremeis the electron mass, δ= 0.681 is Euler’s constant and f= (Zα)2 Σ∞ n=1[n(n2+Z2α2)]−1is the usual Coulomb correction. The ln3term dom- inates at high energy. Other authors have found slightly diﬀ erent results, depending on the details of the calculation. Baur and Bertulani also calculated the probability of pair p roduction at a givenb. With gold at RHIC, this probability is greater than 1 for b= Λ e! The diﬀerential cross section dσ/2πbdbsaturates. The problem is resolved by multiple pair production: a single ion pair small- bconfrontation can produce more than one pair. The the number of pairs is Poisson distrib uted, with the b-dependent mean11. This saturation can also aﬀect calculations of the single pair cross section. Numerous authors have considered non-perturbative e+e−production, usually using the time-dependent Dirac equation. Some auth ors solved the coupled-channel equations numerically. The ions were step ped through their positions. At each step, the coupling from the initial state to a pair-containing ﬁnal states was calculated. An accurate calculation requir es a complete and orthogonal set of states. This turned out to be rather diﬃcul t, and early calculations found results that varied by orders of magnitu de. Baltz and McLerran calculated pair production to all orders12. Their method is similar to the perturbative calculation. They wor ked in light-cone coordinates with Lienard-Wiechert potentials similar to t hose of Bottcher & Streyer. They ﬁrst found the Greens function for the exact wa ve function at the interaction point. The transition amplitude was then co nstructed from the Greens function. The total cross section is this amplitu de, integrated over impact parameter and intermediate transverse momentu m. Their result matches the perturbative result (without Coulomb correcti ons). Recently, Roman Lee and A. Milstein found a problem with the o rder of integration in the Baltz and McLerran paper13. When the order changed, Lee and Milstein the result changed to include the Coulomb corre ction found by Baur & Bertulani (the fterm in Eq. 5). The agreement with perturbation theory is somewhat surpris ing, given the large coupling. However, Baltz and McLerran found that, for multiple 7pair production, their result was smaller than the perturba tive result. Since multiple pair production is naturally a higher order proces s, it’s not surprising that a diﬀerence appears. A related reaction is bound-free pair production where the e lectron is pro- duced bound to one of the nuclei. As with free pairs, perturba tive calculations may be inadequate, and an exact solution to the time-depende nt Dirac equa- tion is desired. This problem has also been tackled perturba tively; here the ﬁnal state consists of a free positron and an electron in an at omic orbital. The cross section to produce an electron bound in an atomic K−shell is3 σ=33πZ8α8¯h2 10m2ec21 exp(2πZα)−1/bracketleftbig ln(Γδ 2)−5 3/bracketrightbig . (6) The stronger Zdependence comes from the electron-nucleus binding energy . Inclusion of higher shells will increase this by about 20%. T his cross section has the form σ=Aln(γ)+B. Extrapolations from lower energy data using this form ﬁnd a cross section about twice as large14. Coupled-channel calculations have been tried on this problem, and produced a wide range of r esults. Also, as with free-production, an all-order solution to the time-de pendent Dirac equa- tion has recently been found, again using light-cone coordi nates15. The result was slightly lower than perturbation theory. The cross sect ion for bound-free production is much lower than for free production, so that dσ/2πbdbis not saturated. The 1-electron atoms produced in this reaction have their mo mentum un- changed, so that they will follow well-deﬁned trajectories . As with nuclear excitation, this can lead to heating of the accelerator magn ets and also allow for extracted beams5. In principle, these non-perturbative aspects of pair produ ction also apply toµ+µ−andτ+τ−production. However, the masses are much larger, so any non-perturbative eﬀects are much smaller. Because mµ>¯h/R A, Eq. 2 applies for heavy lepton production. 4qqﬂuctuations and Vector Meson Production The vacuum ﬂuctuation γ→qqis similar to γ→e+e−; only the ﬁnal state charges and masses are diﬀerent. Just as the virtual e+e−pair can interact with an external Coulomb ﬁeld and become real, the qqpair can interact with an external nuclear ﬁeld and emerge as real vector meson16. This picture is clearest in the target rest frame. The incomi ng photon has a high momentum, and the ﬂuctuation persists for a time τf= ¯h/M, during which it travels a distance known as the formation distance lf= 2¯hk/M2. In 8alternate language, lf= ¯h/p\|\|, wherep\|\|is the momentum transfer required to make the pair real. For e+e−pairs,lfis typically much larger than a single atom; forqqpairs,lfis typically much larger than a single nucleus. So, the ﬂuctuation cannot see the target structure. During it’s lif etime, the ﬂuctuation can interact with the external ﬁeld to become a real pair. Theqqscatters elastically from the a nucleus with atomic number A. This scattering is mediated by the strong force and transfers eno ugh momentum to give the meson its mass. The scattering leaves the photon q uantum num- bersJPCunchanged. This elastic scattering cannot easily be descri bed in terms of quarks and gluons. The most successful description is in terms of the Pomeron17. For hard processes the Pomeron may be thought of as a 2-gluon (quasi-bound) ladder, connected by gluon rungs. Ho wever, for soft processes such as elastic scattering, this picture may be in appropriate. For soft reactions, the best picture is the 40-year old soft-Pom eron diﬀractive pic- ture18. The Pomeron absorbs part of the photon wave function, allow ing aqq to emerge dominant. In this model, the cross section for the reaction A+A→A+A+V may be calculated in a straightforward manner. The starting point is data onγ+p→V+pfrom ﬁxed target experiments and HERA. The forward scattering amplitudes may be parameterized dσ/dt \|t=0=bv(XWǫ+YW−η), wheretis the 4-momentum transfer from the nucleus and here Wis theγp center of mass energy. The ﬁrst term, with ǫ∼0.22, is for Pomeron exchange, while the second is for meson exchange; Pomeron exchange dom inates at high energies. This amplitude factorizes into two parts: the γ→qqamplitude and the elastic scattering amplitude. The ﬁrst part can be deter mined from the partial width for V→e+e−, allowing vector meson production data to ﬁx the scattering amplitude. Vector meson dominance allows us to treat the qq ﬂuctuation as a real vector meson. The optical theorem can be used to ﬁnd the totalVpcross section. The totalVAcross section may be found with a Glauber calculation. This calculation integrates over the transverse plane, sum ming the probability of having 1 or more interactions: σtot(VA) =/integraldisplay d2/vector r/parenleftbig 1−e−σtot(V p)TAA(/vector r)/parenrightbig (7) whereTAA(/vector r) is the nuclear thickness function. These cross sections ri se with Wat low energies, then level oﬀ at an almost constant value. The optical theorem is used to ﬁnd dσ/dt \|t=0for the meson -nucleus scat- tering. Finally, the leptonic width is used to ﬁnd the forwar d amplitude for vector meson production. In the small- σlimit,σtot(Vp)TAA(b= 0)≪1, the 9forward amplitude scales as A2. This limit applies for heavy systems such as cc. Asσtot(Vp) rises, theA−dependence decreases, and for large σtot(Vp), the scaling isA4/3, with the vector meson seeing the front face of the nucleus. The total photonuclear cross section is given by an integrat ion overt: σ(γA→VA) =dσ/dt (γA→VA)\|t=0/integraldisplay∞ tmindt\|F(t)\|2(8) wheretmin=M2 v/4kandF(t) is the nuclear form factor. For a heavy nucleus, F(t) may be ﬁt analytically by a convolution of a hard sphere with a Yukawa potential. Eq. 8 agrees well with data from ﬁxed target experiments. The total cross section is σ(A+A→A+A+V) = 2/integraldisplay dkdNγ dkσ(γA→VA). (9) The factor of 2 is because either nuclei can act as target or em itter. These cross sections are given in Table 2. The implications of this straightforward calculation are s igniﬁcant. The cross sections are huge. With gold at RHIC, ρ0production is 10% of the total hadronic cross section. With lead at LHC, the ρ0cross section is about equal to the hadronic cross section! Heavy ion colliders can act as vector meson factories, with rates comparable to e+e−vector meson machines. The 1010φ produced in 106seconds with calcium beams at LHC is comparable to that expected at a dedicated φfactory. Searches for rare decay modes, CP violation and the like are possible. Also, vector meson spectroscopy w ill be productive; mesons like the ρ(1450),ρ(1700) and φ(1680) will be copiously produced. Fully coherent ﬁnal states will be distinctive. The ﬁnal sta tep⊥is a con- volution of the photon and Pomeron p⊥. Figure 3 shows these contributions. The meanp⊥from the photon is ¯ h/b, considerably smaller than ¯ h/R A. This approach can also ﬁnd the vector meson rapidity distrib ution. The ﬁnal state rapidity y= 1/2 ln(MV/k). So,dσ/dy =k/2dσ/dk and can be determined from Eq. 9. The photon can come from either direct ion, so the totalσ(y) includes contributions for photons from + yand−y.dσ/dy is shown in Fig. 4. 4.1 Interference The observed p⊥spectrum is more complicated than Fig. 3 shows. Either nucleus can emit the photon. The two possibilities are indis tinguishable, and 10dN/dp^2 a) y=0 p^ [GeV/c]dN/dp^2 b) y=-210-310-210-11 10-310-210-11 0 0.05 0.1 0.15 0.2 Figure 3: The vector meson p⊥spectrum (solid line) at y= 0 (a) and y= 2 (b) is the convolution of the photon p⊥(dotted line) and the scattering p⊥transfer (dashed line). therefore, they interfere. In essence, the two nuclei act as a two-source inter- ferometer. The two possible emitters are related by a parity transformation. Vector mesons are negative parity so the two possibilities c ontribute with op- posite signs, producing destructive interference19. The cross section is σ(p⊥,y,b) =A2(p⊥,y,b) +A2(p⊥,−y,b) −2A(p⊥,y,b)A(p⊥,−y,b)cos(φ(y)−φ(−y) +/vector p⊥·/vectorb)(10) whereA(p⊥,−y,b) is the production amplitude and φ(y) is the production phase.Amay be found from the previous section. For pure Pomeron exch ange, the production is almost real. The production phase always c ancels aty= 0, and cancels everywhere unless φdepends on k. Variation is likely with the ρ andωbecause of the meson contribution. For other mesons, it is li kely to be small or negligible. 11ds/dy [mb]Au RHIC r fds/dy [mb] J/Y yds/dy [mb]Ca LHC y050100 -5 -2.5 0 2.5 5 02468 -5 -2.5 0 2.5 5 0255075100 -5 -2.5 0 2.5 502.557.510 -5 0 5 00.20.40.60.8 -5 0 5 0204060 -5 0 5 Figure 4: Rapidity distribution dσ/dy with gold at RHIC (left panels) and calcium at the LHC (right panels) for the ρ0,φandJ/ψ. The solid line is the total, while the dashed line shows the production for a single photon direction. At midrapidity, the interference simpliﬁes to σ(p⊥,y= 0,b) =A2(p⊥,y= 0,b)(1−cos[/vector p·/vectorb]). (11) For a given b,σoscillates with period ∆ p⊥= ¯h/b. Whenp⊥b<¯h, the inter- ference is destructive and there is little emission. The mea nbforρproduction at RHIC is about 40 fermi, rising to 300 fermi at LHC. The impact parameter is unmeasured, so it is necessary to int egrate over allb. This dilutes the interference, except for p⊥<¯h//angbracketleftb/angbracketright. Figure 5 shows the expectedp⊥spectrum with and without interference. The mean impact parameter for ρproduction with gold at RHIC is 40 fermi, far larger than the rho decay distance γβcτ < 1 fermi. The vector 12dN/dp^2 a) Au+Au fb) Si+Si fc) Ca+Ca f p^ [GeV/c]dN/dp^2 d) Au+Au J/Y p^ [GeV/c]e) Si+Si J/Y p^ [GeV/c]f) Ca+Ca J/Y00.20.40.60.81 00.20.40.60.81 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 Figure 5: Meson p⊥spectra, with (solid lines) and without (dashed line) inter ference, at y=0. The top panels are for the φ, and the bottom for the J/ψ, with gold (left) and silicon (center) at RHIC, and calcium at the LHC (right). mesons decay before their wave functions can overlap! Howev er, the decay product do overlap and interfere. The angular distribution s for the two ρ0 sources are the same, so the interference pattern is not aﬀec ted. This process requires a non-local wave function. Consider ρ0→π+π−, withb∼40 fermi. Before the π+waves from the 2 sources can overlap, they must travel ∼20 fermi each, during which time the π−waves will travel 20 fermi in the opposite direction, and the π+andπ−waves will be separated by 40 fermi. So, non- locality is required to produce this interference pattern. Although there is as yet no counterpart to Bell’s inequality , the choice of quantum observable does matter for this system. Consider a s ystem where bis measured. For the π+andπ−, one can measure either the momentum or position. If the momenta of both πare measured, then the interference pattern is observed. If the π+momentum is known, that disallows certain values ofπ−momentum where destructive interference is complete. If th e positions of both πare measured, the production point can be determined, but the interference disappears. If one position and one moment um are observed, neither the interference pattern nor the production point c an be determined. 13The wave function of the system is Ψ(/vector x) = exp(i(/vectork−+/vectork+)·/vector x)/bracketleftbig exp(i(/vectork−+/vectork+)·/vectorRA)−exp(i(/vectork−+/vectork+)·/vectorRB)/bracketrightbig (12) where/vector xis where the vector meson would be if it didn’t decay; in the ve ctor meson rest frame /vector x= 1/2(/vector x++/vector x−) where/vector x+and/vector x−are the position for theπ+andπ−, and/vectork+and/vectork+their momenta. This wave function cannot be factorized: Ψ( π+π−)/negationslash= Ψ(π+)Ψ(π−). Since the π+andπ−are well separated, the wave function is non-local. This system is thus an exampl e of the Einstein- Podolsky-Rosen paradox. 4.2 Multiple Vector Meson Production The vector meson production probability at a given bmay be calculated with the impact-parameter dependent photon ﬂux. This is shown in Fig. 6. At b= 2R, the probability of ρ0production is 1% at RHIC, rising to 3% at LHC. These probabilities are high enough that multiple meson pro duction should be observable. In the absence of quantum or other correlations , multiple meson production should be independent and Poisson distributed. Atb= 2R, the ρ0ρ0probabilities are (1%)2/2 and (3%)2/2 at RHIC and LHC respectively. After integration over b, 1.4 million ρ0ρ0are expected per year at RHIC. Like meson triples should also be produced in observable numbers . Vector mesons are bosons so production of like-meson pairs should be enhan ced for momentum diﬀerencesδp<¯h/R A. The meson follows the photon spin and can be aligned or anti-aligned with the beam direction, so the enhancement is only 50%, so N(pair) ∼= 1 + 0.5 exp(δpRA/¯h). 5 Experimental Status Fixed target measurements have been published for pair prod uction, with and without capture, and nuclear excitation. Due to space limit ations, this writeup will only consider relativistic collisions, with Γ >10. The solid targets, with the nuclei surrounded by their electron clouds, diﬀer from t he stripped ion collisions we focus on here. Measurements of pair productio n in sulfur on heavy ion collisions around Γ = 160 have matched theoretical predictions20. Pair production with capture has also been studied with lead beams14. As was previously mentioned, when scaled to RHIC and LHC energies, this data may exceed current estimates. However, corrections may be need ed for the limited boost of the current experiments. Programs to study a variety of peripheral reactions are unde rway in the STAR collaboration at RHIC and the CMS collaboration at LHC. For most 14b [fm]ds/2pbdb J/YFwr 10-810-710-610-510-410-310-210-1 0 10 20 30 40 50 60 Figure 6: Probability of meson production, with gold at RHIC , as a function of b. reactions, the largest backgrounds are expected to be grazi ng hadronic colli- sions, beam gas interactions, and incoherent photonuclear interactions8. For triggering, debris from upstream interactions, and cosmic ray muons can be important. These backgrounds can be separated from the signals by selec ting events with low multiplicity, typically, 2 or 4, low total p⊥, and zero net charge. Baryon number and strangeness must also be conserved. At the trigger level, signiﬁcant rejection can be achieved b y requiring that the event originate inside the interaction region; this rem oves most of the beam gas events, along with almost all of the upstream interactio ns and cosmic ray muons. Event timing cuts also help reject cosmic ray muons. The STAR detector combines a large acceptance with a ﬂexible trigger21. Charged particles are detected in the pseudorapidity range \|η\|<2 and 2.4< \|η\|<4 by a large central time projection chamber (TPC) and two for ward TPCs. This TPC can also identify particles by dE/dx . Neutral particles are detected by a central barrel ( \|η\|<1) and endcap (1 <η< 2) calorimeter. Two zero degree calorimeters will detect neutrons from nuclear breakup, useful for background rejection. For triggering, a scintillator barrel covering \|η\|<1 and multi-wire pro- portional chambers covering 1 <\|η\|<2 measure charged particle multiplicity on an event by event basis. These detectors have good segment ation, allowing 15Figure 7: Side view of an event collected with the peripheral collisions trigger. The invariant mass andp⊥are consistent with coherent ρ0production. for total multiplicity and topological selection in the tri gger. The trigger has 4 levels, with the earliest level based on ﬁeld programmable gate arrays and the later levels computer based. The ﬁnal selection uses on- line TPC track- ing. Peripheral collisions data will be collected in parall el with central collision data. Simulations show that the planned trigger algorithms should be able to eﬃciently select peripheral events while rejecting enough background enough to minimize deadtime8. STAR took it’s ﬁrst data this summer (2000). The central TPC, scintillator barrel and zero degree calorimeters were operational. Alth ough the trigger was not completely functional, in late August, the collaborati on took about 7 hours of data with a dedicated trigger optimized to select 2-track peripheral events22. The trigger rate of 20-40 Hz was ﬁltered to 1-2 Hz by the ﬁnal tr igger, which reconstructed the tracks on-line. About 20,000 events were written to tape. The initial event selection required a 2-oppositely-charg ed track, primary vertex in the interaction diamond. The tracks were required to be at least slightly acoplanar to eliminate cosmic ray muons, and the ev ent had to have a smallp⊥. About 300 events passed these cuts. This data is now being analyzed for signals from e+e−pair andρ0production - the two processes 16with the largest cross sections. Figure 7 shows an example of aρ0candidate. The CMS collaboration plans to study peripheral collisions with lead and calcium beams at LHC23. Their plans are at a fairly early stage. 6 Conclusions Peripheral collisions of heavy nuclei can probe a wide varie ty of phenomena, including many faces of strong QED. Production of e+e−andqqpairs can probe the electrodynamics of the vacuum. Besides the physic s interest, pe- ripheral collisions aﬀect many other areas, as a tool for had ron spectroscopy, and impacting accelerator design, After many years of theoretical discussion, experimental r esults are begin- ning to become available. Acknowledgements I would like to acknowledge Joakim Nystrand, my collaborato r in the studies of vector mesons. This work was supported by the U.S. DOE, und er Contract No. DE-Ac-03-76SF00098. References 1. G. Baur and L.G. Ferreira Filho, Nucl. Phys. A518 , 786 (1990). 2. R. N. Cahn and J. D. Jackson, Phys. Rev. D42, 3690 (1990). 3. G. Baur, K. Hencken and D. Trautmann, J. Phys. G 24, 1657 (1998). 4. M. Vidovic, M. Greiner and G. Soﬀ, Phys. Rev. C48, 2011 (1993). 5. S. Klein, physics/005032, to appear in Nucl. Instrum. Met h. 6. A. J. Baltz, C. Chasman and S. N. White, Nucl. Instrum. Meth A417 , 1 (1998). 7. K. Hencken, D. Trautmann and G. Baur, Z. Phys. C68, 473 (1995). 8. J. Nystrand and S. Klein, in Proc. Wkshp. on Photon Interactions and Photon Structure , Lund, Sweden, 1998 ed. G. Jarlskog and T. Sj¨ ostrand; nucl-ex/9811007. 9. C. Bottcher and M. R. Strayer, Phys. Rev. D39, 1330 (1989). 10. C. A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1998). 11. K. Hencken, D. Trautmann and G. Baur, Phys. Rev. C59, 841 (1999). 12. A. J. Baltz and Larry McLerran, Phys. Rev. C 58, 1679 (1998). 13. R. Lee, contribution to this conference; R. Lee and A. I. M ilstein, Phys. Rev. A 61, 032103 (2000). 14. P. Grafstr¨ om et al.,Measuurement of electromagnetic cross sections in heavy ion interactions and its consequence for luminosity l ifetimes in ion 17colliders , CERN-SL-99-033 EA. 15. A. J. Baltz, Phys. Rev. Lett. 78, 1231 (1997). 16. S. R. Klein and J. Nystrand, Phys. Rev. C 60, 014903 (1999). 17.Quantum Chromodynamics and the Pomeron , by J. R. Forshaw and D. A. Ross, Cambridge University Press, 1997, is a good discuss ion of the modern Pomeron. For a more traditional approach, see Ref. 6. 18. T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev . Mod. Phys.50, 261 (1978). 19. S. R. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000). 20. R. Baur et al., Phys. Lett. B332 , 471 (1994); C. R. Vanes et al., Phys. Rev. Lett. 69, 1711 (1992). 21. K. H. Ackermann et al., Nucl. Phys. A661 , 681 (1999). 22. J. Seger, presented at the 2000 APS Division of Nuclear Ph ysics Meeting, Oct. 4-7, 2000, Williamsburg, VA. Transparencies are avail able on the web at http://www-rnc.lbl.gov/STAR/conf/talks2000/dnp /seger.pdf. 23. G. Baur et al., hep-ph/9904361. 18
arXiv:physics/0012022v1 [physics.ins-det] 12 Dec 2000Direct measurement of sub-pixel structure of the EPIC MOS CCD on-board the XMM/NEWTON satellite J. HIRAGAa,1H.TSUNEMIaA.D.SHORTbA.F.ABBEYb P.J.BENNIEbM.J.L.TURNERb aDepartment of Earth and Space Science, Graduate School of Sc ience, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 56000 43, Japan CREST, Japan Science and Technology Corporation (JST) bSpace Reserch Centre, University of Leicester, Leicester, LE1 7RH, UK Abstract We have used a mesh experiment in order to measure the sub-pix el structure of the EPIC MOS CCDs on-board the XMM/NEWTON satellite. The EPI C MOS CCDs have 40 µm-square pixels which have an open electrode structure in or der to improve the detection eﬃciency for low-energy X-rays. We ob tained restored pixel images for various X-ray event grades (e.g. split-pixel eve nts, single pixel events, etc.) at various X-ray energies. We conﬁrmed that the open electrode structure results in a di storted horizon- tal pixel boundary. The open electrode region generates bot h single pixel events and vertically split events, but no horizontally split even ts. Because the single pixel events usually show the best energy resolution, we discuss a method of increas- ing the fraction of single pixel events from the open electro de region. Furthermore, we have directly measured the thickness of the electrodes an d dead-layers by com- paring spectra from the open electrode region with those fro m the other regions: electrodes, electrode ﬁnger and channel stop. We can say tha t EPIC MOS CCDs are more radiation hard than front-illumination chips of ACIS o n-board Chandra X-ray Observatory because of their extra absorption thickness ab ove the charge transfer channel. We calcurated the mean pixel response and found tha t our estimation has a good agreement with that of the ground calibration of EPIC M OS CCD. Key words: charge-coupled device, mesh experiment, open electrode st ructure PACS; 07.85.-m, 29.30.Kv 1Partially supported by JSPS Research Fellowship for Young S cientists, Japan. Preprint submitted to Elsevier Preprint1 Introduction Charge-coupled devices (CCDs) in use in X-ray astronomy com bine moderate energy resolution with good spatial resolution[1]. Thanks to these character- istics, they have become a standard X-ray photon counting de tector. When an X-ray photon is photoabsorbed inside a CCD, a number of ele ctrons pro- portional to the incident X-ray energy are liberated. In thi s way, the energy of an X-ray photon can be measured. Because optical photons c an produce only a few electrons, however, CCDs have no energy resolutio n at optical wavelengths. The charge cloud produced by an X-ray photon inside the CCD dr ifts within the depletion region to the bottom of the potential well of th e given pixel, resulting in a detected X-ray event. Due to the diﬀusion proc ess, the charge cloud has a ﬁnite size, which can result in the event being det ected in more than one pixel. X-ray events may be therefore classiﬁed by ‘g rade’ according to the number of pixels in which they are detected. When the enti re charge cloud is collected within one pixel, for example, it is referred to as a ‘single pixel event’. When the cloud splits into an adjacent horizontal pi xel, it is referred to as a ‘horizontally split event’. Similarly, when the cloud s plits into an adjacent vertical pixel, it is referred to as a ‘vertically split even t’. A CCD consists of a two dimensional array of small pixels. The spatial reso- lution is limited by the pixel size which is typically severa l tens of µm. The electrode structure, which comprises layers of poly-silic on and silicon-oxide, results in a non-uniformity of detection eﬃciency over the p ixel. In order to measure the structure of one pixel directly, it is therefore necessary to de- termine the X-ray interaction position on a scale smaller th an the pixel size. Recently, we have developed a new technique which allows us t o specify the X-ray interaction position with sub-pixel resolution usin g a two-dimensional mesh containing small holes (much smaller than the CCD pixel size) which are periodically spaced [2] . The sub-pixel structure of var ious types of CCD (ASCA SIS [3], CHANDRA ACIS [4] and HPK CCD [5]) have been meas ured using this method. The X-ray response of a CCD is very sensitive to the thickness of the gate structures within the pixel and is also non-uniform within t he pixel. The time- averaged output therefore represents a mean pixel response , rather than the response at any given location within the pixel. Thus the CCD response con- sists of various parameters which are diﬃcult to measure sep arately. We have therefore developed the mesh technique in order to measure t he response (pixel structure) of CCDs with sub-pixel resolution. This techniq ue has been previ- ously applied to the CCDs [6] employed in the XMM/NEWTON obse rvatory by Tsunemi et al[7]. In this paper, we report on measurements of the sub- 2pixel structure of the EPIC MOS CCD with substantially impro ved spatial resolution. 2 The EPIC MOS CCD The XMM/NEWTON satellite was developed by the European Spac e Agency (ESA) which comprises a membership of 13 European countries . It was launched into a relatively high-earth orbit in December, 19 99. Among its in- struments are the 3 EPIC imaging spectrometers, which resid e in the focal plane, at the foci of the three mirror modules. All carry sili con CCD detec- tors. One of the cameras utilizes back-illuminated PN CCDs w ith 150 µm square pixels and was developed at the Max Plank Institute [8 ]. The other two cameras carry MOS CCDs (EEV CCD 22s) which were developed pri- marily by the X-ray Astronomy Group at Leicester University and Marconi Applied Technologies (formerly EEV) in the United Kingdom [ 6]. The EPIC MOS CCD is a frame transfer, front-illuminated device. The EPIC MOS CCD is a three phase device. The electrodes, or ga tes (poly1, poly2 and poly3) are shown schematically in Figure1. The thi ckness of deple- tion region is approximately 37 µm with nominal clock voltages and substrate bias [6]. The most important feature of the EPIC MOS CCD is its ‘open ele ctrode structure’. In order to improve the detection eﬃciency at lo w energies, one of the gates, poly3, has been enlarged by partially removing the front face, leaving two ‘holes’ in each pixel. These holes cover 40% of th e pixel area and are separated by a central electrode ‘ﬁnger’ comprising pol ysilicon, oxide and nitride layers. A P-plus dopant is implanted in the etched ar eas, which pins the surface potential to the substrate potential. 3 Experiment 3.1 Experimental Setup A detailed explanation of our mesh experiments may be found i n the literature [2][5]. Figure 2 gives a schematic view of the mesh experimen t. The mesh experiment consists of a CCD, a metal mesh and a pseudo parall el X-ray beam. The metal mesh employed has periodically spaced holes which are much smaller than the pixel size. The hole spacing is an integer mu ltiple of the CCD 3pixel size. The mesh is placed just above the CCD surface, as c lose to the CCD as practically possible. The mesh must have an orientation which is slightly rotated w ith respect to the CCD so that the shadow of the mesh hole on the CCD gradually shifts its position inside the pixel as shown in Fig. 2. In this way, o ver the CCD dimensions, the X-rays passing through the mesh holes perio dically sample the entire pixel. This produces a moire pattern from which th e relative align- ment between the mesh and the CCD may be determined [3]. An X-r ay event detected by the CCD must have come through one of the mesh hole s. Taking into account the hole spacing and the pixel size, we can unequ ivocally deter- mine the hole location for individual X-ray events. We can th erefore calculate the X-ray interaction position within the CCD with sub-pixe l resolution. The accuracy is limited by the eﬀective mesh hole size which is sl ightly bigger than the geometrical shape of the hole due to diﬀraction. The experiment was performed in a CCD test facility at Leices ter University. The EEV CCD 22 has 600 ×600 pixels with each pixel being 40 µm square. A gold mesh was employed which has a thickness of 10 µm and small holes of 2µm diameter. The spacing between the holes was 120 µm; just three times the pixel size. We placed the mesh about 0.5mm above the CCD su rface and rotated it by about 1.◦7 from the CCD. The X-ray generator manufactured by KEVEX Inc. was approximately 3m from the CCD and several ﬂuor escence targets were used, generating characteristic X-rays as wel l as a Bremsstrahlung spectrum. Figure3 gives an example the spectrum obtained by the mesh ex- periment using a Ag target with a voltage of 5kV. There are sev eral char- acteristic emission lines, O-K (0.52keV), Al-K (1.5keV), S i-K (1.8keV) and Ag-L(2.9keV), superposed on a continuum extending up to 5ke V. The CCD operating conditions were almost identical to those employed on the XMM/NEWTON satellite. During frame integration we appl y 8V (high voltage) to poly3 (in Fig.1) only. The other two gates remain ed un-biased. The CCD chip was cooled to −100◦C using liquid nitrogen and was driven using duplicate ﬂight electronics. Tsunemi et al. (1999) performed the mesh experiments using a similar ex- perimental setup [9]. Their experiment used an existing cop per mesh with a thickness of 10 µm and holes of 4 µm diameter. The spacing between the holes was 48 µm which is not a multiple of the pixel size. The eﬀective mesh h ole at that time was 7 µm in diameter. For this experiment, we have improved the conditions by fabricating a new mesh which may be positioned much closer to the CCD surface with holes 2 µm in diameter. In the new conﬁguration, the eﬀective mesh hole gives about a factor of 3 improvement in sp atial resolution over the previous experiment. Furtheremore, the hole spaci ng is much greater, and is an integer multiple of the pixel size, which makes the d etermination of 4the X-ray interaction location within a given pixel much cle arer. 4 Data Analysis and Discussion 4.1 Image Restoration The mesh technique allows us to determine the X-ray interact ion position within the CCD pixel. Furthermore, it samples the entire pix el which enables us to restore the pixel images for the various X-ray energies as well as the various X-ray event-types. In the data analysis, we pick out the characteristic X-ray energies thanks to their good statistics (as summariz ed in table 1 in order of attenuation length in SiO 2). We then construct restored images for various X-ray event-types; (a) single pixel events, (b) ver tically split events, (c) horizontally split events and (d) all X-ray events, as shown in Figure 4. In this ﬁgure, each image represents 2 ×2 pixels of the CCD, with the dashed square corresponding to the pixel size of 40 µm square. Brighter regions correspond to a higher detection eﬃciency. Looking at the pixel image restored using all X-ray events (t he right hand column), we can clearly see absorption features within the p ixel. In particular, the two etched regions in each pixel are clearly visible in th e O-K image, since O-K has the shortest attenuation length of the X-ray en ergies employed. Moving to greater attenuation lengths, the etched region be comes less obvious. It is diﬃcult to see the enhanced region in the Ag-L image whic h is almost free from absorption within the gate structure. In the restored image using single pixel events, we clearly s ee the enhanced region only in the O-K image. Other than the gate structure, w e notice that there are three regions in each pixel: the regions generatin g single pixel events, vertically split events and horizontally split events. Spl it events are generated in the regions within the pixel where the charge cloud splits into adjacent pixels. We can clearly see that the horizontal pixel boundar y (giving rise to vertically split events) is not a straight line but is instea d a wavy line. In contrast to this, the vertical pixel boundary is governed by the channel stops, which include a P-plus implant in order to make an electric po tential ‘barrier’ which results in a normal (i.e. straight line) shape for the p ixel boundary. There are small gaps in the region generating horizontally s plit events. These correspond to the pixel corners where the 3- or 4-pixel event s are generated. The wavy boundary is due to the fact that the poly3 gate is etch ed so that it has a ﬁnger structure as shown in Fig.1. In other words, the po tential generated by poly3 during integration deﬁnes a wavy boundary between a djacent pixels 5rather than a straight line. The shape of the region generati ng vertically split events also depends on the attenuation length of the X-rays i ndicating the depth dependence of the electric ﬁeld inside the CCD. It can b e compared with that expected from the model calculation. 4.2 Spectra from Various Regions inside the Pixel In the mesh experiment, we can identify both the energy and th e interaction position within the pixel for an individual X-ray photon. We can therefore extract X-ray spectra from any region within the pixel. We se lected ﬁve regions within the pixel in order to measure the pixel structure of th e CCD. These are labeled in Figure 5 as ‘ﬁnger’, ‘channel stop’, ‘electro de’, ‘open electrode 1’ and ‘open electrode 2’, respectively. Among them, the cha nnel stop and the ﬁnger are so narrow that the selected regions partially o verlap with the open electrode region due to the ﬁnite spatial resolution of the experiment. Figure 6 shows the spectra for each region using the same data in Fig. 3. We ﬁrst compared spectra for the open electrode 1 and 2. We con ﬁrmed that the ratio shown in Figure 7 is almost constant and independen t of energy. Because these two regions must show the highest eﬃciency at l ow energies, we added them together to generate a standard spectrum for the o pen electrode with which we compare other data. Figure 8 shows the ratio bet ween the spectra for the open electrode and those for other three regi ons:upper panel, middle panel and lower panel are results of electrodes, ﬁnge r and channel stop, respectively. Since these ﬁgures represent the extra absor ption features in each region, we ﬁtted them with a relatively simple absorption mo del. The model contains a normalization, a constant component and absorpt ion features for Si and SiO 2. The normalization is required due to the diﬀerence in area o f each region and the constant comes from the area of overlap wi th the open electrode region. In this way, we obtained the thickness of S i and SiO 2for each region. The best ﬁt results are shown in Fig. 8 by solid lines a nd summarized in Table 2. 4.3 Radiation Hardness In June 1999 the Chandra X-ray Observatory (CXO) was launche d into a a high-earth orbit, similar to that of the XMM/NEWTON satell ite. It is equipped with a CCD camera (ACIS) consisting of ten CCD chips : eight front- illuminated(FI) CCDs and two back-illuminated(BI) CCDs. I n august, it was reported that a substantial degradation of the CCD had occur red [11]. The degree of the degradation is worse than that for the ASCA CCDs , despite that ASCA has been in a low-earth orbit since February, 1993. It should be 6noted that the FI CCDs are heavily damaged while the BI CCDs ar e free from damage. It was concluded that the degradation was due to a rel atively high ﬂux of low energy ( ∼100 keV) protons. The interpretation is that the low energy protons are collected onto the CCD and penetrate the r elatively thin electrode structure leaving traps near the charge transfer channel. The charge transfer channel is a relatively narrow path insi de the pixel that is localized as a ‘notch’ structure [12]. The eﬀective thick ness of the electrode structure above the charge transfer channel is directly mea sured by the mesh experiment and found to be ∼0.3µm thick Si and ∼0.3µm thick SiO 2[4]. Therefore, the high ﬂux of low energy protons penetrates thi s structure and causes permanent damage on the buried channel of the CCD. In t he case of the BI CCD, the charge transfer channel is relatively far away fr om the entrance side, resulting in no damage to these CCDs. The EPIC MOS CCD has a complicated electrode structure. The e ﬀective thickness of the absorber above the open electrode structur e is designed to be∼0.085µm thick SiO 2. This indicates that low energy protons will easily generate traps under the open electrode structure. However , the charge trans- fer channel is just under the ﬁnger structure. Our measureme nts indicate that the ﬁnger structure consists of ∼0.2µm thick Si and ∼0.7µm thick SiO 2. This indicates that the charge transfer channel of the EPIC M OS CCD is bet- ter protected than that of the ACIS CCD. We can say that the EPI C-MOS CCD is more radiation hard than the FI chip of the ACIS. The det ails on the radiation hardness requires more quantitative measuremen t. 4.4 Application X-ray events detected by a CCD are classiﬁed by the number of p ixels over which the resultant charge splits. Due to readout noise and c harge loss around the charge cloud perimeter, single pixel events generally g ive the best energy resolution. One of the main features of this CCD is its open el ectrode region where the low energy eﬃciency is enhanced. We must therefore endeavour to control the operating conditions so that we can increase t he active region generating single pixel events in the open electrode. Fig. 4 clearly shows that vertically split events are genera ted primarily in the open electrode. Furthermore, the shape of this region de pends on the attenuation length of the photon in Si. This must be due to the interaction of the applied voltages on neighbouring electrodes. In the s tandard mode, poly3 is biased during integration while the other two elect rodes are not. We performed an experiment in which we biased both poly3 and p oly1 during integration. Figure 9 shows the restored images for Mo-L(2. 3keV) X-ray events 7again classiﬁed by X-ray event-type as in Fig. 4. It is clear t hat the horizontal pixel boundary for vertically split events has been changed . We have not yet been able to determine which working condition is the best fo r spectroscopic study. This will require further study, taking into account the eﬀect of the thickness of the depletion region. In the mesh experiment, we can study how the primary charge be haves inside the CCD [10]. In some pixel regions, when the primary charge i s photoabsorbed close to the front surface of the device, it produces a tail in the spectrum. This eﬀect is more evident for low energy X-rays since they have a s horter attenu- ation length in Si. In particular, the response of the O-K lin e shows a strong dependence on the interaction position within the pixel. In our experiment, the incident X-ray spectrum contains several emission line s as well as a rel- atively strong continuum which prevents us from studying in detail any tail to the response function. In order to study this, we require a mono-energetic X-ray beam. The present experimental setup does not current ly permit us to generate such mono-energetic X-ray beams with suﬃcient ﬂux to conduct the mesh experiment. 4.5 Detection Eﬃciency We calculated the meadn pixel response that is shown in Figur e 10 with tak- ing into acount our results of each structre within the pixel ;open-electrodes, electrodes, ﬁnger and channel stop. It was calcurated with p arameters mea- sured from the mesh experiment as described in table 2 weight ed with area fractions of each structure. In this calculation, we assume d that there is an extra absorption of ∼0.085µm thick SiO 2and Si 3N4over the pixel since we can not measure the absorption on the open electrode region. These extra absorption play an important role of the absorption feature at the low energy region. We also assume that the depletion region is 37 µm [6] thick that plays an important role at the high energy region. We compared our result with that reported by C. Pigiot et al. 1 999 [13] which performed the ground caribrations of EPIC MOS CCD using mono -energetic X-ray beam at Osay synchrotron facility in France. We found t hat our estima- tion has a good agreement with the data points of thier ground calibration. 5 Conclusion We performed a mesh experiment on a CCD CCD identical to that e mployed in the EPIC MOS imaging spectrometers on-board XMM/NEWTON. We 8were able to obtain restored X-ray images with sub-pixel res olution using X- ray photons of characteristic emission lines: O-K, Y-L, Mo- L, Al-K and Ag-L. All the X-ray events are classiﬁed by their event-types: sin gle pixel event, horizontally split event, vertically split event and all ev ents. There are clear absorption features inside the pixel including electrodes , channel stops etc. The shorter the attenuation length in SiO 2, the clearer these absorption features become. We also conﬁrmed that the horizontal pixel boundary between vertically split events is not a straight line, but is ‘wavy’. The speciﬁc shap e of this wavy line depends on the attenuation length of the X-ray photons i n silicon. This indicates the depth dependence of the electric ﬁeld inside t he CCD. We ob- tained spectra from various regions within a CCD pixel, whic h showed a non- uniformity of the detection eﬃciency. We selected ﬁve regio ns from which we extracted spectra. We then compared the spectrum from the op en electrode region with the other regions in order to measure the absorpt ion features in detail. The electrode structure comprises 0.29 ±0.03µm of Si and 0.94 ±0.05µm of SiO 2while the electrode ﬁnger comprises 0.15 ±0.05µm of Si and 0.73 ±0.02µm of SiO 2. The eﬀective absorption at the channel stop is equivalent to 0.57 ±0.03µm of SiO 2. The charge transfer channel is a relatively narrow path with in the pixel;just below the electrode ﬁnger in the case of EPIC MOS CCD. We found that the extra absorption feature above the charge transfer channel of EPIC MOS CCD is thicker than that of the FI CCD of ACIS( ∼0.3µm of Si and SiO 2) on-board the CXO whose orbit is similar to that of XMM/NEWTON. For the A CIS CCD, it was reported that FI CCDs got permanent damages on the buried channel due to a relatively high ﬂux of low energy protons. Ou r measurement indicates that the EPIC MOS CCD is more radiation hard than th e FI CCD of the ACIS. In the standard working condition of the CCD, the vertically split events are generated mainly in the open electrode region where the dete ction eﬃciency at low energies is enhanced. Since single pixel events usually give better energy resolution than split pixel events, it is preferable to gene rate more single events and less split events in the open electrode region. We conﬁrm ed that the shape of the region generating the vertically split event va ries by changing the operation mode. We can therefore control the CCD so that the o pen electrode region generates more single pixel events rather than split events. We calcurated the mean pixel response based on our measureme nt. Our esti- mation has a good agreement with that performed on the ground calibration of the EPIC MOS CCD using mono-energetic X-ray beam. 9Acknowledgements This research was partially suported by Simitomo Foundatio n. References [1] G. W. Frazer: X-ray Detectors in Astronomy (Cambridge Un iversity Press, Cambridge, 1989) p. 208. [2] H. Tsunemi, K. Yoshita, S. Kitamoto, Jpn. J. Appl. Phys. 36(1997) 2906. [3] K. Yoshita, H. Tsunemi, K. C. Gendreau, G. Pennington and M. W. Bautz, IEEE Trans. Nucl. Sci. ,45, (1998) 915. [4] M. J. Pivovaroﬀ, S. Jones, M. Bautz, S. Kissel, G. Prigozh in, G. Ricker, H. Tsunemi and E. Miyata, IEEE Trans. Nucl. Sci. ,45, (1998) 164. [5] H. Tsunemi, K. Yoshita, J. Hiraga, S.Kitamoto, Jpn. J. Appl. Phys. 37(1998) 2734. [6] A. D. Short, A. Keay, M. J. L. Turner, Proc. SPIE ,3445, (1998) 13. [7] K. O. Mason, G. Bignami, A. C. Brinkman, A. Peacock, Advances in Space Research ,16, (1995) 41. [8] E. Pfeﬀermann et al., Proc. SPIE ,3765, (1999) 184. [9] H. Tsunemi, K. Yoshita, A.D. Short, P.J. Bennie, M.J.L Tu rner and A.F. Abbey, Nucl. Instr. and Meth. A, 437 (1999) 359-366. [10] M.Bautz et al., Nucl. Instr. and Meth. A, 436 (1999) 40-52 [11] S.L.O’Dell et al., Proc. SPIE ,4140, (2000) in press. [12] B. E. Burke et al. IEEE Trans.Electron Devices ,44,(1997) 1633. [13] C.Pigiot et al., Proc. SPIE ,3765, (1999) 10Table 1 Summary of characteristic X-ray energies obtained. Characteristic X-ray Energy [keV] Attenuation Attenuatio n length in Si [ µm] length in SiO 2[µm] O-K 0.52 0.47 0.93 Y-L 1.9 1.4 2.4 Mo-L 2.3 2.2 3.9 Al-K 1.5 7.9 4.2 Ag-L 2.9 4.4 7.9 11Table 2 The extra thickness of various regions within the pixel Selected region thickness of Si [ µm] thickness in SiO 2[µm] Electrodes 0.29 ±0.03 0.94 ±0.05 Finger 0.15 ±0.05 0.73 ±0.02 Channel stop − 0.57±0.03 12/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 mµ40 mµ40etched region channel stop poly 2poly 1 poly 3 Fig. 1. Schematic structure inside the pixel of an EPIC MOS CC D. There are three electrodes: one is partly etched in order to improve detecti on eﬃciency at low energy. CCDmeshX-ray Fig. 2. Schematic view of the mesh experiment showing the ori entation of the mesh with respect to the CCD. The X-ray landing position is restri cted by the mesh hole. 13Fig. 3. Energy spectrum obtained using the whole CCD data. Th ere are several characteristic emission lines, O-K (0.52 keV), Al-K (1.5 ke V), Si-K (1.8 keV) and Ag-L(2.9 keV), superposed on a continuum extending up to 5 ke V. 14O-K(a) (b) (c) (d) O-K Y-L Mo-L Al-K Ag-L Fig. 4. Restored images using characteristic X-ray emissio n lines for various X-ray event-types; (a) single pixel events, (b) vertically split events, (c) horizontally split events and (d) all X-ray events. The results are sorted by the order of the attenuation length in SiO 2. Each image represents 2 ×2 pixels of the CCD, with the dashed square corresponding to the pixel size of 40 µm square. 15/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 finger channel stopelectrode mµ40mµ40poly 1 poly 2 poly 3 open-electrode1 open-electrode2 Fig. 5. Five selected regions within the pixel are shown. The y are taking into account the electrodes or channel stop structure to compare spectra to each other. 16channel stopfingerelectrodesopen-electrode1 open-electrode2 Energy [keV]Intensity per unit area (Arbitrary unit) Fig. 6. Spectra from ﬁve selected regions within the pixel. 1700.511.5 1 2 3 4 trans_gate/open.specEnergy (keV)1 2 3 4 Energy [keV]0.51 Fig. 7. Ratio between the spectrum obtained at the open elect rode 1 and that obtained at the open electrode 2. 180.250.50.751 1 2 3 4 Energy (keV)1 0.75 0.5 0.25 1 2 3 4 Energy [keV] 0.20.40.6 1 2 3 4 Energy (keV)1 2 3 4 Energy [keV]0.6 0.4 0.40.50.6 1 2 3 4 Energy (keV)1 2 3 4 Energy [keV]0.6 0.5 0.4 0.3 Fig. 8. Ratios between the spectrum of the open electrodes an d those of selected regions: upper, middle and lower panels represent results o f electrodes, ﬁnger and channel stop, respectively. Solid lines represent the best ﬁt absorption model. 19Mo-L Fig. 9. Same as for Fig.4 for Mo-L (2.3 keV) using the data with another cloking pattern during integration time. Fig. 10. Mean pixel response calcurated based on our results . 20
arXiv:physics/0012023v1 [physics.gen-ph] 12 Dec 2000GEODESIC LINES IN THE GRAVITATIONAL FIELD OF NONLINEAR COSMIC STRINGS L. M. Chechin1and T. B. Omarov Astrophysics Institute, National Academy of Sciences, Almaty, 480068, Kazakstan Received 24 March 2000 Abstract We brieﬂy review the equations of motion and the space-time i n- terval due to the nonlinear cosmic string that have been deri ved in ref. [3] for the ﬁrst time. The diﬀerent types of isotropic and non isotropic geodesic lines in the gravitational ﬁeld of nonlinear cosmi c string have been analyzed in detail. 1E-mail: chel@aﬁ.academ.alma-ata.su . 01 Introduction It is well known that cosmic strings were appeared at the very early stages of the Universe evolution. The cosmic string ”sources” are t he scalar ﬁelds that were, in turn, produced due to the vacuum phase transiti ons [1]. One of the lagrangians that describes these ﬁelds is the Higgs’s lagrangian for the complex scalar ﬁeld χ(x), L=∂αχ∗∂αχ+m2χ∗χ−λ(χ∗χ)2, (1) where λis coupling constant describing the ﬁeld self-interaction . The la- grangian is essentially nonlinear. This is the reason that, in general, cosmic strings are nonlinear as well. To describe the strings, let us study the thread-like matter that consists of the set of inﬁnitely thin crossless threads and a fully con tinuous spacetime. The energy-momentum tensor of this thread-like matter can b e written as Tαβ=µuαuβ−tαβ. (2) The second term can be interpreted as the stress tensor that a ppeared due to the oscillations of each thread. These oscillations occur i n view of the string’s tension, or due to elastic force. We should stress, in a compl ete analogy with the classical mechanics [2], that the elastic force must inc lude not only linear term but also a nonlinear one. By virtue of the nonlinear term , the stress tensor of the thread-like matter has the following form: tαβ=µlαlβ(1−ε2 2lγlγ), (3) whereε2 2is the corresponding coupling constant of the order of λ. Inserting eqs. (2) and (3) into the energy-momentum consrevational la w and using the standard notation, we get the equations of motion of nonline ar cosmic string [3], ¨xα−x′′α(1−ε2 2x′γx′ γ) = 0. (4) The nonlinear character of cosmic string leads to the appear ance of some additional terms in the metric coeﬃcients that describe the spacetime interval in the vicinity of such a string. 1Indeed, inserting eqs. (2) and (3) into the Einstein’s gravi tational ﬁeld equations, putting that cosmic string along the z-axis and specifying the results of [3], we get the 4-dimensional interval in quasi-c artesian coordinates in the following form: ds2= (1−2γµε2lnr r0)dx02− [1−8γµ(1 +ε2 4) lnr r0](dx12+dx22)− (5) [1 + 2γµε2lnr r0]dx32. For the case ε= 0, the expression (5) recovers the standard Vilenkin’s spa ce- time interval [4]. Deducing of Vilenkin’s metric ansatz for the rectilinear ma ssive cosmic string has stimulated a number of research papers devoted th eir cosmological and astrophysical applications; e.g., motion of a test part icle, nonrelativistic [4] and relativistic [5], and light ray propagation in the so litary cosmic string background. Moreover, in all subsequent generalizations of this metric the behaviour of geodesic lines has been examined as well. Particularly, t he dynamics of test particle in the gravitational ﬁeld of cosmic string wit h kink has been studied in [6]; light deﬂection caused by a cosmic string car rying a wave pulse was investigated in [7]; the motion of light rays and te st particles in the gravitational ﬁeld of a semiclassical cosmic string was considered in [8]; the dynamics of test particles in the gravitational ﬁeld of c osmic string pass- ing through a black hole, through a domain wall was analyzed i n [9]; the inﬂuence of the current-carrying cosmic string background on the light rays propagation was treated in [10]. In this paper, in accordance to the general logic of investig ation of the new types of cosmic string metrics, brieﬂy discussed above, we s tudy geodesics lines in the gravitational ﬁeld of nonlinear cosmic string. 22 General form of a geodesic line in the non- linear cosmic string background Let us write down the spacetime interval (5) in quasi-cylind rical coordinates, ds2= (1−2γε2lnr r0)dx02− (1−8γµ(1 +ε2 4) lnr r0)(dr2+r2dϕ2)− (6) (1 + 2 γµε2lnr r0)dz2. Upon the admited coordinate transformations, ¯x0= x0, ¯r=r{1 + 4γµ(1 +ε2 4)(1−lnr r0)}, ¯ϕ= ϕ, ¯z= z  (7) and retaining corrections terms of the order of γµε2, the expression (6) can be written in the form ds2= (1−2γµε2ln¯r ¯r0)d¯x02− d¯r2−¯r2[1−8γµ](1 +ε2 4)d¯ϕ2−d¯z2. (8) Introducing the new angle variable, ¯Φ = [1 −4γµ](1 +ε2 4)¯ϕ (9) and omitting, for simplify, the minus sign at all the coordin ates, it is easy to verify that the interval (8) looks like a quasi-newtonian on e. Namely, ds2= (1−2γµε2lnr r0)dx02− dr2−r2dΦ2−dz2. (10) 3We have accounted that the angle Φ varies in accord to 0≤Φ≤2πb, (11) where b= [1−4γµ](1 +ε2 4) . The above form is the reason why the interval (10) describes the quasi-newtonian modiﬁed conical spacet ime. The direct procedure of ﬁnding lagrangian from the arbitrar y spacetime interval has been presented in [11]. Using this procedure, f or the test particle that moves in z=const plane in the gravitational ﬁeld with metric (10), we get L= (1−2γµε2lnr r0)(dx0 ds)2− (dr ds)2−r2(dΦ ds)2. (12) As this lagrangian does not explicitly depend neither on x0, nor on Φ, then the derivatives of Lwith respect to dx0/dsanddΦ/dsgive two ﬁrst integrals of motion: the total energy, (1−2γµε2lnr r0)dx0 ds=C1=E, (13) and the total angular momentum, r2dΦ ds=C2=M, (14) respectively. Moreover, as the lagrangian (12) does not explicitly depend onstoo, then another integral of motion exists, namely, (1−2γµε2lnr r0)(dx0 ds)2− (dr ds)2−r2(dΦ ds)2=C3=C. (15) So, from eqs. (13) and (14) we have the 4-velocity components , dx0 ds=E 1−2γµε2lnr r0, (16) 4dΦ ds=M r2. (17) Substituting eqs. (16) and (17) into (15) and introducing ne w variable u= 1/r, we obtain the general form of the equations of geodesics in t he nonlinear cosmic string background, (du dΦ)2+u2=−C M2+E2 M2(1 + 2 γµε2lnr0u). (18) 3 Null geodesic lines in the nonlinear cosmic string background Let us consider the basic types of the null geodesic lines in t he nonlinear cosmic string spacetime characterized by the interval (10) . a) Hyperbolic light rays propagation . For any null geodesic line the con- stantCis equal to zero. Then from (18) we have, approximately, (du dΦ)2+u2=E2 M2(1−2γµε2lnr0u). (19) Diﬀerentiating the above expression with respect to Φ we get d2u dΦ2+u=−γµε2E2 M21 u. (20) We look for the solution of this equation in the form u=u0+u1+···, (21) where u0describes the free motion while u1describes a perturbation, up to the ﬁrst order accuracy. For the nonperturbed motion we have simple trajectory. Name ly, the equation d2u0 dΦ2+u0= 0 (22) has the solution u0=E Mcos Φ. (23) 5It describes the straight-line trajectory that passes thro ugh the coordinate origin at the distanceM E. Next, for the perturbed motion we can write the equation of tr ajectory as d2u1 dΦ2+u1=−γµε2E Mcos−1Φ, (24) and its solution is u1=−γµε2E M(Φ sin Φ + cos Φ ln cos Φ) . (25) Hence, the general solution of the equation of motion (18) is u=E M(cos Φ −γµε2Φ sin Φ) − γµε2E Mcos Φ ln cos Φ . (26) Now, it is easy to ﬁnd the light deﬂection angle from eq. (26). We note that in the case of straight-line trajectory the polar angle varies from −π 2b toπ 2b. That is why the radius-vector can be rotated up to the full an gle πb. The angle values Φ = ±π 2bgive us u= 0, i.e. r=±∞. All the above relations are valid for the nonperturbed trajectory (22). F or the perturbed trajectory (24), the same angle values lead to a small, but no nzero, value of u. Therefore, the last term is equal to zero if Φ >±π 2b. So, let us denote the angle at which u= 0 as ±π 2b+δ, where δis a small angle. Substituting it into the eq. (26) we obtain, approxim ately, δ−γµε2π/2 = 0. (27) Therefore, the angle δis found as δ=γµε2π/2 (28) and the full deﬂection angle of the light rays in the nonlinea r cosmic string background is ∆Θ = 2 δ=πγµε2. (29) 6b) Radial light rays propagation . In this case from eq. (15) we approxi- mately have dx0= (1 + γµε2lnr r0)dr. (30) Integrating this expression from 0 to arbitrary point rgives us x0=r[1−γµε2(1−lnr r0)]. (31) We can use this relation to calculate the Shapiro’s eﬀect in t he gravitational ﬁeld of nonlinear cosmic string. Indeed, the lapse time corr esponding to the nonlinearity of cosmic string can be easily obtained, ∆x0 ncs=γµε2(r2−r1) lnr2−r1 r0. (32) To estimate the magnitude of the eﬀect let us suppose that the nonlinear cosmic string is placed between Earth and Mercury, in the pla ne which is normal to the planets plane of motion. In this case, r2−r1≈1,4a.u.≈ 2,21011m. Assuming γµ∼10−6andr0∼10−30m, that refers to string line mass density and Compton wavelength corresponding to G UT models, accordingly, ε∼10−7[3], we get ∆ tncs∼10−15sec. One can see that this time interval is too small to be detected experimentally; no te that Shapiro’s eﬀect due to gravitational ﬁeld of the Sun is of the order of 10−4sec. 4 Nonisotropic geodesic lines in the gravita- tional ﬁeld of nonlinear cosmic string Finally, let us consider the main types of timelike geodesic lines in the non- linear cosmic string spacetime. a) Hyperbolic motion . For the case of nonisotropic timelike geodesic lines, we have C= 1. Inserting this value into eq. (18) we obtain, within the adopted accuracy, (du dφ)2+u2=−1 M2+ E2 M2(1−2γµε2lnr0u). (33) 7Formally, this equation diﬀers from that of the isotropic ge odesic line (19). However, diﬀerentiating it with respect to Φ leads to the equ ation that fully coincides with eq. (20), d2u dΦ2+u=−γµε2E2 M21 u. (34) In the following, we deal only with the nonperiodic terms in e q. (26). There- fore, the resulting solution is u=E M(cos Φ −γµε2Φ sin Φ) ≈ E Mcos(1−γµε2)Φ. (35) It is easy to see that the above timelike trajectory is a littl e bit diﬀererent from the straightlike one, eq. (22). This diﬀerence is descr ibed by the angle ∆Θ = πγµε2(36) between the asymptotics of trajectories. This expression i s completely equiv- alent to the angle (29) that had been calculated above. We not e that in the case of linear cosmic srting the deﬂection angles for test pa rticle and light ray are equal to each other, too [9]. b) Radial motion . For this type of motion Φ = const. Hence, from eq. (15) we get v=dr dx0=±(E2−1 E2)1/2(1− 2−E2 1−E2γµε2lnr r0), (37) which is the particle’s free-fall velocity determined by cl ock of inﬁnitely far observer. The local motionless observer determs the velocity in a diﬀe rent way, w=dr dτ=dr dx0(1 +γµε2lnr r0) = (E2−1 E2)1/2(1−1 1−E2γµε2lnr r0). (38) 8Ifr→r0, i.e. on the surface of nonlinear cosmic string, then both of the velocities, (37) and (38), are equal to their newtonian e xpression vn= (E2−1 E2)1/2. The last relation means that the velocity is constant, due t o the free motion in ﬂat space. It should be pointed out that the second term in eq. (37) may be equal to zero, at some conditions. Namely, for the speciﬁc value of the radial coordinate, rlim=r0exp (1−E2 γµε2(2−E2)) (39) we have v= 0. However, the spatial region rlim≤r≤r0can not be reached by the test particle, in its free motion. Hence, it can move on ly in the region r0≤r≤ ∞. c) Circular motion . In this case r=const and from eq. (15) we immedi- ately have ω=rdΦ dx0=±(E2−1 E2)1/2(1− 2−E2 1−E2γµε2lnr r0). (40) This is the particle’s circular velocity determined by cloc k of inﬁnitely far observer. For the local motionless observer, the circular v elocity is, approxi- mately, Ω =rdΦ dτ=rdΦ dx0(1 +γµε2lnr r0) = (E2−1 E2)1/2(1−1 1−E2γµε2lnr r0). (41) It is easy to see from the last two expressions that both of the m tend to the newtonian value, Ω = (E2−1 E2)1/2, asr→r0. This means that the particle’s circular velocity is staying constant in the ﬂat space. More over, the circular velocity is equal to zero if r=rlim; see eq. (39). However, the test particle cannot move in the ﬁnite region rlim≤r≤r0, in this case too. 95 Conclusion We have examined the isotropic and nonisotropic geodesic li nes in the grav- itational ﬁeld of nonlinear cosmic string. Nonlinear terms in the equation of motion imply some new dynamical eﬀects, in both the light ray s propagation and the test particle motion. We have found additional deﬂec tion angle for the null and timelike geodesic lines; see eqs. (29) and (36). As to the timelike geodesics, we have found equivalence of the radial and circu lar velocities, in the newtonian approximation. We conclude this Section with the following remark, Dynamic al processes in the vicinity of cosmic strings are not limited to the light rays and test particle propagation. Research of the reciprocal dynamics of cosmic strings (oscillating cosmic strings [12]) is important in studying the early Universe cosmology. So, the next step in investigation of the nonline ar cosmic strings interactions could be dynamics of the test thread, in the non linear cosmic string background. Acknowledgment This work was done under the partial support within the progr am ”Organi- zation and evolution of natural structures”, National Acad emy of Sciences, Kazakstan. 10References [1] A. D. Linde. Particle Physics and Inﬂationary Cosmology (Harwood, Switzerland, 1990). [2] V. V. Novojilov. Foundations of the Nonlinear Elastic Theory (Moscow, Gostechizdat, 1948) (in Russian); G. B. Whitham. Linear and Nonlinear Waves . (New York, London, Sydney, Toronto, 1974). [3] L. M. Chechin, T. B. Omarov. Hadronic J. 22(1999) 197. [4] A. Vilenkin. Phys. Rev. D 23 (1981) 852. [5] J. R. Gott. Astrophys. J. 288(1985) 422. [6] D. Garﬁnkle, T.Vachaspati. Phys. Rev. D 37 (1988) 2537. [7] D. N. Vollik, W.G.Unruh. Phys. Rev. D 42 (1990) 2621. [8] A. Banerjee, N.Banerjee. Phys. Lett. A 160 (1991) 119. [9] S. Chakraborty, L.Biswas. Class. Quant. Grav. 13(1996) 2153. [10] P. Peter, D. Puy. Phys. Rev. D 48 (1993) 5546. [11] V. Fock. The Theory of Space, Time and Gravitation (Macmillan, New York, 1964). [12] T. B. Omarov, L. M. Chechin. Gen. Relat. Grav. 31(1999) 443. 11
