Split (1)

train · 15.8k rows

text stringlengths 8 1.32M ⌀
THE RF SOURCE OF THE 60-MEV LINAC FOR THE KEK/JAERI JOINT PROJECT S.Fukuda, S. Anami, C. Kubota, M. Kawamura, S. Yamaguchi, M. Ono, H. Nakanishi, KEK, Tsukuba, 305-0035 Japan S. Miyake, M. Sakamoto, Electron Tube and Device Division , Toshiba, Ohtawara, 324-8550 Japan, Abstract The construction of the 60-MeV proton linac has started as a low-energy front of the KEK/JAERI Joint Project for a high-intensity proton accelerator facility at KEK. The accelerating frequency is 324 MHz. Five UHF klystrons are used as an rf source; their ratings have a maximum power of 3 MW, a beam pulse width of a 700 µsec (an rf pulse width is 650 µsec) and a repetition rate of 50 pps. We have manufactured a proto-type rf source (a power-supply system with a modulating-anode pulse modulator and prototype klystrons). In this paper, the specifications and developments of the rf source, including the WR-2300 waveguide system, are summarized. During the manufacturing process, strong oscillations due to back-going electrons from the collector were observed. This phenomenon was analyzed both experimentally and theoretically . We have tested up to an output power of nearly 3 MW, and succeeded to test the DTL hot-model structure. 1 INTRODUCTION The KEK/JAERI Joint Project, a currently proposed accelerator complex, comprises a 600-MeV proton linac, a 3-GeV rapid-cycling synchrotron and a 50-GeV synchrotron [ 1]. The 600-MeV linac comprises a 200- MeV, 324-MHz low- β section, a 972-MHz section from 200 MeV to 400 MeV, and a 972-MHz super-conducting section from 400 MeV to 600 MeV. The construction of the 60-MeV linac in KEK has been started as a low- energy front of this KEK/JAERI Joint Project. The 60- MeV linac comprises a negative hydrogen ion source, a 3-MeV RFQ linac, a 50-MeV DTL and a SDTL [2]. The accelerating frequency is 324 MHz. High-power and high-duty rf-sources are required for these structures, and a 324-MHz klystron with a modulating-anode has been adopted as the rf-power source. Although the maximum rating of the klystron is 3 MW for 650 µsec rf pulse duration and a 50 pps repetition rate, the actual working level of the klystron is 2 MW. Because this working rf- power must be well controlled both in amplitude and phase, the klystron is required to generate a saturated output power of 2.5 MW at the same duty and be operated at the 2-MW output power level by controlling the driving power. Since a klystron with a modulating-anode is used, the high-voltage power supply at the test station consists of an old JHF dc-cathode voltage supply and a newly developed pulse-modulating anode voltage supply. Newly developed power supplies [3] are now installed in the new building. Since a 324-MHz klystron is the lowest-frequency one in practical use, we manufactured a high-power test model of a co-axial window [4] and a high-power beam test tube composed of an electric gun and a collector [5] to confirm the technical feasibility. Prototype klystrons were manufactured and tested from 1999 and high-power tests of the klystrons, the hot model of a DTL structure and WR-2300 waveguide components were successfully performed. A low-power system of the klystron (the driving system) was also developed. This low-power line is required for a precise feedback system for the amplitude and the phase control to establish operation at the unsaturated output power region by controlling the driving power. Table 1: Specifications of the 324-MHz klystron . Item Unit Max. Working (Sat.) Operating frequency MHz 324 Peak output power MW 3.0 2.5 Beam pulse width µs 700 RF pulse width µs 650 (flat top: 620) Repetition rate pps 50 RF duty % 3.25 Beam current A 50 45 Beam voltage kV 110 102 Mod. anode voltage kV 93 86 Beam micro- perveance 1.37 Efficiency % 55 Gain dB 50 Input / Output port N-type / WR-2300 RF window coaxial ceramic window Mounting position horizontal Focusing electromagnet focusing 2 ANODE MODULATOR FOR THE 324-MHZ KLYSTRON Since the 324-MHz klystron was planed to bemounted horizontally with its oil tank socket, it was necessary to develop a new independent anode modulator. This was installed in a compact oil tank; one pair of filaments in the high-voltage cathode potential and a modulating anode connection are connected to the klystron socket. A DC high voltage was supplied from the old DC power supply to this anode modulator through a high-voltage cable in the test station. A hard- tube switching device (TH-5188) and associated G1, G2 power supplies were assembled as a module; further, it is easy to replace. This modulator produced a pulse with a rise time of 32 µsec and a fall time of 183 µsec. Using the test beam tube mentioned before, this modulator was operated up to the full rating of about 93 kV at a cathode voltage of 110 kV with a pulse width of 750 µsec and a repetition rate of 50 pps. Recently, a high-voltage transistor switch was developed to replace the hard tube, and was successfully tested at KEK. 3 DRIVER AND HIGH-POWER WAVEGUIDE SYSTEM Although the driving system for the 324-MHz klystron in the test station simply comprised a signal generator, a pulse modulator, an attenuator and an amplifier, a feedback control system was recently constructed and tested in the low-power driving system [7]. This was done because the high precision control of the rf amplitude of less than 1% and a phase of less than 1 degree is required in order to mate the injection requirements to the 3-GeV synchrotron so that Δp/p is less than 0.1%. The WR-2300 rectangular waveguide system comprises a directional coupler, a 3-dB hybrid, a phase- shifter of the triple-stub type, a high-power circulator and a co-axial (WX-203D) to a rectangular-waveguide transition. Since the output power from a klystron is fed to two accelerating structures, the high-power WR2300 rectangular waveguide system must have a precise 3dB- hybrid power-divider and a phase shifter in one port after the power divider due to phase and amplitude requirements. A design refinement is now progressing. A high-power waveguide circulator was also developed in order to protect the klystron from any large reflection power at the rising time and the falling time of the pulse. These components have been successfully tested using the high-power klystron. 4 PERFORMANCE OF THE 324-MHZ KLYSTRON 4.1 Klystron Developments The high-power klystron with a modulating anodewas adopted as the rf source while considering the entire project scheme[4]. The specifications are given in Table 1. The operating frequency, 324 MHz, was lower than that of the CERN 1MW tube, which was the lowest- frequency klystron practically used in a large-scale accelerator; therefore, as a first step, a high-power model of a co-axial window and a beam test tube were manufactured and tested in 1998. The former was tested up to 2-MW output power using the 432-MHz JHP test facility. The beam test tube was operated up to 110-kV dc cathode voltage, 720 µsec beam pulse width and 50 pps repetition [5]. The prototype klystrons were then developed and tested from 1999 to 2000 and used for high-power tests of the waveguide components and accelerator structure. This klystron is 4.5m long, from the bottom of the oil tank to the top of the collector, and was mounted horizontally. Figure 1 shows a picture of the test station at KEK. Figure 1: High-power test of the klystron at KEK. During the first test, strong spurious oscillations were observed. These occurred under high-voltage operation in the range of 65-72 kV and higher than 90 kV without any driving input power. The operating frequency was nearly the same as the operating one. These oscillations were concluded to be caused by back-going electrons from the collector, since the some of the oscillation s ceased when a weak deflecting magnetic field was applied in the collector region. Since this klystron ’s operating frequency is low, the drift-tube radius is large and the aspect ratio of diameters of a drift-tube to a collector is small compared with the other frequency- band klystron. From a simulation using the EGS4 code, it became clear that the back-going electrons could be decreased by using a larger diameter collector[6]. Several experiments with different collector shapes were performed to eliminate these oscillations and the associated unstable phenomena with an input drive power. In order to investigate the output power characteristics, the Rieke diagram was measured by changing the reflection from the load using a triple -stub- type phase shifter as the reflection device. This was also useful to optimize the klystron circuit parameters. 4.2 Characteristics of the Klystron So far, three klystrons were built and tested at KEK. After investigating the back-going electrons from the collector, the spurious oscillation was successfully eliminated and the klystron data were measured. The latest test was performed up to the 107-kV cathode voltage. A limitation came from arcing of the switching tetrode in the modulating anode power supply. Figure 2 shows the output-power characteristics with the function of the applied voltage. Figure 3 shows the output characteristic with the function of the input drive power (Figure 3 left) and that with the function of the frequency of the input drive signal (Figure 3 right). Nearly 3-MW output power and the efficiency of 52% were obtained when the anode voltage was applied at a value 10% higher than the nominal operating condition at a cathode voltage of 106.6kV. Lower-gain operation was preferable for more stable output characteristics. For more stable operation and higher efficiency, further design development might be considered.5 SUMMARY A test of the rf source of the 60-MeV linac was successfully performed at KEK. The 324-MHz high- power klystron exceeded the saturated output power of 2.5 MW, the required power of the working point. The power supply and the anode modulator were also operated up to their full ratings. High-power waveguide components and an accelerator structure test are under development. From the fall of 2000, construction of the 60-MeV linac for the KEK/JAERI Joint Project will be conducted in a new building. REFERENCES [1] Y. Yamazaki et al., “Accelerator Complex for the Joint Project of JAERI/NSP and KEK/JHF ”, 12th Symp. on Acc. Sci . and Tech., Wako, Japan, 1999. [2] Y. Yamazaki et al., “The Construction of the Low- Energy Front 60-MeV Linac for the JAERI/KEK Joint Project ”, presented in this Conference. [3] M. Ono et al., “Power Supply System for 324 MHz Klystron of the JHF Proton Linear Accelerator ”, 12th Symp. on Acc. Sci . and Tech., Wako, Japan, 1999. [4] S. Fukuda et al., “Development of a High-Power VHF Klystron for JHF ”, APAC ’98, Tsukuba, Ibaraki , Japan, 1998. [5] M. Kawamura et al., “High-power test of a klystron beam-test-tube and an anode modulator ” (in Japanese) , the 24th Linear Acc. Meeting in Japan, Sapporo, Hokkaido, Japan, 1999. [6] Z. Fang et al., “Simulation of Back-going Electrons from a Collector of Klystron ”, presented in this Conference. [7] S. Yamaguchi et al., “Feedback Control for 324 MHz Klystron ”(in Japanese), the 25th Linear Acc. Meeting in Japan, Himeji, Hyogo, Japan, 2000. Figure 2: Output-power characteristics along with the function of the beam voltage.Figure 3: Output power characteristics with the function of the input power (left) and with the drive frequency (right).00.511.522.53 7580859095100105110Output Power(MW) Beam Voltage(kV)M.Anode Voltage 110% M.Anode Voltage 90%M.Anode Voltage 100% 00.511.522.53 0102030405060 5060708090100110Output Power(MW) Efficiency(%) Beam Voltage(kV)Efficiency Saturated Output Power 11.21.41.61.822.22.42.6 051015202530Output Power(MW) Input Power(W)V=106.6kV 0.511.522.53 322.5 324 325.5Saturated Power Pd=21.2WOutput Power(MW) Frequency(MHz)
HIGH-BRIGHTNESS BEAMS FROM A LIGHT SOURCE INJECTOR: THE ADVANCED PHOTON SOURCE LOW-ENERGY UNDULATOR TEST LINE LINAC* G. Travish, S. Biedron, M. Borland, M. Hahne, K. Harkay, J. W. Lewellen, A. Lumpkin, S. Milton, N. Sereno, Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439 * Work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38Abstract The use of existing linacs, and in particular light source injectors, for free-electron laser (FEL) experiments is becoming more common due to the desire to test FELs at ever shorter wavelengths. The high-brightness, high- current beams required by high-gain FELs impose technical specifications that most existing linacs were not designed to meet. Moreover, the need for specialized diagnostics, especially shot-to-shot data acquisition, demands substantial modification and upgrade of conventional linacs. Improvements have been made to the Advanced Photon Source (APS) injector linac in order to produce and characterize high-brightness beams. Specifically, effort has been directed at generating beams suitable for use in the low-energy undulator test line (LEUTL) FEL in support of fourth-generation light source research. The enhancements to the linac technical and diagnostic capabilities that allowed for self-amplified spontaneous emission (SASE) operation of the FEL at 530 nm are described. Recent results, including details on technical systems improvements and electron beam measurement techniques, will be discussed. The linac is capable of accelerating beams to over 650 MeV. The nominal FEL beam parameters used are as follows: 217 MeV energy; 0.1–0.2% rms energy spread; 4–8 µm normalized rms emittance; 80–120 A peak current from a 0.2–0.7 nC charge at a 2–7 ps FWHM bunch. 1 OVERVIEW The low-energy undulator test line (LEUTL) self- amplified spontaneous emission (SASE) free-electron laser (FEL) project has as its primary goal the identification and study of issues relevant to linac-based fourth-generation x-ray light sources [1], including verifying the behavior of the SASE FEL with varying electron beam parameters. Therefore, good characterization and control of the electron beam is critical to the success of the project. The LEUTL project has taken a conservative approach towards producing a drive beam for the SASE FEL. The project began with the available APS injector and made numerous improvements and additions to the system over the past 2+ years to allow for the production, preservation, and measurement of high-brightness beams. A description of the system, starting at the head of the linac, follows. In collaboration with the Accelerator Test Facility (ATF) at Brookhaven National Laboratory (BNL) a 1.6- cell, S-band photocathode (PC) rf gun and emittance compensation solenoid [2] was installed at the near-optimal drift distance to the existing linac. The photoinjector replaced an existing low peak current, large energy spread DC thermionic gun. The ATF PC gun has been well characterized and proven in several installations around the world, and is intended to produce beams of over 100-A peak current with emittances below 5 µm, as opposed to the DC gun that produced < 75-A peak current and > 100-µm emittances. A drive laser for the photocathode gun was purchased commercially, installed into a laser room constructed adjacent to the linac tunnel, and integrated into the APS control system [3]. An optical transport line delivers the laser either into the APS linac enclosure or into an rf test area adjacent to the laser room. The APS injector linac is constructed from standard SLAC-type traveling-wave disk-loaded linac sections. One linac section immediately following the gun is powered by a single klystron; the remaining twelve linac sections are grouped into three sectors, each of which is powered by a single klystron equipped with SLED cavities. Two thermionic-cathode rf guns provide redundant injection capability for the APS storage ring [4], and one of the guns has performance sufficient to allow limited FEL studies. Improvements made to the linac for LEUTL (as well as APS operations) include rearranging the focusing lattice and realigning all linac sections and magnets. Upgrades made to the rf system improved the power stability from over 7% (pk-pk over several seconds) to better than 2%, and phase stability from several degrees to about 2 degrees over short time periods (seconds). Further improvements to the focusing lattice and the rf system are ongoing. The APS linac and LEUTL transport lines contain four energy spectrometers, a three-screen emittance measurement region, and several metal foils for providing light to optical transition-radiation (OTR) and coherent transition-radiation (CTR) diagnostics. The undulator hall proper contains additional diagnostics stations for both the electron beam and the photon beam generated by the FEL interaction. The linac diagnostics are described in detail in section 4, but first we discuss the project goals and design parameters. 2 PROJECT GOALS The LEUTL FEL operates in high-gain SASE mode, generating light from 530 nm down to less than 100 nm. A principal portion of the FEL studies is being devoted to experimental verification of scaling laws derived from SASE FEL theory, such as the variation of output power with beam parameter changes. Theoreticalcomparison requires both high-resolution beam characterization and good beam stability over long and short time scales. Significant technical effort and study time are dedicated to improving and understanding the linac in order to achieve the above goals. The technical development program also focuses on additional aspects of accelerator technology anticipated to be useful for future light source construction. As an example, a magnetic bunch compression system [5] has been installed into the linac with parameters similar to the BC-1 compressor in the LCLS design [6]. The LEUTL compressor will allow for the study of the onset of coherent synchrotron radiation and other effects as the compression is varied. The bunch compressor will soon be used along with the LEUTL SASE FEL to verify the theoretical tradeoffs between high peak currents and emittance. 3 DESIGN PARAMETERS The necessary electron beam parameters are straightforward to calculate from the undulator specifications (a planar undulator with a period of 3.3 cm, a normalized field of 3, and an effective average betafunction of 1.5 m at 217 MeV) and the desired FEL characteristics (maximum gain at 530 nm). Following the conservative approach described in the previous section, the initial demands on the electron beam were kept to within state of the art (see Table 1). Table 1: Design beam parameters for the LEUTL FEL Parameter Value Energy 217 MeV Energy Spread (rms) 0.1% Charge 1 nC Bunch Length (FWHM) 5–7 ps Peak Current 150 A Emittance (normalized rms) 5 µm Since achieving the performance listed in Table 1 is nontrivial, a number of diagnostics were installed to properly characterize the beam and provide guidance for improvements. 4 DIAGNOSTICS Prior to the LEUTL project, the APS linac was only required to provide a low-brightness, modest current beam for injection [7]. As such, the linac diagnostics consisted of modest resolution devices used for tuning up the beam. After significant upgrade efforts, the APS linac now incorporates many diagnostics for determining various beam parameters throughout the system. The diagnostics used for characterizing the LEUTL beam are described below in roughly the order they appear on the beamline: An Integrating Current Transformer (ICT), located just after the PC gun, measures beam charge; resolution is better than 10 pC, and accuracy is estimated to be 10% at nominal charges (1 nC). YAG screens with charge-injection device (CID) cameras are located after the PC gun and at a few points along the accelerator. The screens are used to determinebeam size, with an approximate resolution and accuracy of 50 µm (at present charge densities). Beam position monitors (BPMs) provide beam trajectory information along the linac and transport line with a resolution of 50 µm (25 µm with improved electronics, better than 10 µm with the BPMs used in the transport lines after the linac). The BPM data are used in a trajectory control law (that employs a previously measured response matrix of steering magnets and BPMs) to actively correct the beam's average trajectory. The control law is especially useful in maintaining a trajectory that best minimizes and compensates for transverse wakes. Bending magnets located in linac sections L3 (midway) and L5 (at the end of the linac) are used for determining beam energy and energy spread at ~135–160 MeV and ~217–650 MeV, with resolutions of 0.033%/pixel and 0.05%/pixel, respectively. Additional bend spectrometers are installed midway in the long transport line between the linac and undulators, as well as after the undulators. Bunch length measurements can be made in one of three ways. One method uses an insertable mirror at the end of the linac to generate OTR that is transported to a streak camera with a resolution of about 1 ps. Bunch length is also measured using the last accelerating section (L5) and the so-called zero phasing technique [8]. Finally, an optical high resolution spectrometer applied to the FEL output light has been used to indirectly obtain the bunch length [9]. Emittance measurements are primarily made using a three-screen arrangement, with drift spaces between the screens. The three-screen system is accurate, helps with betafunction matching, and does not suffer from space charge effects or quadrupole calibration errors. A software utility automates the entire process. Presently the system uses YAG screens located several meters after the linac, but OTR mirrors may be employed in the near future to avoid saturation effects and improve resolution. While it is difficult to estimate the accuracy of the system, a resolution of better than 1 µm in emittance is possible when the beam is properly matched. The above diagnostics were the primary tools used to characterize the photoinjector-generated beam. In addition, fluorescent screens and CID cameras are placed along the linac/LEUTL line. These are used in conjunction with a digitizing image analysis system to measure the beam's position and transverse profile. Finally, diagnostics on the drive laser measure pulse energy (joulemeter with a resolution of better than 1 µJ out of 500 µJ), spot size (CCD camera with a resolution and accuracy of 100 µm with improvements planned), and pulse length (single- shot autocorrelator with a resolution of better than 100 fs and a total system accuracy of perhaps 20%) on a shot-to- shot basis. 5 MEASURED PERFORMANCE Using the above described diagnostics, the beam is well characterized each run shift, often several times per shift. The measurements are made concurrently when possible (i.e., charge and emittance), and in rapid succession whennecessary (i.e., energy spread then emittance). Moreover, the measurements tabulated here represent repeatable data taken under the same linac conditions used to run the FEL. While the capability of saving and restoring machine settings has long existed at the APS [10], the performance levels demanded of the linac by LEUTL required substantial improvements in readback systems, power supplies, magnet degaussing routines, and rf system controls. Perhaps the most significant improvement made to the system over the past two years is this ability to restore and reproduce beam characteristics with minimal manual adjustments. Referring to Table 2, the energy spread measurements are perhaps the least certain due to optical resolution limits in our present spectrometers. The bunch length measurements are typically made using the zero-phasing technique mentioned above (and agree with the less frequently used streak measurements). Finally, the emittance measurements have the largest variation due to trajectory jitter and wakefield effects, as we discuss in the next section. Table 2: Measured parameters for the LEUTL beam when tuned for FEL operation Parameter Value Diagnostic Energy 217 MeV Bend magnets Energy Spread (FWHM) < 0.15% Bend magnets Charge 0.5 nC ICT Bunch Length (FWHM) 3–4 ps Zero phase Peak Current ~140 A Calculated Emittance (norm. rms) < 7 µm 3 screen 6 ISSUES Having followed a conservative and proven approach with the photoinjector, it was originally thought that the moderately high-brightness beams required for LEUTL would be straightforward to produce. However, due to the inherent sensitivity of the photoinjector to the solenoid field, laser spot size, and input rf, it became clear that simply duplicating past efforts was insufficient. Indeed, a number of constraints specific to LEUTL implied that a suitable operating regime would have to be discovered. Finding a good operating point for the photoinjector has been complicated by the need to traverse a number of accelerator structures before reaching a complete set of diagnostics. Below are some of the major issues that limited beam quality during the operating periods presented in this paper. Cathode nonuniformity: While scanning the laser across the cathode surface, it was observed that some structure, as projected by the beam onto a YAG screen, remained stationary. These observations imply that the structure arose from some nonuniformity of the cathode, not the laser. Visual inspection using a 70 degree (off normal) cathode port supports the above observations: the cathode center appears pitted and concentric machining rings are evident. Wakefields: Proper compensation of the transverse wakefield is critical to preserving the beam emittance through the long linac. Compensation involves findingthe correct trajectory through the linac. Typically, wakefield compensation is used; however, the jitter and drifts in the various systems means that the transverse wakefields are only partially compensated and for only a portion of the shots. In addition to the above challenges, it is felt that the first linac section (right after the gun) has an undetermined defect that makes it impossible to find a trajectory that does not suffer from severe phase steering. Longitudinal wakefields are compensated by simple rf phasing, and have proven to be much less problematic than the transverse wakes. Jitter: Transverse trajectory jitter (pointing error) greater than the beam diameter is often observed at the end of the linac. Most of the jitter appears to come from the injector. A significant portion of the jitter may be caused by the pointing jitter of the drive laser (which at the time of the measurements reported here was not relay imaged onto the cathode). The remainder of the jitter is caused by rf amplitude and phase jitter in combination with the possible linac section problem described above. Appropriate rf power and phase levels can be varied to find a “sweet spot” where the effects of jitter are reduced; however, the appropriate parameter set changes with beam charge, launch phase, emittance, etc. 7 NEXT STEPS Driven by the dictates of the FEL and by work of interest to future fourth-generation light-sources, the LEUTL photoinjector will be pushed to produce lower-emittance beams with higher peak currents. The magnetic bunch compressor—currently undergoing commissioning—will further increase the peak current. By reducing the jitter and drift of the various support systems (rf, power supplies, laser, etc.), a more optimal operating regime should be found. An improved resolution energy spectrometer is also being designed. Longer term improvements in the rf and laser subsystems are also being considered. While the initial lasing goal of the LEUTL FEL has been met [11], the primary goal is to understand the FEL SASE process. As such, characterizing the input electron beam will continue to be paramount. REFERENCES [1]S.V. Milton, Proc. EPAC2000, to be published. [2]M. Babzien et al., Phy. Rev. E 57, 6093 (1998). [3]G. Travish et al., Proc. FEL 1999, to be published. [4]J.W. Lewellen, Proc. PAC 1999, 1979, (1999). [5]M. Borland et al., “A Highly Flexible Bunch Compressor for the APS LEUTL FEL,” these proceedings. [6]LCLS Design Study Report, SLAC-R-521, 1998. [7]M. White et al., Proc. PAC 1995, 1073 (1996). [8]D.X. Wang, Proc. LINAC 1996, 303, (1997). [9]V. Sajaev et al., Proc EPAC2000, to be published. [10] R. Soliday et al., “Automated Operation of the APS Linac Using the Procedure Execution Manager,” these proceedings. [11] S.V. Milton et al., Phy. Rev. Lett. 85, 988 (2000).
arXiv:physics/0008042v1 [physics.acc-ph] 12 Aug 2000THEFLAT BEAMEXPERIMENTAT THEFNAL PHOTOINJECTOR D. Edwards,H. Edwards, N.Holtkamp,S. Nagaitsev,J.Santuc ci, FNAL∗ R. Brinkmann,K.Desler, K. Fl¨ ottmann,DESY-Hamburg I. Bohnet,DESY-Zeuthen, M.Ferrario, INFN-Frascati Abstract Atechniqueforproductionofanelectronbeamwithahigh transverseemittanceratio,a“ﬂat”beam,hasbeenproposed by Brinkmann, Derbenev, and Fl¨ ottmann.[1] The cathode of an RF-laser gun is immersed in a solenoidal magnetic ﬁeld; as a result the beam emitted from a round laser spot hasa netangularmomentum. Subsequentpassagethrough a matched quadrupole channel that has a 90 degree differ- ence in phase advance between the transverse degrees of freedom results in a ﬂat beam. Experimental study is un- derway at the Fermilab Photoinjector. Thus far, transverse emittance ratios as high as 50 have been observed,and the resultsareinsubstantialagreementwithsimulation. 1 INTRODUCTION Two years ago, Ya. Derbenev invented an optics maneu- verfortransforminga beamwith a highratioof horizontal to vertical emittance—a “ﬂat beam” —to one with equal emittancesinthetransversedegrees-of-freedom—a“round beam”.[2] High energyelectron coolingat the TeV energy scale wasthemotivation. Last year, R. Brinkmann and K. Fl¨ ottmann of DESY joinedwithDerbenevinapaperthatreversestheprocess— obtain a ﬂat beam from a round beam produced from the cathode of an electron gun.[1] This could be a signiﬁcant steptowardtheeliminationorsimpliﬁcationoftheelectro n damping ring in a linear collider design. The other major step in that process is the delivery of polarized electrons in the ﬂat beam, and this is an R&D challenge beyond the scopeoftheworkreportedhere. Theintentofthepresentexperimentwastodemonstrate the round-to-ﬂat transformation, compare the results with simulation, and verify that the demonstration was not ob- scured by other processes. In the following sections, we presentasimpliﬁedversionofthetransformation,describ e the experimental setup, present the results, and comment onfutureplans. 2 PRINCIPLE Supposethatthecathodeofanelectrongunisimmersedin auniformsolenoidalﬁeldofmagnitude Bz. Forthesakeof this argument, assume that the thermal emittance is negli- gibleandignoreRFfocusinginthegun. Thentheparticles juststreamalongtheﬁeldlinesuntiltheendofthesolenoid ∗The Fermi National Accelerator Laboratory is operated unde r con- tract with the USDepartment of Energyisreached,atwhichpointthebeamacquiresanangularmo- mentum. A particle with initial transverse coordinates x0, y0acquiresangulardeﬂections. With momentum p0at the solenoidend,thestate oftheparticlebecomes  x x′ y y′  0= x0 −ky0 y0 kx0  where k≡1 2Bz (p0/e). Next pass the beam through an alternating gradient quadrupole channel. Assume that the channel is repre- sented by an identity matrix in the x-direction and has an additional 900phaseadvancein y. We gettheoutputstate  x x′ y y′ = 1 0 0 0 0 1 0 0 0 0 0 β 0 0 −1 β0  x0 −ky0 y0 kx0  = x0 −ky0 kβx0 −1 βy0 β=1/k→ x0 −ky0 x0 −ky0 . Inthelaststepabove,with β= 1/k,theparticlesendup withequaldisplacementsin xandyandtravellingatequal angles in xandy. This describes a ﬂat beam inclined at an angle of 450to the coordinate axes. Change to a skew- quadrupolechannel,andtheﬂatbeamcanbealignedalong eitherthehorizontalorverticalaxis. This idealized example is only meant to illustrate the principle. The essential points about the quadrupolechan- nel are the π/2difference in phase advance between the transverse degrees-of-freedom, and the match of the Courant-Snyderparameters.[3] This may be accomplished with as few as three quadrupoles. Of course, in practice, RF focusingﬁeldsinthegunandinaboostercavity,space charge,andso oncannotbeignored. With the inclusion of thermal emittance, Brinkmann, Derbenev, and Fl¨ ottmann[1] speak of an achievable emit- tance ratio of order 102or more for a beam with normal- ized emittance√ǫx·ǫy≈1µm per nC of bunch charge. Theexpressionforthe emittanceratiois ǫx ǫy≈4k2σ2 c σ′2cRF Gun w Solenoidscathode Skew Quad Triplet TransformerFlat Beam Round BeamOTR Screens & Slits SC Tesla Cavity Figure1: Veryschematicrenditionofthelayoutat Fermilab relatedto thisexperiment. where now in the deﬁnition of k,Bzremains the ﬁeld on the cathode, but p0is the momentum at entry to the quadrupolechannel,and σc,σ′ care the standarddeviations of the distribution in displacement and angle at the cath- ode. The resulting vertical emittance would be 0.1 µm, in the range of interest for a linear collider. Liouville’s Theoremremainsineffectforthe4-dimensionaltransverse emittance, but the angular momentum provides the lever by which emittance may be moved from one degree-of- freedomtoanother. 3 THE FERMILAB PHOTOINJECTOR ENVIRONMENT The photoinjector at Fermilab is well suited to this sort of experiment. The RF gun delivers electrons with a ki- netic energy of (typically) 3.8 MeV. The superconducting boostercavityraisesthe electronenergyto17MeV. The solenoid is composed of three separately excited coils permitting ﬁelds at the cathode in the range 0 to 2.7 kG. The coil immediatelyupstream of the cathode, the “bucker”, is normally excited with current opposite to that of the next coil, the “primary” to produce zero ﬁeld at the cathode. Downstream, the combination yields solenoidal focusing, which can be adjusted with the third coil, the “secondary”. The secondary has little effect on the ﬁeld onthecathode. Followingtheboostercavity,about8metersofbeamline are available for experiments. There are 11 quadrupoles that are easily movedaboutor rotated into the skew orien- tation. A dozen view screens are situated on the line, and there are three locations where slits are installed for emit - tance measurement. The laser can operate at a variety of pulse lengths up to 12 ps, the setting that we used. Bunch charge as high as 10 nC is available. We operated at no higherthan 1 nCin orderto reducespace chargeeffectsas much as possible. The layout as related to this experiment is sketchedinFig. 1. 4 PROCEDURE Thesolenoidcoilsweresettoproduceaﬁeldatthecathode intheexpectedrange,about0.75kG.Usingthelanguageof the precedingsection,thismeantsettingthe buckertozero current and controlling the cathode ﬁeld with the primary. The beam was observed at the location of the two screensimmediately downstream of the booster cavity, and by ad- justmentofthesecondarycoil,thebeamspotwasmadethe samesizeatthesetwoplaces. Inotherwords,abeamwaist was produced. At this stage, the beam has a round shape onthescreens. The simple argumentofSec. 2 is nolongervalidforde- termination of the βfor the match, because the solenoid ﬁeld is not uniform and the RF focusing and acceleration must be taken into account. Making use of linearity, axial symmetry, and the conservation of canonical angular mo- mentum between the cathode and the waist yields for the valueof βat entrytothequadrupolechannel β=σ2 w σ2c2(pw/e) Bc wherethesubscripts candwrefertothecathodeandwaist respectively and the σ’s characterize the radii of the beam spots. TheotherCourant-Snyderparameterinvolvedinthe match, α, is zero due to the choice of a waist as the match point. Given preliminary values for the matching parameters, an (asymmetric) skew triplet was set up. Flat beam pro- ﬁleswererathereasilyachievedbyadjustmentofavailable tuningparameters,includingthelaunchphasefromtheRF gun. The latter proved to be particularly important, a cir- cumstancethatisyetto beexplained. 5 RESULTS Thetransformationshouldwork— it’slineardynamics— anditdoes. Thematchandphasedifferencewereachieved with three skew quadrupoles. The beamimage on an OTR screen 1.2m downstreamofthe third quadrupoleisshown in Fig. 2; the beam width is an order of magnitude larger than the height. A critical observation is that the beam re- main ﬂat as it drifts farther downstream. That it does is demonstratedinFig.3neartheendofthebeamlineat3.6m fromthethirdquadrupole. In Fig. 2 there is a hint of an s-shape, which likely in- dicates that spherical aberrations (e.g. space charge) are at work. If the solenoid ﬁeld on the cathode is varied up or down from the matched condition the beam apparently rotates clockwise or counterclockwise as it drifts, indica t- ing that the angular momentum is no longer completely cancelled. Of course, it isn’t a real rotation — there’s no torque—it’s ashear.Figure 2: Beam proﬁle on OTR screen 1.2 m downstream ofthe thirdskew quadrupole. Figure 3: Beam proﬁle on OTR screen 3.6 m downstream of thethirdskew quadrupole. Darkcurrentis visibleto the rightofthemainbeamimage. In these ﬁgures, the beam is ﬂat in the horizontal plane. The OTR screens are viewed from the side, and so a beam that is ﬂat horizontally presents a depth of ﬁeld problem for best emittance analysis. So in later stages of the ex- periment, the beam was made ﬂat in the vertical plane. From slit data in this orientation, the measured ratio of emittances is about 50: ǫx≈0.9µm,ǫy≈45µm, with the one degree-of-freedom normalized emittance deﬁned byǫ2=γ2(v/c)2(/angbracketleftx2/angbracketright/angbracketleftx′2/angbracketright − /angbracketleftxx′/angbracketright2). We feel that this is a good result for an initial experiment. The horizontal emittance measurement is resolution limited, as illustrat ed inFig.4whereinasequenceofslitimagesissuperimposed in order to form a distribution. The standard deviation of the narrow distribution is comparable to a single pixel of the CCD cameraviewingthescreen. The product of the emittances is higher than that usual in operation with round beams; typically, the emittance in each transverse degree-of-freedom is about 3 to 4 µm. However, there is no reason to believe that the emittance compensationnormallyinusewouldbeeffectiveunderthe conditionsofthisexperiment. The simulations[4],[5] carried out prior to the measure- ments provideduseful guidance,but were not perfect. The prediction of spot size just downstream of the gun worked ﬁne. But to achieve the match to the quadrupoles, the solenoidrequiredadjustment. Inorderto obtainagreementbetweenthe locationofthe beam waist downstream of the booster cavity, a modiﬁca-020406080100120 0102030405060 140 150 160 170 180 190 200 210 220L6, L8slit X 063000-1840, ~50mic/picl6-dc,rot-5 l8 Slit X projl6-dc,rot-5, Xprojl8 Slit X proj picsigma= 7.7pic sigma=1.2 pic Figure 4: Projectionof imagesusedin emittancemeasure- mentat slit locationanddownstreamofslit system. tion of the focusing characteristics of this device was re- quired. In the Chambers approximation[6], its demagni- ﬁcation is a factor of 5, so its treatment is sensitive to a number of factors, e.g. the exact ﬁeld proﬁle. It will be worthwhiletomeasurethetransfermatrixthroughthecav- ity experimentally. 6 CONCLUSIONS The round-to-ﬂat transformation has been veriﬁed, with a demonstratedemittanceratio of a factor of 50 between the two transverse degrees-of-freedom. Further work will be neededto restorethe emittance compensationnecessary to thedeliveryoflowtransverseemittance,andthatisthesub - ject of a follow-on experiment, in the direction suggested byBrinkmann,DebenevandFl¨ ottmannintheirEPAC2000 paper.[7]Thepredictivecapabilityofthesimulationsise n- couraging thus far, and the results reported here indicate directionsforimprovement. 7 ACKNOWLEGEMENTS Supportof the Fermilaband DESY managementsis grate- fully acknowledged. Thanks to Jean-Paul Carneiro, Mark Champion, Michael Fitch, Joel Fuerst and Walter Hartung fortheirinvaluablehelpintheoperation. 8 REFERENCES [1] R. Brinkmann, Ya. Derbenev, K. Fl¨ ottmann, “A Flat Beam Electron Source for Linear Colliders”, TESLA Note 99-09, April1999. [2] Ya. Derbenev, “Adapting Optics for High Energy Electron Cooling”,UniversityofMichigan,UM-HE-98-04,Feb.1998. [3] A. Burov and S. Nagaitsev, “Courant-Snyder Parameters o f Beam Adapters”, FermilabTM-2114, June 2000. [4] K.Fl¨ ottmann, ASTRAuser manual, www.desy.de/ ∼mpyﬂo/Astra dokumentation. [5] S.Nagaitsev, private communication. [6] E.Chambers, StanfordHEPLnote, 1965. [7] R.Brinkmann, Y.Derbenev, K.Fl¨ ottmann,EPAC2000, Vienna, June 2000.
arXiv:physics/0008043v1 [physics.acc-ph] 13 Aug 2000MULTISCALE ANALYSISOF RMS ENVELOPEDYNAMICS A. Fedorova,M.Zeitlin,IPME,RAS, St. Petersburg, V.O. Bol shojpr.,61,199178,Russia∗† Abstract We present applications of variational – wavelet approach to different forms of nonlinear (rational) rms envelope equations. We have the representation for beam bunch os- cillations as a multiresolution (multiscales) expansion i n the baseofcompactlysupportedwaveletbases. 1 INTRODUCTION In this paper we consider the applications of a new nume- rical-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysisto the nonlinear root-mean-square (rms) envelope dynamics [1]. Suchapproachmaybe usefulin all modelsinwhichit is possible and reasonable to reduce all complicated prob- lems related with statistical distributions to the problem s described by systems of nonlinear ordinary/partial differ - ential equations. In this paper we consider an approach based on the second momentsof the distributionfunctions forthe calculationofevolutionofrmsenvelopeof a beam. The rms envelopeequations are the most useful for analy- sis ofthe beamself–forces(space–charge)effectsand also allow to consider both transverse and longitudinal dynam- ics of space-charge-dominatedrelativistic high–brightn ess axisymmetric/asymmetric beams, which under short laser pulse–drivenradio-frequencyphotoinjectorshavefasttr an- sition from nonrelativistic to relativistic regime [1]. An al- ysis of halo growth in beams, appeared as result of bunch oscillations in the particle-core model, also are based on three-dimensional envelope equations [2]. From the for- malpointofviewwemayconsiderrmsenvelopeequations after straightforward transformations to standard Cauchy form as a system of nonlineardifferential equationswhich are not more than rational (in dynamical variables). Be- cause of rational type of nonlinearities we need to con- sidersomeextensionofourresultsfrom[3]-[10],whichare based on applicationof wavelet analysis techniqueto vari- ational formulation of initial nonlinear problems. Wavele t analysis is a relatively novel set of mathematical methods, which gives us a possibility to work with well-localized bases in functional spaces and give for the general type of operators(differential,integral,pseudodifferential) in such bases the maximum sparse forms. Our approach in this paper is based on the generalization [11] of variational- wavelet approach from [3]-[10], which allows us to con- sider not only polynomial but rational type of nonlineari- ties. ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmlOurrepresentationforsolutionhasthefollowingform z(t) =zslow N(t) +/summationdisplay j≥Nzj(ωjt), ω j∼2j(1) which correspondsto the full multiresolutionexpansionin all time scales. Formula(1)givesusexpansionintoa slow partzslow Nand fast oscillating parts for arbitraryN. So, we maymovefromcoarsescalesofresolutiontotheﬁnestone forobtainingmoredetailedinformationaboutourdynami- calprocess. TheﬁrsttermintheRHSofequation(1)corre- spondsonthegloballeveloffunctionspacedecomposition to resolution space and the second one to detail space. In thiswaywegivecontributiontoourfullsolutionfromeach scale of resolution or each time scale. The same is correct forthecontributiontopowerspectraldensity(energyspec - trum): we can take into account contributions from each level/scaleofresolution. Inpart2wedescribethediffere nt forms of rms equations. In part 3 we present explicit ana- lyticalconstructionforsolutionsofrmsequationsfrompa rt 2, which are based on our variational formulation of ini- tial dynamical problems and on multiresolution represen- tation [11]. We giveexplicitrepresentationforall dynami - cal variables in the base of compactly supported wavelets. Our solutions are parametrized by solutions of a number ofreducedalgebraicalproblemsfromwhichoneisnonlin- ear with the same degree of nonlinearity and the rest are the linearproblemswhichcorrespondtoparticularmethod of calculationof scalar productsof functionsfromwavelet basesandtheirderivatives. 2 RMSEQUATIONS BelowweconsideranumberofdifferentformsofRMSen- velope equations, which are from the formal point of view not more than nonlinear differential equations with ratio- nal nonlinearities and variable coefﬁcients. Let f(x1, x2) be the distribution function which gives full information about noninteracting ensemble of beam particles regard- ingtotracespaceortransversephasecoordinates (x1, x2). Then we may extract the ﬁrst nontrivial bit of ‘dynamical information’fromthesecondmoments σ2 x1=< x2 1>=/integraldisplay /integraldisplay x2 1f(x1, x2)dx1dx2 σ2 x2=< x2 2>=/integraldisplay /integraldisplay x2 2f(x1, x2)dx1dx2(2) σ2 x1x2=< x1x2>=/integraldisplay /integraldisplay x1x2f(x1, x2)dx1dx2 RMS emittance ellipse is given by ε2 x,rms =< x2 1>< x2 2>−< x1x2>2. Expressions for twiss parameters arealso basedonthesecondmoments.We will consider the following particular cases of rms envelope equations, which described evolution of the mo- ments (1) ([1],[2] for full designation): for asymmetric beams we have the system of two envelope equations of the secondorderfor σx1andσx2: σ′′ x1+σ′ x1γ′ γ+ Ω2 x1/parenleftbiggγ′ γ/parenrightbigg2 σx1= (3) I/(I0(σx1+σx2)γ3) +ε2 nx1/σ3 x1γ2, σ′′ x2+σ′ x2γ′ γ+ Ω2 x2/parenleftbiggγ′ γ/parenrightbigg2 σx2= I/(I0(σx1+σx2)γ3) +ε2 nx2/σ3 x2γ2 The envelopeequation for an axisymmetric beam is a par- ticularcase ofprecedingequations. Also we have related Lawson’s equation for evolution of the rms envelope in the paraxial limit, which governs evolutionofcylindricalsymmetricenvelopeunderexterna l linearfocusingchannelofstrenghts Kr: σ′′+σ′/parenleftbiggγ′ β2γ/parenrightbigg +Krσ=ks σβ3γ3+ε2 n σ3β2γ2,(4) where Kr≡ −Fr/rβ2γmc2, β≡νb/c=/radicalbig 1−γ−2 According [2] we have the following form for envelope equations in the model of halo formation by bunch oscil- lations: ¨X+k2 x(s)X−3K 8ξx Y Z−ε2 x X3= 0, ¨Y+k2 y(s)Y−3K 8ξy XZ−ε2 y Y3= 0,(5) ¨Z+k2 z(s)Z−γ23K 8ξz XY−ε2 z Z3= 0, where X(s), Y(s), Z(s) are bunch envelopes, ξx, ξy,ξz= F(X, Y, Z ). AftertransformationstoCauchyformwecanseethatall this equations from the formal point of view are not more than ordinary differential equations with rational nonlin - earities and variable coefﬁcients (also,b we may consider regimesinwhich γ,γ′arenotﬁxedfunctions/constantsbut satisfy some additional differential constraint/equatio ns, but thiscase doesnotchangeourgeneralapproach). 3 RATIONALDYNAMICS Our problems may be formulated as the systems of ordi- narydifferentialequations Qi(x)dxi dt=Pi(x, t), x= (x1, ..., x n),(6) i= 1, ..., n, max ideg P i=p,max ideg Q i=q with ﬁxed initial conditions xi(0), where Pi, Qiare not more than polynomialfunctionsof dynamicalvariables xjandhavearbitrarydependenceoftime. Becauseoftimedi- lation we can consider only next time interval: 0≤t≤1. Let usconsidera set offunctions Φi(t) =xid dt(Qiyi) +Piyi (7) anda set offunctionals Fi(x) =/integraldisplay1 0Φi(t)dt−Qixiyi\|1 0, (8) where yi(t) (yi(0) = 0) are dual (variational)variables. It isobviousthattheinitial systemandthesystem Fi(x) = 0 (9) are equivalent. Of course, we consider such Qi(x)which donotleadtothesingularproblemwith Qi(x),when t= 0 ort= 1,i.e.Qi(x(0)), Qi(x(1))/ne}ationslash=∞. Nowwe considerformalexpansionsfor xi, yi: xi(t) =xi(0) +/summationdisplay kλk iϕk(t)yj(t) =/summationdisplay rηr jϕr(t),(10) where ϕk(t)are useful basis functions of some functional space ( L2, Lp, Sobolev, etc) corresponding to concrete problem and because of initial conditions we need only ϕk(0) = 0,r= 1, ..., N, i = 1, ..., n, λ={λi}={λr i}= (λ1 i, λ2 i, ..., λN i),(11) where the lower index i corresponds to expansion of dy- namical variable with index i, i.e. xiand the upper index rcorrespondsto the numbersof terms in the expansion of dynamicalvariablesin the formalseries. Thenwe put(10) into the functional equations (9) and as result we have the following reduced algebraical system of equations on the set ofunknowncoefﬁcients λk iofexpansions(10): L(Qij, λ, α I) =M(Pij, λ, β J), (12) where operators L and M are algebraization of RHS and LHS of initial problem (6), where λ(11) are unknownsof reducedsystemofalgebraicalequations(RSAE)(12). Qijare coefﬁcients (with possible time dependence) of LHS of initial system of differential equations (6) and as consequencearecoefﬁcientsofRSAE. Pijare coefﬁcients (with possible time dependence) of RHS of initial system of differential equations (6) and as consequence are coefﬁcients of RSAE. I= (i1, ..., i q+2), J= (j1, ..., j p+1)are multiindexes, by which are labelled αIandβI— othercoefﬁcientsofRSAE (12): βJ={βj1...jp+1}=/integraldisplay/productdisplay 1≤jk≤p+1ϕjk,(13) wherepisthe degreeofpolinomialoperatorP (6) αI={αi1...αiq+2}=/summationdisplay i1,...,i q+2/integraldisplay ϕi1...˙ϕis...ϕiq+2, (14)where q is the degree of polynomial operator Q (6), iℓ= (1, ..., q+ 2),˙ϕis= dϕis/dt. Now,whenwesolveRSAE(12)anddetermineunknown coefﬁcients from formal expansion (10) we therefore ob- tain the solution of our initial problem. It should be noted if we consideronly truncatedexpansion(10)with N terms then we have from (12) the system of N×nalgebraical equations with degree ℓ=max{p, q}and the degree of this algebraicalsystem coincideswith degreeofinitial di f- ferential system. So, we have the solution of the initial nonlinear(rational)problemintheform xi(t) =xi(0) +N/summationdisplay k=1λk iXk(t), (15) where coefﬁcients λk iare roots of the corresponding re- duced algebraical(polynomial)problemRSAE (12). Con- sequently, we have a parametrization of solution of initial problem by solution of reduced algebraical problem (12). The ﬁrst main problem is a problem of computations of coefﬁcients αI(14),βJ(13) of reduced algebraical sys- tem. These problems may be explicitly solved in wavelet approach. The obtained solutions are given in the form (15), where Xk(t)are basis functions and λi kare roots of reduced system of equations. In our case Xk(t)are obtained via multiresolution expansions and represented by compactly supported wavelets and λi kare the roots of corresponding general polynomial system (12). Our con- structions are based on multiresolutionapproach. Because afﬁne group of translation and dilations is inside the ap- proach, this method resembles the action of a microscope. We havecontributionto ﬁnal resultfromeachscale of res- olution from the whole inﬁnite scale of spaces. More ex- actly, the closed subspace Vj(j∈Z)corresponds to level j of resolution,or to scale j. We considera multiresolution analysis of L2(Rn)(of course, we may consider any dif- ferent functional space) which is a sequence of increasing closed subspaces Vj:...V−2⊂V−1⊂V0⊂V1⊂V2⊂... satisfyingthefollowingproperties: /intersectiondisplay j∈ZVj= 0,/uniondisplay j∈ZVj=L2(Rn), So, onFig.1wepresentcontributionsto bunchoscillations fromﬁrst5scalesorlevelsofresolution. Itshouldbenoted that such representations (1), (15) for solutions of equa- tions (3)-(5) give the best possible localization properti es in correspondingphase space. Thisis especiallyimportant because our dynamical variables corresponds to moments ofensembleofbeamparticles. In contrast with different approaches formulae (1), (15) do not use perturbation technique or linearization proce- duresandrepresentbunchoscillationsviageneralizednon - linearlocalizedeigenmodesexpansion. We wouldliketothankProf. J.B.RosenzweigandMrs. MelindaLaraneta(UCLA) andProf. M. Regler(IHEP,Vi- enna) for nice hospitality, help and support during UCLA ICFA WorkshopandEPAC00.0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−101 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.200.2 Figure 1: Contributions to bunch oscillations: from scale 21to25. 4 REFERENCES [1] J.B.Rosenzweig, Fundamentals of Beam Physics,e-ver- sion: http://www.physics.ucla.edu/class/99F/250Rosen zwe- ig/notes/ L. Seraﬁni and J.B. Rosenzweig, Phys. Rev. E 55, 7565, 1997. [2] C. Allen, T. Wangler, papers in UCLA ICFA Proc., Nov., 1999, WorldSci.,2000. [3] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [4] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [5] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [8] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262 [9] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [10] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [11] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095. 6papers inProc. EPAC00,Vienna, 2000.
arXiv:physics/0008044 13 Aug 2000 -1.5-1.0-0.50.00.51.01.5 -270 -180 -90 0 90δT0 δT2 Fig.1Approximated sawtooth energy gain function δT2 and the original function δT0. Fig. 2 Cross sections of x- and y-vanes. Horizontal and vertical axes denote z and r/r 0, respectively. Lc/r0=1.5, z2=0, A=.05, A20=.25. Broken line shows the ordinary vanes.DFQ--Double Frequency RFQ Y. Iwashita NSRF, ICR, Kyoto University, Gokanosho, Uji, Kyoto 611-0011 JAPAN Abstract RFQ with a harmonic higher order quadrupole mode is studied. Assuming that we superpose a higher order mode with twice the frequency of the fundamental mode, a sawtooth waveform is approximated. In such a case, the bunching function is enhanced while the transverse stability is modified. Second order longitudinal harmonic component is also required to enable the effect. 1 INTRODUCTION Some bunchers use sawtooth wave forms for better bunching factor. Application of such sawtooth wave form to an RFQ may shorten a length of RFQ. Because of the orthogonality of the trigonometric functions, we also have to introduce a longitudinal harmonics. Energy gain in a period is described in the next section. Section 3 shows a potential function generating such electric field and an example of its vane shape. RF defocusing factor is calculated in section 4. The transverse stability is discussed in section 5. Rough resonator examples can be seen in section 6. 2 BUNCHING ENHANCEMENT Let us assume that the electric field Ez on a beam axis in an RFQ cell with length of Lc is expressed as: Ezt,z( )=E0sinωt+φ( )+Hisiniωt+φ+φi( )( ) ( ) sinkz( )+jAj0sinj kz+zj( )( ) ( ),(1) where ω is angular frequency of RF and k=π/Lc. The second terms of time and spatial harmonics are newly introduced in addition to the two term potential function [1,2]. The energy gain δT for an ion with relationship ωt=kz is obtained by integrating Eq.(1) from z=0 to 2Lc : δT=q Ezkz/ω,z( ) dz 02Lc∫, (2) where q is the charge of the ion. Because of the orthogonality, we need a condition i=j for the harmonics effect. In order to enhance the bunching function in an RFQ, a condition i=j=2 will be taken. It should be noted that the range of the integration is twice the range of the conventional RFQ case. Thus, the energy gain in the period is: δT2=qE0cosφ+2A20H2cos 2φ+φ2−πz2 ( )( ) ( ) .(3) In order to approximate the sawtooth function, φ2-πz2 and A20H2 have to be - π/4 and 1/4, respectively. Figure 1 shows the resulted energy gain as a function of φ.3 POTENTIAL FUNCTION The potential function that generates such longitudinal harmonics is expressed by: U r,ψ,z( )=V 2{r r0  2 cos2ψ+ A Iokr( )coskz( )+A20Io2kr( )cos 2kz+z2( ) ( ) }(4) whereA=m2−1 m2I0ka( )+I0mka( ) and a is the minimum radius at z=0[1]. The electric fields are: Er=−V r02rcos2ψ−kAV 2{I1kr( )coskz −2A20I12kr( )cos 2k z+z2( )( ) }, Eψ=V r02rsin2ψ, Ez=kAV 2I0kr( )sinkz+2A20I02kr( )sin 2k z+z2( )( ) ( ) .(5) Cross sections of vanes can be obtained by contours: U x,0,z( )=V 2 for x vane and U y,π 2,z  =V 2 for y vane. An example of the vane shapes with Lc/r0=1.5, A=0.05, A20=0.25 and zn=0 are shown in Fig. 2. Because of theACOSD1X71 .DAT -0.3 -0.2 -0.1 0. -0.3 -0.2 -0.1 0.ACOS.DATcos 2 Limit Limit=5max Lc 43 =5max Lc4 3-0.3 -0.2 -0.1 0.-0.3 -0.2 -0.1 0.3.3.54.4.55.5.56. =5max Lc ACOSD1 .DAT ACOSD2 .DATcos 2 + sin 4 432cos 2 + (sin 4 )/2 LimitLimit=5max Lc 43 3.3.54.4.55.5.56. 3.3.54.4.55.5.56. 3.3.54.4.55.5.56.0.71 (cos 2 + sin 4 ) BB B B Fig. 6Stability diagram for A20=1,1/,1/3 and 0.second order term, x and y vanes are not symmetric. It can be seen that y-vane is separated at z=0 and z=3 Lc, which should not be harmful in this particular case, because of the small distance. The condition A20H2=1/4 implies H2=1. Figure 3a shows the time dependent term of Eq.(1). Figure 3b shows the corresponding longitudinal distribution. It should be noted that the peak value is higher than the single sine wave, which will be discussed later. Larger A20 term in short Lc region makes the contour lines complex, which means that the A20 cannot be large at such region. As shown in Fig.2, the vane tip position sa at (y,z)=(0,0), is modified from the original value a by the A20 term. A20 is plotted as functions of the coefficients at m=1.1, Lc/r0=1.1,1,2, and 1.5 in Fig. 4, where m is the commonly used modulation parameter. -2-1012 -270 -180 -90 0 90Cos Cos + Sina) -2-1012 0 0.5 1 1.5 2Two term potential Three term potential Z/LckAVx2 Ezb) Fig. 3a) Time dependent term in the example, b) the longitudinal distribution. 0.8 0.85 0.9 0.95 1s00.511.52 A20 Lc/r0=1.2Lc/r0=1.5Lc/r0=2 Fig. 4A20 as functions of s. 4 RF DEFOCUSING The averaged RF defocusing force Dx is obtained by integrating Ex with time dependent factor over a period. Ex and Ey are given by: Ex Ey  =cosψ−sinψ sinψcosψ  Er Eψ  . (6) Thus, Dx is given by:Dx=q Exsinkz+φ( )+H2sin 2kz+φ+φ2( )( ) ( ) dz 02Lc∫ 2Lc =−qk2AVx 4sinφ−4A20H2sin 2kz2−2φ−2φ2 ( ) ( ) ,(7) where the Ex function are approximated up to first order of x around the beam axis. Dy is given by the same form as Dx. Figure 5 shows Dx as a function of φ. The RF defocusing term is zero at the synchronous phase of 90° (bunching operation) because of the flat region in δT2. -2.0-1.00.01.02.0 -270 -180 -90 0 90Conventional buncher Sawtooth buncherqk2AVx 4x Fig. 5RF defocusing terms for the approximated sawtooth function and the conventional sine function. 5 TRANSVERSE STABILITY The stability of the transverse motion is obtained by Hill's equation: d2x dη2+Bcos2πη+A20sin4πη ( ) +Δ ( ) x=0, whereη=z 2βλ,B=qλ2V m0c2r02.(8) Figure 6 shows the stability regions for cases of A20=1, 1/2, 0 and A20=1 with B scaled by 71%. The last one corresponds to preserving the total RF power withf=433.1MHzf=865.4MHz Beam AxisCapacitive load Vane86.4mm r0=3mm Fig. 7An example of harmonic cavity for a four-vane DFQ. Beam axis is bottom left. L1L2 Cgap C2i1 i2 Fig. 9The equivalent circuit for the DFQ 0.20.250.30.350.40.450.50.5501234 i1 i2Fundamental Mode Second Mode Fig. 10Current ratios i1/i2 as functions of u.assumption of the equal shunt impedance for the second order mode. Because of the extra force term, the focusing strength increases with A20 for constant amplitude of the fundamental mode. 6 CAVITY EXAMPLE Figure 7 shows an example of the harmonic cavity for a four-vane DFQ. The right lower half of the figure shows the fundamental mode and the left upper one shows the second order mode. This particular example exhibits rather small shunt impedance (less than half) compared with that of a conventional single-mode-RFQ[3]. Figure 8 shows another example for four-rod-DFQ that has the frequencies of 143 and 290MHz. These are shown only for the possibility of the harmonic resonators, where the geometries are not fully optimized. The equivalent circuit for such resonators is shown in Fig. 9. Using following notations: ω1=1 L1Cgap,ω2=1 L2C2,u=L2 L1andw=ω2 ω1,(9) and condition that the ratio of two resonant frequencies is two, u and w should hold following equation: w=5+9−16u( ) 41+u( )( ). (10) The current ratios i1/i2 for both the modes is shown inFig.10. u should be chosen with power supply specifications. This knowledge would be helpful for the design of a real cavity. One more scheme is to apply fundamental RF to horizontal electrodes and to apply second order mode to another ones. This scheme is under investigation. 7 DISCUSSION Lc/r0 at an entrance is large for a low frequency RFQ (as used for heavy ions), which allows larger A 20 value. The sparking issue in the superposed RF wave form is not clear, but seems easier for lower frequencies [4,5,6]. The complex vane shapes may be approximated by trapezoids, because only the longitudinal higher order mode having the corresponding RF mode can affect the longitudinal motion. The transverse motion will not be changed much as long as the quadrupole component is preserved. This can be extended to an IH-DTL (inter- digital H type) with electro-focusing fingers, where the gap centers shift alternatively. Because of the wide stability region in the synchrotron oscillation, the synchronous phase may be at 0° or more, which makes the accelerator section short. The RF power consumption is just a sum of that for each mode, in spite of the high peak field. Because two additional parameters are added (phase and amplitude for the second RF), focusing characteristics may be adjusted independently. If z-dependence in the quadrupole field is added, more focusing force may be available with penalty of multipole effects. The determination of the cell parameters is complex compared with the conventional RFQ. The multiple RF feed technique is to be established[7]. Fig. 84-rod-DFQREFERENCES [1]R. H. Stokes, K. R. Crandall, J. E. Stovall and D. A. Swenson, "RF QUADRUPOLE BEAM DYNAMICS", IEEE Trans. Nucl. Sci. NS-26, No.3, 1979, pp.3469-3471 [2]N. Tokuda, "RFQ Linac", Lecture note for High Energy Accelerator Seminar OHO'96, KEK, (1996) in Japanese. [3]Y. Iwashita, et at, "7MeV PROTON LINAC", Proc. of 15th Linac Conf., 1990, Alburquerque, pp.746-748 [4]W.D.Kilpatrick, "Criterion for Vacuum Sparking Designed to Include Both rf and dc", Rev. Sci. Instr. 28, p824 (1957). [5]K. W. Shepard, et al., "A LOW-FREQUENCY RFQ FOR A LOW-CHARGE-STATE INJECTOR FOR ATLAS", Proc. of 18th Linac Conf., 1996 Geneva Switzerland, pp.68-70 [6]A. Morita and Y. Iwashita, “Kilpatrick’s Sparking Limit for General RF Wave Forms”, Beam Science and Technology 5, NSRF, ICR, Kyoto University, pp. 25-27 [7]Y. Iwashita, "Multi-Harmonic Impulse Cavity", Proc. 1999 Particle Accelerator Conference, NY, p.3645, and Y. Iwashita, "Superposition of Multiple Higher Order Modes in A Cavity", Proc. EPAC 2000, THP5A02, in print.
arXiv:physics/0008045v1 [physics.acc-ph] 13 Aug 2000MULTIRESOLUTIONREPRESENTATIONFORORBITALDYNAMICS IN MULTIPOLARFIELDS A. Fedorova,M.Zeitlin,IPME,RAS, V.O. Bolshojpr.,61,199 178,St. Petersburg, Russia∗† Abstract We presentthe applicationsofvariation– wavelet analysis topolynomial/rationalapproximationsfororbitalmotion in transverseplane fora single particlein a circularmagneti c latticeincasewhenwetakeintoaccountmultipolarexpan- sion up to an arbitrary ﬁnite number and additional kick terms. We reduce initial dynamical problem to the ﬁnite number (equal to the number of n-poles) of standard alge- braicalproblems. Wehavethesolutionasamultiresolution (multiscales)expansioninthebaseofcompactlysupported wavelet basis. 1 INTRODUCTION Inthispaperweconsidertheapplicationsofanewnumeri- cal-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysistoth e orbital motion in transverse plane for a single particle in a circularmagneticlatticeincasewhenwetakeintoaccount multipolar expansion up to an arbitrary ﬁnite number and additional kick terms. We reduce initial dynamical prob- lemtotheﬁnitenumber(equaltothenumberofn-poles)of standard algebraical problems and represent all dynamical variablesas expansionin the bases of maximally localized in phasespace functions(wavelet bases). Wavelet analysis is a relatively novel set of mathematical methods, which gives us a possibility to work with well-localized bases in functional spaces and gives for the general type of opera- tors(differential,integral,pseudodifferential)insuc hbases the maximum sparse forms. Our approach in this paper is basedonthegeneralizationofvariational-waveletapproa ch from [1]-[8],whichallows usto considernot onlypolyno- mial but rational type of nonlinearities [9]. The solution hasthefollowingform z(t) =zslow N(t) +/summationdisplay j≥Nzj(ωjt), ω j∼2j(1) which correspondsto the full multiresolutionexpansionin all time scales. Formula(1)givesusexpansionintoa slow partzslow Nand fast oscillating parts for arbitraryN. So, we maymovefromcoarsescalesofresolutiontotheﬁnestone forobtainingmoredetailedinformationaboutourdynami- calprocess. TheﬁrsttermintheRHSofequation(1)corre- spondsonthegloballeveloffunctionspacedecomposition to resolution space and the second one to detail space. In thiswaywegivecontributiontoourfullsolutionfromeach scale of resolution or each time scale. The same is correct ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmlforthecontributiontopowerspectraldensity(energyspec - trum): we can take into account contributions from each level/scale of resolution. Starting in part 2 from Hamilto- nian of orbital motion in magnetic lattice with additional kicks terms, we consider in part 3 variational formulation for dynamicalsystem with rational nonlinearitiesand con- struct via multiresolution analysis explicit representat ion for all dynamical variables in the base of compactly sup- portedwavelets. 2 PARTICLE INTHE MULTIPOLAR FIELD Themagneticvectorpotentialofamagnetwith 2npolesin Cartesian coordinatesis A=/summationdisplay nKnfn(x, y), (2) where fnisahomogeneousfunctionof xandyoforder n. Therealandimaginarypartsofbinomialexpansionof fn(x, y) = (x+iy)n(3) correspond to regular and skew multipoles. The cases n= 2ton= 5correspond to low-order multipoles: quadrupole, sextupole, octupole, decapole. The corre- spondingHamiltonian([10]fordesignation): H(x, px, y, p y, s) =p2 x+p2 y 2+ /parenleftbigg1 ρ(s)2−k1(s)/parenrightbigg ·x2 2+k1(s)y2 2(4) −Re /summationdisplay n≥2kn(s) +ijn(s) (n+ 1)!·(x+iy)(n+1)  Then we may take into accountarbitrarybut ﬁnite number oftermsinexpansionofRHS ofHamiltonian(4)andfrom ourpointofviewthecorrespondingHamiltonianequations of motions are not more than nonlinear ordinary differen- tial equations with polynomial nonlinearities and variabl e coefﬁcients. Also we may add the terms corresponding to kicktypecontributionsofrf-cavity: Aτ=−L 2πk·V0·cos/parenleftbig k2π Lτ/parenrightbig ·δ(s−s0)(5) or localized cavity V(s) =V0·δp(s−s0)withδp(s− s0) =/summationtextn=+∞ n=−∞δ(s−(s0+n·L))atposition s0. Fig.1and Fig.2presentﬁnitekicktermmodelandthecorresponding multiresolutionrepresentationoneachlevelofresolutio n.00.10.20.30.40.50.60.70.80.9105001000150020002500 Figure1: Finitekickmodel. 00.10.20.30.40.50.60.70.80.91−10−9−8−7−6−5−4−3−2−1 Figure2: Multiresolutionrepresentationofkick. 3 RATIONALDYNAMICS Theﬁrst mainpartofourconsiderationissomevariational approachto thisproblem,which reducesinitial problemto the problem of solution of functional equations at the ﬁrst stage and some algebraical problems at the second stage. We havethesolutioninacompactlysupportedwaveletba- sis. Multiresolution expansion is the second main part of our construction. The solution is parameterized by solu- tions of two reduced algebraical problems, one is nonlin- ear and the second are some linear problems, which are obtained from one of the next wavelet constructions: the method of Connection Coefﬁcients (CC), Stationary Sub- divisionSchemes(SSS). 3.1 VariationalMethod Our problems may be formulated as the systems of ordi- narydifferentialequations Qi(x)dxi dt=Pi(x, t), x= (x1, ..., x n),(6) i= 1, ..., n, max ideg P i=p,max ideg Q i=q with ﬁxed initial conditions xi(0), where Pi, Qiare not more than polynomialfunctionsof dynamicalvariables xj andhavearbitrarydependenceoftime. Becauseoftimedi- lation we can consider only next time interval: 0≤t≤1.Let usconsidera set offunctions Φi(t) =xid dt(Qiyi) +Piyi (7) anda set offunctionals Fi(x) =/integraldisplay1 0Φi(t)dt−Qixiyi\|1 0, (8) where yi(t) (yi(0) = 0) are dual (variational)variables. It isobviousthattheinitial systemandthesystem Fi(x) = 0 (9) are equivalent. Of course, we consider such Qi(x)which donotleadtothesingularproblemwith Qi(x),when t= 0 ort= 1,i.e.Qi(x(0)), Qi(x(1))/ne}ationslash=∞. Nowwe considerformalexpansionsfor xi, yi: xi(t) =xi(0) +/summationdisplay kλk iϕk(t)yj(t) =/summationdisplay rηr jϕr(t),(10) where ϕk(t)are useful basis functions of some functional space ( L2, Lp, Sobolev, etc) corresponding to concrete problem and because of initial conditions we need only ϕk(0) = 0,r= 1, ..., N, i = 1, ..., n, λ={λi}={λr i}= (λ1 i, λ2 i, ..., λN i),(11) where the lower index i corresponds to expansion of dy- namical variable with index i, i.e. xiand the upper index rcorrespondsto the numbersof terms in the expansion of dynamicalvariablesin the formalseries. Thenwe put(10) into the functional equations (9) and as result we have the following reduced algebraical system of equations on the set ofunknowncoefﬁcients λk iofexpansions(10): L(Qij, λ, α I) =M(Pij, λ, β J), (12) where operators L and M are algebraization of RHS and LHS of initial problem (6), where λ(11) are unknownsof reducedsystemofalgebraicalequations(RSAE)(12). Qijare coefﬁcients (with possible time dependence) of LHS of initial system of differential equations (6) and as consequencearecoefﬁcientsofRSAE. Pijare coefﬁcients (with possible time dependence) of RHS of initial system of differential equations (6) and as consequencearecoefﬁcientsofRSAE. I= (i1, ..., i q+2), J= (j1, ..., j p+1)are multiindexes, by which are labelled αIandβI— other coefﬁcients of RSAE (12): βJ={βj1...jp+1}=/integraldisplay/productdisplay 1≤jk≤p+1ϕjk,(13) wherepisthe degreeofpolinomialoperatorP (6) αI={αi1...αiq+2}=/summationdisplay i1,...,i q+2/integraldisplay ϕi1...˙ϕis...ϕiq+2, (14)where q is the degree of polynomial operator Q (6), iℓ= (1, ..., q+ 2),˙ϕis= dϕis/dt. Now,whenwesolveRSAE(12)anddetermineunknown coefﬁcients from formal expansion (10) we therefore ob- tain the solution of our initial problem. It should be noted if we consideronly truncatedexpansion(10)with N terms then we have from (12) the system of N×nalgebraical equations with degree ℓ=max{p, q}and the degree of this algebraicalsystem coincideswith degreeofinitial di f- ferential system. So, we have the solution of the initial nonlinear(rational)problemintheform xi(t) =xi(0) +N/summationdisplay k=1λk iXk(t), (15) where coefﬁcients λk iare roots of the corresponding re- duced algebraical(polynomial)problemRSAE (12). Con- sequently, we have a parametrization of solution of initial problem by solution of reduced algebraical problem (12). The ﬁrst main problem is a problem of computations of coefﬁcients αI(14),βJ(13) of reduced algebraical sys- tem. These problems may be explicitly solved in wavelet approach. Next we consider the constructionof explicit time solu- tion for our problem. The obtained solutions are given in the form (15), where Xk(t)are basis functionsand λi kare rootsofreducedsystemofequations. Inourcase Xk(t)are obtainedviamultiresolutionexpansionsandrepresentedb y compactlysupportedwaveletsand λi karetherootsofcorre- spondinggeneralpolynomialsystem(12)withcoefﬁcients, whicharegivenbyCCorSSS constructions. Accordingto the variationalmethodto givethe reductionfrom differen- tialtoalgebraicalsystemofequationsweneedcomputethe objects αIandβJ[1],[9]. Our constructions are based on multiresolution appro- ach. Because afﬁne group of translation and dilations is inside the approach, this method resembles the action of a microscope. We havecontributiontoﬁnalresultfromeach scale of resolution from the whole inﬁnite scale of spaces. More exactly, the closed subspace Vj(j∈Z)corresponds to level j of resolution, or to scale j. We consider a mul- tiresolution analysis of L2(Rn)(of course, we may con- sideranydifferentfunctionalspace)whichisasequenceof increasingclosedsubspaces Vj: ...V−2⊂V−1⊂V0⊂V1⊂V2⊂...(16) satisfyingthefollowingproperties: /intersectiondisplay j∈ZVj= 0,/uniondisplay j∈ZVj=L2(Rn), On Fig.3 we present contributions to solution of initial problemfromﬁrst 5scalesorlevelsofresolution. We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help and supportduringUCLA ICFAWorkshop.0 50 100 150 200 250 300−505x 10−3 0 50 100 150 200 250 300−0.0100.01 0 50 100 150 200 250 300−202x 10−3 0 50 100 150 200 250 300−505x 10−3 0 50 100 150 200 250 300−202x 10−3 0 50 100 150 200 250 30000.0050.01 Figure3: Contributionstoapproximation: fromscale 21to 25. 4 REFERENCES [1] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [2] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [3] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [4] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [5] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262 [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [8] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [9] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095 [10] Bazzarini, A.,e.a.,CERN94-02.
arXiv:physics/0008046v1 [physics.acc-ph] 13 Aug 2000NONLINEAR BEAMDYNAMICS AND EFFECTSOF WIGGLERS A. Fedorova,M. Zeitlin,IPME, RAS, V.O.Bolshojpr.,61,199 178,St. Petersburg, Russia∗ † Abstract We present the applications of variational–wavelet ap- proachfortheanalytical/numericaltreatmentoftheeffec ts of insertiondevicesonbeamdynamics. We investigatethe dynamical models which have polynomial nonlinearities and variable coefﬁcients. We construct the corresponding waveletrepresentationforwigglersandundulatormagnets . 1 INTRODUCTION In this paper we consider the applications of a new nume- rical-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysisto the treatment of effects of insertion devices on beam dy- namics. Ourapproachinthispaperisbasedonthegeneral- izationofvariational-waveletapproachfrom[1]-[8],whi ch allowsustoconsidernotonlypolynomialbutrationaltype of nonlinearities[9]. We presentsolutionvia fullmultire s- olutionexpansionin all time scales, which givesusexpan- sion into a slow part and fast oscillating parts. So, we may move from coarse scales of resolution to the ﬁnest one for obtaining more detailed information about our dynamical process. In this way we give contribution to our full solu- tion from each scale of resolution or each time scale. The same is correct for the contribution to power spectral den- sity (energy spectrum): we can take into account contri- butions from each level/scale of resolution. Starting from formulationofinitial dynamicalproblems(part2)we con- struct in part 3 via multiresolution analysis explicit repr e- sentation for all dynamical variables in the base of com- pactly supportedwavelets. Then in part 4 we considerfur- therextensionofourpreviousresultstothecaseofvariabl e coefﬁcients. 2 EFFECTS OFINSERTION DEVICES ONBEAM DYNAMICS Assumingasinusoidalﬁeldvariation,wemayconsiderac- cording to [10] the analytical treatment of the effects of insertion devices on beam dynamics. One of the major detrimental aspects of the installation of insertion devic es istheresultingreductionofdynamicaperture. Introducti on of non-linearities leads to enhancement of the amplitude- dependent tune shifts and distortion of phase space. The nonlinearﬁeldswillproducesigniﬁcanteffectsatlargebe - tatronamplitudessuchasexcitationofn–orderresonances . The components of the insertion device vector potential ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmlused for the derivation of equations of motion are as fol- lows: Ax= cosh(kxx)cosh(kyy)sin(ks)/(kρ)(1) Ay=kxsinh(kxx)sinh(kyy)sin(ks)/(kykρ) withk2 x+k2 y=k2= (2π/λ)2,whereλistheperiodlength of the insertion device, ρis the radius of the curvature in the ﬁeldB0. After a canonical transformation to betatron variables, the Hamiltonian is averaged over the period of theinsertiondeviceandhyperbolicfunctionsareexpanded to thefourthorderin xandy(orarbitraryorder). Thenwe havethefollowingHamiltonian: H=1 2[p2 x+p2 y] +1 4k2ρ2[k2 xx2+k2 yy2] +1 12k2ρ2[k4 xx4+k4 yy4+ 3k2 xk2x2y2](2) −sin(ks) 2kρ[px(k2 xx2+k2 yy2)−2k2 xpyxy] We have in this case also nonlinear (polynomial with de- gree 3) dynamical system with variable (periodic) coefﬁ- cients. After averaging the motion over a magnetic period we havethefollowingrelatedequations ¨x=−k2 x 2k2ρ2/bracketleftBig x+2 3k2 xx3/bracketrightBig −k2 xxy2 2ρ2(3) ¨y=−k2 y 2k2ρ2/bracketleftBig y+2 3k2 yy3/bracketrightBig −k2 xx2y 2ρ2 3 WAVELET FRAMEWORK The ﬁrst main part of our consideration is some varia- tionalapproachtothisproblem,whichreducesinitialprob - lem to the problem of solution of functional equations at the ﬁrst stage and some algebraical problems at the sec- ond stage. Multiresolution expansion is the second main part of our construction. Because afﬁne group of trans- lation and dilations is inside the approach, this method resembles the action of a microscope. We have contri- bution to ﬁnal result from each scale of resolution from the whole inﬁnite scale of increasing closed subspaces Vj: ...V−2⊂V−1⊂V0⊂V1⊂V2⊂.... The solution is parameterized by solutions of two reduced algebraical problems, one is nonlinear and the second are some linear problems, which are obtained by the method of Connec- tion Coefﬁcients (CC)[11]. We use compactly supported wavelet basis. Let ourwaveletexpansionbe f(x) =/summationdisplay ℓ∈Zcℓϕℓ(x) +∞/summationdisplay j=0/summationdisplay k∈Zcjkψjk(x)(4)Ifcjk= 0forj≥J, thenf(x)has an alternative ex- pansion in terms of dilated scaling functions only f(x) =/summationtext ℓ∈ZcJℓϕJℓ(x). This is a ﬁnite wavelet expansion, it can be written solely in terms of translated scaling functions. To solve our second associated linear problem we need to evaluate derivatives of f(x)in terms of ϕ(x). Let be ϕn ℓ= dnϕℓ(x)/dxn. We consider computation of the wavelet - Galerkin integrals. Let fd(x)be d-derivative of functionf(x), then we have fd(x) =/summationtext ℓclϕd ℓ(x), and valuesϕd ℓ(x)canbeexpandedintermsof ϕ(x) ϕd ℓ(x) =/summationdisplay mλmϕm(x), (5) λm=∞/integraldisplay −∞ϕd ℓ(x)ϕm(x)dx, whereλmare wavelet-Galerkin integrals. The coefﬁcients λmare 2-termconnectioncoefﬁcients. In generalwe need to ﬁnd (di≥0) Λd1d2...dn ℓ1ℓ2...ℓn=∞/integraldisplay −∞/productdisplay ϕdi ℓi(x)dx (6) ForRiccaticaseweneedtoevaluatetwoandthreeconnec- tioncoefﬁcients Λd1d2 ℓ=/integraldisplay∞ −∞ϕd1(x)ϕd2 ℓ(x)dx, (7) Λd1d2d3=∞/integraldisplay −∞ϕd1(x)ϕd2 ℓ(x)ϕd3 m(x)dx According to CC method [11] we use the next construc- tion. When Nin scaling equation is a ﬁnite even positive integerthefunction ϕ(x)hascompactsupportcontainedin [0,N−1]. Foraﬁxedtriple (d1,d2,d3)onlysome Λd1d2d3 ℓm are nonzero: 2−N≤ℓ≤N−2,2−N≤m≤ N−2,\|ℓ−m\| ≤N−2. ThereareM= 3N2−9N+7 suchpairs (ℓ,m). LetΛd1d2d3beanM-vector,whosecom- ponents are numbers Λd1d2d3 ℓm. Then we have the ﬁrst re- duced algebraical system : Λsatisfy the system of equa- tions(d=d1+d2+d3) AΛd1d2d3= 21−dΛd1d2d3, (8) Aℓ,m;q,r=/summationdisplay papaq−2ℓ+par−2m+p Bymomentequationswehavecreatedasystemof M+d+ 1equationsin Munknowns. Ithasrank Mandwecanob- tain unique solution by combination of LU decomposition andQR algorithm. Thesecondreducedalgebraicalsystem givesusthe2-termconnectioncoefﬁcients( d=d1+d2): AΛd1d2= 21−dΛd1d2, A ℓ,q=/summationdisplay papaq−2ℓ+p(9) For nonquadraticcase we have analogouslyadditional lin- earproblemsforobjects(6). Solvingtheselinearproblemswe obtainthe coefﬁcientsof reducednonlinearalgebraical system and after that we obtain the coefﬁcients of wavelet expansion(4). Asaresultweobtainedtheexplicittimeso- lution of our problem in the base of compactly supported wavelets. OnFig.1wepresentanexampleofbasiswavelet function which satisﬁes some boundary conditions. In the following we consider extension of this approach to the case ofarbitraryvariablecoefﬁcients. 4 VARIABLECOEFFICIENTS In the case when we have the situation when ourproblems (2),(3)aredescribedbyasystemofnonlinear(rational)di f- ferential equations, we need to consider also the extension of our previous approach which can take into account any type of variablecoefﬁcients(periodic,regularorsingula r). We can produce such approach if we add in our construc- tion additional reﬁnement equation, which encoded all in- formation about variable coefﬁcients [12]. According to our variational approach we need to compute only addi- tionalintegralsoftheform /integraldisplay Dbij(t)(ϕ1)d1(2mt−k1)(ϕ2)d2(2mt−k2)dx,(10) wherebij(t)are arbitrary functions of time and trial func- tionsϕ1,ϕ2satisfy thereﬁnementequations: ϕi(t) =/summationdisplay k∈Zaikϕi(2t−k) (11) If we consider all computations in the class of compactly supportedwaveletsthenonlyaﬁnitenumberofcoefﬁcients do not vanish. To approximate the non-constant coefﬁ- cients, we need choose a different reﬁnable function ϕ3 alongwithsomelocalapproximationscheme (Bℓf)(x) :=/summationdisplay α∈ZFℓ,k(f)ϕ3(2ℓt−k),(12) whereFℓ,kare suitable functionals supported in a small neighborhood of 2−ℓkand then replace bijin (10) by Bℓbij(t). In particular case one can take a characteristic functionandcanthusapproximatenon-smoothcoefﬁcients locally. To guarantee sufﬁcient accuracy of the resulting approximationto (10)it is importantto havethe ﬂexibility of choosing ϕ3different from ϕ1,ϕ2. In the case when D issome domain,we canwrite bij(t)\|D=/summationdisplay 0≤k≤2ℓbij(t)χD(2ℓt−k),(13) whereχDis characteristic function of D. So, if we take ϕ4=χD, which is again a reﬁnable function, then the problem of computation of (10) is reduced to the problem ofcalculationofintegral H(k1,k2,k3,k4) =H(k) =/integraldisplay Rsϕ4(2jt−k1)· ϕ3(2ℓt−k2)ϕd1 1(2rt−k3)ϕd2 2(2st−k4)dx(14)The key point is that these integrals also satisfy some sort ofreﬁnementequation[12]: 2−\|µ\|H(k) =/summationdisplay ℓ∈Zb2k−ℓH(ℓ), µ =d1+d2.(15) Thisequationcanbeinterpretedastheproblemofcomput- ing an eigenvector. Thus, we reduced the problem of ex- tensionofourmethodtothecaseofvariablecoefﬁcientsto the same standard algebraical problem as in the preceding sections. So, the general scheme is the same one and we have onlyonemoreadditionallinearalgebraicproblemby which we can parameterizethe solutionsof corresponding probleminthesame way. OnFig.2wepresentapproximatedconﬁgurationandon Fig.3thecorrespondingmultiresolutionrepresentationa c- cordingtoformula(4). 0 0.05 0.1 0.15 0.2 0.25−0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25 Figure1: Basiswavelet withﬁxedboundaryconditions 00.10.20.30.40.50.60.70.80.91−100−50050100150 Figure2: Approximatedconﬁguration We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help and supportduringUCLA ICFAWorkshop. 5 REFERENCES [1] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [2] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol-00.10.20.30.40.50.60.70.80.91−10−9−8−7−6−5−4−3−2−1 Figure3: Multiresolutionrepresentation ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [3] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [4] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [5] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tionsofNonlinearAcceleratorProblems. CP468,48(Amer- ican Instituteof Physics,1999). Los Alamos preprint, physics/990262 [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [8] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [9] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint physics/0003095 [10] A. Ropert,CERN98-04. [11] A.Latto,H.L.Resnikoff and E.Tenenbaum, Aware Techni - cal Report AD910708, 1991. [12] W. Dahmen, C. Micchelli, SIAM J. Numer. Anal. ,30, 507 (1993).
arXiv:physics/0008047v1 [physics.acc-ph] 13 Aug 2000SPIN-ORBITAL MOTION:SYMMETRYAND DYNAMICS A. Fedorova,M.Zeitlin,IPME,RAS, St. Petersburg, V.O. Bol shojpr.,61,199178,Russia∗† Abstract We present the applications of variational–wavelet ap- proach to nonlinear (rational) model for spin-orbital mo- tion: orbital dynamics and Thomas-BMT equations for classical spin vector. We represent the solution of this dy- namical system in framework of periodical wavelets via variationalapproachandmultiresolution. 1 INTRODUCTION Inthispaperweconsidertheapplicationsofanewnumeri- cal-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysistoth e spin orbital motion. Wavelet analysis is a relatively novel set of mathematical methods, which gives us a possibil- ity to work with well-localized bases in functional spaces andgiveforthegeneraltypeofoperators(differential,in te- gral,pseudodifferential)insuchbasesthemaximumsparse forms. Our approachin this paper is based on the general- izationofvariational-waveletapproachfrom[1]-[8],whi ch allowsustoconsidernotonlypolynomialbutrationaltype ofnonlinearities[9]. Thesolutionhasthefollowingform z(t) =zslow N(t) +/summationdisplay j≥Nzj(ωjt), ω j∼2j(1) which correspondsto the full multiresolutionexpansionin all time scales. Formula(1)givesusexpansionintoa slow partzslow Nand fast oscillating parts for arbitraryN. So, we maymovefromcoarsescalesofresolutiontotheﬁnestone forobtainingmoredetailedinformationaboutourdynami- calprocess. TheﬁrsttermintheRHSofequation(1)corre- spondsonthegloballeveloffunctionspacedecomposition to resolution space and the second one to detail space. In thiswaywegivecontributiontoourfullsolutionfromeach scale of resolution or each time scale. The same is correct forthecontributiontopowerspectraldensity(energyspec - trum): we can take into account contributions from each level/scaleofresolution. Inpart2weconsiderspin-orbitalmotion. Inpart3start- ingfromvariationalformulationweconstructviamultires - olution analysis explicit representation for all dynamica l variables in the base of compactly supported periodized wavelets. In part 4 we consider results of numerical cal- culations. 2 SPIN-ORBITALMOTION Let us consider the system of equations for orbital motion and Thomas-BMT equation for classical spin vector [10]: ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmldq/dt=∂Horb/∂p, dp/dt=−∂Horb/∂q,ds/dt= w×s,where Horb=c/radicalbig π2+m0c2+eΦ, w=−e m0cγ(1 +γG)/vectorB (2) +e m3 0c3γG(/vector π·/vectorB)/vector π (1 +γ) +e m2 0c2γG+γG+ 1 (1 +γ)[π×E], q= (q1,q2,q3),p= (p1,p2,p3)are canonical position and momentum, s= (s1,s2,s3)is the classical spin vec- tor of length ¯h/2,π= (π1,π2,π3)is kinetic momen- tum vector. We may introduce in 9-dimensional phase spacez= (q,p,s)the Poisson brackets {f(z),g(z)}= fqgp−fpgq+ [fs×gs]·sandtheHamiltonianequations aredz/dt={z,H}withHamiltonian H=Horb(q,p,t) +w(q,p,t)·s. (3) Moreexplicitlywe have dq dt=∂Horb ∂p+∂(w·s) ∂p dp dt=−∂Horb ∂q−∂(w·s) ∂q(4) ds dt= [w×s] We will consider this dynamical system in [11] via invari- ant approach, based on consideration of Lie-Poison struc- tures on semidirect products. But from the point of view which we used in [9] we may consider the similar approx- imations and then we also arrive to some type of polyno- mial/rationaldynamics. 3 VARIATIONALWAVELET APPROACH FORPERIODICTRAJECTORIES Westartwithextensionofourapproachtothecaseofperi- odic trajectories. The equations of motion corresponding to our problems may be formulated as a particular case of the general system of ordinary differential equations dxi/dt=fi(xj,t),(i,j= 1,...,n),0≤t≤1, where fiare not more than rational functions of dynamical vari- ablesxjand have arbitrary dependence of time but with periodicboundaryconditions. Accordingtoourvariationa l approachwe havethesolutioninthe followingform xi(t) =xi(0) +/summationdisplay kλk iϕk(t), x i(0) =xi(1),(5)whereλk iare the roots of reduced algebraical systems of equations with the same degree of nonlinearity and ϕk(t) corresponds to useful type of wavelet bases (frames). It shouldbenotedthatcoefﬁcientsofreducedalgebraicalsys - tem are the solutionsof additional linear problemand also dependonparticulartypeofwavelet constructionandtype ofbases. Our constructions are based on multiresolution appro- ach. Because afﬁne group of translation and dilations is inside the approach, this method resembles the action of a microscope. We have contribution to ﬁnal result from each scale of resolution from the whole inﬁnite scale of spaces. More exactly, the closed subspace Vj(j∈Z)cor- respondsto level j of resolution,or to scale j. We consider a r-regular multiresolution analysis of L2(Rn)(of course, wemayconsideranydifferentfunctionalspace)whichisa sequenceofincreasingclosedsubspaces Vj: ...V−2⊂V−1⊂V0⊂V1⊂V2⊂... (6) Thenjustas Vjisspannedbydilationandtranslationsof thescalingfunction,so Wjarespannedbytranslationsand dilationofthemotherwavelet ψjk(x), where ψjk(x) = 2j/2ψ(2jx−k). (7) All expansions,whichweused,arebasedonthefollowing properties: L2(R) =V0∞/circleplusdisplay j=0Wj (8) We needalso toﬁndingeneralsituationobjects Λd1d2...dn ℓ1ℓ2...ℓn=∞/integraldisplay −∞/productdisplay ϕdi ℓi(x)dx, (9) but now in the case of periodic boundaryconditions. Now we consider the procedure of their calculations in case of periodic boundary conditions in the base of periodic wavelet functions on the interval [0,1] and corresponding expansion (1) inside our variational approach. Periodiza- tionproceduregivesus ˆϕj,k(x)≡/summationdisplay ℓ∈Zϕj,k(x−ℓ) (10) ˆψj,k(x) =/summationdisplay ℓ∈Zψj,k(x−ℓ) So,ˆϕ,ˆψare periodic functions on the interval [0,1]. Be- causeϕj,k=ϕj,k′ifk=k′mod(2j), we may consider only0≤k≤2jand as consequence our multiresolution hastheform/uniondisplay j≥0ˆVj=L2[0,1]withˆVj= span {ˆϕj,k}2j−1 k=0 [12]. Integration by parts and periodicity gives useful relations between objects (9) in particular quadratic case (d=d1+d2): Λd1,d2 k1,k2= (−1)d1Λ0,d2+d1 k1,k2, (11) Λ0,d k1,k2= Λ0,d 0,k2−k1≡Λd k2−k1So, any 2-tuple can be represented by Λd k. Then our sec- ond additional linear problem is reduced to the eigenvalue problem for {Λd k}0≤k≤2jby creating a system of 2jho- mogeneousrelations in Λd kand inhomogeneousequations. So, if we have dilation equation in the form ϕ(x) =√ 2/summationtext k∈Zhkϕ(2x−k), then we have the following ho- mogeneousrelations Λd k= 2dN−1/summationdisplay m=0N−1/summationdisplay ℓ=0hmhℓΛd ℓ+2k−m,(12) or in such form Aλd= 2dλd, whereλd={Λd k}0≤k≤2j. Inhomogeneousequationsare: /summationdisplay ℓMd ℓΛd ℓ=d!2−j/2, (13) where objects Md ℓ(\|ℓ\| ≤N−2)can be computed by re- cursiveprocedure Md ℓ= 2−j(2d+1)/2˜Md ℓ, (14) ˜Mk ℓ=<xk,ϕ0,ℓ>=k/summationdisplay j=0/parenleftbiggk j/parenrightbigg nk−jMj 0,˜Mℓ 0= 1. So, we reduced our last problem to standard linear alge- braical problem. Then we use the methods from [9]. As a resultwe obtainedforclosedtrajectoriesoforbitaldynam - ics the explicit time solution (1) in the base of periodized wavelets(10). 00.10.20.30.40.50.60.70.80.91−0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25 Figure1: Periodicwavelet 4 NUMERICALCALCULATIONS In this part we considernumericalillustrationsof previou s analytical approach. Our numerical calculations are based onperiodiccompactlysupportedDaubechieswaveletsand relatedwaveletfamilies(Fig.1). Alsoinourmodellingwe addednoiseasperturbationtoourspinorbitconﬁgurations . On Fig. 2 we present accordingto formulae(2),(6)con- tributionstoapproximationofourdynamicalevolution(to p row on the Fig. 3) starting from the coarse approximation, corresponding to scale 20(bottom row) to the ﬁnest one corresponding to the scales from 21to25or from slowto fast components (5 frequencies) as details for approxi- mation. Then on Fig. 3, from bottom to top, we demon- strate the summation of contributions from corresponding levels of resolution given on Fig. 2 and as result we re- store via 5 scales (frequencies)approximationour dynam- ical process(top row on Fig. 3 ). The same decomposi- tion/approximation we produce also on the level of power spectraldensityinthe processwith noise(Fig.4). 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.500.5 Figure2: Contributionstoapproximation: fromscale 21to 25(withnoise). 0 50 100 150 200 250−505 0 50 100 150 200 250−505 0 50 100 150 200 250−505 0 50 100 150 200 250−202 0 50 100 150 200 250−101 0 50 100 150 200 250−0.500.5 Figure 3: Approximations: from scale 21to25(with noise). 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500200400 051015202530354045500200400 Figure4: Powerspectraldensity: fromscale 21to25(with noise) We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help and supportduringUCLA ICFAWorkshop.5 REFERENCES [1] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [2] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [3] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [4] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [5] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262 [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [8] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [9] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095 [10] V. Balandin, NSF-ITP-96-155i. [11] A.N.Fedorova, M.G.Zeitlin,inpress [12] G. Schlossnagle, J.M. Restrepo and G.K. Leaf, Technica l Report ANL-93/34.