arXiv:physics/0012024v1 [physics.comp-ph] 12 Dec 2000Solution of Poisson’s equation for ﬁnite systems using plan e wave methods Alberto Castro and Angel Rubio Departamento de F´ ısica Te´ orica, Universidad de Valladol id, E-47011 Valladolid, Spain M. J. Stott Physics Department, Queen’s University, Kingston, Ontari o, Canada K7L 3N6 (February 2, 2008) Abstract Reciprocal space methods for solving Poisson’s equation fo r ﬁnite charge distributions are investigated. Improvements to previous proposals are pre- sented, and their performance is compared in the context of a real-space den- sity functional theory code. Two basic methodologies are fo llowed: calcula- tion of correction terms, and imposition of a cut-oﬀ to the Co ulomb potential. We conclude that these methods can be safely applied to ﬁnite or aperiodic systems with a reasonable control of speed and accuracy. PACS numbers: 31.15.-p, 02.70.-c, 71.15.-m Typeset using REVT EX 1I. INTRODUCTION Density-functional theory1,2in its time-dependent3as well as ground or time-independent forms has proved to be an eﬃcient method for treating electro n-electron interactions and has been applied successfully to ﬁnite systems such as clust ers,4to bulk systems or surfaces, and to aperiodic systems such as defects.5However, the high computational cost of treating large systems places a practical limit on the size of systems that can be studied. The use of pseudopotentials6,7enhances the performance of this sort of calculation by avoiding an explicit treatment of the Kohn-Sham orbitals as sociated with the core. Further- more, the smoothness of the resulting valence pseudowavefu nctions allows the use of a plane wave basis for describing them, and consequently also the el ectronic density. A plane wave basis is particularly attractive because it allows use of th e fast fourier transform (FFT), for rapid and memory eﬃcient transformations. A discrete but truncated set of plane waves based on the recip rocal lattice is one natural basis set for a periodic system. However, for ﬁnite systems, or more generally, for systems lacking periodicity such as defects in solids, the use of a di screte set of plane waves will generate periodic images of the ﬁnite cell to be studied. In t he case of a ﬁnite system this leads to a problem in the calculation of the electrostatic po tential due to the electrons, the so-called Hartree potential, due to the long range of the Cou lomb interaction. Nevertheless, discrete plane wave basis sets are often used for ﬁnite syste ms because of the great eﬃciency of the FFT, and errors in the Hartree potential due to periodi c images are usually ignored, or reduced by increasing the size of the supercell. These spu rious eﬀects might seriously aﬀect the calculated equilibrium structure and dynamics of weakly bounded molecules or clusters, eg. water. However, several methods have been pro posed recently for treating this problem.8–13 Our purpose here is to compare four methods for solving Poiss on’s equation for ﬁnite systems. One of them is an iterative, real-space method base d on ﬁnite diﬀerences and conjugate-gradients minimization, which obviously doesn ’t suﬀer from the problems related to periodic images. The other three use discrete plane wave b asis sets and FFT’s, but treat the cell-to-cell interactions in diﬀerent ways. Two of thes e plane wave methods impose a cut-oﬀ to the Coulomb interaction in real space, and have be en described elsewhere.11 However, for one of these, which uses what we term a cubic cut- oﬀ, we have found signiﬁcant improvements which we believe will be of interest to practit ioners. The third plane wave method has been developed and tested by us, but we have found a close relationship between it and the local moment countercharge (LMCC) method propose d by Schultz,12and also to the Fourier analysis with long range forces (FALR) method of Lauritsch and Reinhard.8 However, our scheme is formulated more generally, and allow s for better control of errors. In order to compare the performance of the diﬀerent methods w e have studied some exactly soluble model systems, and NaCl and Na+2 10molecules which, because of their polar or charged nature, are diﬃcult to treat with plane wave metho ds. Of particular interest is the way the computational time scales with system size. Al though all the plane wave methods scale as a few times NlogN, whereNis the number of real-space mesh points, the proportionality factor varies substantially from method t o method. We shall compare the speed and memory requirements of the methods, and how these s cale with the size of the systems. 2The plane wave schemes we discuss are intended mainly to deal with neutral or charged molecules or clusters in free space and could be implemented directly in existing ab initio plane wave or real-space codes, but they could also be used in those LCAO basis set codes which base the calculation of the Hartree potential on the FF T. General subroutines for calculating the Hartree potential using these methods are a vailable upon request from the authors or can be downloaded from the web page.14 Short theoretical descriptions of the plane wave methods ar e given in section II where we emphasize the improvements we have developed. Section II I presents and compares the results for the Hartree potential calculated using the d iﬀerent methods, and concluding remarks are made in the ﬁnal section. Atomic units are used un less otherwise stated. II. THEORETICAL DESCRIPTION OF PLANE-WAVE METHODS A. Uncorrected calculations The solution of Poisson’s equation, ∇2VH+ 4πn= 0, which goes to zero at inﬁnity, for a charge density n, localized within a cell Cof volume Ω, is given by VH[n,r] =/integraldisplay dr′n(r′) \|r′−r\|. (1) Within the cell nandVHmay be expressed as Fourier series: n(r) = Ω−1/summationdisplay GeiGr˜n(G) where ˜n(G) =/integraldisplay drn(r)e−iGr, and similarly for VH, and where the Gvectors are reciprocal vectors of the lattice formed by repeating the cell C. If the Fourier coeﬃcients, ˜ n(G), are negligible for Glarger than some cut-oﬀ so that the sums over Gmay be truncated, then the ˜n(G) and then(r) points are related through a discrete Fourier transform. T his amounts to approximating the integral over the cell in the deﬁnition of ˜n(G) by the trapezium rule, a point to which we shall return later. However, n(r) given by the Fourier sum is periodic so that the straightforward substitution into Eq. (1) gives a potential: V[n,r] =4π Ω/summationdisplay G/negationslash=0˜n(G) G2eiGr, (2) which diﬀers from VH. But the merit of V is that it can be calculated using the the ve ry eﬃcient FFT with its NlogN scaling, and so we now modify or apply corrections to Eq. (2) so that it can be used to obtain VH. Two aspects of Vgiven by Eq. (2) require attention. •TheG=0component in Eq. (2) is arbitrarily set to zero. For a charged system, this corresponds physically to introducing a uniform compensat ing charge background, b, so that the system is electrically neutral. For a neutral sys tem, this means that the boundary condition, V(r→ ∞) = 0, is not satisﬁed (which, of course, also happens in the charged case). •Vis the potential due to the charge distribution nin the central cell plus that due to the images of n−bin all other cells. 3Since we are dealing with the electron charge distribution, there is obviously a net charge. The fact that the whole system ( cores+ electrons) may or may not be charged, is irrelevant for the discussion presented in this paper. However, it migh t be important if the calculations are total-energy supercell calculations. In this case, the system of ion cores is also treated using reciprocal space, so that a background of opposite sig n has to be added. If the ﬁnite system is neutral, the eﬀect of the backgrounds cancel s. The spurious eﬀect of higher multipoles, however, will remain. The distinction is impor tant, though, because the uniform background introduces an important error in the total energ y of orderO(L−1),Lbeing the size of the cell, whereas the leading eﬀect of the presence of the multipoles is the dipole-dipole term, behaving like O(L−3). This is shown in the calculations presented in next sectio n. B. Multipoles-corrections method We can start to deal with the cell-to-cell interaction, by el iminating the eﬀects of net charge. This can be done by subtracting from the original cha rge distribution, n, an auxiliary charge distribution naux, so that no net charge remains. The potential VHthen becomes: VH[n] =VH[n−naux] +VH[naux]. (3) The termVH[n−naux] can be treated using the FFT techniques, and then the correc tion VH[naux], calculated explicitly in real space, added on. This metho d is especially convenient if the Fourier components of nauxcan be calculated analytically, so that the expression becomes: VH[n]≈V[n]−ψ, (4) whereV[n] follows the deﬁnition given in Eq. (2), and ψ=V[naux]−VH[naux] is a function which can be calculated analytically. The sign ≈denotes that the eﬀect of higher multipoles is still included. The choice of nauxis arbitrary; it could be a uniform density, or a gaussian centered on the origin, in both cases the function ψcan be calculated analytically.15 This procedure can be easily generalized, to account for the interaction of higher multi- poles. We merely need to add an auxiliary charge distributio n which mimics the multipoles whose eﬀect we want to subtract. This procedure is called by S hultz12local moment counter- charge method (LMCC). Shultz accounts for the monopole and d ipole corrections through a superposition of localized Gaussian charge distribution s constructed to have the same net charge and dipole moment. Higher multipoles can similarly b e accounted for by the super- position of additional Gaussian distributions, but the pro cedure becomes complicated. A straightforward approach which is more easily generalized introduces an auxiliary charge distribution in the form: naux(r) =∞/summationdisplay l=0l/summationdisplay m=−lnlm(r), (5) where nlm(r) =Mlm2l+2 a2l+3√π(2l+ 1)!!rle−(r/a)2Ylm(r), (6) 4andMlmis the multipole moment of ngiven by Mlm=/integraldisplay drn(r)rlYlm(r). (7) The width parameter ais to be chosen so that nauxis negligible at the cell boundary. If high order moments are required acould be taken to be l-dependent, decreasing somewhat withl. Note that the l= 0 term in Eq. (5) corrects for the net charge as described abo ve. The auxiliary density is localized within a cell and has the s ame multipole moments as n. We can now correct for the presence of the periodic images of n, and obtain for the required Hartree potential in the central cell: VH[n,r] =V[n,r]−∞/summationdisplay l=0l/summationdisplay m=−lMlmψlm+V0, (8) whereV0is a constant shift yet to be determined, and the functions ψlm, which are inde- pendent of the charge distribution n, are given by ψlm(r) =(4π)2 Ωil (2l+ 1)!!/summationdisplay G/negationslash=0Gl−2e−a2G2/4Ylm(G)e−iGr−√π2l+3 (2l+ 1)!!1 rl+1Il(r/a)Ylm(r).(9) The ﬁrst term in ψlmis the periodic potential due to nlmin every cell, and the second term subtracts the eﬀect of nlmin the central cell. The function Ilis: Il(x) =/integraldisplayx 0dtt2le−t2. (10) The procedure for obtaining VHis to calculate and store the functions ψlmonce and for all for as many of the multipoles as are needed to achieve the d esired precision, then for the particular charge distribution Vis calculated using FFTs, the Mlmare computed, and Eqs. (8) and (9) used to obtain VHwithin the central cell apart from the constant shift. The shiftV0is chosen so that the boundary condition VH(r→ ∞) = 0 is satisﬁed. This we accomplish by computing the average value over the surfac e of the cell of the corrected but unshifted potential VHobtained by simply putting V0= 0 in Eq. (8). For a cubic cell we have:16 VH=1 6Ω2/3/integraldisplay CdsVH(r) =1 6Ω2/3/integraldisplay Cds/integraldisplay dr′n(r′) \|r−r′\|. (11) Sincenis zero at the boundary we may interchange the order of integr ation in Eq. (11) to give: VH=/integraldisplay dr′n(r′)u(r′), (12) where u(r′) =1 6Ω2/3/integraldisplay Cds1 \|r−r′\|(13) 5is the potential inside the cube due to a unit charge uniforml y distributed over the cube surface. As such, by Gauss’s theorem and symmetry, uis constant inside the cell and has the valueα/Ω1/3withα= 1.586718 for a cube. If nintegrates to z, then we have ﬁnally: VH=αz Ω1/3. (14) andV0should be chosen so that VHhas this value. The calculation of the correct shift to satisfy the boundary condition requires the computation of the surface average of V, but since the Kohn-Sham orbitals are unaﬀected by the addition o f a constant to the potential, this only needs to be performed at the end of a self-consisten t cycle.17 C. Cut-oﬀ methods We now review two established methods,9,11based on imposing a cut-oﬀ on the Coulomb interactions. They are exact, but need a bigger cell, which i s a computational drawback as we shall see. Let us deﬁne a new cell D, which includes C. This new cell will deﬁne new coeﬃcients GD, which are the reciprocal vectors of the lattice formed by re peatingD. Retrieving Eq. (2), we realize that the function V[n] can also be expressed as: V[n,r] =/integraldisplay dr′n(p)(r′)1 \|r′−r\|, (15) wheren(p)is the function formed by the sum of nand all its periodic repetitions in the superlattice. We now introduce a truncated Coulomb potential, with the fol lowing properties: f(r−r′) =/braceleftbigg1 \|r−r′\|,forrandr′both belonging to the same image of C. 0,forrandr′belonging to diﬀerent images of C.(16) It is easily seen that: /integraldisplay dr′n(p)(r′)f(r′−r) =VH[n,r] (17) for every r∈C. And thus we can calculate VHas we did for Vin Eq. (2): VH(r) =4π Ω/summationdisplay G/negationslash=0˜n(G)˜f(G)eiGr, (18) where ˜f(G) =/integraltext drf(r)e−iGr. Two choices for Dandfhave been given. One of them uses a spherical shape9for the cut-oﬀ of the Coulombic interaction, and the other a cubic sh ape,11based on the assumption that the original cell Cis cubic itself. 61. Spherical cut-oﬀ method LetLCbe the length of the side of C. We will deﬁne a larger cubic cell of side LD= (1 +√ 3)LC, centered on C. For this choice of Dwe deﬁne next the truncated Coulomb interaction: f(r−r′) =/braceleftBigg 1 \|r−r′\|,\|r−r′\|<√ 3Lc 0,\|r−r′\|>√ 3Lc.(19) Deﬁned in this way, fmeets the required conditions expressed in Eq. (16), becaus e any two points belonging to Care always closer than√ 3LC, and any two points belonging to diﬀerent images of Care always farther away than√ 3LC. The Fourier transform of nhas to be calculated numerically in the larger cell D; however that offis easily obtained analytically: F{f}(G) = 4π1−cos(G√ 3LC) G2. (20) 2. Cubic cut-oﬀ method The former proposal is exact; but a very large cell is needed, which increases the time to evaluate the FFTs. Reducing LDintroduces spurious interactions and thus spoils the pre- cision of the calculations, but if extremely precise calcul ation are not needed, a compromise could be reached. Our aim now is to reduce LDbut maintain an accurate evaluation of VH. We take the larger cell to have LD= 2LCand the cut-oﬀ Coulomb interaction to be: f(r−r′) =/braceleftbigg1 \|r−r′\|,r−r′∈D 0,otherwise.(21) Ifrandr′belong toC,r−r′belongs toD. And if randr′belong to diﬀerent images ofC, then r−r′will not belong to D. Thus again fis correctly deﬁned. The Fourier transform of this function fhas to be calculated numerically, and we face here two drawbacks: the function has a singularity at the ori gin, and is not analytic at the boundary. Jarvis et al11dealt with the ﬁrst problem by integrating and averaging the singularity over a grid unit, which may not be adequate. The second proble m appears to have been overlooked. In any event, their treatment of the Mg atom usin g the cubic cut-oﬀ method converges poorly compared with the results with the spheric al cut-oﬀ, and they declare a preference for the spherical cut-oﬀ despite the larger cell size required. However, we shall show how to overcome these diﬃculties so that the cubic cutoﬀ method, to be preferred because of its smaller cell size, can be used with great preci sion. The singularity at r=r′can be treated as follows: /integraldisplay Ddr1 re−iGr=/integraldisplay Ddrerf(r/a) re−iGr+/integraldisplay Ddr1−erf(r/a) re−iGr, (22) 7whereais chosen small enough so as to make 1 −erf(r/a) negligible at the cell boundaries. The second term can be calculated analytically /integraldisplay dr1−erf(r/a) re−iGr=4π G2{1−e−G2a2/4} (23) and the numerical integration reduces to the ﬁrst term, whic h is free of singularities. Even so, this term cannot be calculated by simply applying an FFT b ecause the repeated function, although periodic, is not analytic at the boundary. Use of th e FFT amounts to using the trapezium rule for the integration, which is exact for a peri odic analytic function, but leads to substantial errors when there are discontinuous derivat ives as we have in this case. We evaluated the integral using a second-order Filon’s method ,19which proved to be eﬀective. Other procedures18(Simpson’s, Romberg’s...) could have been used - they are al l rather slow if accurate results are to be obtained, but this calculation needs to be done only once for a cubic cell. If we denote by I[D(L)](n1,n2,n3) the integral in Eq. (22) for a cubic box of side L and frequency indices ( n1,n2,n3), it is clear that I[D(L)](n1,n2,n3) =L2I[D(1)](n1,n2,n3). III. RESULTS A. Exactly soluble systems It is interesting to see the eﬀect of the various multipoles t hat a charge distribution might have by using the multipoles-correction method on an e xactly soluble system. We have studied systems consisting of superpositions of Gauss ian charge distributions placed at various points Riwithin the cubic cell of side L: n(r) =/summationdisplay iziexp(−\|r−Ri\|2 a2 i) a3 iπ3/2. (24) We have investigated the eﬃciency with which the multipole c orrections remove the eﬀects of the images of the charge distributions in other cel ls. This has been done as functions of L, as for a large enough cell the results for the p eriodic system should become exact, but at rates depending on the order of the multipoles. The results are shown in Fig. 1 for the cases in which there are (i) no corrections (by which w e mean that only the constant to meet the proper boundary conditions is added to the raw pot ential obtained from the Fourier transform), (ii) monopole corrections, (iii) mono pole + dipole corrections, and (iv) monopole + dipole + quadrupole corrections. The following p oints are noteworthy. •There is a serious, roughly 10%, error in the total energy whe n the Hartee potential is uncorrected. Although this is not a consideration in superl attice calculations provided the system is neutral, it is an important matter in real space calculations when the Hartree potentials due to the electrons alone is calculated in reciprocal space. •The time for the calculations behaves roughly as O(L3logL), but with irregularities. The eﬃciency of the FFT algorithm depends on the prime factor ization of the number of points to be transformed. The original FFT was developed f or powers of two, but now algorithms exist with more ﬂexibility.20,21We have used the FFTW package,22 with support for all the primes involved in our calculations . 8•Adding the quadrupole corrections does not seem to improve t he accuracy of the result, nor is theL-dependence improved. This is because the interaction ener gy between the dipole of the charge distribution in the central cell and oct upoles in other cells, has the sameL-dependence and order of magnitude as the quadrupole-quadr upole energy. Consequently, although the potential will be improved by ad ding to it the quadrupole corrections, there could be no signiﬁcant improvement in th e total energy if the system has a strong dipole. In general, it can be shown that the error in the electrostatic energy due to the presence of an lmultipole in the charge distribution in the central cell, an d l′multipoles in all other cells goes to zero like L−(l+l′+1), or in some special cases faster due to symmetry (for instance, if l= 0 andl′≤3, orlandl′have diﬀerent parity). Thus, adding octupole corrections to the potential will not change the L-dependence of the total energy if the system is charged because of the int eraction of the monopole with thel= 4 multipole. Our calculations below on the Na+2 10cluster provide an interesting example of this behaviour. B. Real systems We have performed several electronic structure calculatio ns on real systems to assess the performance of the methods. We have used a real-space code,24in which a superlattice and plane waves are only used to accelerate the solution of Poiss on’s equation for the electron charge distribution. In this type of approach a correction f or the net charge is always needed irregardless of whether the molecule or cluster itself is ch arged or neutral. Furthermore, in this approach the value of the multipoles will depend on the p osition of the molecule with respect to the centre of the cell. In order to minimize the mul tipole corrections the centre of charge of the system of ions should be placed at the centre o f the cell. If this is not done in real space calculations the errors caused by cell-to-cel l interactions could be magniﬁed. In order to illustrate the eﬀects we take the center of charge as the cell center for one of our test cases, and not for the other. As for other details of the c alculations we used density- functional theory with the local-density approximation fo r exchange and correlation, and Troullier-Martins23nonlocal, norm-conserving pseudopotentials. Our ﬁrst choice for a realistic system was the NaCl molecule, also treated by Shultz12 and Jarvis et al,11because of its strong dipole moment (experimental value of 9 .0D in the gas phase, as reported by Nelson et al.25) In this case, the center of charge of the system of ions is placed at the center of the cell. The equilibrium bond -length was calculated: (i) using the spherical cut-oﬀ method which is exact with a large enoug h cutoﬀ, and (ii) using the multipoles correction and correcting only for the monopole term so as to show the inﬂuence of the dipole-dipole interactions which are ignored. Our ca lculated “exact” value is 2.413 ˚A, whereas the result ignoring the dipole-dipole interaction s is 2.448 ˚A. Next, we investigated the performance and accuracy of the me thods by determining the errors in the total energy and electric dipole moment, wh ile monitoring the calculation times. We compared results for the energy and dipole moment a gainst those obtained with the spherical cutoﬀ method with a cut-oﬀ radius of√ 3LC, grid parameter (0.2 ˚A) and cell size (L=10˚A). In this way an electric dipole moment of 8.4551D was obtai ned. Each of the four methods was then used to converge the electronic gro und state of the molecule for successive values of a “control parameter” for speed and acc uracy: 9•For the real-space, conjugate-gradients method this param eter was the order of the diﬀerence formula used to evaluate derivatives. •For the spherical cut-oﬀ method, we note that, if the electro n density is well localized within the Ccell, the need for the full cut-oﬀ radius,√ 3LC, may be relaxed and a correspondingly smaller Dcell used, introducing some error but accrueing time savings. We have investigated the eﬀect of using a reduced cu t-oﬀ radius, rcut-oﬀ, through a control parameter, α, which is the ratio of the DandCcubic cell edges: α=LD LC= 1 +rcut-oﬀ LC(25) so thatα= 1 +√ 3 is the minimum value for which exact results are guaranteed . •TheDcell size can also be reduced in the case of the cubic cut-oﬀ me thod, and the control parameter is again α=LD LCwhereα≥2 guarantees exact results. •For the multipoles correction method, the order of the multi poles corrected for is the control parameter. In Fig. 2 we illustrate the results obtained for each of the me thods. Both cut-oﬀ methods are presented in the same column as they use the same control p arameter, although the ranges of values are diﬀerent. 1. The real-space method is signiﬁcantly slower than the oth er methods for the same accuracy, and a case can be made for using reciprocal space me thods for calculating the Hartree potential in what are otherwise real-space code s. However, enhancements of the conjugate-gradients method are possible through pre conditioning and multigrid techniques.26,27 2. The cut-oﬀ methods reach acceptable accuracy much below t he values for αwhich guarantee exact results: 1 +√ 3 and 2 respectively, for the spherical and cubic cut- oﬀ. This to be expected when the charge distribution is well l ocalized within the cell. However, it is clearly demonstrated that, for a given accura cy, the size of the auxiliary cell is smaller for the cubic cut-oﬀ method, and as a result, t he calculation time is also shorter. 3. The multipoles-correction method already gives good acc uracy if the dipole interac- tions are corrected for (5 ×10−5eV error in the energy, and 10−4D error in the electric dipole). Without the dipole correction, the error in the ene rgy is 0.085eV, and in the dipole is 0.17D, which give an indication of the size of er rors to be expected when supercell calculations are performed for neutral molecule s and no corrections are made. We have also performed calculations on the Na+2 10cluster containing the same number of valence electrons as the NaCl molecule. Results are simil ar to those presented for NaCl, but some diﬀerences should be reported. In this case the cent er of charge was not placed at the center of the cell, consequently, although the cluste r has a calculated net dipole of 104.5D, the electronic dipole responsible for the errors is a much l arger 10.2D. The cluster was positioned in the cell so that the charge density occupie d most of the cell, allowing an optimally small cell. As a result, to achieve similar accura cy as for the NaCl molecule, we should expect the need for (i) larger cut-oﬀ lengths for the c ut-oﬀ methods, and (ii) higher multipole corrections. In Fig. 3 we show the error in the energy obtained by using the m ultipoles correction method. It can be seen how the inclusion of the dipole correct ion yields a much less satis- factory error in the energy than for NaCl. Furthermore, for t he reasons given earlier, the removal of the quadrupole-quadrupole, dipole-octupole, a nd octupole-octupole terms does not signiﬁcantly improve the accuracy. Only by including al l corrections to the potential up to fourth order multipole moments do we obtain a comparable r esult for the energy. The calculation time, which is also shown in the ﬁgure, is beginn ing to increase sharply by fourth order as further corrections are added. In Fig. 4 we present, as well, the results for the error in the t otal energy and the calcu- lation time for the cubic-cutoﬀ method. Comparison with Fig . 2 conﬁrms that the energy converges much less rapidly as a function of αthan for the NaCl molecule. IV. CONCLUSIONS. We have studied some of the methods which have been proposed r ecently for solving Poisson’s equation in reciprocal space for electronic stru cture calculations on ﬁnite systems. We also propose a method based on multipole corrections. Tes t calculations have been performed to assess the performance of the methods. We concl ude that reciprocal-space methods can be accurate enough for ﬁnite or aperiodic system s, and their eﬃciency is a signiﬁcant improvement over that of real-space methods. Tw o basic reciprocal-space meth- ods have been investigated: one which imposes a cut-oﬀ on the Coulomb potential, and one based on the removal of the spurious eﬀects through a mult ipole expansion. Both yield satisfactory results, and comparable eﬃciency. The former approach has been already been surveyed by Jarvis et al .11There are two possibilities for the cut-oﬀ function, one, the spherical c ut-oﬀ, was highlighted for having superior convergence with the plane wave cut-oﬀ of the recip rocal lattice. However, we have identiﬁed and corrected problems with the other possib ility, the cubic cut-oﬀ, which eliminates the poor behaviour, and makes this cut-oﬀ method the better of the two because smaller FFT’s are allowed. The method based on multipoles corrections was initiated by Shultz,12but we have developed a scheme which we think is more general. Its perfor mance is more predictable than that of the cut-oﬀ methods, which are sensitive to the ch oice of the cut-oﬀ length. The reason for the sensitivity is that the cut-oﬀ length determi nes the size of a larger auxiliary cell and the number of grid points over which FFT calculation s are performed, and the FFT is sensitive to the prime number decomposition of the num ber of points. On the other hand, the speed and accuracy of our “multipoles correction” method, are adequate for most applications, and can be easily controlled by choosing the o rder of corrections applied. All the methods have been presented assuming a cubic cell. Ho wever, generalizations to other cell shapes are possible if the geometry of the syste m requires it. The multipoles correction method is immediately applicable to any cell. Cl early the spherical cut-oﬀ method 11would be ineﬃcient for elongated cells because the radius of the cut-oﬀ sphere is determined by the longest dimension of the cell. But, the cubic cut-oﬀ me thod can easily be generalized to other cell shapes, at the cost of more, and more lengthy cal culations of the Fourier transforms of the truncated Coulomb interaction. We have made a simple implementation of the solvers within th e self-consistent frame- work, but smarter algorithms can be developed, since not all the iterations of a self-consistent calculation need be done with the same accuracy. For example , signiﬁcant improvements in eﬃciency can be gained if, for a given method, the iteratio ns are started with a fast but inexact solver through appropriate choice of the control pa rameter, but improved as self- consistency is approached. Moreover, methods could be comb ined using, for instance, the real-space method for the last few iterations because of its eﬃciency when a good starting point is known. ACKNOWLEDGMENTS We are pleased to acknowledge useful discussions with J. A. A lonso. We also acknowledge ﬁnancial support from JCyL (Grant: VA 28/99) and DGES (Grant No. DGES-PB98-0345) and European Community TMR contract NANOPHASE: HPRN-CT-20 00-00167. A. C. acknowledges ﬁnancial support by the MEC, and hospitality b y Queen’s University, where most of this work has been done, during a research visit. M. J. S. acknowledges the support of the NSERC of Canada, of Iberdrola through its visiting pro fessor program, and of the Universidad de Valladolid where this work began. 12REFERENCES 1P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 2W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 3E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984). 4A. Rubio, J. A. Alonso, X. Blase and S. G. Louie, Int. J. Mod. Ph ys. B11, 2727 (1997). 5M. Brack, Rev. Mod. Phys. 65, 677 (1993). 6W. E. Pickett, Comput. Phys. Rep. 9, 115 (1989). 7M. L. Cohen, Solid. State Commun. 92, 45 (1994); Phys. Scri. 1, 5 (1982). 8G. Lauritsch and P.-G. Reinhard, Int. J. Mod. Phys. C 5, 65 (1994). 9G. Onida et al, Phys. Rev. Lett. 75, 818 (1995). 10G. Makov and M. C. Payne, Phys. Rev. B 51, 4014 (1995). 11M. R. Jarvis, I. D. White, R. W. Goodby and M. C. Payne, Phys. Re v. B56, 14972 (1997). 12P. A. Shultz, Phys. Rev. B 60, 1551 (1999). 13P. A. Shultz, Phys. Rev. Lett. 84, 1942 (2000). 14http://www.fam.cie.uva.es/˜arubio. 15G. Hummer, J. Electrostatics 36, 285 (1996). 16Although a cubic cell is used for simplicity, extension to a r ectangular parallelpiped is possible. This could be important for molecular dynamics si mulation with variable cell shape. 17In fact, some numerical diﬃculties arise when calculating t his constant. A better approach is splitting the function into periodic and aperiodic parts , and performing the integration only for the periodic part, which should integrate to zero. 18W. H. Press, S. A. Teukolsky, W. T Vetterling and B. P. Flanner y,Numerical Recipes (Cambridge University Press, New York). 19L. N. G. Filon, Proceedings of the Royal Society of Edinburgh 49, 38 (1928). 20A. V. Oppenheim and R. W. Schafer, Discrete Signal Processing (Prentice-Hall, Engle- wood Gliﬀs NV,1989). 21P. Duharnel and M. V´ etterli, Signal Processing 19, 259 (1990). 22M. Frigo and S. G. Johnsons, in Proceedings of the IEEE International Conference on Acoustic Speech and Signal Processing, Seattle, Washingto n, 1998 , vol. 3, p. 1381. 23N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). 24G. F. Bertsch, J.-I. Iwata, A. Rubio and K. Yabana, Phys. Rev. B62, 7998 (2000). 25R. D. Nelson, D. R. Lide, A. A. Maryott, National Reference Data Series - National Bureau of Standards (NRDS-NBS 10). 26M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joann opoulos, Rev. Mod. Phys.64, 1045 (1992). 27T. L. Beck, Rev. Mod. Phys. 72, 1041 (2000). 13FIGURES 10 12 14 16 18 Side of box (Å)00.0050.010.0150.020.025∆E (eV)00.0050.010.015∆E (eV)00.511.5∆E (eV)10152025303540∆E (eV) 0123 Time (s) 01234 Time (s) 02468 Time (s) 0510 Time (s)No corrections: ∆E(L)=αL-1 Monopole corrections: ∆E(L)=αL-3 Dipole corrections: ∆E(L)=αL-5 Quadrupole corrections: ∆E(L)=αL-5 FIG. 1. Error in the electrostatic energy for the system of Ga ussian charges, Eq. (24), (continuous line) and total time of calculations (dashed li ne), for the indicated order used of the multipoles correction. 1410-410-310-210-1100∆E (eV) Conjugate-Gradients Cut-off methods Multipole corrections 10-410-310-210-1100∆z (D) 1 2 3 4 Order of Laplacian1.03.09.027.0t (s) 1 1.2 1.4 1.6 αNone0 1 Order of corrections FIG. 2. Error in the electrostatic (ﬁrst row), electric dipo le (second row) and time of cal- culations (third row) for the NaCl molecule, using the metho ds indicated, as a function of the respective “control parameter” (see text). For the cut-oﬀ m ethods, crosses refer to the spherical cut-oﬀ method, and circles to the cubic cut-oﬀ method. All sc ales are logarithmic. 150 1 2 3 4 Order of corrections10-510-410-310-210-1100101∆E (eV) 0510 t (s) FIG. 3. Error in the electrostatic energy (continuous line) for the Na+2 10cluster, using the multipoles correction method, as a function of the order of c orrections included in the calculations. Also shown is the time of calculation for each case (dashed li ne). Note that the time scale is not logarithmic. 1 1.2 1.4 1.6 1.8 α10-510-410-310-210-1100101∆E (eV) 0510 t (s) 16FIG. 4. Error in the electrostatic energy for the Na+2 10cluster using the cubic cut-oﬀ method, as a function of its “control parameter”. Also shown is the time of calculation for each case (dashed line). Note that the time scale is not logarithmic. 17
arXiv:physics/0012025v1 [physics.gen-ph] 14 Dec 2000Did 20th century physics have the means to reveal the nature o f inertia and gravitation? Vesselin Petkov Physics Department, Concordia University 1455 de Maisonneuve Boulevard West Montreal, Quebec H3G 1M8 vpetkov@alcor.concordia.ca (or vpetkov@sympatico.ca) 13 December 2000 Abstract At the beginning of the 20th century the classical electron t heory (or, perhaps more appropriately, the classical electromagnetic mass theory) - the ﬁrst physical theory that dared ask the question of what inertia and mass were - was gaining momentum and there were hopes that physics would be ﬁnally able to explain their origin. It is argued in this paper that if that promising rese arch path had not been inexplicably abandoned after the advent of relativity and quantum mechanics, the co ntemporary physics would have revealed not only the nature of inertia, mass, and gravitation, but most impor tantly would have outlined the ways of their manipulation. Another goal of the paper is to try to stimulat e the search for the mechanism responsible for inertia and gravitation by outlining a research direction, which demonstrates that the classical electromagnetic mass theory in conjunction with the principle of equivalenc e oﬀers such a mechanism. NOTE: This paper presents only conceptual discussions of ri gorously obtained results which are available at: http://alcor.concordia.ca/˜vpetkov/papers/ 1 Introduction According to an eastern proverb the darkest place is beneath the lantern. The meaning of a proverb could hardly be more profound and more suitable for fundamental concepts : Nature has given us her greatest secrets as self- evident phenomena. Everyone ”knows” what existence, space , time, mass, inertia, gravitation, etc. are; we even have complex theories dealing with those concepts. If, howe ver, we want to explain their nature we ﬁnd ourselves in the same situation in which St. Augustine found himself wh en tried to explain the nature of time: ”What then is time? If no one asks me, I know; if I wish to explain it to one t hat asketh, I know not” [1]. We are about to enter the 21st century but our understanding o f the origin of inertia, mass, and gravitation still remains what has been for centuries - an outstanding pu zzle. All theories constituting the contemporary physics are concerned mostly (if not only) with the descript ion of those phenomena. Even the modern theory of gravitation, general relativity, which provides a consi stent no-force explanation of gravitational interaction o f bodies following geodesic paths, is helplessly silent on th e nature of the very force we identify as gravitational - the force acting upon a body deviated from its geodesic path b eing at rest on the Earth’s surface. The classical electromagnetic mass theory appears to have b een gradually forgotten and now many physicists believe that physics cannot say anything about the origin of inertia, mass, or gravitation. Even scientists directly involved in the eﬀorts to discover the Higgs boson [2] (belie ved to be responsible for endowing particles with mass) such as Claude Detraz, one of the two research director s atCERN , think that ”Mass is a very important property of matter, and we have nothing in our current theory that says even a word about it”. In what follows we will see whether this is really the case. Here we shall follow the tradition established by the classi cal electron theory (i.e. the classical electromagnetic mass theory) and will study the inertial and gravitational p roperties of the simplest charged particle - the classical electron. There are two reasons why the classical electron i s studied: 1(i) There is no quantum mechanical model of the electron - qua ntum mechanics describes only its state not the electron itself (later we will make an attempt to outline the basis of the quantum electrodynamical formulation of the electromagnetic mass theory), and (ii) It is quite natural to complete the classical electroma gnetic mass theory ﬁrst before making the transi- tion to a quantum description of the electron inertial prope rties. This has never been done since the classical electromagnetic mass theory was virtually abandoned when t he theory of relativity and quantum mechanics were formulated. To abandon a promising theory that has never bee n proven wrong is an unprecedented case in physics. This unforgivable neglect is truly beyond one’s comprehens ion because the electromagnetic mass theory is even now the only theory that addresses the origin of inertia and i nertial mass in accordance with the experimental evidence of the existence of electromagnetic inertia and of the electromagnetic origin of some of the mass of charged particles. Moreover, the classical electromagnet ic mass theory predicted that the (electromagnetic) mass increases with the increase of velocity (yielding the corre ct expression) and that the relationship between energy and mass is E=mc2- all this before the theory of relativity. 2 Classical electromagnetic mass theory In 1881 Thomson [3] ﬁrst realized that a charged particle was more resistant to being accelerated than an otherwise identical neutral particle and conjectured that inertia ca n be reduced to electromagnetism. Due mostly to the works of Heaviside [4], Searle [5], Lorentz [6], Poincar´ e [ 7], Abraham [8], Fermi [26], Mandel [10], Wilson [11], Pryce [12], Kwal [13], and Rohrlich [14] this conjecture was developed into a theory (the classical electromagnetic mass theory of the electron) in which inertia is a local pheno menon originating from the interaction of the electron charge with itself (i.e. its own electromagnetic ﬁeld) [15] . According to the classical model of the electron its charge i s uniformly distributed on a spherical shell. Such a model, however, cannot explain why the electron is stable s ince the negatively charged spherical shell tends to blow up due to the mutual repulsion of the diﬀerent ”parts” of the charge. This diﬃculty, known as the stability problem of the electron, has two sides - computati onal and conceptual. In the beginning of the century it appeared that the stability problem did lead to a computatio nal diﬃculties with the famous 4 /3 factor stubbornly appearing in the diﬀerent expressions for the electromagne tic mass. Several authors [9]-[14] independently showed that the 4 /3 factor had been caused by incorrect calculations, not by th e model itself. This implies that there is no real problem with the stability of the electron. We do not k now why. What we do know, however, is that if there were such a problem it would inevitably show up in all ca lculations of the electromagnetic mass which is not the case. The conceptual diﬃculty with the classical model of the elec tron - its failure to explain what prevents the electron charge from blowing apart - has never been satisfac tory resolved by the classical electromagnetic mass theory [17]. This is viewed as an indication that that model i s not entirely correct. There are three reasons, however, which demonstrate that the classical model of the e lectron works. (i) The very existence of the radiation reaction force is evidence that there is interaction (repul sion) between the diﬀerent parts of the electron charge since ”The radiation reaction is due to the force of the charg e on itself - or, more elaborately, the net force exerted by the ﬁelds generated by diﬀerent parts of the charge distri bution acting on one another” [18] (in the case of a single radiating electron the presence of a radiation react ion implies interaction of diﬀerent parts of the electron). This may indicate that the charge of the real electron can ind eed be modeled by a small spherical shell which, however, may mean that the electron charge is not continuously existing as smeared out on the shell; only if the electron charge were occupying the whole shell at every i nstant the stability problem would arise [19]. (ii) The calculations of the electron electromagnetic mass (ass uming a spherical distribution of its charge) yield the correct expression for the mass. (iii) A more important indi cation that the classical model of the electron cannot be discarded as inadequate is that the classical electromag netic theory is the onlytheory that correctly predicts that at least part of the electron’s inertia and mass are elec tromagnetic in origin; as Feynman put it: ”There is deﬁnite experimental evidence of the existence of electr omagnetic inertia - there is evidence that some of the mass of charged particles is electromagnetic in origin” [22 ]. At the beginning of the century many physicists recognized ” the tremendous importance, which the concept of electromagnetic mass possesses for all of physics: It is the basis of the electromagnetic theory of matter” (E. Fermi [23]). Therefore, it would have been natural to develop furt her the theory of electromagnetic mass by taking into account the relevant new results in physics achieved in this century. Instead, it had been inexplicably abandoned: ”The state of the classical electron theory reminds one of a h ouse under construction that was abandoned by 2its workmen upon receiving news of an approaching plague. Th e plague in this case, of course, was quantum theory. As a result, classical electron theory stands with m any interesting unsolved or partially solved problems” (P. Pearle [24]). It is clear that what Fermi, Feynman, and Pearle are saying is important but not crucial (one can always ﬁnd favourable quotes). What is crucial is what the classical el ectromagnetic mass theory itself is saying on inertia, mass, and gravitation when the principle of equivalence is t aken into account. The mechanism responsible for the electron’s inertia and ma ss according to the classical electromagnetic mass theory is the following. The repulsion of the charge element s of an electron in uniform motion in ﬂat spacetime cancels out exactly and there is no net force acting on the ele ctron. If, however, the electron is accelerated the repulsion of its volume elements becomes unbalanced and as a result it experiences a net self-force Fselfwhich resists its acceleration - it is precisely this resistance t hat we call inertia (for a detailed description of why the repulsion of the diﬀerent parts of an accelerating electron becomes unbalanced see [22, p. 28-5]). The self-force is opposing the external force that accelerates the electron ( i.e. its direction is opposite to the electron’s accelerati on a) and turns out to be proportional to a: Fself=−ma, where the coeﬃcient of proportionality mrepresents the inertial mass of the electron and is equal to E/c2where E is the energy of the electron ﬁeld (therefore the elec tron inertial mass is electromagnetic in origin). This is an amaz ing result for three reasons: (i) it reveals that both inertia and mass have electromagnetic origin (the mass min the expression for the self-force is electromagnetic since it is simply the mass that corresponds to the energy of t he electron’s electric ﬁeld through the relation E=mc2; (ii) it demonstrates that inertia is a local phenomenon (co ntrary to Mach’s hypothesis that the local property of inertia has a non-local origin [25]) and (iii) it constitutes the ﬁrst derivation of Newton’s second law F=ma[19] - a law that is considered so fundamental that after Newt on postulated it no one has attempted to derive it. Therefore, the classical electromagnetic mass theory does say not only a word, but oﬀers a detailed mechanism explaining the origin of inertia and mass of charged particl es: it is the unbalanced repulsion of the volume elements of the charge of an accelerating electron that gives rise to t he electron’s inertia and inertial mass. For an observer accelerating with an electron its electric ﬁ eld is distorted. Unlike uniform velocity, acceleration is absolute and the distorted electric ﬁeld of an accelerati ng charge is one of the means by which the observer can detect his acceleration. Therefore, in terms of the distort ed electric ﬁeld of an accelerating charge and avoiding the use of the controversial concept ”parts of an elementary charge”, one can equivalently say that an electron’s inertia and inertial mass originate from the interaction of its charge with its own distorted electric ﬁeld. The interaction of the charge of an uniformly moving electron wi th its own Coulomb (undistorted) ﬁeld produces no net force acting on the electron as a whole; that is why an el ectron moving with constant velocity oﬀers no resistance to its uniform motion [27]. The electromagnetic mass theory has been not only gradually forgotten; its status is now even more awkward - those who mention it regard the electron mass as electromagn etic only in part as if the 4 /3 factor in the expression for the electromagnetic mass has not been accounted for (it i s that factor that was considered an indication that not the entire electron mass was electromagnetic). It clear ly follows from the classical electromagnetic mass theory and special relativity that after the removal of the 4 /3 factor the entire mass of the electron should be electromagnetic in origin [28]. 3 Electromagnetic mass theory and the principle of equivale nce The classical electromagnetic mass theory oﬀered a mechani sm accounting for the origin of inertia and inertial mass, but before the formulation of the equivalence princip le by Einstein it appeared that that theory does not explain the origin of the passive gravitational mass and doe s not aﬀect gravitation at all. The equivalence principle, however, postulated that the inertial mass (the measure of r esistance that a body oﬀers when accelerated) is equal to the (passive) gravitational mass (the measure of resista nce that a body oﬀers when being prevented from falling in a gravitational ﬁeld); L. von E¨ otv¨ os’ experiments had a lready conﬁrmed that equality. The equivalence principle requires that inertial and gravitational masses be equal bu t does not provide any insight into what is the origin of the gravitational mass in a gravitational ﬁeld. The answe r to this question is that it is a spacetime anisotropy around massive bodies that is responsible for the force an el ectron on the Earth’s surface is subjected to and its gravitational mass. It manifests itself in the anisotropy o f the velocity of electromagnetic signals (for short - the velocity of light). To explain what is the origin of the passi ve gravitational mass according to the electromagnetic mass theory and to shed some light into the basis of the equiva lence principle here is a brief description of what 3happens to an electron in an accelerated reference frame and a frame of reference supported in a gravitational ﬁeld. 3.1 An electron in an accelerated reference frame Na For an observer at rest in an inertial reference frame I the el ectromagnetic ﬁeld of an accelerating electron is distorted due to the electron’s accelerated motion. As the a ccelerated motion is absolute the electron’s electric ﬁeld will be also distorted for an observer at rest in an accel erating (non-inertial) reference frame Nain which the electron is at rest. The distortion of the electron’s ﬁeld fo r the inertial observer in I is caused by the electron’s accelerated motion. For the non-inertial observer in Na, however, the electron is at rest and therefore there is no (accelerated) motion of the electron that can account for th e distortion of its ﬁeld as determined by the observer inNa. What causes the deformation of the electron’s ﬁeld in Nais the anisotropic velocity of light there; Na is an accelerating frame and it is the anisotropy in the propa gation of light (and its manifestations such as the distorted electron ﬁeld) which allow an observer in Nato determine from within Nathat it is an accelerating (non-inertial) frame (for a more detailed discussion why th e velocity of light in a non-inertial frame is anisotropic see [28] and [29]). 3.1.1 An electron falling in Na Imagine that an inertial observer Iis observing an electron ﬂoating inside a spacecraft which m oves with a constant velocity with respect to I(so both the spacecraft and the electron move by inertia oﬀer ing no resistance to their motion). Let’s now assume the spacecraft start to ac celerate with an acceleration a, i.e. it becomes a non-inertial frame Na(an observer in the spacecraft will be also called Na). For the inertial observer Inothing happens to the electron - it continues to move by inertia unti l the spacecraft’s ﬂoor reaches it. For an observer in the spacecraft, however, the electron is falling toward the ﬂoor with an acceleration a (for Iit is the spacecraft’s ﬂoor that approaches the electron). Obviously, there is a pr oblem here - as the non-resistant motion by inertia is absolute, both observers ( IandNa) should agree that the electron is moving by inertia inside t he spacecraft which does not appear to be the case since for Nathe electron is accelerating toward the ﬂoor (which implies that there is force that accelerates it). Despite the fact th at today’s physics regards the force accelerating the electron in Naas ﬁctitious, it has never explained why the electron oﬀers n o resistance to its accelerated motion as viewed in Na. When the anisotropic velocity of light in Nais taken into account in the calculation of the electric ﬁeld of the falling in Naelectron it turns out that at every instant the electron ﬁeld is the Coulomb (not distorted) ﬁeld (here the instantaneous ﬁeld is considered in order to separate th e Lorentz contraction of the ﬁeld and the distortion due to acceleration) [28]. Therefore, for an observer in Nathe motion of the falling (accelerating) electron will not be resistant since its electric ﬁeld is not distorted. Th is means that the free electron in Nafalls with an acceleration ain order to compensate the anisotropy in the propagation of l ight in Naand to prevent its ﬁeld from being distorted; in other words, the falling electron o ﬀers no resistance to its accelerated motion in Naand therefore moves by inertia while falling in Na. In such a way, as expected, both the inertial observer Iand the non-inertial observer Naagree that the electron in Nais moving by inertia oﬀering no resistance to its motion. 3.1.2 An electron at rest in Na Now consider the moment when the spacecraft’s ﬂoor reaches t he ﬂoating electron as seen by the inertial observer I. The electron starts to accelerate and its motion is no longe r non-resistant; its ﬁeld gets distorted and a self- forceFself=−maoriginating from the unbalanced repulsion of the electron’ s charge ”elements” (caused by the accelerated motion of the electron) starts to oppose its acc eleration (i.e. the deformation of its ﬁeld); here again mis the electromagnetic mass of the electron which is the mass corresponding to the energy of the electron ﬁeld E(m=E/c2). What can an observer in Na(in the spacecraft) say about the electron on the spacecraft ’s ﬂoor? At ﬁrst, it appears that the electron ﬁeld is not distorted with respect toNasince it is at rest in Nawhich would mean that no force is acting on the electron. If this were the case, there would be a problem again: the inertial and the non-inertial observers would diﬀer on whether the elect ron is subjected to a force; as the existence of a force is an absolute fact all observers should recognize it. That p roblem disappears when the anisotropic velocity of light is taken into account in the calculation of the electro n ﬁeld in Na. Due to an unnoticed up to now Li´ enard- Wiechert-like contribution to the potential of a charge in a non-inertial reference frame [28], [29] (caused by the 4anisotropic velocity of light in such frames) the electric ﬁ eld of the electron in Nais as distorted as the ﬁeld seen by the inertial observer I. Therefore, the non-inertial observer Nawill also ﬁnd that the electron is subjected to the purely electric self-force Fself=−ma, originating from the anisotropic velocity of light in Nawhich disturbs the balance of the mutual repulsion of the ”elements” of the e lectron charge. As seen from the expression for the self-force it coincides with what we call the inertial force ; hence it follows that the inertial force is electromagneti c in origin. The non-inertial observer in Nasees that when the falling electron reaches the ﬂoor of the sp acecraft it can no longer compensate the anisotropy in the propagation of li ght in Na(by falling with an acceleration a), its ﬁeld gets distorted which gives rise to the self-force Fself. 3.2 An electron in a gravitational ﬁeld Consider now an electron at rest in the Earth’s gravitationa l ﬁeld. The Newtonian theory of gravitation tells us that the electron is subjected to a gravitational force - its weight F=mg. What does general relativity say about that force? Nothing. The gravitational ﬁeld in general rela tivity is a manifestation of spacetime curvature and (unlike the electromagnetic ﬁeld) is not a force ﬁeld (which means that ”there is no gravitational force in general relativity” [30]). A body falling toward the Earth is repres ented by a geodesic worldline which means that no force is acting on it. If a body is on the Earth’s surface, howe ver, its worldline is no longer geodesic and it is subjected to a force whose nature is an open question in gener al relativity [29]. This fact alone (not to mention the issue of the represented by a pseudo -tensor energy and momentum of the gravitational ﬁeld [31]) is a suﬃcient reason for a thorough re-examination of the foundations of g eneral relativity. And this is urgently needed since, as we shall see bellow, there are strong arguments indicatin g that the correct interpretation of the formalism of general relativity should be in terms of anisotropic, not cu rved spacetime. 3.2.1 An electron at rest in the Earth’s gravitational ﬁeld One of the formulations of the equivalence principle states that what is happening in a non-inertial reference frame Nawhich accelerates with an acceleration aalso happens in a non-inertial reference frame Ngat rest in a gravitational ﬁeld characterized by an acceleration g=−a. One of the results Einstein obtained by analyzing the principle of equivalence is that in two elevators - one ac celerating (frame Na) and another at rest in the Earth’s gravitational ﬁeld (frame Ng) - light bends when propagating perpendicularly to the acce lerations aand g, respectively. If one considers light propagating paralle l and anti-parallel to aandg, it turns out that the average velocity of light in NaandNgis anisotropic: the velocity of a light ray from the elevator ’s ceiling toward the ﬂoor is slightly greater that the velocity of light propa gating in the opposite direction. Interestingly, that expression for the average anisotropic velocity of light fo llows from the expression of the velocity of light in a gravitational ﬁeld obtained by Einstein in 1911 but abandon ed when the calculations of the deﬂection of light by the Sun (based on that expression) predicted a wrong value fo r the deﬂection angle. A careful analysis of the propagation of light in the Einstein thought experiment inv olving the two elevators demonstrates that his 1911 expression for the velocity of light in a gravitational ﬁeld has been prematurely discarded [28]. Due to the anisotropic velocity of light in Ngthe electric ﬁeld of an electron at rest in Ngdistorts, the balance of the mutual repulsion of the electron charge ”elements” is disturbed which in turn gives rise to a self-force Fself which tries to restore the balance in the mutual repulsion. T he self-force turns out to be Fself=mg, where m=E/c2represents the passive gravitational mass of the electron a ndEis the energy of its ﬁeld. As the electric self-force Fselfis precisely equal to the gravitational force F=mg, the classical electromagnetic mass theory predicts that the gravitational force acting on an electron on the Earth’s surface is purely electromagnetic in origin which means that its passive gravitational mass is al so electromagnetic in origin. This is an important result since it demonstrates that the se lf-forces Fself=−mainNaandFself=mgin Nghave precisely the same origin: in both cases it is the anisotropic velocity of light (electromagnetic signals) that gives rise to the electric force Fselfby distorting the electric ﬁeld of the electron at rest in NaandNgwhich in turn disturbs the balance in the repulsion of its charge ”e lements”. What we call the inertial mass m(inFself =−ma) and the passive gravitational mass m(inFself=mg) are precisely the same thing: mis the measure of the resistance an electron oﬀers when its ﬁeld is being dis torted. In the case of an accelerating electron it is its acceleration (i) that distorts its ﬁeld as seen by an iner tial observer and (ii) that causes the anisotropy in the propagation of light in Nawhich in turn distorts the electron ﬁeld as observed by a non- inertial observer in Na. 5Similarly, the distortion of the ﬁeld of an electron at rest o n the Earth’s surface (i.e. at rest in Ng) is caused by the anisotropic velocity of light in Ng. 3.2.2 An electron falling in the Earth’s gravitational ﬁeld The self-force Fselfacting on an electron at rest on the Earth’s surface arises on account of its distorted electric ﬁeld (caused by the anisotropic velocity of light in Ng) which disturbs the balance in the mutual repulsion of the electron charge ”elements”. Fselftries to prevent the electron ﬁeld from distorting and to res tore the repulsion balance. If we allow Fselfdo its job by removing the obstacle beneath the electron, it w ill start to fall and it will fall in such a way that the distortion of its ﬁeld is eliminate d, the repulsion balance is restored and the self-force Fselfceases to exist. The calculation of the electric ﬁeld of an el ectron left on itself in a gravitational ﬁeld shows that the only way for the electron to compensate the anisotro py in the propagation of light in the gravitational ﬁeld and to prevent its ﬁeld from being distorted is to fall wi th an acceleration g[28]. Therefore, a free electron in a gravitational ﬁeld will move by inertia (without resist ance) only if it falls with an acceleration g. This result sheds light on the fact that in general relativity the motion of a body falling toward a gravitating center is regarded as inertial (non-resistant) and is represented by a geodesic worldline. Therefore, the electromagnetic mass theory gives an elegant answer to the question why an ele ctron is falling in a gravitational ﬁeld and no force is causing its acceleration. The result that the electric ﬁeld of an electron falling in th e Earth’s gravitational ﬁeld at any instant is the Coulomb ﬁeld, which means that no self-force is acting on the electron, also demonstrates that a falling electron does not radiate - its electric ﬁeld is the Coulomb ﬁeld and th erefore does not contain the radiation r−1terms [28]. If the electron is prevented from falling its electric ﬁeld d istorts, the self-force Fselfappears and tries to force the electron to move (fall) in such a way that its ﬁeld be comes the Coulomb ﬁeld; as a result the self-force disappears. The behaviour of the classical electron in a gravitational ﬁ eld is fullyaccounted for by the classical electro- magnetic mass theory and the equivalence principle: the ani sotropic velocity of light in Ng(in an elevator at rest in the Earth’s gravitational ﬁeld) (i) gives rise to a self-force acting on an electron at rest in Ng(whose worldline, according to general relativity, is deviated from its geodesic status) by distorting the elec tric ﬁeld of the electron which in turn disturbs the balance in the mutual repulsion of its charge ”elements”, and (ii) makes a free electron fall in Ngwith an acceleration gin order to preserve its Coulomb ﬁeld and therefore to balance the repulsion of its charge ”elements”. No force i s acting upon a falling electron (whose worldline is geodesic) but if it is prevented from falling (i.e. deviated from its geodesic path) the mutual repulsion of the ”elements” of its charge becomes unbalanced which results i n a self-force trying to force the electron to fall. General relativity does not provide an explanation of the na ture of the force acting on a body at rest in a gravitational ﬁeld whose worldline is not geodesic. It appe ars that general relativity cannot provide such an explanation at all since ”there is no gravitational forcein general relativity” [30]; this fact constitutes not only an open question but a crisis in general relativity. The classi cal electromagnetic mass theory in conjunction with the principle of equivalence provides a natural answer to the qu estions (i) why a free electron in a gravitational ﬁeld is falling by itself (with no force acting upon it) and why its worldline is geodesic, and (ii) why an electron at rest in a gravitational ﬁeld is subjected to a force and why its wor ldline is not geodesic: the worldline of an electron which preserves the shape of its Coulomb ﬁeld is geodesic and represents a free non-resistantly moving electron; if the ﬁeld of an electron is distorted, its worldline is not g eodesic and the electron is subjected to a self-force on account of its own distorted ﬁeld. In a spacetime region where the propagation of light is isotr opic a free electron does not resist its motion only if it moves with uniform velocity (which means that its elect ric ﬁeld is not distorted - it is the Coulomb ﬁeld [33]); in this case the electron’s worldline is a straight ge odesic line. If the electron is prevented from moving with constant speed its ﬁeld distorts and the electron resists it s acceleration (i.e. it resists the distortion of its ﬁeld); in this case the worldline of the accelerating electron is ne ither geodesic nor straight. In a spacetime region where the propagation of light is anisotropic (i.e. in an elevator on the Earth’s surface) the motion of a free electron is non-resistant (preserving the Coulomb shape of its ﬁeld) on ly if it falls with an acceleration g; in this case the electron’s worldline is geodesic but not a straight line. If the electron is prevented from falling (i.e. from moving 6by inertia in an anisotropic region of spacetime) its ﬁeld di storts and the electron resists the deformation of its ﬁeld; in this case its worldline is neither geodesic nor stra ight. The anisotropic velocity of light in both NaandNgis responsible for the fall of a free electron and the appearance of a self-force when the electron is prevented fr om falling in NaandNg. Therefore, it is the anisotropy in the propagation of light in NaandNgthat makes the two non-inertial reference frames NaandNgequivalent. In such a way, the equivalence principle is a straightforwar d corollary of the anisotropic propagation of light in Na andNg. The anisotropy in the velocity of light in the accelerating reference frame Nais caused by the frame’s acceleration. What is the origin of the anisotropic velocit y of light in the non-inertial reference Ng(at rest on the Earth’s surface) will be discussed bellow. 4 Spacetime curvature or spacetime anisotropy? We have seen that the anisotropy in the propagation of light f ully accounts for the behaviour of the classical electron in a gravitational ﬁeld. It appears that no curvatu re of spacetime is needed. In order to see whether this is really the case let us ﬁrst consider what causes the gravit ational attraction of two electrons (no matter how negligible it is). As we have seen the electron inertial and passive gravitatio nal masses are entirely electromagnetic in origin. As it is believed that all three masses - inertial, passive gr avitational, and active gravitational - are equal, it follows that the electron active gravitational mass is full y electromagnetic in origin as well. And since it is only the charge of the electron that represents it (there is no mec hanical mass), it follows that the active gravitational mass of the electron is represented by its charge. Therefore it is the electron charge that causes its gravity and the anisotropic velocity of light in the electron’s neighborho od. The question now is: ”Is the electron’s gravitational ﬁeld a manifestation of a spacetime curvature around the ele ctron?” or more precisely: ”Does the electron’s charge create a curvature which in turn causes the anisotropic velo city of light in the electron’s vicinity?” We have seen that it is the anisotropy in the velocity of light alone that f ully and consistently explains the fall of an electron toward the Earth and the self-force acting on an electron at r est on the Earth’s surface. In addition to the electric repulsion of two electrons ( e1ande2) in open space, they also attract each other through the anis otropy in the velocity of light around each of them: e1falls toward e2in order to compensate the anisotropy caused by e2and vice versa. Therefore, the anisotropy in the propagation of light in the electrons’ vicinity is completely suﬃcient to explain their (gravitational) attraction and no additio nal spacetime curvature hypothesis is necessary. This is an indication that, according to the classical electromagn etic mass theory, what the electron’s charge creates is not a spacetime curvature; it is a spacetime anisotropy whic h causes the anisotropic velocity of light. In such a way, an electron’s gravitational ﬁeld according to the elec tromagnetic mass theory turns out to be the spacetime anisotropy in the neighborhood of the electron originating from its charge. An anisotropic-spacetime re-interpretation of general re lativity appears more trouble-free than its current curved-spacetime interpretation: (i) there will be no prob lem with the force acting on a body whose worldline is not geodesic; (ii) the problem with the existence of gravi tational energy will be solved (there is energy, but electromagnetic); (iii) the equivalence of an acceleratin g frame and a frame at rest in gravitational ﬁled will be explained along with the equivalence of inertial and passiv e gravitational mass; (iv) it will be also explained what tells a body to move non-resistantly (by inertia) or to oﬀer r esistance: a free body will move by inertia if the electric ﬁelds of its charges are the Coulomb ﬁeld; if the bod y’s charges’ ﬁelds are distorted, the body’s motion is with resistance - it resists the deformation of its charges’ electric ﬁelds. 5 Toward a quantum electrodynamical formulation of the elec tro- magnetic mass theory Although the lack of a quantum model of the electron makes it i mpossible to formulate the classical electromagnetic mass theory in terms of quantum mechanics, it appears that it s quantum electrodynamical formulation may be possible. According to the electromagnetic mass theory the inertial a nd gravitational forces acting on the classical electron come from its self-interaction with its distorted ﬁeld. In quantum electrodynamics (QED) the electric ﬁeld of a charge is the swarm of virtual photons that are const antly being emitted and absorbed by the charge. The distorted ﬁeld of a non-inertial electron in QED will be r epresented by the anisotropy in the velocity of the 7virtual photons comprising the electron ﬁeld. As the recoil an electron is subjected to every time it emits or absorbs a virtual photon depends on the photon’s velocity th e anisotropy in the virtual photons’ velocity will disturb the balance of the recoils (which cancel out exactly if there is no anisotropy). This means that in QED too the interaction of a non-inertial electron with its dist orted ﬁeld also gives rise to a self-force which coincides with the inertial force in the case of an accelerating electr on and with the gravitational force in the case of an electron supported in a gravitational ﬁeld. It turns out that for the QED formulation of the electromagne tic mass theory it does not matter whether an electron will be regarded as a point or a sphere - in both cases the self-force originates from the unbalanced recoils of the virtual photons being absorbed by the electron; the re coils of the emitted virtual photons cancel out since the photons are always emitted with speed = c. The QED formulation of the electromagnetic mass theory make s it possible to compare the electromagnetic mass approach to inertia with the zero-point ﬁeld (ZPF) appr oach according to which inertia originates from the interaction of an electron charge with the virtual photo ns of the zero-point ﬂuctuation of the electromagnetic vacuum [34]. If it can be stated with certainty that in QED the interaction of an electron with the virtual photons comprising its own electric ﬁeld is diﬀerent from the electr on’s interaction with the zero-point virtual photons then the two approaches to inertia are also diﬀerent. There a re two reasons which seem to indicate that the ZPF inertia may be only a small contribution to an electron’s ine rtia caused by the unbalanced recoils from the virtual photons of its ﬁeld: (i) The ZPF approach cannot explain gravitation and the orig in of the gravitational mass. The electromagnetic resistance oﬀered by an accelerating charge as explained by the ZPF approach can be viewed as the Lorentz force caused by interaction of the magnetic component of the electromagnetic ZPF with the charge. This d eals with the charge’s inertial mass. However, when the charge is at re st on the Earth’s surface there is NO magnetic component of electromagnetic ZPF and therefore there is no c ontribution to the charge’s gravitational mass. One may speculate that inertial and gravitational masses ar e not equal at the quantum level which is, of course, something that should be studied. It becomes clear from here why the ZPF contribution to inertia may be a small correction at best: if there is no ZPF contribution to t he gravitational mass, a ZPF contribution to the inertial mass should be extremely small; otherwise, a great er diﬀerence between the two masses would have been observed by now. (ii) As the classical electromagnetic mass theory is the onl y classical theory (supported by experimental evidence) that deals with inertia it is natural to expect tha t it should be the QED version of that theory that accounts for inertia in QED. On the other hand, the ZPF approa ch to inertia does not have a classical analog which seems to disqualify it as a theory describing the major contribution to inertia (this would not be so if there were no classical theory dealing with inertia). As inertia and gravitation have predominantly macroscopic manifestations it appears certain that these phe- nomena should possess not only a quantum but a classical desc ription as well. This expectation is corroborated by the fact that such a description already exists - the elect romagnetic mass theory which yields the correct expressions for the inertial and gravitational masses of th e classical electron. Therefore, the chances of any modern theory of inertia (and gravitation) can be evaluated by seeing whether it can be considered a quantum generalization of the classical electromagnetic mass theo ry. 6 Is all the mass electromagnetic? We have seen that both the inertial and the passive gravitati onal masses of the classical electron are fully elec- tromagnetic in origin. If we now ask what about the inertial a nd gravitational masses of the real electron? Are they electromagnetic in origin as well? An argument against regarding the entire mass as electromag netic is that strong and weak interactions should also contribute to the mass. This argument, however, does no t apply to the electron for two reasons: (i) the electron does not participate in strong interactions, and ( ii) a free electron does not participate in any weak interactions either (only the volume elements of its charge interact electromagneticly). This argument, however, is quite relevant when the nature of mass of the other elementary charged particles is discussed. As the issue of the strong and weak contribution t o the mass is an open one and needs a separate study, let us outline an argument demonstrating that at least the st rong interaction does not contribute to the mass. As we have seen the unbalanced repulsion of the charge ”elements” of the classical electron gives ris e to a self-force and its mass. Unbalanced attraction of opposite charges results in the reduction of the charges’ mass [28]. This 8is true not only for electric forces. Early attempts by Poinc ar´ e [7] to resolve the stability problem in the classical electromagnetic mass theory resulted in the introduction o f unknown attraction forces (called Poincar´ e stresses) that balance the repulsion of the charge ”elements” of the cl assical electron. As it turned out that those attraction forces had a negative contribution to the mass the problemat ic 4/3 factor was reduced to 1. Therefore, due to (i) the fact that the forces of strong interaction are attrac tion forces and (ii) the strength of strong interaction (over two orders of magnitude greater than the electromagne tic interaction) one can expect a signiﬁcant negative contribution to the mass of a charged particle (compared to t he electromagnetic contribution). If it turns out that the strong interaction does contribute to the mass, we w ill face a major crisis in physics - it will not be clear what compensates the negative contribution to the mass that originates from the strong interaction. On the other hand, however, the strong and weak interactions as fundamental forces should make a contribution to the mass (as the electromagnetic interaction does) [35] a nd if they do not, then we might be forced to re-examine their very nature as separate fundamental interactions. If it turns out that the strong and weak interactions make no c ontribution to the mass then the mass of all particles will prove to be entirely electromagnetic in o rigin. It should be noted, however, that a fully elec- tromagnetic mass implies that there are no elementary neutr al particles (with non-zero rest mass) in nature. A direct consequence from here is that only charged particle s or particles that consists of charged constituents possess inertial and passive gravitational mass. Stated an other way, it is only elementary charges that comprise a body; there is no such fundamental quantity as mass. It is ev ident that in this case the electromagnetic mass theory predicts zero neutrino mass and appears to be in conﬂi ct with the apparent mass of the Z0boson which is involved in the weak interactions. The resolution of this apparent conﬂict could lead to either restricting the electromagnetic mass theory (in a sense that not the entire m ass is electromagnetic) or re-examining the facts believed to prove (i) that the Z0boson is a fundamentally neutral particle (unlike the neutr on), and (ii) that it does possess inertial and gravitational mass if truly neutr al. Another argument that the mass of a particle is fully electro magnetic in origin comes from the velocity dependence of the mass. It is a corollary of the classical ele ctromagnetic mass theory that the electromagnetic mass rises with velocity inversely as/parenleftbig 1−v2/c2/parenrightbig1/2[22, p. 28-3]. And instead of viewing the result that all the mass depends on velocity discovered by the special theory of relativity as a serious indication that all the mass is electromagnetic, inexplicably the whole issue of electr omagnetic mass has been practically abandoned. If we assume that the mass of a body consists of several kinds of mas ses (electromagnetic, mechanical, strong and weak) we have to answer the question how all of them obey the same law of velocity dependence? 7 Conclusions We have seen that inertia, inertial mass, gravitation, pass ive gravitational mass, and the equivalence of the two masses of the classical electron are fully accounted for by t he electromagnetic mass theory. The self-force to which a non-inertial electron is subjected on account of its own distorted electric ﬁeld unambiguously indicates that the inertial and passive gravitational masses of the cl assical electron are electromagnetic in origin. This result provides a straightforward explanation of the equiv alence of inertial and gravitational mass. The inertial and passive gravitational masses of the electron are the samething - the mass that corresponds to the energy stored in its electric ﬁeld. However, the inertial and passive grav itational masses of the electron manifest themselves as such - as a measure of the electron’s resistance to being acce lerated - only if it is subjected to an acceleration (kinematic or gravitational). This resistance originates from the unbalanced mutual repulsion of the volume elements of the electron. According to the electromagnetic mass theory it is the elect ron charge that causes the anisotropy in the prop- agation of light in the electron’s vicinity. That anisotrop y completely accounts for the (gravitational) attraction of two electrons. This is an indication that the Riemann curv ature tensor should be regarded as describing an anisotropy (not curvature) of spacetime. An important open question of the electromagnetic mass theory is how a charge causes the anisotropy of spacetime around itself. It should be stressed that the electromagnetic mass theory i s not just a hypothesis; it is a valid physical theory since (i) it is based on ﬁrm experimental evidence (the exper imental fact that at least part of the mass of charged particles is electromagnetic in origin; there is no other th eory that accounts for this fact), and (ii) it is a further natural development of the classical electron theory in con junction with the principle of equivalence. It should be speciﬁcally emphasized that even if it turns out that only part of the mass is electromagnetic in origin it still follows from the electromagnetic mass theor y that inertia, inertial mass, gravitation, and gravitatio nal 9mass (passive and active) are in part electromagnetic in ori gin. All results outlined here could have been obtained at least e ighty years ago when Fermi [9] initiated the approach of studying the classical electromagnetic mass th eory in conjunction with general relativity (more speciﬁcally, with the equivalence principle). Unfortunat ely, he later turned to atomic physics and perhaps deprived our century from solving the mystery of inertia and gravitat ion. I believe that the answer to the question posed in the title is now clear - Yes, the 20 thcentury physics did have the means to reveal the nature of inertia and gravitatio n. References [1] St. Augustine, Confessions, Book 11(11 .14.17); http://ccat.sas.upenn.edu/jod/augustine.html (se e the Sec- tion ”Texts and Translations”). [2] See: http://press.web.cern.ch/Press/Releases00/PR 08.00ELEPRundelay.html [3] J. J. Thomson, Phil. Mag., 11, 229 (1881). [4] O. Heaviside, The Electrician, 14, 220 (1885). [5] G. F. C. Searle, Phil. Mag., 44, 329 (1897). [6] H. A. Lorentz, Proceedings of the Academy of Sciences of A msterdam, 6, 809 (1904); Theory of Electrons, 2nd ed. (Dover, New York, 1952). [7] H. Poincar´ e, Compt. Rend., 140, 1504 (1905); Rendicont i del Circolo Matematico di Palermo 21, 129 (1906). [8] M. Abraham, The Classical Theory of Electricity and Magn etism, 2nd ed. (Blackie, London, 1950). [9] E. Fermi, Nuovo Cimento, 22, 176 (1921); Phys. Zeits., 23 , 340 (1922); Rend. Acc. Lincei (5), 31, 184; 306 (1922); Nuovo Cimento, 25, 159 (1923). [10] H. Mandel, Z. Physik, 39, 40 (1926). [11] W. Wilson, Proc. Phys. Soc., 48, 736 (1936). [12] M. H. L. Pryce, Proc. Roy. Soc., A168, 389 (1938). [13] B. Kwal, J. Phys. Rad., 10, 103 (1949). [14] F. Rohrlich, Am. J. Phys., 28, 639 (1960); Classical Cha rged Particles, (Addison-Wesley, New York, 1990). [15] On the historical development of the classical electro magnetic mass theory see [14] and [1]. [16] J. W. Butler, Am. J. Phys. 37, 1258 (1969). [17] In order to account for the stability of the classical el ectron Poincar´ e [7] assumed that part of the electron mass (regarded as mechanical) originated from forces (know n as the Poincar´ e stresses) holding the electron charge together and that it was this mechanical mass that com pensated the 4 /3 factor (reducing the electron mass from 4 /3mtom). However, the 4 /3 factor, as discussed above, turned out to be an error in the calculations of electromagnetic mass as shown in [9]-[14]. As there remained nothing to be compensated (in terms of mass), if there were some unknown attraction forces (the Poincar´ e stresses) responsible for holding the electron charge together, their negative contribution to the electron mass would result in reducing it from m to 2 /3m. This made the stability problem even more puzzling - on the o ne hand, a spherical electron tends to disintegrate due the repulsion of the diﬀerent parts of th e spherical shell; on the other hand, however, an assumption that there is a force that prevents the electro n charge from blowing up leads to a wrong expression for its mass. Obviously, there is an implicit ass umption in the classical model of the electron that leads to such a paradox - it is assumed that at every instant th e electron charge occupies the whole spherical shell (see [19]). [18] D. J. Griﬃths, Introduction to Electrodynamics, 2nd ed ., Prentice Hall, New Jersey, 1989, p. 439. 10[19] It is not impossible for an elementary charge to have a spherical but not continuous distribution. Such a possibility follows from a work [20] which has received litt le attention so far. By bringing the idea of atomism to its logical completion (discreteness not only in space bu t in time as well - 4 −atomism), it is argued in that work that a quantum-mechanical description of the elec tron itself (not only of its state) is possible if the electron is represented not by its worldline (as determi nistically described in special relativity) but by a set of four-dimensional points (modeled by the energy-mome ntum tensor of dust - in this case a sum of delta functions) scattered all over the spacetime region in which the wave function of the electron is diﬀerent from zero. The 4 −atomism hypothesis gives an insight into two questions: (i) how an elementary charge can have ”parts” and still remain an elementary charge, and (ii) why t here is no stability problem despite that the ”parts” of an electron repel one another: since for 1 second a n electron is represented by 1020four-dimensional points (according to the 4 −atomism hypothesis) at one instant the electron exists as a s ingle point carrying a greater (bare) charge, but for one second, for example, the re will be 1020such points occupying a spherical shell that manifest themselves as an electron whose eﬀectiv e charge is equal to the elementary charge. The 4−atomistic model of the electron appears to overcome the diﬃc ulties of both a purely particle and a purely wave models of the quantum object and may be a candidate for wh at Einstein termed ”something third” (neither a particle nor a wave). For a brief description of wh y neither the purely particle nor the purely wave models of the quantum object can be accepted see reference [2 1]. [20] A. H. Anastassov, ”The Theory of Relativity and the Quan tum of Action (4 −Atomism)”, Doctoral Thesis, Soﬁa University, 1984 (unpublished); ”Self-Contained Pha se-Space Formulation of Quantum Mechanics as Statistics of Virtual Particles”, Annuaire de l’Universit e de Soﬁa ”St. Kliment Ohridski”, Faculte de Physique, 81, 1993, pp. 135-163. [21] Here are two of the most serious problems with a purely pa rticle and purely wave models of the electron. (i) If the s-electron in the hydrogen atom is regarded as a particle, i.e . as localized (its charge being localized) somewhere above the proton, then the hydrogen atom should po ssess a dipole moment in its s-state. Both quantum mechanics and the experiment show that this is not th e case. One may picture the electron in the s-state as so rapidly orbiting the proton that what is experim entally measured is the average value of the dipole moment over the measurement time. And since there is a spherical symmetry in the s-state all dipole moments cancel out exactly - the average value is zero. To ver ify that hypothesis Madelung calculated the orbital velocity of the electron that would ensure that all d ipole moments during the measurement cancel out. It turned out that the electron orbital velocity should be several orders of magnitude greater than the velocity of light. This shows that the electron charge shoul d be somehow uniformly distributed around the proton. (ii) A system of n”particles” cannot be represented by a pure wave since that w ave cannot be a real wave in the real space - it is a wave in a space of 3 ndimensions. [22] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Le ctures on Physics, Vol. 2, Addison-Wesley, New York, 1964, p. 28-10. The Feynman Lectures on Physics is n ot the only physics textbook that stresses that experimental fact; see, for example, R. Stevenson and R . B. Moore, Theory of Physics, W. B. Saunders, Philadelphia and London, 1967 (p. 590: ”there is experiment al evidence for the existence of electromagnetic mass”). [23] E. Fermi, Z. Physik, 23, 340-346 (1922), quoted by P. Moy lan, Amer. J. Phys., 63, 818 (1995). [24] P. Pearle, in D. Teplitz, ed. Electromagnetism: Paths t o Research, Plenum Press, New York, 1982, p. 213. [25] E. Mach, Science of Mechanics, 9th ed., Open Court, Lond on, 1933. Around 1883 Mach argued that inertia was caused by all the matter in the Universe (no matter how dis tant it may be) thus assuming that inertia had a non-local cause. [26] The self-force Fself=−mais traditionally called inertial force. According to Newto n’s third law the external forceFthat accelerates the electron and the self-force Fselfhave equal magnitudes and opposite directions: F=−Fself. Therefore F=mawhich means that Newton’s second law is derived on the basis o f Maxwell’s electrodynamics and Newton’s third law. [27] Strictly speaking, in the framework of classical elect rodynamics the explanation of the self-force acting on an accelerating charge appears possible only in terms of rep ulsion of charges, not in terms of interaction of a charge and a distorted electric ﬁeld. 11[28] V. Petkov, Ph. D. Thesis, Concordia University, Montre al, 1997; for an account of why all arguments against regarding the entire mass of the classical electron as electromagnetic in origin have been an- swered see also: ”Acceleration-dependent electromagneti c self-interaction eﬀects as a basis for inertia and gravitation”(http://xxx.lanl.gov/abs/physics/990901 9) [29] V. Petkov, What is general relativity silent on? (http: //xxx.lanl.gov/abs/gr-qc/0005084). [30] J. L. Synge, Relativity: the general theory, Nord-Hola nd, Amsterdam, 1960, Ch. III. Sec. 3. [31] The formalism of general relativity refuses to yield an appropriate mathematical (tensor) expression for the energy and momentum of gravitational ﬁeld; instead a pseudo -tensor to model gravitational energy and momentum is used. The problem with a pseudo-tensor is that it cannot represent a real physical quantity; this implies that there is no gravitational energy and momen tum. Such a conclusion appears to be fully in line with the way general relativity describes gravitation - as a manifestation of spacetime curvature, not as a force ﬁeld which possesses energy and momentum; gravitational ﬁl ed in general relativity can be regarded as an energy-less and momentum-less geometric ﬁeld. However, so me people think that there is a real problem with such a conclusion since the experimental evidence seems to d emonstrate the existence of gravitational energy and momentum - it is suﬃcient to mention only the tidal electr ic power stations converting what appears to be gravitational energy into electric energy. That exper imental evidence looks completely diﬀerently from the viewpoint of the electromagnetic mass theory which reve als that gravitation is electromagnetic in origin (at least in part) - the tidal electric power stations conver t electric energy into electric energy (as in the cases in which mechanical energy is converted into mechanical ene rgy). There is an attempt to explain the fact a pseudo-tensor is used for modeling gravitational energy a nd momentum: ”At issue is not the existence of gravitational energy, but the localizability of gravitati onal energy. It is not localizable” [32]. That is confusion - it is claimed that two mutually excluding things are true: ( i) there is gravitational energy, and (ii) there is no (force) gravitational ﬁeld which means that there is no gr avitational energy. [32] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation , Freeman, San Francisco, 1973, p.467. [33] An inertial observer with respect to whom an electron mo ves with constant speed will see its ﬁeld deformed (Lorentz-contracted), but this does not mean that the elect ron resists its motion with uniform velocity. The observed deformation of the electron ﬁeld gives rise to the i ncrease of the electron mass as seen by the inertial observer. [34] B. Haisch, A. Rueda and H. E. Puthoﬀ, Phys. Rev. A 49, 678 ( 1994); A. Rueda and B. Haisch, Phys. Lett. A 240, 115 (1998); A. Rueda and B. Haisch, Found. Phys. 28, 105 7 (1998). [35] H. Stephani, General Relativity, 2nd ed., (Cambridge U niversity Press, Cambridge, New York, 1990), p. 69. 12
arXiv:physics/0012026v1 [physics.class-ph] 14 Dec 2000In Memory of my Beloved Mama, Vlasova Alexandra V., 1922-2000 CHARGED BROWN PARTICLE: THE MORE RETARDATION IS - THE LOWER IS THE EFFECTIVE TEMPERATURE Alexander A. Vlasov High Energy and Quantum Theory Department of Physics Moscow State University Moscow, 119899 Russia The Brownian motion of a charged particle with ﬁnite size (de scribed by Sommerfeld model) is considered. It is found out that due t o radiation reaction: (1) the eﬀective temperature of such particle is l ower, and (2) the acceleration of the average velocity is smaller, then that f or classical Brown particle without electric charge. 03.50.De Sommerfeld particle [1] is the model of a charged particle of ﬁnite size - sphere with uniform surface charge Qand mechanical mass m. In nonrel- ativistic approximation such sphere obeys the equation (se e [2] and works, cited there): m˙/vector v=/vectorFext+η[/vector v(t−2a/c)−/vector v(t)] (1) herea- radius of the sphere, η=Q2 3ca2, /vector v=d/vectorR/dt, /vectorR- coordinate of the center of the shell, /vectorFext- some external force. The term in RHS of eq.(1), proportional to η, describes the eﬀect of radiation reaction (eﬀect of retardation). This model is a good tool to consider eﬀects of radiation reac tion, free of problems of classical point-like Lorentz-Dirac descripti on (see, for ex.[3]). A. In this paper we consider Sommerfeld particle in the role of t he Brown 1particle, i.e. particle (with radiation reaction), moving in a stochastic path under action of some external stochastic force /vectorFstoch. For simplicity we take that: (1) the viscosity of the surrounding medium is zero; (2) the statistical average of /vectorFstoch-/vectorF0- is non zero and constant in time t; (3) particle moves in one dimension. Under these assumptions the Langevin equation for Sommerfe ld particle takes the form m˙v=Fstoch+η[v(t−2a/c)−v(t)] (2) or for statistical average value < v > ofv m˙< v > =< F stoch>+η[< v > (t−2a/c)−< v > (t)] (3) For dimensionless variables y=< v > /M, x =ct/M (M-scale factor) equation (3) takes the form ˙y=f+γ·[y(x−δ)−y(x)] (4) here γ=Q2M 3a2mc2, f=F0M mc2, δ=2a M Classical analog of equation (4) for Brown point particle on e can get taking γ= 0 in (4). Equation (4) for f=const has the exact solution y=y0+k·x (5) withk=f/(1 +γδ) and initial velocity y0. Following the theory of Brownian motion ( there are many text books on Brownian motion, see, for ex., [4]), the dispersion D=<(v−< v > )2> for surrounding medium without viscosity can be function of time: D= D(t). The form of D(t) strongly depends on the form of correlation function of stochastic force Fstochand its concrete realization in computer program ( if correlation function is compact enough, then D(t) for ”not very large moments of time t” is proportional to time: D(t)∼t- Einstein formula for 2Brownian motion). If the dependence D=D(t) is known, one can try to ﬁnd the form of correlation function, but this is not our goal . Do not going into details, we can say that time average of Dwith respect to the whole time of ”observation” Tmust be proportional (in dimensionless variables) to the eﬀective temperature θand inverse to the mass of particle: ¯D∼θ mc2. (6) Solution (5) describes motion of charged particle with cons tant accelera- tion. This result seems to be unusual. Indeed, following classical electrodynamics, particle wi th acceleration must radiate. Then one could expect that radiation, due to energy loss, leads to radiation damping of particle motion. Instead, we see that after statistical averaging, radiatio n reaction in Som- merfeld form only changes the value of the eﬀective force, ac ting on particle, making it smaller. One can interpret this in the following way. Radiation react ionFradin Sommerfeld model is Frad=γ·[y(x−δ)−y(x)] on trajectory (5) it is nonzero and equals to −γδk. Ifδ- is small we can expand the force Fradin powers of δ: Frad=−kδ˙y+k/2(δ)2¨y+... (7) First term in (7) is the eﬀective electromagnetic mass of the particle, mul- tiplied by acceleration with sign minus. On trajectory (5) t his term equals to −γδk. Second term in (7) is the radiation force in classical Loren tz form - on trajectory (7) it is zero. Thus radiation eﬀects after stati stical averaging lead only to change in eﬀective particle mass (mechanic + electro magnetic) - it becomes greater and the particle becomes more ”inertial” (m ore ”retarded”). It looks like the energy of self-electromagnetic ﬁeld of a ch arged particle does not radiate away, the ”electromagnetic fur-coat” does not get thin, thus particle becomes ”heavier” (in comparison with particle of zero charge). This eﬀect can also make the dispersion (6), i.e. the eﬀectiv e Brownian temperature smaller. 3B. Dispersion (6), also as the particle motion, we investigate d numerically. The particle mass mand size awe take close to that of classical electron, this yields γδ= 1 in (4). The stochastic force Fstochwe extract step by step from the known pro- cedure (see, for ex.,[5]): φn+1={K·φn}; here brackets {...}denotes fractional part of ..., and (Fstoch)n·(M/mc2) = 10+3·(φn−φ0) withK= 100000 /3 and φ0= 0.5007645. The results of numerical calculations can be summed in the fo llowing way: (1) Sommerfeld particle in the role of the Brown particle has the eﬀective temperature θSlower, than that for classical Brown particle θB:θS< θB. (2) The more is the retardation (i.e. the greater is γin (4) ), the greater is the diﬀerence between θBandθS. (3) The acceleration kof the average velocity of Sommerfeld particle is smaller than that of classical Brown particle without elect ric charge. These results are illustrated in Fig. 1, where y0= 0.1;γ= 2000 .0;δ= 2a/M = 1/2000.0; observation ”time” is 10 .0005, with respect to it the average value of stochastic force is f= 4.881996 ·10−4and the average dispersions are ¯DS= 4.4359 ·10−8,¯DB= 4.8496 ·10−8, i.e.θS< θB. The upper curve - is the path of classical Brown particle (hor izontal axis - is ”time” x), the lower - is the path of Sommerfeld particle in the role of Brown particle. The acceleration of the average velocities for these partic les diﬀers by multiplier 2, as the consequence of the exact solution (5) ( f or Brown particle we have γ= 0, and k=f, for Sommerfeld - γδ= 1 and k=f/2). These results conﬁrm numerically our considerations, made before. 4REFERENCES 1. A.Sommerfeld, Gottingen Nachrichten, 29 (1904), 363 (19 04), 201 (1905). 2. L.Page, Phys.Rev., 11, 377 (1918). T.Erber, Fortschr. Ph ys., 9, 343 (1961). P.Pearle in ”Electromagnetism”,ed. D.Tepliz, (Pl enum, N.Y., 1982), p.211. A.Yaghjian, ”Relativistic Dynamics of a Char ged Sphere”. Lecture Notes in Physics, 11 (Springer-Verlag, Berlin, 199 2). F.Rohrlich, Phys.Rev.D, 60, 084017 (1999). 3. Alexander A.Vlasov, physics/9905050, physics/9911059 , physics/0004026. 4. I.A.Kvasnikov, ”Thermodynamics and statistical physic s. Part 2”. Moscow, Moscow State University, 1987. 5. R.Z.Sagdeev, G.M.Zaslavsky, ”Introduction to nonlinea r physics”. Moscow, Nauka, 1988. 59.97e-21.01e-11.02e-11.03e-11.04e-11.05e-1 0.00e01.00e02.00e03.00e04.00e05.00e06.00e07.00e08.0 0e09.00e01.00e1 Fig. 1 6
arXiv:physics/0012027v1 [physics.plasm-ph] 14 Dec 2000Dynamical properties and plasmon dispersion of a weakly degenerate correlated one-component plasma V. Golubnychiy1, M. Bonitz1, D. Kremp1, and M. Schlanges2 1Fachbereich Physik, Universit¨ at Rostock, Universit¨ ats platz 3, 18051 Rostock, FRG 2Institut f¨ ur Physik, Ernst–Moritz-Arndt Universit¨ at Gr eifswald, Domstr. 10a, 17489 Greifswald, FRG Abstract Classical Molecular Dynamics (MD) simulations for a one-co mponent plasma (OCP) are presented. Quantum eﬀects are included in the form of the Kelbg potential. Results for the dynamical structure factor are c ompared with the Vlasov and RPA (random phase approximation) theories. The i nﬂuence of the coupling parameter Γ, degeneracy parameter ρΛ3and the form of the pair interaction on the optical plasmon dispersion is investiga ted. An improved analytical approximation for the dispersion of Langmuir wa ves is presented. Typeset using REVT EX 1I. INTRODUCTION The model of a classical one-component plasma has - due to its simplicity - been widely investigated both theoretically and with various numerica l and simulation methods, see e.g. [1–3] and [4,5], respectively. Since the pioneering numeri cal work of Brush, Sahlin, Teller [6], the thermodynamic and dynamic characteristics of the c lassical OCP have been stud- ied in detail. In particular, the dependence of the properti es on the coupling parameter Γ = 4 πe2/(¯rkBT), where ¯ r= (3 4πρ)1/3is the mean interparticle distance and ρthe density, have been investigated up to very large values of Γ [7,8]. Amo ng the most important ther- modynamic results is the prediction of crystallization at v alues of Γ of the order of 172-180 [9,10]. Furthermore, investigations of the dynamic proper ties of strongly correlated classical plasmas have indicated that the wave number dependent plasm on dispersion changes from monotonic growth, common for weakly coupled plasmas, to a de creasing dispersion around Γ≈3 [1]. On the other hand, there is growing interest in the dynamic pr operties of dense quantum plasmas, in particular in astrophysics, laser plasmas and c ondensed matter. While the case of strong degeneracy (strong quantum limit) and weak coupli ng at very high densities is well described by the random phase approximation (RPA, see e .g. [11–14]), the properties atintermediate coupling and degeneracy remain poorly explored. Especially, one is interested in the dynamic plasma behavior in cases where the average kin etic energy is of the same order as the mean potential energy, i.e. Γ ∼1, where collisionless theories such as the RPA fail, e.g. [12,13,15]. For these situations, quantum molecular d ynamics (QMD) simulations [16] are the appropriate numerical approach which, however, is y et lacking the required eﬃciency. For weakly degenerate plasmas, with ρΛ3≤1, where Λ is the DeBroglie wave length (see below), it is expected that one can perform much simpler clas sical MD simulations using eﬀective quantum pair potentials, e.g. [3,17]. These poten tials can be derived from the 2- particle Slater sum using Morita’s method. It is the aim of th is paper to explore this MD approach in detail, especially for the analysis of the optic al (Langmuir) plasmon dispersion. 2It is natural to start this analysis with OCP–simulations be cause they have the advan- tage of the absence of a collapse of oppositely charged parti cles at small distances. On the other hand, the existence of a homogeneous background of opp ositely charged particles leads to some additional technical diﬃculties compared to 2-comp onent systems, due to restricted carrier rearrangement causing less eﬀective screening of t he Coulomb interaction. One major problem of MD simulations of dynamical properties is that th e behavior at small wave num- bers is diﬃcult to investigate. The reason is that large box- sizes are required which, for the analysis of high density plasmas, translates into large par ticle numbers. The current increase of available computer power gives one the possibility to inv estigate size-dependent properties like the density-density correlations /angbracketleftρ/vectork(0)ρ−/vectork(t)/angbracketrightfor smaller k-vectors than before. In this paper, we are able to present accurate results for the dynami cal properties of the OCP, such as the dynamical structure factor and the wave vector disper sion of Langmuir oscillations. Our simulations for intermediate values of the coupling par ameter, Γ = 1 . . .4, show an in- teresting dispersion: the frequency increases up to a maxim um and, for large wave numbers, decreases again. Further, we investigate the role of quantu m eﬀects by comparing simula- tions with the Coulomb potential and an eﬀective quantum pai r potential (Kelbg potential [17]) for the region of small and intermediate coupling. We f ound that quantum diﬀraction eﬀects have noticable inﬂuence on the behavior of the optica l dispersion curves. Increase of the degeneracy leads to a softenig of the dispersion ω(k), especially at intermediate wave vectors. II. DYNAMICAL PROPERTIES OF THE OCP A. Statistical approach A central quantity to determine the dynamic properties of ch arged many-particle systems is the frequency-dependent dielectric function ǫ(/vectork, ω) which, for the OCP, is given by ǫ(/vectork, ω) = 1−UC(/vectork)Π(/vectork, ω). (1) 3HereUC(/vectork) is the spatial Fourier transform of the Coulomb potential, UC(k) = 4πe2/k2, and Π(/vectork, ω) is the longitudinal polarization function. Thus, many-bo dy eﬀects enter the dielectric function via Π. There exist many approximations for the latt er function, the simplest one being mean-ﬁeld theories which neglect short-range correl ation eﬀects, i.e. collisions between the particles. For the classical OCP, the mean-ﬁeld result i s the Vlasov polarization: ΠVlasov(/vectork, ω) =−1 m/integraldisplayd3v (2π)3/vectork ω−/vectork/vector v+iδ∂F(/vector v) ∂/vector v. (2) Hereδ→+0, indicating the retarded (causal) character of the polar ization and the dielectric function. Further, Fis the distribution function. The Vlasov polarization appl ies only to classical plasmas, where the wave character of the particle s can be neglected. Quantum eﬀects are important if the interparticle distance or the De bye radius become comparable to the DeBroglie wave length Λ = h/√2πmk BT. Therefore, quantum diﬀraction eﬀects should show up in the dielectric properties at large wave numbers. T he quantum generalization of the Vlasov polarization is the RPA polarization function gi ven by ΠRPA(/vectork, ω) =−/integraldisplayd3p (2π¯h)3f(/vector p)−f(/vector p−¯h/vectork) ¯hω+p2 2m−(/vector p+¯h/vectork)2 2m+iδ. (3) In this paper we consider only plasmas in equilibrium, so Fandfare the Maxwell and Fermi function, respectively. One readily conﬁrms that, in the limit of long wavelengths, /vectork→0, indeed the RPA result (3) goes over to the Vlasov polarizat ion function (2). An important quantity which follows from dielectric function (1) via the ﬂuctuation-dissipation theorem is the dynamical structure factor S(/vectork, ω) S(/vectork, ω) =−kBT πUC(k)ωIm1 ǫ(/vectork, ω), (4) which shows the frequency spectrum of density ﬂuctuations f or a given value of /vectork. As mentioned above, the mean ﬁeld expressions (2) and (3) neg lect short-range correla- tions and are, therefore, valid only for weakly coupled plas mas, Γ ≪1. There exist many theoretical concepts to go beyond the RPA which are based on q uantum kinetic theory, density functional theory and other approaches. This is bey ond the scope of this paper, see 4e.g. Ref. [12,13,15] and references therein. Here, we consi der the alternative approach to the OCP at ﬁnite coupling which is based on molecular dynamic s simulations. B. Molecular dynamics approach to the dynamical properties The dielectric and dynamical properties of an interacting m any-particle system are easily accessible from the density-density correlation function which is deﬁned as A(/vectork, t) =1 N/angbracketleftρ/vectork(t)ρ−/vectork(0)/angbracketright, (5) where Nis the number of particles. ρ/vectork(t) is the Fourier component of the density, ρ/vectork(t) =N/summationdisplay i=1ei/vectork/vector ri(t), (6) which is computed from the trajectories /vector ri(t) of all particles. The dynamical structure factor is just the Fourier transform of the density-density correl ation function (5) S(/vectork, ω) =1 2π+∞/integraldisplay −∞dt eiωtA(/vectork, t). (7) Equation (7) can be directly compared to formula (4) and, thu s, allows for a comparison of the simulation results with the statistical theories. Furt hermore, Eq. (7) allows to investigate the inﬂuence of quantum eﬀects on the dynamical properties a nd plasmon dispersion of the OCP. Variations of the interaction potential (see below ) directly aﬀect the particle trajectories and, via Eqs. (5)–(7), the dynamical structur e factor. III. DETAILS OF THE MD-SIMULATIONS The simulations have been performed in a cube of length Lcontaining Ninteracting electrons on a uniform positive background. For this system , we solved Newton’s equations of motion containing all pair interactions which are derive d from a total potential U(r), see below. As an algorithm of motion we used a second-order sc heme in form of the Swope algorithm [18]. Since our simulations are performed in the m icrocanocial ensemble, the mean 5kinetic energy may change. Therefore, to maintain the chose n value of temperature and Γ, we applied scaling (renormalization) of all velocities at e very second step. A central goal of our simulations was to study the inﬂuence of quantum eﬀects. We, therefore, performed several simulations which used eithe r a Coulomb potential or an eﬀec- tive quantum pair potential (see below). To permit ﬂexibili ty in the use of the potential, Uwas divided into a short-range and a long-range part, U=Usr+Ulr, where quantum eﬀects inﬂuence only Usr, whereas the behavior at large distances, Ulr, is dominated by the long-range Coulomb interaction. Let us ﬁrst describe the tr eatment of the long-range term. A. Long-range interaction The long-range interaction was computed in standard way usi ng periodic boundary con- ditions and the Ewald summation procedure [20,21]. As a resu lt, the long-range potential is given by the Coulomb interaction in the main box and all ima ge cells: Ulr(/vector r) = 4πe2N/summationdisplay i/negationslash=jVEwald(/vector rij), (8) VEwald(/vector r) =nx,ny,nz≤1/summationdisplay /vector n=0erfc [√π\|(/vector r+/vector nL)/L\|] \|/vector r+/vector nL\|+nx,ny,nz≤5,n2≤27/summationdisplay /vector n/negationslash=0exp(−πn2) cos(2 π/vector n/vector r/L ) πn2L−1 L,(9) where erfc is the complementary error function, L- the side length of the simulation cell and/vector n- a vector of integer numbers which labels the periodic image s of the simulation box. In this expression, the ﬁrst term corresponds to a poten tial of particles with Gaussian broadened charge distribution around the electrons with a w idth of√π, the second one - the compensating Gaussian distributions, and the last one a ccounts for the inﬂuence of the homogeneous background. It turns out that the second term in (8) can be reduced to 2 loops (one over the particles and one over the vectors /vector nin the reciprocal space) and is not very time consuming. The more complicated part is the ﬁrst term which c ontains three loops. In case of a two-component plasma, a proper choice of the width of the Gaussian distribution and use of periodic boundary conditions greatly simpliﬁes this ter m due to cancellations. In contrast, for an OCP, the background cancels the interactions only par tially, “statically”. As a result, 6convergence of the sum is worse, and one needs to take into acc ount all ﬁrst neighboring image cells (total of 26) at every time step. The contributio n of all neighboring cells except for the main one (0 <\|/vector n\| ≤√ 3) was computed, before the start of the simulations and stored in 3-dimensional tables for the potential and forces . During the simulations, we used 3D-bilinear interpolation at every step to obtain the value s of the potential and forces for intermediate distances. We found that 100 grid points in eve ry direction are adequate, so the total size of the table was 106elements. The particle interactions inside the main ( /vector n= 0) cell were evaluated directly at every time step without mini mum image convention. B. Short-range interaction. Quantum eﬀects Let us now discuss the short-range potential. As has been sho wn by Kelbg and co-workers [17,19], quantum eﬀects can be treated eﬃciently by an eﬀect ive pair potential, the Kelbg potential: UKELBG (r, T) = 4πe2/parenleftBigg1−exp(−r2/λ2) r+√π λerfc(r/λ)/parenrightBigg (10) where λ=Λ√ 2π. As a consequence of quantum eﬀects, this potential diﬀers f rom the Coulomb potential at small distances r≤λand is ﬁnite at r= 0. Further, it is temperature-dependent via the thermal DeBroglie wavelength. The Kelbg potential c an be regarded as the proper quantum pair potential following from the two-particle Sla ter sum S2without exchange eﬀects: lnS2=−UKELBG kT+O(Γ2). (11) It treats quantum diﬀraction eﬀects exactly, up ﬁrst order i n Γ. Frequently other quantum pair potentials have been used, including the Deutsch poten tial [22], which has the same value at r= 0 but diﬀers from the Kelbg potential at intermediate dista nces. As was mentioned by Hansen [23], symmetry eﬀects do not have a big in ﬂuence on the dynamical properties (although they give a major contribution to the s tatic properties, especially for 7the light mass components). Using the Kelbg potential (10), we can immediately separate the short-range part of the interaction, Usr(r, T) = 4πe2/parenleftBigg−exp(−r2/λ2) r+√π λerfc(r/λ)/parenrightBigg , (12) which has been calculated together with the ﬁrst sum of Eq. (9 ) using the interpolation table. The Kelbg potential contains just the lowest order quantum c orrections (lowest order in e2) and is, thus, accurate at small coupling, Γ <1. Nevertheless, we expect that it correctly reproduces the inﬂuence of quantum eﬀects also at intermedi ate coupling, Γ ≤5. Further improvements are straightforward, e.g. by including excha nge eﬀects or by evaluating the full two-particle Slater sum. We note that the described num erical procedure applies to such improved quantum pair potentials as well, even if they are no t given analytically. C. Thermodynamic and dynamical quantities Solving Newton’s equations with forces derived from the tot al potential Usr+Ulr, we computed thermodynamic and static quantities, such as tota l energy and pair distribution function in usual manner. The results will be presented in th e next section. Here we discuss some details on computation of the dynamical properties, as they require much more eﬀort and computation time in order to achieve suﬃcient accuracy. To obtain useful results for the dynamical structure factor , requires simulation results in a suﬃciently broad range of wave numbers and frequencies. Natural units of the wave number and frequency are 1 /¯rand the plasma frequency ωpl=/radicalBig 4πe2ρ/m, respectively, which will be used in the following. The minimum wave number kmindepends on the size Lof the simulation box and thus, for a given density or couplin g parameter, on the number of particles N. One readily veriﬁes that kmin= 2π/L= 2π(ρ/N)1/3or, using dimensionless wave numbers, qmin=kmin¯r= (6π2/N)1/3. Clearly, to reduce kminrequires an essential increase of the number of particles in the simulation. 8The simulation accuracy can be further increased by taking a dvantage of the isotropy of the plasma in wave vector space. Indeed, in equilibrium, t he density-density correlation function and dynamical structure factor should only depend on the absolute value of the wave vector. On the other hand, the simulations yield slight ly diﬀerent results for diﬀerent directions of the wave vector. Averaging over all results co rresponding to the same absolute value of /vectorkallows to reduce the statistical error. For example, the min imum wave number kmincorresponds to directions of /vectorkalong either the x-, y- or z-axis, cf. Eq. (6), so we can use the average of the three. The next larger value is√ 2kmin, corresponding to the diagonals in the x-y, x-z and y-z planes. The third value,√ 3kmin, corresponds to the space diagonal and is not degenerate; consequently it carries the largest s tatistical error. This is the main reason for the ﬂuctuations of the numerical results for the w ave vector dispersion, see for example Fig. 5. Finally, to resolve the collective plasma oscillations, th e duration of the simulations has to be much larger than the plasma period. Also, increased simul ation times leads to a reduction of the noise. We found that times of the order of 250 plasma per iods are adequate. IV. NUMERICAL RESULTS We have performed a series of simulations for varying values of Γ and ρΛ3, using the Coulomb and Kelbg potential. Also, time step and particle nu mber have been varied until a satisfactory compromise between accuracy and simulation e ﬃciency has been achieved. The parameters of the runs chosen for the ﬁgures below are summar ized in Table I. We mention that kinetic energy conservation in all runs (if velocity sc aling was turned oﬀ) did not exceed 0.1%. Also, the results for the total energy (not shown), in cas e of the Coulomb potential, agree very well with data from the literature. We ﬁrst consider the pair distribution function g(r) for varying interaction potentials and parameter values. Fig. 1 shows g(r) for three values of the coupling parameter, Γ = 0.5,1,4. As expected, the Coulomb pair distribution function is cl ose to the Debye-H¨ uckel 9limit for small coupling, with increasing Γ, the deviations , especially around r= ¯r, grow systematically. The Kelbg pair distributions practically coincide with the Coulomb functions forr >0.6 ¯rbut deviate from the latter at small distances of the order of the thermal DeBroigle wave length Λ where quantum eﬀects are important. Clearly, with increasing degeneracy, the ratio Λ /¯rincreases, and the deviations extend to larger distances an d grow in magnitude. With increasing Γ, the deviations become smal ler since Coulomb eﬀects dominate the behavior at small distances. Let us now turn to the dynamical properties. In case of an OCP, charge and mass ﬂuc- tuations are identical because of the rigid opposite charge background. In our simulations, we have calculated the density-density correlation functi on (5) and, by numerical Fourier transformation, obtained the dynamical structure factor S(q, ω) for several (from 6 to 10, de- pending on the simulation) wave numbers, the values of which are determined by the size of the simulation box L (see above). The value of the smallest wa ve number is given in Table I. The frequency dependence of S(q, ω) for several wave vectors is presented in Figs. 2-4 for the Coulomb and Kelbg potentials. Also, the results of the mean- ﬁeld models are shown. The peak of the structure factor is related to the optical plasmo n (Langmuir mode) of the elec- trons, its position shows the plasmon frequency Ω( k), its width - the damping of the mode. In the limit k→0, Ω(k)→ωplfor all models. For increasing wave numbers, the width of the peak grows steadily, and it merges with the continuum of s ingle-particle excitations, e.g. [13,14], therefore, no results for larger wave numbers are s hown. Consider now the results for the plasmon dispersion more in d etail, cf. Fig. 5. First, we discuss the mean ﬁeld results (4) which are calculated using the Vlasov and RPA polariza- tions, Eqs. (2) and (3), respectively. The Vlasov result was computed using the formulas given in the review of Kugler [24], and for the RPA, a code was d eveloped which accu- rately evaluates the pole integration in Eq. (3). Both appro ximations show the same general trend for small and intermediate wave numbers: with increas ing wave number, the plasmon frequency and the damping increase. At large q, the dispersion exhibits a maximum and de- creases again. In all situations, the RPA yields a slightly s maller frequency than the Vlasov 10result, whereas the damping values are very close to each oth er. Let us now turn to the simulation results. The Coulomb and Kel bg simulations have been performed for exactly the same parameters, except for N and run time (cf. Table I). (Notice that, in contrast to the Kelbg case, the Coulomb simu lations depend only on Γ which can be achieved by various combinations of density and tempe rature). Comparison of the two simulations shows, cf. Fig. 2, that the results for the st ructure factors are very similar in case of small Γ. Peak positions and widths as well as the low and high frequency tails are very close to each other. The reason is obvious: since the pot entials (and pair distributions, cf. Fig. 1) diﬀer only at a small interparticle distances of t he order of Λ, diﬀerences in the structure factor would show up only at k >2π/Λ, which is about an order of magnitude larger than the wave numbers shown in Fig. 2. There, the plasm on peak has already a width of the order of the frequency and no longer describes a well-d eﬁned collective excitation. It is now interesting to compare the simulation results to th e theoretical approximations. The ﬁrst observation is that the simulation peaks are signiﬁ cantly broader, cf. Fig. 2. This is obvious since the simulations fully include interpartic le correlations missing in the mean- ﬁeld results. Consequently, the plasmon damping contains c ollisional damping in addition to the Landau damping (which is the only damping mechanism in the mean-ﬁeld models). Correspondingly, the plasmon peaks in the simulations are s hifted to smaller frequencies. This eﬀect grows with increasing wave number as well as with i ncreasing coupling (see also Fig. 5). We note that, in our simulations, this shift is obser ved for all wave numbers, which is in contrast to the result of Hansen [see Fig. 9 of Ref. [23] f orq= 0.6187]. In other words, the plasmon dispersion curves from the MD simulations are lo wer than the mean-ﬁeld result for all wave vectors /vectork, which is seen more clearly in Fig. 5. As expected the MD curve s for the structure factor are much closer to the RPA than to the Vla sov result. In Fig. 5 we plot the optical plasmon dispersion curves for th ree values of the coupling parameter for the Vlasov and RPA dispersions together with t he simulation results. We further show the well-known analytical approximation to th e Langmuir dispersion, 11ω(q) =ωpl/parenleftBigg 1 +q2 Γ/parenrightBigg1/2 . (13) Clearly, this predicts a monotonically increasing dispers ion. However, this approximation is valid only for k <1/rDand for Γ <1. Let us now consider the simulation results which do not have this limitation. In Fig. 5 we show the MD results for a Coulomb potential and for the Kelbg potential for three values of the degeneracy paramete r,ρΛ3= 0.1,0.5,1.0.One clearly sees that, for these parameters, the dispersion is positive ,dω(q)/dq > 0, up to wave numbers of the order of one over the mean interparticle distance. For larger q, the dispersion changes sign. This general trend is observed for the Coulomb and the K elbg potential. On the other hand, with increasing quantum eﬀects, ρΛ3, the deviations between the two potentials are growing, which becomes more pronounced as Γ increases, cf. t he curves for Γ = 1 and Γ = 4: the dispersion in case of the Kelbg potential shows a softer i ncrease with increasing wave number and reaches a lower maximum approximately at the same wave number as in the Coulomb case. We mention that this sign change of the dispers ion has not been reported by Hansen [1]. Comparing the simulations with the mean-ﬁeld results, we again see that the MD dispersions proceed lower than the mean ﬁeld results, and this eﬀect grows with increasing Γ and increasing wave number. Once more, we conﬁr m that the RPA dispersion is much closer to the MD result than the Vlasov dispersion, at le ast for Γ ≤0.5. [As mentioned above, the simulation results for the dispersion show certa in statistical ﬂuctuations due to the varying accuracy of the results for the diﬀerent wave num bers]. Let us now consider the plasmon damping more in detail. Fig. 6 , shows the damping (full width at half maximum of the plasmon peak of the structu re factor) as a function of wave number. It is interesting to compare with the familiar a nalytical expression from the Vlasov theory, e.g. [26], δ(κ) =/radicalbiggπ 8√ 1 + 3κ2 κ3e−1 2κ2−3 2 (14) where κ≡krDis the dimensionless wave number in units of the inverse the D ebye radius rDgiven in Table I. Formula (14) is derived under the condition that the damping is much 12smaller then the frequency [ δ(q)≪ω(q)], and is limited to small wave numbers κ≪1. As expected, the damping given by formula (14) which is only Lan dau damping, is much smaller than the damping found in the simulations, as the latter cont ain the full collisional damping also. Obviously, for small coupling and small q, Eq. (14) shows the correct trend. However, deviations increase rapidly with growing coupling paramet er. Furthermore, the simulations which are not limited to small wave numbers, show a qualitati vely diﬀerent behavior at largeq: a monotonic increase of the damping. Interesingly, with in creasing Γ the damping is reduced, cf. Figs. 6a,b. Finally, we try to extend the analytical result for the plasm on dispersion, Eq. (13), to larger Γ and to include quantum eﬀects. To this end, we used th e MD data with the Kelbg potential to construct an improved ﬁt of the form ω(q) =ωpl(1 +aq2+bq4)1/2. The result is shown in Fig. 7 for Γ = 1 and Γ = 4. Due to the large ﬂuctuations in the dispersion data and the increasing damping for large wave numbers, we us ed a weighted ﬁt where the smallest q−values had the largest weight and the statistical errors of t he individual points have been taken into account. Table II contains the resultin g ﬁt parameters. According to this data both parameters aandbdepend on Γ and ρΛ3. The parameter ais close to 1 /Γ in agreement with Eq. (13), but with increasing Γ, deviations a re growing, compare Table II. We see no systematic inﬂuence of quantum eﬀects on the parame terafor Γ = 1. Noticeable eﬀects show up for Γ = 4, where increased degeneracy leads to a reduction of the coeﬃcient a. The second ﬁt parameter allows to qualitatively reproduce the change of the sign of the dispersion. The overall agreement is satisfactory for wave numbers up to the inverse mean interparticle distance up to which the plasmons are compara tively weakly damped. V. DISCUSSION We have presented classical molecular dynamics simulation s of the dielectric properties of a one-component plasma at intermediate coupling and dege neracy, Γ ≤4 and ρΛ3≤1. While classical MD simulations can be extended to very large values of Γ, they have lim- 13ited applicablility to quantum plasmas. We used, as an eﬀect ive quantum pair potential, the Kelbg potential which correctly describes quantum diﬀr action eﬀects for small Γ. In general, we found that the simulation results for the dielec tric properties and the plasmon dispersion with the Coulomb and the Kelbg potential are rath er close, but start to deviate from each other as Γ increases. Nevertheless, the use of the K elbg potential is preferable. It correctly treats close collisions, i.e. the two-particle i nteraction at distances smaller than the DeBroglie wavelength. This is of even higher importance in t he case of two-component plas- mas where the Kelbg potential allows to avoid the collapse of oppositely charged particles. Therefore, the present investigation should be important f or future work on two-component plasmas. Finally, we mention that the Kelbg potential is onl y the ﬁrst term of a Γ expan- sion. Therefore, for Γ >1 the account of higher order corrections to the quantum diﬀr action eﬀects is necessary. Work on this subject is in progress. 14REFERENCES [1] J.P. Hansen, Phys. Rev. A 8, 3096 (1973) [2] J.M. Caillol, D. Levesque, J.J. Weis, and J.P. Hansen, J. Stat. Phys 28, 325 (1982) [3] J. Ortner, F. Schautz, and W. Ebeling, Phys. Rev. E 56, 1 (1997) [4] R.K. Moudgil, P.K. Ahluwalia, and K. Tankeshwar, Phys. R ev. B54, 8809 (1996) [5] W. Sch¨ ulke, K. H¨ oppner, and A. Kaprolat, Phys. Rev. B 54, 17464 (1996) [6] S.G. Brush, H.L. Sahlin, and E. Teller, J. Chem. Phys. 45, 2102 (1966) [7] W.L. Slattery, E.D. Dollen, and H.E.DeWitt, Phys. Rev. A 26, 2255 (1982) [8] S. Ogata, S. Ichimaru, Phys. Rev. A 36, 5451 (1987) [9] R.T. Farouki, and S. Hamaguchi, Phys. Rev. E 47, 4330 (1993) [10] S. Ichimaru, “Statistical Plasma Physics” Vol. II, Addison-Wesley Publishing Company, 1994; G.S. Stringfellow, H.E. DeWitt, and W.L. Slattery, Ph ys. Rev. A 41, 1105 (1990) [11] D. Pines, and Ph. Nozieres, “The Theory of Quantum Liqui ds”, Benjamin, New York 1966 [12] G.D. Mahan, “Many-Particles Physics”, Plenum Press, N ew York/ London 1990 [13] W.D. Kraeft, D. Kremp, W. Ebeling, and G. R¨ opke, “Quant um Statistics of Charged Particle Systems” (Plenum, London, New York, 1986) [14] M. Bonitz, “Quantum Kinetic Theory”, B.G. Teubner, Stu ttgart/Leipzig 1998 [15] N. Kwong, and M. Bonitz, Phys. Rev. Lett. 84, 1768 (2000) [16] See e.g. D. Klakow, C. Toepﬀer, and P.-G. Reinhard, Phys . Lett. A 192, 55 (1994); V.S. Filinov, J. Mol. Phys. 88, 1517, 1529 (1996) [17] G. Kelbg, Ann. Physik (Leipzig) 12, 219 (1963); 13, 354 (1964); 14, 394 (1964) 15[18] W.C. Swope, H.C. Andersen, P.H. Berens, and K.R. Wilson , J. Chem. Phys. 76, 637, (1982) [19] W. Ebeling, H.J. Hoﬀmann, and G. Kelbg, Contr. Plasma Ph ys.7, 233 (1967) [20] B.R.A. Nijboer, and F.W. De Wette, Physica XXIII , 309 (1957) [21] M.J.L. Sangster, and M. Dixon, Advances in Physics 25, 247 (1976) [22] C. Deutsch, Phys. Lett. A 60, 317 (1977) [23] J.P. Hansen, I.R. McDonald, and E.L. Pollock, Phys. Rev . A11, 1025 (1975) [24] A.A. Kugler, J. Stat. Phys. 8, 107 (1973) [25] G. Zwicknagel, PhD thesis, University of Erlangen 1994 [26] A.F. Aleksandrov, L.S. Bogdankievich, A.A. Rukhadze, “Principles of Plasma Electro- dynamics“, Springer, 1984 16TABLES TABLE I. Parameters of the molecular dynamics simulations w ith the Kelbg potential. Num- bers in parentheses refer the runs with Coulomb potential. ΓρΛ3ρ, [cm−3]T, [K] ωpl, [fs]−1rD/¯r N kmin¯r run time, [ Tpl] 0.5 0.1 9.12·10211.126 ·1055.387 0.816 500(250) 0.491(0.619) 515(341) 0.5 0.5 2.28·10233.292·10526.940 400(250) 0.529(0.619) 429(429) 1.0 0.1 1.14·10212.228·1041.905 0.577 250 0.619 290(327) 1.0 0.5 2.85·10228.23·1049.524 250 0.619 682(682) 1.0 1.0 1.14·10231.31·10519.048 250 0.619 477 4.0 0.1 1.78·10191.76·1030.238 0.289 250 0.619 570(227) 4.0 1.0 1.78·10218.17·1032.381 250 0.619 716 TABLE II. Fit parameters of the Langmuir dispersion curves s hown on Fig. 7. The ﬁt equation was taken in the form of ω(q)/ωpl= (1 + aq2+bq4)1/2. Parameters of the ﬁt for Γ = 1 and ρΛ3= 0.1 are less reliable, because of the absence of data for big w ave vectors, cf. Table I. Γ ρΛ3a b 1.0 0.1 1.013 ±0.031 -0.260 ±0.023 1.0 0.5 1.074 ±0.041 -0.288 ±0.013 1.0 1.0 0.975 ±0.055 -0.259 ±0.018 4.0 0.1 0.169 ±0.015 -0.034 ±0.006 4.0 1.0 0.121 ±0.007 -0.025 ±0.003 17FIGURES =0.5 =1.0 =4.00.20.40.60.81.0g(r) Kelbg;3= 0.5Kelbg;3= 0.1CoulombDH 0.20.40.60.8g(r) 0.0 0.6 1.2 1.8 2.4 3.0 r,r0.20.40.60.81.0g(r)0.0 0.15 0.3 0.45 0.60.120.240.360.48Kelbg;3=1.0 FIG. 1. Pair distribution functions for Γ= 0.5 (upper ﬁgure) , Γ= 1.0 (middle ﬁgure), 4.0 (lower ﬁgure), and ρΛ3= 0.1, 0.5, 1.0 for systems with Coulomb and Kelbg potential. Further, the Debye-H¨ uckel (DH) limit is shown (solid line). Line styles are the same in all three ﬁgures. Inset in the middle Fig. shows the inﬂuence of the degeneracy at sma ll distances. The result for Γ = 4.0 ,ρΛ3= 0.1 with Kelbg potential are not distinguishable from the C oulomb result and are not plotted. 18q=0.619 q=0.872 q=1.0720.10.20.30.40.5S(q,)RPAVlasovKelbgCoulomb 0.050.10.150.2 0.5 0.75 1.0 1.25 1.5 1.75 2.0 ,pl0.050.10.150.2 FIG. 2. Dynamical structure factor for an OCP at Γ = 1 and ρΛ3= 0.1 from MD simulations with Coulomb and Kelbg potentials. In addition, Vlasov and R PA results are shown. The wave numbers are shown in the ﬁgures in units of ¯ r, i.e.q=k¯r. 19q=0.491 q=0.6950.00.080.160.240.320.4S(q,)RPAVlasovKelbg 0.5 0.75 1.0 1.25 1.5 1.75 2.0 ,pl0.060.120.180.24 FIG. 3. Same as Fig. 2, but for Γ = 0.5 and ρΛ3= 0.1. The values of the wave numbers diﬀer from Fig. 1 due to the diﬀerent particle numbers, cf. Table I. q=0.529 q=0.7480.00.080.160.240.320.4S(q,)RPAVlasovKelbg 0.5 0.75 1.0 1.25 1.5 1.75 2.0 ,pl0.060.120.180.24 FIG. 4. Same as Fig. 3, but for ρΛ3= 0.5. 20=4.0 =1.0 =0.50.42 0.84 1.26 1.68 2.11.01.11.2(q),pl eq.(13)RPA ;3=0.1Kelbg;3=1.0Coulomb 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.00.881.11.321.54(q),pl Vlasoveq. (13)RPA ;3=0.5Kelbg;3=1.0Coulomb 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 q,1/ r1.21.41.6(q),pl eq.(13)RPA ;3= 0.5Kelbg;3= 0.5Coulomb FIG. 5. Optical plasmon dispersion for various coupling and degeneracy parameters from MD simulations with Coulomb and quantum potentials. Also show n are results of the Vlasov and RPA approximations, and of the analytical approximation of Eq. (13). For Γ = 4.0 and ρΛ3= 0.1 (upper graph) the MD simulations with Kelbg potential and th e RPA curve are not shown since they almost coincide with the Coulomb simulation and the Vla sov curve, respectively. 21=4.0 =1.0a b0.0 0.45 0.9 1.35 1.8 2.250.00.40.81.21.6(q),pl 0.0 0.3 0.6 0.9 1.2 1.5 q,1/r0.00.20.40.60.81.01.21.4(q),pl eq.(14)Kelbg;3=1.0Kelbg;3=0.5Kelbg;3=0.1 FIG. 6. Damping of Langmuir waves from MD simulations with th e Kelbg potential for various values of Γ and ρΛ3. Solid lines are the analytical small damping limit of the Vl asov theory, Eq. (14). 22=4.0 =1.00.42 0.84 1.26 1.68 2.11.01.041.081.121.161.2(q),pl®t;3=1.0®t;3=0.1Kelbg;3=1.0Kelbg;3=0.1 0.4 0.8 1.2 1.6 2.0 q,1/ r0.91.01.11.21.31.41.5(q),pl ®t,3=1.0®t,3=0.5®t,3=0.1Kelbg;3=1.0Kelbg;3=0.5Kelbg;3=0.1 FIG. 7. Dispersion of Langmuir oscillations from MD simulat ions with the Kelbg potential for various values of the coupling and degeneracy. Symbols are M D results, lines the best ﬁts to the low wave number part ( q <1/¯r), the ﬁt formula and parameters are given in Table II. 23
arXiv:physics/0012028v1 [physics.optics] 14 Dec 2000Anisotropic charge displacement supporting isolated phot orefractive optical needles Eugenio DelRe Advanced Research, Pirelli Cavi e Sistemi, Viale Sarca 222, 20126 Milan, Italy Alessandro Ciattoni Dipartimento di Fisica, Universit `adell’Aquila, and INFM, Unit `adi Roma 1, Rome, Italy Aharon J. Agranat Department of Applied Physics, Hebrew University of Jerusa lem, Jerusalem 91904, Israel (July 24, 2013) The strong asymmetry in charge distribution supporting a si ngle non-interacting spatial needle soliton in a paraelectric photorefractive is directly obse rved by means of electroholographic readout. Whereas in trapping conditions a quasi-circular wave is sup ported, the underlying double-dipolar structure can be made to support two distinct propagation mo des. Far from being a peculiarity of low dimensional sys- tems, solitary waves and solitons have been widely doc- umented in bulk three-dimensional environments [1]. In biased photorefractives, nonlinear visible optical waves have been shown to undergo self-trapping both as ex- tended one-dimensional waves, in the form of slab- solitons [2], and as conﬁned two-dimensional spatial beams, needle-solitons [3]. These are self-funneled micron-sized beams of light that propagate through the bulk dielectric without suﬀering diﬀraction or distortion . Needles, in their richer higher-dimensional environment, have led to a substantial advance in our phenomenolog- ical investigation of nonlinear dynamics, expanding the scope of possible soliton-based applications [4] [5]. Whereas both slabs and needles emerge in the same physical system, a biased photorefractive sample, their underlying nonlinear nature is rather diﬀerent [6]. For slabs, the entire physical system, and thus, consequently, the optical nonlinearity, depends only on the transverse beam direction along which the external ﬁeld is applied (say the x direction), whereas the system is fully invariant for spatial translations in the second orthogonal trans- verse direction y. This reduces slab soliton description to that associated with a saturated Kerr-like nonlinearity [7]. For needles, on the contrary, the higher dimension- ality of the optical beam, whose quasi circular symmetry suggests an isotropic self-action [3], is inherently at odd s with the screening nonlinearity, whose one basic driv- ing mechanism is the x directed external bias ﬁeld. A simpliﬁed description of needles, tracing the steps that lead to a local Kerr-like understanding of slabs, is simply not possible [8]. Given the complexity of the higher- dimensional interaction, the theoretical interpretation of needles is largely based on numerical integration. What emerges is a picture in which nonlocal nonlinear eﬀects [9], as opposed to local conventional paradigm Kerr-like phenomenology, play a central role. An understanding ofthese requires an explicit distinction between the under- lying space-charge ﬁeld distribution Esc, which mediates self-action, and the propagating light ﬁeld Eopt. The space-charge distribution simply does not have a local relationship to the optical ﬁeld [8]. The numerical so- lution of the full boundary-value problem indicates that the highly anisotropic screening conﬁguration allows the formation of needles only through an equally anisotropic local space-charge, characterized by the appearance of two distinct lateral ﬁeld lobes in the x direction, absent in the second transverse direction y [6] [9]. This dou- ble dipolar ﬁeld distribution induces, as a consequence, a complicated needle supporting index pattern that has little to do with a mere self-written graded-index waveg- uide (excluding the possibility of a simple linear interpre - tation [10]). For system parameters far from the soliton supporting conﬁguration, this anisotropy leads to an ob- servable asymmetric beam distortion, but the question naturally arises as to how these lobes manifest their ex- istence when the optical beam is actually a needle-like solitary wave. Repulsion of mutually incoherent needles provides in- direct evidence of the lobe-like charge distribution [11]. However, no direct experimental evidence of charge anisotropy has yet been reported. The main reason lies in the fact that photorefractive solitons are generally ob- served in ferroelectric samples. In these crystals there is no direct way of isolating the contribution of charge displacement from the ﬁnal guiding structure. Read- out with non-photorefractively active light can lead to no substantial increase in knowledge on the underlying charge pattern, short of performing precise bulk inter- ferograms or far ﬁeld soliton transforms [9]. Direct in- vestigation of the space-charge residue with a probe is furthermore hampered by the fact that the lobes are ac- tually antiguiding [6] [9]. In this Letter we give direct evidence of this nonlocal 1ﬁeld structure. This is made possible by the quadratic electro-optic response of paraelectrics, that allows the electro-holographic separation of optical phenomenology from the underlying space-charge ﬁeld [5]. Experiments are carried out in a sample of photore- fractive 3.7xx4.7yx2.4zmm KLTN (potassium-lithium- tantalate-niobate) [12] , biased along the x axis (of size L=3.7mm), and kept at a constant temperature T=20◦C. The x-polarized cw TEM 00λ=532nm beam from a diode- pumped doubled NdYag laser is focused on the input facet of the sample and launched along the z axis. As it propagates in the sample, it diﬀracts, passing from an initial intensity I= \|Eopt\|2full-width-half-maximum (FWHM) in the x and y directions ∆ x∼=∆y∼=10µmto a broadened intensity distribution of ∆ x∼=∆y∼=20µm (see Fig1.(a),(b)). The application of the external con- stant bias V on the x electrodes makes photoexcited free charges drift, leading to an inhomogeneous ﬁeld screen- ing. The electro-optic response of the paraelectric sam- ple is ∆ n=−(1/2)n3g11ǫ2 0(ǫr−1)2(V/L)2(E/(V/L))2≡ −∆n0E2, where n∼=2.4 is the zero-ﬁeld index of refrac- tion,g11≡gxxxx∼=0.12m4C−2is the dominant com- ponent of the quadratic electro-optic tensor g ijkl(and thus tensorial eﬀects are neglected), ǫ0is the vacuum dielectric constant, ǫr∼=9·103(at T ∼=20◦C) is the rela- tive sample low frequency dielectric constant, Eis the x component of the electric ﬁeld resulting from screening, ∆n0∼=2.8·10−4, and E ≡E/(V/L). The spatially mod- ulated index distribution allows needle formation (see Fig.1(c)). The needle, that shows a slight anisotropy in the output intensity distribution, is trapped and stable in time for an external bias voltage of V=0.85kV and a ratio of peak intensity I pto the dark artiﬁcial illumina- tion I b(obtained by illuminating the sample with a co- propagating y polarized plane wave of equal wavelength) of u2 0≡Ip/Ib∼=2.6. Annulling the externally applied voltage V, i.e., setting V=0, gives an index modulation ∆nV=0=−∆n0E2 sc,onlydue to the charge displace- ment, where evidently Esc≡ E-1. The resulting index pattern has a guiding structure for regions in which Esc passes through a minimum . Given that the lobes repre- sent an excess of screening in the x direction [6] [9], there are two points, i.e., x 1and x 2, along the x axis, located to the left and right of the needle peak, in which ∆ nV=0 forms a guiding ”hump”. Along the y axis, this hump will follow the shape of the lobe. In order to investigate ∆ nV=0without modifying the space-charge distribution, we launch into the sample the same beam leading to the needle, but attenuated so as to have a much lower intensity. This guarantees that the characteristic time scale of charge displacement induced by the probe, τd, is much longer than any characteristic observation time. For typical µW intensity beams, τd∼1 min. Results, shown in Fig.1(d), clearly indicate the anisotropic lobe structure in the form of a split diﬀrac-tion pattern in the x direction. The slight asymmetry in the diﬀraction pattern is a consequence of needle self- bending, that inevitably distorts the diﬀractive read-out phase. x profile 0 20 40 60 80/c109m0 20 40 60 80 /c109m 0 20 40 60 80/c109m0 20 40 60 80/c109m 0 20 40 60 80/c109m0 20 40 60 80/c109m Fig.2□DelRe□et□al.x□profile y□profile (a) (c)(b) (d)20 m/c109 xy 0 20 40 60 80/c109m0 20 40 60 80/c109m FIG. 1. Electroholography of a single photorefractive nee- dle. (a) Image and proﬁles of input transverse intensity dis - tribution; (b) Linear diﬀraction with nonlinear charge sep - aration turned oﬀ (V=0); (c) Self-trapping distribution fo r V=0.85kV; (d) Read-out for V=0. A similar phenomenology has been observed for tran- sient quasi-steady-state needles, where I b=0, blocking beam evolution in the trapped regime, i.e., before the needle has decayed. The two light lobes are a signature of the lobes pre- dicted by numerical integration of the full Kukhtarev model and constitute direct proof that the nonlinear- ity supporting needle trapping in biased photorefrac- tives is not the saturated Kerr-like ∆ n∝1/(1 +I/Ib)2 that allows slab formation. More precisely, whereas the lobes are notpresent in the slab case (and are not merely ”negligible”), they play a fundamental role in needle trapping [13]. Although needles have been doc- umented in various conditions, it is legitimate to ask whether the nonlocal space-charge ﬁeld structure, and thus index modulation, can actually support circular- symmetric solitary waves. The mathematical answer is no [14]. However, the anisotropic space-charge struc- turecansupport waves that are to all practical purposes circular-symmetric. For the conditions investigated ex- 2perimentally, we ﬁnd the space-charge distribution by solving the simpliﬁed associated electro-static problem, i.e.,∇·[(I+Ib)E+(KbT/q)∇I]=0, where E[8] is the in- ternal electric ﬁeld vector, assuming a given Gaussian in- tensity distribution. The resulting index pattern is shown in Fig.(2a). Propagating the very same ﬁeld distribu- tionEopt(whose intensity is I, shown in Fig.(2b)) into this pattern, gives results shown in Fig.(2c)-(2d). The intensity pattern does not suﬀer discernible distortion. This means that the exact nonlinear behaviour is well described by this approximate linear approach, and thus we can conclude that quasi-circular needles can be sup- ported by the anisotropic pattern. (a) (c) (d)(b) /c68n(x,y)0 -2.5□10 -5□10. .-4 -4 40 -40040 -400 40-40 0y( m)/c109 x( m)/c1093 2 01 20 -20020-200 y( m)/c109 x( m)/c109I/Ib 3 2 01 20 -20020-200 y( m)/c109 x( m)/c109I/Ib3 2 01 20 -20020-200 y( m)/c109 x( m)/c109I/Ib FIG. 2. Self-consistency of a needle solitons trapped in the anisotropic nonlinear index pattern. (a) Anisotropic inde x pattern; (b) input intensity distribution; (c)-(d) Intens ity af- ter 2mm and 4.5mm propagation, respectively, for the exper- imental situation described above. One basic consequence of these ﬁndings is that the anisotropy underlying a photorefractive needle leads not to one, but to threespatially separated index structures, that can be made to alternatively guide light depending on the applied external voltage in the read-out phase. This would not have been possible had the nonlinear re- sponse been local, as in the one-dimensional case [5]. The electroholographic read-out would have implied a tran- sition from a localized single mode structure (the nee- dle) to a delocalized ”doughnut-like” guiding pattern. To demonstrate this, we investigate the guiding capabilities at V=0. We were able to show the two guided modes launching, in sequence, the probe beam into one of the two lateral guiding humps of the ∆ nV=0pattern, i.e., in x1andx2. Results are shown in Fig.3. We did not ob- serve any directional coupling between the modes, this clearly being a consequence both of the distance between the humps, the probe wavelength, propagation length, and the presence of the antiguiding central pattern.Fig.3□DelRe□et□al. xy (d)(c)(b)(a) (b)(c) (d) 0 20 40 60 80/c109m 40 m/c1090 20 40 60 80 /c109m FIG. 3. Double hump guiding structure. (a) Two diﬀerent input beams, shifted by approximately ±10µm; (b-c) Guided beam in the two humps; (d) Linear diﬀraction for unshifted beam. The work of E.D. was partially carried out during pre- vious activity at Fondazione Ugo Bordoni. The work of A.C. was funded by the Istituto Nazionale Fisica della Materia through the PAIS2000 SESBOM project. Re- search carried out by A.J.A. is supported by a grant from the Ministry of Science of the State of Israel. [1] G.I. Stegeman and M. Segev, Science 286, 1518 (1999); M. Segev and M. Stegeman, Phys. Today 51, 42 (1998). [2] M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys.Rev.Lett. 68, 923 (1992). [3] M. Shih, M. Segev, G.C. Valley, G. Salamo, B. Crosig- nani, P. Di Porto, Electron. Lett. 31, 826 (1995). [4] S. Lan, M. Shih, G. Mizell, J. A. Giordmaine, Z. Chen, C. Anastassiou, J. Martin, and M. Segev, Opt. Lett. 24, 1145 (1999); S. Lan, E. DelRe, Z.G. Chen, M.F. Shih, M. Segev, Opt. Lett. 24, 475 (1999). [5] E. DelRe, M. Tamburrini, A.J. Agranat, Opt. Lett. 25, 963 (2000). [6] A.A. Zozulya and D.Z. Anderson, Phys.Rev. A 51, 1520 (1995). [7] M. Segev, M. Shih, G.C. Valley, J.Opt.Soc.Am.B 13, 706 (1996). [8] B. Crosignani, P. Di Porto, A. Degasperis, M. Segev, and S. Trillo, J.Opt.Soc.Am.B 14, 3078 (1997). [9] C.M. Gomez Sarabia, P.A. Marquez Aguilar, J.J. Sanchez Mondragon, S. Stepanov, and V. Vysloukh, J.Opt.Soc.Am.B 13, 2767 (1996). [10] A.W. Snyder, D.J. Mitchell, L. Poladian, and F. Ladouceur, Opt.Lett. 16, 21 (1991). [11] W. Krolikowski, M. Saﬀman, B. Luther-Davies, and C. Denz, Phys.Rev.Lett. 80, 3240 (1998). [12] A.J. Agranat, R. Hofmeister, and A. Yariv, Opt. Lett. 17, 713 (1992). [13] There is no fundamental diﬀerence between anisotropy in paraelectrics and ferroelectrics. [14] M. Saﬀman and A.A. Zozulya, Opt.Lett. 23, 1579 (1998). 3
arXiv:physics/0012029v1 [physics.class-ph] 14 Dec 2000Do we need to recourse to Amp` ere-Neumann electrodynamics t o explain wire fragmentation in the solid state? A. Lukyanov, S. Molokov Coventry University, School of Mathematical and Informati on Sciences, Priory Street, Coventry CV1 5FB, U.K. Abstract Exploding wires are widely used in many experimental setups and pulsed power systems. However, many aspects of the process of wire fragme ntation still remain unclear. If the current density is not too high, the wire may b reak up in the solid state. The experiments have shown that the wires break in ten sion due to longitudinal forces of unknown nature. In a series of papers, see [2, 3, 4, 5] Graneau argued that neit her mechanical vi- brations induced by the electromagnetic pinch force nor the rmal expansion could have been responsible for the wire disintegration because they w ere too weak. To explain the phenomenon he appealed to the obsolete Amp` ere force law as opposed to the con- ventional Biot-Savart force law. Graneau argued that the Am p` ere force law would lead to a longitudinal tension in the wire, although his calc ulations may have been in error on this point. Therefore, Graneau’s explanations i nduced a controversy in electrodynamics with a number of authors arguing proandcon. Previous theoretical and numerical investigations served to provide a search for these forces without recourse to unconventional electrody namics have identiﬁed the pinch eﬀect and thermal expansion as a source of strong longi tudinal vibrations. But the tensile stress component has been proved to be present on ly in the wires having free ends. Thus the mechanism doesn’t give a satisfactory explan ation of the phenomenon in the wires with clumped ends. In this investigation, we use a simpliﬁed magneto- thermo-elastic model to study ﬂexural vibrations induced b y high pulsed currents in wires with clumped ends on account of their role in the disint egration process. Several aspects are studied, namely (i) the buckling instability du e to simultaneous action of the thermal expansion and the magnetic force, (ii) the ﬂexur al vibrations induced in initially bent wires. It is shown that the induced ﬂexural vi brations are strong enough to lead to the breaking of the wire in a wide range of parameter s. 1 Introduction The phenomenon of wire fragmentation in the solid state by hi gh pulsed currents was studied experimentally by Nasilowski [1], and Graneau [2, 3, 4]. The y observed that a suﬃciently strong electric current would shatter a thin metal wire unde r a broad range of conditions (various wire material and geometry, diﬀerent current type s, etc.). As a result of the explo- sion the wires fragmented into 2-100 pieces with apparent si gns of longitudinal tensile stress. 1The experimental results, as well as (sometimes controvers ial) early attempts to explain the phenomenon have been reviewed by Graneau [5] , Hong [6] and Mo lokov&Allen [7]. It is obvious that an electric current passing through a meta l wire induces stress waves. Their origins are in (i) the thermal expansion owing to the vo lumetric Joule heating, and (ii) the Lorentz-, or pinch-, force. These forces depend on the ma gnitude and the rate of change of the current passing through the wire. They grow as either o f these parameters increases. Ternan [8] suggested that in Graneau’s experiments with wir es with free ends standing stress waves may be induced as a result of thermal expansion. Using a simple one-dimensional model he showed that the resulting stresses were suﬃcient to lead to a fracture. This mechanism of fragmentation of wires with free ends has b een explored in more detail by Molokov&Allen in [7]. They employed a magneto-thermo-el astic model of the stress-wave propagation, and solved the problem numerically on assumpt ion of axisymmetric nature of vibrations . Their conclusion was that the characteristi c value of both compressive and tensile longitudinal stresses obtained for the aluminium w ire with radius a= 0.6 mm carrying current I= 5 kA and having free ends (the parameters relevant to the exp eriment in [3]) would be about 33 MPa per unit length of the wire. The stress ha s been shown to grow linearly with the wire length and thus, for suﬃciently long w ires, the resultant stress exceeds the ultimate strength of aluminium. This is 75 MPa at 100◦C going down to 17 MPa at 320◦C [9]. Although the mechanism does give large value of tensile stre sses, it cannot explain the fragmentation of wires with clamped ends (as in the experime nts [1, 4]) because within the axisymmetric model in this case the stresses can be only comp ressive [7]. It is clear, however, that in the wires with clamped ends ﬂexural vibrations may be induced. The apparent signs of these modes have been observed in the Graneau’s experimen ts on the wires with ﬁrmly clamped ends. The resultant wire shape, after breaking, cle arly indicated on that, see photographs in [4]. In this paper we carry out a special study of ﬂexural vibratio ns to account for their role in fragmentation of wires with clamped ends. In section 2, we present the formulation of 2the problem and discuss simpliﬁcations of the model. In sect ion 3, the linear stability of the system is considered. In section 4, we carry out a qualitativ e analysis of expected stresses. In section 5, numerical algorithm and results are presented . 2 Formulation Consider a wire which is ﬁrmly clamped between two points loc ated along the z-axis. Let the distance between these points be L. We restrict our attention to the case of plane motion, and denote the deﬂection of the wire from the axis by Xwhich is a function of zand time t. The deﬂection of the wire in the initial undeformed state is denoted by X0. Possible three-dimensional modes, not considered here, may only inc rease stresses, especially at the clamped ends. Our theoretical consideration is based on a simpliﬁed model valid for suﬃciently long or suﬃciently thin wires. In this model the deﬂection Xof the wire is assumed to be much lower than the wire length, i.e. \|X\|/L≪1. Since large deﬂections of the order of the wire length are not expected in the wire explosion experiments, t his restriction is not expected to be very stringent. Another small parameter used in the model is the ratio a/L << 1, where ais the wire radius. This condition is usually well satisﬁed. Plastic deformations are not considered here although they may be important in the vicini ty of the melting point. Thus all deformations are assumed to be elastic. Then the deﬂection Xobeys the following equation ([10]): ρS∂2X ∂t2+E∂2 ∂z2J∂2X ∂z2−E∂2 ∂z2J∂2X0 ∂z2−G∂2X ∂z2−Fx−∂Cy ∂z= 0 (1) where ρis the material density, Sis the wire cross-section which may be a function of z, Jis the eﬀective moment of inertia of the wire cross-section ( for a circular cross-section J=πa4/4,where ais the wire radius), Eis the Young modulus, Gis the force along the z axis applied at the wire ends, Fxis the distributed external force per unit length along the xdirection and Cyis the distributed moment of force along the ydirection per unit length. For the physical situation considered here, the force Gcan be represented as 3G=ES{(˜l−˜l0)/L−α[T(t)−T0]}, The second term is a compressive force due to the thermal expa nsion, αbeing the linear ex- pansion coeﬃcient. The ﬁrst, nonlinear term is a retarding f orce due to the wire deformation from the initial state, where ˜l−˜l0=1 2/integraldisplayL 0/parenleftBigg∂X ∂z/parenrightBigg2 −/parenleftBigg∂X0 ∂z/parenrightBigg2 dz (2) is the increment of the wire length due to the deﬂection. When X=X0,˜l=˜l0and the retarding force equals to zero, as required. The boundary conditions are those for the clamped ends, name ly X=∂X ∂z= 0 at z= 0 and z=L. (3) The initial conditions are X=X0t= 0. (4) 2.1 The magnetic force In long, thin wires under consideration a/L << 1, uniform current is directed perpendicular to the cross-section. Thus any possible forces of magnetic n ature act only in the cross-section plane. These forces create no moment and thus Cy= 0 in our case. The net distributed force Fcan be calculated directly from F=/integraldisplay j×BdS. (5) The integral in (5) is taken over the wire cross-section area . The magnetic ﬁeld is given by the Biot-Savart law B=µ0 4π/integraldisplayj×R R3dV′, (6) where dV′is the volume element on the wire, jis the current density, and the vector R=r−r′ is directed to the ﬁeld point as usual. To calculate the xcomponent of the force, Fx, one can split the integral (6) over the wire length z′into two parts B=Bin+Bex. The ﬁrst part of the integral Binis taken over the 4range \|z′−z\|<∆, where a << ∆<< ξ L,ξLbeing the characteristic length scale of the function X(z). The second part Bexis taken over the rest of the wire length ∆ <\|z′−z\|< L, whereby the volume integral reduces to the integration over the wire length only. That is, Bex=µ0 4π/integraldisplayI×R R3dl′, where Iis the total current. The ﬁrst integral can be evaluated in ge neral with the assump- tion of a uniform current distribution and in the approximat iona << ξ L,\|X\|/ξL<<1. The ﬁrst inequality has been implicitly introduced in fact when we set the length ∆, the second inequality just represents the fact that the wire is slightl y deﬂected from a straight line. The ﬁrst integration asymptotically gives the expression for t he net force as follows {Fx}in=−µ0I2 4π∂2X ∂z2{ln(2∆ /a)−3/4}+O((X/ξ L)2,∆/ξL,(a/∆)2). (7) That is the force is proportional to the local wire curvature [11]. The second part cannot be integrated in a general case and one needs to make certain ass umptions about the wire form and thus about the function X(z) itself. In the case of an inﬁnitely long wire with periodic lateral perturbations X=/tildewiderXcos(kz),k= 2π/ξL, one gets {Fx}ex=−µ0I2 4π∂2X ∂z2{−C+ 1/2−ln(k∆)}+O(k∆,(X/ξ L)2,(a/∆)2), (8) where C≃0.577 is the Eiler’s constant and we have substituted Xk2=−∂2X ∂z2. Combining two parts we obtain Fx=−µ0I2 4π∂2X ∂z2{ln(2/ka)−C−1/4}. (9) Note, that the expression (9) is identical to the formula obt ained in [12] in a particular case of small periodic deﬂections \|X\|<< a. It is clear that the assumption of an inﬁnitely long wire to ca lculate the force, Fx, is not very rigorous, since in experiments the circuit is always cl osed. This implies that in addition to the magnetic force by the wire current itself, there may be a component generated by the currents ﬂowing in the external circuit. This eﬀect can b e taken into consideration by substituting an appropriate circuit form, in other words th e function X(z) in the second integral. 5Alternatively, this force can be obtained by means of the vir tual work principle. To estimate how strong the eﬀect could be, we will use a general e xpression for the inductance of a closed circuit Λ =µ02πLsln(Ls/as), where Lsis the circuit characteristic length and asis the radius of the circuit wire, see [13]. Then varying the cir cuit length Lsone can calculate the subsequent variation of the associated magnetic energy Em=1 2ΛI2. Then, equalising the net force acting on the circuit FNwith the variation of the energy we get FNδLs=µ0 4πI2(ln(Ls/as) + 1)δLs. (10) An average force acting on a unit length is f=FN/L. Then, for the average force from (10) we get f≃µ 4π0I2(ln(Ls/as) + 1)/Ls. The ﬁrst, logarithmic part of the net force represents contribution from the end eﬀects, see (7). The se cond part gives the integrated force from the circuit as a whole. If L << L s, one can assume that the force is constant throughout the wire length in the ﬁrst approximation. Then, this ”constant” force acting on a wire of length Lwill create longitudinal stress σ/bardbl=µ2 0I4 60480ln(Ls/as)2 E(πa2)4L6 L2 s, see the problem in [10] on page 93. For a copper wire with L= 30 cm, a= 0.6 mm and the circuit with I= 5 kA, Ls= 20 m and as= 1 cm the stress is amounting to σ/bardbl≃8.4 MPa. This is only an estimation, and the eﬀect needs special analysis since th e geometry of external circuits is usually unknown. Moreover, using a symmetrical circuit wit h two loops from both sides of the wire one can completely compensate the external magneti c force. We leave this question for further detailed investigation a nd neglect the contribution from the external circuit in our analysis. Further we will use for mula (9). 