arXiv:physics/0008048v1 [physics.acc-ph] 13 Aug 2000LOCAL ANALYSIS OF NONLINEAR RMSENVELOPE DYNAMICS A. Fedorova,M.Zeitlin,IPME,RAS, St. Petersburg, V.O. Bol shojpr.,61,199178,Russia∗† Abstract We present applications of variational – wavelet approach to nonlinear (rational) rms envelope dynamics. We have thesolutionasamultiresolution(multiscales)expansion in the baseofcompactlysupportedwaveletbasis. 1 INTRODUCTION In this paper we consider the applications of a new nume- rical-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysisto the nonlinear root-mean-square (rms) envelope dynamics [1]. Suchapproachmaybe usefulin all modelsinwhichit is possible and reasonable to reduce all complicated prob- lems related with statistical distributions to the problem s described by systems of nonlinear ordinary/partial differ - ential equations. In this paper we consider an approach based on the second momentsof the distributionfunctions forthe calculationofevolutionofrmsenvelopeof a beam. The rms envelopeequations are the most useful for analy- sis ofthe beamself–forces(space–charge)effectsand also allow to consider both transverse and longitudinal dynam- ics of space-charge-dominatedrelativistic high–brightn ess axisymmetric/asymmetric beams, which under short laser pulse–drivenradio-frequencyphotoinjectorshavefasttr an- sition from nonrelativistic to relativistic regime [2]. Fr om the formal point of view we may consider rms envelope equationsafterstraightforwardtransformationstostand ard Cauchyformasasystemofnonlineardifferentialequations which are not more than rational (in dynamical variables). Because of rational type of nonlinearities we need to con- sider some extension of our results from [3]-[10], which are based on application of wavelet analysis technique to variationalformulationofinitialnonlinearproblems. Wavelet analysis is a relatively novel set of mathemat- ical methods, which gives us a possibility to work with well-localized bases in functional spaces and give for the general type of operators (differential, integral, pseudo d- ifferential) in such bases the maximum sparse forms. Our approachinthispaperisbasedonthegeneralization[11]of variational-wavelet approach from [3]-[10], which allows ustoconsidernotonlypolynomialbutrationaltypeofnon- linearities. In part 2 we describe the different forms of rms equa- tions. In part 3 we present explicit analytical construc- tion for solutions of rms equations from part 2, which are based on our variational formulation of initial dynamical problems and on multiresolution representation [11]. We ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmlgive explicit representation for all dynamical variables i n the base of compactly supported wavelets. Our solutions are parametrized by solutions of a number of reduced al- gebraical problems from which one is nonlinear with the samedegreeofnonlinearityandtherestarethelinearprob- lems which correspondto particularmethod of calculation ofscalarproductsoffunctionsfromwaveletbasesandtheir derivatives. In part 4 we considerresults of numerical cal- culations. 2 RMSEQUATIONS BelowweconsideranumberofdifferentformsofRMSen- velope equations, which are from the formal point of view not more than nonlinear differential equations with ratio- nal nonlinearities and variable coefﬁcients. Let f(x1, x2) be the distribution function which gives full information about noninteracting ensemble of beam particles regard- ingtotracespaceortransversephasecoordinates (x1, x2). Then we may extract the ﬁrst nontrivial bit of ‘dynamical information’fromthesecondmoments σ2 x1=< x2 1>=/integraldisplay /integraldisplay x2 1f(x1, x2)dx1dx2 σ2 x2=< x2 2>=/integraldisplay /integraldisplay x2 2f(x1, x2)dx1dx2(1) σ2 x1x2=< x1x2>=/integraldisplay /integraldisplay x1x2f(x1, x2)dx1dx2 RMS emittance ellipse is given by ε2 x,rms =< x2 1>< x2 2>−< x1x2>2. Expressions for twiss parameters arealso basedonthesecondmoments. We will consider the following particular cases of rms envelope equations, which described evolution of the mo- ments (1) ([1],[2] for full designation): for asymmetric beams we have the system of two envelope equations of the secondorderfor σx1andσx2: σ′′ x1+σ′ x1γ′ γ+ Ω2 x1/parenleftbiggγ′ γ/parenrightbigg2 σx1= (2) I/(I0(σx1+σx2)γ3) +ε2 nx1/σ3 x1γ2, σ′′ x2+σ′ x2γ′ γ+ Ω2 x2/parenleftbiggγ′ γ/parenrightbigg2 σx2= I/(I0(σx1+σx2)γ3) +ε2 nx2/σ3 x2γ2 The envelopeequation for an axisymmetric beam is a par- ticularcase ofprecedingequations. Also we have related Lawson’s equation for evolution of the rms envelope in the paraxial limit, which governs evolutionofcylindricalsymmetricenvelopeunderexterna llinearfocusingchannelofstrenghts Kr: σ′′+σ′/parenleftbiggγ′ β2γ/parenrightbigg +Krσ=ks σβ3γ3+ε2 n σ3β2γ2,(3) where Kr≡ −Fr/rβ2γmc2, β≡νb/c=/radicalbig 1−γ−2 After transformations to Cauchy form we can see that all this equations from the formal point of view are not morethanordinarydifferentialequationswithrationalno n- linearities and variable coefﬁcients (also,we may conside r regimesinwhich γ,γ′arenotﬁxedfunctions/constantsbut satisfysomeadditionaldifferentialconstraint/equatio n,but thiscase doesnotchangeourgeneralapproach). 3 RATIONALDYNAMICS Theﬁrst mainpartofourconsiderationissomevariational approachto thisproblem,which reducesinitial problemto the problem of solution of functional equations at the ﬁrst stage and some algebraical problems at the second stage. We havethesolutioninacompactlysupportedwaveletba- sis. An example of such type of basis is demonstrated on Fig. 1. Multiresolution representation is the second main Figure1: Waveletsat differentscalesandlocations. part of our construction. The solution is parameterized by solutionsoftwo reducedalgebraicalproblems,oneis non- linear and the second are some linear problems, which are obtained from one of the standard wavelet constructions: the method of Connection Coefﬁcients (CC) or Stationary SubdivisionSchemes(SSS). So, our variational-multiresolution approach [11] gives us possibilityto constructexplicitnumerical-analytica lso- lution for the following systems of nonlinear differential equations ˙z=R(z, t)orQ(z, t)˙z=P(z, t),(4) where z(t) = (z1(t), ..., z n(t))is the vector of dynamical variables zi(t), R(z, t)isnotmorethanrationalfunctionofz, P(z, t), Q(z, t)are not more than polynomial functions ofz andP,Q,Rhavearbitrarydependenceoftime. Thesolutionhasthefollowingform z(t) =zslow N(t) +/summationdisplay j≥Nzj(ωjt), ω j∼2j(5)which correspondsto the full multiresolutionexpansionin all time scales. Formula(5)givesusexpansionintoa slow partzslow Nand fast oscillating parts for arbitraryN. So, we maymovefromcoarsescalesofresolutiontotheﬁnestone forobtainingmoredetailedinformationaboutourdynami- calprocess. Theﬁrst termintheRHSofrepresentation(5) corresponds on the global level of function space decom- position to resolution space and the second one to detail space. In this way we give contribution to our full solu- tion from each scale of resolution or each time scale. The same is correct for the contribution to power spectral den- sity (energyspectrum): we can take into accountcontribu- tionsfromeachlevel/scaleofresolution. So,wehavethesolutionoftheinitialnonlinear(rational) problemintheform zi(t) =zi(0) +N/summationdisplay k=1λk iZk(t), (6) where coefﬁcients λk iare roots of the corresponding re- duced algebraical (polynomial) problem [11]. Conse- quently, we have a parametrization of solution of initial problembysolutionofreducedalgebraicalproblem. So, the obtained solutions are given in the form (6), where Zk(t)arebasisfunctionsand λi karerootsofreduced system of equations. In our case Zk(t)are obtained via multiresolution expansions and represented by compactly supported wavelets and λi kare the roots of reduced poly- nomialsystem with coefﬁcients,whichare givenbyCC or SSS constructions. EachZj(t)isarepresentativeofcorrespondingmultires- olutionsubspace Vj,whichisamemberofthesequenceof increasingclosedsubspaces Vj: ...V−2⊂V−1⊂V0⊂V1⊂V2⊂... (7) Thebasisineach Vjis ϕjl(x) = 2j/2ϕ(2jx−ℓ) (8) whereindices ℓ, jrepresenttranslationsandscalingrespec- tively or action of underlying afﬁne group which act as a “microscope” and allow us to construct corresponding so- lutionwithneededlevelofresolution. It should be noted that such representations (5),(6) for solutions of equations (2),(3) give the best possible local - ization propertiesin correspondingphase space.This is es - pecially important because our dynamical variables corre- spondstomomentsofensembleofbeamparticles. 4 NUMERICALCALCULATIONS In this part we considernumericalillustrationsof previou s analytical approach. Our numerical calculations are based on compactly supported Daubechies wavelets and related waveletfamilies. OnFig.2wepresentaccordingtoformu- lae(5),(6)contributionstoapproximationofourdynamica l evolution (top row on the Fig. 3) starting from the coarse0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−101 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.200.2 Figure2: Contributionstoapproximation: fromscale 21to 25. 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−202 0 50 100 150 200 250−0.500.5 0 50 100 150 200 250−0.200.2 Figure3: Approximations: fromscale 21to25. approximation, corresponding to scale 20(bottom row) to the ﬁnest one corresponding to the scales from 21to25 or from slow to fast components(5 frequencies) as details for approximation. Then on Fig. 3, from bottom to top, we demonstrate the summation of contributions from cor- respondinglevelsofresolutiongivenonFig.2andasresult werestorevia5scales(frequencies)approximationourdy- namicalprocess(toprowonFig.3 ). Wealsoproducethesamedecomposition/approximation on the level of power spectral density (Fig. 4). It should be noted that complexity of such algorithms are minimal regarding other possible. Of course, we may use differ- ent multiresolution analysis schemes, which are based on different families of generating wavelets and apply such schemes of numerical–analytical calculations to any dy- namical process which may be described by systems of ordinary/partialdifferentialequationswith rationalno nlin- earities[11]. We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help and supportduringUCLA ICFAWorkshop. 5 REFERENCES [1] J.B.Rosenzweig, Fundamentals of Beam Physics,e-ver- sion: http://www.physics.ucla.edu/class/99F/250Rosen zwe- ig/notes/ [2] L. Seraﬁni and J.B. Rosenzweig, Phys. Rev. E 55, 7565, 1997.05101520253035404550012x 104 05101520253035404550012x 104 051015202530354045500500010000 051015202530354045500500010000 051015202530354045500510 051015202530354045500510 Figure4: Powerspectraldensity: fromscale 21to25. [3] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [4] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [5] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [8] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262 [9] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [10] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [11] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095
arXiv:physics/0008049v1 [physics.acc-ph] 13 Aug 2000MULTISCALE REPRESENTATIONS FORSOLUTIONSOF VLASOV-MAXWELL EQUATIONSFORINTENSE BEAMPROPAGATION A. Fedorova,M.Zeitlin,IPME,RAS, V.O. Bolshojpr.,61,199 178,St. Petersburg, Russia∗† Abstract We present the applications of variational–wavelet ap- proach for computingmultiresolution/multiscalereprese n- tation for solution of some approximations of Vlasov- Maxwellequations. 1 INTRODUCTION Inthispaperweconsidertheapplicationsofanewnumeri- cal-analytical technique which is based on the methods of localnonlinearharmonicanalysisorwaveletanalysistoth e nonlinearbeam/acceleratorphysicsproblemsdescribedby some formsof Vlasov-Maxwell(Poisson)equations. Such approach may be useful in all models in which it is possi- ble and reasonable to reduce all complicated problems re- latedwithstatisticaldistributionstotheproblemsdescr ibed by systems of nonlinear ordinary/partial differential equ a- tions. Wavelet analysis is a relatively novel set of math- ematical methods, which gives us the possibility to work withwell-localizedbasesinfunctionalspacesandgivesfo r the general type of operators (differential, integral, pse u- dodifferential) in such bases the maximum sparse forms. Our approach in this paper is based on the generalization ofvariational-waveletapproachfrom[1]-[8],whichallow s ustoconsidernotonlypolynomialbutrationaltypeofnon- linearities[9]. Thesolutionhasthefollowingform(relat ed formsinpart3) u(t,x) =/summationdisplay k∈ZnUk(x)Vk(t), (1) Vk(t) =Vk,slow N (t) +/summationdisplay j≥NVk j(ω1 jt), ω1 j∼2j Uk(x) =Uk,slow N(x) +/summationdisplay j≥NUk j(ω2 jx), ω2 j∼2j which correspondsto the full multiresolutionexpansionin all time/spacescales. Formula (1) gives us expansioninto the slow part uslow N andfast oscillatingpartsforarbitraryN.So, wemaymove from coarse scales of resolution to the ﬁnest one for ob- tainingmoredetailedinformationaboutourdynamicalpro- cess. TheﬁrsttermintheRHSofformulae(1)corresponds onthegloballeveloffunctionspacedecompositiontoreso- lutionspaceandthesecondonetodetailspace. Inthisway we give contribution to our full solution from each scale of resolutionoreach time/spacescale. The same is correct forthecontributiontopowerspectraldensity(energyspec - trum): we can take into account contributions from each ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmllevel/scale of resolution. Starting in part 2 from Vlasov- Maxwell equations we consider in part 3 the generaliza- tion of our approach based on variational formulation in the biorthogonalbasesofcompactlysupportedwavelets. 00.20.40.60.81 00.20.40.60.81−0.2−0.100.10.20.3 Figure1: Base wavelet. 2 VLASOV-MAXWELL EQUATIONS Analysis based on the non-linear Vlasov-Maxwell equa- tions leds to more clear understanding of the collecti- ve effects and nonlinear beam dynamics of high inten- sity beam propagation in periodic-focusing and uniform- focusing transport systems. We consider the following formofequations([11]forsetupanddesignation): /braceleftBig∂ ∂s+x′∂ ∂x+y′∂ ∂y−/bracketleftBig kx(s)x+∂ψ ∂x/bracketrightBig∂ ∂x′− /bracketleftBig ky(s)y+∂ψ ∂y/bracketrightBig∂ ∂y′/bracerightBig fb= 0, (2) /parenleftBig∂2 ∂x2+∂2 ∂y2/parenrightBig ψ=−2πKb Nb/integraldisplay dx′dy′fb.(3) The corresponding Hamiltonian for transverse single-par- ticle motionisgivenby ˆH(x,y,x′,y′,s) =1 2(x′2+y′2) +1 2[kx(s)x2+ky(s)y2] +ψ(x,y,s).(4) Related Vlasov system describes longitudinal dynamics ofhighenergystoredbeam[12]: ∂f ∂T+v∂f ∂θ+λV∂f ∂v= 0, (5) ∂2V ∂T2+ 2γ∂V ∂T+ω2V=∂I ∂T(6) I(θ;T) =/integraldisplay dvvf(θ,v;T). (7)3 VARIATIONALAPPROACHIN BIORTHOGONALWAVELETBASES Now we consider some useful generalization of our varia- tional wavelet approach. Because integrand of variational functionalsisrepresentedbybilinearform(scalarproduc t) itseemsmorereasonabletoconsiderwaveletconstructions [13] which take into account all advantages of this struc- ture. Theactionfunctionalforloopsin thephasespaceis F(γ) =/integraldisplay γpdq−/integraldisplay1 0H(t,γ(t))dt (8) The critical points of Fare those loops γ, which solve the Hamiltonian equationsassociated with the Hamiltonian H and hence are periodic orbits. Let us consider the loop space Ω =C∞(S1,R2n), whereS1=R/Z, of smooth loops inR2n. Let us deﬁne a function Φ : Ω →Rby setting Φ(x) =/integraldisplay1 01 2<−J˙x,x>dt −/integraldisplay1 0H(x(t))dt, x ∈Ω (9) Computing the derivative at x∈Ωin the direction of y∈ Ω, weﬁnd Φ′(x)(y) =/integraldisplay1 0<−J˙x− ▽H(x),y>dt (10) Consequently, Φ′(x)(y) = 0for ally∈Ωiff the loopxis a solution of the Hamiltonian equations. Now we need to takeintoaccountunderlyingbilinearstructureviawavele ts. We started with two hierarchical sequences of approxima- tions spaces [13]: ...V−2⊂V−1⊂V0⊂V1⊂V2..., .../tildewideV−2⊂/tildewideV−1⊂/tildewideV0⊂/tildewideV1⊂/tildewideV2...,and as usu- ally,W0is complement to V0inV1, but now not neces- sarily orthogonal complement. New orthogonality condi- tions have now the following form: /tildewiderW0⊥V0, W 0⊥ /tildewideV0, V j⊥/tildewiderWj,/tildewideVj⊥Wjtranslates of ψspanW0, translates of ˜ψspan/tildewiderW0. Biorthogonality conditions are< ψ jk,˜ψj′k′>=/integraltext∞ −∞ψjk(x)˜ψj′k′(x)dx=δkk′δjj′, whereψjk(x) = 2j/2ψ(2jx−k). Functionsϕ(x),˜ϕ(x−k) form dual pair: < ϕ(x−k),˜ϕ(x−ℓ)>=δkl, < ϕ(x−k),˜ψ(x−ℓ)>= 0for∀k,∀ℓ.Functionsϕ,˜ϕ generate a multiresolution analysis. ϕ(x−k),ψ(x−k) are synthesis functions, ˜ϕ(x−ℓ),˜ψ(x−ℓ)are analysis functions. Synthesisfunctionsarebiorthogonalto analys is functions. Scaling spaces are orthogonal to dual wavelet spaces. Two multiresolutionsare intertwining Vj+Wj= Vj+1,/tildewideVj+/tildewiderWj=/tildewideVj+1. These are direct sums but not orthogonal sums. So, our representation for solution has nowtheform f(t) =/summationdisplay j,k˜bjkψjk(t), (11) where synthesis wavelets are used to synthesize the func- tion. But ˜bjkcome from inner products with analysiswavelets. Biorthogonalityyields ˜bℓm=/integraldisplay f(t)˜ψℓm(t)dt. (12) So, now we can introduce this more useful construction into our variational approach. We have modiﬁcation only onthelevelofcomputingcoefﬁcientsofreducednonlinear algebraical system. This new construction is more ﬂexi- ble. Biorthogonalpointofviewismorestableundertheac- tion oflargeclassof operatorswhileorthogonal(onescale for multiresolution) is fragile, all computations are much moresimplerandweacceleratetherateofconvergence. In all types of (Hamiltonian) calculation, which are based on some bilinear structures (symplectic or Poissonian struc- tures,bilinearformofintegrandinvariationalintegral) this frameworkleadstogreatersuccess. So, we try to use wavelet bases with their good spatial and scale–wavenumber localization properties to explore the dynamics of coherent structures in spatially-extended , ’turbulent’/stochasticsystems. Aftersomeansatzesandr e- ductionswearrivefrom(2),(3)or(5)-(7)tosomesystemof nonlinear partial differential equations [10]. We conside r application of our technique to Kuramoto-Sivashiinsky equation as a model with rich spatio-temporal behaviour [14] (0≤x≤L,ξ=x/L,u(0,t) =u(L,t), ux(0,t) =ux(L,t)): ut=−uxxx−uxx−uux=Au+B(u) ut+1 L4uξξξξ+1 L2uξξ+1 Luuξ= 0 (13) Let be u(x,t) =N/summationdisplay k=0M/summationdisplay ℓ=0ak ℓ(t)ψk ℓ(ξ) =/summationdisplay ak ℓψk ℓ,(14) whereψk ℓ(ξ),ak ℓ(t)arebothwavelets. Variationalformulation /parenleftBigg/summationdisplay k,ℓ/braceleftBig ˙ak ℓψk ℓ+1 L4ak ℓψk′′′′ ℓ+1 L2ak ℓψk′′ ℓ +1 L/summationdisplay p,qak ℓap qψk ℓψp′ q/bracerightBig ,ψr s/parenrightBigg = 0 (15) reduces(13)toODEandalgebraicalone. Mrk sℓ˙ar s=/summationdisplay k,ℓLrk sℓak ℓ+/summationdisplay k,ℓ/summationdisplay p,qNrpk sqℓap qak ℓ Mrk sℓ=/parenleftbig ψk ℓ,ψr s/parenrightbig (16) Lrk sℓ=1 L2(ψr′ s,ψk′ ℓ/parenrightbig −1 L4(ψr′′ s,ψk′′ ℓ) Nrpk sqℓ=1 L(ψr s,ψp qψk′ ℓ) Inparticularcase on V2\V0we have:  ˙a0 ˙a1 ˙a2 =/bracketleftBigg L/bracketrightBigg a0 a1 a2 +0102030405060 0204060−0.06−0.04−0.0200.020.040.060.08 Figure2: Thesolutionofeq.(13)  ca0a1−ca0a2+da2 1−da2 2 −ca2 0−da0a1+ℓa0a2−fa1a2−fa2 2 ca2 0−ℓa0a1+da0a1+da0a2+fa2 1+fa1a2  Thenincontrastto[14]weapplyto(16)methodsfrom[1]- [9] and arrive to formula (1). The same approach we use forthegeneralnonlinearwaveequation utt=uxx−mu−f(u), (17) where f(u) =au3+/summationdisplay k≥5fkuk(18) Accordingto[2],[10]wemayconsideritasinﬁnitedimen- sional Hamiltonian systems with phase space =H1 0×L2 on[0,L]andcoordinates: u,v=ut, then H=1 2<v,v> +1 2<Au,u> +/integraldisplayπ 0g(u)dx A=d2 dx2+m, g =/integraldisplay f(s)ds ut=∂H ∂v=v (19) vt=−∂H ∂u=−Au−f(u) or˙u(t) =J∇K(u(t)) Thenanzatzes: u(t,x) =U(ω1t,...,ω nt,x) (20) u(t,x) =/summationdisplay k∈ZnUk(x)exp(ik·ω(k)t) u(t,x) =S(x−vt) u(t,x) =/summationdisplay k∈ZnUk(x)Vk(t) and methods [1]-[10] led to formulae (1). Resulting mul- tiresolution/multiscale representation in the high-loca lized bases (Fig.1) is demonstrated on Fig.2, Fig.3. We would like to thank Professor James B. Rosenzweig and Mrs. MelindaLaranetafornicehospitality,helpandsupportdur - ingUCLA ICFA Workshop.0102030405060 0204060−1−0.500.51 Figure3: Thesolutionofeq.(17) 4 REFERENCES [1] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [2] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31, 101 (Kluwer, 1998). [3] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [4] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [5] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262 [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [8] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [9] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095 [10] A.N.Fedorova and M.G.Zeitlin,inpress [11] R.Davidson,H.Qin,P.Channel,PRSTAB,2,074401,1999 [12] S.Tzenov, P.Colestock, Fermilab-Pub-98/258 [13] A. Cohen, I. Daubechies and J.C. Feauveau, Comm. Pure. Appl. Math. ,XLV,485 (1992). [14] Ph. Holmes e.a.,Physica D86, 396, 1995
arXiv:physics/0008050v1 [physics.acc-ph] 13 Aug 2000QUASICLASSICAL CALCULATIONS INBEAM DYNAMICS A. Fedorova,M.Zeitlin,IPME,RAS, V.O. Bolshojpr.,61,199 178,St. Petersburg, Russia∗† Abstract We presentsomeapplicationsofgeneralharmonic/wavelet analysisapproach(generalizedcoherentstates,waveletp a- ckets) to numerical/analytical calculations in (nonlinea r) quasiclassical/quantumbeam dynamicsproblems. (Naive) deformation quantization, multiresolution representati ons andWignertransformarethekeypoints. 1 INTRODUCTION In thispaperwe considersomestartingpointsin the appli- cations of a new numerical-analytical technique which is based on the methods of local nonlinear harmonic analy- sis (wavelet analysis, generalized coherent states analys is) tothequantum/quasiclassical(nonlinear)beam/accelera tor physics calculations. The reason for this treatment is that recently a number of problems appeared in which one needs take into account quantum properties of par- ticles/beams. We mention only two: diffractive quan- tumlimitsofaccelerators(achievabletransversebeamspo t size) and the description of dynamical evolution of high density beamsby using collectivemodels[1]. Ourstarting point is the generalpoint ofview of deformationquantiza- tion approach at least on naive Moyal/Weyl/Wigner level (from observables to symbols) (part 2). Then we present some useful numerical wavelet analysis technique, which givesthemostsparserepresentationfortwomainoperators (multiplication and differentiating)in any Hilbert space of states. Wavelet analysis is a some set of mathematical methods, which givesusthe possibility to work with well- localized bases (Fig.1) in functional spaces and gives for the general type of operators (differential, integral, pse u- dodifferential) in such bases the maximum sparse forms. The approach from this paper is related to our investiga- tion of classical nonlinear dynamics of accelerator/beam problems [2]-[10]. The common point is that any solu- tionwhichcomesfromfullmultiresolutionexpansioninall time scalesgivesusexpansionintoa slow partandfast os- cillatingparts. So,wemaymovefromcoarsescalesofres- olution to the ﬁnest one for obtaining more detailed infor- mation about our dynamical process. In this way we give contribution to our full solution from each scale of resolu- tion or each time scale. The same is correct for the contri- butiontopowerspectraldensity(energyspectrum): wecan takeintoaccountcontributionsfromeachlevel/scaleofre s- olution. Because afﬁne group of translations and dilations (or more general group, which acts on the space of solu- tions) is inside the approach(in wavelet case), thismethod ∗e-mail: zeitlin@math.ipme.ru †http://www.ipme.ru/zeitlin.html; http://www.ipme.nw. ru/zeitlin.htmlresembles the action of a microscope. We have contribu- tion to ﬁnal result from each scale of resolution from the whole inﬁnite scale of spaces. Besides afﬁne group sym- metry,in part3 we considermodelling,basedonveryuse- ful and quantum oriented Wigner transform/function ap- proach (corresponding to Weyl-Heisenberg group), which explicitlydemonstratesquantuminterferenceof(coheren t) states. Figure1: Localizedcontributionsto beammotion. 2 QUASICLASSICAL EVOLUTION Let us consider classical and quantum dynamics in phase space Ω =R2mwith coordinates (x,ξ)and generated by Hamiltonian H(x,ξ)∈C∞(Ω;R). IfΦH t: Ω−→Ωis (classical) ﬂow then time evolution of any bounded clas- sical observable or symbol b(x,ξ)∈C∞(Ω,R)is given bybt(x,ξ) =b(ΦH t(x,ξ)). LetH=OpW(H)and B=OpW(b)are the self-adjoint operators or quantum observablesin L2(Rn),representingtheWeylquantization ofthesymbols H,b[12] (Bu)(x) =1 (2π¯h)n/integraldisplay R2nb/parenleftbiggx+y 2,ξ/parenrightbigg · ei<(x−y),ξ>/¯hu(y)dydξ, whereu∈S(Rn)andBt=eiHt/¯hBe−iHt/¯hbe the Heisenbergobservableorquantumevolutionoftheobserv- ableBunder unitary group generated by H.Btsolves theHeisenbergequationofmotion ˙Bt= (i/¯h)[H,B t].Let bt(x,ξ; ¯h)is a symbol of Btthen we have the following equationforit ˙bt={H,bt}M, (1) with the initial condition b0(x,ξ,¯h) =b(x,ξ). Here {f,g}M(x,ξ)is the Moyal brackets of the observablesf,g∈C∞(R2n),{f,g}M(x,ξ) =f♯g−g♯f, wheref♯g is the symbol of the operator product and is presented by the compositionofthesymbols f,g (f♯g)(x,ξ) =1 (2π¯h)n/2/integraldisplay R4ne−i<r,ρ>/ ¯h+i<ω,τ>/ ¯h ·f(x+ω,ρ+ξ)·g(x+r,τ+ξ)dρdτdrdω. Forourproblemsitisusefulthat {f,g}Madmitstheformal expansioninpowersof ¯h: {f,g}M(x,ξ)∼ {f,g}+ 2−j·/summationdisplay \|α+β\|=j≥1(−1)\|β\|·(∂α ξfDβ xg)·(∂β ξgDα xf), whereα= (α1,...,α n)isamulti-index, \|α\|=α1+...+ αn,Dx=−i¯h∂x. So, evolution(1)for symbol bt(x,ξ; ¯h) is ˙bt={H,bt}+1 2j/summationdisplay \|α\|+β\|=j≥1(−1)\|β\|·(2) ¯hj(∂α ξHDβ xbt)·(∂β ξbtDα xH). At¯h= 0this equation transforms to classical Liouville equation ˙bt={H,bt}. (3) Equation (2) plays a key role in many quantum (semiclas- sical) problem. Our approach to solution of systems (2), (3) is based on our technique from [11] and very useful linear parametrization for differential operators which w e present now. Let us consider multiresolution representa- tion...⊂V2⊂V1⊂V0⊂V−1⊂V−2.... Let T be an operator T:L2(R)→L2(R), with the ker- nelK(x,y)andPj:L2(R)→Vj(j∈Z)is projec- tion operators on the subspace Vjcorresponding to j level of resolution: (Pjf)(x) =/summationtext k< f,ϕ j,k> ϕ j,k(x). LetQj=Pj−1−Pjis the projection operator on the subspaceWjthen we have the following ”microscopic or telescopic” representation of operator T which takes into account contributions from each level of resolution from different scales starting with coarsest and ending to ﬁnest scales[13]:T=/summationtext j∈Z(QjTQj+QjTPj+PjTQj).We rememberthatthisisaresultofpresenceofafﬁnegroupin- side this construction. The non-standard form of operator representation [13] is a representation of an operator T as a chainoftriples T={Aj,Bj,Γj}j∈Z, actingonthesub- spacesVjandWj:Aj:Wj→Wj,Bj:Vj→Wj,Γj: Wj→Vj,where operators {Aj,Bj,Γj}j∈Zare deﬁned asAj=QjTQj, B j=QjTPj,Γj=PjTQj.The operatorTadmitsarecursivedeﬁnitionvia Tj=/parenleftbigg Aj+1Bj+1 Γj+1Tj+1/parenrightbigg , whereTj=PjTPjandTjworks onVj:Vj→Vj. It should be noted that operator Ajdescribes interaction on the scale jindependently from other scales, opera- torsBj,Γjdescribe interaction between the scale j andall coarser scales, the operator Tjis an ”averaged” ver- sion ofTj−1. We may compute such non-standard repre- sentations of operator d/dxin the wavelet bases by solv- ing only the system of linear algebraical equations. Let rℓ=/integraltext ϕ(x−ℓ)d dxϕ(x)dx,ℓ∈Z.Then, the representa- tion ofd/dxis completely determined by the coefﬁcients rℓor by representation of d/dxonly on the subspace V0. The coefﬁcients rℓ,ℓ∈Zsatisfy the usual system of lin- ear algebraical equations. For the representation of op- eratordn/dxnwe have the similar reduced linear system of equations. Then ﬁnally we have for action of operator Tj(Tj:Vj→Vj)onsufﬁcientlysmoothfunction f: (Tjf)(x) =/summationdisplay k∈Z/parenleftBigg 2−j/summationdisplay ℓrℓfj,k−ℓ/parenrightBigg ϕj,k(x), whereϕj,k(x) = 2−j/2ϕ(2−jx−k)iswaveletbasisand fj,k−1= 2−j/2/integraldisplay f(x)ϕ(2−jx−k+ℓ)dx are wavelet coefﬁcients. So, we have simple linear para- metrization of matrix representationof our differential o p- eratorinwaveletbasisandofthe actionofthisoperatoron arbitrary vector in our functional space. Then we may use suchrepresentationin allquasiclassical calculations. 3 WIGNERTRANSFORM Accordingto Weyl transform(observable-symbol)state or wave function corresponds to Wigner function, which is analogofclassicalphase-spacedistribution. If ψ(x,t),x∈ Rnsatisﬁes theSchroedingerequation i¯h∂tψ=−(¯h2/2)△ψ+Vψ (4) andWistheWignertransformof ψ W(t,x,ν) =/integraldisplay e−iνy¯ψ(t,x+ (¯h/2)y)· ψ(t,x−(¯h/2)y)dy, (5) then W satisﬁes the pseudo-differential ( ψDO) Wigner equation ∂tW+υ∂xW−(i/¯h)P(V)W= 0,(6) whereψDO operator P(V)is P(V)f(x,ν) =1 (2π)n/integraltexte−iνy/bracketleftBig V(x+¯h 2y)− V(x−¯h 2y)/bracketrightBig ·/parenleftBig/integraltext eiyξf(x,ξ)dξ/parenrightBig dy (7) Inquasiclassicallimit ¯h→0theoperator P(V)converges to−∂xV·∂ν. We considerit in[11]. OnFig.2wepresent calculations [14] of Wigner transform for beam motion, represented by four gaussians, which explicitly demon- stratesquantuminteferenceinthe phasespace. We givemoredetailsin[11]. We would like to thank Professor James B. Rosenzweig and Mrs. Melinda Laraneta for nice hospitality, help and supportduringUCLAICFA Workshop.0425849Linear scaleEnergy spectral density−0.500.51Real partSignal in time 20 40 60 80 100 12000.050.10.150.20.250.30.350.40.45WV, lin. scale, Threshold=5% Time [s]Frequency [Hz] Figure2: Wignertransform 4 REFERENCES [1] C.Hill,LosAlamospreprint: hep-ph/0002230, S.Khan,M . Pusterla, physics/9910026. [2] A.N.Fedorova and M.G.Zeitlin,’WaveletsinOptimizati on and Approximations’, Math. and Comp. in Simulation ,46, 527, 1998. [3] A.N.FedorovaandM.G.Zeitlin,’WaveletApproachtoMe- chanical Problems. Symplectic Group, Symplectic Topol- ogy and Symplectic Scales’, New Applications of Nonlin- ear and Chaotic Dynamics in Mechanics , 31,101 (Kluwer, 1998). [4] A.N. Fedorova and M.G. Zeitlin, ’Nonlinear Dynamics of Accelerator via Wavelet Approach’, CP405, 87 (American Institute of Physics,1997). Los Alamos preprint, physics/9710035. [5] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, ’Wavelet Ap- proach to Accelerator Problems’, parts 1-3, Proc. PAC97 2, 1502, 1505, 1508 (IEEE,1998). [6] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Proc. EPAC98, 930, 933 (Instituteof Physics,1998). [7] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Variational Ap - proach in Wavelet Framework to Polynomial Approxima- tions of Nonlinear Accelerator Problems. CP468, 48 ( American Instituteof Physics, 1999). Los Alamos preprint, physics/990262[8] A.N. Fedorova, M.G. Zeitlin and Z. Parsa, Symmetry, HamiltonianProblemsandWaveletsinAcceleratorPhysics. CP468, 69 (AmericanInstitute of Physics,1999). Los Alamos preprint, physics/990263 [9] A.N. Fedorova and M.G. Zeitlin, Nonlinear Accelerator ProblemsviaWavelets,parts1-8,Proc.PAC99,1614,1617, 1620,2900,2903,2906,2909,2912(IEEE/APS,NewYork, 1999). Los Alamos preprints: physics/9904039, physics/9904040, physics/9904041, physics/9904042, physics/9904043, phy - sics/9904045, physics/9904046, physics/9904047. [10] A.N. Fedorova and M.G. Zeitlin, Los Alamos preprint: physics/0003095. [11] A.N.Fedorova, M.G.Zeitlin,inpress [12] D. Sternheimer,Los Alamos preprint: math/9809056. [13] G. Beylkin, R.Coifman, V. Rokhlin, Comm. Pure Appl. Math.,44, 141, 1991 [14] F. Auger, e.a., Time-frequency Toolbox, CNRS/Rice Uni v., 1996
arXiv:physics/0008051v1 [physics.atom-ph] 13 Aug 2000Some possibilities for laboratory searches for variations of fundamental constants Savely G. Karshenboim∗ D. I. Mendeleev Institute for Metrology (VNIIM), St. Petersburg 198005, Russia and Max-Planck-Institut f¨ ur Quantenoptik, 85748 Garching, Germany† October 31, 2013 Abstract We consider diﬀerent options for the search for possible var iations of the fundamental constants. We give a brief overview of the results obtained with several method s. We discuss their advantages and disadvantages with respect to simultaneous variations of all constants in both time and space in the range 108−1010yr. We also suggest a few possibilities for the laboratory searc h. Particularly, we propose some experiments with the hyperﬁne structure of hydrogen, deuterium and ytte rbium–171 and of some atoms with a small magnetic moment. Other suggestions are for some measuremen ts of the ﬁne structure associated with the ground state. Special attention is paid to the interpretati on of the hfsmeasurements in terms of variations of the fundamental constants. 1 Introduction There is no physical reason to expect that the “fundamental c onstants” are really constant quantities. Indeed, their variations have to have a cosmological scale. Since th e publishing of a famous Dirac paper [1] a number of diﬀerent hypotheses on their possible variations have be en suggested as well as a number of trial to search for these. A review of old models and searches could be found e . g. in Ref. [2]. We live in an expanding universe and an accepted contemporary picture of the histor y of our universe assumes that there were a few phase transitions with spontaneous breaking of some symmet ries during the early stages of evolution (inﬂation model) [3]. We consider here only atomic and nuclear properties and the v ariations of the fundamental constants that can be determined from changes in such properties. We do not d iscuss variation of the gravitational constant (a review on that can be found in Refs. [2, 4]). There are two reas ons for that. First of all, in looking for variations of nuclear and atomic properties (magnetic moments, masses , decay rates) it is not possible to consider strong, weak and electromagnetic eﬀects independently. Next, from a theoretical point of view, we have to expect that the variations of strong, weak and electromagnetic couplin g constants are strongly correlated. In contrast to that, any investigation for possible variations of the grav itational interaction can be done separately. 1.1 Variation of the constants and atomic spectroscopy Any search for a possible variation of the fundamental const ants can be actually performed measuring some atomic, molecular and nuclear properties and it is necessar y to discuss relations between them. There are three very diﬀerent kinds of atomic and molecular transitions, av ailable for precision spectroscopy: ∗E-mail: sek@mpq.mpg.de; ksg@hm.csa.ru †The summer address 1•Gross structure is associated with transitions between levels with diﬀeren t values of the principal number n. They have a scaling behaviour as α2mec2/horRy. It has to be remembered that for any atom with two or more electrons, one has to introduce an eﬀective quant um number n∗, which is a function of the orbital momentum ( l). The nonrelativistic behaviour α2mec2/his actually associated with n∗(l) and, hence, in non-hydrogen-like atoms, transitions between le vels with diﬀerent value of lare also a part of the gross structure. •Atomic ﬁne structure , or separations of levels with the same n∗(l) and diﬀerent spin-orbit coupling (e. g. diﬀerent jor diﬀerent sum of spin of valence electrons), has a scaling b ehaviour α4m c2/horα2Ry. That is part of relativistic corrections, which have the same ord er of magnitude. •Atomic hyperﬁne structure is a splitting due to a magnetic moment of the nucleus ( µ) and it is proportional toα4m c2/h(µ/µb) orα2(µ/µb)Ry. Comparing frequencies with diﬀerent scaling behaviour, on e can look for variations of the fundamental constants and nuclear magnetic moments. An important part o f those comparisons is the so-called absolute frequency measurements. An absolute measurement of a frequ ency is actually its comparison to the hfsof Cs, because of the deﬁnition of second via the Cs hfsinterval νhfs(133Cs) = 9 9192 631 .770 kHz (exactly) . (1) Another possibility is to study some corrections. E. g. the s tudy of relativistic corrections to the atomic gross structure or hfscan yield to detection of possible variations of the ﬁne stru cture constants, while investigations of the electronic-vibrational-rotational lines can give l imitations for the variation of the proton-to-nuclear mass ratio. Due to the importance of the value in Eq.(1) let us to di scuss corrections to the hfsin some detail. The value of the hyperﬁne splitting in an atomic state can be pres ented in the form νhfs=c0α2µ µbRyFrel(α), (2) where µstands for the nuclear magnetic moment, c0is a constant speciﬁc for the atomic state and all dependence onαis contained in a function Frel(α), which is also speciﬁc for the state. The function is associ ated with the relativistic corrections and Frel(0) = 1. The corrections are more important for high Zand in the case of low Z(e. g. for the hydrogen hfs) they are negligible. 1.2 Variable particle, atomic and nuclear properties There are three possibilities for the variation of fundamen tal properties, which can be studied within spectro- scopic methods: •eﬀects due to electromagnetic interactions and value of ele ctromagnetic coupling constant α(study of 1s−2s, comparison of the gross and ﬁne structure, study of relativ istic corrections); •eﬀects of quark-quark and quark-gluon strong interactions and nucleon properties like µp,µn,mpetc (study of the hfsof hydrogen and light nuclei, or molecular electronic-vibr ational-rotational lines); •nuclear properties (moments and masses) due to the structur e of heavy nuclei. Some of those properties (like magnetic dipole and electric quadrupole moments) can be studied by means of atomic and molecular spectroscopy. Nuclear decay rates and scattering cross sec tions of nuclear collisions can be investigated by other methods. We mainly consider here applications of the precision spect roscopy. Experiments with the hfsof heavy atoms (e. g. Cs, Rb, Yb+or Hg+) cannot have any clear interpretation: any of the values are proportional to a non-relativistic matrix element, but with signiﬁcant rel ativistic corrections (see Eq.(2)). The non-relativistic value is proportional to a magnetic moment of the nucleus, wh ich includes the moments of one or two valence protons and neutrons and some contribution of an internal nu clear motion. The last is not a pure eﬀect of the strong interactions and an inﬂuence of electromagnetic int eractions can be estimated from the proton-neutron asymmetry. If we keep in mind a general picture, we have to expect a kind of “grand uniﬁcation” theory and there has to be some direct relations between coupling constants for w eak, electromagnetic and strong interactions (see e. g. [5, 3]). Hence, all fundamental constants (i. e. α,αsandαw) are expected to vary within about the same 2rate. Indeed, diﬀerent atomic, molecular or nuclear proper ties can have quite diﬀerent sensitivities to variations of the coupling constants. It is necessary to emphasise that it is incorrect to think tha t some values like proton-electron mass ratio or nuclear g-factor are expected to be relatively constant, while the el ectromagnetic coupling constant αvaries, as is expected in the interpretation of some papers. The bare electron mass is a result of the interaction with the Higgs ﬁeld [6] and it has signiﬁcant quantum electrodyna mical corrections due to the renormalization. The bareu- andd-quark masses also appear from the Standard model as a result of interaction with the Higgs sector and are a few Mev. However, the actual masses of the proton and neutron are determined by the masses of the constituent quarks, which are about 300 Mev and those values are completely a result of the dynamic eﬀects of the QCD in a strong coupling range. The same is true with the pr oton and neutron magnetic moments. Such a value, appearing as a result of the strong interactions, is deﬁnitely a function of the strong coupling constant, which is expected to vary together with the electromagnetic and weak coupling constants. Let us consider the electron mass in more detail. A standard i nterpretation of quantum electrodynamics (QED) is that it has to be possible to explain all low-energy ( with respect to the Planck mass Mp) physical phenomena, using only few eﬀective low-energy parameters ( like renormalized electron mass and charge etc). The origin of those values is not important. However, in the c ase of the search for variations of the constants it is necessary to study the origin of the low-energy paramet ers, like the electron mass. The bare electron mass m0is to be renormalized due to the QED eﬀects. In the one-loop ap proximation on can ﬁnd me=m0/bracketleftBigg 1 +3 4α πln/parenleftbiggΛ m0/parenrightbigg2/bracketrightBigg , (3) where the Λ is an eﬀective cut-oﬀ of the ultraviolet divergen ce. If the cut-oﬀ is associated with the Planck scale Λ =Mp∼1.2·1019Gev∼2.4·1021me, the logarithmic correction is about 20% of the leading term . One can see that, even in the case of a constant value of m0, the actual electron mass mehas to vary with α. From a theoretical point of view we can rather expect a primar y variation of some parameters which are not visible directly in our low-energy world and this forces some secondary variations of the coupling constants. E. g. in the inﬂatory universe (see Ref. [3] for detail), some eﬀective Higgs potentials depend on the average temperature, which is a function of the time. That have led to some phase transition in the past, when the hot universe was cooling. We think that even in our “cool” uni verse we can expect some slow time variation of those Higgs potentials and perhaps their long-scale space v ariation. The variations of the constants is to be a direct consequence of that. Another and more sophisticated idea [7] was proposed due to t he so-called Kaluza-Klein theories, which are associated with a world of 4+ Ndimensions. In contrast to “our” 4 dimensions, the extra spa tialNdimensions form a compact manifold with a radius Rkkabout Planck length ( ∼10−33cm). While our four dimensional universe is expanding, Marciano suggested that value of Rkkis also not constant. However, in the Kaluza-Klein theory this value is associated with the coupling constants of our world. So, we expect that the variation of diﬀerent coupling consta nts has to have the same scale. However, it is necessary to remember that due to the strong coupling any e ﬀective parameters coming from the strong interactions, can have a kind of random variation. We cannot often know which quantity is varied more (or less) rapidly than the fundamental constants. That is why we have t o try with several ways as diﬀerent as possible. No model on a possible dependence of the values of the fundame ntal constants has not been assumed in our paper. However, we must underline, that we consider a pic ture of simultaneous variations of all coupling constants, which particularly determine all property of pa rticles, nuclei, atoms and molecules. We have not yet speciﬁed a term “variations”. To our mind ther e are actually two possibilities for those: •A time and/or space variation over all the universe with a cos mological scale ( T∼1010yr and L∼c·1010 yr). •Time and/or space ﬂuctuations over some less, but signiﬁcan tly cosmological, scale ( T∼108−109yr and L∼c·(108−109) yr). Such a ﬂuctuation of the gravitation interaction was consid ered [8, 9, 10, 11, 12, 13] due to a periodicity in the galaxies distributions in the direction of the Galactic nor th and south poles [14]. The ﬂuctuations of gravitation is also one of the explanations of possible variations to the solar year [12]. The variation can be induced e. g. by cooling of the universe a nd a variation of some eﬀective Higgs potentials. We have no a priori estimation for the speed of th e variations. We expect the cosmological time and space scale ( TandL), but we have no idea on the amplitude of the possible variati ons. The expected 3amplitude particularly depends on our assumption if we expe ct some kind of a primary direct variation of the constants, or their variations are a consequence of variati ons of some other parameters. E. g. let us pretend that the primary variation is due to the compactiﬁcation rad iusRkkand there is no variation of the coupling constants on the unperturbed level. Nevertheless, the vari ations have to appear due to the renormalization and the time-dependence is to be of the relative order αlnRkk(t). Particularly, following QED one can ﬁnd in the one-loop approximation an expression for the ﬁne structure constant α=α0 1−1 3α0 π/summationtext jcjq2 jln/parenleftbig Λ/mj/parenrightbig2, (4) where Λ ≃¯h/cR kk, the sum is over all fundamental charged particles (leptons , quarks, W-bosons, Higgs particles etc), mjstands for their masses, qjis for their charges. The coeﬃcient cjis dependent on their spin and particularly for 1/2 it is equal to one. A signiﬁcant vari ation of Rkk(e. g. 1%) can induce a variation of αonly on a level between 10−4−10−5. The extra αand the logarithm reduce the amplitude of the variation dramatically. However, it is important to note, that the var iation of the mass in Eq.(3) with Λ = Rkk(t) and of the charge in Eq.(4) are to be of the same origin and of about the same order. That example shows that the variation ∆ α/αcan be small and that the mass variation ∆ m/m and the coupling constant variation can be of the same order and vary simultaneously. Actually comparing Eqs.(3) and (4) one can note that the variation of the mass can be larger, smaller or of the same order of magni tude as the variation of the coupling constant. Mass variations smaller or of the same order as the coupling c onstant have been discussed above. The larger mass variation can appear if we suggest that there is no direc t variation of the coupling constants, but the masses vary e. g. due to some variation of the eﬀective Higgs p otential of the Standard Model. The α-variation has to appear from Eq.(4) because of the m-variation in the logarithm. We conclude that a priori it is n ot feasible to dismiss the mass variation and to consider only v arying coupling constants. 2 Non-laboratory search for variations of the fundamental c onstants 2.1 Geochemical data and nuclear properties Nuclear reactions (collisions, decay etc) often involve so me relatively small diﬀerences of large contributions. E. g. to understand if any isotope is stable for a particular c hannel of decay, one has to compare the initial and ﬁnal binding energy. A problem of the stability is a probl em of this diﬀerence, which is sometimes quite small. Relatively small shifts in particle masses or coupli ng constants can make the decay of a stable isotope possible, or disturb an allowed decay. The variation of the c oupling constant of strong, electromagnetic or weak interactions can be weakly limited, but the geochemical est imations take advantage of long term comparisons. The study of the abundance of some isotopes allows one to make a comparisons over a geophysical scale of 109 yr. 2.1.1 Geochemical data Some estimations of possible variations of the coupling con stants of strong, weak and electromagnetic interac- tions from geophysical, or rather, geochemical data were pe rformed in Refs. [15, 16, 18, 17, 19, 2, 21, 22, 23] (see review in Refs. [2, 4] for more references). Some examin ations also include data on the abundance of some isotopes in the meteorites, and so the results are, in part, a strochemical ones. The typical limits are 1 α∂α ∂t<5·10−13−3·10−15yr−1, 1 αs∂αs ∂t<5·10−11−2·10−11yr−1 and1 αw∂αw ∂t<10−10yr−1. (5) One problem with the interpretation of those data is the auth ors assumed of Refs. [2, 18, 20, 21, 22] that only coupling constants vary, while the masses of the proton, the neutron and the electron are constant. Conversely, we expect those to vary as well, and some nuclear eﬀects are se nsitive to that variation. Particularly, the β-decay must be very sensitive to their diﬀerence mn−mp−me. 4We should remember that often the violation of the isotopic i nvariance and particularly the small diﬀerence of the proton and neutron masses mn−mp mp∼0.14% (6) is associated with electromagnetic eﬀects. Next, it was assumed that it is possible to look for a variatio n of some particular constant (e. g. α) while others are really constant. We cannot accept such an evaluat ion, but nevertheless we would like to underline, that the nuclear property can be very sensitive to a variatio n of the constants, because the decay rates are strongly dependent on the transition energy, which is actua lly a small diﬀerence of two larger energies of initial and ﬁnal states. Both include contributions of the electrom agnetic interactions and the diﬀerence can be quite sensitive to these. Another important problem is timing. Geochemical clocks ar e based on the study of the abundance of some long-living isotopes and others associated with them. A lon g lifetime is a result of a small value of the transition either matrix element or energy. Both, being small, are sens itive to the same variation of the constants. To the best of our knowledge there have been no discussions on a corr elation between the clock and the variation. 2.1.2 Geochemical data from Oklo reactor Shlyakhter [24] introduced two important elements in the st udy of nuclear reactions. First, he pointed out that laboratory investigations of nuclear property can als o give reasonable limitations (see section 3.1). His other idea was due to the recently discovered Oklo Fossil rea ctor in Gabon (West Africa). That is a natural ﬁssion reactor (see review in Refs. [25, 26]), and condition s for its existence are very narrow. Investigating those conditions and the local abundance of diﬀerent isotop es (particularly Sm) it is possible to derive some limitations such that it had operated 1.7 billions years ago for a period from 0.6 to 1.5 millions years. The limits from Ref. [24] are 1 α∂α ∂t<1·10−17yr−1, 1 αs∂αs ∂t<1·10−18yr−1 and 1 αw∂αw ∂t<4·10−12yr−1. (7) One evidence of the operation of the nuclear reactor in the pa st was isotope compositions of some elements like samarium, europium and gadolinium. Some of their isoto pes (149Sm,151Eu,155Gd and157Gd) are strong neutron absorbers and they have been found in very small quan tities with respect to the natural abundance. They have simply been burned by the ﬂux of thermal neutrons. A study of such isotopes can give information on the fundamental constants at a time when the fossil reacto r was operating. Particularly, the limitations in Eqs.(7) have appeared because of the resonance 149Sm + n→150Sm + γ , (8) which has an energy of only 97.3 meV. Two isotopes of samarium (147 and 149) have been studied. The halftime of the isotopes is presented in Table 11. The cross sections of reaction in Eq.(8) and a similar one fo r 147Sm + n→148Sm + γ , (9) diﬀer by about two orders of magnitude because of the resonan ce. Studying the147Sm/149Sm ratio one can deduce a possible variation of the position of the resonance from when the reactor was operating to the present day. The strength of the limitations in Eqs.(7) has three rea sons: •the sensitivity of the abundance of samarium isotopes to the position of the resonance; •the fact that the resonance energy ( ∼100 meV) has to be compared with a well of the nuclear potentia l (∼50 Mev) i. e. it is 5 ·108times larger then the energy; •large time separation ( ∼2·109yr). 5Isotopes Halftime Natural Neutron separation abundance energy Sn[keV] 147 1.06·1011yr 15.0% 6342(3) 148 7·1013yr 11.3% 8141.5(6) 149 >2·1015yr 13.8% 5871.6(9) 150 stable 7.4% 7985.7(7) Table 1: Properties of some samarium isotopes. After publication of Ref. [24] the Oklo data have been re-eva luated by a number of authors [26, 28, 4, 29, 30]. The evaluations were concentrated on the samarium abundanc e. Particularly, it was pointed out in Ref. [30] that for the limitations for the strong coupling constant it is unlikely to be appropriate to compare the position of the resonance to the well of the nucleon-nucleon potentia l, which is essentially of use only in the few-body problem. In the case of many-body eﬀect (like the resonance) it was suggested to consider a neutron separation energy Sn(see Table 1) as a characteristic reference value. The latte r is 6–8 Mev and signiﬁcantly smaller than the well ( ∼50 Mev). Investigations of other isotopes were not as eﬀective in set ting limits. Some estimations due to europium were presented by Shlyakhter, while gadolinium was studied in Ref. [30]. But the studies did not yield such strong limitations as the investigations of the samarium is otopes. Results for the variation of the ﬁne structure constant from the Oklo reactor study are collected in Table 2. The most recent estimates are [29] 1 α∂α ∂t=−1.4(54)·10−17yr−1, 1 αw∂αw ∂t<1·10−11yr−1, (10) and [30] 1 α∂α ∂t<1.0·10−17yr−1, 1 αs∂αs ∂t<1.3·10−18yr−1. (11) Those results cannot be used directly, because of the same re asons as those for other geochemical data. Partic- ularly, the timing was based on the abundance of strong absor bers of thermal neutrons and any inﬂuence of the variation of the constants on utilized cross sections was no t investigated. Variation of the ﬁne structure constant was analyzed only with respect to the static Coulomb interac tion energy of the proton in the nucleus. However, we have to expect that a small part of the proton and neutron ma sses is a result of electromagnetic interactions and hence any variations of αlead to a shift in these masses and of the kinetic energy. Inde ed only a small part of the kinetic energy is of this electromagnetic origin. How ever the entire kinetic energy is much larger than the static Coulomb interaction and eventually both electromag netic contributions can be compatible. Concerning two diﬀerent recent results on the ﬁne structure constant we note that in Ref. [30] more recent and accurate data on the samarium isotopic composition were used, while in Ref. [29] the estimation of the temperature was more secure. We also have to point out that it is unclear how much the neutron ﬂux during the operation time of the reactor used in Refs. [29, 30] is sen sitive to the possible variation of the constants. We think a proper way is to determine the ﬂux and a possible var iation simultaneously. 2.1.3 Nucleosynthesis Some estimations due to Big Bang nucleosynthesis have also b een performed [31, 32] in a similar way to the geochemical study. A possible variation of the ﬁne structur e constant is not larger than (1 −2)·10−12yr−1. 1Nuclear data (and particularly in Tables 1, 7 and 11) have bee n taken from Ref. [27], when the reference is not speciﬁed. 6∂lnα/∂t Ref. <1·10−17yr−1[24] <2·10−18yr−1[28] −1.4(54)·10−17yr−1[29] <1.0·10−17yr−1[30] Table 2: Limits for the αvariation from Oklo. This cosmological estimate takes an advantage of a large tim e separation between the epoch of nucleosynthesis and the present day which is about the lifetime of the univers e, i. e. about 1010yr. 2.2 Astrophysical data 2.2.1 Absorption lines in quasar spectra An advantage of astrophysical studies is a possibility of a l ong term comparison. A typical astrophysical time associated with extragalactic sources (quasars) is up to 1010yr. Due to the long reference time the accuracy of spectroscopic measurements need not be high. It is also po ssible to look for corrections (like me/mpterms in H2spectrum). A key point of any astrophysical study is a compar ison of observed lines with a data base of lines collected under laboratory conditions in order to det ermine a value of the redshift λobs=λ0/parenleftbig 1 +z/parenrightbig (12) and, hence, a time separation between the epoch of the absorp tion and the epoch of the observation t(z). This depends on the choice of evolution parameters of the univers e and can vary by a factor two for the same z. When the lines are identiﬁed, one can try to interpret spectr oscopic data in terms of variations of transition frequencies. It is quite important to determine the redshif t and look for the variation simultaneously. When one does this separately it is equivalent to an assumption on the stability of particular transitions. E. g. in Ref. [33], the authors used the redshift from the observation of s ome astrophysical data on the hydrogen hfsline, and so they actually assumed within the evaluation that the α2µp/µbRyis a stable value2and this led to some misinterpretation. A comparison of frequencies with diﬀerent scaling behaviou rs is described in Sect. 1.2. Particularly, Savedoﬀ [35] pointed out this application for the comparison of atom ic lines. Thompson ﬁrst noted that molecular spectra could be used to examine the variation of the nuclear masses [ 36]. An analysis of absorption lines of molecular hydrogen in a quasar spectrum can possibly provide a limit fo r the variation of the proton-to-electron mass ratio. Such an evaluation is based on the Born-Oppenheimer a pproximation of the molecular spectrum νmol= Ry/bracketleftbigg ce+cv/parenleftBigme M/parenrightBig1/2 +cr/parenleftBigme M/parenrightBig +. . ./bracketrightbigg , (13) where ciare dimensionless parameters of order O(1). The dominate term ceis determined by the electronic structure, the second is due to the vibrational excitations , while the third one is associated with the rotational levels. Comparing levels with the same electronic structur e (the same a), it is possible to study two other terms and to limit the variation of the electron-to-proton mass ra tio from the H 2spectrum. Value Mis associated with some nuclear mass, particularly in the case of diatomic molecules it is the reduced mass of two nuclei. It is important for applications that the coeﬃcients cican be found both theoretically and experimentally. The latter is possible by studying diﬀerent isotopes, particul arly the reduced mass Mfor H 2, HD, D 2etc varies enough to allow this (see e. g. Ref. [33]). An examination of the molecular lines, or a comparison of the rotational and vibrational transitions with the gross structure yields a variation of me/mp. In contrast to this a comparison of the rotational terms to t he 2That was pointed out in Ref. [34]. 7hfsin the hydrogen atom yields a limit for the α2gpvariation. Analysis of the molecular lines was performed in Ref. [37, 38], while in Refs. [39, 40, 33, 41, 34] the authors p referred to compare the rotational and vibrational transitions with the hfsof the hydrogen atom. The most accurate results are collecte d in Table 3. A comparison of the gross structure to the ﬁne structure of so me ions presented in Refs. [42, 43, 44, 45, 46, 40, 33, 47, 38, 48, 49] gave limitations on the variations of α2. Variations of the same value can be found after studying the relativistic corrections. The most recent res ults were obtained in [50], where an evaluation of data for some of Fe+and Mg+lines was performed. The variation of the ﬁne structure cons tant was obtained from a study of relativistic corrections, calculations for whic h were presented separately in Ref. [51, 52]. Thehfsof atomic hydrogen was examined with respect to the gross str ucture in Refs. [43, 40] and with respect to the ﬁne structure in Refs. [43, 33]. The former of t hose examinations is for possible variations of α2(µp/µb), while the latter is for µp/µb. The strongest astrophysical limitations are summarized i n Table 3, where we give variations for actually measured values. Refe rences to previous, less precise results can be found in quoted articles and in Ref. [44]. Value Ref. ∂ln(<value >)/∂t [yr−1] me/mp [37] <3·10−13 me/mp [40] −0.8(35)·10−14 me/mp [47] <2·10−14 me/mp [38] 9(6)·10−15 α2[46] <6·10−14 α2[45] <4·10−14 α2[40] 1.0(35)·10−14 α2[33] <3·10−14 α2[38] <5·10−15 α2[50, 51] 0.4(51)·10−15 α2[48] <5·10−15 α2gp [33] <2·10−14 α2gp[34, 41] <2·10−15 α2µp/µb[43] <2·10−14 α2µp/µb[40] −1.0(13)·10−15 Table 3: Possible variations of the constants from astrophysics. 2.2.2 Background radiation An analysis of microwave background radiation data to be obt ained in the near future is expected to give a limitation for a variation of the ﬁne structure constant of 1 0−12−10−13yr−1[53, 54]. 3 Laboratory search The limitations from Eq.(10) and Table 3 are stronger than po ssible in any laboratory experiments. However, to our mind, the most reliable limitations can be achieved on ly under laboratory conditions and there are a few very diﬀerent ways to determine limits for possible variati ons of the fundamental constants. 83.1 Laboratory nuclear data As has been mentioned, Shlyakhter noted that the laboratory study of some nuclear properties can give reason- able limitations for variations of the constants [24]. He co nsidered some very low-lying resonances, the energies of which are extremely small diﬀerences of large quantities and those must be sensitive to small variations of parameters. The positions of some of these resonances had be en known with enough accuracy for about 10 years and an estimate 1 αs∂αs ∂t<4·10−12yr−1(14) was obtained [24]. The diﬀerence between this approach and o thers was that the most sensitive values were studied, whereas others investigated easily available dat a from geochemistry. Such a test is free of the timing problem, although, Schlyakhter’s analysis has been perfor med under the assumption of particle masses stability. Unfortunately, to the best of our knowledge this idea has not been developed further. Actually the estimations [24] from laboratory data were competitive with ordinary ge ochemical data examined that time. However, that is not ture in the case of the Oklo reactor. We think that it is necessary to examine the data base of low-l ying resonances. Even, if there is no progress in measurement, the limitation in Eq.(14) is reduced by a fac tor of about 3.5. That is a result of adjustment of analysis by Shlyakhter, who claimed in 1976, that the positi ons of the resonances had not been shifted for 10 years. 3.2 Clock comparison Another example for a laboratory search would be a compariso n of diﬀerent clocks looking for any variation during a relatively short time ( ≤1 year) [55, 57]. Ref. [55] presented a one-year comparison o f hyperﬁne structure of Cs to the ﬁne structure of24Mg. Another limit on a possible variation of the ratio of the f requencies from the same authors [56] is 2 .6·10−13yr−1, though, the data seems to be the same. They neglected relati vistic corrections in their evaluation and gave some interpretati on based on that. Another recent comparison of the Hg+clock (based on the hfstransition) and hydrogen clock for 140 days [57] led to a limit of 1 .7·10−14yr−1. The original result was presented in terms of the ﬁne struct ure constant which was derived from relativistic corrections t o the nonrelativistic formula. The authors of Ref. [57] underlined the importance of relativistic eﬀects. We would like to point out that the treatment of the Cs hfs in Ref. [55] is just the opposite: while here the relativisti c corrections were neglected [55], the others believe that the corrections are crucially important [57]. The resu lts from diﬀerent clock comparisons are collected in Table 4. We give here the limits for possible variations of th e ratio of the frequencies and dismiss any original interpretations. We include also an H–Cs comparison for 1 ye ar at the PTB [60] with a result 5 .5·10−14yr−1 and at U. S. Navy Observatory [61], as has been interpreted in Ref. [57]. We would also like to mention a result of Ref. [62] because the clock was quite diﬀerent from others. A 12-days comparison of a Cs clock to a new standard based on and SCSO (superconductivity-cavity stabilized oscillator [63]) was performed. The frequency of the standard depends on its size, which is taken as proportional to the Bohr radius. This is correct in a nonrelativistic approximation, and it is not quite clea r how to estimate the relativistic corrections. Transitions Ref. ∂ln(<value >)/∂t [yr−1] hfsof Cs to SCSO [62] <1.5·10−12 fsof24Mg to hfsof Cs [55] −2.5(23)·10−13 hfsof H to hfsof Cs [60] <5·10−14 hfsof H to hfsof Cs [61] <5·10−14 hfsof Hg+tohfsof H [57] <2.7·10−14 Table 4: Clock comparisons and possible variations in frequencies o f the standards. 9Ref. [57] in the most important one in the table, because the h ydrogen-to-cesium comparisons are taken from there and because of discussions on the relativistic eﬀ ects. We discuss the interpretation of the hyperﬁne separation in Sect. 5, but here we comment on some statements of Ref. [57]. •The correcting function for the relativistic eﬀects Frel(Zα) used there was not quite correct. The authors did not give enough explanations and to brieﬂy discuss their evaluation we would like to mention a few points: - The relativistic correcting function Frel(Zα) was given with some analytic expression, while the non- relativistic term was possible to ﬁnd only within an empiric al formulae. Indeed, that is inconsistent. The function was expected to be valid for any alkali atoms inc luding hydrogen-like and Li-like atoms. The result for Frel(Zα) isn-independent and it was claimed to be valid for S1/2levels, while rather it should be n-dependent (see e. g. Ref. [58]). Particularly, the result u sed in Ref. [57] is in dis- agreement with both 1 sand 2sresults for a hydrogen-like atom with a nuclear charge Z[58]. In the case of low- Zthe results for a hydrogen-like atom are Frel(Zα)≃1 +11 6(Zα)2+. . ., [57], (15) F1s(Zα)≃1 +3 2(Zα)2+. . . , [58], (16) and F2s(Zα)≃1 +17 8(Zα)2+. . ., [58]. (17) - Actually, the calculation of the Casimir correction Frel(Zα) =3/radicalbig 1−(Zα)21 3−4(Zα)2(18) has been performed under the condition that the relativisti c corrections can appear only when the electron is close to the nucleus and hence they are proportio nal to a squared value of the wave function at the origin [59]. That is correct for heavy ( Z≫1) and slightly charged ( z≪Z, where zis an eﬀective charge for a valence electron) alkali atoms. Indee d that is incorrect for hydrogen. Actually, the corrections for hydrogen are small, and it is enough to re produce a correct order of magnitude. - Recalculation in Ref. [52] gave results for ∂ln(Frel(Zα))/∂α, which are diﬀerent from the Casimir calculation ∂ln/parenleftBig Frel(Zα)/parenrightBig ∂lnα=(Zα)2/parenleftBig 11−12(Zα)2/parenrightBig /parenleftBig 1−(Zα)2/parenrightBig /parenleftBig 3−4(Zα)2/parenrightBig (19) within about 10%. The results are 2.30 for Hg+and 0.83 for Cs [52] instead of 2.2 and 0.74 [57]. Actu- ally the mercury ion is not an alkali one, but since all subshe lls are closed the Casimir approximation has to work and that has been conﬁrmed by the many-body calcul ation [52]. •The authors assumed that there are no corrections to the nucl earg-factor which depend on the strong coupling constant. Actually that means that the magnetic mo ment of any nucleus is to be understood in terms of a pure kinematic description (spin and orbital co ntributions) with high accuracy. That is deﬁnitely not the case (see Sect. 5 for detail). The correcti ons do not grow with increase of the nuclear charge Z, but nevertheless they are large enough for a number of value s ofZin a broad range. Particularly, for tritium ( Z= 1) such eﬀects shift a value of the nuclear magnetic moment b y 7% (cf. Eq.(18) with Z= 26). Even in the case of a pure kinematic model, it is incorre ct to neglect variations of gpandgn, which contribute diﬀerently to magnetic moments of hydroge n, rubidium, cesium and mercury (see Sect. 5 for detail). •The hydrogen maser was tried intensively as a candidate for t he primary frequency standard about 30 years ago. A crucial problem was low long-time stability and it has not been improved up-to-now. That means that the maser frequency can disagree with the transit ion frequency and varies with the time because of diﬀerent eﬀects, particularly, a wall-shift. Co mparison of anything with the hydrogen maser itself makes no sense. Actually, the authors of Refs. [60, 61 ] make no statement on a possible variation of any transition frequency, and the interpretation in Table 4 is from Ref. [57]. 10The best limits for the annual variations of the constants fr om the clock (see Table 4) are on the level of a few units in 1014but it is not quite clear if any direct interpretation of such a comparison is actually possible. A clock is a device designed to maintain some frequency in the most stable way. An equality of the maintained frequency to any atomic transition frequency is not necessa ry, and, actually, it is not quite clear, if any clock frequency agrees with the transition one within an accuracy on the level of its reproducibility. We expect that, if it were to agree it would be no problem to have better limita tions by just searching for a longer time. In particular, we expect that variations of a frequency of an y maser standards should also be determined by the cavity size (cf. SCSO standard [62]). The frequency of th e hydrogen hfstransition is not itself important in some sense for the hydrogen maser. When the hfsfrequency and the size of the resonator are inconsistent the standard cannot work, and when they are consistent (with in the line width) everything is determined by the cavity. That is indeed a reason why there are a lot of possi bilities of drifts for the frequency from the maser standard. Such a maser, without any tuning of the cavity size is called a passive maser . On the other hand in the case of active maser there is an adjustment of the cavity size to the hydrogen hfsstudying the eﬃciency of the maser. In this case the dependence on the variation of the constants is more complicated. It is known that hydrogen masers can have high short-time sta bility. But there is no a priori statement applicable to any particular hydrogen maser, after a rather preliminary study of these. The study assumes some comparisons with either another known standard, or a wide re presentative ensemble of masers. In both cases, any later comparison with that maser assumes that is consist ent with some other standard and that the stability property (e. g. short-term stability) does not vary with tim e. This is an indirect comparison with something else via the maser. In the case of a 140 day comparison the mase r stability and agreement between the maser frequency and the hydrogen hfsare questionable and the stability may only be a result of pre liminary study of the maser frequency with respect to some other standard. We think that the clock comparison can give reliable limits o nly in two cases: •The clock frequency is expressed in terms of the transition f requency. But that means, that with the clock comparison one can measure the transition frequency as well . We consider some of possibility for search with the precise frequency measurements below (see Sect. 3. 3). •There are a number of diﬀerent standards with the same transi tion (like in the case of Cs). They should ﬁrst be compared with each others and then we can estimate pos sible deviations from some eﬀective frequency that depends on the transition rather than on the c lock. This is not as secure as a direct determination of the transition frequency, but it is more or less reliable. In the case where some particular transition is applied in on ly a single clock, it is absolutely unclear, which drifts or ﬂuctuations are properties of the clock and which a re properties of the transition. 3.3 Precise frequency measurements Precision spectroscopy provides us with other ways to searc h for the variation of the fundamental constants. The most straightforward method is to obtain a high accuracy and to compare two results (let us say, one taken a year after the other). It is also possible to make a com parison of results obtained now with some relatively old ones for the frequency of atomic or molecular transitions measured better than 10−11. Tables of the most accurately measured values of any transition frequ encies are presented below. Table 5 contains the best radiofrequency results, while Table 6 is for the optica l transitions. The tables contain the results obtained from 1967 to the present with a fractional uncertainty below 10−11. One can see that some older results are competitive with the newer ones. In the tables we give a possi ble limit of variation of the frequency if the new experimental value is to be obtained in the year 2000 and with some higher accuracy. If the precision is about the same, one should also take into consideration an uncerta inty in the newer measurement. So, the ﬁnal limit has to be larger than that given in the tables by a factor betwe en√ 2 (if the uncertainties are independent) and 2 (when they are strongly correlated). In our evaluation we c onsider the date of publication as the date of the measurement, whereas they are slightly diﬀerent and some sh ifts may arise from this. The hydrogen hfsis presented in Table 5 with a value from review [64]. We discu ss the original results in Sect. 6.2. The tables mainly indicates some opportunities f or experiments in the near future. Except for the hydrogen hfsonly one value in Table 5 has been accurately and independent ly measured twice (namely, the hfs intervals in the171Yb+atom). A comparison of two171Yb+measurements ([70] and [71]) gives the variation of the frequencies ∂ ∂tln/parenleftbiggνhfs(171Yb+) νhfs(Cs)/parenrightbigg ≃ −1(2)·10−13yr−1, (20) 11Atom Frequency ( ν) Ref. Fractional ∂lnν/∂t [kHz] uncertainty ( δ) [yr−1] H 1 420 405.751 766 7(9) [64],∼1970 6.4·10−132.2·10−14 D 327 384.352 521 5(17) [65], 1972 5.2·10−121.8·10−13 T 1 516 701.470 773(8) [66], 1967 5.3·10−121.6·10−13 9Be+1 250 017.678 096(8) [67], 1983 6.4·10−123.8·10−12 87Rb 6 834 682.610 904 29(9) [68], 1999 1.3·10−141.3·10−14 133Ba+9 925 453.554 59(10) [69], 1987 10·10−127.7·10−13 171Yb+12 642 812.118 471(9) [70], 1993 7.5·10−131.1·10−13 12 642 812.118 466(2) [71], 1997 2.5·10−138.3·10−14 173Yb+10 491 720.239 55(9) [72], 1987 8.6·10−126.6·10−13 199Hg+40 507 347.996 841 6(4) [73], 1998 1.1·10−140.53·10−14 Table 5: The most precise measurements of the ground state hyperﬁne s tructure interval and a possible level of limit for the variation of their frequencies. Values for ∂lnν/∂thave been calculated under the condition that the frequency will be re-measured in the year 2000 with a bett er accuracy. The variation of the frequencies is assumed with respect to the cesium hfs. if we suppose that the time separation is 4 years. It is important to note, that even a single laboratory study c an give a relatively secure result. In some experiments several traps were used and so they contain an in dependent measurement in part. Since most of the recent precision investigations have the building of a n ew standard as a target, some long-term monitoring of the measured frequency was often performed. Unfortunate ly, the published data are rather incomplete, but we expect that some limits on a level between 10−12and 10−13yr−1will be available after complete publications of studies giving most of the recent results in Tables 5 and 6. The most precise comparison with one of the results that is al ready known for some time can be performed for the hydrogen ground state hyperﬁne structure interval. The possibility to reach a good result has almost been missed. However, in the case of new experiments, like fo r rubidium or mercury, it may easily be a shift of 1–2 sigma afterwards. On the other hand, some new results a re going to be presented soon: for the Rb hfs (better than 10−14) and for the 1 s−2stransition in hydrogen (a few units in 1014). That means that in the case of any delay the measurement could be not compatible. It is also important to underline that so-called variation- of-constants experiments check diﬀerent possibilities associated with drifts of primary and secondary standards. A one-year comparison, which can usually be realized in a laboratory, is not the same as a kind of “world wide” compa rison over years. The hydrogen hfsinterval is a value which can be measured in a number of diﬀerent laborator ies now and which was in the past also studied in a few diﬀerent places. Two radio-frequency measurements (namely for tritium and b arium) were performed for unstable isotopes (T1/2(T) = 12 .3 yr and T1/2(Ba) = 10 .5 yr) and this indicates that the search for appropriate tran sitions should not be limited to stable isotopes only. We do not mention the r adioactivity of rubidium-87 which has a halftime of 4.8·1010yr, comparable with the age of the universe. We also have to mention an experiment with the ground state hy perﬁne structure of9Be+in a strong magnetic ﬁeld. A splitting between ( MI=−3/2, MJ= +1/2) and ( −1/2,+1/2) was determined [81] ν= 303 016 .377 265 070(57) kHz . (21) The measurement was performed at a ﬁeld of about 0 .8194Tand the splitting was found at its magnetic-ﬁeld- independent point. The fractional uncertainty is 1 .9·10−13and it was believed [82] that this may be reduced to about 1 ·10−13. A further measurement of the splitting in the year 2000 is to give a limit of the variation of the Be-frequency with respect to the cesium standard on leve l of 1.3·10−14. 12Transition Frequency ( ν) Ref. Relative ∂lnν/∂t [kHz] uncertainty [yr−1] H, 1s−2s 2 466 061 413 187.34(84) [74], 1997 3.4·10−131.1·10−13 H, 2s−12d 799 191 727.402 8(67) [75], 1999 8.4·10−128.4·10−12 D, 2s−12d 799 409 184.967 6(65) [75], 1999 8.1·10−128.1·10−12 40Ca,3P1−1S0455 986 240 493.95(43) [76], 1996 9.4·10−132.4·10−13 455 986 240 494.13(10) [77], 1999 2.5·10−132.5·10−13 88Sr+, 5S−4D 444 779 044 095.4(2) [78], 1999 4.5·10−134.5·10−13 CH4, E-line 88 373 149 028.53(20) [79], 1998 2.3·10−121.1·10−12 Table 6: The most accurate optical measurements and a possible level of limit for the variation of the frequency. Values for ∂lnν/∂thave been found under the condition that a measurement of the frequency will be repeated in the year 2000 with a higher precision. In Tables 5 and 6 and Eq.(21) in Fig. 1 we collect all limits of t he variations of frequency available in 2000 in the case of a repetition of the measurements. 4 Some new options for precise comparison of frequencies 4.1 Hyperﬁne structure For a while the hyperﬁne splitting of atomic levels has been a quantity available for the most precise measure- ments. Comparison of hfsin diﬀerent atoms can give us precise information on the vari ation of the nuclear magnetic moment rather than on the ﬁne structure constant. •We start with the hydrogen hfsproject. The hyperﬁne structure interval in the ground stat e of the hydrogen atom was frequently measured (see Sects. 6.1 and 6. 2). For a preliminary estimation we accept a value νhfs(H) = 1420 405 .751 766 7(9) kHz , (22) which used to be presented in reviews (see e. g. Ref. [64]) as a ﬁnal result for the hfsseparation. The fractional uncertainty is about 6 parts in 1014and on being divided by 30 years that gives 2 ·10−14yr−1. If the accuracy is now the same this should rather be multipli ed by√ 2. •Let us consider brieﬂy a possible variation of the deuterium hyperﬁne separation, which was measured in 1972 [65] with an uncertainty of 5 .2·10−12. The relative accuracy is much worse than for H (cf. Eq.(22)) . However, the magnetic moment of a deutron (see Table 11) incl udes a large cancellation µ(D)≃µp+µn=\|µp\| − \|µn\| (23) between the proton ( µp= 2.793µn) and neutron ( µn=−1.913µn) contributions and this hfsvalue might be very sensitive to the variation of eﬀective parameters of the strong interactions. •Thehfsof the ground state in the171Yb+ion can also provide a limit for the variation per year on a lev el of a few units in 1014. The most precise measurements are presented in Fig. 2, wher e the open circles are for preliminary results [83, 84], while the full ones are for ﬁnal values [70, 71]. The disadvantages (in comparison to the hydrogen case) are: a less strong limit for the variation with less reliability (the result was reached with a high accuracy in t wo laboratories, but the precision was diﬀerent by a factor 3). 131965 1970 1975 1980 1985 1990 1995 200001020 Year of publication Limitation δ/∆t [ 10-14 yr-1 ] HTD Be+Yb+ Yb+ RbHg+CaH Ca Figure 1: Strongest possible limits on the variation of the frequenci es available in 2000. The limit is deﬁned as a fractional uncertainty of the frequency divided by the t ime from the publication to 2000. •Generally, study of nuclei with small magnetic moments are e xpected to lead to sensitive tests for possible variations of the proton or neutron magnetic moment. We give a list of stable nuclei with small magnetic moments in Table 7. The tungsten183 74W has a halftime of 1 .1·1017yr and we include it in the table. Up to now, there is no way to measure the nuclear magnetic mome nt with an accuracy on the level 1011or better. For nuclei with spin 1/2, our proposal is to measure t hehfsof a neutral atom or ion and to search for a variation of that value. In the case of spin 3/2, a value o f thehfsinterval is essentially aﬀected by the nuclear quadrupole moment. An exception is41K, where the quadrupole term is also small and it is worthwhile studying the hfs. Fortunately, the stable nuclei with the smallest magnetic moments (namely, 57Fe,103Rh and187Os) have spin 1/2 and we hope that the accurate study of the hfsof the nuclei with a small magnetic moment is possible. The nuclear magnetic moment of radionuclides can be even sma ller, for example, for198 81Tl (\|µ\|<10−3µn, I= 2−,T1/2= 5.3(5) h),153 62Sm (µ=−0.022µn,I= 3/2−,T1/2= 46 h) and192 79Au (µ=−0.009(2) µn, I= 1−,T1/2= 4.9 h). Our proposal for such isotopes is to measure a value of th e shielded magnetic moment of the nucleus, investigating ions with coupled elec trons only (Hg-, Os-, W-, Hf-, Ba-, Xe-like etc). Particularly, Hg-, Ba- and Xe-like ions have complete subsh ells and this is an advantage for the study of 198Tl. We hope that a method developed in Ref. [85] to study a boun d electron g-factor in H-like ions can be applied here. By achieving an uncertainty of about 10−9for the magnetic moment, the limit for the variation of gpis expected to be on the level of 3 ·10−13yr−1. We should mention that not all magnetic moments of radionuclides with a halftime longer than 10 days are known [27] and it might happen, that some of them are even smaller than 10−3µn. 4.2 Fine structure When the ﬁne structure (proportional to α2Ry) is determined, it can be compared with the gross structure (proportional to Ry) and thus yielding a direct limit for a variation of the ﬁne st ructure constant α. The gross structure can be taken from measurements with neutral hydro gen and calcium atoms, and with strontium and indium ions and, maybe, in the future with other atoms. •To date there are no competitive results on the fs. The best result for the atomic ﬁne structure has been 141993 1995 19970.460 Hz0.465 Hz0.470 Hz0.475 Hz0.480 Hz0.485 Hz Year of publication Frequency νHFS - 12 648 812 124 Hz Figure 2: Precise study of the hfsinterval in the ground state of171Y b. reached for the Ba+ion [86]3 νfs(138Ba+,5d2D3/2−5d2D5/2) = 24 012 048 317 .17(44)kHz . (24) Note that this is for the fsof excited states. We think that some higher accuracy can be a chieved by studying the ﬁne structure associated with the ground state . If the subshell with valence p-electrons (or d) is open, the lowest excited states are due to the ﬁne structu re. The frequency can lie in radio-frequency range and the lines are very narrow. That is because of two rea son: the E1 transition is not allowed for P−Ptransitions and the decay rate is proportional to some power of the low transition frequency. Thus the lowest levels are split due to relativistic eﬀects only a nd they can be measured accurately. This is another way of measuring the ﬁne structure precisely. In com bination with the gross structure, one can reach a limit for α. The interpretation of such a comparison is simpler than in t he case of hfs. A similar way is to study the relativistic correction for the gross str ucture is of use for astrophysical data [35]. In Table 84we give a list of the rftransition of the low-lying ﬁne structure in some neutral at oms, while those for diﬀerent ions are collected in Table 9. Unfortunately, there are no systems in the tables which can b e easily studied. In the case of neutral atoms in Table 8, laser cooling is hard to apply because of the metas table ﬁne structure levels. Finding levels which are insensitive to the magnetic ﬁeld is a problem for io ns. These are (23P0) in N+and (3d4−5D0) in V+. Finding proper means of detection of the fstransition can be another problem for a few-ions trap experiments. •It is also possible to ﬁnd an optical or infrared transition f or the low-lying ﬁne structure. Let us mention a transition νfs(Pb,63P0−63P2)≃1.32 eV (25) in neutral lead, which can be studied by means of two-photon D oppler-free spectroscopy (the wave length of each photon is 1.88 µm). The excited level3P2lives for 2.6 sand it is narrow enough to reach an accurate result. It has to be mentioned that in the case of neu tral lead, calculations using jjcoupling are competitive with those using LSone. Actually, a clear separation of non-relativistic and r elativistic physics is only possible for LScoupling. LScoupling means that one must ﬁrst ﬁnd a non-relativistic energy level with n∗(L) and next to take into account the (relativistic) spin eﬀect s. For jjcoupling the (relativistic) spin eﬀects for individual electrons are mo re important than a (non-relativistic) interaction of their orbital momenta. We expect large relativistic corr ections to the ﬁne structure. One possible experiment is presented in Fig. 3. The ground st ate (6p2 3P0) of one of the spinless isotopes of lead (204Pb, natural abundance 1.4%;206Pb, 24%; or208Pb, 52%) is excited by two photons ( λ= 1.878µm) 3In Refs. [55, 56] a clock, based on ﬁne structure in Mg, was com pared with a cesium clock. However, the authors gave no resul t of the fstransition frequency. As I was informed by A. Godone it was ex pected that the corrections to the fswere well understood at least on level of 10−12. 4Atomic data (and particularly that in Tables 8, 9 and 10) have been taken from Ref. [80], unless otherwise speciﬁed. 15Natural Nuclear Magnetic ZIsotope abundance spin and moment parity [µn] 715N 0.37 % 1/2−-0.2831884(5) 1941K 6.7 % 3/2+0.2148701(2) 2657Fe 2.2 % 1/2−0.09044(7) 3989Y 100 % 1/2−-0.1374154(4) 45103Rh 100 % 1/2−-0.08840(2) 47107Ag 52 % 1/2−-0.11357(2) 109Ag 48 % 1/2−-0.13056(2) 64155Gd 15 % 3/2−-0.2591(5) 157Gd 16 % 3/2−-0.3398(7) 69169Tm 100 % 1/2+-0.2316(15) 74183W 14 % 1/2−0.11778476(9) 76187Os 1.6 % 1/2−0.06465189(6) 77191Ir 37 % 3/2+0.1507(6) 193Ir 63 % 3/2+0.1637(6) 79197Au 100 % 3/2+0.145746(9) Table 7: Isotopes with small nuclear spin. to a metastable3P2level with a lifetime of 2.6 s. The line is narrow because of th e metastability and of Doppler-free excitation. Detection of the3P2level can be done using an additional one-photon excitation (λ= 0.283µm) to a 6 p7s3P0state and measuring the ﬂuorenscence ( λ= 0.406µm). •Alkali atoms have simple spectra and that is an advantage for both experiment [55, 76, 78, 86, 88] and theory [59, 52]. Measurement of the ﬁne structure of such a sy stem as a test for the variations of αwas proposed by Jungmann [87] (cf. Ref. [35]) particularly for C a+and Sr+ions. Similar measurement can be performed for In+. All these atoms are now a subject of some investigations as a part of eﬀorts to design new optical standards. Let us discuss shortly the indium cas e. An accurate result for the ﬁne structure may be obtained by considering the 5 s5p−3PJlevels in the115In+ion. The fsinterval of excited levels cannot usually be measured precisely. For an indium ion it ma y however be determined as the diﬀerence between two gross transitions (5 s2−1S0−5s5p−3PJ)ith diﬀerent J. Indeed, since the fsinterval is about 1% of the transition frequency for the gross structure, one c an expect that the fractional accuracy is not very high. The advantage is that in the case of very accurate m easurements of 5 s2−1S0−5s5p−3P0 for diﬀerent PJ-states, it may be possible to go beyond the accuracy of stand ards. As an example, let us recall the results on the 1 s−2stransition in the hydrogen atom [74] and on the hydrogen-deu terium isotopic shift of the 1 s−2sfrequency [90]. By comparing the two frequencies, it was pos sible to detect a drift of the standard used and to reduce the absolute uncerta inty to 150 Hz for the isotopic shift, while for hydrogen this was 850 Hz. The absolute measurement in the indium ion (5 S−5P0transition) now gives [88] 1 267 402 452 914(41) kHz (uncertainty is 3 .2·10−11) and the result is soon to be improved. •Another approach to compare the gross and ﬁne structure may p ossibly be realized in an atomic system, where the levels with diﬀerent nandllie close each to other. This may be in the case of an accidenta l cancellation of the Ryandα2Ryterms. An example of such a cancellation can be seen in the spe ctrum of the Ag atom. Two excited multiplets, 4 d105pand 4d95s2(one of the 4 d95s2lines is quite narrow ∼1 Hz), are split slightly. The splitting comes from the gross s tructure. However, it is comparable with the 16Z Atom Level Energy Lifetime Nuclear spin 5B(22P0 1/2)22P0 3/20.457 THz 3.2·107s 3 (10B), 3/2 (11B) 6C(23P0)23P10.492 THz 1.3·107s 0 (12C), 1/2 (13C) 23P21.30 THz 3.7·106s 14 Si(33P0)33P12.31 THz 1·105s0 (28Si,30Si), 1/2 (29Si) Table 8: Low-lying rfﬁne structure of neutral atoms. Z Atom Level Energy Nuclear spin 6 C+(22P0 1/2) 22P0 3/21.90 THz 0 (12C), 1/2 (13C) 7 N+(23P0) 23P1 1.46 THz 1 (14N), 1/2 (15N) 23P2 3.92 THz 21 Sc+(3d4s−3D1) 3d4s−3D2 2.03 THz 7/2 (21Sc) 3d4s−3D3 5.33 THz 22Ti+(3d24s−4F3/2)3d2(3F)4s−4F5/22.82 THz 0 (46Ti,48Ti,50Ti), 3d24s−4F7/2 6.77 THz 5/2 (47Ti), 7/2 (49Ti) 23 V+(3d4−5D0) 3d4−5D1 1.08 THz 6 (50V), 7/2 (51V) 3d4−5D2 3.20 THz 3d4−5D3 6.26 THz 3d4−5D4 10.17 THz Table 9: Low-lying rfﬁne structure of single-charged ions. internal structure of the multiplets, which is due to the rel ativistic corrections (ﬁne structure). For such a cancellation, a value of energy splitting between levels f rom diﬀerent multiplets is quite sensitive to a variation of α. Measuring the splitting with relatively low accuracy, it i s possible to reach a strong limit. A similar idea for the 4 f105d6s−4f95d26slines in the Dyatom was proposed in Ref. [51, 52]. In some sense, the search for the accidental degeneration is quite c lose to approaches using nuclear data, in which the smallness of some diﬀerences is widely utilized (see e. g . Ref. [24]). Let us discuss conditions needed for success in such an exper iment. We consider the spectrum of the Ag atom as an example and some properties of low-lying levels in that atom are presented in Fig. 4 and Table 10. The conditions for a precise measurement of a transition sen sitive to possible variations of the constants are: –The ﬁne structure terms should be of the same order of magnitu de as the gross structure contributions. In the silver ion this is true: the splitting of the two 5 p2P0 Jlevels is about the same value as the separation between one of them (5 p2P0 3/2) and one of the 4 d95s22DJlevels (4 d95s22D5/2). The other ﬁne structure splitting (between Dlevels) is larger than the P−Dinterval. –The non-relativistic ( Ry) and relativistic terms α2Ryhave to have a diﬀerent sign. This is very likely because of the sandwich sequence of levels P−D−P−Dwith the center of gravity of the Dlevel lying above that for the Pstates. That means that the non-relativistic contribution for the Dstates is likely higher than for the Pstates. As far as we are interested in the lower Dlevel, we 170.00.1005.00.1031.00.1041.50.1042.00.1042.50.1043.00.1043.50.1044.00.1044.50.104 Energy of levels [cm-1] 6p2 3P06p2 3P2 (τ = 2.6 s)6p7s 3P10 (τ = 6 ns) 6p2 3P16p7s 3P00 6p2 1D26p2 1S0 λ = 1.878 µmλ = 0.406 µmλ = 0.283 µm Figure 3: The energy levels and a scheme of the experiment on the ﬁne str ucture of neutral lead. can expect that the relativistic contribution to that is neg ative. Indeed, if silver qualiﬁed for other conditions it would be studied theoretically before perfor ming the experiment. The relativistic eﬀects shift and split levels. We can estimate an enhancement of the sensitivity assuming that there are no shifts, but only splittings. We expect this should work for a preliminary estimation. In such a case a non-relativistic correction can be found from the separat ion of the centers of gravitation of the P andDlines: ∆NR(2P0 1/2−2D5/2) =−55.89 THz = ARy , (26) while the relativistic corrections are obtained from the sh ift of the energy level from the position of the center of gravity ∆Rel(2P0 3/2−2D5/2) = 62 .82 THz = B α2Ry . (27) The separation eventually is equal to ∆(2P0 3/2−2D5/2) = 62 .82 THz −55.89 THz = 6 .93 THz . (28) A variation of the Rydberg constant can be neglected and one c an ﬁnd ∂ln(∆/Ry) ∂lnα≃2B α2Ry ∆≈2·10. (29) The factor 2 is the common factor because any relativistic co rrection depends on α2and 10 is an estimation of the enhancement due to the accidental degener ations. 18875900925950975100010251050Energy from ground state [ THz ] 2P1/22P3/22D3/2 2D5/2 Ground state 2S1/2 Figure 4: Scheme of low-lying levels of the neutral Ag atom. The hyperﬁ ne structure is not shown. –For accurate measurement it may be important to use laser coo ling. An important conditions for this is lack of the hyperﬁne structure. Unfortunately both s table isotopes (107Ag and109Ag) have some magnetic moment and the ground state of any stable isoto pe is usually split into two states. The laser cooling is possible but rather complicated. –It may also be important to apply two-photon Doppler-free ex citation to produce one of the two levels, splitting of which contains a cancellation between relativistic and non-relativistic term. This is necessary because it is somehow possible to cool the groun d state, but not excited states. We have to eliminate Doppler eﬀects due to excitation. The Dstates in the silver atom can be excited by means of the two-photon transitions. –Both levels have to be narrow. The Dlevel (4 d95s22D5/2) is very narrow with a width of about 1 Hz, but both Pstates are very broad. –A precise measurement of the small splitting due to the cance llation must be possible. Generally this means that one must be to induce a single-photon transition. The one-photon transition between states 4 d95s22D5/2and 5p2P0 3/2lies at 7 THz. It can be induced but it is hard to measure such a transition frequency precisely. One can see that the conditions can be realized in some atomic systems and it is necessary to search for them. One should note that in the case of ions the condition fo r choice of nuclear spin is diﬀerent. It may be more important to have some states that are insensitive to a magnetic ﬁeld (the whole moment Fof a system of electrons plus nucleus must be integer and states w ithmF= 0 are alowed). Our suggestion is to look for a relatively small enhancement but with an appropriate possibility for precise measurements. The proposal with Dyin Refs. [51, 52] is rather for a great enhancement without an y accurate spectroscopy. 5 Nuclear magnetic moments and interpretation of the freque ncy comparisons A frequency comparison involves atoms that are very diﬀeren tin nature and one must be prepared to interpret results. Most precise results are for the hfs(see Table 5) and we mainly discuss the hfstransitions. Since 19Ground state: 4 d105s2S1/2 Excited states Level Energy Lifetime 4d105p2P0 1/2885.94 THz 8·10−9s 4d95s2 2D5/2906.64 THz 0.2 s 4d105p2P0 3/2913.55 THz 7·10−9s 4d95s2 2D3/21040.70 THz 9·10−5s Fine structure Multiplet Thefs Center of splitting gravity 4d105p2P0 j 27.60 THz 904.35 THz 4d95s2 2Dj134.06 THz 960.24 THz Table 10: Properties of low-lying excited states of the neutral silve r atom. The hfsis neglected. any absolute measurements assume a comparison to the cesium hyperﬁne separation this is also important for interpreting the absolute optical measurements (see Table 6). 5.1 Magnetic moments To the leading order the hfsinterval can be presented in the form of Eq.(2). There are two diﬀerent factors important for the comparison: magnetic moments and the rela tivistic corrections. The magnetic moments and some other nuclear properties are collected in Table 11. Some atoms, hyperﬁne separation in which was measured accurately, but less precisely than one part in 1011, are also included: •νhfs(43Ca+) = 15 199 862 .858(2) kHz [91]; •νhfs(113Cd+) = 3 225 608 .286 4(3) kHz [92]; •νhfs(131Ba+) = 9 107 913 .699 0(5) kHz [69]; •νhfs(135Ba+) = 7 183 340 .234 9(6) kHz [93]; •νhfs(137Ba+) = 8 037 741 .667 7(4) kHz [94]. Most of these are stable, expect cadmium ( T1/2(113Cd) = 9 ·1015yr) and the lightest barium ( T1/2(131Ba) = 12 d). The sign of the magnetic moment of131Ba was presented in Ref. [27] as unknown and we follow Ref. [69 ]. A discussion on the value of the nuclear magnetic moments is a lso important because of our proposal to look for variations of small moments. All nuclei in Table 11 and 7 have an odd value of A, while Zis even for iron, gadolinium, osmium and tungsten (Table 7) and calcium, cadmium, barium, ytterbium and mercury (Table 11) and odd for all others. An even value of Zindicates that the nuclear magnetic moment is associated wi th the neutron magnetic moment and in the case of odd Zthe moment is due to the proton one. Let us start with small mom ents. Some of these can be understood using a simple model (the Schmidt model), w hile assuming that the magnetic moment of the nucleus—like a moment of an electron in a hydrogen-like a tom—includes a spin part and an orbital part µa(I) =µI/I=µs aS+µl aL, (30) 20Nuclear Magnetic ZNucleus spin and moment parity [µn] H 1/2+2.793 D 1+0.857 T 1/2+2.979 49Be 3/2−-1.178 2043Ca 7/2−-1.318 3787Rb 3/2−2.751 48113Cd 1/2+-0.622 55133Cs 7/2+2.582 56131Ba 1/2+-0.708 133Ba 1/2+-0.772 135Ba 3/2+0.838 137Ba 3/2+0.938 70171Yb 1/2−0.494 173Yb 5/2−-0.680 80199Hg 1/2−0.506 Table 11: Properties of some nuclei important for precise microwave s pectroscopy. where I=S+L,µs p= 2µp,µs n= 2µn,µl p=µnandµl n= 0. We should remember that the values of the spin terms ( µs a) originate from dynamic eﬀects of Quantum Chromodynamics ( QCD) in the strong coupling regime and so they sensitive to a variation of the QCD coupling const ant (αs). The Schmidt model leads to some relatively small values in a few cases: •for odd Z,I= 1/2,L= 1, in particular, N, Y, Rh and Ag in Table 7 (a cancellation be tween spin and orbit contributions) µ=4−gp 6µn∼ −0.26µn, (31) where gp= 2·2.793. . .; •for odd Z,I= 3/2,L= 2, particularly, K, Ir and Au in Table 7 (a cancellation betw een spin and orbit contributions): µ=3 10/parenleftBig 6−gp/parenrightBig µn∼0.12µn; (32) •for even Z,I= 1/2,L= 1, in particular, iron, osmium and tungsten in Table 7, ytte rbium–171 and mercury in Table 11 and153 62Sm, discussed in Sect. 4.1 (the result is relatively small be cause of the coeﬃcient 1/3): µ=−gn 6µn∼0.64µn, (33) where gn=−2·1.913. . .. In all other cases presented in Tables 11 and 7 the value of the magnetic moment is not smaller than one in units of the nuclear magneton µn. In some cases the agreement between the Schmidt values and t he actual ones is a 10% level, but in other cases the actual values are signiﬁ cantly smaller and that is a result of the nuclear 21eﬀects and, hence, small magnetic moments are sensitive to t hese. Comparison of Eqs.(31), (32) and (33) shows that the nuclei with odd Zand odd Acan be more interesting because the magnetic moment is small partly due to cancellations between the spin and orbit contributio ns, which are sensitive to variation of gp. We collect the Schmidt values for diﬀerent nuclear spin µs(I=L±1/2) =I/parenleftbigg gl±gs−gl 2l+ 1/parenrightbigg µn, (34) where gb=µb/µnandµbfor a proton and neutron are deﬁned above, in Table 12. Spin ( I) and Schmidt value of the magnetic moment ( µS) of odd- Anuclei l I=l+ 1/2 I=l−1/2 I Magnetic moment [ µn] I Magnetic moment [ µn] OddZ Even Z OddZ Even Z 01/2+gp/2≃2.793 gn/2≃ −1.913 - - - 13/2−gp/2 + 1≃3.793 gn/2≃ −1.913 1/2−1/3/parenleftbig 2−gp/2/parenrightbig ≃ −0.264 1/3/parenleftbig −gn/2/parenrightbig ≃0.638 25/2+gp/2 + 2≃4.793 gn/2≃ −1.913 3/2+3/5/parenleftbig 3−gp/2/parenrightbig ≃0.124 3/5/parenleftbig −gn/2/parenrightbig ≃1.148 37/2−gp/2 + 3≃5.793 gn/2≃ −1.913 5/2−5/7/parenleftbig 4−gp/2/parenrightbig ≃0.862 5/7/parenleftbig −gn/2/parenrightbig ≃1.366 49/2+gp/2 + 4≃6.793 gn/2≃ −1.913 7/2+7/9/parenleftbig 5−gp/2/parenrightbig ≃1.717 7/9/parenleftbig −gn/2/parenrightbig ≃1.488 Table 12: Magnetic moment, spin and parity of nuclei in the Schmidt mod el (Eqs.(30) and (34)). Comparison of the Schmidt model with the most important isot opes for precision measurements and vari- ations of the constants is presented in Fig. 5. Here we collec t the Schmidt values (two lines) and actual values of the magnetic moments of the isotopes from Table 11 ( ﬁlled circles), other stable or long-lived iso- topes associated with simple atomic spectra of neutral or si ngle-charged ions (open circles), and the isotopes from Table 7 with small magnetic moments (triangles). One nu cleus of this kind (87Sr, 9/2+,µ=−1.094µn, µs(9/2+) =−1.931µn)) is not included in the ﬁgure because of its large spin. From the ﬁgure one sees that the agreement between real value s and the simple Schmidt model is not perfect. This means that nature of nuclear spin are more comp licated and the eﬀects of nuclear interaction are signiﬁcant. However, the model contains some important phy sics: e. g. the model predicts simple relations between the nuclear parity and magnetic moment. Only one iso tope in Fig. 5 has inconsistent values of the parity and spin (169Tm, 1/2+,µ=−0.23µn). There is no simple model for even A, when there are two valence particles—a proton and neutron— and a contribution of orbital motion. For deuterium the proton an d neutron contributions have diﬀerent signs (23). Two radionuclides discussed in Sect. 4.1,198 81Tl,192 79Au, have even A. 5.2 Relativistic corrections Now let us discuss the hfsintervals in Table 5. First, we consider the relativistic co rrections, which have been in part discussed in Sect 3.2. They can be calculated within usi ng many-body perturbation theory (see e. g. Ref. [52]). An approximation of Eqs.(18) and (19) is a good one for alkali atoms (all but ytterbium and mercury in Table 5) under the conditions: •High nuclear charge: Z≫1. •Atoms are neutral or slightly charged: Z≫z, where zis an eﬀective charge of a compound nucleus for the valentce electron, i. e. z= 1 for neutral atoms and z= 2 for single-charged ones. Note that, Eq.(18) is incorrect for hydrogen and its isotopes and for the Li-lik e ion of beryllium presented in Table 5. 22a: odd Z b: even Z 172 3/2 5/2 7/2-2.50-2.00-1.50-1.00-0.500.000.501.001.502.00 1/2 3/2 5/2 7/2-0.50.51.52.53.54.55.56.5 I = l + 1/2 I = l - 1/2 I = l - 1/2 I = l + 1/2 small moments small moments T H 39K87Rb 23K7Li 85Rb137Cs 133Cs 3He199Hg 171Yb137Ba 135Ba 41Ca43Ca9Be201Hg173Yb 25Mg113Cd111Cd 133Ba67Zn 131BaNuclear magnetic moment µ [µN] Nuclear spin I Figure 5: Magnetic moment of some isotopes: actual values and the Schm idt model. Nuclei with small magnetic moments are listed in Table 7. •Accuracy is expected to be on a 5 −15% level and a comparison of two atoms with a small charge diﬀ erence (Z1,2≫ \|Z1−Z2\|) will not give accurate results. As already mentioned, due to a discussion on the interpretat ion of results in Ref. [57], any atom with a closed subshell and one valence electron satisﬁes the above condit ions. Alkali atoms actually form only one example of such atomic systems. Relativistic corrections are also important for the gross a nd ﬁne structure. They have a relative order of (Zα)2. In the simplest case, namely for alkali atoms, they were dis cussed in Ref. [52], where some results were obtained for Ca and Sr+and in Ref. [95] for In+. It seems that relativistic eﬀects are less important for th e optical transitions than for the hfsbecause of relatively small numerical coeﬃcients and an ext ra 1/n∗factor. In the hydrogen atom for both hfsand 1s−2stransitions the corrections are negligible. 5.3 Hyperﬁne structure Thehfsof atoms with nuclei with even Zand odd Ais sensitive to the neutron magnetic moment. In particular the Schmidt model predicts the magnetic moment of mercury an d ytterbium–171 (see Eq.(33)) within 15% uncertainty. In contrast the hfsof atoms with odd Zand odd Adepends on a value of the proton magnetic moment. An important point however is the size of nuclear eﬀe cts of the strong and electromagnetic interactions. They can be estimated by a deviation from the Schmidt value. I n the case of Cs the nuclear eﬀects increase the value by a factor of about 1.5. This means that any interpreta tion of comparisons with Cs cannot neglect the strong interactions. In the case of even-to-even or odd-to-odd comparisons, in pa rticular for H–Rb, and Be–Yb–Hg, we expect an information on the variation of the constants due to the nu clear eﬀects and the relativistic corrections. Comparison of odd-to-even hfsyields to variation of gnwith respect to gp. This is important, because there is no reason to expect that a value of gn/gpis relatively stable, while the constants are varying. Our p oint of 23view is diﬀerent from that in Ref. [57], where the H–Hg+comparison was examined assuming that a variation of the nuclear g-factors can be neglected. We have not mentioned the Cs hfsbecause here the interpretation is slightly diﬀerent. Alth ough133Cs is an isotope with odd Z, its comparison with H or Rb is sensitive to a variation of gp, because the gpcontribution to the nuclear gfactor has a negative sign in contrast to H and Rb. For example, the sensitivity of the H-to-Cs comparison to the gpvariation is determined by the value of ∂ ∂lngplnµs(Cs) µs(H)=−10 10−gp≃ −2.3, (35) and for the Rb-to-Cs comparison one can ﬁnd ∂ ∂lngplnµs(Cs) µs(Rb)=−12gp/parenleftbig 2 +gp/parenrightbig /parenleftbig 10−gp/parenrightbig≃ −2.0. (36) In contrast, the H-to-Rb comparison is actually insensitiv e to any variations of the proton g-factor: ∂ ∂lngplnµs(Rb) µs(H)=−2 2 +gp≃ −0.26. (37) Magnetic moment Relativistic Atom Naive ( µs) Actual ( µ) correction Eq. for µs[µn][µn][µn]µ/µsF(α)∂lnF/∂lnα H gp/2 2.79 2.79 1.00 1.00 0.00 D (gp− \|gn\|)/20.88 0.86 0.98 1.00 0.00 9Be+gn/2 -1.18 -1.91 0.62 1.00 0.00 87Rb gp/2 + 1 3.73 2.75 0.74 1.15(10) 0.30(6) 133Cs7 18(10−gp)1.72 2.58 1.50 1.39(7) 0.83, [52] 171Yb+−gn/6 0.64 0.49 0.77 1.78 (9) 1.42(15) 199Hg+−gn/6 0.64 0.51 0.80 2.26(12) 2.30, [52] Table 13: Hyperﬁne structure properties. Relativistic corrections , if not speciﬁed, are calculated from Eqs.(18) and (19). We estimate the uncertainty for Fas 5% and for the derivative as 10% by comparison with Ref. [52 ]. In the case of hydrogen and beryllium, the Casimir correctio n is not appropriate, but Eqs.(15), (16) and (17) lead to negligible shifts. We summarize some properties of hfsintervals of some atoms in Table 13. The naive value ( µs) is the Schmidt one ((see Table 12) for any atom, except deuterium. T he deuterium naive value is determined from Eq.(23). It is not quite clear if the Schmidt model works, but we expect it to be appropriate for preliminary estimations and we consider deviations from the naive value µ/µs−1 as a correction due to nuclear interactions. For Cs, the corrections shift the value by 50% and unsurprisi ngly that is the largest contribution in Table 13. Cesium has only one stable isotope and this indicates that th e nuclear conﬁguration is not strongly bounded and so diﬀerent nuclear core polarization eﬀects or an admix ture of excited states can be signiﬁcant. We have included the beryllium ion in the Table mainly due to the meas urement [81] in the magnetic ﬁeld. In the leading non-relativistic approximation this value is prop ortional to the hfsinterval at zero ﬁeld. The relativistic corrections are diﬀerent, but they are small enough. Relativistic corrections are calculated using Casimir app roach of Eqs.(18) and (19), except for derivatives for Cs and Hg, taken from Ref. [52]. One reason, why derivativ es are relatively large is that the function F is always a function of α2. However, one can expect the same for the strong interaction s. One can see that 24the relativistic corrections for alkali atoms are of the sam e sign and it must be a partial cancellation for the comparison of two diﬀerent hfs. In contrast, the nuclear eﬀects shift a value of the nuclear magnetic moment in diﬀerent directions for diﬀerent atoms and an enhancemen t (e. g. for the Rb–Cs hfs) is possible. We can however hope that for a representative enough set of atoms th ey can be considered rather as statistical errors. Clearly if we would like to have a clear interpretation of som e frequency comparisons, we have to choose one of two options: •We can measure only the gross and ﬁne structure with understa ndable relativistic eﬀects and reach a limit for the variation of α. •We can involve the hfsin the comparison. In this case we must be able to estimate nuc lear eﬀects and their variations. The magnetic moment can be close to the Schmidt v alue accidently due to some cancellation of contributions with diﬀerent nature and it can be quite sen sitive to the variations of the constants. So it is necessary to have small and understandable corrections d ue to the nuclear eﬀects. It is also necessary to do a few comparisons in order to be able to transform variatio ns of the frequencies to variations of α,gp, gnandµn/µb. One should emphasize a signiﬁcant diﬀerence between the st udy of atoms with odd Aand either odd or even Z. The even Znuclear magnetic moments are mainly proportional to the sam e value (gn), which can be corrected by nuclear eﬀects. A comparison bet ween two even Zhfsis insensitive to any variations gn. Conversely investigation of only odd Zatoms should provide enough information on gp because they contain spin and orbit contributions which dif erently depend on nuclear quantum numbers (L,SandJ). In principle, there is one more option search available. One can look for two isotopes (with odd Zand odd A, but diﬀerent nuclear spins or parity) of one element with ma gnetic moments close to their Schmidt values. The ratio of the hfsintervals should be free oﬀ any relativistic and many-body a tomic corrections and could be not too sensitive to nuclear eﬀects. In this case the ratio of thehfsshould be close to a ratio of their respective Schmidt values, i. e. a simple function of gp. Unfortunately we have been not able to ﬁnd an appropriate pa ir of the isotopes for this study. 5.4 Time structure of the measurements We should point out a timing problem in the absolute frequenc y measurements. Some results have been obtained via a direct comparison to the primary cesium clock, i. e. dir ectly compared to the Cs hfs. Others have been compared indirectly and the “time structure” of such an expe riment can be quite complicated and includes some intermediate steps with comparisons between diﬀerent seco ndary standards. An example is a measurement of the 2s−8sand 2s−8dtransitions in hydrogen and deuterium atoms by the Paris gro up. The measurements [96] involved a comparison in 1997 of the hydrogen and deuterium l ines with some secondary standard, gauged in 1985 [97]. The standard was recalibrated in 1999 [98]. Such a time structure is reasonable for a determination of the Rydberg constant and the hydrogen Lamb shift, but it is not appropriate to search for a variation of the constants. In Tables 5 and 6 we assume that publication ti me is the time for a direct comparison with a primary Cs standard. For the hydrogen and deuterium hyperﬁn e structure that is approximately correct. In some cases that is not so. The other time structure problem is in a comparison with seco ndary standards, like a hydrogen maser. Some preliminary studies estimated possible deviations of its frequency with respect to some known standards or with respect to some average value of an ensemble of masers . In both case it is not clear how to interpret any comparison with a speciﬁc maser at some particular time. 6 The hyperﬁne separation in the hydrogen atom 6.1 Historical remarks The hydrogen hfswas studied for a few generations of experiments. The ﬁrst re sults with an accuracy of about 1012were reached about 35 years ago and some of them are presented in Table 14. Most experiments [99, 100, 101, 102] were devoted to a measurement of the maser frequency, while previously measured value for the wall-shift using another maser in Ref. [112] was accepte d. In Ref. [101] no explicit value of uncertainty was claimed. From discussions in that paper we estimate it as 3 ·10−12for the NRC cesium standard and 1 .7·10−12 for the maser. 25Frequency ( ν) Ref. to Year Relative Ref. to [kHz] frequency uncertainty wall-shift 1 420 405.751 786(2) [99], 1966 1965 1.4·10−12[103] 1 420 405.751 756(3) [100], 1968 1966 2.1·10−12[103] 1 420 405.751 758(2) [100], 1969 1967 1.4·10−12[103] 1 420 405.751 776(5) [101], 1968 1968 3.5·10−12[103] 1 420 405.751 777(3) [102], 1970 1968 2.1·10−12[103] Table 14: Some early precise measurements of the hydrogen hyperﬁne se paration. The references are given with the year of publication, while the year of the experiment is g iven additionally. One of the ﬁrst really accurate experiments was performed by NBS [104] and in part by Harvard University team [105]. It was pointed out [104] that the wall-shift and t he frequency have to be determined in the experiments for the same masers. This generation of experim ents [106, 104, 105, 107, 108, 109, 110, 111, 112] is discussed in the next section. When we speak about two gene rations we refer to an ideology, rather than to a time-frame. Some other experiments [115, 116, 117, 118, 11 9, 120] performed in that time or slightly earlier were not so precise (see Fig. 6). Let us mention Ref. [115] whe rein an experiment that measured both the frequency and the wall-shift was described. However, only t wo bulbs were used. As was noted in Ref. [104] another important condition for appropriate results is the use of a large number of bulbs of diﬀerent size (e. g. in Ref. [104] 11 bulbs were and in Ref. [105] the number was 18) . 1960 1970 19800.725 Hz0.750 Hz0.775 Hz0.800 Hz0.825 Hz0.850 Hz Time of measurementFrequency νHFS - 1 420 405 751 Hz Figure 6: Precise experimental results on the hydrogen hyperﬁne sepa ration. Filled circles are for those when the wall-shift was measured together with the frequency, wh ile the open ones are for frequency measurements using a value of the wall-shift found separately for another maser. 266.2 Thirty years ago About thirty years ago a number of precise results for the hfsinterval in the ground state of the hydrogen atom were published [106, 104, 107, 108, 109, 110, 111, 112, 113, 1 14]. That was due to a trial to use the hydrogen maser as a primary frequency standard. The results are colle cted in Table 15. Until the publication of results Eq.(21) of experiment on the hfsof the beryllium ion ﬁfteen years ago [81], the value of the hy perﬁne separation in the hydrogen atom had been the most precisely measured phy sical quantity. # Frequency ( ν) Ref. Year Relative Cs standard Comment [kHz] uncertainty 11 420 405.751 778(4) [106], 1969 1969 28·10−13LSRH commerc. 2 bulbs 21 420 405.751 769(2) [104], 1970 1969–1970 14·10−13NBS primary Exp. 1, 12 bulbs [105] 31 420 405.751 767(2) [104], 1970 1969–1970 14·10−13NBS primary Exp. 2, 18 bulbs 41 420 405.751 768(2) [104], 1970 1969–1970 14·10−13NBS primary Exp. 1 & 2 51 420 405.751 767(1) [107], 1971 1970 7.0·10−13NPL primary 6 bulbs 61 420 405.751 770(3) [108], 1971 1970–1971 21·10−13NRC primary 5 bulbs 71 420 405.751 767(3) [109], 1973 1970 21·10−13NPL primary 6 bulbs 81 420 405.751 768(2) [110], 1974 1974 14·10−13LORAN C, USNO Flexible bulb 91 420 405.751 770(3) [111], 1974 1972 21·10−13TOP, TAF Wall-shift [105] 101 420 405.751 771(6) [112], 1978 1975–1976 41·10−13LORAN C 6 bulbs 111 420 405.751 768(2) [113], 1980 1979 14·10−13LORAN C 5 bulbs 121 420 405.751 768(3) [113], 1980 1979 21·10−13LORAN C 5 bulbs 131 420 405.751 773(1) [114], 1980 1978 7·10−13TAF Flexible bulb, zero wall-shift Table 15: The most precise measurements of the hydrogen hyperﬁne stru cture interval. The references are given with the year of publication, while the year of the expe riment is given additionally. The abbreviations are: NBS—former National Bureau of Standards (USA), NPL—National Physical Laboratory (UK), NRC— National Research Council (Canada), LORAN C —a navigation system signal controlled by a cesium standard , USNO —U. S. Naval Observatory, TOP—cesium clock from the Paris Observatory, TAF—the French atomic time. The experiments were performed using hydrogen masers and tw o key values were measured simultaneously: the frequency of the maser and the wall-shift. Result #9 is an exception: the authors utilized a value of the wall-shift from Ref. [105]. Result #5 of Ref. [107] with the s mallest uncertainty was actually a preliminary presentation. The ﬁnal value of the NPL experiment (result # 5) had an accuracy [109] three times lower. One of the papers contains a report on two independent measur ements [104], quoted as experiment 1 and experiment 2, and we give both results in the table, as well as an average value presented by the authors of Ref. [104]. Experiment 2 is a pure NBS experiment. In experim ent 1 the maser frequency of a maser from Harvard university was measured with respect to the NBS prim ary cesium standard [104], while the wall-shift of the same maser was measured by the Harvard people [105]. Th e same result for the wall-shift [105] was accepted by the authors of Ref. [111] for another maser, the f requency of which they measured. Values #11 and #12 were presented in Ref. [113]. That paper is devoted to a measurement of the wall-shift and the unperturbed maser frequency (result #11) at the Shanghai Bu reau of Metrology. It also contains a reference to an unpublished result (#12) for the unperturbed hydrogen frequency of two hydrogen maser by the Shaanxi Asrtonomical Observatory. In this two bulbs from the SBM wer e used. The last result in Table (# 13) is based on a diﬀerent idea. Authors studied the temperature depende nce of the wall-shift and they found the the shift 27vanished at some temperature T0. The hfsinterval was afterwards determined from a maser frequency a tT0. The value in Table 5 (Eq.(22)) calculated by Ramsey [64] is an average of results #4 and #5, and so cannot be actually accepted. However, the results in Table 15 are co nsistent (apart from the earliest one) and one can calculate some average values. For instance the average val ues can be calculated over results ##1–3, 6–13 (a wideset) or over ##4, 6, 7, 10 and 11 (a conservative set). The wide set includes all original results, while the conservative one contains only values published in refe reed journals. We consider results #1 and #7 as preliminary and also exclude result #8 because the wall-shi ft and the frequency werenot measured during the same experiment. As a preliminary estimation, one can assume that the uncerta inties are independent and one ﬁnds: νwide= 1 420 405 .751 7704(6) kHz (38) and νcons= 1 420 405 .751 7683(12) kHz . (39) The fractional uncertainty of the average value is (4–9) ·10−13and that is approximately as in Table 5. In fact there has to be some correlation between the systematic erro rs. However since most of the results are consistent, any averaging cannot yield a value that is less precise than o ne of the results, namely ν(#4) = 1420 405 .751 768(2) kHz . (40) We think this result should be used for comparisons because i t is the most accurate and reliable result in Table 15. Result #4 was published in a refereed journal [104] (in co ntrast to result #8 [110]) and it was based on two independent measurements of the unperturbated hydrogen hfsfrequency with respect to the NBS primary Cs standard. The largest number of bulbs was used to determine t he wall-shift using two independent experiments [105, 104]. The result is in fair agreement with other result s in Table 15, apart from results #1 and # 13. The discrepancy with the former is not important, because resul t #1 does not agree with others on a one-sigma level, but rather within three sigma. We also should mention that Ref. [108] refers to a private communication by one of the authors of Ref. [106] (Menoud, 1971) wherein a ne w LSRH value of hydrogen hfs ν(LSRH ) = 1 420 405 .751 764(10) kHz , (41) is quoted. This is less precise than result #1 but agrees with other results. Two results in Table 15 are more precise than the value in Eq.(40). One of them (# 5) [107] was l ater corrected by the authors [109]. The other (result # 13) was obtained using very diﬀerent methods. We do not include it in the conservative set since it is inconsistent with most other results of this set. To conclud e, we present in Fig. 7 an overview of experiments done about 30 years ago. Most experimental results measured the wall-shift together with the frequency. Some details on the crucial results can be found in the Appendix. One should note that 30 years ago the accuracy of the primary c esium standards was not so high as now and this was an important source of uncertainty. The results in T able 15 are presented together with the name of the primary cesium standard used. We expect that it is possible t o study variations of those standards which were frequently discussed due to international comparisons. We also expect that the reference set for comparison with experiments possible in the year 2000 will to be between theconservative andwideones. Particularly, the main sources of uncertainty in experiments 1 and 2 in Ref. [10 4] are, in part, independent. 6.3 Why it is important to do experiment with hydrogen now The hydrogen atom can now be used in the search for a limit for t he variations of fundamental constants on a level that is comparable with most other possible projects . If the Paris group have really reached a level of accuracy ( δν/ν≃2.4·10−15) reported [121] as a preliminary result νhfs(87Rb) = 6 834 682 .610 904 333(17) kHz , (42) then only the work on the Rb clock and the Rb hfscan provide a signiﬁcantly more precise test for the variati ons. Usually, it takes some time to really remove all sources of sy stematic errors after the ﬁrst announcement of a performed measurement with a signiﬁcantly better accuracy . The ratio of Rb and Cs hfs may also be less sensitive to the variations, and therefore it is not enough t o compare two hfstransitions only. It is necessary to have several diﬀerent data sets. In any case, the study of the hydrogen hfshas a number of advantages, as are listed below: 281970 1980 1990 20000.755 Hz0.760 Hz0.765 Hz0.770 Hz0.775 Hz0.780 Hz Time of measurementFrequency νHFS - 1 420 405 751 Hz ?? Figure 7: Hydrogen hyperﬁne structure as measured thirty years ago. R esult with measurement of the wall-shift and the frequency for the same masers. Open circles are for pr eliminary results (conference proceedings, private communications) and results, corrected after publication s. Full circles are for the ﬁnal results. The question marks are for the possible 2000 measurements. •This is the only accurate comparison over a long interval (30 yr). Only the171Yb+hfsand the9Be+in the magnetic ﬁeld (with a shorter time separation) is compar able to this. The other results obtained ﬁve or more years ago cannot provide a search for a variation of th e constants on a level below 10−13yr−1. •The result was obtained 30 years ago independently in a few di ﬀerent laboratories (see Table 15) after long studies over a number of years. Only ytterbium–171 was m easured in two laboratories, whereas other transitions from Tables 5 and 6 have been studied in only one p lace for each atom. •Most of the measurements of the hydrogen hfswere performed by direct comparisons with primary Cs clocks and thus they have simple time structures. •Nowadays, this measurement can be also done in a number of lab oratories. •There is no direct problem due to the accuracy of any primary f requency standard because of the relatively low accuracy needed for the measurement. •The measurement should be as precise as possible. If a level b etter than 10−14is achieved, a repetition in a few years can yield another strict limit for the variation o f the constant. Thus the measurement in the year 2000 can be the “second” measurement of a 30-years test a nd the “ﬁrst” measurement of a few-years test of the variation. •Combining the variation of the H–Cs hfswith a variation of 1 s−2sin the hydrogen atom with respect to the Cs hfs, one can achieve an estimation for a variation of the hydroge nhfswith respect to the Rydberg constant. The former limit is going to be about (2 −5)·10−14yr−1. The accuracy of a determination of the 1 s−2sfrequency in the hydrogen atom with respect to the Cs hfshas been improved and the uncertainty in 1999 [122] is about 6 ·10−14yr−1. This indirect comparison between the hfsand the 1 s−2s frequency in hydrogen can be interpreted without any diﬃcul ties arising from the relativistic corrections or unclear nuclear eﬀects. Only properties of fundamental p articles are to be compared: αandµp/µb. The hydrogen experiment must have systematic errors, which are quite diﬀerent from the short-term com- parisons. The value of µpis on for the simplest for interpreting. The result can be rep roduced in diﬀer- ent laboratories. The expected limit for the variation of th e hydrogen hfslies between 2 ·10−14yr−1and 295·10−14yr−1. This depends on the possibility of averaging the values fro m Table 15. Some additional anal- ysis of the old data [106, 104, 108, 109, 110, 111, 105, 112] is needed. This would include the more recent data of international comparisons on the national primary s tandards utilized in the hydrogen experiments [106, 104, 108, 109, 110, 111, 112]. In 2000 only the Rb hfscan provide a signiﬁcantly better result. From pub- lished values, we estimate the limits from possible experim ents to approximately 1 .3·10−14yr−1(for Rb) and 1.1·10−14yr−1(for Hg+). The latter is diﬀerent from that presented in Table 5 becau se we expect signiﬁcant correlations between the uncertainties of the original mea surement and the repetition. 7 Conclusions The strongest published limits (better than 10−16yr−1) on the variation of fundamental constants arise from geophysical data. However, any direct use of such estimatio ns is not possible because of a incorrect interpretation of the data. The best astrophysical data gives limits betwee n 10−14and 10−15yr−1and they are only slightly better than laboratory limits. Cosmological methods (such as investigation of nucleosynthesis and microwave background radiation) are far less precise, but they may be i mportant if the variations at the beginning of the evolution of the universe were faster. The clock comparison s lead to limit on a level between 10−13and 10−14 yr−1, although, their interpretation is not quite clear. We summ arize the diﬀerent methods used to search for the variations in Table 16. The data contain the amplitude an d velocity of the variations and the time and space separations where appropriate. The scale of the space variations due to geochemical studies is estimated from the absolute motion of the Earth (i. e. a motion with resp ect to the frame of the microwave background radiation frame). Method ∆lnα/∆t ∆α/α ∆t ∆l/c Geochemical (Oklo) 10−17yr−110−82·109yr 106yr Astrophysical 10−15yr−110−5109−1010yr109−1010yr Cosmological 10−13yr−110−31010yr 1010yr Laboratory 10−14yr−110−141 yr H/Cs hfs (2−5)·10−14yr−1(6−15)·10−1330 yr Table 16: Comparison of diﬀerent search for the variations. We believe that the most reliable limits can be reached under laboratory conditions by comparing two results for the same transition, each result obtained in diﬀerent ti me. In 2000, there are a number of possibilities to reach results a limit of a few units of 10−14yr−1. The most secure one will be that involving the hydrogen hyperﬁne splitting. The limit for a possible variation of th e ratio of µCs/µpis expected to be about (1 −2)·10−14 yr−1. The α-variation due to the relativistic corrections is to be limi ted by (2 −5)·10−14yr−1. The ytterbium limit for the variation of gn/gpis expected to be 6 .5·10−14yr−1. We need to mention, that the preliminary results on the second ytterbium experiment were in fact publ ished in 1995, but these were later corrected [71]. A signiﬁcant part of the measurements was performed by the en d of 1995, while the remainder was done in mid 1996. If we accept that the experiment was done in 1995, th e potential limit is 4 ·10−14yr−1. For the deuterium hfs, if we assume a variation of gponly, the limit is about 5 .5·10−14yr−1, whereas if we only consider gn-variations it is 8 .1·10−14yr−1. The motivation to study the deuterium hfsis that it is the only value is sensitive to gp− \|gn\|, amongst those known for a while in Table 5. In Table 16 one can note that the astrophysical, cosmologica l and geochemical data are not quite sensitive to any ﬂuctuations of the constants in T∼108yr and/or L∼c·108yr. Astrophysical data yield good limits for the value of ∆ln α/∆z, but any interpretation of the redshift zin terms of the time separation such as e. g. ∆t=t0/parenleftbigg 1−1 (1 +z)3/2/parenrightbigg , (43) where t0= 1.5·1010yr, is actually only an estimation. It is known that the Hubbl e velocity-distance law is not a strong one and this means that for any particular case, t here is no well-established connection between 30the redshift (associated with the velocity) and the time t(z) (associated with the distance). This also means that would even be hard to detect a ﬂuctuation with T≤109yr. Since the interpretation is actually based on a lack of such ﬂuctuations the astrophysical and geochemi cal limits are actually weaker than presented in the table. Another problem is the correlation between space and time variations. If we suppose e. g. that the constants increase with time and distance, we have to expect a signiﬁcant cancellation between the time and space variations (actually we only study some distant objec ts from the past at a distance ∆ l=c∆t). Thence the astrophysical data can be insensitive to some of thse cor relations. It must also be mentioned that some particular astrophysical data rather indicate in existenc e of some variations of the constants. E. g. a few points in Fig. 1 of Ref. [50] conﬁrm a variation in α. However, there are a lot of data points plotted and, statist ically, perhaps those few are not important. nevertheless, the ﬁt as sumes that there is only a slow drift in time and that there are neither space variations and nor ﬂuctuations on a scale shorter than 1010yr. We think these points need to be re-examined in order to understand if the eﬀ ect is purely statistical. In contrast to examinations of the astrophysical and geoche mical data, a laboratory experiment can de- termine the derivatives of the constants and is also sensiti ve to the ﬂuctuations. There should also be no cancellation between space and time deviations: no space va riation is involved because of the small absolute velocity of Earth. There are four basic dimensionless constants which determi ne any atomic spectra (in unit of Ry):α,me/mp, gpandgn. When the frequency is measured in units of the Rydberg const ant, the nuclear magnetic moment comes in units of the Bohr magneton µb. Some of these constants enter into the equations for the ene rgy levels with nuclear magnetic moments constructed of the proton spi n contribution µp=gpme/mpµb, the neutron (spin) contribution µn=gnme/mpµband the proton orbital contribution me/mpµb. The last combination can also appear as a Dirac part of the proton spin contributio n for relativistic corrections. These are known to split the Dirac part and the anomalous part of the magnetic mo ment. This simple, four-constant description is perturbed by nuclear eﬀects. We would also like to underline that there are two kinds of sea rches for the variations. One is for clear interpretation and it involves transitions without the inﬂ uence of nuclear eﬀects or with only a small inﬂuence. This is valid for secure limits on the possible variations of the constants. The other kind involves studying transitions which may be expected to be extremely sensitive to some variations, but for which there in no clear interpretation. Particularly, the search for sensit ive values includes some geochemical study (positions of low-lying resonances) and some laboratory investigations . Astrophysical studies are appropriate in the search for var iations of α,me/mpandgp. The limits for gn cannot be obtained from astrophysics. If nuclear eﬀects for some magnetic moments calculated, the laboratory investigation can give some limits for a variation of me/mpandgn. A precision laboratory study of transitions which are not perturbed by nuclear eﬀects, can give a limit fo r the ﬁne structure constant and for the ratio gpme/mpµn. The neutron magnetic moment cannot be studied successfull y in this way, because of the relatively low accuracy of the deutron hfsand the inﬂuence of the nuclear eﬀects on most magnetic momen ts of heavier nuclei. The proton magnetic moment can only be investigated by using the hfsof atomic hydrogen, otherwise the nuclear eﬀects are involved. We hope that a measurement o f the hyperﬁne separation in the hydrogen atom will be performed and that a reliable limit on the possible va riation of the fundamental constant will be reached. Acknowledgements The idea of studying the ﬁne structure in order to search for a variation of the ﬁne structure constant was learnt by me from Klaus Jungmann [87] and all related possibilities arise from discussions with him. Opportunity, provided with the hydrogen hyperﬁne structure have been fre quently discussed with Ted H¨ ansch. Possibilities of studying the ﬁne structure of diﬀerent neutral atoms were discussed with Martin Weitz, while for ions the problem was discussed with Ekkehard Peik. I was happy to lear n from them that some search for the ﬁne structure associated with the ground state suggested in the paper can be really performed. I am very grateful to all of them. A preliminary version of this work was present ed at a seminar at the PTB and I thank A. Bauch, B. Fisher and F. Riehle for comments and useful discus sions. I would like also to thank A. Godone, R. Holzwarth, J. Reichert, M. Plimmer and D. A. Varshalovich fo r stimulating discussions and useful references. I am thankful to Sile Nic Chormaic for vastly improving the leg ibility of this paper. The work was supported in part by the Russian State Program ‘Fundamental Metrology’ a nd NATO grant CRG 960003. 31A Precision measurements of the hyperﬁne separation in the h y- drogen atom Here we give a brief description of the most important, accur ate experiments on the hfsas presented in Table 15. •Value #2 [104] is the result of an experiment done by NBS and Ha rvard university. The wall-shift was measured in early 1969 [105] and the frequency at the end of 19 69. –The result is 1 420 405.751 769 1(24) kHz. Three sources of unc ertainty were considered important. –The most signiﬁcant of these is the wall-shift, because the p roperties of the bulb could vary with time: δ= 1.9·10−6kHz. –The accuracy capability of the NBS cesium standard gave δ= 0.7·10−6kHz. –The frequency dispersion of the standard since it was calibr ated was obtained as δ= 1.3·10−6kHz. •Value #3 (another result from the same paper [104]) had a diﬀe rent budget of errors. This is a pure NBS experiment, with simultaneous measurements of the wall-sh ift and the frequency. –The result for the frequency was found to be 1 420 405.751 766 7 (18) kHz. The uncertainty due to the cesium standard was the same: –the accuracy capability of the standard led to δ= 0.7·10−6kHz; –The frequency dispersion since the calibration was δ= 1.3·10−6kHz. –The largest contribution to the uncertainty other than thos e mentioned was due to a comparison of the maser to the cesium standard: δ= 1.0·10−6kHz. •Value #6 [108] was an NRC experiment. –The result was 1 420 405.751 770(3) kHz. –The uncertainty of the comparison properly was only δ= 0.7·10−6kHz. –The variation of the maser frequency during the experiment l ed to δ= 2.1·10−7kHz. –The instability of the cesium standard frequency gave δ= 1.4·10−6kHz. •An NPL experimental result (#5) has been frequently quoted a s the most precise one [107]. However, the ﬁnal NPL value (#7) has not been the best one. –The result is 1 420 405.751 766 2(30) kHz and there are three so urces of uncertainties. –The measurement itself gave a statistical error δ= 1.4·10−6kHz. –The accuracy of the cesium standard itself gave δ= 1.4·10−6kHz. –Due to the same problem a comparison of the NPL cesium standar d with the international time cannot be perfect and the uncertainty was δ= 1.4·10−6kHz. •The result (#8) of an experiment in Ref. [110] has been never p ublished in a refereed journal to the best of our knowledge, so we cannot consider this as a ﬁnal result. However, it is important because they used a very diﬀerent approach. In contrast to the other experimen ts the wall-shift was measured by using of a ﬂexible bulb. –The result was found to be 1 420 405.751 768 0(20) kHz. –The maser uncertainty was δ= 1.4·10−6kHz. –The comparison itself was a little more accurate: δ= 1.2·10−6kHz. –The uncertainty due to the wall-shift was estimated as δ= 0.4·10−6kHz. •Result #11 (1 420 405.751 768(2) kHz) is an average over 5 hydr ogen masers with a dispersion of their frequencies between 1 420 405.751 765 9(17) kHz and 1 420 405. 751 769 3(16) kHz. The budgets of errors for particular masers have not been presented. 32•Result #13 (1 420 405.751 773(1) kHz) is not in fair agreement with most other precise values. The method was also quite diﬀerent from others as well. First, a n ew maser was a developed using a ﬂexible bulb. Next, the dependence of the wall-shift on the temperat ure was studied and it was found that the wall-shift vanished at a temperature T0. The result was found by interpolating the maser frequency t oT0. •We do not consider result #10 (1 420 405.751 771(6) kHz) in det ail, because the accuracy is slightly lower than for forementioned results and the statistical we ight for any evaluations is small enough. The uncertainty of a measured value of 1 420 405.751 771 0(58) kHz was mainly due to the comparisons of the maser frequency to the local rubidium standard and the stand ard to the international time scale. 33References [1] P. A. M. Dirac, Nature 139(1937) 323. [2] F. J. Dyson in Aspects of Quantum Theory (Cambridge Univ. Press, Cambridge, 1972) p. 213; in Current Trends in the Theory of Fields (AIP, New York, 1983), p. 163. [3] A. Linde in Three Hundred Years of Gravitaion (Ed. by S. W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987), p. 604; S. K. Blau and A. H. Guth, ibid., p. 524; G. B¨ orner, The Early Universe (Springer-Verlag, 1993). [4] P. Sisterna and H. Vucetich, Phys. Rev. D 41(1990) 1034. [5] P. Landgacker, Phys. Rep. 72(1981) 185. [6] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGrow-Hill, New York, 1980); C. Kiesling, Tests of the Standard Theory of Electroweak Interactions (Springer-Verlag, 1988). [7] W. J. Marciano, Phys. Rev. Lett. 52(1984) 489. [8] T. C. Hill, P. J. Steinhardt and M. S. Turner, Phys. Lett. B 252(1990) 343. [9] M. Morikawa, Astrophys. J. 362(1990) L37. [10] R. G. Crittenden and P. J. Steinhardt, Astrophys. J. 395(1992) 360. [11] D. Sudarsky, Phys. Lett. B 281(1992) 98. [12] P. D. Sisterna and H. Vucetich, Phys. Rev. Lett. 72(1994) 454. [13] M. Salgado, D. Sudarsky and H. Quevedo, Phys. Rev. D 53(1996) 6771; Phys. Lett. B 408(1997) 69. [14] T. Broadhurst, R. Ellis, D. Koo and A. Szalay, Nature 343(1990) 726. [15] D. H. Wilkinson, Phil. Mag. 3(1958) 582. [16] P. J. Pebbles and R. H. Dicke, Phys. Rev. 128(1962) 2006. [17] R. Gold, Phys. Rev. Lett. 20(1968) 219. [18] F. Dyson, Phys. Rev. Lett. 19(1967) 1291. [19] A. Peres, Phys. Rev. Lett. 19(1967) 1293. [20] B. Broulik and J. S. Treﬁl, Nature 232(1971) 246. [21] P. C. W. Davies, J. Phys. A 5(1972) 1296. [22] M. Lindner et al., Nature 320(1986) 246. [23] A. S. Barabash, JETP Letters 68(1998) 1. [24] A. I. Shlyakhter, Nature (London) 264(1976) 340. See also the preprints: A. I. Shlyakhter, LNPI N. 260 (1976); ATOMKI Report A/1 (1983). [25] M. Maurette, Ann. Rev. Nucl. Part. Sci. 26(1972) 319; P. K. Kuroda, Origin of the Chemical Elements and the Oklo Phenomen (Springer-Verlag, Berlin, 1982). [26] Yu. V. Petrov, Sov. Phys. Usp. 20(1977) 937. [27] R. B. Firestone, Table of Isotopes (John Wiley & Sons, Inc., 1996). [28] J. M. Irvine, Contemp. Phys. 24(1983) 427. [29] T. Damour and F. Dyson, Nucl. Phys. B 480(1994) 596. 34[30] Y. Fujii et al., hep-ph/9809549. [31] E. W. Kolb, Phys. Rev. D 33(1986) 869. [32] L. Bergstr¨ om et al., astro-ph/9902157. [33] D. A. Varshalovich and A. Y. Potekhin, Astronomy Lett. 22(1995) 1. [34] M. J. Drinkwater et al., Mon. Not. Astron. Soc. 295(1998) 457. [35] M. P. Savedoﬀ, Nature 178(1956) 688. [36] R. I. Thompson, Astrophys. Lett. 16(1975) 3. [37] D. A. Varshalovich and S. A. Levshakov, JETP Lett. 58(1993) 237. [38] D. A. Varshalovich et al., astro-ph/9607098. [39] A. D. Tubbs and A. M. Wolfe, Astrophys. J. 236(1980) L105. [40] L.L Cowie and A. Songalia, Astrophys. J. 453(1995) 596. [41] M. J. Drinkwater et al., astro-ph/9709227. [42] J. N. Bachall and M. Schmidt, Phys. Rev. Lett. 19(1967) 1294. [43] A. M. Wolfe, R. L. Brown and M. S. Roberts, Phys. Rev. Lett .37(1976) 179. [44] S. A. Levashev, Mon. Not. R. Astr. Soc. 269(1994) 339. [45] A. Y. Potekhin and D. A. Varshalovich, Astron. Astrophy s. Suppl. Ser. 104(1994) 89. [46] D. A. Varshalovich, V. E. Panchuk and A. V. Ivanchik, Ast ronomy Lett. 20(1994) 771. [47] D. A. Varshalovich and A. Y. Potekhin, Space Sci. Rev. 74(1995) 259. [48] A. V. Ivanchik et al., astro-ph/9810166. [49] A. Y. Potekhin et al., astro-ph/9804116. [50] J. K. Webb et al., Phys. Rev. Lett. 82(1999) 884. [51] V. Dzuba et al., Phys. Rev. Lett. 82(1999) 888. [52] V. A. Dzuba et al., Phys. Rev. A 59(1999) 230. [53] M. Kaplinghat et al., astro-ph/9810133v2. [54] S. Hannestad, astro-ph/9810102v2. [55] A. Godone et al., Phys. Rev. Lett. 71(1993) 2364. [56] A. Godone et al., IEEE Trans. IM- 45(1996) 261. [57] J. D. Prestage et al., Phys. Rev. Lett. 74(1995) 3511. [58] G. Breit, Phys. Rev. 35(1930) 1477. [59] H. B. G. Casimir, On the Interaction Between Atomic Nuclei and Electrons (Freeman, San Francisco, 1963); C. Schwarz, Phys. Rev. 97(1955) 380. [60] N. A. Demidov et al., in Proceedings of 6th European Frequency and Time Forum (Noordwijk, 1992), p. 409. [61] L. A. Breakiron in Proceedings of 25th Annual Precise Time Interval Applicati ons and Planning Meeting (NASA Conf. Publ. No. 3267), p. 401. 35[62] J. P. Turneaure and S. R. Stein, in Atomic Masses and Fundamental Constants , 5 (Ed. by. J. H. Sandars and A. H. Wapstra, Plenum Press, 1976), p. 636; J. P. Turneaure et al., Phys. Rev. D 27(1983) 1705. [63] S. R. Stein and J. P. Turneaure, Electronic Lett. 8(1972) 312. [64] N. Ramsey, in Quantum Electrodynamics (Ed. by T. Kinoshita, World Sci., Singapore, 1990), p. 673; H yp. Interactions 81(1993) 97. [65] D. J. Wineland and N. F. Ramsey, Phys. Rev. 5(1972) 821. [66] B. S. Mathur et al., Phys. Rev. 158(1967) 14. [67] J. J. Bollinger et al., in Laser Spectroscopy, VI (ed. by H. P. Weber and W. L¨ uthy, Springer, 1983) p. 169. [68] S. Bize et al., Europhys. Lett. 45(1999) 558. [69] H. Knab et al., Europhys. Lett. 4(1987) 1361. [70] Chr. Tamm et al., Appl. Phys. B 60(1995) 19. [71] P. T. Fisk et al., IEEE Trans UFFC- 44(1997) 344; P. T. Fisk, Rep. Prog. Phys. 60(1997) 761. [72] A. M¨ unch et al., Phys. Rev. 35(1987) 4147. [73] D. J. Berkland et al., Phys. Rev. Lett. 80(1998) 2089. [74] Th. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedelli , M. Weitz and T. W. H¨ ansch, Phys. Rev. Lett. 79(1997) 2646. [75] C. Schwob, L. Jozefowski, B. de Beauvoir, L. Hilico, F. N ez, L. Julien and F. Biraben, Phys. Rev. Lett. 82 (1999) 4960. [76] H. Schnatz et al., Phys. Rev. Lett. 76(1996) 18. [77] F. Riehle et al., IEEE Trans. IM- 48(1999) 613; in Trapped Charged Particles and Fundamental Physics (Ed. by D. H. E. Dubin and D. Schneider, AIP, 1999), p. 348. [78] J. E. Bernard et al., Phys. Rev. Lett. 82(1999) 3228. [79] P. S. Ering et al., Opt. Comm. 151(1998) 229. [80] A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer-Verlag, 1985). [81] J. J. Bollinger et al., Phys. Rev. Lett. 54(1985) 1000. [82] J. J. Bollinger et al., IEEE Trans. IM- 40(1991) 126. [83] A. Bauch et al., in Proceedings of 7th European Frequency and Time Forum (Neuchˆ atel, 1993), p. 531. [84] M. J. Sellars et al., Proceedings of the 1995 IEEE Frequency Control Symposium/ , p. 66. [85] K. Hermanspahn, W. Quint, S. Stahl, M. T¨ onges, G. Bolle n, H.-J. Kluge, R. Ley, R. Mann, and G. Werth, Hyp. Inter. 99(1996) 91; M. Diedrich, H. H¨ aﬀner, N. Hermanspahn, M. Immel, H.-J. Klu ge, R. Ley, R. Mann, W. Quint, S. Stahl, J. Verd´ u, and G. Werth , Phys. Scripta T 80(1999) 437; in Trapped Charged Particles and Fundamental Physics (Ed. by D. H. E. Dubin and D. Schneider, AIP, 1999), p. 43. [86] A. A. Madej et al., in Proceedings of 5th Symposium on Frequensy Standards and Met rology (Ed. by J. C. Berquist, World Sci., 1996) p. 169. [87] K. Jungmann, unpublished. [88] J. von Zanthier et al., Opt. Comm. 166(1999) 57. 36[89] J. von Zanthier, Th. Becker, C. Schwedes, E. Peik, H. Wal ther R. Holzwarth, J. Reichert, Th. Udem, T. W. Hnsch, A. Yu. Nevsky, P. Pokasov, and S. N. Bagayev, in preparation . [90] A. Huber, Th. Udem, B. Gross, J. Reichert, M. Kourogi, K. Pachucki, M. Weitz and T. W. H¨ ansch, Phys. Rev. Lett. 80(1998) 468. [91] F. Arbes et al., Z. Phys. D 31(1994) 27. [92] U. Tanaka et al., Phys. Rev. A 53(1996) 3982. [93] W. Becker and G. Werth, Z. Phys. A 311(1983) 41. [94] R. Blatt and G. Werth, Phys. Rev. A 25(1982) 1476. [95] V. A. Dzuba and V. V. Flambaum, physics/9908047. [96] B. de Beauvoir et al., Phys. Rev. Lett. 78(1997) 440. [97] A. Clairon et al., IEEE Trans. IM- 34(1985) 265; Metrologia 25(1988) 9. [98] G. D. Rovera and O. Acef, IEEE Trans. IM- 48(1999) 571. [99] R. Vessot et al., IEEE Trans IM- 15(1966) 165. [100] G. Becker and B. Fischer, PTB-Mitt. 78(1968) 177. [101] A. G. Mungall, Metrologia 4(1968) 87. [102] A. R. Chi et al., Proc. IEEE (Letters) 58(1970) 142. [103] J. Vanier and R. F. C. Vessot, Metrologia 6(1970) 52. [104] H. Hellwig et al., IEEE Trans. IM- 19(1970) 200. [105] P. W. Zitzewitz et al., Rev. Sci. Instr. 41(1970) 81. [106] Ch. Menoud and J. Racine, Z. Angew. Math. Phys. 20(1969) 578; in Proc. Colloque International de Chronom´ etrie (Paris, 1969), p. A8-1. [107] L. Essen et al., Nature 229(1971) 110. [108] D. Morris, Metrologia 7(1971) 162. [109] L. Essen et al., Metrologia 9(1973) 128. [110] V. S. Reinhard and J. Lavanceau, in Proceedings of the 28th Annual Symposium on Frequency Contr ol (Fort Mammouth, N. J., 1974), p. 379. [111] P. Petit et al., Metrologia 10(1974) 61. [112] J. Vanier and R. Larouche, Metrologia 14(1976) 31. [113] Y. M. Cheng at al., IEEE Trans. IM- 29(1980) 316. [114] P. Petit, M. Desaintfuscien and C. Audoin, Metrologia 16(1980) 7. [115] S. B. Crampton et al., Phys. Rev. Lett. 11(1963) 338. [116] J. Vanier et al., IEEE Thans. IM- 13(1964) 185. [117] H. E. Peters et al., Appl. Ohys. Lett. 6(1965) 34. [118] H. E. Peters and P. Kartaschoﬀ, Appl. Ohys. Lett. 6(1965) 35. [119] E. H. Johnson and T. E. McGunigal, NASA Tech. Note TND-3 292 (1966). [120] M. Bangham, Proc. Colloque International de Chronom´ etrie (Paris, 1969), p. A7-1. [121] P. Laurent et al., to be published in Proceedings of ICOL’99 . [122] M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, Th. Udem, M. Weitz, T. W. Hnsch, P. Lemonde, G. Santarelli, M. Abgrall and C. Salomon and A. Clairon, to be published . 37
DESIGN OF A 3 GHZ ACCELERATOR STRUCTURE FOR THE CLIC TEST FACILITY (CTF 3) DRIVE BEAM G. Carron, E. Jensen, M. Luong), A. Millich, E. Rugo, I. Syratchev, L. Thorndahl, CERN, Geneva, Switzerland ) now at CEA-Saclay, DSM/DAPNIA/SEA, Gif-sur-Yvette, FranceAbstract For the CLIC two-beam scheme, a high-current, long- pulse drive beam is required for RF power generation.Taking advantage of the 3 GHz klystrons available at theLEP injector once LEP stops, a 180 MeV electronaccelerator is being constructed for a nominal beamcurrent of 3.5 A and 1.5 /G50s pulse length. The high current requires highly effective suppression of dipolar wakes.Two concepts are investigated for the acceleratingstructure design: the “Tapered Damped Structure”developed for the CLIC main beam, and the “Slotted Iris– Constant Aperture” structure. Both use 4 SiC loads percell for effective higher-order mode damping. A full-sizeprototype of the TDS structure has been built and testedsuccessfully at full power. A first prototype of the SICAstructure is being built. 1 INTRODUCTION The power generation in the CLIC two-beam scheme, described in detail in [1], relies on drive beamaccelerators (DBAs) operated at 937 MHz (30 GHz/32).The DBA for CTF 3 will operate at 3 GHz. This allowsre-use of S-band equipment from the LEP injector whichwill be available once LEP stops. To maximize RF generation efficiency, the drive beam accelerator will be operated at almost 100 % beamloading, i.e. over the length of each accelerating structure,the accelerating gradient will decrease to almost zero. The CTF 3 drive beam accelerator will consist of approximately 20 accelerating structures, each of 32 cellsand a total length of 1.3 m. It will operate in 2 /G53/3 mode and at a moderate accelerating gradient of 7 MV/m. Two types of structure are being studied: the Tapered Damped Structure (TDS) has originally been designed forthe CLIC main accelerator and was scaled down infrequency by a factor 10. This study is well advanced.This structure has, however, the disadvantage of its size(outer diameter 430 mm). Since some structures at theupstream end will have to fit into focusing solenoids, weare now also studying a slotted iris structure with an outerdiameter of only 174 mm as an alternative. This approachis interesting also in view of the scaling to 937 MHz, andhas the additional feature of a large constant iris apertureand consequently lower short-range transverse wakefields. We refer to the latter as SICA (Slotted Iris – Constant Aperture). 2 GENERAL DESCRIPTION Both the TDS and the SICA structure are based on classical S-band cells. For the generic geometry, seeFig.1. Detuning and modulation of the group velocity(from 5 to 2.5 % over the length of the structure) areimplemented by iris variation in the TDS (keeping thenose-cone size x at zero), and by nose-cone variation in the SICA structure (keeping a constant at 17 mm). In both cases, b is adjusted for the correct phase advance of /G4F/3 at the operating frequency. The first cell has the same dimensions in both designs ( a = 17 mm, x = 0). The main difference between the two approaches is the coupling of the higher-order modes (HOMs) to the SiCloads. TDS uses wide openings in the outer cell wall,coupled to 4 waveguides with an axial E-plane, and witha cut-off above the operating frequency serving as high-pass filter for the HOMs. SICA on the other hand relieson geometrical mode-separation by 4 thin radial slotsthrough the iris, coupling dipole modes to a ridgedwaveguide with its E-plane in the azimuthal direction. InFigure 1: parameters of the cell geometry. The nose-cone size x is zero for the TDS structure. Nose-cone sizes in different SICA cells are shown dotted.both cases, the SiC absorbers are wedge-shaped for good matching. Both designs allow highly effective dipole mode damping. Fig. 2 shows the time domain MAFIAsimulations of the transverse wake for a Gaussian bunch with a /G56 of 2.5 mm, assuming the damping waveguides to be matched. These results were confirmed by frequency-domain calculations with HFSS, taking the properties ofSiC into account: The first dipole mode of the TDS had aQ of 18, that of the SICA structure had a Q of 5. 3 TDS The cells are supplemented with four 32 mm wide damping waveguides against transverse and longitudinalHOMs. The cell wall thickness is about 20 mm and theextruded copper waveguides are brazed into openings inthe cell wall as shown in Fig. 3. The extruded waveguidesconstitute convenient housings at their outer extremitiesfor the SiC absorbers that can be inserted through 16 mmmini-flanges after the final brazing of the structure. Theabsorbers will then either have been clamped or brazedonto metal holders. By introducing the SiC wedges after the final structure brazing, thermal strains on SiC bondscan be avoided and exchangeability is obtained. The prototype was brazed in 5 parts in a vertical position (2 couplers and the main body in 3 units). Duringthat operation the damping waveguides with prebrazedend flanges were also bonded with the cells. Finally the 5parts were brazed horizontally. For future TDSs a single, uncomplicated vertical braze is foreseen at eutectic temperature, joining cells, dampingwaveguides and couplers. To avoid deformations (duringthe brazing) of the lowermost cells, the cell wall thicknesswill be increased to 35 mm, the total structure weightbeing ~160 kg. Figure 3: Brazed TDS ready for power testing Figure 4: Bead-pull measurement: an error of about 40 º between the 1st and last cell is measured in this reflection diagram, corresponding to a phase slip of ±10 º between electron and wave over 32 cells. The loss in accelerating efficiency is less than 1 %. The cells are not equipped with dimple tuners.-6.00-4.00-2.000.000E+0 02.004.006.00 0.000E+0 00.500 1.00 1.50 2.00 2.50FRAME: 1 01/06/00 - 17:17:01 VERSION[V4.024] WGD32.DRD *** CLIC 3 GHZ DBA T D S *** A: 1.341E+01 MM, B: 3.653E+01 MM. G.C., L.T., E.J., JUNE 2000 X COMPONENT OF WAKE POTENTIAL IN V [INDIRECT CALC.] OP-:4024 #1DGRAPH ORDINATE: WAKET COMPONENT: X FIXED COORDINATES: DIM...........MESHLINE X 11 Y 1 ABSCISSA: GEOMWAKE [BASE OF WAKET] REFERENCE COORDINATE: S VARY..........MESHLINE FROM 0 TO 3001 -6.00-4.00-2.000.000E+002.004.006.00 0.000E+00 0.500 1.00 1.50 2.00 2.50FRAME: 1 02/06/00 - 08:31:25 VERSION[V4.024] SICA32.DRD SI: A-3,10,2, B-5,CELL-6,15 A: 1.700E+01 MM, B: 3.703E+01 MM.FOR REF., E.J., JUNE 2000 X COMPONENT OF WAKE POTENTIAL IN V [INDIRECT CALC.] OP-:4024 #1DGRAPH ORDINATE: WAKET COMPONENT: X FIXED COORDINATES: DIM...........MESHLINE X 11 Y 1 ABSCISSA: GEOMWAKE [BASE OF WAKET] REFERENCE COORDINATE: S VARY..........MESHLINE FROM 0 TO 2994 Figure 2: MAFIA time domain simulation of the dipole wake in the 32nd cell of TDS (top) and SICA structure (bottom). Abscissa in m behind bunch centre, ordinate inV/(2 pC)/m/mm. Note the smaller short-range wake fieldin the SICA structure.3.1 Low-level measurements and power tests Figs. 4 and 5 give measured low-level results for the brazed 32-cell TDS. Fig. 4 shows a measurable deviationfrom the ideal 240 º phase advance per cell of thereflection coefficient, but even this error would lead toacceptable 1 % loss of accelerating efficiency. Thematching of the input and output couplers over abandwidth of 10 MHz is documented in Fig. 5. After brazing, the TDS was submitted to RF conditioning and reached the nominal power of 40 MWin less than 1 week, and subsequently 52 MW (themaximum power level available at CERN). 4 SICA In the tapered damped structure, the waveguides between the accelerating cells and the SiC loads serve ashigh-pass filters, below cut-off for the accelerating mode,but transparent for higher-order modes, in particular thefirst dipole mode (at approximately 4.1 GHz). In order for this filter to work effectively, substantial waveguidelength is required. This lead to an outer diameter of the3 GHz TDS of approximately 430 mm. As opposed to the “filter” type mode selection of the TDS, the SICA structure uses “geometric” mode selection[2]: a small radial slot in the iris does not intercept radialnor axial surface currents, so it will not perturb theaccelerating TM 01 mode (nor any other TM0n mode). Dipole modes however have azimuthal currentcomponents which are intercepted and will thus induce avoltage across the slot. If this slot continues radially,cutting the outer cell wall, it can be considered as awaveguide, the cut-off of which can be made small byusing a ridged waveguide. Another concern was the short-range transverse wake, the strength of which is dominated by the iris aperturealone and hardly affected by detuning or damping. So weintroduced nose-cones, varying in size from cell to cell toobtain the same linear variation of the group velocity (5to 2.5 %) as in the conventionally detuned structure,keeping the iris aperture constant at Ø 34 mm. Theoverall detuning of the first dipole mode is slightlysmaller than for the TDS, but sufficient. All other HOMs,longitudinal and transverse, show a larger detuning. The ridged waveguides are machined into the cell, the slots by wire etching, the waveguide by milling. The SiCwedges are fixed with a specially designed clamp,avoiding thermal strains during brazing. Water coolingchannels and dimple tuning are provided. The completestructure is brazed vertically and in one pass. The SICA structure leads to a very compact design and its outer diameter of 174 mm allows the reuse of focusingsolenoids from the LEP preinjector which have a bore of180 mm. A disadvantage of the SICA structure is an increased ratio of surface field to accelerating gradient, about 30 %higher than for the TDS structure at the downstream end.A further 20 % increase of the surface field will occur atthe edges of the slots. Another critical issue is currentlyunder study: since geometric mode selection is relying onsymmetry, the demands on the symmetry of the structureare high. A quantitative sensitivity analysis is under way. ACKNOWLEDGEMENTS The couplers are modified versions from the DESY S-Band collider structures. Thanks to DESY also foradvice on brazing methods. REFERENCES [1] J.P. Delahaye, 37 co-authors: “CLIC – A Two-Beam e+e- Linear Collider in the TeV Range”, this conference. [2] H. Deruyter, 11 co-authors: “Damped and Detuned Accelerator Structures”, SLAC-PUB-5322, 1990.Figure 6: Cutaway view of 4 SICA cells. The outer diameter is 174 mm. The round holes are cooling watertubes; dimple tuning is provided (not shown). Figure 5: Reflection measurement: the input reflectionloss is below –30 dB in a usable bandwidth of 10 MHz.
arXiv:physics/0008053v1 [physics.ed-ph] 14 Aug 2000The Physicist’s Guide to the Orchestra Jean-Marc Bonard D´ epartement de Physique, ´Ecole Polytechnique F´ ed´ erale de Lausanne, CH-1015 Lausanne EPFL, Switzerland Email: jean-marc.bonard@epﬂ.ch An experimental study of strings, woodwinds (organ pipe, ﬂute, clarinet, saxophone and recorder), and the voice was u n- dertaken to illustrate the basic principles of sound produc tion in music instruments. The setup used is simple and consists of common laboratory equipment. Although the canonical examples (standing wave on a string, in an open and closed pipe) are easily reproduced, they fail to explain the majori ty of the measurements. The reasons for these deviations are outlined and discussed. INTRODUCTION Being a clarinet enthusiast, I wanted to share the rich- ness of music (and the physicist’s way of approaching it) to engineering students within the frame of their general physics classes. The basics are reported in nearly every physics textbook and many details are provided in more specialized contributions [1–6]. I could however not ﬁnd a comparison of the waveforms and spectra of diﬀerent instruments in any of these sources. I therefore measured and compared this data by myself. The whole orchestra being too ambitious for a start, I restricted myself to two instrument families, namely strings and woodwinds (and the former are given a rather perfunctory treatment) and included the voice. The waveform and spectra presented in the ﬁrst part illus- trate the behavior of standing waves on strings and in closed and open pipes (variations 1–2), as well as high- light the basic diﬀerences between the timbre of the in- struments (variations 3–5). In the second part, we will note that although the instruments are easy to identify, variations between models or performers remain hard to assess (variation 6). Furthermore, the simple models fail to explain the characteristics of some instruments (vari- ations 6–8). THEME Physicists have been studying the spectrum of musi- cal instruments at least since the 1940s. The ﬁrst avail- able methods were based on heterodyne analysis [7] or on sonographs [1,8]. The development of the Fast Fourier Transform (FFT) algorithm coupled with the apparition of fast and relatively cheap microprocessors has greatlyfacilitated the task of the musically inclined physicists. Quite sophisticated analysers have been realized [9] but setups based on commercial instruments work just as well for basic analysis [10,11]. The waveforms have been acquired with the setup pre- sented in Figure 1. It consists of a condenser microphone connected directly to a 125 MHz LeCroy9400 digital os- cilloscope. The spectrum was calculated by the data ac- quisition program LabView by FFT (this task can be directly performed on most modern oscilloscopes). On most ﬁgures, the time axis of the waveforms has been scaled and aligned for easier comparison. The spectra are given over 10 or 20 harmonics with ticks correspond- ing to multiples of the fundamental frequency. The waveforms and corresponding spectra analyzed here represent the continuous part of the sound only, which is only a small part of the character of a note. The dependence on the type of attack, duration or strength was not considered, and would be a fascinating study in itself. It is also clear that a more profound analysis would require a careful consideration of instrument and microphone position, as well as a calibration of the room acoustics and of the microphone response [7]. None of these points has been taken into account. FIG. 1. The experimental setup. VARIATION 1: VIBRATING STRINGS A string instrument in its simplest form is composed of a stretched string over a resonance body that transmits the vibration of the string to the air. Despite the sim- plicity of the vibrating system string instruments show a phenomenal diversity of timbre [12]. This arises from the variety of excitation as the string can be plucked (guitar, harp, harpsichord), bowed (violin, viola.. . ), or struck (piano). The resonance body plays also a great role as is attested by the timbre diﬀerence between a guitar and a banjo. When a continuous transverse wave is generated on a stretched string of length l, a standing wave forms fol- lowing the superposition of the waves after reﬂection at the stops. Simple considerations show that the only al- lowed modes of vibration correspond to wavelengths of λ= 2l/nwhere n≥1 [2–5], which forms a harmonic se- ries [13] (inset of Figure 2). The vibration of a string will be a superposition of the diﬀerent modes with varying 1amplitudes determined by the mode of excitation, time since the excitation etc. One of the most simple string instrument, the string sonometer, is formed of a simple stretched string over a box-shaped resonance body. Figure 2 shows the sound produced by a plucked and bowed sonometer tuned to A 2 [14] and demonstrates the richness of the sound produced by a vibrating string as many intense upper harmonics are detected. During bowing for example, the 5th, 16th and 23rd harmonics are stronger than the fundamental. The point and manner of excitation along the string (what physicists call the initial conditions) inﬂuences de - cisively the timbre: the spectra displayed in Figure 2 dif- fer markedly, especially in the higher modes. By pluck- ing a string, we impose an initial condition such that the shape of the string is triangular with zero and maximal displacement at the stops and at the position of the ﬁn- ger, respectively. The relation between position and in- tensity of the excited harmonics can be easily predicted (but is not easy to reproduce experimentally). This re- lation is not as simple for the bowed string, since the bow imparts both displacement (a fraction of mm) and velocity ( ∼0.5 m/s) [15]. Finally, we always consider that the properties of the vibrating string are ideal. A real string has however some stiﬀness, which causes an increase of the frequency of the higher modes with respect to an ideal string [4]. This can be detected on Figure 2, especially for the plucked string (presumably because of the larger displacement). As a consequence, the harmonics of a string are systematically sharp, be it for plucked [11], bowed [16] or struck strings such as in a piano [9,17]. FIG. 2. Waveforms and corresponding spectra of a sonome- ter (A 2, 110 Hz) plucked and bowed at 1 /10 of the string length, with in inset the ﬁrst three vibration modes. The thick line above the waveforms indicates the period of an os- cillation. VARIATION 2: VIBRATING AIR COLUMNS The principle of wind instruments is a bit more com- plicated than that of strings. The vibrating medium is the air inside a pipe that acts as a resonator where a standing wave can form. The ends of the pipe determine the boundary conditions. At a closed end, the amplitude of vibration is zero and the pressure variation is maximal (the displacement of the air and the resulting variation of pressure are in anti-phase). Conversely, the pressure will remain constant at the end of an open pipe and the standing wave shows a pressure node and a displacement antinode. This is schematically shown in the insets of Figure 4. As a consequence, a complete series of harmonics can form in a pipe of length lopen at both ends with wave-lengths equal to 2 ·l/nwithn≥1. If the pipe is closed at one end, the wavelength of the fundamental corresponds to four times the length of the pipe and only the odd har- monics of wavelengths equal to 4 ·l/(2n+ 1) with n≥0 are allowed [2–5]. The vibration of the air can be excited by diﬀerent means. The most simple one is to produce an edge tone by steering an airjet over an edge [like the top of a bottle or the edge of the organ ﬂue pipe shown in Figure 3(a)]. The edge forms an obstacle for the jet and generates pe- riodic vortices at the mouth of the instrument. The vor- tices produce in turn periodic displacements of the air molecules. When the edge forms the upper portion of a pipe, the edge tone is locked by resonance to the modes of the pipe. The pitch can then only be changed by increas- ing the frequency of the vortices (i.e., by blowing faster) to lock the edge tone in a higher mode (which is exactly how ﬂautists change the register and reach higher octaves with their instruments). Such an edge excitation acts like an open end, as the vortices induce air displacement but no pressure variations. The other means of excitation in wind instruments in- volve a mechanical vibration: that of the performer’s lips for brass instruments or of a reed for woodwinds. Sim- ilarly to the edge tones, the vibration of the lips or of the reed is locked to the resonances of the pipe. The simple reed of the clarinet [see Figure 3(b)] and of the saxophone, and the double reed of the oboe and bas- soon, acts actually as a pressure regulator by admitting periodically air packets into the pipe. A reed is there- fore equivalent to a closed end as it produces a pressure antinode. FIG. 3. Excitation systems for woodwinds: (a) the edge of a ﬂue organ pipe, and (b) the reed and mouthpiece of a clarinet. FIG. 4. Waveform and corresponding spectra of a closed and open organ ﬂue pipe (B ♭3, 235 Hz and B ♭4, 470 Hz, re- spectively). The timescale of the upper waveform has been divided by two with respect to the lower waveform. The in- sets show the ﬁrst three vibration modes for the variation of the pressure. We can verify the above principles with a square wooden ﬂue organ pipe of 0.35 m length. The excitation system is reproduced in Figure 3(a) and acts as an open end, and the other end can be either closed or open. As shown on Figure 4, the fundamental of the closed pipe is found at 235 Hz, which corresponds well to a wavelength ofλ=v/f= 4·0.35 = 1 .4 m with v= 330 m/s. The waveform is nearly triangular, and the even harmonics are far weaker than the odd. The same pipe with its end open sounds one full octave higher (the wavelength of the fundamental is shorter by a factor of 2) and displays a complete series of harmonics. 2VARIATION 3: TUTTI FIG. 5. Waveform of a violin, recorder, ﬂute, clarinet, sax- ophone and of the author singing the french vowel “aa” (as in sat) (A 4, 440 Hz for the former and A 3, 220 Hz for the latter three instruments). The timescale of the upper waveforms has been divided by two with respect to the lower waveforms. FIG. 6. Spectra corresponding to the waveforms of Fig- ure 5. We are now ready to study the behaviour of most strings and woodwinds. Figures 5 and 6 show the wave- forms and spectra of six diﬀerent instruments. Table I also summarizes the characteristics of the woodwinds studied here. A quick glance shows numerous disparities between the instruments, and we will try now to under- stand these timbre variations and their origin in more detail. TABLE I. Characteristics of the woodwinds studied in this work. instrument bore excitation ﬂute cylindrical edge clarinet cylindrical single reed saxophone conical single reed recorder cylindrical edgeVARIATION 4: THE VIOLIN The violin produces a very rich sound with at least 20 strong harmonics and complex waveforms, as was the case for the string sonometer in Figure 2. The strongest mode is not the fundamental, but the 7th harmonic in the case of Figure 6. VARIATION 5: THE FLUTE As can be seen on Figure 5, woodwinds show simple waveforms and spectra when compared to string instru- ments. The ﬂute (ﬂauto traverso) is a textbook exam- ple of an wind instrument with open ends as the pipe is (nearly) cylindric over the whole length. The most salient feature of the ﬂute is the limited number of harmon- ics (∼7) with an intensity that decreases monotonously [18,19]. The timbre is also very similar for the ﬁrst two registers (not shown here). VARIATION 6 (MENUETTO): THE CLARINET We have seen that a pipe closed at one end shows only odd harmonics: the clarinet, with its simple reed and (nearly) cylindric bore, should be a prototype of such a pipe. At ﬁrst sight, this is indeed the case. In the low reg- ister (Figure 5 for an A 3[21]), the odd harmonics are clearly the strongest modes. The even harmonics, al- though present, are strongly attenuated (at least up to the 6th harmonic). There are other marked diﬀerences with the ﬂute. First, the sound is far richer in higher har- monics. Second, the waveform varies considerably with the pitch as displayed on Figure 7. The contents of higher harmonics strongly decreases from 20 for the A 3, to 9 and 5 for the A 4and A 5. The contribution of the even har- monics becomes also increasingly important. The third mode remains more intense than the second for the A 4, but this is not the case anymore for A 5. FIG. 7. Waveform and corresponding spectra of a clarinet (A3, A4, A5at 220, 440 and 880 Hz, respectively). The timescale of the second and third waveform have been divided by two and four with respect to the lower waveform. The clarinet shows thus a fascinating behavior: it re- sponds like a pipe closed at one end in the lower reg- ister but gives a sound with strong even harmonics in the higher registers. The timbre varies therefore as the pitch is increased, with a very distinctive sound for each register. This is due to several facts. First, the bore of the clarinet is not perfectly cylindric but has tapered and slightly conical sections [2]. Second, the ﬂared bell, the constricting mouthpiece [Figure 3(b)] and the toneholes 3(even if they are closed) perturb signiﬁcantly the stand- ing waves. Finally, for wavelengths comparable to the diameter of the toneholes, the sound wave is no longer reﬂected at the open tonehole but continues to propa- gate down the pipe. This corresponds a frequency of ∼1500 Hz in typical clarinets [2,20], and the sound will show increasing amounts of even harmonics with increas- ing pitch, as found on Figure 7. Figure 7 leaves out one important feature. The clar- inet does not change from the ﬁrst to the second regis- ter by an octave (i.e., by doubling the frequency), but by a duodecime (tripling the frequency). This feature is due to the excitation system alone (the reed acts as a closed end), as can be easily demonstrated by replacing the mouthpiece of a ﬂute (or of a recorder) with a clarinet mouthpiece mounted on a section of pipe such that the overall length of the instrument remains identical. The instrument sounds a full octave lower and changes reg- isters in duodecimes, not in octaves. The reverse eﬀect can be demonstrated by mounting a ﬂute mouthpiece on a clarinet. Trio I: timbre quality It appears from Figure 5 that it is quite easy to rec- ognize an instrument family by its waveform or spectra. It would be tantalizing if one could also recognize one clarinet from another, for example to choose and buy a good instrument. Figure 8 shows the spectra of my three clarinets play- ing the same written note [21], with the same mouth- piece, reed, embouchure and loudness. The upper curve correspond to my ﬁrst instrument, a cheap wooden B ♭ student model. The two lower curves were obtained with professional grade B ♭and A clarinets. I can identify each instrument by playing a few notes from the produced sound and from muscular sensations in the embouchure and respiratory apparatus. At ﬁrst glance, the spectra of the three clarinets are readily comparable. Closer inspection shows that the spectra begin to diﬀer from the 10th harmonic on! There are actually far less variations in relative intensities be - tween the two B ♭instruments than between the two pro clarinets. The pro B ♭seems to be slightly richer in har- monics than the student model. The A has no strong harmonics beyond the 11th. This leads to the conclusion that the B ♭and A clarinets are (slightly) diﬀerent in- struments (many clarinetists will agree with that point). The measured diﬀerences between two B ♭clarinets re- main however quite subtle despite the huge and easily audible diﬀerence in timbre.FIG. 8. Spectra of the written C 4of a student B ♭and a professional grade B ♭and A clarinet played with the same mouthpiece and reed (sounding B ♭3, 235 Hz, and A 3, 220 Hz, respectively). Trio II: tone quality Is it possible to tell apart a good from a bad tone? This question is of utmost importance for every musician to obtain the desired tone quality. Figure 9 shows two clarinet tones obtained on the same instrument, with the same mouthpiece and reed. The ﬁrst is a good tone: one could describe it as fullbodied, agreeable to the ear. The second is a beginner’s tone: emitted with a closed throat and weak. The diﬀerence is instantly audible but diﬃcult to quantify from Figure 9. The variations appear again in the higher harmonics: the bad tone show no harmonics beyond the 12th, which is at least ﬁve modes less than the good tone. FIG. 9. Spectra of a good and of a bad sound on the clar- inet (B ♭3, 235 Hz). It is quite astonishing that the quality of the sound is determined by the presence (or absence) of high har- monics with amplitudes that are at least 40 dB (a fac- tor 104) weaker than the fundamental! Musicians are sensitive to very subtle eﬀects which are diﬃcult to (a) link to a physically measurable value and (b) to quantify precisely. Conventional statistics have proven ineﬀectiv e for classifying the sound quality: interestingly, eﬀectiv e solutions based on neural networks have been recently demonstrated [22]. VARIATION 7: THE SAXOPHONE Can one predict the spectra of the saxophone, a single reed instrument with a truncated conical bore, by ex- trapolation from the previous observations? The sax is a wind instrument, which would imply a limited number of harmonics, and a spectra mainly composed of odd har- monics because of the reed. A short glance at Figures 5 and 6 shows that both predictions are wrong. The even harmonics are as strong as the odd [23]. The sound re- mains very rich in harmonics even in the higher registers, far more than for the clarinet, and the timbre changes only slightly between the ﬁrst and the second register. The saxophone does not behave at all like a clarinet! The main reason is the form of the bore: in a cone, the standing waves are not plane but spherical [2,23–25]. This has profound implications for the standing wave pat- tern [24]. In short, the intensity of a wave travelling down or up the pipe is in ﬁrst approximation constant along the pipe, which implies that the amplitude scales with 4the inverse of the distance to the cone apex. The waves interfer to form a spherical standing wave with pressure nodes separated by the usual half-wavelength spacing, but with an amplitude that varies as the inverse of the distance to the cone apex. This is true for a closed as well as an open end [24]. A conical bore shows therefore a complete harmonic series, be it excited with a reed, the lips or an edge [25]! It would seem also that the coni- cal pipe of the saxophone favors the higher harmonics as compared to the cylindric bore of the clarinet. VARIATION 8: THE RECORDER The predictions for the saxophone were wrong, so let’s try again with another instrument – the recorder for ex- ample. That should be easy: the bore is nearly cylin- drical, it is excited by an edge and should therefore be similar to the open organ ﬂue pipe. I expected a limited number of harmonics and a full harmonic series. Fig- ure 5 shows that I was wrong again and this puzzled me greatly. The alto recorder indeed has a limited number of harmonics, and a similar timbre in the two registers. But it shows the spectrum of a closed pipe – the even harmonics are more suppressed than for the clarinet – and despite that it changes registers in octaves ! What is the explanation for the odd behaviour of the recorder? The player generates an airjet by blowing into a rectangular windcanal, which is then cut by the edge (see Figure 3). It appears from calculations that the po- sition of the edge relative to the airjet inﬂuences critical ly the intensity of the diﬀerent harmonics [26]. When the edge cuts the side of the jet, the full harmonic series is observed. The even harmonics are however completely absent when the edge is positioned in the center of the jet, as is the case for most modern recorders (among those the one I used). This of course does not aﬀect the modes of resonance of the instrument: the second harmonic can be excited easily by increasing the speed of the airjet, which raises the pitch by an octave. It follows also that I have been very lucky with the open organ ﬂue pipe – which follows the expected behaviour shown on Figure 4 thanks to a favorable position of the edge with respect to the airjet [26]! VARIATION 9: A CAPELLA We perform frequently with a peculiar and versatile musical instrument, namely our voice. Few instruments have such varied expressive possibilities and ability to change the timbre and loudness. From the point of view of musical acoustics, the voice is a combination of a string and a wind instrument. A pipe, the vocal tract, is ex- cited by the vibration of the vocal cords that generate a complete harmonic series as is usual for vibrating strings(see Figure 6). The timbre is however determined by the shape of the vocal tract that acts as a resonator. De- pending on the position of the tongue and on the mouth opening, the position and width of the formants of the vocal tract (the broad resonances, indicated in Figure 10) can be varied and some harmonics produced by the vo- cal cords are favored with respect to others [3,27]. Note that vocal tract and vocal cords are independent of each other, which implies that the timbre of the voice will change with the pitch for a given tract shape as the har- monics are shifted towards higher frequencies while the position of the formants remains constant. The eﬀect of the vocal tract shape is displayed in Fig- ure 10 for three vowels sung at the same pitch (the for- mants are also indicated). The tongue is placed closed to the palate to produce the “ii”: it is nearly sinusoidal with weak upper harmonics. The ﬁrst formant peaks around 200 Hz and decreases rapidly. The second and third for- mant around 2000 and 3000 Hz are however easily visible. The “ou” results from a single formant with a maximum around 300 Hz and a slowly decreasing tail: the wave- form is more complex and richer in higher harmonics, giving a ﬂute-like sound. The “aa” is obtained with an open tract and is far more complex. The most intense harmonic is the third because of the relatively high posi- tion ( ∼800 Hz) and large width of the ﬁrst formant. The second and the third formant are as intense as the ﬁrst and give a signiﬁcant amount of higher harmonics to the sound. FIG. 10. Waveform and corresponding spectra of the au- thor singing an A 3(220 Hz) on three diﬀerent vowels: the french “ii” (as in this), “ou” (as in shoe) and “aa” (as in sat) . The formants are indicated for each spectrum by a dotted line in linear scale. FINALE We have seen that the physicist’s approach to musical instruments opens fascinating and complex possibilities. The classical examples (closed and open pipe, for exam- ple) are easy to reproduce, but one steps quickly into ter- ritory uncharted by the classical physics textbook, which makes the exploration all the more exciting. It remains also that instruments are easy to identify by their timbre, but that it is quite diﬃcult to tell two diﬀerent models from one another and to classify the quality of the pro- duced sound. It may be even more diﬃcult (not to say impossible) to examine the quality of an interpretation and to understand why well-played music touches us so deeply. Musical acoustics is a beautiful subject to teach at ev- ery level. Music appeals to everybody and a lot of stu- dents play or have played at some stage an instrument: this makes often for lively demonstrations in front of the 5class. It involves both wave mechanics and ﬂuid me- chanics in quite complex ways, and a simple experimen- tal setup can oﬀer direct and compelling insights in the physics of sound production. I hope that this excursion in the basic physics of musical instruments will motivate some of the readers to include the subject in their curricu- lum and that it may provide helpful material for those who already do. ACKNOWLEDGMENTS I thank heartily the diﬀerent people that either lent me their instrument or that took some time to come and play in the lab: Ariane Michellod (ﬂute), S´ everine Michellod (recorders), Stephan Fedrigo (violin – hand- crafted by the performer!) and Lukas B¨ urgi (saxophone). I am also greatly indebted to Paul Braissant, Bernard Eg- ger and Yvette Fazan, who maintain and expand an im- pressive collection of physics demonstration experiments at EPFL, and who are never put oﬀ by the sometimes strange requests of physics teachers. [1] Leipp E 1975 Acoustique et musique (Paris: Masson) [2] Benade A H 1976 Fundamentals of musical acoustics (New York: Dover Publications) [3] White H E and White D H 1980 Physics and music (Philadelphia: Sauders College) [4] Rossing T D 1990 The Science of Sound, second edition (Reading: Addison-Wesley) [5] Fletcher N H and Rossing T D 1991 The Physics of Mu- sical Instruments (Berlin: Springer) [6] Taylor C 1992 Exploring music (Bristol: Institute of Physics Publishing) [7] Benade A H and Larson C O 1985 J. Ac. Soc. Am. 78 1475 [8] Lalo¨ e S and Lalo¨ e F 1985 Pour la science Mai73 [9] Brown J C 1996 J. Ac. Soc. Am. 991210 [10] Matsres M F and Miers R E 1997 Am. J. Phys. 65240 [11] Smedley J E 1998 Am. J. Phys. 66144 [12] The timbre is usually deﬁned as the quality of tone dis- tinctive of a particular singing voice or musical instru- ment, and is essentially determined by the relative inten- sity of the diﬀerent harmonics [13]. [13] The frequency ratios given by the modes of vibration on a string are pleasing to the ear and form the basis of western music (e.g., between the ﬁrst and second mode, 2:1 or an octave; the second and third mode, 3:2 or a ﬁfth). The series of modes with wavelengths λ= 2l/nand corresponding freqencies f=v/(2l/n) (where vis the velocity of the wave) is hence called the harmonic series, and the term mode and harmonic is used interchangeably. [14] The names of the notes refer to the tempered scale, formed of 12 semitones per octave with a frequency ra-tio between each semitone of12√ 2 = 1.059464. The A 4is deﬁned as the tuning A at 440 Hz, and the index of the note gives the octave. For further information on scales and intervalls see, e.g., Ref. [3]. [15] Broomﬁeld J E and Leask M J M 1999 Eur. J. Phys. 20 L3 [16] Halter C and Zuﬀerey J-C 2000 Regard scientiﬁque sur le monde artistique du son (´Ecole Polytechnique F´ ed´ erale de Lausanne) [17] Fletcher H 1964 J. Ac. Soc. Am. 36203 [18] Klein W L and Gerritsen H J 1975 Am. J. Phys. 43736 [19] Smith J R, Henrich N and Wolfe J 1997 Proc. Inst. Acous- tics19315 [20] Benade A H and Kouzoupis S N 1988 J. Ac. Soc. Am. 83292 [21] The clarinet is a transposing instrument, meaning that a written A 3willnotsound at the pitch an A 3. Most of to- day’s clarinets are either B ♭or A instruments, implying that a written C sound like a B ♭or an A, respectively. When not speciﬁed, the pitch given in the text corre- sponds to the actual sounding pitch. [22] Fasel I R, Bollacker K D and Ghosh J 1999 in Pro- ceedings of IJCNN’99 - International Joint Conference on Neural Networks vol. 3 (Piscataway: IEEE) p 1924 – also accessible at http:// www.ece.utexas.edu/ ∼fasel/ ICJNN draft/ICJNN Final.pdf [23] Benade A H and Lutgen S J 1988 J. Acoust. Soc. Am. 831900 [24] Ayers R D, Eliason L J and Mahgerefteh D 1985 Am. J. Phys.53528 [25] Boutillon X and Valette C 1992 J. Phys. IV C1-2 C1-105 [26] Fletcher N H and Douglas L M 1980 J. Acoust. Soc. Am. 68767 [27] Sundberg J 1982 Scientiﬁc American March 77 6This figure "figure1.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure2.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure3.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure4.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure5.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure6.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure7.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure8.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure9.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1This figure "figure10.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0008053v1
LIFETIME TESTING 700 MHZ RF WINDOWS FOR THE ACCELERATOR PRODUCTION OF TRITIUM PROGRAM K. A. Cummings, M. D. Borrego, J. DeBaca, J. S. Harrison, M. B. Rodriguez, D. M. Roybal, W. T. Roybal, S. C. Ruggles, P. A. Torrez, LANL, Los Alamos, NM 87545, USA G. D. White, Marconi Applied Technologies, Chelmsford, England Abstract Radio frequency (RF) windows are historically a point where failure occurs in input-power couplers foraccelerators. To understand more about the reliability ofhigh power RF windows, lifetime testing was done on 700MHz coaxial RF windows for the Low EnergyDemonstration Accelerator (LEDA) project of theAccelerator Production of Tritium (APT) program. The RFwindows, made by Marconi Applied Technologies(formerly EEV), were tested at 800 kW for an extendedperiod of time. Changes in the reflected power, vacuum,air outlet temperature, and surface temperature weremonitored over time. The results of the life testing aresummarized. 1 INTRODUCTION A lifetime window test stand has been set up to learn more about the failure mechanisms and lifetime of RFwindows for the Accelerator Production of Tritium (APT)program. Two 700 MHz RF windows prototypes wereordered from the vendor, Marconi Applied Technologiesand were conditioned in less than 20 hours and then testedto 800 kW for 4 hours. Based on these test results, 14additional windows were ordered for window life timetesting. Due to problems encountered with the initial highpower testing of these additional windows, not as muchlife time test data was collected as planned. Two windowswere tested for 245 hours and 155,513 kW hours on thestand. The other windows that were life tested for less timeencountered various problems . This paper summarizes thelife data obtained on the two windows which ran for anextended period of time, and it also summarizes theproblems encountered with the other windows . 2 EXPERIMENTAL APPARATUS AND PROCEDURE 2.1 Window Geometry The windows are a coaxial geometry using an AL995 alumina ceramic. As shown in Figure 1, the windowassembly consists of 1/2 WR 1150 waveguide on thevacuum side, a T-Bar transition into the coaxial region ofthe alumina ceramic, and another T-Bar to transition to fullheight WR 1500 waveguide on the air side. Theconfiguration shown in Figure 1 is referred to as a righthanded configuration. The air side waveguide may berotated 180 degrees around the axis of the coax to yield aleft handed configuration. Both the right handed and the left handed configurations are used to feed power to theCCDTL on the APT accelerator. The vacuum sidewaveguide is copper plated stainless steel. The inner andouter coax is copper and the air side waveguide isaluminum. The windows are cooled with 40 CFM of air cooling and 2 GPM of water cooling. The air inlet is on the air side Tbar. The air then flows through the T bar on the air side,down the inner conductor, and then exits the innerconductor through a series of exit holes and flows acrossthe alumina ceramic. The air exits the coaxial regionthrough a series of exit holes on the outer conductor and isvented to the ambient environment. The water circuitcools the vacuum side T bar region and the vacuumwaveguide. Figure: 1 EEV 700 MHz RF Window 2.2 Test Stand Description The test stand was designed to test up to eight windows in series into a matched load. The windows are set up inpairs in a back to back configuration with a section of 1/2height WR 1150 waveguide between them. Each windowpair is isolated by a pair of waveguide switches, so in theevent of a window problem, the affected pair can beisolated and the RF testing can continue. The test stand iscapable of passing 1 MW CW RF power. The test stand isconfigured so it may be operated remotely because it is inthe beam tunnel for the Low Energy DemonstrationAccelerator (LEDA) and access is not permitted whenLEDA is producing beam. 2.3 Diagnostic Equipment The RF window test stand includes many diagnostics. The vacuum pressure is measured at three places in eachwindow pair and is interlocked with the RF power. The pressure at which the interlock is set may be adjustedduring testing. Each RF window has two fiber optic arcdetectors, one on the air side and one on the vacuum side.The arc detectors are also interlocked with the RF powerand will shut off the RF power for 1.6 seconds upondetection of an arc. An analog voltage is displayedcorresponding to the intensity of light detected. Thisvoltage is used to detect a window that may not be arcing,but instead displaying a low intensity glow. The water, airinlet and air exit temperatures are monitored andinterlocked with the RF power. In addition, thetemperature of the outer coax in the region of the ceramicis also monitored and recorded. The position of thewaveguide switches are interlocked. If the position of awaveguide switch is changed during testing, the RF poweris shut off. If the switches in a given pair are not set in thesame position, the RF power is also disabled. ALabView® program is used to record and archive the outercoaxial and air outlet temperatures, vacuum pressure,number of arcs, and RF power. A DataHighway® systemis used to link information from the window test stand inthe beam tunnel to information at the klystron used toprovide RF power to the test stand. An EPICS interfacehas been set up to remotely operate the test stand and theklystron and to archive the data. 3 EXPERIMENTAL RESULTS 3.1 Initial High Power Acceptance Test Results When the windows first arrive at LANL they undergo high power acceptance testing. This testing consists ofconditioning the windows to 800 kW and then running at800 kW for 4 hours. Once they pass the high poweracceptance tests, the windows are then moved to the lifetest stand. This section summarizes the results of theinitial high power acceptance testing. Fourteen additional windows were ordered from EEV. So far, a total of nineteen windows have been received atLANL. Of these nineteen windows, six windows passedthe high power acceptance tests. Of the six windows, twodeveloped vacuum leaks and two were sent back to EEV tohave additional water cooling added in the area of thevacuum side T bar because of high temperatures in thisregion. The remaining two windows, serial #12 and #13,are on the life test stand at this time. Three windows arcedexcessively and were sent back to EEV. Four windowsshowed glowing and/or heating problems, and all four ofthese windows were grit blasted. Two of the four thenpassed the high power acceptance tests, however, they laterdeveloped vacuum leaks. The third of the four windowsbroke after grit blasting and the fourth window was sentback to EEV for additional cooling modifications. Anotherwindow developed a vacuum leak on a Conflat ® flange. This information is summarized in Table 1.Table 1: Summary of High Power Acceptance Testing Description Quantity of Windows Passed the High Power Tests 2 Arced 3 Glowed and/or Heated 4 Broken due to Air Cooling problem 2 Developed Vacuum leaks 3 Damaged in shipping 2 3.2 Lifetime Window Test Results Currently the RF windows have undergone 408.14 MW hours of life time testing. The breakdown for the totalnumber of MW hours is summarized in Table 2. Serial#04 and #06 were removed in April and sent back to thevendor to have the additional cooling modifications addedin the T-bar region. Serial #09 and #11 both developedvacuum leaks in the T-bar region and were sent back to thevendor as well. Thus, most of the life time test data hasbeen acquired on Serial #12 and #13. Table 2: Cumulative MW-Hours from Life Tests MW- Window Serial Number Hours 04 06 09 11 12 13 M o 11-99 16.77 16.77 16.77 16.77 n 12-99 0000 t 1-00 0.38 0.38 0.38 0.38 h 2-00 0000 & 3-00 14.26 14.26 38.36 38.36 Y 4-00 43.73 43.73 e 6-00 41.21 41.21 a 7-00 18.19 18.19 r 8-00 14.03 14.03 Total31.41 31.41 17.15 17.15 155.51 155.51 30405060708090 0 100 300 500 700 900 Power (kW)Temperature (Deg C) Figure 2. Steady State Temperature vs. Power The steady state temperatures of the outer coaxial surface at various power levels are shown in Figure 2. Thesetemperatures are on windows that did not show any arcing,glowing or heating problems. During the high poweracceptance testing the temperatures of the windows arecompared to this data as a tool to indicate any problems.The variations in the coaxial surface temperature, air out temperature, amount of arcing, and vacuum pressure areexamined over time. As shown in Figure 3, the coaxialsurface temperature is plotted at various power levelsduring each month for serial numbers 12 and 13. Asexpected from Figure 2, the average temperature increaseswith the average power, however, there is no correlation oftemperature, either increasing or decreasing with time.Figure 4 shows no correlation between the air outtemperature and test time; however, the noticeable increasein the air out temperature with power can be easily seen.The vacuum pressure remained very constant throughoutthe testing and there were no significant number of arcs. Temperature (Deg C) ower (kW)00 50 00 00 00 50 50510 00 00 5505 2-April 3-April 2-June 3-June 2-July3-July 2-August 3-August Figure: 3 Average Coaxial Surface Temperatures ower (kW)00 50 00 00 00 50 5000Temperature (Deg C)0 0 0 0 02-April 3-April 2-June3-June 2-July 3-July 2-August 3-August Figure: 4 Average Air Out Temperatures 4 DISCUSSION The problems encountered in the initial high power testing include an air cooling problem, shipping damage,vacuum leaks, arcing, and glowing and/or heating. The aircooling problem was caused by inadequate air flow due to a mis-adjusted orifice. This cooling problem is easilyexplainable and fixed. The shipping damage is also easilyexplained. The interesting problems are the vacuum leaks,glowing and/or heating, and arcing. The vacuum leaks inthe region of the T-bar are thought to be unrelated to theglowing or the grit blasting. The leaks are thought to bedue to high stresses caused by a large thermal gradient inthe region of the T-bar. This problem was corrected byadding a copper insert and additional cooling in the regionof the T bar. One window had a vacuum leak on aConflat® flange which was due to a poor weld. The glowing and/or heating and arcing are caused by surfacecontaminates and or defects[1]. The glowing and heatingare grouped together because if one problem is seen, theother one usually accompanies it. The glow is a blue orpurple color and the window temperatures are 10 to 20degrees Celsius higher. The temperature rise and theoptical emission are indicators multipacting[2,3]. If asignificant glow occurs, an analog voltage can be indicatedon the arc detectors. The intensity of the glow increaseswith the RF power level. Windows that are arcing usuallyexhibit a very good vacuum and do not display highertemperatures than normal. After the current amount of life testing, no definitive conclusions can be made about the lifetime of the windowbecause there were no trends in the surface and airtemperatures, the amount of arcing, or the vacuumpressure with time. Some improvements were made in thethermal mechanical design of the window by addingadditional cooling. Making the vacuum side waveguide outof copper would be a design improvement becauseincreasing in conductivity would improve the thermalmechanical properties of the window. 5 CONCLUSIONS Unknown variables about manufacturing and surface preparation processes have led to a low initial high powertesting success rate. Other problems encountered (i.e., aircooling problem, shipping damage, vacuum leaks) can becorrected and/or fixed. More life test data is needed todetermine the life time of the windows REFERENCES [1] K. A. Cummings, `Theoretical Predictions and Experimental Assessments of the Performance ofAlumina RF Windows ’, Ph.D. Thesis, University of California, Davis, CA, June 1998. [2] Y. Saito, N. Matuda, S. Anami, `Breakdown of Alumina RF Windows ’, Rev. Sci. Instrum., Vol. 60, No. 7, July 1989. [3] Y. Saito, N. Matuda, S. Anami, A. Kinbara, G. Horikoshi, J. Tanaka, `Breakdown of Alumina RFWindows ’, IEEE Trans. on Elect. Insul. , Vol. 24, No. 6, Dec 1989.
Funneling with the Two -Beam RFQ* H. Zimmermann, A. Bechtold, A. Schempp, J. Thibus, IAP, Frankfurt, Germany Institut für Angewandte Physik, Johann Wolfgang Goethe -Universität, Robert -Mayer -Straße 2 -4, D-60054 Frankfurt am Main, Germany Abstract New high current accelerator facilities like proposed for HIDIF or ESS require a beam with a high brilliance. These beams can not be produced by a single pass rf-linac. The increase in brightness in such a driver linac is done by several funneling stages at low e nergies, in which two identically bunched ion beams are combined into a single beam with twice the frequency current and brightness. Our Two -Beam -RFQ funneling experiment is a setup of two ion sources, a two beam RFQ, a funnel deflector and beam diagnostic equipment to demonstrate funneling of ion beams as a model for the first funneling stage of a HIIF driver. The progress of the funneling experiment and results of simulations will be presented. 1 INTRODUCTION The beam currents of linacs are limited by sp ace charge effects and the focusing and transport capability of the accelerator. Funneling is doubling the beam current by the combination of two bunched beams preaccelerated at a frequency f 0 with an rf -deflector to a common axis and injecting into anothe r rf-accelerator at frequency 2f 0, as indicated in fig. 1. Fig 1: Principle of funneling. By the use of the two -beam RFQ the two beams are brought very close together while they are still radially and longitudinally focused. Additional discrete element s like quadrupole -doublets and -triplets, debunchers and bending magnets, as they have been proposed in first funneling studies, are not necessary [1,2,3]. A short rf - funneling deflector will be placed around the beam crossing position behind the RFQ (fig. 2). The layout of Work supported by the BMBF the proposed HIDIF -injector with two -beam RFQs in front of the first and second funneling sections is shown in figure 3. The HIDIF linac starts with 16 times 3 ion sources for three different ion species t o allow so -called „telescoping“ at the final focus [5]. With four funneling stages the frequency has been increased from 12.5 MHz to 200 MHz accordingly [6]. Fig. 2: Experimental set -up of the two -beam funneling experiment. For studies of the new two -beam RFQ structure and the rf-deflector, the first two -beam funneling experiments have been done with He+-ions at low energies to facilitate ion source operation and beam diagnostics. Fig. 3: Layout of the 12.5...200MHz linac system for 400 mA of Bi+. Two small multicusp ion sources and electrostatic lenses, built by LBNL (Lawrence Berkeley National Laboratory) [7,8] are used. The ion sources and injection systems are attached directly on the front of the RFQ with an angle of 76 mrad, the angle of the b eam axes of the two - beam RFQ. Figure 2 shows a scheme of the experimental set -up of the two -beam funneling experiment with a slit -grid emittance measurement device on the right side . 2 ION SOURCES INJEC TION SYSTEMS AND TWO -BEAM RFQ Two multicusp ion sourc es have to deliver two identically ion beams. This operation has been tested on an emittance measurement device. The measured emittances of both ion -sources show differences up to 30% [4]. The two -beam RFQ consists of two sets of quadrupole electrodes, whe re the beams are bunched and accelerated with a phase shift of 180° between each bunch, driven by one resonant structure. With the use of identical RFQ electrode designs for both beam lines, the electrodes of one beam line are installed with a longitudinal shift of 2.55 cm (i.e. βλ/2 at final energy) to achieve the 180° phase shift between the beam bunches of each beam line. The measured normalized 90% RMS -emittance of the two beams are equal within 6 % [4]. 3 FUNNELING -DEFLECT ORS To bend the beam to a com mon axis we use two types, the singlegap and the multigap funnel deflector. Fig. 4: Schemes of the single - and multigap funnel deflectors. The arrows show the electric field during different periods. a) a one cell singlegap deflector , b) a three cell mu ltigap deflector, c) multigap deflector with one central drift tube. The deflectors are mounted at the point of beam crossing, which is 52 cm behind the RFQ. This device is like a plate capacitor, oscillating with the same resonant frequency as the RFQ. T he singlegap deflector is shown in Figure 5. Fig. 5: Figure of the singlegap funnel deflector . The deflector discs are mounted at water -cooled stems. The length is about 2 m. The angle between the two beam axis is 76 mrad. The singlegap funnel deflect or bends this angle down to zero by a voltage, which is in our experiment about 25 kV. Figure 6 shows a simulated beam bending with the singlegap funnel deflector for one beam. The deflector bends the beam from an average angle of 38 mrad down to zero. Fig. 6: Simulation of the beam bending in the singlegap funnel deflector . The angle is bending down from 38 mrad down to zero. Fig. 7: Emittance measurement while the singlegap deflector is switched off. Fig. 8: Emittance measurement while the single gap deflector is switched on. The two emittances are merged into a common beam. Figures 7 and 8 show two emittance measurements. If the funnel deflector is switched off, the beam drifts through the deflector and we measure the beam -angle of 76 mrad. Figur e 8 shows an emittance measurement with the singlegap funnel deflector switched on. The two beams are merged into a common beam. Fig. 9: Simulation of funnelling Fig 9 shows a simulation of beam bending of two beam s. The angle is reduced from 37.5 mrad d own to zero. The “banana form” of the simulated emittance is caused by inhomogeneous electric fields [10]. In a multigap funnel deflector the bending is done by many gaps, which reduce the bending voltage. Figure 10 shows a beam simulation for the multigap funnel deflector with nine gaps. Figure 11 shows an emittance measurement with the multigap funnel deflector. Fig. 10: Simulation of the beam bending by the multigap funnel deflector. Fig. 11: Emittance measurement while the multigap deflector is swit ched on. The two beams are merged to a single beam. CONCLUSION The measured emittances demonstrate, that both funnel deflectors brought the two beams to a common axis. The form of the measured ellipses show however, that we have to improve the matching of the beam to the RFQ to reduce the beam radius and phase width. As long as there is no matching section at the end of the RFQ electrodes, the beam is too large to measure emittance growing during funneling with our emittance measurement device. REFERENCES [1] K. Bongardt and D. Sanitz, Funneling of Heavy Ion Beams, Primary Report, Kernforschungszentrum Karlsruhe, 11 04 02P14C (September 1982) [2] J.F. Stovall, F.W. Guy, R.H. Stokes and T.P. Wangler, Beam Funneling Studies at Los Alamos, Nucl. Instr, and Me th. A278 (1989) p.143 [3] K.F. Johnson, O.R. Sander, G.O. Bolmer, J.D. Gilpatrick, F.W. Guy, J.H. Marquardt, K. Saadatmand, D. Dandoval and V. Yuan, A Beam Funnel Demonstration: Experiment and Simulation, Particle Accelerators, Vols. 37 -38 (1992) p. 261 [4] H. Zimmermann, A. Bechtold, A. Schempp, J. Thibus, PAC99, IEEE 99CH36366 (1999) p. 55 [5] M. Basco, M. Churazov, D. Koshkarev, Fusion Engineering and Design 32 -33 (1996) p. 73. [6] A. Schempp, The injector for the HIDIF driver linac, Nuclear Instruments and Methods in Physics Research A415 (1998) 209 -217 [7] K.N. Leung, Multicusp Ion Sources, Rev. Sci. Instrum. 65(4) (1994) p. 1165. [8] R. Keller in: The Physics and Technology of Ion Sources, Edited by I. G. Brown, Wiley -Interscience Publication, New Yo rk. [9] A. Schempp, Design of Compact RFQs, Proc. Linear Accelerator Conference 1996, CERN 96 -07, p. 53. [10] J. Thibus, Intrep 2000 -10
OVERVIEW OF THE APT ACCELERATOR DESIGN* J. F. Tooker, R. Bourque, D. Christiansen, J. Kamperschroer, G. Laughon, M. McCarthy, M. Schulze, General Atomics, San Diego, CA 92186 Abstract The accelerator for the APT Project is a 100 mA CW proton linac with an output energy of 1030 MeV. A High Energy Beam Transport (HEBT) conveys the beam to a raster expander, that provides a large rectangular distribution at a target/blanket (T/B) assembly. Spallation neutrons generated by the proton beam in the T/B reacts with Helium-3 to produce tritium. The design of the APT linac is an integrated normal-conducting (NC)/superconducting (SC) proton linac; the machine architecture has been discussed elsewhere [1]. The NC linac consists of a 75 keV injector, a 6.7-MeV 350-MHz RFQ (radio frequency quadrupole), a 96-MeV 700-MHz CCDTL (coupled-cavity drift-tube linac), and a 700-MHz CCL (coupled-cavity linac), with an output energy of 211 MeV. This is followed by a SC linac , that employs 700- MHz elliptical niobium 5-cell cavities to accelerate the beam to the final energy. The SC linac has two sections, optimized for beam velocities of β =0.64 and β =0.82. Each section is made up of cryomodules containing two, three, or four 5-cell cavities, driven by 1-MW 700-MHz klystrons. The singlet FODO lattice in the NC linac transitions to a doublet focusing lattice in the SC linac, with conventional quadrupole magnets in the warm inter-module spaces. This doublet lattice is continued in the HEBT. An overview of the current linac design will be presented. 1 INTRODUCTION Figure 1 shows the architecture of the APT 1030 MeV accelerator. It is a normal conducting accelerator up to 211 MeV. A 75-KeV injector generates the cw proton beam and acceleration continues through a RFQ, a CCDTL, and a CCL to 212 MeV. This is followed by a superconducting accelerator to the final energy. The first SC section has 102 five-cell medium- β niobium cavities optimized for β = 0.64. At the end of this section, the proton beam energy is 471 MeV. The second section is a high- β section that has five-cell niobium cavities optimized for β = 0.82. To attain an output energy of 1030 MeV, 140 cavities are needed. A HEBT directs the beam around a 90 degree bend to a 45 degree switchyard, where the beam goes straight to a beam stop for tuning or to the target/blanket. *Supported by DOE Contract DE-AC04-96AL99607Figure 1: Architecture for APT 1030 MeV accelerator 2 ACCELERATOR DESIG N The following sections describe the design of the various stages of the APT Linac. 2.1 Low Energy Linac The LE linac consists of the injector, a 350-MHz RFQ, a 700-MHz CCDTL, and a 700-MHz CCL. 2.1.1 Injector The 2.8-m long injector has a radio frequency (RF)- driven ion source that produces a 110-mA cw proton beam at 75 keV [1]. A low energy beam transport that has two solenoid magnets matches the proton beam into the acceptance of the RFQ. 2.1.2 RFQ The RFQ is an eight-meter long structure built of four resonantly coupled segments tuned for 350 MHz [1]. The RFQ accepts the 75-keV, 110-mA beam from the injector and produces a 6.7-MeV, 100-mA beam. It is driven by three 1.2-MW klystrons. The RFQ has been operated in the Low Energy Demonstration Accelerator of APT and its performance is discussed elsewhere [2,3]. 2.1.3 CCDTL A 700-MHz CCDTL [4] accepts the 100-mA beam from the RFQ and accelerates it to 96 MeV. The CCDTL cavities are grouped into six resonant structures called supermodules that span a length of 112.8 m. The first supermodule consists of a series of side-coupled 2- gap drift-tube linac (DTL) cavities with the quadrupole magnets [5] of the FODO lattice located between them. The focusing lattice begins with a period of 8- βλ to match the beam from the RFQ and transitions to 9- βλ at 9 MeV to provide additional space for the quadrupole magnets and beam diagnostics. Module two is made up of DTL cavities with two drift tubes, forming a series of three-gap cavities connected by coupling cells. Modules 3 to 6 consist of two-cavity, two-gap segments. To maintain strong transverse focusing, the quadrupole magnets in the FODO lattice continue with the same 9- βλ periodicity. The first module is energised by a single one MW klystron. The other five supermodules are energised by up to five klystrons. 2.1.4 CCL The CCL is composed of five supermodules spanning a length of 110.4 m. Each is made up of a string of seven-cell segments side-coupled to form a single resonant structure energised by up to seven one MW klystrons. The singlet 9- βλ FODO lattice is continued throughout the CCL The Coupled-Cavity Tuning (CCT) code [6] has been developed to help design the CCL cavities (and later the CCDTL cavities). This code automates the RF calculations using CCLFISH and iterates the geometry of the accelerating cavity, the geometry of the coupling cavity, and the separation between them to achieve the correct frequencies and coupling. The resultant geometry can then be fed into a CAD program to generate the drawings. Four cold models are planned along the length of the CCL to validate the code. Figure 2 shows one of the CCL cold models designed with CCT. Three-D RF analysis and coupled RF/structural analyses are also being performed on this cw RF structure [7] to address cavity sizing and the effects of high RF power densities. Figure 2: Cold Model of CCL Segments 283 & 284 2.2 High Energy Linac The superconducting cavities of the HE linac are contained in cryomodules that provide the thermal insulation and connection to the cryogenics system to maintain the cavities at their operating temperature of 2.15 K. In the HE linac, the FODO lattice of the LE linac transitions to a doublet lattice, consisting of normal-conducting quadrupole magnets [8] located in the warm regions between the cryomodules. The first six cryomodules in the medium- β section contain twocavities each (see Figure 3), providing a shorter 4.877-m focusing period. This was done to improve the match from the LE linac, Figure 3: Cross-section of two-cavity cryomodule. The remaining 30 cryomodules in the medium- β section contain three cavities each, with a longer 6.181-m focusing period. Each cryomodule in the medium- β section is powered by a single 1-MW klystron. Each cavity is fed by a single RF coaxial power coupler, so that the RF power from each klystron is divided by two or three, with up to 420 kW per coupler. The high- β section is a series of 35 four-cavity cryo- modules with the room temperature doublet quadrupole magnets in the warm regions between them. This section has a period of 8.540 m. Each high- β cryomodule is powered by two 700 MHz klystrons. The power from each klystron is split by two to feed a single RF coaxial power coupler on each of two cavities. 2.3 High Energy Beam Transport The doublet lattice of the HE linac is continued along the transport line of the HEBT. There are ten periods (85.4 m) prior to a 90 degree bend. If the dipole magnets of this bend are de- energized, the beam goes straight to a 0.1% duty cycle beam stop for tuning. If they are energized, the beam is then directed to a 45 degree switchyard. Here the beam can go straight to a beam stop capable of handling 2% of the full beam power. It is used during commissioning, start up, and tuning of the accelerator. The 45° bend in the switchyard can then direct the beam from this beam stop to the target/blanket assembly. The beam is expanded onto the target by a beam raster system [9], which sweeps the beam uniformly over the 19-cm wide by 190-cm high tungsten target. 2.4 RF Power System Three 1.2-MW, 350-MHz klystrons are used to energise the RFQ. Only two are required to accelerate the beam in the RFQ; the third is a spare so that the linac can continue to operate if one of the 350-MHz RF power systems fails. The power from each klystron is split by two, feeding six iris couplers in the RFQ cavity. This is shown in Figure 4. Figure 4: Three waveguide feeds to RFQ Cavity. The supermodules in the LE linac and the superconducting cavities in the HE linac are powered by 1-MW, 700-MHz klystrons. There are 52 klystrons in the LE linac, 36 klystrons in the medium- β section of the HE linac, and 70 klystrons in the high- β section of the HE linac for the 1030 MeV accelerator. The power from each klystron is split by two, except for two locations. The first module of the CCDTL is fed by a single klystron, where the power is split by four. For the three- cavity cryomodule, the power is split by three, with each cavity fed by a single power coupler, as shown in Figure 5. Figure 5: RF power splitting to 3-cavity cryomodule. 2.5 Cryo genics System The cryogenic system [10] supplies helium cooling to maintain the niobium cavities at 2.15 K. This system provides to the 1030 MeV accelerator approximately 15 kW of refrigeration at 2.15 K for the superconducting RF cavities and approximately 64 kW of refrigeration between 5 K and 30 K for the thermal intercepts on the RF power couplers, thermal shields in the cryomodules, and the cryo-distribution system. The cryogenic distribution line contains two sets of supply and return lines, and runs the length of the HE linac. Sets of four U-tube transfer lines connect each cryomodule to this distribution line. The distribution line is supplied by a cryogenics plant, where three, semi-independent helium refrigerators provide the closed-loop helium cooling. Each refrigerator contains a 4-K coldbox using gas- bearing turbine expanders, a 2-K coldbox with cold compressors to generate the sub-atmospheric conditions within the cryomodule, warm helium compressors for gas compression, liquid and gas storage, and appropriate instrumentation and controls. REFERENCES [1]J. Tooker, et al., “Overview of the APT Accelerator Design,” Proc. 1999 Particle Accelerator Conf., New York City, March-April 1999 [2] L.M. Young, et. al., “High Power Operations of LEDA,” these proceedings. [3] M. Schulze, et. al., “Beam Emittance Measurements of the LEDA RFQ,” these proceedings. [4] R. Wood, et. al, “Status of Engineering Development of CCDTL for Accelerator Production of Tritium,” Proc. 1998 Int. Linac Conf., Chicago (August 1998). [5] S. Sheynin, et. al, “APT High Energy Linac Intertank Assembly Design,” Proc. 1999 Particle Accelerator Conf., New York City, 1999 [6] P.D. Smith, “CCT, a Code to Automate the Design of Coupled Cavities,” these proceedings. {7} G. Spalek, et. al., “Studies of CCL Structures with 3D Codes,” these proceedings. [8] S. Sheynin, “APT High Energy Linac Intertank Assembly Design,” Proc. 1999 Particle Accelerator Conf., New York City, 1999. [9] S. Chapelle, et. al, “Testing of a Raster Magnet System for Expanding the APT Proton Beam,” Proc. 1999 Particle Accelerator Conf., New York City, 1999 [10] G. Laughon, “APT Cryogenic System,” Proc. of the 1999 Cryogenic Engineering Conf., Montreal, Canada, July, 1999 .
0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8(m) -14 -12 -10 -8 -6 -4 -2 0 1 2 3 4 5 6 7* * 0 5 10 15 20 0 5 10 15 20Systematic Exact Value SystematicRandom Random Error (Percent) 0 2 4 6 8 10 12 14 16 0 5 10 15 20 2530 Shots 30 Shots60 Shots 60 Shots90 Shots 90 Shots180 Shots 180 Shots Percent Random ErrorRECONSTRUCTION OF INITIAL BEAM CONDITIONS AT THE EXIT OF THE DARHT II ACCELERATOR* Arthur C. Paul, LLNL, Livermore, CA 94550, USA Abstract We consider a technique to determine the initial beam conditions of the DARHT II accelerator by measuring the beam size under three different magnetic transport settings. This may be time gated to resolve the parameters as a function of time within the 2000 nsec pulse. This technique leads to three equations in three unknowns with solution giving the accelerator exit beam radius, tilt, and emittance. We ®nd that systematic errors cancel and so are not a problem in unfolding the initial beam conditions. Random uncorrelated shot to shot errors can be managed by one of three strategies: 1) make the transport system optically de-magnifying; 2) average over many individual shots; or 3) make the ran- dom uncorrelated shot to shot errors suf®ciently small. The high power of the DARHT II beam requires that the beam transport system leading to a radius measuring apparatus be optically magnifying. This means that the shot to shot random errors must either be made small (less than about 1%) or that we average each of the three beam radius determinations over many individual shots. 1 THE DARHT II BEAMLINE The DARHT II beamline[1] consists of a series of transport solenoid lens and a kicker system to chop the beam to be sent to the X-ray converter target. Between the accelerator exit and the kicker is a series of three solenoids, lens S0, S2, and S3. Lens S3 matches the beam to the kicker system. Solenoid S2 is used in con- junction with an insertable beam dump, the "shuttle dump", to blow the beam up to a point that the density of energy deposition in the dump is small enough to allow the dumps survival. Solenoid S0, between the accelerator and S2, is used to generate different beam transport conditions for unfolding the initial beam condi- tions at the exit of the accelerator. A viewing port just in front of S2 is used to measure the beam radius. The beam must be several cm in radius to allow the survival of the viewing foil. The beam exiting this foil has been scattered to the point, that solenoid S2 and the large beam emittance induced by scattering in the viewing foil is suf®cient to diverge the beam on the shuttle dump. 2 THE PROCEDURE We de®ne several terms that will be used in this ------------------------------- * This work performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.work in order to avoid ambiguity. A"shot"is a 2 usec beam pulse from the accelerator. At minimum, three shots are required to re-construct the initial beam condi- tions. A measurement of radius can be either a single radius measurement or the average value of many indi- vidual measurements. To avoid confusion, we will use the term "determination" to be the measured radius value used in the procedure of beam reconstruction. Three radius determinations are required to re-construct the beam parameters. These three determinations require three shots if each determination is made using a single shot, or 3N shots if each determination is the average of N radius measurements for each determination. The procedure of re-construction unfolds from the determina- tions the beam emittance, initial radius and tilt. A single unfolding yields a value for the beam emittance, initial radius and tilt. Several unfoldings can be averaged to give an improved value of these parameters. Consider solenoid S2 set to 7.5 kG to expand the beam onto the shuttle dump almost independent of how the beam address S2. Consider a viewing foil to be inserted into the beamline at the pump port just in front of S2. This is the location were we make the radius measurements. Take the point for beam re-construction to be located 0.1111 meters beyond the exit of the accelerator. The transport of the beam from the re- construction point to the view port is then given by a ®eld free region L1, solenoid lens S0, and ®eld free region L2. V=      0001 001L2 0100 1L200            kS2−SC−kSCC2 −SC−k−1S2C2k−1SC −kSCC2−kS2SC C2k−1SCSCk−1S2            0001 001L1 0100 1L100           y′yx′x     2.1 R=        kS2kL2S2−SC−kSCC2−kL2SC −SC+kL2S2(kL1L2−k−1)S2−(L1+L2)SCC−L1kS(L1+L2)C+k−1S−L1L2kS −kSCC2−kL2SC−kS2SC−kL2S2 C2−kL1SC(L1+L2)C2+(k−1−kL1L2)SCSC−kL1S2(k−1−kL1L2)S2+(L1+L2)SC         Here L 1, L2are the drifts between the re-construction point and S0, and S0 and the view port respectively, C=cosθ, S=sinθ, k=B⁄(2Bρ) , andθ =kLsare the solenoid focusing terms. Three such transformations are required, one each at a given setting of solenoid S0, say 0, 3.5, and 4.5 kG. Consider three shots with the initial beam conditions of r=0.5 cm, tilt=0, and emittance 3.0 cm-mr, the nominal matched design values. For the three S0 settings above, ®gure 1, the beam at the view port would be        √  σ44√  σ33√  σ22√  σ11        =     5.7922.3065.7922.306          10.3501.66910.3501.669          18.7053.69218.7053.692     mrcmmrcm 2.3 √  σ11is the horizontal projection of the beam envelope, √  σ33is the vertical projection of the beam envelope. Here the beam radius is √  σ11=√  σ33as the beam is round and in its principle coordinate system, x=y=r. 3 THE SIGMA MATRIX Letσbe the matrix characterizing the phase space ellipse bounding all particles in the beam. Let R be the linear transformation matrix from the point of beam reconstruction to the location of the beam size measure- ment. A point (x,x') on the phase ellipse is given by σ22x2−2σ12xx′ + σ11x′2=det(σ) 3.1 (x,x′)  σ21σ11 σ22σ21  −1  x′x =1 3.2 In the principle coordinate system of a round beam represented by the vector space ( x, x′, y, y′)Twe have x=y and x′ =y′. If the beam is un-correlated between x and y, thenσ31= σ32= σ41= σ42=0 , and from sym- metryσ13= σ23= σ14= σ24=0 , but possibly tilted in the horizontal and vertical phase space, then σ21andσ43 would be non-zero, and eq(3.2) becomes (x,x′,y,y′)     00σ21σ11 00σ22σ21 σ43σ3300 σ44σ4300     −1    y′yx′x   =1 3.3 The four dimensional linear transformation matrix R=     R41R31R21R11 R42R32R22R12 R43R33R23R13 R44R34R24R14     3.4 transforms the sigma matrix by the similarity transform σ =RσoRT3.5 De®ne the initial reconstruction sigma matrix elements ofσofor a round beam to be: a≡ σ110= σ330, b≡ σ210= σ430, c≡ σ220= σ440. The square of the measured round beam size r2≡ σ11≡ σ33is given by r2= R112+R132 a+2 R11R12+R13R14 b+ R122+R142 c3.6 Here we have explicitly expanded eq(3.5) in terms of matrix eq(3.4) to represent σ11in terms of the initial beam sigma elements (a,b,c) and the values of the transformation matrix. Three sets of radius determina- tions, r 1, r2, and r3allow reconstruction of the initial beam parameters, a, b, and c.r32=C31a+C32b+C33cr22=C21a+C22b+C23cr12=C11a+C12b+C13c 3.7 C11,C12,.... are the known combinations of the transfor- mation matrix elements for the jth determination. Cj1= R112+R132 j, Cj2= R11R22+R13R14 j, Cj3= R122+R142 j Inverting we have the desired initial reconstructed beam parameters.       σ22oσ21oσ11o      =detC1_ ____     C21C32−C31C22−C21C33+C23C31C22C33−C23C32 −C11C32+C31C12C11C33−C13C31−C12C33+C32C13 C11C22−C12C21−C11C23+C13C21C12C23−C13C22           r32r22r12       4 RECONSTRUCTION Lets consider a case of beam reconstruction using three shots with each radius determination subject to a systematic error. As each radius determination would have the same error, the reconstructed initial beam radius should systematically have that same error, and the beam emittance being an area should have twice that error. De®ne the systematic measured radius error to be δr, then εoδε_ __=2Rδr_ __ 4.1 Lets now consider a case of un-correlated random errors in the three shots used to reconstruct the beam initial conditions. If the beam size is magni®ed by the "optics"of the lens system, then a measurement error δr will be magni®ed by the system magni®cation M. As the three measurements are un-correlated, the magni®ed errors will not cancel and we expect, for small errors that the emittance error should grow as εoδε_ __=2MRδr_ __ 4.2 Note for small Mδr, that the emittance curve is a straight line with slope M times that of the systematic error curve, ®gure 2. For largeMδrthe emittance curve parallels the systematic error curve. The transition between these two regimes appears to be some fractional power of the parameter Mδr. εoε_ __=1+a zn−b z2n+....... 4.3 z≡MRδr_ __ 4.4 Figure 2 shows the unfolded beam emittance as a func- tion of the magnitude of the un-correlated random error in the beam size determination. The shape of the curve is approximated by eq(4.3) with a=0.2, b=0.004, and n=2/3. Let M be some measure of the optical magni®cation of the system and δrbe the radius error.εoε_ __=1+0.2z2⁄3−0.004 z4⁄34.5 Consider our example with M=5.39 δr⁄R=10%,εo=8.44, a=0.2, n=2/3, then εoδε_ __=anMzn−1 Rδr_ __ 4.6 δε =1.90εo=16.0 cm−mr almost a 100% error in the reconstructed emittance. 5 NUMBER OF REQUIRED SHOTS We consider two strategies for rendering the beam emittance from N shots. The ®rst uses one unfolding of the emittance based on three radius determinations were each radius determination is the average of N/3 shots. Call this scenario A. The second strategy is based averaging N/3 emittance unfoldings each of which are the result of three radius determination with each radius determination consisting of a single shot. Call this scenario B. The ®t to the emittance error curves are represented by eq(5.1) ε =a1x+a2xn5.1 x is the error in percent and εis the emittance in cm-mr. This equation is used to ®t scenarios A and B. Scenario A is well represented by a 0.72 power law. Scenario B is represented by a 0.5 - 0.6 power law. Note that the lower bounds, are straight lines for large errors given a lower bound to the emittance independent of the value of the random error, ®gure 3. The reason that making many emittance determi- nations with out averaging the radius values gives an better value for the average value of the unfolded beam emittance is that values of emittance that by the luck of the draw (random number sequence) that are negative or zero are averaged out by the positive random values. In scenario A, were we unfold just one emittance but aver- age the radius determinations yield a negative or zero value for some random sequences. With just one unfold- ing there are no positive values to average this unfor- tunate value. 6 CONCLUSIONS The beam emittance is related to the area in phase space occupied by the particles comprising the beam. The reconstruction of the emittance by a radius measure- ment with error δrshould yield an error in the emit- tance of at most 2δr. With a system of optical magni®cation M the error is 2Mδr. Use of the shuttle dump diagnostic on DARHT II to determine the beam emittance to within a factor of two using a minimum number of shots requires either 1) the random un- correlated shot to shot errors be less than about 1%, or 2) we average 30 to 60 shots using scenario B with ran- dom errors some where in the range of 5 to 10 percent.7 REFERENCES [1] "DARHT-II Downstream Beam Transport Beamline ", G.A.Westenskow, L.R.Bertolini, P.T.Duffy, A.C.Paul this proceedings. Figure 1. Over plot of the three beam envelopes for the example of beam reconstruction with solenoid S0 at 0, 3.5, and 4.5 kG. The radius measurement is made at view port located at 7.20 meters. Figure 2. Unfolded beam emittance vs random error in the beam radius measuremet. The exact value and the range of values for a systematic error is also shown. The radial magni®cation of 5.4 ampli®es the error. Figure 3. Required number of shots for a given emit- tance range vs radial error using averaging of emittance value method. Nominal beam radius 0.95 cm, tilt -0.35, and emittance of 8.44 cm-mr.
arXiv:physics/0008058v1 [physics.gen-ph] 14 Aug 2000QUANTUM STRING FIELD THEORY AND PSYCHOPHYSICS Denis V. Juriev ul.Miklukho-Maklaya 20-180, Moscow 117437 Russia (e-mail: denis@juriev.msk.ru) physics/0008058 The quantum string ﬁeld theoretic structure of interactive pheno mena is discussed. This note continues the author’s researches on the boundary of experimental mathematics, psychophysics and computer science, which we re initiated about ten years ago. Precisely, it is devoted to the unraveling of quan tum string ﬁeld theoretic (general aspects of this theory are discussed in the book [1] and its mathematical formalism based on the inﬁnite dimensional geometry is expo sed in [2]) structures in the picture described in two previous notes [3]. The resul ts may be signiﬁcant for the constructing of a very important bridge between fund amental theoretical high-energy physics and modern psychophysics. The interac tive game theoretic sur- rounding of the least may essentially enrich the quantum str ing ﬁeld theory by new original features, which will be interesting for pure mathe maticians. Such alliance may be interesting to the theoretical physicists as supplyi ng their sophisticated constructions with a very simple and inexpensive experimen tal veriﬁcation. I. Interactive phenomena: experimental detection and analys is [3] 1.1. Experimental detection of interactive phenomena. Let us consider a natural, behavioral, social or economical system S. It will be described by a set {ϕ}of quntities, which characterize it at any moment of time t(so that ϕ=ϕt). One may suppose that the evolution of the system is described by a diﬀerential equation ˙ϕ= Φ(ϕ) and look for the explicit form of the function Φ from the exper imental data on the system S. However, the function Φ may depend on time, it means that the re are some hidden parameters, which control the system Sand its evolution is of the form ˙ϕ= Φ(ϕ, u), where uare such parameters of unknown nature. One may suspect that s uch parameters are chosen in a way to minimize some goal function K, which may be an integrodiﬀerential functional of ϕt: K=K([ϕτ]τ≤t) Typeset by AMS-TEX(such integrodiﬀerential dependence will be brieﬂy notate d asK=K([ϕ]) below). More generally, the parameters umay be divided on parts u= (u1, . . . , u n) and each partuihas its own goal function Ki. However, this hypothesis may be conﬁrmed by the experiment very rarely. In the most cases the choice of parameters uwill seem accidental or even random. Nevertheless, one may suspe ct that the controls uiareinteractive , it means that they are the couplings of the pure controls u◦ iwith theunknown or incompletely known feedbacks: ui=ui(u◦ i,[ϕ]) and each pure control has its own goal function Ki. Thus, it is suspected that the system Srealizes an interactive game . There are several ways to deﬁne the pure controls u◦ i. One of them is the integrodiﬀerential ﬁltration of the cont rolsui: u◦ i=Fi([ui],[ϕ]). To verify the formulated hypothesis and to ﬁnd the explicit f orm of the convenient ﬁltrations Fiand goal functions Kione should use the theory of interactive games, which supplies us by the predictions of the game, and compare the predictions with the real history of the game for any considered FiandKiand choose such ﬁltrations and goal functions, which describe the reality better. One m ay suspect that the dependence of uionϕis purely diﬀerential for simplicity or to introduce the so- called intention ﬁelds , which allow to consider any interactive game as diﬀerentia l. Moreover, one may suppose that ui=ui(u◦ i, ϕ) and apply the elaborated procedures of a posteriori analysis and predictions to the system. In many cases this simple algorithm eﬀectively unravels the hidden interactivity of a complex system. However, more sophisticated procedure s exist [3]. Below we shall consider the complex systems S, which have been yet represented as the n-person interactive games by the procedure described above . 1.2. Functional analysis of interactive phenomena. To perform an analysis of the interactive control let us note that often for the n-person interactive game the interactive controls ui=ui(u◦ i,[ϕ]) may be represented in the form ui=ui(u◦ i,[ϕ];εi), where the dependence of the interactive controls on the argu ments u◦ i, [ϕ] andεiis known but the ε-parameters εiare the unknown or incompletely known functions ofu◦ i, [ε]. Such representation is very useful in the theory of intera ctive games and is called the ε-representation . One may regard ε-parameters as new magnitudes, which characterize the syst em, and apply the algorithm of the unraveling of interactivity t o them. Note that ε- parameters are of an existential nature depending as on the s tatesϕof the system Sas on the controls. Theε-parameters are useful for the functional analysis of the in teractive controls described below. 2First of all, let us consider new integrodiﬀerential ﬁltrat ionsVα: v◦ α=Vα([ε],[ϕ]), where ε= (ε1, . . . , ε n). Second, we shall suppose that the ε-parameters are ex- pressed via the new controls v◦ α, which will be called desires: εi=ε(v◦ 1, . . . , v◦ m,[ϕ]) and the least have the goal functions Lα. The procedure of unraveling of interac- tivity speciﬁes as the ﬁltrations Vαas the goal functions Lα. 1.3. SD-transform and SD-pairs. The interesting feature of the proposed de- scription (which will be called the S-picture ) of an interactive system Sis that it contains as the real (usually personal) subjects with the pu re controls uias the impersonal desires vα. The least are interpreted as certain perturbations of the ﬁrst so the subjects act in the system by the interactive cont rolsuiwhereas the desires are hidden in their actions. One is able to construct the dual picture (the D-picture ), where the desires act in the system Sinteractively and the pure controls of the real subjects are hidden in their actions. Precisely, the evolution of the system is g overned by the equations ˙ϕ=˜Φ(ϕ, v), where v= (v1, . . . , v m) are the ε-represented interactive desires: vα=vα(v◦ α,[ϕ]; ˜εα) and the ε-parameters ˜ εare the unknown or incompletely known functions of the states [ ϕ] and the pure controls u◦ i. D-picture is convenient for a description of systems Swith a variable number of acting persons. Addition of a new person does not make any i nﬂuence on the evolution equations, a subsidiary term to the ε-parameters should be added only. The transition from the S-picture to the D-picture is called theSD-transform . TheSD-pair is deﬁned by the evolution equations in the system Sof the form ˙ϕ= Φ(ϕ, u) =˜Φ(ϕ, v), where u= (u1, . . . , u n),v= (v1, . . . , v m), ui=ui(u◦ i,[ϕ];εi) vα=vα(v◦ α,[ϕ]; ˜εα) and the ε-parameters ε= (ε1, . . . , ε n) and ˜ ε= (˜ε1, . . . ,˜εm) are the unknown or incompletely known functions of [ ϕ] and v◦= (v◦ 1, . . . , v◦ m) oru◦= (u◦ 1, . . . , u◦ n), respectively. Note that the S-picture and the D-picture may be regarded as c omplementary in the N.Bohr sense. Both descriptions of the system Scan not be applied to it simultaneously during its analysis, however, they are comp atible and the structure of SD-pair is a manifestation of their compatibility. 3II. Quantum string field theoretic structure of interactive phenomena 2.1. The second quantization of desires. Intuitively it is reasonable to con- sider systems with a variable number of desires. It can be don e via the second quantization. To perform the second quantization of desires let us mention that they are deﬁned as the integrodiﬀerential functionals of ϕandεvia the integrodiﬀerential ﬁltrations. So one is able to deﬁne the linear space Hof all ﬁltrations (regarded as classical ﬁelds) and a submanifold Mof the dual H∗so that His naturally identiﬁed with a subspace of the linear space O(M) of smooth functions on M. The quantized ﬁelds of desires are certain operators in the space O(M) (one is able to regard them as unbounded operators in its certain Hilbert completion). Th e creation/annihilation operators are constructed from the operators of multiplica tion on an element of H⊂ O(M) and their conjugates. To deﬁne the quantum dynamics one should separate the quick a nd slow time. Quick time is used to make a ﬁltration and the dynamics is real ized in slow time. Such dynamics may have a Hamiltonian form being governed by a quantum Hamil- tonian, which is usually diﬀerential operator in O(M). IfMcoincides with the whole H∗then the quadratic part of a Hamiltonian de- scribes a propagator of the quantum desire whereas the highe st terms correspond to the vertex structure of self-interaction of the quantum ﬁ eld. If the submani- foldMis nonlinear the extraction of propagators and interaction vertices is not straightforward. 2.2. Quantum string ﬁeld theoretic structure of the second q uantization of desires. First of all, let us mark that the functions ϕ(τ) and ε(τ) may be regarded formally as an open string. The target space is a pro duct of the spaces of states and ε-parameters. Second, let us consider a classical counterpart of the evolu tion of the integrodif- ferential ﬁltration. It is natural to suspect that such evol ution is local in time, i.e. ﬁltrations do not enlarge their support (as a time interval) during their evolution. For instance, if the integradiﬀerential ﬁltration depends on the values of ϕ(τ),ε(τ) forτ∈[t0−t1, t0−t2] at the ﬁxed moment t0it will depend on the same values forτ∈[t−t1, t−t2] at other moments t > t 0. This supposition provides the reparametrization invariance of the classical evolution. Hence, it is reasonable to think that the quantum evolution is also reparametrization invariant. Reparametrization invariance allows to apply the quantum s tring ﬁeld theoretic models to the second quantization of desires. For instance, one may use the string ﬁeld actions constructed from the closed string vertices (n ote that the phase space for an open string coincides with the conﬁguration space of a closed string) or string ﬁeld theoretic nonperturbative actions. In the least case t he theoretic presence of additional ”vacua” (minimums of the string ﬁeld action) is v ery interesting. 2.3. Additional ﬁelds and virtual subjects. Often quantum string ﬁeld the- ory claims an introduction of additional ﬁelds (such as boso nised ghosts). Let us consider such ﬁelds in the D-picture. In D-picture desires have their own ε-parameters and depend on the pure con- trols of subjects. These pure controls may be obtained from t heε-parameters of desires via integrodiﬀerential ﬁltrations. One is able to a pply such ﬁltrations to the 4additional ﬁelds. There are two possibilities. First, the r esult is expressed via the known pure controls. Second, the result is a new pure control of avirtual subject . Certainly, any experimental detection of virtual subjects is extremely interesting. III. Conclusions Thus, the quantum string ﬁeld theoretic structure of intera ctive phenomena is described. Possible qualitative eﬀects, which are produce d by this structure and conﬁrm its presence, are emphasized. Perspectives are brie ﬂy speciﬁed. References [1] Green M.B., Schwarz J.H., Witten E., Superstring theory. Camb ridge Univ.Press, Cam- bridge, 1988. [2] Juriev D., Inﬁnite dimensional geometry and quantum ﬁeld t heory of strings. I-III. AGG 11(1994) 145-179 [hep-th/9403068], RJMP 4(3) (1996) 18 7-314 [hep-th/9403148], JGP 16 (1995) 275-300 [hep-th/9401026]; String ﬁeld theory an d quantum groups. I: q-alg/9708009. [3] Juriev D., Experimental detection of interactive phenomen a and their analysis: math.GM/0003001; New mathematical methods for psychophys ical ﬁltering of experi- mental data and their processing: math.GM/0005275. 5
THE MECHANICAL DESIGN FOR THE DARHT-II DOWNSTREAM BEAM TRANSPORT LINE* G. A. Westenskow, L. R. Bertolini, P.T. Duffy, A. C. Paul Lawrence Livermore National Laboratory, Livermore, CA 94550 USA Abstract. This paper describes the mechanical design of the downstream beam transport line for the second axis of theDual Axis Radiographic Hydrodynamic Test (DARHT II)Facility. The DARHT-II project is a collaborationbetween LANL, LBNL and LLNL. DARHT II is a20-MeV, 2000-Amperes, 2- µsec linear induction accelerator designed to generate short bursts of x-rays forthe purpose of radiographing dense objects. The down-stream beam transport line is approximately 20-meter longregion extending from the end of the accelerator to thebremsstrahlung target. Within this proposed transport linethere are 15 conventional solenoid, quadrupole and dipolemagnets; as well as several speciality magnets, whichtransport and focus the beam to the target and to the beamdumps. There are two high power beam dumps, which aredesigned to absorb 80-kJ per pulse during acceleratorstart-up and operation. Aspects of the mechanical designof these elements are presented. 1 INTRODUCTION We are working on the engineering design of the downstream beam transport components of the DARHT IIAccelerator [1]. Beam transport studies for this designhave been performed [2]. Figure 1 shows the proposedlayout for the elements in the system during earlycommissioning of the downstream components. Thebeamline from the exit of the accelerator to the target isabout 20 meters long. In the accelerator the pulse length isabout 2 µsec. However, only four short (20 to 100 nsec) pulses separated b ’y about 600 nsec are desired at the bremsstrahlung target. The function of the kicker systemis to "kick" four shorter pulses out from the main 2- µsec. The kicker includes a bias dipole operated so that the non-kicked parts are deflected off the main line into the mainbeam dump, while the kicked pulses are sent straightahead. Focusing elements between the kicker and theseptum would complicate operation. Therefore, to achievea narrow beam waist at the septum, solenoid S3 must"throw" a waist to the septum. The first 3 meters ofbeamline allow the beam to expand from its 5-mm * This work was performed under the auspices of the U.S. Department of Energy by University of CaliforniaLawrence Livermore National Laboratory under contractNo. W-7405-Eng-48.matched radius in the accelerator to 2.25-cm at solenoid S3. The system is designed to have a 20% energyacceptance to transport the main beam and most of theleading and falling edge of the pulse exiting theaccelerator. The proposed system using a quadrupolemagnet [2] allows for a larger beam pipe radius than themore conventional septum dipole magnet studied earlier.This increases the energy acceptance of the transport lineto the main beam dump. Work on the kicker system has been described elsewhere[3]. After the septum, there are four Collinsstyle quadrupole magnets to restore the beam to a roundprofile. Experience has shown ejecta material from thetarget can travel the length of the accelerator. To keep thismaterial from reaching the injector and accelerator cellswe have included a rotating wheel debris blocker that doesnot allow direct line-of-sight between the target and theupstream elements for times shortly after the pulse. Finally the beam will be pinched to a tight focus at the target to provide an intense spot of x-rays for radiographicpurposes. Work on the target is also presented in theseproceedings [4]. 2 TRANSPORT ELEMENTS The magnets within the DARHT II transport line are all water-cooled conventional dc electromagnets (except thebias dipole and the kicker sextupole corrector). Themagnets are listed in Table 1. The transport solenoids have external iron shrouds with water-cooled copper coils. Solenoid coils are wound intoindividual two-layer “pancake ” coils. Each magnet has an even number of these pancake coils. The pancakes areinstalled in an A-B-A-B orientation to minimize axialfield errors. The inside diameters of the solenoid coils aresized large enough to fit over the outside diameter of thebeam tube flanges. The septum quadrupole and dipole magnets have solid iron cores with water-cooled copper coils. The SeptumQuadrupole Magnet is a four-piece, solid-core construc-tion. The Collins Quadrupole Magnets are two-piece solidcores with non-magnetic support. The dipole magnet is athree-piece, solid-core, “C” magnet. The alignment requirements for the transport magnets are +0.4-mm positional tolerance and + 3-mrad angular tolerance. Steering corrector coils will be installed undermany of the magnetic elements to allow for smallalignment errors or stray magnetic fields.Collins Hor. Quad S1 SolenoidS2 SolenoidS3 Solenoid Shuttle DumpKicker, Bias Dipole,Sextupole Cor.Septum QuadCollins Ver. QuadCollins Waist QuadS4 SolenoidFinal Focus SoloenoidConverter Region End of the acceleratorCollins X QuadDebris blockerS5 SoloenoidStairs Septum Dipole(oriented down)Main Dump (oriented down) Figure 1: Layout of the transport elements. Figure 2: Picture of a Collins Quadrupole. Figure 3: Picture of the Septum Quadrupole being characterised during a rotating coil test.Transport Elements in the Downstream Beamline. Magnet TypeMagnet NameMax. Field (kG) or Gradient (kG/m)Bore or Gap (cm) Solenoid S1 8 27 Solenoid S2 8 27 Solenoid S3 2.5 27 Dipole Bias Dipole 0.009 41 PulsedDipoleKicker -0.009 (equivalent)12.8 Sextupole Sextupole Corrector16 gauss @20.5 cm radius41 Quadrupole Septum Quadrupole8.0 38 Quadrupole Collins-H 10.0 12 Quadrupole Collins-V 10.0 12 Quadrupole Collins-W 10.0 12 Quadrupole Collins-X 10.0 12 Solenoid S4 2.5 27 Solenoid S5 2.5 27 Solenoid Final Focus Solenoid~5 ~13 Dipole Septum Dipole1.0 16 Table 1: Magnet specifications3 VACUUM SYSTEM The vacuum chambers for the DARHT II Transport Line are circular beam pipes constructed from 304Lstainless steel. The region from the end of the acceleratorthrough the septum has a 16-cm bore diameter. From theseptum to the target, the bore diameter is reduced to9.55 cm. The vacuum seals are made with conflat styleknife-edge flanges with annealed copper gaskets. The useof all-metal seals is driven by the potential requirement toin situ bake the transport vacuum system. In situ bake-out may be required to minimize adsorbed gas on the beamtube walls, which may be desorbed by beam halo scrapingthe walls. The vacuum design requirement for thetransport line is 10 -7 Torr pressure. Figure 4 shows a side view of the septum vacuum chamber. The chamber resides in the region were thebeamline splits between the line going to the target and theline going to the main dump. The chamber is formed bytwo aluminium halves that are then welded together at themidplane. Septum Quadrupole to target to main dumpSeptum DipoleCollins-H Quad.Collins-V Quad. Septum Vacuum Chamber Figure 4: Arrangement of the transport elements around the septum. Horizontal view. The main part of the pulseenters the Septum Quadrupole o ff-axis and is bent into the Septum Dipole, it is then bent further into the main dump.The kicked portion goes straight ahead though the CollinsQuadrupoles. 4 BEAM DUMPS There are two beam dumps included in the DARHT II downstream transport system; a main beam dump, and ashuttle dump. The purpose of the shuttle dump is to allowaccelerator operations while personnel are working in thetarget area outside the accelerator hall. The shuttle dumpwill have a composite absorber, made up of a 3-inch thickgraphite block, backed by 12-inches of tungsten. Therewill be additional shielding surrounding the beam stop toabsorb radiation.The main beam dump absorbs the portion of the beam that is not deflected by the kicker system. The normalhorizontal beam size at the main beam dump is 8 cm.However, the start-up parameters for the beam will not bewell known. We must therefore provide some safetymargin. First consideration is to keep the instantaneoustemperature (temperature at the end of the 2- µsec pulse) of the impact area below the damage point for thematerial. At 1 pulse per minute repetition rate we canmanage the average temperature increase. We also desireto keep the neutron yield low to minimize activation ofcomponents and simplify radiation shielding. Theconstruction of the beam dump must be compatible withhigh vacuum as explained in the previous section. 5 DIAGNOSTICS Throughout the beamline there are beam position monitors (BPMs) to measure the location and angle oftrajectory of the beam. The BPMs mount between theflanges of adjacent transport beam tubes. The accuratetransverse location of the BPMs is critical to the operationof the transport line and it is their positional requirements,which set the alignment tolerances for the beam linevacuum system. To satisfy the BPM requirements, thetransport line beam tubes must be aligned to within 0.2mm offset. Each beam tube will be manufactured withfiducials, which allow the survey crew to measure andposition external to the centerline. During commissioning rather than going into the "target" region, we will have beam diagnostics on thestraight ahead beamline. They will include a spectrometerfor measuring beam energy, and a “pepper-pot ” for measuring beam emittance. The septum vacuum chamber has several ports near the septum region to allow characterisation of the plasmagenerated during pulses when the beam is spilt on nearbywalls. REFERENCES [1] M.J. Burns, et al., “DARHT Accelerators Update and Plans for Initial Operation ”, Proc. of the 1999 Particle Accelerator Conference, New York, NY, (1999). [2] A. Paul, et al., “The Beamline for the Second Axis of the Dual Axis Radiographic Hydrodynamic TestFacility ”, Proc. of the 1999 Particle Accelerator Conference, New York, NY, (1999). [3] Y.J. Chen, et al., “Precision Fast Kickers for Kiloampere Electron Beams ”, Proc. of the 1999 Particle Accelerator Conf., New York, NY, (1999). [4] S. Sampayan, et al., “Beam-Target Interactions Experiments For Bremsstrahlung ConvertersApplications ”, Proc. of the XX Intern. Linac Conference, Monterey, CA, (2000).