2.2 The temperature behaviour The temperature behaviour with time is governed by the passi ng current. It can be calculated according to the direct Joule heating of the wire material fr om ρcv∂T ∂t=j2/σ, (11) neglecting by the process of thermal conductivity. This ass umption would hold for thermally isolated wires with uniform current distribution. In expre ssion (11), jis the current density, 6ρandσare the density and the electrical conductivity of the metal ,cvis the speciﬁc heat. From the above equation using the well-known inverse depend ence of the conductivity on temperature σ=σ0T0Tone can obtain T=T0exp(γT/integraldisplayt 0f(t′)2dt′), where γT=j2 0σ0T0ρcvis the characteristic temperature rise-time, j0is maximum current density. The function f(t) deﬁnes current rise time proﬁle. 3 Linear stability To investigate possible instabilities in the system, we ﬁrs t carry out a linear stability analysis assuming for a moment that all the parameters, such as curren t and wire temperature, are constants independent of time. Even though, in a real situat ion, they vary quite fast with time and numerical methods must be involved to solve the syst em, our simpliﬁed analysis will form a basis for qualitative interpretation and unders tanding of the results. 3.1 Stability of an initially straight wire Consider an initially straight wire by letting X0= 0. General analysis of the system (1) shows that two basic types of buckling instabilities may dev elop. They are due to thermal expansion, expressed by the force Gapplied at the wire ends [10], and the magneto-elastic buckling instability [14] expressed by force Fx. In both cases, the instability has a threshold character. When either the force Gor the current Iexceed some critical value, new stable states appear. The initially stable state X= 0 then becomes unstable and buckling occurs. We are looking for non-trivial stationary solutions of the l inearized system (1), (3), (4). They are given by the equation EJXIV−GX′′−Fx= 0, (12) with the boundary conditions X=X′= 0 at z= 0 and z=L. (13) 7Here, the temperature, T, and the total current, I, are assumed to be independent of time, and so are the functions G=−αES[T−T0] and Fx=−µ0I2 4πX′′{ln(2/ka)−C−1/4}. To obtain a solution to (12) it is convinient to use a complete set of orthogonal functions {Xi}, i≥1 deﬁned by the problem XIV i+λ2 iX′′ i= 0 X′ i(0) = X′ i(L) =Xi(0) = Xi(L) = 0(14) with the orthogonality given by /integraltextL 0X′′ iX′′ kdz=  0, i /ne}ationslash=k λ4 i2L, i =k /integraltextL 0X′ iX′ kdz=  0, i /ne}ationslash=k λ2 i2L, i =k(15) This set of functions is commonly used in the theory of stabil ity and buckling of elastic columns [15]. The eigenvalues of the boundary-value problem (14) satisfy the dispersion equation 1−cos(λiL) =λiL 2sin(λiL) (16) So, one can see that the set of functions consists of two subse ts. The ﬁrst subset is deﬁned by the eigenvalues given by cos(λiL) = 1 sin(λiL) = 0.(17) That is λi=π·(i+ 1)/L, i = 1,3,5. . .. In this case the associated eigenfunctions are given by Xi= cos( λiz)−1. (18) And the second subset is deﬁned by cos(λiL) =4−(λiL)2 4 + (λiL)2 sin(λiL) =4λiL 4 + (λiL)2.(19) 8The associated eigenfunctions are given by Xi= cos( λiz)−1 + 2z−2 λiLsin(λiz). (20) The ﬁrst eigenvalue of this subset is equal to λ2= 8.986819 /L. Further eigenvalues are approximately given by λi≃π·(i+ 1)/L, i = 4,6,8. . .. Now, with the help of (15) one can obtain from (12) a criterion when the instability ﬁrst appears α(T−T0)ES−µ0 4πI2(ln(λ1a) + 0.14)> EJλ2 1. (21) The ﬁrst term on the left hand side of (21) is responsible for t he thermal expansion eﬀect, while the second term represents magneto-elastic buckling . It should be noted that while we are considering both eﬀects simultaneously, for paramet ers relevant to wire explosion experiments these terms have diﬀerent orders of magnitude. The criterion obtained shows when the ﬁrst eigenmode of (1) a nd the system as a whole become unstable, λ1= 2π/Lbeing the corresponding eigenvalue. In a similar manner, substituting other eigenvalues λiforλ1, one can obtain respective criteria for higher modes as well. The set (14) being very usefull in the linear stability analy sis of system (1) doesn’t seem to be very relevant to study the increments of the instabilit y since /integraldisplayL 0XiXkdz/ne}ationslash= 0, i/ne}ationslash=k. But, it would be interesting to obtain an estimate of the incr ements of the instability. For this purpose, we will substitute a solution to (1) in the form X=Ai(t)Xi, i.e assuming that only one mode is present. Then, for the increment γi, (Ai(t)∼exp(γit)), one gets γ2 i=E ρλ2 i 2L /integraldisplayL 0X2 idz/braceleftbigg Ξ−λ2 iJ S/bracerightbigg (22) Ξ =α(T−T0)−µ0I2 4πES{0.14 + ln( aλi)} or since/integraltextL 0X2 idz≃L γi≃/radicalBigg E ρλi{Ξ−λ2 iJ S}1/2(23) 9As is seen from (23), the increment has a maximum at λi=λextdeﬁned by the equation λ2 ext=S Jα(T−T0) 2−µ0I2 8πEJ{0.64 + ln( aλext)}. (24) As we will see further, for the range of parameters used in the wire explosion experiments, the contribution from the terms due to magnetic force can be n eglected during the initial stage and within this approximation, using explicitly J=πa4/4 and S=πa2, λext=/radicalBigg 2α(T−T0) a2. (25) And, since λi≃π·(i+ 1)/L, iext=/radicalBigg 2α(T−T0) a2L π−1. (26) On the other hand, the maximal λlimat which instability may exist is obtained from (23) to give λlim≃2/radicalBigg α(T−T0) a2(27) with the same accuracy. From (27), since λi≃π·(i+ 1)/L, one gets for the maximal mode number ilim≃2/radicalBigg α(T−T0) a2L π−1 (28) Thus the dominant mode lies somewhere between the ﬁrst unsta ble mode and the last one with the maximal increment γext=/radicalBigg E ρα(T−T0) a. (29) From the expression (26) one can see that higher modes are lik ely to become dominant in the instability spectrum with increase in the wire length . As a result, the wire shape can take a rather intricate form. It seems that the instability o n higher modes has been observed experimentally by Graneau, [4]. At the experimental condit ions he used, i.e. aluminium wire, L= 1 m, a= 0.6 mm, from (25) one gets λext≃294 or the corresponding mode number iext= 93 at T=Tmelt. The photographs of the wire shape after the current had been switched oﬀ showed clearly the appearance of at least 5 b ulges on a small part of the wire. 10We will refer further to the sets of parameters presented in T able I corresponding to diﬀerent wire lengths and diﬀerent currents. One should not e that two sets, [B] and [E], are relevant for wires and currents used in the experiments [4] a nd [1] respectively. Table.I material a L I P1 P2 P3 Aaluminium 0.6 mm 0.05 m 5 kA 1.19×1036.11 114 Baluminium 0.6 mm 1 m 5 kA 1.19×10313.6 0.28 Caluminium 0.6 mm 1 m 2 kA 1.19×1032.18 0.28 Daluminium 0.6 mm 0.3 m 8 kA 1.19×10334.8 0.28 Ecopper 0.5 mm 1 m 500 A 1.84×1030.14 0.24 For all the represented sets, the criterion (21) is fulﬁlled well enough. Indeed, we have put in the Table I the corresponding values of the terms in (21 ), designating them as P1, P2andP3from left to right. The estimations have been done at T=Tmelt,Tmeltbeing the melting temperature; Tmelt= 660◦C for aluminium and Tmelt= 1085◦C for copper. The characteristic increment (29) at the same conditions, i n the case [B] for instance, is γmax≃105sec−1, which is much greater than the temperature increment γT≃7×102sec−1. Thus, even though the estimation has been done in the linear a pproximation, it is clear that the instability has suﬃcient time to develop before the wire reaches the melting point. In all the cases, as is seen, the major contribution is from th e thermal expansion eﬀect, while the inﬂuence of the magnetic-force terms can be neglec ted during the initial stage (this, of course, can be estimated directly from (1) as well). Two te rms,P1andP2, for instance in case [B], become equal only at T−T0= 6.6◦C. The time it takes for the temperature to be driven through this range is ∆ t≃30µsec. The characteristic time for the instability to develop for this temperature diﬀerence is much longer, γ−1 max= 800 µsec. 3.2 Stability of an initially bent wire Consider a wire which has an initial form X0=/summationtext iA0 iXi, where the functions {Xi}is the set (14) and A0 iare given weight coeﬃcients. One needs to stress, that with t he accuracy the system (1) was derived, \|X0\|> amust always be the case. Performing analysis similar 11to that leading to equation (21) and neglecting the magnetic force for the sake of simplicity, for a mode kone gets λ2 k(Ak−A0 k)EJ−AkES(α∆T) = 0. (30) From (30) one can see that there is always a stationary state w hich diﬀers from X0at any nonzero value of the parameter α∆T. Thus, those modes which have A0 k/ne}ationslash= 0 are always unstable. The result is not surprising, since from the physi cal point of view it is obvious that a constantly heated wire will change its form owing to ex pansion. On the other hand, those modes which have A0 k= 0 become unstable at some value of the parameter α∆T, so that buckling instability is still possible on these modes. 4 Qualitative stress analysis. Once the instability occurs, all the potential energy compr ised into the compressed wire can be quickly released. The characteristic value of the longit udinal stress τzzwhich may be accumulated during the preconditioning is within the inter val between the maximum value τmax zz=α(Tmelt−T0)E∼103MPa and the minimum value τmin zz=EJ Sλ2 0≃0.3÷30 MPa corresponding to the onset of the instability. The estimati on has been done for an aluminium wire of radius a= 0.6 mm and the length spanning the range 1 m > L > 0.05 m. One can see, that the maximum value itself is very high, about 10 times higher than the ultimate strength value. On the other hand, the minimum valu e can be quite small depending on the wire length. The average transverse stress is given by < τxx>=G SX′−EJ SX′′′. For small deﬂections, \|X\|/ξL≪1, it is in general smaller then the longitudinal one. From the qualitative point of view it is evident that the maxi mal stress energy can be accumulated if the temperature rises suﬃciently quickly. T he temperature rise time must be shorter or comparable with the rise time of the instability. The most dramatic result might be expected if the wire is heated up to the temperature just sl ightly below the melting point. Then the current is switched oﬀ thus allowing the instabilit y to develop without melting the material. On the other hand, if the current is low and, as a con sequence, the temperature rise time is low too, then all the accumulated energy can be re leased at the onset of the 12instability by the ﬁrst mode which is becoming ﬁrst unstable . This case corresponds to the lower limit of the estimated stress value τmin zz. Thus, already now, from the linear analysis, it is obvious th at many scenarios of the instability are possible. Dynamically, many modes can be ex cited simultaneously while the temperature Tincreases from T0up to the melting point Tmelt. Even though the ﬁrst mode becomes ﬁrst unstable it might happen that further othe r modes play dominant role creating rather complex dynamical behaviour. Any particul ar pattern, of course, depends on temporal characteristics such as the ratio between instabi lity increment and the temperature rise time. 5 Numerical results Now, there are several further questions. What character wi ll the instability have on the nonlinear stage? How high longitudinal and transverse stre sses could be obtained in this situation? To answer these questions we solve the nonlinear equation (1) numerically. 5.1 Numerical algorithm The numerial method of the solution of equation (1) with the b oundary conditions (3) is based on the complete set of orthogonal functions (14). By expanding any solution to (1) into the series of the functi ons{Xi}, i.eX=/summationtext iAiXi, the partial diﬀerential equation turns into a system of ODEs with respect to time. They are /summationdisplay jBijd2Aj dt2+Aiλ4 i 2+Aiλ2 i 2/braceleftBig βa(˜l−˜l0−α[T(t)−T0]) +βB(0.14 + ln( λia/L))/bracerightBig = 0 (31) where the coeﬃcients are βa= 4L2/a2,βB=µ0I2L2 4πEJ. The symmetric matrix Bijis given by Bij=/integraltext1 0XiXjdz. The length increment is ˜l−˜l0=1 4/summationtext j(A2 j−A0 j2)λ2 j. The functions X,X0 and the variable zhave been normalised on Landthas been normalised by t0=/radicalbigg ρS EJL2. In the numerical simulations, the inﬁnite series have been t rancated at the maximal unstable mode deﬁned by (28). 135.2 Flexural vibrations of initially straight wires In the case of initially straight wires the initial conditio ns are Ai(0) = 0. To excite the instability one needs to seed some initial noise level. To si mulate noise present an additional ﬂuctuation force in the equation (1), δF=δF0δf(t)/summationtext iexp(−ξ2 i/ξ2 1)Xi, has been added, with δF0= 4×10−4newton. The function −1< δf(t)<1 has been calculated by means of a random number generator at each step of calculations over ti me. This force can give rise to deﬂection of an aluminium wire with L= 1 m and a= 0.6 mm from a straight line by approximately X∼0.1a. The current proﬁle in the simulations has been taken in the fo rm I(t) =I0sinh(t/t0)/cosh(t/t0), where t0is the current rise time. In all runs t0= 30µsec. This time is about twice as high as the skin time for aluminium wires with a= 0.6 mm, tskin=a2σµ0≃17µsec. We also kept T0= 300◦K and calculations stopped once the tem- perature had reached the melting point T=Tmelt. We designated this moment by t=te. We have performed simulations for diﬀerent conditions: diﬀ erent wire lengths, diﬀerent currents and diﬀerent wire materials. All the sets used are p resented in Table I. As the ﬁrst example, let’s consider a short aluminium wire ca rrying 5 kA current, case [A]. For these parameters from the linear analysis one would expect a few unstable modes to develop, ilim= 6. The results obtained are presented in Fig.1. We have plot ted in the ﬁrst two frames the deﬂection Xas a function of zand the spectrum < A(t)2 i>both taken at t= te. In the last two frames, we have plotted the longitudinal τzzand transverse < τxx>stress components as functions of time. The spectrum has been calcu lated by means of averaging over time around t=te, i.e. < A(te)2 i>=/integraltextte te−∆A(t)2 idt, where ∆ is chosen to be greater than the period of nonlinear vibrations. Both the spectrum, Fig.1b and the wire shape, in the form being close to a simple arc, Fig.1a, show that the ﬁrs t mode dominates during the nonlinear stage of the instability. Temporal behaviour of t he longitudinal and transverse stress components demonstrates developed nonlinear ﬂexur al vibrations, Fig.1c and Fig.1d. The appearance of the instability is clearly seen at t≃900µsec in both ﬁgures. The observed longitudinal stress has both compressive and tensile compo nents. The instability appeared 14after strong compression, the maximal compressive stress b eing 429 MPa. The maximal tensile longitudinal stress is amounting to τzz≃244 MPa. This value is well above the ultimate stress value for aluminium. The observed transver se stress, as has been expected, is lower then the longitudinal one, < τxx>≃55 MPa. Let’s consider now a longer wire, case [B]. This case is relev ant to the conditions of the Graneau’s experiments [4]. One might expect, in accordence with our qualitative analysis, more active modes in the instability spectrum to develop wit h respect to the previous case, ilim= 129. Indeed, from Fig.2a and Fig.2b one can observe that the spectrum becomes very rich with the major contribution from mode n= 27 at the melting point. The tensile longitudinal and transverse stresses are amounting to τzz= 122 MPa and < τxx>= 92 MPa in this case, see Fig.2c and Fig.2d. Both of them are well abov e the ultimate strength of aluminium. Thus, one might expect the ﬁrst wire break to ap pear just after buckling occured, at t= 1000 µsec. Let’s investigate now the eﬀect of variations of current. In case [C], which is a replica of case [B] but with lower current, two modes n= 5 and n= 7 become dominant, Fig.3a. This fact indicates, if we compare two spectra in cases [B] and [C] , that the main channel of the accumulated energy to release is going now through the modes with lower numbers. Thus, one would anticipate a lower accumulated stress according t o our qualitative stress analysis. Indeed, we have observed that both tensile longitudinal and transverse stresses reduced to τzz= 70 MPa and < τxx>= 23 MPa, Fig.3b and Fig.3c. On the other hand, an increase of the current leads to higher r esultant tensile stress value. In case [D] which is similar to set [B] but with a higher current, the longitudinal tensile stress is τzz≃233 MPa in maximum, Fig.4. It should be noted that because of f aster heating the preliminary compression before the instabilit y developed is much more stronger, namely 851 MPa, and the instability occurred just before the melting point. It appears that at higher current the wire will be molten before the buckling occurs. Let’s consider now the Nasilovski’s experiment with a long c opper wire, case [E]. This experiment has been carried out at quite a low current I= 500 A in comparison with the 15all previously considered cases. As a result, one might expe ct low stress values compared to previous cases. Also, a few ﬁrst modes must be dominant. Inde ed, that is just the case, see Fig.5a. The observed values of stress indeed become quite lo w. The longitudinal stress is reaching τzz≃27 MPa in maximum, Fig.5b, while the transverse stress is jus t about 4 MPa. It means that the wire can be broken only if it was heated up eno ugh. 5.3 Flexural vibrations of initially bent wires In the above, we considered somewhat idealised situation, w hen the undeformed wire had the shape of a straight rod. In reality, the wire might have had in itially any form. For instance, in the case of horizontal positioning of the wire it might be b ent by gravitational force. Once the initial dislocation becomes grater than the radius, the character of the instability changes as has been discussed. Indeed, if X0/ne}ationslash= 0 the initial state is unstable from the beginning. But, if the temperature risetime is shorter than the period o f ﬂexural vibrations then those modes which contribute into X0=/summationtext iA0 iXican be excited directly. Moreover, buckling on the other modes is still possible. To estimate the stresses d eveloped simulations for case [B] have been performed but with initial conditions given by X0=A0 1X1withA0 1= 20a. That is, the wire was initially shaped like the ﬁrst mode of the set (14 ). The choice of the amplitude A0 1follows from estimations for a horizontally positioned wir e of this length. The wire in this case has a form X0=qz2(z−L)2 24EJ, see the problem in [10] page 93, qbeing the wire weight per unit length. Thus, the maximal displacement is ex pected to be X0max∼18a. The results of simulations are shown in Fig.6. As before, one can see developed nonlinear vibrations. The regular character of the vibrations points out to the fact that only one single mode i= 1 is active in this case. This means that buckling on the othe r mode didn’t occur. The amplitude of tensile longitudinal stress is amou nting for this particular case to τzz≃50 MPa. So, a suﬃciently heated wire can be broken in this case as well. 16Conclusions When an electric current passes through a thin metal wire wit h clamped ends, ﬂexural elastic stress waves are induced owing to the Joule heating and the el ectromagnetic force. The Joule heating leads to thermal expansion of the wire material, whi ch is the dominant mechanism of the excitation of vibrations. Under realistic experimen tal conditions the electromagnetic force is of minor importance. Flexural vibrations in an initially straight wire may be exc ited as a result of the buckling instability. The energy accumulated in the wire during the i nitial stage in the form of a compressive stress is suddenly released. As a result, high t ensile stress appears, which is suﬃciently high to cause the fracture of the wire within 1 ms f or all the cases considered. The number of modes of the instability that are excited depen ds on the current magnitude and wire length. Depending on these parameters it is possibl e to excite just a single mode, which has a form of an arch, or actually any number of modes. If a wire is slightly curved, which is a more realistic case th an that of the straight wire, the buckling instability is still possible. The amplitude o f modes, that are initially present in the curved wire, grows. The other modes can still be excited i n a rapidly heated wire owing to buckling instability. The magnitude of tensile stress in duced in a curved wire is clearly lower, but is still suﬃcient to induce a wire fracture on a mil lisecond timescale. Three-dimensional eﬀects, which may be induced by the exter nal circuit, suspensions at the clamped ends, imperfections of the wire cross-section o r a wire material, will increase the magnitude of tensile stress. These eﬀects will lead to th e coupling between ﬂexural, longitudinal, and torsional modes. This study is beyond the scope of the present paper. The experimental evidence, in general, is supportive of the mechanism described above. However, no direct comparison is possible, since previous e xperiments were of exploratory nature. The model developed favours the view that the phenomenon of t he wire fragmentation in the solid state can be explained without resorting to cont roversial Amp` ere-Neumann electrodynamics. 17Acknowledgements This work has been supported by Engineering and Physical Sci ences Research Council grant No. GR/M07403. The authors are grateful to Prof. J. Allen and Dr. D. Wall for usefull discussions. A. Lukyanov would like to express his gratitud e to O.V. Umnova for support. References [1] J. Nasilovski, Unduloids and striated disintegration o f wires Exploding wires 1964, v.3, ed. W.G. Chase and H.K. Moore, (New York: Plenum), p.295. [2] P. Graneau, Longitudinal magnet forces? J. Appl. Phys. 1 984, v.55, p.2598. [3] P. Graneau, Ampere tension in electric conductors, IEEE Trans. Magn., 1984, v.20, p.444. [4] P. Graneau, Wire explosions, Phys. Lett. A, 1987, v.120, p.77. [5] P. Graneau, Ampere-Neumann Electrodynamics of Metals, 1994, Palm Harbor: Hadronic. [6] J. S. Hong, Electromagnetic forces in ﬂexible systems, D Phil Thesis University of Oxford, 1994. [7] S. Molokov, J. E. Allen, The fragmentation of wires carry ing electric current, J. Phys. D, 1997, v.30, p.3131. [8] J.G. Ternan, Stresses in rapidly heated conductors, Phy s. Lett. A, 1986, v.115, p.230. [9] Smithells Metals Reference Book 7th ed., editors E.A. Br andes and G.B. Brook, London, Butterworth 1998. [10] L.D. Landau, E.M. Lifshitz, Theory of elasticity , 1970, (Oxford: Pergamon Press), p.89. [11] W.B. Thompson, ”An Introduction to Plasma Physics”, Pe rgamon Press, 1964, p.107. 18[12] S. Chattopadhyay, F. Moon, Magnetoelastic buckling an d vibration of a rod carrying electric current, J. Appl. Mech., 1975, December, p.809. [13] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media , 1984, (Oxford: Perg- amon Press), p.123. [14] M.A. Leontovich and V.D. Shafranov, ”The stability of a ﬂexiable conductor in a lon- gitudinal magnetic ﬁeld”. Plasma physics and the problem of thermonuclear fusion reac- tions, v.1, Pergamon Press, 1961. [15] W.F. Chen and T. Atsuta, Theory of Beam-Columns , V.1. 1976, McGraw-Hill, Inc. 19Captions Fig.1a The wire displasement X/Las a function of z/L, case [A]. Fig.1b The spectrum of the wire vibrations at the melting poi nt< A2 i>as a function of mode number i, case [A]. Fig.1c The longitudinal stress τzzas a function of time, case [A]. Fig.1d The transverse stress < τxx>as a function of time at z/L= 0.9 atz/L= 0.9, case [A]. Fig.2a The wire displasement X/Las a function of z/L, case [B]. Fig.2b The spectrum of the wire vibrations at the melting poi nt< A2 i>as a function of mode number i, case [B]. Fig.2c The longitudinal stress τzzas a function of time, case [B]. Fig.2d The transverse stress < τxx>as a function of time at z/L= 0.9, case [B]. Fig.3a The spectrum of the wire vibrations at the melting poi nt< A2 i>as a function of mode number i, case [C]. Fig.3b The longitudinal stress τzzas a function of time, case [C]. Fig.3c The transverse stress < τxx>as a function of time at z/L= 0.9, case [C]. Fig.4 The longitudinal stress τzzas a function of time, case [D]. Fig.5a The spectrum of the wire vibrations at the melting poi nt< A2 i>as a function of mode number i, case [E]. Fig.5b The longitudinal stress τzzas a function of time, case [E]. Fig.6 Flexural vibrations of an initially bent wire, longit udinal stress as a function of time. X0=A0 1X1,A0 1= 20a. The wire and current parameters are relevant for case [B]. 20/c19 /c17/c19 /c19 /c19 /c17/c21 /c19 /c19 /c17/c23 /c19 /c19 /c17 /c25 /c19 /c19 /c17 /c27 /c19 /c20 /c17 /c19 /c19 /G93/G18/G47/c16/c19 /c17 /c19 /c25/c16/c19 /c17 /c19 /c23/c16/c19 /c17 /c19 /c21/c19 /c17/c19 /c19/c19 /c17/c19 /c21/G3/G59/G18/G47 /c41/c76/c74/c17/c20/c68/c19 /c20 /c21 /c22 /c23 /c24 /c25 /c26 /G3/G76/c19 /c40 /c14 /c19/c21 /c40 /c16/c23/c23 /c40 /c16/c23/c25 /c40 /c16/c23/c27 /c40 /c16/c23/G31/G36/G76/G21/G33 /c41 /c76/c74/c17/c20/c69/c19 /c24 /c19 /c19 /c20 /c19 /c19 /c19 /c20 /c24 /c19 /c19 /G87/G15/G3/G62µ/G86/G64/c16/c23 /c19 /c19/c16/c21 /c19 /c19/c19/c21 /c19 /c19τ/c93/c93/G15/G3/G62/G48/G51/G68/G64/G3 /c41/c76/c74/c17/c20/c70/c19 /c24 /c19 /c19 /c20 /c19 /c19 /c19 /c20 /c24 /c19 /c19 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c22 /c19/c19/c22 /c19/c25 /c19/G3/G31τ/c91/c91/G33/G15/G3/G62/G48/G51/G68/G64 /c41/c76/c74/c17/c20/c71/c19 /c17 /c19 /c19 /c19 /c17 /c21 /c19 /c19 /c17/c23 /c19 /c19 /c17/c25 /c19 /c19 /c17/c27 /c19 /c20 /c17/c19 /c19 /G93/G18/G47/c16/c19 /c17/c19 /c20/c16/c19 /c17/c19 /c19/c19 /c17/c19 /c19/c19 /c17/c19 /c19/c19 /c17/c19 /c20/G59/G18/G47 /c41/c76/c74/c17/c21/c68/c19 /c20 /c19 /c21 /c19 /c22 /c19 /c23 /c19 /c24 /c19 /c25 /c19 /c26 /c19 /G76/c19 /c40 /c14 /c19/c20 /c40 /c16/c25/c21 /c40 /c16/c25/G31/G36/c76 /c21/G33 /c41/c76/c74/c17/c21/c69/c19 /c23 /c19 /c19 /c27 /c19 /c19 /c20 /c21 /c19 /c19 /c20 /c25 /c19 /c19 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c24 /c19 /c19/c16/c23 /c19 /c19/c16/c22 /c19 /c19/c16/c21 /c19 /c19/c16/c20 /c19 /c19/c19/c20 /c19 /c19/c21 /c19 /c19τ/c93/c93/G15/G3/G62/G48/G51/G68/G64 /c41 /c76/c74/c17/c21/c70/c19 /c23 /c19 /c19 /c27 /c19 /c19 /c20 /c21 /c19 /c19 /c20 /c25 /c19 /c19 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c27 /c19/c16/c25 /c19/c16/c23 /c19/c16 /c21 /c19/c19/c21 /c19/c23 /c19/c25 /c19/c27 /c19/G31τ/c91/c91/G33/G3/G15/G3/G62/G48/G51/G68/G64 /c41/c76/c74/c17/c21/c71/c19 /c20 /c19 /c21 /c19 /c22 /c19 /c23 /c19 /G76/c19 /c40 /c14 /c19/c20 /c40 /c16/c24/c21 /c40 /c16/c24/c22 /c40 /c16/c24/c23 /c40 /c16/c24/G31/G36/c76 /c21/G33 /c41/c76/c74 /c17 /c22 /c68/c19 /c21 /c19 /c19 /c19 /c23 /c19 /c19 /c19 /c25 /c19 /c19 /c19 /c27 /c19 /c19 /c19 /c20 /c19 /c19 /c19 /c19 /c20 /c21 /c19 /c19 /c19 /G87/G76/G80/G72/G3/G62µ/G86/G72/G70/G64/c16/c21 /c19 /c19/c16/c20 /c19 /c19/c19/c20 /c19 /c19τ/c93/c93/G15/G3/G62/G48/G51/G68/G64 /c41/c76/c74/c17/c22/c69/c19 /c23 /c19 /c19 /c19 /c27 /c19 /c19 /c19 /c20 /c21 /c19 /c19 /c19 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c22 /c19/c16/c21 /c19/c16/c20 /c19/c19/c20 /c19/c21 /c19/c22 /c19/G31τ/c91/c91/G33/G3/G15/G3/G62/G48/G51/G68/G64 /c41 /c76 /c74 /c17/c22/c70/c19 /c21/c24/c19 /c24/c19/c19 /c26/c24 /c19 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c28/c19/c19/c16/c27/c19/c19/c16/c26/c19/c19/c16/c25/c19/c19/c16/c24/c19/c19/c16/c23/c19/c19/c16/c22/c19/c19/c16/c21/c19/c19/c16/c20/c19/c19/c19/c20/c19/c19/c21/c19/c19/c22/c19/c19τ/c93/c93/G15/G3/G62/G48/G51/G68/G64 /c41/c76/c74/c17/c23/c19 /c17 /c19 /c19 /c23 /c17/c19 /c19 /c27 /c17/c19 /c19 /c20 /c21 /c17/c19 /c19 /G76/c19 /c40 /c14 /c19/c21 /c40 /c16/c24/c23 /c40 /c16/c24/c25 /c40 /c16/c24/G31/G36/c76 /c21/G33 /c41 /c76 /c74 /c17 /c24 /c68/c19 /c40 /c14 /c19 /c20 /c40/c14 /c23 /c21 /c40/c14 /c23 /c22 /c40/c14 /c23 /c23 /c40/c14 /c23 /c24 /c40/c14 /c23 /c25 /c40 /c14 /c23 /c26 /c40 /c14 /c23 /G87/G15/G3/G62µ/G86/G72/G70/G64/c16/c25/c19/c16/c23/c19/c16/c21/c19/c19/c21/c19/c23/c19τ/c93/c93/G15/G3/G62/G48/G51/G68/G64 /c41 /c76/c74/c17/c24/c69/c19 /c20 /c19 /c19 /c19 /c21 /c19 /c19 /c19 /G87/G15/G3/G62µ/G86/G64/c16 /c21 /c19 /c19/c16 /c20 /c19 /c19/c19/c20 /c19 /c19τ/G93/G93/G15/G3/G62/G48/G51/G68/G64/G3 /c41 /c76 /c74 /c17 /c25
arXiv:physics/0012030v1 [physics.comp-ph] 14 Dec 2000MONTECARLO METHODS:APPLICATION TO HYDROGEN GAS AND HARD SPHERES BY MARK DOUGLASDEWING B.S., MichiganTechnologicalUniversity,1993 M.S., UniversityofIllinoisat Urbana-Champaign, 1995 THESIS Submittedin partialfulﬁllmentoftherequirements forthedegreeofDoctorofPhilosophyin Physics intheGraduateCollegeofthe UniversityofIllinoisatUrbana-Champaign, 2001 Urbana, IllinoisMONTECARLO METHODS:APPLICATION TO HYDROGEN GAS AND HARD SPHERES Mark DouglasDewing,Ph.D. Department ofPhysics UniversityofIllinoisatUrbana-Champaign, 2001 David M.Ceperley, Advisor Quantum Monte Carlo (QMC) methods are among the most accurat e for computing groundstatepropertiesofquantumsystems. Thetwomajorty pesofQMCweuseareVari- ational Monte Carlo (VMC), which evaluates integrals arisi ng from the variational princi- ple, and Diffusion Monte Carlo (DMC), which stochastically projects to the ground state from a trial wave function. These methods are applied to a sys tem of boson hard spheres toget exact, inﬁnitesystemsizeresultsforthegroundstat eat severaldensities. The kinds of problems that can be simulated with Monte Carlo m ethods are expanded throughthedevelopmentofnewalgorithmsforcombiningaQM Csimulationwithaclassi- calMonteCarlosimulation,whichwecallCoupledElectroni c-IonicMonteCarlo(CEIMC). The new CEIMC method is applied to a system of molecular hydro gen at temperatures rangingfrom 2800Kto4500Kand densitiesfrom0.25 to0.46g/ cm3. VMC requires optimizing a parameterized wave function to ﬁn d the minimumenergy. WeexamineseveraltechniquesforoptimizingVMCwavefunct ions,focusingontheability tooptimizeparameters appearing intheSlaterdeterminant . Classical Monte Carlo simulations use an empirical interat omic potential to compute equilibriumpropertiesofvariousstatesofmatter. TheCEI MCmethodreplaces theempir- ical potential with a QMC calculation of the electronic ener gy. This is similar in spirit to the Car-Parrinello technique, which uses Density Function al Theory for the electrons and moleculardynamicsforthenuclei. ThechallengesinconstructinganefﬁcientCEIMCsimulatio ncentermostlyaroundthe noisyresultsgeneratedfromtheQMCcomputationsoftheele ctronicenergy. Weintroduce two complementary techniques, one for tolerating the noise and the other for reducing it. The penalty method modiﬁes the Metropolis acceptance ra tio to tolerate noise without introducing a bias in the simulation of the nuclei. For reduc ing the noise, we introduce the two-sided energy difference method, which uses correla ted sampling to compute the energy changeassociatedwithatrialmoveofthenuclearcoo rdinates. Unlikethestandard reweightingmethod,itremains stableas theenergy differe nceincreases. iiiAcknowledgments First I would liketo thank my advisor, David Ceperley, for su pportingmein thisresearch, for teaching these Monte Carlo methods to me, and for being av ailable and patient when answeringquestions. I would also like to thank the graduate students and postdocs in the group who have helped me and provided useful and interesting discussions. And special thanks to Tadashi OgitsufortheuseofhisDFT code. I am grateful to my parents for their support during my colleg e and graduate school pursuits. Ienjoyed therefreshing summervisitsto theirfa rm. Thanks to mybrotherLuke, whoselivingin Colorado wasconvenientforskitrips. During my time here, I have beneﬁted greatly from friendship s and relationships with peopleinGraduateIntervarsityChristianFellowship,Gra ceCommunityChurch,andIllini LifeChristianFellowship. Theyhavegivenmeagreatdealof strengthandencouragement when Ineeded it. My work was supported by the computational facilities at NCS A, by a Graduate Re- search Trainee fellowship NSF Grant No. DGE93-54978, and by NSF Grant No. DMR 98-02373. Andﬁnally,apologiestomycat,Lucy,fornotgivingherenou ghattentionwhileﬁnish- ingthiswork. ivContents 1 Introduction ...................................... ........................ 1 1.1 ThesisOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Monte CarloMethods ................................. .................... 4 2.1 BasicMonteCarlo Integration . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 MetropolisSampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 ClassicalMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 VariationalMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.1 Two LevelSampling . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 DiffusionMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 StatisticalErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.7 WaveFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 PeriodicBoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Energy Difference Methods........................... ..................... 20 3.1 DirectDifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 3.2 Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Bennett’sMethodforFree Energy Differences . . . . . . . . . . . . . . . . 22 3.4 Two-SidedSampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5.1 DiffusionMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5.2 BindingEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 WaveFunction Optimization.......................... ..................... 31 4.1 Energyvs. VarianceMinimization . . . . . . . . . . . . . . . . . . . . . . 32 4.2 FixedSampleReweighting . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 NewtonMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 StochasticGradient Approximation . . . . . . . . . . . . . . . . . . . . . 36 v4.5 GradientBiased Random Walk . . . . . . . . . . . . . . . . . . . . . . . . 40 4.6 Comparisonofmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0 4.7 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5 Coupled SimulationMethods .......................... .................... 43 5.1 PenaltyMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.1.1 Othermethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1.2 Handlingnoisydata . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Pre-rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 5.3 TrialMoves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 SingleH 2molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6 HardSpheres ....................................... ...................... 51 6.1 Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 FiniteSizeEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.3 DistributionFunctionsand CondensateFraction . . . . . . . . . . . . . . . 59 7 Hydrogen.......................................... ....................... 62 7.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.3 Pressureand KineticEnergy . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.4 IndividualConﬁgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.6 Simulationanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.7 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8 Conclusions....................................... ........................ 71 A Determinant Properties............................. ....................... 72 B Elements ofthe Local Energy.......................... .................... 74 C Cusp Condition ..................................... ...................... 76 References......................................... .......................... 77 Vita............................................... ........................... 85 viList of Tables 2.1 Timings for Li 2molecule using the standard sampling method. All times arein secondsonan SGI Origin2000. . . . . . . . . . . . . . . . . . . . . 1 0 2.2 Timings for Li 2molecule using the two level sampling method. All times arein secondsonan SGI Origin2000. . . . . . . . . . . . . . . . . . . . . 1 0 2.3 Timings for the system of 32 H 2molecules in a periodic box using the standardsamplingmethod. Alltimesarein secondson aSun Ul tra5. . . . 10 2.4 Timings for a system of 32 H 2molecules in a periodic box using the two levelsamplingmethod. Alltimesare insecondson aSun Ultra 5. . . . . . 11 2.5 Comparison of energies and variances for various forms f or orbitals and Jastrowfactors fora singleH 2molecule. . . . . . . . . . . . . . . . . . . 18 2.6 Valuesofvariationalparameters for H 2. . . . . . . . . . . . . . . . . . . . 19 2.7 Valuesofvariationalparameters for wavefunctionE . . . . . . . . . . . . 19 5.1 Efﬁciency of classical Monte Carlo for moving several pa rticles at once. The table on the left is for low density system at rs=3.0 and T=5000K. Thetableontherightisforahighdensitysystemat rs=1.8andT=3000K. Thelargest valuesoftheefﬁciencyare shownin boxes. . . . . . . . . . . . 49 5.2 ResultsofCEIMCforisolatedH 2moleculeatT=5000K. . . . . . . . . . . 50 6.1 Variationalparameters forhard spheregas . . . . . . . . . . . . . . . . . . 53 6.2 Energyextrapolatedto inﬁnitesystemsize(in unitsof¯h2 mσ2) . . . . . . . . 58 6.3 Condensatefraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 7.1 Pressurefrom simulationsand shockwaveexperiments . . . . . . . . . . . 67 7.2 Energyfromsimulationsandmodels,relativetothegrou ndstateofaniso- lated H 2molecule. The H 2column is a single thermally excited molecule plusthequantumvibrationalKE. . . . . . . . . . . . . . . . . . . . . . . 6 7 7.3 Average molecular H 2bond length. The H 2column is a single thermally excitedmoleculeinfree space. . . . . . . . . . . . . . . . . . . . . . . . . 67 vii7.4 Simulation quantities ordered according to average noi se level,βσ. The time column is the time for a single quantum step in minutes on an SGI Origin2000. Nis thenumberofmoleculesinthesimulation. . . . . . . . 70 viiiList of Figures 2.1 EfﬁciencyofVMC.ThegraphontheleftisforLi 2. Thegraphontheright isfor32 H 2molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Examples of statistical data analysis using reblocking . The error in the graph on the left has converged, while the error in the graph o n the right hasnot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Optimizedelectron-electron Jastrowfactorfordiffer entforms. . . . . . . . 18 3.1 TwoH 2moleculesin aparallel conﬁguration . . . . . . . . . . . . . . . . 26 3.2 Energy difference (left) and the estimated statistical error (on logscale) (right)fortwoH 2moleculesinaparallelconﬁguration,startingfromd=2.5 Bohr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Finitesamplesizebiasin theenergy differenceofLi 2. . . . . . . . . . . . 27 3.4 Errorinenergy difference ofLi 2usingDMC(top)andVMC(bottom) . . . 29 3.5 ErrorinVMCbindingenergy ofH 2-H2system . . . . . . . . . . . . . . . 30 4.1 ExamplesusingtheNewtoniterationwithvaryingamount sofnoise. . . . 36 4.2 Examples of SGA. The graph on the top shows the convergenc e of one variationalparameterforseveralSGAalgorithms. Thegrap honthebottom showstheresultingenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9 4.3 Optimizationmethodsapplied to(a) SingleH 2(b)8 H 2’s (c) 16 H 2’s (d) 32H2’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.1 CEIMC program outlines. Boxes indicate quantum computa tions. The dashed box indicates a quantity saved from a previous comput ation. The topalgorithmis incorrect. Thebottomalgorithmiscorrect . . . . . . . . . 47 5.2 Exampleson aLennard-Jonespotentialwithsyntheticno ise. . . . . . . . . 48 5.3 H 2bondlengthdistribution. . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.1 Timestep errorfor(a) ρ=0.01 (b)ρ=0.05 (c)ρ=0.1 (d)ρ=0.2 . . 54 6.2 S(k)for(a) ρ=0.05(b)ρ=0.2 . . . . . . . . . . . . . . . . . . . . . . . 55 ix6.3 VMCﬁnitesizeeffects for(a) ρ=0.01(b)ρ=0.05(c)ρ=0.1(d)ρ=0.2 56 6.4 DMCﬁnitesizeeffects for(a) ρ=0.01(b)ρ=0.05(c)ρ=0.1(d)ρ=0.2 57 6.5 Energyvs. density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.6 Pairdistributionfunctionforseveraldensities . . . . . . . . . . . . . . . . 60 6.7 Singleparticledensitymatrixforseveraldensities . . . . . . . . . . . . . 60 6.8 Condensatefraction vs. density . . . . . . . . . . . . . . . . . . . . . . . 61 7.1 Electronicenergyforseveralconﬁgurationscomputedb yseveralmethods. Theenergy isrelativetoan isolatedH 2molecule. . . . . . . . . . . . . . . 66 7.2 Proton pair distribution function g(r)for (a)rs=2.1 and T=4530 K (b) rs=2.202and T=2820K . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3 Theprotonpairdistributionfunction, g(r),closeto rs=1.8and T =3000K. 68 xGuideto Notation CEIMC Coupled Electronic-IonicMonteCarlo DMC DiffusionMonteCarlo DFT DensityFunctionalTheory GBRW GradientBiased RandomWalk LDA Local DensityApproximation(in DensityFunctionalThe ory) PIMC Path IntegralMonteCarlo QMC QuantumMonteCarlo SGA StochasticGradient Approximation VMC VariationalMonteCarlo αGeneral variationalparameter. aInstantaneousacceptance probabilityinpenaltymethod. a0Bohrradius,unitoflengthin atomicunits. 1a0=5.29×10−9m. A(s→s′)Acceptance probabilityinMetropolismethod. βInversetemperature,1 kBT. δQMCestimateofan energy difference. ΔExactenergydifference. AlsothesamplingboxsizeintheMe tropo- lisalgorithm. dBond length. xiDSlaterdeterminant. ζA variationalparameter inH 2orbitals. fA trial wave function (also denoted ψT). Also the relative noise parameter. ηAdditionalnoiserejection ratio. ELThelocalenergy ofa trialwavefunction. θvVibrationaltemperature. hStep sizeparameterinSGA andGBRW. H A many-bodyHamiltonian. Ha Hartree, unitofenergy inatomicunits. 1Ha=27.21 eV. G(R→R′,τ)Green’s functionpropagatorinDMC. kBBoltzmannfactor. KKineticenergy. λ¯h2/2m,wheremis theparticle’smass. n0Condensatefraction. NThenumberofparticlesin asystem. π(s)Probabilitydistributionto besampledin aMarkovprocess. PPressure, oraprobabilitydistributioninthetwo-sidedme thod. P(s→s′)Transitionprobabilityin aMarkovchain. QNormalizationintegral. ρDensity. ρ1(r)Singleparticledensitymatrix. rs/parenleftbig3 4πn/parenrightbig1/3, wherenistheelectron numberdensity. rijTheseparationbetween particle jand particle i. xiiRThecoordinatesofalltheparticles inamany-bodysystem. σNoise level (variance or standard error). Also the hard sphe re diameter. sStatein conﬁgurationspace. τDMCtimestep. TTemperature. T(s→s′)Samplingdistributionin Metropolismethod. u(r)Jastrowfactorina many-bodywavefunction. V Volumeofthesimulationcell, orpotentialenergy. V Alternate notation for potential energy (for formulas that have bothpotentialenergy and volume). wWeightfactorin correlated samplingmethods. wlWidthofH 2orbital. ψTA trialwavefunction(alsodenoted f). φ0Theexactmanybodyground statewavefunction. φA singleparticleorbital. xiiiChapter1 Introduction The ﬁrst computer simulations of a condensed matter system u sed the simplest potential, the hard sphere (Metropolis et al., 1953). As computers and simulationsprogressed, more sophisticated and realistic potentials came into use. Thes e potentials are parameterized andthenﬁttoreproducevariousexperimentalquantities. B othMolecularDynamics(MD) and Monte Carlo (MC) methods are used to generate ensemble av erages of many-particle systems. These potentialsoriginatefrom themicroscopicstructure of matter, described in terms of electrons, nuclei, and the Schr¨ odinger equation. But th e many-body Schr¨ odinger equa- tion is too complicated to solve directly, so some approxima tions are needed. The one electron approximation is a successful approach, where a si ngle electron interacts with an externalpotential(ie,thenuclei)andwithameanﬁeldgene ratedbyalltheotherelectrons. This is done by Hartree-Fock (HF) or with Density Functional Theory (DFT) (Parr and Yang, 1989). DFT is in principle exact, but contains an unkno wn exchange and correla- tion functional that must be approximated. The most common o ne is the Local Density Approximation(LDA). Theseﬁrstprinciplescalculationsareusedinﬁttingthepo tentials,whicharethenused in an MC or MD computation. But the problem of transferabilit y still remains. Empirical potentialsare onlyvalidinsituationsforwhichtheyhaveb een designedand ﬁtted. In 1985, Car and Parrinello introduced their method, which r eplaced the empirical po- tential with a DFT calculation done ‘on the ﬂy’ (Car and Parri nello, 1985). They did a molecular dynamics simulation of the nuclei of liquid silic on and then computed the den- sity functional energy of the electrons at every MD step. To i mprove the efﬁciency of the computation of the DFT energy, they introduced a new iterati ve method for solving the DFT equations. It has been a very successful method, with the original paper being cited over2300timessinceitspublication. 1Previously,theDFTequationshadbeen solvedbyeigenvalue methods. Buteigenvalue problems can also be regarded as optimization problems, whe re an energy functional is minimized. Car and Parrinello used an idea similar to simula ted annealing, but they used molecular dynamics to movethrough parameter space, rather than Monte Carlo. This had theeffectofmakingtheequationsofmotionsimilarbetween theelectronicproblemandthe nuclearproblem,withtheonlydifferencebeingintherelat ivemasses. Sincetheelectronic problem was not real electron dynamics, the electron mass do es not correspond to any physicalquantity,andisonlyaparametercontrollingthec onvergenceoftheelectronicpart of the simulation. Since then, other iterativemethods have been introduced, usually based on theConjugate-Gradientmethod(Payne et al.,1992). A briefreviewofapplicationsoftheCar-Parrinello method to liquidproblemsis given by Sprik (2000). This review also mentions that LDA and some o ther functionals are not good enough to accurately simulate water (there are improve d functionals that are accept- able). Anotherreview of molecular dynamicsby Tuckerman an d Martyna(2000) includes material on treating the nuclei classically and also using p ath integrals to treat the nuclei quantummechanically,donebyMarx and Parrinello(1996). QuantumMonteCarlo(QMC)methodshavedevelopedasanother meansforaccurately solving the many body Schr¨ odinger equation (Hammond et al., 1994; Anderson, 1995; Ceperley and Mitas, 1996). The success of QMC lies partly in t he fact these methods explicitlyincludecorrelationamongtheelectrons,which can notbedonedirectlywiththe oneelectronmethods. ParticularlywiththeLocalDensityA pproximation(LDA),DFThas knowndifﬁcultiesin handlingelectron correlation (Gross manetal., 1995). In the spirit of the Car-Parrinello method, we integrate a Cl assical Monte Carlo sim- ulation of the nuclei with a QMC simulation for the electrons . This we call Coupled Electronic-Ionic Monte Carlo (CEIMC). There are some chall enges in constructing an ef- ﬁcient method. The ﬁrst problem we encounter is that the results of a QMC simu lation are noisy. The QMCenergy hassomeuncertaintyassociatedwithit,and itco uldbiastheclassicalpartof the simulation. We could run the QMC simulation until the noi se is negligible, but that is very time-consuming. A better way is use the penalty method, which modiﬁes the usual MCformulastobetolerant ofnoise. Theelectronsareassumedtobeintheirgroundstate,bothin theCar-Parrinellomethod and in our CEIMC method. There are two internal effects that c ould excite the electrons - coupling to nuclear motion and thermal excitations. In the ﬁrst case, we make the Born- Oppenheimerapproximation,wherethenucleiaresomuchmor emassivethantheelectrons thattheelectrons areassumedto respondto nuclearmotioni nstantaneously,andso stayin 2their ground state. We neglect any occupation of excited sta tes of the electrons due to couplingtonuclearmotion. In the case of thermal excitation, let us examine several rel evant energy scales. If we consider a gas of degenerate electrons at a density of n=0.0298 electrons per cubic Bohr (i.e.rs=/parenleftbig3 4πn/parenrightbig1/3=2.0), the Fermi temperature is about 140,000K. The gap between the groundstateandtheﬁrstexcitedstateofahydrogenmolecul eatequilibriumbonddistance is about 124,000K.As long as our temperatures are well below this (and they are), and we arenotattoohighpressures(pressuredecreasesthegap),t hethermaloccupationofexcited statescan beneglected. Hydrogenisthemostabundantelementintheuniverse,makin ganunderstandingofits properties important,particularly for astrophysicalapp lications. Modelsof theinteriors of thegasgiantplanetsdependsonaknowledgeoftheequationo fstateofhydrogen(Hubbard and Stevenson, 1984; Stevenson, 1988). Hydrogen is also the simplest element, but it still displaysremarkablevarietyinitspropertiesand phasedia gram. Ithas severalsolidphases at low temperature, and the crystal structure of one of them ( phase III) is not fully known yet. At high temperature and pressure the ﬂuid becomes metal lic, but the exact nature of thetransitionisnot known. Computer simulation can also be used to obtain results on mod el systems. We will examine the hard sphere Bose gas, a simple and important mode l. For this model, all the approximations we make are controllable, and we will loo k at how to deal with those approximationsand obtainexactresultsforthismodel. 1.1 ThesisOverview Chapter 2 is an introduction to the basic classical and quant um Monte Carlo techniques we will be using. Chapter 3 presents an improvedQMC method fo r computingthe energy difference between two systems. Chapter 4 is an examination of parameter optimization, which is essential in VMC. We present various methods for min imizing the energy, and givesomecomparisonsbetween them. Successful CEIMC simulationsare based on thepenalty metho dfor toleratingnoisein the Metropolis method, which is detailed in Chapter 5. Some a dditional details are dis- cussed,andan exampleofCEIMCappliedtoasingleH 2moleculeisgiven. Theresultsof computations of the ground state energy of the boson hard sph ere model are presented in Chapter 6. In Chapter 7, the CEIMC simulation method is appli ed to ﬂuid molecular hy- drogen. Wepresentdataforafewstatepointsandperformsom eanalysisofthesimulation itself. 3Chapter2 MonteCarlo Methods Monte Carlo integration methods are very useful for evaluat ing the basic integrals of sta- tisticalandquantumphysics. In asystemwith Npparticles,theseintegralshavetheform /an}bracketle{tO/an}bracketri}ht=/integraltextdRπ(R)O(R)/integraltextdRπ(R)(2.1) whereRis a 3Npdimensional vector, π(R)is a probability distribution, and O(R)is the observableorquantityofinterest. Theseintegralshavetw oimportantcharacteristics: high dimensionality and the integrands are sharply peaked - only small parts of phase space contributesigniﬁcantlyto theintegral. Thehighdimensionalitymakesagridbasedschemeimpractic alintwoways. First,sup- posewehavea300dimensionalintegral(100particlesimula tion),andwant10gridpoints in each dimension. Even this crude integration requires fun ction evaluations at 10300grid points! Second,considerthetrapezoidalrule(asaconcret eexample)in ddimensions. The error using Nsamples will go asO(N−2/d). As we will show, the error in Monte Carlo integration goes asO(N−1/2). The Monte Carlo error is independent of the dimensional- ity whereas the grid based method depends on it strongly. For these high dimensionality problems,MonteCarlo isonlypractical choice. 2.1 BasicMonte CarloIntegration Consideran integraloftheform I=/integraldisplay1 0f(x)dx. (2.2) To evaluate by Monte Carlo, compute f(x)atNpoints sampled uniformly from [0,1]. An approximationto Iisgivenby I≈¯f=1 NN ∑ i=1f(xi) (2.3) 4Theestimateofthestatisticalerrorin ¯fwillbe σI=σf/√ N (2.4) whereσ2 fis thevariance, and isgivenby σ2 f=1 (N−1)N ∑ i=1/parenleftbig f(xi)−¯f/parenrightbig2(2.5) Thustheerrorgoes asO(N−1/2). The error bounds can be improved by sampling more points, or b y reducing the vari- ance,σ2 f. The lattercan beaccomplishedwith importancesampling. C onsidersomeprob- ability,P(x), thatisan approximationto f(x). WriteEq. (2.2)as I=/integraldisplay1 0P(x)f(x) P(x)dx. (2.6) Theestimateof Iisobtainedby sampling Npointsfrom P(x)and computing I≈N ∑ i=1f(xi) P(xi)(2.7) IfPis a good approximation to f, then the variance of the sum in Eq. (2.7) is much less thanthevarianceofthesumin Eq. (2.3). The fact that the integrands of interest are sharply peaked, as mentioned previously, makes importance sampling a necessity. The most useful type of importance sampling for theseproblemsis theMetropolismethod. 2.2 MetropolisSampling The Metropolis method (Metropolis et al., 1953) uses a Markov process to generate sam- ples from a normalized probability distribution, π(R)//integraltextdRπ(R). These samples are then used toestimateEq. (2.1)by ¯O=1 N∑ iO(Ri) (2.8) Forgeneralityin thefollowingsection,wewilldenotethes tateofthesimulationby s. A Markov process takes a transition probability between sta tes,P(s→s′), and con- structs a series of state points s1,s2,...(called a chain). An important characteristic of a Markov process is that the choice of the next state point in th e chain depends only on the current statepoint,notany previousstatepoints. 5The Metropolis method constructs a transition probability such that generated state pointsaresampledfromthedesireddistribution. Forthist owork,thetransitionprobability mustsatisfyergodicity. ThismeanstheMarkovchainmustev entuallybeabletoreach any stateinthesystem. A sufﬁcientconditionfor satisfyinger godicityisdetailed balance, π(s)P(s→s′) =π(s′)P(s′→s). (2.9) The generalized Metropolis method breaks the transition pr obability into the product of two pieces - an a priorisampling distribution T(s→s′)and an acceptance probability A(s→s′). TheMetropolischoicefortheacceptance probabilityis A(s→s′) =min/bracketleftbigg 1,π(s′)T(s′→s) π(s)T(s→s′)/bracketrightbigg (2.10) The procedure is to sample a trial state, s′, according to T(s→s′)and evaluate Eq. (2.10). The acceptance probability is compared with a unifo rm random number on [0,1]. IfAis greater than the random number, the moveis accepted, s′becomes the new sand is used in the average in Eq (2.8). Otherwise the move is rejecte d,sis not changed, and is reused intheaverage. The original Metropolisprocedure chooses a trial position uniformlyinsidea box cen- tered around thecurrent point, s′=s+y (2.11) whereyisauniformrandomnumberon [−Δ/2,Δ/2],withΔbeinganadjustableparameter. In thiscase, Tisuniformand willcancel outofEq. (2.10). An important measure of the Metropolis procedure is the acce ptance ratio - the ratio of accepted moves to the number of trial moves. It can be adjus ted by the choice of Δ. If the acceptance ratio is too small, state space will be expl ored very slowly because very few moves are accepted. If the acceptance ratio is high, it is likely that the trial moves are too small and once again, diffusion through state space w ill be very slow. Balancing theseconsiderationsleadstothestandardruleofthumbtha ttheoptimalacceptanceratiois around 50%. A betterconsiderationismaximizationoftheefﬁciency, ξ=1 σ2T(2.12) whereTis thecomputertimetaken to getan error estimateof σ. 62.3 ClassicalMonteCarlo Theprobabilitydistributionwewish tosampleistheBoltzm anndistribution π(s)∝exp[−V(s)/kT]. (2.13) The ﬁrst simulationsof this type were done with the hard sphe re potential (Metropolis et al., 1953; Wood and Jacobson, 1957). Later simulationsused Len nard-Jones potentials, and thenothertypesofempiricalpotentials. TheMetropolisproceduresamplesonlythenormalized π(s). Averagesoverthisdistri- bution are readily computed, but quantities that depend on t he value of the normalization are difﬁcult to compute. In classical systems,this include s quantitiessuch the entropy and the free energy. There are techniques, however, for computi ng the free energy difference between twosystems. 2.4 VariationalMonteCarlo Variational Monte Carlo (VMC) is based on evaluating the int egral that arises from the variational principle. The variational principle states t hat the energy from applying the Hamiltonian to a trial wave function must be greater than or e qual to the exact ground stateenergy. Typicallythewavefunctionisparameterized andthenoptimizedwithrespect to those parameters to ﬁnd the minimum energy (or minimum var iance of the energy). MonteCarlo isneededbecausethewavefunctioncontainsexp licittwo(orhigher)particle correlationsand thisresultsinanon-factoring highdimen sionalintegral. Theenergy is writtenas E=/integraltextdRψT(R)HψT(R)/integraltextdRψT(R)2=/integraltextdR\|ψT(R)\|2EL(R) /integraltextdR\|ψT(R)\|2(2.14) whereEL=HψT ψT, and is called the local energy. Other diagonal matrix eleme nts can be evaluated in a similarfashion. Off diagonal elements can al so be evaluated, but with more effort. McMillan (1965) introduced the use of Metropolis sa mpling for evaluating this integral. A typical form of the variational wave function is a Jastrow f actor (two body correla- tions)multipliedby twoSlater determinantsofonebodyorb itals. ψT=exp/bracketleftBigg −∑ i<ju(rij)/bracketrightBigg Det/parenleftBig S↑/parenrightBig Det/parenleftBig S↓/parenrightBig (2.15) 7TheJastrowfactor, u,willcontainelectron-nucleusandelectron-electroncor relations. Ap- pendixBhas detailson thederivativesthat enterintothelo cal energy. As two electrons or an electron and a nucleus get close, there is a singularity in the Coulomb potential. That singularity needs to be canceled by kinetic energy terms in the wave function. This requirement is known as the cusp conditi on. Details are given in AppendixC. Techniques for the efﬁcient handling of the determinants we re developed by Ceperley et al.(1977). The VMC algorithm is implemented so that only single electron trial moves are proposed. This causes a change in only one column of the Sl ater matrix. The new determinant and its derivatives can be computed in O(N)operations, given the inverse of theoldSlatermatrix. Thisinverseiscomputedonceatthebe ginningofthesimulationand then updated whenever a trial move is accepted. The update ta kesO(N2)operations. By comparison, computing the determinant directly takes O(N3)operations. This technique creates asituationwherethereismorework doneforanaccep tance thanforarejection. A loweracceptance ratio will be faster, sincefewer updates n eed to be performed. Details of theupdatingprocedureandsomeotherpropertiesofdetermi nantsaregiveninAppendix A. Optimization of the parameters in the wave function is a larg e topic, so we will defer the discussionuntil a later chapter. However, we will make o ne observation here. If ψTis an eigenstate, the local energy becomes constant and any MC e stimate for the energy will havezero variance. This zero-varianceprincipleallowsse arching for optimumparameters by minimizing the variance rather than minimizing the energ y. In principle this is true for anyeigenstate,not justthegroundstate. 2.4.1 TwoLevelSampling A multilevel sampling approach can be used to increase the ef ﬁciency of VMC (Dewing, 2000). Multilevelsamplinghasbeenusedextensivelyinpat hintegralMonteCarlo(Ceper- ley, 1995). The general idea is to use a coarse approximation to the desired probability function for an initial accept/reject step. If it is accepte d at this ﬁrst level, a more accurate approximation is used, and another accept/reject step is ma de. This continues until the move is rejected or until the most detailed level has been rea ched. This method increases the speed of the calculation because the entire probability function need not be computed everytime. Consider splitting the wave function into two factors - the s ingle body part ( D) and the two body part ( e−U). Treat the single body part as the ﬁrst level, and the whole w ave functionas thesecondlevel. 8First, atrialmove, R′, isproposedand accepted withprobability A1=min/bracketleftbigg 1,D2(R′) D2(R)/bracketrightbigg (2.16) If it is accepted at this stage, the two body part is computed a nd the trial moveis accepted withprobability A2=min/bracketleftbigg 1,exp[−2U(R′)] exp[−2U(R)]/bracketrightbigg (2.17) It can be veriﬁed by substitution that this satisﬁes detaile d balance in Eq. (2.9). After an acceptance at this second level, the inverse Slater matri ces are updated, as described previously. We compared the efﬁciency between the standard sampling met hod and the two level samplingmethodontwotestsystems: asingleLi 2moleculeinfreespace, andacollection of32H 2moleculesinaperiodicboxofside19.344a.u. (r s=3.0). Thewavefunctionsare taken fromReynolds et al.(1982), andwillbedescribed inSection 2.7. The step size, Δ, is the obvious parameter to adjust in maximizing the efﬁcie ncy. But wecanalsovarythenumberofstepsbetweencomputationsof EL. TheMetropolismethod produces correlated statepoints(seemoreonserial correl ationsin Section 2.6),so succes- sive samples of ELdo not contain much new information. In these tests we sample dEL everyﬁvesteps. Results for the different sampling methods with Li 2are shown in Tables 2.1 and 2.2. The Determinant Time and Jastrow Time columns include only t he time needed for com- putingthewavefunctionratiointheMetropolismethod,and notthetimeforcomputingthe local energy. The total time column does include the time for computing the local energy. Theefﬁciency isalsoshownontheleft inFigure2.1. For the two level method with Li 2, the second level acceptance ratio is quite high, indicatingthesinglebodypart isagood approximationtoth ewholewavefunction. Results for the collection of H 2molecules are given in Tables 2.3 and 2.4. The efﬁ- ciencyis alsoshownontherightgraph inFigure2.1. Comparing the maximum efﬁciency for each sampling method, t wo level sampling is 39% more efﬁcient than standard sampling for Li 2, and 72% more efﬁcient for the collec- tionofH 2’s. 2.5 DiffusionMonteCarlo Diffusion Monte Carlo (DMC) is a method for computing the gro und state wave function. It typically takes an order of magnitude more computing time than VMC, and is most efﬁcient when usedin conjunctionwithagoodVMCtrialfunct ion. 9Table2.1: TimingsforLi 2moleculeusingthestandard samplingmethod. Alltimesarei n secondson an SGIOrigin2000. Acceptance Determinant Jastrow Total ΔRatio Time Time Time ξ 1.0 0.610 48.3 340 516 1190 1.5 0.491 48.1 340 508 1680 2.0 0.407 48.2 340 503 1460 2.5 0.349 48.2 339 499 1070 3.0 0.307 48.2 339 496 800 Table2.2: TimingsforLi 2moleculeusingthetwolevelsamplingmethod. Alltimesarei n secondson an SGIOrigin2000. FirstLevel Second Level Total Acc. ΔAcc. Ratio Acc. Ratio Ratio Time ξ 1.0 0.674 0.899 0.606 400 1580 1.5 0.543 0.894 0.485 347 2430 2.0 0.447 0.897 0.401 304 2340 2.5 0.379 0.902 0.342 276 1910 3.0 0.331 0.906 0.300 256 1400 Table2.3: Timingsforthesystemof32H 2moleculesinaperiodicboxusingthestandard samplingmethod. Alltimesarein secondson aSun Ultra5. Acceptance Determinant Jastrow Total ΔRatio Time Time Time ξ 2.0 0.606 167 1089 2015 0.61 3.0 0.455 167 1085 1891 1.22 4.0 0.338 166 1084 1794 1.23 5.0 0.250 166 1080 1722 1.06 6.0 0.185 164 1080 1668 1.02 7.0 0.139 162 1084 1629 0.76 10Table 2.4: Timings for a system of 32 H 2molecules in a periodic box using the two level samplingmethod. Alltimesarein secondson aSun Ultra5. First Level Second Level TotalAcc. Total ΔAcc. Ratio Acc. Ratio Ratio Time ξ 2.0 0.740 0.795 0.589 1804 0.59 3.0 0.598 0.728 0.436 1421 1.77 4.0 0.468 0.681 0.319 1185 2.11 5.0 0.357 0.649 0.232 994 1.55 6.0 0.370 0.627 0.169 849 1.87 7.0 0.204 0.609 0.124 740 1.46 80012001600200024002800 1 1.5 2 2.5 3ξ Δstandard sampling two level sampling 0.40.60.811.21.41.61.822.2 2 3 4 5 6 7ξ Δstandard sampling two level sampling Figure 2.1: Efﬁciency of VMC. The graph on the left is for Li 2. The graph on the right is for32 H 2molecules. 11Formally, DMC and related methods work by converting the dif ferential form of the Schr¨ odingerequationintoanintegralequationandsolvin gthatintegralequationbystochas- tic methods. From another point of view, the Schr¨ odinger eq uation in imaginary time and the diffusion equation are very similar, enabling one to use a random process to solve the imaginary time Schr¨ odinger equation. This similarity was recognized early and was pro- posed as a computational scheme in the early days of computin g (Metropolis and Ulam, 1949). Unfortunately,withoutimportancesampling,itis v eryinefﬁcient computationally. Thegroundstatewavefunctioncan beobtainedbytheproject ion φ0=lim τ→∞e−τ(H−E0)ψT (2.18) whereE0is the ground state energy. This can be seen by expanding ψTin energy eigen- states, e−τ(H−E0)ψT=e−τ(H−E0)∑ iφi =∑ ne−τ(En−E0)φn. (2.19) At largeτ, the contribution from the excited states will decay expone ntially, and only the groundstatewillremain. Tomakeapracticalcomputationme thod,wewritetheprojection inthepositionbasisas ψ(R′,t+τ) =/integraldisplay dRψ(R,t)G(R→R′;τ) (2.20) whereG=/an}bracketle{tR′\|e−τH\|R/an}bracketri}htandf(R,t)is the wave function after some time t. This equation is iterated toget to thelarge timelimit. The fullyinteract ing,many-bodyGreen’s function is too hard to compute, so the various methods differ in how th ey approximate the full projector. In particular, DMC makes a short time approximat ion, and the resulting pieces have natural interpretations in terms of a diffusion proces s with branching. The name ProjectorMonteCarlo orGreen’sFunctionMonteCarloisoft enappliedtothesemethods. Perhaps unfortunately,thenameGreen’s FunctionMonteCar lo (GFMC)is alsoappliedto aspeciﬁctechniquethatusesa spatialdomaindecompositio nfortheGreen’sfunction. ForamoredetailedpresentationofDMC,withimportancesam pling,wemostlyfollow Reynolds et al.(1982). We startwiththeSchr¨ odingerequationinimaginar ytime −∂φ(R,t) ∂t=/bracketleftbig −λ∇2+V(R)−ET/bracketrightbig φ(R,t) (2.21) whereλ=¯h2/2m. ImportancesamplingisaddedbymultiplyingEq. (2.21)bya knowntrial functionψT. Theresult,writtenintermsofthe“mixeddistribution” f(R,t) =φ(R,t)ψT(R), is −∂f(R,t) ∂t=−λ∇2f+(EL(R)−ET)f+λ∇·(fFQ(R)) (2.22) 12whereEListhelocal energy,as in VMC,and FQ=2∇ψT/ψT(called thequantumforce). Onceagain, thesolutionfor fin termsofaGreen’s functionis f(R′,t+τ) =/integraldisplay dRf(R,t)G(R→R′;τ) (2.23) Forsufﬁcientlyshorttimes,wecanignorethenon-commutiv ityofthekineticandpotential terms in the Hamiltonian, e−τH≈e−τTeτV. The explicit form for the short time Green’s functioninthepositionbasisis G(R→R′;τ) = (4πλτ)−3N/2Gbranch(R→R′;τ)Gdrift(R→R′;τ)(2.24) Gbranch(R→R′;τ) =exp[−τ{¯E−ET}] (2.25) Gdrift(R→R′;τ) =exp/bracketleftBig −/braceleftbig R′−R−λτFQ(R)/bracerightbig2/4λτ/bracketrightBig (2.26) where¯E= [EL(R)+EL(R′)]/2. The algorithm is started by generating a collection of conﬁg urations (“walkers”), usu- ally sampled from ψT. Equation (2.23) proceeds by applying a drifting random wal k to each particle. Thenewpositionofthe ithparticleisgivenby /vectorr′ i=/vectorri+λτ/vectorFQ(R)+√ 2λτχ (2.27) whereχis a normally distributed random variable with zero mean and unit variance. In a simple interpretation of Eq. (2.23), this would always be th e new position. But consider the case if ψTbecomes the true ground state, φ0. The branching term is then constant and the algorithm becomes similar to VMC. In this case we want to sample the correct distribution for any τ. This is done by adding a Metropolis rejection step, where th e trial moveis accepted withprobability A=min/bracketleftbigg 1,ψT(R′)2G(R→R′,τ) ψT(R)2G(R′→R,τ)/bracketrightbigg (2.28) Each conﬁguration is thenweighted by Gbranch. Because ofrejections in thepreviousstep, thetime, τ,in Eq. (2.25)shouldbereplaced by τeff, whichis τeff=/an}bracketle{tr2 accepted/an}bracketri}ht /an}bracketle{tr2 total/an}bracketri}htτ (2.29) where /an}bracketle{tr2 accepted/an}bracketri}htisthemeansquaredisplacementoftheacceptedelectronmov esand /an}bracketle{tr2 total/an}bracketri}ht isthemean squaredisplacementofall theproposedelectron moves. Theweightingisdonebyaddingorremovingconﬁgurationsfr omthecollection(branch- ing). Thisisdonebycomputingthemultiplicity M=int(Gbranch+y), whereyisarandom numberon [0,1]. Thismultiplicityisthenumberofcopiesofthisconﬁgurat ionthatshould 13be retained in the collection of walkers. If it is zero, the co nﬁguration is deleted from the collection. If it is one, the conﬁguration remains as is. If i t is greater than one, additional copiesofthisconﬁgurationare added tothecollection. The number of walkers in the collection is kept roughly const ant by adjusting ET. In particular, thetrial energy isadjustedaccording to ET=E0+κln(P∗/P) (2.30) whereE0isthebestguessforthegroundstateenergy, Pisthecurrentpopulation, P∗isthe desired population,and κisafeedback parameter. Theenergy is computedbyaveraging thelocal energy overthe distributionofwalkers. Once the transients have decayed away, subsequent steps are part of the ground state dis- tribution. Theprogramis thenrun forhoweverlongisnecess ary to gatherstatisticsforthe energy and otherestimators. There is a problem with DMC for estimating quantities other t han the energy. The expectation value is not averaged over the ground state, but over the mixed distribution φ0ψT. Thiscan bepartlycorrected byusingtheextrapolatedesti mator, /an}bracketle{tφ0\|A\|φ0/an}bracketri}ht ≈2/an}bracketle{tφ0\|A\|ψT/an}bracketri}ht−/an}bracketle{tψT\|A\|ψT/an}bracketri}ht (2.31) . Gettingthecorrectestimator(alsocalledapureestimator )requires”forwardwalking”, so named because the weight needed, φ0/ψT, is related to the asymptotic number of chil- dren of each walker (Liu et al., 1974). This can be implemented by storing the value of the estimator and propagating it forward with the walker for a given number of steps (Ca- sullerasand Boronat, 1995). 2.5.1 Fermions In all these methods, some quantity is treated as a probabili ty, which requires that it be positive. In VMC this quantity is \|ψT\|2, which is always positive. For DMC, we sample fromψTφ0,whichcanbenegativeifthefermionnodesof ψTarenotthesameasthenodes ofφ0. The simplest cure is to ﬁx the nodes of the ground state to be t he same as ψT. This is known as the ﬁxed-node approximation. It is implemented i n the DMC algorithm by rejectingmovesthatwouldchangethesignofthedeterminan tofψT. 140.0010.0020.0030.0040.0050.0060.0070.0080.009 02468101214σ Blocking Level00.0050.010.0150.020.0250.030.035 012345678σ Blocking Level Figure 2.2: Examples of statistical data analysis using reblocking. Th e error in the graph on theleft has converged,whiletheerrorin thegraphon ther ight hasnot. 2.6 StatisticalErrors TheformulaforthevariancegiveninEq. (2.5)assumesthatt herearenoserialcorrelations in the data. However, the Metropolis sampling method produc es correlated data, which mustbeconsideredwhen estimatingthestatisticalerror. Correlations in data are quantiﬁed by the autocorrelation f unction, deﬁned for some estimator, E, as C(k) =1 (N−k)N−k ∑ i=1(Ei−¯E)(Ei+k−¯E) (2.32) Theautocorrelationtime, κ,is computedas κ=1+2 σ2cutoff ∑ k=1C(k) (2.33) Thissumtendstobequitenoisy. As aheuristicstrategy,wec an approximate κbytheﬁrst place where the autocorrelation function drops below 10%. T he true variance of the mean isthesimplevarianceoftheindividualdatapointsmultipl iedbyκ. Another way to estimatethe true error is by reblocking (Flyv bjerg and Petersen, 1989; Nightingale,1999). Atthesecondlevel,taketheaverageof every2datapoints. Nowcom- putethevarianceofthissetofdatathathas N/2points. Continuethisprocedurerecursively until the variance stops changing. Nightingale (1999) give s a well-deﬁned procedure for computingwhen thatoccurs. Figure2.2 showssomeexamplepl otsoferrorvs. reblocking level. Onthelefthandgraphweseetheexpectedplateauinth eerrorestimate. Ontheright hand graph there is no plateau, indicatingthat there is not e nough datato reliably estimate theerror. 152.7 WaveFunctions Forourstudiesofmolecularhydrogen,westartedwiththewa vefunction ψIIIfromReynolds (Reynolds et al.,1982). TheJastrowfactors are uee=−∑ i<jaerij 1+beerij une=∑ i,αZαanriα 1+benriα(2.34) whereZis the nuclear charge and bis the variational parameter. The cusp conditions are satisﬁed by setting an=1 andae=1/2. As noted in Appendix C, having the correct cusp conditionforparallelspinsdoesnotaffecttheenergymuch ,sothesamevaluefor aeisused for parallel and antiparallel electron spins. The b′sfrom the two types of Jastrow factors are foldedintoasingleparameter, β=a/b2. Theorbitalsareﬂoating Gaussians,withtheform φl(r) =exp/bracketleftbigg−(r−cl)2 w2 l/bracketrightbigg (2.35) whereclis the center of orbital, and wlis a free parameter. In molecular hydrogen, clwill beﬁxed atthebondcenter. Theorbitalscan begeneralized to beanisotropic, φl(r) =exp/bracketleftbig −(r−cl)T·RTΓR·(r−cl)/bracketrightbig (2.36) whereΓis a diagonal tensor and Ris a rotation matrix. There are two parameters - the widthalongthebonddirection(rotatedsoas tobethez-axis ), andthewidthperpendicular tothebonddirection. Theelementsof Γare deﬁned tobe (1/w2 xy,1/w2 xy,1/w2 z). Finally, additional energy reduction was found for the isol ated H2molecule by multi- plyingtheorbitalby (1+ζ\|r−cl\|), whereζisavariationalparameter. 