arXiv:physics/0008060v1 [physics.atom-ph] 14 Aug 2000Theoretical study of the absorption spectra of the sodium di mer H.-K. Chung, K. Kirby, and J. F. Babb Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Abstract Absorption of radiation from the sodium dimer molecular sta tes correlating to Na(3s)-Na(3s) is investigated theoretically. Vibratio nal bound and contin- uum transitions from the singlet X1Σ+ gstate to the ﬁrst excited A1Σ+ uand B1Πustates and from the triplet a3Σ+ ustate to the ﬁrst excited b3Σ+ gand c3Πgstates are studied quantum-mechanically. Theoretical and experimen- tal data are used to characterize the molecular properties t aking advantage of knowledge recently obtained from ab initio calculations, spectroscopy, and ultra-cold atom collision studies. The quantum-mechanica l calculations are carried out for temperatures in the range from 500 to 3000 K an d are com- pared with previous calculations and measurements where av ailable. PACS numbers: 33.20.-t, 34.20.Mq, 52.25.Qt Typeset using REVT EX 1I. INTRODUCTION Vast amounts of experimental spectroscopic data on the elec tronic states and ro- vibrational levels of the sodium dimer are available and man y theoretical studies have been performed. For example, Ref. [1] presents an extensive bibl iography summarizing a variety of work dating from 1874 to 1983. Nevertheless, recent devel opments in atom trapping and cold atom spectroscopy have led to improved atomic and molec ular data through combi- nations of cold collision data, photoassociation spectros copy, and magnetic ﬁeld induced Feshbach resonance data [2–8]. Now that very reliable information is available, calculati ons of absorption spectra at high temperatures become feasible. Absorption coeﬃcients in ab solute units for a gas of sodium atoms and molecules at temperatures from 1070 to 1470 K were m easured over the range of wavelengths from 350 to 1075 nm by Schlejen et al. [9]. They performed semiclassical calculations involving the relevant molecular singlet and triplet transitions, however, those previous calculations do not fully reproduce their experim ental spectra [9]. The present work is concerned with the absorption involving two ground N a (3s) atoms and a ground Na (3s) atom and an excited Na (3p) atom, corresponding to tra nsitions between the singlet transitions from the X1Σ+ gstate to the A1Σ+ uand B1Πustates and the triplet transitions from the a3Σ+ uto the b3Σ+ gand c3Πgstates. We assembled and evaluated the available data for the molecular system and calculated quantum-mecha nically the absorption spectra at temperatures between 500 and 3000 K. II. ABSORPTION COEFFICIENTS The thermally averaged absorption coeﬃcients kνfor molecular spectra at wavelength ν are obtained from the product of the thermally averaged cros s sections and the molecular density [10]. In turn, the molecular density can be expresse d in terms of the atomic density squared and the chemical equilibrium constant [11]. In the p resent study, we use the atomic density-independent reduced absorption coeﬃcient,kν n2a, where nais atomic density. Four possible types of vibrational transitions between two electronic states can be iden- tiﬁed: bound-bound (bb), bound-free (bf), free-bound (fb) and free-free (ﬀ) and quantum- mechanical expressions for the reduced absorption coeﬃcie nt can be derived. The radial wave function φfor a bound level, with vibrational and rotational quantum n umbers, vand J, is obtained from the Schr¨ odinger equation for the relativ e motion of the nuclei, 2d2φvJΛ(R) dR2+/parenleftBigg 2µEvJΛ−2µV(R)−J(J+ 1)−Λ2 R2/parenrightBigg φvJΛ(R) = 0, (1) where V(R) is the potential for the relevant electronic state labeled by the projection Λ of the electronic orbital angular momentum on the internuclea r axis, µis the reduced mass of the nuclei, and Eis the eigenvalue of the bound level or the continuum energy. For the temperatures of interest here, T≤3000 K, the bound-bound reduced absorption coeﬃcient from a vibration-rotation state of the lower elec tronic state ( v′′, J′′,Λ′′) to the vibration-rotation state of the upper electronic state ( v′, J′,Λ′) is [10,12–14] kbb ν n2 a=C(ν) hf(kBT) exp(De/kBT) ×/summationdisplay v′′J′′/summationdisplay v′ωJ′′(2J′′+ 1) exp( −Ev′′J′′/kBT)\|/angbracketleftφv′′J′′Λ′′\|D(R)\|φv′J′Λ′/angbracketright\|2g(ν−¯ν),(2) where νis the frequency, ¯ νis the transition energy of the bound-bound transition, C(ν) =(2−δ0,Λ′+Λ′′) 2−δ0,Λ′′8π3ν 3c(3) and [15] f(kBT) =(2Sm+ 1) (2Sa+ 1)2/bracketleftBiggh2 2πµk BT/bracketrightBigg3/2 , (4) SmandSaare spin multiplicities for, respectively, the Na molecule and the Na atom, and kB is Boltzmann constant. The Q-branch approximation ( J′=J′′) is used and the line-shape function g(ν−¯ν) is replaced by 1 /∆ν. In evaluating Eq. (2) at some νion the discretized frequency interval, all transitions within the frequency r angeνi−1 2∆νtoνi+1 2∆νare summed to give a valuekbb(νi) n2a. The nuclear spin statistical factor ωJfor7Na2withI=3 2is [I/(2I+ 1)] =3 8for even Jand [( I+ 1)/(2I+ 1)] =5 8for odd J. The bound-free absorption coeﬃcient from a bound level of th e lower electronic state (v′′J′′Λ′′) to a continuum level of the upper electronic state ( ǫ′J′Λ′) is kbf ν n2a=C(ν)f(kBT) exp(De/kBT) ×/summationdisplay v′′J′′ωJ′′(2J′′+ 1) exp( −Ev′′J′′/kBT)\|/angbracketleftφv′′J′′Λ′′\|D(R)\|φǫ′J′Λ′/angbracketright\|2. (5) The continuum wave function is energy normalized. For a free -bound transition and free-free transition, respectively, kfb ν n2 a=C(ν)f(kBT) ×/summationdisplay v′J′ωJ′′(2J′′+ 1) exp( −ǫ′′/kBT)\|/angbracketleftφǫ′′J′′Λ′′\|D(R)\|φv′J′Λ′/angbracketright\|2(6) 3and kff ν n2a=C(ν)f(kBT) ×/summationdisplay J′/integraldisplay dǫ′ωJ′′(2J′′+ 1) exp( −ǫ′′/kBT)\|/angbracketleftφǫ′′J′′Λ′′\|D(R)\|φǫ′J′Λ′/angbracketright\|2. (7) III. MOLECULAR DATA The adopted singlet X1Σ+ g, A1Σ+ u, and B1Πupotentials and the diﬀerences of the upper potential and the lower X1Σ+ gpotential (diﬀerence potentials or transition energies) a re plotted in Fig. 1. The adopted triplet a3Σ+ u, b3Σ+ g, and c3Πgpotentials and the diﬀerence potentials are plotted in Fig. 2. In the remainder of the sect ion details on the construction of the potentials are given. We use atomic units throughout. A. The X1Σ+ gpotential ForR <4a0, we adopted a short range form aexp(−bR), with a= 2 702 514 .0 cm−1and b= 2.797 131 ˚A−1as given by Zemke and Stwalley [16]. Over the range of 4 < R < 30a0we used the Inverse Perturbation Analysis (IPA) potential giv en by van Abeelen and Verhaar [7] which is consistent with data from photoassociation spectr oscopy, molecular spectroscopy, and magnetic-ﬁeld induced Feshbach resonances in ultra-co ld atom collisions. For the long range form, we used −C6/R6−C8/R8−C10/R10−AR7 2α−1exp(−2αR), (8) where C6= 1 561 [2], C8= 111 877, C10= 11 065 000 [17], A=1 80, and α= 0.626 [7,18]. To ﬁt the very accurate dissociation energy, 6 022 .0286(53) cm−1, recently measured by Jones et al.[19], a point at the potential minimum 5 .819 460 a0was added. The short and long-range data were smoothly connected to the IPA values. Vibrational eigenvalues calculated with our adopted potential agree for v≤44 to within 0.1 cm−1with published Rydberg-Klein- Rees (RKR) values [16]. Our ﬁnal potential yielded an s-wave scattering length of 15 a0in satisfactory agreement with the accepted value of 19 .1±2.1a0[7]. B. The A1Σ+ upotential We used ab initio calculations given by Konowalow et al. [20] for values of Rover the range 3 .8a0< R < 4.75a0. We combined the RKR potential values over the range 42.522 19 ˚A< R < 7.204 14 ˚A given by Gerber and M¨ oller [21] with the RKR potential values over the range of 7 .260 536 ˚A< R < 261.327 403 ˚A given by Tiemann, Kn¨ ockel, and Richling [22,23]. The data was connected to the long range fo rm, −C3/R3−C6/R6−C8/R8, (9) with the values of C3= 12.26,C6= 4 094 and C8= 702 500 [24]. For R <3.8a0, the form aexp(−bR) +cwas used with the parameters a= 0.9532, b= 0.5061 and c= 0.104696 computed to smoothly connect to the RKR points. The adopted p otential yields a value of De= 8 297 .5 cm−1using Te= 14 680 .682 cm−1[21] and the atomic asymptotic energy of 16 956 .172 cm−1[25]. The calculated eigenvalues reproduce the input RKR va lues to within 0.4 cm−1. For the transition frequencies measured by Verma, Vu, and S twalley [26] and by Verma et al.[1] over a range of vibrational bands we ﬁnd typical agreemen t to about 0.4 cm−1 forJ′values up to 50 increasing to 1 cm−1forJ′= 87. We also have good agreement with less accurate measurements by Itoh et al. [27]. One precise transition energy measurement is available: In a determination of the dissociation energy of the sodium molecule Jones et al.[19] measured the value 18762.3902(30) cm−1for the v′= 165 , J′= 1 to v′′= 31, J′′= 0 transition energy. Our value of 18762.372 cm−1is in excellent agreement. C. The B1Πupotential The RKR potential of Kusch and Hessel [28] was used1over the range of 2 .655 5671 ˚A< R <5.173 513 4 ˚A. For the values of Rin the ranges 2 .581˚A< R < 2.646 0268 ˚A and 5.251 918 4 ˚A< R < 11.0˚A, we took the potential values from Tiemann [30]. We also too k his long-range form, C3/R3−C6/R6+C8/R8−aexp(−bR), (10) withC3= 6.1486, C6= 6490 .5,C8= 868135 .2,a= 23.7011, and b= 0.7885. For R <2.581˚A, the form aexp(−bR) +cwas used with the values a= 14.97332, b= 1.42983 andc= 0.0121935 chosen to give a smooth connection with the data from Tiemann. The B1Πupotential exhibits a barrier that has been studied extensiv ely [21,29–31] and the maximum value occurs around R= 13a0(6.9˚A) as shown in Fig. 1. We took De= 1For this reference, we correct an apparent typographical er ror of 4 .30978 ˚A with 4 .33978 ˚A ob- tained by comparison with RKR potential of Demtr¨ oder and St ock [29]. 52 676.16 cm−1using Te= 20 319 .19 cm−1from Kusch and Hessel [28] and the barrier energy 371.93 cm−1measured from dissociation given by Tiemann [30]. The calcu lated energy 23 393 .524 cm−1of the v′= 31, J′= 42 state with respect to the X1Σ+ gstate potential minimum compares well to the measured value, 23 393 .650 cm−1. Quasibound levels from v′= 24 to v′= 33 for the several J′values observed by Vedder et al. [32] are reproduced to within 0 .1 cm−1and calculated transition frequencies compare well, to wit hin 0.5 cm−1, with those measured by Camacho et al. [33]. D. The a3Σ+ upotential RKR potentials are available from Li, Rice, and Field [34] an d Friedman-Hill and Field [35] and a hybrid potential was constructed by Zemke an d Stwalley [16] using var- ious available data. An accurate ab initio study was carried out by Gutowski [3] for R values between 2 and 12.1 ˚A and the resulting potential has well depth 176 .173 cm−1and equilibrium distance 5.204 ˚A. Our adopted potential consists of Gutowski’s potential con nected to the long-range form given in Eq. (8) with the values for C6,C8,C10andαthe same as for the X1Σ+ gstate, but withA=−1 80. For R <2˚A the short range form aexp(−bR) was used with a= 1.4956 andb= 0.79438 chosen to smoothly connect to the ab initio data. Our adopted potential yields an s-wave scattering length of 65 a0in agreement with the value 65 .3±0.9 of van Abeleen and Verhaar [7]. Recently, a potential alternative to Gutowski’s was presented by Hoet al. [36]. For the present study the two potentials are comparabl e—we will explore their diﬀerences in a subsequent publication. E. The b3Σ+ gand c3Πgpotentials We are unaware of empirical excited state triplet potential s but ab initio calculations are available from Magnier et al. [37], Jeung [38] and Konowalow et al. [20]. Comparing the available potentials, we found for the b3Σ+ gstate that the experimentally [39] determined Teof 18 240 .5 cm−1andDeof 4 755 cm−1are closest to Magnier’s calculated values ( Teof 18 117 cm−1andDeof 4 740 .7 cm−1) compared with Jeung’s ( Teof 18 400 cm−1andDeof 4 702.4 cm−1) and Konowalow et al.’s (Deof 4 599 cm−1). Also, we found Magnier’s potential gave the best agreement with experimental measurements [40 ] of the term diﬀerences of the a3Σ+ u(v′′)→b3Σ+ g(v′) vibrational transitions. For the b3Σ+ gstate and, in the absence of experimental data for the c3Πgpotential, we adopted Magnier’s calculated potentials ove r 6the range of Rvalues 5 < R < 52a0. Over the range of Rvalues 4 .25< R < 5a0, we used potentials by Konowalow et al. [20]. For the b3Σ+ gand c3Πgadopted potentials, the long- range form was taken from Marinescu and Dalgarno [24] for R >52a0and for R <4.25a0, we used the form aexp(−bR) where the values are a= 55.7864 and b= 1.75934 for the b3Σ+ gpotential and a= 2.67691 and b= 0.91547 for the c3Πgpotential. F. Transition dipole moment functions We used for the singlet transitions the ab initio calculations of Stevens et al.[41] over the range 2 < R < 12a0. ForR >12 the transition dipole moment functions were approximate d bya+b/r3, where a= 3.586 4 and b= 284 .26 for X1Σ+ g→A1Σ+ utransitions and a= 3.5017 andb=−142.13 for X1Σ+ g→B1Πutransitions. The parameter values for awere selected to match the short-range parts and those for bwere from Marinescu and Dalgarno [24]. The X1Σ+ g→A1Σ+ udipole moment function was scaled with a factor of 1.008, as d iscussed in Sec. IVA below. For the triplet transitions the ab initio calculations of Konowalow et al.[42] were used over the range 4 < R < 100a0. IV. RESULTS A. Lifetimes In order to evaluate our assembled potential energy and tran sition dipole moment data we calculated lifetimes of ro-vibrational levels of the A1Σ+ uand B1Πustates and compare with prior studies. Lifetimes for levels of the A1Σ+ ustate have been measured [26,43–45] and calcu- lated [26,46]. In Fig. 3 we present a comparison of rotationa lly resolved lifetimes for levels of the A1Σ+ ustate measured by Baumgartner et al. [45] with the present calculations. In evaluating the lifetimes, we used the procedures described in Ref. [10]. When the transition dipole moment function of Ref. [41] is multiplied by a factor of 1.008, agreement is generally very good over the range 0 to 3500 cm−1of available term energies. Our calculations are also in good agreement with the rotationally unresolved mea surement of Ducas et al. [43] and the calculations using diﬀerent molecular data by Pardo [46]. Rotationally resolved lifetimes for the B1Πustate have been measured by Demtr¨ oder et al. [47]. Demtr¨ oder et al. found that the lifetimes for the B1Πustate are sensitive to the slope of the transition dipole moment function in the ran ge of internuclear separation 7from, roughly, 4 < R < 10a0and they obtained an empirical value for the function that we found to be in good agreement with the transition dipole mo ment function of Stevens et al.[41]. Using the transition dipole moment function of Steven set al., in turn, we ﬁnd good agreement between our calculated lifetimes and experiment al lifetime measurements [47], as shown in Fig. 4. The ab initio dipole moment of Konowalow et al. [42] was found not to reproduce the experimental lifetimes. B. Absorption Coeﬃcients Absorption spectra in the far-line wings of the Na(3s)-Na(3 p) resonance lines are investi- gated in terms of singlet and triplet molecular transitions . The blue wing consists of radiation from X1Σ+ g→B1Πuand a3Σ+ u→c3Πgtransitions and the red wing from X1Σ+ g→A1Σ+ u and a3Σ+ u→b3Σ+ gtransitions. There are few experimental studies [9,14,48] and that of Schlejen et al. [9] is most relevant to our work. In this section, we compare o ur calculated absorption coeﬃcients with the measurements of Schlejen et al. [9]. The theoretical spectra are assembled from four molecular b and spectra over the wave- length range 450–1000 nm excluding the region 589 ±2 nm around the atomic resonance lines. The far line wings are calculated using Eqs. (2)–(7), with the data from Sec. III. In the calculations, all the vibrational levels including qua si-bound levels with rotational quan- tum numbers up to 250 are included. The maximum internuclear distance that is used for integration of the transition dipole matrix element is appr oximately 100 a0and the Numerov integration used to obtain the energy-normalized continuu m wave function is carried out to 100a0at which the wave function is matched to its asymptotic form. The bin size ∆ νused for Eq. (2) was 10 cm−1simulating the experimental resolution. Results for absol ute absorp- tion coeﬃcients computed with the quoted atomic densities a nd temperatures of Schlejen et al[9] and shown in Fig. 5 compare very well with the four experim ental spectra given by Schlejen et al, given in Figure. 5 of Ref. [9]. The spectra show clearly that as temperature increases, certain satellite features grow more apparent a t 551.5 nm and 804 nm. These satellites will be discussed in greater detail later in this section. The present calculations reproduce ﬁne-scale ro-vibrational features present but u nresolved in the measurements of Ref. [9]. We also have calculated reduced absorption coeﬃcients at te mperatures up to 3000 K using the bin size ∆ νof 1 cm−1for Eq. (2). The contributions of the four molecular bands to the reduced absorption coeﬃcients are shown in Fig. 6 for t hree temperatures 1000 K, 2000 K and 3000 K. As can be seen by comparing columns (a) and (b ) in Fig. 6, the singlet 8transitions contribute more to the reduced absorption coeﬃ cients in the far line wings and the triplet transitions contribute more near the atomic res onance lines. We found that for singlet transitions bound-bound and bound-free transitio ns are dominant over free-bound and free-free transitions for the temperature range T≤3000 K, thus accounting for the “grassy” structure in Fig. 6(a). However, the free-bound an d free-free contributions increase rapidly with temperature. In contrast to the singlet transi tions, the triplet transitions arise mainly from free-bound and free-free transitions due to the shallow well of the initial a3Σ+ u state. Hence, the reduced absorption coeﬃcients in Fig. 6(b ) do not exhibit much structure. Because the density of bound molecules decreases rapidly wi th increasing temperature, the reduced absorption coeﬃcient in the line wings due to the sin glet transitions also decreases rapidly with increasing temperature. It should be noted tha t the scale of the reduced absorption coeﬃcient at 1000 K is two orders of magnitude lar ger than the scale shown for T= 2000 K and 3000 K. Woerdman and De Groot [48] derived the reduced absorption co eﬃcient at 2000 K from a discharge spectra. The measured values of 5 ±1×10−37cm−1at 500 nm and 10 ±1× 10−37cm−1at 551 .5 nm are well-reproduced by our values of, respectively, 5 ×10−37cm−1 and 11 ×10−37cm−1calculated with the bin size ∆ νof 5 cm−1simulating the experimental resolution obtained by Woerdman and De Groot [48]. The molecular absorption spectra contain “satellite” feat ures around the energies where the diﬀerence potentials possess local extrema [49,50]. Fo r Na 2the satellite frequencies have been studied [9,14,48,51] and the energies have been calcul ated using ab initio methods [42]. In the present work, we investigate the satellites arising f rom a3Σ+ u→c3Πg, X1Σ+ g→A1Σ+ u, and a3Σ+ u→b3Σ+ gtransitions with measured maximum intensities at, respect ively, the wavelengths 551 .5 nm, 804 nm, and 880 nm. The calculated extrema of the diﬀeren ce potentials adopted in the present study occur at wavelength s at 548 nm, 809 nm and 913 nm, however in the quantum-mechanical approach there is no well -deﬁned singularity. We can use the quantum-mechanical theory to study satellite features in more detail and as a function of temperature. In Fig. 7 we show calculated red uced absorption coeﬃcients at three temperatures for the a3Σ+ u→c3Πgand X1Σ+ g→A1Σ+ utransitions. The rich ro-vibrational structure in the X1Σ+ g→A1Σ+ usatellite feature arises because the dominant contributions are from bound-bound transitions; the struc ture is not reproduced by semi- classical theories [9]. In contrast, the smooth, structure less a3Σ+ u→c3Πgsatellite feature is due mainly to free-free transitions, and consequently, the decrease of the satellite intensity with temperature is less severe. The slight discrepancy bet ween the calculated wavelength of 550 nm and the measured wavelength of 551.5 nm [48,52,53] f or the peak intensity is 9probably due to remaining uncertainties in the triplet pote ntials [36,54]. We also investigated the a3Σ+ u→b3Σ+ gsatellite which is far weaker in intensity at T≤3000 K than the X1Σ+ g→A1Σ+ uand a3Σ+ u→c3Πgsatellites. The a3Σ+ u→b3Σ+ g satellite arises primarily from free-bound transitions. T he population density of atom pairs with high continuum energies in the initial a3Σ+ ustate increases with temperature, see Eq. (6), and more ro-vibrational levels in the b3Σ+ gstate are accessible through absorption of radiation, as can be seen from the potential curves shown i n Fig. 2(a). As a result, this satellite feature exhibits an increase in intensity with te mperature. In Fig. 8 calculated absorption coeﬃcients for temperatures 1000, 1500, 2000 an d 3000 K are plotted. The satellite feature intensity was measured at 1470 K by Schlej enet al. [9]. They observed a primary peak at 880 nm and a secondary peak at 850 nm, compared to our calculated values at 1500 K of 890 nm and 860 nm, respectively. The 10 nm discrepa ncy in both peaks is probably a result of uncertainties in the short range parts o f our adopted a3Σ+ uand b3Σ+ g potentials. Our calculations also demonstrate that the wav elengths of the peaks change with temperature, see Fig. 8, and that the primary peak from quant um-mechanical calculations is less prominent than that obtained from semiclassical calcu lations exhibited in Figures 6(c) and 6(d) of Schlejen et al. [9]. Our calculated reduced absorption coeﬃcients appear t o be in excellent agreement with the reduced absorption coeﬃcie nts interpolated from Figures 6(a) and 6(b) of Schlejen et al. [9] using their quoted Na densities. V. CONCLUSIONS We have carried out quantum-mechanical calculations of the reduced absorption coeﬃ- cients in sodium vapor at high temperatures. Accurate molec ular data are an important ingredient. Comparisons with experiments [9,48] are good, but the theory is not limited by the previous experimental resolution. Future work [55] w ill focus on comparisons of the present theory and experiments currently on-going in our gr oup [56]. ACKNOWLEDGMENTS We thank R. Cˆ ot´ e and A. Dalgarno for helpful communication s and E. Tiemann for generously supplying us with additional unpublished data. We are grateful to Dr. G. Lister, Dr. H. Adler, and Dr. W. Lapatovich of OSRAM SYLVANIA Inc. and Dr. M. Shurgalin, Dr. W. Parkinson, and Dr. K. Yoshino for helpful discussions . This work is supported in part by the National Science Foundation under grant PHY97-2 4713 and by a grant to the 10Institute for Theoretical Atomic and Molecular Physics at H arvard College Observatory and the Smithsonian Astrophysical Observatory. 11FIGURES 4 8 12 16 20 24 internuclear distance (a0)051015202530potential energy (103 cm−1) 4 8 12 16 20 24 internuclear distance (a0)1214161820difference potential (103 cm−1) X1Σg+A1Σu+B1Πu A1 Σu+−X1 Σg+B1 Πu−X1 Σg+(a) (b) FIG. 1. (a) Adopted potentials V(R) for the X1Σ+ g, A1Σ+ u, and B1Πuelectronic states. (b) Diﬀerence potentials VB1Πu(R)−VX1Σ+ g(R) and VA1Σ+ u(R)−VX1Σ+ g(R). 4 8 12 16 20 24 internuclear distance (a0)051015202530potential energy (103 cm−1) 4 8 12 16 20 24 internuclear distance (a0)101214161820difference potential (103 cm−1) a3Σu+b3Σg+c3Πg b3 Σg+−a3 Σu+c3 Πg−a3 Σu+(a) (b) FIG. 2. (a) Adopted potentials V(R) for the a3Σ+ u, b3Σ+ gand c3Πgstates. (b) Diﬀerence potentials Vb3Σ+ g(R)−Va3Σ+ u(R) and Vc3Πg(R)−Va3Σ+ u(R). 120 1000 2000 3000 4000 term energy (cm−1)11.81212.212.412.612.813lifetime (ns) measurements calculations FIG. 3. Comparisons of calculated lifetimes of A1Σ+ uro-vibrational levels with experimental measurements [45]. 0 1000 2000 3000 4000 term energy (cm−1)6.97.17.37.57.7lifetime (ns)measurements calculations FIG. 4. Comparisons of calculated lifetimes of B1Πuro-vibrational levels with experimental measurements [47]. 13400 500 600 700 800 900 1000 1100 wavelength (nm)00.050.100.050.100.050.100.050.1absorption coefficient (cm−1) 1070K1180K1340K1470K FIG. 5. Absolute values of the absorption coeﬃcient are show n for four diﬀerent temperatures for a comparison with experimental spectra reported in Figu re. 5 of Ref. [9]. The calculations were performed with bin size ∆ ν= 10 cm−1. 14500 600 700 8000200400600800500 600 700 800051015reduced absorption coefficient (10−37cm5) 500 600 700 8000510(a) singlet bands 500 600 700 800 wavelength (nm)0200400600800500 600 700 800051015500 600 700 8000510(b) triplet bands 500 600 700 8000200400600800500 600 700 800051015500 600 700 8000510(c) all bands (2) (1) (3)(4) (2) (2)(1) (1)(4) (4)(3) (3) FIG. 6. Contributions to the reduced absorption coeﬃcient a t 1000 K (bottom plots), 2000 K (center plots) and 3000 K (top plots) from molecular band rad iation from (a) the singlet bands, X1Σ+ g→A1Σ+ u(1) and X1Σ+ g→B1Πu(2) transitions, and (b) the triplet bands, a3Σ+ u→b3Σ+ g(3) and a3Σ+ u→c3Πg(4) transitions. The total of the singlet and triplet bands i s shown in (c). Note that the scale for the reduced absorption coeﬃcient at 1000 K is very much greater than the scale at 2000 K and 3000 K. The calculations were performed with bin size ∆ ν= 1 cm−1. 15785 795 805 815 wavelength (nm)0123reduced absorption coefficient (10−36 cm5) 545 550 555 560 wavelength (nm)0123reduced absorption coefficient (10−36 cm5) 1000K 3000K2000K 1000K 2000K 3000K(a) (b) FIG. 7. (a) Calculated reduced absorption coeﬃcients for th e a3Σ+ u→c3Πgsatellite for three temperatures. (b) Calculated reduced absorption coeﬃcien ts for the X1Σ+ g→A1Σ+ usatellite for three temperatures. 830 840 850 860 870 880 890 900 910 920 wavelength (nm)00.511.52reduced absorption coefficient (10−38 cm5) 2000K 1500K 1000K3000K FIG. 8. Reduced absorption coeﬃcients near satellite struc tures from a3Σ+ u→b3Σ+ gbands for four temperatures. Note that the scale is two orders of ma gnitude smaller than in Fig. 7. 16REFERENCES [1] K. K. Verma, J. T. Bahns, A. R. Rajaei-Rizi, and W. C. Stwal ley, J. Chem. Phys. 78, 3599 (1983). [2] P. Kharchenko, J. F. Babb, and A. Dalgarno, Phys. Rev. A 55, 3566 (1997). [3] M. Gutowski, J. Chem. Phys. 110, 4695 (1999). [4] A. J. Moerdijk and B. J. Verhaar, Phys. Rev. A 51, R4333 (1995). [5] R. Cˆ ot´ e and A. Dalgarno, Phys. Rev. A 50, 4827 (1994). [6] E. Tiesinga, C. Williams, P. D. Julienne, K. M. Jones, P. D . Lett, and W. D. Phillips, J. Res. Natl. Inst. Stand. Technol. 101, 505 (1996). [7] F. A. van Abeelen and B. J. Verhaar, Phys. Rev. A. 59, 578 (1999). [8] A. Crubellier, O. Dulieu, F. Masnou-Seeuws, H. Kn¨ ockel , and E. Tiemann, Eur. Phys. D6, 211 (1999). [9] J. Schlejen, C. J. Jalink, J. Korving, J. P. Woerdman, and W. M¨ uller, J. Phys. B 20, 2691 (1987). [10] H.-K. Chung, K. Kirby, and J. F. Babb, Phys. Rev. A 60, 2002 (1999). [11] C. Garrod, Statistical mechanics and thermodynamics (Oxford University Press, New York, 1995). [12] K. M. Sando and A. Dalgarno, Mol. Phys. 20, 103 (1971). [13] R. O. Doyle, J. Quant. Spect. Rad. Trans. 8, 1555 (1968). [14] L. K. Lam, A. Gallagher, and M. M. Hessel, J. Chem. Phys. 66, 3550 (1977). [15] F. H. Mies and P. S. Julienne, J. Chem. Phys. 77, 6162 (1982). [16] W. T. Zemke and W. C. Stwalley, J. Phys. Chem. 100, 2661 (1994). [17] M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. R ev. A49, 982 (1994). [18] B. M. Smirnov and M. I. Chibisov, Sov. Phys. JETP. 21, 624 (1965). [19] K. M. Jones, S. Maleki, S. Bize, P. D. Lett, C. J. Williams , H. Richling, H. Kn¨ ockel, E. Tiemann, H. Wang, P. L. Gould, and W. C. Stwalley, Phys. Rev. A 54, R1006 (1996). 17[20] D. D. Konowalow, M. E. Rosenkrantz, and D. S. Hochhauser , J. Chem. Phys. 72, 2612 (1980). [21] G. Gerber and R. M¨ oller, Chem. Phys. Lett. 113, 546 (1985). [22] E. Tiemann, H. Kn¨ ockel, and H. Richling, Z. Phys. D 37, 323 (1996). [23] E. Tiemann, 1998, private communication. [24] M. Marinescu and A. Dalgarno, Phys. Rev. A 52, 311 (1995). [25] NIST Atomic Spectroscopic Database, http://physics. nist.gov/PhysRefData/contents- atomic.html (1999). [26] K. K. Verma, T. Vu, and W. C. Stwalley, J. Mol. Spectr. 85, 131 (1981). [27] H. Itoh, Y. Fukuda, H. Uchiki, K. Yamada, and M. Matsuoka , J. Phys. Soc. Japan 52, 1148 (1983). [28] P. Kusch and M. M. Hessel, J. Chem. Phys. 68, 2591 (1978). [29] W. Demtr¨ oder and M. Stock, J. Mol. Spect. 55, 476 (1975). [30] E. Tiemann, Z. Phys. D 5, 77 (1987). [31] J. Keller and J. Weiner, Phys. Rev. A 29, 2943 (1984). [32] H. Vedder, G. Chawla, and R. Field, Chem. Phys. Lett 111, 303 (1984). [33] J. J. Camacho, J. M. L. Poyato, A. M. Polo, and A. Pardo, J. Quant. Spectrosc. Rad. Transfer 56, 353 (1996). [34] L. Li, S. F. Rice, and R. W. Field, J. Chem. Phys. 82, 1178 (1985). [35] E. J. Friedman-Hill and R. W. Field, J. Chem. Phys. 96, 2444 (1992). [36] T.-S. Ho, H. Rabitz, and G. Scoles, J. Chem. Phys. 112, 6218 (2000). [37] S. Magnier, P. Milli´ e, D. O., and F. Masnou-Seeuws, J. C hem. Phys 98, 7 (1993). [38] G. Jeung, J. Phys. B 16, 4289 (1983). [39] A. F¨ arbert and W. Demtr¨ oder, Chem. Phys. Lett. 264, 225 (1997). [40] A. F¨ arbert, J. Koch, T. Platz, and W. Demtr¨ oder, Chem. Phys. Lett. 223, 546 (1994). [41] W. J. Stevens, M. M. Hessel, P. J. Bertoncini, and A. C. Wa hl, J. Chem. Phys. 66, 1477 18(1977). [42] D. D. Konowalow, M. E. Rosenkrantz, and D. S. Hochhauser , J. Mol. Spect. 99, 321 (1983). [43] T. W. Ducas, M. G. Littman, M. L. Zimmerman, and D. Kleppn er, J. Chem. Phys. 65, 842 (1976). [44] J. P. Woerdman, Chem. Phys. Lett. 53, 219 (1978). [45] G. Baumgartner, H. Kornmeier, and W. Preuss, Chem. Phys . Lett. 107, 13 (1984). [46] A. Pardo, preprint, 1999 (unpublished). [47] W. Demtr¨ oder, W. Stetzenbach, M. Stock, and J. Witt, J. Mol. Spect. 61, 382 (1976). [48] J. P. Woerdman and J. J. de Groot, Chem. Phys. Letter 80, 220 (1981). [49] K. M. Sando and J. C. Wormhoudt, Phys. Rev. A 7, 1889 (1973). [50] J. Szudy and W. E. Baylis, J. Quant. Spectr. Rad. Trans. 15, 641 (1975). [51] J. J. de Groot, J. Schlejen, J. P. Woerdman, and M. F. M. De Kieviet, Phillips J. Res. 42, 87 (1987). [52] J. J. de Groot and J. A. J. M. van Vliet, J. Phys. D 8, 651 (1975). [53] D. Veˆ za, J. Rukavina, M. Movre, V. Vujnovi´ c, and G. Pic hler, Opt. Comm. 34, 77 (1980). [54] E. Tiemann, 2000, private communication. [55] J. F. Babb, M. Shurgalin, H.-K. Chung, K. Kirby, W. Parki nson, and K. Yoshino, tbd, in preparation. [56] M. Shurgalin, W. H. Parkinson, K. Yoshino, C. Schoene, a nd W. P. Lapatovich, Meas. Sci. Technol. 11, 730 (2000). 19
arXiv:physics/0008061v1 [physics.acc-ph] 15 Aug 2000PRODUCTION AND STUDIESOF PHOTOCATHODESFOR HIGH INTENSITYELECTRONBEAMS E. Chevallay,S. Hutchins,P. Legros,G. Suberlucq, H.Traut ner, CERN, Geneva,Switzerland Abstract For short,high-intensityelectronbunches,alkali-tellu rides have proved to be a reliable photo-cathodematerial. Mea- surements of lifetimes in an RF gun of the CLIC Test Fa- cility II at ﬁeld strengths greater than 100 MV/m are pre- sented. Before and after using them in this gun, the spec- tral response of the Cs-Te and Rb-Te cathodes were deter- minedwiththehelpofanopticalparametricoscillator. The behaviour of both materials can be described by Spicer’s 3-step model. Whereas during the use the threshold for photo-emissioninCs-Tewasshiftedtohigherphotonener- gies,thatofRb-Tedidnotchange. Ourlatestinvestigation s on the stoichiometric ratio of the components are shown. Thepreparationofthephoto-cathodeswasmonitoredwith 320 nm wavelength light, with the aim of improving the measurement sensitivity. The latest results on the protec- tion of Cs-Te cathodesurfaceswith CsBr againstpollution are summarized. New investigationson high mean current productionarepresented. 1 INTRODUCTION In the CTF II drive beam gun, Cs-Te photocathodes are used to produce a pulse train of 48 electron bunches, each 10pslongandwithachargeofupto10nC[1]. InCTF,the main limit to lifetime is the available laser power, which requires a minimal quantum efﬁciency (QE) of 1.5% to produce the nominal charge. Although Cs-Te photocath- odesarewidelyused,acompleteunderstanding,especially of their aging process, is still lacking. Spectra of the QE against exciting photons may help to understand the phe- nomenon. 2 MEASUREMENTS OFQEAGAINST PHOTONENERGY According to Spicer [2], the spectra of the quantum efﬁ- ciency (QE) of semiconductors with respect to the energy ofthe excitingphotons( hν)canbe describedas: QE=c1(hν−ET)3 2 c2+ (hν−ET)3 2, (1) where ETisthethresholdenergyforphotoemission,c 1and c2areconstants. 2.1 TheOPO To measure the spectral response of photocathodes, wave- lengths from the near UV throughout the visible are nec- essary. To attain these, an OpticalParametrical Oscillator was built [3]. A frequency-tripled Nd:YAG laser pumpsa BetaBarium Borate (BBO) crystal in a double-pass con- ﬁguration, as shown in Fig.1. The emerging signal-beam, with wavelengths between 409nm and 710nm, is fre- quency doubled in two BBO crystals. The wavelengths obtained are between 210nm and 340nm. The idler-beam deliverswavelengthsbetween710nmand 2600nm. HR SignalIdler SignalPumpbeam HR 355nmPR SignalBBO BBO BBO Figure1: OPOschemewith followingdoublingcrystals Themeasurementsofthespectralresponseofphotocath- odesweremadein theDC-gunofthephotoemissionlabat CERN [4], at a ﬁeld strength of about 8 MV/m. Spectra weretakenshortlyaftertheevaporationofthecathodema- terials ontothe coppercathodeplug,as well as after use in the CTF IIRF-gun[5] atﬁeldsoftypically100MV/m. 2.2 CesiumTelluride To be able to interpret the spectra in terms of Spicer’s the- ory, it was necessary to split the data into 2 groups, one at “low photonenergy”and oneat ”highphotonenergy”,see Fig.2. Then, the data can be ﬁtted well with two indepen- dent curves, following Eq.(1), which give two threshold energies. ForatypicalfreshCs-Tecathode,thehighenergy thresholdis3.5eV,thelowoneis1.7eV,asshowninFig.2, upper curve. This might be a hint that two photo-emissive phases of Cs-Te on copper exist. Several explanations are possible: ThecoppermightmigrateintotheCs-Te,creating energy levels in the band gap; or possibly not only Cs 2Te, butalsootherCs-Tecompoundsmightformonthesurface and these might give rise to photoemission at low photon energy. A hint to this might be that the ratio of evaporated atoms of each element is not corresponding to Cs 2Te, see below. Afteruse,wefoundthatnotonlythecompletespectrum shifted towards lower quantumefﬁciency, but also that the photoemission threshold for high QE increased to 4.1eV, which is shown in Fig.2, lower curve. One might expect that the photocathodeis poisoned by the residual gas, pre- ventinglow-energyelectronsfromescaping. However,be-1.E-061.E-041.E-021.E+001.E+02 1 2 3 4 5 6 Photon energy [eV]Quantum Efficiency [%] Figure 2: Spectra of a Cs-Te photocathodeon copper. Be- fore(squarepoints)andafteruse(roundpoints)intheCTF IIdrivebeamgun cause typical storage lifetimes are of the order of months, the effectmust beconnectedtoeitherthe laserlight,orthe electricalﬁeld. We alsoproduceda Cs-Tecathodeonathingoldﬁlmof 100nm thickness. As shown in Fig.3, the shoulder in the lowenergyresponsedisappeared. Itisdifﬁculttoﬁtacurve for the Spicer model to the low energy data. The “high” photoemission threshold is at 3.5eV. At the moment, this cathodeisin usein theCTF II gunandwill be remeasured in the future. In terms of lifetime, this cathode is compa- rable to the best Cs-Te cathodes, as it hasalreadyoperated for20daysintheRF-gun. 1.E-061.E-041.E-021.E+001.E+02 1 2 3 4 5 6 Photon energy [eV]Quantum Efficiency [%] Figure 3: Cs-Te cathode, evaporated on a gold ﬁlm of 100nm thickness. The round points are not used for the ﬁt. 2.3 RubidiumTelluride Asanewmaterialpresentedﬁrstin[1],wetestedrubidium- telluride. WetookspectraofQEbeforeandafteruseinthe CTF II gun, as for Cs-Te. Remarkably, with this material, there was no shift in the photoemission threshold towards higher energies, but only a global shift in QE, see Fig.4. This might be due to the lower afﬁnity of rubidium to the residual gas. Detailed investigations are necessary to cla r- ifythis.1.E-061.E-041.E-021.E+001.E+02 1 2 3 4 5 6 Photon energy [eV]Quantum Efficiency [%] Figure 4: Spectra of a Rb-Te cathode before use (square points) and after use (round points) in the CTF II drive beamgun 2.4 CoatingwithCsBr Long lifetimes for Cs-Te cathodesare achievedonly when theyareheldunderUHV( ≤10−8mbar). Otherphotocath- ode materials like K-Sb-Cs are immunized against gases like oxygen by evaporating thin ﬁlms of CsBr onto them [6]. Therefore, we evaporated a CsBr ﬁlm of 2nm thick- ness onto the Cs-Te. Fig.5 shows the spectrum before the CsBr ﬁlm (square points) and after it (round points). The 1.E-061.E-041.E-021.E+001.E+02 1 2 3 4 5 6 Photon energy [eV]Quantum Efficiency [%] Figure5: Spectraofa Cs-TecathodewithoutCsBr (square points)andwith CsBr coating(roundpoints) QEat266nmdroppedfrom4.3%to1.2%. Inaddition,the photoemissionthresholdwasshiftedfrom3.9eVto4.1eV. A long-term storage test showed no signiﬁcant difference between uncoated and coated cathodes. More investiga- tions will determine the usefulness of these protectivelay - ers. 2.5 Preparationwith otherwavelengths In order to increase the sensitivity of the on-line QE mea- surement during evaporation of the photocathodes, we monitoredtheprocesswithlightatawavelengthof320nm. We did not see any signiﬁcant improvement in sensitivity, notablyin thehighQE region.3 STOICHIOMETRICRATIO Film thicknessesare measuredduringthe evaporationpro- cessbyaquartzoscillator[4]. Typicalthicknessesforhig h quantumefﬁcienciesat λ= 266nmare10nmoftellurium and around 15nm of cesium. This results in a ratio of the numberof atoms of each species of NTe/NCs= 2.85, far from the stoichiometricratio of 0.5 for Cs 2Te. It is known thattelluriuminteractsstronglywithcopper[7],sothatn ot all of the evaporatedtelluriumis available fora compound with subsequently evaporated cesium. Therefore, we used also Mo and Au as substrate material. However, the ratio between the constituents necessary for optimum QE, did not change signiﬁcantly. Another reason might be that in- steadofCs 2Te,Cs 2Te5iscatalyticallyproducedonthesur- face. Thiscompound,aswellassomeothers,wasfoundto be stable[8]. 4 LIFETIME INCTF II Lifetime in CTF depends on parameters like maximum ﬁeld strength on the cathode, vacuum and especially ex- tracted charge. Typically, a cathode is removed from the gun, if the QE falls below 1.5%. As shown in Fig.6, life- time does not depend on the initial QE; a cathode having an initial QE of 15% (round points) lasted as long as one with 5% (triangles). 03691215 0 10 20 30 40 50 60 Lifetime [days]Quantum Efficiency [%] Figure6: Lifetimein CTF oftwodifferentcathodes 5 HIGHCHARGETEST As shown in Table1, the averagecurrent producedin CTF II is nearly a factor 10000 lower than what is required for the CLIC drive beam. A test to produce 1mC is under preparationinthe photoemissionlaboratoryat CERN. The exact reproduction of the CLIC pulse structure would re- quire the CLIC laser, which is still in the design stage in Table 1: Comparisonofcathoderelevantparameter CTF II CLIC3TeV ”Test1mC” Current 0.008mA 75mA 1mA Power 0.072mW 35W 300mWa collaboration between Rutherford Appleton Laboratory andCERN.Atestwhichiscompatiblewithourcurrentin- stallationistheproductionof1mAofdccurrent,whichre- quiresaUVlaserpowerof300mWatthecathode. Forthis test,wewillilluminatethecathodewithpulsesof100nsto 150ns pulse length, at repetition rates between 1kHz and 6kHz. As Table1 shows, this is a factor 1000 more aver- agecurrentthaninCTFII,andalsodemonstratesthebasic ability of the cathodes to produce the CTF 3 drive beam (I=26µA).CLIC is still a factor 75away. We are currently searchingforwaysto producehigherchargesaswell. 6 CONCLUSION Measurements of QE against photon energy are routinely made after productionand after use of photocathodes. We havedemonstratedthatbothlowenergyandhighenergyre- sponsesagreewellwithSpicer’stheory. Agoldbufferlayer reduces the low energy response of Cs-Te cathodes. More workisneededto understandthemeasurementsofthe sto- ichiometricratioofCs-Te. Coatingwith 2nmCsBr signif- icantly decreased the quantum efﬁciency, without improv- ing thestoragelifetime. Forthe high-chargedrivebeamof CLIC,itisstillnecessarytodemonstratethecapabilities of Cs-Te, forwhichﬁrst testswill be donesoon. 7 REFERENCES [1] E.Chevallay,J.Durand,S.Hutchins,G.Suberlucq,H.Tr aut- ner, “Photo-cathodes for the CERN CLIC Test Facility”, CERNCLICNote 373, Proceedings of Linac 1998 [2] W.E. Spicer, “Photoemissive, Photoconductive and Optical Absorption Studies of Alkali-Antimony Com- pounds”,Physical Review, Vol. 112, No.1, (1958), 114-122 [3] H.Trautner,“SpectralResponseofCesiumTellurideand Ru- bidium TelluridePhotocathodes forthe Productionof Highl y Charged ElectronBunches”, (Doctoral Thesis), 1999, CERN CLICNote 428 [4] E. Chevallay, J. Durand, S. Hutchins, G. Suberlucq, M. W¨ urgel, “Photocathodes tested in the dc gun of the CERN photoemission laboratory”, Nuclear Instruments Methods in Physics Research Section A, Vol. 340, (1994) 146-156, CERNCLICNote 203 [5] R. Bossart, H.H. Braun, M. Dehler, J.-C. Godot, “A 3 GHz Photoelectron Gun for High Beam Intensity”, Proc. FEL Conf.,New York, 1995 [6] E. Shefer, A. Breskin, R. Chechik, A. Buzulutskov, B.K. Singh, M. Prager, “Coated photocathodes for visible pho- ton imaging with gaseous photomultipliers”, Nuclear Instr u- mentsMethodsinPhysicsResearchSectionA,vol.433,no.1- 2, (1999) 502-506 [7] A. Di Bona, F. Sabary, S.Valeri, P.Michelato, D.Sertore , G. Suberlucq,“Auger and x-ray photemission spectroscopy study on Cs 2Te photocathodes”, J. Appl. Phys., Vol. 80, No. 5, (1996), 3024-3030 [8] G. Prins, E.H.P. Cordfunke,“Compounds in the system Cs- Te at room temperature”,Journal of the Less-Common Met- als,Vol.104, (1984), L1-L3
/G24/G03/G05/G2B/G3C/G25/G35/G2C/G27/G05/G10/G37/G3C/G33/G28/G03/G36/G3C/G36/G37/G28/G30/G03/G29/G32/G35 /G24/G31/G03/G24/G26/G2B/G35/G32/G30/G24/G37/G2C/G26/G03/G29/G32/G26/G38/G36/G2C/G31/G2A/G03/G32/G29/G03/G2C/G32/G31/G36 /G3C /G11/G3C/G58 /G11/G03/G25/G52/G4A/G47/G44/G51/G52/G59/G4C/G46/G4B/G0F/G03 /G3A /G11/G39/G11/G03/G31/G48/G56/G57/G48/G55/G52/G59/G4C/G46/G4B/G0F /G30/G52/G56/G46/G52/G5A/G03/G28/G51/G4A/G4C/G51/G48/G48/G55/G4C/G51/G4A/G03/G33/G4B/G5C/G56/G4C/G46/G56/G03/G2C/G51/G56/G57/G4C/G57/G58/G57/G48/G03/G0B/G37/G48/G46/G4B/G51/G4C/G46/G44/G4F/G03/G38/G51/G4C/G59/G48/G55/G56/G4C/G57/G5C/G0C /G2E/G44/G56/G4B/G4C/G55/G56/G4E/G52/G48/G03/G56/G4B/G11/G03/G16/G14/G0F/G03/G30/G52/G56/G46/G52/G5A/G0F/G03/G14/G14/G18/G17/G13/G1C/G03/G35/G58/G56/G56/G4C/G44 /G24/G45/G56/G57/G55/G44/G46/G57 /G2C/G57/G03 /G4C/G56/G03 /G4E/G51/G52/G5A/G51/G0F/G03 /G57/G4B/G44/G57/G03 /G44/G57/G03 /G4B/G4C/G4A/G4B/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G52/G49/G03 /G4C/G52/G51/G56/G03 /G0B/G50/G52/G55/G48/G03 /G57/G4B/G44/G51 /G14/G13/G03 /G30/G48/G39/G03 /G49/G52/G55/G03 /G4F/G4C/G4A/G4B/G57/G03 /G4C/G52/G51/G56/G0C/G03 /G57/G4B/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G45/G5C/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G49/G4C/G48/G4F/G47/G56/G03 /G4C/G56/G03 /G50/G52/G56/G57/G03 /G48/G49/G49/G48/G46/G57/G4C/G59/G48/G0F/G03 /G5A/G4B/G48/G55/G48/G44/G56/G03 /G44/G57/G03 /G56/G50/G44/G4F/G4F /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G0B/G4F/G48/G56/G56/G03 /G57/G4B/G44/G51/G03 /G14/G03 /G30/G48/G39/G0C/G03 /G10/G03 /G57/G4B/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G45/G5C/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G49/G4C/G48/G4F/G47/G56/G11/G03 /G24/G57/G03 /G4C/G51/G57/G48/G55/G50/G48/G47/G4C/G44/G57/G48/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G52/G49/G03 /G4C/G52/G51/G56/G03 /G0B/G49/G55/G52/G50 /G14/G03 /G57/G52/G03 /G14/G13/G03 /G30/G48/G39/G0C/G03 /G57/G4B/G48/G03 /G48/G49/G49/G4C/G46/G4C/G48/G51/G46/G5C/G03 /G52/G49/G03 /G48/G44/G46/G4B/G03 /G52/G49/G03 /G57/G4B/G48/G56/G48/G03 /G57/G5C/G53/G48/G56/G03 /G52/G49/G03 /G44 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G4C/G56/G03 /G55/G48/G47/G58/G46/G48/G47/G11/G03 /G37/G4B/G48/G03 /G44/G58/G57/G4B/G52/G55/G56/G03 /G52/G49/G49/G48/G55/G03 /G57/G52/G03 /G58/G57/G4C/G4F/G4C/G5D/G48/G03 /G49/G52/G55/G03 /G57/G4B/G4C/G56 /G53/G58/G55/G53/G52/G56/G48/G03 /G57/G4B/G48/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G03 /G57/G5C/G53/G48/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G56/G03 /G46/G52/G51/G56/G4C/G56/G57/G4C/G51/G4A /G52/G49/G03/G56/G48/G54/G58/G48/G51/G46/G48/G03/G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G44/G51/G47/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G4F/G48/G51/G56/G48/G56/G0F /G5A/G4B/G4C/G46/G4B/G03 /G44/G55/G48/G03 /G4F/G52/G46/G44/G4F/G4C/G5D/G48/G47/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G56/G44/G50/G48/G03 /G56/G53/G44/G46/G48/G11/G03 /G29/G52/G55/G03 /G50/G44/G4E/G4C/G51/G4A/G03 /G49/G4C/G48/G4F/G47/G56 /G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G4F/G48/G51/G56/G48/G56/G03 /G4C/G57/G03 /G4C/G56/G03 /G52/G49/G49/G48/G55/G48/G47/G03 /G57/G52/G03 /G58/G57/G4C/G4F/G4C/G5D/G48/G03 /G35/G29 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G11/G03 /G37/G4B/G48/G03 /G35/G29/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G03 /G44/G55/G48/G03 /G46/G55/G48/G44/G57/G48/G47 /G47/G58/G48/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G55/G48/G4F/G48/G59/G44/G51/G57/G03 /G46/G52/G51/G49/G4C/G4A/G58/G55/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G47/G55/G4C/G49/G57/G03 /G57/G58/G45/G48/G56/G03 /G4C/G51/G03 /G57/G4B/G48 /G59/G4C/G46/G4C/G51/G4C/G57/G5C/G03 /G52/G49/G03 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G44/G53/G11/G03 /G24/G56/G03 /G44/G03 /G55/G48/G56/G58/G4F/G57/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G4A/G48/G51/G48/G55/G44/G4F /G57/G4B/G48/G52/G55/G48/G57/G4C/G46/G44/G4F/G03 /G44/G51/G44/G4F/G5C/G56/G4C/G56/G03 /G44/G51/G47/G03 /G46/G44/G4F/G46/G58/G4F/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G46/G52/G51/G46/G55/G48/G57/G48/G03 /G59/G48/G55/G56/G4C/G52/G51/G56 /G52/G49/G03 /G53/G55/G44/G46/G57/G4C/G46/G44/G4F/G03 /G55/G48/G44/G4F/G4C/G5D/G44/G57/G4C/G52/G51/G56/G03 /G4C/G57/G03 /G4C/G56/G03 /G56/G4B/G52/G5A/G51/G03 /G57/G4B/G44/G57/G03 /G57/G4B/G48/G03 /G4A/G4C/G59/G48/G51/G03 /G56/G5C/G56/G57/G48/G50 /G4B/G44/G56/G03 /G44/G47/G59/G44/G51/G57/G44/G4A/G48/G03 /G44/G56/G03 /G46/G52/G51/G57/G55/G44/G56/G57/G48/G47/G03 /G57/G52/G03 /G45/G5C/G03 /G4E/G51/G52/G5A/G51/G03 /G44/G51/G44/G4F/G52/G4A/G56/G03 /G4C/G51/G03 /G57/G4B/G48 /G49/G4C/G48/G4F/G47/G03 /G52/G49/G03 /G50/G48/G47/G4C/G58/G50/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G11/G03 /G37/G4B/G48/G03 /G59/G48/G4F/G52/G46/G4C/G57/G5C/G03 /G45/G44/G51/G47/G03 /G52/G49/G03 /G4C/G52/G51/G56/G0F/G03 /G4C/G51 /G5A/G4B/G4C/G46/G4B/G03 /G57/G4B/G48/G03 /G56/G57/G44/G45/G4F/G48/G03 /G50/G52/G57/G4C/G52/G51/G03 /G4C/G56/G03 /G48/G51/G56/G58/G55/G48/G47/G0F/G03 /G4C/G56/G03 /G48/G56/G56/G48/G51/G57/G4C/G44/G4F/G4F/G5C /G46/G4B/G44/G51/G4A/G48/G47/G0F/G03 /G44/G51/G47/G03 /G57/G4B/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G45/G48/G46/G52/G50/G48/G56/G03 /G05/G44/G46/G4B/G55/G52/G50/G44/G57/G4C/G46/G05/G11/G03 /G37/G4B/G48 /G46/G44/G4F/G46/G58/G4F/G44/G57/G4C/G52/G51/G56/G03 /G4B/G44/G59/G48/G03 /G45/G48/G48/G51/G03 /G53/G48/G55/G49/G52/G55/G50/G48/G47/G03 /G49/G52/G55/G03 /G53/G55/G52/G57/G52/G51/G56/G03 /G5A/G4C/G57/G4B /G59/G48/G4F/G52/G46/G4C/G57/G4C/G48/G56/G03 /G13/G11/G18/G10/G14/G11/G13/G03 /G08/G03 /G52/G49/G03 /G4F/G4C/G4A/G4B/G57/G03 /G59/G48/G4F/G52/G46/G4C/G57/G5C/G0F/G03 /G44/G57/G03 /G57/G4B/G48/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G56/G03 /G52/G49 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G49/G4C/G48/G4F/G47/G03 /G58/G53/G03 /G57/G52/G03 /G18/G13/G03 /G4E/G39/G03 /G53/G48/G55/G03 /G56/G54/G58/G44/G55/G48/G03 /G46/G50/G0F/G03 /G44/G51/G47/G03 /G57/G4B/G48 /G4A/G55/G44/G47/G4C/G48/G51/G57/G56/G03 /G52/G49/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G49/G4C/G48/G4F/G47/G03 /G58/G53/G03 /G57/G52/G03 /G14/G03 /G37/G12/G50/G11/G03 /G03 /G24/G46/G4B/G55/G52/G50/G44/G57/G4C/G46/G4C/G57/G5C /G52/G49/G03 /G57/G4B/G4C/G56/G03 /G56/G5C/G56/G57/G48/G50/G03 /G44/G4F/G4F/G52/G5A/G56/G03 /G57/G52/G03 /G4F/G52/G5A/G48/G55/G03 /G44/G03 /G55/G44/G47/G4C/G44/G57/G4C/G59/G48/G03 /G44/G46/G57/G4C/G59/G44/G57/G4C/G52/G51/G03 /G52/G49 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03/G44/G51/G47/G03/G57/G52/G03/G55/G48/G47/G58/G46/G48/G03 /G47/G44/G50/G44/G4A/G48/G03 /G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G52/G47/G48/G03 /G56/G58/G55/G49/G44/G46/G48/G56/G03 /G45/G5C /G4F/G52/G56/G57/G03/G4C/G52/G51/G56/G11 /G14/G03/G03/G2C/G31/G37/G35/G32/G27/G38/G26/G37/G2C/G32/G31 /G2C/G51/G03 /G52/G58/G55/G03 /G53/G44/G53/G48/G55/G03 /G3E/G14/G40/G0F/G03 /G5A/G48/G03 /G53/G55/G48/G56/G48/G51/G57/G48/G47/G03 /G57/G4B/G48/G03 /G55/G48/G56/G58/G4F/G57/G56/G03 /G52/G49 /G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G44/G4F/G03 /G55/G48/G56/G48/G44/G55/G46/G4B/G03 /G52/G49/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G59/G44/G55/G4C/G44/G45/G4F/G48/G03 /G53/G55/G52/G57/G52/G51/G03 /G4F/G4C/G51/G44/G46/G0F /G5A/G4B/G4C/G46/G4B/G03 /G46/G52/G51/G56/G4C/G56/G57/G56/G03 /G52/G49/G03 /G44/G03 /G46/G4B/G44/G4C/G51/G03 /G52/G49/G03 /G4C/G51/G47/G48/G53/G48/G51/G47/G48/G51/G57/G4F/G5C/G03 /G47/G55/G4C/G59/G48/G51/G03 /G52/G51/G48/G10 /G4A/G44/G53/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G46/G44/G59/G4C/G57/G4C/G48/G56/G11/G03 /G24/G03 /G53/G52/G4F/G5C/G44/G5B/G4C/G44/G4F/G03 /G55/G48/G56/G52/G51/G44/G57/G52/G55/G03 /G0B/G33/G35/G0C /G52/G49/G49/G48/G55/G48/G47/G03 /G44/G56/G03 /G44/G03 /G50/G44/G4C/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G48/G4F/G48/G50/G48/G51/G57/G11/G03 /G2C/G51/G03 /G57/G4B/G4C/G56 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G0F/G03 /G45/G48/G44/G50/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G4B/G44/G56/G03 /G45/G48/G48/G51/G03 /G53/G55/G52/G59/G4C/G47/G48/G47/G03 /G45/G5C/G03 /G50/G48/G44/G51/G56 /G52/G49/G03/G48/G4F/G48/G46/G57/G55/G52/G56/G57/G44/G57/G4C/G46/G03/G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G03/G5A/G4C/G57/G4B/G03/G59/G44/G55/G4C/G44/G45/G4F/G48/G03/G53/G52/G57/G48/G51/G57/G4C/G44/G4F/G11 /G37/G4B/G48/G03 /G46/G52/G55/G55/G48/G46/G57/G03 /G46/G4B/G52/G4C/G46/G48/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G49/G52/G55/G03 /G56/G58/G46/G4B /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03 /G4C/G56/G03 /G59/G48/G55/G5C/G03 /G4C/G50/G53/G52/G55/G57/G44/G51/G57/G11/G03 /G03 /G24/G57/G03 /G44/G03 /G4F/G44/G55/G4A/G48/G03 /G59/G44/G55/G4C/G44/G57/G4C/G52/G51/G03 /G52/G49 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03 /G48/G51/G48/G55/G4A/G5C/G0F/G03 /G4C/G57/G03 /G4C/G56/G03 /G51/G48/G46/G48/G56/G56/G44/G55/G5C/G03 /G57/G52/G03 /G53/G55/G52/G59/G4C/G47/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G49/G52/G55 /G44/G03 /G5A/G4C/G47/G48/G03 /G55/G44/G51/G4A/G48/G03 /G52/G49/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G03 /G59/G48/G4F/G52/G46/G4C/G57/G4C/G48/G56/G11/G03 /G2C/G57/G03 /G4C/G56/G03 /G4E/G51/G52/G5A/G51/G0F/G03 /G57/G4B/G44/G57/G03 /G57/G4B/G48 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G45/G5C/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G0B/G28/G34/G0C/G03 /G03 /G49/G4C/G48/G4F/G47/G56/G03 /G4C/G56/G03 /G48/G49/G49/G48/G46/G57/G4C/G59/G48 /G44/G57/G03 /G56/G50/G44/G4F/G4F/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G0B/G4F/G48/G56/G56/G03 /G57/G4B/G44/G51/G03 /G14/G03 /G30/G48/G39/G0C/G11/G03 /G24/G57/G03 /G4B/G4C/G4A/G4B/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G52/G49 /G4C/G52/G51/G56/G03 /G0B/G50/G52/G55/G48/G03 /G57/G4B/G44/G51/G03 /G14/G13/G03 /G30/G48/G39/G03 /G49/G52/G55/G03 /G4F/G4C/G4A/G4B/G57/G03 /G4C/G52/G51/G56/G0C/G03 /G57/G4B/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G45/G5C /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G0B/G30/G34/G0C/G03 /G49/G4C/G48/G4F/G47/G56/G03 /G4C/G56/G03 /G50/G52/G55/G48/G03 /G48/G49/G49/G48/G46/G57/G4C/G59/G48/G11/G03 /G24/G57/G4C/G51/G57/G48/G55/G50/G48/G47/G4C/G44/G57/G48/G03 /G48/G51/G48/G55/G4A/G4C/G48/G56/G03 /G52/G49/G03 /G4C/G52/G51/G56/G03 /G0B/G49/G55/G52/G50/G03 /G14/G03 /G57/G52/G03 /G14/G13/G03 /G30/G48/G39/G0C/G03 /G57/G4B/G48 /G48/G49/G49/G4C/G46/G4C/G48/G51/G46/G5C/G03/G52/G49/G03/G48/G44/G46/G4B/G03/G52/G49/G03/G57/G4B/G48/G56/G48/G03/G57/G5C/G53/G48/G56/G03/G52/G49/G03/G44/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4C/G56/G03/G55/G48/G47/G58/G46/G48/G47/G11 /G2C/G51/G03 /G53/G55/G48/G56/G48/G51/G57/G03 /G53/G44/G53/G48/G55/G0F/G03 /G57/G4B/G48/G03 /G44/G58/G57/G4B/G52/G55/G56/G03 /G52/G49/G49/G48/G55/G03 /G57/G52/G03 /G58/G57/G4C/G4F/G4C/G5D/G48/G03 /G57/G4B/G48 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G03 /G57/G5C/G53/G48/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G56/G03 /G46/G52/G51/G56/G4C/G56/G57/G4C/G51/G4A/G03 /G52/G49/G03 /G56/G48/G54/G58/G48/G51/G46/G48 /G52/G49/G03/G28/G34/G03/G44/G51/G47/G03/G30/G34/G03/G4F/G48/G51/G56/G48/G56/G03/G4F/G52/G46/G44/G4F/G4C/G5D/G48/G47/G03/G4C/G51/G03/G57/G4B/G48/G03/G56/G44/G50/G48/G03/G56/G53/G44/G46/G48/G11 /G15/G03/G03/G24/G31/G03/G24/G31/G24/G2F/G3C/G37/G2C/G26/G24/G2F/G03/G24/G31/G24/G2F/G3C/G36/G2C/G36/G03/G32/G29/G03/G24 /G36/G37/G24/G25/G2C/G2F/G2C/G37/G3C/G03/G27/G2C/G24/G2A/G35/G24/G30 /G2C/G57/G03 /G4C/G56/G03 /G4E/G51/G52/G5A/G51/G03 /G57/G4B/G44/G57/G03 /G4C/G51/G03 /G44/G03 /G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50 /G46/G52/G51/G56/G4C/G56/G57/G4C/G51/G4A/G03 /G52/G49/G03 /G4C/G47/G48/G51/G57/G4C/G46/G44/G4F/G03 /G30/G34/G0A/G56/G0F/G03 /G44/G51/G03 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G03 /G52/G51/G03 /G44 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03/G47/G4C/G44/G4A/G55/G44/G50/G03/G50/G52/G59/G48/G56/G03/G49/G55/G52/G50/G03/G44/G03/G46/G48/G51/G57/G55/G44/G4F/G03/G53/G44/G55/G57/G03/G57/G52/G03 /G44/G03 /G45/G52/G58/G51/G47/G44/G55/G5C /G44/G57/G03 /G44/G03 /G59/G44/G55/G4C/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G03 /G48/G51/G48/G55/G4A/G5C/G11/G03 /G2C/G51/G03 /G46/G52/G51/G57/G55/G44/G55/G5C/G03 /G57/G52/G03 /G57/G4B/G4C/G56 /G56/G4C/G57/G58/G44/G57/G4C/G52/G51/G0F/G03 /G44/G51/G03 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G03 /G52/G51/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G52/G49/G03 /G44 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G03 /G56/G5C/G56/G57/G48/G50/G03 /G50/G52/G59/G48/G56/G03 /G5A/G4C/G57/G4B/G4C/G51/G03 /G44/G03 /G46/G48/G51/G57/G55/G44/G4F/G03 /G53/G44/G55/G57/G03 /G52/G49/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C /G47/G4C/G44/G4A/G55/G44/G50/G11/G03 /G2F/G48/G57/G03 /G58/G56/G03 /G56/G4B/G52/G5A/G03 /G4C/G57/G03 /G58/G56/G4C/G51/G4A/G03 /G44/G03 /G56/G4C/G50/G53/G4F/G48/G03 /G57/G4B/G48/G52/G55/G48/G57/G4C/G46/G44/G4F/G03 /G50/G52/G47/G48/G4F /G5A/G4B/G48/G51/G03 /G56/G53/G44/G46/G48/G03 /G55/G48/G4A/G4C/G52/G51/G56/G03 /G52/G46/G46/G58/G53/G4C/G48/G47/G03 /G45/G5C/G03 /G30/G34/G03 /G4F/G48/G51/G56/G48/G56/G03 /G44/G51/G47/G03 /G35/G29/G03 /G28/G34/G0A/G56 /G44/G55/G48/G03 /G46/G52/G4C/G51/G46/G4C/G47/G48/G11/G03 /G37/G4B/G4C/G56/G03 /G53/G44/G55/G57/G4C/G46/G58/G4F/G44/G55/G03 /G46/G44/G56/G48/G03 /G46/G44/G51/G03 /G45/G48/G03 /G55/G48/G44/G4F/G4C/G5D/G48/G47/G03 /G44/G56/G03 /G47/G55/G4C/G49/G57 /G57/G58/G45/G48/G56/G03 /G4B/G44/G59/G4C/G51/G4A/G03 /G35/G29/G10/G48/G4F/G48/G46/G57/G55/G52/G47/G48/G56/G03 /G46/G52/G51/G57/G44/G4C/G51/G4C/G51/G4A/G03 /G53/G48/G55/G50/G44/G51/G48/G51/G57 /G50/G44/G4A/G51/G48/G57/G56/G11/G03 /G38/G56/G4C/G51/G4A/G03 /G51/G52/G57/G44/G57/G4C/G52/G51/G56/G03 /G52/G49/G03 /G35/G48/G49/G11/G03 /G3E/G15/G40/G0F/G03 /G57/G4B/G48/G03 /G48/G54/G58/G44/G57/G4C/G52/G51/G56/G03 /G49/G52/G55/G03 /G44 /G57/G55/G44/G51/G56/G59/G48/G55/G56/G48/G03 /G50/G52/G57/G4C/G52/G51/G03 /G4C/G51/G03 /G57/G4B/G4C/G56/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G03 /G56/G5C/G56/G57/G48/G50/G03 /G46/G44/G51/G03 /G45/G48/G03 /G5A/G55/G4C/G57/G57/G48/G51 /G4C/G51/G03/G57/G4B/G48/G03/G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A/G03/G49/G52/G55/G50 ()222 221qTΩ−=− βγν /G0F/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G14/G0C () ()∫+ Ω −=+ Lsq dzzhTmZeT 0 0222 22 2cot 1 µ ϕ σβγν /G0F/G03/G03/G03/G0B/G15/G0C /G5A/G4B/G48/G55/G48/G03 s qZeEϕβλµπsin2 02=Ω /G0F/G03 cTλ= /G0F/G03 0µ /G03 /G4C/G56/G03 /G53/G48/G55/G50/G48/G44/G45/G4C/G4F/G4C/G57/G5C /G52/G49/G03 /G49/G55/G48/G48/G03 /G56/G53/G44/G46/G48/G0F/G03 ()Lbπ σ2cot= /G0F/G03 sϕ /G03 /G4C/G56/G03 /G44/G03 /G53/G4B/G44/G56/G48/G03 /G52/G49/G03 /G44/G51 /G48/G54/G58/G4C/G4F/G4C/G45/G55/G4C/G58/G50/G03 /G4C/G52/G51/G03 /G0B/G4C/G51/G03 /G05/G46/G52/G56/G4C/G51/G48/G05/G03 /G55/G48/G49/G48/G55/G48/G51/G46/G48/G03 /G56/G5C/G56/G57/G48/G50/G0C/G0F () GLxHLxHLdzzhLeff eff eff d/d/ = = ≈∫/G0F/G03 λ /G03 /G4C/G56/G03 /G57/G4B/G48 /G5A/G44/G59/G48/G4F/G48/G51/G4A/G57/G4B/G03 /G4C/G51/G03 /G44/G03 /G49/G55/G48/G48/G03 /G56/G53/G44/G46/G48/G0F/G03 c /G03 /G4C/G56/G03 /G56/G53/G48/G48/G47/G03 /G52/G49/G03 /G4F/G4C/G4A/G4B/G57/G0F/G03 0/mZe /G57/G4B/G48/G03 /G55/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G4C/G52/G51/G03 /G46/G4B/G44/G55/G4A/G48/G03 /G57/G52/G03 /G4C/G57/G56/G03 /G50/G44/G56/G56/G0F/G03 β /G03 /G4C/G56/G03 /G44/G03 /G55/G48/G4F/G44/G57/G4C/G59/G48/G03 /G4C/G52/G51 /G59/G48/G4F/G52/G46/G4C/G57/G5C/G0F/G03 lb2 /G03 /G4C/G56/G03 /G57/G4B/G48/G03 /G4A/G44/G53/G03 /G46/G52/G48/G49/G49/G4C/G46/G4C/G48/G51/G57/G0F/G03 effL /G03 /G4C/G56/G03 /G44/G51/G03 /G48/G49/G49/G48/G46/G57/G4C/G59/G48 /G4F/G48/G51/G4A/G57/G4B/G03 /G52/G49/G03 /G44/G03 /G50/G44/G4A/G51/G48/G57/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G0F/G03 G /G03 /G4C/G56/G03 /G57/G4B/G48/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G03 /G52/G49 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G49/G4C/G48/G4F/G47/G03 /G52/G49/G03 /G50/G44/G4A/G51/G48/G57/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G0F/G03 )(22γν− /G03 /G44/G51/G47 )(22γν+ /G03 /G44/G55/G48/G03 /G57/G4B/G48/G03 /G46/G52/G52/G55/G47/G4C/G51/G44/G57/G48/G03 /G44/G5B/G48/G56/G03 /G52/G49/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50 /G47/G48/G57/G48/G55/G50/G4C/G51/G48/G47/G03/G44/G46/G46/G52/G55/G47/G4C/G51/G4A/G03/G57/G52/G03/G55/G48/G49/G11/G03/G3E/G15/G40/G11 /G37/G52/G03 /G48/G44/G56/G5C/G03 /G44/G03 /G4A/G55/G44/G53/G4B/G4C/G46/G44/G4F/G03 /G4C/G51/G57/G48/G55/G53/G55/G48/G57/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G52/G58/G55/G03 /G44/G51/G44/G4F/G5C/G56/G4C/G56/G0F/G03 /G57/G4B/G48 /G48/G54/G58/G44/G57/G4C/G52/G51/G56/G03/G0B/G14/G10/G15/G0C/G03/G46/G44/G51/G03/G45/G48/G03/G55/G48/G5A/G55/G4C/G57/G57/G48/G51/G03/G4C/G51/G03/G57/G4B/G48/G03/G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A/G03/G49/G52/G55/G50 ()s22sinϕβ γν ED=−/G0F/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G16/G0C ()[ ]C EBD + =+s22cosϕβ γν/G0F/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G17/G0C/G5A/G4B/G48/G55/G48/G03 )/()1(22 02cm ZeD βλπ− = /G0F/G03 ()Lb Bπ2cot= /G0F π µ/eff0GLcC= /G11 22γν+ 1.2 0.80.4 0 8 6 4 203.0=β 3.0=β1 cos−=µ 1 cos+=µ 22γν−MQL Hybrid /G29/G4C/G4A/G58/G55/G48/G03/G14/G1D/G03/G24/G03/G37/G5C/G53/G4C/G46/G44/G4F/G03/G48/G5B/G44/G50/G53/G4F/G48/G03/G49/G52/G55/G03/G44/G51/G03/G57/G55/G44/G4D/G48/G46/G57/G52/G55/G5C/G03/G52/G49 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03/G53/G52/G4C/G51/G57/G03/G52/G51/G03/G44/G03/G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03/G47/G4C/G44/G4A/G55/G44/G50/G03/G49/G52/G55/G03/G44/G03/G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F /G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G45/G5C/G03/G30/G34/G03/G4F/G48/G51/G56/G48/G56/G03/G44/G51/G47/G03/G45/G5C/G03/G05/G4B/G5C/G45/G55/G4C/G47/G05/G03/G56/G5C/G56/G57/G48/G50/G11 /G2F/G48/G57/G03 /G58/G56/G03 /G44/G51/G44/G4F/G5C/G5D/G48/G03 /G57/G4B/G48/G56/G48/G03 /G48/G54/G58/G44/G57/G4C/G52/G51/G56/G03 /G44/G57/G03 /G44/G03 /G46/G52/G51/G56/G57/G44/G51/G57 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03 /G55/G44/G57/G48/G03 /G0B const=E /G0C/G0F/G03 /G57/G4B/G48/G03 /G5A/G44/G59/G48/G4F/G48/G51/G4A/G57/G4B/G03 /G52/G49/G03 /G35/G29 /G49/G4C/G48/G4F/G47/G56/G03/G0B const=λ /G0C/G03/G44/G51/G47/G03/G44/G51/G03 /G4C/G47/G48/G51/G57/G4C/G46/G44/G4F/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56 /G0Bconsteff=L /G0F/G03 const=G /G0C/G11/G03 /G2C/G57/G03 /G4C/G56/G03 /G56/G48/G48/G51/G03 /G57/G4B/G44/G57/G03 /G57/G4B/G48/G03 /G57/G48/G55/G50/G03 D /G47/G52/G48/G56/G03 /G51/G52/G57/G03 /G59/G44/G55/G5C/G03 /G44/G57/G03 /G51/G52/G51/G55/G48/G4F/G44/G57/G4C/G59/G4C/G56/G57/G4C/G46/G03 /G46/G52/G51/G47/G4C/G57/G4C/G52/G51/G56/G0F/G03 /G4C/G11/G48/G11 const≈D /G11 /G29/G52/G55/G03 /G44/G03 /G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F/G03 /G30/G34/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G52/G58/G57/G03 /G35/G29 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G0F/G03 /G57/G4B/G48/G03 /G49/G4C/G55/G56/G57/G03 /G57/G48/G55/G50/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G55/G11/G4B/G11/G56/G11/G03 /G52/G49/G03 /G48/G54/G11/G03 /G0B/G15/G0C/G03 /G4C/G56/G03 /G48/G54/G58/G44/G4F /G57/G52/G03 /G5D/G48/G55/G52/G11/G03 /G24/G51/G03 /G57/G55/G44/G4D/G48/G46/G57/G52/G55/G5C/G03 /G52/G49/G03 /G44/G51/G03 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G03 /G52/G51/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C /G47/G4C/G44/G4A/G55/G44/G50/G03 /G5A/G4C/G57/G4B/G03 /G46/G52/G52/G55/G47/G4C/G51/G44/G57/G48/G56/G03 };{2222γνγν +− /G03 /G4C/G56/G03 /G53/G44/G55/G44/G4F/G4F/G48/G4F /G57/G52/G03 /G57/G4B/G48/G03 )(22γν− /G10/G44/G5B/G4C/G56/G03 /G49/G55/G52/G50/G03 /G44/G03 /G46/G48/G51/G57/G48/G55/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G57/G52/G03 /G44 /G45/G52/G58/G51/G47/G44/G55/G5C/G11 /G29/G52/G55/G03 /G44/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G03 /G35/G29/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G0F /G57/G4B/G48/G03 /G49/G4C/G55/G56/G57/G03 /G57/G48/G55/G50/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G55/G11/G4B/G11/G56/G11/G03 /G52/G49/G03 /G48/G54/G11/G03 /G0B/G15/G0C/G03 /G55/G48/G47/G58/G46/G48/G56/G03 /G44/G57/G03 /G4C/G51/G46/G55/G48/G44/G56/G4C/G51/G4A /G52/G49/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G03 /G59/G48/G4F/G52/G46/G4C/G57/G5C/G03 β /G11/G03 /G25/G5C/G03 /G44/G51/G03 /G52/G53/G57/G4C/G50/G44/G4F/G03 /G46/G4B/G52/G4C/G46/G48/G03 /G52/G49 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G0F/G03 /G44/G03 /G57/G55/G44/G4D/G48/G46/G57/G52/G55/G5C/G03 /G52/G49/G03 /G44/G51/G03 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G03 /G52/G51/G03 /G44 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03/G47/G4C/G44/G4A/G55/G44/G50/G03/G46/G44/G51/G03/G56/G57/G44/G5C/G03/G5A/G4C/G57/G4B/G4C/G51/G03/G44/G03/G46/G48/G51/G57/G55/G44/G4F/G03/G53/G44/G55/G57/G11 /G2F/G48/G57/G03 /G58/G56/G03 /G46/G52/G51/G56/G4C/G47/G48/G55/G03 /G44/G03 /G51/G58/G50/G48/G55/G4C/G46/G44/G4F/G03 /G48/G5B/G44/G50/G53/G4F/G48/G11/G03 /G37/G4B/G48/G03 /G48/G54/G58/G44/G57/G4C/G52/G51/G03 /G0B/G17/G0C /G46/G44/G51/G03/G45/G48/G03/G5A/G55/G4C/G57/G57/G48/G51/G03/G4C/G51/G03/G57/G4B/G48/G03/G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A/G03/G49/G52/G55/G50 DCLb+ ⋅ −=+ )/2cot(cot)(s2222πϕγνγν /G11/G03/G03/G03/G03/G03/G03/G03/G0B/G18/G0C /G24/G57/G03 m 5.1=λ /G0F/G03 MV/m3=E /G0F/G03 2cots=ϕ /G0F 75.1)/2cot( =Lbπ /G0F/G03/G57/G4B/G48/G03/G48/G54/G11/G03/G0B/G18/G0C/G03/G45/G48/G46/G52/G50/G48/G56 DC+−=+ )(5.322 22γνγν /G11/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G19/G0C /G03/G29/G52/G55/G03 /G44/G51/G03 /G4C/G51/G53/G58/G57/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G03 03.0=β /G0F/G03 /G4F/G48/G57/G03 /G58/G56/G03 /G53/G58/G57 722=+γν /G11/G03/G03/G38/G56/G4C/G51/G4A/G03 /G48/G54/G11/G03 /G0B/G19/G0C/G0F/G03 /G5A/G48/G03 /G52/G45/G57/G44/G4C/G51/G03 5.3=DC /G11/G03 /G29/G52/G55/G03 /G44/G51 /G52/G58/G57/G53/G58/G57/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G03 3.0=β /G0F/G03 /G4F/G48/G57/G03 /G58/G56/G03 /G53/G58/G57/G03 /G44/G51 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G03 /G4C/G51/G03 /G44/G03 /G46/G48/G51/G57/G48/G55/G03 /G52/G49/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50/G0F/G03 /G4C/G11/G48/G11/G03 /G44/G57 422=+γν /G11/G03 /G2C/G57/G03 /G46/G44/G51/G03 /G45/G48/G03 /G56/G4B/G52/G5A/G51/G03 /G57/G4B/G48/G03 /G50/G44/G5B/G4C/G50/G58/G50/G03 /G59/G44/G4F/G58/G48/G03 /G52/G49 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G49/G4C/G48/G4F/G47/G03/G44/G57/G03/G44/G03/G50/G44/G4A/G51/G48/G57/G03/G56/G58/G55/G49/G44/G46/G48/G03/G50/G58/G56/G57/G03/G45/G48/G03 /G48/G54/G58/G44/G4F/G03 /G57/G52/G03 /G14/G11/G16/G03 /G37 /G49/G52/G55/G03 /G53/G55/G52/G57/G52/G51/G56/G11/G03 /G2C/G57/G03 /G46/G44/G51/G03 /G45/G48/G03 /G55/G48/G44/G4F/G4C/G5D/G48/G47/G03 /G49/G52/G55/G03 /G53/G48/G55/G50/G44/G51/G48/G51/G57/G03 /G50/G44/G4A/G51/G48/G57/G56 /G50/G44/G47/G48/G03 /G49/G55/G52/G50/G03 /G56/G44/G50/G44/G55/G4C/G58/G50/G10/G46/G52/G45/G44/G4F/G57/G03 /G44/G4F/G4F/G52/G5C/G11/G03 /G37/G55/G44/G4D/G48/G46/G57/G52/G55/G4C/G48/G56/G03 /G52/G49 /G52/G53/G48/G55/G44/G57/G4C/G51/G4A/G03 /G53/G52/G4C/G51/G57/G56/G03 /G52/G51/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G44/G55/G48/G03 /G56/G4B/G52/G5A/G51/G03 /G4C/G51 /G29/G4C/G4A/G11/G14/G11/G16/G03/G24/G03/G31/G38/G30/G28/G35/G2C/G26/G24/G2F/G03/G30/G24/G37/G35/G2C/G3B/G03/G24/G31/G24/G2F/G3C/G36/G2C/G36/G03/G32/G29 /G24/G03/G35/G28/G24/G2F/G03/G36/G3C/G36/G37/G28/G30 /G2F/G48/G57/G03 /G58/G56/G03 /G44/G51/G44/G4F/G5C/G5D/G48/G03 /G44/G03 /G55/G48/G44/G4F/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4B/G48/G51/G03 /G30/G34/G03 /G4F/G48/G51/G56/G48/G56/G03 /G44/G51/G47/G03 /G35/G29 /G28/G34/G03 /G4F/G48/G51/G56/G48/G56/G03 /G52/G46/G46/G58/G53/G5C/G03 /G47/G4C/G49/G49/G48/G55/G48/G51/G57/G03 /G56/G53/G44/G46/G48/G03 /G55/G48/G4A/G4C/G52/G51/G56/G11/G03 /G2C/G57/G03 /G46/G44/G51/G03 /G45/G48 /G53/G48/G55/G49/G52/G55/G50/G48/G47/G03 /G58/G56/G4C/G51/G4A/G03 /G57/G55/G44/G51/G56/G49/G48/G55/G03 /G50/G44/G57/G55/G4C/G46/G48/G56/G03 /G3E/G15/G10/G17/G40/G11/G03 /G37/G4B/G48/G03 /G51/G52/G57/G44/G57/G4C/G52/G51 /G44/G47/G52/G53/G57/G48/G47/G03 /G4C/G51/G03 /G35/G48/G49/G11/G3E/G16/G0F/G17/G40/G03 /G4C/G56/G03 /G58/G56/G48/G47/G03 /G49/G52/G55/G03 /G44/G03 /G47/G48/G56/G46/G55/G4C/G53/G57/G4C/G52/G51/G03 /G52/G49/G03 /G57/G4B/G48 /G57/G55/G44/G51/G56/G59/G48/G55/G56/G48/G03 /G50/G52/G57/G4C/G52/G51/G03 /G4C/G51/G03 /G53/G48/G55/G4C/G52/G47/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G49/G4C/G48/G4F/G47/G56/G11/G03 /G37/G4B/G48 /G4F/G4C/G51/G48/G44/G55/G03 /G50/G52/G57/G4C/G52/G51/G03 /G52/G49/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G56/G03 /G4C/G56/G03 /G47/G48/G56/G46/G55/G4C/G45/G48/G47/G03 /G45/G5C/G03 /G57/G4B/G48/G03 /G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A /G48/G54/G58/G44/G57/G4C/G52/G51/G03 /G49/G52/G55/G03 /G47/G4C/G50/G48/G51/G56/G4C/G52/G51/G4F/G48/G56/G56/G03 /G47/G4C/G56/G53/G4F/G44/G46/G48/G50/G48/G51/G57/G56/G0F/G03 x /G03 /G49/G55/G52/G50/G03 /G57/G4B/G48 /G45/G48/G44/G50/G03/G44/G5B/G4C/G56/G0F/G03 z  =+=+ ,0)(,0)( 2222 yzPdzydxzPdzxd/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G1A/G0C /G5A/G4B/G48/G55/G48/G03 )()()()(2zhzgAzP Λ+ −=ϕ /G03 /G44/G51/G47 )()()()(2zhzgAzP Λ− −=ϕ /G03/G44/G55/G48/G03 /G55/G48/G44/G4F/G03 /G49/G58/G51/G46/G57/G4C/G52/G51/G56/G03 /G47/G48/G56/G46/G55/G4C/G45/G4C/G51/G4A /G44/G51/G03 /G56/G57/G55/G58/G46/G57/G58/G55/G48/G03 /G52/G49/G03 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G10/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G46/G4B/G44/G51/G51/G48/G4F/G11/G03 /G37/G4B/G48 /G47/G4C/G50/G48/G51/G56/G4C/G52/G51/G4F/G48/G56/G56/G03 /G46/G52/G48/G49/G49/G4C/G46/G4C/G48/G51/G57/G56/G03 ) (ϕA /G03 /G44/G51/G47/G03 2Λ /G03 /G47/G48/G56/G46/G55/G4C/G45/G48 /G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G44/G51/G47/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G48/G49/G49/G48/G46/G57/G56/G0F/G03/G55/G48/G56/G53/G48/G46/G57/G4C/G59/G48/G4F/G5C/G11/G03/G37/G4B/G48/G5C/G03/G44/G55/G48 ϕ λββ πϕ sin)1()(32 0223 cmL eEAs m−= /G0F/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G1B/G0C 22 022 2 022 21 1 ss ss cmL Ee cmL He ββ ββ −′=−′=Λ /G0F/G03/G03/G03/G03/G03/G03/G03/G03/G03/G0B/G1C/G0C /G5A/G4B/G48/G55/G48/G03 E′ /G44/G51/G47/G03 H′ /G03 /G44/G55/G48/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G56/G03 /G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G44/G51/G47 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 /G4F/G48/G51/G56/G48/G56/G0F/G03 /G55/G48/G56/G53/G48/G46/G57/G4C/G59/G48/G4F/G5C/G0F/G03 mE /G03 /G4C/G56/G03 /G44/G51 /G44/G50/G53/G4F/G4C/G57/G58/G47/G48/G03 /G52/G49/G03 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G49/G4C/G48/G4F/G47/G0F/G03 ) (zg /G44/G51/G47/G03 ) (zh /G03 /G44/G55/G48 /G55/G48/G44/G4F/G03 /G49/G58/G51/G46/G57/G4C/G52/G51/G56/G03 /G47/G48/G56/G46/G55/G4C/G45/G4C/G51/G4A/G03 /G44/G03 /G47/G4C/G56/G57/G55/G4C/G45/G58/G57/G4C/G52/G51/G03 /G52/G49/G03 /G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G44/G51/G47 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G49/G52/G55/G46/G48/G56/G11 /G29/G52/G55/G03 /G50/G44/G57/G55/G4C/G5B/G03 /G44/G51/G44/G4F/G5C/G56/G4C/G56/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G57/G55/G44/G51/G56/G59/G48/G55/G56/G48/G03 /G50/G52/G57/G4C/G52/G51/G0F/G03 /G4C/G57/G03 /G4C/G56 /G51/G48/G46/G48/G56/G56/G44/G55/G5C/G03 /G57/G52/G03 /G4E/G51/G52/G5A/G03 /G50/G44/G57/G55/G4C/G46/G48/G56/G03 /G47/G48/G56/G46/G55/G4C/G45/G4C/G51/G4A/G03 /G48/G4F/G48/G50/G48/G51/G57/G44/G55/G5C /G56/G48/G46/G57/G4C/G52/G51/G56/G03 /G52/G49/G03 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G10/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G46/G4B/G44/G51/G51/G48/G4F/G11/G03 /G2C/G51/G03 /G52/G58/G55/G03 /G46/G44/G56/G48/G0F /G57/G4B/G48/G03 /G50/G44/G57/G55/G4C/G46/G48/G56/G03 /G49/G52/G55/G03 /G44/G03 /G47/G55/G4C/G49/G57/G03 /G56/G53/G44/G46/G48/G03 drifta /G0F/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G44/G53 aga /G0F/G03 /G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G44/G51/G47/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03 dqa /G03 /G44/G51/G47/G03 fqa /G44/G55/G48/G03 /G58/G56/G48/G47/G11/G03 /G2F/G48/G57/G03 /G58/G56/G03 /G44/G56/G56/G58/G50/G48/G03 /G57/G4B/G44/G57/G03 /G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G49/G52/G55/G46/G48/G56/G03 /G44/G55/G48 /G46/G52/G51/G46/G48/G51/G57/G55/G44/G57/G48/G47/G03 /G4C/G51/G03 /G44/G03 /G4A/G44/G53/G03 /G46/G48/G51/G57/G48/G55/G03 /G3E/G16/G40/G11/G03 /G37/G4B/G48/G51/G03 /G57/G4B/G48/G03 /G50/G44/G57/G55/G4C/G46/G48/G56/G03 /G44/G55/G48 /G48/G5B/G53/G55/G48/G56/G56/G48/G47/G03/G4C/G51/G03/G57/G4B/G48/G03/G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A/G03/G49/G52/G55/G50 , cosh sinh/sinhcosh , 11 , cossin/sincos , 10N1 dq agfqd drift   ΛΛ= =  Λ−Λ= = q qqqaNADaqqqqa aε /G03/G0B/G14/G13/G0C /G5A/G4B/G48/G55/G48/G03 N qqεΛ= /G0F/G03 /G44/G51/G47/G03 dε /G0F/G03 /G03 qε /G03 /G44/G55/G48/G03 /G57/G4B/G48/G03 /G51/G52/G55/G50/G44/G4F/G4C/G5D/G48/G47 /G4F/G48/G51/G4A/G57/G4B/G03 /G52/G49/G03 /G56/G48/G46/G57/G4C/G52/G51/G56/G0F/G03 N /G03 /G4C/G56/G03 /G57/G4B/G48/G03 /G51/G58/G50/G45/G48/G55/G03 /G52/G49/G03 /G4A/G44/G53/G56/G03 /G5A/G4C/G57/G4B/G4C/G51/G03 /G44 /G46/G4B/G44/G51/G51/G48/G4F/G03/G53/G48/G55/G4C/G52/G47/G11 /G37/G4B/G48/G51/G03 /G50/G44/G57/G55/G4C/G46/G48/G56/G03 /G47/G48/G56/G46/G55/G4C/G45/G4C/G51/G4A/G03 /G48/G4F/G48/G50/G48/G51/G57/G44/G55/G5C/G03 /G56/G48/G46/G57/G4C/G52/G51/G56/G03 /G44/G55/G48 /G50/G58/G4F/G57/G4C/G53/G4F/G4C/G48/G47/G03 /G44/G51/G47/G03 /G44/G03 /G55/G48/G56/G58/G4F/G57/G4C/G51/G4A/G03 /G57/G55/G44/G51/G56/G49/G48/G55/G03 /G50/G44/G57/G55/G4C/G5B/G03 []ija /G03 /G4C/G56 /G44/G51/G44/G4F/G5C/G5D/G48/G47/G03 /G44/G46/G46/G52/G55/G47/G4C/G51/G4A/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G4E/G51/G52/G5A/G51/G03 /G46/G52/G51/G47/G4C/G57/G4C/G52/G51/G03 /G52/G49/G03 /G50/G52/G57/G4C/G52/G51 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03/G3E/G16/G40/G1D/G03/G03 ()12 12211 <+<− aa /G11 /G36/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G52/G49/G03 /G44/G03 /G57/G55/G44/G51/G56/G59/G48/G55/G56/G48/G03 /G50/G52/G57/G4C/G52/G51/G03 /G4B/G44/G56/G03 /G45/G48/G48/G51/G03 /G44/G51/G44/G4F/G5C/G5D/G48/G47/G03 /G49/G52/G55 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G10/G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G46/G4B/G44/G51/G51/G48/G4F/G03 /G56/G4B/G52/G5A/G51/G03 /G4C/G51/G03 /G29/G4C/G4A/G11/G15/G03 /G44/G57T/cm 1 =′H /G0F/G03 kV/cm 50 =′E /G0F/G03 /G44/G51/G47/G03 0.05=pβ /G11/G03 /G3A/G48/G03 /G4B/G44/G59/G48 /G46/G52/G51/G56/G4C/G47/G48/G55/G48/G47/G03 /G57/G5A/G52/G03 /G56/G5C/G56/G57/G48/G50/G56/G11/G03 /G37/G4B/G48/G03 /G49/G4C/G55/G56/G57/G03 /G52/G51/G48/G03 /G4C/G56/G03 /G44/G03 /G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F /G30/G34/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G52/G58/G57/G03 /G35/G29/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G03 /G44/G51/G47/G03 /G57/G4B/G48 /G56/G48/G46/G52/G51/G47/G03/G52/G51/G48/G03/G4C/G56/G03/G57/G4B/G48/G03/G53/G55/G52/G53/G52/G56/G48/G47/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03 /G56/G5C/G56/G57/G48/G50/G56/G11/G03 /G35/G29/G03 /G28/G34 /G49/G4C/G48/G4F/G47/G03 /G4C/G56/G03 /G53/G55/G52/G59/G4C/G47/G48/G47/G03 /G45/G5C/G03 /G57/G58/G55/G51/G4C/G51/G4A/G03 /G52/G49/G03 /G48/G4F/G4F/G4C/G53/G57/G4C/G46/G44/G4F/G03 /G44/G53/G48/G55/G57/G58/G55/G48/G56/G03 /G52/G49 /G51/G48/G4C/G4A/G4B/G45/G52/G55/G4C/G51/G4A/G03 /G47/G55/G4C/G49/G57/G03 /G57/G58/G45/G48/G56/G03 /G45/G5C/G03 o90 /G11/G03 /G33/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03 /G52/G49/G03 /G57/G4B/G48/G56/G48 /G56/G5C/G56/G57/G48/G50/G56/G03/G44/G55/G48/G03/G53/G55/G48/G56/G48/G51/G57/G48/G47/G03/G4C/G51/G03/G37/G44/G45/G4F/G48/G03/G14/G03/G44/G51/G47/G03/G15/G11 IV I II A IV′ III′I′ B/G26A′ II′ L894123 567A′A /G29/G4C/G4A/G58/G55/G48/G03/G15/G1D/G03/G24/G03/G4A/G48/G51/G48/G55/G44/G4F/G03/G59/G4C/G48/G5A/G03/G52/G49/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G10/G49/G52/G46/G58/G56/G4C/G51/G4A /G46/G4B/G44/G51/G51/G48/G4F/G03/G44/G51/G47/G03/G4C/G57/G56/G03/G53/G44/G55/G57/G4C/G57/G4C/G52/G51/G4C/G51/G4A/G03/G4C/G51/G57/G52/G03/G48/G4F/G48/G50/G48/G51/G57/G44/G55/G5C/G03/G56/G48/G46/G57/G4C/G52/G51/G56/G03/G49/G52/G55 /G44/G03/G50/G44/G57/G55/G4C/G5B/G03/G44/G51/G44/G4F/G5C/G56/G4C/G56/G03/G0B/G58/G53/G53/G48/G55/G03/G47/G55/G44/G5A/G4C/G51/G4A/G0C/G0F/G03/G44/G51/G47/G03/G44/G51/G03/G48/G5B/G44/G50/G53/G4F/G48/G03/G52/G49/G03/G44 /G57/G48/G46/G4B/G51/G4C/G46/G44/G4F/G03/G55/G48/G44/G4F/G4C/G5D/G44/G57/G4C/G52/G51/G03/G52/G49/G03/G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G5A/G4C/G57/G4B /G51/G58/G50/G45/G48/G55/G4C/G51/G4A/G03/G52/G49/G03/G53/G48/G55/G50/G44/G51/G48/G51/G57/G03/G50/G44/G4A/G51/G48/G57/G56/G03/G0B/G4F/G52/G5A/G48/G55/G03/G47/G55/G44/G5A/G4C/G51/G4A/G56/G0C/G11 /G24/G03/G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G53/G55/G52/G59/G4C/G47/G48/G56/G03/G44/G51/G03/G44/G47/G47/G4C/G57/G4C/G52/G51/G44/G4F/G03 /G47/G48/G4A/G55/G48/G48 /G52/G49/G03 /G49/G55/G48/G48/G47/G52/G50/G03 /G49/G52/G55/G03 /G44/G51/G03 /G52/G53/G57/G4C/G50/G4C/G5D/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G11 /G37/G4B/G48/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G56/G03 /G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G44/G51/G47/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G4F/G48/G51/G56/G03 /G46/G44/G51/G03 /G45/G48 /G59/G44/G55/G4C/G48/G47/G03/G49/G55/G52/G50/G03/G5D/G48/G55/G52/G03/G57/G52/G03/G44/G03/G50/G44/G5B/G4C/G50/G58/G50/G03/G59/G44/G4F/G58/G48/G11 /G24/G57/G03 /G4C/G47/G48/G51/G57/G4C/G46/G44/G4F/G03 /G4F/G48/G51/G4A/G57/G4B/G56/G03 /G52/G49/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G44/G53/G56/G0F/G03 /G59/G4C/G48/G5A/G03 /G52/G49/G03 /G44 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G5A/G4C/G4F/G4F/G03 /G46/G4B/G44/G51/G4A/G48/G03 /G47/G58/G48/G03 /G57/G52/G03 /G44/G03 /G55/G48/G47/G58/G46/G57/G4C/G52/G51/G03 /G52/G49/G03 /G44 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G03 /G49/G52/G55/G03 /G35/G29/G03 /G48/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03 /G4F/G48/G51/G56/G48/G56/G0F/G03 /G45/G48/G46/G44/G58/G56/G48 βE′∝Λ2/G11/G03 /G29/G52/G55/G03 /G48/G5B/G44/G50/G53/G4F/G48/G0F/G03 /G44/G57/G03 005.0=β /G0F/G03 /G57/G4B/G48/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C /G47/G4C/G44/G4A/G55/G44/G50/G03 /G4C/G56/G03 /G51/G48/G44/G55/G03 /G57/G4B/G48/G03 /G56/G44/G50/G48/G03 /G44/G56/G03 /G44/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G5A/G4B/G48/G51/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46 /G4F/G48/G51/G56/G48/G56/G03 /G44/G55/G48/G03 /G44/G45/G56/G48/G51/G57/G11/G03 /G24/G57/G03 /G4C/G51/G46/G55/G48/G44/G56/G4C/G51/G4A/G03 β /G0F/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50 /G45/G48/G46/G52/G50/G48/G56/G03 /G46/G4F/G52/G56/G48/G03 /G57/G52/G03 /G44/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G49/G52/G55/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48 /G56/G5C/G56/G57/G48/G50/G03 /G5A/G4C/G57/G4B/G52/G58/G57/G03 /G35/G29/G03 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G11/G03 /G29/G4C/G4A/G58/G55/G48/G03 /G16/G03 /G56/G4B/G52/G5A/G56/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C /G47/G4C/G44/G4A/G55/G44/G50/G56/G03 /G52/G49/G03 /G44/G03 /G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F/G03 /G50/G44/G4A/G51/G48/G57/G4C/G46/G10/G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G44/G51/G47/G03 /G44/G03 /G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03 /G47/G4C/G44/G4A/G55/G44/G50/G03 /G52/G49/G03 /G44/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G10 /G57/G5C/G53/G48/G03 /G56/G5C/G56/G57/G48/G50/G03 /G44/G57/G03 /G47/G4C/G49/G49/G48/G55/G48/G51/G57/G03 β /G11/G03 /G37/G4B/G48/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G03 /G52/G49 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G03/G4C/G56/G03/G58/G56/G48/G47/G03/G44/G56/G03/G44/G51/G03/G52/G55/G47/G4C/G51/G44/G57/G48/G11 /G24/G51/G03 /G44/G53/G53/G4F/G4C/G46/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03 /G56/G5C/G56/G57/G48/G50/G03 /G48/G5B/G57/G48/G51/G47/G56/G03 /G44 /G59/G48/G4F/G52/G46/G4C/G57/G5C/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G48/G47/G03 /G4C/G52/G51/G56/G03 /G5A/G4C/G57/G4B/G52/G58/G57/G03 /G44/G03 /G46/G4B/G44/G51/G4A/G48/G03 /G52/G49/G03 /G49/G52/G46/G58/G56/G4C/G51/G4A /G56/G5C/G56/G57/G48/G50/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G11/G03 /G2C/G57/G03 /G53/G55/G52/G59/G4C/G47/G48/G56/G03 /G44/G03 /G53/G52/G56/G56/G4C/G45/G4C/G4F/G4C/G57/G5C/G03 /G52/G49 /G52/G53/G57/G4C/G50/G4C/G5D/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G44/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03 /G4C/G51/G03 /G57/G48/G55/G50/G56/G03 /G52/G49/G03 /G44/G03 /G55/G44/G47/G4C/G44/G4F /G50/G52/G57/G4C/G52/G51/G11/G24/G26/G2E/G31/G32/G3A/G2F/G28/G27/G2A/G28/G30/G28/G31/G37/G36 /G37/G4B/G48/G03 /G44/G58/G57/G4B/G52/G55/G56/G03 /G48/G5B/G53/G55/G48/G56/G56/G03 /G57/G4B/G44/G51/G4E/G56/G03 /G57/G52/G03 /G30/G55/G11/G03 /G39/G11/G2E/G58/G48/G59/G47/G44/G03 /G49/G52/G55/G03 /G4B/G48/G4F/G53 /G5A/G4C/G57/G4B/G03/G51/G58/G50/G48/G55/G4C/G46/G44/G4F/G03/G46/G44/G4F/G46/G58/G4F/G44/G57/G4C/G52/G51/G56/G11 /G37/G44/G45/G4F/G48/G03/G14/G1D/G03/G33/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03/G52/G49/G03/G44/G03/G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F/G03/G50/G44/G4A/G51/G48/G57/G4C/G46/G10 /G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G56/G5C/G56/G57/G48/G50/G03/G5A/G4C/G57/G4B/G52/G58/G57/G03/G35/G29/G03/G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56 /G36/G48/G46/G57/G4C/G52/G51 /G31/G52/G11 /G37/G5C/G53/G48/G03/G52/G49/G03/G56/G48/G46/G57/G4C/G52/G51 /G37/G4B/G48/G03 /G56/G48/G46/G57/G4C/G52/G51 /G4F/G48/G51/G4A/G57/G4B/G0F/G03/G46/G50 /G14 /G14/G12/G15/G03/G52/G49/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G16 /G15 /G27/G55/G4C/G49/G57/G03/G56/G53/G44/G46/G48 /G14 /G16 /G37/G4B/G4C/G51/G10/G4A/G44/G53 /G13 /G17 /G27/G55/G4C/G49/G57/G03/G56/G53/G44/G46/G48 /G14 /G18 /G27/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G19 /G19 /G27/G55/G4C/G49/G57/G03/G56/G53/G44/G46/G48 /G14 /G1A /G37/G4B/G4C/G51/G10/G4A/G44/G53 /G13 /G1B /G27/G55/G4C/G49/G57/G03/G56/G53/G44/G46/G48 /G14 /G1C /G14/G12/G15/G03/G52/G49/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G16 /G03/G37/G44/G45/G4F/G48/G03/G15/G1D/G03/G33/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03/G52/G49/G03/G44/G03/G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03/G56/G5C/G56/G57/G48/G50 /G36/G48/G46/G57/G4C/G52/G51 /G31/G52/G11 /G37/G5C/G53/G48/G03/G52/G49/G03/G56/G48/G46/G57/G4C/G52/G51 /G2F/G48/G51/G4A/G57/G4B/G0F /G46/G50 /G14 /G14/G12/G15/G03/G52/G49/G03/G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G49/G52/G46/G11/G03/G4F/G48/G51/G56 /G16 /G15 /G28/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03/G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G14 /G16 /G37/G4B/G4C/G51/G10/G4A/G44/G53 /G13 /G17 /G28/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G14 /G18 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G19 /G19 /G28/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G14 /G1A /G37/G4B/G4C/G51/G10/G4A/G44/G53 /G13 /G1B /G28/G4F/G48/G46/G57/G55/G4C/G46/G44/G4F/G03/G47/G48/G49/G52/G46/G58/G56/G4C/G51/G4A/G03/G4F/G48/G51/G56 /G14 /G1C /G14/G12/G15/G03/G52/G49/G03/G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G49/G52/G46/G11/G03/G4F/G48/G51/G56 /G16 008.0=β010.0=β2magnΛ A6 4 2 0 2-1224 005.0=β s quadrupolemagnetic /G29/G4C/G4A/G58/G55/G48/G03/G16/G1D/G03/G37/G4B/G48/G03/G56/G57/G44/G45/G4C/G4F/G4C/G57/G5C/G03/G47/G4C/G44/G4A/G55/G44/G50/G03/G49/G52/G55/G03/G44/G03/G46/G52/G51/G59/G48/G51/G57/G4C/G52/G51/G44/G4F /G50/G44/G4A/G51/G48/G57/G4C/G46/G03/G54/G58/G44/G47/G55/G58/G53/G52/G4F/G48/G56/G03/G44/G51/G47/G03/G05/G4B/G5C/G45/G55/G4C/G47/G05/G10/G57/G5C/G53/G48/G03/G56/G5C/G56/G57/G48/G50/G11 /G35/G28/G29/G28/G35/G28/G31/G26/G28/G36 /G3E/G14/G40 /G24/G11/G36/G4B/G44/G4F/G51/G52/G59/G0F/G03 /G25/G11/G25/G52/G4A/G47/G44/G51/G52/G59/G4C/G46/G4B/G0F/G03 /G24/G11/G31/G48/G56/G57/G48/G55/G52/G59/G4C/G46/G4B/G0F/G03 /GB3/G36/G50/G52/G52/G57/G4B /G2C/G52/G51/G03 /G28/G51/G55/G4A/G48/G5C/G03 /G37/G58/G51/G4C/G51/G4A/G03 /G2C/G51/G03 /G2F/G4C/G51/G48/G44/G55/G03 /G24/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/GB4/G0F/G03 /G33/G55/G52/G46/G11 /G3B/G39/G2C/G2C/G2C/G03 /G2C/G51/G57/G11/G03 /G2F/G4C/G51/G48/G44/G55/G03 /G24/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03 /G26/G52/G51/G49/G48/G55/G48/G51/G46/G48/G0F/G03 /G2A/G48/G51/G48/G59/G44/G0F /G24/G58/G4A/G58/G56/G57/G03/G14/G1C/G1C/G19/G11 /G3E/G15/G40 /GB3/G2F/G4C/G51/G48/G44/G55/G03 /G24/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G56/GB4/G03 /G12/G48/G47/G4C/G57/G48/G47/G03 /G45/G5C/G03 /G27/G11/G39/G11/G03 /G2E/G44/G55/G48/G57/G51/G4C/G4E/G52/G59/G03 /G48/G57 /G44/G4F/G11/G12/G03/G30/G52/G56/G46/G52/G5A/G0F/G03/G2A/G52/G56/G44/G57/G52/G50/G4C/G5D/G47/G44/G57/G0F/G03/G14/G1C/G19/G15/G03/G0B/G4C/G51/G03/G35/G58/G56/G56/G4C/G44/G51/G0C/G11 /G3E/G16/G40 /G24/G11/G27/G11/G39/G4F/G44/G56/G52/G59/G0F/G03 /GB3/G37/G4B/G48/G52/G55/G5C/G03 /G52/G49/G03 /G4F/G4C/G51/G48/G44/G55/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G56/GB4/G0F /G30/G52/G56/G46/G52/G5A/G0F/G03/G24/G57/G52/G50/G4C/G5D/G47/G44/G57/G0F/G03/G14/G1C/G19/G18/G03/G0B/G4C/G51/G03/G35/G58/G56/G56/G4C/G44/G51/G0C/G11 /G3E/G17/G40 /G32/G11/G24/G11/G39/G44/G4F/G47/G51/G48/G55/G0F/G03 /G24/G11/G27/G11/G39/G4F/G44/G56/G52/G59/G0F/G03 /G24/G11/G39/G11/G36/G4B/G44/G4F/G51/G52/G59/G0F/G03 /GB3/G2F/G4C/G51/G48/G44/G55 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G56/GB4/G0F/G03 /G30/G52/G56/G46/G52/G5A/G0F/G03 /G24/G57/G52/G50/G4C/G5D/G47/G44/G57/G0F/G03 /G14/G1C/G19/G1C/G03 /G0B/G4C/G51 /G35/G58/G56/G56/G4C/G44/G51/G0C/G11
arXiv:physics/0008063v1 [physics.gen-ph] 15 Aug 2000Fractal Statistics B.G. Sidharth∗ Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India) Abstract We consider the recent description of elementary particles in terms of Quantum Mechanical Kerr-Newman Black Holes, a descripti on which provides a rationale for and at the same time reconciles the B ohm- hydrodynamical formulation on the one hand and the Nelsonia n stochastiic formulation on the other. The Boson-Fermion divide is discu ssed, and it is pointed out that in special situations, anomalous stat istics, rather than Bose-Einstein or Fermi-Dirac states, can be encounter ed. 1 Introduction In a recent model[1, 2] Fermions, in particular the electron s have been treated as, what may be called Quantum Mechanical Kerr-Newman Black Holes (QMKMBH), a treatment that leads to a successful interpreta tion of several hitherto inexplicable features, both in Particle Physics a nd Cosmology[3, 4, 5, 6]. In the hydrodynamical formulation these QMKNBH can be considered to be vortices bounded by the Compton wavelength in a descrip tion that leads to a harmonious convergence with Nelson’s stochastic theory also[7] (Cf.Discussion). In this picture, Bosons would be, not vort ices, but rather streamlines[8]. In what follows, we shall show that within this framework it i s possible to explain the divide between Fermi-Dirac and Bose-Einstein s tatistics, as also examine special situations where there would be fractal sta tistics, that is Bosonisation of Fermions and Fermionisation of Bosons. 0∗Email:birlasc@hd1.vsnl.net.in; birlard@ap.nic.in 12 Usual Statistics We ﬁrst observe that the above vortex and streamline descrip tion provides an explanation for the Fermionic and Bosonic statistics. In deed, letnKbe the occupation number for the energy or momentum state deﬁne d byK. For Fermions nK= 0 or 1, where as nKcan be arbitrary for Bosons. The reason is that Fermions are bounded by the Compton wavelengt h. That is, they are localised, a description which requires both negat ive and positive energy solutions[9], which infact is expressed by zitterbe wegung eﬀects. The localisation in space automatically implies an indetermin acy of energy or momentum K. Thus, though an energy or momentum state, in practical terms implies a small spread ∆ K, it is not possible to cram Fermions which also have a momentum energy interdeterminacy spread, arbit rarily into this state. On the other hand Bosons are not bound by the Compton waveleng th vor- tices, and so have sharper momentum states, so that any numbe r of them can be crammed into the state K(which really is blurred by the Uncertainity Principle indeterminacy of ∆ K). Another way of expressing these facts are by saying that the Fermionic wave function in space is weak, b ut not the Bosonic wave function[10], the latter fact being symptomat ic of a ﬁeld or an interaction. The above considerations immediately follow from a recent d escription in terms of quantized fractal space time[11]. In this case we ha ve a non com- mutative geometry given by [x,y] = 0(l2),[x,px] =ı¯h[1 +l2] (1) wherelis the Compton wavelength. It is precisely the space quantiz a- tion at the Compton scale that leads to the Dirac matrices and their anti- commutation relations. Bosons on the other hand would be bound states of the Fermions (Cf.[12] and other references), or alternatively they would be a super po sition of vortices leading to a streamline like description[8]. 23 Fractal Statistics However there could be certain special situations in which t he above space localised and momentum space localised description of Ferm ions and Bosons gets blurred, in which case anomalous or fractal statistics would come into play. This could happen, for example when the Compton wavele ngthlof the Fermion becomes very small, that is the particle is very mass ive. In this case the non commutativity of the geometry referred to above in (1 ) disappears and we return to the usual commutation relations of non relat ivistic Quan- tum Mechanics, that is a description in terms of the spinless Schrodinger equation. Indeed in this case, vbeing the velocity of the particle v/cwould be small and the Dirac equation tends to the Schrodinger equa tion[9]. There would thus be a Bosonisation eﬀect. This would also be e xpected at very low temperatures, for example below the Fermi temperat ure, when the energy spread of the Fermions would itself be small, rather a s in the case of Bosons and anomalous behaviour, for example on the lines of t he superﬂuid- ity ofHe3[13] can be expected. For very light Fermions, for example Neutrinos, the Compton wavelength be very large, but in this case the double connectivity of the QM KNBH dis- appears and the observed anomalous features of the Neutrino show up, as discussed elsewhere[1, 14]. Indeed with recent developments in nano technology and thin ﬁlms, we are able to consider one dimensional and two dimensional Fermio ns, in which case, Bosonisation eﬀects show up as discussed elsewhere[1 3]. In any case in the one dimensional and two dimensional cases, the Dirac e quation be- comes a two component equation, without an invariant mass[1 5], while at the same time, handedness shows up[16]. It is worth mentioni ng here that the Dirac matrices can have only even dimensionality, corre sponding to the above anomalous two component Dirac spinors, and the usual D irac bispinors of the three dimensional theory. We now consider in greater detail two illustrative situatio ns where anomalous behaviour shows up. 1. Nearly Mono Energetic Fermions (Cf.[17]): Our starting point is the well known formula for the occupati on number of a Fermion gas[18] ¯np=1 z−1ebEp+ 1(2) 3where,z′≡λ3 v≡µz≈zbecause, here, µ≈1 (Cf. Appendix); v=V N,λ=/radicaltp/radicalvertex/radicalvertex/radicalbt2π¯h2 m/b b≡/parenleftbigg1 KT/parenrightbigg ,and/summationdisplay ¯np=N (3) Let us consider particular a collection of Fermions which is somehow made nearly mono-energetic, that is, given by the distribution, n′ p=δ(p−p0)¯np (4) where ¯npis given by (2). This is not possible in general - here we consider a special hy pothetical sit- uation of a collection of mono-energetic particles in equil ibrium which is the idealization of a contrived experimental set up. For exampl e, the following is a mono energetic equilibrium distribution function: fαexp [−ρ(\|/vector v\| − \|/vector v0\|)2],ρ≡m/2KT >> 1, Infact one can show that the above function is consistent wit h the Bolitzmann equation for a collection of particles under a uniform magne tic ﬁeld/vectorB: Infact ∂f ∂t= 0, while one can easily show that /vector∆f= 0,/vector∆/vector v[exp[−ρ(\|/vectorV\| − \|/vectorV0\|)2]α[(\|/vectorV\| − \|/vectorV0\|)f/vectorV] Further the force is given by /vectorFα/vectorV×/vectorB so that the force term in the Boltzmann equation viz., /vectorF m·/vector∆/vector vfα/vectorV×/vectorB·/vectorV= 0 Finally ∂f ∂t= 0 meansdH dt= 0 that is (∂f ∂t)coll= 0 4Thus the above distribution function which becomes mono ene rgetic forλ>> 1 satisﬁes the Boltzmann equation (Cf.ref.[18]); when ρ→ ∞ we get the aboven′ pin this case. By the usual formulation we have, N=V ¯h3/integraldisplay d/vector pn′ p=V ¯h3/integraldisplay δ(p−p0)4πp2¯npdp=4πV ¯h3p2 01 z−1eθ+ 1(5) whereθ≡bEp0. It must be noted that in (5) there is a loss of dimension in mome ntum space, due to theδfunction in (4) - infact such a fractal two dimensional situa tion would in the relativistic case lead us back to the anomalous b ehaviour already alluded to[19]. In the non relativistic case two dimensions would imply that the coordinate ψof the spherical polar coordinates ( r,ψ,φ ) would become constant,π/2 infact. In this case the usual Quantum numbers landmof the spherical harmonics[20] no longer play a role in the usua l radial wave equation d2u dr2+/braceleftBigg2m ¯h2[E−V(r)]−l(l+ 1) r2/bracerightBigg u= 0, (6) The coeﬃcient of the centrifugal term l(l+ 1) in (6) is replaced by m2as in Classical Theory[21]. To proceed, in this case, KT=< E p>≈Ep), so that,θ≈1. But we can continue without giving θany speciﬁc value. Using the expressions for vandzgiven in (3) in (4), we get (z−1eθ+ 1) = (4π)5/2z′−1 p0; whence z′−1A≡z′−1/parenleftBigg(4π)5/2 p0−eθ/parenrightBigg = 1, (7) where we use the fact that in (3), µ≈1 (Cf.Appendix). A number of conclusions can be drawn from (7). For example, if , A≈1,i.e., p0≈(4π)5/2 1 +e(8) 5whereAis given in (7), then z′≈1. Remembering that in (3), λis of the order of the de Broglie wave length and vis the average volume occupied per particle, this means that the gas gets very densely packe d for momenta given by (8). Infact for a Bose gas, as is well known, this is th e condition for Bose-Einstein condensation at the level p= 0 (cf.ref.[18]). On the other hand, if, A≈0(that is(4π)5/2 e≈p0) thenz′≈0. That is, the gas becomes dilute, or Vincreases. More generally, Equation (7) also puts a restriction on the e nergy (or mo- mentum), because z′>0, viz., A>0(i.e.p 0<(4π)5/2 e) But ifA<0,(i.e.p 0>(4π)5/2 e) then there is an apparent contradiction. The contradiction disappears if we realize that A≈0, or p0=(4π)5/2 e(9) (corresponding to a temperature given by KT=p2 0 2m) is a threshold momen- tum (phase transition). For momenta greater than the thresh old given by (9), the collection of Fermions behaves like Bosons. In this case , the occupation number is given by ¯np=1 z−1ebEp−1, instead of (2), and the right side equation of (7) would be giv en by′−1′ instead of +1, so that there would be no contradiction. Thus i n this case there is an anomalous behaviour of the Fermions. The Bosonic behaviour of Fermions can be understood in a simp le way from a diﬀerent standpoint. Let us consider the case described be fore equation (5), of a collection of Fermions with m >> 1, and consequently all having nearly the same momentum /vector p. In this case, it is possible to choose a Lorentz 6frame in which all the particles are nearly at rest, i.e., v/c≈0. As mentioned, it is known that the Dirac equation describing spin 1 /2 particles goes over into the Schrodinger equation describing spinless particl es[9]. Eﬀectively, we have a collection of Bosons. 2. Degenerate Bosons: We could consider a similar situation for Bosons also (Cf.[2 2] where an equa- tion like (4) holds. In this case we have equations like (8) an d (9): p0≈(4π)5/2 1.4e−1(10) p0≈(4π)5/2 e(11) ((11) is the same as (9), quite expectedly). At the momentum g iven by (10) we have a densely packed Boson gas rather as in the case of Bose Einstein condensation. On the other hand at the momentum given by (11) we have inﬁnite dilution, while at lower momenta than in (11) there i s an anomalous Fermionisation. Finally it may be pointed out that at very high temperatures, once again the energy - momentum spread of a Bosonic gas becomes large, and F ermioni- sation can be expected, as in indeed has been shown elsewhere [23]. In any case at these very high temperatures, we approach the Classi cal Maxwell Boltzmann situation. 4 Discussion 1. It was mentioned in Section 1 that in the QMKNBH descriptio n there is a reconciliation between Nelson’s stochastic theory and the Bohm-Hydrodynamical approach, once it is realized that the diﬀusion constant of N elson’s theory is related to the Compton wavelength, and that the non local B ohm po- tential gives the energy of the particle (Cf.[1, 7] for detai ls). Infact in the Nelsonian theory, space time is a non diﬀerenciable manifol d, there being a Double Weiner process which leads to the usual solinoidal ve locity given by /vector v=/vector∇S mas also an osmotic velocity given by /vector u=¯h 2m/vector∇R Rwhere the Quan- tum Mechanical wave function ψ=Reı ¯hS.We get identical expressions in 7the Bohm-hydrodynamical approach also. Indeed if there wer e no double Weiner process, then in the Nelsonian theory /vector uwould vanish and so would ∇2RandQ, that is the Compton wavelength vortex and the mass and the energy of the particle would both disappear. 2. Interestingly in the QMKNBH-Hydrodynamical vortex pict ure as in the usual Quantum Theory of addition of angular momentum, we can recover the fact that the sum or bound state of two such vortices or spin ha lf particles would indeed give Bosons. This can be seen as follows, from the theory of vortices[24]. The velocity distribution is given by v= Γ/2πr (12) In the case of the QMKNBH vortex we have to use in (12) v=candr= ¯h/2mc, the Compton wavelength of the particle. So we have Γ =h mwhich is also the diﬀusion constant of Nelsonian Theory. If we consider two parallel spinning vortices separated by a distanced, then the angular velocity is given by ω=Γ πd2 whence the spin of the system turns out to be h, that is in usual units the spin is (1), wth states ±1. There is also the case where the two above vortices are anti pa rallel. In this case there is no spin, but rather there is the linear velocity given by v= Γ/2πd This corresponds to the spin 1 case with state 0. Together, the two above cases give the 3 ,−1,0,+1 states of spin 1 as in the Quantum Mechanical Theory. It must be noteed that the distance dbetween vortices could be much greater than the Compton wavelength scale, so that the wave function of the Boson in the above description would be extended in space in compar ison to the Fermionic wave function, as pointed out in the text. References [1] Sidharth, B.G., 1997 Ind.J. of Pure and Applied Phys., 35 , pp456-471. 8[2] Sidharth, B.G., 1998 Int.J.Mod.Phys.A., 13(15), pp259 9-2612. [3] Sidharth, B.G., 1999 Mod.Phys.Lett.A., Vol.14 No.5, pp 387-389. [4] Sidharth, B.G., Gravitation & Cosmology, Vol.4, No.2, 1 998. [5] Sidharth, B.G., 1997 Mod.Phys.Lett.A., Vol.12 No.32, p p2469-2471. [6] Sidharth, B.G., International Journal of Theoretical P hysics, 37 (4), 1307-1312, 1998. [7] Sidharth, B.G., ”Space Time as a Random Heap”, to appear i n Chaos, Solitons & Fractals. [8] Sidharth, B.G., 1999 in Proceedings of Eighth Marcel Gro ssmann Meet- ing on General Relativity, T. Piran and Remo Ruﬃni (Eds.), Wo rld Scientiﬁc, Singapore,pp479-481. [9] J.D. Bjorken and S.D. Drell, ”Relativistic Quantum Mech anics”, McGraw-Hill Inc., New York, 1964. [10] Schiﬀ, L.I., ”Quantum Mechanics”, McGraw-Hill Book Co mpany, Sin- gapore, 1968, p.490. [11] Sidharth, B.G., 2000 Chaos, Solitons and Fractals, 11 ( 8), p1269-1278. [12] Finkelstein, D., Saller, H., and Tang, Z., ”Gravity Par ticles and Space Time”, Eds. P. Pronin, G. Sardanashvily, World Scientiﬁc, S ingapore, 1996. [13] Sidharth, B.G., Journal of Statistical Physics, 95(3/ 4), May 1999. [14] Sidharth, B.G., ”From the Neutrino to the edge of the Uni verse”, to appear in Chaos, Solitons & Fractals. [15] A. Zee, ”Unity of Forces in the Universe”, Vol.II, World Scientiﬁc, Sin- gapore, 1982, and several papers reproduced and cited there in. [16] Heine, V.,”Group Theory in Quantum Mechanics”, Pergam on Press, Oxford, 1960, p.364. 9[17] Sidharth, B.G., ”A Note on Degenerate Bosons”, CAMCS Te chnical Report 95-04-07b. [18] Huang, K., 1975 ”Statistical Mechanics”, Wiley Easter n, New Delhi,pp230-237. [19] Sidharth, B.G., ”Interstellar Hydrogen” to appear in C haos, Solitons & Fractals. [20] Powell, J.L., & Crasemann, B., ”Quantum Mechanics”, Na rosa Publish- ing House, New Delhi, 1988. [21] Goldstein, H., 1966 ”Classical Mechanics”, Addison-W esley, Reading, Mass., p76. [22] xxx.lanl.gov/quant-ph/abs 9506002. [23] Sidharth, B.G., Bull.Astr.Soc.India, 25, 1997. [24] Donnelly, R.J., ”Quantized Vortices in Helium II”, Cam bridge Univer- sity Press, Cambridge, 1991, pg.13. 10
arXiv:physics/0008064 15 Aug 2000CLIC – A TWO-BEAM MULTI-TeV e /G72/G72 LINEAR COLLIDER J.P. Delahaye and I. Wilson for the CLIC study team: R. Assmann, F. Becker, R. Bossart, H. Braun, H. Burkhardt, G. Carron, W. Coosemans, R. Corsini, E. D'Amico, S. Doebert, S. Fartoukh, A. Ferrari, G. Geschonke, J.C. Godot, L. Groening, G. Guignard, S. Hutchins, B. Jeanneret, E. Jensen, J. Jowett, T. Kamitani, A. Millich, P. Pearce, F. Perriollat, R. Pittin, J.P. Potier, A. Riche, L. Rinolfi, T. Risselada, P. Royer, F. Ruggiero, D. Schulte, G. Suberlucq, I. Syratchev, L. Thorndahl, H. Trautner, A. Verdier, W. Wuensch, F. Zhou, F. Zimmermann, CERN, Geneva, Switzerland, O. Napoly , SACLAY, France. ABSTRACT The CLIC study of a high-energy (0.5 - 5 TeV), high-luminosity (1034 - 1035 cm-2 sec-1) e/G72 linear collider is presented. Beam acceleration using high frequency (30 GHz) normal-conducting structures operating at high accelerating fields (150 MV/m) significantly reduces the length and, in consequence, the cost of the linac. Using parameters derived from general scaling laws for linear colliders, the beam stability is shown to be similar to lower frequency designs in spite of the strong wake-field dependency on frequency. A new cost-effective and efficient drive beam generation scheme for RF power production by the so-called "Two-Beam Acceleration" method is described. It uses a thermionic gun and a fully-loaded normal-conducting linac operating at low frequency (937 MHz) to generate and accelerate the drive beam bunches, and RF multiplication by funnelling in compressor rings to produce the desired bunch structure. Recent 30 GHz hardware developments and CLIC Test Facility (CTF) results are described. Figure 1: Overall layout of the CLIC complex. 1 INTRODUCTION The Compact Linear Collider (CLIC) covers a centre-of-mass energy range for e/G72 collisions of 0.5 - 5 TeV [1] with a maximum energy well above those presently being proposed for any other linear collider [2]. It has been optimised for a 3 TeV e/G72 colliding beam energy to meet post-LHC physics requirements [3] but can be built in stages without major modifications. An overall layout of the complex is shown in Fig.1. In order to limit the overall length, high accelerating fields are mandatory and these can only be obtained with conventional structures, by operating at a high frequency. The RF power to feedthe accelerating structures is extracted by transfer structures from high-intensity/low-energy drive beams running parallel to the main beam (Fig. 2). A single tunnel, housing both linacs and the various beam transfer lines without any modulators or klystrons, results in a very simple, cost effective and easily extendable configuration. Figure 2: One main-beam and drive-beam module. 2 MAIN PARAMETERS The main-beam and linac parameters are listed in Table 1 for two colliding beam energies. The luminosity L normalised to the RF power, PRF, depends on a small number of parameters in both low ( /G38<<1) and high beamstrahlung ( /G38>>1) regimes: 2/12/1 1 nyRF b fB RFUPL /G48/G4B /G47/G76/G1F/G1F /G38 and 2/12/12/12/12/3 1 nyzRF b yfB RFUPL /G48 /G56/G4B /G45/G47/G76/G21/G21 /G38 (1) where /G47B, is the mean energy loss, /G4BRF b the RF-to-beam efficiency and Uf, /G56z, /G45y, /G48ny the beam energy, bunch length, vertical beta function and normalised vertical beam emittance at the I.P. respectively [4]. The parameters have been derived from general scaling laws [4] covering more than a decade in frequency. These scaling laws, which agree with optimised linear collider designs, show that the beam blow-up during acceleration can be made independent of frequency for equivalent beam trajectory correction techniques. As a consequence, and in spite of the strong dependence of wakefields on frequency, CLIC whilst operating at a high frequency but with a low charge per bunch N, a short bunch length /G56z, strong focussing optics and a high accelerating gradient G, preserves the vertical emittance as well as low frequency linacs. The RF-to-beam transfer efficiency is optimised by using a large number of bunches and by choosing anAcc.Struct.229 MW 229 MW 223 cm Main Linac AcceleratorQUAD QUADDrive Beam Decelerator Acc.Struct.Acc.Struct.Acc.Struct.Decel. Structure BPMDecel. Structure 229 MW 229 MW DETECTORS 624 m DRIVE BEAM DECELERATOR e- e - FINAL FOCUSe - e+ /G4A/G03 /G4A e -e+ e + MAIN LINAC LASERFINAL FOCUSe - MAIN LINAC LASER DRIVE BEAM GENERATION COMPLEX ~ 460 MW/m 30 GHz RF POWER MAIN BEAM GENERATION COMPLEXoptimum accelerating section length. In spite of the reduced charge per bunch and the high gradient, excellent RF-to-beam efficiency is obtained because the time between bunches is shorter and the shunt impedance of the accelerating structures is higher. Table 1: Main beam and linac parameters Beam parameters at IP 1 TeV3 TeV Luminosity (1034cm-1s-1) 2.710.0 Mean energy loss (%) 11.231 Photons /electrons 1.12.3 Coherent pairs per crossing 3×1066.8×108 Repetition rate (Hz) 150 100 e/G72 / bunch 4 /G751094 /G75109 Bunches / pulse 154 154 Bunch spacing (cm) 20 20 /G48n (10-8 rad.m) H/V 130/268/2 Beam size (nm) H/V 115/1.7543/1 Bunch length ( /G50m) 30 30 Accel. gradient (MV/m) 150 150 Two-linac length (km) 10 27.5 Accelerating structures 14140 42940 Power / section (MW) 229 229 Number of 50 MW klystrons364 364 Klystron pulse length ( /G50s) 33.392 RF-to-beam efficiency (%) 24.424.4 AC to beam efficiency (%) 9.89.8 AC power (MW) 150 300 0.0010.010.11 0.8 0.85 0.9 0.95 1L/L0 per bin E/E00.5 TeV 1 TeV 3 TeV 5 TeV Figure 3: Luminosity distribution with energy. (L0: total integrated luminosity, E0: max. energy) Up to 1 TeV, where the beamstrahlung parameter /G38 <1, the beam parameters are chosen to have a small /G47B. To limit the power consumption above 1 TeV, /G48ny is reduced and /G38 allowed to be >>1. In this regime (see Eq.1), high frequency linacs are very favourable because /G56z is small. As a consequence, even with /G38 >>1, neither the L spectrum (Fig. 3), nor the number of emitted gammas which increase the background in the detector, significantly deteriorate with energy [1] (see Table 1). The number of e/G72 pairs generated per crossing however increases significantly with energy.3 MAIN LINAC The effects of the strong 30 GHz wakefields (WT) can be kept moderate by choosing N to be small (4×109 at all energies) and /G56z at the lower limit that is permitted by the momentum acceptance of the final focus. With a high gradient G and strong focusing, the single-bunch blow-up /G27/G48ny can be kept below /G7C100 /G08 at all energies (Fig. 4) [5]. 55.566.577.588.59 02004006008001000120014001600vertical emittance [nm] E [GeV]single bunch multi bunch Figure 4: Emittance variation along the main linac. To obtain the values of L given in Table 1 a very small injected /G48ny of 5×10-9 rad.m is assumed. Limiting the overall /G27/G48ny relies in part on the use of bumps which are created locally at 5-10 positions along the linac by mis-aligning a few upstream cavities. The effects of these bumps are used to minimise the local /G48ny (Fig.4). Without these bumps, dispersive effects are /G7C10 times weaker than WT effects. The average lattice /G45-function starts from /G7C 4- 5 m and is scaled approx. as (energy)0.5. The FODO lattice is made up of sectors with equi-spaced quadrupoles of equal length and normalised strength, with matching insertions between sectors. The RF cavities and quadrupoles are pre-aligned to 10 and 50 /G50m respectively using a stretched-wire positioning system. The misalignments of the beam position monitors (BPMs) are measured as follows [6]. A section of 12 quadrupoles is switched off, and with the beam centred in the two end BPMs of this section, the relative mis-alignment of the other monitors are measured with an accuracy of 0.1 /G50m. The beam trajectory and ground motion effects are corrected by a 1-to-1 correction. BNS damping is achieved by running off the RF-crest by 6o to 10o. Multiple bunches are required to obtain high luminosities. The multi-bunch emittance blow-up /G27/G48ny is /G7C 20%. To make the 154-bunch train stable requires a strong reduction of the transverse wakefields induced by the beam in the accelerating structures. A new Tapered Damped Structure (TDS) [7] has been designed. Each of the 150 cells is damped by its own set of four radialwaveguides (Fig. 5) giving a Q of 16 for the lowest dipole mode. A simple linear tapering of the iris dimension provides a de-tuning frequency spread of 2 GHz (5.4%). The waveguides are terminated with short silicon carbide loads [8]. Figure 5: A cut-away view of the CLIC TDS Calculations of the transverse wakefields in this structure indicate a short-range level of about 1000 V/(pC /G98mm /G98m) decreasing to less than 1 % at the second bunch and with a long-time level below 0.1 %. Figure 6 : Comparison of measured ASSET wakefield levels and theory. A 15 GHz scale model of this structure has been tested in the ASSET Test Facility at SLAC. The measured wakefield levels shown in Fig. 6 are in excellent agreement with the theoretical predictions [9]. The 7.6 GHz signal at the 1% level is not related to the structure but to a beam-pipe/structure transition. The recent observation of surface damage at relatively low accelerating gradients (~65 MV/m) and with short pulses (16 ns) in these high group velocity structures is a cause of concern. It is not clear at the moment whether this can be attributed to the geometry of the structure or to other contributing factors such as vacuum level or conditioning procedure.4 THE RF POWER SOURCE The overall layout of the CLIC RF power source scheme for a 3 TeV centre-of-mass collider is shown in Fig. 7. The RF power for each 624 m section of the main linac is provided by a secondary low-energy high-intensity electron beam which runs parallel to the main linac. The power is generated by passing this electron beam through energy-extracting RF structures in the so-called “Drive Beam Decelerator”. For the 3 TeV c.m. collider there are 44 drive beams (22 per linac). Each drive beam has an energy of 1.18 GeV and consists of 1952 bun ches with a spacing of 2 cm and a maximum charge per bunch of 16 nC. These 22 drive beams, spaced at intervals of 1248 m, are produced as one long pulse by one of the two drive beam generators. By initially sending this drive beam train in the opposite direction to the main beam, different time slices of the pulse can be used to power separate sections of the main linac. Figure 7 : Layout of RF power source The drive beam is generated as follows [10]. All the bunches (for 22 drive beams) are first generated and accelerated with a spacing of 64 cm as one long continuous train in a normal-conducting fully-loaded 937 MHz linac operating at a gradient of 3.9 MV/m. This 7.5 A 92 /G50s continuous beam can be accelerated with an RF/beam efficiency /G7C 97%. After acceleration the continuous train of 42944 bun ches is split up into 352 trains of 122 bunches using the combined action of a delay line and a grouping of bunches in odd and even RF buckets. These trains are then combined in a 78 m circumference ring by interleaving four successive bunch trains over four turns to obtain a distance between bunches at this stage of 8 cm. A second combination using the same method is subsequently made in a similar, larger 312 m circumference ring, yielding a final distance between bunches of 2 cm. The power-extracting structures consist of four periodically-loaded rectangular waveguides coupled to a circular beam pipe. Each 80 cm long structure provides 458 MW of 30 GHz RF power, enough to feed two accelerating structures. For stability in the drive beam decelerator, these structures MAIN LINAC /G03COMBINER RINGSINJECTOR /G1A /G1B/G03 /G50 /G16 /G14 /G15/G03/G50/G16 /G1C/G03 /G50 DRIVE BEAM ACCELERATOR 937 MHz - 1.18 GeV - 3.9 MV/m/G27 /G28 /G2F /G24 /G3C BUNCH COMPRESSOR182 Klystrons 50 MW 92 microsec Main Beamhave to be damped to reduce long-range transverse wakefield effects. Two drive-beam accelerator options are presently being studied: to use re-circulation to reduce the installed RF power, and to use a single accelerator to produce the drive beams for both the e- and e+ linacs. 5 MAIN BEAM INJECTORS The main beam injector complex is located centrally (see Fig.1). To reduce cost the same linacs accelerate both electrons and positrons on consecutive RF pulses. The positrons are produced by standard technology already in use at the SLC (SLAC Linear Collider) but with improved performance due to the larger acceptance of the L-band capture linac [11]. The electron and positron beams are damped transversely in specially designed damping rings for low emittances [12]. The damping rings are made up of arcs based on a Theoretical Minimum Emittance (TME) lattice and straight sections equipped with wigglers. The positrons are pre-damped in a pre- damping ring. A specific design for a 3 TeV collider is underway but has not yet been completed. The aim is to provide normalised emittances of 5×10-7 and 5×10-9 rad.m. in the horizontal and vertical planes respectively, at the entrance to the main linac. The bunches are compressed in two stages in magnetic chicanes [13], the first one after the damping ring using 3 GHz structures, the second one just before injection into the main linac with 30 GHz structures. An option to use a common injector linac for both main beams and drive beams is being studied. 6 THE BEAM DELIVERY AND INTERACTION POINT Studies of the beam delivery section consisting of a final-focus chromatic correction section and a collimation section for the 3 TeV collider have only just started and for the moment there is no consistent design. A large crossing angle (20 mrad total) is required [14] to suppress the multi-bunch kink instability created by parasitic collisions away from the main interaction point (IP). This however means that crab-cavities will have to be used to avoid a reduction in luminosity. Although the final-focus design is at a very preliminary stage, an optics has been found (see Fig.8) which looks promising [15]. It consists of horizontal and vertical chromatic correction sections followed by a final transformer. 80% of the ideal luminosity is obtained for a 1% full- width flat energy spread of the beams. The rms spot sizes in both planes are 20-30% larger than expected from the simple calculation using the emittance and the beta function at the IP. Peak beta functions reach 1000 km. The length per side is 3.1 km. The design allocation for the total beam delivery section (final focus plus collimation) is 2 /G755 km.Figure 8: Final focus optics for a 3 TeV collider. The feasibility of maintaining 1 nm beam sizes in collision in the presence of ground movement and component jitter has to be investigated. The use of position feedback systems within the 130 ns pulse are being considered [16]. The consequences of the large mean energy loss (31%) and energy spread (100%) which are produced by the very strong beam-beam forces at the IP in the vertical plane (even when the beams miss each other by 10-20 /G56) must be carefully studied. Extraction of a spent beam with 100% energy spread and with a large beam divergence is a concern and will make bending and focussing without beam loss particularly challenging. The total energy carried by the 6.8×108 coherent pairs at 3 TeV is about 40 Joules, and extracting the particles without producing losses in the detector will be a challenge. Background levels in the detector may also dramatically increase due to their sheer number. 7 TEST FACILITIES The first CLIC Test Facility (CTF1) operated from 1990 to 1995 and demonstrated the feasibility of two-beam power generation. 76 MW of 30 GHz peak power was extracted from a low-energy high-intensity beam and this power was used to generate a gradient in the 30 GHz structure of 94 MV/m for 12 ns. A second test facility (CTF2) [17] is now being operated. The 30 GHz part of this facility is equipped with a few-microns-precision active-alignment system. The 48-bunch 450 nC drive beam train is generated by a laser-driven S-band RF gun with a Cs2Te photo-cathode. The beam is accelerated to 40 MeV by two travelling-wave sections operating at slightly different frequencies to provide beam loading compensation along the train. After bunch compression in a magnetic chicane, the bunch train passes through four power extraction and transfer structures, each of which powers one 30 GHz accelerating section (except the third which powers two) with 16 ns long pulses. The single probe beam0.0 1000. 2000. 3000. 4000. s (m)CLIC 3-TeV Final Focus (-15.0x-50.0) 0.00.10.20.30.40.50.60.70.80.91.0β(m) [*10^6] 0.00.010.020.030.040.050.060.070.080.090.10 D (m)βxβyDxDybunch is generated by an RF gun with a CsI+Ge photo-cathode. It is pre-accelerated to 50 MeV at S- band before being injected into the 30 GHz accelerating linac. The drive beam RF gun has produced a single bunch of 112 nC and a maximum charge of 755 nC in 48 bun ches. The maximum charge transmitted through the 30 GHz modules was 450 nC. A series of cross-checks between drive beam charge, generated RF power, and main beam energy gain have shown excellent agreement. The maximum RF power generated by one 0.5 m structure was 27 MW. The highest average accelerating gradient was 59 MV/m and the energy of an 0.7 nC probe beam has been increased by 55 MeV. Unexpected surface damage was found at these field levels and further studies are needed to find the cause. The tests were made under particularly bad vacuum conditions which for the moment makes vacuum a prime suspect. Extremely high gradients were obtained [18] by powering a 30 GHz single-cell resonant cavity directly by the drive beam. The cavity operated without breakdown at a peak accelerating gradient of 290 MV/m. When pushed further, the cavity started to breakdown at surface-field levels around 500 MV/m. The breakdown manifested itself as a field extinction of the decaying pulse at different times in the pulse. At the end of the test when the cavity was breaking down continuously, surface field levels as high as 750 MV/m were obtained. A new facility (CTF3 – see Fig.9) is under study [19] in collaboration with LAL (France), LNF (Italy) and SLAC (USA), which would test all major parts of the CLIC RF power scheme. To reduce cost, it is based on the use of 3 GHz klystrons and modulators recuperated from the LEP Injector Linac (LIL). Figure 9: CTF3 schematic layout. The drive beam is generated by a thermionic gun and is accelerated by twenty 1.3 m long fully-loaded 3 GHz structures operating at 7 MV/m with an RF-to- beam efficiency of 96%. The power is supplied by ten 30 MW klystrons and compressed by a factor 2.3 to give a peak power at each structure of 69 MW. The beam pulse is 1.4 /G50s long with an average current of3.5 A. The bunches are initially spaced by 20 cm (two 3 GHz buckets) but after two stages of frequency multiplication they have a final spacing of 2 cm. This bunch train, with a maximum charge of 2.3 nC per bunch, is then decelerated by four 0.8 m long transfer structures in the 30 GHz drive beam decelerator from 184 MeV to 125 MeV. Each transfer structure provides 458 MW. The main beam is accelerated from 150 MeV to 510 MeV by eight 30 GHz accelerating structures operating at a gradient of 150 MV/m. 8 CONCLUSION The CLIC Two-Beam scheme is an ideal candidate for extending the energy reach of a future high- luminosity linear collider from 0.5 TeV up to 5 TeV c.m. The high operating frequency (30 GHz) should allow the use of high accelerating gradients (150 MV/m) which shorten the linacs (27.5 km for 3 TeV) and reduce the cost. This level of gradient however has yet to be demonstrated, and the recent unexpected structure damage at much lower field levels is a cause for concern. The effects of the high transverse wakefields have been compensated by a judicious choice of bunch length, charge and focusing strength, such that the emittance blow-up is made independent of the frequency of the accelerating system for equivalent beam trajectory correction techniques. The two-beam RF power source based on a fully-loaded normal-conducting low-frequency linac and frequency multiplication in combiner rings is an efficient, cost- effective and flexible way of producing 30 GHz power. The feasibility of two-beam power production has been demonstrated in the CLIC Test Facilities (CTF1 and CTF2). A third test facility is being studied to demonstrate the newly-proposed drive beam generation and frequency multiplication schemes. REFERENCES [1] J.P. Delahaye, 26 co-authors, PAC’99, New York. [2] G. Loew, SLAC-R-95-471, 1995. [3] J. Ellis, E. Keil, G. Rolandi, CERN-EP/98-03. [4] J.P. Delahaye, 3 co-authors, CERN/PS/97-51. [5] D. Schulte, Proc. EPAC’98, Stockholm, [6] D. Schulte, T. Raubenheimer, PAC’99, New York. [7] I. Wilson and W. Wuensch, this conference. [8] M. Luong, 2 co-authors, PAC’99, New York. [9] I. Wilson , 9 co-authors, EPAC’2000, Vienna. [10] H. Braun, 16 co-authors, CERN Report 99-06. [11] L. Rinolfi, CLIC note 354, 1997. [12] J.P. Potier, L. Rivkin, Proc. PAC’97, Vancouver. [13] E. D’Amico, 2 co-authors, EPAC’98, Stockholm. [14] O. Napoly, CERN-SL-99-054 AP (Aug. 99). [15] F. Zimmermann, 4 co-authors, EPAC’2000. [16] D. Schulte, this conference. [17] R. Bossart, 15 co-authors, CERN/PS 98-60. [18] W. Wuensch, 3 co-authors, EPAC’2000, Vienna. [19] CLIC Study Team, CERN/PS 99-47 LP (July 99).X 5 Combiner Ring 84 mX 2 Delay 42 m Drive Beam Injector Main Beam Injector 150 MeVMain Beam Accelerator 8 Accelerating Structures 30 GHz - 150 MV/m - 0.3 mDrive Beam Decelerator 4 Transfer Structures - 30 GHz 10 Modulators/Klystrons with LIPS (x2.3) 3 GHz - 30 MW - 6.7 /G50s Drive Beam Accelerator ~ 15 m ~ 10 m0.51 GeV125 MeV3.5 A - 2100 b of 2.33 nC 184 MeV - 1.4 /G50s 35 A - 184 MeV 140 nsHigh Power Test Stand20 Accelerating structures 3 GHz 7 MV/m 1.3m
PHASE ROTATION, COOLING AND ACCELERATION OF MUON BEAMS: A COMPARISON OF DIFFERENT APPROACHES G. Franchetti, S. Gilardoni, P. Gruber, K. Hanke, H. Haseroth, E. B. Holzer, D. Küchler, A. M. Lombardi, R. Scrivens, CERN, Geneva, Switzerland Abstract Experimental and theoretical activities are underway at CERN [1] with the aim of examining the feasibility ofa very-high-flux neutrino source ( ≈10 21 neutrinos/year). In the present scheme, a high-power proton beam (some4 MW) bombards a target where pions are produced. Thepions are collected and decay to muons under controlledoptical condition. The muons are cooled and acceleratedto a final energy of 50 GeV before being injected into adecay ring where they decay under well-definedconditions of energy and emittance. We present the most challenging parts of the whole scenario, the muon capture, the ionisation-cooling andthe first stage of the muon acceleration. Differentschemes, their performance and the technical challengesare compared. 1 INTRODUCTION A next generation neutrino source (Neutrino Factory) should have the following advantages with respect totoday's sources: a higher flux (10 21 neutrinos/year vs. the present 1011), a higher energy (50 GeV vs. the present 20 GeV for the CERN Neutrinos to Gran Sasso project)and the possibility to control the flavour of the neutrinobeam. A beam with the aforementioned characteristicscan be obtained by bombarding a target with high-energy protons, collecting the produced pions andallowing them to decay in a controlled environmentbefore accelerating the muons to the required 50 GeV.The muons decaying in a storage ring, whose straightsections are pointed towards the detectors, will produceneutrinos with the desired characteristics. The design ofthe pion collection system and the first stage of the muonacceleration (front-end) are extremely challenging due tothe large emittances involved. 2 FRONT-END DESIGN To meet the requirements of a Neutrino Factory, 1020 to 1022 muons at 50 GeV, within a transverse emittance of 1.5 cm ⋅rad and a longitudinal emittance of 0.06 eV ⋅s must circulate in the decay ring. Internationalcollaborations, for engineering and radio-protectionconsiderations, have standardised on the power on targetof 4 MW. In the CERN reference scheme, a proton driver [2] would be built to deliver 4 MW of 2.2 GeV protons inbursts of 3.3 µs repeated at 75 Hz, i.e. 10 23 protons/year (notice that through this paper an operational year isassumed to consist of 10 7 seconds). From these figures it follows that the system downstream of the target should have a yield of at least0.001 muon/proton for a minimum Neutrino Factory. A typical phase space plot of the pions after production is shown in Fig. 1: the beam radius is 30 cm,the divergence 200 mrad and the kinetic energy rangesfrom 0.05 to 1 GeV. The peak of the production isaround 0.1 GeV kinetic energy, and 75% of the pionshave energies between 0.050 and 0.3 GeV. For the casepresented in Fig. 1, 0.017 π + are produced for each 2.2 GeV proton on a thin (26 mm) mercury target. Thesedata are linearly extrapolated to a 300 mm target. Figure 1: Transverse phase space and energy histogram of a pion beam produced by a 2.2 GeV proton beamimpinging on a 26 mm Hg target immersed in a 60 cmbore 20 T solenoid (FLUKA calculations). The plots areat 4 m from the target. In the upper plot the acceptanceof the decay ring is indicated for reference. Insufficient pions fall within the energy-spread acceptance of any conventional accelerator. A reductionof energy spread by phase rotation can be envisaged inthree possible ways: • Apply it to the pions just after the target : this allows the full benefit of the defined timestructure of the pions, but it implies the use ofhigh-gradient rf cavities in a high radiation area.Further complications arise due to the decay ofpions in an rf field. • Allow the pions to decay (drift of tens of meters) before they enter an rf cavity with moderategradients. • Allow the pions to decay and let the muons drift for some hundreds of meters in order to build up astrong correlation between time and energy andthen match the energy spread with a quasi-dcdevice (e.g. an induction linac). The two latter approaches [3,4] will be described indetail in the next sections. 2.1 The rf approach to phase rotation The parameters are presented in Table1. The pions decay in a 30 m long channel and are focused by a 1.8 T solenoid. At the end of the decaychannel, the beam enters a series of 44 MHz cavities andthe energy spread of the particles with kinetic energy inthe range 100-300 MeV is reduced by a factor two. Afirst cooling stage, employing the same rf cavities and 24cm long H 2 absorbers, reduces the transverse emittance in each plane by a factor 1.4 while keeping the averageenergy constant. After the first cooling stage, the beam isaccelerated to an average energy of 300 MeV. The beamphase extent, as well as the reduced physical dimensionsof the beam, allows the continuation of the cooling with88 MHz cavities. 40 cm long absorbers are used in this section. At the end of cooling, the emittance is reducedby a factor 4 in each transverse plane. The system iscontinued at 88 MHz, and 176 MHz until the energy of 2GeV -suitable for injection in a Recirculating LinacAccelerator (RLA)- is reached. The system works alsowith 40 MHz, 80 MHz, 200 MHz. 2.2 The induction linac (IL) approach The parameters are presented in Table 2. The pions are transported through a tapered solenoid of initial field of 20 T, as favoured by the “Neutrino Factory and Muon Collider Collaboration ” [5]. At 1.46 T the solenoid is continued for a distance of 200 m wherethe pion to muon decay occurs, and the strongmomentum and time correlation is formed. A high performance induction linac is then used to correct the energy spread of the beam. The inductionlinac is formed of 25 cm cells pulsed with a maximumvoltage of 500 kV. The electrostatic field distributionwas calculated using POISSON. A solenoid is placedinside each of the cells, producing an on-axis field of1.46 T, rising to 1.9 T near the outer limits of the beamchamber, for which a large aperture of 60 cm is required.The linac produces an average gradient of 2 MV/m for adistance of 50 m. The correction affects the muons withkinetic energies between 120 and 310 MeV and resultsin a 330 ns long macro bunch. To obtain a bunchedbeam for cooling, a series of 176 MHz cavities areemployed over a distance of 17 m followed by a further 17 m of drift space. The βλ/2 cavities are operated at 2 MV/m with 0.5 m solenoids between them, againproviding an average on-axis field of 1.46 T. A coolingsection follows in which the beam could be reduced intransverse emittance by a factor 3. Table 1: RF solution, main parameters Decay Rotation Cooling I Accel. Cooling II Accel Length [m] 30 30 46 32 112 ≈450 Diameter [cm] 60 60 60 60 30 20 B - f i e l d [ T ] 1 . 81 . 82 . 02 . 02 . 62 . 6 Frequency [MHz] - 44 44 44 88 88-176 G r a d i e n t [ M V / m ] 2224 4 - 1 0 Kin Energy [MeV] 200 280 300 2000 Table 2: Induction Linac solution, main parameters Decay Rotation Bunching Cooling Length [m] 200 50 37 80 Diameter [cm] 60 60 60 68 B-field [T] 1.4 1.4 2.0 3 Frequency [MHz] - I.L. 176 176 Gradient [MV/m] ±2 2 15 Kin Energy [MeV] - 110 110 110 2.3 Particle budget The particle budget for the two solutions studied is shown in Fig. 2. The number of muons delivered to theRLA is comparable for the two schemes and, assumingthe CERN proton driver and target, the number ofmuons/year reaches 10 21 in both. Figure 2: Particle budget along the rf and IL scheme. In Fig. 2 we can identify the bottlenecks and the possible improvements to the designs. In the rf solution, thetransition between phase rotation and the first stage ofcooling should be smoothed in order to limit the losses(longitudinal losses, particles falling outside the bucket):interlacing of the two sections as well as the use ofhigher harmonics will be studied in the future. Besides,the result of rf tests should give a guideline for themaximum allowable gradient. Figure 3: Emittance evolution in the rf solution For the IL solution the improvement is needed in the bunching section before the cooling (50% efficiency atthe moment). In Fig. 3 the emittance evolution along the rf front end can be followed. An emittance increase by a factor 1.5 isgenerated during the decay. No effort has yet beendedicated to limiting this growth. 2.4 Comparison of the two approaches Both the rf and IL approach deliver a sufficient number of neutrinos. The difference between them liesin the feasibility of the main components and on theconstraints imposed on the proton driver. The InductionLinac parameters are very demanding and the limitationin the number of bunches/burst from the proton driver(12) would result in very high space-charge forces in thecompressor ring. Conversely, the rf solution could acceptany number of bunches from the accumulator (up to amax of 140). Preliminary calculations on a 44 MHz anda 88 MHz cavity show that an average power of 0.06 and0.032 MW/cavity would suffice to provide the necessaryfield. The total average power (15 MW for the wholephase rotation and cooling rf system) is alsoconsiderably less than that needed for the IL alone (75MW, scaling from [5]; although this can be reduced to37 MW with a better core material). 3 CONCLUSIONS Two possible schemes for the front-end of a neutrino factory have been explored at CERN. Both of themdeliver the required number of neutrinos. The technicalchallenges, the CERN rf expertise [6] and the bettermatch to the CERN proton driver have led to the choiceof the rf scheme as the CERN reference scenario. REFERENCES [1] H. Haseroth (for the NFWG), “Status of Studies for a Neutrino Factory at CERN ”, and R. Cappi et al., “Design of a 2 GeV Accumulator-Compressor for a Neutrino Factory ”, EPAC2000, Vienna. [2] M. Vretenar, “A High-Intensity H− Linac at CERN Based on LEP-2 cavities ”, these proceedings. [3] A.M. Lombardi, “The 40-80 MHz Scheme ” ,CERN- NUFACT-Note34. [4] R. Scrivens, “Example Beam Dynamic Designs for a Neutrino Capture and Phase Rotation Line Using50m, 100m and 200m Long InductionLinacs ”,CERN-NUFACT-Note14. [5] N. Holtkamp, D. Finley (Eds.), “A Feasibility Study of a Neutrino Source Based on a Muon StorageRing ”, FERMILAB pub-00/08-E, (2000). [6] R. Garoby, D. Grier, E. Jensen, CERN. A. Mitra, R.L. Poirier, TRIUMF, “The PS 40 MHz Bunching Cavity ”, PAC ’97, Vancouver, 1997.
arXiv:physics/0008066 15 Aug 2000Superconducting H-mode Structures for Medium Energy Beams R. Eichhorn and U. Ratzinger GSI, Planckstr. 1, 64291 Darmstadt, Germany and IAP, Goethe Universität Frankfurt, Robert-Mayer-Str. 2-4, 60325 Frankfurt, Germany Abstract Room temperature IH-type drift tube structures are used at different places now for the acceleration o f ions with mass over charge ratios up to 65 and velocitie s between 0.016 c and 0.1 c. These structures have a high shunt impedance and allow the acceleration of very intense beams at high accelerating gradients. The overall power consumption of room temperature IH- mode structures is comparable with superconducting (sc) structures up to 2 MeV/u. With the KONUS [1] beam dynamics, the required transversal focusing elements, e.g. quadrupole triplets can be placed ou tside of multicell cavities, which is favourable for buil ding sc H-mode cavities. The design principles and consequences to the geometry compared to room temperature (rt) cavities will be described. The re sults gained from numerical simulations show that a sc multi-gap H21(0)-mode cavity (CH-type) can be an alternative to the sc spoke-type or reentrant cavit y structures up to beam energies around 150 MeV/u. Th e main cavity parameters and possible fabrication opt ions will be discussed. 1 INTRODUCTION Linacs based on rt H-mode cavities (RFQ and drift tube structures) are used today in the velocity ran ge from β=0.002 up to β=0.1. RF power tests show the capability of IH-cavities to stand about 25 MV/m on - axis field. Beside these high accelerating gradient s H- mode cavities allow the acceleration of intense bea ms [2]. One aspect of the investigations started at GS I and IAP is to extend the velocity range of the H-mode cavities up to β=0.5 by using the H21(0) or CH-mode. Many future projects (the Accelerator Driven Transmutation Project ADTP[3], the European Spallation Source ESS[4] or the Heavy Ion Inertial Fusion HIIF [5]) are based on the availability of efficient accelerating cavities with properties lik e mentioned above, which additionally could be operat ed in cw mode. It is commonly accepted that above an energy of 200 MeV/u superconducting cavities are superior to rt structures. By combining the advanta ges of CH-mode cavities with the benefits of superconductivity, effective ion acceleration at hi gh duty cycle will be possible. For high current proto n beams the injection energy will be around 10 MeV, while for heavy ions the injection energy may becom eas low as 1 MeV/u. The CH-structure is efficient fo r beam energies up to 150 MeV/u. This paper describes the properties of H-mode cavities and especially the CH-type. Important is t he application of the KONUS beam dynamics [1], resulting in long, lens free accelerating sections housed in individual cavities. This opens the superconduct ing option, as the magnetic field of cavity internal quadrupoles cannot be easily shielded well enough t o avoid frozen current contributions. The results from numerical simulations of three H21(0)-type cavities with different resonant frequencies and velocity profiles will be reported and first mechanical construction layouts will be presented. 2 H-MODE CAVITIES The IH-DTL (Interdigital H-mode or H11(0)) has become a standard solution for heavy ion accelerati on. The CERN Pb-injector installed in 1994 [6] is one CH-DTL H21(0)4-vane RFQ H210 100 - 400 MHz β < 0.12 250 - 800 MHz β < 0.5 Figure 1: The CH-mode structure family: The main direction of the magnetic RF field is oriented para llel and anti parallel with respect to the beam axis.example. The new high current injector at GSI [2,7] consists of the first IH-DTL designed for very heav y ions with A/q = 65 including space charge effects o f considerable strength at the design current of I/em A = 0.25·A/q. An important property of H-type DTLs is their high acceleration efficiency, i.e. they provi de a high shunt impedance, especially at low β-values up to 0.2. The H210-mode is already used in accelerator physics: the 4 vane-RFQ is well established for proton and l ight ion acceleration [8]. A competitive H-mode drift tu be cavity in the velocity range from β = 0.05 to 0.5 may become the H21(0)-mode CH structure, which can be deduced from the 4-vane RFQ by cutting down the vanes around the aperture and replacing the electro des by drift tubes. These drift tubes are connected by two stems with the girders of identical RF potential (s ee fig. 1). The dipole mode that usually causes troubl e during the tuning procedure of the 4-vane RFQ is sh ort- circuited by the stems along the whole cavity. The analytically estimated shunt impedance assuming sli m drift tubes without quadrupoles is about a factor 1 .4 higher compared to the IH-cavity. Numerical simulations have shown that CH-cavities for resonan ce frequencies up to around 800 MHz can be realised. T his allows to close the velocity gap between RFQs and Coupled Cavity Linacs (CCL). 3 DESIGN OF A SUPERCONDUCTING CH-CAVITY The CH-cavity exceeds by far the mechanical rigidity of IH-tanks. This opens the possibility to develop superconducting multi-cell cavities [9]. So far only 2- and 3-cell sc structures were realised for low beam velocities. The different prototype cavities discussed are modules of an accelerator design investigated for h igh current proton acceleration. The parameters of thes e prototype cavities are given in tab.1. The RF behavior of the resonators was studied with an analytical model [10] that allows a first optimi zation step of the fundamental cavity parameters. The consequent numerical simulations of the resonators were done using the MAFIA package [11]. All parameters predicted with the analytical model were confirmed by MAFIA within ±10 % deviation. As a first step of the design process, only two accelerating gaps have been computed. Starting with a rt design the drift tubes and the shape of the stem s were optimized. Special care was taken on the magnetic surface field trying to keep it well below the BCS- Limit of Niobium. By increasing the stem cross section, t he magnetic surface field could be reduced by aprox. a factor of two, reaching up to 30 mT in the actual design.Table 1: Parameters and expected performance of the prototype cavities frequency (MHz) 352 433 700 particles protons injection energy (MeV) 10.9 10.9 130 eff. voltage gain (MV/m) 6.7 particle velocity (v/c) 0.17 0.17 0.5 mode H21(0) (CH) gap number 18 18 10 length of the cavity (m) 1.43 1.01 1.07 drift tube aperture (mm) 25 25 10 transit time factor 0.8-0.85 tank radius (mm) 185 130 105 stored energy (J) 12.6 4.1 4.3 Emax/Eacc 4.1 3.9 5.2 Bmax/Eacc (mT/(MV/m)) 5.8 3.7 8.8 R/Q0 (kΩ/m) 2.7 4.7 2.6 geometric factor ( Ω) 163 160 208 Q-factor (4K,Nb) 4.3·109 2.7·109 1.4·109 diss. power (4K,Nb) (W) 6.5 4.0 13.8 shunt impedance (M Ω/m) 44 68 39 Q-factor (rt, copper) 16400 14600 14800 This optimization process caused an increase in the capacitive load and thus lowered the shunt impedanc e by 20 %, which is no drawback in case of a sc cavit y. The maximum electric field (27.5 MV/m) has been found at the drift tube ends. After this optimization step the whole cavity was computed to study the impacts on the field flatness . As the magnetic flux bends from one to the neighboring sector at its ends (see also fig. 1) one expects hi gh magnetic surface fields in this region. The girder undercuts, used in rt IH-mode and 4-vane cavities successfully to create the zero mode are not the ri ght choice for a sc cavity. Therefore a careful redesi gn had to be performed. The lowest surface fields were yie lded by combining two modifications: The tank radius is increased by 17 % between the end flange and the fi rst drift tube stem. To assure the field flatness, i.e. a constant accelerating field along the whole cavity, the local capacity at the cavity ends had to be further increased. This was attained by forming thicker ste ms. As a measure to reduce the surface currents at the cavity end flanges, which may be sealed by an indiu m joint, half drift tubes were introduced. Figure 2 shows the layout of the cavity. Table 1 summarizes the results of the simulations. Comparin g these parameters to that of existing cavities [12] displays the potential of CH-cavities. 4 FABRICATION OPTIONS Up to now only two different options in fabricating the cavity are considered:- Like it is done with the normal conducting IH- mode resonators, one fabricates the drift tubes and the stems out of bulk copper mounted inside a copper plated steel tank. The deposition of the superconducting surface layer is done by sputtering with niobium. Though the behavior of thin niobium films is well known and meet the requirements, it might need much effort for sputtering this complex geometry with the needed precision. The progress in this technology made elsewhere [13] shows however the great potential of this option. - On the other hand one can think of building the cavity out of bulk niobium components. In that case there exists a lot of experience in electron beam welding. The first superconducting RFQ, which has currently past the cold low level tests successfully, has been built this way [14]. The third possibility, namely leadplating a copper resonator has been abandoned because of the limited performance of cavities build this way, as for exam ple in Legnaro [15]. Even though the magnetic surface f lux of our resonator is below the BCS-limit for lead, electric surface fields of up to 28 MV/m seems to b e beyond the limit of that technology. 5 OUTLOOK The normal conducting H-mode cavities showed their suitability for ion acceleration at low energ ies in many accelerator laboratories. Our investigations indicate that CH-mode cavities are well suited to d esignsuperconducting resonators. The results of the numerical simulation of such a cavity are very promising. Using state of the art technology the fabrication of a superconducting CH-mode cavity should be possible [13,14]. The design, constructio n and rf test of an 352 MHz prototype cavity are scheduled at IAP, Frankfurt University. REFERENCES [1] U. Ratzinger, Nucl. Intr. Meth., A 415 (1998) 22 9. [2] U. Ratzinger, Proc. Int. Linac. Conf., Geneva, CERN 96-07 (1996) 288. [3] R.A. Jameson et al., Proc. Europ. Part. Acc. Con f., Berlin, (1992) 230. [4] ESS-CDR, eds.: I. Gardener, H. Lengeler and G. Rees, ESS-Report 95-30-M (1995). [5] U. Ratzinger, Nucl. Intr. Meth., in press. [6] H.D. Haseroth, Proc. Int. Linac Conf., Geneva, CERN 96-07 (1996) 283. [7] W. Barth et al., this conference [8] K.R. Crandall et al, Proc. Int. Linac Conf., Montauk, BNL51134 (1979) 205. [9] R. Eichhorn and U. Ratzinger, Proc. 9th RF Supercond. Workshop, Santa Fe (1999) in press. [10]U. Ratzinger, CAS 2000 Seeheim (2000) submitted [11]T. Weiland, MAFIA - A Three Dimensional Electromagnetic CAD System for Magnets, RF Structures and Transient Wake Field Calculations (1998). [12]K. Shepard, Nuc. Instr. Meth. A 382 (1996) 125. [13]V. Palmieri et al., LNL-INFN (REP) 149/99. [14]G. Bisoffi et. al., Proc. Europ. Part. Acc. Con f., Vienna ,(2000) in press. [15]A.-M. Porcelatto et al., Proc. 9th RF Supercond. Workshop, Santa Fe (1999) in press. Emax = 27.5 MV/m Bmax = 38 mTBstem = 30 mTBflange = 4 mT Bgirder = 20 mT Increased Tank RadiusThicker StemsHalf Drift Tubes Figure 2: Three dimensional view of a 352 MHz CH mo de cavity for β = 0.17. The cavity dimensions can be found in tab. 1. Also shown are some field values calculated with MAFIA, especially the maximum values of the m agnetic (38 mT at both resonator ends) and of the electric field (27.5 MV/m on the drift tube surface).
AN INJECTOR FOR THE CLIC TEST FACILITY (CTF3) H. Braun, R. Pittin, L. Rinolfi, F. Zhou, CERN, Geneva, Switzerland B. Mouton, LAL, Orsay, France R. Miller, D. Yeremian, SLAC, Stanford, CA94309, USA Abstract The CLIC Test Facility (CTF3) is an intermediate step to demonstrate the technical feasibility of the keyconcepts of the new RF power source for CLIC. CTF3will use electron beams with an energy range adjustablefrom 170 MeV (3.5 A) to 380 MeV (with low current).The injector is based on a thermionic gun followed by aclassical bunching system embedded in a long solenoidalfield. As an alternative, an RF photo-injector is also beingstudied. The beam dynamics studies on how to reach thestringent beam parameters at the exit of the injector arepresented. Simulations performed with the EGUN codeshowed that a current of 7 A can be obtained with anemittance less than 10 mm.mrad at the gun exit.PARMELA results are presented and compared to therequested beam performance at the injector exit. Sub-Harmonic Bunchers (SHB) are foreseen, to switch thephase of the bunch trains by 180 degrees from even toodd RF buckets. Specific issues of the thermionic gun andof the SHB with fast phase switch are discussed. 1 INTRODUCTION The CTF3 construction [1] will start in 2001. It is foreseen in 3 stages and the injector will also have 3stages [2] of development. The first one is called"Preliminary stage". The injector is the same as thepresent one used for LIL [3] except for the thermionicgun. The second one is called "Initial stage". The injectorhas all new components (gun, bunching, focusing, andmatching systems) and should deliver the nominal currentwith all RF buckets filled. The third one is called"Nominal stage". The injector is identical to the previousone except that SHBs will be added. Even and odd trainsare generated with every other bucket filled. The conceptsof fully-loaded linac and SHB with fast phase switch areexplained in [1]. However the process produces unwantedsatellite bunches. The Preliminary stage is beingimplemented [4], so this paper focuses on the Nominalstage. 2 BEAM PARAMETERS 2.1 Target parameters at injector exit The Preliminary stage is based on the existing LIL injector and therefore the beam parameters are based onexperimental measurements. For the Nominal stage, the charge in the satellite bunches should be as small aspossible in order to get maximum RF generationefficiency and minimum radiation problems. The bunchlength should be as short as possible to get both goodbeam stability (wake fields) in the following Drive Beamaccelerator, and flexibility for the bunch compressor. Theemittance should be minimised (linac acceptance,funnelling process in the combiner ring, deceleratorlinac). The single-bunch incoherent energy spread shouldbe small in order to minimise bunch lengthening afterbunch compression. The energy spread over the pulse,including beam-loading effects, should also be minimum.This will reduce the phase error after the bunchcompression and therefore increase the 30 GHzgeneration efficiency. Table 1 summarises the beamparameters requested at the injector exit for the 3 stages. Table 1: Target parameters at injector exit Parameters Unit Pre. Init. Nom. Beam energy MeV 4 /G74 20 /G74 20 Beam pulse /G50s 2.53 1.54 1.54 RF pulse /G50s /G74 3.8 /G74 1.6 /G74 1.6 Beam current A 0.3 3.5 3.5 Gun current A 1 7 7 Charge/bunch nC 0.1 1.17 2.33 Bunch spacing m 0.1 0.1 0.2 Bunches/pulse 100 4200 2100 Charge/pulse nC 10 4893 4893 Charge/satellite % - - /G64 5 Bunch length ps- fwhh 7 /G64 12 /G64 12 Bunch length mm-rms 0.9 /G64 1.5 /G64 1.5 Normalised rms emittances mm.mrd 50 /G64100 /G64100 Energy spread (Single bunch) MeV /G64 0.5 /G64 0.5 /G64 0.5 Energy spread (on flat-top) MeV - /G64 1 /G64 1 Charge variation bunch-to-bunch % /G64 20 /G64 2 /G64 2 Charge flatness (on flat-top) % - /G64 0.1 /G64 0.1 Beam rep. rate Hz 50 5 5 RF rep. rate Hz 100 30 30 /G7C20 MeV 140 keVCTF 3 Injector Nominal stage Quadrupole tripletGun SHB 10.4 MW 1.5 GHz 2 x 9 MeVQuadrupole tripletBeam matching, beam diagnostics and collimation B1 S1 S2PB1 SNLSNLA40 MW 3 GHz40 MW 3 GHz AA SHB 3SHB 2 Figure 1: Layout of CTF3 injector. 2.2 Layout Figure 1 shows the layout for the Nominal stage. After the thermionic gun (140 kV), there are 3 sub-harmonicbunchers SHB (1.5 GHz), one pre-buncher PB1 (3 GHz),one buncher B1 (3 GHz, 6-cells TW) and 2 acceleratingstructures S1/S2 (3 GHz, 32-cells TW). All componentsbetween the gun and the injector exit are embedded in asolenoidal field (SNL). A matching section with beamcollimation and beam diagnostics is located between theinjector and the Drive Beam accelerator. 3 EGUN SIMULATIONS For the Preliminary stage, the new thermionic gun provides 2 A at 90 kV. A classical Pierce gridded gun,called CLIO [5], is proposed. It has a thermoelectronicdispenser cathode with an emitting surface of 0.5 cm 2, grid-cathode spacing of 0.15 mm, cathode-anode distanceof 24 mm and anode hole diameter of 8 mm. Theelectrode geometry was modelled using the EGUN code[6]. The beam radius does not exceed 10 mm between theanode and 125 mm downstream of the anode, where acapacitive electrode allows beam current measurements.The normalised emittance is 7 mm.mrad, 62 mmdownstream of the cathode. For the Initial and Nominalstages, the thermionic gun should provide 7A at 140 kV,with 150 kV voltage capability. A thermionic gun of theSLAC-type is proposed. It has a dispenser cathode withan emitting surface of 2 cm 2, cathode-anode distance of 45 mm and an electrode angle 45o. Figure 2 shows the field lines and electron trajectories. Simulations in aspace-charge-limited regime with thermal effect(1223 oK) give a normalised emittance of 9.5 mm.mrad at z = 120 mm for a beam current of 7.4 A (perveance0.128 /G50P). However, PARMELA simulations start at the anode exit. At this place, the normalised emittance is7 mm.mrad. The maximum electric field on the contour is less than 10 MV/m. Figure 2: EGUN results for the CTF3 gun. 4 PARMELA SIMULATIONS 4.1 Longitudinal beam dynamics Extensive simulations [7,8] are performed with PARMELA code at CERN, LAL and SLAC. They startfrom the gun exit with an initial normalised emittance of7 mm.mrad, and 140 keV kinetic energy of the referenceparticle. The total number of particles is 6000 over arange of 6 S-band cycles. One of the issues is to make thesatellite charge (in a 20 o window) less than 5% of the main bunch (in a 20o window). Results are based on beam dynamics simulations assuming 400 kW RF power for 3SHBs. Figure 3 shows the phase spaces obtained at theinjector exit. The bunch length at the end of the injector isclose to 10 ps (fwhh) and about 82% of the particles arecaptured in 20 o. Studies are going on based on a cluster of 3 sub-harmonic bunchers with 3 different frequencies:1.5 GHz, 3 GHz, 4.5 GHz. Preliminary results, notdiscussed here, with a bunch length of 10 ps and anormalised rms emittance of 40 mm.mrad give 3% chargein the satellites. Figure 3: PARMELA results. Left two figures: Particle number versus phase (degrees).Right top: Beam size: y (cm) versus x (cm).Right bottom: Energy spread (keV) versus particlenumber. 4.2 Transverse beam dynamics To simulate the transverse beam dynamics correctly, different mesh sizes are chosen for different regions.Emittance variations are studied as a function of thestrength and shape of the solenoidal magnetic field. Anoptimum is found for a value of 0.1 T along the 2accelerating structures, with a maximum value of 0.2 T inthe buncher. Table 2 compares the simulation results withthe goal. Table 2: Comparison between simulations and requirements Parameters Simul. Goal Energy (MeV) 20 /G74 20 Satellite charge (%) 5 /G64 5 Bunch length (fwhh, ps) 10 /G64 12 Bunch length (fw, ps) 20 --- Energy spread (fwhh, MeV) 0.25 0.5 Charge/bunch (nC) 2.51 2.33 RF power for 3 SHBs (kW) 400 Minim. Normalised rms emittance (mm.mrad) (B= 0.1 T)33 /G64 100 5 GUN AND SUB-HARMONIC ISSUES For the Preliminary stage the CLIO gun will replace the present one in the LIL tunnel. For the followingstages (150 kV, 7A) voltage and current stability of /G64 0.1% are requested on the flat-top. To obtain such performances, high voltage capacitors will be installed ona modified SLAC-type gun. Under these conditions thestored energy will be 1 kJ and a gun protection systemhas to be designed. Concerning the SHBs 1.5 GHz, the PARMELA optimisation gives a gradient of 0.4 MV/m over a gap of4 cm. This will require a voltage of 16 kV. Based onHFSS simulation [9], a power of 124 kW is needed at theinput of each SHB. For 3 SHBs a total power of 372 kWwill be necessary. A fast phase switch of the order of 3 to4 ns is envisaged for the SHBs. It could be difficult tobuild a SHB with very low Q (~10) and to provide a400 kW power supply at 1.5 GHz with large bandwidth. 6 CONCLUSION The Preliminary stage of the CTF3 injector is being implemented. For the Nominal stage, a configuration ofthe injector has been found which fulfills therequirements of CTF3. The Initial stage should be easy toimplement since it is a simple version of the Nominalstage. However, several steps will still be needed in orderto improve the beam performance at the injector exitbefore starting the design of the RF cavities. ACKNOWLEDGEMENTS The authors would like to thank I. Syratchev and R. Corsini for many improvements in the design.Stimulating discussions with J. Le Duff, G. Bienvenu, J.Gao, Y. Thiery (LAL) and R. Ruth (SLAC) are alsoacknowledged. REFERENCES [1] CLIC study team, "Proposal for future CLIC studies and a new CLIC Test Facility (CTF3)", CLIC Note402, July 1999. [2] L. Rinolfi (Ed.), “Proceedings of the fourth CTF3 collaboration meeting”, CLIC Note 433, May 2000. [3] A. Pisent, L. Rinolfi, "A new bunching system for the LEP Injector Linac (LIL)", CERN PS 90-58 (LP),July 1990. [4] R. Corsini, A. Ferrari, J.P. Potier, L. Rinolfi, T. Risselada, P. Royer, "A low charge demonstration ofelectron pulse compression for the CLIC RF powersource", this conference. [5] JC Bourdon et al, "Commissioning the CLIO injection system", SERA 91-23, LAL/LURE, Orsay, January1991. [6] W. Hermannsfeldt, "EGUN an electron optics and gun design ", SLAC, Report 331, October 1988. [7] F. Zhou, H. Braun, "Optimisation of the CTF3 injector with 2 SHBs and 3SHBs", CTF3 Note 2000-01, February 2000. [8] Y. Thiery, J. Gao, J. LeDuff, "Design studies for a high current bunching system", SERA 2000-130,LAL, Orsay, May 2000. [9] I. Syratchev, private communication.
arXiv:physics/0008068 15 Aug 2000PULSED NEUTRON SOURCE USING 100-MEV ELECTRON LINAC AT POHANG ACCELERATOR LABORATORY* G. N. Kim#, H. S. Kang, Y. S. Lee, M. H. Cho, I. S. Ko, and W. Namkung Pohang Accelerator Laboratory, POSTECH, Pohang 790-784, Korea Abstract The Pohang Accelerator Laboratory operates an electron linac for a pulsed neutron source as one of the long-term nuclear R&D programs at the Korea Atomic Energy Research Institute. The designed beam parameters are as follows; The nominal beam energy is 100 MeV, the maximum beam power is 10 kW, and the beam current is varied from 300 mA to 5A depends on the pulse repetition rate. The linac has two operating modes: one for the short pulse mode with the repetition rates of 2 - 100 ns and the other for the long pulse mode with the 1 µs repetition rate. We tested an electron linac based on the existing equipment such as a SLAC-5045 klystron, two constant gradient accelerating sections, and a thermionic RF-gun. We investige the characteristics of the linac, and we report the status of the pulsed neutron source facility including a target system and the time-of- flight paths. 1 INTRODUCTION The nuclear data project as one of the nation-wide nuclear R&D programs was launched by the KAERI in 1996 [1]. Its main goals are to establish a nuclear data system, to construct the infrastructure for the nuclear data productions and evaluations, and to develop a highly reliable nuclear data system. In order to build the infrastructure for the nuclear data production, KAERI is to build an intense pulsed neutron source by utilizing accelerator facilities, technologies, and manpower at the Pohang Accelerator Laboratory (PAL). The PAL proposed the Pohang Neutron Facility (PNF), which consists of a 100-MeV electron linac, a water-cooled Ta target, and at least three different time-of-flight (TOF) paths [2]. We designed a 100-MeV electron linac [3] and constructed an electron linac based on experiences obtained from construction and operation of the 2-GeV linac at PAL. In this paper, we describe the characteristics of the electron linac, and then we present the status of the Pohang Neutron Facility (PNF). ___________________ *Work supported in part by POSCO, MOST, and KAERI # Email: gnkim@postech.ac.kr ; Joint appointment at the Institute of High Energy Science, Kyungpook National Univ., Taegu , Korea.2 CHARACTERISTICS OF E-LINAC 2.1 Construction of electron linac We constructed an electron linac for the various R&D activities of the neutron facility by utilizing the existing components and infrastructures at PAL on December 1997 [4]. The design beam parameters of the electron linac are as follows; The beam energy is 100 MeV with the energy spread less than 1%, the peak current is 20A, the beam pulse width is 6-µs, and the rms normalized emittance is less than 30 πmm-mrad. The electron linac consists of a thermionic RF-gun, an alpha magnet, four quadrupole magnets, two SLAC-type accelerating structure, a quadrupole triplet, and a beam-analyzing magnet. A 2-m long drift space is added between the first and second accelerating structures to insert an energy compensation magnet or a beam transport magnet for the future FEL research. The RF-gun is a one-cell cavity with a tungsten dispenser cathode of 6 mm diameter. The RF-gun produced an electron beam with an average current of 300 mA, a pulse length of 6-µs, and aproximate energy of 1 MeV [5]. The measured rms emittance for the beam energy of 1 MeV was 2.1 πmm- mrad. The alpha magnet is used to match the longitudinal acceptance from the RF- gun to the first acceleration structure. Electrons move along a α-shaped trajectory in the alpha magnet, and the bending angle is 278.6o. The higher energy electron has a longer path length than the lower energy electron, thus the length of electron beams is not lengthened or is shortened in the beam transport line from the RF-gun to the first accelerating structure. Four quadrupole magnets are used to focus the electron beams in the beam transport line from the thermionic RF- gun to the first accelerating structure. The quardupole triplet installed between the first and the second accelerating structure is used to focus the electron beam during the transport to the experimental beam line at the end of linac. There are three beam current transformers (BCT) and three beam profile monitors for beam instrumentation. The BCT is the toroidal shape of ferrite core 25 turns-wound by 0 .3 mm diameter enameled wire. The beam-analyzing magnet has a bending angle of 30 degrees and zero pole-face rotation. The main components of RF system consist of a SLAC 5045 klystron and an 80-MW modulator, and RF waveguide components, etc. Two branched waveguides through a 3 dB power divider from the main klystron waveguide are connected accelerating structures. A highpower phase-shifter is inserted in the waveguide line of the second accelerating structure. The branched waveguide with a 10 dB power divider from the main waveguide is connected to the RF-gun cavity through a high-power phase-shifter/attenuator and a high-power circulator. The circulator is pressurized with dry nitrogen at 20 psig. Two waveguide windows isolate the circulator from the evacuated waveguide line. 2.2 Beam Acceleration After the RF-conditioning of the accelerating structures and the wave-guide network, we tested the beam acceleration. The maximum RF power from a SLAC 5045 klystron was up to 45 MW. The RF power fed to the RF-gun was 3 MW. The maximum energy is 75 MeV, and the measured beam currents at the entrance of the first accelerating structure and at the end of linac are 100 mA and 40 mA, respectively. The length of electron beam pulses is 1.8 µs, and the pulse repetition rate is 12 Hz. The measured energy spread is /GE41% at its minimum. The energy spread was reduced by adjustment of the RF phase for the RF-gun and by optimization of the magnetic field for the alpha magnet. 3 STATUS OF NEUTRON FACILITY The Pohang Neutron Facility consists of a 100-MeV electron linac, a photo-neutron target, and at least three different time-of-flight (TOF) paths. 3.1 Photo-neutron Target High-energy electrons injected in the target produce gamma rays via bremsstrahlung, these gamma rays then generate neutrons via photonuclear reactions. We are considering tantalum rather than fissile material because the technology for handling and the target characteristics are well known [6]. The neutron yield depends sensitively on the materials and the target geometry. The target system is desgined using the MC simulation codes, EGS4 and MCNP4. The target system, 4.9-cm in diameter and 7.4-cm in length, is composed of ten sheets of Ta plates, and there is 0.15-cm water gap between them, in order to cool the target effectively [7]. The estimated flow rate of the cooling water is about 5 liters per minute in order to maintain below 45 oC. The target housing is made of Titanium. The conversion ratio obtained from the MCNP4 code from a 100-MeV electron to neutrons is 0.032. The neutron yield per kW beam power at the target is 2.0×1012 n/sec, which is about 2.5% lower than the calculated value based on the Swanson’s formula [8]. Based on this study, we constructed a water-cooled Ta- target system. 3.2 Time-of-Flight Path The pulsed neutron facility based on the electron linac is a useful tool for high-resolution measurements ofmicro-scopic neutron cross-sections with the TOF method. In the TOF method, the energy resolution of neutrons depends on the TOF path length. Since we have to utilize the space and the infrastructures in the laboratory, the TOF paths and experimental halls are placed perpendicular to the electron linac. We constructed a 15 m long TOF path perpendicular to the electron linac. The TOF tubes were made by 15 and 20 cm diameter of the stainless pipes. 3.3 Neutron Production Experiment The neutron production facility PNF was operated from January 2000. We measured the neutron TOF spectra for several samples with the pulsed neutron beam. The experimental arrangement for the neutron TOF spectrum measurement is shown in Fig. 1. The target is located in the position where the electron beam hits the center of the photoneutron target, which is also aligned with the center- line of the TOF tube. The target was installed at the center of a water modulator, 50 cm in diameter and 30 cm long aluminum cylindrical shape, to moderate the fast neutrons. A lead block, 20cm x 20cm x 10 cm, was inserted in front of the TOF tube to reduce the gamma-flash generated by the electron burst from the target. The sample was placed at the midpoint of the flight path. There is 1.8 m thick concrete between the target room and the detector room. For the neutron TOF spectrum measurement, a 6Li glass scinitillator BC702, 12.5 cm in diameter and 1.5 cm in thickness was mounted on an EMI-93090 photomultiplier, and it is used as a neutron detector. In order to monitor the neutron intensity during the measurement, a BF3 proportional counter, 1.6 cm in diameter and 5.8 cm long was placed inside the target room at a distance of about 6Fig. 1. Experimental arrangement for TOF spectra measurement m from the target. The BF3 counter was inserted in a 30.5 cm diameter polyethylene sphere and surrounded by 5 cm thick borated polyethylene plates and Pb bricks to shield thermal neutrons and gamma flash. During the experiment, the electron linac was operated with a repetition rate of 12 Hz, a pulse width of 1.5 µs, a peak current of 30 mA, and electron energy of 60 MeV. The total operated period is about two weeks from March 2000. During this period, we spend most of time to check the radiation level around the facility and to reduce the electronic noises originated from the RF source. The block diagram of the data acquisition system is also shown in Fig. 1. The TOF signal from the 6Li- ZnS(Ag) scintillator was connected through an ORTEC- 113 pre-amplifier to an ORTEC-571 amplifier, the amplifier output was fed into a discriminator input and used a stop signal of a 150 MHz time digitizer (Turbo MCS). The lower threshold level of the discriminator was set to 30 mV. The Turbo MCS was operated as a 16,384-channel time analyzer. The channel width of the time analyzer was set to 0.5 µs. The 12 Hz trigger signal for a modulator of an electron linac was connected to an ORTEC-550 single channel analyzer (SCA), the output signal was used as a start signal of a Turbo MCS. The Turbo MCS is connected to a personal computer. The data were collected, stored and analyzed on this computer. In order to determine the flight path distance for our facility, we used neutron TOF spectra for Sm, Ta, W, and Ag sample runs. A Cd filter of 0.5 mm in thickness was used to suppress thermal neutrons. The samples were placed at the midpoint of the flight path. The resonance energy E and the channel number I in Table 2 are used to find the flight path length L in the following equation by the method of least squares fitting. tEtLI Δ+ ×Δ×=τ3.72 In the above equation, Δt is the channel width of the time digitizer and set to 0.5 µs. The delay time τ is the time difference between the start signal from the RF trigger and the real zero time. The flight path length L is determined from the fitting. As shown in Fig. 2, the results of the fit are: L=10.81±0.02 m and τ=0.87 µs. 4. SUMMARY The nuclear data project was launched by KAERI from 1996 in order to support nuclear R&D programs, medical and industrial applications. We have constructed and tested an electron linac for the pulsed neutron facility by utilizing the existing components and infrastructures at PAL. The characteristics of accelerated electron beams are about 75 MeV of energy, 12 Hz of repetition rate, 1.8µs of pulse width and about 40 mA of peak current. We made a 15-m TOF path perpendicular to the linac inorder to test a Ta-target system and a data acquisition system. We tested the neutron production and measured the neutron TOF spectra. Using the resonance energy of TOF spectra for various samples, we measured the TOF path length of the neutron facility. REFERENCES [1] Ministry of Science and Technology of Korea (1999), “1997 Request for Project on Long-Term Nuclear R&D Program.” [2] G. N. Kim, J. Y. Choi, M. H. Cho, I. S. Ko, W. Namkung and J. H. Chang, “Proposed neutron facility using 100-MeV electron linac at Pohang Accelerator Laboratory,” Proc. Nuclear Data for Science and Technology, Trieste, May 19- 24, 1997, p. 556, Italian Physical Society, Bologna, Italy (1997). [3] J. Y. Choi, H. S. Kang, G. N. Kim, M. H. Cho, I. S. Ko and W. Namkung, “Design of 100-MeV Electron Linac for Neutron Beam Facility at PAL,” Proc. Of 1997 KAPRA Workshop (Seoul, Korea, Jun. 26-27, 1997), 88 (1997). [4] H. S. Kang, J. Y. Choi, Y. J. Park, S. H. Kim, G. N. Kim, M. H. Cho, and I. S. Ko, “Beam Acceleration Result of Test Linac,” Proc. 1st Asian Particle Accelerator Conf. (Tsukuba, Japan, Mar. 23-27, 1998) 743 (1998). [5] H. S. Kang, G. N. Kim, M. H. Cho, W. Namkung, and K. H. Chung, IEEE Trans. Nucl. Sci. 44, 1639 (1997). [6] J. M. Salome and R. Cools, Nucl. Instr. Meth. 179, 13 (1981). [7] W. Y. Baek, G. N. Kim, M. H. Cho, I. S. Ko and W. Namkung, “Design of the Photoneutron Target for the Pulsed Neutron Source at PAL,” Proc. Workshop on Nuclear Data Production and Evaluation (Pohang, Korea, Aug. 7-8, 1998). [8] W. P. Swanson, “Radiological Safety Aspects of the operation of Electron Linear Accelerators,” IAEA Tech. Rep. 188 (1979). Fig. 2 A fit of the flight path length to resonance energies
Collector Failures on 350 MHz, 1.2 MW CW Klystrons at the Low Energy Demonstration Accelerator (LEDA) D. Rees, W. Roybal, J. Bradley III, LANL, Los Alamos, NM 87545, USA Abstract We are currently operating the front end of the accelerator production of tritium (APT) accelerator, a 7 MeV radio frequency quadrapole (RFQ) using three, 1.2 MW CW klystrons. These klystrons are required and designed to dissipate the full beam power in the collector. The klystrons have less than 1500 operational hours. One collector has failed and all collectors are damaged. This paper will discuss the damage and the difficulties in diagnosing the cause. The collector did not critically fail. Tube operation was still possible and the klystron operated up to 70% of full beam power with excellent vacuum. The indication that finally led us to the collector failure was variable emission. This information will be discussed. A hydrophonic system was implemented to diagnose collector heating. The collectors are designed to allow for mixed-phase cooling and with the hydrophonic test equipment we are able to observe: normal, single- phase cooling, mixed-phase cooling, and a hard boil. These data will be presented. The worst case beam profile from a collector heating standpoint is presented. The paper will also discuss the steps taken to halt the collector damage on the remaining 350 MHz klystrons and design changes that are being implemented to correct the problem. 1 INTRODUCTION The APT 350 MHz, 1.2 MW, CW klystrons operating on LEDA are similar in design to the English Electric Valve (EEV) 352 MHz klystrons at CERN, APS, and ESRF with one difference. The APT klystrons were designed to allow for the steady-state dissipation of the full DC beam power (95 kV, 20 A, 1.9 MW) in the collector. This requirement was intended to mitigate AC grid transients which could result from loss of accelerator beam since the APT accelerator is heavily beam loaded or from interlock-induced, short-term, accelerator outages. Each klystron was factory tested to this requirement and operated at LEDA without an interlock to limit the time the collector was subject to the full beam power in the event the RF drive to the klystron is interrupted. Three klystrons of this type provide power to a RFQ. Approximately 2.4 MW of RF power is required from the klystrons for the RFQ. Three klystrons are connected to the RFQ which acts as the power combiner. The system is designed so that only two of the three klystrons are required for operation. Waveguide switches are included in each waveguide run so that a failed klystron can be removed. The switch reflects a short circuit at theappropriate phase back to the RFQ and the other two klystrons can be used to continue operations. A picture of the klystron is included in Fig. 1. Figure 1: EEV 350 MHz APT klystron. 2 DISCOVERY OF DAMAGE 2.1 Initial Indications The first indication of a problem was an audible boiling in the collector. The collector is a mixed-phase collector which allows for boiling and recondensation within the collector grooves. The audible boiling was not this mixed-phase boiling. The noise was the result of large steam bubbles being transported at the high water velocities through the collector cooling pipes and metering. We notified the vendor of the boiling and were told it was nothing to worry about. This boiling was noticeably worse on one of the three klystrons, but was observable on all klystrons for high collector powers. In order to keep the audible boil to a minimum we had been operating the klystrons at the minimum beam power required for RFQ operations. Earlier in the day we had increased the klystron beam power from approximately 1.1 MW to 1.5 MW to allow for increases in RFQ beam current and provide more margin for field control. We had operated several hours at this new operating point with neligible ion pump current (less than .2 uA) when we experience a large vacuum event that triggered both the high voltage interlock (which triggers at 10 uA ion pump current) and the filament power interlock (which triggers at 100 uA of ion pump current). We suspended operations and attempted to recondition the klystron. Although we were able to recondition the klystron, as we operated at higher beam power we began to observe variable cathode emission. 2.2 Variable Emission As we reconditioned the klystron, we discovered that as we increased beam power, cathode current would vary byFigure 2: Strip chart record illustrating variable emission with changes in collector power. several amps when increasing or decreasing the RF drive. We could also cause this same variation by small changes in cathode voltaage. Our system operates with a mod-anode regulator tube that maintains the mod-anode voltage at a fixed setpoint relative to the cathode voltage and we carefully monitored this voltage, as well as the filament current, to verify neither of these factors were causing the change in cathode current. Finally, we did a series of tests where we triggered our arc detectors which inhibit RF drive to the klystrons. Fig. 2 shows the results of one of these tests. In the figure, the RF power is observed to go to zero and a coresponding plunge in cathode current is observed. When the RF power is reestablished, the cathode current is observed to slowly return to its previous value. The filament current and mod-anode voltage relative to cathode voltage is observed to be constant over the variation in cathode current. From this data we concluded copper vapor from the collector was poisoning the cathode emission as we increased the thermal load in the collector. It was also interesting that the vac-ion pump power supply did not register an increase in vacuum and indicated very low current throughout this test. 2.3 Damage Assessment Based on this result and conversations over the duration of our testing, EEV had two klystron engineers visit our site. We removed the end of the collector water jacket from the klystron and observed the damage shown in Fig. 3. From Fig. 3 it is observed that several portions of the collector had become quite hot and dimples had resulted from a combination of the heat, water pressure, and vacuum forces. The general collector shape had also distorted. It was now elliptical in cross section rather than round. Also, the collector end had drooped and the bottom of the conical shaped portion of the collector was actually resting on the water jacket. 2.4 Short Term Solution We inspected the collectors of all the klystrons and all showed various levels of damage ranging from minimal to the significant damage shown in Fig. 3. All collectors showed indications of droop and three of the four collectors were becoming elliptical rather than round in cross section. All collectors had less than 1000 high voltage hours and one had only 60 high voltage hours since delivery. Of the four klystrons, one klystron had a bucking coil in place to minimize magnetic field penetration into the collector. This klystron had the least damage. Two other klystrons had this bucking coil installed but not electrically connected to the magnetic circuit. The fourth klystron did not have the bucking coil installed. We were dismayed to discover the variation in the magnetic circuit of the klystrons, but were encouraged by the fact that the klystron with the bucking coil operable showed the least damage. We took the following short-term steps to return the klystrons to service due to programmatic pressure. Figure 3: External view of collector damage.The klystron in Fig. 3 was returned to the factory for rebuild still under vacuum and operable at low beam powers. The collector jackets of the remaining three klystrons were adjusted to center the collectors in the water jackets in order to compensate for the droop, and supports were welded on the inside of the water jacket to support the end of the collector to avoid further droop. An acoustical measurement system was installed on the collector to validate the effect of the bucking coil and to characterize safe regions of operation. The sensor was a microphone on the collector cooling return. These results are presented in Fig. 4. The four curves in Fig. 4 show some unexpected results. The mixed phase boiling that the collector was designed to accommodate manifests as an increase in the higher frequency audio components of Fig. 4. This increase can be used as a qualitative indication of peak collector heat load. From Fig. 4 it can be seen that the localized collector heat load at 77 kV, 14.5 A is actually higher than at 85 kV, 17.1 A. This suggests that the beam power is better distributed over the collector at 85 kV. The plots in Fig. 4 were measured without RF power with the full beam power dissipated in the collector. Fig. 4 also shows that application of the bucking coil lowers the localized collector heat load. Using this tool we were able to define an allowable safe operating region and also regions to avoid. Operation with three klystrons at an 85 kV voltage provides sufficient power for the RFQ with full accelerator beam current and sufficient control margin. The data in Fig. 4 were measured on the klystron that had the second most significant amount of collector damage. The other two klystrons showed a similar behavior but not the same magnitude of accoustical variation. For all klystrons, operation between 70 kV and 80 kV resulted in audio indications of increased localized heating. In addition to the steps taken above, we also implemented a beam interlock such that above 70 kV of beam voltage, 100 kW of RF power was required or an interlock would remove high voltage within 15 seconds. We would have preferred a shorter interlock period, but the 15 second interval was a compromise with the RFQ operations team and with EEV. 3 LONG TERM SOLUTIONS EEV has redesigned the collector so it is capable of achieving the initial requirements. The details provided below are intentionally vague to protect proprietary information. Additional modeling was conducted using a recently released, new version of one of the beam dynamics codes originally used in the collector design. The beam was discovered to penetrate further into the collector than originally thought. Additional length has been added to the collector in the new collector design some. The conical collector end has been changed to a shape which reduces the thermal stress. The bucking coilis now a required part of the design. The water jacket design has been changed to maintain a high flow velocity over the entire collector. The supports on the end of the collector implemented as part of the short term solution have become a part of the design. We also conducted x- ray measurements of the klystron internal to the lead shielding and discovered that the beam alignment could be improved. These results were duplicated by EEV. EEV is taking steps to improve beam alignment. Figure 4: Collector audio profile. 4 CONCLUSION We discovered the initial collector design to be inadequate for satisfying our requirements. We have worked closely with EEV to diagnose and modify the collector design. The first 350 MHz klystron with the new collector design is scheduled for factory testing in August, 2000. The factory acceptance tests of the collector have been significantly increased and an inspection of the collector is conducted at the conclusion of the testing. After delivery of this initial tube, another klystron from Los Alamos will be rebuilt to include the new collector. The remaining two klystrons at Los Alamos had only slight collector damage and will continue to be used on LEDA. As part of diagnosing the collector problem, we developed a useful and interesting tool to characterize localized collector heat load using an acoustical measurement. This tool led to the unexpected result that lower beam powers had higher localized collector heat loads than operation a higher beam powers. We have operated the three remaining klystrons for slightly over 1000 cumulative hours since the collector problem was discovered. The two slightly damaged klystrons show no sign of further damage. The remaining klystron with the most significant damage is showing signs of further degradation. We hope that it will continue to meet the RFQ needs until the replacement tube arrives in early September.