2.8 PeriodicBoundaryConditions Theeffectsofaninﬁnitesystemcanbeapproximatedbyimpos ingperiodicboundarycon- ditions on a ﬁnite system. Every particle in the system then h as an inﬁnite number of im- ages. Inter-particle distances are calculated using the mi nimum image convention, which usesonly thedistancetotheclosestimage. Careneedstobetakenwiththewavefunctionwhenusingthemi nimumimageconven- tion. As the inter-particle distance crosses over from one i mage to another there can be a 16discontinuityin thederivativeofthewavefunction,leadi ngto adeltafunctionspikeinthe energy. Ifthisisnotaccountedfor,theVMCenergycanbecom elowerthanthetrueground state because this delta function term in the energy has been neglected. Additionally, the Gaussianorbitalscanlowertheirenergybyhavingawidthco mparabletoorlargerthanthe boxsize. Then sectionsoftheorbitalwithlarge kineticene rgy areoutside L/2, and do not get counted in the integral. This can be ﬁxed by summing over i mages, or by insuring the wavefunctionshavethecorrect behaviorat ±L/2. Weusethelattersolution. Theorbitalsaremultipliedbyacutofffunctionthatensure sitsvalueandﬁrstderivative are zero at thebox edge. Thefunctionweuseis fc(r) =1−exp/bracketleftbig −γc(r−rm)2/bracketrightbig (2.37) wherermisﬁxed at L/2 andγcisa variationalparameter. The Jastrow factors are constructed so that they obey the cor rect cusp conditions as r→0 and so that the ﬁrst and second derivatives are zero at rm≤L/2. The simplest functionthatsatisﬁes theseconditionsis acubicpolynomi al. Lety=r/rm. Then u(y) =a1y+a2y2+a3y3, (2.38) wherea1= (cuspvalue )∗rm,a2=−a1, anda3=a1/3. Variational freedom is gained by varying rm, and by adding a general function multiplied by y2(y−1)3to preserve the boundary conditions. We choose a sum of Chebyshev polynomia lsas the general function (Williamson et al.,1996). ThefullJastrowfactoris then u(y) =a1(y−y2+1 3y3)+y2(y−1)3∑ ibiTi(2y−1), (2.39) wherermandthebiarevariationalparameters. WeuseﬁveChebyshevpolynomia lsforthe electron-electron part and another ﬁve for the electron-nu clear part. We optimized one set ofrmandbiparametersforallelectron-electronpairsinanyparticul arsystem,andanother set ofparameters for allelectron-nuclearpairs. Comparisonsoftheenergyandvarianceofvariouscombinati onsofformsfortheorbital and Jastrowfactors are shownin Table2.5. The variationalp arameters are givenin Tables 2.6 and 2.7. A comparison of the electron-electron Jastrow f actors is shown in Figure 2.3. Theirshortrangebehaviorissimilar,butthelongrangebeh aviordiffersbetweenthetypes ofJastrowfactors. The quality of wave functions is often measured by the percen t of correlation energy recovered. For H 2, the HF (no correlation) energy is −1.1336 Hartrees and the exact (full correlation) energy is −1.17447 Ha. Sun et al.(1989) compared a number of forms for electron correlation functions. Their best value recovere d 80% of the correlation energy. 1700.20.40.60.811.21.41.6 012345678uee(r) rPade cubic cubic + Chebyshev Figure2.3: Optimizedelectron-electron Jastrowfactorfordifferent forms. Table2.5: Comparisonofenergiesandvariancesforvariousformsforo rbitalsandJastrow factors forasingleH 2molecule. ΨOrbital Jastrow Energy Variance % CE A Isotropic simplePad´ e −1.1598(4)0.046 64.0(9) B Anisotropic simplePad´ e −1.1643(2)0.040 75.0(6) C Anisotropic+ ζsimplePad´ e −1.1653(2)0.033 77.5(5) D Anisotropic+ ζ cubic −1.1688(2)0.039 86.1(6) E Anisotropic+ ζcubic+Chebyshev −1.1702(2)0.046 89.6(5) Usingoneoftheseforms,butwithbetteroptimization,Huan getal.(1990)recovered84% ofthecorrelationenergy. Snajdr etal.(1999)obtained93%ofthecorrelationenergyusing a Linear Combination of Atomic Orbitals (LCAO) form with 1s, 2s and 2p orbitals, and usingtheJastrowfactors ofSchmidtand Moskowitz(1990). Thevariance ishigherwiththosewavefunctionsinvolvingt hecubicpolynomial,even though the energy is lower. I believe this is mostly likely be cause the cubic polynomial does not have the correct 1 /rbehavior at large r, but the simple Pad´ e form does. This long range behavior contributes little to the average of the energy, but it contributes more signiﬁcantlytothevariance. 18Table2.6: Valuesofvariationalparameters forH 2. ΨOrbitalparameters Jastrowparameters Aw=2.74 β=9.913 Bwxy=2.514,wz=2.977 β=10.002 Cwxy=2.416,wz=2.833,ζ=0.0445β=9.958 Dwxy=2.357,wz=2.628,ζ=0.248e-erm=5.404 e-nrm=5.376 Table2.7: Valuesofvariationalparameters forwavefunctionE Component Parameters Orbital wxy=2.299wz=2.515 ζ=0.301 Electron-electronJastrow rm=6.281b0=-1.012b1=0.193 b2=0.619b3=0.025b4=0.138 Electron-nuclearJastrow rm=5.329b0=-2.084b1=0.153 b2=0.952b3=1.217b4=1.027 19Chapter3 EnergyDifferenceMethods Very often it is the difference in energy between two systems that is of interest, and not theabsoluteenergy of asinglesystem. Fora quantitysuch as the bindingenergy, wewant thedifference between theenergy of themoleculeand theene rgy ofthefree atoms. In our CEIMC simulations, we want the change in energy from moving a few nuclei. In VMC optimization, we want to know the change in energy from modif ying some of the wave functionparameters. Correlatedsamplingmethodscanprovideamoreefﬁcientapp roachtocomputingthese energy differences. But the widely used reweighting method has some drawbacks. We will introduce a new method that alleviates some of the drawb acks of reweighting while retainingitsadvantages. 3.1 DirectDifference Thesimplest,andmoststraightforwardwayofcomputingthe differenceinenergybetween two systems is to perform independent computations of the en ergy for each system. Then theenergy differenceand errorestimateare simply ΔE=E1−E2 (3.1) σ(ΔE) = =/radicalBig σ2 1+σ2 2(3.2) This method is simple and robust, but has the drawback that th e error is related to the error in computing a single system. If the systems are very si milar, either in variational parameters or in nuclear positions, the energy difference i s likely to be small and difﬁcult to resolve, since σ1andσ2are determined by the entire system. Similarities between t he systemscan beexploitedwithcorrelated sampling. 203.2 Reweighting Reweightingisthesimplestcorrelated samplingmethod. ΔE=E1−E2 =/integraltextdRψ2 1EL1/integraltextdRψ2 1−/integraltextdRψ2 2EL2/integraltextdRψ2 2 =/integraltextdRψ2 1EL1/integraltextdRψ2 1−/integraltextdRψ2 1/parenleftBigψ2 2 ψ2 1/parenrightBig EL2 /integraltextdRψ2 1/parenleftBigψ2 2 ψ2 1/parenrightBig (3.3) An estimateof ΔEforaﬁnitesimulationis ΔE≈1 N∑ Ri∈ψ2 1/bracketleftbigg EL1(Ri)−w(Ri)EL2(Ri) ∑iw(Ri)/bracketrightbigg (3.4) wherew=ψ2 2/ψ2 1. Thesameset ofsamplepointsis usedforevaluatingbothter ms. Reweighting works well when ψ1andψ2are not too different, and thus have large overlap. As the overlapbetween them decreases, reweightin ggets worse dueto large ﬂuc- tuationsintheweights. Thiseffectcanbequantiﬁedbycomp utingtheeffectivenumberof pointsappearing in thesumin Eq. (3.4), whichis Neff=∑iw2 i (∑iwi)2(3.5) Eventually, one or a few large weights will come to dominate t he sum, and the effective numberofpointswillbeverysmall,andthevariancein ΔEwillbevery large. Particularly perniciousis thecase whenthenodesdifferbe tween thetwo systems. The denominatoroftheweightcaneasilybeverysmall,causinga verylargeweightvalue. This is encountered when using reweighting to optimize orbital p arameters in VMC (Barnett et al.,1997). In Eq. (3.4) we derived reweighting by drawing points from ψ1and computing the properties of both systems from them. It could also be derive d by drawing points from ψ2 as well. We can computetheenergy differenceboth waysand ta ketheaverage. This gives usthesymmetrizedreweightingmethod, ΔE=1 2N∑ Ri∈ψ2 1/bracketleftBigg EL1(Ri)−wx(Ri)EL2(Ri) 1 N∑iwx(Ri)/bracketrightBigg +1 2N∑ Ri∈ψ2 2/bracketleftBigg wy(Ri)EL1(Ri) 1 N∑iwy(Ri)−EL2(Ri)/bracketrightBigg (3.6) wherewx=ψ2 2/ψ2 1andwy=ψ2 1/ψ2 2. 213.3 Bennett’sMethodfor FreeEnergyDifferences First let us digress to discuss computation of the normaliza tion integral. It was mentioned earlier that the Metropolissamplingmethod makes it difﬁcu lt to extract information about thenormalizationintegral,whichthepartitionfunctioni ntheclassicalcase. Bennett(1976) demonstratedamethodforﬁndingthefreeenergydifference betweentwosystems. Wewill describehismethodintermsofa ratioofnormalizations. We can compute the ratio of two normalizations, Q1andQ2, in a fashion very similar toreweighting. Q1/Q2=/integraldisplay dRψ2 1(R)/slashBig/integraldisplay dRψ2 2(R) (3.7) =/integraldisplay dRψ2 2(R)ψ2 1(R) ψ2 2(R)/slashBig/integraldisplay dRψ2 2(R) (3.8) ≈∑ Ri∈ψ2ψ2 1(Ri) ψ2 2(Ri)(3.9) This is a one-sided estimate, because it only uses samples fr om system two to compute properties of system one. Note that this sum is the same as the sum of the weights used in reweightingin Eq. (3.4). Bennett improvedonthisone-sided estimate,startingwith an identitywrittenas Q1/Q2=Q1/integraltextdRψ2 2Wψ2 1 Q2/integraltextdRψ2 1Wψ2 2(3.10) whereWis an arbitrary weight function. He found the optimum Wby minimizing the varianceofthefree energy difference, W∝1 Q1ψ2 2+Q2ψ2 1(3.11) LetQ=Q1/Q2. InsertingEq. (3.11)intoEq. (3.10),weget 1=/integraltextdRψ2 2 Q2ψ2 1//parenleftbig ψ2 2+ψ2 1/Q/parenrightbig /integraltextdRψ2 1 Q1ψ2 2//parenleftbig Qψ2 2+ψ2 1/parenrightbig (3.12) Letxrepresent theconﬁgurationssampledfrom ψ1andytheconﬁgurationssampledfrom ψ2. Theﬁnitesampleversionofthisequationis ∑ iQψ2 2(xi) ψ2 1(xi)+Qψ2 2(xi)=∑ iψ2 1(yi) ψ2 1(yi)+Qψ2 2(yi)(3.13) 22Thevalueof Qcan befound byasimpleiteration Qn+1=Qn ∑yψ2 1(yi) ψ2 1(yi)+Qnψ2 2(yi) ∑xQψ2 2(xi) ψ2 1(xi)+Qnψ2 2(xi)  (3.14) The iteration is started with Q0=1 and stopped when the correction factor in brackets is sufﬁciently close to one. Typically convergence takes less that ten iterations, but if Qis muchlarger orsmallerthanoneitcan takemoreiterations. We have written these formulas assuming that the number of sa mple points from each system is the same. Bennett derived them for case with differ ing numbers of samples in each sum, and found the best variance was usually very near an equal ratio of computer time spent on each system. In our case the systems are of equal complexity,so this means usingequalnumbers ofpointsisoptimal,orvery nearly so. Byproperlycombininginformationfrombothsystems,wecan getamuchbetter(lower variance)estimateoftheratiooftheirnormalizationstha nifwehadusedinformationfrom onlyasinglesystem(one-sidedsampling). 3.4 Two-SidedSampling We can apply this notion to computing the energy difference b etween two quantum sys- tems. Consider sampling from some distribution that contai ns information about both ψ1 andψ2. Thesimplestsuch distributionis P=1 2/bracketleftbiggψ2 1 Q1+ψ2 2 Q2/bracketrightbigg (3.15) Theenergy difference can bewrittenas ΔE=/integraltextdRψ2 1EL1 Q1−/integraltextdRψ2 2EL1 Q2 =/integraldisplay dR P/parenleftbiggψ2 1EL1 Q1P/parenrightbigg −/integraldisplay dR P/parenleftbiggψ2 2EL2 Q2P/parenrightbigg (3.16) In theﬁnitecase, wehave ΔE≈1 2N∑ Ri∈x,yψ2 1(Ri)EL1(Ri) Q1P−ψ2 2(Ri)EL2(Ri) Q2P(3.17) It is important to note that the sum covers samples taken from bothψ1andψ2. The sum includesboth“direct”terms(eg. ψ1andEL1evaluatedatconﬁgurationssampledfrom ψ1) and “cross”terms(eg. ψ1andEL2evaluatedat conﬁgurationssampledfrom ψ2). 23Thedenominatoroftheﬁrst term inEq. (3.17)is Q1P=1 2/bracketleftbigg ψ2 1+Q1 Q2ψ2 2/bracketrightbigg (3.18) Theratio Q=Q1/Q2iscomputed by theBennett method. The denominatorofthesec ond term can becomputedsimilarly. One major feature of the two-sided method is that it reproduc es reweighting in the large overlap regime, and the direct method in the low overla p regime. In the intermediate regime,itsmoothlyjoinsthetwo limits. To show this, ﬁrst consider the case where the two systems are very different and the wavefunctionshavelowoverlap. Here ψ2 1(yi)andψ2 2(xi)willbesmall. ExpandEq. (3.17) intoitsfourterms ΔE=1 2N∑ xψ2 1(xi)EL1(xi) 1 2/bracketleftbig ψ2 1(xi)+Qψ2 2(xi)/bracketrightbig−1 2N∑ yψ2 2(yi)EL2(yi) 1 2/bracketleftbig ψ2 1(yi)/Q+ψ2 2(yi)/bracketrightbig /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright direct +1 2N∑ yψ2 1(yi)EL1(yi) 1 2/bracketleftbig ψ2 1(yi)+Qψ2 2(yi)/bracketrightbig−1 2N∑ xψ2 2(xi)EL2(xi) 1 2/bracketleftbig ψ2 1(xi)/Q+ψ2 2(xi)/bracketrightbig /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright cross(3.19) Each denominator will have one large term ( ψ2 1(xi)orψ2 2(yi)) and one small term ( ψ2 1(yi) orψ2 2(xi)). ThevalueofQismoderatecomparedtothewavefunctions,s oitwillnotaffect the relative sizes of these terms. Always having one large te rm in the denominator means there will never be any excessively large contributions to t he sum resulting from division by a small value, as happens in reweighting. The cross terms h ave a small value( ψ2 1(y)or ψ2 2(x)) in the numerator, and so vanish. The large terms in the denom inators in the direct termscancel the ψ2’sin thenumerator,and weareleft with ΔE≈1 N∑ xEL1(xi)−1 N∑ yEL2(yi) (3.20) whichis justthedirect method. Now for the case where the systems are very similar and have la rge overlap. Recall from Eq. (3.9)thatwecan writeone-sidedestimatesforQ as Q=1 N∑ ywy(yi) =1/slashBig1 N∑ xwx(xi) (3.21) wherewy=ψ2 1(yi)/ψ2 2(yi)andwx=ψ2 2(xi)/ψ2 1(xi). WritethefourtermsofEq. (3.17)ina differentorder ΔE=1 N∑ xEL1(xi) 1+Qwx(xi)−Q N∑ xwx(xi)EL2(xi) 1+Qwx(xi) +1 QN∑ ywy(y1)EL1(yi) 1+wy(yi)/Q−1 N∑ yEL2(yi) 1+wy(yi)/Q(3.22) 24Approximatethedenominatorofeachtermbytwo,replacethe leadingQ’swiththeappro- priateone-sidedapproximation,and weget ΔE≈1 2N∑ x/bracketleftBigg EL1(xi)−wx(xi)EL2(xi) 1 N∑xwx(xi)/bracketrightBigg +1 2N∑ y/bracketleftBigg wy(yi)EL1(yi) 1 N∑xwy(yi)−EL2(yi)/bracketrightBigg (3.23) whichis thesymmetricversionofreweightinggivenin Eq. (3 .6). Duetocomputationalconsiderations,itisusefultodivide Eqns(3.13)and(3.17)by ψ2 1 orψ2 2as appropriate, and work with the resulting ratios wx=ψ2 2/ψ2 1andwy=ψ2 1/ψ2 2, as wasdoneinEq(3.22). Thevaluesofthewavefunctionscaneas ilyoverorunderﬂowdou- ble precision variables. It is best to use the log of the wave f unction, take differences, and then exponentiate. Furthermore, an arbitrary normalizati on of the wave functions makes no physical difference, but can result in very large or small numbers, even after taking the difference ofthe logarithms. This problemis ameliorated b y subtracting theaverage value of the log of the wave function from the individual values. So metimes this is not enough, however,andthevalueoftheenergydifferenceexceedsther angerepresentableinadouble precision variable, indicated by NaN (Not a Number). In this case, the overlap is clearly very small and the two-sided method should give the same resu lts as the direct method. The program checks for the energy difference being NaN, and i f so, it substitutes the di- rect methodresult(thedatacollectedforthetwo-sidedmet hodisasupersetofthatneeded for the direct method). Having done this, the subroutine com puting the two-sided energy difference will always return a reasonable answer, an impor tant consideration for a core routinein aprogram. 3.5 Examples The ﬁrst example is of two H 2molecules in a parallel orientation as shown in Figure 3.1. The bond lengths are at equilibrium, 1.4 Bohr, and the starti ng separation between the moleculesis d=2.5Bohr. The energy difference between that conﬁguration and conﬁgu rations with other inter- molecular distances was computed using the direct method, t he two-sided method, and reweighting. The resulting energy differences are shown on the left in Figure 3.2. Note that reweighting gets the wrong answer at large separations . This is most likely due to a ﬁnitesamplesizebias. Moreimportantis theerror in thaten ergy difference, shownon the right in Figure 3.2. Note that both reweighting and the two-s ided method have errors that 25y x1.4d Figure3.1: Two H2moleculesinaparallel conﬁguration 00.511.52 -2-101234Δ E Move distanceDirect Reweighting Two-sided 0.00010.0010.01 -2-101234Energy σ Move distanceDirect Reweighting Two-sided Figure 3.2: Energy difference(left) and theestimated statisticalerr or (on logscale)(right) fortwoH 2moleculesinaparallel conﬁguration,startingfromd=2.5 B ohr. drop to zero as the overlap increases. This graph also clearl y shows the properties of the two-sidedmethodmentionedpreviously,behavinglikerewe ightingatsmallchangesinthe separation (large wave function overlap), and smoothly cro ssing over to the direct method forat largechanges intheseparation(smallwavefunctiono verlap). Reweighting and the two-sided method may give biased result s because there are a ﬁnitenumberofsamplepointsinthesumsinEqns. (3.6)and(3 .17). Totestforthis,asum of a given length is repeated many times and the average energ y difference for that length computed. The test for a bias was performed on a Li 2molecule. The energy difference was computedbetween a bond length of4.5 Bohr and the equilib riumbond length of 5.05 Bohr. Figure3.3showstheresultsfordifferentnumbersofp ointsinthesum. Reweighting shows a much larger ﬁnite sample size bias than the two-sided method, which has almost none. 26-0.012-0.01-0.008-0.006-0.004-0.00200.0020.0040.0060.0080.01 020406080100120140160180200E(d=4.5) - E(d=5.05) Number of points in sumDirect Reweighting Two-sided Correct Figure3.3: Finitesamplesizebias intheenergy differenceofLi 2. 3.5.1 DiffusionMonte Carlo Using the two-sided method (or reweighting, for that matter ) with DMC is slightly more complicated. Thereweightingtransformationapplied toth ebasicDMCiterationgives f1(R′;t+τ) =/integraldisplay dR f(R;t)G1(R→R′;τ) (3.24) =/integraldisplay dR f(R′;t)G2(R→R′;τ)G1(R→R′;τ) G2(R→R′;τ)(3.25) Theweight w=G1/G2mustbecomputedoverseveraliterations. Theﬁnalweightus edin thecorrelated samplingformulasis aproductoftheweights ofeveryiteration. Theweightfactorisnotquiteright,duetotherejectionste p. Sincetherejectionratiofor DMC is very small ( <1%), ignoring theissueshould not introduce a large error. U mrigar and Filippi(2000)givea moresophisticatedmethodfordeal ingwithrejections. The ﬁxed-node condition also has to be obeyed, and conﬁgurat ions that cross a node whileprojectinghavetheirweightset tozero. A version of reweightingwas implementedby Wells (1985) as t he differential Green’s functionMonteCarlomethod(actuallyDMC).Heusedtheresp onsetoanexternalﬁeldto determine the dipole moment of LiH. The same trial function w as used, so the drift term was the same between both systems. Only the branching term wa s different; that entered as aweightfactor. Inourcase, thetrialfunctionandthenuc learpositionsmaybedifferent between thetwosystems. ThetopofFigure3.4showsthedifferenceinDMCenergiesusi ngthevariousmethods. 27The energy difference was computed starting from the equili brium bond length of 5.05 Bohr. Partly because of the need to project for several DMC st eps, the two-sided method has a fairly small range where it does better than the direct m ethod (compared with the range for VMC). For comparison, the VMC results are shown at t he bottom of Figure 3.4. There are two lines in the DMC graph for the direct method. The implementation had a limitationwhere onlyone projectionto accumulatetheweig hts wouldoccur at a time. We used30stepsintheprojection,andconsequentlycouldonly gettheweightsonceevery30 steps. Thelimiteddatalineiscomputedfrom datacollected onceevery 30steps(thesame amount of data available to the correlated methods) and is th e line the two-side method joins on to. The full data line used all the data available in t he simulation and so has a lowerstatisticalerror. 3.5.2 Binding Energy Tocomputethebindingenergy,letthenon-interactingsyst embeψ2,andthefullyinteract- ingsystembe ψ1. Thenuclearpositionsarethesameforbothsystems,andthe appropriate interactiontermsare setto zero forthenon-interactingsy stem. A pair of H 2molecules in a parallel conﬁguration was used as the test sys tem. The binding energy we are interested in is that of the interactin g molecules minus separate H 2 molecules(and nottheseparateatoms). EB=E((H2)2)−2E(H2) (3.26) There is a problem in that the electrons in the fully interact ing system can switch moleculesandhavenoeffectonthecomputation,buttheseco nﬁgurationsareveryunlikely in the non-interacting system. This leads to an artiﬁcially small overlap. The solution in this symmetric case is to restrict the domain of integration . The electron coordinates are ordered alongthex-axisso that x1<x2andx3<x4. To seethatthisrestrictionis exact,considertheintegral /integraldisplay∞ −∞dx1/integraldisplay∞ −∞dx2f(\|x1−x2\|)g(x1,x2) (3.27) wherefcorresponds to the electron-electron Jastrow factor and gis symmetric under the interchange of x1andx2and corresponds to the electron-nucleus Jastrow factor and the square of the Slater determinant. Change variables to R= (x1+x2)/2 andr=x1−x2. Nowwehave /integraldisplay∞ −∞dR/integraldisplay∞ −∞dr f(\|r\|)g(R+r/2,R−r/2) (3.28) Theintegralover riseven,andweonlyneedtointegrateoverhalfoftheinterva l(r<0or r>0), whichcorrespondsto therestrictions x1<x2orx1>x2. 280.00010.0010.010.11 4.24.34.44.54.64.74.84.95Error in Energy Difference Bond Lengthdirect, full data direct, limited data reweighting two-sided 0.00010.0010.01 4.24.34.44.54.64.74.84.95Error in Energy Difference Bond Lengthdirect reweighting two-sided Figure3.4: Errorinenergy differenceofLi 2usingDMC(top)and VMC(bottom) 290.00010.0010.010.11 0246810121416Error in Binding Energy H2 - H2 distanceTwo-sided, Restricted Reweighting, Restricted Two-sided, Unrestricted Reweighting, Unrestricted Direct Figure3.5: ErrorinVMCbindingenergy ofH 2-H2system Figure 3.5 shows the error in the VMC binding energy for vario us intermolecular dis- tances. Withoutrestrictingthedomainofintegration,rew eightingperformspoorly,andthe two-sided method reproduces the results of the direct metho d. With the restricted domain, thecorrelated methodsperform quitewell. 30Chapter4 WaveFunctionOptimization VariationalMonteCarlo(VMC)dependscruciallyontheopti mizationofparametersinthe wave function to ﬁnd the minimumenergy. The general problem of function optimization is a well-studied area. For a general introduction to variou s optimization techniques, see Presset al.(1992). For more in-depth work, consult Polak (1997), Denni s and Schnabel (1983),orOrtegaand Rheinboldt(1970). The main difﬁcultly in applying these techniques to optimiz ing VMC wave functions is noise - we only get stochastic estimates for function valu es or gradients. Glynn (1986) describes several strategies for optimizationin the prese nce of noise. We will dividethese intothreecategories. The ﬁrst strategy is to convert the problem into a nearby smoo th, non-noisy problem, andsolvethatprobleminstead. Fixedsamplereweightingta kes thisapproach bysampling someset ofconﬁgurationsandoptimizingwithjustthesecon ﬁgurations. The second approach is to reduce the noise to negligible leve ls, and proceed with reg- ular optimization techniques. This is possible with a Newto n method, where the ﬁrst and second derivatives of the function are computed, and the num ber of iterations needed for convergencehopefullyissmall. Thethirdapproachistouseamethodtailoredtohandlenoise . TheStochasticGradient Approximation(SGA)is suchamethod. Alsosomewhatinthisc ategory,wewillexamine a method that is essentially a biased random walk, and the mov es are accepted or rejected based onthewhetherornottheenergy decreases. These approaches will be compared on several problems of dif ferent sizes to see how theyscale. WewilluseasingleH 2molecule,andcollectionsof8,16,and32H 2molecules ina boxas trialproblems. 314.1 Energyvs. VarianceMinimization There is a choice of objective functions - either the energy o r the variance of the energy can be minimized. Under certain circumstances, variance mi nimization is more stable that energy minimization. For the reweighting method, this is deﬁnitely true, but it may not be the case for the other optimization techniques. It is g enerally held that variance optimization would produce better values for observables o ther than energy (Williamson etal.,1996),butthismaynotalwaysbethecase(Snajdr etal.,1999;SnajdrandRothstein, 2000). The argument is that the variance is more sensitive to parts of the wave function that do not contribute to the energy. As we have seen in Chapte r 2, having incorrect long range behavior in the H 2wave function does not affect the energy much, but does cause thevariancetorise. Inotherwords,varianceminimization shouldyielda“smoother”wave function,whichshouldthenhavebetternon-energypropert ies. 4.2 FixedSampleReweighting Fixed sample reweighting with minimization of the variance was popularized by Umrigar et al.(1988), and has been used extensively since then. The curren t state of the art is described byKent(Kent etal., 1999;Kent,1999). The core of the method is the single sided reweighting method described in Chapter 3. A numberofconﬁgurations are sampledfrom adistribution withvariationalparameters α0. Theenergy at an arbitrary valueofthevariationalparamet er,α, iscomputedby E(α) =∑ iw(Ri;α)EL(Ri;α)/∑ iw(Ri;α) (4.1) wherew(Ri;α) =ψ2(Ri;α)/ψ2(Ri;α0). Alternatively,onecould computethevarianceby A(α) =∑ iw(Ri;α)(EL(Ri;α)−ET)2/∑ iw(Ri;α) (4.2) whereETcan either be the weighted average energy (4.1) or it could be a guess at the desired energy. The weights in these expressions can get very large when the v ariational parameters move far from the sampled value α0, and especially when the parameters that affect the nodes are adjusted. Then just a few conﬁgurations will domin ate the sum, and the energy estimator can often give meaningless low values. The varian ce estimator, however, will remain more stable in this situation. For either estimator, the best ﬁx is to regenerate the conﬁgurations being used when the parameters move too far aw ay fromα0. This can be used inconjunctionwiththeenhancementsdescribed below. 32A second advantageofthevarianceestimatoris thattheweig htscan bemodiﬁedwith- out changing the locationof theminimum(Kent et al., 1999). The sameis not truefor the energy. Theproblem ofa few large weightscan be solvedby lim itingthemto a maximum value(Filippiand Umrigar,1996),ormoresimplyby justset tingthemallto one(Schmidt and Moskowitz, 1990). Barnett et al.(1997) tame the ﬂuctuating weights by sampling from apositivedeﬁniteguidingfunction. In thiscases, ETshouldbeset toabest guess,or slightlybelow,becausetheenergy estimatorwillnotberel iable. Further increases in stability can be gained by limiting out liers in Eq. (4.2). Large outlyingvalueshaveadisproportionatelylargeeffectont hevariance,buttheircontribution is not that meaningful. Kent et al.(1999) gives a procedure choosing a cutoff that will reduceitseffectasthenumberofsamplesincreases. Weused asimplerapproach,removing from thesumanyvaluesgreater than5 standard deviationsfr om theaverage. Another efﬁciency improvement can be exploited when the Jas trowUis linear in the variationalparameters. Thenthevariationalparametersc anbefactoredoutofthesumover interparticle distances, and the value of that sum can be sto red. Fixed sample reweighting has been applied mostly to optimizing Jastrow factors, and n ot parameters in the Slater determinant, so this results in a dramatic time savings. In o ur case we have both Jastrow and determinantal parameters, and the code spends about 40% of its time computing the Jastrowfactor. Thispercentagewilldecreaseasthesystem sizeincreases,sincetheJastrow computation isO(N2)but the determinantal part requires matrix work that is O(N3). We did not implement this improvement, so bear in mind when peru sing the results that the reweightingtimecould bereduced, probablyby30%. An additional advantage of reweighting is that, since it is s olving a smooth problem, an off-the-shelf minimizer can be used. We used the DSMNF gen eral minimizer from the PORT library, which uses only the function values and doe s not need any derivatives. Routines to minimize sums of squares are also available, but we did not try them. The ﬁxed samplereweightingalgorithmisthen: Generate aset of conﬁgurationsand minimize the variance with this set. Generate a new set of conﬁguratio ns using the new variational parameters and ﬁnd the minimum variance again. Repeat for se veral steps to ensure con- vergence. 4.3 NewtonMethod The Newton method makes use of the ﬁrst and second derivative s. We can approximate a functionnearitsminimumas aquadraticsurface f(x)≈f(x0)+(x−x0)T·b+(x−x0)T·A·(x−x0) (4.3) 33wherebi=∂f ∂xandAij=∂2f ∂xi∂xjis the Hessian matrix. The location of theminimumis then givenby x0=x−A−1·b (4.4) Since weare likelyto startin aregion where fis not quadratic,this stepis iterated several times. This procedure, along with analytic evaluation of the deriv atives, was applied to VMC energy minimization by Lin et al.(2000). Analytic derivatives of the local energy with respect to determinantal parameters are given by Bueckert et al.(1992), but these were used inthecontextofareweightingminimization. Recall theVMCenergy is computedby E=/an}bracketle{tEL/an}bracketri}ht=/integraldisplay dRψ2(α)EL(α)/slashbigg/integraldisplay dRψ2(α) (4.5) WewantthederivativesofEwithrespecttovariousvariatio nalparameters, α. Thesecould be computed with ﬁnite differences and reweighting, but it i s better to do some analytical work onthisexpressionﬁrst. Linetal.(2000)usesomeGreen’srelationstoeliminateexplicitder ivativesofthelocal energy (Ceperley et al.,1977), and derivethefollowingexpressionforthegradien t, ∂E ∂αm=2/bracketleftbig /an}bracketle{tELψ′ ln,m/an}bracketri}ht−/an}bracketle{tEL/an}bracketri}ht/an}bracketle{tψ′ ln,m/an}bracketri}ht/bracketrightbig (4.6) where ψ′ ln,m=∂lnψ ∂αm=1 ψ∂ψ ∂αm. (4.7) Theyalso givetheexpressionfortheHessian, ∂2E ∂αm∂αn=2/braceleftbig /an}bracketle{tELψ′′ ln,m,n/an}bracketri}ht−/an}bracketle{tEL/an}bracketri}ht/an}bracketle{tψ′′ ln,m,n/an}bracketri}ht +2/bracketleftbig /an}bracketle{tELψ′ ln,mψ′ ln,n/an}bracketri}ht−/an}bracketle{tEL/an}bracketri}ht/an}bracketle{tψ′ ln,mψ′ ln,n/an}bracketri}ht/bracketrightbig −/an}bracketle{tψ′ ln,m/an}bracketri}ht∂E ∂αn−/an}bracketle{tψ′ ln,n/an}bracketri}ht∂E ∂αm +/an}bracketle{tψ′ ln,mE′ L,n/an}bracketri}ht/bracerightbig (4.8) where E′ L,n=∂EL ∂αn(4.9) and ψ′′ ln,m,n=∂2lnψ ∂αm∂αn. (4.10) 34Computing the ﬁrst derivatives of the wave function analyti cally is relatively easy. Computing the second derivatives with respect to parameter s in the Jastrow factor is also easy analytically. For parameters that appear in the determ inant, however, second deriva- tivesaremoredifﬁcult. Forthisreasonwecomputemostofth eﬁrstderivativesanalytically, andusethesetocomputethesecondderivativeswithasimple ﬁnitedifferencescheme. The ﬁrst derivativeof the local energy was computed with ﬁnite d ifferences. The derivativeof theorbital cutoffparameter, which is thesamefor all theor bitals,was also computedwith ﬁnitedifferences. An advantage of the Newton approach over the gradient-only a pproaches is that it has informationabouthowbigofstepshouldbetaken,whereasth estepsizeisaparameterthat mustbetunedin thegradient-onlymethods. Thedrawback, th ough,isagreater sensitivity to noise. The gradient and Hessian must be sufﬁciently accur ate, or the Newton iteration will get wildly wrong results. More precisely, it is the non- linear process of taking the inverseinEq. (4.4)thatcausestheproblem. Furthermore,t hissensitivitytonoiseincreases withthenumberofparameters. Another problem is parameter degeneracy, or near degenerac y. This will make the Hessian singular, or nearly so. Even if it not exactly singul ar, being nearly singular is the equivalent of dividing by a small number, which will also gre atly magnify the effects of noise. The usual solution is use of the Singular Value Decomp osition (SVD). See Press et al.(1992) or Kincaid and Cheney (1991) for a description of the a lgorithm. A more detailed look at ”regularization” (of which the SVD is one me thod) is in Hansen (1998). TheSVD startsby decomposingamatrixas A=PDQ (4.11) wherePandQare unitary matrices and Dis a diagonalmatrix. The elementsof Dare the eigenvalues of ATA. For our square, symmetric matrix, these are are the squares of the eigenvalues of A. We can also take PandQto be the eigenvectors of A. The utilityof the SVD is seenwhen wewritetheinverseof Aas A−1=QTD−1PT(4.12) IfAis singular, then at least one of its eigenvalues is zero. In t his case, zero eigenvalues also indicate parameter degeneracy, so it’s not really nece ssary to move in the directions corresponding to the zero eigenvalues. To avoid moving in th ese these directions, and to stabilizetheinverse,set1 /diinEq. (4.12)tozero when diissmallerthansomecutoff. With the eigenvalue decomposition we have an additional tec hnique - negative eigen- values correspond to uphill directions and mean we are at a sa ddle point or are far from a 35-1.16-1.15-1.14-1.13-1.12-1.11-1.1 0 0.5 1 1.5 2Energy (Ha/molecule) Time (Hours)Ns=4000 Ns=8000 Ns = 8000, no SVD Ns=16000 Ns=32000 Ns = 32000, no SVD Figure4.1: ExamplesusingtheNewtoniterationwithvaryingamountsof noise. region where the quadratic approximation is good.1The simplest way of handling this is to ignore negative eigenvalues. So we remove small positive and all negative eigenvalues when solvingEq. (4.4). Some examples of this Newton iteration with 8 H 2molecules are shown in Figure 4.1. NsisthenumberofsamplesusedincomputingthegradientandHe ssian. Unlessotherwise noted, the SVD method for solving Eq. (4.4) was used with remo val of eigenvalues less than0.01. Otherrunswithoutusingregularizationarenotshownbe causetheydivergevery drastically. 4.4 StochasticGradientApproximation TheStochasticGradientApproximation(SGA)wasdesignedb yRobbinsandMunro(1951) to handle optimization with noisy gradients. It was ﬁrst app lied to VMC optimization by Harjuet al.(1997). TheSGA iterationcan bewritten as αi=αi−1−hγi∇αE(αi−1). (4.13) wherehisa stepsizeparameterand γiis somespeciallychosen series. 1MuchofthecomplexityofcurrentNewtonandquasi-Newtonop timizationmethodsisindecidinghowto movetheparameterswhenthequadraticapproximationisnot good. Typicallyitinvolvesalineminimization inthe gradientdirectionorsomesortofbacktracking. 36There are some conditions on γithat must be satisﬁed in order for this iteration to converge. Theyare γi>0 (4.14) ∞ ∑ i=1γi=∞ (4.15) ∞ ∑ i=1γ2 i<∞ (4.16) The condition given by Eq. (4.15) allows the iteration to rea ch anywhere in parameter space. The condition in Eq. (4.16) is needed so the effects of noise will eventually be damped out. An obvious choice for γiis 1/i. For more discussionon these conditionsand forsomeconditionson theobjectivefunction,seeYoung(19 76)and Tsypkin(1971). Wecananalyzetheconvergenceinthelimitingcaseofnonois einonedimension. First letusmakeacontinuousversionofEq. (4.13)byletting γ(t) =dt/tanddα=α(t)−α(t− dt). Then inthe dt→0 limit,theSGA iterationis dα dt=−h t∇αf(α(t)) (4.17) Now let us assume that fhas a quadratic form, f(α) =1 2Aα2+Bα+f0, with a mini- mumatα=−B/A. NowEq. (4.17)is dα dt=−h t[Aα+B] (4.18) Thesolutionis α(t) =−B/A+a0t−hA(4.19) wherea0is a constant of integration. So we see that it will converge t o the solution at t→∞, witharatethatis controlledbythecurvatureofthepotent ialandourchoiceof h. Now consider generalizing to the case where γ(t) =dt/tδ. Our continuous equation is then dα dt=−h tδ[Aα+B] (4.20) Thesolutionis α(t) =−B/A+a0exp/bracketleftBig −hAt1−δ/(1−δ)/bracketrightBig (4.21) Weseethatthesmaller δis,thefastertheconvergence. Iftherewerenonoise, δ=0would indeed bethebest choice. Now let us represent the noise in the gradient with an additiv e noise term, η(t). Then Eq. (4.20)is dα dt=−h tδ[Aα+B+η(t)] (4.22) 37Previously we considered case where noise was negligible. N ow consider the case where thenoisedominates,so Eq. (4.22)becomes dα dt=−η(t) tδ(4.23) Thesolutionistheintegral α(t) =−/integraldisplayT dtη(t) tδ(4.24) To look at convergence, we need to compute the variance of αintegrated over the noise. Takethenoisetohaveaprobabilitydistribution P(x,t)withzeromeanandvariance σ. The varianceof αisthen σ2(α) =−/integraldisplay∞ −∞dx/integraldisplayT dt P(x,t)x2 t2δ(4.25) Ifwetake P(x,t)to haveno dependenceon t,theintegralsfactorand weget σ2(α) =−σ2 1−2δ1 T2δ−1(4.26) Hereweseethatlargervaluesof δleadtofasterconvergenceofthenoise. Sincesmaller values of δlead to faster convergence of the non-noisy problem, we need an intermediate valueofδto balancetheseeffects. Onevariation,suggestedbyNemirovksyandYudi(1983),ist ouseδ=1/2andusethe cumulativeaverage of the variational parameters. This val ue ofδviolates the condition in Eq. (4.16),butthisconditionistheretoinsurethenoisypa rtconverges. Insteadweusethe cumulativeaveragingprocess toremovethenoise. Another acceleration technique involves monitoring the si gn of the gradient (Tsypkin, 1971). Far from the minimum the gradient will not often chang e sign between successive steps. Close to the minimum, the noise will eventually domin ate, and the gradient will change sign more often. The acceleration procedure is to onl y updateγiwhen the sign of the gradient changes. This also has the advantage of adjusti ng the convergence of each parameterseparately. Inpractice,startingtheseriesat γ1=1tendstomaketheﬁrststepshaveadramatically largereffectontheparametersthansubsequentsteps. Ofte n,theﬁrstfewstepswouldmove the parameters very far from the minimum, and then the iterat ion will take a long time to converge. In thiswork westarted theseries at i=10 tominimizethiseffect. We tried several of these SGA variants on the box of 8 H 2molecules. We used h=3 whenγi=1/√ iandh=10whenγi=1/i. Thiswaytheinitialstepsizes(giveby hγi)were similar. Figure 4.2 shows the convergence of one of the varia tional parameters ( rmfor the electron-electron Jastrow). The convergence of the energy is also shown. We see that the twoaccelerated methodsconvergefasterthanthesimpleSGA . 3844.24.44.64.85 10 100 1000 10000 100000e-e rm iγ = 1/i γ = 1/i1/2 γ = 1/i1/2,cumulative average γ = 1/i, accelerated -9.275-9.27-9.265-9.26-9.255-9.25 100 1000 10000 100000Energy iγ = 1/i γ = 1/i1/2,cumulative average γ = 1/i, accelerated Figure 4.2: Examples of SGA. The graph on the top shows the convergence of one varia- tionalparameterforseveralSGA algorithms. Thegraph onth ebottomshowstheresulting energy. 394.5 GradientBiasedRandomWalk We introduce a new method that is made possible by the two-sid ed energy difference method in Chapter 3. Using this, it is relatively easy to dete rmine whether a change of thetrial parameters lowers theenergy or not. This determin ationcan be ﬁtted onto a num- berofmethods,evenarandomwalk. Weevaluatethegradienta ndmakeatrialmoveinthe gradientdirection,similartotheSGA.UnliketheSGA,them oveisacceptedonlyifitlow- erstheenergy. Sincethegradientisnoisy,weareeffective lymakingarandomwalkthatis biasedin thegradientdirection,hencethenameGradient Bi ased Random Walk(GBRW). Thetrial moveis αT=αi−1−h∇αE(αi−1). (4.27) wherehis randomly chosen from [0,hmax]. To provide some simple adaptivity, hmaxis adjusted during the run. If a trial moveis rejected, hmaxis decreased via multiplicationby somefactor,usually0.5or0.6. Ifatrialmoveisaccepted,i tisincreasedbymultiplyingby thereciprocal ofthatsamefactor. Currently thelevelof convergenceofthis methodiscontrol led by howwell theenergy difference is computed. In other words, once the energy diff erences are of the same size as the estimated error, it simply ﬂuctuates. There are sever al possibilities for making a convergent method. The ﬁrst is to take the cumulative averag e of the parameters, or add a damping parameter as in the SGA. The second is to increase th e number of samples to computetheenergy difference(and sodecrease thenoise)at each iteration. 4.6 Comparisonofmethods We test the various optimization methods and compare their r un times. The test systems are an isolated H 2molecule, and 8,16, and 32 molecules in a box at rs=3.0, a fairly low density. Each system has 12 parameters in the Jastrow fac tor, and 3 determinantal parameters per molecule, plus one more for the box cutoff (wh ich is the same for all the orbitals). Thus we have 15, 37, 61, and 109 variational param eters, respectively. For the starting parameters,we set the Jastrow cutoff to rm=4.0, the orbital widths to 2 .0, the orbitalboxcutoffto1 .0,and alltheotherparameters tozero. TheNewtonmethodusedtheregularizationmethodwithacuto ffof0.01forN=8,16 and a cutoff of 0 .1 forN=32. No regularization was used for N=1. The SGA method usedγi=1/√ iandparameteraveraging. Reweightingused16000conﬁgurat ionsforN=1 and 1000 conﬁgurations for N=8 and 16. We did not attempt reweighting on the largest system. 40-1.171-1.169-1.167-1.165-1.163-1.161 00.2 0.4 0.6 0.8 1Energy Time (hr)SGA GBRW Reweighting Newton -1.16-1.158-1.156-1.154-1.152-1.15-1.148-1.146-1.144 0123456789Energy (Ha/molecule) Time (hr)SGA GBRW Reweighting Newton (a) (b) -1.17-1.16-1.15-1.14-1.13-1.12-1.11-1.1 0 5 10 15 20 25Energy (Ha/molecule) Time (hr)SGA GBRW Reweighting Newton -1.17-1.16-1.15-1.14-1.13-1.12-1.11-1.1 020406080100120140Energy (Ha/molecule) Time (hr)SGA GBRW Newton (c) (d) Figure 4.3: Optimizationmethods applied to (a) SingleH 2(b) 8 H 2’s (c) 16 H 2’s (d) 32 H2’s The best way to compare these methods would be to run them all m any times starting from different random number seeds. The average of the resul ting distributionwould give the average quality of each method, and the spread of the dist ribution would indicate the stability. However,thisistime-consumingandinstead,as aﬁrst approximationwepresent theresultsforasinglerunofeachmethodinFigure4.3. Thet imesareinhoursonanAMD Duron 600 Mhz (which is approximately 1/2 to 2/3 the speed of a 195 Mhz R10000 in an SGIOrigin). For the single molecule, it is clear that the Newton method is the best method. The reweighting method also performs well, and the two gradient methods take longer to con- verge. Asthesystemsizeincreases,however,thegradientm ethodsdobetter,withtheSGA methoddoingthebest. The Newton method in particular has difﬁculty with stabilit y as the system size in- creases. It needs to be run long enough so the noise is small en ough that it does not affect 41theresults. The reweighting method performs surprisingly poorly on the larger systems. Looking more closely at the results of reweighting for the N=8 case, we get a total energy of −9.244(2)Ha and a variance of 0 .30. From the SGA we get E=−9.275(3)Ha and a variance of 0 .42. From the GBRW we get E=−9.268(2)Ha and a variance of 0 .36. It appears that the problem is with the variance minimization a nd we have a case where the minimumvariance solutionis not thelowestenergy solution . Althoughon the scale ofthe total energy, the difference between reweighting and the SG A is only 0 .3%. On the scale ofthecorrelationenergy in theisolatedmolecule,thisdif ferenceis about10%. 4.7 FutureWork We have compared a few basic methods for VMC parameter optimi zation. Many more improvementsand modiﬁcationscouldbeconceivedand tried . Currently we ran these with set numbers of iterations and num bers of samples, then looked at the results, and perhaps made adjustments and trie d again. What would be very helpful is some sort of adaptivity - adjusting the number of s amples or even the type of methodas theoptimizationproceeds inorderto ensureconve rgence. So far the gradient-only methods seem to have the advantage, but have the disadvan- tage that they require a step size be set manually. In order to generate a trial step size automatically, a secant updating method could be tried, whe re successive gradient evalu- ations are used to build up an approximate inverse Hessian (D ennis and Schnabel, 1983). These methods are often superior to using the actual Hessian (Presset al., 1992) , but it is notclear howthepresence ofnoisewillaffect thealgorithm . Finally, it would be instructive to perform these compariso ns on systems containing atomswithhigheratomicnumber. 42Chapter5 Coupled Simulation Methods Thereareseveralissueswehavetodealwithwhenconstructi nganefﬁcientCEIMCsimula- tion. TheﬁrstisnoisefromtheQMCevaluationoftheenergy. Wewilldiscussamodiﬁca- tion to the Metropolis acceptance ratio, called the penalty method, that will accommodate noise. Next we will examine some of the details involved in a C EIMC simulation, and ﬁnally giveresultsforasingleH 2molecule. 5.1 PenaltyMethod TheMetropolisacceptanceratio,fromChapter2,ismin [1,exp(−Δ)],whereΔ=β[V(s′)− V(s)]. TheQMCsimulationwillyielda noisyestimatefor Δ, which wedenoteas δ The exponential in the acceptance ratio is nonlinear, so tha t/an}bracketle{texp(−δ)/an}bracketri}ht /ne}ationslash=exp(/an}bracketle{t−δ/an}bracketri}ht). Thenoisewill introducea bias intoour acceptance ratio for mula. To avoid thisbias in our simulations,wecaneitherrununtilthenoiseisnegligible ,orwecantryﬁndamethodthat toleratesnoise. Typical energy differences for moves in our simulations are on the order of .01−.05 Ha. If we want an error level of 10% (statistical error of .001 Ha)1it would take about 7 hours of computer time for a system of 16 H 2molecules. We need to perform hundreds of these steps as part of the classical simulation, so clearl y a method that could tolerate highernoiselevelswould beverybeneﬁcial. Thepenaltymet hodofCeperley and Dewing (1999) does this, and our simulations run with noise levels o n the order of .01 Ha, which onlytakes about4 minutesofcomputertime. In thepenaltymethod,westartwithdetailed balance, writt enas A(s→s′) =A(s′→s)exp[−Δ]. (5.1) 1Theusualerrorlevelconsideredchemicalaccuracyis1kcal /mol= .0016Ha 43To deal with noise, we would like to satisfy detailed balance on average, We introduce an instantaneous acceptance probability, a(δ), that is a function of the estimated energy difference. The average acceptance probability is the inst antaneous one averaged over the noise, A(s→s′) =/integraldisplay∞ −∞dδP(δ;s→s′)a(δ) (5.2) Thedetailedbalance equationwewouldliketosatisfyis the n /integraldisplay∞ −∞dδP(δ;s→s′)/bracketleftBig a(δ)−e−Δa(−δ)/bracketrightBig =0 (5.3) Supposethenoiseisnormallydistributedwithvariance, σ. Then P(δ) = (2σ2π)−1/2exp/bracketleftbigg −(δ−Δ)2 2σ2/bracketrightbigg (5.4) A simplesolutionto Eq. (5.3)is a(δ) =min/bracketleftbigg 1,exp(−δ−σ2 2)/bracketrightbigg (5.5) Theextra −σ2/2termcausesadditionrejectionsoftrialmovesduetonoise . Forthisreason itis calledthepenaltymethod. To verify that the solution in Eq. (5.5) satisﬁes detailed ba lance (5.3), let us compute theaverageacceptance probability A(Δ) =1√ 2σ2π/integraldisplay∞ −∞dδe−(δ−Δ)2/2σ2min/bracketleftBig 1,e−δ−σ2/2/bracketrightBig =1√ 2σ2π/integraldisplay−σ2/2 −∞dδe−(δ−Δ)2/2σ2+1√ 2σ2π/integraldisplay∞ −σ2/2dδe−(δ−Δ)2/2σ2e−δ−σ2/2 =1√ 2σ2π/integraldisplay−σ2/2−Δ −∞dδ′e−δ′2/2σ2+1√ 2σ2π/integraldisplay∞ σ2/2−Δdδ′′e−Δe−δ′′2/2σ2 =1 2erfc((σ2/2+Δ)/2σ2)+1 2e−Δerfc((σ2/2−Δ)/2σ2) where we have made the substitutions δ′=δ−Δandδ′′=δ′+σ2. This expression for A(Δ)willsatisfydetailedbalance, A(Δ) =e−ΔA(−Δ). In practice, both the energy difference and the error are bei ng estimated from a ﬁnite set of data. Assumewe have nestimates for theenergy difference, y1,...,yn. Estimates for themeanand varianceare givenby δ=1 nn ∑ i=1yi (5.6) χ2=1 n(n−1)n ∑ i=1(yi−δ)2(5.7) 44and wehave Δ=/an}bracketle{tδ/an}bracketri}htandσ2=/an}bracketle{tχ2/an}bracketri}ht. The average acceptance ratio can be written as integral over δandχ2. The probability distributionfortheestimatederrorisachi-squareddistr ibution. Anasymptoticsolutioncan beformedbyexpanding a(δ,χ2)andperformingtheintegralstogettheaverageacceptance ratio. Thisis setequal toa powerseries forexp (−σ2/2),and bymatchingpowersof σwe get the coefﬁcients for the original series for a(δ,χ2). This series can by summed to get a Bessel function,hencewecallit theBessel acceptance form ula. It isconvenientto expand thelogoftheBesselacceptanceformulainpowersof χ2/n. TheBesselacceptanceformula isthen a(δ,χ2,n) =min[1,exp(−δ−uB)] (5.8) where uB=χ2 2+χ4 4(n+1)+χ6 3(n+1)(n+3)+χ8(5n+7) 8(n+5)(n+3)(n+1)2)+··· (5.9) Note that as ngets large, only the ﬁrst term is important, which is just the regular penalty method. 5.1.1 Other methods There is another method for handling noise, originally prop osed by Kennedy, Kuti, and Bhanot (Kennedy and Kuti, 1985; Bhanot and Kennedy, 1985), t hat uses a power series expansion of exp [−δ]to construct an unbiased acceptance ratio. It has an advanta ge over thepenalty method in that it does not assumeany particulard istributionfor thenoise. The method has a major drawback in that it depends on the value of δnot becoming too large, and not just the error estimate for δ. This could severely restrict the maximum steps sizes for moving the nuclei in our simulations. Methods for dealin g with this restriction has recently been addressed by Lin et al.(1999) and Bakeyev and de Forcrand (2000), but we didnotexploretheseextensions. 5.1.2 Handling noisy data Using noisy data requires care in handling. Particularly, i nappropriate reuse of any single estimated value can lead to biased results. For instance, in a classical simulation the en- ergy difference would be computed, and one of the two energie s involved would be used in accumulatingtheaverageenergy. See thetopof Figure5.1 foran outlineof sucha sim- ulation. However, this leads to a bias when noisy energies ar e involved. This can be seen by considering a negative ﬂuctuation in the energy of the tri al move. This will make the 45energy difference smaller (or more negative) and hence more likely to be accepted. Thus the negative ﬂuctuations would be preferentially added to t he accumulated average, and biastheresultdownward. This program outline could be corrected by computing a new va lue for the energy in the average. However, there is another arrangement that is m ore amenable to the energy differencemethodsof Chapter3. Thecomputationoftheener gy used intheaverage isthe same quantity needed for the old energy in the next iteration . So that computation can be movedto thenextiteration,as shownon thebottominFigure5 .1. Several points are illustrated with a system of two particle interacting via a Lennard- Jonespotentialwith ε=0.1andσ=1.52. Thetemperatureofthesystemwas3160K( β= 100). Noise was simulated by adding a Gaussian random variab le with known variance to every energy computation. Results for several algorithm s versus noise level are shown in Figure 5.2. The top curve shows the bias that results from h aving no penalty method. The middle curve is the correct method, which we see is indepe ndent of the noise level. The bottom curve demonstrates the bias from reusing the ener gy involved in making the accept/reject decision. The noise level of a system can be characterized by the relati ve noise parameter, f= (βσ)2t/t0, wheretis the computer time spent reducing the noise, and t0is the computer time spent on other pursuits, such as optimizing the VMC wave function or equilibrating theDMCruns. A small fmeans littletimeis beingspenton reducingnoise,where ala rge fmeans much time is being spent reducing noise. The efﬁciency of a CEIMC simulation can be written in terms of this parameter. Our paper gives an e xample of a double well potential, and ﬁnds the noise level that gives the maximum ef ﬁciency. Generally it falls aroundβσ=1−2, with theoptimialnoiselevelincreasing as therelativen oiseparameter increases. Theoneexceptionoccurswhencomputingtheﬁrst moment,whichissensitiveto crossingthebarrierbetweenthedoublewells. Thesecrossi ngsareassistedbyanincreased noiselevel,hencetheoptimalnoiselevelismuchhigher. 5.2 Pre-rejection We can apply multi-level sampling ideas (see section 2.4.1 f or an application to VMC) to our CEIMC simulations as well. The idea is to use an empirical potential to ”pre-reject” movesthatwouldcause particlestooverlapand berejected a nyway. In this case, the trial move is proposed and accepted/reject ed based on a classical po- 2Hereσisthe lengthscale fortheLJpotential 46Computeold Energy loopoverClassical steps loopoverNumberofmolecules maketrialmove(translationand rotationofH 2molecule) ComputetrialEnergy acceptance probability =min/bracketleftbig 1,exp/parenleftbig −βΔE−(βσ)2/2/parenrightbig/bracketrightbig accept/reject trial move ifaccept, set oldEnergy =trialEnergy UseupdatedoldEnergy inaverage end loop end loop loopoverClassical steps loopoverNumberofmolecules maketrialmove(translationand rotationofH 2molecule) ComputeoldEnergy Computetrial Energy acceptance probability =min/bracketleftbig 1,exp/parenleftbig −βΔE−(βσ)2/2/parenrightbig/bracketrightbig accept/reject trial move Useold Energyin average end loop end loop Figure5.1: CEIMCprogramoutlines. Boxesindicatequantumcomputatio ns. Thedashed boxindicatesaquantitysavedfromapreviouscomputation. Thetopalgorithmisincorrect. Thebottomalgorithmis correct. 47-0.1-0.095-0.09-0.085-0.08-0.075 00.511.522.533.544.5Energy Noise level ( βσ)w/o penalty w/ penalty, reusing energy w/ penalty, new energy Figure5.2: ExamplesonaLennard-Jones potentialwithsyntheticnoise . tential A1=min/bracketleftbigg 1,T(R→R′) T(R′→R)exp(−βΔVcl)/bracketrightbigg (5.10) whereΔVcl=Vcl(R′)−Vcl(R)andTisthesamplingprobabilityforamove. Ifitisaccepted at thisﬁrst level,theQMCenergy differenceiscomputedand accepted withprobability A2=min[1,exp(−βΔVQMC−uB)exp(βΔVcl)] (5.11) whereuBisnoisepenalty. ComparedtothecostofevaluatingtheQMCenergydifference ,computingtheclassical energy difference is free. Reducing the number of QMC energy difference evaluations is valuableinreducing thecomputertimerequired. In Chapter7,usingthepre-rejection techniquewithaCEIMC -DMCsimulationresults in a ﬁrst level (classical potential) acceptance ratio of 0. 43, and a second level (quantum potential) acceptance ratio of 0.52. The penalty method rej ects additional trial moves be- cause of noise. If these rejections are counted as acceptanc es (ie, no penalty method or no noise), then the second level acceptance ratio would be 0.71 . The classical potential is a fairlygoodrepresentationfortheDMCpotential,andwecan usethattoreducethenumber ofDMCenergy differenceevaluationsneeded. 5.3 TrialMoves Molecularmovesare separated intotranslation,rotationa nd bondlengthchanges. 48Table 5.1: Efﬁciency of classical Monte Carlo for moving several parti cles at once. The table on the left is for low density system at rs=3.0 and T=5000K. The table on the right is for a high density system at rs=1.8 and T=3000K. The largest values of the efﬁciency are shownin boxes. Δ Nm0.8 1.6 3.0 4.0 11.4 4.9 9.6 11.6 22.4 8.1 15.3 17.4 45.6 12.5 16.917.7 86.9 14.8 18.014.8 169.522.27.4 3211.9 14.8 2.7Δ Nm0.4 0.8 1.2 1.6 2.0 174 134 99 236191 243 149 179121 114 4118170141 66 23 8172128 24 16155 52 3239 TheSilvera-Goldmanpotentialisusedastheempiricalpote ntialforpre-rejectingtrans- lational moves. Anisotropic potentials were tried for pre- rejecting rotational moves, but they did not work very well. It is not clear whether this was fr om the the potentials being derivedforisolatedH 2-H2interaction,orfrom inaccuracy inthetrialwavefunction. Bondstretchingmoveswerepre-rejectedusingthe,essenti allyexact,H 2intramolecular potentialofKolosandWolniewicz(1964). Thenewbondlengt hissampleduniformlyfrom a box of size Δbaround the current position. Because of phase space factors we need to includeasamplingprobabilityof T(R) =1/R2in theacceptanceformula. The trial move for classical Monte Carlo is usually presente d as either moving one particle at a time, or all of the particles at once. However, w e can move other numbers of particlesaswell. Table5.1showstheefﬁciencyforaclassi calsystemwith32H 2molecules for two densities and temperatures. On the left is a low densi ty system with rs=3.0 and at a temperature of 5000K, and on the right higher density sys tem with rs=1.8 and a temperature of 3000K. For the lower density system, the high est efﬁciency occurs when moving half the molecules at a time. Relatively high efﬁcien cy can also be found moving 2, 4 or 8 at a time as well. For the higher density system, the mo st efﬁcient regime shifts towardssmallerstepsizes and fewernumberofparticles mov edata time.3 3Theseresultsarenotgenerallyapplicabletoclassical MCs imulations,sincemuchmoreefﬁcientimple- mentationsarepossibleforsystemsinteractingwith atwo b odypotential. 49Table5.2: ResultsofCEIMC forisolatedH 2moleculeat T=5000K. Energy Virial /an}bracketle{tr/an}bracketri}ht/radicalbig /an}bracketle{tr2/an}bracketri}ht exact-1.1630 0.0 1.57 1.60 VMC-1.159(1) -0.009(6) 1.56(2) 1.58(2) DMC-1.163(2) -0.015(6) 1.58(2) 1.60(2) 00.20.40.60.81 1 1.5 2 2.5 3 Rexact VMC DMC Figure5.3: H2bondlengthdistribution. 5.4 SingleH 2molecule The CEIMC method was applied to a single H 2molecular in free space, at a temperature of 5000K. Exact results are obtained by integratingthepote ntial of Kolosand Wolniewicz (1964). Results for the energy, pressure, and ﬁrst and secon d moments of the bond length are giveninTable5.2. TheVirial columnis computedby Virial = [2/an}bracketle{tK/an}bracketri}ht+/an}bracketle{tV/an}bracketri}ht] (5.12) Thisisrelated by thevirialtheoremto theforce onthenucle i,which shouldbezero foran isolatedmolecule. In Chapter7, wewillseethisexpression used tocomputethepressure. As wewouldexpect,theVMCenergy is higherthan theexact ene rgy. Theotherquan- tities are close to their expected values. Histograms of the bond length distribution are shown in Figure 5.3. Here again, both VMC and DMC reproduce th e exact distribution well. 50Chapter6 HardSpheres A system of hard spheres is the simplest non-ideal many body s ystem to study. Statis- tical Monte Carlo techniques and molecular dynamics were ﬁr st applied to hard spheres (Metropolis et al., 1953; Alder and Wainwright, 1957). Additionally,some of t he ﬁrst ap- plications of ﬁeld theoretic methods to condensed matter sy stems were on hard spheres. Morerecently,theachievementofBECintrappedatomicgase shasrenewedinterestinthe theoryofthehard sphereBosegas(Dalfovo et al.,1999). This chapter is a VMC and DMC study of a homogeneous, boson har d sphere ﬂuid. Because they are bosons,there is no ﬁxed nodeapproximation in DMC, and theresult can be made essentially exact. The approximations we must contr ol are ﬁnite size effects and timesteperrorinDMC. ApplyingperturbationtheorytotheBosehardspheregasyie ldsthelowdensityexpan- sion, E=2πρ[1+C1√ρ+C2ρlnρ+C3ρ+...] (6.1) whereC1=128/(15√π)andC2=8(4π/3−√ 3). Meanﬁeldtheorygivesthelinearterm. Straight forward application of perturbation theory yield s theC1term, as was done by Huang, Yang, and Lee (Lee et al., 1957; Huang and Yang, 1957). The next higher or- der of perturbation theory diverges, but this was solved by i ncluding the depletion of the condensateby Beliaev (1958) toget C2. Wu (1959)obtained thesameresultsvia aresum- mation technique. Hugenholtz and Pines (1959) also obtaine d the logarithmicterm. They also obtainedthefunctionalform fortheseries, which incl udestermsoftheform ρn/2and ρn/2log(ρ). Renormalization group techniques have recently been appli ed to examine this diver- gence(Castellani etal.,1997;Braaten andNieto,1997a,b). Inaddition,Braaten an d Nieto have calculated C3(Braaten and Nieto, 1997a,b). This term is also ﬁrst term tha t depends onmorethatthetwobody s-wavescatteringlength. Itwouldrequireasolutiontothet hree 51body scattering problem, which makes an explicit computati on of that coupling constant difﬁcult. Hansenet al.(1971) used VMC on an 864 particle system to ﬁnd the ﬂuid-soli d tran- sition density. Kalos et al.(1974) used the more accurate Green’s Function Monte Carlo (GFMC) to calculate the energy of the solid and liquid phase n ear freezing to determine thefreezing density. Theyused256particles. In theliquid statetheycomputedfourpoints rangingin densityfrom 0.16to 0.27. Recently Giorgini et al.(1999) did DMC calculations on the homogeneous Bose gas, with various potentials, including the hard sphere potenti al. They used 500 particles and therewasnomentionofwhatDMCtimestepwasused. Thesecalc ulationswerealsoused tomakeaﬁttingtotheextendedformofEq. (6.1)(usingterms uptoρ5/2)byBoronat etal. (2000). Therehavebeenotherattemptstogetanequationofstatebyu singvariousﬁttingtech- niques to combine the low density results and the GFMC result s (Aguilera-Navarro et al., 1987; Keller et al., 1996). As noted by Keller et al.(1996), the earlier work had an error and used only half the energy of the actual GFMC results. A new attempt at ﬁtting the variousfunctionalformswas notdone,but anewvaluefor C3was estimated. TheHamiltonianforthissystemis H=−1 2∑ i∇2 i+∑ i<jv(\|ri−rj\|) (6.2) where v(r) =/braceleftbigg∞r<σ 0r>σ(6.3) Wehaveset ¯ h=m=σ=1 in allthesecalculations. 6.1 Wavefunction Anapproximatewavefunctionforthebosongroundstatethat weuseforatrialfunctionis ψ=∏ i<jf(rij) (6.4) The individual functions are very similar to the ones used fo r hydrogen. The correlation functionhasamaximumrange, rmax,beyondwhich fisconstant. Inordertobecompatible with periodic boundary conditions, we require rmax≤L/2. The “cusp” condition is that the wave function must vanish linearly when two spheres get c lose,f(r→σ)∝(r−σ). (Unliketheelectroniccase, theslopeisnotﬁxed.) 52Table6.1: Variationalparameters forhard spheregas ρxmax b0b1b2 b3 b4 .21.92 -0.3209 0.25395 0.5624 0.0145 -.05123 .12.68 0.9173 1.995 0.8147 0.2269 0.0345 .052.68 -0.33 0.674 -0.12 0.056 0.0 .015.9152 -1.95 0.86267 -1.2982 -0.08135 -0.3152 A change of variables will simplify these expressions. Let x=r−σ,xmax=rmax−σ andy=x/xmax. Nowylies in the range [0,1]. In these variables, the boundary conditions onfare f(y=0) =0 f(y=1) =1 (6.5) f′(y=1) =f′′(y=1) =0 (6.6) (6.7) and thewavefunctionis f(y) =3(y−y2)+y3+y(y−1)34 ∑ i=0biTi(2y−1) (6.8) whereTiare Chebyshevpolynomials,and biarevariationalparameters. The parameters for each density are given in Table 6.1. They w ere obtained from op- timizationof the smallest system ( N=40). Then those same parameters were used for all systemsizesat aparticulardensity. Thecost ofcomputingofthewavefunctionand local energy is dominatedby calculat- ing theN(N−1)/2 interparticledistances. There are techniques forimprov ingthe scaling of computations of short range interactions to achieve O(N). We used the cell method (Allen and Tildesley, 1987). The simulation box is divided i nto cubic cells and a list is made of all the particles in each cell. For simplicity, consi der the case where each cell is larger than the cutoff distance, rmax. Then a particle in a cell will have a non-zero inter- action only with the particles in the same cell and with parti cles in the neighboring cells. Particles in cells further away can be ignored. There is an ov erhead in computing and maintainingtheselists. Weused thecell methodonsystemsw ith500particles andlarger. Thereisadeﬁciencywiththetrialwavefunctionthatleadst oundersamplingwhenthree particlesareincloseproximity. InDMC,thisleadstoalarg enumberofbranchingwalkers to compensate for the undersampling, which invariably caus es problems with maintaining a stable population. One solution is to use a guiding functio n, which differs from the trial 530.089160.08920.089240.089280.089320.08936 00.020.040.060.080.1Energy τextrapolated to τ=0 0.66460.66480.6650.66520.66540.66560.66580.666 00.010.020.030.040.05Energy τextrapolated to τ=0 GFMC (a) (b) 1.7971.7981.7991.81.8011.8021.803 00.0020.0040.0060.0080.01Energy τextrapolated to τ=0 5.645.6455.655.6555.665.665 00.002 0.004 0.006 0.008Energy τextrapolated to τ=0 (c) (d) Figure6.1: Timesteperror for(a) ρ=0.01 (b)ρ=0.05 (c)ρ=0.1 (d)ρ=0.2 wave function, for the diffusion and branching. Then a weigh t,ψT/ψG, is associated with each sample point. In this case the simplest guiding functio n is to use is ψG=ψα T. We foundthat α=0.9was sufﬁcientto makethepopulationofwalkers stable. The DMC timestep errors should be local, and hence the same fo r all system sizes. We did timestepextrapolationon systemswith N=40 particles. The timesteperrors were foundto belinearin τ. Theextrapolationsto τ=0 areshowninFigure6.1. Green’s Function Monte Carlo (GFMC) uses a different decomp osition of the Green’s functionthatDMC.TheprincipleadvantageofGFMCisthatis hasnotimesteperror. We ran GFMC at ρ=.05 andN=40 to verify the timestep errors. The GFMC data point is shown at τ=0 in Figure 6.1. The importance sampling in GFMC is not as effe ctive as in DMC, hence the variance is larger, and GFMC is less efﬁcient t han DMC. GFMC has an efﬁciency of about 7, whereas DMC has an efﬁciency of 240 at τ=.002and an efﬁciency of 570 at τ=.007. Even with computingat several timesteps to extrapolat e to zero, DMC ismoreefﬁcient thatGFMC. 5400.20.40.60.811.2 00.5 11.5 22.5 3S(k) kVMC DMC, mixed est DMC, extrapolated est 00.10.20.30.40.50.6 00.5 11.5 22.5 3S(k) kVMC DMC, mixed est DMC, extrapolated est (a) (b) Figure6.2: S(k)for(a) ρ=0.05(b)ρ=0.2 6.2 FiniteSizeEffects The main contribution to ﬁnite size effects in the energy is t he long wavelength phonons. Thefunctionalform for theircontributiondepends on thesm allkbehaviorofthestructure factor,S(k). Theenergy can bewrittenas E=4π/integraldisplay kbk2dkε(k) (6.9) wherekb=2π/Lis the small kcutoffdue to the ﬁnite box size. Theenergy of thephonon excitations at small kis proportional to S(k)(Feynman and Cohen, 1956). For a classical liquid,S(k)∝1+O(k2). ForaBoseﬂuid, S(k)shouldbeproportionalto kandS(k→0) = 0. The VMC wave function has no long range part, and so we expect i t to behave like a classicalﬂuidatsmall k. IntegratingEq(6.9),weget E∝k3 b,whichisthesameasscalingby 1/Nforaﬁxeddensity. Amorerigorousderivationofthisscalin gisgivenbyLebowitzand Percus (1961). The DMC algorithm should pick up the correct l ong wavelength behavior, leading to an S(k)that is linear in k. Integrating Eq. (6.9) we get E∝k4 b. This then gives us1/N4/3scaling. Thesmall kbehaviorforS(k)canbeseennicelyfor ρ=0.05,showninFigure6.2a. The graphshowsthattheVMCstructurefactorappearsquadratic asexpected. TheDMCmixed estimator shows the S(k)behaving linearly, but still not headed to zero. The extrapo lated estimatorlookslikeitovercorrects and lowersS(k) toomuc h. We did calculations for systems with 40, 108, 256, and 500 par ticles. For ρ=.2, additional VMC runs with N=103andN=104particles were done, and they are shown on thegraph. TheVMCenergy nicelyﬁts the1 /Nbehavior, as shownin Figure6.3. 550.090.09020.09040.09060.09080.0910.09120.09140.0916 00.005 0.010.015 0.020.025Energy 1/Nextrapolated to N= ∞ 0.6820.6840.6860.6880.690.692 00.005 0.010.015 0.020.025Energy 1/Nextrapolated to N= ∞ (a) (b) 1.861.8621.8641.8661.8681.871.8721.8741.876 00.005 0.010.015 0.020.025Energy 1/Nextrapolated to N= ∞ 6.016.026.036.046.056.06 00.005 0.010.015 0.020.025Energy 1/Nextrapolated to N= ∞ (c) (d) Figure6.3: VMCﬁnitesizeeffects for(a) ρ=0.01(b)ρ=0.05(c)ρ=0.1 (d)ρ=0.2 560.08920.08940.08960.08980.09 0 0.002 0.004 0.006 0.008Energy 1/N4/3extrapolated to N= ∞ 0.6650.6660.6670.6680.6690.670.671 0 0.002 0.004 0.006 0.008Energy 1/N4/3extrapolated to N= ∞ (a) (b) 1.7981.81.8021.8041.8061.8081.811.812 0 0.002 0.004 0.006 0.008Energy 1/N4/3extrapolated to N= ∞ 5.645.655.665.675.685.695.7 0 0.002 0.004 0.006 0.008Energy 1/N4/3extrapolated to N= ∞ (c) (d) Figure6.4: DMCﬁnitesizeeffects for(a) ρ=0.01(b)ρ=0.05(c)ρ=0.1 (d)ρ=0.2 57Table6.2: Energy extrapolatedtoinﬁnitesystemsize(inunitsof¯h2 mσ2) ρ VMC DMC Giorgini et al.(1999) .26.0546(6) 5.67(1) .11.8744(4) 1.809(1) 1.8130(35) .050.6917(1) 0.6690(4) 0.6690(5) .019.144(2)×10−28.9896(8)×10−28.980(5)×10−2 TheDMCﬁnitesizeextrapolationisshowninFigure6.4. Athi gherdensities,theDMC energydoesnotappeartohavea1 /N4/3dependence(oreven1 /Ndependence). However, the data is too sparse and noisy to make a good determination a s to what the functional form should be, so we ﬁt it to 1 /N4/3. The ﬁnal inﬁnite system results are given in Table 6.2. The long wavelength excitations will take a long time to samp le, and their effect on the energy (and S(k)) may only be apparent with very long runs. And the time needed to sample them will increase with box size. The larger box siz es may be insufﬁciently converged, causing the energy to be too high. This may explai n the apparent curvature in thewrongdirection. There are several approaches for resolving the problem at la rger box sizes. The ﬁrst is simply to perform even longer runs to see if the energy drop s. Similarly, data for more systemsizeswouldbehelpfulinoutliningthefunctionalfo rmoftheﬁnitesizedependence. Finally, explicit long range correlations could be added to the wave function, of the form proposedby Chesterand Reatto (1966). TheenergyversusdensityisshowninFigure6.5,relativeto theﬁrstorderterm,which is linear in the density. The low density exansion up to the C1term is also shown, as well as uptoC3,usingtheﬁtted valueof73 .296(Keller et al.,1996). Itisclear fromthegraph, and was noted by Hugenholtz and Pines (1959), that these addi tional terms by themselves do nothelptheexpansion. Boronatet al.(2000) treated C2andC3as adjustable parameters and added two addi- tionalterms, ρ5/2log(x)andρ5/2. Theygetaverygoodﬁttothehighdensitydata,asseen inFigure6.5. The results of Giorgini et al.(1999) are also given in Table 6.2 for the three common densitiescomputed. Theirdataand oursagree withintheerr orbars. 580123456 0.001 0.01 0.1E/2π ρ ρσ3low density expansion up to C1low density expansion up to C3fit of Boronat et al. this work Figure6.5: Energy vs. density 6.3 DistributionFunctionsandCondensateFraction The two particle correlation function for all densities was calculated with a system size of 256particles. Theresultsfor g(r)usingtheextrapolatedestimatorareshowninFigure6.6. Weseetheliquidshell structuredevelopingas thedensityi ncreases. Thesingleparticledensitymatrixistheprojectionofaman y-bodywavefunctiononto asingleparticlespace. It isdeﬁned by ρ1(r,r′) =/integraldisplay dr2...drNψ(r,r2,...,rN)ψ∗(r′,r2,...,rN) (6.10) In the homogeneous case, ρ1only depends on the distance between randr′. The large rbehavior of ρ1(or thek=0 behavior of its Fourier transform, n(k)) is the condensate fraction. Atzerotemperature,themanybodywavefunctioni sinthegroundstate. Because ofinterations,notalltheparticlesareinthesinglebodyz eromomentumstate( k=0plane wavestatein thiscase). We used amethod for sampling ρ1(r)which was givenby McMillan(1965). The con- densatefractionwasobtainedbyintegratingthesinglepar ticledensitymatrixfordistances greaterthan somecutoff, rc, chosen tobewhere ρ1(r)had reached a plateau. The condensate fraction is given in Table 6.3 and shown in Fig ure 6.8. Also shown in the ﬁgure is the GFMC result of n0=0.095(1)atρ=0.2. The low density expansion is givenby n0=1−8 3√πρ1/2(6.11) 5900.20.40.60.811.21.4 0 0.5 1 1.5 2 2.5 3g(r) r ρ1/3ρ=0.2 ρ=0.1 ρ=0.05 ρ=0.01 Figure6.6: Pairdistributionfunctionforseveraldensities 00.20.40.60.811.2 0 1 2 3 4 5ρ1(r) r ρ1/3ρ=0.2 ρ=0.1 ρ=0.05 ρ=0.01 Figure6.7: Singleparticledensitymatrixforseveraldensities 60Table6.3: Condensatefraction ρVMC DMC(mixed) Extrapolated .20.1009(5)0.0876(3)0.0743(8) .10.307(2)0.2960(5)0.285(2) .050.563(2)0.5401(4)0.517(8) .010.834(1)0.826(1)0.818(2) 00.10.20.30.40.50.60.70.80.91 0 0.05 0.1 0.15 0.2n0 ρσ3This work Low density expansion GFMC fit of Boronat et al Figure6.8: Condensatefractionvs. density Similartotheirtreatmentoftheenergy,Boronat etal.(2000)addedtwoadditionalterms, ρ andρ3/2. Their ﬁt does a good job at higherdensities, where, as expec ted, thelow density expansionmissesthefullextentofthedepletion. 61Chapter7 Hydrogen Hydrogenhasbeenthesubjectofmanyexperimentalandtheor eticalstudies. Theoretically, its simple electronic structure make it a favorable ﬁrst tar get for various methods. Exper- imentally, hydrogen has been compressed by shock waves and a lso with a diamond anvil cell. We will present some CEIMC simulations and compare the results with those from oneofthegas gunshockwaveexperiments. 7.1 Experiment Thehighpressureexperimentsfallintotwocategories-tra nsientcompressionfromashock wave or staticcompression from a diamond anvil cell. The sho ck wave experiments reach highertemperaturesandpressures,butobtainmorelimited data. Ahigh-velocityprojectile hits a stationary target, inducing a shock wave in the target . The target is analyzed by the Hugoniot relations, derived by treating the shock wave a s an ideal discontinuity and applying conservation of mass, momentum, and energy across it (Zel’dovich and Raizer, 1966). Therelationsare then P−P0=ρousup (7.1) ρ=ρ0us/(us−up) (7.2) E−E0=1 2(V0−V)(P+P0) (7.3) whereE0,P0,V0, andρ0are the initialenergy, pressure, volume, and density,resp ectively. The velocity of the shock wave is usandupis the velocity of the projectile driving the shock. There are a numberof methods for accelerating a projectile ( Cable, 1970), but the two mostprominentmethodsforhydrogentargetsare thetwo stag elightgasgun(Nellis et al., 621983; Holmes et al., 1995; Weir et al., 1996; Nellis et al., 1999) and a large laser (Silva et al.,1997;Collins et al.,1998;Celliers et al., 2000). Recent advances make it possible to measure the temperature by light emission from thesamplesduringcompression(Holmes etal.,1995). Measurementoftheconductivityis alsopossible,usedin recent experimentsto detect metalli chydrogen(Weir et al., 1996). The diamond anvil cell (DAC) is used to generate large static pressures. It has been usedto studytheﬂuidphaseandseveralsolidphases (Maoand Hemley,1994). It has also been used to determine the melting curve for hydrogen up to 50 0K (Diatschenko et al., 1985;Datchi et al.,2000). 7.2 Theory Free energy models are typically based on the chemical pictu re, where molecules, atoms and various types of ions are all treated as different specie s of particles. Solving the phys- ical picture, where the only fundamentalparticles are elec trons and protons, is much more difﬁcult (see Path Integral Monte Carlo below). The free ene rgy of the various phases is constructed from a variety of ﬁts to experimental data, equa tion of state data from refer- ence systems (Lennard-Jones and hard sphere), and empirica l and theoretical interaction potentials. One of the best known models is that of Saumon and Chabrier (Sa umon and Chabrier, 1991, 1992). Extensive tables for astrophysical use were pu blished by Saumon et al. (1995). Another model was developed by Kitamura and Ichimar u (1998), to study the plasma(metal-insulator)transition. ThePath Integral MonteCarlo (PIMC) methodis in principlet hebest methodforsim- ulations, since it treats both the electrons and protons qua ntum mechanically at non-zero temperature (Pierleoni et al., 1994; Magro et al., 1996; Militzer and Ceperley, 2000; Mil- itzer, 2000). The only major uncontrolled approximation is the location of the electron nodes, with problems and a solution similar to the ﬁxed node m ethod in DMC. Militzer and Pollock(2000)havemadeprogressin improvingthenodal structureusedin thesecal- culations. PIMC is based on breaking up a thermal density mat rix into a product of high temperaturecomponents,andconsequentlyitworkswellath ightemperatureandbecomes less efﬁcient as the temperature decreases. About 5000K is c urrently the lower limit for PIMC calculations. Our CEIMC simulation technique should m ake a nice complement to thePIMCmethod. Therehavealso beenpathintegralstudiesusingempiricalp otentials,inorderto exam- ine the quantum effects of the nuclei on the system (Wang et al., 1996, 1997; Cui et al., 631997;Chakravarty, 1999). The Car-Parrinello method has been used to simulate this sys tem (Hohl et al., 1993; Kohanoff etal.,1997;PfafenzellerandHohl,1997;Galli etal.,2000). Atlowtemperature, ititnecessarytotreatthenucleiwithpathintegrals(Bier mannetal.,1998;Kitamura etal., 2000). Some studies used LDA with the Γpoint approximation (using only one k-point fortheintegralovertheBrillouinzone), whichis notsufﬁc ient toconvergetheanisotropic behavior of the potential (Mazin and Cohen, 1995), and gives rise to unphysical planar structures(Kohanoff et al.,1997). 