THE SPALLATION NEUTRON SOURCE (SNS) LINAC RF SYSTEM* Michael Lynch, William Reass, Daniel Rees, Amy Regan, Paul Tallerico, Los Alamos National Laboratorys, Los Alamos, NM, 87544, USA Abstract The SNS is a spallation neutron research facility being built at Oak Ridge National Laboratory in Tennessee [1]. The Linac portion of the SNS (with the exception of the superconducting cavities) is the responsibility of the Los Alamos National Laboratory (LANL), and this responsibility includes the RF system for the entire linac. The linac accelerates an average beam current of 2 mA to an energy of 968 MeV. The linac is pulsed at 60 Hz with an H- beam pulse of 1 ms. The first 185 Mev of the linac uses normal conducting cavities, and the remaining length of the linac uses superconducting cavities [2]. The linac operates at 402.5 MHz up to 87 MeV and then changes to 805 MHz for the remainder. This paper gives an overview of the Linac RF system. The overview includes a description and configuration of the high power RF components, the HV converter/modulator, and the RF controls. Issues and tradeoffs in the RF system will be discussed, especially with regards to the use of pulsed superconducting cavities. Figure 1: SNS Linac configuration. 1 OVERVIEW OF SNS LINAC The SNS Linac is a nominal 968 MeV, 2 mA average, H- accelerator (Figure 1). The system provides 52 mA peak at 1 ms pulse width and a 60 Hz repetition rate. The beam is chopped with a 68% chopping factor, for an average beam during the pulse of 36 mA. The accelerator begins with an injector and RFQ (from Berkeley), then a Drift Tube Linac (DTL) to 86.8 MeV, followed by a Coupled-Cavity Linac (CCL) to 186 MeV. Los Alamos is responsible for the DTL and CCL structures and all of the linac RF systems. The remaining part of the accelerator uses superconducting cavities (from Jefferson Laboratory), using ß=0.61 cavities to 375 MeV, then ß=0.81 cavities to 968 MeV. The RFQ and DTL operate at 402.5 MHz, and the CCL and superconducting cavities operate at 805 MHz. The different structures and frequencies require that 3 different types of klystrons be used. The RFQ and DTL will use 2.5 MW peak klystrons at 402.5 MHz. The CCL *Work supported by the US Department of Energywill use 5 MW peak klystrons at 805 MHz, and the superconducting cavities will use 550 kW klystrons at 805 MHz. The types and quantities and applications are listed in Table 1. Table 1: Types and quantities of klystrons and HV systems for the SNS Linac H- Energy 968 MeV Average beam during pulse 36 mA Pulse Width 1 ms Rep Rate 60 Hz Klystrons 402.5 MHz, 2.5 MW pk (includes 1 for RFQ , 6 for DTL) 805 MHz, 5 MW pk (includes 4 for CCL, 2 for HEBT) 805 MHz, 0.55 MW pk, SC7 6 92 HV Converter/Modulators (16 Total)1 for each 5 MW klystron or pair of 2.5 MW klystrons (except 1 for RFQ and first 2 DTL’s & 1 for 2 HEBT cavities) 1 for 11 or 12 0.55 MW klystrons Figure 2 shows a block diagram of the RF systems for the superconducting portion of the accelerator. Depending on the power required in the particular portion of the linac, each grouping consists of either 11 or 12 klystrons per HV system. Klystron 11, 12HV System, 11 or 12 klystrons RF Controls 1/klystron Klystron Gallery Accelerator Tunnel Accelerator Module 1Klystron 1 RF Controls 1/klystron6-klystron Transmitter Electronics Accelerator Module 11 or 126-klystron Transmitter ElectronicsCirculator Figure 2: RF System block diagram for superconducting cavities. For room temperature cavities, each klystron has its own RF controls and Transmitter electronics. In order to standardize as much as possible, the HV systems are designed to provide power to klystrons for ‘approximately’ 5 MW of RF. Therefore, the 402.5 MHz klystrons are configured with 2 klystrons per power supply. The CCL uses one klystron per HV system, andInjector 2.5 MeVRFQ 968 MeVDTL 86.8 MeVTo Ring CCL402.5 MHz 805 MHz SC, ß=0.61 SC, ß=0.81 186 MeV 375 MeVHEBTthe superconducting klystrons are combined in groups of 11 or 12 per HV system (depending on the amount of power needed for that portion of the linac). Transmitters provide support electronics and interlocks to 2 klystrons (402.5 MHz), 1 klystron (CCL), or 6 klystrons (superconducting systems). Each klystron along the entire linac has its own feedback/feedforward RF control system. The power required along the superconducting portion of the linac is shown in Figure 3. Since the SNS is a proton machine, the power required in each cavity varies, depending on the cavity ß and the velocity of the particles. Since one HV supply provides power to 11 or 12 klystrons, each klystron in that group has the same saturated power level, set by the highest power requirement in the group. 0100000200000300000400000500000600000 161116212631364146515661667176818691P-gen P-sat Superconducting Cavity Number Figure 3: The required power level and the saturated power setting for each klystron in the superconducting portion of the SNS Linac. 2 RF SYSTEM COMPONENTS 2.1 Transmitter The transmitter includes the klystron support tank and all of the support and interlock electronics for the high power RF systems. The transmitter includes the HV metering, the klystron filament power supply, the solid state driver amplifier, the klystron vac-ion pump power supply, the solenoid supplies, the interlocks for water flow, water temperature, and air cooling, the accelerator window cooling diagnostics, the AC distribution, and the user interface, consisting of the programmable controller and user display. 2.2 High Voltage (HV) System Line Filter, Step-Down Transformer, SCR Controller, Capacitor Bank StorageIGBT Inverter 3-Ø Boost Transformer, Rectification, FilterFeedback/Feedforward Controller13.8 kV, 3-Ø3000 VDC 20 kHz, 3-Ø Pulsed HV to Klystrons HV FeedbackCap Bank Feedback SCR Ø ControlPWM Control Figure 4: HV Converter/ModulatorThe HV System provides everything from the input 13.2 kVAC line to the klystron, including the electronics to provide pulsed HV to the klystron cathode. Los Alamos is developing a unique design that provides conversion of HV-AC to HV-DC as well as the high voltage pulse modulation needed for SNS [3]. The design is very conservative with space and appears to be a very cost-effective approach as well. A block diagram of the converter-modulator is shown in Figure 4. The design uses standard 60 Hz technology to convert the incoming 13.2 kVAC to 3 kVDC. It then uses IGBT’s, operating at 20 kHz switching speeds into a 3-phase inverter step-up circuit and a high voltage rectifier/filter, to convert the low voltage DC into the required pulsed, high voltage DC for the klystrons. The system is very versatile in that the only changes needed between the 5 MW klystron application (135 kV, 70 A) and the 550 kW klystron application (75 kV, 130 A) are the turns-ratio and peaking-capacitor value in the 3-phase high voltage step- up transformer. The system employs pulse-width modulation (PWM) of the IGBT switching to maintain regulation. The control of this PWM as well as the SCR phase control is achieved by a combination of both feedback and feedforward. 2.3 RF Controls Linac CavityDigital ControllerKlystron Resonance Control LOLO50 MHz ReferenceHigh Power Protect 50 MHz Cavity FeedbackUpconverter Downconverter Figure 5: RF Controls for SNS The RF control system has to control RF systems consisting of both room temperature and superconducting cavities, frequencies of both 402.5 and 805 MHz, and klystrons of 2.5 MW, 5 MW, and 550 kW [4,5]. The control system is based on the controls originally developed for the Ground Test Accelerator at LANL, and then recently adapted for the Accelerator for Production of Tritium. The system is based on VXI hardware. A block diagram of the control system is shown in Figure 5. The original GTA system did all of the actual control electronics in analog circuitry. For APT , the system used a combination of digital (for low frequency control) and analog (for higher speed control). Recent advances in the speed of digital electronics has led us to pursue an all- digital control system for use on SNS. In addition it will make extensive use digital signal processors (DSP’s) which will allow changes in the control algorithms by reprogramming the DSP’s rather than changing the hardware.We have also developed an extensive model of the control system, including such things as klystron saturation, loop delay, microphonics, Lorentz-force detuning, and beam and power supply noise [6]. The model has been developed in MATLAB/SIMULINK, and allows us to optimize the design approach before producing any hardware. An example of one modeling result is shown in Figure 6. Figure 6: Modeling results for SNS superconducting cavity showing the effect of iterative learning control. Each successive pulse shows a reduced error due to improved feedforward correction. 3 PULSED OPERATION OF A SUPERCONDUCTING PROTON ACCELERATOR Until recently, superconducting cavities have been used only in CW applications. Pulsed operation of a superconducting accelerator has been addressed recently in work at DESY on the Tesla facility, in Japan for the Joint Project, and at various places for the European Spallation Source (ESS). The SNS is adding to that work and will make advances especially with respect to the use of pulsed superconducting cavities for proton acceleration. The original superconducting design for SNS had external Q’s of approximately 5e5 and Eacc of about 11.9 MV/m. The resultant effect of Lorentz detuning and microphonics was relatively small. As the design of SNS has progressed, however, the external Q’s have increased (to about 7e5) and Eacc has increased (to 14.2 MV/m). This means that the effects of Lorentz detuning and microphonics, while not at the Tesla level, have become a more serious problem. Current external bandwidths are about 550 Hz, and the expected microphonics and Lorentz detuning total is 400 to 500 Hz. Because it is a proton accelerator, SNS has the added problem of nonzero synchronous phase (to maintain longitudinal bunching). In addition, the synchronous phase of the machine varies from one cavity to the next (over a range of –26.5° to –15°). To minimize the effect of reflected power, one typically detunes the cavity so thatthe reactive effect of the nonzero synchronous phase is cancelled by the cavity detuning. With Lorentz forces and microphonics, it is impossible to maintain exact cavity detuning. The system must therefore be designed to handle the reflected power, both in terms of the RF components and the power capability in the RF source. 4 TECHNOLOGY DEVELOPMENT AND ALTERNATIVE CONFIGURATIONS Other superconducting applications have spent several years in technology development activities. The SNS is scheduled to begin commissioning in 2005, with first operation in 2006. There is little time for development. All 105 RF systems must be designed, built, integrated, installed, and commissioned in only 5 Years. This puts great emphasis on using what is available, on designing a robust system that can accommodate changes, and on using dependable models for the development. Many different configurations of the linac and RF systems were considered in terms of the cost and performance. The current baseline, as presented in this paper, represents the necessary technology advancements to meet the performance requirements while staying within the very tight budget and schedule limitations. In addition, as in most modern accelerator applications, there is always the need to consider other applications for the system in order to make best use of the large capital investment. In addition to the basic 60 Hz application, we are considering the repercussions of interleaving a 10 Hz pulse pattern for a second neutron target. This has implications for the power generation and handling, for the feedforward control systems, and for the definition of component requirements as we begin making the procurements. The primary consideration at this point is to ensure that we do not preclude the option of the interleaved pulses and the second target. REFERENCES [1]R.L. Kustom, “An Overview of the Spallation Neutron Source Project”, Linac 2000, August 2000. [2]J. Stovall, et al, “Superconducting Linac for the SNS”, Linac 2000, August 2000. [3]W.Reass, et al, “Progress on the 140 kV, 10 MW Peak, 1 MW Average, Polyphase Quasi-Resonant Bridge, Boost Converter/Modulator for the SNS Klystron Power System”, Linac 2000, August 2000. [4]A. Regan, et al, “Design of the Normal Conducting RF Control System”, Linac 2000, August 2000. [5]Y. Wang, et al, “Analysis and Synthesis of the SNS Superconducting RF Control System”, Linac 2000, August 2000. [6]S. Kwon, et al, “An Iterative Learning Control of a Pulsed Linear Accelerator”, Linac 2000, August 2000. First Pulse Fourth Pulse
BEAM EMITTANCE MEASUREMENTS FOR THE LOW-ENERGY DEMONSTRATION ACCELERATOR RADIO-FREQUENCY QUADRUPOLE* M. E. S chulze , General Atomics , San Diego, CA 92186 , USA J. D. G ilpatrick , W. P. L ysenko , L. J. R ybarcyk , J. D. S chneider , H. V. S mith, Jr., and L. M. Y oung , Los Alamos National Laboratory, Los Alamos, NM 87545, USA * Work supported under DOE Contract DE-AC04-96AL99607Abstract The Low-Energy Demonstration Accelerator (LEDA) radio-frequency quadrupole (RFQ) is a 100% duty factor (CW) linac that delivers >100 mA of H+ beam at 6.7 MeV. The 8-m-long, 350-MHz RFQ structure accelerates a dc, 75-keV, 110-mA H+ beam from the LEDA injector with >90% transmission. LEDA [1,2] consists of a 75-keV proton injector, 6.7-MeV, 350-MHz CW RFQ with associated high-power and low-level rf systems, a short high-energy beam transport (HEBT) and high-power (670-kW CW) beam stop. The beam emittance is inferred from wire scanner measurements of the beam profile at a single location in the HEBT. The beam profile is measured as a function of the magnetic field gradient in one of the HEBT quadrupoles. As the gradient is changed the spot size passes through a transverse waist. Measurements are presented for peak currents between 25 and 100 mA. 1 INTRODUCTION The primary objective of LEDA is to verify the design codes, gain fabrication knowledge, understand beam operation, measure output beam characteristics, learn how to minimize the beam-trip frequency, and improve prediction of costs and operational availability for the APT accelerator. The configuration of the LEDA RFQ accelerator is shown in Figure 1. This paper presents the analysis of quad-scan measurements of the output beam from the 6.7 MeV RFQ. Figure 1. LEDA configuration for RFQ commissioning.A schematic of the LEDA HEBT [3] showing the location of beamline magnets and diagnostics is given in Figure 2. The function of the LEDA HEBT is to characterize the properties of the beam and transport the beam with low losses to a shielded beamstop. The beamline magnets consist of four quadrupoles and two sets of X-Y steering magnets. The HEBT contains beam diagnostics that allow measurement of pulsed-beam- current, dc-beam-current, and bunched-beam-current as well as transverse centroid, longitudinal centroid (i.e., beam energy from time-of-flight and beam phase), and transverse beam profile (wire scanner and video fluorescence) [4]. Q #4 Q #3 Q #2 Q #1 SM #1 SM #2 Figure 2. Layout of HEBT beamline optics and diagnostics. Beam direction is from left to right. Quad-scan measurements were made using only the first two quadrupole magnets. For the horizontal scan, the first quadrupole in the HEBT was fixed while the second quadrupole was varied over a range between 4.7 T/m and 10.7 T/m. For the vertical scan, the second quadrupole in the HEBT was fixed while the first quadrupole was varied over a range between -7.0 T/m and -13.0 T/m. In both cases the two downstream quadrupoles were off. The resulting beam size measured at the wire profile monitor passes through a minimum as the gradient is varied. In each scan, beam profile measurements were typically made at nine different settings. Profile measurements were performed at beam peak currents of 25, 50, 75 and 94 mA. Many of these measurements were repeated over a period of one month.InjectorRFQ Beamstop WaveguideHEBT2 ANALYSIS The analysis of the quad scan data utilized the beam optics code LINAC [5] and only addresses the rms properties of the distribution. The Twiss parameters α, β, and ε at the exit of the RFQ were adjusted to fit the rms widths of the beam distributions taken during the quad scans. A six dimensional waterbag distribution was assumed although the analysis is generally independent of the distribution. Initially, the Twiss parameters were adjusted to fit the predicted distribution from RFQ simulations (PARMTEQM or nominal) for a 94 mA beam current. Beginning with these Twiss parameters, the sensitivity to ε, α, and β was analysed at 94 mA for the two transverse planes. No attempt was made to study the longitudinal properties of the distribution. The data sets ( rms width vs. quadrupole gradient) were combined and analyzed to obtain a polynomial fit as shown in Figures 3a and 3b. The data sets represent measurements made on three occasions. 3a. HORIZONTAL SCAN – 94 mA 3b. VERTICAL SCAN – 94 mA Figures 3a and 3b. Measured values of the rms beam width (red triangles) and corresponding fits (long blue dashes) at 94 mA. The black curves (small dashes) show the nominal RFQ output beam and the green (solid) curves shows the best fit to the data using LINAC. From the polynomial fits to the data the rms beam width was calculated at nine gradient settings. The LINAC code was then run at each of these gradient settings for a specific set of Twiss parameters. The Twiss parameters were then optimized by minimizing the sum of the squared errors (SSE) between the fit andthe results of the nine runs. Typically, the SSE was less than 1.0 mm2. Figures 3a and 3b show the fits to the data in the horizontal and vertical planes at 94 mA. The black curve labelled nominal represents the RFQ beam simulation from PARMTEQM [6]. The green curve represents the best fit by adjusting the Twiss parameters according to the label (e.g. 110e 82b 107a means 1.10 times the nominal emittance, 0.82 times the nominal beta and 1.07 times the nominal alpha). These parameters indicate that the horizontal beam size is about 5% larger at the RFQ exit with a 20% higher divergence than that predicted by PARMTEQM. In comparison, the vertical beam size is about 25% larger at the RFQ exit with a divergence consistent with that predicted by PARMTEQM. The same analysis was performed on data taken at 25 mA. The results are shown in Figures 4a and 4b. The analysis shows the horizontal beam emittance to be about 90% of the 94 mA emittance predicted by PARMTEQM with the horizontal beam size about 12% smaller at the RFQ exit and the divergence about 4% higher. The vertical beam distribution has an emittance close to the design prediction at 94 mA while the beam size and divergence are about 75% of the nominal design. 4a. HORIZONTAL SCAN – 25 mA 4b. VERTICAL SCAN – 25 mA Figures 4a and 4b. Measured values of the rms beam width (red triangles) and corresponding fits (long blue dashes) at 25 mA. The black curves (small dashes) show the nominal RFQ output beam and the green (solid) curves shows the best fit to the data using LINAC It should be noted that the beam measurements made at 25 mA were not made with the injector matched to the RFQ. Figure 4b shows many different fits to the vertical02468101214 4.5 5.5 6.5 7.5 8.5 9.5 10.5Q2 (T/m)RMS Width (mm)Measured Xrms Fit 110e 82b 107a nominal 0246810121416 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 Q1 (T/m)Y rms (mm)25 mA (4-4) 25 mA (3-29) Nominal 106e 55b 62a 106e 50b 62a 100e 55b 67a 105e 60b 67a 100e 55b 63a02468101214 4.5 5.5 6.5 7.5 8.5 9.5 10.5 Q2 (T/m)Xrms (mm)25 mA (3-29) Nominal 90e 85b 120a 024681012141618 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 Q1 (T/m)RMS Width (mm)Measured Yrms Fit 134e 115b 90a nominalrms spot size that produces similar SSE’s. The overall variation in the Twiss parameters is only about 5%. A similar analysis has been performed for a quad scan taken at 75 mA peak current. Table 1 summarizes these results as well as the results at 25 and 94 mA. Table 1: Normalized Beam Emittance and Twiss Parameters from Quad Scan Analysis Current Nominal 94 mA 75 mA 25 mA εxn π(mm-mrad)0.229 0.253 0.216 0.207 αx 1.671 1.785 1.913 1.919 βx meters0.436 0.357 0.379 0.371 εyn π(mm-mrad)0.234 0.314 0.268 0.258 αy -2.750 -2.483 -2.479 -1.711 βy meters0.778 0.892 0.649 0.427 3 DISCUSSION During commissioning it was observed that the RFQ fields had to be increased 5-10% above the design value for optimal transmission [7]. The effect of this higher field has been studied and does not result in significant changes in the predicted PARMTEQM beam distributions. The effects of beam neutralization were investigated for the 94 mA data. A small reduction in the SSE was observed in the horizontal quad scan data consistent with about 10% neutralization. Analysis of the vertical quad scan data with 10% neutralization resulted in Twiss parameters that were within 5% of those presented in Table 1. A rigorous error analysis was not performed in the analysis of the quad scan data. The different errors that contribute to the analysis uncertainty have been estimated. These errors result from beam jitter (1.0 mm typically), measurement reproducibility (0.5 mm), background subtraction (< 0.5 mm), analysis uncertainty (5%), and quadrupole gradient fluctuations (1-2%). The effect of beam jitter is most pronounced in the minimum of the horizontal quad scan data where it is observed that the simulations are consistently lower than the measurements. A systematic uncertainty in the analysis results from the use of the LINAC code to infer the Twiss parameters. LINAC includes non-linear space-charge effects, which are essential to modelling the beam transport. TRACE- 3D [8], which includes only linear space charge, was found to be inadequate in describing the beamdistributions. Beam profiles produced by LINAC were generally in good agreement with the measured beam profiles for the lower quadrupole gradients before the minimum. After the minimum the measured beam distributions exhibited significant tails which are not well reproduced by LINAC even though the rms widths are in close agreement. One difficulty in reproducing the exact shape of the distribution is attributed to the large aspect ratio (>5) between the horizontal and vertical beam widths when the beam is at a waist [9,10]. The details of the beam distribution from the RFQ are also unknown. IMPACT, a more sophisticated PIC beam-optics code, has been used to analyze the quad scan distributions [9]. This analysis gives essentially the same rms widths as LINAC but with significantly better agreement in predicting the shape of the profiles. 4 SUMMARY The rms output beam parameters from the 6.7 MeV LEDA RFQ have been inferred from quad scan measurements using the LINAC beam optics code. Analysis of the data presented in this paper continues. We are now preparing to intentionally introduce and measure the beam halo in a 52-magnet FODO lattice [11]. This measurement will also allow for an independent measure of the beam emittance.REFERENCES [1] J. D. Schneider, “Operation of the Low-Energy Demonstration Accelerator: the Proton Injector for APT,” Proc. PAC99 (New York, 29 March - 2 April 1999) pp. 503-507. [2] L. M. Young et al. , "High Power LEDA Operations," Proc. LINAC2000 (Monterey, 21-25 August 2000) (to be published). [3] W. P. Lysenko et al. , “High Energy Beam Transport Beamline for LEDA," Proc. LINAC98 (Chicago, 24-28 August 1998) pp. 496-498. [4] J. D. Gilpatrick, et al., “LEDA Beam Diagnostics Instrumentation: Measurement Comparisons and Operational Experience“, BIW ‘00, Boston, MA, May 8-11, 2000. [5] K. R. Crandall and T. P. Wangler, Private Communication. [6] K. R. Crandall et al., “RFQ Design Codes”, Los Alamos National Laboratory report LA-UR-96-1835 (revised February 12, 1997). [7] L. J. Rybarcyk et al. , "LEDA Beam Operations Milestone and Observed Transmission Characteristic," Proc. LINAC2000 (Monterey, 21-25 August 2000) (to be published). [8] D. P. Rushtoi, W. P. Lysenko and K. R. Crandall, “Further improvements in TRACE-3D”, Proc. PAC97 (Vancouver, 12-16 May 1997) pp. 2574-2576. [9] W. P. Lysenko et al. , "Determining the Phase-Space Properties of the LEDA RFQ Output Beam,” Proc. LINAC2000 (Monterey, 21-25 August 2000) (to be published). [10] L. M. Young, Private Communication. [11] P. L. Colestock et al. , "The Beam Halo Experiment at the LEDA Facility: a First Test of the Core-Halo Model,” Proc. LINAC2000 (Monterey, 21-25 August 2000) (to be published).
arXiv:physics/0008072 15 Aug 2000NEW DEVELOPMENTS IN LINEAR COLLIDERS FINAL FOCUS SYSTEMS P.Raimondi, A. Seryi, SLAC,Stanford,CA Abstract The length, complexity and cost of the presen t Final Focus designs for linear colliders grows very quick ly with the beam energy. In this letter, a novel final focu s system is presented and compared with the one proposed for NLC [1]. This new design is simpler, shorter and ch eaper, with comparable bandwidth, tolerances and tunabilit y. Moreover, the length scales slower than linearly wi th energy allowing for a more flexible design which is applicable over a much larger energy range. 1 INTRODUCTION The main task of a linear collider final focus system (FFS) is to focus the beams to the small sizes requ ired at the interaction point (IP). To achieve this, the FF S forma a large and almost parallel bem at the entrance of the final doublet (FD) which contains two or more strong quadrupole lenses. For the nominal energy, the beam size at the IP is then determined by εβσ=where ε is the beam emittance and β is the betatron function at the IP (typically about 0.1-1mm). However, for a beam with an energy spread σE (typicall 0.1-1%), the beam size is diluted by the chromaticity of these strong lenses. The chromaticity scales as L/β, where L* (typically 2-4 m) is the distance from the IP to the FD, and thus the ch romatic dilution of the beam size σEL/β is very large. The design of a FFS is therefore driven primarily by the neces sity of compensating the chromaticity of the FD. Figure 1: Optics of the traditional Final Focus for the NLC showing horizontal and vertical betatron and dispersion functions. Focusing and defocusing quadrupoles are indicated as up and down bars on th e magnet plot above the optics, bends are centered. In an “traditional” final focus system (SLC [1], FF TB [2] or the new linear collider designs) the chromaticit y is compensated in dedicated chromatic correction secti ons (CCX and CCY) by sextupoles placed in high dispersi onand high beta regions. The geometric aberrations generated by them are canceled by using them in pai rs with an identity transformation between them. As an example, the “traditional” design of the NLC Final Focus [3] with mL2=,mmx10=β and mmy12 . 0=β is shown in Fig.1. The advantage of the traditional FFS is i ts separated optics with strictly defined functions an d straightforward cancellation of geometrical aberrat ions. This makes such a system relatively simple for desi gn and analysis. The major disadvantage of the “traditional” final f ocus system is that the chromaticity of the FD is not lo cally compensated. As a direct consequence there are intr insic limitations on the bandwidth of the system due to t he unavoidable breakdown of the proper phase relations between the sextupoles and the FD for different ene rgies. This precludes the perfect cancellation of the chro matic aberrations. Moreover, the system is very sensitive to any disturbance of the beam energy in between the sourc es of chromaticity, whether due to longitudinal wake-fiel ds or synchrotron radiation. In particular, the bends in the system have to be long and weak to minimize the additional energy spread generated. In addition, th e phase slippage of the off-momentum particles drastically limits the dynamic aperture of the system. Therefore very long and problematic collimation sections are required i n order to eliminate these particles that would otherwise h it the FD and/or generate background in the detector. The collimation section optics itself also becomes a source of aberrations since very large beta and dis persion functions are required. As a result of all these limitations, the length of the beam delivery system becomes a significant fraction of t he length of the entire accelerator, and scaling to hi gher energies is difficult. 2 “IDEAL” FINAL FOCUS SYSTEM Taking into account the disadvantages of the tra ditional approach, one can formulate the requirements for a more “ideal” final focus: 1) The chromaticity should be corrected as locally as possible. 2) The number of bends should be minimized. 3) The dynamic aperture or, equivalently, the preservation of the linear optics should be as larg e as possible. 4) The system should be as simple as possible. 5) The system should be optimized for flat beams. It is straightforward, starting from the IP, to bui ld such a system: 1) A Final Doublet is required to provide focusing.2) The FD generates chromaticity, so two sextupoles interleaved with these quadrupoles and a bend upstream to generate dispersion across the FD will locally cancel the chromaticity. 3) The sexupoles generate geometric aberrations, so tw o more sextupoles in phase with them and upstream of the bend are required. 4) In general four more quadrupoles are needed upstream to match the incoming beta function (see the schematic in Fig.2). Figure 2: Optical layout of the new final focus. The second order geometric aberrations are cancelle d when the x and y-pairs of sextupoles are separated by transfer matrices MF and MD: DDDDDD M FFFFFF MD F 1000 00001000 ; 1000 00001000 4321 4321= = Where all nonzero parameters are arbitrary. In orde r to cancel the second order chromatic aberrations, the sextupole integrated strengths KS have to satisfy the equations: 13 2 13 2 SD SD SF SF KD K KF K − = − = ’3 341 ’3 1221 1ηξ ηξξ Dy SD Fxx SFRK RK =+= (1) EdEdxxd x xx’2 21 = = ξ ξξ x and x’ are the beam coordinates at the IP, ξx1 is the horizontal chromaticity of the system upstream of t he bend, ξx2 is the chromaticity downstream, ξy is the vertical chromaticity. RF and RD are the transfer matrices defined in Fig.2. The angular dispersion at the IP, η’, is necessarily nonzero in the new design, but can be s mall enough that it does not significantly increase the beam divergence. Half of the total horizontal chromatici ty of the whole final focus must be generated upstream of the bend in order for the sextupoles to simultaneously cancel the chromaticity and the second order dispersion. The horizontal and vertical sextupoles are interlea ved, so in general they can generate third order geometric aberrations according to: 122 122 12 1222 ϕFDSFSD RRKKU= 122 342 34 3444 ϕFDSFSD RRKKU= ⋅ − =SFSDKK U 21 1244 ( ) [ ]3434123412 122 342 122 122 34 4 ϕ ϕFFDD FD FD RRRR RRRR − + 1244 3224UU=ϕ12 and ϕ34 are the elements of the transfer matrix between SF1 and SD1. The beam spot sizes dilutions from U3444 and U1222 are small if the last quadrupole is defocusing and give n the typical flat beam parameters like in Tab.1. U1244 and U3224 can be made to vanish by properly chosing the trans fer matrices between the sextupoles. Similar constraint s hold for third order chromo-geometric aberrations. All t hese constraints can be satisfied with the simple system described above. A system with the same demagnifica tion as the NLC FF and comparable optical performance ca n be built in a length of about 300m. Table 1: Beam parameters Beam energy, GeV 500 Normalized emittances γεx / γεy (µm)4 / 0.06 Beta functions βx / βy at IP (mm) 9.5 / 0.12 Beam sizes σx / σy at IP (nm) 197 / 2.7 Beam divergence θx / θy at IP (µrad)21/23 Energy spread σE (10-3) 3 Dispersion η’x at IP (mrad) 5.4 Figure 3: Optics of the new NLC Final Focus System showing horizontal and vertical betatron and disper sion functions similarly to Fig.1 3 BANDWIDTH The new FF system has potentially much better performance than the traditional design. The “minim al” optics concept can be further improved by adding mo re elements to minimize residual aberrations. An addit ional bend upstream of the second sextupole pair decrease s chromaticity through the system. An additional sex tupole upstream and in phase with the last one further red uces third order aberrations in x-plane. Such system has vanishing aberrations up to third order, the residu al higher order aberrations can be further minimized by using decapoles. In particular the fourth order aberratio ns generated by the interleaved horizontal and vertica l sextupoles can be reduced with a decapole placed ne ar the closest quadrupole to the IP. The new optics is sho wn in Fig.3. The flat beam parameters are given in Tab.1. The new system has an L=4.3m, which is more than twice the original value. This allows the use of large bore superconducting quadrupoles and simplifies the desi gn of the detector. Although the chromaticity is doubled due to the larger L, the performance of the system is still better than for the original NLC FF design.Figs. 4a/b compare the bandwidth of the NLC FF and the new design in the IP phase. Figs.5 show the bandwid th in the FD phase. The bandwidth is derived from the variation of the beta function and the beam sizes a s they actually contribute to luminosity, which is determi ned by tracking. The beam size bandwidth is narrower than the beta function bandwidth because of higher order cro ss- plane chromatic aberrations. While the IP bandwidth for these two systems is comparable, the FD bandwidth i s much wider for the new FF. Figure 4a: IP bandwidth of the traditional NLC Fina l Focus. Normalized beta functions and normalized luminosity equivalent beam sizes vs energy offset ∆E/E, and normalized luminosity vs rms energy spread σE. Figure 4b: IP bandwidth of the New NLC Final Focus with definitions as in Fig. 4 Figure 5: FD bandwidth of the Traditional and New N LC Final Focus. Normalized betatron functions at the f inal doublet versus energy offset ∆E/E. Figs.6a/b show the chromaticity through the two sys tems. The new FF one is much smaller and goes through muc h fewer optical elements and much shorter distance. T his greatly benefits the chromatic properties of the ne w system and more than compensate for the disadvantag e ofhaving the horizontal sextupole pair interleaved wi th the vertical pair. Figure 6a: Horizontal and vertical chromaticity thr ough the NLC Final Focus Figure 6b: Horizontal and vertical chromaticity thr ough the NEW Final Focus 4 BACKGROUND The chromatic aberrations as one of the main so urces of the background in the detector have been extensi vely investigated in SLC. As an example, Fig.7 shows the background in the SLD drift chamber as a function o f the chromatic aberration in the doublet faceEdEdxx dT’’2 226= Figure 7: Background in the SLD drift chamber vs th e chromatic aberration T226 The design value for this aberration was 700rad, la ter additional sextupoles were added to minimize it, re ducing the background of about a factor 2. Many more high order chromatic aberrations were generated by the sextupo les inthe chromatic correction section. It has been obser ved both in the SLC-FFS and in FFTB that simply turning the sextupoles off, eliminates most of the background a nd the beam loss monitors signals in the whole beam line. For SLD-SLC the background was one of the major limiting factors in achieving high luminosity. For future colliders this problem is greatly enhanced: design parameters require beam currents nearly thirty time s larger than in SLC at ten times the energy. A rough estimate of the background could be made us ing the following assumption: 1) The background is mainly determined by chrom o- geometric high order aberrations originated in the CCS’s as observed in SLC, for a given aberration it shoul d scale roughly as: ()r q yp xm yn x EdE N Background IPIP FD FD ) () ()()( σσσσ∝ being n m p q and r determined by the order and phase of the aberration, N is the beam charge Assuming similar aberrations than SLC-FF, the small er beam sizes across the FD than what SLC had and the increased beam current, we should expect about 3 ti mes more background than in SLC for a given relative be am collimation. 2) The radiation levels generated in the collim ation section should be comparable to the SLC ones. This requirement comes simply by the requirement that th e area should maintain residual radiation levels acce ptable for human intervention. The possible relative colli mation then cannot greatly exceed just about a 3 ⋅10-3 of what done at SLC. From 1) and 2) the expected background could then b e about a thousand times higher than in SLC. Clearly this requires a formidable improvement in the collimatio n section altogether or a minimization of the sources of background in the FFS. The new FF offers a possible solution to this probl em. Fig.10 shows the halo particle distribution at the face of the final doublet for the traditional FF and for th e new FF. The beam is very distorted in the traditional FF, v ery similarly the SLC-FFS, while the nonlinear terms ar e still negligible for the new FF. The nonzero dispersion across the FD in the new sys tem has little affect on the dynamic aperture. In addit ion, the design aperture of the NLC final doublet is about ra=10mm while for the new FF with twice longer L this aperture can be as large as ra=40mm. Therefore the collimation requirements for the new FF may be rela xed by a factor of at least one hundred in the IP phase , and by a factor of at least 3 for the FD phase and energy without increasing particle losses at the FD. Due to the shorter length of the system, there woul d also be less regeneration of the beam halo in the final focus itself from beam-gas scattering, reducing an additi onal source of background. Given the fewer elements and less bends, it could a lso be possible to build the system with a larger bore ape rture everywhere (40mm), further improving the beam stay- clear and the vacuum in the system. Figure 8:Beam at the entrance of the final doublet for the traditional NLC FF and for the new FF. Particles of the incoming beam are placed on a surface of an ellipso id with dimensions Nσ (x,x’,y,y’,E) = (800,8,4000,40,20) times larger than the nominal beam sizes. 5 TOLERANCES AND TUNING The effects of magnet displacements on IP beam offsets for the NLC-FF and the new FF are shown in Figs.9a/ b respectively. The two systems are relatively simila r, most of the contribution to the beam offset at the IP is caused by the FD motion. Fig.10 show the effect of a parti cular model of ground motion on the vertical beam offset at the IP for the two systems. The main contribution comes from the FD and probably it could be much smaller in the new FF because the longer L, thus allowing a more rigid support for the magnets. Figure 9a: Traditional FF horizontal (black bars) a nd vertical (white bars) beam offset at the IP for eve ry magnet unitary displacement. Figure 9b: New FF horizontal (black bars) and verti cal (white bars) beam offset at the IP for every magnet unitary displacement. The tuning of the new FFS is very much similar to t he old schemes. The main first order aberrations come from quadrupole and skew quadrupoles components generate dby sextupole offsets and can be routinely minimized by optimizing spot sizes and luminosity by “knobs” bui lt with sextupole movers. In general for the new FF th e luminosity dilutions in the horizontal plane are ab out 50% worst, since the larger chromaticity required (eq. 1) in the system, however the dominant dilutions come from th e vertical plane and those are better mainly due to t he simpler scheme with less elements. Higher order spu rious aberrations, due to unavoidable lattice errors, can be minimized with additional magnets placed in conveni ent locations. 1 10 1000.00.20.40.60.81.01.2Ideal Optical Anchor ON/OFFG round motio n "S LAC 2 am"; fe edback f0=6 Hz; id eal q uad sup ports. O FF ONONOFF Traditional FFS, L = 2m SFD = 8m 1 10 1000.00.20.40.60.81.01.2rms offset ∆Y, nm rms offset ∆Y, nm New FFS, v.01, L* = 4.3m SFD = 8m Frequency, Hz Figure 10: Frequency spectra of the vertical beam r ms offset at the IP from ground motion for the NLC-FFS and the new-FFS. The main contribution being from the f inal doublet (Ideal Optical Anchor OFF) 6 SCALING WITH EMITTANCE AND ENERGY. To maintain optimal performance of the system with larger incoming beam emittances, the bend field mus t increase like B0∝√ε. The increased field is necessary to hold constant the contribution of high order aberra tions to the IP beam size, as well as the contribution of th e IP angular dispersion η’IPσE to the beam divergence. The dependence of the luminosity on beam energy is shown in Fig.11. A fixed length FFS has a wide rang e of energies where it could operate, especially if the bend field is rescaled. Scaling to higher energies is very favorable wi th the new design. For a wide range of parameters, the IP spot size dilution is dominated by the energy spread cre ated by synchrotron radiation in the bends. This scales lik e: ( )52 72 32’3 ’’ 2 3 3 25 ’ LL L yIP IPB y B yy γ εη ηηγεηγ σσ         ∝ ∝∆ η’B is the angular dispersion produced by the bends, t he bend length is assumed to be proportional to the to tal length of the system L. The terms in the parenthesis are constant if the IP angular dispersion is proportion al to the beam divergence and if we conservatively assume tha t the normalized emittance will be the same at higher ene rgies. In this case the length of the system scales with e nergy as10 7γ∝L . If, however, the achievable normalizedemittance scales approximately inversely with energ y, as is assumed in [4], then the scaling is5 2γ∝L. In this case, with the new design, the FF for a 3 TeV cente r of mass energy collider could be only about 500 m long . The beam also emits synchrotron radiation in the quadrupoles, which becomes more of a problem at hig her energies. This can be reduced in the new design bec ause the larger bandwidth allows the FD quadrupoles to b e lengthened to minimize the synchrotron radiation th ey generate. For the presented optics, the dependence of the luminosity on beam energy is shown in Fig.16. If th e beam parameters from [4] are assumed, this FF can operate almost up to 5TeV center of mass energy. Figure 11: Luminosity vs beam energy for the new FF , bend field optimized at each energy, with and witho ut synchrotron radiation. The “1TeV” parameters corres pond to Tab.1, the “5TeV” set corresponds to [4] with: γεx/y =50/1⋅10-8m, β* x/y=9.5/0.14mm, σE=0.2%, σx/y(2.5TeV/Beam)=31/0.54nm. 7 CONCLUSION We have developed a new Final Focus system that has better properties than the systems so far considere d and built. It is much shorter, providing a significant cost reduction for the collider. The system has similar bandwidth and several orders of magnitudes larger dynamic aperture. This reduces the backgrounds and relaxes the design of the collimation section. It i s also compatible with an L* which is twice as long as that in the traditional NLC FF design, which simplifies enginee ring of the Interaction Point area. Finally, its favorab le scaling with beam energy makes it attractive for multi-TeV colliders. We believe that further improvements of the performance of the system are possible. 8 BIBLIOGRAPHY [1] NLC ZDR Design Group, “A Zeroth-Order Desig n Report for the Next Linear Collider”, SLAC Report-4 74 (1996). [2] J.J. Murray et al., “The Completed Design o f the SLC Final Focus System”, IEEE Proceedings, (1988). [3] J. Irwin et al., “The optics of the Final Focus Test Beam”,IEEE Proceedings, New York, (1991), p. 2058. [4] J.P. Delahaye, et al., “A 30 GHz 5-TeV Line ar Collider'”, PAC Proceedings, (1997), p. 482.
CCT, A CODE TO AUTOMATE THE DESIGN OF COUPLED CAVITIES * P. D. Smith, General Atomics, Los Alamos, NM 87544, USA Abstract The CCT (Coupled Cavity Tuning) code automates the RF calculations and sizing of RF cavities for the CCL (Coupled Cavity Linac) structures of APT. It is planned to extend the code to the CCDTL (Coupled Cavity Drift Tube Linac). The CCT code controls the CCLFISH code, a member of the Poisson Superfish series of codes [1]. CCLFISH performs RF calculations and tunes the geometry of individual cavities, including an accelerating cavity (AC) and a coupling cavity (CC). CCT also relates the AC and CC by means of equations that describe the coupling slot between cavities. These equations account for the direct coupling, the next nearest neighbor coupling between adjacent AC’s, and the frequency shifts in the AC and CC caused by the slot. Given design objectives of a coupling coefficient k, the pi/2 mode frequency, and the net frequency of the CC, the CCT code iterates to solve for the geometry of the AC and CC and the separation distance between them (this controls the slot size), satisfying the design objectives. The resulting geometry is used to automate CAD drawing preparation. The code can also be used in “as- designed” mode to calculate the frequencies and coupling of a specified configuration. An overview of the code is presented. 1 REASONS FOR DEVEL OPMENT The development of CCT was undertaken to speed up the design and tuning of coupled cavities in the Low Energy Linac of APT. The Low Energy Linac, which accelerates a cw beam of protons to 212 MeV, is subdivided into 341 segments, each of which has a unique cavity design (AC, CC, and slot). The design objective is to specify the frequencies of the AC and CC and the coupling constant k and to solve for the cavity geometry that achieves this. Each AC has a specified length and gap (defined by the Parmila code). The AC diameter is tuned using CCLFISH. Each CC has a specified length and diameter. The CC post gap is tuned using CCLFISH. The geometry of the slot between each AC-CC pair is a function of the unknown AC and CC dimensions and the separation distance between cavities. The slot geometry influences both the coupling coefficient k and the net frequencies of the AC and CC. Therefore, to obtain a consistent design solution, it is necessary to iterate between CCLFISH tuning *Work supported by the APT project, U.S. DOE contract DE-AC04-96AL89607calculations and slot coupling calculations, as will be explained in more detail in section 2.4. The design was formerly done by manually iterating between CCLFISH runs and slot calculations, using a spread sheet to control the process. If each segment were designed this way, up to 6 weeks per segment and 40 man years for the complete accelerator could be required. Fortunately, one can simplify this by designing fewer segments, tuning the cavities and the slot insertion with the aid of cold models, and interpolating the tuned geometry between segments, as is now standard practice. With the CCT code, a lower cost approach can be used. Now, each design calculation can be performed in minutes rather than weeks, so it is possible to tune each segment analytically. Instead of interpolating the geometry, the approach is to interpolate empirical factors used in the coupling calculations. The empirical factors may be estimated fairly accurately with fewer, simpler cold models and the use of 3-D RF calculations to supplement the experiments [2]. Regardless of the strategy to be employed for cold models and interpolation, the CCT code has enabled a significant reduction in effort per design calculation, and it has permitted us to perform many more design calculations than would otherwise be possible, to investigate different design approaches and tuning. 2 DESCRIPTI ON OF THE CCT CODE 2.1 Geometry CCT models half of an AC, half of a CC, and the coupling slot between them ( Fig. 1). Centerline CC SlotCoupling Cavity (CC) CC Post Accelerating Cavity (AC)Beam AxistoLC LC Figure 1: CCT cavity geometryThe half cavities are nominally symmetrical, representing an infinite, biperiodic structure. Asymmetry can be introduced by an option to specify the frequency shift of the “opposite” coupling slot manually. In the accelerator, the cavity plates (Fig. 1) are assembled with adjacent AC’s back-to-back, creating a CC between them. The AC’s communicate magnetically through their common CC, giving rise to “next-nearest- neighbor coupling” between them. This causes the π/2 mode frequency to be lower than the average AC frequency (see Eq. 2). Adjacent coupling cavities, however, are rotated 180 ° apart, so the next-nearest- neighbor coupling between two CC’s is negligible. The RF model used by CCLFISH (which is called by CCT) is axisymmetric, i.e., the slot is absent. The frequency shift and coupling effects of the slot are calculated separately by CCT. To do this, CCT must first calculate the geometry of the slot (length, width, and chamfer thickness) from the dimensions of the cavities and their center-to-center distance. Although the actual slot is a three-dimensional surface intersection, the key slot dimensions (length and width) can be calculated exactly using two-dimensional trigonometry. The chamfer thickness of the slot is calculated by an approximation that is corrected with the aid of a 3 -D CAD model. The slot geometry calculations are too complicated to present here in detail – several combinations of surface intersections are possible. Fig. 2 illustrates the geometry of the slot width calculation for one case. Slot width YDHi YZJ ( Xmin, Ymin ) Line-to-arc( Xmax, Ymax ) Line-to-arc XRCO Figure 2: Example illustrating calculation of slot width For the case shown in Fig. 2, the following equations are used to calculate the slot width: ()2 2 min iH YD RCO X −− −= iH YD Y − =min ZJ X −=max 2 2 maxZJ RCO Y − += ( )( )2 min max2 min maxY Y X X widthSlot −+− =2.2 Accounting for Frequency Shifts The CCT code accounts for the following frequencies and frequency shifts. 1.The frequency calculated by CCLFISH in the absence of a coupling slot, SFf. 2.A finite mesh correction term, meshfΔ , subtracted from #1 to obtain the “real” frequency with no slot. 3.The frequency shift, slotfΔ , caused by the presence of (usually) two coupling slots in the cavity. 4.The net frequency, netf , adjusted for the effect of the slot, given by slot mesh SF netf f f f Δ− Δ− = (1) 5.The accelerating mode π/2 mode frequency for the structure is the net frequency of the AC adjusted for the next-nearest-neighbor coupling, given by kkf f ffslot mesh SF −Δ−Δ−= 12/p(2) where kk is the next-nearest-neighbor coupling coefficient. The value of kk is negative. The values of 2/pf and meshfΔ are specified as input to the CCT code. slotfΔ and kk are found iteratively by CCT code using the solution logic explained in 2.4. 2.3 Slot Coupling Equations The effects of the coupling slot are based on theoretical approaches presented by Gao [3] (for the frequency shifts and the direct coupling, k) and Greninger [4] (for the next nearest neighbor coupling, kk). These, in turn, are based on the Slater perturbation theory. The theoretical equations are adjusted by empirical “A” factors to account for the fact that the perturbation theory is not exact. Following is a summary of the equations used. The coupling coefficient k between an AC and a CC that are connected by an elliptical slot is given by: ( ) elecctric magnetic kk kAk + = where kA is an empirical correction factor, tH ccaccc ac magnetice UUHH eE eKe Lkamp −     −  =)( )( 2 3 0 02 03 0 tE cc acccac electrice UUEE eE LW Lkaep −        −=)(1 2 3 02 3 0W W, L, and t are the full width, full length, and thickness respectively of the slot, E and H are electric field magnetic field in a particular cavity evaluated at a location central to the slot, U is the stored energy in the cavity, and 22 01 LWe −= . )(0eK and )(0eE are complete elliptic integrals of the first and second kind, defined as follows:()∫−≡2/ 02 2 00 sin 1)(p jj edeK ()∫−≡2/ 02 2 00 sin 1 )(p jjd e eE , and Ea and Ea are decay rates for evanescent modes in an elliptical waveguide (details not given here). The frequency shift for a particular cavity, either the AC or the CC, is given by ( ) electric magnetic ff f Af Δ+ Δ=Δ where fA is an empirical factor, different for the AC and CC, tHslotno magneticeUH eE eKe Lf fa mp −     −  = Δ2 0 02 03 0 _)( )( 2 12 tEslotno electriceUE L LW eEf fa ep −             =Δ2 3 2 00 _2 )( 12 slotnof_ refers to the CCLFISH frequency with correction for a finite mesh. Other variables are defined above. Values applicable to the AC or CC are used as appropriate. The next-nearest-neighbor coupling coefficient kk between two adjacent accelerating cavities is given by tH ACAC kke UxH eE eKeL A kka mp 2 322 0 02 03 0 )( )(2 18−     +   −= where kkA is an empirical factor, and x is the distance between the centers of two adjacent slots in the CC. 2.4 Solution Logic The logic used to obtain a consistent, coupled solution for both the AC and the CC is as follows: 1.Specify values for the cavity dimensions, including those that will later be adjusted by CCLFISH. 2.Assume values of the frequency shifts caused by the slots and the next-nearest-neighbor coupling. 3.Starting with the des ign objective π/2 mode frequency and CC net frequency, work backward through Eqs. (1) and (2) to calculate the shifted CCLFISH target frequencies, SFf. 4.Run CCLFISH to tune the AC and CC to meet these shifted target frequencies. This modifies the cavity diameters or gaps. 5.Assume a center-to-center distance between the tuned cavities. 6.Given the geometries of the tuned cavities and the assumed center-to-center distance, calculate the geometry of the knife edge slot.7.If the slot chamfer is specified in terms of a thickness on the “major diameter” of the slot, solve (by iteration) for the radial chamfer that will produce the specified thickness. 8.Apply the chamfer to the slot by increasing its length and width dimensions. 9.Given the calculated slot geometry and the fields and energies from CCLFISH, calculate the coupling coefficients k and kk. 10.Compare the calculated k with the design target k. Determine another guess for the center-to-center distance. Go to Step 6. Iterate on the center-to- center distance until k meets its design objective. 11.Given the converged slot, calculate new frequency shifts ACslotf,Δ and CCslotf,Δ . Go to Step 3. Iterate until the frequencies have converged. 3 EXPERIENCE USING CCT The CCT code has greatly increased our productivity in designing coupled cavities. A CCL cold model that was designed using the code is shown in Fig. 3. Figure 3: Example of cold model designed with CCT Agreement between design and experiment has been good, although the perturbation theory has limitations. Improved accuracy is expected when we use 3-D analysis to refine estimates of the “A” factors. REFERENCES [1]J. H. Billen and L. M. Young, "POISSON SUPERFISH," Los Alamos National Laboratory report LA-UR-96-1834 (revision March 14, 2000). [2]G. Spalek, D. Christiansen, C. Charman, P. Greninger, P. Smith, R. DeHaven, L. Walling, "Studies of CCL Accelerating Structures with 3-D Codes," these proceedings. [3]J. G ao, "Analytical Formulas for the Resonant Frequency Changes due to Opening Apertures on Cavity Walls," Nuclear Instruments and Methods in Physics Research A311 (1992) 437-443, North- Holland Publishing Company. [4] P. Greninger, "Next Nearest Neighbor Coup ling from Analytical Expressions for the APT Linac," Proc. 1999 Particle Accelerator Conf., New York City (March-April, 1999).
High-Power Testing of the APT Power Coupler∗ E. N. Schmierer, K. C. D. Chan, D. C. Gautier, J. G. Gioia, W. B. Haynes, F. L. Krawczyk, M. A. Madrid, D. L. Schrage, J. A. Waynert, LANL, Los Alamos, NM 87545, USA B. Rusnak, LLNL, Livermore, CA 94550, USA ∗ Work supported by US Department of energyAbstract For the baseline APT (Accelerator Production of Tritium) linac design, power couplers are required to transmit 210-kW of CW RF power to the superconducting cavities. These APT couplers operate at 700 MHz, have a coaxial design and an adjustable coupling to the superconducting cavities. Since May 1999, we have been testing couplers of this design on a room-temperature test stand. We completed tests at transmitted-power and reflected-power conditions up to 1 MW. We also tested the couplers with a portion of the outer conductor cooled by liquid nitrogen. Under this latter condition, we studied the effects of condensed gases on coupler performance. The results of these tests indicate that the APT couplers are capable of delivering more than 500 kW to the cavities. We are in the process of increasing the baseline coupler design requirement to 420 kW of transmitted power to take advantage of this successful development. In this paper, we describe the results of our high-power coupler tests. 1 INTRODUCTION The APT power coupler is one of the challenges for the APT (Accelerator Production of Tritium) linac, which has a high-energy superconducting (SC) section spanning 210–1030 MeV [1]. Due to a high current of 100 mA, a large amount of CW RF power must be transmitted through these couplers to the SC cavities. In 1997, 180 kW was the highest CW power transmitted through couplers to a particle beam, which was limited by multipacting and cooling to remove RF losses. At that time, a coupler requirement of 210 kW (at 700 MHz) was chosen for the APT couplers as a reasonable extrapolation of the technology, and two couplers would be needed to feed each cavity operating at a modest gradient of 5 MV/m. Figure 1 shows the APT coupler design. Five design features were particularly employed to enhance its capability to robustly transmit high CW RF power: (1) A coaxial-type RF window with two ceramic disks was chosen to minimize window failures and their impacts; (2) Warm windows were chosen and located away from line of sight of the beam to minimize multipacting; (3) A coaxial line with a large diameter was chosen to increase the power thresholds of the multipacting band; (4) A large vacuum port at the coupler was included to insure goodvacuum at the coupler and window; and (5) Extensive RF and thermal simulations were performed to reduce RF loss and to optimize cooling. Figure 1. Schematic diagram of the APT power coupler Details of the coupler design and results of low-power tests can be found in Ref. [2] and [3] respectively, and will not be repeated here. In this paper, we describe the high-power tests that we performed in the last twelve months. 2 HIGH-POWER TESTS The high-power tests were performed on a specially designed test stand (Figure 2) [5]. The test stand was supplied with up to 1 MW of 700-MHz RF power from a klystron. RF power was transmitted from one coupler, through a copper cavity, to a second coupler, and then to a water-cooled RF load. The couplers and the copper cavity were well matched to provide 100% transmission. This arrangement allowed the testing of two couplers simultaneously. Since the test stand was at room temperature, the test conditions were slightly different from those conditions that would be if the couplers were in a cryomodule. Also, the inner conductors were cooled by air on the test stand instead of gaseous helium as is planned for the APT plant operation. The test stand was equipped to perform three types of tests: (1) Transmitted-Power Capability; (2) Totally- Reflected Power; and (3) Condensed-Gas Effect. Details and results of these tests are described in Sections 2.1 to 2.3. We tested couplers with fixed and adjustable couplings. The adjustability of a coupler is provided by tip bellows made of BeCu. Couplers with fixed coupling, which donot have the tip bellows, are expected to offer more robust operation, but less flexibility during the APT linac operation where a range of beam currents will be accelerated. Before testing, the components of the couplers were cleaned in a Class-1000 clean area. The outer conductors were rinsed with deionized-water (DI) and allowed to dry in the cleanroom. Using Gauge-32 copper wool, the inner conductors were scrubbed with DI water and methanol. Following scrubbing, they were rinsed with DI water and air-dried. The parts were assembled in the cleanroom and then transported to the coupler test stand. The components were inserted into the copper cavity and the RF window assemblies attached. Figure 2. Test Stand for APT couplers 2.1 Transmitted-Power Capability Test In the first test, RF power was transmitted through the couplers into the RF load. During conditioning, the power level was raised in steps to reach the highest power level possible. The size of the steps was defined by increases in vacuum level caused by increased heating and outgassing of the couplers. With a baseline vacuum of 8 x 10-9 Torr, the increase in vacuum was kept to below 1 x 10-6. During these tests, both fixed and adjustable couplers reached 1 MW of CW power. 2.2 Totally-Reflected Power Test In the reflected-power test, the RF load was replaced with a “sliding short,” which reflected all the forward power and set up a standing-wave pattern in the couplers. The total reflection of power was to simulate a situation during APT plant operation where the beam might be lost, reflecting nearly all the RF power. This test was conducted with the fixed couplers only. During testing, the maxima and minima of the standing wave were moved to five locations over a quarter wavelength along the system by adjusting the sliding-short position. For a sliding-short position corresponding to full current operation, the tests reached 950 kW CW. For othersliding-short positions, we reach a power level of 550 kW CW and a power level of 850 kW at a duty cycle of 50%. 2.3 Condensed Gas Test Experiences of other laboratories have shown that gas condensation on power couplers enhances multipacting by changing the secondary emission coefficients of coupler surfaces and introducing vacuum bursts released by RF heating. To investigate the condensed-gas effects, we cooled the outer conductor by passing liquid nitrogen through the passages of the coupler heat exchanger originally designed for liquid helium. Figure 3 shows the measured temperature profiles at different power levels and the calculated temperature profile at 210 kW for the cryomodule and on test bed. Water vapor and CO2 were condensed during these tests. Test results showed no enhancement of multipacting. 050100150200250300350 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 distance from quarter-wave stub shorting plate (m)temperature (K)Calculated CM, 210 kW Calculated RTTB, 210 kW 0 kW 210 kW 350 kW start of OC taper end of OC taper Figure 3. Cold test temperature profile of outer conductor 3 OBSERVATIONS DURING TESTS In this section, we summarize our observations during the tests. These observations were interesting to high- power coupler operations. Their resolution will lead to future tests and design improvements. 3.1 Bellows Failures and Thermal Analysis While testing power couplers with adjustable coupling, the BeCu tip bellows failed on two occasions after reaching a transmitted power level of 750 kW. Inspection of the failed tip bellows indicated that over-heating caused the failures. The proposed explanation of overheating was supported by thermal calculations. The calculations showed that because of the stagnation of cooling air in the convolutions and the lower thermal and RF electrical conductivities of BeCu (factors of four and two respectively compared to pure copper), the temperature of the tip bellows could be 800˚C higher than the rest of the inner conductor. This is much higher than the design temperature and would lead to failure of BeCu material. Thermal simulations further showed that the temperature of the bellows could be effectively reduced by 400˚C if wecopper-plated the BeCu bellows and used helium gas as coolant (instead of air) during the test as is specified in the plant design. 3.2 Electron Etching All power couplers tested exhibited evidence of electron activity on the surface of the inner conductors. Figure 4 shows a pattern of markings indicative of such activity. The patterns varied in concentration and position with test as well as with power level. At low power levels ( ≤500 kW), there were only shallow dendrical patterns. Above 500 kW, the patterns were deeper and more concentrated. Although the electron activity did not affect coupler operation, we are concerned with degradation of the components with time and deposition of copper from the inner conductor surface onto RF windows and cavities. We are planning additional tests to find the threshold of this “electron etching” and its dependence on other factors. Figure 4. Markings indicative of electronic activities 3.3 Outgassing During Power Conditioning During conditioning, partial pressures of residual gases were measured. Figure 5 shows these pressures as a function of conditioning time along with power levels during the testing of an adjustable coupler. Initially, these pressures increased with power level. At 500 kW, the pressures of CO2 and water decreased while the pressure of hydrogen kept increasing. The residual-gas pressures during cold tests are shown in Figure 6. As can be seen, the partial pressures increased starting from time 10:00 when RF power of 500 kW was applied. The partial pressures decreased starting at time 11:30 when the RF power level was reduced to 350 kW, and condensed residual gases were released starting at time 13:40 when liquid nitrogen ran out and the outer conductor warmed up. 4 SUMMARY AND CONCLUSION We successfully tested the performance of the APT power couplers at high-power with respect to transmitted power capability, totally reflected power, and condensed- gas effects on multipacting. Data indicates that the APT power-coupler design is capable of handling power much higher than the APT requirement of 210 kW. We are inthe process of changing the APT power-coupler requirement to 420 kW to reduce the number of couplers by a factor of two, which will lead to a reduction in the cost of the linac by $60M. We are presently conducting a second set of tests to insure that we can run the couplers robustly at a power level of 420 kW. 1.E-081.E-071.E-061.E-05 0:00 6:00 12:00 18:00 0:00 6:00 12:00 cumulative conditioning time (hours)partial pressure (Torr) 01002003004005006007008009001000 transmitted power (kW)Hydrogen WaterPower(CO2, CO, N2) Figure 5. Residual gas vs. time compared with power level during conditioning 1.E-101.E-091.E-081.E-071.E-06 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 time (hours)partial pressure (Torr) 100200300400500 transmitted power (kW) & flange temperature (K)Power Hydrogen Water(CO2, CO, Flange temp Figure 6. Residual gas vs. time compared with power level and outer conductor flange temperature during cold test REFERENCES [1]G. P. Lawrence, High-Power Proton Linac for APT; Status Of Design And Development, Proc. 19th Intern. Linac Conf., August 1998, Chicago, USA, p.26. [2]E. N. Schmierer, et al., Development of the SCRF Power Coupler for the APT Accelerator, Proc. 1999 Part. Accel. Conf., May 1999, New York City, USA, p.977. [3]E. N. Schmierer, et al., Testing Status of the Superconducting RF Power Coupler for the APT Accelerator, Presented at the 9th RF Superconductivity Workshop, November 1999, Santa Fe, NM, USA. [4]J. Gioia, et al., A Room Temperature Test Bed for Evaluating 700-MHz RF Windows and Power Couplers for the Superconducting Portion of the APT Accelerator, Proc. 1999 Part. Accel. Conf., May 1999, New York City, USA, p.1396.
arXiv:physics/0008075v1 [physics.optics] 16 Aug 2000BIHEP-TH-2000-2 Possibility of a Light Pulse with Speed Greater than c Xian-jian Zhou Institute of High Energy Physics, Academia Sinica P.O.Box 918-4, Beijing 100039 People’s Republic of China Abstract In two models it is shown that a light pulse propagates from a v acuum into certain media with velocity greater than that of a light in a vacuum (c). By numerical calculation the propagating properties of suc h a light are given. Recently L. J. Wang and his collaborators in their experimen t [1] have found that the group velocity of a laser pulse in particularly prep ared atomic caesium gas can much exceed that of light in a vacuum (c). Here we will expl ore the problem from a theoretical piont of view. Many years ago A. Sommerfel d rigorously proved [2] that the velocity of light pulse can not exceed c in absorp tion media. We call this conclusion as basic theorem afterwardz. In the following we will ﬁrst examine why usually the basic theorem holds and in what condition it will be violated. Then two models are proposed, where with precise numerical calculat ion we will show that the basic theorem is indeed violated. Therefore the propert ies of light propagating in media of the models are given. We believe that these proper ties may appears in more realistic models. 1Let a light pulse propagating along x axis toward its positiv e direction in a vacuum ( x <0), at time t = 0 arriving at x = 0, and then entering into a mediu m (x≥0) afterwards. At x = 0, the amplitude of the light pulse chang es with time t as f(t) =  F(t)t≥0 0 t <0.(1) Usually f(t) can be rigorously expressed in Fourier integra tion as f(t) =Re/integraldisplay∞ 0A(n)e−intdn . (2) For simplicity suppose there is no reﬂection of light at x = 0. The amplitude of the light pulse entering into the medium ( x≥0) is [2] f(t, x) =Re/integraldisplay∞ 0A(n)e−int+ikxdn , (3) where k=nµ(n)/candµ(n) is the complex refractive index of the medium, which depends on the frequency of incident light (dispersion). In vacuum ( x <0), the amplitude of the light pulse is g(t, x) = Re/integraldisplay∞ 0A(n)e−int+inx/cdn (x <0) =f(t−x/c) =  F(t−x/c)t−x/c≥0 0 t−x/c < 0.(4) The shape of the light pulse propagating in a vacuum does not c hange because all its Fourier components in (4) propagate with the same veloci ty c and do not decay. In particular, all these components completely cancel each other in the space-time region t−x/c < 0 (or θ=t c/x < 1) as long as the light pulse propagates in a vacuum. One may think that such a cancellation may not occur w hen light pulse propagates in a medium because of dispersion and absorption . But Sommerfeld proved that such cancellation also occurs when light pulse p ropagates in absorption media, which is just the basic theorem. In his proving Sommer feld let f(t) =  sin ν t t ≥0 0 t <0,(5) so that (3) becomes [2] f(t, x) =1 2πRe/integraldisplay ce−i(nt−nµ(n)x/c) /(n−ν)dn , (6) 2where µ(n) is taken as µ2= 1 +a2 n2 0−2iρn−n2, (7) which is the Lorentz-Lorenz refraction formula. a2, n0, ρare the constants of the medium. n0, ρrepresent the characteristic absorption frequency, dampi ng constant of the medium. Usually ρ >0, light propagating in the medium decays. The integration path c in (6) is shown in Fig.1, which is along the real axis of n from +∞to−∞through n=νby a small semicircle in the upper half of the complex plane. µ(n) in (7) has branch pionts: U1,2=−iρ±/radicalBig n2 0−ρ2, where µ =∞; N1,2=−iρ±/radicalBig n2 0+a2−ρ2, where µ = 0. We joing U1toN1andU2toN2by two branch lines, which lie in the lower half of the complex plane, when ρ >0. Because there is no singularity and branch lines of the integrand of (6) in the upper half plane, one can replace t he integration path c by u in Fig.1, which is parallel to the real axis in the upper h alf plane. When u moves to inﬁnity in the upper half plane, µ→1 and when t−x/c < 0, then (6) = 0, which is just the basic theorem. But when ρ <0 (this is our model 1), the branch lines U1N1andU2N2lie in the upper half plane and the integration path u is not equivalent to the path c in (6). Now the equivalent path u should be taken as u1+u2+u3+u4in Fig.1. By the same argument above, integration along u1vanishs. Integrations along a pair of u2cancel each other. The remaining integations along u3andu4usually do not vanish because the branch lines lie in them. Th erefore when ρ <0 (model 1), the basic theorem does not hold. In the following we will do numerical integration of (6) to show that the basic theorem i s indeed violated and to see what happens. ρ <0 means propagation of light in the medium is gain-assisted light propagation. From (6) we get f(t, x) =1 2Re[ieγω(¯ν)] +1 2πRe/integraldisplay∞ 0[eγw(−z+¯ν)−eγw(z+¯ν)]dz z, (8) where γw(¯n) =−int+inµx/c ;γ=xn0/c,¯n=n/n0,¯ν=ν/n0,¯a2=a2/n2 0,¯ρ=ρ/n0, all of them are dimensionless. Let w(z) =X(z) +iY(z). (9) 3When z >> 1, X(z)∼ −¯a2¯ρ/z2, Y (z)∼z(1−θ)−¯a2/(2z), (10) where θ=ct/x. In reference [2], the typical values of parameters are given as n0= 4×1016s−1, a2= 1.24n2 0, ρ= 0.07n0, (11) where the medium is solid or liquid. For gas a2is about 1.001. The numerical integration for (8) is diﬃcult when \|γY(z)\|becomes very large and hence eγY(z)is a very fast oscillatory function of z. In fact when n0= 4×1016s−1 andx= 1cm, γ =4 3×106is very large. But if γ <100 for example, i.e., x < 7.5×10−5cm, we can do numerical integration of (8) with high precision. Now the integration of (8) is divided into sum of/integraltextR 0and/integraltext∞ R. With R∼100 for example, fast oscillatory integrand appears in/integraltext∞ R, when z >> 1. Using (10) Re(eγω(z)) =e−γ¯a2¯ρ/z2/braceleftBigg cos[γz(1−θ)]cosγ¯a2 2z+sin[γz(1−θ)]sinγ¯a2 2z/bracerightBigg , where the fast oscillatory factors are seperated in forms of sin and cos functions. Mathematica can do that kind of integration with high precis ion. As an example, let ¯a2= 1.24,¯ρ= 0.07,¯ν= 10, γ= 1, θ= 0.98 in (8), one may get f(t, x) = −1.02057×10−13, which should vanish exactly due to the basic theorem ( ρ >0, θ < 1). Considering the amplidude of incident light pulse is 1, 1 0−13is a very high accuracy of calculation. Now let ¯ ρ=−0.07 and other parameters unchanged, f(t, x) in (8) is 0.0630255, which is a deﬁnite evidence to show that t he basic theorem is violated indeed when ρ <0. Because γ=xn0/c, θ=tc/x, f (t, x) may be looked as a function of γandθ, h(γ, θ). Let us ﬁx γ= 1, i.e., x= 0.75×10−6cm, andθchange from (-7) to 4, i.e., the time t from −7/n0to 4/n0, the amplitude of light pulse h(1, θ) in the medium is shown in Fig.2, where ¯ a2= 1.24,¯ρ=−0.07, γ= 1,¯ν= 10( ν= 10n0). Again when ρ <0, h(1, θ) does not vanish and the basic theorem is violated. In Fig.2 even if t <0, the amplitude h(1, θ) still does not vanish, which means that before the incident light pulse arrives at the medium, the li ght in the medium is already produced. For convenience, we call the light produc ed in the medium when θ <1 as fastlight and that when θ >1 as normal light. When the basic theorem 4holds, the fastlight vansishes. Some maximal values of ligh t amplitude near θ= 1 and their corresponding θvalues are listed in table 1, where two characteristics are shown: (1) The amplitude of fastlight is oscillatory and decays as θ→ −∞ ; (2) The period for each oscillation for θless and near 1 are roughly around θ= 6, which means its frequency is near the characteristic fre quency n0of the medium (the corresponding period is θ= 2π). This near equality is due to the fact that the Fourier components of the light pulse having freqen cies equal to or near n0 are most gain-assisted when ρ <0. When θ >1, the normal light soon oscillates with the frequency νof the incident light pulse (corresponding period θ= 0.2π) and its amplitude is a little bit lager than 1(1.00089). When ¯ ν= 1 with other parameters ¯a2= 1.24,¯ρ=−0.07, γ= 1 unchanged the two properties (1) and (2) remain, but the amplitude of the fastlight increases to 6.14 near θ= 1. The amplitude of normal light inceases to 7.31 with frequency ν=n0. Model 2: the ρin (7) depends on frequency n as ρ(n) =  ρ1ℓ−b < n < ℓ +b ρ2n≤ℓ−b or n ≥ℓ+b ,(12) which is not continuous and therefore µ(n) and the integrand in (6) are not the analystic functions of n. We still can use numerical calcula tion to see whether the basic theorem holds: (a) ¯a2= 1.24, ρ1=−0.07, ρ2= 0.07, ℓ= 1, b= 0.01,¯ν= 1, γ= 1. Now light amplitude in the medium are gain-assisted when (1 −b)n0< n < (1+b)n0and decays otherwise. Fastlight with above two properties appears aga in and its amplitude near θ= 1 is 3.81, while that of normal light is about 4.5; (b)ρ1= 0.02 with other parameters unchanged as in (a). Now light in the whole frequency range in the medium decays. Still fastlight remains with the two properties, but its amplitude becomes small (0.0757) near θ= 1. So we may conclude that if µ(n) in (6) is an analystic function of n with sin- gularities (such as poles or branch lines) appearing in the u pper half plane of n, or µ(n) is not an analystic function of n at all, the basic theorem in general may not hold and fastlight appears. The Fourier components of a ligh t pulse now will not cancel each other in the medium in the space-time region t−x/c < 0. This is why fastlight appears. 5How to measure the velocity of the light pulse when fastlight appears in the medium? Suppose a light pulse produced in a source at t= 0, propagating a distance ℓ1in a vacuum, then going through a medium with thickness ℓ2. Just after the medium a light pulse detector is put. Usually the fa stlight appears in the medium after the light pulse is produced and the amplitud e of the fastlight is inceasing when the light pulse is approaching to the medium. If the amplitude of fastlight is able to become large enough to trigger the detec tor at t=t1, one may takev= (ℓ1+ℓ2)/t1as the velocity of the light pulse propagating from the sourc e to the detector, which certainly exceeds c. Although the models proposed above are not completely reali stic, we believe that production of the fastlight and some its properties in the mo dels may remain in a more realistic model, which is under investigation now. We would like to thank professors Gu Yi-fan and Dong Fang-xia o for their ben- eﬁcial discussion. References [1] L. J. Wang, A. Kuzmich and A. Dogariu, Nature, 20 July 2000, Vol. 406, No.6793, p.277. Gain-assisted superluminal light propagation. [2] A. Sommerfeld, Ann. Phys., 44, 177(1914). L. Brillouin, Ann. Phys., 44, 203(1914). L. Brillouin, Wave propagation and group velocity. New York : Academic Press, 1960. Table 1. Some maximal values of light amplitude h(1, θ)near θ= 1 (¯a2= 1.24,¯ρ=−0.07, γ= 1and¯ν= 10in model 1) θ -46.86 -40.70 -34.57 -28.44 -22.35 -16.29 -10.27 h(1, θ)0.00909 0.0125 0.0172 0.0232 0.0316 0.0413 0.0580 -4.59 0.8714 1.1508 1.77912 2.40744 3.03575 0.0742 0.0635 1.0089 1.0089 1.0089 1.0089 6Figure 1: The integration paths of (6). -6 -4 -2 2 4 -1-0.50.51 Figure 2: Figure 2. h(1, θ) (¯a2= 1.24,¯ρ=−0.07, γ= 1 and ¯ ν= 10 in model 1.) 7
arXiv:physics/0008076v1 [physics.optics] 16 Aug 2000Energy focusing inside a dynamical cavity K. Colanero and M. -C. Chu Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. We study the exact classical solutions for a real scalar ﬁeld inside a cavity with a wall whose motion is self-consistently determined by the pressure of t he ﬁeld itself. We ﬁnd that, regardless of the system parameters, the long-time solution always becomes nonadiabatic and the ﬁeld’s energy concentrates into narrow peaks, which we explain by means of a simple mechanical system. We point out implications for the quantized theory. The dynamics of conﬁned cavity ﬁelds interacting with the cavity wall is of great interest for the understand- ing of a variety of problems such as hadron bag models [1], sonoluminescence [2], cavity QED [3] and black hole radiations [4]. Previous works have mostly approached the problem assuming an externally imposed wall mo- tion, neglecting the eﬀects of the radiation pressure, or used the adiabatic approximation [5,6]. In this paper we study, without any approximation, the dynamics of a real scalar ﬁeld inside a cavity, the wall of which moves ac- cording to the combined force of a static potential V(R) and the ﬁeld pressure. This system bears important re- semblances to more complicated ones, such as the Dirac and electromagnetic ﬁelds, since they can be partially or completely cast in the form of a wave equation. Moreover the classical solutions should be a good approximation to the quantized ﬁelds at least in the case of a large number of ﬁeld quanta. As initial condition for the ﬁeld we al- ways consider a normal mode of the static cavity. This is in fact a common situation in the study of many physical systems. We ﬁnd that in general the system evolves nonadia- batically, and the ﬁeld energy concentrates into narrow peaks. This phenomenon can be understood with the help of a simple classical mechanical system. In the present work we use natural units and hence the action Sis dimensionless as are the velocities. This sim- ply means that, although we are dealing with a classical system, for convenience the action is taken in units of ¯ h. In one space dimension and with the ﬁeld only inside the cavity, the system is deﬁned by the action S=/integraldisplayt 0dt′/braceleftBigg 1 2M˙R2−V(R) +/integraldisplayR 0dx1 2/bracketleftbig φ2 t′−φ2 x/bracketrightbig/bracerightBigg . (1) ImposingδS= 0 under any variation of the dynamical variables that vanishes at t′= 0 andt′=twe obtain: M¨R+∂V(R) ∂R−1 2/bracketleftbig φ2 t−φ2 x/bracketrightbig x=R= 0, (2) φtt−φxx= 0 0 ≤x<R , (3)φx= 0 at x= 0, φx=−˙Rφt atx=R .(4) Notice the dependence on ˙Rof the boundary condi- tions. Ifφ(R) = 0 is imposed, the total energy, which is conserved for a static cavity, is no longer constant for ˙R/negationslash= 0. Eq. 3 is satisﬁed by φ(x,t) with φ(x,t) =G(t−x) +G(t+x) (5) and the positive sign between the two G’s ensures that the ﬁrst boundary condition of Eqs. 4 is met. Substitut- ing Eq. 5 in the second of Eqs. 4 we obtain: G′(t+R(t)) =1−˙R(t) 1 +˙R(t)G′(t−R(t)). (6) For prescribed wall motion, G(z) for anyzcan be found by using Eq. 6 and the null line method [5]. It is assumed that the cavity is static for t≤t0with a lengthR(t0). This is equivalent to saying that there is a static zone z≤z0=t0+R(t0), in which G(z) is an- alytically known. One can ﬁnd the values of G(z > z 0) outside the static zone by ﬁrst solving the algebraic equationz=teqv+R(teqv) forteqvand then ﬁnding z−≡teqv−R(teqv). This process, which is equivalent to constructing a null line connecting the points zand z−, can be repeated many times until a point zsin the static zone is reached. The values of G(z) andG(zs) are related through Eq. 6. However in the case under study, we do not have, in general, a static zone, and we need to verify that knowing the initial conditions of the system is enough to implement the above method. We will show that in order to ﬁnd φ(x,t+dt) with 0≤x≤R(t+dt), it is necessary and suﬃcient to know G(z) andG′(z) fort−R(t)≤z≤t+R(t) andR(t′) for t≤t′≤t+dt. That is just what is required in order to have a unique solution of the system of two second order equations (2) and (3). Sinceφ(x,t+dt) =G(t+dt−x) +G(t+dt+x), we need to ﬁnd G(z) andG′(z) fort+dt−R(t+dt)≤ z≤t+dt+R(t+dt). Now we have two cases: either z≤t+R(t) orz>t+R(t). 1In the ﬁrst case it is also true that z≥t+dt−R(t+dt)≥t−R(t) as long as ˙R≤1,i.e.in all physical situations, so that we already have the solution. In the second case we have to solve the equation z=teqv+R(teqv), as explained previously. We have t+R(t)≤teqv+R(teqv)≤t+dt+R(t+dt), which, with ˙R≥ −1, implies t≤teqv≤t+dt. Having found teqvwe can derive G′(z) from Eq. 6 be- cause, with zeqv≡teqv−R(teqv), t−R(t)≤zeqv≤t+R(t) \|˙R\| ≤1, so that again we have the necessary information to de- termine the evolution of the ﬁeld. G(z) can then be ob- tained by the numerical integration of G′(z). Note how- ever that while ˙R= 1 still admits a solution for the ﬁeld, ˙R=−1 doesn’t, because the boundary condition requires G′(t+dt−R(t+dt)) =G′(t+ 2dt−R(t)) = 0, which in general is inconsistent. Evolving backward in time, i.e.withdt<0, the opposite would be true. Using the procedure above we have studied the case withV(R) =1 2K(R−R0)2, solving, step by step, Eq. 2 numerically by a standard ﬁnite diﬀerence method. As initial condition for the ﬁeld we choose the funda- mental mode of the static cavity with Eqs. 4 as the b.c., R(t0) =R0, and ˙R(t0) = 0: /braceleftbiggφ= sinωt0cosωx , φt=ωcosωt0cosωx ,w≡π R0. (7) For convenience we deﬁne the dimensionless parameters αandβ:α≡M/ω,β≡Ω/ω=/radicalbig K/M/ω , and we set the amplitude of the initial ﬁeld to be 1. In the case of a wall initially at rest and with a large mass compared to the initial energy of the ﬁeld, we expect the dynamics not to depart considerably from the adiabatic one, that is, the wall’s motion should be well approximated by the solution of Eq. 2, with the ﬁeld’s pressure term replaced by its static wall counterpart and the solution of Eq. 3 by φ(x,t) = sinω(t)tcosω(t)x w (t)≡π R(t).(8) In order to check the reliability of our numerical im- plementation of the algorithm, we ﬁrst considered a large mass of the wall ( α= 1000/π,β= 1/(10π√ 2)). We ver- iﬁed that the total energy is very well conserved and the motion of the wall is well reproduced by the solution of Eq. 2 with the static wall solution for the ﬁeld pressure.We then used a smaller mass keeping Kconstant, i.e.α= 100/πandβ= 1/(π√ 20). As shown in Fig. 1, both the wall motion and the ﬁeld energy density become nontrivial. An interesting feature is the concentration of the energy density, shown in Fig. 1 c. This is con- ﬁrmed by the plot of the energy density at two instances t= 349R(to) andt= 697R(to) in Fig. 2 compared with the static cavity solution. The two peaks at t= 697R(to) move in opposite directions, and their widths decrease in time. This phenomenon is even more evident with α= 10/πandβ= 1/(π√ 2) (Fig. 3a), showing a com- plex distribution of the peak locations and heights. The total energy of the system is the same in all cases. Even for the case in Fig. 3b ( α= 1000/π,β= 1/(10π√ 2)), for which we observed the adiabatic evo- lution lasting for a long time after t0, we can still, letting the system evolve long enough, observe the squeezing of the ﬁeld energy density in spite of the slow motion of the wall. Keeping Kconstant we found that the time at which the focusing of the energy starts increases roughly linearly with M. This suggests that, as one takes into ac- count the backreaction of the ﬁeld on the wall motion, the long-time dynamics always becomes nonadiabatic. We have veriﬁed that this remains true also changing the boundary conditions so that the ﬁeld equals zero at the boundaries. We believe that the origin of this phenomenon lies in the mechanism of energy exchange between the wall and the ﬁeld. To explain it we give the following qualita- tive argument. Let’s consider the interaction between the wave inside the cavity and the wall. At some in- stance, the peak of the wave will hit the wall, which can be moving either outward or inward. In the former case, there will be a transfer of energy from the ﬁeld to the wall, and the speed of the wall will increase slightly. The wave- fronts following the peak will lose more and more energy to the wall, since the wall moves faster with each succes- sive collision. As a result the spatial width of the energy distribution decreases. When the wall moves inward, the wave gains energy from the wall, and the wavefronts fol- lowing the peak gain less because the wall moves slower with each successive collision. Again the width of the waveform decreases. After some time, this eﬀect leads to a drastic concentration of energy into narrow peaks. Our argument depends only on kinematics and should therefore be applicable not only to waves but many other systems, such as a set of particles bouncing back and forth in a dynamical cavity. For simplicity we consider the dynamics of a set of massless non-interacting parti- cles, each having momentum and energy pi,\|pi\|(c= 1). Inside the cavity they move unperturbed at the speed of light. If a particle bounces on the static wall, its momen- tum changes sign. The movable wall is subjected to a harmonic potential V(R) =1 2K(R−R0)2. The particle momentum p′′ iand the wall velocity v′′after an inter- action, which is assumed to be instantaneous, are easily 2derived from energy and momentum conservation: v′′=/radicalbig (1 +v′)2+ 4p′/M−1, p′′=p′+M(v′−v′′),(9) wherev′andp′are the wall velocity and particle mo- mentum before the collision. The above equations are derived assuming that the sign of p′′is always opposite to the sign of p′, which is true as long as the speed of the wall is less than 1 and 2 M(1−v′)>p′(p′>0). We consider ﬁrst a set of 1000 particles all with the same initial momentum pi= 0.01/R(t0) and a wall ini- tially at rest with M= 1000/R(t0) and Ω = 1 /R(t0). Already after a few interactions with the wall we could observe a regular transfer of energy from the last to the ﬁrst particles to hit the wall. In Fig. 4a we show the momenta of the particles after a time t= 3221R(t0) as a function of their position. For clarity only positive mo- menta are plotted. It is remarkable that the ﬁrst particle to hit the wall has gained more than one tenth of the total energy of the system. The above is a very special situ- ation which however demonstrates the process of energy transfer among particles. We then extend this simple mechanical model to the case of an inﬁnite number of particles labeled with a con- tinuous index k, each having position q(k) and momen- tump(k)dk. In this way we can deﬁne an energy density: E(x,t)≡/integraldisplay dk\|p(k,t)\|δ(q(k,t)−x). (10) Not surprisingly E(x,t) satisﬁes the wave equation in- side the cavity. We numerically simulate such a system choosing 2000 particles. Initially, we put two particles at each of the 1000 uniformly separated sites, and the pairs have opposite momenta p(k) =±10(π2cos2πq(k) + 1). In Fig. 4b we plot E(0,t)R2(t), which is evidently similar to Fig. 1, although the details of the evolution depend on how the particles or the ﬁeld interact with the wall. After a long time we observe the formation of many smaller peaks in the energy density. Further work is needed to understand the problem of the t→ ∞ evo- lution of the system. For the scalar ﬁeld an important situation to study is when Ω = π/R0,i.e., when the wall motion is in reso- nance with the ﬁeld inside the cavity. We have computed the solutions of Eqs. 2 and 3 for various masses of the wall. In Fig. 5 we plot the wall’s position and the ﬁeld en- ergy density at x= 0 vs. time in the case of α= 1000/π, R(t0) =R0and˙R(t0) = 0.1. In this case we choose t0=R0/2 so that ˙φ= 0 and the initial functions Eq. 7 satisfy the boundary condition Eqs. 4 with ˙R(t0)/negationslash= 0. Besides the beats in the wall motion, two features are important. One is the fact that the wall continues to re- turn to its initial position after a time T=R0. This is diﬀerent from the case of non-resonant wall parameters where the back reaction of the ﬁeld changes the frequencyof the wall motion. Another remarkable eﬀect, as a con- sequence, is the appearance of narrow peaks typical of a resonantly driven wall motion [5,7,9]. This indicates the possibility of transferring a large amount of energy to the ﬁeld even with an external, non-resonant, driving force[8,10]. As long as the frequency of the cavity wall is Ω =π/R0, it is enough to push the wall at the instances marked by the arrows in Fig. 5, and this frequency de- pends on the mass of the wall and can be much smaller than Ω; increasing the mass decreases the frequency of energy exchange between wall and ﬁeld. This fact might help to by-pass the experimental diﬃculty of achieving a resonant driving force, i.e. at frequency Ω, on a mirror in order to produce high frequency photons [8]. We have veriﬁed that for a small mass, α= 10/π, the wall period remains close to T= 2R0so that the motion is still resonant [7]. In Ref. [7] it has been shown that the method of null lines can also be applied to waves inside an oscillating spherical cavity for any value of the angular momentum. However, when considering a self-consistent wall motion, the spherical symmetry is achieved only in the case of s- waves, for which the radial ( φ) and angular parts can be separated. Deﬁning ψ≡rφ, so thatψsatisﬁes the one- dimensional wave equation, we can apply the null lines method. The boundary condition for φ, derived from the action similarly to Eqs. 4, is: ˙Rφt(R(t),t) =−φr(R(t),t), (11) which however for ψtranslates to: ˙Rψt(R(t),t) =ψ(R(t),t) R(t)−ψr(R(t),t). (12) If we want φto be ﬁnite at r= 0 then we must re- quireψ= 0 atr= 0, which is satisﬁed by writing ψ=G(t−r)−G(t+r). Eq. 12 becomes: G′(t+R(t))−ηG(t+R(t)) =γ=−1−˙R 1+˙RG′(t−R(t))−ηG(t−R(t)),(13) withη≡1/R/parenleftBig 1 +˙R/parenrightBig . An eﬀective way to solve Eq. 13 numerically forG(t+R(t)) is to deﬁne z=t+R(t) and to approximate ηandγwith a constant value between zandz−dzfor a small enough dz. Integrating Eq. 13 betweenzandz−dzwe obtain: G(z) =/bracketleftbigg G(z−dz) +γ η/bracketrightbigg eηdz−γ η, (14) which turns out to be more accurate than standard nu- merical integration. The force of the s-wave ﬁeld on the wall is Fφ= 2πR2(t)/bracketleftbig φ2 t−φ2 r/bracketrightbig . For ˙R(t0) = 0 we set as initial condi- tions for the ﬁelds: 3/braceleftBigg φ(r,t0) =cosωt0 R2 0sinωr ωr, φt(r,t0) =−ωsinωt0 R2 0sinωr ωr,(15) whereω≃4.4934/R(t0) is chosen such that φ(r,t0) sat- isﬁes Eq. 11 with ˙R(t) = 0. As in the 1D case, we observe the formation of high energy density regions, although in 3D, this process is much slower. In Fig. 6 we plot the energy density at r=R(t) vs. time for α= 5/4.4934 and β=√ 8/4.4934. These values of the parameters produce a completely non-adiabatic evolution. For larger Mor smallerKwe have to evolve the system for a much longer time in order to observe the formation of high energy peaks. However we have veriﬁed that imposing φ= 0 atr=R(t) the peaks appear much earlier and the dy- namics is very similar to the one-dimensional situation. With resonant wall parameters, Ω = π/R(t0), the fea- tures observed in 1D remain in 3D. With the b.c. Eq. 11 it is also possible to have resonances with Ω equal to the diﬀerence between the frequencies of the nthmode and the fundamental mode of the cavity. However such an Ω is close tonπ/R(t0) ifnis large, and such resonances are not easily distinguishable from the geometric ones [7]. In summary we have applied the null lines method to study the dynamics of a scalar ﬁeld inside a cavity whose wall is subjected to a harmonic force and the pressure due to the scalar ﬁeld. We have found that the long time evolution of the system is always non-adiabatic, re- gardless of the parameters of the system. In particular there is an interval of time when the ﬁeld develops nar- row packets in energy density that bounce back and forth inside the cavity, which can be understood by means of a simple mechanical analog consisting of a set of mass- less particles bouncing inside a one-dimensional box with a movable wall. Such a system conﬁrms our hypothesis that the wall motion provides a mechanism of energy transfer from low to high energy regions. We have veri- ﬁed that the focusing of energy is a robust phenomenon, being insensitive to the type of potential for the wall and the presence of an external driving force. For a quantized ﬁeld previous works [8] have shown that in the case of a prescribed slow wall motion no pho- ton production is achieved. Our results strongly suggest that the back-reaction of the ﬁeld may change signiﬁ- cantly the evolution of the system. In particular the sec- ond derivative of the wall position, which is one of the quantities that determine the number of quanta [8], can be much larger than in the adiabatic case, as it can be seen from the slope of ˙Rin Fig.1. If the initial number of fundamental mode quanta is large, the peaks in energy density in the classical solution can imply the production of several high energy quanta. We have also studied the special situation in which the wall frequency is equal to the fundamental frequency of the static cavity ﬁeld. Remarkably the frequency of the wall motion does not change due to the ﬁeld pressure, and thus narrow peaks typical of a resonantly driven wallmotion are produced. A large amount of energy may be transferred to the ﬁeld by providing mechanical energy to the wall when the amplitude of the oscillation reaches its minimum. This fact might help to by-pass the experi- mental diﬃculty of achieving a resonant driving force on a mirror in order to produce high frequency photons [8]. In a further work we would like to address the problem of whether periodical solutions are admitted for this kind of system and for which values of parameters. We would like to thank Dr. C. K. Law for his in- terest in the paper and valuable discussions. This work is partially supported by a Hong Kong Research Grants Council grant CUHK 312/96P and a Chinese University Direct Grant (Project ID: 2060093). [1] P. Hasenfratz and J. Kuti, Phys. Rep. 40, 75 (1978). [2] B. P. Barber et al., Phys. Rep. 281, 65 (1997), and ref- erences therein. [3] G. T. Moore, J. Math. Phys. 11, 2679 (1970); P. W. Milonni, The Quantum Vacuum (Academic Press, New York, 1993); N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982). [4] S. W. Hawking, Nature 248, 30 (1974); Com- mun. Math. Phys. 43, 199 (1975). [5] C. K. Cole and W. C. Schieve, Phys. Rev. A 52, 4405 (1995), and references therein. [6] P.Meystre et al., J. Opt. Soc. Am. B 2, 1830 (1985) [7] K. W. Chan, U. M. Ho, P. T. Leung, and M.-C. Chu, The Chinese University of Hong Kong Preprint, 2000 (unpub- lished); K. W. Chan, Master Thesis, The Chinese Uni- versity of Hong Kong (unpublished), 1999. [8] V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 (1996), and references therein. [9] C. K. Law, Phys. Rev. Lett. 73, 1931 (1994). [10] A. Lambrecht, M. T. Jaekel and S. Reynaud, Phys. Rev. Lett. 77, 615 (1996). 4300 310 320 330 340t/R(t0)0510 ε(0,t) R2(t)−0.10−0.050.000.05R(t)0.81.01.21.41.61.8 R(t)/R(t0)non adiabatic result adiabatic approximation .a) b) c) FIG. 1. a) Wall position, b) wall velocity, c) energy density of the ﬁeld at x= 0 for α= 100 /π,β= 1/(π√ 20). 0 0.2 0.4 0.6 0.8 1 x/R(t)0510ε(x,t) R2(t)t/R(t0)=0 t/R(t0)=349 t/R(t0)=697 FIG. 2. Spatial distribution of energy density at t/R(to) = 0 (dot-dashed), 349 (dashed line), and 697 (solid line) for α= 100 /π,β= 1/(π√ 20). 1980 1985 1990 1995 2000t/R(t0)02468ε(0,t) R2(t) 30 35 40 45 5001020a) b) FIG. 3. Energy density of the ﬁeld at x= 0 for: a) α= 10/π,β= 1/(π√ 2), b) α= 1000 /π,β= 1/(10π√ 2). Notice the time intervals.635 640 645 650t/R(t0)0.020.040.060.080.10.12ε(0,t) R2(t)0 0.2 0.4 0.6 0.8 1x/R(t0)00.511.5pi R(t0) −> a) b) FIG. 4. Classical particles in a dynamical cavity, with M= 1000 /R(t0), Ω = 1 /R(t0), and initial momenta 0.01/R(t0). a) Particle momenta at t= 3221 R(t0). b) Gen- eralized energy density at x= 0. 0 20 40 60 80t/R(t0)0204060ε(0,t) R2(t)0.960.9811.021.04R(t)/R(t0)a) b) FIG. 5. a) Wall position and b) energy density in a reso- nant cavity with α= 1000 /πandβ= 1. 30 35 40 45 50t/R(t0)0510152025ε(R(t),t) R4(t) FIG. 6. Energy density in a spherical cavity at r=R(t) vs. time for α= 5/4.4934 and β=√ 8/4.4934. 5