7.3 PressureandKineticEnergy Thepressureiscomputedbya virialestimatorbased on thepo tentialand kineticenergies P=1 3V[2/an}bracketle{tK/an}bracketri}ht+/an}bracketle{tV/an}bracketri}ht] (7.4) whereVis the volume andVis the potential energy. In these MC simulations, only the kineticenergyoftheelectronsisexplicitlycomputed. The kineticenergyofthenucleimust alsobeadded. We are only considering hydrogenin the molecularstate, and further assumethat rota- tionalandvibrationalmotioncanbeseparated. Thecharact eristictemperatureforquantum effectsforrotationalmotionisabout85KforH 2(LandauandLifshitz,1980). Atoursim- ulation temperatures, we can use the classical expression f or the rotational kinetic energy, Erot=2kT. The characteristic vibrational temperature for H2isθv=6100 K, so it is necessary to usethequantumexpressionfor thevibrationalkineticener gy. It is Evib=θv e−θv/T−1(7.5) ForD2, thecharacteristictemperatureshouldbeafactorof√ 2 lower. Ofcourse,theseexpressionsareonlyvalidforafreemolecu le. Totrulytreatthekinetic energy of the nuclei correctly in the interacting system, pa th integrals should be used for thenuclei. 7.4 IndividualConﬁgurations WetookseveralconﬁgurationsfromPIMCsimulationsat5000 Kattwodensities( rs=1.86 andrs=2.0),andcomparedtheelectronicenergyusingVMC,DMC,DFT-L DA,andsome 64empirical potentials. The DFT-LDA results were obtained fr om a plane wave code using an energy cutoffof60 Rydbergs, and usingthe Γpointapproximation(Ogitsu,2000). The empirical potentials are the Silvera-Goldman (Silvera and Goldman, 1978) and the Diep-Johnson (Diep and Johnson, 2000a,b). To these we ad ded the energy from the Kolos(KolosandWolniewicz,1964)intramolecularpotenti altogettheenergyasafunction of bond length variations. The Silvera-Goldman potential w as obtained by ﬁtting to low temperature experimental data, with pressures up to 20 kbar , and is isotropic. The Diep- Johnsonpotentialisthemostrecentinanumberofpotential sfortheisolatedH 2-H2system. It was ﬁt to the results of accurate quantum chemistry calcul ations for a number of H 2-H2 conﬁgurations. It isan anisotropicpotential. The energies relative to an isolated H 2molecule are shown in Figure 7.1. The ﬁrst thing we notice is that the classical potentials are more acc urate than VMC or DFT. The Silvera-Goldman mostly does a good job of reproducing the DM C results.1Some of the failures of theSG potential can be attributed to the lack of a nisotropy. The isolated H 2-H2 potential (Diep-Johnson) has much weaker interactions, co mpared with interactions in a densersystem. ThePIMCmethoditselfgivesanaverageenergyofabout0 .07(3)Haforbothdensities. Improvementsinthefermionnodesappeartolowertheenergy (MilitzerandPollock,2000; Militzer and Ceperley, 2000), although the error bars are st ill quite large. There are also corrections to some internal approximations that lower the energy by an additional 0.02 Ha (Militzer, 2000). These effects combined seem to bring th e PIMC energy in rough agreement withtheDMCenergy. We used the Silvera-Goldman potential for pre-rejection. A s seen in the Figure 7.1, it resembles the DMC potential even though it lacks anisotro py. A hybrid potential was created by Cui et al.(1997), taking the isotropic part from a potential that was ﬁ t to high density,andcombiningthatwiththeanisotropicpartfromo neoftheisolatedH 2-H2poten- tials. Wedidnot pursuethisapproach forconstructinga bet terpotentialforpre-rejection. 7.5 Results We obtained results from simulations at three state points, two of which can be compared with the gas gun data of Holmes et al.(1995). The pressure is given in Table 7.1, with resultsfromthegasgunexperiments,theSaumon-Chabrierm odel,fromsimulationsusing theSilvera-Goldmanpotential,andfromourCEIMCsimulati ons. Thesestatepointsarein the ﬂuid molecular H 2phase. For the gas gun experiments, the uncertainties in the mear- 1It shouldbenotedthat wearetakingtheSG potentialfarfrom the temperaturerangeit wasﬁt to. 650.020.030.040.050.060.070.080.09 0 2 4 6 8 10 12Energy (Ha/molecule) 1-6, rs=1.86 7-12, rs=2.0DMC DFT-LDA VMC Silvera-Goldman Diep-Johnson Figure 7.1: Electronic energy for several conﬁgurations computed by se veral methods. Theenergy isrelativetoan isolatedH 2molecule. sured temperatures are around 100-200K. The experimental u ncertainties in the volume and pressure were not given, but previous work indicates tha t they are about 1-2% (Nellis et al.,1983). We did CEIMC calculations using VMC or DMC for computing the u nderlying elec- tronic energy, which are the ﬁrst such QMC calculations in th is range. The simulations at rs=2.1andrs=1.8weredonewith32molecules,andthesimulationsat rs=2.202were done with 16 molecules. We see that the pressures from VMC and DMC are very similar, and thatfor rs=2.1weget goodagreement withexperiment. There is a larger discrepancy with experiment at rs=2.202. The ﬁnite size effects are fairly large, especially with DMC. We also did simulations a trs=2.1 with 16 molecules andobtainedpressuresof0 .264(3)MbarforCEIMC-VMCand0 .129(4)MbarforCEIMC- DMC. The Silvera-Goldman potential showed much smaller ﬁni te size effects than the CEIMC simulations, so we that the electronic part of the simu lation is largely responsible fortheobservedﬁnitesizeeffects. The energies for all these systems are given in Table 7.2. The energy at rs=2.1 with 16 molecules for CEIMC-VMC is 0 .0711(4)Ha and for CEIMC-DMC is 0 .0721(8)Ha. The average molecular bond length is given in Table 7.3, and w e see the bond length is compressed relative to the free molecule. The proton-proto n distribution functions com- paring CEIMC-VMC and CEIMC-DMC are shown in Figure 7.2. The V MC and DMC distribution functions look similar, with the ﬁrst large in tramolecular peak around r=1.4 and theintermolecularpeak around r=4.5. 66Table7.1: Pressurefrom simulationsand shockwaveexperiments rsV(cc/mol) T(K) Pressure(Mbar) Gasgun S-C S-G CEIMC-VMC CEIMC-DMC 2.100 6.92 4530 0.234 0.213 0.201 0.226(4) 0.225(3) 2.202 7.98 2820 0.120 0.125 0.116 0.105(6) 0.10(5) 1.800 4.36 3000 - - 0.528 - 0.433(4) Table 7.2: Energy from simulationsand models,relativeto theground s tate ofan isolated H2molecule. The H 2column is a single thermally excited molecule plus the quant um vibrationalKE. rsV(cc/mol) T(K) Energy (Ha/molecule) H2S-C S-G CEIMC-VMC CEIMC-DMC 2.100 6.92 4530 0.0493 0.0643 0.0689 0.0663(8) 0.0617(2) 2.202 7.98 2820 0.0290 0.0367 0.0408 0.0305(8) 0.0334(9) 1.800 4.36 3000 0.0311 - 0.0722 - 0.055(1) Table7.3: AveragemolecularH 2bondlength. TheH 2columnisasinglethermallyexcited moleculein freespace. rsT(K) Averagebondlength(Bohr) H2CEIMC-VMC CEIMC-DMC 2.100 4530 1.550 1.431(1) 1.413(3) 2.202 2820 1.486 1.443(1) 1.429(6) 1.800 3000 1.492 - 1.410(1) 00.511.522.533.544.5 01234567g(r) rDMC VMC 00.511.522.533.544.5 0123456g(r) rDMC VMC (a) (b) Figure 7.2: Proton pair distribution function g(r)for (a)rs=2.1 and T=4530 K (b) rs= 2.202and T=2820K 6700.511.522.533.5 0 1 2 3 4 5 6g(r) rCEIMC-DMC Hohl, et al. Figure7.3: Theprotonpairdistributionfunction, g(r),closeto rs=1.8andT =3000K. The CEIMC-VMC simulationsat rs=1.8 and 3000 K never converged. Starting from aliquidstate,theenergy decreased theentiresimulation. Lookingattheconﬁgurationsre- vealedtheywereformingaplane. Itisnotclearwhetheritwa stryingtofreeze, orforming structures similarto thosefound in DFT-LDA calculationsw ith insufﬁcientBrillouinzone sampling(Hohl etal.,1993;Kohanoff etal.,1997). TheCEIMC-DMCsimulationsdidnot appeartohaveanydifﬁculty,soisseemstheVMCbehaviorwas duetoinadequaciesofthe wavefunction. Hohlet al.(1993)didDFT-LDA simulationsat rs=1.78and T=3000K, which isvery close to our simulationsat rs=1.8. The resulting proton-proton distributionfunctions are compared in Figure 7.3. The discrepancy between CEIMC and LD A in the intramolecular portion of the curve has several possible causes. On the CEIM C side, it may be due to an insufﬁciently long run or due to the molecular nature of th e wave function, which does not allow dissociation. The deﬁciencies of LDA may account f or it preferring fewer and less tightly bound molecules. LDA is known to overestimate t he bond length of a free hydrogenmolecule(Hohl etal.,1993),whichwouldaccountfortheshiftedlocationofthe bondlengthpeak. 7.6 Simulationanalysis We also recorded some diagnosticinformation about the work ings of the simulation, such astheaveragenoiselevel,therelativenoiseparameter f= (βσ)2t/t0,andaquantitycalled 68theadditionalnoiserejectionratio, η. When amoveisrejected withthepenaltymethod,it is useful to recompute the acceptance decision with the same random numberand without the noise penalty. If the move would have been accepted witho ut the noise penalty, it is considered a rejection due to noise (as opposed to a reject ion due to a trial move that increasestheenergy). Thiscanbeusedtomonitortheeffect softhenoiseonthesimulation. Theadditionalnoiserejection ratiois deﬁned as η=Nnoiserej Nnoiserej+Naccept(7.6) Ifηissmall,thenoiseis causingfew additionalrejections. If ηis 1/2,thenoiseis causing as many moves to be rejected as accepted. As η→1, the noise is causing many moves to berejected. Table7.4showthenoiselevel (βσ),therelativenoiseparameter, f,theadditionalnoise rejection ratio, a ratio of the error level for the direct met hod and the two-sided method, andthetimeforasinglequantumstep. Lookingat f,weseeitissmallforVMCandlarge for DMC. This is because of VMC optimization takes a proporti onately larger amount of timeintheVMCrun thanin theDMCrun. We used the two-sided method for computing energy differenc es of trial moves with VMC, but only used the direct method with DMC. The column head edσ2 d/σ2 tsshows how the efﬁciency of computingthe energy difference is impr ovedby using the two-sided method. This improvement is only in the energy difference pa rt of the total time, the op- timization time is unaffected (and is a large part of the run t ime, since the fparameter is small). In Chapter 3, there was an example showingthat the tw o-sided method was not as effectiveforDMCasforVMC.ButinthesesimulationstheDMC runshaveamuchlarger fparameter, so even small reductions in the noise level would have an impact on the run time. Some of DMC energy differences had values of noise many times greater than the average, which may be due to an instability in the DMC algorit hm. We removed these outliersincomputingtheaveragenoiselevel. We tried the method for even lower temperatures with a simula tion at T=800 K and rs=1.8and ithadapromisingstart,butafterawhiletheacceptanc e ratiodroppedandwe wereunabletoget anyusabledata. 7.7 FutureWork The ﬁnite size effects in DMC need to be resolved. Using Ewald sums for computing the Coulomb interaction might help alleviate some of the ﬁnite s ize effects. Extending the 69Table 7.4: Simulation quantities ordered according to average noise l evel,βσ. The time column is the time for a single quantum step in minutes on an SG I Origin 2000. N is the numberofmoleculesinthesimulation. rsT(K) N QMC βσfη σ2 d/σ2 tstime(min) 2.100 4530 16 VMC 0.68 0.17 0.11 2.2 18 2.202 2820 16 VMC 0.70 0.27 0.13 3.2 21 2.100 4530 32 VMC 0.90 0.29 0.16 2.3 70 1.800 3000 32 VMC 0.91 0.30 0.15 7.7 89 2.100 4530 16 DMC 1.62 2.28 0.28 - 76 2.100 4530 32 DMC 1.74 5.30 0.29 - 440 2.202 2820 16 DMC 2.02 5.33 0.40 - 92 1.800 3000 32 DMC 2.42 13.1 0.42 - 510 wave function to allow for dissociated molecules and to prov idefor ionization would help makethesimulationsmoreaccurate, particularlyat higher temperatures and pressures. 70Chapter8 Conclusions In this work we have developed new methods for increasing the scope of QMC calcula- tions, and for increasing their efﬁciency. Variational Mon te Carlo depends on optimizing parameters,butthepresenceofnoisemakesitdifﬁcult. Weh aveexaminedseveraldifferent kinds of optimizationapproaches and compared them. Furthe r work shouldimprovethese methodsevenmore. The boson hard sphere model is an important theoretical mode l. We have performed “computational experiments” to obtain the ground state ene rgy of this model. The effects oflongrangecorrelationontheenergyaremaskedbythecurr entuncertaintyintheinﬁnite system size results. However, if more accurate results are d esired, the nature of the long rangecorrelationsand theireffect ontheenergy willneed t o bemoreclearly resolved. As a method for including increasingly more detailed and acc urate physical effects in our simulations, we have developed the Coupled Electronic- Ionic Monte Carlo method. The central idea is simple, but several supporting developm ents were needed to make it computationally feasible. The penalty method enables use o f energy differences with a noise level of approximately kBT, rather than needing noise smaller than some fraction ofkBTto avoid bias. The two sided energy difference method can sta bly compute these energy differences. The CEIMC method was applied to a system of molecular hydroge n at a few state points. It showspromiseforgeneratingaccurate simulatio nresults. 71AppendixA DeterminantProperties TheelementsoftheSlater matrixare Dij=φj(ri). TheSlater determinantlookslike /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1(r1)...φn(r1) ......... φ1(rn)...φn(rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A.1) Weassumethatthesingleparticleorbitalsdependonlyonas inglecoordinate(ie,noback- ﬂow). The determinant of a matrix can be computed using the expansi on by cofactors. This expands the determinant of an N×Nmatrix into a sum of Ndeterminants of (N−1)× (N−1)matrices. As a recursive algorithm for computing the determ inant, it is not very efﬁcient, but for theoretical analysis, it is very useful fo r isolatingthe inﬂuence of a single roworcolumn. Deﬁne thecofactors ofa matrix Mto be cij= (−1)i+j/vextendsingle/vextendsingleMij/vextendsingle/vextendsingle (A.2) where the matrix formed by cijis called the cofactor matrix. The matrix Mijis an(N− 1)×(N−1)matrix formed by removing row iand column jfromA. The determinant of Acan thenbewritten as \|A\|=∑ jakjckj=∑ iaikcik (A.3) fork=1...N. The transpose of the cofactor matrix is called the adjoint o fA. Now the adjointisrelated totheinverseby adjA=\|A\|A−1(A.4) To compute the ratio of determinants, expand the determinan t ofD(r′ k)in cofactors aboutthe kthrow. Notethatthenthecofactors havenodependenceon r′ k. 72/vextendsingle/vextendsingleD(r′ k)/vextendsingle/vextendsingle=∑ iφi(r′ k)cki =∑ iφi(r′ k)\|D(rk)\|(D−1(rk))ik /vextendsingle/vextendsingleD(r′ k)/vextendsingle/vextendsingle \|D(rk)\|=∑ iφi(r′ k)(D−1(rk))ik If a move is accepted, the inverse matrix can be updated in O(N2)time (rather than O(N3)for recomputing the inverse). The formula for updating an in verse if only a sin- gle row (or column) changes was given by Sherman and Morrison (1951). Let qbe the ratio of determinants given above. Row kmerely needs to be updated to reﬂect the new determinant, D−1 kj=D−1 kj/q. Theotherrowsare updatedas D−1 ij=D−1 ij−D−1 ik q∑ lD−1 ljφl(r′ k)i/ne}ationslash=k (A.5) 73AppendixB Elementsof theLocalEnergy Thewavefunctionhas theform ψT=Dexp[−U] (B.1) whereDistheproductofaspinup and aspindownSlaterdeterminanta nd U=∑ i<ju(rij). (B.2) Thelocalenergy isthen EL=1 2∇2U−1 2∇U·∇U−1 2/parenleftbigg∇2D D/parenrightbigg +/parenleftbigg∇D D/parenrightbigg ·∇U+V (B.3) In DiffusionMonteCarlo, weneed thequantumforce, FQ=∇ln\|ψ\|2. FQ=2/parenleftbigg∇D D/parenrightbigg −2∇U (B.4) ThederivativesoftheJastrowfactors are ∇kUee=∑ i/ne}ationslash=ku′ ee(rik)ri−rk rik(B.5) ∇kUne=M ∑ α=1u′ ne(rkα)rk−Rα rkα(B.6) ∇2 kUee=∑ i/ne}ationslash=k2 riku′ ee(rik)+u′′ ee(rik) (B.7) ∇2 kUne=M ∑ α=12 rkαu′ ne(rkα)+u′′ ne(rkα) (B.8) For the gradient with respect to particle k, expand the determinant by cofactors about row k. Thenthecofactors haveno rkdependence. ∇k\|d\| \|d\|=∑ i[∇kφi(rk)]d−1 ik(B.9) 74∇2 k\|d\| \|d\|=∑ i/bracketleftbig ∇2 kφi(rk)/bracketrightbig d−1 ik(B.10) 75AppendixC Cusp Condition WhentwoCoulombparticlesgetclose, thepotentialhas a1 /rsingularity. Thewavefunc- tion must have the correct form to cancel this singularity. F irst, consider an electron and a nucleus. Therelevantpart oftheSchr¨ odingerequationis /bracketleftbigg −1 2M∇2 n−1 2∇2 e−Ze2 r/bracketrightbigg ψ=Eψ (C.1) whereMisthenuclearmassand Zisthenuclearcharge. Assumethat M≫me, sotheﬁrst term can beignored. Writethesecond termin sphericalcoord inatesand weget −1 2ψ′′−1 r/parenleftbig Ze2ψ+ψ′/parenrightbig =Eψ (C.2) In order for the singularity to cancel at small r, the term multiplying 1 /rmust vanish. So wehave 1 ψψ′=−Ze2(C.3) Ifψ=e−crwemusthave c=Ze2. Forthecaseoftwoelectrons, theSchr¨ odingerequationis /bracketleftbigg −1 2∇2 1−1 2∇2 2+e2 r12/bracketrightbigg ψ=Eψ (C.4) Switchingtorelativecoordinates r12=r1−r2givesus /bracketleftbigg −∇2 12+e2 r12/bracketrightbigg ψ=Eψ (C.5) Electrons with unlike spins (no antisymmetry requirement) have an extra factor of 1 /2 in thecuspconditioncompared withtheelectron-nucleuscase . So wehave c=−e2/2. In the antisymmetriccase, the electrons will be in a relativ epstate, reducing the cusp conditionby1 /2, soc=−e2/4. Having thecorrect cuspfor likespin electronsgainsvery little in the energy or the variance, since the antisymmetry requirement keeps them apart anyway. 76References Aguilera-Navarro, V. C., S. 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arXiv:physics/0012031v1 [physics.atom-ph] 15 Dec 2000Enhancement of bichromatic high-harmonic generation with a high-frequency ﬁeld C. Figueira de Morisson Faria1and M. L. Du1,2 1Max Planck Institut f¨ ur Physik komplexer Systeme, N¨ othni tzer Str. 38, D-01187 Dresden, Germany 2Institute of Theoretical Physics, P.O.Box 2735, Beijing, C hina (February 20, 2014) Using a high-frequency ﬁeld superposed to a linearly po- larized bichromatic laser ﬁeld composed by a wave with fre- quency ωand a wave with frequency 2 ω, we show it is pos- sible to enhance the intensity of a group of high harmonics in orders of magnitude. These harmonics have frequencies about 30% higher than the monochromatic-cutoﬀ frequency, and, within the three-step-model framework, correspond to a set of electron trajectories for which tunneling ionizati on is strongly suppressed. The observed enhancement suggests that the high-frequency ﬁeld provides an additional mech- anism for the electron to reach the continuum. The addi- tional high frequency ﬁeld permits the control of this group of harmonics leaving all other sets of harmonics practicall y un- changed, which is an advantage over schemes involving only bichromatic ﬁelds. 32.80.Rm, 32.80.Qk, 42.65.Ky, 42.50.Hz Within the last few years, the perspective of obtaining eﬃcient laser sources in the extreme ultraviolet (XUV) regime has led to the proposal of several schemes for con- trolling the harmonic spectrum of an atom subject to a strong laser ﬁeld ( I∼1014W/cm2), using for instance additional static ﬁelds [1], ultrashort pulses [2], bichro - matic driving ﬁelds [3–6], or additional conﬁning poten- tials [7]. These schemes are based on the “three-step model” [8], which describes very well the spectral fea- tures observed in experiments. These features are the “plateau”, where harmonics of roughly the same inten- sities exist, and the “cutoﬀ”, where the harmonic signal suddenly decreases in orders of magnitude. According to this model, the generation of high-order harmonics in strong laser ﬁelds corresponds to a dynamical process in which an electron leaves an atom at an instant t0via tun- neling ionization, this electron in the continuum is then accelerated by the driving ﬁelds, and, if it comes back and recombines with the parent ion at a time t1, a high energy photon will be generated. The harmonic energy is given by Ω = \|ε0\|+Ekin(t0,t1),withEkin(t0,t1) and \|ε0\|being, respectively, the kinetic energy of the electron upon return and the ionization potential of its parent ion. The cutoﬀ corresponds to the maximum electron kinetic energy. By manipulating the diﬀerent steps of the dynamical process, one can in principle control high- harmonic generation. If the purpose is to increase the cutoﬀ energy, very eﬃ- cient schemes exist. They involve either additional static ﬁelds [1], or atoms placed in conﬁning potentials [7]. In both cases one introduces an additional force which mod- iﬁes the propagation of the electron after injection, resul t-ing in an increased cutoﬀ energy. However, such schemes may require extremely high static ﬁelds [9], or appropri- ate solid-state materials, whose existence is still under investigation [10]. Another scheme for manipulating high harmonics in- volves bichromatic driving ﬁelds [3–6]. By varying the frequencies, the relative phase, and the intensities of the two driving waves, one can modify the propagation of the electron in the continuum, and even the “ﬁrst step”, i.e., the tunneling ionization. The experimental imple- mentation of this bichromatic scheme should be easier compared to the previous ones. In the bichromatic case, however, no simple expres- sion for the cutoﬀ energy exists [4], such that it is not straightforward to predict whether the plateau can be ex- tended. There exist several local maxima for Ekin(t0,t1), but they do not necessarily correspond to an abrupt in- tensity decrease in the harmonic signal. A very illustra- tive example is a bichromatic ﬁeld consisting of a wave with frequency ωand its second harmonic [4,5]. The ad- dition of the second driving wave causes a splitting in the monochromatic-ﬁeld cutoﬀ energy εmax=\|ε0\|+ 3.17Up, Upbeing the ponderomotive energy. As a direct conse- quence, there is a double-plateau structure in the har- monic spectra, with an upper and a lower cutoﬀ, whose energies are higher and lower than εmax, respectively. The upper cutoﬀ corresponds to the absolute maximum forEkin(t0,t1).However, it appears in the spectrum only as a small shoulder due to the relative low harmonic in- tensity. The lower cutoﬀ, on the other hand, is related to a decrease of orders of magnitude in the harmonic yield, being therefore the one eﬀectively measured in experi- ments (see, e.g., [5,6] for a more complete discussion). One can explain the intensity diﬀerence in this double- plateau structure in terms of the width of the eﬀective po- tential barrier through which the electron tunnels. This barrier is given by Veﬀ=V(x)−xE(t0), whereE(t0) is the ﬁeld at the electron emisson time and V(x) the atomic potential. For the upper cutoﬀ, the atomic po- tential is not as much distorted by the ﬁeld as for the lower cutoﬀ. This results in a considerably wider ef- fective potential barrier and, consequently, much weaker harmonics. In this paper we consider again the case of a linearly polarizedω−2ωﬁeld, as in [4,5]. However, this time our aim is to increase the intensities of the upper-cutoﬀ- harmonics close to the intensities of those belonging the lower cutoﬀ, eﬀectively extending the cutoﬀ energy be- yondεmax=\|ε0\|+ 3.17Up. For this purpose, one must 1provide an additional mechanism for the electronic wave packet to reach the continuum, thus compensating the weak tunneling ionization. We demonstrate that this can be done with an additional driving wave. This third wave has a relative high frequency but low intensity compared to the bichromatic ﬁeld. The nature of the third wave is such that it alters the electron injection into the con- tinuum and, at the same time, it does not appreciably modify the ponderomotive energy and the cutoﬀ. This ﬁeld conﬁguration is similar to that used in [11], where the inﬂuence of the background ﬁeld on x-ray-atom scat- tering processes was investigated. We restrict ourselves to a one-dimensional model, which still describes high-harmonic generation with lin- early polarized ﬁelds well in qualitative terms. We solve the time-dependent Schr¨ odinger equation id dt\|ψ(t)/angbracketright=/bracketleftbiggp2 2+V(x)−p·A(t)/bracketrightbigg \|ψ(t)/angbracketright (1) numerically, for an atom initially in the ground state, with binding potential V(x) subject to a laser ﬁeld E(t) =−dA(t)/dt. Atomic units are used throughout. The external laser ﬁeld is taken as E(t) =E01sin(ω1t) +3/summationdisplay i=2E0isin(ωit+φ1i),(2) withE0i, ωiandφ1ibeing the ﬁeld amplitudes, frequen- cies and the phases with respect to the ﬁrst driving wave, respectively. In this paper, we choose ω1=ω, ω2= 2ω, E03≪E01,E02andω3≫ω.Unless stated otherwise, the relative phase φ13is set to zero. The binding poten- tial is taken as VG(x) =−αexp(−x2/β2), (3) which is a widely used expression for modelling short- range potentials. The harmonic spectra are calculated from the dipole acceleration ¨ x=/angbracketleftψ(t)\| −dV(x)/dx+ E(t)\|ψ(t)/angbracketright[12]. The kinetic energy of the electron upon return is taken as Ekin(t0,t1) =1 2[A(t1)−A(t0)]2(4) and the ponderomotive energy is given by Up=1 2/integraldisplayT 0A2(t)dt=3/summationdisplay i=1E2 0i 4ω2 i. (5) To ﬁrst approximation, if the third driving wave is much weaker than the others and its frequency is much higher thanω, the contribution from the additional ﬁeld to the ponderomotive energy (5) and to the kinetic energy (4) can be neglected. More speciﬁcally, for the parameters used in this paper, we observed, after solving the clas- sical equations of motion of an electron in a ﬁeld (2),that forE03/E02≤1 andω3>5ωthe inﬂuence of the third driving wave on these two latter quantities was not signiﬁcant. As a starting point, we shall discuss the existence of the enhancement in question. We will restrict ourselves to varying the relative phase φ12between the two low- frequency driving waves and the high-frequency ﬁeld pa- rameters. The bichromatic ﬁeld strengths are similar to these in [5], namely E01= 0.1 a.u.,E02= 0.032 a.u, which give the intensity ratio I2ω/Iω= 0.1.For these ﬁeld parameters, the high-harmonic spectrum displays a clear double-plateau structure, with a lower and an up- per cutoﬀ (c.f. Fig. 1). The lowest frequency is taken as ω= 0.057 a.u.,which is typically used in experiments. The ground-state energy was taken as \|ε0\|= 0.57 a.u., which roughly corresponds to the argon ionization poten- tial and, unless stated otherwise, we took α= 1.15 a.u. andβ= 1 a.u.in (3). This gives a model-atom with a single bound state. 010203040506070-4-3-2-1012 (b) φ12 = 0.3π E03=0 E03=0.002 E03=0.006 E03=0.012 E03=0.02log10Harmonic Yield (arb. units) Harmonic Order-4-3-2-1012 (a) φ12 = 0 FIG. 1. Harmonic spectra for an atom subject to the three-color ﬁeld (2), with strong-ﬁeld amplitudes E01= 0.1 a.u.,E02= 0.032 a .u., high-frequency ﬁeld strengths 0 ≤E03≤0.02 a.u., frequencies ω1= 0.057 a.u., ω2= 0.114 a .u.andω3= 0.57 a.u., and relative phase φ13= 0, for φ12= 0 (part (a)) and φ12= 0.3π(part (b)). The lower cutoﬀ (near harmonic order 50) and the upper cut- oﬀ (near harmonic order 62) are marked with arrows. The lower-cutoﬀ energy depends more sensitively on φ12than the upper-cutoﬀ energy [5]. The inﬂuence of the third driving wave on the har- monic spectra is shown in Fig. 1, for relative phases 2φ12= 0 andφ12= 0.3πand several ﬁeld strengths E03. The relative phase φ13was set to zero in both cases, and ω3was chosen to be ten times ω1. Apart from the strong enhancement around ω3(near harmonic order 10), the main diﬀerences between the case with a third wave and the bichromatic case ( E03= 0) are displayed in the so- called “upper-cutoﬀ” harmonics, which lie, for the phases in question, roughly at εu=\|ε0\|+ 3.8Up, with the cor- responding harmonic orders near N= 62. The inten- sity of this group of harmonics can be changed signiﬁ- cantly when the ﬁeld strength of the third wave is in- creased. Whereas for φ12= 0 these changes are quite irregular, for φ12= 0.3πthey appear as very pronounced enhancements, which reach three orders of magnitude. They make the intensities of the upper-cutoﬀ and lower- cutoﬀ harmonics comparable, eﬀectively extending the harmonic production region. This occurs already for a third wave whose intensity is only a few percent of that of the bichromatic ﬁeld. A particularly interesting feature of the scheme is that it aﬀects mainly the group of harmonis in the upper cut- oﬀ, leaving other groups of harmonics practically unaf- fected. This is in contrast with the purely bichromatic case, for which a change in the relative phase φ12leads to changes in the whole plateau structure [4,5]. In analysing the enhancement eﬀects, we recall that the presence of the high-frequency third wave does not modify the pon- deromotive energy and the classical trajectories. Thus, the three-step model leads us to the conclusion that a high-frequency induced process is injecting the electron into the continuum. This interpretation is supported by the diﬀerences between Figs. 1(a) and 1(b). For φ12= 0, there is still enough tunneling to compete with the high- frequency induced process, such that the quantum in- terference between both processes leads to irregular in- tensity variations. For φ12= 0.3π, on the other hand, tunneling is strongly suppressed, such that the electron reaches the continuum mainly due to the high-frequency induced process [5]. In the following we understand how the observed en- hancements depend on the relative phase φ13, on the ﬁeld strength E03, on the frequency ω3, and whether the atomic potential has any inﬂuence on it. 0.000 0.005 0.010 0.015 0.020 0.025 0.030-2.8-2.4-2.0-1.6-1.2-0.8-0.4 ω3=0.285,N=61 ω3=0.456,N=60 ω3=0.570,N=58 ω3=0.684,N=59log10Harmonic Yield (arb. units) E03(a.u.)FIG. 2. Harmonic yields as functions of the high-frequency ﬁeld strength E03, for the same ﬁeld as in Fig. 1, with φ12= 0.3πand 0 .285 a.u.≤ω3≤0.684 a.u.. The harmonic orders displayed in the ﬁgure correspond to the most eﬃcient enhancements obtained. The behavior with respect to φ13gives further infor- mation about the nature of the eﬀect due to the third wave. We veriﬁed that the upper-cutoﬀ harmonics re- main unchanged when φ13is varied. This result rules out the possibility that the injection can be described by an eﬀective potential barrier produced by the three waves together. Rather it suggests that the third wave is playing a very diﬀerent role than the bichromatic ﬁelds. Another very important issue is the dependence of the enhancements on the ﬁeld strength E03. A closer inspec- tion of Fig. 1(b) suggests that there is a saturation inten- sity for the upper-cutoﬀ harmonics (in the ﬁgure, close toE03= 0.01 a.u.). Furthermore, these harmonics are unequally enhanced, the maximum enhancement occur- ring, for the example in question, at the harmonic order N= 58. This maximum can be slightly displaced, de- pending on the frequency ω3. In Fig. 2, we show explic- itly the behavior of this maximally enhanced harmonic with respect to the ﬁeld strength E03, for several frequen- ciesω3. This ﬁgure shows that the enhancement eﬀect increases quickly in the weak ﬁeld region but it even- tually saturates when reaching the strong ﬁeld region, which means that upper-cutoﬀ harmonics can not be in- deﬁnitely enhanced by increasing the strength of the high frequency wave. 5.006.007.008.009.0010.0011.0012.00-6-5-4-3-2-10 N=58 N=59 N=60log10Harmonic Yield (arb.units) ω3/ω1 FIG. 3. Harmonic yields of neighboring upper-cutoﬀ har- monics as functions of the frequency ratio ω3/ω1, for a driving ﬁeld as in the previous ﬁgures and E03= 0.015 a.u.. The ion- ization threshold is at ω3= 10ω1. In Fig.3 we show the dependence of the enhanced har- monics with respect to the frequency ω3. The ﬁgure dis- plays major peaks at integer frequency ratios ω3/ω1and sub-peaks between the integers. The enhancements are specially strong at the integers, where the frequencies of the three driving waves are commensurate. The strong 3enhancements at the integers can be understood as a con- sequence of the high-harmonic generation process: since the production of harmonics takes place within many cy- cles of the driving ﬁeld, one expects it to be most eﬃcient when this ﬁeld is periodic. Fig.3 also demonstrates that the observed enhancements are not related to threshold eﬀects, since they occur when the frequency of the third wave is either below or above the threshold. The investigations presented above are from our nu- merical calculations for the short-range potential (3), withαandβchosen in such a way that it has a single bound state. To rule out the possibility that the eﬀect is due to an artifact introduced by this model potential, we performed similar calculations for several potentials wit h diﬀerent values for depth αand widthβ. We also stud- ied the soft-core potential VC=−α/bracketleftbig x2/β2+ 1/bracketrightbig−1/2. We observed that the enhancements are present in all situa- tions, being however stronger for short-range potentials. This is related to the fact that tunneling ionization is more eﬃciently suppressed in the short-range case, such that it does not compete with the high-frequency induced process. For the gaussian potential used, we obtain very pronounced enhancements for β≤2 a.u., which is well within the experimental range. In conclusion, we have shown that, with an additional high-frequency ﬁeld, we are able to enhance a group of very high harmonics of a bichromatic driving ﬁeld con- sisting of a wave of frequency ωand its second harmonic. These harmonics have energies up to \|ε0\|+ 4Up, which is about 30% higher than the monochromatic cutoﬀ fre- quencyεmax=\|ε0\|+ 3.17Up. They correspond to a set of electron trajectories for which tunneling is not very pronounced in the bichromatic case. The high-frequency ﬁeld provides an additional mechanism for the electron to reach the continuum, resulting in the enhancement of this group of harmonics. This eﬀect is particularly strong when the relative phase between the two strong driving waves is chosen such that tunneling ionization is strongly suppressed. Such a case is provided by taking φ12= 0.3π[5], for which an appreciable enhancement is already obtained with high-frequency ﬁelds whose intensities are only a few percent of that of the bichromatic ﬁelds. This phase con- trol is already experimentally feasible [6]. Another pre- rrequisite for pronounced enhancement eﬀects concerns the frequency of the third wave, which must be an inte- ger multiple of the fundamental frequency of the bichro- matic ﬁeld. Furthermore, our theoretical studies show that the enhancements are always present for very diﬀer- ent potentials, but they also suggest that a short-range potential gives stronger enhancements and therefore is a more appropriate choice for a possible experimental con- sideration. Acknowledgements: We are grateful to A. Fring and to J.M. Rost for useful discussions.[1] M. Q. Bao and A. F. Starace, Phys. Rev. A 53, R3723 (1993); A. Lohr, W. Becker, and M. Kleber, Laser Phys. 7, 615 (1997); B. Wang, X. Li, and P. Fu, J. Phys. B 31, 1961 (1998); D. B. Miloˇ sevi´ c and A. F. Starace, Phys. Rev. A 60, 3160 (1999);Phys. Rev. Lett. 82, 2653 (1999); Laser Phys. 10, 278 (2000). [2] A. de Bohan, Ph. Antoine, D. B. Miloˇ sevi´ c, and B. Pi- raux, Phys. Rev. Lett. 81,1837(1998); A. de Bohan, Ph. Antoine, D. B. Miloˇ sevi´ c, G. L. Kamta, and B. Piraux, Laser Phys. 9, 175 (1999). [3] See, e.g., D. B. Miloˇ sevi´ c, W. Becker and R. Kopold, Phys. Rev. A 61, 063403 (2000); D. B. Miloˇ sevi´ c and W. Becker, Phys. Rev A 62, 011403 (2000) and references therein. [4] C. Figueira de Morisson Faria, M. D¨ orr, W. Becker, and W. Sandner, Phys. Rev. A 60, 1377(1999); C. Figueira de Morisson Faria, W. Becker, M. D¨ orr, and W. Sandner, Laser Phys. 9, 388 (1999). [5] C. Figueira de Morisson Faria, D. B. Miloˇ sevi´ c, and G. G. Paulus, Phys. Rev. A 61, 063415 (2000). [6] See, e.g., U. Andiel, G. D. Tsakiris, E. Cormier, and K. Witte, Europhys. Lett. 47, 42 (1999). [7] C. Figueira de Morisson Faria and J. M. Rost, Phys. Rev. A62, 051402(R)(2000). [8] M. Yu. Kuchiev, Pis’ma Zh. Eksp. Teor. Fiz. 45, 319(1987) (JETP Lett 45(7), 404 (1987)); P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993); K. C. Kulander, K. J. Schafer, and J. L. Krause in: B. Piraux et al. eds.,Pro- ceedings of the SILAP conference , (Plenum, New York, 1993); M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier and P. B. Corkum, Phys. Rev. 49, 2117 (1994); W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 41, 4112 (1990) and 50, 1540 (1994). [9] See, e.g., G. B. Lubkin, Phys. Today 51(10), 21(1998); Yu. B. Kudasov et al, Pis’ma Zh. Eksp. Teor. Fiz. 68, 326 (1998)(JETP Lett. 68, 350 (1998)) for discussions on the strongest existing magnetic ﬁelds. [10] For the ﬁrst experimental evidence of materials which c an survive the required intensity range, consult M. Lenzner, J. Kr¨ uger, S. Sartania, Z. Cheng, Ch. Spielmann, G. Mourou, W. Kautek, and F. Krausz, Phys. Rev. Lett. 80, 4076 (1998). [11] D. B. Miloˇ sevi´ c and F. Ehlotzky, Phys. Rev. A 56, 3879 (1997); ibid. 57, 2859 (1998); D. B. Miloˇ sevi´ c and A.F. Starace, J. Phys. B 32, 1831 (1999). [12] See, e.g., K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992); J. L. Krause, K. Schafer, and K. Kulander, Phys. Rev. A 45, 4998 (1992). 4
Causal explanation for observed superluminal behavior of microwave propagation in free space W. A. Rodrigues, Jr.(1,2), D. S. Thober(1), A. L. Xavier, Jr(1,) (1)Center for Research and Technology, CPTec-UNISAL, Av. A. Garret, 267, 13087-290 Campinas, SP Brazil (2)Institute of Mathematics, Statistics and Scientific Computation, IMECC-UNICAMP CP 6065, 13083-970, Campinas, SP Brazil In this paper we present a theoretical analysis of an experiment by Mugnai and collaborators where superluminal behavior was observed in the propagation of microwaves. We suggest that what was observed can be well approximated by the motion of a superluminal X wave. Furthermore the experimental results are also explained by the so called scissor effect which occurs with the convergence of pairs of signals coming from opposite points of an a nnular region of the mirror and forming an interference peak on the intersection axis traveling at superluminal speed. We clarify some misunderstandings concerning this kind of electromagnetic wave propagation in vacuum. ()Corresponding author: xavier@cptec.br PACS:42.25.Bs In this note we make some comments on a recent letter by Mugnai, Ranfagni and Ruggieri [1] claiming the observation of superluminal behavior in microwave propagation for long distances in air. Our comments are of theoretical nature and about possible explanations for the measured effect. From the theoretical point of view, we start by analyzing the following statement quoted in [1] and attributed to [2]: Yet, there is no formal proof, based on Maxwell equations ME that no electromagnetic wave packet can travel faster than the speed of light. First of all we note that every physical wave (satisfying ME) produced by a physical device (antenna) must necessarily have a beginning and (possibly) an end in time. If this is the case, we say that the electromagnetic field configuration produced when ready has compact support in the time domain. Such a configuration is called electromagnetic pulse. Observe also that if the pulse generated by the device does not spread with an infinity speed, then when it is ready, let us say at 0=t it must occupy a finite region in space. In an appropriate reference frame, we can then write that at 0=t , the signal has support only for Rx≤/G72, where R is the maximum linear dimension involved. ME are a hyperbolic system of partial differential equations [3]. Moreover, each one of the components of the free electromagnetic field solves a homogeneous wave equation ( HWE ). It is then possible to prove under very general conditions (strictly hyperbolic Cauchy problem) that, if an electromagnetic pulse has field components and first time derivatives with compact support in space at 0=t , then the time evolution of such field components must be null [4] for tcRx+≥/G72. As usual c is the parameter that appears in the HWE satisfied by any of the components of the electromagnetic field. This result can be called finite propagation speed theorem FSPT . We emphasize here that the FSPT implies that the front of the pulse travels with maximum speed c (in some cases we can prove that it indeed travels with speed c) but it does not fix any minimum speed for the lateral boundary of the signal spread. This is a very important result since it enables the project of antennas for sending well focused waves. However, it is important to stress that perfect W. A. Rodrigues, Jr., D. S. Thober, A. L. Xavier, Jr Causal explanation for observed superluminal 2 focusing is impossible for any finite energy solution of ME [5]. Note also, that it is not possible to prove an analogous of the FSPT for waves that do not have compact support in the space domain (or in time domain). When ME are applied to the description of wave motion in dispersive media with losses or gains and under special conditions, the propagation o wave packets may exhibit superluminal (or even negative) groups velocities. However, Sommerfeld and Brillouin, showed long ago (see [6]) that electromagnetic pulses travel with front velocity c even in dispersive media. In fact they concluded that the concept of group velocity can not be applied to superluminal wave motion and this statement has been repeated in many textbooks since then, as, e.g., in [7,8]. However, their conclusion is misleading since it is now well known that superluminal groups velocities can be observed (see, e.g.[9-11]). Even negative group velocities have been observed [12,13]. The explanation for some of these superluminal (or negative) group velocities observed in dispersive media and also in microwave tunneling is found in the reshaping phenomenon [14-16]. However there are some claims that this is not the case [17]. The basic argument in favor of genuine superluminality, [17], is that a real wave packet must have compact support in the frequency domain because: ...signals with an infinite spectrum are impossible, since Planck has shown in 1900 that the minimum energy of a frequency component is ω/G68 . Lack of space prevent us from discussing this argument here. Instead we recall that Fourier theory implies that a signal with compact support in the frequency domain is unlimited in the time domain, i.e., has no fronts so that it is impossible to define a front velocity for it and only the group velocity has physical meaning. Following this reasoning, we cannot endorse the point of view of [17] which seems also the one adopted in [1], simply because it implies the existence of wave packets in the time range ∞<<∞− t , i.e., even before the antenna was turned on. The concepts presented above, although of fundamental character, are not well known as they should be. We now state a result that at first sight seems to contradict what has been said above, that is: ME (and also all the other linear relativistic wave equations) possess arbitrary speed solutions ( ∞<≤v0 ) that are undistorted progressive waves ( UPWs ) even in free space (for a review, see [16] and also [18-19]). These UPWs solutions, like plane wave solutions of ME, have infinite energy, and classical electromagnetic theory implies that they are only convenient approximations to what can be really built in the physical world. There exists therefore a crucial distinction between solutions of ME and physical realizable solutions of that equations. Once an exact UPW solution is known, it is possible to launch pulses that are finite aperture approximations ( FAA) to that UPW , i.e., pulses obtained through the well known Rayleigh-Sommerfeld approximation [20]. As already stated, there are subluminal, luminal and superluminal UPWs . A FAA to a subluminal ( superluminal ) UPW pulse is theoretically shown to have a peak travelling at subluminal ( superluminal ) group velocity. This phenomenon was predicted and observed for the first time in experiments with acoustic waves, where sub and superluminal means scv< or scv> respectively (sc the sound speed appearing in the corresponding HWE ). In [18] it was predicted that the phenomenon could be observed for electromagnetic superluminal X waves ( SEXWs ). In fact Saari and Reivelt produced a FAA to a SEXW pulse in the optical range [21]. For clarity we emphasize that all of the theoretical (analytical and numerical simulations) studies of real cases of FAA to exact superluminal UPWs have shown that the peak indeed travels at cv≥ in some particular circumstances ( c is the medium characteristic velocity). The initial front of the pulse (which travels at c) is reached by the peak after some propagation time when the pulse loses its lateral W. A. Rodrigues, Jr., D. S. Thober, A. L. Xavier, Jr Causal explanation for observed superluminal 3 confinement and starts to behave like an ordinary spherical wave. After these long but necessary preliminaries, we can now present specific criticisms to the contents of [1]. Recall that we can generate solutions of ME with the Hertz potential method [7,16]. The Hertz potential for a free ME solves a HWE . If we take the form used in [18] , e.g., a magnetic Hertz potential zm Φ=Π in a fixed direction, say the z- direction taken as the propagation direction, then Φ satisfies a HWE . A simple solution of that equation in cylindrical coordinates ),,( zϕρ is ()[] , exp)( ),,(0 zkti J ztz−−Ω= Φ ω ρ ρ (1) () . /2 2 2 zk c− =Ωω (2) In Eq.(1) 0J is the zeroth order Bessel function and Ω is a separation constant (see, e.g., [18] for details). The second equation above establishes the dispersion relation. However, in this case the electric component of the Maxwell field cannot have the same form as the Hertz potential in Eq.(1). As can be easily verified by direct computation, an electric Hertz potential, ze Φ=Π (see [7]) with Φ as in Eq.(1) naturally generates a longitudinal electric field composed of three terms, one of which with a 0J dependence, but that is not definitively the case of the experiment in [1] that reports a horn antenna emitting a TE wave. More importantly, the Hertz potential in (1) and its associated electromagnetic fields are not superluminal. Only the phase velocity is superluminal, the group velocity remains subluminal. This interpretation, as showed in [18] is indeed correct, since it is possible to find a Lorentz reference frame where the solution represents a standing wave. Also, for the acoustical case, as reported in [18], a Bessel beam, i.e., a FAA to the wave packet of the form []()zik TF Jz B exp)( )(1 0 ωρ−Ω=Φ (3) where )(ωT is an appropriate transfer function and 1−F is the inverse Fourier transform, is such that its peak was reported to travel at subluminal speed (i.e., with scv<). So, no FAA to an electromagnetic Bessel beam of the form of Eq.(3) can show any superluminality. If we accept the data in [1] as correct, we must conclude that no FAA to an electromagnetic Bessel beam was observed. But what kind of wave could produce the superluminal effect described in [1]? A proposed answer is a particular kind of SEXW (superluminal electromagnetic X-wave) pulse. X like pulses were firstly reported by Lu and Greenleaf in the acoustical case [22,23]. Their speeds have been measured in an experiment reported in [18], where also the mathematical theory of SEXWs and computer simulations for their behavior were presented. A straightforward solution can be generated through the following magnetic Hertz potential zm Φ=Π [18] (after writing θsenk=Ω and θcosk kz= , such that kc=/ω ) () ct zik X e kJkBdk zt− −∞ /Gf2= Φθθρ ρcos 0 0) sen()( ),,( , (4) where )(kB is a frequency distribution function. Theoretically, these waves are UPWs and move with superluminal speed θcos/cv= . Now, a pulse, i.e., a wave packet with compact support in the time domain (which is exactly what the authors of [1] used in their experiment) satisfying the boundary conditions at 0=z , [] ,0),,(,) ( )( ),,( 00 =∂Φ∂−Θ−Θ= Φ =− zXti X zztTt t ezt ρρω (5) (where )(tΘ is the Heaviside function) is such that its front propagates with the speed c, while the peak propagates with superluminal speed θcos/cv= . Of course, the phenomenon cannot last for ever and disappears when the peak catches the front traveling at the speed of light. These statements can be proved using methods analogous to the ones employed by Brillouin and Sommerfeld [6] and we have corroborated W. A. Rodrigues, Jr., D. S. Thober, A. L. Xavier, Jr Causal explanation for observed superluminal 4 their validity with complete computer simulations which will be reported elsewhere. In order to give a quantitative description of the experiment in [1], a simulation under the correct experimental conditions is necessary, which means to reproduce a FAA to the pulse generated by the horn antenna and reflected by the spherical mirror (see Figure 1 in [1]). The first simulation we present are based on a purely geometrical description of the rays in the apparatus and reproduce expected values in [1] of superluminal velocities. This geometrical simulation naturally includes the spherical aberration of the mirror. Mugnai, Ranfagni and Ruggieri signal velocity for each point along the symmetry z-axis is determined as follows: consider the annular source located on the mirror focal plane projecting rays onto the mirror. Figure 1: Schematic representation of the launcher (horn antenna) and mirror. The two point source model for the scissor effect is seen on the left separated by distance a. In Figure 1 the annular source is represented by two point sources, the annular slit as seen edged-on. Reflected rays cross the z-axis at different points depending on the aperture angle of the source rays (axicon angle). Each ray takes a specific time to travel from the reflection point on the mirror surface to the crossing point on the z-axis. Detectors are placed at different positions (distant L from each other) on the z- axis (not between mirror and source) and we simply calculate the time difference T between the rays reaching these different detectors. The signal velocity is then given by the derivative of the curve L-T. The axicon angle obtained by adjusting the diameter of the circular slit changes the pattern of time distribution along the z-axis, the larger the angle the more pronounced the superluminal effect. Our first simulation shown in Figure 2 for axicon angles 016=θ and 023=θ gives an increase of about 4% and 8% for the signal velocity, respectively. These numbers are in good agreement with the theoretical expectations described in [1]. Figure 2: Signal velocity as a function of the distance from mirror (z-axis) under the geometrical model for the two axicon angles of [1]. However the experimental results in [1] exceeds such numbers mainly for 023=θ and we do agree that diffraction effects could be responsible for the observed velocity in this case. Our belief is supported by more accurate simulations (based on finite difference method) of wave propagation including diffraction of rays due to source shadowing. However, we can make another approximate model that fits the measured effect for 023=θ . Recall that a X- wave can be approximately expanded as an integral over the polar angle ϕ (in the 0=z plane) of plane wave pairs [21] emitted from points of a circle in the plane with angles ϕ and W. A. Rodrigues, Jr., D. S. Thober, A. L. Xavier, Jr Causal explanation for observed superluminal 5 ϕπ− and traveling at speed c. The effect observed in [1] shows a varying propagation speed )(zv of the peak that cannot be explained solely by a single X wave (whose superluminal speed is constant ). If we admit however a sequence of spherical wave points emanating from simultaneous sources (an annular region) on the mirror and interfering on the z-axis, the dependence of v on z can be quantitatively explained. This is indeed the base of the so called scissor effect . For simplicity we admit a virtual annular source with diameter a placed somewhere on a plane behind (or in front of) the mirror. If t is the time counted since the production of the spherical pulse, then the scissor speed on the z-axis is given by 2 2 24/)( c atcttv −= , (6) and therefore the distance covered by the main scissor peak along z-axis until time t is /Gf2=zt tz dttv ttL 1)( ),(1 (7) where cat 2/1≥ is some reference time. We can therefore fit a curve to the experimental results in [1] based on 3 parameters: An offset in z, an offset in time and the distance between the sources, a. Another parameter would be the position along z of the virtual source, but for simplicity, we choose this position at 0=z . Since the interference peak is on the z-axis, it is only locally similar to a X-wave. There is no simple relation between the axicon angle of the local X pulse - which changes along z - and the axicon angle used in [1] on the assumption that it was indeed a Bessel pulse (as imagined by the authors of [1]). A larger circular slit radius (for the real source on the mirror focal plane) simply implies a larger separation of the virtual sources in our model used to fit the experimental results. Numerical simulations are shown in Figure 3 and are reasonably well in agreement with the data in [1]. We conclude the paper stressing that the experiment in [1] shows explicitly that a kind of generalized reshaping phenomenon occurs under appropriate conditions even for pulses propagating in free space (in the case of [1] we recall that air, the medium where the propagation occurs, is transparent to microwaves). There is no question of principle involved in the experiment. The approximated SEXWs produced by the experiment are of compact support in the time domain and their fronts propagate always with the speed c. Only the peak of these pulses travels at superluminal speed being detected instead of the fronts due to the limited detection threshold. The peak however does not causally connect source to detector, leaving relativity theory intact. Also, the phenomenon of superluminal motion cannot last indefinitely. In fact it lasts until the peak catches the front, defining the maximum distance (called field depth of order θcot)2/(a , see [18]) for which a SEXW wave is reasonably focused. A simple explanation for the superluminal motion reported in [1] is given by the interference of spherical wave fronts on the symmetry axis. The interference pattern builds the superluminal peak and constitutes the well known scissor effect. Figure 3: Fit of the delay time measurements as a function of distance L along the z-axis using the scissor effect model. 10=a cm. Triangles represent measured data. Left: 016=θ , Right: 023=θ . W. A. Rodrigues, Jr., D. S. Thober, A. L. Xavier, Jr Causal explanation for observed superluminal 6 Acknowledgements One of the authors (ALXJr) would like to thank financial support by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) under contract number 00/03168-0. References [1] D. Mugnai, A. Ranfagni and R. Ruggieri, Phys. Rev. Lett. 84, 4830 (2000). [2] S. Bosanac, Phys. Rev. A 28, 577 (1983). [3] R. Courant and D. Hilbert, Methods of Mathematical Physics , vol. 1 (J. Wiley and Sons, New York, 1996). [4] M. E. Taylor, Pseudo Differential Operators (Princeton Univ. Press, Princeton, 1981). [5] T. T. Wu and H. Lehmann, J. Appl. Phys. 58, 2064 (1985). [6] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960). [7] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). [8] J. D. Jackson, Classical Electrodynamics, third edition (J. Wiley and Sons, New York, 1999). [9] A. Enders and G. Nimtz, Phys. Rev. B 47, 9605 (1993) [10] A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 2071 (1994). [11] A. M. Steinberg, P. G. Kwiat and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993). [12] E. L. Bolda, J. C. Garrison and R. Y. Chiao, Phys. Rev. A 49, 3938 (1994). [13] L. J. Wang, A. Kumzmich and A. Dogariu, Nature 406, 277 (2000). [14] C. G. B. Garret and D. E. McCumber, Phys. Rev. A 1, 305 (1970). [15] T. Emig, Phys. Rev. E 54, 5780 (1996). [16] J. E. Maiorino and W. A. Rodrigues, Jr., What is Superluminal Wave Motion? , electronic book (at http:www.cptec.br, Sci. and Tech. Mag. 4, 1999) [17] G. Nimtz, Europhys. J. B 7, 523 (1999). [18] W. A. Rodrigues, Jr. and J. Y. Lu, Found. Phys. 27, 435 (1997). [19] E. C. de Oliveira and W. A. Rodrigues, Jr., Ann. Phys. (Berlin) 7, 654 (1998). [20] P. M. Morse and H. Feshback, Methods of Theoretical Physics , Part 1 (McGraw-Hill, New York, 1953). [21] P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997). [22] J. Y. Lu and J. L. Greenleaf, IEEE Transact. Ultrason. Ferroelec. Freq. Cont. 39, 19 (1992). [23] J. Y. Lu and J. L. Greenleaf, IEEE Transact. Ultrason. Ferroelec. Freq. Cont. 39, 441 (1992).
arXiv:physics/0012033v1 [physics.gen-ph] 16 Dec 2000TWENTY-SEVEN LINES ON A CUBIC SURFACE AND HETEROTIC STRING SPACETIMES Metod Saniga Astronomical Institute, Slovak Academy of Sciences, SK-05 9 60 Tatransk´ a Lomnica, The Slovak Republic (E-mail: msaniga@astro.sk – URL: http://www.ta3.sk/ /tildewidemsaniga) Abstract It is hypothetized that the algebra of the conﬁguration of tw enty-seven lines lying on a general cubic surface underlines the dimensional hierarch y of heterotic string spacetimes. 1. Introduction The correct quantitative elucidation and deep qualitative understanding of the observed dimensionality and signature of the Universe represent, un doubtedly, a crucial stepping stone on our path towards unlocking the ultimate secrets of t he very essence of our being. Although there have been numerous attempts of a various degr ee of mathematical rigoros- ity and a wide range of physical scrutiny to address this issu e, the subject still remains one of the toughest and most challenging problems faced by co ntemporary physics. In this contribution, we shall approach the problem by raising a somewhat daring hypothesis that the dimensional aspect of the structure of spacetime ma y well be reproduced by the algebra of a geometric conﬁguration as simple as that of the l ines situated on a cubic surface in a three-dimensional projective space. 2. The Set of Twenty-Seven Lines on a Cubic Surface It is a well-known fact that on a generic cubic surface, K3, there is a conﬁguration of twenty-seven lines /1/. Although this conﬁguration is geom etrically perfectly symmetric as it stands, it exhibits a remarkable non-trival structure when intersection/incidence re- lations between the individual lines are taken into account . Namely, the lines are seen to form three separate groups. The ﬁrst two groups, each compri sing six lines, are known as Schlaﬂi’s double-six . This is indeed a remarkable subset because the lines in eith er group are not incident with each other, i.e. they are mutually skew , whereas a given line from one group is skew with one and incident with the remaining ﬁve lines of the other group. The third group consists of ﬁfteen lines, each one being inci dent with four lines of the Schlaﬂi set and six other lines of the group in question. The b asics of the algebra can simply be expressed as: 27 = 12 + 15 = 2 ×6 + 15 . (1) There exists a particularly illustrative representation o f this algebra. The representa- tion is furnished by a birational mapping between the points ofK3and the points of a projective plane, P2/1/. Under such a mapping, the totality of the planar section s ofK3 has its counterpart in a linear, triply-inﬁnite aggregate ( the so-called web) of cubic curves 1inP2. Each cubic of the aggregate passes via six, generally disti nct points B i(i=1,2,...,6); the latter are called the base points of the web. And the twent y-seven lines of K3are projected on P2as follows. The six lines L(+) i(the ﬁrst group of Schlaﬂi’s double-six) are sent into (the neighbourhood of) the points B i. Other six lines L(−) j(j=1,2,...,6; the second Schlaﬂi’s group) answer to the six conics Qj(B1,B2,...,B j−1,Bj+1,...,B 6), each passing via ﬁve of the base points. Finally, the remaining ﬁf teen lines of the third group have their images in ﬁfteen lines Lij, joining the pairs of base points B iBj,i/negationslash=j. 3. An Algebra-Underlined Heterotic String Spacetime Now, let us hypothetise that the dimensional hierarchy of th e Universe is underlined by the above-discussed simple algebra, indentifying formall y each line of K3with a single dimension of a heterotic string spacetime. The total dimens ionality of the latter would then be 27instead of 26 /2/. Further, we stipulate that the group of ﬁft een lines answers to the ﬁrst set of compactiﬁed dimensions of heterotic strin gs. We are thus left with DS=12 dimensions corresponding to Schlaﬂi’s double-six, and surmise that this “Schlaﬂi” spacetime is a natural setting for the M-theory, or, in fact, for the F-theory /3/; because our algebra also implies that 12 = 2 ×6 = 2×(5 + 1) = 10 + 2! And what about the four macroscopic dimensions familiar to o ur senses? A hint for their elucidation may lie in the following observation. As e xplicitly pointed out, each line in the third group is incident with just fourlines of the double-six. Let us assume that one of the ﬁfteen lines in this group has a special standing am ong the others; then also the corresponding four Schlaﬂi’s lines have a distinguishe d footing when compared with the rest in their group, and the same applies to the four dimen sions they correspond to... To conclude, it is worth mentioning that our hypothesis gets a signiﬁcant support from a recent ﬁnding by El Naschie /4/, based on the so-called Cant orian fractal-space ap- proach, that the exact Hausdorﬀ dimension of heterotic stri ng spacetimes is 26.18033989, i.e.greater than 26. References 1. A. Henderson: 1911. The 27 lines upon a cubic surface , Cambridge University Press, Cambridge. 2. M. B. Green, J. H. Schwartz and E. Witten: 1987, Superstring theory , Cambridge University Press, Cambridge. 3. M. Kaku: 1999, Introduction to superstrings and M-theory , Springer Verlag, New York. 4. M. S. El Naschie: 2001, “The Hausdorﬀ dimensions of hetero tic string ﬁelds are D(−)=26.18033989 and D(+)=10,” Chaos, Solitons &Fractals ,12, 377–379. 2
arXiv:physics/0012034v1 [physics.gen-ph] 16 Dec 2000Several New Cases Violating Time Reversal Symmetry In the Processes of Particle Interactions Mei Xiaochun ( Institute of Theoretical Physics in Fuzhou, No.303, Build ing 2, Yinghu Garden, Xihong Road, Fuzhou, 350025, P.R.Chian, E-mail: fzbgk@pub 3.fz.fj.cn ) Abstract It is pointed that we now actually have no enough proofs to pro ve that time reversal symmetry is universally obeyed in the processes of particle interactio ns. The analyses show that the time reversal symmetry is obviously violated at least in the processes of p article’s decays, especially in the processes of strange particle’s decays, and some resonance states wit h only one middle resonance particle to appear, as well as particle pair’s annihilations. So the phe nomena of violating time reversal symmetry may exist commonly in the particle’s interactions. PACS number: 1130 According to the current viewpoint, except a few cases of K0particle decays, time reversal symme- try is considered universally tenable in the interaction pr ocesses of micro-particles. However, it should be pointed out that only a few experiments show the symmetry o f time reversal with low precision at present. Corresponding to so many processes of particle i nteractions, it is not enough for us to consider the symmetry of time reversal as a commonly obeyed l aw in the particle physics. The real situations may not be as what we believe now. It can be pointed out that the time reversal symmetry is obviously violated at least in the processes particle’s d ecays, especially in strange particle’s decays, and some resonance state with only one middle resonance part icle to appear, as well as particle pair’s annihilations. So we have to re-examine the conclusion abou t time reversal symmetry. We only discuss the experimental problems in this paper and will discuss the theoretical problems in the paper titled ” Time reversal symmetry not exist actually in the theories o f particle interactions”. In order to discuss the problems strictly, we ﬁrst deﬁne and c lassify the processes of time reversals. The processes can be classiﬁed as both the determinative and the statistical processes of time reversals. 1.The determinative processes of time reversals The determinative processes of time reversals can still be d ivided into both the determinately reversible processes and determinately irreversible proc esses In the determinative processes, let particles A and B collid e each other, then C and D particles are certainly produced. The determinately reversible proc ess of time reversal of A+B→C+Dis deﬁned as follows. When the velocities of the particles C and D are reversed accurately, they would move along the completely same paths as in the positive proce ss so that they would collide to each other. Then particles A and B would be certainly produced aga in. After that, A and B particles would also move along the completely same paths as they do in t he positive process and then apart each other. These processes are called as the determinately reversible processes of time reversals. If they can not do so, the process is called as the determinately irreversible proces of time reversal. The concept of the determinately reversible or irreversible pr ocess of time reversal is meaningful for the process of a single particle’s decay, so it is also meaningfu l for a system composed of a large number of particles. It is obvious that there is no any experiment to show the exist ence of the determinately irreversible or reversible process of time reversals in the interactions of micro-particles up to now. First, in the nature processes, a particle’s velocity can not reverse aut omatically, so the process does not exist in nature. Next, we now can not accurately reverse a particle ’s velocity by artiﬁcial method. No laboratory in the world now can do it. Therefore, both the det erminately reversible and determinately 1irreversible processes of time reversals are actually unve riﬁed at least on the resent experimental level. What we have done now is the statistical processes of time rev ersals shown as follows. 2.The statistical processes of time reversal For the statistical process of time reversal of A+B→C+D, we do not reverse the velocities of particles C and D directly then make them collide along the co mpletely same orbits as in the positive process. Substitute for it, we take other C and D particles ob tained from another method and make them collide each other along the diﬀerent paths. After coll isions, A and B particles may be produced or may not. For the process in which A and B particles are produ ced, if the transition amplitudes ofA+B→C+DandC+D→A+Bare the same, we call them the statistically reversible (or symmetry) processes of time reversal. Otherwise, we called them the statistically irreversible processes. The concept of statistically reversible or irreversible pr ocesses of time reversal is only meaningful for the system composed of a large number of particles. It is mean ingless for the process of a single particle. It is obvious that all have been done in the current experimen ts are the statistically processes of time reversal, so we only discuss them below. But the theoret ical discussions are beneath determinative and statistical processes, for in the theoretical discussi on of time reversal, the directions of particle’s moments and spins are always accurately reversed, but the tr ansition amplitudes are always calculated in the statistical forms. Some time we call the statistical p rocess of time reversal as the time reversal process directly for simpliﬁcation below. The time reversal of a single particle’s decay and the double particle’s collision are discussed individually below. For a single particle’s decay, we have f ollowing situations: (1).The statistically irreversible processes of time reve rsals Suppose the transition amplitude of A particle decaying int o B and C particle is SA→BC, and the transition amplitude of the reversal process to produce A pa rticle by the collision of B and C particles isSBC→A. IfSA→BC=SBC→A, we call the process as the statistically reversible (or sym metry) process of time reversal. If SA→BC/negationslash=SBC→A, we call the process as the statistically irreversible (or non-symmetry) process of time reversal. Suppose the transi tion probabilities of positive and reversal processes in the unit time are dWA→BCanddWBC→Aindividually, according to the deﬁnition, we have dWA→BC= lim T→∞1 T¯/summationdisplay \|SA→BC\|2dp3 Adp3 Bdp3 C (1) dWBC→A= lim T→∞1 T¯/summationdisplay \|SBC→A\|2p3 Adp3 Bdp3 C (2) The transition probability WA→BCof positive process is a measurable quantity with the relati on τ−1=WA→BCHereτis particle’s lifetime. But for the reversal process, what c an be measured in the experiment is the cross-section of collision Σ BC→A. We have the relation dΣBC→A=1 IdWBC→A=δ4(p−q)πK 2B−1JEA¯/summationdisplay \|MBC→A\|2d3pA (3) In whichIis the unit ﬂow strength, J=/radicalbig (pB·pC)2−M2 BM2c,MBC→Ais the invariable amplitude, the four-dimension moments p=pA,q=pB+pC,p0=EA,q0=EB+EC,EA,EBandECare the energies of particle A, B and C. K is the product of Fermion ’s masses. If three are no Fermions in the process, K=1. The integral of Eq.(3) can be written as ΣBC→A=δ(EA−EB−EC)πK 2B−1JEA¯/summationdisplay \|MBC→A\|2 pA→pB+pC(4) Considering the law of energy conservation, we have EA=EB+ECand getδ(EA−EB−EC)→ ∞ according to the nature of δfunction. If MBC→A/negationslash= 0, we have Σ BC→A→ ∞. This is obviously impossible. In order to let Σ BC→Aﬁnite, we have to suppose MBC→A→0 orSBC→A→0. The resultMBC→A→0 coordinates with the experimental facts. In fact, no any ex periment has reported 2that in the double particle’s collision only one particle is produced in its ﬁnal state up to now (in spite of the collisions producing a boson in its middle state .). Even though the process two particles colliding and forming a single particle may be possible, the possibility would be very little so that it can almost be regarded as zero. We can take the process of π0meson decaying into double photons π0→2γas an example(1). In this case we have¯/summationtext\|Mπ0→2γ\|2=m4/2. In the center of mass frame, π0meson is at lest, Eπ=m= 2Eγ,J= 2Eγ\|pγ\|= 2E2 γ. If the process is reversible, we can get from Eq.