arXiv:physics/0008077v1 [physics.ins-det] 16 Aug 2000Annealing of radiation induced defects in silicon in a simpliﬁed phenomenological model S. Lazanuaand I. Lazanub aNational Institute for Materials Physics, P.O.Box MG-7, Bu charest-Magurele, Romania, electronic address: lazanu@alpha1.inﬁm.ro bUniversity of Bucharest, Faculty of Physics, P.O.Box MG-11 , Bucharest-Magurele, Romania, electronic address: ilaz@s cut.ﬁzica.unibuc.ro Abstract The concentration of primary radiation induced defects has been previously esti- mated considering both the explicit mechanisms of the prima ry interaction between the incoming particle and the nuclei of the semiconductor la ttice, and the recoil energy partition between ionisation and displacements, in the frame of the Lind- hard theory. The primary displacement defects are vacancie s and interstitials, that are essentially unstable in silicon. They interact via migr ation, recombination, an- nihilation or produce other defects. In the present work, th e time evolution of the concentration of defects induced by hadrons in silicon is mo delled, after irradiation. In some approximations, the diﬀerential equations represe nting the time evolution processes could be decoupled. The theoretical equations so obtained are solved an- alytically for a wide range of particle ﬂuences and/or for a w ide energy range of the incident particles, for diﬀerent temperatures, and the corresponding stationary solutions are presented. PACS : 61.80.Az: Theory and models of radiation eﬀects 61.70.At: Defects formation and annealing processes Key words: radiation damage, hadrons, atom displacements, kinetics o f defects, annealing processes 1 Introduction A point defect in a crystal is an entity that causes an interru ption in the lattice periodicity. In this paper, the terminology and deﬁnitions in agreement with M. Lannoo and J. Bourgoin [1] are used in relation to defects. Preprint submitted to Elsevier Preprint 21 February 2014Vacancies and interstitials are produced in materials expo sed to irradiation in equal quantities. In radiation ﬁelds, after the interactio n between the incoming particles and the target, mainly two classes of degradation eﬀects are observed: surface and bulk damage, the last one due to the displacement of atoms from their sites in the lattice. For electrons and gammas, the eﬀe cts are dominantly at the surface, while heavy particles (pions, protons, neut rons, ions) produce both types of damages. In silicon, vacancies and interstitials are essentially un stable and interact via migration, recombination, annihilation or produce other d efects. The system evolves toward natural equilibrium. The problem of the annealing of radiation induced defects in semiconduc- tor materials is old. Several models, empirical or more theo retic, have been previously proposed to explain these phenomena; see for exa mple [2–8] and references cited therein. In the present paper, the time evolution of the primary radia tion induced defects in silicon is studied, in the frame of a simpliﬁed phe nomenological model based on direct interactions between the primary indu ced defects and the impurities present in the material, assuming that the an nealing phenomena start after irradiation. 2 Equations for the kinetics of radiation induced defects. G eneral formulation The physical model under investigation is the following: eq ual concentrations of vacancies and interstitials have been produced by irradi ation, in much greater concentrations than the corresponding thermal equ ilibrium values, corresponding to each temperature. Both the pre-existing d efects and those produced by irradiation, as well as the impurities, are assu med to be randomly distributed in the solid. An important part of the vacancies and interstitials annihilate. The sample contains certain concentrations of impurities which can trap interstitials and vacancies respectively, and form st able defects. In the present paper, vacancy-interstitial annihilation, interstitial migration to sinks, vacancy and interstitial impurity complex formatio n as well as divacancy formation are considered. The sample could contain more imp urities that trap vacancies or interstitials, and in this case all processes a re to be taken into account. The following notations are used: V- monovacancy concentration; I- free interstitial concentration, J1- total impurity ”1” concentration (impurity ”1” 2traps interstitials and forms the complex C1);C1- interstitial-impurity con- centration: one interstitial trapped for one complex forme d;J2- total impu- rity ”2” concentration (impurity ”2” traps vacancies and fo rms the complex C2);C2- vacancy-impurity concentration: one vacancy trapped for one com- plex formed; V2- divacancy concentration. All concentrations are express ed as atomic fractions. This picture could be written in terms of chemical reactions by the simple kinetic scheme: V+IK1→annihilation (1) IK2→sinks (2) J1+IK3−→←−K4C1 (3) J2+VK5−→←−K6C2 (4) V+VK7−→←−K8V2 (5) The corresponding diﬀerential equations are: dV dt=−K1V I−K5V(J20−C2) +K6C2−K7V2+K8V2 (6) dI dt=−K1V I−K2I−K3I(J10−C1) +K4C1 (7) dC1 dt=K3I(J10−C1)−K4C1 (8) 3dC2 dt=K5V(J20−C2)−K6C2 (9) dV2 dt=1 2K7V2−1 2K8V2 (10) IfNis the total defect concentration, expressed as atomic frac tion: N=V+I+ 2V2+C1+C2 (11) then it satisﬁes the diﬀerential equation: dN dt=−2K1V I (12) The initial conditions, at t= 0, are: at the end of the irradiation, there are equal concentrations of interstitials and vacancies; I0=V0; the concentrations of impurities are J10andJ20respectively, there are no complexes in the sample: C10=C20= 0 and no divacancies V20= 0. The reaction constants K1andK3are determined by the diﬀusion coeﬃcient for the interstitial atom to a substitutional trap, and ther eforeK1=K3: K1= 30νexp (−Ei1/kBT) (13) where Ei1is the activation energy of interstitial migration and νthe vibra- tional frequency. The reaction constant in process (2) is pr oportional to the sink concentration α: K2=ανλ2exp (−Ei1/kBT) (14) withλthe jump distance. 4K4= 5νexp/parenleftbiggEi1+B1 kBT/parenrightbigg (15) K5= 30νexp (−Ei2/kBT) (16) withEi2the activation energy for vacancy migration; K6= 5νexp/parenleftbigg −Ei2+B2 kBT/parenrightbigg (17) K7= 30νexp (−Ei2/kBT) (18) K8= 5νexp/parenleftbigg −Ei2+B3 kBT/parenrightbigg (19) where B1is the binding energy of C1,B2the binding energy of C2andB3is the corresponding binding energy of divacancies. Due to the mathematical diﬃculties to solve analytically th e complete diﬀer- ential equation system, some simpliﬁcations are necessary . 3 Hypothesis, approximations and discussions The interstitials are much more mobile in silicon in respect to vacancies, and are characterised by an activation energy of migration a fac tor of 2 times smaller. This fact permits the introduction of the followin g hypothesis: the ﬁrst two processes are the most rapid, and we introduce two time sc ales: in the ﬁrst one, the processes (1) and (2) are studied, and interstitial concentration decays much rapidly than vacancy concentration. A cut-oﬀ conditio n forIis imposed (aptimes decrease of interstitial concentration). The vacanc y concentration determined by this procedure is the initial value for the pro cesses (4) and (5), and will be denoted by Vi. The process (3) is considered less important, and is neglected in the following discussion. 5So, for the ”ﬁrst stage”, after some simple manipulations, f rom (1) and (2), the equation: dI dt= 1 +K2 K1V(20) has been obtained, with the solution: I=V+K2 K1lnV V0(21) and: t=ln/bracketleftBig 1 +K1V(t) K2ln(V(t))/bracketrightBig K2ln (V(t))(22) Imposing the cut-oﬀ condition for the concentration of inte rstitials, both Vi and the characteristic time could be found. As speciﬁed, Viis used as initial vacancy concentration for the second step in the analysis, w here the equations (4) and (5) are considered. This system of equations, expres sing the kinetics of divacancy and vacancy-impurity formation, have no analy tical solution. These processes are governed by the initial concentrations of vacancies and impurities. If the impurity that traps vacancies is phosphorus, the limi ting cases cor- respond to low initial doping concentration (high resistiv ity, uncompensated materials) and very high impurity concentration (low resis tivity), respectively. In both cases the equations could be decoupled, and analytic al solutions are possible. So, if the formation of vacancy-impurity complexes is not so important, the main process responsible for the decay of vacancy concentra tion is divacancy production. In this case, the time evolution of the vacancy c oncentration is described by the law: V(t) =1 4K7  −K8+RK8+4K7Vi R+ tanh/parenleftBig tR 4/parenrightBig 1 +K8+4K7Vi Rtanh/parenleftBig tR 4/parenrightBig  (23) 6where: R≡/radicalBig K8(K8+ 8K7Vi) (24) while the increase of the divacancy concentration is given b y: V2(t) =Vi−V(t) 2(25) The stationary solution for V(t) is given by: lim t→∞V(t) =1 4K7(R−K8) (26) For n-type high doped Si, the process described by eq. (4) is t he most probable. IfJ0is the initial concentration of impurities, and the initial concentration of complexes is zero, than: V(t) =1 2K5  −K6+K5(Vi−J0) +R∗K6+K5(Vi−J0) R∗ + tanh/parenleftBig tR∗ 2/parenrightBig 1 +K6+K5(Vi−J0) R∗ tanh/parenleftBig tR∗ 2/parenrightBig  (27) with: R∗≡/radicalBig K2 6+K2 5(Vi−J0)2+ 2K5K6(Vi+J0) (28) and with the stationary solution: lim t→∞V(t) =1 2K5[K5(Vi−J0) +R∗−K6] (29) and 7C=Vi−V (30) 4 Results and physical interpretations In Figure 1, the concentration of primary defects per unit ﬂu ence (CPD) induced by pions and protons in silicon is presented as a func tion of the kinetic energy of the particle. For pions, the curves have been calcu lated in [9,10] and proton data are from [11,12]. The diﬀerence between the two energy dependencies comes fro m the pecu- liarities of the behaviour of pions and protons in the intera ction with the semiconductor material, and these are underlined in refere nces [13,14]. The process of partitioning the energy of the recoil nuclei (pro duced due the in- teraction of the incident particle with the nucleus, placed in its lattice site) in new interaction processes, between electrons (ionisati on) and atomic mo- tion (displacements) has been considered in the frame of the Lindhard theory. The CPD for protons presents an abrupt decrease at low energi es, followed by a minimum and a plateau at higher energies. For pions, there e xists a large maximum in the energy range where the resonance is produced, followed by a slight monotonic decrease at higher energies. Some local m axima are also presents at high energies, but with less importance. The CPD multiplied by the ﬂuence is the initial value of the co ncentration of vacancies and interstitials, and in the forthcoming discus sion it is expressed, as speciﬁed, as atomic fraction. The development of the model will be illustrated, without lo ss of generality, on pions induced defects. The diﬀerence between the activation energies Ei1andEi2respectively justiﬁes the introduction of two time scales, and the separate study o f processes (1), (2) and (4), (5) respectively. In ﬁgure 2, the time evolution of the concentration of vacanc ies, intersti- tials, divacances and vacancy-impurity complexes produce d in silicon by 1015 pions/cm2irradiation are presented as a function of the kinetic energ y of the particles. The following values of the parameters have been used: Ei1= 0.4 eV,Ei2= 0.8 eV, B1= 0.2 eV, B2= 0.2 eV, B3= 0.4 eV, ν= 1013Hz,λ2= 1015cm2,α= 1010cm−2, and all curves are calculated for the temperature T = 293 K. Figures 2a and 2b correspond to vacancy-interstitial annih ilation and to in- 8terstitial migration to sinks. A p=100 times decrease of the concentration of interstitials conduces to a speciﬁed concentration of vaca ncies, Vi, that is the initial concentration for the second step. The time scale of these processes is of the order of the second. Figures 2c and 2d correspond to divacancy formation by proce ss (5) from the vacancies remained after annihilation, in the case vacancy -impurity complex formation is neglected. The case of high initial impurity (phosphorus) concentrati on, where divacancy formation is neglected in respect to complex formation, is p resented in ﬁgures 2e and 2f respectively. Some explicit considerations must be done about the formati on of vacancy- impurity complexes. The mechanisms supposed above can be us ed both for boron and for phosphorus impurities. In this case, the corre sponding processes are [8]: Bi+V→[Bi−V] (31) and respectively: Ps+V→[Ps−V] (32) While the complex formed by boron is unstable and self anneal s bellow room temperature, the interaction between a Vand a Psleads to the formation of anEcentre which is stable in the same conditions. Interactions between interstitial oxygen (another very st udied impurity in the last time) and free vacancies is described as a higher order p rocess (third order in the [8] and fourth power in [15]). If the process is stopped as a ﬁrst order one, the time evolution of the concentrations is not diﬀeren t from the case of phosphorus, studied before. If two or more impurities that trap vacancies are considered as existing si- multaneously in silicon, the system of coupled equations mu st be solved. Only numerical solutions for particular cases are possible. Coming back to the ﬁrst step of the process, it is to note that t he decrease of vacancy concentration (for p= 100 diminish of interstitial concentration) is much more important for higher initial vacancy concentra tions. This idea 9is illustrated in Figure 3, where the ratio V/V0is represented as a function of particle ﬂuence, for pions of 150 MeV kinetic energy. The wei ght of the anni- hilation process in respect to interstitial migration to si nks increases abruptly with the ﬂuence. At low and intermediate ﬂuences, up to 1014pions/cm2, the annihilation has a low importance. This curve is temperatur e independent. On the other side, the characteristic time of the ﬁrst proces s step, correspond- ing to a p=100 times decrease of interstitial concentration, is a fun ction on the initial vacancy concentration and on the temperature. I ts dependence on pion ﬂuence, for 150 MeV kinetic energy of pions, is represen ted in Figure 4, for -20oC , 0oC and 20oC temperatures, respectively. It could be seen that, up to 1016pions/cm2, the characteristic time is independent on the ﬂuence, and for higher ﬂuences a decrease of the characteristic time is t o be noted and a ﬂuence dependence is obtained. The characteristic times for the diﬀerent processes estima ted in this work are in general agreement with the experimental data obtained by Z. Li and co- workers [16] after neutron irradiated silicon. This way, du e to the fact that the characteristic times of diﬀerent physical process, a direc t correspondence with the microscopic mechanisms in the semiconductor material c ould be done. 5 Summary The time evolution of the primary concentration of defects i nduced by hadrons after irradiation process is modelled. Vacancy-interstitial annihilation, interstitial migrat ion to sinks, vacancy and interstitial impurity complex formation as well as divacan cy formation are considered. Always it is possible to decouple the time evolu tion of impurity concentrations into two steps, the ﬁrst one involving vacan cy-interstitial an- nihilation and interstitial migration to sinks, the second vacancy-complex and divacancy formation. The equations corresponding to the ﬁrst step are solved anal ytically for a wide range of particle ﬂuences and for a wide energy range of incid ent particles, and for diﬀerent temperatures: -20oC, 0oC and 20oC. The approximations that permit to decouple the diﬀerential equations repre- senting the time evolution processes in the second step have been studied, and the processes have been treated separately. The concomitan t consideration of more processes in this last step is possible only numericall y. 106 Acknowledgements The authors are very grateful to Professor Gh. Ciobanu from t he Bucharest University for helpful discussions during the course of thi s work. References [1] M. Lannoo, J. Bourgoin, ”Point Defects in Semiconductors” , Springer Series in Solid State Science 2, Eds. M. Cardona, P. Fulde, H.-J. Que isser, Springer- Verlag 1981. [2] A. C. Damask, G. J. Dienes, Phys. Rev. 125(1962) 444. [3] G. J. Dienes, A. C. Damask, Phys. Rev. 125(1962) 447. [4] G. J. Dienes, A. C. Damask Phys. Rev. 128(1962) 2542. [5] A. C. Damask, G. J. Dienes, Phys. Rev. 120(1960) 99. [6] M. Moll, H. Feick, E. Fretwurst, G. Lindstrom, T. Schultz , Nucl. Phys. (Proc. Suppl.) 44B(1998) 468. [7] S. J. Bates, C. Furetta, M. Glaser, F. Lemeilleur, C. Soav e, E. Leon-Florian, Nucl. Phys. (Proc. Suppl.) 44B(1998) 510. [8] I. Tsveybak, W. Bugg, J. A. Harvey, J. Walker, IEEE Trans. Nucl. Sci. NS-39 (1992) 1720. [9] S. Lazanu, I. Lazanu, Nucl. Instr. and Meth. in Phys. Res. A 419 (1998) 570. [10] S. Lazanu, I. Lazanu, U. Biggeri, E. Borchi, M. Bruzzi, N ucl. Phys. (Proc. Suppl.) 61B(1998) 409. [11] E. Burke, IEEE Trans. Nucl. Sci. NS-33 (1986) 1276. [12] A. van Ginneken, preprint Fermi National Accelerator L aboratory, FN-522, 1989. [13] I. Lazanu, S. Lazanu, Nucl. Instr. and Meth. in Phys. Res .A 432 (1999) 374. [14] I. Lazanu, S. Lazanu, E. Borchi, M. Bruzzi, Nucl. Instr. and Meth. in Phys. Res.A 406 (1998) 259. [15] H. Reiss, Journal of App. Phys. 30(1959) 141. [16] Z. Li, W. Chen, H. W. Kraner, Nucl. Instr. and Meth. in Phy s. Res. A 308 (1991) 585. 11Figure captions Figure 1: Energy dependence of the concentration of primary defects on unit ﬂuence, induced by pions (continuous line) and protons (das hed line) in silicon. Figure 2: Dependence of the defect concentrations (atomic f raction) on time and pion kinetic energy, after 1015pions/cm2irradiation. a) Interstitial concentration versus time and pions kineti c energy in the case only vacancy-interstitial annihilation and interstitial migration to sinks are considered. b) Vacancy concentration versus time and pion kinetic energ y in the same condition as in 2a). c) Vacancy concentration versus time and pion kinetic energ y when divacancy formation is the only process considered. d) Divacancy concentration versus time and pion energy, in t he same condi- tions as in 2c). e) Concentration of vacancies as a function of time and pion k inetic energy, forJ20=4.5 1018atoms/cm3, when vacancy-impurity formation is considered. f) Same as e) for the concentration of complexes. Figure 3: Ratio of vacancy concentration on initial vacancy concentration after ap=100 times decrease of interstitial concentration versus t he ﬂuence of 150 MeV kinetic energy of pions, in the case when vacancy-inters titial annihilation and interstitial migration to sinks are considered. Figure 4: Characteristic time corresponding to a p=100 times decrease of inter- stitial concentration versus the ﬂuence of 150 MeV kinetic e nergy pions, for -20oC, 0oC and +20oC temperatures. Only vacancy-interstitial annihilation and interstitial migration to sinks are considered. 1202004006008001000 00.5101234x 10-9 Kinetic Energy [MeV]Time [sec]Interstitial Concentration (atomic fr.)02004006008001000 00.511.522.533.54x 10-9 Kinetic Energy [MeV]Time [sec]Vacancy Concentration (atomic fr.)55.566.577.58 0500100001234x 10-9 log10(t [sec])Kinetic Energy [MeV] Vacancy Concentration [atomic fr.]55.566.577.58 05001000012x 10-9 log10(t [sec])Kinetic Energy [MeV] Divacancy Concentration [atomic fr.]11.522.533.54 05001000123x 10-9 log10(t [sec])Kinetic Energy [MeV] Vacancy Concentration [atomic fr.]11.522.533.54 0500100000.511.522.5x 10-9 log10(t [sec])Kinetic Energy [MeV]Complex Concentration [atomic fr.]10 010 110 210 310 410 5 Particle Kinetic Energy [MeV]10 -310 -210 -110 010 1CPD on unit fluence (1/cm)10 1410 1510 1610 1710 18 Fluence [part./cm2]0.00.20.40.60.81.0Vi/V010 910 1010 1110 1210 1310 1410 1510 1610 1710 18 Fluence [part./cm2]012345Characteristic time [s] - 20OC 0OC 20OC
arXiv:physics/0008078v1 [physics.acc-ph] 16 Aug 2000LATESTDEVELOPMENTSFROM THE S-DALINAC∗ M. Brunken, H.Genz, M. Gopych,H.-D. Graef, S. Khodyachykh, S. Kostial,U. Laier, A. Lenhardt, H. Loos,J.Muehl,M. Platz, A.Richter, S. Richter, B. Schwei zer, A. Stascheck, O. Titze, S. Watzlawik(TU Darmstadt),S. Doebert(CERN) Abstract The S-DALINAC is a 130 MeV superconducting recircu- lating electron accelerator serving several nuclear and ra - diation physics experiments as well as driving an infrared free-electron laser. A system of normal conducting rf res- onators for noninvasive beam position and current mea- surementwasestablished. Forthemeasurementofgamma- radiationinside theacceleratorcavea systemof Compton- diodes has been developed and tested. Detailed investiga- tions of the transverse phase space were carried out with a tomographical reconstruction method of optical transitio n radiation spots. The method can be applied also to non- Gaussianphasespacedistributions. Theresultsareingood accordance with simulations. To improve the quality fac- tor of the superconducting 3 GHz cavities, an external 2K testcryostat was commissioned. The inﬂuence of electro- chemicalpolishingandmagneticshieldingiscurrentlyun- der investigation. A digital rf-feedback system for the ac- celerator cavities is being developed in order to improve the energyspreadofthebeamfromtheS-DALINAC. 1 INTRODUCTION A comprehensive discussion of the layout and the proper- ties of the recirculating superconducting electron accele r- ator S-DALINAC is given in [1]. The electrons are emit- ted by a thermionic gun and then accelerated electrostat- ically to 250 keV. A normal conducting 3 GHz chopper- prebuncher system creates the required 3 GHz time struc- ture of the beam. An additional subharmonic 600 MHz chopper/buncherallowsfora10MHzbunchrepetitionrate forFELoperation. Thesuperconductinginjectorlinaccon- sistsofone2-cellcapturecavity( β=0.85),one5-cellcavity (β=1),andtwo20-cellcavitiesoperatedinliquidheliumat 2 K. The electron beam behind the injector with a max- imum energy of 10 MeV can either be directed to a ﬁrst experimental site or it can be injected into the main linac. There,eight20-cellcavitiesprovideanenergygainofupto 40MeV.Whenleavingthemainlinac,thebeamcanbeex- tractedtotheexperimentalhalloritcanberecirculatedan d reinjected one or two times. The maximum beam energy after three passes through the linac amounts to 130 MeV. AninfraredFELwithwavelengthsbetween3and10 µmis driven by the electron beam with an energy from 25 up to 50MeV. Forthedifferentexperiments,abeamcurrentfromsome ∗Supported by DFG under contract no. FOR 272/2-1 and Graduiertenkolleg ”Physik und Technik von Beschleunigern ”nAupto60 µAcanbedelivered. Inthesubharmonicinjec- tion mode, a peak current of 2.7 A can be passed through the FELundulator. 2 BEAM- ANDPOSITION MONITORS A combination of normal conducting TM 010- and TM 110- cavitiesasdisplayedinﬁg.1wasrecentlydevelopedforthe S-DALINAC to measure the beam intensity and position. Thecavitiesarefabricatedfromstainlesssteel, theyhave a common centerpiece and two covers which connect to the beam line. The rf outputs use ceramic feedthroughs. The monitorsareoperatedatloadedQsoflessthan1000. Thus, theyneednofrequencyortemperaturestabilization. Figure1: Nonintercepting3GHz rfmonitor. Thesensitivityis 15nW/( µA)2fortheintensitymonitor and15pW/(mm µA)2forthepositionmonitor. Forthede- tectionoftheratherlowsignals,lockintechniquesareuse d. Dedicated electronicsclose to the monitor convertthe sig- nal to a dc voltage, enabling even the measurement of a 0.1 mm beam position change at a beam current of 1 µA or a 10 nA currentchange. Seven monitorunits have been installed indifferentsectionsof theaccelerator. Themon i- torsignalscanbedisplayedgraphicallyintheS-DALINAC controlroom. 3 COMPTON-DIODES Foradetailedexaminationofeffectsofthebremsstrahlung background in the accelerator cave on accelerator system components,amonitoringsystemhasbeenconstructedand is currentlybeingtested. Thelayoutof thebremsstrahlung monitors(also referredto as Compton-diodes)is shown in ﬁg. 2. They consist of an inner lead electrodeand an outerFigure2: Layoutofa Compton-diode. aluminiumelectrodeinsulatedbyplexiglas. Duetothedif- ferent Compton cross sections of the electrodes, a photon beam penetrating the monitor creates a small current be- tween the electrodes, typically several pA for a dose rate of 10 mSv/hr. This current is converted to a voltage, am- pliﬁed and read out via ADCs. The linearity of the output voltage over the photon ﬂux was demonstrated at a radia- tion physics setup behind the injector. The electrons were targetedontoacopperbremsstrahlungconverter,theresul t- ing gamma beam was collimated by a copper collimator. Figure 3showsthe monitoroutputvoltageas a functionof the electron current on the converter target. The Compton diodes are very rugged and form a ﬂexible system which can monitor any location outside the beam pipe. Thus ra- diationimpactonacceleratorcomponentscanbemeasured andbeamlossescanbedetected. Figure3: LinearityoftheCompton-diodeshowninﬁg.2.4 TRANSVERSE PHASE SPACE TOMOGRAPHY The method of transverse phase space tomography[2] has beenappliedtotheelectronbeambehindtheinjectorofthe S-DALINAC. The setup shown in ﬁg. 4 consists of an op- tical transition radiation (OTR) target, a CCD camera and a PC with a framegrabber board. Two quadrupoles have beenusedtochangethebeamtransportmatrixaccordingly. A computercode written in the InteractiveData Language (IDL) reconstructs the transverse phase space with a to- mographical algorithm. The advantage of this method is thecapabilityofreconstructingthephasespacedistribut ion withoutassuminganyparticularshape. Figure4: Set upforphasespacetomography. The accuracyof the reconstructionalgorithmwas tested by simulations. A total of 18 projections of a non- symmetric distribution interpolated to 90 projections lea d to a reconstruction result with an emittance error of less than 15%. First measurements with an 8 MeV electron beam showed good agreement of the so determined emit- tancewith theonefromthecommonmethod. 5 Q-VALUEOFTHEACCELERATOR CAVITIES The acceleratorcavities used at the S-DALINACare oper- ated at 2 K,the frequencyofthe π-mode,usedforacceler- ationis2.997GHz. Thedesignparametersofthe1mlong 20-cellcavitiesconsistingofniobium(RRR=280)assumed anunloadedqualityfactorof3 ·109andanacceleratinggra- dientof5MV/m. Almostallgradientsachievedduringrou- tine operation exceed this design criteria, some resonator s reach up to 10 MV/m. On the other hand, although differ- entpreparationtechniqueshavebeentested,currentlynon e of the cavities has achieved a Q-value signiﬁcantly higher than 1 ·109. The reduction of the Q-values in comparison with the design criteria increases the dissipated power per cavityfrom4.2to12.6W.Asaconsequencethemaximum energy of the S-DALINAC in the cw-mode is limited by the installed He-refrigeratorpower. A measurement of thequalityfactorasafunctionoftemperaturehasrevealedtha t the resonators have a residual resistance of 276 n Ωcom- paredtotheBCS-resistance of50n Ωat2 K. Figure5: Layoutoftheexternal2 Ktestcryostat. In order to ﬁnd an explanation for this behaviour a ver- tical 2K testcryostatwas turnedintooperation(see ﬁg.5). This test setup allows to perform systematic studies with- out interfering with accelerator operation. We intend to develop an improved magnetic shielding for the cavities which takes the constraints of the complicated geome- try (couplers, tuners) better into account than the present shielding. Additionally,systematicstudiesontheinﬂuen ce of different surface and material preparation methods on the Q-valueareplanned. 6 DIGITALRF-CONTROLSYSTEM The superconducting accelerator cavities have to be con- trolled to an rf phase error of less than 1◦and a relative amplitude error of less than ±1·10−4. The present analog control system fullﬁlls the phase speciﬁcations, but it doe s not quite meet the amplitude speciﬁcationsand it does not allowtheuseofmoderndigitalcontrolmethodsordetailed control data analysis. Figure 6 displays the schematic lay- outofanewdigitalcontrolsystemwhichiscurrentlyunder developmentincooperationwithDESY,Hamburg[3]. The 3 GHz signal extractedfroma sc cavity is converteddown to an intermediate frequency of 250 kHz. An ADC sam- plesthissignal at a rateof1 MHz yieldinga complexﬁeld vector. A digital signal processor (DSP) using techniques like feed forward tables creates a new output ﬁeld vector. This vectoris convertedby DACsand mixedup to 3 GHz, ampliﬁedbyklystronsandfedintothecavity. Theremain- ing energy spread of the electron beam should be smaller bya factorofthreewiththe newsytem. 7 CONCLUSION AttheS-DALINAC,severalimprovementsweremadewith respecttobeamdiagnostics. Especiallytherfintensityan d position monitors as well as the Compton diodes will give substantialaidinlinacoperation. Thetomographicalphas eVector ModulatorKlystron Cavity LO +250□kHz DAC DACReference Oscillator3□GHz 250□kHz 1□MHz TimerGainImRe Feed ForwardSet□PointDSP +-ADC x Figure6: Digitalrf-controlsystem. space reconstruction will provide more detailed informa- tion on the electron beam structure. The studies on cavity Q-values will hopefully result in a higher average Q, thus enablingahigherachievablebeamenergy. Thenewdigital rfsystem shouldreducetheenergyspreadofthebeamand improvethe stabilityofacceleratoroperation. 8 REFERENCES [1] A.Richter,OperationalExperienceattheS-DALINAC,Pr oc. ofthe5thEurop.ParticleAcceleratorConf.,Eds.S.Myerset al.,IOPPublishing, Bristol,(1996) 110 [2] C.B.McKee,P.G.O’Shea,J.M.J.Madey,Phasespacetomog - raphyofrelativisticelectronbeams,Nucl.Instr.andMeth ods inPhysicsResearch A358, (1995) 264 [3] T. Schilcher, Vector Sum Control of Pulsed Accelerating Fields in Lorentz Force Detuned Superconducting Cavities, DESYPrintTESLA98-29, (1998)
/G2F/G2C/G31/G24/G26/G03 /G2F/G38/G28/G15/G13/G13/G11/G03/G29/G2C/G35/G36/G37/G03/G37/G28 /G36/G37/G2C/G31/G2A/G03/G35/G28/G36/G38/G2F/G37/G36 /G36/G11/G31/G11/G03/G27/G52/G4F/G5C/G44/G0F/G03/G3A/G11/G2C/G11/G03/G29/G58/G55/G50/G44/G51/G0F/G03/G39/G11/G39/G11/G03/G2E/G52/G45/G48/G57/G56/G0F/G03/G28/G11/G30/G11/G03/G2F/G44/G5D/G4C/G48/G59/G0F/G03/G3C/G58/G11/G24/G11/G03/G30/G48/G57/G48/G4F/G4E/G4C/G51/G0F /G39/G11/G24/G11/G03/G36/G4B/G59/G48/G57/G56/G0F/G03/G24/G11/G33/G11/G03/G36/G52/G58/G50/G45/G44/G48/G59/G0F/G03/G2D/G2C/G31/G35/G0F/G03/G27/G58/G45/G51/G44/G0F/G03/G35/G58/G56/G56/G4C/G44 /G37/G4B/G48/G03/G39/G28/G33/G33/G10/G18/G03/G37/G48/G44/G50/G0F/G03/G25/G2C/G31/G33/G0F/G03/G31/G52/G59/G52/G56/G4C/G45/G4C/G55/G56/G4E/G0F/G03/G35/G58/G56/G56/G4C/G44/G14 /G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03 /G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03 /G14/G03/G37/G4B/G48/G03/G49/G58/G4F/G4F/G03/G4F/G4C/G56/G57/G03/G52/G49/G03/G57/G4B/G48/G03/G39/G28/G33/G33/G10/G18/G03/G37/G48/G44/G50/G03/G4C/G56/G03/G4C/G51/G03/G57/G4B/G48/G03/G55/G48/G49/G48/G55/G48/G51/G46/G48/G56/G03/G3E/G18/G40/G11/G24/G45/G56/G57/G55/G44/G46/G57 /GB3/G2F/G38/G28/G15/G13/G13/GB4/G03/GB1/G03/G15/G13/G13/G03 /G30/G48/G39/G03/G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G4F/G4C/G51/G44/G46/G03 /G4C/G56/G03 /G45/G48/G4C/G51/G4A/G03 /G46/G55/G48/G44/G57/G48/G47/G03 /G44/G57 /G2D/G2C/G31/G35/G03 /G44/G56/G03 /G44/G03 /G47/G55/G4C/G59/G48/G55/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G47/G03 /G51/G48/G58/G57/G55/G52/G51/G03 /G56/G52/G58/G55/G46/G48/G03 /GB3/G2C/G35/G28/G31/GB4/G03 /G3E/G14/G40/G11 /G37/G4B/G48/G03 /G56/G53/G48/G46/G4C/G44/G4F/G03 /G49/G58/G4F/G4F/G10/G56/G46/G44/G4F/G48/G03 /G49/G44/G46/G4C/G4F/G4C/G57/G4C/G48/G56/G03 /G49/G52/G55/G03 /G57/G48/G56/G57/G4C/G51/G4A/G03 /G57/G4B/G48/G03 /G50/G44/G4C/G51 /G56/G5C/G56/G57/G48/G50/G56/G03 /G52/G49/G03 /G2F/G38/G28/G10/G15/G13/G13/G03 /G0B/G29/G36/G37/G29/G0C/G03 /G44/G55/G48/G03 /G58/G56 ed /G03 /G44/G57/G03 /G2D/G2C/G31/G35/G0F/G03 /G25/G2C/G31/G33/G0F /G30/G28/G33/G4B/G2C/G03 /G44/G51/G47/G03 /G3C/G48/G55/G33/G4B/G2C/G03 /G3E/G15/G40/G11/G03 /G37/G4B/G48/G03 /G59/G48/G55/G4C/G49/G4C/G46/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G4F/G4C/G51/G44/G46 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G4C/G56/G03 /G53/G55/G52/G59/G4C/G47/G4C/G51/G4A/G03 /G44/G57/G03 /G57/G4B/G48/G03 /G39/G28/G33/G33/G10/G18 /G53/G55/G48/G4C/G51/G4D/G48/G46/G57/G52/G55/G03 /G46/G52/G51/G56/G57/G55/G58/G46/G57/G48/G47/G03 /G44/G57/G03 /G25/G2C/G31/G33/G03 /G3E/G16/G40/G11/G03 /G37/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A /G56/G5C/G56/G57/G48/G50/G03 /G52/G49/G03 /G2F/G38/G28/G15/G13/G13/G03 /G4C/G51/G46/G4F/G58/G47/G48/G56/G03 /G57/G5A/G52/G03 /G36/G10/G45/G44/G51/G47/G03 /G0B/G15/G1B/G18/G19/G03 /G30/G2B/G5D/G0C /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G56/G03 /G52/G49/G03 /G03 /G16/G03 /G50/G03 /G4F/G52/G51/G4A/G11/G03 /G37/G4B/G48/G03 /G56/G48/G46/G57/G4C/G52/G51/G56/G03 /G44/G55/G48 /G46/G52/G51/G51/G48/G46/G57/G48/G47/G03 /G5A/G4C/G57/G4B/G03 /G50/G52/G47/G58/G4F/G44/G57/G52/G55/G03 /G45/G44/G56/G48/G47/G03 /G52/G51/G48/G03 /G18/G13/G17/G18/G03 /G4E/G4F/G5C/G56/G57/G55/G52/G51 /G0B/G36/G2F/G24/G26/G03 /G53/G55/G52/G47/G58/G46/G57/G4C/G52/G51/G0C/G11/G03 /G37/G4B/G48/G55/G48/G03 /G44/G55/G48/G03 /G36/G2F/G28/G27/G10/G56/G5C/G56/G57/G48/G50/G56/G03 /G49/G52/G55/G03 /G57/G4B/G48 /G50/G58/G4F/G57/G4C/G53/G4F/G5C/G4C/G51/G4A/G03 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G03 /G35/G29/G03 /G53/G52/G5A/G48/G55/G11/G03 /G37/G4B/G48/G03 /G49/G4C/G55/G56/G57/G03 /G55/G48/G56/G58/G4F/G57/G56/G03 /G52/G49/G03 /G57/G4B/G48 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G57/G48/G56/G57/G03 /G52/G51/G03 /G57/G4B/G48/G03 /G39/G28/G33/G33/G10/G18/G03 /G53/G55/G48/G4C/G51/G4D/G48/G46/G57/G52/G55/G03 /G44/G55/G48 /G53/G55/G48/G56/G48/G51/G57/G48/G47/G11/G03 /G37/G4B/G48/G03 /G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G45/G48/G44/G50/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G58/G53/G03 /G57/G52/G03 /G1C/G15/G03 /G30/G48/G39/G03 /G44/G51/G47 /G46/G52/G51/G56/G48/G54/G58/G48/G51/G57/G4F/G5C/G03 /G44/G59/G48/G55/G44/G4A/G48/G03 /G55/G44/G57/G48/G03 /G52/G49/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G48/G4F/G48/G46/G57/G55/G52/G51 /G45/G48/G44/G50/G03 /G50/G52/G55/G48/G03 /G57/G4B/G44/G51/G03 /G16/G13/G03 /G30/G48/G39/G12/G50/G03 /G5A/G48/G55/G48/G03 /G44/G46/G4B/G4C/G48/G59/G48/G47/G03 /G44/G49/G57/G48/G55 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03/G4C/G51/G03/G52/G51/G48/G03/G56/G48/G46/G57/G4C/G52/G51/G11 /G14 /G2C/G31/G37/G35/G32/G27/G38/G26/G37/G2C/G32/G31 /G2C/G51/G03 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G47/G03 /G51/G48/G58/G57/G55/G52/G51/G03 /G56/G52/G58/G55/G46/G48/G03 /GB3/G2C/G35/G28/G31/GB4 , /G03 /G57/G4B/G48/G03 /G5A/G48/G4F/G4F/G10/G4E/G51/G52/G5A/G51 /G45/G52/G52/G56/G57/G48/G55/G0F/G03 /G48/G10 γ /G10/G51/G03 /G56/G46/G4B/G48/G50/G48/G03 /G0B/G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G45/G48/G44/G50/G03 /G0E/G03 /G50/G58/G4F/G57/G4C/G53/G4F/G5C/G4C/G51/G4A /G57/G44/G55/G4A/G48/G57/G0C/G03 /G5A/G4C/G4F/G4F/G03 /G45/G48/G03 /G58/G56/G48/G47/G11/G03 /G37/G4B/G48/G03 /G57/G44/G55/G4A/G48/G57/G03 /G46/G52/G51/G56/G4C/G56/G57/G56/G03 /G52/G49/G03 /G57/G58/G51/G4A/G56/G57/G48/G51 /G46/G52/G51/G59/G48/G55/G57/G48/G55/G03/G56/G58/G55/G55/G52/G58/G51/G47/G48/G47/G03/G45/G5C/G03/G44/G03/G53/G4F/G58/G57/G52/G51/G4C/G58/G50/G03/G0B/G33/G58 /G15/G16/G1C /G0C/G03/G46/G52/G55/G48/G11 /G37/G4B/G48/G03 /G2F/G38/G28/G10/G15/G13/G13/G03 /G57/G55/G44/G59/G48/G4F/G4C/G51/G4A/G03 /G5A/G44/G59/G48/G03 /G4F/G4C/G51/G44/G46/G03 /G46/G52/G51/G46/G48/G53/G57/G4C/G52/G51/G03 /G4C/G56 /G47/G48/G56/G4C/G4A/G51/G48/G47/G03 /G45/G5C/G03 /G57/G4B/G48/G03 /G25/G58/G47/G4E/G48/G55/G03 /G2C/G51/G56/G57/G4C/G57/G58/G57/G48/G03 /G52/G49/G03 /G31/G58/G46/G4F/G48/G44/G55/G03 /G33/G4B/G5C/G56/G4C/G46/G56 /G0B/G25/G2C/G31/G33/G0F/G03/G31/G52/G59/G52/G56/G4C/G45/G4C/G55/G56/G4E/G0C/G03/G3E/G14/G0F/G15/G0F/G16/G40/G03/G0B/G56/G48/G48/G03/G37/G44/G45/G4F/G48/G03/G14/G0C/G11 /G37/G44/G45/G4F/G48/G03/G14/G1D/G03/G37/G4B/G48/G03/G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03/G52/G49/G03/G2F/G38/G28/G10/G15/G13/G13/G03/G0B/G53/G55/G52/G4D/G48/G46/G57/G0C /G25/G48/G44/G50/G03/G44/G59/G48/G55/G44/G4A/G48/G03/G53/G52/G5A/G48/G55 /G14/G13/G03/GB1/G03/G14/G15/G03/G4E/G3A /G28/G4F/G48/G46/G57/G55/G52/G51/G03/G48/G51/G48/G55/G4A/G5C /G15/G13/G13/G03/G30/G48/G39 /G33/G58/G4F/G56/G48/G03/G46/G58/G55/G55/G48/G51/G57 /G14/G11/G18/G03/G24 /G26/G58/G55/G55/G48/G51/G57/G03/G53/G58/G4F/G56/G48/G03/G47/G58/G55/G44/G57/G4C/G52/G51 /G15/G18/G13/G03/G51/G56 /G33/G58/G4F/G56/G48/G03/G55/G48/G53/G48/G57/G4C/G57/G4C/G52/G51/G03/G55/G44/G57/G48 /G14/G18/G13/G03/G2B/G5D /G24/G59/G48/G55/G44/G4A/G48/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03/G4A/G55/G44/G47/G4C/G48/G51/G57 ∼ /G03/G16/G18/G03/G30/G48/G39/G12/G50 /G32/G53/G48/G55/G44/G57/G4C/G52/G51/G03/G49/G55/G48/G54/G58/G48/G51/G46/G5C /G15/G1B/G18/G19/G03/G30/G2B/G5D /G2F/G48/G51/G4A/G57/G4B/G03/G52/G49/G03/G57/G4B/G48/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03/G56/G48/G46/G57/G4C/G52/G51 /G16/G03/G50 /G34/G58/G44/G51/G57/G4C/G57/G5C/G03/G52/G49/G03/G57/G4B/G48/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03/G56/G48/G46/G57/G4C/G52/G51/G56 /G15 /G37/G4B/G48/G03 /G50/G44/G4C/G51/G03 /G48/G4F/G48/G50/G48/G51/G57/G56/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G4F/G4C/G51/G44/G46/G03 /G44/G51/G47/G03 /G50/G58/G4F/G57/G4C/G53/G4F/G5C/G4C/G51/G4A /G57/G44/G55/G4A/G48/G57/G03 /G5A/G4C/G4F/G4F/G03 /G45/G48/G03 /G50/G52/G58/G51/G57/G48/G47/G03 /G52/G51/G03 /G57/G4B/G55/G48/G48/G03 /G4F/G48/G59/G48/G4F/G56/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G45/G58/G4C/G4F/G47/G4C/G51/G4A/G03 /G0B/G56/G48/G48 /G29/G4C/G4A/G11/G03/G14/G0C/G11/G29/G4C/G4A/G58/G55/G48/G03/G14/G1D/G03/GB3/G2C/G35/G28/G31/GB4/G03/G51/G48/G58/G57/G55/G52/G51/G03/G56/G52/G58/G55/G46/G48/G03/G4F/G44/G5C/G52/G58/G57/G11 /G15/G03/G03/G28/G3B/G33/G28/G35/G2C/G30/G28/G31/G37/G24/G2F/G03/G35/G28/G36/G38/G2F/G37/G36 /G24/G51/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G5C/G56/G57/G48/G50/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G4F/G4C/G51/G44/G46/G03 /G4C/G56/G03 /G46/G52/G51/G56/G57/G55/G58/G46/G57/G48/G47/G03 /G44/G57/G03 /G57/G4B/G48 /G25/G58/G47/G4E/G48/G55/G03 /G2C/G51/G56/G57/G4C/G57/G58/G57/G48/G03 /G3E/G16/G0F/G17/G40/G11/G03 /G37/G4B/G48/G03 /G44/G53/G53/G4F/G4C/G48/G47/G03 /G57/G48/G46/G4B/G51/G52/G4F/G52/G4A/G5C/G03 /G52/G49 /G49/G44/G45/G55/G4C/G46/G44/G57/G4C/G52/G51/G03 /G56/G4B/G52/G58/G4F/G47/G03 /G4A/G58/G44/G55/G44/G51/G57/G48/G48/G03 /G44/G03 /G56/G57/G44/G45/G4F/G48/G03 /G46/G52/G51/G57/G4C/G51/G58/G52/G58/G56/G03 /G52/G53/G48/G55/G44/G57/G4C/G52/G51 /G52/G49/G03 /G57/G4B/G48/G03 /G47/G48/G59/G4C/G46/G48/G56/G03 /G5A/G4C/G57/G4B/G03 /G03 /G44/G03 /G4B/G4C/G4A/G4B/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G03 /G44/G57/G03 /G44/G03 /G4B/G4C/G4A/G4B /G35/G29/G03 /G53/G52/G5A/G48/G55/G03 /G4F/G48/G59/G48/G4F/G11/G03 /G36/G52/G0F/G03 /G57/G4B/G48/G03 /G44/G4C/G50/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G57/G48/G56/G57/G03 /G5A/G44/G56/G03 /G57/G52/G03 /G44/G46/G4B/G4C/G48/G59/G48/G03 /G44 /G47/G48/G56/G4C/G55/G48/G47/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G03 ∼ /G03 /G16/G18/G03 /G30/G48/G39/G12/G50/G11/G03 /G37/G4B/G48/G03 /G57/G48/G56/G57/G03 /G5A/G44/G56 /G46/G44/G55/G55/G4C/G48/G47/G03 /G52/G58/G57/G03 /G44/G57/G03 /G57/G4B/G48/G03 /G4C/G51/G4C/G57/G4C/G44/G4F/G03 /G53/G44/G55/G57/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G39/G28/G33/G33/G10/G18/G03 /G53/G55/G48/G4C/G51/G4D/G48/G46/G57/G52/G55 /G3E/G17/G0F/G18/G40/G11/G03 /G2C/G51/G03 /G57/G4B/G52/G56/G48/G03 /G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G56/G0F/G03 /G57/G4B/G48/G03 /G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G4A/G58/G51/G03 /G0B/G56/G48/G48/G03 /G37/G44/G45/G4F/G48/G03 /G15/G0C /G44/G51/G47/G03 /G52/G51/G4F/G5C/G03 /G52/G51/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G03 /G52/G49/G03 /G16/G03 /G50/G03 /G4F/G52/G51/G4A/G03 /G5A/G48/G55/G48/G03 /G58/G56/G48/G47/G11 /G37/G4B/G48/G03 /G50/G44/G4C/G51/G03 /G46/G52/G51/G47/G4C/G57/G4C/G52/G51/G56/G03 /G5A/G48/G55/G48/G03 /G46/G4F/G52/G56/G48/G03 /G57/G52/G03 /G52/G53/G48/G55/G44/G57/G4C/G52/G51/G44/G4F/G03 /G52/G49/G03 /G57/G4B/G481-st level2-nd level Target hallElectron gun Multiplying target5045 klystronS-band buncher Acc. section #1 e-beam lineModulator #1 Modulator #2 05 m5045 klystron Acc. section # 2 n-beams channels/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G52/G55/G03 /G53/G55/G52/G4D/G48/G46/G57/G03 /G49/G52/G55/G03 /GB3/G2C/G35/G28/G31/GB4/G0F/G03 /G51/G44/G50/G48/G4F/G5C/G03 /G57/G4B/G48/G03 /G46/G58/G55/G55/G48/G51/G57/G03 /G52/G49/G03 /G57/G4B/G48 /G4C/G51/G4D/G48/G46/G57/G48/G47/G03 /G45/G48/G44/G50/G03 /G4C/G56/G03 /G51/G52/G57/G03 /G4F/G48/G56/G56/G03 /G57/G4B/G44/G51/G03 /G14/G11/G18/G03 /G24/G03 /G49/G52/G55/G03 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G03 /G5A/G4C/G47/G57/G4B /G15/G18/G13/G03/G51/G56/G11/G03/G37/G4B/G4C/G56/G03/G45/G58/G51/G46/G4B/G03 /G47/G58/G55/G44/G57/G4C/G52/G51/G03 /G4C/G56/G03 /G44/G4F/G55/G48/G44/G47/G5C/G03 /G57/G4B/G48/G03 /G56/G44/G50/G48/G03 /G44/G56/G03 /G57/G4B/G48/G03 /G52/G51/G48 /G52/G49/G03/G52/G53/G48/G55/G44/G57/G4C/G52/G51/G44/G4F/G03/G53/G44/G55/G57/G03/G52/G49/G03/G35/G29/G03/G53/G52/G5A/G48/G55/G03/G53/G58/G4F/G56/G48/G11 /G38/G51/G49/G52/G55/G57/G58/G51/G44/G57/G48/G4F/G5C/G0F/G03 /G57/G4B/G48/G03 /G55/G48/G53/G48/G57/G4C/G57/G4C/G52/G51/G03 /G55/G44/G57/G48/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G56/G48/G57/G58/G53/G03 /G5A/G44/G56 /G4F/G4C/G50/G4C/G57/G48/G47/G03 /G57/G52/G03 /G18/G03 /G2B/G5D/G03 /G45/G48/G46/G44/G58/G56/G48/G03 /G52/G49/G03 /G55/G44/G47/G4C/G44/G57/G4C/G52/G51/G03 /G53/G55/G52/G57/G48/G46/G57/G4C/G52/G51/G03 /G52/G49 /G48/G51/G59/G4C/G55/G52/G51/G50/G48/G51/G57/G11/G03 /G24/G51/G52/G57/G4B/G48/G55/G03 /G53/G48/G46/G58/G4F/G4C/G44/G55/G4C/G57/G5C/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G57/G48/G56/G57/G03 /G5A/G44/G56/G03 /G57/G4B/G48 /G44/G45/G56/G48/G51/G46/G48/G03/G52/G49/G03/G57/G4B/G48/G03/G35/G29/G03/G45/G58/G51/G46/G4B/G48/G55/G03/G0B/G4C/G57/G03/G5A/G44/G56/G03/G51/G52/G57/G03/G55/G48/G44/G47/G5C/G0C/G11 /G37/G4B/G48/G03 /G58/G51/G45/G58/G51/G46/G4B/G48/G47/G03 /G45/G48/G44/G50/G03 /G5A/G44/G56/G03 /G56/G57/G48/G48/G55/G48/G47/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G4C/G51/G53/G58/G57/G03 /G52/G49/G03 /G57/G4B/G48 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G11/G03 /G37/G4B/G48/G03 /G35/G29/G03 /G53/G58/G4F/G56/G48/G03 /G52/G49/G03 /G58/G53/G03 /G57/G52/G03 /G17/G1A/G03 /G30/G3A/G03 /G53/G52/G5A/G48/G55 /G4C/G56/G03 /G47/G4C/G55/G48/G46/G57/G48/G47/G03 /G49/G55/G52/G50/G03 /G18/G13/G17/G18/G03 /G4E/G4F/G5C/G56/G57/G55/G52/G51/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G4C/G51/G53/G58/G57/G03 /G52/G49/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A /G56/G57/G55/G58/G46/G57/G58/G55/G48/G03 /G57/G4B/G55/G52/G58/G4A/G4B/G03 /G1A/G15/G03 /G5B/G03 /G16/G17/G03 /G50/G50/G03 /G59/G44/G46/G58/G58/G50/G03 /G5A/G44/G59/G48/G4A/G58/G4C/G47/G48/G03 /G44/G51/G47 /G36/G2F/G28/G27/G10/G46/G52/G50/G53/G55/G48/G56/G56/G52/G55/G11/G03/G37/G4B/G48/G03/G35/G29/G03/G53/G52/G5A/G48/G55/G03/G46/G52/G4F/G4F/G48/G46/G57/G48/G47/G03 /G45/G5C/G03 /G35/G29/G03 /G4F/G52/G44/G47/G03 /G52/G49 /G57/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G03 /G5A/G44/G56/G03 /G50/G48/G44/G56/G58/G55/G48/G47/G03 /G45/G52/G57/G4B/G03 /G5A/G4C/G57/G4B/G03 /G44/G51/G47 /G5A/G4C/G57/G4B/G52/G58/G57/G03 /G45/G48/G44/G50/G03 /G4F/G52/G44/G47/G11/G03 /G37/G4B/G48/G03 /G56/G4B/G44/G53/G48/G03 /G52/G49/G03 /G35/G29/G03 /G53/G52/G5A/G48/G55/G03 /G53/G58/G4F/G56/G48 /G57/G55/G44/G51/G56/G49/G48/G55/G55/G48/G47/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G35/G29/G03 /G4F/G52/G44/G47/G03 /G4C/G56/G03 /G56/G4B/G52/G5A/G51/G03 /G4C/G51/G03 /G29/G4C/G4A/G11/G03 /G15/G11/G03 /G25/G48/G56/G4C/G47/G48/G56/G0F/G03 /G57/G4B/G48 /G45/G48/G44/G50/G03/G46/G58/G55/G55/G48/G51/G57/G03/G53/G58/G4F/G56/G48/G03/G53/G55/G52/G49/G4C/G4F/G48/G56/G03/G44/G55/G48/G03/G4A/G4C/G59/G48/G51/G03/G4C/G51/G03/G57/G4B/G4C/G56/G03/G49/G4C/G4A/G58/G55/G48/G0F/G03/G57/G52/G52/G11 /G29/G4C/G4A/G58/G55/G48/G03/G15/G1D/G03/G37/G4B/G48/G03/G56/G4B/G44/G53/G48/G03/G52/G49/G03/G35/G29/G03/G53/G52/G5A/G48/G55/G03/G57/G55/G44/G51/G56/G49/G48/G55/G55/G48/G47/G03/G57/G52/G03/G57/G4B/G48/G03/G4F/G52/G44/G47 /G45/G52/G57/G4B/G03/G5A/G4C/G57/G4B/G03/G44/G51/G47/G03/G5A/G4C/G57/G4B/G52/G58/G57/G03/G45/G48/G44/G50/G03/G4F/G52/G44/G47/G03/G44/G51/G47/G03/G03/G57/G4B/G48/G03/G56/G4C/G4A/G51/G44/G4F/G56/G03/G52/G49 /G46/G58/G55/G55/G48/G51/G57/G03/G53/G58/G4F/G56/G48/G03/G49/G55/G52/G50/G03/G57/G4B/G48/G03/G4A/G58/G51/G03/G5A/G44/G4F/G4F/G03/G46/G58/G55/G55/G48/G51/G57/G03/G50/G52/G51/G4C/G57/G52/G55/G11 /G37/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G03 /G52/G53/G48/G55/G44/G57/G48/G56/G03 /G5A/G4C/G57/G4B/G03 /G44/G03 /G56/G57/G55/G52/G51/G4A/G03 /G46/G58/G55/G55/G48/G51/G57 /G4F/G52/G44/G47/G0F/G03 /G5A/G4B/G4C/G46/G4B/G03 /G4F/G48/G44/G47/G56/G03 /G57/G52/G03 /G44/G51/G03 /G48/G56/G56/G48/G51/G57/G4C/G44/G4F/G03 /G46/G4B/G44/G51/G4A/G48/G03 /G52/G49/G03 /G57/G4B/G48 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G49/G4C/G48/G4F/G47/G03 /G44/G51/G47/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G48/G47/G03 /G45/G48/G44/G50/G03 /G46/G4B/G44/G55/G44/G46/G57/G48/G55/G4C/G56/G57/G4C/G46/G56/G11 /G37/G4B/G48/G56/G48/G03 /G46/G52/G51/G47/G4C/G57/G4C/G52/G51/G56/G03 /G44/G55/G48/G03 /G57/G5C/G53/G4C/G46/G44/G4F/G03 /G49/G52/G55/G03 /G57/G4B/G48/G03 /G45/G48/G44/G50/G03 /G5A/G4C/G57/G4B/G03 /G4B/G4C/G4A/G4B /G48/G51/G48/G55/G4A/G5C/G03/G46/G52/G51/G57/G48/G51/G57/G11 /G37/G44/G45/G4F/G48/G03/G15/G1D/G03/G30/G44/G4C/G51/G03/G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03/G52/G49/G03/G57/G4B/G48/G03/G48/G4F/G48/G46/G57/G55/G52/G51/G03/G56/G52/G58/G55/G46/G48 /G28/G4F/G48/G46/G57/G55/G52/G51/G03/G4A/G58/G51/G03/G4B/G4C/G4A/G4B/G03/G59/G52/G4F/G57/G44/G4A/G48 /G14/G15/G13/G03÷ /G03/G14/G1A/G13/G03/G4E/G39 /G28/G4F/G48/G46/G57/G55/G52/G51/G03/G4A/G58/G51/G03/G53/G58/G4F/G56/G48/G47/G03/G46/G58/G55/G55/G48/G51/G57 /G14/G11/G13/G03÷ /G03/G15/G11/G19/G03/G24 /G26/G58/G55/G55/G48/G51/G57/G03/G53/G58/G4F/G56/G48/G03/G47/G58/G55/G44/G57/G4C/G52/G51 /G15/G18/G13/G03÷ /G03/G16/G14/G13/G03/G51/G56/G48/G46 /G33/G58/G4F/G56/G48/G03/G55/G48/G53/G48/G57/G4C/G57/G4C/G52/G51/G03/G55/G44/G57/G48 /G14/G03÷ /G03/G18/G03/G2B/G5D /G2C/G51/G03 /G44/G47/G47/G4C/G57/G4C/G52/G51/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03 /G4F/G4C/G56/G57/G48/G47/G03 /G44/G45/G52/G59/G48/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G53/G55/G48/G56/G48/G51/G57 /G56/G48/G55/G4C/G48/G56/G03/G52/G49/G03/G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G56/G0F/G03 /G57/G4B/G48/G03 /G56/G4C/G4A/G51/G44/G4F/G56/G03 /G49/G55/G52/G50/G03 /G45/G48/G44/G50/G03 /G50/G58/G4F/G57/G4C/G53/G52/G56/G4C/G57/G4C/G52/G51 /G46/G52/G4F/G4F/G48/G46/G57/G52/G55/G03 /G0B/G25/G30/G26/G0C/G03 /G53/G4F/G44/G46/G48/G47/G03 /G44/G57/G03 /G57/G4B/G48/G03 /G52/G58/G57/G53/G58/G57/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G14/G1B/G13 ° /G10 /G50/G44/G4A/G51/G48/G57/G4C/G46/G03 /G56/G53/G48/G46/G57/G55/G52/G50/G48/G57/G48/G55/G03 /G5A/G48/G55/G48/G03 /G44/G4F/G56/G52/G03 /G50/G48/G44/G56/G58/G55/G48/G47/G11/G03 /G37/G4B/G48/G03 /G46/G52/G4F/G4F/G48/G46/G57/G52/G55 /G4C/G56/G03 /G53/G48/G55/G49/G52/G55/G50/G48/G47/G03 /G44/G56/G03 /G44/G03 /G56/G48/G57/G03 /G52/G49/G03 /G46/G4B/G44/G55/G4A/G48/G03 /G46/G52/G4F/G4F/G48/G46/G57/G4C/G52/G51/G56/G03 /G44/G51/G47/G03 /G44/G4F/G4F/G52/G5A/G56/G03 /G52/G51/G48 /G57/G52/G03 /G50/G48/G44/G56/G58/G55/G48/G03 /G57/G4B/G48/G03 /G46/G4B/G44/G55/G4A/G48/G03 /G5A/G4C/G57/G4B/G03 /G57/G4B/G48/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G55/G48/G56/G52/G4F/G58/G57/G4C/G52/G51/G03 ≈ /G14/G11/G1B/G03/G30/G48/G39/G11 /G37/G4B/G48/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G56/G53/G48/G46/G57/G55/G58/G50/G03 /G52/G45/G57/G44/G4C/G51/G48/G47/G03 /G49/G55/G52/G50/G03 /G57/G4B/G48/G03 /G56/G4C/G4A/G51/G44/G4F/G56 /G49/G55/G52/G50/G03/G25/G30/G26/G03/G4C/G56/G03/G53/G55/G48/G56/G48/G51/G57/G48/G47/G03/G4C/G51/G03/G29/G4C/G4A/G11/G03/G16/G11 /G29/G4C/G4A/G58/G55/G48/G03/G16/G1D/G03/G33/G44/G55/G57/G4C/G46/G4F/G48/G03/G47/G48/G51/G56/G4C/G57/G5C/G03/G47/G4C/G56/G57/G55/G4C/G45/G58/G57/G4C/G52/G51/G03/G59/G56/G11/G03/G48/G51/G48/G55/G4A/G5C/G11 /G37/G4B/G48/G03 /G50/G44/G5B/G4C/G50/G58/G50/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G48/G47 /G45/G48/G44/G50/G03 /G4C/G56/G03 /G1C/G15/G03/G30/G48/G39/G0F/G03 /G57/G4B/G48/G03 /G50/G4C/G51/G4C/G50/G58/G50/G03 /G4C/G56/G03 /G17/G1A/G03/G30/G48/G39/G11/G03 /G37/G52/G03 /G52/G58/G55 /G52/G53/G4C/G51/G4C/G52/G51/G0F/G03 /G57/G4B/G48/G03 /G4F/G52/G5A/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G57/G44/G4C/G4F/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G45/G48/G44/G50/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G56/G53/G48/G46/G57/G55/G58/G50 /G4C/G56/G03 /G46/G44/G58/G56/G48/G47/G03 /G45/G5C/G03 /G57/G4B/G48/G03 /G44/G45/G56/G48/G51/G46/G48/G03 /G52/G49/G03 /G35/G29/G03 /G45/G58/G51/G46/G4B/G48/G55/G03 /G47/G58/G55/G4C/G51/G4A/G03 /G57/G4B/G48/G03 /G57/G48/G56/G57 /G53/G48/G55/G4C/G52/G47/G11 /G37/G4B/G48/G03 /G50/G48/G44/G56/G58/G55/G48/G47/G03 /G53/G58/G4F/G56/G48/G47/G03 /G46/G58/G55/G55/G48/G51/G57/G03 /G49/G55/G52/G50/G03 /G57/G4B/G48/G03 /G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G4A/G58/G51/G03 /G4C/G56 /G15/G11/G19/G03 /G3A /G0F/G03 /G5A/G4B/G4C/G46/G4B/G03 /G46/G52/G55/G55/G48/G56/G53/G52/G51/G47/G56/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G51/G58/G50/G45/G48/G55/G03 /G52/G49/G03 /G48/G4F/G48/G46/G57/G55/G52/G51/G56/G03 /G4C/G51 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G03 /G52/G49 /G03 /G18/G11/G15 /GC2/G14/G13/G14/G15/G03 /G44/G57/G03 /G57/G4B/G48/G03 /G53/G58/G4F/G56/G48/G03 /G47/G58/G55/G44/G57/G4C/G52/G51/G03 /G52/G49/G03 /G16/G14/G13/G03 /G51/G56/G48/G46/G11 /G37/G4B/G48/G03 /G46/G44/G4F/G46/G58/G4F/G44/G57/G48/G47/G03 /G51/G58/G50/G45/G48/G55/G03 /G52/G49/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48/G56/G03 /G44/G49/G57/G48/G55/G03 /G57/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51 /G4C/G51/G03 /G52/G51/G48/G03 /G56/G48/G46/G57/G4C/G52/G51/G03 /G4C/G56/G03 /G14/G15/G14/G13 /G1A/G18 /G14⋅ /G11 /G11/G03 /G37/G4B/G58/G56/G0F/G03 /G44/G53/G53/G55/G52/G5B/G4C/G50/G44/G57/G48/G4F/G5C/G03 /G19/G19/G08/G03 /G52/G49 /G53/G44/G55/G57/G4C/G46/G4F/G48/G56/G03 /G5A/G48/G55/G48/G03 /G4F/G52/G56/G57/G03 /G47/G58/G48/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G49/G44/G46/G57/G03 /G57/G4B/G48/G03 /G35/G29/G03 /G45/G58/G51/G46/G4B/G48/G55/G03 /G5A/G44/G56/G03 /G51/G52/G57 /G44/G59/G44/G4C/G4F/G44/G45/G4F/G48/G11 /G24/G57/G03 /G57/G4B/G48/G03 /G53/G55/G48/G56/G48/G51/G57/G03 /G57/G4C/G50/G48/G0F/G03 /G44/G49/G57/G48/G55/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03 /G4C/G51/G03 /G52/G51/G48/G03 /G56/G48/G46/G57/G4C/G52/G51 /G5A/G4C/G57/G4B/G52/G58/G57/G03/G35/G29/G03/G45/G58/G51/G46/G4B/G48/G55/G03/G5A/G48/G03/G4B/G44/G59/G48/G03/G49/G52/G4F/G4F/G52/G5A/G4C/G51/G4A/G03/G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G1D • /G2E/G4F/G5C/G56/G57/G55/G52/G51/G03/G52/G58/G57/G53/G58/G57/G03/G35/G29/G03/G53/G52/G5A/G48/G55/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G17/G1A/G03/G30/G3A/G11 • /G37/G52/G57/G44/G4F/G03 /G46/G4B/G44/G55/G4A/G48/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G48/G47/G03 /G45/G48/G44/G50/G03 /G0B/G4C/G51/G03 /G52/G51/G48/G03 /G53/G58/G4F/G56/G48 /G46/G5C/G46/G4F/G48/G0C/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G1A/G14/G13 /G1B /G15−⋅ /G11 /G26/G11 • /G37/G52/G57/G44/G4F/G03/G03/G51/G58/G50/G45/G48/G55/G03/G03/G03/G52/G49/G03/G03/G03/G53/G44/G55/G57/G4C/G46/G4F/G48/G56 /G03/G03/G03/G03/G03/G03/G03/G4C/G51/G03/G03/G52/G51/G48/G03/G03/G53/G58/G4F/G56/G48/G03/G46/G5C/G46/G4F/G48/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G14/G15/G14/G13 /G1A/G18 /G14⋅ /G11 /G11 • /G24/G46/G46/G48/G4F/G48/G55/G44/G57/G48/G47/G03/G45/G48/G44/G50/G03/G53/G58/G4F/G56/G48/G47/G03/G46/G58/G55/G55/G48/G51/G57/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G13/G11/G1C/G03/G24/G11 • /G37/G4B/G48/G03/G44/G59/G48/G55/G44/G4A/G48/G03/G52/G58/G57/G53/G58/G57/G03/G45/G48/G44/G50/G03/G48/G51/G48/G55/G4A/G5C/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G1B/G1A/G03/G30/G48/G39/G11 • /G37/G4B/G48/G03/G45/G48/G44/G50/G03/G48/G51/G48/G55/G4A/G5C/G03/G46/G52/G51/G57/G48/G51/G57/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G15/G19/G11/G14/G03/G2D/G52/G58/G4F/G48/G11 • /G37/G4B/G48/G03/G50/G44/G5B/G4C/G50/G44/G4F/G03/G52/G58/G57/G53/G58/G57/G03/G45/G48/G44/G50/G03/G48/G51/G48/G55/G4A/G5C/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G1C/G15/G03/G30/G48/G39/G11 • /G37/G4B/G48/G03/G44/G59/G48/G55/G44/G4A/G48/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03/G55/G44/G57/G48 /G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G50/G52/G55/G48/G03/G57/G4B/G44/G51/G03/G16/G13/G03/G30/G48/G39 /m. /G16/G03/G26/G32/G31/G36/G2C/G27/G28/G35/G24/G37/G2C/G32/G31 /G38/G56/G4C/G51/G4A/G03 /G57/G4B/G48/G56/G48/G03 /G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G44/G4F/G03 /G55/G48/G56/G58/G4F/G57/G56/G03 /G5A/G48/G03 /G46/G44/G51/G03 /G46/G44/G4F/G46/G58/G4F/G44/G57/G48/G03 /G57/G4B/G48 /G45/G48/G44/G50/G03 /G48/G51/G48/G55/G4A/G5C/G03 /G59/G56/G11/G03 /G57/G4B/G48/G03 /G57/G4C/G50/G48/G03 /G52/G49/G03 /G49/G4F/G4C/G4A/G4B/G57/G03 /G57/G4B/G55/G52/G58/G4A/G4B/G03 /G44/G03 /G56/G4C/G51/G4A/G4F/G48 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G56/G48/G46/G57/G4C/G52/G51/G03 /G44/G51/G47/G03 /G50/G44/G4E/G48/G03 /G48/G56/G57/G4C/G50/G44/G57/G4C/G52/G51/G56/G03 /G49/G52/G55/G03 /G03 /G57/G4B/G48 /G56/G4C/G57/G58/G44/G57/G4C/G52/G51/G03/G5A/G4C/G57/G4B/G03/G57/G4B/G48/G03/G50/G44/G4C/G51/G03/G53/G55/G52/G4D/G48/G46/G57/G03/G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G1D • /G2E/G4F/G5C/G56/G57/G55/G52/G51/G03/G18/G13/G17/G18/G03/G52/G58/G57/G53/G58/G57/G03/G35/G29/G03/G53/G52/G5A/G48/G55/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G19/G16/G03/G30/G3A/G11 • /G28/G4F/G48/G46/G57/G55/G52/G51/G03/G4A/G58/G51/G03/G53/G58/G4F/G56/G48/G47/G03/G46/G58/G55/G55/G48/G51/G57/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G14/G11/G18/G03/G24/G11 • /G26/G58/G55/G55/G48/G51/G57/G03/G53/G58/G4F/G56/G48/G03/G47/G58/G55/G44/G57/G4C/G52/G51/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G15/G18/G13/G03/G51/G56/G48/G46/G11 • /G35/G48/G53/G48/G57/G4C/G57/G4C/G52/G51/G03/G55/G44/G57/G48/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G14/G18/G13/G03/G2B/G5D/G115 6 7 8 9 10010203040 IBEAM (beam "out") IBEAM (beam "on")P LOAD (beam "out") P LOAD (beam "on")P det [mW] t[mksec]40 50 60 70 80 90 100012345678910P5045 = 47 MW τbeam = 310 nsec Eav= 87 MeV Emin = 47 MeVEmax = 92 MeVdNe/dE x 10-10 particles E[MeV] /G37/G44/G45/G4F/G48/G03 /G16/G03 /G56/G4B/G52/G5A/G56/G03 /G57/G4B/G48/G03 /G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G44/G4F/G03 /G44/G51/G47/G03 /G48/G56/G57/G4C/G50/G44/G57/G48/G47 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03 /G52/G49/G03 /G57/G4B/G4C/G56/G03 /G46/G44/G4F/G46/G58/G4F/G44/G57/G4C/G52/G51/G11/G03 /G37/G4B/G48/G03 /G52/G58/G57/G53/G58/G57/G03 /G45/G48/G44/G50/G03 /G53/G44/G55/G57/G4C/G46/G4F/G48 /G48/G51/G48/G55/G4A/G5C/G03 /G59/G56/G11/G03 /G57/G4C/G50/G48/G03 /G44/G46/G46/G52/G55/G47/G4C/G51/G4A/G03 /G57/G52/G03 /G57/G4B/G48/G56/G48/G03 /G53/G44/G55/G44/G50/G48/G57/G48/G55/G56/G03 /G4C/G56/G03 /G56/G4B/G52/G5A/G51/G03 /G4C/G51 /G29/G4C/G4A/G11/G03/G17/G11 /G29/G4C/G4A/G58/G55/G48/G03/G17/G1D/G03/G32/G58/G57/G53/G58/G57/G03/G45/G48/G44/G50/G03/G53/G44/G55/G57/G4C/G46/G4F/G48/G03/G48/G51/G48/G55/G4A/G5C/G03/G59/G56/G11/G03/G57/G4C/G50/G48/G11 /G37/G44/G45/G4F/G48/G03/G16/G1D/G03/G37/G4B/G48/G03/G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G44/G4F/G03/G44/G51/G47/G03/G48/G56/G57/G4C/G50/G44/G57/G48/G47/G03/G53/G44/G55/G44/G50/G48/G57/G48/G55/G56 /G44/G46/G46/G52/G55/G47/G4C/G51/G4A/G03/G57/G52/G03/G57/G4B/G48/G03/G45/G48/G44/G50/G03/G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G52/G51/G03/G4C/G51/G03/G44/G03/G56/G4C/G51/G4A/G4F/G48/G03/G56/G48/G46/G57/G4C/G52/G51 /G28/G5B/G53/G48/G55/G4C/G50/G48/G51/G57 /G28/G56/G57/G4C/G50/G44/G57/G4C/G52/G51 /G32/G58/G57/G53/G58/G57/G03/G18/G13/G17/G18/G03/G4E/G4F/G5C/G56/G57/G55/G52/G51 /G35/G29/G03/G53/G52/G5A/G48/G55/G17/G1A/G03/G30/G3A /G19/G16/G03/G30/G3A /G35/G48/G53/G48/G57/G4C/G57/G4C/G52/G51/G03/G55/G44/G57/G48 /G15/G11/G18/G03/G2B/G5D /G14/G18/G13/G03/G2B/G5D /G25/G48/G44/G50/G03/G53/G58/G4F/G56/G48/G03/G46/G58/G55/G55/G48/G51/G57 /G13/G11/G1C/G03/G24 /G14/G11/G18/G03/G24 /G25/G48/G44/G51/G03/G53/G58/G4F/G56/G48/G03/G47/G58/G55/G44/G57/G4C/G52/G51 /G16/G14/G13/G03/G51/G56/G48/G46 /G15/G18/G13/G03/G51/G56/G48/G46 /G24/G59/G48/G55/G44/G4A/G48/G03/G48/G4F/G48/G46/G57/G55/G52/G51 /G45/G48/G44/G50/G03/G48/G51/G48/G55/G4A/G5C/G1B/G1A/G03/G30/G48/G39 /G14/G13/G16/G03/G30/G48/G39 /G25/G48/G44/G50/G03/G48/G51/G48/G55/G4A/G5C/G03/G46/G52/G51/G57/G48/G51/G57 /G15/G19/G11/G14/G03/G2D/G52/G58/G4F/G48 /G16/G1B/G11/G19/G03/G2D/G52/G58/G4F/G48 /G24/G59/G48/G55/G44/G4A/G48/G03/G45/G48/G44/G50/G03/G53/G52/G5A/G48/G55 /G10 /G18/G11/G1B/G03/G4E/G3A /G17/G03/G03/G26/G32/G31/G26/G2F/G38/G36/G2C/G32/G31 /G24/G51/G03 /G44/G59/G48/G55/G44/G4A/G48/G03 /G44/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G4A/G55/G44/G47/G4C/G48/G51/G57/G03 /G52/G49/G03 /G50/G52/G55/G48/G03 /G57/G4B/G44/G51/G03 /G16/G13 /G30/G48/G39/G12/G50/G03 /G5A/G44/G56/G03 /G44/G46/G4B/G4C/G48/G59/G48/G47/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G46/G52/G51/G47/G4C/G57/G4C/G52/G51/G56/G03 /G46/G4F/G52/G56/G48/G03 /G57/G52/G03 /G57/G4B/G48 /GB3/G2C/G35/G28/G31/GB4/G03 /G53/G55/G52/G4D/G48/G46/G57/G11/G03 /G37/G4B/G48/G03 /G48/G5B/G53/G48/G55/G4C/G50/G48/G51/G57/G44/G4F/G03 /G47/G44/G57/G44/G03 /G44/G4F/G4F/G52/G5A/G03 /G58/G56/G03 /G57/G52/G03 /G48/G5B/G53/G48/G46/G57 /G5A/G4C/G57/G4B/G03 /G46/G52/G51/G49/G4C/G47/G48/G51/G46/G48/G03 /G57/G4B/G44/G57/G03 /G57/G4B/G48/G03 /G4B/G4C/G4A/G4B/G03 /G53/G55/G52/G4D/G48/G46/G57/G03 /G53/G52/G5A/G48/G55/G03 /G52/G49/G03 /G57/G4B/G48 /G48/G4F/G48/G46/G57/G55/G52/G51/G03 /G45/G48/G44/G50/G03 /G4C/G51/G03 /G57/G4B/G48/G03 /G47/G48/G56/G4C/G4A/G51/G48/G47/G03 /G2F/G38/G28/G10/G15/G13/G13/G03 /G4F/G4C/G51/G44/G46/G03 /G5A/G4C/G4F/G4F/G03 /G45/G48 /G52/G45/G57/G44/G4C/G51/G48/G47/G11 /G35/G28/G29/G28/G35/G28/G31/G26/G28/G36 /G3E/G14/G40/G03 /G24/G11/G03 /G2E/G44/G50/G4C/G51/G56/G4E/G5C/G0F/G03 /G48/G57/G03 /G44/G4F/G11/G0F/G03 /G05/G2F/G38/G28/G15/G13/G13/G03 /G10/G03 /G27/G55/G4C/G59/G48/G55/G03 /G2F/G4C/G51/G44/G46/G03 /G29/G52/G55 /G2C/G51/G57/G48/G51/G56/G48/G03 /G35/G48/G56/G52/G51/G44/G51/G57/G03 /G31/G48/G58/G57/G55/G52/G51/G03 /G36/G53/G48/G46/G57/G55/G52/G50/G48/G57/G48/G55/G03 /G0B/G2C/G35/G28/G31/G0C/G05/G0F /G33/G55/G52/G46/G48/G48/G47/G4C/G51/G4A/G56/G03 /G52/G49/G03 /G2F/G2C/G31/G24/G26/G1C/G19/G03 /G26/G52/G51/G49/G48/G55/G48/G51/G46/G48/G11/G03 /G2A/G48/G51/G48/G59/G44/G0F /G24/G58/G4A/G58/G56/G57/G03/G15/G19/G10/G16/G13/G0F/G03/G14/G1C/G1C/G19/G0F/G03/G53/G53/G11/G03/G18/G13/G1B/G10/G18/G14/G13/G11/G03/G26/G28/G35/G31/G03/G0B/G14/G1C/G1C/G19/G0C/G11 /G3E/G15/G40/G03 /G36/G11/G03 /G27/G52/G4F/G5C/G44/G0F/G03 /G48/G57/G03 /G44/G4F/G11/G0F/G03 /G05/G2F/G4C/G51/G44/G46/G03 /G2F/G38/G28/G03 /G10/G03 /G15/G13/G13/G03 /G37/G48/G56/G57/G03 /G29/G44/G46/G4C/G4F/G4C/G57/G4C/G48/G56/G05/G11 /G33/G55/G52/G46/G48/G48/G47/G4C/G51/G4A/G56/G03 /G52/G49/G03 /G57/G4B/G48/G03 /G3B/G2C/G3B/G03 /G2C/G51/G57/G48/G55/G51/G44/G57/G4C/G52/G51/G44/G4F/G03 /G2F/G4C/G51/G44/G46 /G26/G52/G51/G49/G48/G55/G48/G51/G46/G48/G03 /G2F/G2C/G31/G24/G26/G1C/G1B/G0F/G03 /G24/G58/G4A/G58/G56/G57/G03 /G15/G16/G10/G15/G1B/G0F/G03 /G14/G1C/G1C/G1B/G0F/G03 /G53/G53/G11/G03 /G18/G15/G10 /G18/G17/G11/G03/G24/G31/G2F/G11/G03/G24/G55/G4A/G52/G51/G51/G48/G0F/G03/G2C/G4F/G4F/G4C/G51/G52/G4C/G56/G0F/G03/G38/G36/G24/G03/G0B/G14/G1C/G1C/G1B/G0C/G11 /G3E/G16/G40/G03/G24/G11/G39/G11/G03/G24/G4F/G48/G5B/G44/G51/G47/G55/G52/G59/G0F/G03/G48/G57/G03/G44/G4F/G11/G0F/G03 /G05/G28/G4F/G48/G46/G57/G55/G52/G51/G10/G53/G52/G56/G4C/G57/G55/G52/G51/G03 /G33/G55/G48/G4C/G51/G4D/G48/G46/G57/G52/G55 /G52/G49/G03 /G39/G28/G33/G33/G10/G18/G03 /G26/G52/G50/G53/G4F/G48/G5B/G05/G11/G03 /G33/G55/G52/G46/G48/G48/G47/G4C/G51/G4A/G56/G03 /G52/G49/G03 /G2F/G2C/G31/G24/G26/G1C/G19 /G26/G52/G51/G49/G48/G55/G48/G51/G46/G48/G11/G03 /G2A/G48/G51/G48/G59/G44/G0F/G03 /G24/G58/G4A/G58/G56/G57/G03 /G15/G19/G10/G16/G13/G0F/G03 /G14/G1C/G1C/G19/G0F/G03 /G53/G53/G11/G03 /G1B/G15/G14/G10 /G1B/G15/G16/G11/G03/G26/G28/G35/G31/G03/G0B/G14/G1C/G1C/G19/G0C/G11 /G3E/G17/G40/G03 /G39/G11/G28/G11/G03 /G24/G4E/G4C/G50/G52/G59/G0F/G03 /G48/G57/G11/G03 /G44/G4F/G11/G03 /G05/G37/G48/G56/G57/G03 /G52/G49/G03 /G28/G4F/G48/G46/G57/G55/G52/G51/G03 /G2F/G4C/G51/G44/G46/G03 /G49/G52/G55 /G39/G28/G33/G33/G10/G18/G03 /G33/G55/G48/G10/G4C/G51/G4D/G48/G46/G57/G52/G55/G05/G11/G03 /G33/G55/G52/G46/G48/G48/G47/G4C/G51/G4A/G56/G03 /G52/G49/G03 /G28/G33/G24/G26/G03 /G15/G13/G13/G13 /G26/G52/G51/G49/G48/G55/G48/G51/G46/G48/G11/G03 /G39/G4C/G48/G51/G51/G44/G0F/G03 /G15/G19/G10/G16/G13/G03 /G2D/G58/G51/G48/G03 /G15/G13/G13/G13/G11/G03 /G0B/G5A/G44/G56 /G53/G55/G48/G56/G48/G51/G57/G48/G47/G03/G44/G51/G47/G03/G5A/G4C/G4F/G4F/G03/G53/G58/G45/G4F/G4C/G56/G4B/G0C/G11 /G3E/G18/G40/G03/G39/G11/G28/G11/G03 /G24/G4E/G4C/G50/G52/G59/G0F/G03 /G24/G11/G39/G11/G03 /G24/G4F/G48/G4E/G56/G44/G51/G47/G55/G52/G59/G0F/G03 /G24/G11/G39/G11/G03 /G24/G51/G57/G52/G56/G4B/G4C/G51/G0F/G03 /G30/G11/G36/G11 /G24/G59/G4C/G4F/G52/G59/G0F/G03 /G33/G11/G24/G11/G03 /G25/G44/G4E/G0F/G03 /G32/G11/G3C/G58/G11/G03 /G25/G44/G5D/G4B/G48/G51/G52/G59/G0F/G03 /G3C/G58/G11/G30/G11 /G25/G52/G4C/G50/G48/G4F/G56/G4B/G57/G48/G4C/G51/G0F/G03 /G27/G11/G3C/G58/G11/G03 /G25/G52/G4F/G4E/G4B/G52/G59/G4C/G57/G5C/G44/G51/G52/G59/G0F/G03 /G24/G11/G2A/G11/G03 /G26/G4B/G58/G53/G5C/G55/G44/G0F /G31/G11/G36/G11/G03 /G27/G4C/G4E/G44/G51/G56/G4E/G5C/G0F/G03 /G24/G11/G35/G03 /G29/G55/G52/G4F/G52/G59/G0F/G03 /G35/G11/G2E/G4B/G11/G03 /G2A/G44/G4F/G4C/G50/G52/G59/G0F/G03 /G35/G11/G2A/G11 /G2A/G55/G52/G50/G52/G59/G0F/G03/G2E/G11/G39/G11/G03 /G2A/G58/G45/G4C/G51/G0F/G03 /G36/G11/G30/G11/G03 /G2A/G58/G55/G52/G59/G0F/G03 /G3C/G48/G11/G24/G11/G03 /G2A/G58/G56/G48/G59/G0F/G03 /G2C/G11/G39/G11 /G2E/G44/G5D/G44/G55/G48/G5D/G52/G59/G0F/G03 /G39/G11/G27/G11/G03 /G2E/G4B/G44/G50/G45/G4C/G4E/G52/G59/G0F/G03 /G36/G11/G31/G11/G03 /G2E/G4F/G5C/G58/G56/G46/G4B/G48/G59/G0F/G03 /G31/G11/G24/G11 /G2E/G4C/G56/G48/G4F/G48/G59/G44/G0F/G03 /G39/G11/G2C/G11/G03 /G2E/G52/G4E/G52/G58/G4F/G4C/G51/G0F/G03 /G30/G11/G25/G11/G03 /G2E/G52/G55/G44/G45/G48/G4F/G51/G4C/G4E/G52/G59/G0F/G03 /G24/G11/G24/G11 /G2E/G52/G55/G48/G53/G44/G51/G52/G59/G0F/G03 /G39/G11/G2C/G11/G03 /G2E/G52/G53/G5C/G4F/G52/G59/G0F/G03 /G24/G11/G31/G11/G03 /G2E/G52/G56/G44/G55/G48/G59/G0F/G03 /G31/G11/G2E/G4B/G11/G03 /G2E/G52/G57/G0F /G31/G11/G31/G11/G03 /G2F/G48/G45/G48/G47/G48/G59/G0F/G03 /G33/G11/G39/G11/G03 /G2F/G52/G4A/G44/G57/G46/G4B/G48/G59/G0F/G03 /G24/G11/G31/G11/G03 /G2F/G58/G4E/G4C/G51/G0F/G03 /G33/G11/G39/G11 /G30/G44/G55/G57/G5C/G56/G4B/G4E/G4C/G51/G0F/G03 /G2F/G11/G24/G11/G03 /G30/G4C/G55/G52/G51/G48/G51/G4E/G52/G0F/G03 /G24/G11/G24/G11/G03 /G31/G4C/G4E/G4C/G49/G52/G55/G52/G59/G0F/G03 /G24/G11/G39/G11 /G31/G52/G59/G52/G4E/G4B/G44/G57/G56/G4E/G4C/G0F/G03 /G39/G11/G30/G11/G03 /G33/G44/G59/G4F/G52/G59/G0F/G03 /G2C/G11/G2F/G11/G03 /G33/G4C/G59/G52/G59/G44/G55/G52/G59/G0F/G03 /G32/G11/G39/G11 /G33/G4C/G55/G52/G4A/G52/G59/G0F/G03 /G39/G11/G39/G11/G03 /G33/G52/G47/G4F/G48/G59/G56/G4E/G4C/G4E/G4B/G0F/G03 /G36/G11/G2F/G11/G03 /G36/G44/G50/G52/G4C/G4F/G52/G59/G0F/G03 /G39/G11/G36/G11 /G36/G48/G59/G48/G55/G4C/G4F/G52/G0F/G03 /G39/G11/G27/G11/G03 /G36/G4B/G48/G50/G48/G4F/G4C/G51/G0F/G03 /G36/G11/G39/G11/G03 /G36/G4B/G4C/G5C/G44/G51/G4E/G52/G59/G0F/G03 /G25/G11/G24/G11 /G36/G4E/G44/G55/G45/G52/G0F/G03 /G24/G11/G31/G11/G03 /G36/G4E/G55/G4C/G51/G56/G4E/G5C/G0F/G03 /G25/G11/G30/G11/G03 /G36/G50/G4C/G55/G51/G52/G59/G0F/G03 /G24/G11/G31/G11 /G36/G58/G47/G44/G55/G4E/G4C/G51/G0F/G03 /G27/G11/G33/G11/G03 /G36/G58/G4E/G4B/G44/G51/G52/G59/G0F/G03 /G24/G11/G36/G11/G03 /G37/G56/G5C/G4A/G44/G51/G52/G59/G0F/G03 /G3C/G58/G11/G39/G11 /G3C/G58/G47/G4C/G51/G0F/G03 /G31/G11/G2C/G11/G03 /G3D/G4C/G51/G48/G59/G4C/G46/G4B/G11/G03 /G05/G37/G48/G56/G57/G03 /G52/G49/G03 /G24/G46/G46/G48/G4F/G48/G55/G44/G57/G4C/G51/G4A/G03 /G36/G57/G55/G58/G46/G57/G58/G55/G48 /G49/G52/G55/G03/G39/G28/G33/G33/G10/G18/G03/G33/G55/G48/G4C/G51/G4D/G48/G46/G57/G52/G55/G05/G0F/G03/G0B/G57/G4B/G4C/G56/G03/G03/G46/G52/G51/G49/G48/G55/G48/G51/G46/G48/G0C/G113,4 3,5 3,6 3,7 3,8 3,9020406080100120140 Emax = 113 MeV ΔE = 27 MeV PBEAM av. = 5.9 kWEmax = 103 MeV ΔE = 7 MeV PBEAM av. = 5.7 kWEmax = 124 MeV (beam "out")E [MeV]P5045 = 63 MW Ibeam = 1.5 A τbeam= 250 nsec t[mksec]
arXiv:physics/0008080 16 Aug 2000A SUPER-CONDUCTING LINAC INJECTOR FOR THE BNL-AGS∗ D. Raparia, A.G. Ruggiero, BNL, Po Box 5000, Upton, NY 11973, USA ∗ Work performed under the auspices of the US Department of EnergyAbstract This paper reports on the feasibility study of a proton Super-Conducting Linac (SCL) as a new injector to the Alternating Gradient Synchrotron (AGS) of the Brookhaven National Laboratory (BNL). The Linac beam energy is in the range of 1.5 to 2.4 GeV. The beam intensity is adjusted to provide an average beam power of 4 MW at the top energy of 24 GeV. The repetition rate of the SCL-AGS facility is 5 beam pulses per second. 1 INTRODUCTION It has been proposed to upgrade the AGS accelerator complex to provide an average proton beam power of 4 MW at the energy of 24 GeV. The facility can be used either as a proton driver for the production of intense muon and neutrino beams, or/and as a pulsed high-energy spallation neutron source. The upgrade requires operation of the accelerator at the rate of five cycles per second, and a new injector for an increase of the AGS beam intensity by a factor of three. In a separate technical report [1] we have described the methods and the requirements to operate the BNL-AGS accelerator facility at the rate of 5 proton beam pulses per second. The major requirement is an extensive addition and modification of the present power supply system. At the same time, a new injector to the AGS, capable to operate at the rate of 5 beam pulses per second, and to provide a three-fold increase of the present beam intensity, is required. The present injector made of the 200-MeV Linac and of the 1.5-GeV AGS-Booster will not be able to fulfill the goals of the upgrade. The proposed new injector is a 1.5 - 2.4 GeV SCL with an average output beam power of 250 - 400 kW. The largest energy is determined by the need to control beam losses due to stripping of the negative ions that are used for multi-turn injection into the synchrotron. The high- energy case is to be preferred if one wants to reduce the effects of space charge, for a sufficiently small beam transverse size that fits the AGS acceptance. The duty cycle is about a half percent. The paper describes the preliminary design of the SCL. It is composed of three parts; a front end, that is a 60-mA negative-ion source, followed by a 5-MeV RFQ, a room temperature Drift-Tube Linac (DTL) that accelerates protons to 150 MeV, and the SCL proper. This in turn is made of four sections, each with its own energy range,and different cavity-cryostat arrangement. The four sections are labeled: Low-Energy (LE), Medium Low- Energy (MLE), Medium High-Energy (MHE), and High- Energy (HE). 2 THE NEW INJECTOR The two different Linac energy cases are compared in Table 1. AGS performance during multi-turn injection is also summarized in the same Table. Both cases correspond to the same average beam current of 0.17 mA, that yields the same average beam power of 4 MW at the top energy of 24 GeV. The repetition rate of 5 beam pulses per second is assumed in both cases; that gives the same intensity of 2.1 x 1014 protons accelerated per AGS cycle; that is a factor of 3 higher then the intensity routinely obtained with the present injector. At the end of injection, that takes about 317-330 turns, the space-charge tune depression is Δν = 0.4 at the energy of 1.5 GeV, and Δν = 0.2 at 2.4 GeV, assuming a bunching factor (the ratio of beam peak current to average current), during the early part of the acceleration cycle, of 4. Also, with the same normalized beam emittance of 250 π mm-mrad, the actual beam emittance is smaller at 2.4 GeV, ε = 73 π mm- mrad, versus 104 π mm-mrad at 1.5 GeV. Obviously, the effective acceptance of the AGS at injection is to be larger than these beam emittance values. The front-end of the Linac is made of an ion source operating with a 1% duty cycle at the repetition rate of 5 pulses per second. The beam current within a pulse is 60 mA of negative-hydrogen ions. The ion source seats on a platform at 35-60 kVolt, and is followed by a 5-MeV RFQ that works at 201.25 MHz. The beam is pre- chopped by a chopper located between the ion source and the RFQ. The beam chopping extends over 75% of the beam length, at a frequency matching the accelerating rf (4.11-4.28 MHz) at injection into the AGS. Moreover, the transmission efficiency through the RFQ is taken conservatively to be 80%, so that the average current of the beam pulse in the Linac, where we assume no further beam loss, is 36 mA. The combination of the chopper and of the RFQ pre- bunches the beam with a sufficiently small longitudinal extension so that each of the beam bunches can be entirely fitted in the accelerating rf buckets of the following DTL that operates at either 402.5 or 805 MHz. The DTL is a room temperature conventional Linac that accelerates to 150 MeV.Table 1. Injector and AGS Parameters for the Upgrade Linac Aver. Power, MW 0.25 0.40 Kinetic Energy, GeV 1.5 2.4 β 0.9230 0.9597 Momentum, GeV/c 2.2505 3.2037 Magnetic Rigidity, T-m 7.5068 10.6862 AGS Circumference, m 807.12 807.12 Revol. Frequency, MHz 0.3428 0.3565 Revolution Period, µs 2.9169 2.8053 Bending Radius 79.832 79.832 Injection Field 0.9403 1.3386 Protons per Turn, x 10116.563 6.312 Number of injected Turns 317 330 Beam Pulse Length, ms 0.9259 0.9259 Duty Cycle, % 0.4630 0.4630 Emittance, π mm-mrad 104. 73. Space-Charge Δν 0.41 0.21 3 THE SUPER-CONDUCTING LINAC The SCL accelerates the proton beam from 150 MeV to 1.5 or 2.4 GeV. The configuration and the design procedure of the SCL is described in detail in [2]. It is typically a sequence of a number of identical periods as shown in Figure 1. Each period is made of a cryo-module of length Lcryo and of an insertion of length Lins. The insertion is needed for the placement of focusing quadrupoles, vacuum pumps, steering magnets, beam diagnostic devices, bellows and flanges. It can be either at room temperature or in a cryostat as well. Here we assume that the insertions are at room temperature. The cryo-module includes M identical cavities, each of N identical cells, and each having a length NLcell, where Lcell is the length of a cell. Cavities are separated from each other by a drift space d. An extra drift of length Lw may be added internally on both sides of the cryo-module to provide a transition between cold and warm regions. Thus, Lcryo = MN Lcell + (M − 1) d + 2 Lw (1) There are two symmetric intervals: a minor one, between the two middle points A and B, as shown in Figure 1, that is the interval of a cavity of length NLcell + d; and a major one, between the two middle points C and D, that defines the range of a period of total length Lcryo + Lins. Thus, the topology of a period can be represented as a drift of length g, followed by M cavity intervals, and a final drift of length g, where g = Lw + (Lins − d ) / 2 (2) The choice of cryo-modules with identical geometry, and with the same cavity/cell configuration, is economical and convenient for construction. But there is, nonetheless, a penalty due to the reduced transit-time-factors when a particle crosses cavity cells, with length adjusted to a common central value β0 that does not correspond to theparticle instantaneous velocity. To minimize this effect the SCL is divided in four sections, each designed around a different central value β0, and, therefore, with different cavity/cell configuration. The cell length in a section is fixed to be Lcell = λβ0 / 2 (3) where λ is the rf wavelength. We assume the same operating frequency of 805 MHz for the entire SCL, so that λ = 37.24 cm. The major parameters of the four sections of the SCL are given in Tables 2 and 4. Transverse focussing is done with a sequence of FODO cells with half-length equal to that of a period. The phase advance per cell is 90o. The rms normalized betatron emittance is 0.3 π mm-mrad. The rms bunch area is 0.5 π oMeV. The rf phase angle is 30o. The cost estimate (with no contingency) for each section of the SCL has been made assuming the cost and rf parameters shown in Table 3. The total expected cost is around 300 M$, including also the front-end and the room-temperature DTL. The length of the Linac depends on the average accelerating gradient. The local gradient has a maximum value that is limited by three causes: (1) The surface field limit at the frequency of the 805 MHz is 26 MV/m. For a realistic cavity shape, we set a limit of a 13 MV/m on the axial electric field. (2) There is a limit on the power provided by rf couplers that we take here not to exceed 400 kW, including a contingency of 50% to avoid saturation effects. (3) To make the longitudinal motion stable, we can only apply an energy gain per cryo-module that is a relatively small fraction of the beam energy in exit of the cryo-module. The conditions for stability of motion have been derived in [2]. The proposed mode of operation is to operate each section of the SCL with the same rf input power per cryo- module. This will result to some variation of the actual axial field from one cryo-module to the next. If one requires also a constant value of the axial field, this may be obtained by adjusting locally the value of the rf phase. Lcryo Period Cryo-Module Lins Insertion Cavity C d D A B Topology of a Period g g C Cavity A B D Figure 1: Configuration of a Proton SCL Accelerator.The number of cells and cavities varies from insertion to insertion. The number of couplers varies for 1 to 2. The total length of the injector including the front-end and the DTL is expected to be about 500 meters. It is proposed to build the entire SCL in two stages. During the first stage the final energy if 1.5 GeV, and the Linac is made of the three first sections. In a second stage the High-Energy section is added for the final energy of 2.4 GeV, if indeed this should result to be necessary. Table 2. General Parameters of the SCL Linac Section LEMLEMHE HE Av. Beam Power, kW 50133250400 Av. Beam Current, mA 0.1670.1670.1670.167 In. Kin. Energy, GeV 0.1500.3000.8001.500 Fin. Kin. Energy, GeV 0.3000.8001.5002.400 Frequency, MHz 805805805805 Protons / Bunch x 1091.491.491.491.49 Temperature, oK 2.02.02.02.0 Cells / Cavity 4444 Cavities / Cryo-Module 4888 Cavity Separation, cm 32323232 Cold-Warm Trans., cm 30303030 Cavity Int. Diam., cm 10101010 Length of Warm Ins., m 1.0791.0791.0791.079 Acc. Gradient, MeV/m 12.211.912.912.4 Cavities / Klystron 4844 rf Couplers / Cavity 1122 Negative ion stripping during transport down the SCL has been found to be very negligible. But a final 30o bend, before injection into the AGS, could be of a concern [3]. To control the rate of beam loss by stripping to a 10-4 level, the bending field should not exceed 2.6 kGauss over a total integrated bending length of 15 m, in the 1.5 GeV case. At 2.4 GeV, the bending field is 1.9 kGauss, and the integrated bending length about 30 m. Table 3. Cost (’00 $) and Other Parameters AC-to-rf Efficiency 0.45For pulsed mode Cryo. Efficiency 0.004At 2.0 oK Electricity Cost 0.05$ / kWh Linac Availability 75% of yearly time Normal Cond. Cost 150k$ / m Superconducting Cost 500k$ / m Tunnel Cost 100k$ / m Cost of Klystron 2.50$ / W of rf Power Cost of Refrig. Plant 2k$ / W @ 2.0 oK Cost of Elect. Distrib. 0.14$ / W of AC Power There are two problems in the case of the pulsed-mode of operation of the SCL. First, there are Lorentz forces that deform the cavity cells out of resonance. They can be controlled with a thick cavity wall strengthened to the outside by supports. Second, there is an appreciable period of time to fill the cavities with rf power before the maximum gradient is reached [2]. During the filling time, extra power is dissipated also before the beam isinjected into the Linac. The extra amount of power is the ratio of the filling time to the beam pulse length. Table 4. Summary of the SCL Design Linac Section LEMLE MHE HE Energy: in out, GeV0.15 0.300.30 0.800.80 1.501.50 2.40 Velocity, β: in out0.5066 0.65260.6526 0.84180.8418 0.92300.9230 0.9597 Cell Reference β00.530 0.680 0.850 0.935 Cell Length, cm 9.8712.66 15.83 17.41 Total No. of Periods 912 13 15 Length of a period, m 4.218 7.971 8.984 9.490 FODO ampl. func., βQ, m14.40 27.21 30.67 32.40 Total Length, m 37.96 95.65116.79 142.35 Coupler rf Power, kW () 300 375 255 270 En. Gain/Period, MeV 16.67 41.67 56.67 60.00 Total No. of Klystrons 912 26 30 Klystron Power, kW () 1200 3000 2040 2160 Cavity Filling Time, ms 0.30 0.18 0.12 0.12 Z0T02, ohm/m 271.8 447.4 699.0 845.8 Q0 x 1095.5 7.0 8.7 9.6 Ave. Dissip. Power, kW 0.009 0.012 0.009 0.008 Ave. HOM-Power, kW 0.0016 0.0042 0.0046 0.0053 Ave. Cryog. Power, kW 0.152 0.430 0.527 0.644 Ave. Beam Power, MW 0.025 0.083 0.117 0.150 Ave. rf Power, MW () 0.050 0.149 0.197 0.248 AC Power for rf, MW () 0.110 0.331 0.438 0.552 AC Power for Cryo., MW 0.038 0.107 0.132 0.161 AC Power, MW () 0.148 0.439 0.570 0.713 Efficiency, % () 16.9 19.0 20.5 21.0 Capital Cost ’00 M$: Rf Klystron () Electr. Distr. () Refrig. Plant Warm Structure Cold Structure Tunnel0.124 0.021 0.303 1.619 14.126 3.9040.373 0.061 0.860 2.104 41.351 9.6730.493 0.080 1.055 2.266 51.381 11.7870.621 0.100 1.288 2.590 63.085 14.343 0.06167.061 Cost, ’00 M$ () 20.096 54.422 67.061 82.027 Operat, ’00 M$/y () 0.049 0.144 0.187 0.234 (*) Including 50% rf power contingency. REFERENCES [1] I. Marneris and A.G. Ruggiero, “Running the AGS MMPS at 5 Hz, 24 GeV”. BNL Report C-A/AP/12. January 21, 2000. [2] A. G. Ruggiero, “Design Considerations on a Proton Superconducting Linac”. BNL Report 62312. August 1995. [3] A. G. Ruggiero, “Negative-Ion Injection by Charge Exchange at 2.4 GeV”. BNL Report 62310. September 1995. [4] The program is available by making request to one of the Authors.
arXiv:physics/0008081 16 Aug 2000A SUPER-CONDUCTING LINAC DRIVER FOR THE HFBR∗ J. Alessi, D. Raparia, A.G. Ruggiero, BNL, Po Box 5000, Upton, NY 11973, USA ∗ Work performed under the auspices of the US Department of EnergyAbstract This paper reports on the feasibility study of a proton Super-Conducting Linac (SCL) as a driver for the High- Flux Breader Reactor (HFBR) at Brookhaven National Laboratory (BNL). The Linac operates in Continuos Wave (CW) mode to produce an average 10 MW of beam power. The Linac beam energy is 1.0 GeV. The average proton beam intensity in exit is 10 mA. 1 INTRODUCTION A proton SCL has been proposed to drive an internal solid target placed at the core of the de-commissioned HFBR at BNL. The purpose is to take advantage of the HFBR instrumentation and facility for the generation of a continuous, high-intensity neutron source. The SCL driver accelerates protons to 1 GeV, operates in CW mode, and generates an average beam power of 10 MWatt. The average beam current is 10 mA, and the total length about 300 m. The Linac is made of three parts: a Front-End, that is a 15 mA ion source, followed by a 5- MeV RFQ, a room temperature 150-MeV Drift-Tube Linac (DTL), and the SCL proper. This is made of two sections: the Medium-Energy Section (MES) that accelerates protons to 300 MeV, and the High-Energy Section (HES) that accelerates to 1 GeV. The selected operating frequency is 805 MHz. A spreadsheet program has been used for the design and cost estimate. The design has shown that the accelerator is feasible, can be built in a relatively short period of few years, and has an estimated total cost of about 300 M$. 2 REQUIREMENTS OF THE DRIVER The accelerator driver of the HFBR facility is a proton SCL about 300 m long with a straight-line geometry. The proton beam aims directly to the core, that is the center of the HFBR facility. The actual location of the accelerator that does not interfere with other facilities and utilities remains to be investigated. The SCL requirements are: 1-GeV Proton Energy, 10- MWatt Beam Power, and CW Mode of Operation. The Beam Current at the exit of the SCL is 10 mA. Acceleration of positive-ions (protons) is assumed. A preliminary design has shown that the accelerator is feasible, can be build in a relatively short period of few years, and has a total cost of about 300-400 M$. Aspread-sheet program [1] was used for the design and cost estimate, developed at the time of the design of the Accelerator for Tritium Production (APT) Linac [2, 4]. The requirements of the Linac driving the HFBR are similar to those of a Linac already investigated for a different type of application (Energy Amplifier) [3]. We have based our estimates essentially on that design. The accelerator is made of four major parts, shown in Figure 1: the Front-End, a room-temperature Low-Energy Section, the SCL proper, and the Transport to the Target. Front-End MES SCL Target Low-Energy HES Transport Section Figure 1. Layout of the 1-GeV, 10-MW SCL The Front-End is made of an Ion Source placed on a platform at 35-50 kVolt. It has a continuous beam output of 15 mA. It is followed by a 201.25-MHZ RFQ which focus, bunch and accelerate the beam to about 3-5 MeV. At the exit, the beam bunches are compressed sufficiently to be squeezed within the rf buckets of the Low-Energy Section which operates at 402.5 or 805 MHz. Because of the relatively low beam current, and the absence of stringent requirements on the beam emittance and momentum spread, space-charge effects are not expected to play a relevant role. As a consequence, no major beam losses are expected in the RFQ. A transmission of 80% is conservatively assumed, and the beam intensity at the exit of the RFQ is 12 mA. We allow another 80% overall transmission efficiency, that is a 20% beam loss, during the transfer of the beam through the rest of the accelerator, all the way down to the Target. At the exit of the SCL, and on the Target, the beam intensity is then 10 mA. 3 LINAC DESIGN In a proton Linac there is a large variation of beam velocity, in our case from β = 0.08 at 3 MeV to β = 0.875 at 1 GeV. The Low-Energy section cannot be made of half-wavelength super-conducting rf cavities, though quarter-wavelength super-conducting linear accelerators do exist and are successfully operational. We prefer to adopt here a room-temperature Low-Energy Section, made for instance of a conventional DTL similar to theone operating as injector to the Alternating Gradient Synchrotron at BNL. We shall also adopt a 150 MeV energy for this section to ease the design and manufacturing of the rf cavities in the early part of the SCL proper. Other solutions are of course possible, and they should be examined in a more careful and detailed design. We shall not attempt here to say more about the Front-End and the Low-Energy Section, except noting that likely the cost of both together is in the range 50- 100 M$ (year 2000). Thus, the SCL proper begins at 150 MeV and ends at 1 GeV. The corresponding variation of velocity is from β = 0.5066 to β = 0.8750. Since the length of the rf cavity cells is L = βλ/2, it should in principle vary between 9.5 and 16.4 cm, with λ = 37.3 cm, the rf wavelength at 805 MHz, the chosen operating rf frequency of the SCL. To optimize the accelerating gradient, and the transit time factor, it would be desirable to manufacture cavities with cells varying in length as the beam accelerates. This may not be economical, and we prefer [2-4] to manufacture rf cavities all with the same cell length. This simplifies the design, and reduces the cost, at the expense of a modest reduction of the transit time factor. Here we assume that the SCL is divided in two subsections each operating at two intermediate values of velocity. A super-conducting MES, from 150 to 300 MeV, has the cavity cell length adjusted to the intermediate value β = 0.525. The HES, from 300 MeV to 1 GeV, is designed with the intermediate value β = 0.69. The layout of the SCL is described in [2, 3]. For more details see also the related paper [5], where a SCL operating in pulsed mode is described. It is made of a sequence of identical periods each consisting of a Warm Insertion for the location of focussing quadrupoles, steering magnets, vacuum pumps, and instrumentation, and of a Cryo-Module including a number of cavities all with the same number of individual cells. One or more cavities are powered by a rf coupler connected to Klystrons, the rf power source. The general parameters of the SCL are given in Table 1. Cost and other parameters used for our estimate are shown in Table 2. The overall design is summarized in Table 3. The cost items are quoted in year 2000 dollars. There are four cells in one cavity, each 9.78 and 12.85 cm long, respectively in the MES and HES. There are also six and eight cavities in one cryostat, separated by 32 cm from each other. The cold-to-warm transitions at both ends of the cryostat are 30 cm long. With a 1-meter long warm insertion, this makes a period 5.546 and 7.951 m long, respectively in the MES and HES. To focus the transverse motion of the beam a FODO sequence of quadrupole magnets is assumed, with a quadrupole located every period.Table 1. General Parameters of the 10-MW, 1-GeV SCL MES HES Beam Power (CW) 3.6 MW 10 MW Beam Current 12 mA 10 mA In. Kinetic Energy 150 MeV 300 MeV Fin. Kinetic Energy 300 MeV 1.0 GeV Frequency 805 MHz 805 MHz Protons / Bunch 3.73 x 1083.10 x 108 Temperature 2.0 oK 2.0 oK Cells / Cavity 4 4 Cavities / Period 6 8 Cavity Separation 32 cm 32 cm Cold-Warm Trans. 30 cm 30 cm Cavity Int. Diameter 10 cm 10 cm Length of Warm Ins. 1.00 m 1.00 m Accel. Gradient 7.382 MeV/m 11.234 MeV/m Cavities / Klystron 6 8 rf Couplers / Cavity 1 1 Rf Phase Angle 30o30o Focussing Method FODO FODO Phase Advance / cell 90o90o Norm. rms Emitt. 0.3 π mm mrad 0.3 π mm mrad Rms Bunch Area 0.5 π o MeV 0.5 π o MeV We have also assumed a cryogenic temperature of the rf cavities of 2 oK. One Klystron groups together all the cavities in one Cryo-Module. The power of one Klystron is divided in 6 or 8 rf couplers, one for each cavity. To avoid saturation of the Klystrons, a 35% rf power contingency has been included above the requirement for the normal mode of operation. Table 2. Cost (’00 $) and Other Parameters AC-to-rf Efficiency 0.585For CW mode Cry. Efficiency 0.004At 2.0 oK Electricity Cost 0.05$ / kWh Linac Availability 75% of yearly time Normal Cond. Cost 150k$ / m Superconducting Cost 500k$ / m Tunnel Cost 100k$ / m Cost of Klystron 1.7$ / W of rf Power Cost of Refrig. Plant 2k$ / W @ 2.0 oK Cost of Electrical Distr. 0.14$ / W of AC PowerTable 3. Summary of the 1.0-GeV SCL Design LES HES Energy: in out150 MeV 300 MeV 300MeV 1GeV Velocity, β: in out0.5066 0.65260.6526 0.8750 Cell Reference β0 0.525 0.69 Cell Length, cm 9.78 12.85 Total No. of Periods 10 18 Length of a period, m 5.546 7.951 FODO ampl. func., βQ, m 18.94 27.15 Total Length, m 55.46 143.13 Coupler rf Power, kW () 40.5 67.5 En. Gain/Period, MeV 15 40 Total No. of Klystrons 10 18 Klystron Power, kW () 243 540 Z0T02, ohm/m 266.7 460.6 Q0 x 1095.41 7.1 Dissipated Power, kW 0.89 3.02 HOM-Power, kW 0.048 0.079 Cryogenic Power, kW 1.17 3.72 Beam Power, MW 1.8 7.0 Total rf Power, MW () 2.43 9.45 AC Power for rf, MW () 4.16 16.16 AC Power for Cryo., MW 0.29 0.93 Total AC Power, MW () 4.45 17.09 Efficiency, % () 40 41 Capital Cost ’00 M$: Rf Klystron () Electr. Distr. () Refrig. Plant Warm Structure Cold Structure Tunnel4.13 0.623 2.331 1.650 22.731 5.54616.07 2.392 7.441 2.850 62.563 14.313 Total Cost, ’00 M$ () 37.01 105.63 Oper. Cost, ’00 M$/y () 1.461 5.614 (*) Including 35% rf power contingency.REFERENCES [1] The program is available by making request to one of the Authors. [2] A. G. Ruggiero, “Design Considerations on a Proton Superconducting Linac”. BNL Report 62312. August 1995. [3] A.G. Ruggiero, “A Superconducting Linac as the Driver of the Energy Amplifier”. BNL 63527, UC-414 AGS/AD/97-1. October 1996. [4] A Feasibility Study of the APT Superconducting Linac. Edited by K.C.D. Chan. April 1996. Los Alamos National Laboratory, LA-UR-95-4045. [5] D. Raparia, A.G. Ruggiero, “A Super-Conducting Linac Injector to the BNL-AGS”. Contribution to this Conference.