(4) Σ2γ→π0=δ(EA−EB−EC)πm 4(5) The result can be veriﬁed by means of experiments. But consid ering two facts, the result can be considered impossible immediately without the any experim ent. First, we have Σ BC→A→ ∞ because ofδ(EA−EB−EC)→ ∞. This is completely impossible. In order to avoid this diﬃcu lty, we should have¯/summationtext\|Mπ0→2γ\|2/negationslash=¯/summationtext\|M2γ→π0\|2→0 , so that Σ 2γ→π0becomes ﬁnite. Next, it has never been obtained that ψ0meson can be formed by the collision of two free photons (The d etails will be discussed later.). So the process π0→2γviolates the symmetry of time reversal obviously. The similar situation is the process η→2γ. Though we have not done the time reversal experiment of a single particle’s decays described above now, we can concl ude by means of the theoretical analyses that the processes of a single particle decaying into two or m ore particles are irreversible for time reversal. Otherwise we would have to face serious contradic tion in theory. We can discuss this conclusion more directly from another an gle. Suppose we have an isolated system composed of unstable particles at beginning, for exa mple, a large number of free neutrons. After long enough time, most of neutrons in the system would d ecay into protons, electrons and neutrinos. Then, let those protons, electrons and neutrino s collide to each other. (Because we only discuss the statistical process of time reversal, not the de terminate process, it is unnecessary for us to reverse particle’s velocity directly.). It is obvious that the system is completely impossible to return to its original pure neutron state. (2). The statistically and completely irreversible proces ses of time reversals The statistically and completely irreversible processes o f time reversals is deﬁned meaningful for both a single particle and a system composed of a large number of particles. In the reversal processes, we do not directly reverse the velocities of B and C particles coming from A particle’s decay and let them collide. Instead, we use B and C particles coming from ot her resources and collide them each other. After that, we ﬁnd that A particle is never produced. T he obvious examples of statistically and completely irreversible processes of time reversals ar e the processes of strange particle’s decays. For example Λ0→π−+p,n+π0,p+e−+ ¯νe, K0→π0+π0,π++π−,µ++µ−,Σ0→Λ0+γ (6) All of these processes above can be achieved through natural or artiﬁcial forms. But no any opposite process shown below has been ﬁnd in nature or in laboratories in the world up to now ψ−+p→Λ,n+ψ0→Λ,p+e−+ ¯νe→Λ ψ0+ψ0→K0,ψ++ψ−→K0,µ++µ−→K0(7) These processes are regarded to violate the law of strange nu mber conservation and impossible to exist. The strange particles always produce associatively through strong interaction but decay alone through weak interaction. For example, the producing proce sses of Λ and K0particles ψ−+p→Λ +K0 ψ−+p0→Σ0+K0(8) Therefore, if the transition amplitudes of non-strange par ticle’s decays are considered to be tending to zero, the transition amplitudes of strange particle’s deca ys should be considered zero strictly. The law 3of strange number conservation forbids the existence of rev ersal processes of strange particle’s decays. The similar situations are the processes of charmed particl e’s decays, but it needs not to discuss any more here. (2).The other situations. There are two kinds of experiments used to study the time reve rsal problems of single particle’s processes at present. One is to measure neutron’s electrica l dipole moment. If electromagnetic in- teraction is symmetry for time reversal, the neutron’s elec trical dipole moment should be zero. The current experiments show that the measurement value is µ<1.0×10−25ecm. But this result can only show that the existing process of stable neutron’s existenc e seems time reversal symmetry. It can not show that the decay process of an unstable particle is also ti me reversal symmetry. Another experiments for the time reversal of a single partic le’s processes are to measurement the phase angles of decay particles. Because the calculatio ns of phase angles invoice the theory of interaction, it is diﬃcult for us to know what is the theoreti cal eﬀect and what is the experimental eﬀect. These two kinds of experiments are indirect. From the m we can not decide whether or not time reversal is symmetry for micro-particle’s decays. The refore, we can get conclusion from discussion above that for the decay processes of micro-particles, the r eversibility of determinative time reversal is only an unveriﬁed suppose, and the reversibility of stati stical time reversal is actually impossible. The collision processes of double particles are discussed a s follows. They can also be divided into several classes below. (1).The statistically reversible processes of time revers al. Suppose the transition amplitude of producing B and C partic les by the collision of A and B particles is SAB→CD, the transition amplitude that A and B particles are produce d by the collision of B and C particles is SCD→AB. If the invariable amplitudes are the same, i.e., MAB→CD=MCD→AB, we say that the process has reached the detail balance and cal l the process as the statistically reversible (or symmetrical) process of time reversal. Otherwise, the p rocess is considered as the statistically irreversible (or non-symmetry) process of time reversal. A t present it is generally thought that the collision processes of double micro-particles are reversi ble or symmetrical for time reversals. But only a few experiments support this conclusion actually shown as follows a). The processes of strong interaction with(2) 24Mg+d↔25Mg+p and 24Mg+α↔27Al+p (9) The experiments support the symmetry of time reversal with a bout 0.5 per cent precision. b). The process of electromagnetic interaction with(4) n+p↔d+γ (10) Only 20 per cent precision supports the symmetry of time reve rsal. The precision seems too low. c). The experiments used to measurement πmeson’s spin(3) ψ++d↔p+p (11) Because there exist so many collisions of double particles, only based on such a few experiments, it is not enough for us to aﬃrm that all collision processes of double particles are reversible for time reversal. (2)The statistically irreversible processes of time rever sa It can be pointed out that there exists a kind of processes of d ouble particle’s s that obviously violate the symmetry of time reversal. They are the processe s in which particle pairs collide, annihilate and produce new particles with only one resonant particle ap pearing in the middle state. For example e++e−→ρ0(770)→π+π−,µ+µ−,e+e−(12) 4Now we analysis the process e++e−→ρ0(770)→ψ++ψ−in detail. Because the process involves strong interaction with πmesons in the ﬁnal state, the accurate form of transition amp litudeSe→π can not be obtained, so that the section of the process can not be calculated. However, because the initial and ﬁnal states are deﬁnite, we can always write the c ross-section of collision in the center -of-mass frame as follows dσe→π=m2 e/radicalbig s−4m2π 16π2s/radicalbig s−4m2e¯/summationdisplay \|Me→π\|2dΩ(θ,ϕ) (13) Heremeis electron and positive electron’s mass mπis positive and negative meson’s mass, sis the total energy of the system in the center -of-mass frame. For t he process of time reversal ψ†+ψ−→ρ0(770)→e†+e−(14) The cross-section of collision can be write as dσπ→e=m2 e/radicalbig s−4m2e 16π2s/radicalbig s−4m2π¯/summationdisplay \|Mπ→e\|2dΩ(θ,ϕ) (15) After the average of the initial state and the sum of the ﬁnal s tate are taken, we have individually ¯Σ\|Me→π\|2=1 4/summationdisplay r,s\|Me→π\|2 ¯Σ\|Mπ→e\|2=/summationdisplay r,s\|Mπ→e\|2(16) If the process is reversible for time reversal, we have Me→π=Mπ→eso dσπ→e dσe→π=4(s−4m2 e) s−4m2π(17) Suppose the middle particle ρ0is at lest, its static mass is mρ= 770MeV so√s= 770MeV , Taking me= 0.51MeV m π= 139.6MeVthe ratio between the diﬀerential cross-sections of positi ve and opposite processes is dσπ→e dσe→π≃4.14 (18) That is to say, the diﬀerential cross-section of reversal pr ocess is 4.14 times more than that of positive process. Besides, there is another relation dWfi=Ifid/summationdisplay fi=Ifiδ4(P−Q)dσfid4Q (19) The relative ﬂow intensities of incident particles in the ce nter-of-mass frame are individually Ie→π= lim V→∞Je→π VE2e= lim V→∞/radicalbig s(s−4m2e) 2VE2e(20) Iπ→e= lim V→∞Jπ→e VE2π= lim V→∞/radicalBig s(s−4m2 pi) 2VE2 pi(21) Because of Ee=Eπin the center-of-mass frame, by using Eq.(17) the ratio of th e transition proba- bilities between the positive and opposite processes is dWπ→e dWe→π=Iπ→edσπ→e Ie→πdσ e→=4/radicalbig s−4m2e/radicalbig s−4m2π(22) 5Similarly, we have dWπ→e dWe→π≃4.27 (23) That is to say that the transition probability of reversal pr ocess is 4.27 times more than that of the positive process. However, these results are completely impossible. It can be known from Meson Particle Listings (5)that the decay branching ratio of ρ0→π†π−nears 100 e++e−→ψ(3770) →D¯D,e+e−(24) e++e−→ω(782)→π+π−π0,π+π−,π0γ,e+e−(25) e++e−→φ(1020) →K+K−,π+π−π0,π0γ,e+e−(26) e++e−→J/ψ(3100) →hadrons,e+e−(27) In the processes, the decay branching ratio of π(3770) →D¯Dnears 100 Another kind of the processes in which the symmetry of time re versal is seriously violated are that the particle pairs of low energy annihilate into radiation. Taking the annihilation process of electron paire++e−→2γas an example, we have the cross-section of electron pair ann ihilating into two photons in the center-of-mass frame(6) σe→γ=πr2 0m2 4VE2e[3−V2 Vln\|1 +V 1−V\|+2(V2−2)] (28) Hereγ0is electron’s classical radius, is electron’s mass, Eeis electron’s energy in the center-of- mass frame, V=/radicalbig 1−m2/E2eis electron’s speed in the center-of-mass frame. Let m= 0.5MeV, Ee= 1MeV, we can get σe→γ= 0.32πr2 0. For the reversal process γ+γ→e++e−, suppose the transition amplitudes of positive and opposite process es are the same, according to quantum electrodynamics, the cross-section of photon pair collide s and transforms into electron pair in the center-of-mass frame is(5) σγ→e=πγ2 0m2 2ω2[(2 +2m2 ω2−m4 ω4)ln\|ω m+/radicalbigg ω2 m2−1\| −/radicalbigg 1−m2 ω2(1 +m2 ω2)] (29) Letω= 1MeV, we can get σγ→e= 0.53πr2 0. The result show that σγ→e≃1.67σe→γ>σe→γ, that is to say, the cross-section of photon pair collides and tran sforms into electron pair is 1.67 times more than that of electron pair annihilating into two photons. Th is conclusion is unimaginable. It is easy for the electron pair of low energy annihilating into photon s. But up to now it has no any report to show that two free photons can collide and transform into ele ctron pair with bigger possibility than that in the process of electron pair annihilating into photo ns. What has been observed in the current experiment is the process γ+Z→e†+e−+Z, in whichγis a photon of high energy, Zis a heavy atomic nucleus. It is supposed in the process that a free phot on collides with an imaginary photon then electron pair is produced, not two free photons collide . In fact, even though two free photons can collide and transform into electron pair, its probability s hould be very low, otherwise our university would not like what we now see mainly to compose of positive pa rticles. We can discuss this problem further. The scatter of two photo ns is a four-order process. Let ω represent photon’s energy, mrepresent electron’s mass, r0represent electron’s classical radium, by the calculation of re-normalization(6), we know that its cross-section directs ratio to ω6in the low energy process when ω<<m σL=πr2 0α2 π256×139 5×902ω6 m2≃10−6πr2 0ω6 m6(30) 6Hereα=e2/4π= 1/137. Suppose ω/m= 0.01we getσL= 10−18πr2 0. So the probability of collision of two low energy photons is very small, almost can be seen as z ero. On the other hand, in the high-energy process with ω>>m , the scattering cross-section directs ratio to ω−2with σH=bπr2 0m2 ω2(31) Herebis a constant without dimension. The order of magnitude of co nstantbis estimated as follows. It can be known from Eq.(30) that the scattering cross-secti on increases with the increase of photon’s energy under the condition of low energy, but decreases with the increase of photon’s energy under the condition of high energy. So it must exist an energy value on this value the scattering cross section does not increase or decrease with energy’s change. For simp liﬁcation, taking this energy as the middle value withω=m, and letσH=σL, we can get b= 10−6, and have σH= 10−6πr2 0m2 ω2(32) Under the condition of high energy, suppose ω/m= 100we can get σH= 10−10πr2 0. This is also very small, almost can be regarded as zero, though it is quite big t hanσL. On the other hand, we known that the orders of magnitudes of th e cross-section of electron pair’s scatter and annihilation are the same with σ∼πr2 0m2/E2. So the orders of magnitudes of the cross-section of photon pair’s scatter and annihilation sh ould also be the same. But according to the estimation above, if the process of electron pair annihi lating into photons is reversible, according to Eq.(29), the orders of magnitudes of the cross-section of two high energy photons ( ω/m= 100) annihilating into electron pair is about σ∼10−4πr2 0. This is about 106times more than that of the scatter of two photons. The result is unimaginable. From the cases shown above, it is reasonable for us to conside r that in the other processes of particle’s reactions, no matter what kind of interaction, s trong or weak or electromagnetic interactions, the symmetry of time reversal would be widely violated more o r less. The situation seems like the law of parity conservation several decades ago, the symmetry hy pothesis of time reversal in the processes of particle physics is an unveriﬁed one, and in fact, it may be co mpletely wrong. So we should re-examine this hypothesis by the further theoretical and experimenta l researches. References 1.Zhou Guoxing, Quantum Field Theory, Scientiﬁc Publishin g House, 1441980. 2.W.G.Weitkamp, D.W.Sstorm, D.C.Shreve, W.J.Braith Wait e, D.Bodansky, Phys. Rev. 165, 1233 (1968).W.Von Witsch, A.Richter, P.Von Brentano, Phys. Rev. Letter. 19,524,(196 7). 3.R.Dubin, H.Loar, J.Steinberger, Phys. Rev. 83,646,(195 1), R.Marshak, Phys.Rev. 82,313,(1951). 4.Zhang Naisheng, Particle Physics, Scientiﬁc Publishing House, 35 (1987). 5.Meson Particle Listings The European Physical Journal, C 33661998. 6... , .. , Quantum Electrodynamics, Scientiﬁc Publishing House, 341, 334, 56 4 (1964). 7
arXiv:physics/0012035v1 [physics.gen-ph] 16 Dec 2000Time Reversal Symmetry Not Exist Actually In the Theories of Particle Interactions Mei Xiaochun ( Institute of Theoretical Physics in Fuzhou, No.303, Build ing 2, Yinghu Garden, Xihong Road, Fuzhou, 350025, P.R.Chian, E-mail: fzbgk@pub 3.fz.fj.cn ) Abstract It is pointed out that the judgment condition of time reversa l symmetry in the processes of particle interaction is that the Hamiltioians of systems satisfy the relation ˆH(t) =ˆH∗(−t). But this condition can not be satisﬁed actually in the current interaction theo ries, so the symmetry of time reversal does not exist actually in the current theories of particle inter actions. In the current proof of time reversal symmetry, the relation ˆH(t) =ˆH∗(−t) is supposed to exist in advance, then the translation relat ions between the time reversal operator and the ﬁeld quantities a re deduced. So the current proof about the symmetry of time reversal is not a real one. PACS number: 1130 In quantum mechanics, the transformation of time reversal i s carried out in the light of following procedure. Let t→ −tin the Schordinger’s equation. i∂ ∂tψ(x,t) =ˆH(t)ψ(x,t) (1) and suppose that the Hamiltonian ˆHis unchanged when t→ −twith ˆH(t) =ˆH(−t). Thus, we get. −i∂ ∂tψ(x,−t) =ˆH(t)ψ(x,−t) (2) Then, take the complex conjugation of Eq.(2) and suppose ˆH∗(t) =ˆH(t) again, we have i∂ ∂tψ∗(x,−t) =ˆH(t)ψ∗(x,−t) (3) Comparing Eq.3 with Eq.1, it can been seen that ψ∗(x,−t) andψ(x,t) satisfy the same equation. So we deﬁne time reversal operator Tas follows when it is acted on the wave function Tψ(x,t) =ψ∗(x,−t) (4) In this way, ψ∗(x,−t) represents the wave function of time reversal process. Eq. (4) shows that the operator of time reversal is an antiunitary operator, which can be deﬁned as generally T(λψ1+Xψ2) =λ∗Tψ1+X∗Tψ2 (5) Because wave function ψ(x,t) andψ∗(x,−t) satisfy the same equation, we have ψ∗(x,−t) =bψ(x,t). Herebis a constant. We can take b= 1 for simpliﬁcation. But ψ(x,−t)/negationslash=ψ(x,t) in general. It seems that the positive process is diﬀerent from the reversa l process. However, because the wave function can not be directly measured, what can be done is the possibility density ρ. Suppose ρ(x,t) =ψ∗(x,t)ψ(x,t) is the possibility density of the positive process, the pos sibility density of the time reversal process is ρ(x,−t) =ψ∗(x,−t)ψ(x,−t). If the motion equation of quantum me- chanics is unchanged under the time reversal, we have ψ∗(x,−t) =ψ(x,−t) so that the possibility densities of the positive and opposite processes become the same withρ(x,−t) =ψ∗(x,−t)ψ(x,−t) = ψ(x,t)ψ∗(x,t) =ρ(x,t). Therefore, the process is considered as the reversible or symmetrical for 1time reversal. Otherwise we have ρ(x,−t)/negationslash=ρ(x,t), and the process is considered as irreversible or unsymmetrical. The premise of ρ(x,−t) =ρ(x,t) is that the Hamiltonian of the system should satisfy the relation ˆH∗(−t) =ˆH(t). IfˆH∗(−t)/negationslash=ˆH(t)the process is irreversible for time reversal. Therefore, the relation ˆH∗(−t) =ˆH(t) is a criterion to decide whether or not the process reversib le or irreversible for a quantum system obeying the Schrodinger’s equation (1) . There are two kinds of schemes about time reversal in the theo ry of particle physics at present (1)One is to deﬁne the operator of time reversal as an antiunitar y operator shown in Eq.(5). The transformation relations of scalar ﬁeld, vector ﬁeld and sp inor ﬁeld under time reversal are deﬁned as follows individually. Tψ(x,t)T−1=ψ(x,−t) (6) Tψ(x,t)T−1=ir1r3ψ(x,−t) =σ2ψ(x,−t) (7) TAµ(x,t)T−1=−Aµ(x,−t) (8) From the formulas above, we can prove that the Hamiltonian of electromagnetic interaction and the motion equation Hm=−AµJµ i∂ ∂t\|t=Hm\|t (9) are unchanged under the time reversal. Here jµ=ie 2(¯ψrµψ−ψτrτ µ¯ψτ) (10) Aµandψsatisﬁes the motion equations below (∇2−∂2 t)Aµ=−jµ (11) [rµ(∂µ−ieAµ) +m]ψ= 0 (12) ¯ψ[rµ(∂µ+ieAµ)−m] = 0 (13) Another scheme of time reversal is the so-called Wigner tran sformation. The antiunitary operator is also involved. The transformation form is more complex. B ut it is unnecessary for us to discuss it here any more. It is commonly believed that the time reversal invariabilit y in the current theories of particle interactions has been proved. However, it can be seen that th is kind of proof is not a real one. Because in the current proof, the relation ˆH(t) =ˆH∗(−t) is supposed to exist in advance without providing the proof of the relation, then the translation re lations of the ﬁelds are deduced. That is to say, the theory has been supposed unchanged in advance und er time reversal. However, what we should do is to prove the relation ˆH(t) =ˆH∗(−t) at ﬁrst if the theory is invariable under time reversal considering the fact that in quantum theory of ﬁeld, the moti on equation (9) is completely the same as that in quantum mechanics shown in Eq.(1). As shown below, however, it can easy be proved that this condition can not be satisﬁed actually in the current th eories of particle interactions. By using the relations r∗ µ=rτ µ,r4r=−rr4and considering the four-dimension quantity Aµ= (A,iA0) with A∗=A,A∗ 4=−A4, we have A∗ µ(¯ψrµψ)∗=A∗·(ψ†r4rψ)∗+A∗ 4(ψ†r4r4ψ)∗ =A·ψτr4r∗ψ∗ −A4ψτr4r4ψ∗ =−A·ψτr∗r4ψ∗−A4ψτr∗ 4r4ψ∗=−Aµψτrτ µ¯ψτ(14) Similarly, we can get A∗ µ(ψτrτ µ¯ψτ)∗=−Aµ¯ψrµψTherefore, we have H∗ m(−t) =−A∗ µ(−t)j∗ µ(−t) 2=ie 2A∗ µ(−t)[¯ψ(−t)rµψ(−t)−ψτ(−t)rτ µ¯ψτ(−t)]∗ =ie 2Aµ(−t)[¯ψ(−t)rµψ(−t)−ψτ(−t)rτ µ¯ψτ(−t)] =−Hm(−t) (15) We will prove that −Hm(−t)/negationslash=Hm(t). Because the calculations of concrete problems are based o n the interaction representation, we also do in the same repre sentation. It is known that the Hamiltonia can also be written as Hm=−ieN(¯ψˆAψ) =−ie(Aµrµ)αβ(¯ψ(+) αψ(+) β+¯ψ(+) αψ(−) β−ψ(+) β¯ψ(−) α+¯ψ(−) αψ(−) β) (16) The quantized ﬁelds can be written as Aµ(x,t) =A(+) µ(x,t) +A(−) µ(x,t) =4/summationdisplay σ=1Aσ(+) µ(x,t) +4/summationdisplay σ=1Aσ(−) µ(x,t) (17) Aσ(+) µ(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3k√ 2ωǫσ µ(k)α(+) σ(k)e−i(k·x−ωt)(18) Aσ(−) µ(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3k√ 2ωǫσ µ(k)ασ(k)ei(k·x−ωt)(19) ¯ψ(x,t) =¯ψ(+)(x,t) +¯ψ(−)(x,t) =2/summationdisplay r=1[¯ψ(+) r(x,t) +¯ψ(−) r(x,t)] (20) ψ(x,t) =ψ(+)(x,t) +ψ(−)(x,t) =2/summationdisplay r=1[ψ(+) r(x,t) +ψ(−) r(x,t)] (21) ¯ψ(+) r(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3p/radicalbiggm E¯ur(p)b+ r(p)e−i(p·x−Et)(22) ψ(−) r(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3p/radicalbiggm Eur(p)br(p)ei(p·x−Et)(23) ¯ψ(−) r(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3p/radicalbiggm E¯νr(p)dr(p)ei(p·x−Et)(24) ψ(+) r(x,t) =1 (2π)3/2/integraldisplay+∞ −∞d3p/radicalbiggm Eνr(p)d+ r(p)e−i(p·x−Et)(25) Put the formulas shown above into Eq.16, we obtain at last Hm(x,t) =i4/summationdisplay σ=1em 2(2π)9/2/integraldisplay /integraldisplay /integraldisplayd3kd3p1d3p2√2ωE1E2[rµǫσ µ(k)]αβ ×[¯urα(p1)νrβ(p2)α+ σ(k)b+ r(p1)d+ r(p2)e−i[(p1+p2+k)·x−(E1−E2+ω)t] +¯urα(p1)µrβ(p2)α+ σ(k)b+ r(p1)br(p2)e−[(p1−p2+k)·x−(E1−E2+ω)t] −νrβ(p2)µrα(p1)α+ σ(k)d+ r(p2)br(p1)ei[(p1−p2−k)·x−(E1−E2−ω)t] −¯νrα(p1)µrβ(p2)α+ σ(k)dr(p1)br(p2)ei[(p1+p2−k)·x−(E1+E2−ω)t] +¯µrα(p1)νrβ(p2)b+ r(p1)d+ r(p2)ασ(k)e−i[(p1+p2−k)·x−(E1+E2−ω)t] 3+¯µrα(p1)µrβ(p2)b+ r(p1)br(p2)ασ(k)e−i[(p1−p2−k)·x−(E1−E2−ω)t] −νrβ(p2)µrα(p1)d+ r(p2)br(p1)ασ(k)ei[(p1−p2+k)·x−(E1−E2+ω)t] −¯νrα(p1)µrβ(p2)dr(p1)br(p2)ασ(k)ei[(p1+p2+k)·x−(E1+E2+ω)t]] (26) The formula (26) can be simpliﬁed as Hm(x,t) =/integraldisplay /integraldisplay /integraldisplay d3kd3p1d3p2[A1(k,p1,p2,x)ei(E1+E2+ω)t+A2(k,p1,p2,x)ei(E1−E2+ω)t +A3(k,p1,p2,x)e−i(E1−E2−ω)t+A4(k,p1,p2,x)e−i(E1+E2−ω)t +A5(k,p1,p2,x)ei(E1+E2−ω)t+A6(k,p1,p2,x)ei(E1−E2−ω)t +A7(k,p1,p2,x)e−i(E1−E2+ω)t+A8(k,p1,p2,x)e−i(E1+E2+ω)t(27) So according to Eq.(15), we have H∗ m(x,−t) =−Hm(x,−t) =−/integraldisplay /integraldisplay /integraldisplay d3kd3p1d3p2× [A1(k,p1,p2,x)e−i(E1+E2+ω)t+A2(k,p1,p2,x)e−i(E1−E2+ω)t +A3(k,p1,p2,x)ei(E1−E2−ω)t+A4(k,p1,p2,x)ei(E1+E2−ω)t +A5(k,p1,p2,x)e−i(E1+E2−ω)t+A6(k,p1,p2,x)e−i(E1−E2−ω)t +A7(k,p1,p2,x)ei(E1−E2+ω)t+A8(k,p1,p2,x)ei(E1+E2+ω)t] (28) Therefore, we get Hm(x,t)−H∗ m(x,−t) =Hm(x,t) +Hm(x,−t) =/integraldisplay /integraldisplay /integraldisplay d3kd3p1d3p2 ×[A1(k,p1,p2,x)(ei(E1+E2+ω)t+e−i(E1+E2+ω)t) +A2(k,p1,p2,x)(ei(E1−E2+ω)t+e−i(E1−E2+ω)t) +A3(k,p1,p2,x)(ei(E1−E2−ω)t+e−i(E1−E2−ω)t) +A4(k,p1,p2,x)(ei(E1+E2−ω)t+ei(E1+E2−ω)t) +A5(k,p1,p2,x)(ei(E1+E2−ω)t+e−i(E1+E2−ω)t) +A6(k,p1,p2,x)(ei(E1−E2−ω)t+e−i(E1−E2−ω)t) +A7(k,p1,p2,x)(ei(E1−E2+ω)t+e−i(E1−E2+ω)t) +A8(k,p1,p2,x)(ei(E1+E2+ω)t+e−i(E1+E2+ω)t)] (29) The ﬁrst item in the formula above can be written as /integraldisplay /integraldisplay /integraldisplay d3kd3p1d3p2[A1(k,p1,p2,x)(ei(E1+E2+ω)t+e−i(E1+E2+ω)t)] =−i4/summationdisplay σ=12em (2π)9/2/integraldisplay /integraldisplay /integraldisplayd3kd3p1d3p2√ 2ω√E1E2(rµǫσ µ(k))αβ ×¯urα(p1)νrβ(p2)α+ σ(k)b+ r(p1)d+ r(p2)e−i(p1+p2+k)·xcos(E1+E2+ω)t (30) We only consider the integral relative to k. It can be written as /integraldisplay+∞ −∞dk1dk2dk3/radicalbig 2(k2 1+k2 2+k3 3)1/2(rµεσµ(k))αβα+σ(k)× 4e−i(k1x1+k2x2+k3x3)cos(E1+E2+/radicalBig k2 1+k2 2+k2 3)t (31) In the formula, εσ µ(k) contains the direction vectors n1,n2andk/kSuppose in the ﬁxed coordinate reference system, k=k1i+k2j+k3l, let the direction angles of n1areθ1andϕ1, the direction angles ofn2areθ2andϕ2, we have n1=sinθ1cosϕ 1i+sinθ1sinϕ 1j+cosθ1l (32) n2=sinθ2cosϕ 2i+sinθ2sinϕ 2j+cosθ2l (33) Because n1·k= 0, we have k1sinθ1cosϕ 1+k2sinθ1sinϕ 1+k3cosθ1= 0 (34) For simpliﬁcation, taking θ1= 450, we getsinθ1=cosθ1=√ 2/2. From the formula above, we have sinϕ 1=−k2k3±k1/radicalbig k2−2k2 3 k2−k2 3 cosϕ 1=/radicalBigg 1−(−k2k3±k1/radicalbig k2−2k2 3)2 (k2−k2 3)2(35) Considering n2·k= 0 and n1·n2= 0 , we can also decide θ2andϕ2as done above. In this way, εσ µ(k) can be determined. Putting the relation into Eq.(31), it ca n be seen that the integral function are never equal to zero no the odd function of ki, so Eq.31 is not equal to zero. In the same way, the integrals about p1andp2are also not equal to zero. So we have /integraldisplay /integraldisplay /integraldisplay d3kd3p1d3p2A1(k,p1,p2,x)(ei(E1+E2+ω)t+e−i(E1+E2+ω)t)/negationslash= 0 (36) It can also be proved that the all items in Eq.(29) are not equa l to zero, because production and annihilation operators in each items of Eq.(29) are diﬀeren t, so that the sum of all items in Eq.29is not equal to zero. So we have Hm(x,t)−H∗ m(x,t)/negationslash= 0 i.e.,Hm(x,t)/negationslash=H∗ m(x,t). Therefore, we have proved that the current theory of electromagnetic interact ion can not keep unchanged under the time reversal. In the same way, it can also be proved that in the current unite d electro-weak theory and the quantum chromodynamics theory, we have also Hm(t)/negationslash=H∗ m(−t), so the current theory of particle physics can not keep unchanged under time reversal. But it is unnecessary for us to calculate them any more here. Besides, the current scheme of time reversal has other probl ems shown below. 1.It is meaningless. According to the current scheme, from Eq.5,7and (8), by cons idering the relation(2)σ+ 2(ir4rµ)∗σ2= −ir4rµ, we have Ti¯ψrµψT−1=Tψ†ir4rµψT−1=Tψ†T−T(ir4rµ)T−1 =Tψ†T−1(ir4rµ)∗TψT−1 =ψ†σ+ 2(ir4rµ)∗σ2ψ=−i¯ψrµψ (37) Using Eq.8 again, we have THmT−1=Hm. It seems the form of the Hamiltonion is unchanged under time reversal. However, from Eq.(7) and (8) , we know th at this proof at most gives out the deﬁnition of time reversal with THm(t)T−1=Hm(−t). It is not the proof of the invariability of time reversal. The deﬁnition of the invariability of time revers al isTHm(t)T−1=H∗ m(−t) =Hm(t) for the motion equation with the form shown in Eq.9. So the current pr oof of time reversal invariability in the theories of particle physics is meaningful. 2 It is not identical. 5Let’s consider the transformation of motion equation of fre e spinor ﬁeld. We can write the equation as (rµ∂µ+m)ψ(x,t) = (r· ∇ −ir4∂t+m)ψ(x,t) = 0 (38) Acting time reversal operator on it, and considering the fac t that the operator ∇and the matrix r are commutative, ∂tandr4are also commutative, as well as T∂tT−1=−∂t, we have ∇ ·Trψ(x,t)−T∂tT−1Tir4ψ(x,t) +mTψ(x,t) = 0 (39) By considering Eq.(5) and the calculation method shown in Eq .(37), we get [r∗· ∇+ (ir4)∗∂t+m]Tψ(x,t) = 0 (40) If the transformation Eq (7) and the relation r∗ 4=r4are considered, let Tψ(x,t) =ir1r3ψ(x,t), the formula above becomes (r∗· ∇ −ir4∂t+m)ir1r3ψ(x,−t) = 0 (41) In the formula, let t=−t, as well as considering r∗= (−r1,r2,−r3), we get (r∗· ∇+ir4∂t+m)ir1r3ψ(x,t) =ir1r3(r· ∇+ir4∂t+m)ψ(x,t) (42) It can be written as (r· ∇+ir4∂t+m)ψ(x,t) = 0 (43) By comparing it with Eq.(38) , it is obvious that they are diﬀe rent. So the transformation7can not be concerned with 12 and (13). In order to let the motion equation of free spinor ﬁeld is unch anged under time reversal, the current method is to take the transformation of Eq.(38) as(3) (TrT−1· ∇+iTr4T−1∂t+m)Tψ(x,t) = 0 (44) We have taken Tiψ(x,t) =iTψ(x,t) in the process. In this scheme, the operator of time reversa l is supposed to be an antiunitary operator, so the deﬁnition o f time reversal is diﬀerent from that expressed in Eq.5. Let ψ′(x,t) represent the wave function of time reversal with ψ′(x,t) =Tψ(x,t), and let TrT−1=r Tr4T−1=−r4 (45) the motion equation 38 can keep unchanged under the time reve rsal with (r· ∇ −ir4∂t+m)ψ′(x,t) = 0 (46) From Eq.45 , we can get the concrete form of time reversal oper ator T=ir1r2r3 (47) The relation between these two wave functions, before and af ter time reversals are carried out, is ψ′(x,t) =Tψ(x,t) =ir1r2r3ψ(x,t) (48) This transformation is unique to keep the motion equation un changed under the time reversal for free spinor ﬁeld. However, this transformation is obviously diﬀ erent from Eq. 7. As for non-free ﬁeld, we should consider the existence of the ﬁeld,Aµbut the result is the same. When interaction exists, the time reversal of Eq. 12 is (TrT−1· ∇+iTr4T−1∂t−ieTrT−1·TAT−1−ieTr4T−1TA4T−1+m)Tψ(x,t) = 0 (49) 6Considering the relation TAµT−1=−Aµ, we have (r· ∇ −ir4∂t+ier·A−ier4A4+m)ψ′(x,t) = 0 (50) By comparing it with Eq.42, it can been known that even though the transformation 45is used, we can not yet keep the motion equation unchanged under time rev ersal. As for the Wigner transformation considered as another sche me of time reversal, besides too big diﬀerences between these two schemes so that we can not decid e what kind of scheme represents the real situation, the similar problems shown above are als o exist. The discussion of time reversal problem is very chaotic in the current particle physics. It l acks of identity or united standard. We have to use the diﬀerent transforms in the diﬀerent situation. In fact, according to the Noether theorem each kind of symmetry transformation of a system correspond ing to a conservative quantity. But in the quantum theory of ﬁeld, no conservative quantum can be fo und corresponding to time reversal transformation. This fact actually has shown that the symme try of time reversal does not exist in the theory of particle physics. 1. According to this scheme, the operator of time reversal is anti-unitary. But it is considered in general that only unitary operator can be observable, the anti-unitary operator is meaningless in physics. 2.According to this scheme, after the transformation of tim e reversal is carried out, the producing operators of particles are still the producing operators, t he annihilating operators of particles are still the annihilating operators(1,2)withTα†(p,s)T−1=α†(−p,s)Tb(p,s)T−1=b(−p,s). This result does not represent the real situation. In the processes of ti me reversal, the production and annihilation operators of particles should be exchanged each other. In sum, the invariability of time reversal does not exist act ually in the current theories of particle physics, so the current theory of time reversal should be ree stablished. References (1) Ying Pengcheng, Outline of Quantum Theory of Field, Shan ghai. Scientiﬁc and echnology Publishing House, 1251(1986). (2) Li Chengdao, Particle physics and Field, Shangdong Scie ntiﬁc and Technology Publishing House, 164(1996). (3) Introduce to Quantum Theory of Field, Luo Changxong, Sha ngxi Normal University Publishing House, 145(1986). 7
Invited□Paper.□Published□in:□ The□Einstein□Quarterly:□Journal□of□Biology□and□Medicine ,□15□(1998). Biology□and□Thermodynamics: Seemingly-Opposite□Phenomena□in□Search□of□a□Unified□Paradig m by Shahar□Dolev* and Avshalom□C.□Elitzur□† *□The□Kohn□Institute□for□the□History□and□Philosophy□of□Sc iences,□Tel-Aviv□University, 69978□Tel-Aviv,□Israel E-mail:□shahard@ibm.net †□School□of□Physics□and□Astronomy,□The□Raymond□and□Beverly□ Sackler□Faculty□of□Exact Sciences,□Tel-Aviv□University,□69978□Tel-Aviv,□and□The□Seagram□C enter□for□Soil□and Water□Sciences,□The□Hebrew□University,□76100□Rehovot,□Isra el. E-mail:□cfeli@weizmann.weizmann.ac.il It□is□probably□not□a□coincidence□that□two□of□the□pioneer s□of□thermodynamics,□Helmholtz and□Mayer,□were□physicians.□Thermodynamics□studies□the□tr ansformations□of□energy,□and such□transformations□ceaselessly□take□place□in□all□liv ing□systems□(probably□with□important differences□between□the□states□of□health□and□disease ).□Moreover,□thermodynamics□studies the□elusive□notions□of□order□and□disorder,□which□are□also, □respectively,□the□very□hallmarks of□ life□ and□ death.□ These□ similarities□ suggest□ that□ therm odynamics□ might□ provide□ a unifying□ paradigm□ for□ many□ life□ sciences,□ explaining□ the□ mul titude□ of□ life’s manifestations□on□the□basis□of□a□few□basic□physical□princ iples. In□this□article□we□introduce□some□basic□thermodynamic□c oncepts□and□point□out□their usefulness□for□the□biologist□and□the□physician.□We□hope□to□ show□that□thermodynamics enables□looking□at□the□riddles□of□life□from□a□new□perspec tive□and□asking□some□new□fruitful questions. 1. □The□Second□Law□of□Thermodynamics□and□its□Bearing□on□Biology Thermodynamics□relies□on□three□basic□laws□to□study□th e□transports□of□energy□in□physical systems□and□how□they□can□be□used□to□produce□work.□The□Fi rst□Law□of□Thermodynamics states□that□energy□must□be□conserved.□The□Third□Law□sta tes□that□it□is□impossible□to□reduce a□system’s□temperature□to□the□absolute□zero.□But□the□mo st□interesting□of□the□three□is□the Second□Law.□It□states□that□within□a□closed□system□(tha t□is,□a□system□that□no□energy□can enter□or□leave)□entropy□can□only□increase,□or□(when□it □is□maximal)□remain□constant. 2What□is□entropy?□The□dictionary□tells□us:□“A□measure□ of□the□unavailable□energy□in□a closed□system”.□There□are□several□other,□partly□overla pping□definitions□of□this□important term.□We□will□review□them□with□the□aid□of□the□followi ng□simple□example: Imagine□a□sealed□box□divided□in□its□middle□by□a□partition. □Let□the□right□half□of□the□box□be in□vacuum.□If□we□puncture□a□hole□in□the□partition,□the□gas □will□filtrate□to□the□empty□half until□the□entire□box□is□equally□full□with□gas.□The□filt ration□process□increased□the□entropy□of the□system□in□the□following□senses: 1. □Equilibrium. □The□initial□state□has□low□entropy□since□it□was□far from□ equilibrium□ (dense□ gas□ on□ one□ side,□ vacuum□ on□ the other).□The□final□state□is□of□high□entropy□since□it□has□ an□even distribution□of□heat,□pressure,□etc. 2. □Bound□energy. □Energy□that□can□be□used□to□do□work□is□called “free□energy”□while□energy□that□cannot□be□so□used□is□“b ound”. In□our□example,□free□energy□has□degraded□into□bound□energy. Suppose□that□the□partition□had□been□a□piston.□At□the□init ial state,□ the□ pressure□ of□ the□ gas□ on□ the□ partition□ could□ do mechanical□work.□It□was,□therefore,□free□energy.□At□th e□final state,□ in□ contrast,□ all□ the□ energy□ has□ turned□ into□ chao tic, microscopic□ motions□of□ the□ molecules□ that□ have□ spread□ a ll over□the□box.□This□energy□can□no□longer□be□used□for□work 1□– another□manifestation□of□entropy□increase. 3. □Disorder. □Apparently,□in□our□example,□the□final□state,□where the□gas□is□equally□dispersed□in□the□box,□is□more□ordered□th an the□initial,□unequal□distribution□of□the□gas.□But□actually □it’s□the□other□way□around.□The “household□definition”□of□order□turns□out□to□be□consiste nt□here□with□the□physical□one: house□where□the□clothes,□silverware,□books,□etc,□are□e qually□divided□over□the□living room,□kitchen,□etc.,□is□a□house□that□leaves□much□to□be□de sired.□Order,□therefore,□is□a state□far□from□equilibrium. 4. □Irreversibility. □ The□ spontaneous□ changes□ that□ the□ gas□ in□ the□ box□ underwe nt□ are irreversible.□The□likelihood□that,□by□the□same□accidental □motions□of□the□molecules,□all the□gas□will□return□by□itself□to□the□left□half,□is□extr emely□low.□Each□gas□molecule□has□a probability□of□0.5□to□be□found□in□the□right□half.□Since□we □are□dealing□with□about□10 25 particles□(see□section□3□below),□the□combined□probability□i s□1/2 10 25 □(that’s□10 -10 24 )!□The degree□of□the□unlikelihood□for□a□system□to□return□to□its□ initial□state□is□a□measure□of□its irreversibility,□hence□of□its□entropy. □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ 1□An□exception□to□this□rule□is□the□case□where□we□use□a□cold er□environment□outside□the□box.□In□this case□the□absence□of□equilibrium□between□the□hot□and□cold □reservoirs□enables□us□to□do□work,□just as□at□the□box’s□initial□stage.□But□then,□of□course,□we□ are□not□dealing□with□a□closed□system,□which is□the□case□for□which□the□Second□Law□holds. Figure□ 1□-□Initial□setup Figure□2□-□After puncturing□barrier 35. □Number□of□microstates. □Another□definition□of□entropy□is□based□on□the□differen ce between□the□system’s□macroscopic□and□microscopic□stat es.□An□ordered□system□allows only□a□small□number□of□arrangements□of□its□basic□const ituents.□In□contrast,□there□is□a much□larger□number□of□arrangements□that□make□an□unordered□ system.□In□our□case, there□are□much□more□possible□arrangements□of□the□gas□mol ecules□when□the□gas□is evenly□ spread□ over□ the□ two□ halves,□ while□ the□ ordered□ state □ allows□ much□ fewer arrangements 2.□This□insight□is□the□basis□of□Boltzmann’s□definition□of □entropy,□and□here too,□the□household□definition□accords□well□with□the□physica l□one:□There□are□only□a□few arrangements□that□make□a□ house□“ordered”□and,□unfortunately ,□ numerous□ways□to make□it□disordered! In□summary,□the□Second□Law□states□that□entropy□contin uously□increases.□True,□entropy can□sometimes□be□decreased□within□a□system,□but□only□at□t he□cost□of□energy□investment that□will□increase□entropy□outside□the□system.□And□in□t his□case,□the□system□would□not□be□a closed□one.□It□would□be□ the□system□plus□the□environment □that□constitutes□a□closed□system, and□in□this□closed□system,□again,□the□overall□entropy□ha s□ increased.□To□return□to□the household,□you□can□make□order□in□your□house,□but□this□will□ increase□the□entropy□of□your neighborhood.□And□if□you□make□order□in□the□neighborhood,□y ou□increase□the□entropy□of your□city.□“You□can’t□fight□City□Hall”□is□a□common□wis dom,□and□the□Second□Law□seems to□be□the□ultimate□City□Hall! Having□reviewed□these□definitions□of□entropy,□it□immediate ly□strikes□us□that□they□also hint□at□some□profound□definition□of□the□unique□physical□state □we□call□“life”,□although□in□a very□ peculiar□ way.□ Notice,□ first,□ that□ the□ most□ illu minating□ demonstration□ of thermodynamics’□pertinence□to□the□life□sciences□comes □from□observing□the□processes□to which□the□organism□is□subject□upon□ dying .□All□the□manifestations□of□decay□that□reduce□the living□ tissue□ back□ into□ inorganic□ ashes□ share□ a□ fundamenta l□ physical□ characteristic, namely,□complying□with□the□Second□Law:□The□decomposing□orga nism□goes□back□to□the state□of□equilibrium□(thermal,□chemical,□etc.)□with□i ts□environment.□Being□alive,□then, means□ being□ far□ from□ equilibrium□ with□ the□ environment,□ the reby□ manifesting□ the autonomy □which□is□the□very□hallmark□of□life. Another□aspect□that□makes□entropy□the□opposite□of□the□l iving□state□has□to□do□with□the dynamic□aspect□of□the□Second□Law,□explained□in□the□fift h□definition□above.□Take,□for example,□a□rolling□ball□on□a□rigid,□flat□surface.□Initia lly,□the□ball□harbors□kinetic□energy, but□eventually□friction□will□bring□it□to□a□halt.□Where□ did□the□energy□go?□Since□energy□can never□vanish□(see□the□First□Law)□it□can□only□change□fo rm.□Tracing□the□“lost”□energy,□we will□find□that□the□ball□has□transferred□its□momentum□to□ the□molecules□of□the□underlying surface□and□the□ambient□air.□Doing□so,□it□has□lost□kineti c□energy□while□increasing□the surface’s□molecules’□thermal□motion 3.□All□in□all,□we□can□say□that□ordered□energy□–□the □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ 2□See□section□3. 3□The□rule□is□that□temperature□is□actually□a□measure□of□ the□mean□kinetic□energy□of□the□material’s molecules.□That□is,□the□higher□the□temperature,□the□faster□ the□molecules□go□(Sears,□1963). 4macroscopic□ rigid-body□motion□of□the□ball□–□was□transformed□into□dis ordered□energy□– the□ microscopic, □thermal□motions□of□multitude□of□surrounding□molecules. Here□again,□we□can□see□the□conversion□of□free□energy□ to□a□bound□one.□The□ball’s□original motion□could□have□been□harnessed□to□produce□work□(e.g.□by□tur ning□a□dynamo□to□generate electric□current).□However,□the□energy□that□was□dispe rsed□to□the□background□environment cannot□be□used□any□more.□The□Second□Law,□that□gives□our□wor ld□its□time-arrow,□is□the reason□why□we□ never □observe□the□opposite□process:□We□won’t□believe□a□movi e□that□shows a□motionless□ball□beginning□to□roll□spontaneously□and□then□ac celerating□while□the□table cools□down.□We’ll□rather□claim□that□the□movie□is□running□bac kward.□But□why□is□such□a process□impossible?□After□all,□it□does□not□violate□the□F irst□Law,□as□the□energy□came□from the□microscopic□motions□of□the□surfaces□and□air□molecu les.□Indeed,□such□a□case□is□not absolutely□ impossible,□ but□ rather□ very,□ very□ unlikely:□ It□ would□ take□ more□ than□ the universe’s□lifetime□for□such□an□accident□to□occur□somewhe re.□Practically,□no□one□can□trace these□fractions□of□energy□lost□by□the□rolling□ball□and□ re-collect□them□back□into□a□usable form.□Even□if□such□a□method□existed,□it□would□end□up□consumi ng□more□energy□than□it□has “freed.” In□ intriguing□contrast,□the□ living□ organism□ seems□ to□ exhibit□ exactly□ this□ impossible reversal.□ Magnasco□ (1993)□ has□ shown□ that□ under□ sufficient □ conditions,□ a□ biological microscopic□“engine”□is□capable□of□drawing□net□motion□from □thermal□energy□alone.□But we□would□ like□to□point□out□that□the□ living□ organisms□ can□ do□ m uch□ more.□ Take,□ for example,□the□muscles□operation□during□bending□of□the□arm:□ multitudes□of□microscopic muscle□cells□are□cooperating□by□secreting,□building□and□cro ss-linking□actin□and□myosin filaments□ (Berne□ et□ al .,□ 1993).□ Huge□ amounts□ of□ molecules□ move□ in□ a□ seemingly disordered□ manner,□ but□ somehow□ all□ these□ fractions□ of□ e nergy□ pile□ up□ to□ cause□ a macroscopic, □ordered,□motion□of□the□arm.□The□percise□microscopic□co ntrol□enables□the muscles□to□reach□maximum□efficiency□of□45%□(Berne□ et□al .,□1993),□as□opposed□to□25% efficiency□in□man-made□engines□(Sharpe,□1987). 4 Even□when□no□movement□is□apparent,□the□living□body□fights □entropy□all□the□time□by performing□enormous□microscopic□work:□ion□pumps□keep□the□righ t□concentration□of□ions across□the□cell□membrane,□various□enzymes□check□cell□str ucture□and□the□DNA□strands□for errors,□membrane□proteins□convey□nutrients□in□and□waste□o ut,□complex□system□cooperate to□ keep□ homeostasis,□ etc.□ It□ is□ these□ intracellular□ pro cesses□ that□ later□ converge,□ with amazing□ precision,□ into□ macroscopic□ movements.□ We□ can□ the refore□ formulate□ a thermodynamic□property□that□is□unique□to□living□systems: In□ inanimate□ systems□ the□ microscopic□ motions□ are□ chaotic,□ resulting□ fr om□ the disintegration□of□the□ordered□motion□of□microscopic□bodies.□The□living□sys tem,□in □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ 4□One□might□argue□that□the□cooperation□of□many□microscopic □machines□should□be□less□efficient□in comparison□to□one□macroscopic□machine,□as□the□former□case□i nvolves□greater□friction□between the□machines.□Life,□however,□countered□this□problem□by□the□hi ghly□ordered□arrangement□of□the small□machines□so□as□to□avoid□friction.□The□muscle’s□mo lecules,□for□examle,□are□arranged□along highly□ordered□polimers. 5contrast,□maintains□a□very□coordinated□motion□of□its□microscopic□units,□e nabling□them to□converge□at□the□right□time□into□macroscopic,□ordered□motion□when□needed. No□less□striking□from□the□thermodynamic□viewpoint□is□th e□course□of□development□of□a single□creature,□namely,□its□ ontogeny .□An□oak□tree,□for□example,□begins□its□life□as□a□zygot e smaller□than□millimeter.□Within□a□few□years□it□consum es□basic□chemical□elements□from the□surrounding□air□and□soil,□elements□from□highly□ disordered □sources,□only□to□organize them□into□the□form□of□a□mature,□ordered,□highly□complex□t ree.□Life□has□the□unique□ability to□ act□ against□ the□ normal□ course□ of□ events.□ Instead□ of□ s cattering□ ordered□ motion□ of macroscopic□ objects□ into□ a□ multitude□ of□ tiny,□ disordered□ mo vements□ of□ microscopic molecules,□living□systems□control□the□operation□of□singl e□molecules,□guiding□minuscule amounts□of□energy□and□matter□into□an□enormously□ordered, □macroscopic□system. Note□that□nothing□of□this□violates□the□Second□Law□of□therm odynamics.□Living□creatures are□not□closed□systems,□to□which□the□Second□Law□applies. □Since□there□is□no□free□lunch□in nature,□ living□ creatures□ must□ consume□ energy□ in□ order□ to□ cr eate□ and□ maintain□ their internal□order. This□is□the□answer□given□by□all□textbooks□to□the□apparen t□contradiction□between□the Second□ Law□ and□ life’s□ numerous□ manifestations.□ However, □ while□ this□ explanation□ is correct,□it□is□highly□insufficient.□Nearly□everything□aroun d□us□is□an□open□system,□and□yet chairs□ and□ tables□ do□ not□ become□ alive.□ What□ is□ needed□ is□ a □ study□ of□ the□ particular processes□ by□ which□ very□ special□ and□ unique□ systems,□ namely, □ the□ living□ organisms, exploit□ their□ interactions□ with□ the□ environment□ in□ order □ to□ become□ more□ complex, ingenious□and□beautiful.□In□what□follows□we□propose□some□gui delines□for□such□a□model. 2. □Microstate□vs.□Macrostate In□the□previous□chapter□we□pointed□out□two□scales□by□which □one□can□look□at□a□system.□Let us□examine□these□scales□in□more□detail. 1. □The□microscopic□scale,□where□one□can□examine□the□beha vior□of□individual□molecules. 2. □The□macroscopic□scale,□where□one□sees□the□overall□st ate□of□the□system,□regardless□of its□individual□molecules. 6Thermodynamics□taught□us□that it□ is□not□enough□ to□ look□ at□the macroscopic□ level□ alone.□ One must□ take□ into□ account□ some properties□ of□ the□ microscopic level□too.□Consider,□for□example, the□following□experiment:□There are□two□boxes,□each□with□a□string hanging□ out□ (Fig.□ 3).□ One□ box harbors□a□heavy□rock□connected to□ the□ string,□ while□ the□ other contains□a□spring□connected□to□its string.□When□one□pulls□a□ box’s string,□ he/she□ puts□ energy□ into the□ system.□ Although□ the□ two boxes□ look□ identical□ from□ the outside,□there□is□a□profound□difference□between□their□re actions□to□the□pulling.□Pulling□the spring□of□the□second□box□converts□the□energy□into□a□usa ble,□mechanical□energy.□This□is□a reversible□process□and□the□invested□energy□can□be□retri eved□by□letting□the□spring□recoil. Pulling□the□rock□within□the□other□box□will□convert□the□e nergy□to□noise□and□heat,□forms□that are□ hardly□ usable.□ Only□ peering□ down□ to□ the□ molecular□ scale □ –□ i.e.,□ studying□ the differences□between□the□molecular□structures□of□the□ rock□and□the□spring□–□will□reveal□the difference□between□the□two□cases.□When□thermodynamics □was□constructed,□it□was□realized that□only□one□parameter□is□needed□in□order□to□describe□the□ “usability”□of□the□energy.□That parameter□is□the□entropy. It□was□understood□that□one□must□consider□the□difference□be tween□what□is□visible□to□the naked□ eye□ on□ one□ hand,□ and□ the□ world□ of□ atoms□ and□ molecul es□ on□ the□ other.□ The arrangement□of□a□physical□system□at□the□macroscopic□sca le□was□named□ macrostate .□A system’s□temperature□or□pressure□are□such□macrostates.□ Now□each□such□macrostate□can□be described□by□many□different□arrangements□of□the□system’s□ atoms□and□molecules.□These arrangements□in□the□microscopic□scale□were□named□ microstates .□In□the□previous□section we□have□seen□that□high□entropy□is□a□macrostate□that□i s□compatible□with□many□microstates, in□contrast□to□the□ordered□state. The□biological□significance□of□these□formulations□bec omes□conspicuous□if□we□consider again□the□physical□uniqueness□of□the□ living□state.□If□we□change □the□ microstate□of□ an inanimate□object,□say,□a□rock,□by□exchanging□between□the□ positions□and□momenta□of□some of□its□molecules,□or□even□by□replacing□them□with□others, □the□rock□will□remain□a□rock;□no difference□will□be□noticed.□Think,□however,□of□an□elepha nt□or□a□whale:□these□are□huge systems,□but□altering□ their □microstates□even□slightly,□by□adding□or□subtracting□a□f ew□grams of□some□hormone□or□neurotransmitter,□could□have□drastic□r esults□–□it□may□even□kill□the poor□animal!□Similarly,□a□single□nucleotide□in□the□DNA□c an□have□fatal□consequences□in most□cases□–□or□beneficial□consequences□in□a□few□others .□Such□small□may□even□change□the fate□of□the□entire□biosphere.□All□living□creatures,□the refore,□are□unique□in□that□they□keep their□inner□autonomy□by□maintaining□ Homeostasis .□In□thermodynamic□terms,□organisms A□stone□pulled□by□a□string,□the□energy□is□lost□to□heat A□spring□pulled□by□a□string□can□restore□the□energy Figure□3□–□Reversible□vs.□Irreversible□processes 7preserve□their□microstate.□By□using□feedback□loops,□they□ keep□their□internal□environment within□ those□ narrow□ required□ levels.□ We□ can□ therefore□ add□ another□ thermodynamic characteristic□that□is□unique□to□the□living□state: The□living□organism□constantly□resides□in□a□macrostate□that□is□compatible□ with□a□very narrow□range□of□microstates,□maintaining□this□improbable□state□as□long□as□it□ is□alive. 3. □The□Phase□Space The□thermodynamic□explanation□to□entropy□increase□is□ a□statistical□one.□To□follow□that explanation,□we□have□to□acquaint□ourselves□with□the□not ion□of□ phase□space .□This□is□a□huge mathematical□space,□where□each□point□can□be□assigned□to□a □certain□microstate□of□the□entire system□under□examination.□Actually□the□phase□space□has□ma ny□dimensions 5,□but□as□a model,□a□two-dimensional□ space□ is□sufficient.□The□ multi- dimensional□ structure□of□ the phase□space□is□such□that□when□we□map□the□different□micr ostates□of□our□system□into□it,□all the□states□corresponding□to□a□certain□macrostate□are□ adjacent.□Thus□one□can□divide□the phase□space□into□distinct□regions□corresponding□to□differen t□macrostates. Each□point□in□the□phase□space□describes□ exactly □the□positions□and□velocities□of□ all □the particles□constituting□our□system.□That□means□we□can□app ly□the□laws□of□physics□to□predict how□these□properties□would□change□once□the□system□is□at□ such□a□“point.”□The□consequent microstates□that□would□evolve□from□the□initial□one□wou ld□be□represented□by□new□points□in the□phase□space,□arranged□along□a□curve.□Therefore,□it□is □said□that□the□system□“wanders” through□the□phase□space□as□time□goes□by. As□illustrated□earlier□by□the□household□metaphor,□there□ are□very□few□ordered□states, hence□they□occupy□a□very□small□region□in□phase□space. □The□major□part□of□this□space□(by several□orders□of□magnitude)□represents□unordered□states,□ i.e. ,□states□of□high□entropy.□This principle□can□be□demonstrated□by□the□partitioned□box□ment ioned□in□Section□1.□Following the□puncture□of□the□partition,□each□molecule□of□gas□can□ be□found□anywhere□within□the container.□That□means□that□for□each□molecule,□the□vo lume□that□the□system□now□takes□in□the phase□state□is□twice□as□big□(since□the□molecules□can□be □found□on□a□twice□as□large□volume in□the□ x□direction).□The□phase□space□has□a□distinct□set□of□di mensions□for□ each □molecule, hence□the□total□volume□that□our□system□now□takes□is□tw ice□as□big□ for□each□additional particle .□Multiplying□the□contributions□of□all□the□gas□molecules □we□get□a□factor□of□2 n, where□ n□is□the□number□of□gas□molecules. For□a□1-liter□chamber,□at□1□atmosphere□and□room□temperatur e,□we□can□calculate□ n□using the□classical□equation□for□ideal□gases: P⋅V□=□n ⋅R⋅T Where:□ P=1□Atm.,□V=1□Liter,□T=300 °K,□R=1.362 ⋅10 -28□ Liter ⋅Atm/gm ⋅deg We□get:□ n□ ≅□2.5 ⋅10 25 □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ 5□There□are□six□dimensions□for□ each □particle:□three□position□dimensions□and□three□of□velocity . 8That□ means□ that□ the□ volume□ our□ system□ now□ takes□ up□ in□ th e□ phase□ space□ is approximately□2 10 25 □times□bigger□(that’s□more□than□10 10 24 )!□Since□all□microstates□have equal□probability□to□occur,□the□unordered□state□will□have□such□ a□high□probability□that□it□is only□natural□to□assume□that□the□system□will□never□return □to□the□original,□ordered,□setup without□an□external□aid.□The□classical□thermodynamic□ argument□states□that□if□we□leave□the system□alone□for□a□long□enough□period,□it□might□return□on e□day□to□the□original□state.□But you□have□to□be□ really□patient □to□see□that,□since□the□probability□for□such□an□event□t o□occur□is 1/10 10 24 ! According□to□the□formalism□of□thermodynamics,□entropy □is□proportional□to□the□logarithm of□the□phase□space□volume,□hence□the□entropy□in□the□abo ve□case□has□increased□by□a□factor of□10 24 .□Now□we□can□reformulate□the□Second□Law□in□terms□of□ the□phase□space:□Even□if□a system□begins□at□a□very□small□region□of□the□phase□space □that□represent□an□ordered□state, this□region□is□surrounded□by□huge□areas□of□unordered□states.□ Left□for□itself,□the□system□will most□likely□wander□to□these□latter□regions. Applying□this□relation□between□ micro-□and□ macrostates□to□t he□life□science,□ one□ can estimate□the□amount□of□order□manifested□by□living□systems. □A□protozoan□(single-celled organism)□ would□ be□ highly□ unordered□ had□ its□ chemical□ compositi on□ been□ uniformly mixed.□It□is□the□unequal□distribution□of□its□enzymes,□protein s,□etc.□between□the□protozoan’s highly□differentiated□parts□that□makes□it□so□ordered□and□capa ble□of□performing□its□unique biological□tasks.□A□higher□level□of□organization□is□manif ested□by□the□metazoan□(multi- cellular□organism),□that□have□many□types□of□differentia ted□cells□and□tissues,□and□yet□a higher□level□is□manifested□by□the□ecosystem,□where□nume rous□different□species□maintain□a highly□complex□web□of□dependencies. So,□looking□around□us,□we□can□see□that□our□planet□has□moved□f rom□an□unordered□state□of an□even□ mixture□of□chemicals,□which□prevailed□ four□ billio n□ years□ ago,□ into□the□ very ordered□state□that□characterizes□the□biosphere□today.□S tatistically,□it□seems,□the□odds□for such□a□transition□are□nearly□zero.□Yet,□the□very□fac t□that□this□statement□is□made□by□living creatures□means□that,□long□ago,□the□next-to-impossible□ has □happened.□Let□us□see□how□it actually□took□place. 4. □Life□as□an□Information-Gaining□Process We□submit□that□life’s□secret□in□its□battle□against□“all □odds”□lies□in□its□ability□to□process information.□The□relation□between□entropy□and□informati on,□long□known□to□physicists, offers□a□very□valuable□insight□for□biologists.□To□grasp□ this□profound□relation,□let□us□turn□to the□famous□paradox□associated□with□“Maxwell’s□Demon”□( Leff□&□Rex,□1990). We□shall□present□the□paradox□by□considering□a□setup□similar□ to□that□considered□in□section 1□above,□but□Maxwell□added□a□little□twist□to□it.□Suppose□th at,□after□the□gas□has□spread□to the□entire□box,□we□install□a□little□door□in□the□partit ion□between□the□box’s□two□halves,□with□a tiny□demon□guarding□ it□(Figure□4).□This□demon□ is□very□ smart.□ W henever□ she□ sees□ a molecule□of□gas□reaching□from□the□right□to□the□left□half ,□she□opens□the□door□and□lets□the 9molecule□pass□through.□But□when□a□molecule□tries□to□pass □from□left□to□right,□she□closes□the door.□The□door□is□feather□light□and□perfectly□oiled,□requiri ng□very□small□amount□of□energy to□open□and□close.□As□our□demon□continues□with□her□work,□ she□will□eventually□bring□the system□back□to□the□original,□low-entropy□state□(all□the □gas□concentrated□in□the□left□half). This□ would□ be□ achieved□ with□ a□ negligible□ energy□ investment,□ hence□ with□ negligible entropy□production□outside□the□box.□That□is,□the□demon□ma naged□to□decrease□the□entropy of□our□system□by□a□factor□of□10 24□ without□paying□the□penalty□to□the□universe’s□entropy. “City□Hall”□seems□to□have□been□defeated! The□paradox’s□solution□is□based□on□the□concept□of□informa tion: In□order□to□let□only□the□appropriate□molecules□pass□and□to □stop the□others,□our□demon□needs□information□about□them.□It□tu rns□out that□the□amount□of□energy□needed□to□identify□the□approachi ng molecule□is□such□that□it□will□soon□create□much□more□ entropy□than the□order□gained□by□this□operation. 6 This□ paradox□ highlights□ the□ reciprocal□ relations□ between information□and□entropy,□relations□well□known□from□co mputer science.□Any□generation,□maintenance□and□processing□of□inf ormation□take□a□proportionate cost□in□energy.□Conversely□–□and□this□is□a□formulation□ of□crucial□importance□–□ the□use□of information□allows□saving□energy .□If□we□have□some□information□about□a□system,□we□can increase□the□system’s□order□with□only□marginal□waste□of □energy. The□relevance□of□this□insight□for□biology□is□clear.□Th e□living□cell□must□harness□huge amounts□ of□ information□ for□the□ purpose□of□ fighting□ entropy .□ Using□ precisely□ crafted enzymes,□the□living□cell□is□able□to□achieve□high□efficiency □in□its□numerous□biochemical operations.□Each□enzyme□is□a□kind□of□a□small□Maxwell□d emon□that□uses□the□information gained□during□million□of□years□of□evolution□to□operate□eff iciently□on□it’s□substrate.□This efficiency□is□beyond□comparison□to□the□efficiency□that □we□humans□achieve□in□designing machines□and□computers.□Take,□for□example,□sugar□and□other□c arbohydrates.□Synthesizing them□from□their□common□constituents□–□water□and□carbon □dioxide□–□lies,□in□principle, within□the□reach□of□modern□technology.□However,□the□co st□of□this□production□would□be□so high□that□no□one□would□be□able□to□buy□these□products.□In□annoy ing□contrast,□every□grass leaf□accomplishes□this□task□every□minute□by□using□the□negl igible□energy□of□little□sunlight! To□take□a□more□dramatic□example,□a□tiger□exerts□enormous□ force□to□kill□its□prey.□A Cobra,□in□contrast,□kills□its□prey□by□merely□spitting□i nto□its□eye.□What□is□appalling□(or fascinating)□in□this□act□is□the□apparent□disproportion□bet ween□the□force□exerted□on□the□prey and□the□fatal□result.□The□choice□of□the□appropriate□neur otoxin,□that□matches□the□prey’s synapses□ by□ its□ uncanny□ resemblance□ to□ its□ neurotransmitters, □ and□ the□ precise □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ 6□Maxwell’s□original□example□was□slightly□different□in□ that□there□were□equal□amounts□of□gas□in□the box’s□two□halves,□with□full□equilibrium□between□them.□The□de mon□used□the□door□to□let□only□fast molecules□to□pass□to□one□side□and□slow□molecules□to□the□other ,□until□the□gas□was□divided□into□a cold□half□and□a□hot□half,□in□defiance□of□the□Second□Law.□ However,□the□essential□physical□points are□the□same□in□the□original□example□and□the□one□used□above,□ as□well□as□for□the□paradox’s resolution. Figure□4□-□Maxwell's□demon 10 “knowledge”□of□the□location□of□a□vulnerable□point□to□pene trate□the□prey’s□vascular□system –□this□is□the□information□encoded□in□the□Cobra’s□genes□that □allows□it□to□save□the□energy that□would□be□wasted□by□the□tiger.□But□then,□the□gull’s□eff ortless□gliding,□the□bee’s□honey production,□the□human’s□intelligence□–□in□fact,□every□bio logical□process□–□can□be□equally characterized□by□such□Maxwell-demonic□qualities. Let□ us□ summarize.□ Adaptation,□ by□ definition,□ requires□ info rmation□ about□ the environment□to□which□the□organism□adapts.□Natural□selection□i s□the□process□by□which environmental□information□is□incorporated□into□the□species’ □genome.□Once□evolution□is studied□as□a□process□by□which□organisms□incorporate□more□and□m ore□information□about their□environment□from□generation□to□generation,□the□liv ing□organism□appear□as□a□very unique□ Maxwell□ demon□ that□ achieves□ incredible□ feats□ by□ a□ c lever□ use□ of□ the thermodynamic□affinity□between□information□and□energy. □The□magic□formula□is□simple: Living□organisms□use□little□energy,□but□at□the□right□place□and□at□the□right □time! 5. □Complexity□and□the□Struggle□for□Efficiency So□far,□we□have□treated□the□living□state□as□the□mere□opp osite□of□the□high□entropy□state.□It would□ be□ mistaken,□ however,□ to□ simply□ equate□ “life”□ with□ “ order.”□ A□ third□ term, “complexity,”□is□needed□to□capture□the□uniqueness□of□the□l iving□structure. For□an□intuitive□distinction□between□the□three□terms, □think□of□three□objects□of□the□same size:□a□rock,□a□diamond,□and□a□potato.□The□rock’s□entropy□i s□the□highest□of□the□three□–□it□□is only□an□accidental□assembly□of□minerals.□The□diamond,□i n□contrast,□is□the□most□ordered object,□as□it□is□a□perfect□crystal□of□pure□carbon.□What □about□the□potato?□True,□it□is□much less□ordered□than□the□diamond,□yet□it□is□far□more□ complex .□While□it□lacks□the□diamond’s exact□molecular□structure□and□chemical□purity,□it□is□by□n o□means□as□randomly□assembled as□ the□ rock.□The□ potato□ possesses,□ instead,□ highly□ detailed□ relations□ and□ correlations between□its□numerous□constituents.□Its□cells□resemble□ or□complement□one□another□to□form well-defined□tissues,□and□their□dynamic□operation□reveals □even□more□striking□correlations. When□we□look□at□higher□organisms,□even□at□the□simple□lev el□of□their□external□form,□this complexity□becomes□even□more□striking.□Plants□and□animals □are□never□perfect□spheres, cubes□or□pyramids,□yet□they□manifest□clear□symmetries□ and□exact□proportions□between their□different□parts.□We□can□say□that□complexity□is□ a□form□of□order,□but□of□a□very□special kind:□It□is□a□structure□whose□parts□are□different□from□o ne□another,□yet□they□maintain□very strict□relations,□both□structural□and□dynamic,□between□th em. More□precise□mathematical□formulations□of□complexity □are□discussed□in□detail□elsewhere (Elitzur,□1998),□but□for□our□purpose□the□following□observatio n□suffices:□Complexity,□like order,□cannot□evolve□spontaneously.□On□the□contrary,□it□ tends□to□degenerate□into□entropy just□as□order□does.□Similarly,□its□generation□costs□ene rgy□as□the□generation□of□order□does. The□living□organism□is□clearly□a□very□complex□system.□W e□therefore□face□an□old□problem in□a□new□formulation:□It□is□extremely□unlikely□that□l ife□on□Earth□evolved□against□the□laws of□thermodynamics,□hence□there□must□be□some□guiding□princ iple□that□helped□the□biosphere to□advance,□against□all□odds,□from□the□vast□realms□of□diso rder□into□smaller□and□smaller regions□of□growing□complexity.□That□principle□we□are□looki ng□ for□ must□ be□ powerful 11 enough□to□create□the□magnificent,□diverse,□and□perfectly□a dapted□living□creatures□we□see around□us. Our□ suggestion□ is□ that□ physicists□ are□ already□ familiar□ wi th□ that□ principle,□ yet□ have seldom□noticed□its□relevance□to□biology.□To□comprehend□this□ principle,□let□us□think□of evolution□from□the□thermodynamic□aspect□of□energy□eff iciency: The□ability□of□living□systems□to□increase□complexity□is□not□acc idental.□Complexity□is vital□for□efficiency.□Life□was□therefore□compelled□to□increase □complexity□as□organisms fought□for□survival.□The□course□of□evolution□can□be□rephrased□as□“Survival□ of□the□most efficient.” The□reason□is□simple:□efficient□organisms□require□less□ energy,□thereby□being□able□to survive□tougher□conditions□(hunger,□drought,□etc.).□As□we□sa w□in□the□section□concerning information□ and□ efficiency,□ organisms□ had□ to□ accumulate□ i nformation□ about□ their surroundings□in□order□to□achieve□high□efficiency.□Only□this□ way□could□they□acquire□the efficiency□that□enabled□them□to□survive. This□trend□can□be□demonstrated□by□the□evolution□of□Hemoglobi n□(Lodish□ et□al .,□1995; Dickerson,□1983).□Hemoglobin□is□highly□adapted□to□its□role,□namel y,□transporting□oxygen from□the□lungs□to□the□cells.□The□hemoglobin□molecule□ is□a□tetramer□made□of□four□sub- units,□each□capable□of□carrying□one□oxygen□molecule.□An□ener gy□barrier□should□be□crossed in□order□to□attach□an□oxygen□molecule□to□each□sub-unit.□H owever,□thanks□to□hemoglobin’s unique□structure,□each□oxygen□molecule□captured□by□it□causes□a □geometric□(allosteric) modification□of□the□hemoglobin□molecule,□lowering□the □energy□barrier. The□evolution□of□the□hemoglobin□molecule□that□has□lead□ to□its□present□efficiency□can□be traced□by□studying□the□molecule□that□performs□the□same□tas k□in□more□“primitive”□species such□as□insects□or□cartilaginous□fishes.□It□was□found□that □the□hemoglobin□evolved□out□of□a molecule□ that□ is□ similar□ to□ myoglobin□ (a□ molecule□ th at□ transfers□ oxygen□ within□ the muscles).□The□myoglobin□monomer□is□less□efficient□in□ carrying□oxygen,□having□a□higher energy□barrier.□Each□sub-unit□of□the□hemoglobin□is□a□mo dified□myoglobin□molecule□that was□crafted□during□the□evolution□of□vertebrates.□In□the□ course□of□evolution,□in□order□to increase□the□efficiency□of□oxygen□transfer,□the□simple□ myoglobin□molecule□was□evolved□to the□more□complex□hemoglobin.□The□trend□was□driven□by□the□ need□for□higher□efficiency, which□was□accomplished□by□incorporating□information□about□t he□structure□and□physical qualities□of□the□oxygen□ molecule.□Complexity□ is□the□ mean s□ by□which□efficiency□was increased. We□began□this□section□with□an□intuitive□definition□of□ complexity,□but□we□should□stress again□that□more□objective□measures□have□been□proposed.□Benn ett□(1988;□Lloyd□&□Pagels, 1988)□ gave□ the□ following□ physical□ measure:□ Given□ the□ shortest□ algorithm□ for□ the construction□of□a□certain□structure,□how□much□energy□i s□needed□for□the□computation□of that□algorithm□so□as□to□carry□out□the□construction?□Inte restingly,□both□highly□ordered□and highly□disordered□structures□turn□out□to□have□low□complexit y,□while□living□organisms□turn to□have□the□highest□complexity□when□taking□ into□ account□ the□ degree□of□ computation needed□to□carry□out□the□instruction□of□the□organism’s□DNA.12 Another□approach□has□been□adopted□by□Zotin□and□his□co-worke rs□(Zotin□&□Lamprecht, 1996,□and□references□therein).□Their□work□is□base□on□the□ previously□established□relation between□an□organism’s□oxygen□consumption□and□its□bodily□ mass: where□ 2OQ□is□the□oxygen□consumption□rate□given□in□mW,□ M□is□the□organism’s□mass□in grams□and□ a□and□ k□are□coefficients.□They□argue□that□there□is□a□general□ trend□in□evolution that□leads□to□increasing□values□of□ a.□Indeed,□comparative□values□of□ a□from□a□few□main classes□of□animals□accord□with□this□claim.□In□other□words ,□oxygen□consumption□per□body mass□increases□with□evolution,□in□accordance□with□the□ paleontological□record.□The□data□is admittedly□very□partial□and□insufficient,□but□the□findings □are□exciting□enough□to□warrant further□study.□They□indicate□that□a□simple□thermodynamic□ measure□might□enable□one□to determine□the□degree□of□the□organism’s□complexity. 6. □The□Molecular□Scale It□seems□that□the□high□efficiency□of□living□systems□st ems□from□their□ability□to□control processes□at□the□molecular□scale,□an□accomplishment□ that□no□man-made□machine□has□yet achieved.□This□unique□ability□of□life□to□master□microscopic□me chanisms□is,□in□fact,□not□so much□of□a□surprise,□since□life□ began□ on□the□molecular□scale.□All□life□had□later□to□do,□then , was□to□keep□its□precious□control□at□the□molecular□level .□In□other□words,□a□disadvantage□has been□turned□into□an□enormous□advantage. Let□us□describe□this□radical□shift□in□more□detail.□By□the □simple□laws□of□probability,□life could□ not□have□ begun□ at□the□ macroscopic□ scale.□ The□ probabi lity□ for□ even□ the□ tiniest bacteria□to□be□spontaneously□assembled□out□of□an□occasio nal□ binding□of□a□ myriad□of wandering□molecules□is,□of□course,□practically□zero.□Ho wever,□the□spontaneous□assembly of□a□simple,□self-replicating□molecule□is□much□more□pr obable□considering□the□time□frame given□for□the□emergence□of□life□on□Earth.□The□fact□t hat□life□could□only□begin□at□the□very simple□microscopic□level□must□have□been□a□disadvanta ge□for□the□first□living□systems, whatever□they□were.□They□were□tiny,□simple,□and□hence□ highly□inefficient.□However,□this weakness□eventually□turned□into□an□enormous□advantage:□contr ol□at□the□microscopic□level was□ kept□ even□ when,□ by□ natural□ selection,□ macroscopic□ orga nisms□ evolved,□ granting living□organisms□the□enormous□efficiency□that□man-made□ma chines□are□not□even□close□to achieving□today.□As□ noted□above,□living□ organisms□ control□ chem ical□ reactions□ at□the single-molecule□level,□orchestrating□the□reactions□o f□multitude□of□molecules□to□converge into□macroscopic□processes. But□why□is□efficiency□greater□when□the□system□operates□ at□the□small□scale?□From□the above□thermodynamic□formulations□it□follows□that□a□pro cess□gets□more□efficient□as□it approaches□ reversibility .□Perfectly□reversible□machines,□though□impossible□in□pr actice,□are the□ most□ efficient□ ones.□ Now,□ again□ by□ the□ above□ form ulation,□ machines□ approach reversibility□the□smaller□they□are. , 2k OaM Q=13 Let□us□look□at□the□molecular□basis□of□this□principle.□ Efficiency□decreases□when□energy□is lost□to□the□environment□in□the□form□of□random□molecular□ motions□(heat).□What□is□unique about□life□is□that□the□organism□keeps□energy□loss□low□by□ controlling□the□processes□at□the molecular□scale.□When□each□molecule□is□directed□to□perf orm□its□specific□task,□only□few can□escape□their□destiny□and□lose□energy□to□the□surrounding□e nvironment.□Compare□this□to man-made□machines□–□let□us□take□the□extreme□example□of□th e□most□advanced,□sub-micron computer□chips:□They□rely□on□steering□herds□of□electrons □by□macroscopic□electromagnetic forces□in□the□approximate□direction.□Inevitably,□a□great□ deal□of□them□lose□energy□as□they bump□into□one□another,□hitting□other□molecules□in□their□ vicinity□and□diverging□from□the intended□direction.□Only□focusing□the□reactions□to□the□sin gle□molecules□or□even□single particles,□as□living□organisms□do,□can□minimize□electron□ losses□and□increase□efficiency. It□is□even□more□instructive□to□compare□the□ordinary,□w asteful□technological□process□to one□of□the□greatest□wonders□of□animate□nature,□known□as□ photosynthesis□(Lodish□ et□al ., 1995).□In□this□process□photons□are□caught□by□the□chlorophyll□mol ecules,□initiating□a□chain of□reactions□that□transfers□ single □electrons□from□one□protein□to□another.□At□the□end□of□t he process□several□molecules□of□ATP□and□a□single□molecul e□of□sugar□are□constructed.□When humans□tried□to□get□energy□from□light□by□means□of□photoel ectric□cells,□they□ended□up□with a□process□similar□to□the□micro-chip□described□earlier:□A□ multitude□of□electrons□that□were popped□from□a□semi-conductor□by□ incoming□photons□are□directed□ by □ electromagnetic force□to□the□approximate□direction.□Electron□motion□ove r□the□semi-conductor□is□terribly wasteful,□yielding□an□efficiency□of□only□several□percen ts.□In□order□to□achieve□efficiency that□equals□that□of□plants,□a□pure□crystal□should□be□used, □the□production□of□which□would cost□thousands□of□dollars□(Cheremisionoff□ et□al. ,□1978). 7. □Biotechnology□and□Nanotechnology:□Seeking□the□Efficiency□of□Living□S ystems Admiring□the□incredible□efficiency□of□living□organisms,□sci entists□are□trying□to□exploit□the latter’s□knowledge,□acquired□through□billions□of□ years□of□ evolution,□ for□ technological purposes. Nanotechnology□is□a□new□branch□of□technology□that□trie s□to□achieve□the□efficiency□of living□organisms□by□reducing□the□machinery’s□scale.□Nanotechnol ogy’s□short-term□goal□is the□production□of□micron□sized□machinery.□The□envisioned□ma chines□would□be□built□by assembling□single□atoms□and□molecules□together□to□form□t he□desired□precise□structure. They□will□be□able□to□replace□us□in□unpleasant□chores□such□ as□cleaning□our□environment, cultivating□the□ground□and□even□ medical□ tasks□ such□ as□ chec king□ out□our□ bodies□ and helping□the□immune□system□fight□microbes□and□cancer□(F eynman,□1960).□Such□a□structure is□said□to□be□constructed□ from□the□bottom□up . The□longer-term□aspiration□of□nanotechnology□is□a□gene ric□ assembler □machine□that□will be□able□to□build□from□the□bottom□up□ any □product.□Such□an□assembler□will□re-arrange□single atoms□and□molecules□so□as□to□build□the□desired□product.□One □might□instruct□the□assembler to□construct□tasty□ fillet-mignons □after□emptying□the□garbage□can□into□it.□As□unrealistic□a s□it sounds,□this□dream□is□perhaps□not□much□different□from□the□ common□feat□of□the□growing oak□tree□ mentioned□ earlier.□ Just□ as□ a□ tiny□ seed□ is□ able□ to□ collect□ minerals□ from□ the 14 environment□and□rearrange□them□into□living□tissues,□nanotechn ology□aspires□to□assemble□a variety□of□products□requiring□only□chemical□ ingredients,□ a□ co nstruction□ program,□ and energy□(Drexler,□1992). □Nanotechnology□visionaries□keep□stressing□the□importanc e□of□operating□at□the□small scale□for□increasing□efficiency,□by□the□precise□control□ on□each□molecule□and□atom□in□the process.□They□rely□on□the□natural□examples□we□see□aroun d□us□as□a□proof□for□the□viability□of their□master□plan.□They□also□consider□thermodynamics□ when□calculating□energy□intake, efficiency,□and□energy□dissipation.□Yet□they□neglect□a nother□point□that□is□obvious□from□the thermodynamic□point□of□view,□namely,□the□fundamental□r elation□between□efficiency□and information. The□biological□structures□and□processes□we□see□around□us□w ere□crafted□during□billions□of years.□Each□biochemical□process□in□a□living□cell□was□pro grammed□after□evolution’s□trying an□enormous□number□of□different,□random□pathways.□The□pro cess□has□gradually□equipped the□organism□with□invaluable□information.□In□order□to□roughly□a sses□the□magnitude□and value□of□this□information,□imagine□the□cost□of□a□project □whose□aim□would□be□to□build□a single□ameba□in□the□laboratory,□out□of□the□basic□che mical□elements.□Any□estimate□would give□ a□ cost□ far□ above□ any□ nation’s□ capabilities.□ The□ am eba,□ however,□ does□ it□ with infinitesimal□costs□every□time□it□multiplies,□by□util izing□the□information□already□stored□in its□DNA.□Therefore,□anyone□who□wishes□to□create□a□gene ric□assembler□that□will□be□capable of□producing□ anything □overlooks□the□amount□of□information□needed□for□such□a□proj ect. The□ prospect□ is□ much□ better,□ however,□ for□ a□ technology□ that□ seeks□ to□ exploit□ the information□already□encoded□in□the□genomes□of□existing□o rganisms.□The□myriad□of□species sharing□ our□ Globe,□ of□ which□ only□ a□ tiny□ fraction□ is□ known □ to□ science,□ stores□ an immeasurable□ treasure□ of□ pharmacological,□ agricultural□ an d□ technological□ knowledge, only□waiting□to□be□studied.□A□technology□that□would□take□advan tage□of□this□treasure□is certainly□feasible. 8. □Conclusions In□this□article□we□have□briefly□discussed□some□points□ where□thermodynamics□offers□fresh insights□for□the□life□sciences.□New□questions,□ones□tha t□we□did□ not□even□think□about earlier,□emerge□when□we□look□at□the□miracle□of□life□f rom□the□thermodynamic□perspective. While□we□are□not□sure□about□the□answers,□the□questions□t hemselves□are□important.□Our aim□has□been□only□to□appetize□the□medical□and□life□scient ist□to□become□more□acquainted with□the□growing□literature□dealing□with□this□interdiscipl inary□field□(Elitzur,□1994-1998 and□references□therein).□We□believe□that□the□introduct ion□of□basic□notions□like□entropy, information□ and□ complexity□ can□ add□ both□ depth□ and□ rigor□ to□ sciences□ as□ diverse□ as biochemistry,□genetics,□embryology,□morphology□and□ecol ogy. Unfortunately,□it□is□the□latter□field□in□which□thermody namic□thinking□yields□the□most□far- reaching□conclusions□–□and□the□ones□that□are□ones□most□oft en□ignored.□Human□societies keep□ ignoring□ the□ basic□ thermodynamic□ fact□ that□ any□ incr ease□ in□ a□ human’s□ living standards□entails□a□proportionate□increase□in□the□enviro nment’s□entropy.□Every□member□of Western□society□pollutes□the□environment□with□garbage,□pois onous□gases□and□heat□to□an 15 extent□that□poses□a□serious□threat□to□the□entire□biosphe re.□And□on□the□top□of□it,□mankind□is recklessly□multiplying,□nearing□the□incredible□figure□of□12□bi llion□predicted□to□populate the□globe□by□the□middle□of□the□21st□Century.□This□expansion□t hreatens□to□make□all□the achievements□of□modern□medicine□utterly□impotent.□No□reas onable□scenario□allows□such an□explosion□to□happen□without□all□the□dire□ecological□ consequences□seen□at□the□present□– global□warming,□famines,□diseases,□etc.□–□becoming□much□w orse. Such□calamities□are□inevitable□consequences□of□the□Secon d□Law,□to□which□most□policy makers□are□totally□oblivious.□Not□only□do□we□pollute□the□g lobe□with□our□ever-increasing waste□ products,□ we□ also□ directly□ ruin□ the□ biosphere’s□ incr edible□ complexity.□ Our generation□witnesses□one□of□the□greatest□extinctions□o f□species□that□ever□occurred□on□this globe.□Biodiversity□is□rapidly□shrinking□in□favor□of□t he□monotonous□artificial□environment that□ Homo□sapiens□ creates□everywhere,□namely,□the□arrogant,□human-cent ered□blend□of sky-scrapers,□highways,□malls,□market-chains□and□their□l ike.□Konrad□Lorenz□(1974),□the founder□of□ethology,□a□theoretical□biologist□and□a□physic ian□by□training,□has□once□observed that□the□rapid□expansion□of□human□cities□over□the□glo be□strikingly□resembles□the□growth□of a□cancerous□tumor.□Indeed,□in□both□cases□complexity□is□r uined□by□the□malignant□takeover of□only□few□of□the□living□system’s□components.□While□geneti c□therapy□seeks□to□combat cancer□(by□learning□how□to□operate□at□its□own□small□scale) ,□we□might□be□overlooking□all along□the□very□same□calamity□that□we□bring□on□the□ail ing□tissue□of□which□we□are□all□part. Many□philosophers□have□objected□to□the□attempts□to□explain□biol ogical□phenomena□by physical□ principles.□ “Reductionism”□ has□ become□ synonymous□ wit h□ disrespect□ for□the phenomenon□of□life.□In□this□paper□we□have□tried□to□show□t hat□the□contrary□is□the□case.□Not only□does□thermodynamics□give□a□new□dimension□to□the□lif e□sciences;□it□also□emphasizes what□we□have□intuitively□known□all□along:□That□life□i s□a□state□that□is□very□unique,□ill understood□–□and□precious. Acknowledgments It□is□a□pleasure□to□thank□the□Colton□scholarship□for□it s□generous□support. References 1. □Bennett,□C.□H.□(1988)□Logical□depth□and□physical□complexity.□I n□Herken,□R.□[Ed.] The□ Universal□ Turing□ Machine:□ A□ Half-Century□ Survey .□ Oxford:□ Oxford University□Press. 2. □Berne,□M.,□&□Matthew□N.□L.□(1993)□ Physiology .□St.□Luis,□Missuri:□Mosby-Year Book. 3. □Cheremisionoff□N.□P.,□&□Regino□C.□T.□(1978)□ Principles□and□Application□of□Solar Energy .□Michigan:□Ann□Arbor□Science□Publishers. 16 4. □Dickerson,□I.□G.□(1983)□ Hemoglobin:□Structure,□Function,□Evolution,□and□Pathology . Benjamin/Cummings 5. □Drexler,□ K.□ E.□ (1992)□ Nanosystems:□ Molecular□ Machinery,□ Manufacturing,□ and Computation .□New□York:□Wiley□&□Sons. 6. □Elitzur,□A.□C.□(1994a)□Let□there□be□life:□Thermodynamic□r eflections□on□biogenesis and□evolution.□ Journal□of□Theoretical□Biology ,□ 168 ,□429 −459. 7. □—□(1994b)□The□origin□of□life.□Essay□review.□ Contemporary□Physics ,□ 34 ,□275 −278. 8. □—□(1995)□Life□and□mind,□past□and□future:□Schr
arXiv:physics/0012037v1 [physics.atom-ph] 17 Dec 2000Diﬀerential and partial cross sections of elastic and inela stic positronium-helium-atom scattering Sadhan K. Adhikari Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, 01.405-900 S˜ ao Paulo, S˜ ao Paulo, Brazil (July 26, 2013) Abstract Scattering of positronium (Ps) by helium atom has been inves tigated in a three-Ps-state coupled-channel model including Ps(1s,2s ,2p) states using a re- cently proposed time-reversal-symmetric regularized ele ctron-exchange model potential. Speciﬁcally, we report results of diﬀerential c ross sections for elastic scattering and target-elastic Ps excitations. We also pres ent results for total and diﬀerent partial cross sections and compare them with ex periment and other calculations. PACS Number(s): 34.10.+x, 36.10.Dr Scattering of exotic ortho-positronium atom with long life time (142 ns) by neutral gas atoms and molecules is of fundamental interest in both physi cs and chemistry. Recent high precision measurements of positronium (Ps) scattering by H 2, N2, He, Ne, Ar, C 4H10, and C5H12[1–6] have enhanced theoretical activities [7–11] in this s ubject. Due to internal symmetry the direct static Born potential for elastic and ev en-parity transitions for these processes is zero and exchange correlation plays an importa nt role for a correct description at low energies [10,11]. Recently, we suggested [12] a regularized nonlocal electro n-exchange model potential with a single parameter Cand used it in the successful study of of Ps scattering by H [13,14], He [12,15], Ne [15], Ar [15] and H 2[16,17]. Our results were in agreement with experimental total cross section [1,3], specially at low en ergies for He, Ne, Ar and H 2. In our initial calculations we used a non-symmetric form of the model exchange potential for Ps scattering by H [14], He [12], and H 2[16]. Subsequent studies yielded improved results with a time-reversal symmetric form of the model potential f or Ps scattering by H [13] and H2[17]. For H it was found [13] that the symmetric potential yie lded excellent results for S-wave singlet Ps-H binding and resonance energies in agree ment with accurate variational calculations [18]. The symmetric potential also led to very good results [15] for low-energy cross sections for Ps scattering by He, Ne, Ar, and H 2in excellent agreement with experiment [3]. The problem of Ps-He scattering is of relevance to both exper imentalists and theoreti- cians. Theoretically, it is the simplest of all Ps-scatteri ng problems, which has reliable ex- perimental cross sections. Once a good theoretical underst anding of this system is obtained, we can try to understand the problem of Ps scattering by compl ex atoms and molecules. 1With this objective we reinvestigate the problem of Ps scatt ering by He at higher energies using the time-reversal symmetric form of the exchange pote ntial. We consider the three- Ps-state coupled-channel model with Ps(1s,2s,2p) states f or calculating diﬀerent elastic and inelastic cross sections of Ps-He scattering. We calculate the diﬀerent Ps-He diﬀerential cross sections which are of great interest to experimentali sts [5], in addition to the diﬀerent angle-integrated partial cross sections. The diﬀerential cross sections carry detailed infor- mation about the scattering process. Cross sections for hig her excitations and ionization of Ps are calculated by the Born approximation and added to th e above Ps(1s,2s,2p) cross sections to yield the total cross section which is compared w ith experiment. The theory for the coupled-channel study of Ps-He scatterin g with the regularized model potential has already appeared in the literature [7,12,13, 15]. It is worthwhile to quote the relevant working equations here. For target-elastic Ps-He scattering we solve the following Lippmann-Schwinger scattering integral equation in momen tum space for the total electronic doublet spin state f− ν′,ν(k′,k) =B− ν′,ν(k′,k) −/summationdisplay ν′′/integraldisplaydk′′ 2π2B− ν′,ν′′(k′,k′′)f− ν′′,ν(k′′,k) k2 ν′′/4−k′′2/4 + i0(1) where the Born amplitude, B−, is given by B− ν′,ν(k′,k) =gD ν′,ν(k′,k)−gE ν′,ν(k′,k),where gD andgErepresent the direct and exchange Born amplitudes and f−the scattering amplitude, respectively. The quantum states are labeled by the indices νreferring to the Ps atom. The variables k,k′,k′′etc. denote the appropriate momentum states; kν′′is the on-shell relative momentum of Ps with respect to He in channel ν′′. We use units ¯ h=m= 1 where mis the electron mass. The diﬀerential cross section is given by /parenleftBiggdσ dΩ/parenrightBigg ν′,ν=k′ k\|f− ν′,ν(k′,k)\|2. (2) For He ground state, the space part of the Hartree-Fock (HF) w ave function is given by Ψ(r1,r2) = [ϕ(r1)ϕ(r2)]. The position vectors of the electrons are r1andr2, andϕis taken to be in the form ϕ(r) =/summationtext κaκφκ(r), where φκ(r) are the atomic orbitals. The direct and exchange potentials are, respectively, give n by [12,15] gD ν′,ν(kf,ki) =4 Q2 2−/summationdisplay κ,κ′aκaκ′/integraldisplay φ∗ κ′(r) exp(iQ.r)φκ(r)dr  ×/integraldisplay χ∗ ν′(t) [2i sin( Q.t/2)]χν(t)dt. (3) and gE ν′,ν(kf,ki) =/summationdisplay κ,κ′4aκaκ′(−1)l+l′ Dκκ′/integraldisplay φ∗ κ′(r) exp(iQ.r)φκ(r)dr ×/integraldisplay χ∗ ν′(t) exp(iQ.t/2)χν(t)dt (4) with 2Dκ,κ′= (k2 i+k2 f)/8 +C2[(α2 κ+α2 κ′)/2 + (β2 ν+β2 ν′)/2] (5) where landl′are the angular momenta of the initial and ﬁnal Ps states, the initial and ﬁnal Ps momenta are kiandkf,Q=ki−kf,α2 κ/2 and α2 κ′/2, and β2 νandβ2 ν′are the binding energy parameters of the initial and ﬁnal He orbital and Ps states in atomic units, respectively, and Cis the only parameter of the potential. Normally, the parame terCis taken to be unity which leads to reasonably good result [15,1 7,23]. However, it can be varied slightly from unity to get a precise ﬁt to a low-energy observ able. This variation of Chas no eﬀect on the scattering observables at high energies and t he model exchange potential reduces to the Born-Oppenheimer exchange potential [19] at high energies. In the present study we use the value C= 0.84 throughout. This value of Cleads to a very good ﬁt of the elastic Ps-He cross section with the experiment of Skalsey e t al. [3]. This exchange potential for Ps scattering is considered [12] to be a generalization o f the Ochkur-Rudge exchange potential for electron scattering [20]. After a partial-wave projection, the system of coupled equa tions (1) is solved by the method of matrix inversion. A maximum number of partial wave sJmaxis included in solving the system of coupled equations. The diﬀerential an d angle-integrated partial cross sections so calculated are augmented by Born results for hig her partial waves J > J max. A maximum of 40 Gauss-Legendre quadrature points are used in the discretization of each momentum-space integral. The calculations are performed w ith the exact Ps wave functions and the HF orbitals for He ground state [21]. Although it is re latively easy to obtain converged results for angle-integrated partial cross sect ions, special care is needed to obtain converged results for diﬀerential cross sections at higher energies. Coverged results for partial cross sections are obtained for Jmax= 30 at all energies. For obtaining convergent diﬀerential cross sections, we need to take Jmax= 150 partial waves at 100 eV. However, Jmax= 30 is suﬃcient for obtaining convergent diﬀerential cross sections at 20 and 30 eV. Here we present results of Ps-He scattering using the three- Ps-state model that includes the following states: Ps(1s)He(1s1s), Ps(2s)He(1s1s), an d Ps(2p)He(1s1s). The Born terms for the excitation of He are found to be small and are not consi dered here in the coupled- channel scheme. First, we present the elastic Ps(1s)He(1s1 s) diﬀerential cross section and inelastic diﬀerential cross sections to Ps(2s)He(1s1s) an d Ps(2p)He(1s1s) states at diﬀerent energies. In order to show the general trend of the diﬀerential cross se ctions, we perform cal- culations at the following incident positronium energies: 20, 30, 40, 60, 80 and 100 eV. We exhibit the diﬀerential cross sections for elastic scatt ering at these energies in Fig. 1. In Figs. 2 −3 we show the inelastic cross sections for transition to Ps(2 s)He(1s1s) and Ps(2p)He(1s1s) states. From all these ﬁgures we ﬁnd that, as expected, the diﬀerential cross sections are more isotropic at low energies where only the lo w partial waves contribute. At higher energies more and more partial waves are needed to ach ieve convergence and the diﬀerential cross sections are more anisotropic. The small oscillation of the diﬀerential cross sections at larger angles and energies is due to numerical di ﬃculties. Recently, Garner et al. [5] have provided an experimental es timate of average diﬀerential cross section across the energy range 10 to 100 eV with respec t to any process in Ps-He scattering for forward scattering angles: /angbracketleftdσ/dΩ/angbracketright= (34 ±12)×10−20m2sr−1= (121 ±43)a2 0 sr−1. However, it is not possible to make a meaningful comparison between the present diﬀerential cross sections and the experimental estimate o f Garner et al. 3We calculate the diﬀerent angle-integrated partial cross s ections for Ps-He scattering. In addition to the Ps(1s,2s,2p) cross sections calculated usi ng the coupled-channel method, we also calculate the higher Ps(7 > n > 2)-excitation and Ps-ionization cross sections using the Born approximation with present exchange potential. Th ese results are shown in Fig. 4, where we plot angle-integrated elastic, Ps( n=2) [ ≡Ps(2s+2p)], inelastic Ps(7 > n > 2), and Ps ionization cross sections. The total cross section ca lculated from these partial cross sections is also shown in this plot and compared with the expe riments of Refs. [1,3] and total cross section of the 22-Ps-state R-matrix calculation of Re f. [8]. The agreement between theory and experiment is quite good up to 70 eV. The target-in elastic processes ignored in this work are supposed to play important role at higher energ ies, which may be the cause of detorioration of agreement of present results with exper iment above 70 eV. There exists qualitative disagreement between the present total cross s ection and that of the 22-state calculation of Ref. [8], on which we comment below. As the Ps-He system is of fundamental interest to both theore ticians and experimental- ists, it is appropriate to critically compare our results wi th other theories and experiments. The only other recent experiment on Ps-He is the one by Nagash ima et al. [4], who obtained the cross section of (13 ±4)πa2 0for an average energy of 0.15 eV in striking disagreement with the present calculation yielding 2 .58πa2 0at 0.9 eV as well as with the experiment of Skalsey et al. [3] who obtained (2 .61±0.5)πa2 0at about 0.9 eV. Independent experiment on the measurement [22] of pick-oﬀ q uenching rate of Ps on He can be used [23] to resolve the stalemate. It is argued [23] th at a large low-energy Ps-He elastic cross section implies a large repulsive exchange po tential between Ps and He atoms in the elastic channel. In the presence of a large repulsive p otential it will be diﬃcult for the Ps atom to approach the He atom. Consequently, one will ha ve a small value for the pick-oﬀ quenching rate. From a study of the pick-oﬀ quenchin g rates of diﬀerent models, we concluded [23] that a small low-energy cross section, as obt ained by us, will lead to a large pick-oﬀ quenching rate in agreement with experiment. The la rge low-energy cross sections as obtained in other theoretical models [7,8,10,11] will le ad to a much too small pick-oﬀ quenching rate in disagreement with experiment. This subst antiates that the present low- energy cross section and the experiment of Skalsey et al. [3] are consistent with the pick-oﬀ quenching rate measurement [22]. It would be diﬃcult to reco ncile the low-energy cross section of Nagashima et al. [4] and other theoretical result s [7,8,10,11] with the measurement of the pick-oﬀ quenching rate. We note that a model calculation by G. Peach [24], performed b efore the experiment of Skalsey et al. [3], is also in reasonable agreement with the p resent calculation and low energy experiments. The model of Peach was constructed by ﬁtting to known positron-helium [25] and electron-helium [26] scattering data. In Table I we compare the results of the angle-integrated par tial cross sections to Ps(1s,2s,2p) states of diﬀerent theoretical calculations . The present Ps(1s) Born cross sec- tions are much smaller than the Born-Oppenheimer cross sect ions [19] used as input to close- coupling [7] or R-matrix [8] schemes. There have been diﬀere nt static-exchange calculations on Ps-He since the 1960s [7,8,10,11]. These calculations yi elded similar results and in the static-exchange (SE) column of Table I we quote the recent cr oss sections of Refs. [7,8]. Al- though these SE cross sections are much smaller than the corr esponding Born-Oppenheimer cross sections, they are much larger than those of the presen t calculation. The 22-Ps-state 4R-matrix calculation [8] yields elastic cross sections mar ginally smaller than the SE cross sections, and it seems unlikely that the “converged” R-matr ix calculation will lead to elastic cross sections comparable to the present ones. However, the measured pick-oﬀ quenching rate [22] favors [23] a week exchange potential and small Ps( 1s) cross sections at low ener- gies, and future measurements of low-energy Ps-He elastic c ross sections will decide which of the results are more realistic. Although the present elas tic Ps(1) cross sections are much smaller than those of the R-matrix calculation, the reverse is true for the excitation cross sections to the Ps(2) states as can be found from Table I. The l arge Ps-excitation (and Ps ionization) cross sections of the present calculation and t he small low-energy elastic cross sections are collectively responsible for the constructio n of the pronounced peak in the total cross section as in Fig. 4 near 15 −20 eV in agreement with experiments of Refs. [1] and [3]. This peak is also present in the calculation of Peach [24 ] and is clearly absent in the close-coupling [7] and 22-Ps-state R-matrix analysis [8]. Similar peaks also appear in the total cross section of Ps-H 2and Ps-Ar scattering [5]. To summarize, we have performed a three-Ps-state coupled-c hannel calculation of Ps-He scattering at low and medium energies using a regularized sy mmetric nonlocal electron- exchange model potential recently suggested by us and succe ssfully used in other Ps scatter- ing problems. We present results for diﬀerential cross sect ions at several incident Ps energies between 20 eV to 100 eV for elastic scattering and inelastic e xcitation to Ps(2s,2p)He(1s1s) states. We also present the angle-integrated partial cross sections and compare them with those of other calculations. The present total cross sectio ns are in agreement with data of Refs. [1,3]. However, there is alarming discrepancy betwee n the present cross sections and those of conventional R-matrix [8] and close-coupling [7] c alculations. These latter calcula- tions are in agreement with a recent measurement of low-ener gy cross section by Nagashima et al. [4]. At low energies, the present elastic cross sectio ns are much too smaller compared to those of Refs. [7,8]. However, the present total cross sec tion develops a pronounced max- imum near 15 −20 eV as can be seen in Fig. 4 in agreement with the general expe rimental trend [5]. The cross section of Ref. [8] does not have this beh avior. Although, comparison with the pick-oﬀ quenching measurement data [22] at low-ene rgy favors [23] the results of the present model, further precise measurements of total and Ps (2) excitations at low energies will ﬁnally resolve the stalemate. The work is supported in part by the Conselho Nacional de Dese nvolvimento - Cient´ ıﬁco e Tecnol´ ogico, Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado d e S˜ ao Paulo, and Financiadora de Estudos e Projetos of Brazil. 5Table I: Angle-integrated Ps-He partial cross sections in πa2 0at diﬀerent positronium energies: EB −ﬁrst Born with present exchange; BO −ﬁrst Born with Born-Oppenheimer exchange; SE −static exchange of Refs. [7,8]; 3St −three-Ps-state with present exchange; 22St−22-Ps-state R-matrix calculation of Ref. [8] Energy Ps(1s) Ps(2s) Ps(2p) Ps(1s) Ps(1s) Ps(1s) Ps(1s) Ps(2s) Ps(2p) Ps(2) Ps(2) (eV) EB EB EB BO SE 22St 3St 3St 3St 3St22St 015.82 14.6 13.2 3.34 0.068 15.33 132 14.4 13.0 3.15 0.612 12.11 98 12.9 2.75 1.088 10.04 78 12.1 11.3 2.48 1.7 8.08 59 11.3 2.18 2.448 6.38 44 10.5 9.4 1.88 4.352 3.91 23 9.0 1.26 5 3.39 8.6 7.1 1.00 5.508 3.06 0.070 1.44 0.96 0.071 1.15 1.22 0.24 6 2.79 0.091 1.78 8.1 6.1 0.97 0.083 1.35 1.43 0.42 6.8 2.42 0.100 1.89 12 7.7 0.96 0.074 1.47 1.54 8 1.99 0.097 1.77 7.1 4.8 0.92 0.056 1.45 1.51 0.50 10 1.51 0.080 1.48 6.7 3.8 0.84 0.048 1.29 1.34 0.51 15 0.86 0.048 0.97 3.0 4.8 2.4 0.63 0.042 0.91 0.95 0.44 20 0.56 0.031 0.70 1.7 3.6 1.5 0.46 0.032 0.67 0.70 0.31 30 0.29 0.016 0.43 0.6 2.0 1.0 0.27 0.017 0.42 0.44 0.18 40 0.17 0.0094 0.30 0.22 0.7 0.8 0.17 0.010 0.30 0.31 0.12 50 0.11 0.0061 0.23 0.11 0.0067 0.23 0.24 60 0.079 0.0042 0.18 0.04 0.08 0.077 0.0045 0.18 0.19 80 0.043 0.0023 0.13 0.007 0.01 0.042 0.0024 0.13 0.13 100 0.026 0.0014 0.10 0.001 0.002 0.026 0.0014 0.10 0.10 150 0.010 0.0005 0.06 0.010 0.0005 0.06 0.06 6REFERENCES [1] A. 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Oppenheimer, Phys. Rev. 32361 (1928). [20] M. H. R. Rudge, Proc. Phys. Soc. London 86, 763 (1965); V. I. Ochkur, Zh. Eksp. Teor. Fiz. 45, 734 (1963)) [English Transl. Sov. Phys. JETP 18, 503 (1964).] [21] E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974). [22] B. G. Duﬀ and F. F. Heymann, Proc. R. Soc. London, Ser. A 270, 517 (1962); F. F. Heymann, P. E. Osmon, J. J. Veit, and W. F. Williams, Proc. Phy s. Soc. London 78, 1038 (1961). [23] S. K. Adhikari, P. K. Biswas, and R. A. Sultanov, Phys. Re v. A59, 4829 (1999). [24] G. Peach, unpublished, 1995, as quoted in Refs. [1,8]. [25] J. W. Humberston and R. I. Campeanu, J. Phys. B 13, 4907 (1980). [26] R. K. Nesbet, Phys. Rev. A 20, 58 (1979). 7Figure Caption: 1. Diﬀerential cross section (in units of a2 0) for elastic Ps-He scattering at the following incident Ps energies: 20 eV (dashed-dotted line), 30 eV (das hed-double-dotted line), 40 eV (dashed-triple-dotted line), 60 eV (full line), 80 (long da shed line), and 100 eV (short dashed line). 2. Diﬀerential cross section (in units of a2 0) for inelastic Ps-He scattering to Ps(2s)He(1s1s) state at the following incident Ps energies : 20 eV (dashed-dotted line), 30 eV (dashed-double-dotted line), 40 eV (dashed-triple-dot ted line), 60 eV (full line), 80 (long dashed line), and 100 eV (short dashed line). 3. Diﬀerential cross section (in units of a2 0) for inelastic Ps-He scattering to Ps(2p)He(1s1s) state at the following incident Ps energies : 20 eV (dashed-dotted line), 30 eV (dashed-double-dotted line), 40 eV (dashed-triple-d otted line), 60 eV (full line), 80 (long dashed line), and 100 eV (short dashed line). 4. Partial and total cross sections (in units of 10−16cm2) of Ps-He scattering at diﬀerent Ps energies: Ps(1s) (dashed-triple-dotted line), Ps( n=2) (dashed-dotted line), Ps(7 > n > 2) (dashed-double-dotted line), Ps-ionization (dashed line ), total (full line), total (full line with crosses from Ref. [8]), and data points with error bars from R efs. [1,3]. 80 30 60 90 120 150 180 Scattering angle (degree)Differential cross section (units of a02) 20 eV 30 eV 40 eV 100 eV60 eV 10 1 10-1 10-3 10-5 10-7Fig 1 10-980 eV0 30 60 90 120 150 180 Scattering angle (degree) Differential cross section (units of a02)10-1 10-3 10-5 10-7 10-9 Fig 2 20 eV 30 eV 40 eV 100 eV60 eV 80 eV0 30 60 90 120 150 180 Scattering angle (degree) Differential cross section (units of a02)10 1 10-1 10-3 10-5Fig 3 20 eV 30 eV 40 eV 100 eV60 eV 10-780 eV0 20 40 60 80 100 Positronium Energy (eV)0246Cross Section (units of 10-16cm2)Fig. 4
arXiv:physics/0012038v1 [physics.atom-ph] 17 Dec 2000Dynamical calculation of direct muon-transfer rates from t hermalized muonic hydrogen to C6+and O8+ Renat A. Sultanov1,2and Sadhan K. Adhikari1 1Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, 01405-900 S˜ ao Paulo, S˜ ao Paulo, Brazil 2Department of Physics, Texas A & M University,∗ College Station, Texas 77843 (August 4, 2013) We perform dynamical calculations of direct muon-transfer rates from thermalized muonic hy- drogen isotopes to bare nuclei C6+and O8+. For these three-body charge-transfer reactions with Coulomb interaction in the ﬁnal state we use two-component i ntegro-diﬀerential Faddeev-Hahn-type equations in conﬁguration space using close-coupling appr oximation. To take into account the high polarizability of the muonic hydrogen due to the large charg e of the incident nuclei, a polarization potential is included in the elastic channel. The large ﬁnal -state Coulomb interaction is explicitly included in the transfer channel. The transfer rates so calc ulated are in good agreement with recent experiments. I. INTRODUCTION Mainly motivated by the possibility of muon-catalyzed fusi on of the hydrogen isotopes, theoretical and experimental investigations of exotic atomic and molecular systems invo lving negative muon ( µ−) and reactions in such systems continue to be active ﬁelds of current research [1]. Particu lar attention is devoted to the study of the muon-transfer reaction from muonic hydrogen to other elements XZ+with large positive charge Z, because such reactions may have large cross sections and rates. If the hydrogen is contamina ted by even a small amount of these heavier elements, this may strongly inﬂuence the process of muon-catalyzed fu sion by hydrogen isotopes. Consequently, there has been considerable recent experimental interest in the stud y of the muon-transfer reaction in the collision of the muonic hydrogen by heavier nuclei, e.g. carbon (C6+), oxygen (O8+), neon (Ne10+), argon (Ar18+) [2–6] etc. On the theoretical side, these three-body charge transfer rea ctions involving a heavy transferred particle like muon and a strong Coulomb interaction in the ﬁnal state involving nucl ei, such as carbon and oxygen, continue to be challanging problems calling for new investigations. Experimental study of muon-transfer rates from muonic hydr ogen H µto heavier nuclei, such as Ar18+, Kr36+and Xe54+, have revealed a smooth Zdependence [7]. In these cases the transfer rate λZincreases linearly with Z. Theoretical analyses are also in agreement with this conclu sion [8]. However, it has been found in recent experiment that the predicted monotonic Zdependence of the muon-transfer rate is not valid for all Z[5,9]. For these transfer rates, pronounced ﬂuctuations have been observed for eleme nts up to argon contrary to the smooth Zdependence. The experimental muon transfer rates for reactions like (Hµ)1s+ XZ+→(Xµ)(Z−1)++ H+(1.1) depend in a complicated manner on the charge Z[9]. Here H stands for the hydrogen isotope p(proton) or d(deuteron) and XZ+stands for the target nuclei. Another phenomenon which has not yet found a satisfactory th eoretical explanation is the measured isotope eﬀect, e.g. the trend of the direct muon-transfer rate of reaction ( 1.1) from pµanddµto XZ+, where XZ+represents C6+, O8+[3], Ne10+[9], Ar18+[2], and Xe54+[10]. In cases of O8+, Ar18+and Xe54+the direct-transfer rate decreases with increasing mass of the hydrogen isotope. Theoretical a nalyses [8,11] also support this trend. The experimental results for Ne10[5,9] and sulphur dioxide [3] diﬀer considerably from the th eoretical predictions. In view of this, here we perform quantum dynamical calculati on of muon-transfer rates from pµto C6+and O8+ and from dµto O8+. For this purpose, we use close-coupling approximation to t wo-component Faddeev-Hahn-type dynamical equations [12,13]. We are currently investigati ng muon-transfer rates from muonic hydrogen to other heavier elements for a future publication. ∗Present address 1It is diﬃcult to perform a quantum dynamical calculation of a charge-transfer reaction. In addition, a theoretical study of the problems above is extremely complicated due to t he large charge of these heavy nuclei and the presence of a large number of open channels even at zero incident energ y. The large charge of the nuclei leads to a strong polarization of the muonic hydrogen in the initial state and a strong ﬁnal-state Coulomb repulsion. It is diﬃcult to incorporate these two eﬀects properly in a dynamical calcul ation. This is why there are nodynamical calculations of these muon-transfer rates. Although there is a very large number of open channels in this problem, for a given nuclei the muon is transferred predominantly to a few (muonic) atomic labels of the heavy nu clei [8,11]. For example, muon is captured mostly in the n= 4 states of C6+, andn= 5 states of O8+. Also these transfers take place mostly to the ﬁnal muonic-a tomic states with low angular momenta and transfer rates are negligible f or atomic states with angular momenta l >2. The correct dynamical formulation should include all the important tra nsfer channels and we included them in a previous study on muon transfer with light nuclear targets [14–17]. Howeve r, it is quite impossible now to treat even this reduced number of open transfer channels in a dynamical calculation with heavier targets due to convergence diﬃculties in the presence of the large polarization potential and large ﬁ nal-state Coulomb interaction mentioned above. Hence in the present treatment we use a two-channel model to calculat e transfer to a single ﬁnal state, where we include the elastic and one transfer channels. Diﬀerent sets of equatio ns are used for the diﬀerent ﬁnal states. Eventually, the total transfer rate is calculated by summing the diﬀerent co ntributions. The two-component Faddeev-Hahn-type equation, we use, all ows us to introduce explicitly a polarization potential in the initial channel and the repulsive Coulomb potential i n the ﬁnal channel. This has the advantage of building in the correct asymptotic behavior of the wave function in a l ow-order close-coupling type approximation [18]. Hence as in Ref. [14–16] we make a two-state close-coupling approx imation to the Faddeev-Hahn-type equation in the present study and ﬁnd that a numerical solution using the pre sent scheme leads to very encouraging agreement with experimental transfer rates. In Sec. II we present our formalism. Numerical results for mu on-transfer rates from muonic hydrogen to carbon and oxygen are given in Sec. III and compared with those of oth er investigations. In Sec. IV we present a summary and outlook. II. THEORETICAL FORMULATION The theoretical framework for the present study will be base d on the formalism developed in Ref. [15] which was used for the study of muon transfer from muonic hydrogen atom s to light charged nuclei, such as, He2+and Li3+. Here we shall perform a similar study with heavier charged nu clei, such as, C6+and O8+. The presence of the strong Coulomb interaction and the associated large polarization interaction make the present calculational scheme far more complicated theoretically and numerically compared to tha t of Ref. [15]. The formalism of Ref. [15] is a generalization over that of Ref. [14] for charge-transfer reaction with no ﬁ nal-state Coulomb interaction. In the dynamical equations in Ref. [15] the ﬁnal-state Coulomb interaction is explicit ly included in the transfer channel. In addition, in the pres ent work we explicitly include a polarization potential in the e lastic channel. In a coupled-channel approach for atomic processes, the coupling to inﬁnite number of p-wave states i s responsible for generating the polarization potential [19]. As it is impossible to include all such states in a numer ical scheme, the commonly accepted procedure is to replace these coupling terms by a polarization potential as in Ref. [20]. This idea has been recently used successfully in antiproton-hydrogen and antihydrogen-hydrogen reacti ons [21]. Next we describe the dynamical equations we use based on the close-coupling approximation to Faddeev-Hahn -type two-component equations [14]. We use units e= ¯h=mµ= 1, where mµ(e) is the muonic mass (charge), and denote, the heavy nuclei (C6+or O8+) by1, the hydrogen isotopes by 2and muon by 3. Below the three-body breakup threshold, following two-cl uster asymptotic conﬁgurations are possible in the system 123: (23)−1and (13)−2. These two conﬁgurations correspond to two distinct physical channels, denoted by 1 and 2, respec tively. These conﬁgurations are determined by the Jacobi coordinates ( /vector rj3, /vector ρk):/vector r13=/vector r3−/vector r1, /vector ρ 2= (/vector r3+m1/vector r1)/(1+m1)−/vector r2,/vector r23=/vector r3−/vector r2, /vector ρ 1= (/vector r3+m2/vector r2)/(1+m2)−/vector r1, where /vector ri,mi(i= 1,2,3,) are coordinates and masses of the particles respectively. Let us introduce the total three-body wave function as a sum o f two components Ψ(/vector r1,/vector r2,/vector r3) = Ψ 1(/vector r23, /vector ρ1) + Ψ 2(/vector r13, /vector ρ2), (2.1) where Ψ 1(/vector r23, /vector ρ1) is quadratically integrable over the variable /vector r23, and Ψ 2(/vector r13, /vector ρ2) over the variable /vector r13. The compo- nents Ψ 1and Ψ 2carry the asymptotic boundary condition for channels 1 and 2 , respectively. The second component is responsible for pure Coulomb interaction in the ﬁnal stat e. These components satisfy the following set of two coupled equations [E−H0−V23(/vector r23)−Upol(/vector ρ1)]Ψ1(/vector r23, /vector ρ1) = [V23(/vector r23) +V12(/vector r12)−UC(/vector ρ2)]Ψ2(/vector r13, /vector ρ2), (2.2) [E−H0−V13(/vector r13)−UC(/vector ρ2)]Ψ2(/vector r13, /vector ρ2) = [V13(/vector r13) +V12(/vector r12)−Upol(/vector ρ1)]Ψ1(/vector r23, /vector ρ1), (2.3) where Eis the center-of-mass energy, H0is the total kinetic energy operator, Vij(/vector rij) pair potentials ( i/ne}ationslash=j= 1,2,3), UCis the ﬁnal-state Coulomb interaction given by 2UC(/vector ρ2) =(Z−1)Z′ ρ2, (2.4) withZthe charge of the heavy nuclei and Z′(= 1) the charge of the hydrogen isotope. Here Upolis the polarization potential given by [11] Upol(/vector ρ1) =−9Z2 4ρ4 1, ρ 1>Λ (2.5) and zero otherwise. The value of the cut-oﬀ parameter Λ has to be chosen appropriately (see Sec. III). By adding Eqs. (2.2) and (2.3) we ﬁnd that they are equivalent to the Sch r¨ odinger equation. Distortion potential has been very useful in model and pheno menological description of reaction and scattering in nuclear [22] and atomic physics [23]. Although such distort ion potentials are unnecessary in a complete solution of the Schr¨ odinger equation, they enhance the agreement with exp eriment in a simpliﬁed model description. For example, a long-range polarization (distortion) potential has been routinely used in electron-atom scattering [20,24]. Such a distortion potential arising from the polarization of the m uonic hydrogen isotope due to the bare nuclei is eﬀective in the initial channel and has been included in Eq. (2.2). This p olarization potential for C6+or O8+is much stronger and its eﬀect on cross sections much more pronounced than in t he case of electron scattering. Hence, for obtaining a better agreement with experiment in a model calculation it is prudent to include the polarization potential in the elastic channel. There should also be such a polarization po tential in the ﬁnal channel. However, by far the most important interaction in the ﬁnal channel is the Coulomb pot ential between the proton (or deuteron) and the charged muonic atom ( Xµ)(Z−1)+. This Coulomb (distortion) potential has also been explici tly included in Eq. (2.3). This will help in obtaining a realistic description of the transf er process as we shall ﬁnd in the following. Because of the strong ﬁnal-state Coulomb interaction in the present muon-transfer problems it is very diﬃcult to develop and solve successfully multichannel models based o n Eqs. (2.2) and (2.3) above as in Ref. [15]. Hence, for solving Eqs. (2.2) and (2.3) we expand the wave function comp onents in terms of bound states in initial and ﬁnal channels, and project these equations on these bound states . The expansion of the wave function is given by Ψ1(/vector r23, /vector ρ1)≈f(1) 1s(ρ1) ρ1R(Z′) 1s,µ1(\|/vector r23\|)/4π, (2.6) Ψ2(/vector r13, /vector ρ2)≈f(2) nlL(ρ2) ρ2R(Z) nl,µ2(\|/vector r13\|){YL(ˆρ2)⊗Yl(ˆr13)}00, (2.7) where nlLare quantum numbers of the three-body ﬁnal-state, µ1=m3m2/(m3+m2),µ2=m3m1/(m3+m1), Ylm’s are the spherical harmonics, R(Z) nl,µ i(\|/vector r\|) is the radial part of the hydrogen-like bound-state wave fu nction for reduced mass µiand charge Z,f(1) 1s(ρ1) and f(2) nlL(ρ2) are the unknown expansion coeﬃcients. This prescription i s similar to that adopted in the close-coupling approximatio n. After a proper angular momentum projection, the set of two-coupled integro-diﬀerential equations for the unkn own expansion functions can be written as /bracketleftbigg (k(1) 1)2+∂2 ∂ρ2 1−2M1Upol(/vector ρ1)/bracketrightbigg f(1) 1s(ρ1) =g1/radicalbig (2L+ 1)/integraldisplay∞ 0dρ2f(2) nlL(ρ2) /integraldisplayπ 0dωsinωR(Z′) 1s,µ1(\|/vector r23\|)/parenleftbigg −Z′ \|/vector r23\|+Z \|/vector r12\|−UC(/vector ρ2)/parenrightbigg R(Z) nl,µ2(\|/vector r13\|) ρ1ρ2C00 L0l0Ylm(ν2, π)/√ 4π , (2.8) /bracketleftbigg (k(2) n)2+∂2 ∂ρ2 2−L(L+ 1) ρ2 2−2M2UC(/vector ρ2)/bracketrightbigg f(2) nlL(ρ2) =g2/radicalbig (2L+ 1) /integraldisplay∞ 0dρ1f(1) 1s(ρ1)/integraldisplayπ 0dωsinωR(Z) nl,µ2(\|/vector r13\|)/parenleftbigg −Z \|/vector r13\|+Z \|/vector r12\|−Upol(/vector ρ1)/parenrightbigg R(Z′) 1s,µ1(\|/vector r23\|)ρ2ρ1C00 L0l0Ylm(ν1, π)/√ 4π . (2.9) Herek(1) 1=/radicalBig 2M1(E−E(2) 1s),k(2) n=/radicalBig 2M2(E−E(1) n) with M−1 1=m−1 1+(1+ m2)−1andM−1 2=m−1 2+(1+ m1)−1, E(j) nis the binding energy of pair ( j3) and gj= 4πMj/γ3(j= 1,2),γ= 1−m1m2/((1 + m1)(1 + m2)),CLm L0lmthe Clebsch-Gordon coeﬃcient, ωis the angle between the Jacobi coordinates /vector ρ1and/vector ρ2,ν1is the angle between /vector r23and /vector ρ1,ν2is the angle between /vector r13and/vector ρ2. 3We ﬁnd that after the projection of the Faddeev-Hahn-type eq uations (2.2) and (2.3) on the basis states, the initial- state polarization and the ﬁnal-state Coulomb potentials s urvive on the left-hand-side of the resultant equations (2. 8) and (2.9). The presence of the explicit Coulomb potential in the ﬁnal channel will automatically yield the correct physical Coulomb-wave boundary condition in that channel. The explicit inclusion of the polarization potential, although has no eﬀect on the boundary condition in the initia l channel, substantially improves the the results of the truncated model calculation based on Eqs. (2.8) and (2.9) as we shall see in the next section. To ﬁnd unique solution to Eqs. (2.8) −(2.9), appropriate boundary conditions are to be considere d. We impose the usual condition of regularity at the origin f(1) 1s(0)=0 and f(2) nlL(0)=0. Also for the present scattering problem with 1+(23) as the initial state, in the asymptotic region, two so lutions to Eqs. (2.8) −(2.9) satisfy the following boundary conditions f(1) 1s(ρ1)∼ ρ1→+∞sin(k(1) 1ρ1) +Knl 11cos(k(1) 1ρ1), (2.10) f(2) nlL(ρ2)∼ ρ2→+∞/radicalbig v1/v2Knl 12cos(k(2) 1ρ2−η/2k(2) 1ln 2k(2) 1ρ2−πL/2), (2.11) where Knl ijare the appropriate coeﬃcients. For scattering with 2+ (13) as the initial state, we have the following conditions f(1) 1s(ρ1)∼ ρ1→+∞/radicalbig v2/v1Knl 21cos(k(1) 1ρ1), (2.12) f(2) nlL(ρ2)∼ ρ2→+∞sin(k(2) 1ρ2−η/2k(2) 1ln 2k(2) 1ρ2−πL/2) + (2.13) Knl 22cos(k(2) 1ρ2−η/2k(2) 1ln 2k(2) 1ρ2−πL/2), (2.14) where vi(i= 1,2) is velocity in channel i. The Coulomb parameter in the second transfer channel is η= 2M2(Z− 1)/k(2) n[23]. The coeﬃcients Knl ijare obtained from the numerical solution of the Faddeev-Hah n-type equations. The cross sections are given by σtr 1s→nl=4π k(1)2(Knl 12)2 (D−1)2+ (Knl 11+Knl 22)2, (2.15) where D=Knl 11Knl 22−Knl 12Knl 21. When k(1)→0:σtr 1s→nl∼1/k(1) 1. The transfer rates are deﬁned by λtr 1s→nl=σtr 1s→nlv1N0, (2.16) withv1being the relative velocity of the incident fragments and N0the liquid-hydrogen density chosen here as 4.25×1022cm−3, because λtr(k(1)→0)∼const. In our model approach the total muon transfer rate is λtr tot=/summationdisplay λtr 1s→nl. (2.17) III. NUMERICAL RESULTS We employ muonic atomic unit: distances are measured in unit s ofaµ, where aµis the radius of muonic hydrogen atom. The integro-diﬀerential equations are solved by disc retizing them into a linear system of equations. The integrals in Eqs. (2.8) and (2.9) are discretized using the t rapezoidal rule and the partial derivatives are discretize d using a three-point rule [25]. The discretized equation is s ubsequently solved by Gauss elimination method. As we are concerned with the low-energy limit, only the total angu lar momentum L= 0 is taken into account. Even at zero incident energy, the transfer channels are open and their wa ve functions are rapidly oscillating Coulomb waves. In order to get a converged solution we needed a large number of d iscretization points. More points are taken near the origin where the interaction potentials are large; a smalle r number of points are needed at large distances. First we solved the system of equations without the polariza tion potential in the incident channel. However, the ﬁnal-state Coulomb interaction is correctly represented i n our model. In this case it is relatively easy to obtain the numerical convergence for the system of equations which inc ludes the elastic channel and one transfer channel at a time. Finally, the total transfer cross section is calculat ed by adding the results of diﬀerent two-channel contributi ons. Without the polarization potential, we needed up to 700 disc retization points adequately distributed between 0 and 50aµ. Near the origin we took up to 60 equally spaced points per uni t interval ( aµ). It was more diﬃcult to obtain convergence with the polarizat ion potential. The polarization potential (2.5) is taken to be zero at small distances below the cut oﬀ Λ. In this case to get numerical convergence we had to integrate to very 4large distances −up to 300 aµ. We needed up to 2000 discretization points to obtain conver gence. Again we needed more points near the origin and less at large distances. For e xample, near the origin we took up to 60 equally spaced points per unit length interval aµ; in the intermediate region ( ρ= 10−20aµ) we took up to 15 equally spaced points per unit length interval, and in the asymptotic region ( ρ= 20−300aµ) we took up to 5 equally spaced points per unit length interval. It is well-known that the results for the cr oss section is sensitive to the value of the cut oﬀ Λ of the polarization potential. The short-range potential of the p resent problem extends to about 25 aµ. We considered the polarization potential in the asymptotic region ρ1>Λ≃75aµ. For a small variation of Λ in this region from 75 aµto about 120 aµ, we ﬁnd the transfer cross sections to be reasonably constan t and the reported transfer cross sections of this study are the averages of these cross sections. If Λ is in creased past 120 aµ, the eﬀect of the polarization potential on the cross sections gradually decreases and ﬁnally disapp ears. If Λ is decreased much below 75 aµ, the cross sections become rapidly varying function of Λ and could become unphys ically large. The range of Λ values (here between 75 aµ and 120 aµ) for which the cross sections are slowly varying smooth func tions should increase with the charge of the bare nucleus. We present partial muon-transfer rates λtr nland total transfer rates λtr totcalculated using the formulation of last section. In this work using the model of Sec. II we calculate t he low-energy muon-transfer rates from ( pµ)1sto C6+ and O8+and from ( dµ)1sto O8+. From other theoretical [8,11] investigations it was concl uded that in the case of C6+the transfer takes place predominantly to the n= 4 state and for O8+it happens to the n= 5 state, which is also found to be true in our model calculation. Hence in this w ork we only present rates for the l= 0,1,2 states of then= 4 and 5 orbitals of carbon and oxygen, respectively. The con tribution of the higher angular momentum states to the total transfer cross section is very small. Numerical ly converged results were obtained in these cases. The low energy partial rates λtr 1s→nl/1010sec−1and total rates λtr tot/1010sec−1are presented in Tables I, II, and III together with the results of other theoretical and experimental work s. First we comment on the results in Table I for muon transfer fr om (pµ)1sto C6+. The partial transfer rates without the polarization potential increases with decreasing cent er-of-mass energy Eand saturates to a constant value for E <0.04 eV for 4s, 4p, and 4d states of muonic carbon. In the case of C6+the transfer takes place predominantly to the 4s state. These qualitative behaviors are also true afte r the inclusion of the polarization potential and were also true in a previous theoretical study. However, after the inc lusion of the polarization potential the 4s and 4p transfer rates are enhanced by about a factor of 1.5. The present total transfer rate of 8 .5×1010sec−1is about three times larger than the previous theoretical calculation of 2 .8×1010sec−1[11]. We quote two experimental results in this case: (9 .5±0.5)×1010sec−1[2] and (5 .1±1.0)×1010sec−1[7]. The present theoretical result lies in between these two somewhat conﬂicting experimental results. In the case of muon transfer from ( pµ)1sto O8+, we ﬁnd from Table II that transfer takes place predominantl y to the 5s and 5p states of muonic oxygen. Again the transfer rate s saturate and attain constant values for E <0.04 eV. The transfer rate is higher in the 5s state and lowest in the 5d state. This behavior remain true after the inclusion of the polarization potential, when the transfer rate to the 5p state increases by a factor of more than two whereas the contribution to the 5s state increases by a factor of 1.5. The present total transfer rate of (7 .7±0.5)×1010sec−1is about 1.5 times larger than the previous theoretical calcul ation of 5 .6×1010sec−1[11] and in reasonable agreement with the recent experimental rate of (8 .5±0.2)×1010sec−1[4]. Finally, in the case of muon transfer from ( dµ)1sto O8+, we ﬁnd from Table III that transfer also takes place predominantly to the 5s state of muonic oxygen. The contribu tion to the transfer rate due to 5s state is two times as large as the contribution due to the 5p state. Again the tra nsfer rate saturate and attain a constant value for E <0.04 eV. After the inclusion of the polarization potential the transfer rate to the 5s state increases by a factor of 1.5. The present total transfer rate of (4 .4±0.6)×1010sec−1is in reasonable agreement with the recent experimental rate of 5 .5×1010sec−1[3]. TheZdependence of the transfer rates from a speciﬁc hydrogen iso tope to XZ+has been a subject of interest. Although for large Zthese rates increase linearly with Z, there is no general behavior for small Z. The most recent experimental transfer rates decrease when we move fr om the system pµ−C6+topµ−O8+[2,4]. Through our dynamical calculation we have been able to reproduce thi s behavior. Our calculation is also consistent with the experimentally observed isotope eﬀect, e.g., the transfer rate decreases when we move from pµ−O8+todµ−O8+ [3,4]. IV. CONCLUSION We have studied muon-transfer reactions from muonic hydrog en to carbon and oxygen nuclei employing a full quantum-mechanical few-body description of rearrangemen t scattering by solving the Faddeev-Hahn-type equations using close-coupling approximation. To provide the correc t asymptotic form of the wave function in the transfer channel, the ﬁnal-state Coulomb interaction has been incor porated directly into the equations. We also included a polarization potential at large distances in the initial ch annel. It is shown that in the present approach, the applicat ion of a close-coupling-type ansatz leads to satisfactory resu lts for direct muon-transfer reactions from muonic hydroge n to C6+and O8+. In the case of muon transfer from ( pµ)1sto C6+the present transfer rate to the 4s state of muonic carbon is about 1.5 times larger than that to the 4p state. For muon transfer from ( pµ)1sto O8+, the present 5transfer rate to the 5s state of oxygen is about twice as large as that to the 5p state. The present rates are much larger by factors of about two to three compared to the calcul ation of Ref. [11]. Finally, in the case of muon transfer from ( dµ)1sto O8+, the present transfer rate to the 5s state is large compared t o that to the 5p state. In all cases the inclusion of the polarization potential improves the ag reement with experiment and our ﬁnal transfer rates are in encouraging agreement with recent experiments [2–4]. Th e present rates for oxygen from ( pµ)1sand (dµ)1sare in agreement with the observed isotope eﬀect [3]: the transf er rate increases with the decrease of the mass of the hydrogen isotope. Because of the present promising results for the muon-transfer rates for Z= 6 and Z= 8 it seems useful to make future applications of the present formulati on for larger targets. Calculations involving nuclei of hig her charges (Ne10+, S16+, Ar18+etc.) are in progress. ACKNOWLEDGMENTS We acknowledge the support from Funda¸ c˜ ao de Amparo ˜ a Pesq uisa do Estado de S˜ ao Paulo of Brazil. The numerical calculations were performed on the IBM SP2 Supercomputer of the Departamento de F´ ısica - IBILCE - UNESP, S˜ ao Jos´ e do Rio Preto, Brazil. [1] P. Ackerbauer, J. Werner, W. H. Breunlich, M. Cargnelli, S. Fussy, M. Jeitler, P. Kammel, J. Marton, A. Scrinzi, J. Zmeskal, J. Bistirlich, K. M. Crowe, J. Kurck, C. Petitjean, R. H. Sherman, H. Bossy, H. Daniel, F. J. Hartmann, W. Neumann, G. 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Schellenberg, Hyperﬁne Interact. 82, 513 (1993). [10] A. Bertin, M. Bruno, A. Vitale, A. Placci, and E. Zavatti ni, Phys. Rev. A 7, 462 (1973). [11] S. S. Gershtein, Zh. Eksp. Teor. Fiz. 43, 706 (1962) [Sov. Phys. JETP 16, 501 (1963). [12] Y. Hahn, Phys. Rev. 169, 794 (1968). [13] Y. Hahn and K. M. Watson, Phys. Rev. A 5, 1718 (1972). [14] R. A. Sultanov and S. K. Adhikari, Phys. Rev. A 61, 022711 (2000). [15] R. A. Sultanov and S. K. Adhikari, J. Phys. B 32, 5751 (1999). [16] R. A. Sultanov and S. K. Adhikari, in the Proceedings of t he 16th IUPAP International Conference on Few-Body Problem s in Physics, Nucl. Phys. A, to appear. [17] R. A. Sultanov, W. Sandhas, and V. B. Belyaev, Eur. Phys. J. D5, 33 (1999). [18] R. A. Sultanov, Few Body Syst. Suppl. 10, 281 (1999); R. A. Sultanov, Innovative Computational Methods in Nuclear Many-Body Problems ed H Horiuchi, Y Fujiwara, M Matsuo, M Kamimua, H Toki and Y Sak uragi (Singapore: World Scinetiﬁc) p 131 (1998). [19] L. Castillejo, I. C. Percival, and M. J. Seaton, Proc. Ro y. Soc. (London) A 254, 259 (1960). [20] P. G. Burke and K. Smith, Rev. Mod. Phys. 34, 458 (1962), see page 465. [21] A. Yu. Voronin and J. Carbonel, Phys. Rev. A 57, 4335 (1998). [22] W. Tobocman, Theory of Direct Nuclear Reactions (Oxford University Press, London, 1961). [23] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, London, 1965). [24] R. K. Nesbet, Variational Methods in Electron-Atom Scattering Theory (Plenum Press, New York, 1980); S. K. Adhikari, Variational Principles and the Numerical Solution of Scatt ering Problems (John Wiley and Sons, New York, 1998). [25] M. Abramowitz and I. A. Stegun, in Handbook of Mathematical Functions , (Dover Publications, New-York, 1968), p. 884, eq. (25.3.23), and p. 885, eq, (25.4.1). Table I. Low energy partial λtr 1s→nl/1010sec−1and total λtr tot/1010sec−1muon transfer rates reduced to liquid-hydrogen density N0= 4.25×1022cm−3from muonic hydrogen ( pµ)1sto hydrogen-like excited state of muonic carbon (C µ)5+ n=4. 6Energy Upol(ρ1) = 0 With polarization Theory Experiment E(eV) ( nl)λtr 1s→nl λtr tot λtr 1s→nl λtr tot [11] [7] [2] 4s 3.4 5 .2±0.2 0.04 4 p 2.1 5 .5 3 .2±0.5 8 .5±0.7 2 .8 5 .1±1.0 9 .5±0.5 4d 0.05 0 .1 4s 2.1 2 .8±0.2 0.1 4 p 1.1 3 .2 1 .5±0.2 4 .3±0.4 4d ∼0 ∼0 4s 1.2 1 .6±0.1 0.5 4 p 0.4 1 .6 0 .7±0.1 2 .3±0.2 4d ∼0 ∼0 Table II. Low energy partial λtr 1s→nl/1010sec−1and total λtr tot/1010sec−1muon transfer rates reduced to liquid-hydrogen density N0= 4.25×1022cm−3from muonic hydrogen ( pµ)1sto hydrogen-like excited state of muonic oxygen (O µ)7+ n=5. Energy Upol(ρ1) = 0 With polarization Theory Experiment E(eV) ( nl)λtr 1s→nl λtr tot λtr 1s→nl λtr tot [11] [3] [4] 5s 3.5 5 .5±0.2 0.04 5 p 0.8 4 .35 2 .1±0.2 7 .7±0.5 5 .6 8 .3 8 .5±0.2 5d 0.05 0 .1±0.05 5s 3.1 5 .0±0.2 0.1 5 p 0.7 3 .8 1 .7±0.2 6 .8±0.5 5d 0.02 0 .05 5s 2.0 2 .9±0.1 0.5 5 p 0.2 2 .2 1 .0±0.1 3 .9±0.2 5d ∼0 ∼0 Table III. Low energy partial λtr 1s→nl/1010sec−1and total λtr tot/1010sec−1muon transfer rates reduced to liquid-hydrogen density N0= 4.25×1022cm−3from muonic hydrogen ( dµ)1sto hydrogen-like excited state of muonic oxygen (O µ)7+ n=5. Energy Upol(ρ1) = 0 With polarization Experiment E(eV) ( nl) λtr 1s→nl λtr tot λtr 1s→nl λtr tot [3] 5s 1.9 2 .9±0.3 0.04 5 p 0.8 2 .7 1 .5±0.3 4 .4±0.6 5 .5 5d ≤0.01 ∼0 5s 1.1 1 .8±0.2 0.1 5 p 0.5 1 .6 0 .7±0.2 2 .5±0.4 5d ≤0.01 ∼0 5s 0.7 1 .0±0.2 0.5 5 p 0.1 0 .8 0 .2±0.1 1 .2±0.3 5d ≤0.01 ∼0 7
arXiv:physics/0012039v1 [physics.atm-clus] 17 Dec 2000Positronium atom scattering by H 2in a coupled-channel framework P K Biswas$and Sadhan K Adhikari+ $Departamento de F ´isica-IGCE, Universidade Estadual Paulista 13500-970 Rio Claro, S˜ ao Paulo, Brasil +Instituto de F ´isica Te´ orica, Universidade Estadual Paulista, 01405-900 S˜ ao Paulo, S˜ ao Paulo, Brasil July 13, 2011 Abstract The scattering of ortho positronium (Ps) by H 2has been investigated using a three- Ps-state [Ps(1s,2s,2p)H 2(X1Σ+ g)] coupled-channel model and using Born approximation for higher excitations and ionization of Ps and B1Σ+ uand b3Σ+ uexcitations of H 2. We employ a recently proposed time-reversal-symmetric nonlo cal electron-exchange model potential. We present a calculational scheme for solving th e body-frame ﬁxed-nuclei coupled-channel scattering equations for Ps-H 2, which simpliﬁes the numerical solution technique considerably. Ps ionization is found to have the l eading contribution to target- elastic and all target-inelastic processes. The total cros s sections at low and medium energies are in good agreement with experiment. 11 Introduction With the availability of improved monoenergetic ortho posi tronium (Ps) beam, low-energy collision of exotic ortho Ps atom with neutral atoms and mole cules is of interest in both physics and chemistry due to its vast applicational potential [1]. R ecently, measurements of total ortho-Ps scattering cross section from various atomic and m olecular targets (He, Ne, Ar, H 2, N2, C4H10, and C 5H12) have been carried out at low and medium to high energies [2, 3 , 4]. Experimental investigations are also in progress for the me asurement of pick-oﬀ annihilation of ortho Ps from closed shell atoms [5]. Because of the composit e nature and high polarizability of Ps, a reliable theoretical description of Ps scattering i s far more complicated than electron scattering [6], as excitations and ionization of Ps are expe cted to play a signiﬁcant role for each target states. Besides if the target is vulnerable to excita tions, prediction of total cross section becomes extremely diﬃcult as the number of scattering chann els grow as N2, whereNexcited states of both Ps and target are considered. Due to internal charge and mass symmetry, Ps atom yields zero elastic and even-parity tran- sition potentials in the direct channel and Ps scattering is dominated mainly by exchange corre- lation [7, 8, 9]. This eventually complicates the convergen ce of a conventional coupled-channel description [7]. For Ps scattering with molecular target we have additional complication due to the presence of the nuclear degrees of freedom. The presence of three charge centers complicates the solution scheme of the coupled-channel equations. The d egree of complication can be real- ized from the mathematical analysis of ref. [10] for Ps forma tion in positron-hydrogen-molecule scattering. Because of mathematical complication, this ap proach has not been further pursued in numerical analysis and there exists no successful calcul ation of Ps formation in positron- hydrogen-molecule scattering. We recall that the mathemat ical complication of a dynamical coupled-channel study of Ps formation in positron-hydroge n-molecule scattering is similar to that of Ps-H 2scattering. Here we undertake a theoretical study of Ps scattering by H 2. We employ a hybrid approach of treating the molecular orientational dependence as para meters in the coupled equations and then perform the partial-wave expansion [11]. The resultin g one-dimensional equations are solved at the equilibrium nuclear separation by the matrix- inversion technique and the partial- wave cross sections are numerically averaged over molecula r orientations. The present approach makes the Ps-H 2scattering problem easily tractable. Recently, we suggested a regularized electron-exchange mo del potential [6, 8] and demon- strated its eﬀectiveness in exchange-dominated Ps scatter ing by performing quantum coupled- channel calculations using the ab initio framework of close -coupling method for both simple and complex targets [7, 12, 13, 14, 15]. In our initial calcul ations we used a nonsymmetric model-exchange potential for Ps scattering by H [12], He [8] and H 2[6, 13] and obtained rea- sonably good agreement with experiment on He and H 2. In a subsequent application of Ps scattering by H, it was found that a time-reversal symmetric form of the exchange potential leads to far superior result than the nonsymmetric form both in qualitative and quantitative agreement with accurate variational calculations on H [7]. The symmetric potential also led to very good results for low-energy cross sections for Ps sca ttering by He, Ne, and Ar [16] in excellent agreement with experiment [4]. In view of the above we reinvestigate the problem of Ps scatte ring by H 2using the symmetric exchange potential employing the three-Ps-state [Ps(1s,2 s,2p)] coupled-channel model for elastic 2and Ps(2s,2p) excitations. We solve the coupled-channel eq uations by the above scheme and report partial cross sections for Ps(1s,2s,2p) excitation s. We also calculate cross sections for Ps excitations to 6 ≥n≥3 states and Ps ionization using the ﬁrst Born approximation . A target-elastic total cross section is calculated by adding the above partial cross sections. We use the present symmetric exchange potential to calculate the t arget-inelastic Born cross sections for excitation to B1Σ+ uand b3Σ+ ustates of H 2by Ps impact. We present a total Ps-H 2cross section by adding the above target-elastic and target-inel astic results. In section 2 we present the theoretical formulation in the bo dy-frame ﬁxed-nuclei approx- imation. In section 3 we present the numerical results. Fina lly, in section 4 we present a summary of our ﬁndings. 2 Theoretical Formulation The total wave function Ψ of the Ps-H 2system is expanded in terms of the Ps and H 2quantum states as Ψ(r0,r1,r2,x;R) = A/summationdisplay a,b/bracketleftbigg Fab(s0)ϕa(t0)ψb(r1,r2;R)/bracketrightbigg (1) where s0= (x+r0)/2,t0= (x−r0),x(r0) is the coordinate of the positron (electron) of the Ps atom, r1andr2are the coordinates of the electrons of H 2, 2Ris the internuclear separation of H 2,ϕathe wave function of Ps, ψbthe wave function of H 2, withbdenoting the electronic conﬁguration of H 2andadenoting the quantum state of Ps. Here Adenotes antisymmetrization with respect to Ps- and target-electron coordinates and Fabis the continuum orbital of Ps with respect to the target. The spin of the positron is conserved i n this process and the exchange proﬁle of the Ps-target system is analogous to the correspon ding electron-target system [14]. Primary complication arises in the coupled-channel study i n retaining both the summations over Ps and target states in the coupling scheme. As we shall s ee later that the contribution of the individual target-inelastic channels to cross secti on is one order small compared to that of target-elastic channels. So we exclude the summation ove r target states from the coupling scheme and treat cross section for target excitations separ ately using Born approximation. In this work we are mostly interested to see whether the model ca n accommodate the measured total cross sections of Ps-H 2scattering. So we retain the Ps(1s,2s,2p) excitations in th e coupled- channel model and treat the remaining excitations by Born ap proximation. The ﬁrst Born calculation with present regularized exchange leads to res ults reasonably close to coupled- channel calculation at medium energies. Hence it is expecte d that this calculation should yield a fairly good picture of the total cross section except near t he Ps-excitation thresholds. Projecting the resultant Schr¨ odinger equation on the ﬁnal Ps and target states and averaging over spin, the resulting momentum-space Lippmann-Schwing er scattering equation in the body- frame representation for a particular total electronic spi n stateScan be written as fS a′a(k′,k;R) =BS a′a(k′,k;R)−1 2π2/summationdisplay a′′/integraldisplay dk′′BS a′a′′(k′,k′′;R)fS a′′a(k′′,k;R) k2 a′′/4−k′′2/4 + i0(2) wherefS a′aare the scattering amplitudes, and BS a′athe corresponding Born potentials. ka′′is the on-shell momentum of Ps in the intermediate channel a′′. We use units ¯ h=m= 1 where m 3is the electronic mass. For Ps-H 2target-elastic scattering there is only one scattering equ ation (2) corresponding to total electronic spin S= 1/2. The input potential of (2) is given by B1/2 a′a(k′,k;R) =BD a′a(k′,k;R)−BE a′a(k′,k;R) (3) whereBDis the direct Born potential and BEis the exchange potential. As in the ground electronic state X1Σ+ gof H 2, the total electronic spin is zero, there will contribution of one target electron to BE a′a(k′,k;R) [17]. For the electronic ground state X1Σ+ gof H2we use the wave function of the form ψb(r1,r2;R) ≡1σ(b) g(1)1σ(b) g(2) =N2U0(r1;R)U0(r2;R),whereN= [2(1 + T)]−1/2withTthe overlap integral [18] and U0(r;R) = (δ3/π)1/2[exp(−δ\|r−R\|)+exp( −δ\|r+R\|)] withδ= 1.166 [18, 19]. For Ps we use the exact wave functions, e.g., for 1s state ϕ(r) = exp( −βr)/√ 8πwithβ= 0.5. For the excited states of H 2we take the conﬁgurations ψb′(r1,r2;R)≡[1σ(b′) g(1)1σ(b′) u(2)± 1σ(b′) g(2)1σ(b′) u(1)], where + ( −) corresponds to the spin singlet (triplet) state. All wave f unctions of H 2are taken from ref. [19]. The direct Born potential for Ps transition from state atoa′and H 2transition from btob′ can be rewritten in the following convenient factorized for m [20]: BD a′b′←ab(kf,ki;R) =4 Q2/integraldisplay dtϕ∗ a′(t)[eiQ.t/2−e−iQ.t/2]ϕa(t) ×/integraldisplay dr1dr2ψ∗ b′(r1,r2;R)/bracketleftBigg 2 cos(Q.R)−2/summationdisplay n=1eiQ.rn/bracketrightBigg ψb(r1,r2;R).(4) Next we describe the electron-exchange model potential for Ps-H 2scattering. We develop the model exchange potential from the following term [6, 13] : BE a′b′←ab(kf,ki;R) = −1 π/integraldisplay dxdr0dr1dr2e−ikf.(x+r1)/2ϕ∗ a′(x−r1)ψ∗ b′(r0,r2;R)1 \|r0−r1\| ×ψb(r1,r2;R)ϕa(x−r0)eiki.(x+r0)/2. (5) After removing the nonorthogonality of the initial and ﬁnal wave functions of (5) and some straightforward simpliﬁcation the model exchange potenti al for a general target-elastic transi- tion becomes [6, 8] BE a′b←ab(kf,ki;R) =4(−1)l+l′ (k2 f+k2 i)/8 + (2δ2 b(g)+β2 a+β2 a′)/2/integraldisplay dtϕ∗ a′(t)eiQ.t/2ϕa(t) ×/integraldisplay dr2dr0ψ∗ b(r0,r2;R)eiQ.r0ψb(r0,r2;R) (6) =4(−1)l+l′ (k2 f+k2 i)/8 + (2δ2 b(g)+β2a+β2 a′)/2/integraldisplay dtϕ∗ a′(t)eiQ.t/2ϕa(t) ×/integraldisplay dr21σ(b) g(r2)1σ(b) g(r2)/integraldisplay dr01σ(b) g(r0)eiQ.r01σ(b) g(r0) (7) wherelandl′are the angular momenta of the initial and ﬁnal Ps states and Q=ki−kf. Hereδb(g)andβaare the parameters of the H 2and Ps wave functions in the initial state. The parameter β2 a′corresponds to the ﬁnal-state binding energy of the Ps atom a nd is taken as 4zero while considering exchange for the Ps ionization chann el. For target-inelastic processes, following the prescription outlined in refs. [6, 8] the exch ange potential takes the form BE a′b′←ab(kf,ki;R) =4(−1)l+l′ (k2 f+k2 i)/8 + (δ2 b(g)+δ2 b′(u)+β2a+β2 a′)/2/integraldisplay dtϕ∗ a′(t)eiQ.t/2ϕa(t) ×/integraldisplay dr21σ(b′) g(r2)1σ(b) g(r2)/integraldisplay dr01σ(b′) u(r0)eiQ.r01σ(b) g(r0). (8) In (8) the additional indices bandb′are introduced on the molecular orbitals 1 σgand 1σu to distinguish the initial and ﬁnal states. The model exchan ge potentials (7) and (8) may be considered as a generalization of a similar potential sug gested by Rudge [21] for electron- atom scattering. The nonlocal and time-reversal symmetric exchange potential (7) has a very convenient form and is expressed as a product of form factors of Ps and H 2. Both the direct and exchange amplitudes have been factored out in terms of Ps and target “form-factors” leading to a substantial simpliﬁcation of the theoretical calculat ion. In our previous study of Ps-H 2 scattering [6, 13] the prefactor of the exchange potential w as not time-reversal symmetric. In Eq. (7) we have restored time-reversal symmetry as in ref. [8] wh ich is found to provide signiﬁcant improvement in the results. Although, for electron-molecu le scattering several model potentials are found in the literature [22, 23], there is no other conven ient model exchange potential for Ps scattering. 3 Numerical Procedure and Results In the body-frame calculation the coupled-channel equatio ns are solved at the equilibrium nuclear separation 2 R0= 1.4a0. The polar and azimuthal angles θRandφRofRare taken as parameters in the coupled equations [11]. This reduces (2) t o the following form fR0,θR,φR a′a (k′,k) =BR0,θR,φR a′a (k′,k)−1 2π2/summationdisplay a′′/integraldisplay dk′′BR0,θR,φR a′a′′(k′,k′′)fR0,θR,φR a′′a(k′′,k) k2 a′′/4−k′′2/4 + i0.(9) After standard partial-wave projection the three-dimensi onal coupled scattering equations (9) are ﬁrst reduced to coupled one-dimensional integral equat ions in momentum space. The one- dimensional equations are then discretized by Gauss-Legen dre quadrature rule and solved by the matrix inversion technique. A maximum of forty points ar e used in the discretization of the integrals. The discretized coupled-channel equations are solved for eight to ten discrete values each of polar and azimuthal angles θRandφR. This leads to a convergence of cross sections up to three signiﬁcant ﬁgures. For targets with mor e charge asymmetry and for polar molecules we expect that more points will be required for ang ular averaging. The present averaging amounts to solving the coupled set of scattering e quations sixty four to hundred times. Although this procedure increases the computational (CPU) time, mathematical complications of tedious (and untractable) angular-momentum analysis [1 0] are thus replaced by a tractable and convenient calculational scheme. Finally, the partial -wave cross sections are numerically averaged over molecular orientation using Gauss-Legendre quadrature points for both polar and azimuthal angles θRandφR. Maximum number of partial waves included in the calculatio n is 12. Contribution of higher partial waves to cross section is included by corresponding Born 5terms. These Born cross sections are also numerically avera ged over all molecular orientations in a similar fashion. In ﬁgure 1, we show the angle-integrated target-elastic cro ss sections for elastic, Ps(2s+2p), Ps(3≤n≤6) excitations and ionization of Ps. As expected and observe d in previous calcula- tions [7, 8, 12, 13] with other targets, the contribution of t he Ps ionization channel to the cross section plays a dominant role from medium to high energies as can be seen in ﬁgure 1. The detailed angle-integrated partial cross sections of the Bo rn and three-Ps-state calculation are tabulated in table 1. Near 10 eV, 20 eV and 30 eV the total Born c ross sections for Ps(1s,2s,2p) are found to be nonconvergent by 28%, 9% and 4%, respectively . Near 10 eV, 20 eV and 30 eV the total Born cross sections for Ps( n= 2) excitations are found to be nonconvergent by 18%, 6% and 3%, respectively. At 60 eV the Born and three-Ps-state results are essentially identical. The nonconvergence of Ps( n≥3) excitations calculated using Born approximation are exp ected to lie within the limit set by Ps( n= 2) cross sections above. It is expected from experience that the ionization cross section calculated using Coulomb Born will be more converged than the Ps(n= 2) Born cross sections. Table 1: Ps-H 2partial cross sections in units of πa2 0at diﬀerent positronium energies using the Born approximation and three-Ps-state calculation E (eV) Ps(1s) Ps(1s) Ps(2s) Ps(2s) Ps(2p) Ps(2p) Ps(n≥3)Ps-ion Born 3-St Born 3-St Born 3-St Born Born 0.068 23.72 3.79 0.612 17.56 3.16 1.45 11.84 2.47 3 6.71 1.60 4 5.03 1.17 5 3.94 0.81 6 3.18 1.01 0.098 0.23 3.46 2.28 0 0 7 2.63 1.05 0.101 0.18 3.60 2.52 1.28 0.11 8 2.22 1.03 0.092 0.14 3.36 2.51 1.43 1.15 10 1.65 0.94 0.072 0.090 2.80 2.27 1.29 3.03 12.5 1.21 0.81 0.052 0.068 2.24 1.93 1.05 4.34 15 0.92 0.68 0.041 0.054 1.85 1.65 0.87 4.94 20 0.59 0.49 0.026 0.035 1.34 1.25 0.63 5.18 25 0.40 0.36 0.017 0.023 1.04 0.99 0.49 4.93 30 0.29 0.27 0.012 0.017 0.84 0.81 0.40 4.55 40 0.17 0.16 0.007 0.009 0.60 0.58 0.28 3.80 60 0.07 0.07 0.003 0.004 0.37 0.37 0.17 2.72 Now we concentrate on some target inelastic processes. Each target inelastic transition is accompanied by elastic, excitation and ionization of Ps and hence by an inﬁnite number of possibilities. Here to account for target inelastic proces ses we consider Ps( n= 1→6) discrete excitations and ionization of Ps using the ﬁrst Born approxi mation. Contribution of higher discrete excitations of Ps are expected to be insigniﬁcant a nd are neglected in the calculation. We calculate the cross sections for the inelastic transitio n Ps(1s) + H 2(X1Σ+ g)→Ps* + H 2(B 1Σ+ u) where Ps* represents the ground and the excited states of th e Ps atom. These Born cross 6sections are also calculated at a equilibrium internuclear separation 2 R0= 1.4a0and ﬁnally averaged over angular orientations of the target. In ﬁgure 2 we display the Born contribution of partial cross sections for transition to diﬀerent Ps stat es while the hydrogen molecule is excited to the H 2(B1Σ+ u) state. The total cross section summing the diﬀerent contri butions is also shown for this target-inelastic process. When compa red with the corresponding total cross section calculated with the nonsymmetric exchange po tential [6] (not shown in ﬁgure 2) we ﬁnd marginal change at low energies (below 20 eV) and basic ally no change at medium to high energies. This is quite expected as the eﬀect of exchang e dies out at higher energies and these cross sections are controlled by the direct Born poten tials. Next we calculate the cross sections for the inelastic trans ition Ps(1s) + H 2(X1Σ+ g)→Ps* + H 2(b3Σ+ u) using the ﬁrst Born approximation, where Ps* represents th e ground and the abovementioned excited states of the Ps atom. These cross se ctions are again calculated for equilibrium internuclear separation 2 R0= 1.4a0and averaged over molecular orientations. In ﬁgure 3 we display the contribution of partial cross section s for transition to diﬀerent Ps states while the hydrogen molecule is excited to the H 2(b3Σ+ u) state. The total cross section summing the diﬀerent contributions is also shown for this target-in elastic process. There is signiﬁcant change when we compare this total cross section calculated w ith the symmetric exchange po- tential with our previous result [6] obtained with the nonsy mmetric exchange potential. This is quite expected as this process is purely exchange dominated . The qualitative and quantitative diﬀerences between the two total cross sections reveal the i mportance of using the symmetric exchange potential. In ﬁgure 4, we exhibit the total cross section obtained from t arget-elastic processes (Ps excitation up to n= 6 and Ps ionization) plus target excitations to B1Σ+ uand b3Σ+ ustates [6, 13] and compare with the total cross section measurement s of ortho-Ps scattering from H 2 [2, 4]. The agreement with the experimental cross sections i s highly encouraging. Here, we have considered only two lowest target-inelastic processes and their combined eﬀect on the total cross section. This gives an indication that the inclusion of rema ining important target-inelastic cross sections might give a better agreement with measurement. However, the present theoretical peak in total cross sectio n is shifted to a lower energy compared to experiment. This trend was also found in our calc ulation of Ps-He scattering [7]. In the energy range of 7 to 15 eV, the total cross section overe stimates the measured data. This is due to the fact that Ps( n≥3) excitations and ionization have been treated in the ﬁrst Born approximation framework. The neglect of other tar get-inelastic channels and the ﬁrst Born calculation for higher excitations and, in partic ular, ionization of Ps are supposed to be responsible for the shift in the theoretical peak. A dyn amical calculation for higher Ps excitations and ionization should reduce the theoretical t otal cross section in the intermediate energy region, while the inclusion of further target-inela stic channels will increase the cross section for medium to higher energies. These two eﬀects are e xpected to shift the peak in the total cross section to higher energies and lead to a better ag reement with experiment. A previous ﬁrst Born description of Ps-H 2scattering [24] with Born-Oppenheimer exchange [25] led to unphysically large cross section at low energies . The poor performance of that scheme in Ps scattering compared to electron scattering is d ue to the fact that in the absence of the direct potential (zero for elastic scattering), the B orn-Oppenheimer exchange potential solely determines the cross section. This clearly shows the very unrealistic nature of the Born- Oppenheimer exchange potential at low energies. 7In previous studies [7, 8, 14] of Ps scattering we introduced a parameter Cin the exchange potential for obtaining a more quantitative ﬁt with experim ent essentially by replacing the denominator term in (7) by ( k2 f+k2 i)/8 +C(δ2 a+δ2 a′+β2 b+β2 b′)/2.In the original form the constantC= 1 and we have used this value in this study. However, it can be varied slightly from unity to obtain a precise ﬁt of a low-energy scattering obser vable (experimental or variational), as has been done in some applications of model potentials [23 ]. A variation of Cfrom unity leads to a variation of the average values for square of momen ta [8, 12], which were taken as the binding energy parameters ( δ2,β2etc.) in the expression for the denominator in (7). This variation, in turn, tunes the strength of the exchange p otential at low energies. At high energies this model potential is insensitive to this parame trization and leads to the well-known Born-Oppenheimer form of exchange [25]. 4 Summary To summarize, we have used a time-reversal symmetric form of the nonlocal model potential for exchange and applied it to the study of Ps-H 2scattering. We have also presented a simpliﬁed prescription for performing coupled-channel dynamical sc attering calculation with molecular target using a body-frame ﬁxed-nuclei scheme. With this pre scription we have performed a three-Ps-state coupled-channel calculation of target-el astic Ps-H 2scattering. Higher excitations and ionization of the Ps atom are treated using the ﬁrst Born a pproximation with a regularized exchange. We also calculated cross sections for two target- inelastic excitations of H 2(B1Σ+ u and b3Σ+ u) using ﬁrst Born approximation with present exchange consi dering Ps excitations (n= 1,...,6) and ionization. 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Angle integrated target-elastic Ps-H 2partial cross sections at diﬀerent positronium ener- gies: elastic (solid line) and Ps(2s+2p) excitation (dashe d-dotted line) from three-state coupled- channel model at low energies interpolated to exchange Born at high energies, Ps(6 ≥n≥3) excitation (dashed-double-dotted line), Ps ionization (d ashed line) from exchange Born. 2. Angle integrated target-inelastic Ps-H 2partial and total cross sections at diﬀerent positro- nium energies to H 2(B1Σ+ u) state using the ﬁrst Born approximation: Ps(1s) (dashed-d otted line), Ps(2s+2p) (dashed-double-dotted line), Ps(6 > n > 2) excitation (dotted line), Ps ion- ization (dashed line), and total (full line). 3. Angle integrated target-inelastic Ps-H 2partial and total cross sections at diﬀerent positro- nium energies to H 2(b3Σ+ u) state using the ﬁrst Born approximation: Ps(1s) (dashed-d otted line), Ps(2s+2p) (dashed-double-dotted line), Ps(6 > n > 2) excitation (dotted line), Ps ion- ization (dashed line), total (full line), total cross secti on with the nonsymmetric potential from ref. [6] (dashed-triple-dotted line). 4. Total Ps-H 2cross section at diﬀerent positronium energies: total targ et elastic cross section from ﬁgure 1 (dashed line), total target-elastic (f rom ﬁgure 1) plus target-inelastic (from ﬁgures 2 and 3) cross section (solid line), experiment al data (solid circles from ref. [2], solid square from ref. [4]). 100 20 40 60 80 Energy (eV)024Partial Cross Section (10-16cm2) Ps-IonPs(n=2) Elastic Ps(n=3,..,6)Figure 140 80 120 160 200 Energy (eV)01234Cross Section ( 10-17cm2) Ps(Ion) Ps(n=2) Ps(n=1) Ps(n>2)Figure 2 Total(sym)10 20 30 40 50 Energy (eV)01234Cross Section ( 10-17cm2) Ps(Ion) Ps(n=2)Ps(n=1) Ps(n>2)Figure 3 Total(sym) Total(nonsym)0 20 40 60 Energy (eV)024681012Total Cross Section (10-16cm2)Figure 4

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