arXiv:physics/0008082v1 [physics.data-an] 16 Aug 2000On the Conﬁdence Interval for the parameter of Poisson Distribution S.I. Bityukov3, N.V. Krasnikov1V.A. Taperechkina2 Institute for High Energy Physics, 142284, Protvino Moscow Region, Russia Abstract In present paper the possibility of construction of continu ous ana- logue of Poisson distribution with the search of bounds of co nﬁdence intervals for parameter of Poisson distribution is discuss ed and the re- sults of numerical construction of conﬁdence intervals are presented. PACS number(s): 02.70.Lq, 06.20.Dk Keywords: Statistics, Confidence Intervals, Poisson Dist ribution. 1Institute for Nuclear Research RAS, Moscow, Russia 2Moscow State Academy of Instrument Engineering and Compute r Science 3Email address: bityukov@mx.ihep.su 1Introduction In paper [1] the uniﬁed approach to the construction of conﬁd ence in- tervals and conﬁdence limits for a signal with a background p resence, in particular for Poisson distributions, is proposed. The met hod is widely used for the presentation of physical results [2] though a number of investigators criticize this approach [3] (in particular, this approach a voids a violation of the coverage principle). Series of Workshops on Conﬁdenc e Limits has been held in CERN and Fermilab. At these meetings demands for properties of constructed conﬁdence intervals and conﬁdence limits ha ve been formu- lated [4]. On the other hand, the results of experiments ofte n give noninteger values of a number of observed events (for example, after ﬁtt ing [5]) when Poisson distribution take place. That is why there is a neces sity to search a continuous analogue of Poisson distribution. The present w ork oﬀers some generalization of Poisson distribution for continuous cas e. The generalization given here allows to construct conﬁdence intervals and conﬁ dence limits for Poisson distribution parameter both for integer and real va lues of a number of observed events, using conventional methods. More than, the supposi- tion about continuous of some function f(x, λ) described below allows to use Gamma distribution for construction of conﬁdence interval s and conﬁdence limits of Poisson distribution parameter. In present paper we consider only the construction of conﬁdence intervals. In the Section 1 the generalization of Poisson distribution for the con- tinuous case is introduced. An example of conﬁdence interva ls construction for the parameter of analogue of Poisson distribution is giv en in the Section 2. In the Section 3 the results of construction of conﬁdence i ntervals having the minimal length for the parameter of Poisson distributio n using Gamma distribution are discussed. The main results of the paper ar e formulated in the Conclusion. 1 The Generalization of Discrete Poisson Dis- tribution for the Continuous Case Let us have a random value ξ, taking values from the set of numbers x∈X. Let us consider two-dimensional function f(x, λ) =λx x!e−λ, 2where x≥0λ >0. Assume, that set Xincludes only integer numbers, then discrete function f(x, λ) describes distribution of probabilities for Poisson dist ribution with the parameter λand random variable x Let us rewrite the density of Gamma distribution using uncon ventional notation f(x, a, λ ) =ax+1 Γ(x+ 1)e−aλλx, where ais a scale parameter, x >−1 is a shape parameter and λ >0 is a random variable. Here the quantities of xandλ take values from the set of real numbers. Let a= 1 and as is the convention x! = Γ( x+ 1), then a continuous function f(x, λ) =λx x!e−λ,λ >0,x >−1 is the density of Gamma distribution with the scale parameter a= 1. To anticipate a little, it is indicative of the Gamma distrib ution of pa- rameter λfor the Poisson distribution in case of observed value x= ˆx. Figure 1 shows the surface described by the function f(x, λ). Smooth behaviour of this function along xandλ(see Fig.2) allows to assume that there is such a function −1< l(λ), that/integraldisplay∞ l(λ)f(x, λ)dx= 1 for given value ofλ. It means that in this way we introduce continued analogue of Poisson distribution with the probability density f(x, λ) =λx x!e−λover the area of function deﬁnition, i.e. for x≥l(λ) and λ >0. The values of the function f(x, λ) for integer xcoincide with corresponding magnitudes in the probabil- ities distribution of discrete Poisson distribution. Depe ndences of the values of function l(λ), the means and the variances for the suggested distributio n onλwere calculated by using programme DGQUAD from the library C ERN- LIB [6] and the results are presented in Table 1. This Table sh ows that series of properties of Poisson distribution ( Eξ=λ, Dξ =λ) take place only if the value of the parameter λ >3. It is appropriate at this point to say that/integraldisplay∞ 0f(x, λ)dx=/integraldisplay∞ 0λxe−λ Γ(x+ 1)dx=e−λν(λ). The function ν(λ) =/integraldisplay∞ 0λx Γ(x+ 1)dxis well known and, according to ref. [7], ν(λ) =∞/summationdisplay n=−Nλn Γ(n+ 1)+O(\|λ\|−N−0.5) =eλ+O(\|λ\|−N) 3ifλ→ ∞,\|argλ\| ≤π 2for any integer N. Nevertheless we have to use the function l(λ) in our calculations in Section 2. We consider it as a mathema t- ical trick for easy construction of conﬁdence intervals by n umerically. In principle, we can numerically to transform the function f(x, λ) in the interval x∈(0,1) so that/integraldisplay∞ 0f(x, λ)dx= 1,Eξ=/integraldisplay∞ 0xf(x, λ)dx=λand Dξ=/integraldisplay∞ 0(x−Eξ)2f(x, λ)dx=λfor any λ. In this case we can construct conﬁdence intervals without introducing of l(λ). In Section 3 only assumption about continuous of the functio nf(x, λ) along the variable xare used for construction of conﬁdence intervals of pa- rameter λfor any observed ˆ x. Let us construct a central conﬁdence intervals for the conti nued analogue of Poisson distribution using function l(λ). 2 The Construction of the Conﬁdence Inter- vals for Continued Analogue of Poisson Dis- tribution. Assume that in the experiment with the ﬁxed integral luminos ity the ˆ xevents (ˆxis not necessity integer) of some Poisson process were obser ved. It means that we have an experimental estimation ˆλ(ˆx) of the parameter λof Poisson distribution. We have to construct a conﬁdence interval ( ˆλ1(ˆx),ˆλ2(ˆx)), cov- ering the true value of the parameter λof the distribution under study with conﬁdence level 1 −α, where αis a signiﬁcance level. It is known from the theory of statistics [8], that the value of mean of selected d ata is an unbi- assed estimation of mean of distribution under study. In our case the sample consists of one observation ˆ x. For the discrete Poisson distribution the mean coincides with the estimation of parameter value, i.e. ˆλ= ˆx. This is not true for small value of λin considered case (see Table 1). That is why in order to ﬁnd the estimation of ˆλ(ˆx) for small value ˆ xthere is necessary to introduce correction in accordance with Table 1. Let us cons truct the central conﬁdence intervals using conventional method assuming th at/integraldisplay∞ ˆxf(x,ˆλ1)dx=α 2for the lower bound ˆλ1and 4/integraldisplayˆx l(ˆλ2)f(x,ˆλ2)dx=α 2for the upper bound ˆλ2of conﬁdence interval. Figure 3 shows the introduced in the Section 1 distributions with param- eters deﬁned by the bounds of conﬁdence interval ( ˆλ1= 1.638,ˆλ2= 8.498) for the case ˆ x=ˆλ= 4 and the Gamma distribution with parameters a= 1, x= ˆx= 4. The association between the conﬁdence interval and the G amma distribution is seen from this Figure. The bounds of conﬁden ce interval with 90% conﬁdence level for parameter of continued analogue of P oisson distri- bution for diﬀerent observed values ˆ x(ﬁrst column) were calculated and are given in second column of the Table 2. It is necessary to notic e that the conﬁdence level of the constructed conﬁdence intervals alw ays coincides ex- actly with the required conﬁdence level. As it results from T able 2 that the suggested approach allows to construct conﬁdence interval s for any real and integer values of the observed number of events in the case of the values of parameter λ >3. The Table 2 shows that the left bound of central conﬁdence intervals is not equal to zero for small ˆ x. It is not suitable. Also note that 90% of the area of Gamma distributions with par ameter x= ˆxare contained inside the constructed 90% conﬁdence interva ls for ob- served value ˆ x(for small values of λ <0.3 we have got 88%). It points out the possibility of Gamma distribution usage for conﬁdence inte rvals construction for parameter of Poisson distribution. 3 Shortest Conﬁdence Intervals for Parame- ter of Poisson Distribution. As is follow from formulae for f(x, λ) (see Fig.3) we may suppose that the parameter λof Poisson distribution for the observed value ˆ xhas Gamma distribution1with the parameters a= 1 and x= ˆx. This supposition allows to choose conﬁdence interval of minimum length from all poss ible conﬁdence intervals of given conﬁdence level without violation of the coverage principle. The bounds of minimum length area, containing 90% of the corr esponding Gamma distribution square, were found by numerically both f or integer value of ˆxand for real value of ˆ x. Here we took into account that ˆλ= ˆx, constructed the central 90% conﬁdence interval and, then, found the shor test 90% con- 1The similar supposition is discussed in ref. [9] 5ﬁdence interval for the parameter of Poisson distribution. The results are presented in third column of Table 2. For comparison with the results of conventional procedure [2] of ﬁnding conﬁdence intervals, the results of cal- culations of conﬁdence intervals for integer value of ˆ x[1] are adduced in the Table 2. By this means conﬁdence intervals, got using Gamma d istribution, may be used for real values of ˆ x, even though the ˆ xis negative (ˆ x >−1). Conclusion In the paper the attempt of introducing of continued analogu e of Poisson distribution for the construction of classical conﬁdence i ntervals for the pa- rameter λof Poisson distribution is described. Two approaches (with using of function l(λ) and with using of Gamma distribution) are considered. Con- ﬁdence intervals for diﬀerent integer and real values of num ber of observed events for Poisson process in the experiment with given inte gral luminosity are constructed. As seems the approach with the use of Gamma d istribution for construction of conﬁdence intervals more preferable th an approach with using of function l(λ). Acknowledgments We are grateful to V.A. Matveev, V.F. Obraztsov and Fred Jame s for the interest to this work and for valuable comments. We are th ankful to S.S. Bityukov and V.A. Litvine for useful discussions. We wo uld like to thank E.A.Medvedeva for the help in preparing the paper. Thi s work has been supported by RFFI grant 99-02-16956 and grant INTAS-CE RN 377. References [1] G.J. Feldman and R.D. Cousins, Uniﬁed approach to the classical sta- tistical analysis of small signal, Phys.Rev. D57(1998) 3873 [2] C. Caso et al., Review of particle physics, Eur. Phys.J. C 3, 1-794 (1998) [3] as an example, G. Zech, Classical and Bayesian Conﬁdence Limits, Proceedings of 1st Workshop on Conﬁdence limits, by James, F ed., Lyons, L ed., Perrin, Y ed., CERN, Geneva, January 17-18, 200 0, p.141. 6[4] F. James, Introduction and Statement of the Problem, Proceedings of 1st Workshop on Conﬁdence limits, by James, F ed., Lyons, L ed., P errin, Y ed., CERN, Geneva, January 17-18, 2000, p.1. [5] D. Cronin-Hennessy et al., Observation of B→K±π0andB→K0π0 and Evidence for B→π+π−.Phys.Rev.Lett.D 85(2000) 515. [6] CERN PROGRAM LIBRARY, CERNLIB, Short Writeups, Edition - June 1996, CERN, Geneva, 1996. [7]Higher Transcendental Functions. vol. 3, by Erd´ elyi A. ed.; Bateman H., McGraw-Hill Book Company Inc., 1955, p.217. [8] as an example, Handbook of Probability Theory and Mathematical Statis- tics (in Russian), ed. V.S. Korolyuk, Kiev, ”Naukova Dumka”, 1978 [9]The advanced theory of statistics . v.1Distribution theory. by Kendall, Maurice G; Stuart, Alan; J. Keith Ord; 6th ed. published by Ar nold, A member of the Hodder Headline Group, 338 Euston Road, Londo n, 1994, p.182. 7Table 1: The function l(λ), mean and variance versus λ. λ l(λ)mean ( Eξ)variance ( Dξ) 0.001 -0.297 -0.138 0.024 0.002 -0.314 -0.137 0.029 0.005 -0.340 -0.130 0.040 0.010 -0.363 -0.120 0.052 0.020 -0.388 -0.100 0.071 0.050 -0.427 -0.051 0.113 0.100 -0.461 0.018 0.170 0.200 -0.498 0.142 0.272 0.300 -0.522 0.256 0.369 0.400 -0.539 0.365 0.464 0.500 -0.553 0.472 0.559 0.600 -0.564 0.577 0.653 0.700 -0.574 0.681 0.748 0.800 -0.582 0.785 0.844 0.900 -0.590 0.887 0.939 1.00 -0.597 0.989 1.035 1.50 -0.622 1.495 1.521 2.00 -0.639 1.998 2.012 2.50 -0.650 2.499 2.506 3.00 -0.656 3.000 3.003 3.50 -0.656 3.500 3.501 4.00 -0.647 4.000 3.999 4.50 -0.628 4.500 4.498 5.00 -0.593 5.000 4.997 5.50 -0.539 5.500 5.497 6.00 -0.466 6.000 5.996 6.50 -0.373 6.500 6.495 7.00 -0.262 7.000 6.995 7.50 -0.135 7.500 7.494 8.00 0.000 8.000 7.993 8.50 0.000 8.500 8.496 9.00 0.000 9.000 8.997 9.50 0.000 9.500 9.498 10.0 0.000 10.00 9.999 8Table 2: 90% C.L. intervals for the Poisson signal mean λfor total events observed ˆ x. bounds (Section 2) bounds (Section 3) bounds (ref[1]) ˆx ˆλ1ˆλ2ˆλ1ˆλ2ˆλ1ˆλ2 0.000 0.121E-08 2.052 0.0 2.302 0.00 2.44 0.001 0.205E-08 2.054 0.0 2.304 0.002 0.292E-08 2.056 0.0 2.306 0.005 0.666E-08 2.061 0.0 2.311 0.01 0.307E-07 2.076 0.0 2.320 0.02 0.218E-06 2.098 0.0 2.337 0.05 0.765E-05 2.166 1.66E-05 2.389 0.10 0.137E-03 2.275 2.23E-05 2.474 0.20 0.186E-02 2.490 6.65E-05 2.642 0.30 0.696E-02 2.692 1.49E-04 2.806 0.40 0.161E-01 2.891 2.60E-03 2.969 0.50 0.295E-01 3.084 5.44E-03 3.129 0.60 0.466E-01 3.269 1.35E-02 3.290 0.70 0.673E-01 3.450 2.63E-02 3.452 0.80 0.911E-01 3.629 4.04E-02 3.611 0.90 0.1179 3.804 6.12E-02 3.773 1.0 0.1473 3.977 8.49E-02 3.933 0.11 4.36 1.5 0.3257 4.800 0.2391 4.718 2.0 0.5429 5.582 0.4410 5.479 0.53 5.91 2.5 0.7896 6.340 0.6760 6.220 3.0 1.056 7.076 0.9284 6.937 1.10 7.42 3.5 1.340 7.792 1.219 7.660 4.0 1.638 8.493 1.511 8.358 1.47 8.60 4.5 1.946 9.188 1.820 9.050 5.0 2.264 9.869 2.120 9.714 1.84 9.99 5.5 2.590 10.55 2.453 10.39 6.0 2.924 11.21 2.775 11.05 2.21 11.47 6.5 3.264 11.87 3.126 11.72 7.0 3.609 12.53 3.473 12.38 3.56 12.53 7.5 3.961 13.18 3.808 13.01 8.0 4.316 13.82 4.160 13.65 3.96 13.99 8.5 4.677 14.46 4.532 14.30 9.0 5.041 15.10 4.905 14.95 4.36 15.30 9.5 5.406 15.73 5.252 15.56 10. 5.779 16.36 5.640 16.21 5.50 16.50 20. 13.65 28.49 13.50 28.33 13.55 28.52 9Figure 1: The behaviour of the function f(x,λ) versus λandxiff(x,λ)<1. 10Figure 2: Two-dimensional representation of the function f(x,λ) versus λandx for values f(x,λ)<1. 11Figure 3: The probability densities f(x,λ) of continuous analogous Poisson dis- tribution for λ’s determined by the conﬁdence limits ˆλ1andˆλ2in case of observed number of events ˆ x= 4 and the probability density of Gamma distribution with parameters a= 1 and x= ˆx= 4. 12
arXiv:physics/0008083v1 [physics.chem-ph] 16 Aug 2000Theory of nuclear spin conversion in ethylene P.L. Chapovsky∗and E. Ilisca Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee, Universit´ e Paris 7–Denis Diderot, 2, Place Jussieu, 75251 Paris Cedex 05, FRANCE (February 2, 2008) Abstract First theoretical analysis of the nuclear spin conversion i n ethylene molecules (13CCH 4) has been performed. The conversion rate was found equal ≃3· 10−4s−1/Torr, which is in qualitative agreement with the recently o btained experimental value. It was shown that the ortho-para mixing in13CCH 4is dominated by the spin-rotation coupling. Mixing of only two pairs of ortho- para levels were found to contribute signiﬁcantly to the spi n conversion. 03.65.-w; 31.30.Gs; 33.50.-j; Typeset using REVT EX ∗IA&E, Russian Academy of Sciences, 630090 Novosibirsk, Rus sia; e-mail: chapovsky@iae.nsk.su 1I. INTRODUCTION Nuclear spin isomers of molecules were discovered nearly 70 years ago when ortho and para hydrogen isomers were separated for the ﬁrst time. Alth ough it was realized already at the time of this discovery that many other symmetrical molec ules have nuclear spin isomers too, until recently almost nothing was known about isomers o f molecules heavier than hy- drogen. This happened because of the lack of practical separ ation methods. The separation method based on deep cooling was applicable to hydrogen (and deuterium) but failed in the case of heavier molecules. Presently a few new methods fo r spin isomer separation have been proposed and successfully tested which has advanced th is ﬁeld signiﬁcantly. Yet we are at the very early stage of this research. The list of molec ules for which the nuclear spin isomers have been separated is rather short: H 2[1], CH 3F [2], H 2O [3], CH 20 [4], Li 2[5], H+ 3 [6], and C 2H4[7]. The CH 3F molecules occupies a special place in this list because it i s the only poly- atomic molecule for which the isomer conversion mechanism h as been identiﬁed. The CH 3F spin isomers appeared to be extremely stable surviving ∼109gas collisions. Nevertheless, the isomer conversion was found to be based on ortho-para sta te mixing induced by tiny intramolecular hyperﬁne interactions and interruption of this mixing by collisions. This speciﬁc type of relaxation was proposed to refer to as quantu m relaxation. It is essential for the physics of molecular spin isomers to understand which pr ocesses are responsible for the spin conversion in other molecules. Recently ﬁrst separation of nuclear spin isomers of ethylen e molecules (13CCH 4) has been performed [7]. The conversion rate has been determined and it was shown that the rate increases proportional to the gas pressure, γ/P≃5·10−4s−1/Torr. An understanding of mechanism of the conversion in ethylene needs theoretica l investigation of the process. In the present paper we perform ﬁrst theoretical analysis of the spin conversion in ethy- lene molecules. The purpose is to verify the consistency of t he experimental data on isomer conversion in ethylene with the conversion by quantum relax ation. We consider the same isotope species,13CCH 4, for which the experiment was done [7]. An essential diﬀeren ce of the present study from the well-understood case of CH 3F molecules is that now we have to work with an asymmetric-top molecule. This appeared to be mu ch more complicated than the case of symmetric-top molecules. We would like to point o ut that spin conversion in 2asymmetric tops were considered theoretically previously for the CH 2O and H 20 molecules [8]. II. ISOMER CONVERSION BY QUANTUM RELAXATION Although the spin conversion in molecules by quantum relaxa tion is explained in other publications we will describe brieﬂy its essence here for th e convenience of the reader. At room temperature almost all molecules are situated in their ground electronic and vibrational states. Suppose that these states are separated into two sub spaces which are the nuclear spin ortho and para states, as is shown in Fig. 1 for the particular case of the13CCH 4molecules. Note that some molecules may have more than two nuclear spin i somer forms, e.g., methane, or normal ethylene. On the other hand, number of molecules, e .g., H 2, CH 3F, and13CCH 4 too, have just two type of spin species. The relaxation process which we are going to consider has two main ingredients. First, the ortho and para quantum states of the test molecules are no t completely independent. There is small intramolecular perturbation, ˆV, which is able to mix the ortho and para states. This perturbation can mix, in general, many ortho-p ara level pairs. But in Fig. 1 the mixing is present just for one pair of states as will be the case for13CCH 4, see below. Second, the test molecules are embedded into an environment which is able to induce fast relaxation inside the ortho, or para subspace, but is not abl e to produce direct transitions between these subspaces. This implies that the relevant cro ss-section, σ(ortho\|para) = 0. The relaxation by direct transitions is the process opposit e to the quantum relaxation. The isomer conversion by quantum relaxation consists in the following. Suppose that at the instant t= 0 a test molecule is placed into the ortho subspace. Due to co llisions with surrounding particles, the test molecule starts to perform fast migration along rotational states inside the ortho subspace. This is the familiar rotat ional relaxation. This running up and down along the ladder of the ortho states continues unt il the molecule jumps in the stateαwhich is mixed by an intramolecular perturbation with the energetically close para stateα′. During the free ﬂight after this collision, para state α′will be admixed to the ortho stateα. Consequently, the next collision can transfer the molecul e in other para states and thus localizes it inside the para subspace. Such a mechanism of spin isomers conversion was proposed in the theoretical work by Curl et al [8] (see also [9 ]). 3The quantum relaxation of spin isomers can be quantitativel y described in the framework of the kinetic equation for density matrix [9]. One needs to s plit the molecular Hamiltonian into two parts ˆH=ˆH0+ˆV , (1) where the main part of the Hamiltonian, ˆH0, has pure ortho and para states as the eigen- states; the perturbation ˆVmixes the ortho and para states. If at initial instant the non equi- librium concentration of, say, ortho molecules δρo(t= 0) was created, the system will relax then exponentially, δρo(t) =δρo(0)e−γt, with the rate γ=/summationdisplay a′∈p,a∈o2ΓF(a′\|a) Γ2+ω2 a′a(Wp(α′) +Wo(α));F(a′\|a)≡/summationdisplay ν′∈p,ν∈o\|Vα′α\|2. (2) The sets of quantum numbers α′≡ {a′, ν′}andα≡ {a, ν}consist of the degenerate quantum numbers ν′,νand the quantum numbers a′,awhich determine the energy of the states. In (2) Γ is the decay rate of the oﬀ-diagonal density matrix elem entρα′α(α′∈para;α∈ortho) assumed here to be equal for all ortho-para level pairs; ωa′ais the gap between the states a′anda;Wp(α′) and Wo(α) are the Boltzmann factors of the corresponding states. For the following it is convenient to introduce the strength of mixing ,F(a′\|a), which sums the intramolecular couplings over all degenerate states. III. ROTATIONAL STATES OF ETHYLENE Ethylene is a popular object in high resolution infrared spe ctroscopy. This fortunate circumstance made available rather accurate data on molecu lar parameters and position of molecular rotational levels. The molecular structure, num bering of hydrogen atoms and orientation of molecular system of coordinates are present ed in Fig. 2.13CCH 4is a plane, prolate, nearly symmetric top having the symmetry point gro up C 2v. The characters of the group operations and the irreducible representations are g iven in the Table 1. We give in the Table 1 also two isomorphic groups, molecular symmetry grou p C2v(M) [10] and the point group D 2. The bond lengths and angles which were used in our calculati on are as follows: rCH= 1.087˚A,rCC= 1.339˚A, the angle αHCH=117.40[11]. The fact that the ethylene molecule is an asymmetric top comp licates in two aspect the theoretical analysis of the isomer conversion. First, the e nergy levels and wave functions 4for asymmetric tops can be found only numerically but not ana lytically as is possible for symmetric-top molecules. Second, the rotational quantum n umber k(projection of molecular angular momentum on the molecular symmetry axis) becomes an approximate quantum number in asymmetric tops. Consequently, one should calcul ate numerically more ortho- para matrix elements than in the conversion of symmetric-to p molecules, see, e.g., [9]. Rotational states of13CCH 4can be determined using the Hamiltonian of Watson [12] and the set of molecular parameters from Refs. [13,14]. The r otational Hamiltonian up to sextic order terms has the form [12] ˆH0=1 2(B+C)J2+ (A−1 2(B+C))J2 z−∆J(J2)2−∆JKJ2J2 z−∆KJ4 z +HJ(J2)3+HJK(J2)2J2 z+HKJJ2J4 z+HKJ6 z +1 4(B−C)F0−δJJ2F0−δKF2+hJ(J2)2F0+hJKJ2F2+hKF4 (3) whereJ,Jx,Jy, andJzare the molecular angular momentum operator and its project ions on the molecular axes. The B,C, andAare the parameters of a rigid top which characterize the rotation around x,y, andzmolecular axes, respectively (see, Fig. 2). The rest of para meters account for the centrifugal distortion eﬀects [12]. In Eq. ( 3) the notation was used Fn≡Jn z(J2 x−J2 y) + (J2 x−J2 y)Jn z. (4) The Hamiltonian (3) can be diagonalized for each value of JandMin the basis of the symmetric-top quantum states \|J, k, M > (−J≤k≤J), where J,kandMare the quantum numbers of the angular momentum, its projection on t he molecular symmetry axis and on the laboratory quantization axis, respectively. The Hamiltonian (3) has diagonal inkmatrix elements due to the ﬁrst two lines and matrix elements having \|∆k\|= 2. The calculations of the rotational eigen states can be simpliﬁe d if one uses the Wang basis [15] withK=\|k\|: \|α, p > =1√ 2/bracketleftig \|α >+(−1)J+K+p\|α >/bracketrightig ; 0< K≤J, \|α0, p > =1 + (−1)J+p 2\|α0>;K= 0. (5) Herep= 0,1 and the sets of quantum numbers are α≡ {J, K, M };\|α >≡ {J,−K, M};α0≡ {J, K= 0, M}. Depending on the parity of J,Kandp, the states (5) generate 4 diﬀerent irreducible representations of the molecular symmetry gro up D 2(Table 1). The molecular 5Hamiltonian is full symmetric (symmetry A 1). Consequently, the matrix elements between the states of diﬀerent symmetry disappear. Thus diagonaliz ation of the total Hamiltonian in the basis of (5) is reduced to the diagonalization of four ind ependent submatrices, each for the states of particular symmetry. The rotational states of asymmetric top can be expanded over the basis states (5) \|β, p > =/summationdisplay KAK\|α, p >, (6) where AKstands for the expansion coeﬃcients. The summation index, K, is given explicitly in (6), although AKdepends on the other quantum numbers as well. All coeﬃcients in the expansion (6) are real numbers because the Hamiltonian ( 3) is symmetric in the basis \|α, p > . There are a few schemes for practical classiﬁcation of the ro tational states of asymmetric tops [16]. In this paper we will use the scheme which is somewh at better adapted to the description of nuclear spin isomers. We will designate the r otational states of asymmetric top by indicating p,Jand prescribing the allowed Kvalues to the eigenstates keeping both in ascending order. For example, the eigenstate having p= 0, J= 20, being the third in ascending order will be designated by K= 4, because the allowed Kin the expansion (6) areK= 0,2,4. . .20. (Note the diﬀerence between the two characters KandK). It gives unambiguous notation of rotational states for each of the fo ur species A 1, A2(K-even) and B1, B2(K-odd). This classiﬁcation establishes the connection with the prolate symmetric top for which K=K. One should remember that physical meaning of the quantum nu mber Kis clearly limited. To illustrate this we consider as an exam ple the rotational states of A 1 and A 2symmetry ( K-even) for J= 20. The upper panel in Fig. 3 shows that the energy of the rotational states is not determined solely by Kas it would be for a rigid symmetric top. The graph shows the diﬀerence in energy between rotational s tates A 1and A 2having the sameK-number but diﬀerent quantum numbers p. The states A 1,K= 0 and A 2,K= 0 are omitted from the graph because the latter state does not exis t. As it is seen from the data the splitting is signiﬁcant for low values of Kand rapidly disappears as Kincreases. Note, that the A 1and A 2states for a rigid symmetric-top molecule would be degenera te for all K. The low panel in Fig. 3 illustrates the property of the eigen s tate expansions over the basis states (5). In this panel the squared magnitude of the two AKcoeﬃcients in each eigenstate of the symmetry A 1(J= 20) is given. The ﬁrst one (” K-term” in the Fig. 3) is A2 Khaving 6K=K. These coeﬃcients appeared to be the biggest coeﬃcients in e ach expansion. This adds some physical insight to the proposed classiﬁcation sc heme. The ”second term” is the second biggest coeﬃcient in the expansion. Again, at low Kvalues there are more than one signiﬁcant terms in the expansion (6). As Kgrows, the contribution from just one term becomes predominant. Note that the same graph for a symmetri c-top molecule would show only one signiﬁcant term, AK=K, in the expansions of all eigenstates. IV. ORTHO AND PARA ISOMERS OF ETHYLENE The total molecular wave function is a product of spatial wav e function and spin wave function. The nuclear spin states in13CCH 4are of A 1and B 1species, having the statistical weights 20 and 12, respectively. The ortho spin states (symm etry A 1) have the two hydrogen pairs (H 1-H2and H 3-H4) either both in triplet state, or both in singlet state. The p ara species (symmetry B 1) have one pair of protons in singlet state but the other pair i n triplet state. Each symmetry operation of the C 2vgroup interchange in13CCH 4even number of protons. Consequently, the total wave function must be unch anged under all operations, thus belonging to the representation A 1. In order to have the total wave function being of species A 1one has to have the spatial wave function being of symmetry A 1and B 1as the spin wave functions are. This implies that the rotational st ates A 1and B 1should be only positive (even in parity) but the rotational states A 2and B 2should be only negative (odd in parity). Summarizing this discussion we can write the total states in the13CCH 4molecules. The ortho states can be presented as \|µ >=\|β, p > \|i12, i34, iC>;i12, i34=t, s;K −even. (7) Herei12,i34, and iCdesignate the spin states of the two pairs of protons and the n ucleus 13C (Fig. 2). For ortho molecules the spin states of the proton p airs should be either both triplet, \|t, t >, or both singlet states, \|s, s >. The para states can be presented as \|µ′>=\|β′, p′>\|i′ 12, i′ 34, i′ C>;K′−odd. (8) For para molecules one pair of proton is in singlet, but the ot her in triplet state, thus the proton spin states are \|t′, s′>, or\|s′, t′>. In (7), (8) and further we use unprimed 7parameters for ortho species and primed parameters for para species; p= 0,1 indicates positive, or negative sign of the state, respectively. Eqs. (7), (8) imply that the ortho-para state mixing in ethylene needs coupling of states having diﬀ erent parity of K. The relative position of ortho and para states is important f or the calculation of the conversion rate (see Eq. (2)). In the case of a symmetric-top molecule the ortho-para energy gaps can be expressed as a polynomial of JandK. This gives “accidental” (isolated) resonances between the ortho and para states in symmetric to p. In the case of an asymmetric top one has a phenomenon which can be called collapse of ortho and para states. It appears as a progressive decrease of the ortho-para energy gaps betwee n the states of identical rotational momenta, J, asJincreases. This is illustrated in Fig. 4, which shows the gap s between the 13CCH 4ortho states, K= 0, and the para states, K′= 1, both having positive sign ( p= 0) and the same J. If such sequence of ortho-para level pairs would exist for s ymmetric tops it would dominate the conversion. In the case of asymmetric top s the situation is diﬀerent. We will see below that this sequence of closed ortho-para level pairs do not produce a signiﬁcant contribution to the isomer conversion in13CCH 4. The Boltzmann factors Wo(α) and Wp(α′) in Eq. (2) determines the relative population of rotational states in the ortho and para families, ρα=ρoWo(α);ρα′=ρpWp(α′), (9) where ρoandρpare the total densities of ortho and para molecules, respect ively. The Eqs. (9) imply the equilibrium distributions inside the ortho and pa ra subspaces. This is fulﬁlled with high accuracy even if the ratio ρo/ρpis out of equilibrium because rotational relaxation is on many orders of magnitude faster than the ortho-para conve rsion. The partition functions for ortho and para molecules at room temperature (T=295 K) ar e found to be equal to Zortho= 2.66·104;Zpara= 1.60·104. (10) In the calculation of these partition functions we took into account the degeneracy over M, nuclear spins, including also spin of nucleus13C, parity of states, as well as the restrictions imposed by quantum statistics. 8V. ORTHO-PARA STATE MIXING In the present paper we will consider the ortho-para convers ion induced by the two hyperﬁne interactions, viz., magnetic dipole-dipole inte raction between the molecular nuclei (spin-spin interaction, ˆVSS) and the nuclear spin-rotation interaction, ˆVSR. Thus the total intramolecular perturbation able to mix the ortho and para s tates is ˆV=ˆVSS+ˆVSR. (11) All matrix elements of ˆVare diagonal in parity p. A. Nuclear spin-spin coupling The ortho-para conversion in molecules induced by nuclear s pin-spin coupling were in- vestigated in [8,9] (other references see in the review [17] ). The spin-spin Hamiltonian for the two magnetic dipoles µ1andµ2separated by the distance rhas the form [15] ˆV12=P12T(12) • •ˆI(1)ˆI(2); T(12) ij=δij−3ninj;P12=µ1µ2/r3I(1)I(2)h , (12) where ˆI(1)andˆI(2)are the spin operators of the particles 1 and 2, respectively ;nis the unit vector directed along r;iandjare the Cartesian indices. The second rank tensor T(12)in the Eq. (12) represents a spatial part of the spin-spin inter action. The second rank tensor ˆI(1)ˆI(2)acts on spin variables. The total spin-spin interaction in13CCH 4is composed from the interactions between all pairs of molecular nuclei. One can show that the spin-spin in teractions between the protons 1-3, 2-4, 1-2, and 3-4 have the spatial part which can mix the q uantum states only if they haveKnumbers of the same parity. Consequently, these terms do not contribute to the ortho-para state mixing in ethylene. The complete nuclear s pin-spin interaction in13CCH 4 able to mix the ortho and para states reads VSS=V(14) SS+V(23) SS+V(C1) SS+V(C2) SS+V(C3) SS+V(C4) SS. (13) Here the upper indices refer to the hydrogen nuclei in the mol ecule 1...4, and to the nucleus 13C, which has spin equal 1/2 (see Fig. 2). Calculation of the sp in-spin matrix elements can 9be simpliﬁed by a few observations which we discuss in more de tail for the perturbations V(14) SSandV(23) SS. First, one can proof by applying proper symmetry operation that: < µ′\|V(14) SS\|µ >=< µ′\|V(23) SS\|µ > . (14) Next, the two, out of four, matrix elements between the ortho and para spin states vanish, < t′, s′\|V(14) SS\|s, s >=< s′, t′\|V(14) SS\|s, s >= 0, (15) because in these two cases one has the matrix elements of a vec tor (spin operator) between the states both having zero spin. On the other hand, the remai ning two matrix elements are equal to each other, < t′, s′\|V(14) SS\|t, t >=< s′, t′\|V(14) SS\|t, t > . (16) To summarize, one can conclude that the variety of matrix ele ments for the operators V(14) SSandV(23) SSis reduced to one matrix element, e.g., < β′, p, t′, s′\|V(14) SS\|β, p, t, t > . Obvi- ously, the carbon spin state, which is omitted in this expres sion for simplicity, should be unchanged in this matrix element. Further, we write this matrix element using an expansion ove r symmetric-top states (6) < β′, p, t′, s′\|V(14) SS\|β, p, t, t > =< t′, s′\| /summationdisplay K′,KA′ K′AK< α′, p\|V(14) SS\|α, p > \|t, t > . (17) This expression reduces the calculation of the spin-spin ma trix elements in asymmetric tops to the calculation of symmetric-top matrix elements. Solut ion for the latter can be found in [9], which allows to express the strength of mixing in ethyle ne by ˆVSSas FSS(a′\|a) = (2 J′+ 1)(2 J+ 1)T2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay K>0,qqA′ K+qAK J′2J −K−q q K  +1 + (−1)J+p √ 2A′ 1A0 J′2J −1 1 0 /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (18) Hereq=±1; (: : :) stands for the 3j-symbol and the notation was used T2= 2\|P14T(14) 2,1\|2+ 2\|PC1T(C1) 2,1\|2+ 2\|PC3T(C3) 2,1\|2;T= 46.5 kHz , (19) where P-factors are equivalent to the similar factor in (12); T(mn) 2,1are the spherical compo- nents of the corresponding T-tensor calculated in the molecular frame. The numerical va lue 10ofTwas found using the molecular structure from [18]. The selec tion rules for the spin-spin mixing in ethylene read ∆p= 0; \|∆J\| ≤2, (20) and in addition, parity of K′andKis opposite. B. Spin-rotation coupling Ortho-para conversion in molecules induced by spin-rotati on coupling were studied in [8,19,20] (more references can be found in [17]). Nuclear sp in-rotation coupling in molecules is due to magnetic ﬁelds produced by the molecular electrica l currents. The spin-rotation perturbation can be presented as [21,16,20] ˆVSR=1 2/parenleftigg/summationdisplay iˆI(i)•C(i)•ˆJ+h.c./parenrightigg ;i= 1,2,3,4. (21) For the spin-rotation perturbation relevant to the ortho-p ara mixing, the index ishould refer only to the hydrogen nuclei. The calculation of the second rank spin-rotation tensor Cis rather complicated problem which is not solved yet completely. We will use the following estimation of C. First we note that contribution to the spin-rotation coupling arising fr om the electric ﬁelds in the molecule has been shown to be very small compared to the part having “ma gnetic” origin and can be neglected [22]. Then, we split the C-tensor arising from magnetic ﬁelds of the moving charges into two parts, C=eC+nC, (22) produced by the molecular electrons and nuclei, respective ly. The nuclear contribution,nC, is obtained as a ﬁrst order average in the vibrational ground state, and depends only on the nuclear coordinates. Contrary, the electron part,eC, is a second order series expansion which involves the full electronic spectrum. Fortunately, xzandzxcomponents which are responsible for the mixing of states having ∆ K= 1 in C 2H4are vanishing by symmetry requirements [20]. Therefore, the C-tensor, eﬀective in ortho-para conversion, can be written (in Hz) for thei-th as [22] 11C(i)=/summationdisplay k/negationslash=ibk[(rk•Rk)1−rkRk]•B; bk= 2µpqk/c¯hR3 k, (23) where Rkis the radial vector from the proton H(i)to the charge k;rkis the radial from the center of mass to the particle k;qkare the nuclei’ charges; Bis the inverse matrix of inertia moment. Bis a diagonal matrix having the elements Bxx= 58.6 GHz, Byy= 48.7 GHz, andBzz= 291 .9 GHz. Index kruns here over all nuclei in the molecule except the proton i. The spherical components of the spin-rotation tensor of the rankl(l= 1,2) for the i-th proton calculated in the molecular frame, C(i) l,q, can be determined using Eq. (23). For the ethylene molecular structure from the Ref. [18] and bare nuc lei’ charges these components are C(1) 2,1= 3.8 kHz; C(1) 1,1=−2.9 kHz; C(3) 2,1=−4.1 kHz; C(3) 1,1= 3.2 kHz . (24) The diﬀerence between absolute values of C(1) l,qandC(3) l,qappears because of the shift of the molecular center of mass caused by the bigger mass of13C in comparison with12C. Similar to the previous section, one can reduce the calculat ion of the matrix elements of the spin-rotation coupling in asymmetric tops to the calc ulation of the symmetric-top matrix elements. The latter can be found in [19,20,23]. Thus one has the the strength of mixing due to the spin-rotation coupling in ethylene, FSR(a′\|a) = 2(2 J′+ 1)(2 J+ 1)/summationdisplay i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay K>0,qA′ K+qAKΦ(i;J′, K+q\|J, K) +1 + (−1)J+p √ 2A′ 1A0Φ(i;J′,1\|J,0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (25) Hereq=±1;i= 1,3 denotes the hydrogen nuclei 1 and 3. Note, that the protons 2 and 4 were also taken into account because they produce mixing equ al to that of protons 1 and 3. In (25) the notation was used Φ(i;J′, K′\|J, K) =/summationdisplay l√ 2l+ 1C(i) l,q J′l J −K′q K ×  y(J)(−1)l  J′J l 1 1J  +y(J′)  J J′l 1 1J′   , (26) 12where {: : :}stands for the 6j-symbol; y(J) =/radicalig J(J+ 1)(2 J+ 1). The selection rules for the ortho-para mixing by spin-rotation perturbation in eth ylene read ∆p= 0; \|∆J\| ≤1. (27) And again, parity of K′andKis opposite. VI. CONVERSION RATES AND DISCUSSION The decoherence rate, Γ, is another important parameter for the isomer conversion by quantum relaxation (see Eq. (2)). This parameter is rather d iﬃcult to calculate or to measure. Due to its physical meaning the value of Γ is close to the rotational state population decay, which determines the line broadening in microwave ro tational spectra. Unfortunately, the ethylene does not have pure rotational spectra because i t has no permanent electric dipole moment. Or, to be precise, very small dipole moment in the cas e of13CCH 4. In the present calculations we take the estimation of Γ from t he line broadening in infrared spectra, for which the information is available. R ecently the line broadening was accurately measured for the C 2H4molecules embedded in nitrogen [24]. One should not expect big diﬀerence for the line broadening in the isotope s pecies C 2H4and13CCH 4. From the data [24] one can estimate the decoherence rate as Γ/P= 2·107s−1/Torr. (28) This rate is 10 times smaller than the Γ determined for the iso mer conversion in CH 3F [17]. The decrease of Γ is because ethylene is not a polar molecule a nd thus has its rotational relaxation slower. Probably, the estimation (28) is too low for the case of pure ethylene gas because ethylene has polarizability larger than nitrogen. For the buﬀer gases having large polarizability, like Kr, or SF 6, nearly three times bigger line broadening was found [25]. Nevertheless, we will use the estimation (28) which can be co nsidered as a low limit for Γ. Now we are ready to calculate the conversion rate in ethylene . The ﬁnal expression for the conversion rate, γ, is again the Eq. (2) having the strength of mixing, F(a′\|a) = FSS(a′\|a) +FSR(a′\|a). The results of the calculations are given in the Table 2. Fr om these data we note that there is no appreciable contribution to the conversion from the collapsing ortho-para level pairs presented in Fig. 4. Another observa tion is that the contribution from 13the mixing of states having \|∆K\|>1 is also small. This is the consequence of the fact that the ethylene molecule is rather close to a symmetric top. The main contributions to the conversion come from just two level pairs, both having the sa me quantum numbers ( J′,K′)– (J,K), but diﬀerent p. The most important ortho-para level pair (1,27,7)–(1,28, 6) has the frequency gap ≃1.0 GHz. This level pair is indicated in Fig. 1 from which one c an see that this pair of states is situated at rather high energies. Near ly 10% of the conversion rate is due to the pair (0,27,7)–(0,28,6). This level pair has the sa me strength of mixing as the ﬁrst pair but three times bigger level splitting, ω. The properties of the states (1 , J= 28,K= 6) and (1 , J= 27,K= 7) are illustrated in Fig. 5 which presents the expansion co eﬃcients, AK, for these states. One can note that the biggest coeﬃcient in b oth expansions has the value ofKequal to the value of Kchosen for the designation of the eigenstate in our classiﬁc ation scheme. The total conversion rate in13CCH 4which is the sum over all ortho-para level pairs having J′andJup to 40, was found equal γ/P= 2.7·10−4s−1/Torr. (29) This rate is close to the experimental value (5 .2±0.8)·10−4s−1/Torr [7]. The calculations of the conversion rates were repeated usin g less accurate molecular parameters from [13]. We obtained essentially the same resu lts. The conversion is again determined by the same two level pairs and the value of the tot al conversion rate is 3 .5· 10−4s−1/Torr, which is close to the more precise value (29). Our model allows to determine the pressure dependence of the conversion rate in13CCH 4 at the conditions of the experiment [7]. By comparing the lev el splitting of the most impor- tant ortho-para level pair (1,27,7)–(1,28,6), which is ∼1.0 GHz, with the decoherence rate (28) one can conclude that the case corresponds to the limit Γ ≪ω. From Eq. (2) one can see that in this limit linear grows of the conversion rate ,γ, versus gas pressure should take place. (Note that Γ is proportional to the gas pressure. ) This dependence was indeed observed in the experiment [7]. Magnitude of the calculated conversion rate relies heavily on the determination of the ortho-para level splitting. The gap of the most important le vel pair appeared to be rather large, ≃1.0 GHz, and is unlikely that it has been determined with sign iﬁcant error. This can be proven by comparing this splitting with the one calculate d using less accurate molecular 14parameters [13]. These parameters give the gap 0.914 GHz, wh ich is close to the value given in the Table 2. The two sources of the ortho-para mixing have been analysed. The spin-spin interaction between the molecular nuclei was possible to calculate rath er accurately. The uncertainty of this perturbation is mainly due to small errors in the knowle dge of the molecular structure. On the other hand, our analysis has shown that the spin-spin c oupling contributes less than 1% to the ethylene conversion and thus negligible. Ther e are a few reasons for this reduction. First, the spin-spin tensor has relatively smal l value due to the big distances between interacting nuclei and small magnitude of the magne tic moment of the carbon nucleus. Second, due to high symmetry of the molecular struc ture interactions between many pairs of protons do not contribute to the ortho-para mix ing. Moreover, there are no close ortho-para level pairs at small J, where spin-spin coupling can compete with the spin-rotation coupling. We have taken into account also the spin-rotation perturbat ion. The magnitude of the spin-rotation tensor was estimated and it was demonstrated that the main contribution to the ethylene conversion comes from the spin-rotation mix ing. Strong contribution from the spin-rotation coupling is due to the fact that the streng th of the mixing FSRgrows asJ3in comparison with the slower ( J2) grows of the spin-spin strength of mixing FSS. This makes the “accidental” resonance at big J(1,27,7)–(1,28,6) being so important. An extra cause of the eﬃciency of the spin-rotation coupling is that the spin-rotation tensors of ﬁrst and second rank contribute both to the conversion. On the other hand, not all close ortho-para level pairs contribute to the conversion. In the particular case of the collapsing ortho-para levels (Fig. 4) there is no strong mixing mainly b ecause of interferences between many symmetric-top components with diﬀerent quantum numbe rsK, and also because for these J′=Jpairs the ﬁrst rank spin-rotation tensor does not contribut e. Error in the estimation of Γ introduces systematic uncertai nty into the ﬁnal result. But it does not undermine the model as a whole. First, one can turn around the approach and determine which decoherence rate Γ should be chosen in order to ﬁt the experimental data. In order to reproduce the experimental rate of conversion one s hould take the decoherence rate, Γ/P= 3.9·107s−1/Torr, which is less than two times diﬀerent from our rough es timation (28). Of course, such approach will be justiﬁed only when the spin-rotation perturbation 15in ethylene will be known from an independent source. There i s another point which is worth to mention. Recently it was proposed a new approach to t he isomer conversion in which the knowledge of Γ is not necessary at all [26]. It is bas ed on a fast linear sweep of the molecular levels through the ortho-para resonance. N uclear spin conversion in such experimental arrangement does not depend on Γ and is depende nt on the strength of ortho- para mixing and the population of the mixed states. Finally, it seems interesting to emphasize the diﬀerence in the conversion rates produced by the two pairs of identical rotational quantum numbers (27 ,7)–(28,6) but diﬀerent in parity. This peculiarity arises from the asymmetric top pro perties, where energy of states depend on parity. This in turn has dynamical consequence tha t the spin isomers equilibrate ﬁrst in one parity manifold (here the odd one). VII. CONCLUSIONS The theoretical model for the nuclear spin conversion in eth ylene (13CCH 4) has been developed. For the ﬁrst time a theory of the spin isomer conve rsion in asymmetric tops was possible to compare with experimental data on the isomer con version. We have found that the two experimental results [7]: the magnitude of the ethyl ene (13CCH 4) isomer conversion rate and its pressure dependence, are consistent with the sp in conversion governed by quan- tum relaxation. The ortho-para state mixing is performed in this molecule mainly by the coupling between the protons’ spins and the molecular rotat ion. We have identiﬁed also the two pairs of ortho-para states which are almost completely d etermine the spin conversion in 13CCH 4 ACKNOWLEDGMENTS The authors are indebted to A. Fayt for the possibility to use the latest set of the ethylene molecular parameters prior to publication and to M. Irac-As taud for stimulating discussions. 16REFERENCES [1] K. F. Bonhoeﬀer and P. Harteck, Die Naturwisssenschafte n.17, 182 (1929). [2] L. N. Krasnoperov, V. N. Panﬁlov, V. P. Strunin, and P. L. C hapovsky, Pis’ma Zh. Eksp. Teor. Fiz. 39, 122 (1984), [JETP Lett, 39, 143-146 (1984)]. [3] V. K. Konyukhov, A. M. Prokhorov, V. I. Tikhonov, and V. N. Faˇizulaev, Pis’ma Zh. Eksp. Teor. Fiz. 43, 65 (1986), [JETP Lett. 43, 65 (1986)]. [4] J. Kern, H. Schwahn, and B. Schramm, Chem. Phys. Lett. 154, 292 (1989). [5] R. A. Bernheim and C. He, J. Chem. Phys. 92, 5959 (1990). [6] D. Uy, M. Cordonnier, and T. Oka, Phys. Rev. Lett. 78, 3844 (1997). [7] P. L. Chapovsky, J. Cosl´ eou, F. Herlemont, M. Khelkhal, and J. Legrand, Chem. Phys. Lett.322, 414 (2000). [8] R. F. Curl, Jr., J. V. V. Kasper, and K. S. Pitzer, J. Chem. P hys.46, 3220 (1967). [9] P. L. Chapovsky, Phys. Rev. A 43, 3624 (1991). [10] P. R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, San Francisco, London, 1979). [11] E. Hirota, Y. Endo, S. Saito, K. Yoshida, I. Yamaguchi, a nd K. Machida, J. Mol. Spectrosc. 89, 223 (1981). [12] J. K. G. Watson, J. Chem. Phys. 48, 4517 (1968). [13] M. D. Vleeschouwer, C. Lambeau, A. Fayt, and C. Meyer, J. Mol. Spectrosc. 93, 405 (1982). [14] A. Fayt, private communication, 1999. [15] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1981). [16] C. H. Townes and A. L. Shawlow, Microwave Spectroscopy (McGraw-Hill Publ. Comp., New York, 1955), p. 698. 17[17] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Che m.50, 315 (1999). [18] M. V. Vol’kenshtein, L. A. Gribov, M. A. El’jashevich, a nd B. I. Stepanov, Molecular vibrations (In Russian) (Nauka, Moscow, 1972). [19] K. I. Gus’kov, Zh. Eksp. Teor. Fiz. 107, 704 (1995), [JETP. 80, 400-414 (1995)]. [20] E. Ilisca and K. Bahloul, Phys. Rev. A 57, 4296 (1998). [21] G. R. Gunther-Mohr, T. C. H, and J. H. Van Vleck, Phys. Rev .94, 1191 (1954). [22] K. Bahloul, M. Irac-Astaud, E. Ilisca, and P. L. Chapovs ky, J. Phys. B: At. Mol. Opt. Phys.31, 73 (1998). [23] K. I. Gus’kov, J. Phys. B: At. Mol. Opt. Phys. 32, 2963 (1999). [24] C. Blanquet, J. Warland, and J. Bouanich, J. Mol. Spectr osc.201, 56 (2000). [25] B. Nagels, P. L. Chapovsky, L. J. F. Hermans, G. J. van der Meer, and A. M. Shalagin, Phys. Rev. A 53, 4305 (1996). [26] P. L. Chapovsky, J. Cosl´ eou, F. Herlemont, M. Khelkhal , and J. Legrand, Eur. Phys. J. D (2000), accepted for publication. 18Table 1. The character table for the symmetry groups C 2v, C2v(M), D 2and classiﬁcation of the basis states (5). C2v(M)E (12)(34) E∗(12)(34)∗ortho ortho para C2vE C 2σvσ′ v D2E Rπ zRπ yRπ xK-even K=0K-odd A11 1 1 1 p=0J, p-even, – B21 -1 -1 1 – – p=1 A21 1 -1 -1 p=1J, p-odd, – B11 -1 1 -1 – – p=0 19Table 2. The most important ortho-para levels and their cont ributions to the spin con- version in ethylene. In calculations the molecular paramet ers from [14] were used. The total rate combines the contribution from all ortho-para level pa irs having J≤40. Level pair Energy ω/2π F SS FSR γ/P p′, J′,K′-p, J,K(cm−1) (MHz) (MHz2) (MHz2) (10−4s−1/Torr) 1,27,7–1,28,6 871.53 1006.7 4.2 ·10−24.4 2.49 0,27,7–0,28,6 871.69 -3435.5 4.2 ·10−24.4 0.21 0,4,3–0,6,2 53.72 -3180.6 1.1 ·10−30 3.4 ·10−3 1,26,3–1,24,6 680.68 944.3 3.4 ·10−40 5.6 ·10−4 0,20,1–0,20,0 356.02 1894.4 7.3 ·10−82.4·10−64.8·10−6 Total rate 2.70 2002004006008001000 para orthoVα'α α' α Level energy (cm-1) FIG. 1. The ortho and para states of the13CCH 4molecules. The bent lines indicate the rotational relaxation inside the two subspaces. Vα′αrefers to the intramolecular mixing of the ortho and para states. The indicated pair of states is the most impo rtant one for the spin conversion in 13CCH 4. 21x yz1 23 413C12C FIG. 2. Numbering of atoms in13CCH 4and orientation of the molecular system of coordinates. 2204812162005101520 A1-A2 level splitting (cm-1) 0481216200.00.20.40.60.81.0 Expansion coefficients /G03/G2E /G2E-term Second term FIG. 3. Properties of asymmetric-top rotational states. Upper pan el gives splitting between the states ( 0,J= 20,K) and ( 1,J= 20,K). Low panel gives squared values of the two terms in the expansion (6) for the state ( 0,J= 20,K). “K-term” is A2 K=K. “Second term” is the second biggest coeﬃcient in each expansion. 23010203040100101102103104105 Ortho-para gaps (MHz) J FIG. 4. Collapse of the ortho and para states in ethylene. The ﬁgure s hows the energy gaps between the ortho states ( 0,J,K= 0) and the para states ( 0,J,K= 1). 240 10 20-0.40.00.40.8 Expansion coefficients, AK K FIG. 5. The expansion coeﬃcients, AK, for the states most important for the spin conversion in13CCH 4. (o)–ortho state ( 1,J= 28,K= 6); (•)–para state ( 1,J= 27,K= 7). 25
arXiv:physics/0008084 16 Aug 2000WIRE SCANNERS FOR SMALL EMITTANCE BEAM MEASUREMENT IN ATF H. Hayano, KEK, Tsukuba, Ibaraki, Japan Abstract The wire scanners are used for a measurement of the very small beam size and the emittance in Accelerator Test Facility (ATF). They are installed in the extraction beam line of ATF damping ring. The extracted beam emittance are εx=1.3x10-9 m.rad, εy=1.7x10-11 m.rad with 2x109 electrons/bunch intensity and 1.3GeV energy[1]. The wire scanners scan the beam by a tungsten wire with beam repetition 0.78Hz. The scanning speed is, however, very slow( ~500µm/sec). Since the extracted beam is quite stable by using the double kicker system[2], precision of the size measurement is less than 2µm for 50 - 150µm horizontal beam size and 0.3µm for 8 - 16µm vertical beam size. The detail of the system and the performance are described. 1 INTRODUCTION ATF is a test accelerator to realize a small emittance beam which will be used in an electron positron linear collider. The beam emittance measurement in the ATF extraction line is required for a single bunch and 20 multi- bunched beam which has 2x1010 electrons in each bunch with 2.8ns spacing[3]. The required resolution of the beam size monitors is less than 1µm for the beam of 6 - 7µm vertical size. On the other hand, the horizontal beam size is around 50 -150µm, bigger than vertical one. The wire scanner beam size monitor is the most appropriate monitor for the required performance. The five wire scanners together with a gamma detector at downstream are installed at the region of no dispersion in the extraction line between qudrupole magnets. The beam size from 4 or 5 scanners are used to fit an emittance assuming the optics between the scanners[4]. In order to measure precisely such a small vertical beam size with big horizontal size, the installation of the wire in the beam line and precision of the movement are very important. The pulse-to-pulse stability of the beam is also important for such a scanning monitor. The performance of the wire installation and movement are described together with measurement stability. 2 WIRE SCANNER Fig.1 shows the picture of the wire mover stage and the vacuum chamber. The wire mount shown in Fig.2 is supported by the two arms in the vacuum chamber. A 50µm diameter gold plated tungsten wire is stretched simultaneously to X, Y and U directions which is 45 degree tilted from the X direction. In the other side of the mount, a 10µm diameter gold plated tungsten wire is alsoFig.1: Wire scanner chamber in ATF extraction line. stretched to X, Y, U and 10degree tilted from X direction. The only one wire is directly stretched on the steel mount between precise steel pillars in order to have a precision of the stretched angle. Both end of wire are fixed to copper pillars with holding by solder. The moving direction of the wire stage is 45 degree between X and Y axis has an Fig.2 Wire mount in the wire scanner chamber advantage of that the only one move direction is necessary for 3 axis scans. The angles of the stretching wires are measured by using a microscope with precision mover stage. An example of the measurement is shown in Fig.3. Each measured point are fitted to a linear function and relative angles are calculated. The results are summarized moving direction 45deg 50µm Tungsten wire10µm Tungsten wire 0.5µm-step stepping moter stage 0.5µm resolution digital scalebeam 150in Table 1. The error from the design angle is less than 1 degree. The installation into the beamline was done to have 10µm horizontal wire to sit in the precise horizontal. Using an alignment telescope, tilt angle of the whole wire scanner chamber was adjusted within 0.2 degree. Fig.3:measurement of wire stretching angle. Table 1: measured stretch angle of wire stretch angle [degree]No.1 wire mountNo.2 wire mountNo.3 wire mountNo.4 wire mount 10µm#1 wire -89.745 -90.614 -90.074 -90.062 #2 -135.365 -134.849 -134.994 -134.942 #3 0.000 0.000 0.000 0.000 #4 9.301 9.276 9.628 9.705 50µm#1 wire -89.658 -90.534 -90.322 -90.211 #2 -135.104 -135.626 -134.845 -134.978 #3 0.232 -0.694 -0.099 -0.069 The two arm supports of the wire mount inside the vacuum chamber are fixed to the two stages with two bellows at both ends of support arm tubes. The support tubes are loaded by the vacuum pressure loading from both side of tubes, however, the pressure cancels it out on the movement of the stage. The load of mover stage is a spring recovery force of the bellows only. Furthermore a vibration of the wire mount is reduced by using this double support stage compared with a single end support. As it has many advantages, the double stage mover was adopted. The one end of the wire mover stage is powered by a 5-phase stepping motor stage assembly (Physik Instrumente M-510.10). This stage is driven by a ball bearing spindle of 1mm pitch for one revolution. The stepping motor performs one rotation by 2000 steps. Combining the ball bearing spindle and the precise stepping motor, the resolution of step is 0.5µm and repeatability is less than 0.1µm. As a maximum starting pulse rate is limited to 1 kHz, a 1024 pulse/sec constant pulse rate is used for the stage control for the reason of simplicity. The stepping motor stage assembly has a radiation resistant. Also, the stage position sensor must have a radiation resistant. The Magnescale position sensor is adopted, because it does not have processing electronics near the sensor. It include a magnetized rod with very fine pitch and a pickup coil. The processing electronics is placed outside of the accelerator tunnel. The resolution ofthe Magnescale is 0.5µm for 100mm travel, enough small for the beam size measurement of 10µm. The vibration measurements[5] were done for the same type of the stage using a laser light beam on the wire. The cw laser beam of 70µm diameter was used to simulate stable particle beam. The vibration of wire appears on the absorption change of the photo-detector at downstream of the laser beam. The observed vibration of wire( 50µm diameter ) was always less than 0.3µmp.p. for 55 Hz to 771 Hz clock speed. With higher clock speed for the stepping motor more than 150 Hz reduce the vibration amplitude to 0.2µmp.p.. It is enough small amplitude for ATF beam measurement. 3 WIRE SIGNAL DETECTION A breamsstrahlung gamma-ray is used for the signal of beam and wire interaction. A gamma-ray detector[6] is placed at the last bending magnet in the end of beam diagnostic section. The detector is a cerencov detector which consists of 2mm thick lead plate converter, air light guide and a photo-multiplier tube(PMT). In order to shield the PMT from other noise source such as gamma- ray and neutrons caused by beam loss in other place, the PMT is placed at the floor with lead shield and the 1m long light guide is used between the cerenkov light radiator and the PMT. Since the gamma detection is done by a calorimetric way, low HV voltage is applied to the PMT with light intensity filter in front of the PMT. The intensity filter is made by thin plate with many small holes which reduce the light intensity. The HV voltage is changed from 600V to 950V depending on the beam size and the wire diameter. The wire signal out from the PMT is a pulse signal with around 20ns width detected by a charge sensitive ADC using 100ns gate. Since the ADC is installed at the outside of the tunnel, a 20m coaxial cable is used between the PMT and the ADC. Fig.4: Example of scanned beam profile. 4 BEAM SIZE MEASUREMENT The acquisition of the wire signal is done by the BPM reader task which read the all BPM signal and the beam-15-10-505 -40 -30 -20 -10 0 10 20 30 40wire mount No.1Y 10µm #1 wire Y 10µm #2 wire Y 10µm #3 wire Y 10µm #4 wire Y 50µm #1 wire Y 50µm #2 wire Y 50µm #3 wirey = 23.824 + -0.96114x R= 1 y = 2391.7 + -111.29x R= 0.99916 y = -18.979 + 1.0312x R= 1 y = -9.3429 + 1.4378x R= 1 y = -15.425 + -0.95822x R= 1 y = -1369 + -73.887x R= 0.99967 y = 21.86 + 1.0396x R= 1 Y position [mm] X position [mm]10mm 10mm 10mm 10mm 10mm 10mm50µm wire 10µm wire 00.20.40.60.811.21.4 61.15 61.20 61.25 61.30 61.35MW2X_00APR15_0043normalized gamma signal ave gamma signalnormalized gamma signal stage position[mm]y = m1exp(-(m0-m2)(m0-m2)/... ÉGÉâÅ[ íl 0.044903 1.1045 m1 0.00077479 61.244 m2 0.00042944 0.012509 m3 0.011635 0.058203 m4 NA 0.062325 ÉJÉCÇQèÊ NA 0.99192 Rσy= 8.8 +/- 0.3 µmMW2X 5µm step scan 5 data in each stepintensity monitor synchronously with one ATF beam cycle. As a consequence, the wire signal, the beam intensities and the beam positions are stored into the database for the same beam pass. A fluctuation of the wire signal caused by the beam intensity is corrected by normalizing the wire signal by the extraction beam intensity. As an example, a fluctuation of wire signal is shown in Fig.4 together with its averaged signal. The spread of the distribution which is not so much gives 0.3µm error in the fitted beam size. A position fluctuation is not corrected, because of insufficient resolution of BPM compared with beam fluctuation. The resolution of BPM is around 10 to 20µm, while the position jitter of beam is estimated to 2.4 to 4.4µm by using the wire sitting in beam and using high resolution microwave BPMs which are newly installed in the same region. In case of X, Y and 10 degree wires, stage positions are recorded as an abscissa which must be converted to a real movement in its directions, such as multiplying 1/sqrt(2) for X and Y. The size also must be corrected by the wire diameter. The effect is quadratic of (d/4) where d is a wire diameter. The simulation shows that the corrected beam size is enough accurate down to 2µm beam size. The scanning speed is around 30sec for one profile. The slow scan is coming from both the 0.78Hz beam repetition and the slow stage movement. The scan is made as follows; by setting the wire stage to the initial position at first, then wait the beam passing, get the wire signal together with the beam intensity, go to the next stage position, then repeat again until the end position coming. This means that the shortest scanning time is the beam repetition times number of the scanning points. 5 EMITTANCE MEASUREMENT Beam size measurements at more than three locations in different optical condition are necessary for measuring an emittance. The optics of the wire scanner region is determined by the SAD simulation to have less error in the emittance calculation assuming reasonable measurement error[7]. In case of ATF flat beam, vertical beam size measurement is affected by a small beam tilt or a small wire tilt. The beam tilt happens by an x-y coupling and a residual x, y dispersion. Four skew quadrupole magnets are introduced into the upstream of the wire scanner region to compensate beam tilt. Also, the careful tuning of dispersion suppression less than 10mm is necessary for the vertical emittance measurement. A 10mm dispersion causes about 6µm beam size increment which is comparable to the vertical size. The dispersion is measured by wire scanner with 1mm resolution using peak shift of the scanned profile by changing the ring frequency. The measured Y emittance by changing the beam intensity using one skew quadrupole are shown in Fig.5. The observed emittance growth with beam intensity is larger than the growth ofan intrabeam effect. The further study on this emittance growth is now in progress. Fig.5: measured Y emittance vs. beam intensity 6 SUMMARY The five wire scanners in the ATF extraction line are used to measure the beam size of 50 - 150µm in horizontal and 8 - 16µm in vertical. Since the extracted beam is stable within 4.4µm, the wire scanner measurement is performed by less than 2µm stability in X and 0.3µm in Y. The measured emittance are near the ATF target in case of low beam intensity. The emittance growth with beam intensity is still not understandable. 7 ACKNOWLEDGMENT The author would like to acknowledge Prof. Sugawara, director of KEK organization, Prof. Kihara, director of Accelerator research, Prof. Iwata, Prof. Takata and Prof. Yamazaki for their support of ATF. The author also thank to M. Ross, D. McCormick of SLAC and the member of ATF for their cooperation and useful discussion. REFERENCES [1]J. Urakawa,“Experimental Results and Technical Research at ATF” EPAC, Vienna (June 2000); K. Kubo et. al.,“Beam Tuning for Low Emittance in ATF Damping Ring” EPAC, Vienna (June 2000) [2]T. Imai et. al.,“Double Kicker System in ATF” LINAC2000, Monterey (August 2000) [3]F. Hinode et. al., “ATF Design and Study Report” KEK Internal 95-4, (June 1995). [4]T. Okugi et. al.,“Evaluation of extremely small horizontal emittance” Physical Review Special Topics-Accelerator and Beams, vol.2 022801(1999) S. Kashiwagi et. al., “Diagnosis of the low emittance beam in ATF DR Extraction Line,” EPAC, Stockholm (June 1998) [5]M. Ross et. al., KEK Internal 92-13, (January 1993) [6]D. McCormick et. al., ATF internal report ATF-98-3 (January 1998) [7]K. Kubo, ATF internal report ATF-98-35 (November 1998)0.05.0 10-121.0 10-111.5 10-112.0 10-112.5 10-113.0 10-113.5 10-11 0 2 1094 1096 1098 1091 1010Y emittance by 4 wire scannersY emittance [rad.m] DR beam Intensity [No. of electrons/bunch]single bunch, 1.28GeV, QK2X=0.8A 4/19/00
arXiv:physics/0008085 16 Aug 2000Energy Stability in High Intensity Pulsed SC Proton Linac A. Mosnier, CEA-DAPNIA, Gif-sur-Yvette, France Abstract Spallation sources dedicated to neutron scattering experiments, as well as multi-purpose facilities serving several applications call for pulsed mode operation of a high intensity proton linac. There is general agreement on the superconducting technology for the high-energy part, which offers some advantages, like higher gradient capabilities or operational costs reduction, as compared to room-temperatures accelerating structures. This mode of operation however could spoil the energy stability of the proton beam and needs thus to be carefully studied. First, transient beam-loading effects, arising from the large beam phase slippage along a multi-cell cavity and associated with the finite RF energy propagation, can induce significant energy modulation with a too small cell-to-cell coupling or a too large number of cells. Second, due to beam phase slippage effects along the linac, energy spread exhibits a larger sensitivity to cavity fields fluctuations than relativistic particles. A computer code, initially developed for electron beams has been extended to proton beams. It solves the 6xN coupled differential equations, needed to describe cavity fields and beam-cavity interactions of an ensemble of N cavities driven by one single power source. Simulation examples on a typical pulsed proton linac are given with various error sources, like Lorentz forces or microphonics detuning, input energy offsets, intensity jitters, etc... 1 INTRODUCTION With the aim of studying the energy stability in a high intensity pulsed superconducting proton linac, which could be spoiled by transient beam loading and beam phase slippage effects, we used the two computer codes MULTICELL and PSTAB. The former[1], based on a multi-mode analysis, calculates the systematic energy modulation generated within a multi-cell cavity due to the finite speed of the rf wave propagation; the latter, initially developed for relativistic beams[2] has been extended to low beta beams and can handle all major field error sources (Lorentz forces, microphonics, input energy offsets, beam charge jitter, multiple cavities driven by a single power source, etc) including feedback system and extra power calculation. Due to lack of space, this paper is a shortened version of a longer report[7] and presents a few results of simulation for the case of a typical neutron spallation source, like ESS[3]. The relevant parameters of the SC High Energy Linac, which have been chosen for this study are shown in Table 1. The bunch frequency during beam-on time is assumed, after funnelling of two beams emerging of 352.2 MHzRFQs, to be equal to the 704.4 MHz rf frequency of the SC cavities. Three different cavity types have been selected from input to exit of the linac and their energy gains as a function of the incoming energy are plotted in Fig.1 (synchronous phase of -30° included). Table 1: Typical SC High Energy Linac for ESSInput energy 85 MeV Exit energy 1.333 GeV Peak beam current 107 mA Chopper duty factor 60 % Bunch train period 600 ns Number of bunch trains 2000 RF frequency 704.4 MHz 024681012 100 1000beta = 0.50 beta = 0.65 beta = 0.86Energy gain (MeV) Energy (MeV) Figure 1: Energy gain for the 3 cavity types. The operating accelerating fields G (net energy gain of a particle of constant speed equal to the geometric beta of the cell divided by the iris-to-iris cavity length) correspond to electric and magnetic peak surface fields of about 27 MV/m and 50 mT. The main characteristics of the 3 sectors are shown in Table 2. Longitudinal beam matching between sectors is controlled by adjustment of the synchronous phases of the two interface cryomodules. The resulting zero current phase advance per unit length ranges from about 20° to 4°. The beam power per cavity ranges from 130 kW at the beginning to 680 kW at the end of the linac for the mean beam current of 64.2 mA. Table 2: Characteristics of the 3 sectors Low-βMedium-βHigh-β G (MV/m) 8.5 10.5 12.5 Geometric β 0.5 0.65 0.86 # cells 5 5 5 # cavities /cryom 2 3 4 # cryomodules 16 14 23 Sync phase (deg) - 30 - 27 - 25 Energy (MeV) 85 - 195 195 - 450 450 - 1348 2 TRANSIENT BEAM-LOADING Fig.2 shows for example the energy gain and the phase slippage with respect to a constant velocity particle,which would stay on-crest of the RF wave, along the first cavity of the linac. With an accelerating gradient of 8.5 MV/m, the net energy gain is 2 MeV and the integrated beam phase is -30°. -1,5-1-0,500,511,522,5 -200-150-100-50050100150 00,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8Energy gain (MeV)Bunch Phase (deg) (m) Figure 2: Energy gain and phase slippage along the β=0.50 cavity (G = 8.5 MV/m, Ein = 85 MeV, φb = -30°). This large phase slippage leads to different energy gains and then beam induced voltages between the cells of the cavity. As a result, very large fluctuations of transient beam-loading would be expected from multi-bunch trains. Fortunately, thanks to intercell coupling (about 1% here) RF power propagates from cell to cell and will tend to even the individual cell excitations. Some fluctuation however still remains due to the finite propagation velocity of the RF wave. In addition, with a chopped beam pulse, periodic gaps are cut in the regular bunch train. Modelling the cavity as a single resonator results in a perfect sawtooth-like voltage due to the periodic beam- loading and refilling of the cavity. With the multi-mode analysis, the fluctuation is increased and follows the oscillation caused by the closest mode to the accelerating Pi-mode of the passband. The energy gain modulation is then increased by a factor larger than 3 (Fig.3). -10123 0 1000 2000 3000 4000 5000 6000ΔE (keV) Time / T bunch Figure 3: Bunch energy gain along - chopped beam pulse (bottom: single resonator model; top: multi-mode model). 3 CAVITY FIELD FLUCTUATIONS Due to beam phase slippage effects of proton beams inside and outside the cavities, energy spread exhibits a much larger sensitivity to cavity fields fluctuations than for relativistic electron beams. Furthermore, in case of multiple cavities driven by one single power source, the control of the vector sum of the cavity voltages is not any more as efficient as for relativistic beams since thedynamic behaviour differs from one cavity to another. With N cavities driven by one single klystron, a total of 6×N coupled differential equations per klystron is required. - 3 equations per cavity for beam-cavity interaction Instead of using the crude RF-gap approximation (cosine- like acceleration at the cavity middle, corrected by the transit time factor) we preferred to integrate the exact differential equations in each cavity in order to model properly the beam-cavity interaction. Once the linac configuration has been defined (cavity types, number of cryomodules, design accelerating field and synchronous phase) a reference particle is launched through the linac in order to set the nominal phase of the field with respect to bunch at the entrance of all cavities. - 3×N equations per klystron for cavity field The dynamics of each resonator is described by two first order differential equations, plus another one modelling dynamic cavity detuning by the Lorentz forces [4]. Beam- loading is modelled by a cavity voltage drop during each bunch passage with a magnitude varying from cavity to another. In order to minimize the needed RF power,1)the Qex is set near the optimal coupling (about 5 105) 2)the cavity is detuned to compensate the reactive beam- loading due to the non-zero beam phase3.4 One cavity per klystron The cavity voltages are controlled by modulating the amplitude and phase of the power source via an I/Q modulator. The in-phase and out-of-phase feedback loop gains are set to 100 and 50, respectively. Examples of error sources, which were studied, are reproduced below. → Input energy offset Bunch phase oscillations induced by beam injection errors will upset cavity voltage via beam-loading. Without feedback, the bunches become unstable very soon, while with feedback, cavity voltages are efficiently controlled and the constant field dynamics are recovered (Fig. 4). -20-15-10-5051015 0 5 10 15 20 25 30 35w/o feedback with feedbackΔE (MeV) time (µs) Figure 4: Beam energy deviation at linac exit without and with feedback (input energy error 0.5 %). → Lorentz forces Because of the peak surface fields and of the mechanical rigidity of the structure, Lorentz force parameter increases as the cavity beta decreases. Simulations were carried out with expected values of 16, 8 and 4 Hz/(MV/m)2 for the three β= 0.5, 0.65 and 0.86 cavity types. In order to relaxthe feedback requirements, the cavity must be pre-detuned, such that the resonance frequency equals the operating frequency at approximately half the beam pulse. The total detuning must then be set to the sum of the detunings for Lorentz forces and beam-loading compensations. Fig.5 shows the resulting phase and energy errors of all bunches of the train at the linac exit. The extra power is maximum at beginning of linac (14%) and decreases as energy gain per cavity grows. -0.2-0.100.10.20.3 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012Phase error (deg) Energy error (MeV) time (s) Figure 5: Bunch phase and energy deviations at linac exit (with Lorentz forces only). → Microphonics With mechanical vibrations, feedback loops must be closed during the filling time, following pre-determined amplitude and phase laws, to ensure minimum RF power during the beam pulse. Assuming typical 40 Hz mechanical oscillation with an amplitude of 100 Hz (equivalent to phase fluctuations of ±8°), the increase in energy deviation at linac end is about 50% (Fig.6) and the extra peak power to be paid is about 20%. -0.15-0.1-0.0500.050.10.15 0 200 400 600 800 1000 1200 1400w/o microphonics with microphonicsΔE (MeV) Energy (MeV) Figure 6: Energy deviation of last bunch along the linac (with Lorentz forces and microphonics). 3.5 Multiple cavities per klystron With relativistic electron beams, multiple cavities powered by a single power source can be easily controlled by the vector sum of the cavity voltages [4,5]. However for proton beams, since the dynamic behaviour of low-β cavities varies as the energy increases, even when the vector sum is kept perfectly constant, the individual cavity voltages can fluctuate dramatically. We could however envisage to feed individually the cavities at the low energy part of the SC linac and to feed groups of cavities by single klystrons at the high energy part, where the cavities have closer dynamic properties. Assuming forexample groups of 8 cavities from the beginning of the 2nd sector only (above 200 MeV), Fig. 7 shows the 8 cavity voltages of the last klystron during the beam pulse with Lorentz forces detuning effects. The total energy deviation at linac end is lower than 0.5%. -0,008-0,006-0,004-0,00200,0020,0040,0060,008 0 0,0002 0,0004 0,0006 0,0008 0,001 0,0012ΔVc/Vc time (s) Figure 7: Amplitude of the 8 cavity voltages for the last group (with Lorentz forces only). 4 CONCLUSION The systematic energy modulation, enhanced by transient beam-loading effects within a multi-cell cavity, looks actually harmless (about 10-3) by using SC cavities with low number of cells and not too small intercell coupling (respectively 5 and 1% in this study). Besides, the impact of various error sources on energy stability in a typical SC proton linac has been studied by means of a simulation code, which integrates the coupled differential equations governing both cavity field and beam-cavity interaction. When each cavity has its own RF feedback system, the cavity voltages can be very well controlled, providing energy spreads at the linac end well below the specifications. However, groups of multiple cavities driven by one common klystron and controlled by the vector sum, give rise to significant energy fluctuations and should only be used in high energy part, when the cavity dynamic properties become similar. REFERENCES [1]A. Mosnier, “Coupled Mode Equations for RFtransient Calculations in Standing Wave Structures”,DAPNIA/SEA 94-30 report, 1994, non published. [2]M. Ferrario et al, “Multi-Bunch Energy Spread Induced by Beam-Loading in a Standing WaveStructure”, Particle Accelerators , 1996, Vol.52. [3]The ESS Technical Study, Vol. 3, 1996 [4]A. Mosnier and J.M. Tessier, “Field Stabilization ina Superconducting Cavity powered in Pulsed Mode”,Proc. of EPAC , London, 1994. [5]S. Simrock et al, “Design of the Digital RF Control System for the Tesla Test Facility”, Proc. of EPAC , Sitges, 1996. [6]A. Mosnier et al, “RF Control System for the SC Cavity of the Tesla Test Facility Injector”, Proc. of Part. Acc. Conf. , Vancouver, 1997. [7]A. Mosnier, internal report DAPNIA-SEA 00-08
The Low Energy Beam Transport System of the New GSI High Current Injector L. Dahl, P. Spädtke, GSI Darmstadt, Germany Abstract The UNILAC was improved for high current performance by replacing the Wideröe prestripper accelerator by an RFQ and an IH-type DTL. In addition, one of two ion source terminals was equipped with high current sources of MUCIS- and MEVVA-type. Therefore, a redesign of the LEBT from the ion source to the RFQ entrance was necessary. The new LEBT was installed at the beginning of 1999. The commissioning was carried mainly with a high intense argon beam. The achieved 17 emA Ar+ are 6 emA above the RFQ design intensity. The expected performance of the new LEBT was achieved: full transmission for high current beams, preservation of brilliance along the beam line, isotope separation for all elements, transverse phase space matching to the RFQ linac, and macropulse shaping. In particular, the high degree of space charge compensation was confirmed. 1 INTRODUCTION To fill up the heavy ion synchrotron SIS to its space charge limits for all ion species, prestripper intensities of I = 0.25 ⋅ A/q (emA) for mass over charge ratios up to 65 are neccessary. For this goal the Wideröe accelerator was substituted by a high current IH-RFQ and two IH- type cavities working at 36 MHz [1]. To compensate the calculated transmission loss in the RFQ the beam current in the LEBT was even required 10 % higher, in case of the design ion U4+ the beam current to be transported amounts to 17 emA. Performance goals for the new LEBT were loss free beam transport from the new source types MUCIS and MEVVA to the RFQ, avoiding of emittance growth, mass resolution m/Δm = 220 of the magnet spectrometer, macropulse shaping with rise and fall times Δt ≤ 0.5 µs and exact transverse beam emittancematching to the RFQ acceptance. Furthermore, the LEBT from the switching magnet downstream was required to be 50 Hz pulseable for two ion species operation of the high current injector. 2 DESIGN OF THE HIGH CURRENT LEBT Because the high current sources were installed years before, many high intense beam investigations were carried out at the former beam transport system to get a basis for the ion optical design of the new LEBT [2]. It turned out that there is no evident emittance growth and the beam behavior agrees very well with zero-current simulations. All measurements indicated a space charge compensation by electrons out of the residual gas of at least 95 %. Therefore, the existing LEBT down to the switching magnet was evaluated as basically useful for the transport of high intense beams. It was improved by the introduction of steering magnets near the ion source and an extension of a magnetic quadrupole doublet to a triplet for a flattened dispersion trajectory. Due to the longer new accelerator tanks, the downstream beam line was significantly shortened. It was redesigned also for better dispersion quality, and to perform the transverse beam matching to the narrow RFQ entrance (Fig.1) . Fig. 2 Transverse envelope of a zero current beam Fig. 2 shows the transverse beam envelope calculated by the ellipse transformation code MIRKO based on emittance measurements directly behind the acceleration column of the ion source terminal. The 77.5° magnet spectrometer provides a mass resolution of m/Δm = 220. This allows the isotope separation for all elements and reduces remaining space charge forces in the following system. This LEBT is achromatic only in the way that particles of different energies coincide in a focus at the RFQ entrance but with different angles. Investigations carried out with the multi-particle code PARMT, proved a possible momentum spread of only ± 5 ⋅ 10–4. This value was considered for the new high voltage power supply for preacceleration to an energy of 2.2 keV/u. Beam transport is limited to a space charge analogous to an effective beam current of 0.5 emA only. Within this limit no beam losses occur and the emittance growth stays within the RFQ acceptance.Fig. 1 Mechanical layout of the new LEBTRFQbefore RFQ installation:
COMMISSIONING OF THE 1.4 MeV/u HIGH CURRENT HEAVY ION LINAC AT GSI W. Barth, GSI, Planckstr.1, 64291 Darmstadt, Germany Abstract The disassembly of the Unilac prestripper linac of the Wideröe type took place at the beginning of 1999. An increase of more than two orders of magnitude in particle number for the most heavy elements in the SIS had to be gained. Since that time the new High Current Injector (HSI) consisting of H-type RFQ and DTL- structures for dual beam operation was installed and successfully commissioned. The High Charge Injector (HLI) supplied the main linac during that time. Simultaneously conditioning and running in of the rf- transmitters and rf-structures were done. The HSI commissioning strategy included beam investigation after each transport and acceleration section, using a versatile diagnostic test stand. Results of the extensive commissioning measurements (e.g. transverse emittance, bunch width, beam transmission) behind LEBT, RFQ, Super Lens, IH tank I and II and stripping section will be discussed. An 40Ar1+ beam coming from a MUCIS ion source was used to fill the linac up to the theoretical space charge limit. Routine operation started in November 1999. 1 INTRODUCTION The original Unilac was not dedicated as a synchrotron injector, fulfilling all requirements due to high intensities (especially for mass number above 150). In 1994 the vision of a new High Current Injector (HSI) was drafted. This injector should provide an increase of beam intensities by 2.5 orders of magnitude filling the synchrotron up to its space charge limit for all ions – including uranium [1]. An increase of the accelerating gain by a factor of 2.5 is necessary to accelerate ion species up to maximum A/q-values of 65 (130Xe2+) within the given length of the former Wideröe injector. For a 15mA 238U4+ beam out of the HSI 4⋅1010 U73+ particlesshould be delivered to the SIS during 100µs. This means that the SIS space charge is reached by a 20 turn injection into the horizontal phase space. The required parameters of beam quality are summarised in Tab. 1 for the uranium case. 2 HSI LAYOUT The beam of the new High Current Injector is stripped and injected into the Alvarez accelerator, which is approx. 30 years in operation now. It was predicted by PARMILA-calculation and confirmed by beam tests that the Alvarez accelerates highly space charge dominated ion beams coming from the new HSI without any significant particle loss and without decrease in brilliance [2]. The new injector is illustrated in Fig. 1, a more detailed description of the main acceleration and transport-sections is given in the following. 2.1 Ion Source and LEBT section The HSI is fed by two ion source terminals: One of them is yet since many years housing a Penning ion source (PIG), generating intermediate charge state ion beams with a duty cycle up to 30% (50Hz, 6ms), limited to an A/q value of 24. The other one is upgraded as a high current terminal; optionally the MUCIS source or the MEVVA source is installable, providing short intense macro pulses (≤ 1.2ms, ≤ 16Hz) as required for the linac operation with ion beams of high magnetic rigidity (A/q≤65) [3]. The existing LEBT [4] section down to the switching magnet was surveyed as basically useful for loss-free transport of high intense beams, assuming a high degree of space charge compensation. The section from the switching magnet to the RFQ entrance was rebuilt as a “50Hz-pulseable” beamline, allowing for a two beam operation now. The RFQ-matching condition of a double-waist at a very small transverse beam diameter of 5mm is accomplished by a quadrupole Fig. 1: The HSI as a scheme. Table 1:Specified beam parameters at Unilac and SIS injection, exemplary for a uranium beam. HSI entranceHSI exitAlvarez entranceSIS injection Ion species238U4+ 238U4+ 238U28+ 238U73+ El. Current [mA]16.5 15 12.5 4.6 Part. per 100µs pulse2.6⋅10122.3⋅10122.8⋅10114.2⋅1010 Energy [MeV/u]0.00221.4 1.4 11.4 ΔW/W - ±4⋅10-3±2⋅10-3±2⋅10-3 εn,x [mm mrad]0.3 0.5 0.75 0.8 εn,y [mm mrad]0.3 0.5 0.75 2.5quartet. A beam chopper with a rise time less than 500ns is located close to the entrance of the RFQ. 2.2 36 MHz IH-RFQ and Matching to IH-DTL The 9.35m long 36 MHz IH-RFQ [5] accelerates the ion beam from 2.2 keV/u up to 120 keV/u, where a voltage amplitude of 137 kV and a max. surface field of 28 MV/m is necessary to provide the required accelerating gain. The postulated mechanical precision of electrodes [9] should be better than 0.05 mm along each module, obtaining a high transmission rate. The time for rf- conditioning up to now was less than 500 h. The reduction of a primarily large dark current contribution was observed in so much that 90% of the design field are reached now routinely to obtain optimum operation for U4+ beams. The matching to the IH-DTL is done with a very short (0.8m) 11 cell adapter RFQ [6], with large aperture and a synchronous phase of –900 (Super Lens). The surface field is 26 MV/m at a design vane voltage of 212 kV. 2.3 83 MV IH-DTL The IH-DTL [7] consists of two separate tanks (9.1m and 10.3m long), connected by an intertank section. The final beam energy of IH1 is 0.743 MeV/u, while IH2 accelerates the ion beam to the full HSI-energy of 1.4 MeV/u. The whole DTL is structured into six KONUS (“Kombinierte Nullgrad Struktur”) sections, mainly consisting of a rebuncher section operating at asynchronous phase of -350, followed by the acceleration stage (Φs=00), where the synchronous particle is injected with a surplus in energy resulting in a high accelerating rate. Finally in each section a quadrupole triplet provide the transverse focusing. Tank 1 houses four accelerating sections with three internal triplet lenses as shown in Fig.2. Behind the external triplet tank 2 contains two accelerating sections and one external triplet between. The rf-conditioning of the structures results in lower dark current contributions regarding rf power losses when compared to the more critical RFQ. 2.4 1.4 MeV/u and 11.4 MeV/u stripper section Two quadrupole doublets match the 1.4 MeV/u HSI- beam to the gas stripper [8]. No additionally focusing device is necessary to feed the new charge state separator system. If as a worst case a 15mA U4+ ion beam is stripped, U28+ should purely separated from the neighbouring charge states under extremely high space charge conditions (105 emA total pulse current). After the rebuilt of the stripper region a multi-pulse mode [9] from the different injectors is possible. In the 11.4 MeV/u stripper region [10] of the transfer line to the synchrotron the beam power in case of an intense uranium beam is approximately as high as in the gas stripper section. In order to cope with the potentially disastrous beam load on the stripper foil a magnetic sweeper system had been established, saving the foil, while a significant decrease of beam quality due to stripping effects should not occur. A new short charge state separator system, to be installed directly behind the foil stripper, is still under construction. 3 RF SYSTEM Supplying the 36 MHz structures, rf amplifiers with a peak power of about 2 MW had to be installed, while the 27 MHz rf equipment became dispensable. Additionally redesigned fast amplitude- and phase controls were built along the whole Unilac, enabling the operation with high beam loading. The five 200 kW amplifiers were externally built (TOMCAST AG, Switzerland) and partly utilised as pre amplifiers for the 2 MW end stages and as rf provider for the Super Lens and a rebuncher. The in house developed end stages are powered by the Siemens tetrode RS2074 SK; they feed the RFQ and the two IH cavities. The assembly of all rf-components took place just in time and the whole system shows excellent reliability [11], [12]. 4 BEAM DIAGNOSTIC TEST BENCH A mobile test bench [13] was designed and already used for the stepwise performed beam commissioning of LEBT, RFQ, Super Lens and IH-DTL. It was equipped with in part newly designed beam diagnostics, as shown in Fig.3: four segmented capacitive pick-up probes, Fig. 2: A view into IH1: Four KONUS sections, connected by the three large drift tubes each housing a quadrupole triplet, can be seen.beam transformers, a profile grid in combination with a non-destructive residual gas ionisation profile monitor, a slit-grid emittance measurement device, as well as a first time used pepperpot system and a particle detector for bunch structure observation. At the end of the test bench the beam was dumped in a cooled Faraday cup. 5 TIME SCHEDULE&ACHIEVEMENT Table 2.: HSI-Assembly&Commissioning milestones Dec. 98 Last operation-shift with Wideröe injector Jan.-Feb. 99Disassembly of Wideröe and rf, installation of LEBT section March 99Successful commissioning of LEBT April-May 99Mounting IH-RFQ and first acceleration up to 120 keV/u June 99 Beam tests with Super Lens, achieving 10 mA Ar1+ at RFQ exit July 99 Assembly of IH1, verification of beam acceleration up to 743 keV/u August 99Completing HSI with IH2 and stripper section 2.Sept. 99Proof of acceleration up to 1.4 MeV/u, further on: 80% IH-transmission for highest argon intensities (8 mA) October 99Upgrade of transfer line to SIS and mounting of matching section to Alvarez November 99Establishing three beam operation, complete Alvarez transmission at highest current Since Nov. 99HSI in routine operation February 2000Achievement of the 90%-rf levels, first 1.4 MeV/u U4+ beam (3 mA) The mounting and commissioning of the new injector took place in the first 9 months of 1999 – the assembly was done in 6 steps, each subsequently completed with a “two week beam-commissioning period”. Regular beamoperation started end of November, just in time as scheduled a long time in advance; the milestones of mounting and commissioning are summarised in table 2. 6 BEAM COMMISSIONING RESULTS 6.1 TOF-Measurements After each step the correct beam energy was verified by a time of flight measurement using the signals of two pick-ups in a well-known distance. As an example Fig. 4 shows an oscilloscope view of two signals, received by phase probes, placed after the IH2 structure (1.396 MeV/u). The achievable accuracy ΔW/W is less than +/-0.12%. The evaluation of difference signals allows an on-line monitoring of the beam position for a higher intensity. 6.2 Phase probe signals Considering a max. beam power of 1.3 MW inside a macro-pulse at the HSI-energy of 1.4 MeV/u - leading to melting and evaporation of hit material after some µs - it is necessary to limit the rise time of the macro pulse to a value as short as possible [9]. As shown in Fig.5 the Fig. 4: Phase probe signals behind IH2 structure. Fig. 5: Rise time of the phase probe signal after RFQ [14] Fig. 3: Beam diagnostic test bench during beam commissioning of the 36 MHz IH-RFQ. [13]measured rise time of the phase probe signal after the RFQ is of the same amount as the design value of the chopper (0.5 µs). No significant increase of rise time takes place along the whole HSI, demonstrating good working rf control loops with respect to beam loading of the cavities. Macro-pulse shape investigations were carried out for Ar1+ close to the current limit, as well as for many different other ion species and intensities (for example 6.5 mA 238U4+ from the MEVVA source [15]) and showed no considerable difference. 6.3 Particle loss and transversal emittance Fig. 6 (above) represents the measurement of the Ar1+ current by a beam transformer behind of the RFQ as a function of the injected beam in front of the quadruplet lense. The RFQ-matching was optimised for the high intensity case (18 mA) in such a way that the design current limit of 10 mA was reached [16]. The controlled intensity reduction was done by cutting the horizontal phase space distribution with slits and without any tuning of the RFQ matching, resulting in full particle transmission from 4 mA downwards. The transmission decrease to roughly 55 % at the theoretical current limit (10 mA) can not be completely explained by high transversal input emittance: The emittance measurements resulted in normalised 90%-emittances from 0.25 π⋅mm⋅mrad up to 0.45 mm⋅mrad without anysignificant influence to RFQ-transmission. Mismatch problems due to space charge effects or misalignment inside the RFQ are not ruled out to lead to particle loss. The RFQ is the bottleneck as figured out and emerged by many measurements; the transmission of the IH-DTL is better than 90% over a wide range of beam intensities and ion species (Fig. 6, beneath). For the U4+ case 3 mA after the HSI were reached, while the particle transmission is close to the amount of the space charge dominated argon beam. The high intensity-fluctuation due to the behaviour of the MEVVA source is about ±25 % (30 following macropulses), without a significance influence to the beam emittance. The pulse- reproducibility of position and amount of the transverse emittance at the HSI exit is better than 4 % (verified by pepperpot emittance measurements). The beam loading for instance for RFQ (44 kW) and IH2 (147 kW) measured by the additional rf-power needed for the acceleration is close to the theoretical values. 6.4 Emittance growth Measurements of the transversal emittance for the several energy steps of the HSI (and at 11.4 MeV/u) were done exclusively with a slit-grid device for short pulses. The input emittance was measured before the quadrupole quartet. For 120 keV/u, 750 keV/u and 1.4 MeV/u the beam was transported to a measurement device in the gas stripper region [17]; another device is placed after the Alvarez. Fig. 7 summarises the measured emittance data for an Ar1+ beam with 10 mA at RFQ injection and 6.5 mA at the HSI exit. The Ar10+ current (after stripping and charge state analysis) came up to 7 mA by gas stripper density variation. The measurements agreed to the calculation [18], if a measuring error of about ±15 % is taken into account. 6.5 Bunch shape Bunch shape measurements were done by the use of diamond detectors, whereas the ion beam (here Ar1+)0.010.11 0.0010.010.1110100 Wkin [MeV/u]εrms,norm [mm∗mrad] measured calculated Fig. 7 Measurement of the horizontal emittance along the Unilac020406080100particle transmission [%] LEBT RFQ IH_1 IH_216 mA, Ar1+ (input intensity) 6.5 mA, U4+ (input intensity)0246810 0 4 8 121620 RFQin [mA]RFQout [mA] HSI-RFQ particle transmission16 mA, Ar1+ (input intensity) Fig. 6: Output current of the RFQ as a function of the input intensity (top) and particle transmission along the HSI (bottom) for Ar1+ (MUCIS) and U4+ (MEVVA)passes a thin Au-foil – the “Rutherford“-scattered particles hit the detector below a small angle. The bunch shape is obtained by measuring the arrival time of the particles against a reference [19]. It was even possible to observe the typical “zero current” phase space distribution in longitudinal plane, leading to intensity peaks in the center and at the beginning (resp. at the end) of the measured bunch shape. This effect is still present after accelerating the beam up to 0.743 MeV/u, and after transport to the stripper region (Fig. 8). Regardless the higher defocusing due to space charge forces in the high current case the bunch is shorter and without the significant “low current“-structure. At full HSI energy the beam is very well bunched. Independently from space charge effects along the HSI, the bunch is small enough to be matched to the poststripper by the two rebunchers operated at 36 MHz and 108 MHz, respectively. 7 CONCLUSION The new High Current Injector was mounted and commissioned with great success. The measured beam parameters, as energy, bunch width, energy spread, transverse emittance fit very well to the calculation. The transmission at beam currents up to 40 % of the design intensity is close to 100%. Significant particle losses due to insufficient understanding problems are observable at the space charge limit. Within the intrinsic error bars of measurement the emittance growth in the high current case is as predicted by simulation. So far the rf levels are high enough to provide a stable operation with U4+. First beam experiments with medium uranium intensity, feeding the HSI with the MEVVA beam, showed no significant deterioration of beam quality. Since November 1999 the HSI delivers beam to experiments in routine operation.8 ACKNOWLEDGEMENTS It is hard to believe that converting an act into an idea often turns out to be rather sophisticated (Karl Kraus) – vice versa by the help of numerous people at GSI and at collaborating institutes… REFERENCES [1]U. Ratzinger, The new GSI Prestripper Linac for high current heavy ion beams, LINAC96, Geneva, Switzerland, p. 288 (1996) [2]J. Glatz, et al., high current beam dynamics for the upgraded Unilac, PAC97, Vancouver, Canada, p. 1897, (1997) [3]P. Spädtke, et al., Ion sources for the new High Current Injector at GSI, Geneva, Switzerland, p. 884 (1996) [4] L. Dahl, et al., High-intensity low energy beam transport design studies for the new injector linac of the Unilac, Geneva, Switzerland, p. 134 (1996) [5]U. Ratzinger, et al., The RFQ section of the new Unilac prestripper accelerator at GSI, EPAC96, Sittges, Spain, p. 304, (1996) [6]U. Ratzinger, R. Tiede, A new matcher type between RFQ and IH-DTL for the GSI high current heavy ion prestripper linac, Geneva, Switzerland, p. 128 (1996) [7]U. Ratzinger, et al., Numerical simulation and rf model measurement of the new GSI IH-DTL, PAC97, Vancouver, Canada, p. 2645, (1997) [8]W. Barth, at al., Space charge dominated beam transport in the 1.4 MeV/u Unilac stripper section, Geneva, Switzerland, p. 131 (1996) [9]J. Klabunde, et al., Operational aspects of the high current upgrade at the Unilac, these proceedings [10]J. Glatz, B. Langenbeck, A high duty foil stripper system in the injection line to the heavy ion synchrotron SIS at GSI, EPAC96, Sittges, Spain, p. 2406, (1996) [11]W. Vinzenz, et al., Status of the 36 MHz rf-system for the High Current Injector at GSI, Linac98, Chicago, U.S.A., p. 219, (1998) [12]G. Hutter, et al., The rf-system of the new GSI High Current Linac HSI, this workshop’s proceedings [13]P. Forck, et al., Beam diagnostic developments for the new high current linac of GSI, 9th Workshop on Beam Instruments, Boston, U.S.A., (2000) [14]L. Dahl, et al., The low energy beam transport system of the new GSI high current injector, these proceedings [15]H. Reich, et al., Commissioning of the high current ion sources at the new GSI injector (HSI), these proceedings [16]U. Ratzinger, et al., Commissioning of the IH-RFQ and the IH-DTL for the GSI high current linac, these proceedings [17]W. Barth, P. Forck, The new gas stripper and charge state separator of the GSI high current injector, these proceedings [18]W. Barth, J. Glatz, J. Klabunde, High current transport and acceleration at the upgraded Unilac, Linac98, Chicago, U.S.A., p. 454, (1998) [19]P. Forck, et al., Measurement of the 6 dimensional phase space at the new GSI high current linac, these proceedingsFig. 8: Bunch shape measurements for different beam energies and intensities
THE NEW GAS STRIPPER AND CHARGE STATE SEPARATOR OF THE GSI HIGH CURRENT INJECTOR W. Barth, P. Forck GSI, Planckstr.1, 64291 Darmstadt, Germany Abstract The GSI Unilac was upgraded as a high current injector for the SIS in 1999. Therefore, the stripping section [1] at 1.4 MeV/u, where a beam transport under highest space charge conditions and multi beam operation had to be established, was completely new designed. Results of the commissioning of the stripper section will be presented. The beam transport to the new gas stripper, and the charge state analysis under space charge conditions confirmed the calculations. A 238U4+ beam - coming from a MEVVA ion source and accelerated by the new injector linac- was stripped (with the expected stripping efficiency) to the charge state 28+ and successfully separated by the new spectrometer (15 degree and 30 degree kicker magnets). The transport and matching to the poststripper accelerator under highest space charge conditions was investigated with a 15 emA 40Ar10+ beam. Especially space charge and charge state dependent emittance growth effects in 6d-phase space will be discussed. 1 INTRODUCTION For the upgraded Unilac a new stripper section was designed and installed in 1999. Some additional design features had to be considered: charge state separation and beam transport under highest space charge conditions and multi beam operation with pulsed magnets. 2 SETUP OF THE STRIPPER SECTION The layout is shown in Fig. 1. By two quadrupole doublets the beam is matched to the gas stripper. The new stripper device consists of the interaction zone (super-sonic N2-jet) and three steps of differential pumping, upstream and downstream respectively. Compared to the old stripper [2], the free aperture in the new one is approx. 40% larger, ensuring a small beam size at the analyzing slit without any additional focusing elements. The charge state separator consists of three bending magnets, operating in pulsed mode. With the new 150 fast kicker magnet (inflecting the HLI-beam from the ECR-source to the Unilac-axis) a multi-pulse mode from the different injectors is possible. In the following transport line the longitudinal matching (with two rebunchers at 36 MHz and 108 MHz) and the transversal matching (with a quadrupole doublet and atriplet) to the poststripper accelerator is accomplished. All sensitive elements are protected by diaphragms to handle with the highest beam pulse power along the Unilac (up to 1.4 MW).
HIGH-POWER CIRCULATOR TEST RESULTS AT 350 AND 700 MHZ W. T. Roybal, J. Bradley III, D. Rees, P. A. Torrez, D. K. Warner, LANL, Los Alamos, NM 87545, USA J. DeBaca, (General Atomic) LANL, Los Alamos, NM 87545, USA Abstract The high-power RF systems for the Accelerator Production of Tritium (APT) program require high-powercirculators at 350 MHz and 700 MHz to protect 1 MWContinuous Wave (CW) klystrons from reflected power.The 350 MHz circulator is based on the CERN, ESRF,and APS designs and has performed very well. The 700MHz circulator is a new design. Prototype 700 MHzcirculators have been high-power tested at Los AlamosNational Laboratory (LANL). The first of thesecirculators has satisfied performance requirements. Thecirculator requirements, results from the testing, andlessons learned from this development are presented anddiscussed. 1 INTRODUCTION The APT 350 and 700 MHz klystrons require circulators that are capable of operating indefinitely withup to full reflection at any phase. The circulators werespecified to function properly under these conditions atpower levels up to the maximum CW forward power outof each klystron: 1.2 MW for the 350 MHz klystron and1 MW for the 700 MHz klystron. A 350 MHz circulatorwas easily produced as 352 MHz circulators at thesepower levels are already in use. The APT 350 MHz, 1.2MW circulator produced by Advanced Ferrite Technology,Inc. (AFT) is shown in Figure 1. Figure 1: AFT 350 MHz APT Circulator. Development of the 1 MW, 700 MHz circulator was more challenging. AFT was initially the sole vendor, butwe chose Atlantic Microwave Corporation as a secondvendor after several failures by AFT to meet the high-power operating requirements. Los Alamos NationalLaboratory worked closely with both AFT and AtlanticMicrowave Corporation to develop and test 700 MHzprototypes for APT.2 CIRCULATOR REQUIREMENTS The APT circulator specifications for insertion loss, isolation and waveguide contention flanges for the 350and 700 MHz circulators are shown in Table 1. Table 1: APT Circulator Specifications Requirement 350 MHz 700 MHz Insertion Loss < 0.05 dB < 0.05 dB Isolation > 26 dB > 26 dB Flange WR2300 WR1500 The limits on circulator loss were determined by the required control margin for the APT accelerator and themaximum saturated klystron power. These factorsdictated that the sum of all waveguide and circulator lossesbe less than 5%. This in turn required that the budget ofallowable insertion loss for the circulators be 0.05 dB atthe design center frequency of the RF system. At least 26 dB of isolation is required in order to protect the klystron from excessive reflected power whileoperating at megawatt CW power levels. We specifiedthat the vendors demonstrate this ability while operatingat full power with a short circuit on the output port andover the range of all possible phases of reflected power.The circulator is required to operate under all conditions ofthe specification without arcing. Arc detectors wererequired to verify that no arcing occurred and to protect thecirculator in the event of arcing. Finally, the circulator isrequired to satisfy all requirements from startup to steady-state operating temperature without the need for operatoradjustments to a Temperature Compensating Unit (TCU).A TCU is typically used to adjust the magnet coil currentat the circulator, thereby maintaining the fields at theferrite plates for maximum isolation over a range ofoperating temperatures. 3 TEST RESULTS 3.1 AFT 350 MHz Circulator The AFT 350 MHz circulator was tested at power levels up to 1100 kW with a waveguide shorting plate on theoutput port. A shorting plate on the output port producesa standing wave in the circulator ferrite. Various lengthsof waveguide were inserted between the circulator and theshorting plate to adjust the phase of the standing wave.The maximum voltage stress in the circulator occurswhen this standing wave has an E-field maximum in theferrite at the center of the circulator. We tested thevoltage breakdown (arcing) characteristics by placing the E-field maximum at the center of the circulator. Themaximum dissipated power in the circulator occurs whenthis standing wave has an H-field maximum in the ferriteat the center of the circulator. We added a quarter wavesection of waveguide between the circulator and theshorting plate to obtain an H-field maximum at the centerof the circulator to test the dissipated powercharacteristics. The data taken in the H-field maximumconfiguration showed the highest power dissipation in theferrite, supporting our calculations. As expected, the 350 MHz circulator provided by AFT met all requirements without failure. The circulatorperformed well and demonstrated >26 dB of isolation withless than 0.05 dB of insertion loss at all phases.Continuous operation in both the maximum E and H fieldconfigurations demonstrated reliable voltage standoff atthe voltage maximum and reliable power handlingcapabilities at the maximum steady–state temperature.No arcs were detected over the full range from turn on tosteady-state operation. Once the TCU was properlyadjusted to the optimum setting, the entire set of testswere performed without requiring further adjustment. 3.2 AFT 700 MHz Circulators AFT has delivered three 1.0 MW, 700 MHz circulators for use on the Low Energy Demonstration Accelerator(LEDA). All three circulators have been tested on a teststand and each one has had difficulty meeting the finalhigh-power acceptance test criteria. The problemsencountered included arcing, poor isolation, and poor TCUperformance. A total of 5 high-power testing sessionswere required for the first circulator to meet allspecifications. As a result of this performance, weselected Atlantic Microwave as an alternate vendor. AFT initially approached this development by scaling down the well-demonstrated 350 MHz WR2300 circulatorto be packaged in WR1500 waveguide and operateequivalently at 700 MHz. The prototype met all lowpower requirements on the bench but had problems underhigh power. A picture of one of the three AFT 700 MHzcirculators delivered to LANL is shown in Figure 2. After demonstrating that it could pass all low power tests at AFT, the AFT 700 MHz prototype circulator wasshipped to Los Alamos for high-power testing. Weobtained a sliding short device from Mega Industries forthe full-power, full-reflection circulator tests. Thissliding short installed on the output port of the circulatorallowed us to vary the RF phase of the short continuouslyover a range more than 360 degrees. The first test resulted in continuous arcing at forward power levels greater than 275 kW at E-field maximum.During the testing AFT personnel determined that therewas an incorrect cooling water distribution between theferrite plates and the reduced-height waveguide. Attemptswere made at Los Alamos to restrict the coolant flow tothe waveguide and matching posts in order to increase coolant flow to the ferrite plates. Results from theseefforts were positive but the circulator continued to arc atpower levels of 500 kW. Figure 2: AFT 700 MHz APT Circulator Installed on the Test Stand. The circulator was rebuilt for the second set of high- power tests. The second test showed improvement overthe first test; however, arcing again limited the circulatorto just over 500 kW at E-field maximum. Internalinspection revealed arcing on the threads of a bolt used toattach the ferrite plates to the water connection and to thewaveguide structure. A major redesign and rebuildfollowed these findings. The circulator was fitted with anew design for supporting the plates in place andconnecting the water lines to the ferrite plates. For the third test, it seemed that the arcing issue was resolved. The circulator was tested to full power withoutarcing at almost all phases. Only at the H-fieldmaximum the isolation did not meet specification. Overthe range of sliding short positions ±15 degrees about theH-field maximum postion, the transmitter's klystron-reflected-power interlock tripped off the high voltage anddid not allow further testing. Manual adjustments couldbe made to the TCU to prevent the tripping off of thesystem, but the specificaton required operation withoutmanual adjustment of the TCU. It was also determinedthat the speed of the adjustment of the sliding shortcontributed to the running away of the reflected power tothe klystron. The TCU needed to respond more quickly tosudden changes in phase. The focus was then turned to upgrading the capabilities of the TCU. Improvements such as the addition of a fast-feed-forward loop were added to the electronics to obtainbetter performance. The fourth test demonstrated the 26dB of isolation required, but the circulator arced under fullpower at the H-field max. The arc spot was located on thetop plate and on the waveguide wall. The plate wasreplaced and the waveguide wall arc damage was cleanedfor the fifth high-power test. The fifth and final AFT circulator high-power acceptance test took place without incident. Thecirculator passed all requirements for insertion loss andisolation at high power when presented with a variable sliding short. The sliding short was varied in bothdirections at all possible speeds. Reflected power at theklystron did not exceed 2 kW and no arcs were detected. Aplot of the power dissipated in the circulator versus thephase of the waveguide short during the final AFT test isgiven in Figure 3. Full power from the 1.0 MW klystroncombined with waveguide losses, circulator losses, slidingshort losses, and accuracy of power meters andcalibrations yielded just over 900 kW of forward powerdelivered through the circulator, reflected off the slidingshort, and dissipated at the high-power water load. The two remaining AFT prototype circulators are scheduled for rebuilding for better high-voltage standoffand the TCUs will be upgraded to be equivalent to theTCU that was used with the circulator that passed finalacceptance. 5101520Dissipated Power (KW)050 100 150 200 250 Phase (Degrees)900 KW Figure 3: Plot of Dissipated Power versus Phase for the AFT 700 MHz Circulator Tested at LANL. 3.3 Atlantic Microwave 700 MHz Circulator After the second failed test of the AFT 700 MHz circulator, we had Atlantic Microwave begin to develop acirculator for APT to serve as a backup source in theevent that AFT could not resolve their 700 MHzcirculator problems. The first part of the 3-partdevelopment plan was to design the circulator. Thesecond part was to develop and build a circulator with asingle plate of ferrite material that would be rated for 100kW CW at 700 MHz. The third part would be to deliver afull megawatt CW circulator prototype. Atlantic Microwave had completed the design and delivered the single plate 100 kW circulator to LANL bythe time of the AFT circulator failed its fourth test.Figure 4 shows the Atlantic Microwave 100 kWprototype circulator installed on the test stand. The Atlantic Microwave circulator performed well at its 100 kW power rating and passed all of our testingrequirements at this power level. The circulator was thentested at power levels above its 100 kW rating. First weadjusted the sliding short to produce an E-field maximumin the center of the circulator and raised the RF power in50 kW steps. The circulator was allowed to come tosteady-state temperature at each 50 kW step. The circulator began to arc when the power reached 600 kW.The circulator was then tested with the sliding shortadjusted to produce an H-field maximum in the center ofthe circulator and with the RF power level increased in 50kW steps as before. Under these conditions, the circulatorpower reached only 300 kW before the circulator isolationdegraded to the point that further increases in power werenot possible. Improvements to the isolation can be madeby using a TCU to compensate for temperature effects inthe ferrite. The 100 kW circulator prototype did notinclude a TCU, but the design of the full-scale 1.0 MWcirculator will include a TCU. Figure 4: Atlantic Microwave 700 MHz 100 kW Circulator Installed on the Test Stand. 4 CONCLUSIONS The development of a 700 MHz circulator for APT proved to be far more challenging than the development ofthe 350 MHz circulator. Simply scaling down the welldemonstrated 350 MHz WR2300 circulator resulted in aprototype that met all low power requirements on thebench but did not meet our specifications at high power. Rigorous testing procedures revealed inherent arcing and isolation problems in the initial prototype AFT circulatorand led to substantial changes to the internal design. Thelessons learned in correcting the arcing problems in highfield regions within the circulator structure led to theredesign of the circulator interior for improved voltagestand off. Lessons learned in providing adequatetemperature compensation at higher response rates led tothe development of a faster, smarter TemperatureCompensating Unit. The problems encountered in thetesting phase of the AFT circulator also led us to developan alternative supplier of high power circulators. The high power testing results give us confidence that we can obtain a fully functional and reliable circulator forevery high-power RF system required for APT. A secondsupplier also provides a safety margin in meeting thechallenging production and delivery deadlines required ifthe full APT accelerator becomes fully funded andhundreds of these circulators must be produced.
INDUSTRIAL FABRICATION OF MEDIUM-BETA SCRF CAVITIES FOR A HIGH-INTENSITY PROTON LINAC J.Kuzminski, General Atomics, San Diego, CA 92186, USA K.C.D.Chan, R.Gentzlinger, LANL, Los Alamos, NM 87545 USA P. Maccioni, CERCA, Romans, France Abstract During 1999, four 700-MHz, medium-beta ( β = 0.64), superconducting radio frequency (SCRF) cavities for a high-intensity proton linac project at Los Alamos National Laboratory (LANL) were manufactured by industry. The SCRF cavities were designed by a LANL team in Los Alamos, New Mexico, USA, and manufactured at a CERCA plant in Romans, France. The cavities were made of 4-mm-thick, solid niobium sheets with a residual resistivity ratio (RRR) greater than 250. These niobium sheets were supplied by Wah Chang (USA), Heraeus AG (Germany), and Tokyo Denkai (Japan). The SCRF cavities were shipped to LANL for performance testing. This paper describes the experience gained during the manufacturing process at CERCA. 1 INTRODUCTION Recently, a considerable interest emerged in the high- intensity proton linacs ranging in energy from 1 to 2 GeV because of the variety of possible applications, including the Accelerator Production of Tritium (APT), Accelerator Transmutation of Waste (ATW), and Spallation Neutron Source (SNS) [1]. These applications require powerful proton accelerator drivers capable of delivering to the target a proton beam of up to a hundred megawatts. The high beam power and continuous-wave (CW) regime of operation required by most applications make the accelerator design based on the SCRF technology desirable. Elliptical SCRF cavities are successfully applied to accelerate electrons. However, for particles with β<1, the cavity shape poses some challenging problems, such as a higher likely of multipacting during the cavity operation. To answer these questions an Engineering Design and Demonstration (ED&D) program was initiated at LANL within the APT project. One goal of the ED&D program was to build, deliver, and test a complete cryomodule containing two medium-beta ( β = 0.64), 700 MHz, 5-cell niobium SCRF cavities with helium vessels. Supported by the U.S. Department of Energy under contract No. DE-AC04-AL89607 .Because the APT linac would require a large-scale application of superconducting technology, industrial participation in fabrication of SCRF cavities and other accelerator components is necessary. Burns and Roe Enterprises Inc. with General Atomics (BREI/GA) were selected by the U.S. Department of Energy (DOE) as the prime contractor for the APT project. Under a contract with DOE, BREI/GA coordinated the industrialization part of the ED&D program. 2 SCRF CAVITIES MAN UFACTURING 2.1 Medium-beta SCRF cavity design requirements Design of the β = 0.64, 700-MHz, 5-cell ED&D SCRF cavity underwent many iterations before the final version was released for fabrication [2]. Design work was supported by extensive structural and thermal analyses performed using the ABACUS® finite element code [3]. In addition, an experimental program to determine at 2 K the mechanical properties of Nb, Ti, and their mutual joints was initiated at Florida State University [4]. Information obtained from these measurements was essential to determining the structural requirements, i. e., maximum flaw size in the SCRF cavity and He vessel welds. Figure 1. Cross section of the β = 0.64, 700-MHz, 5 -cell ED&D SCRF cavity with helium vessels (computer rendering). CavityOuter He Vessel TunerInner He Vessel Vent PipeWe present here the main features of the ED&D cavity design. The β = 0.64, 700-MHz, 5-cell ED&D SCRF cavity shown on Figure 1 is made from 4 mm thick solid Nb (RRR grade 250) sheets. There are two ports for power couplers (PC), two ports for higher-order mode (HOM) couplers, and one port for the RF pick-up. To improve RF power transmission to the cavity, the diameter of the PC beam tube was increased to 160 mm. Two tuner flexures are located on the beam tube cut-offs opposite the PC side. These are made from unalloyed grade 2 Ti. The inner helium vessels (Figure 1) made from unalloyed grade 2 Ti are welded to the elliptical heads spun from unalloyed grade 2 Ti and to the beam tube through special Ti edge-welded bellows. The outer vessel has four access ports that may be welded shut after installation of cavity instrumentation. Design requirements for manufacturing of ED&D cavities were specified in the Scope of Work (SOW) that was released to industry. These included the frequency of the fundamental accelerating mode (p-mode) to be 698.75 MHz at room temperature, RF energy density along the cavity axis (E-field flatness) to be within 10%, accelerating gradient E acc = 6.5 MV/m and the unloaded quality factor Q 0 = 5×109 at Eacc = 6.5MV/m. These last two were not a part of technical requirements for cavity acceptance, but were suggested design goals. 2.2 Fabrication of SCRF at CERCA After winning a competitive bid process, CERCA was selected as the fabricator of four ED&D SCRF cavities. The cavities were manufactured in Romans, France. A detailed SOW that specified design requirements was submitted with the contract documents. CERCA provided manufacturing drawings and the manufacturing plan with detailed procedures for each manufacturing step, including an outline of the Quality Assurance plan based on the ISO 9000 approach. These were reviewed and accepted by the technical team from BREI/GA and LANL before cavity fabrication began. Special attention was paid to provide sufficient quality controls at each step of the manufacturing process. All measurements and checks required by the manufacturing plan were performed and signed by the qualified QA technician and documented in the cavity traveller. When a nonconformance occurred, a detailed nonconformance report (NCR) was issued. The work stopped until the corrective action were proposed and accepted by BREI/GA and the LANL team. In addition, periodic visits to the CERCA manufacturing plant in Romans were made to provide oversight and ensure the quality of the manufactured parts. CERCA manufacturing plan called for fabrication of half-cells by spinning. Two half-cells were spun to validate the process. Geometry of each cell was measuredafterwards by a computerized coordinate measuring machine (CCMM) and found to be within specified tolerances of ±0.35 mm. The cavity design requires single pass full-penetration cosmetic underbead electron-beam (EB) welds. The welding parameters were determined for each weld design and demonstrated on specimens of equal thickness. Since the equator weld is the most critical operation of the fabrication process, two spare half-cells were EB welded together at the equator to validate those weld parameters. The RRR of the parent, heat affected zone, and the welded material between high grades of niobium were measured at Oregon State University in Corvallis, Oregon [5] to ensure that the RRR did not degrade more than 10%. No RRR degradation was found within the accuracy of the measurement. [5] Before starting the EB welding of the cavity parts, CERCA was required to provide specimens of high-to- low RRR Nb and low RRR Nb-to-Ti EB welded joints. These specimens were inspected to ensure that a complete penetration was achieved using these parameters. Once welding parameters were established and validated, CERCA was committed to maintain the weld parameters, and change them only after testing and acceptance by BREI/GA and LANL team. Beam tube cut-offs and PC ports were made from Nb grade 250 material, while the HOM ports and RF pick-up tube were made from (reactor) RRR grade 40 material. Stainless steel CONFLAT® flanges were brazed to Nb tubes using a procedure developed by CERCA. Figure 2. Fabrication of ED&D SCRF cavities at CERCA. In the foreground are “dumb-bells” formed by EB-welding of half-cells . In the background, a finished cavity is removed from the EB welder’s chamber. EB-welding of SCRF cavities followed the well- defined sequence described in the manufacturing plan. First, two half-cells were EB-welded (iris weld, performed from inside) to form “dumb-bells” (see Figure 2). Next, dumb-bells were EB-welded together (equator weld) to form an intermediate structure (see Figure 3). A single-pass, full-penetration EB-weld was performed from the outside. Finally, the beam tube cut-offs with end half-cells were EB-welded (equator weld) to form the five-cell cavity. Before each EB-welding operation, parts were thoroughly cleaned (buffered chemical polishing, 1:1:1), rinsed in ultra-pure water, and dried under ultra-pure N 2. Adherence to this procedure was carefully checked and documented on the sign-off sheet in the traveller. Figure 3. Beam tube cut-off and an intermediate part formed by EB-welding of dumb-bells prior to the final EB-welding. Before shipment to LANL, all cavities underwent BCP (1:1:2) that removed 150 µm of Nb from the cavity inner surface. Cavities were filled with ultra-pure N 2 at 1.3 bar, sealed, and shipped to LANL in specially designed containers. 3 SCRF CAVITY PERFO RMANCE All four SCRF cavities manufactured by CERCA were cold-tested at 2 K to determine their performances. Cavities GERMAINE and SYLVIE were tested at Thomas Jefferson National Accelerator Facility [6], and ELEANORE and AYAKO were and are to be tested at LANL respectively [7]. Cavities were tuned to required frequency. RF energy density along the cavity axis (E- field flatness) was demonstrated to be better than 5%, thus exceeding design requirements. Prior to testing, cavities underwent a light, buffered chemical polishing that removed ~20 µm of Nb from the cavity’s inner surface, followed by a high-pressure (90 bar) ultra-pure water rinsing. Preliminary results for ELEANORE, GERMAINE, and SYLVIE are presented on Figure 4. Maximum accelerating gradient E acc reached during tests was limited by the available RF power. As can beseen, performances of all three cavities exceeded the APT design goal. At the time this paper was written, testing of AYAKO was still underway. After testing, the cavities were filled with ultra-pure nitrogen at a pressure of 1.3 bar, sealed, and shipped for the installation of the helium vessels. Figure 4. Performance of ED&D SCRF cavities manufactured by CERCA. ELEANORE and SYLVIE were built from Wah-Chang-supplied Nb and GERMAINE from Nb supplied by Heraeus AG. 4 CONCLUSION During 1999, four SCRF cavities were successfully manufactured by CERCA for the APT project. During this process, a constant interaction between the customer and industrial partners resulted in high-quality product that exceeded the APT design goal. Preliminary results at 2 K show maximum accelerating gradient E acc in excess of 10 MV/m, the highest achieved to date in medium-beta multi-cell SCRF cavities. Industry experience injected into the manufacturing plan allowed simplification of the fabrication process and suggested cost-saving approaches. REFERENCES [1] K.C.D. Chan, G.P. Lawrence, and J.D. Schneider, Development of RF Linac for High-Current Applications, NIM B 139 (1998) 394-400. [2] R.Gentzlinger et al.,“Fabrication of the APT Cavities,” in Proceeding of the 18th Particle Accelerator Conference, New York, 1999. [3] R.Mitchell et al., “Structural Analysis of the APT Superonducting Cavities”, in IX Workshop on RF Superconductivity” Santa Fe, 1999. [4] R.P.Walsh et al., “Low Temperature Tensile and Fracture Toughness Properties of SCRF cavity Structural Material”, in IX Workshop on RF Superconductivity” Santa Fe, 1999. [5] W.H.Warnes , private communication. [6] J.Mammosser , private communication. [7] T.Tajima , private communication . 1.E+081.E+091.E+101.E+11 0 2 4 6 8 10 12 Eacc (MV/m)Q0ELEANORE SYLVIE GERMAINE APT Design Goal SYLVIE with INNER VESSEL
Abstract The operation of a Free Electron Laser (FEL) in the ultra- violet or in the X-ray regime requires the acceleration ofelectron bunches with an rms length of 25 to 50 µm. Thewakeﬁelds generated by these sub picosecond bunches ex-tend into the frequency range well beyond the threshold forCooper pair breakup (about 750 GHz) in superconductingniobium at 2 K. It is shown, that the superconducting cavi-ties can indeed be operated with 25 µm bunches withoutsuffering a breakdown of superconductivity (quench),however at the price of a reduced quality factor and an in-creased heat transfer to the superﬂuid helium bath. This was ﬁrst shown by wakeﬁeld calculations based on the diffraction model [1]. In the meantime a more conven-tional method of computing wake ﬁelds in the time domainby numerical methods was developed and used for thewakeﬁeld calculations [2]. Both methods lead to compara-ble results: the operation of TESLA with 25 µm bunches ispossible but leads to an additional heat load due to thehigher order modes (HOMs). Therefore HOM dampers forthese high frequencies are under construction [3]. Thesedampers are located in the beam pipes between the 9-cellcavities. So it is of interest, if there are trapped modes inthe cavity due to closed photon orbits. In this paper we investigate the existence of trapped modes and the distribution of heat load over the surface ofthe TESLA cavity by numerical photon tracking. 1 INTRODUCTION The use of a numerical photon tracking is justiﬁed by the fact, that we are only interested in photons above 750 GHz,which means a wavelength lower than 0.4 mm. Fromformer investigations two important questions remain: • Are there photons, which remain trapped within the cavity? • Are there cavity surfaces with a pronounced local heat load? The ﬁrst point is of importance for the applicability of pho- ton absorbers located in the beam pipes. The second pointconcerns the risk of quenching of cavities caused by con-centrated photon absorption. 2 PHOTON TRACKING For the calculations we used a numerical model of a mass point bouncing elastically through a two dimensional cutof the TESLA cavity like a classical billiard. The chaoticmotion in these billiards is well investigated and used asmodel for eigenvalues of quantum systems. This is a suit- able model for high frequency photons in a superconduct-ing resonator, because of the very low surface resistance:In the frequency range below 750 GHz, which covers over96% of the HOM energy of a 25 µm bunch, the surface re-sistance of niobium at 2 K is of some µ W [2]. The charac- teristic number of hits n depends on the impedance step at the surface: (1) So many thousands of reﬂections are possible. Above the Cooper pair threshold of 750 GHz the surface resistancerises immediately to 15 m W , but nevertheless some thou- sand reﬂections are possible. Figure 1: Elliptic and circular shape of TESLA cavity Table 1: Geometric parameters of TESLA cavities geometric param. of half cell / [mm]inner cellsend cells (sym.)asym. cavity end cell 1end cell 2 length L 57.692 56 56 57iris radius R 1 35 39 39 39 cell radius R 2 103.3 103.3 103.3 103.3 elliptic radius 1 a 12 10 10 9elliptic radius 2 b 19.0 13.5 13.5 12.8circular radius R C 42.00 40.34 40.30 42.00 total length L T 1,035 1,036 beam pipe len. L B 346ZZ0– ZZ0+----------------Łłæön1 e---= a bR2 R1RC CavityaxisLP1P2 THE DIFFRACTION MODEL AND ITS APPLICABILITY FOR WAKEFIELD CALCULATIONS P. Hülsmann, W.F.O. Müller and H. Klein, Institut für Angewandte Physik, Universität Frankfurt am Main, Germany As can be seen in Fig. 1 the shape of the TESLA cavity is deﬁned by a sequence of straight, round and elliptic graph-ical elements. The geometric data are given in table 1.Fig. 2 shows the geometry of one half 9-cell cavity consist-ing of 42 graphical elements. With a computer program theintersection of the photon track and each wall element iscalculated and sorted with respect to the distance. Each in-tersection between photon trace and cavity contour istreated as a reﬂection process. 3 TRAPPED MODES Following the concepts as described in [4], [5], [6] and [7] we assumed an entirely chaotic behavior for the ellipticshaped TESLA cavity. This was proven by numerical ex-periments with different starting conditions. For example,Fig. 3 shows two photon tracks (black and gray) started inthe upper left corner of the cavity, both in parallel to thebeam axis with slightly different offset. After a few reﬂec-tions the tracks become completely different from eachother. There is no need to distinguish between differentclasses of tracks, since photon tracks change between„whispering gallery orbits“, „bouncing ball orbits“ and or-bits in parallel to the beam axis travelling from cavity tocavity through the beam pipe within some reﬂections. The next investigation is devoted to the problem of photons trapped in one cavity. Therefore in the shadow region ofeach cavity a photon source was established (see Fig. 4).This is the numerical analogon to the experiment with alight bulb in a cavity model made of copper [1]. The number of reﬂections until the escape through the beam pipe takes place is well below 100. Even from themiddle cell (#5) photons leave the cavity within 64 reﬂec-tions in the average (see Fig. 5). Towards the end cells thisnumber is reduced to 23. Thus it is clearly shown, that thereexists no location in the cavity, where photons are trapped. 4 HEAT DISTRIBUTION To study the process of heating a photon was injected into the half cavity. During the ﬂight the photon experiencesmultiple reﬂections with the cavity wall. Fig. 6 shows thetrack of the photon after 1,000 reﬂections. As can be seen in Fig. 7 the large number of reﬂections and their chaotic structure have the beneﬁcial effect, that the ra-diation power, which eventually has to be absorbed by theniobium, is distributed over the whole cavity surface. Theoscillating characteristic in Fig. 7 is caused by the differentshape of the reﬂecting objects: There are straight objectsand objects with curvature alternating with each other, asFigure 2: Deﬁnition of one half TESLA cavity with half beam pipe and symmetry planes (left and mid) 12 345 6 789 10 111213 14 151617 18 192021 4241 40 393837 36 353433 32 313029 28 272625 2422 23 Figure 3: Two tracks with slightly different initial cond. Figure 4: Photon sources in the shadow region of each cell in the center plane at r = 90 mm Figure 5: Reﬂections until escape through beam pipe Figure 6: One photon track consisting of 1,000 reﬂections70 60 50 40 30 20 10 0mean reflections to escape 987654321 number of start cell N=1000 per cell N=100 per cell can be seen in Fig. 1. And in the average the surface seen by the photon is smaller for bent objects. Despite this be-havior the number of hits are equally distributed over thewhole surface: There are about eight reﬂections per mm fora total number of 20,000 reﬂections. Due to the fact, that most of the diffracted radiation hits the ﬁrst iris of the 9-cell cavity in a narrow ring shaped regionclose to the smallest diameter of the iris, a point like photonsource at the iris edge was investigated. The initial condi-tion with an emitting angle of p /100 is shown in Fig. 8: The results in Fig. 9 show clearly, that the starting condi- tion for multiple reﬂections is not of importance for theover all distribution. Only the symmetry planes #1 and #22are hit slightly more often by the photons. This is caused bythe fact, that both surfaces are hit in the very beginning ofthe process, where the motion is still deterministic. After ﬁve or ten reﬂections the motion is dominated by chaoticbehavior. 5 CONCLUSION The results of the numerical calculations have clearly shown, that the results delivered by a former light bulb ex-periment are in quite good agreement but a bit too pessi-mistic caused by the limited reﬂectivity of the coppermodel. Thus no problems caused by trapped photons areexpected during FEL operation of the TESLA collider. Thenumber of reﬂections needed to escape from a cavity isquite small compared to the number of reﬂections neededto absorb the photon in the wall. So the efﬁciency of highfrequency photon absorbers for the THz region in the beampipes can be taken for granted. Furthermore it was shown, that the photon absorption is not concentrated on certain regions, e.g. the iris edges; theheat load is distributed over the whole cavity surface witha ﬂuctuation of 25% only. 6 REFERENCES [1] P. Hülsmann, H. Klein, C. Peschke, W.F.O. Müller: „The In- ﬂuence of Wakeﬁelds on Superconducting TESLA-Cavitiesin FEL-Operation“, Proceedings of the eighth Workshop onSuperconductivity, Abano Terme, Italy, October 1997 [2] R. Brinkmann, M. Dohlus, D. Trines, DESY; A. No- vokhatski, M. Timm, T. Weiland, TU Darmstadt; P. Hüls-mann, Frankfurt University; C.T. Rieck, K. Scharnberg, P.Schmüser, Hamburg University: „Terahertz Wakeﬁelds andtheir Effect on the Superconducting Cavities in TESLA“;EPAC 2000, Wien, June 2000 [3] A. Jöstingmeier, M. Dohlus, N. Holtkamp; TESLA-Report 98-23 and 98-24, September 1998 [4] M.C. Gutzwiller: „Chaos in Classical and Quantum Mechan- ics“; Springer, New York, 1990 [5] H.-D. Gräf, H.L. Harney, H. Lengeler, C.H. Lewenkopf, C. Rangacharyulu, A. Richter, P. Schardt, H.A. Weidenmüller;Phys. Rev. Lett. 69 , 1296 (1992) [6] H.-J. Stöckmann, J. Stein; Phys. Rev. Lett. 64 , 2215 (1990) [7] H. Rehfeld; Diplomarbeit, Institut für Kernphysik, Tech- nische Universität Darmstadt, May 1996 (work supported by DESY / Hamburg and BMBF under contract 06 OF 841) Figure 7: Hits per unit length versus geometric elements marked by numbers in Fig. 2 Figure 8: Point like photon emitter at the iris edge Figure 9: Comparison of hits per unit length for elements as marked in Fig. 2 by numbers10 8 6 4 2 0I1 (reflections per mm) 40 30 20 10 0 element number total 20,000 reflections 0.124681246810I2/I1 (relative refl. per mm) 40 30 20 10 0 element number two runs with 20,000 reflections
12 PM-tubes or CCD SpectrometerdigitizerCurrent GeneratorFlashboard plasmasources cathodeanode lensmirrorplasma cylindrical lensfiber-bundle array X Z YPCdoped material G-10 plateepoxy resin Gas doping Surface dischargeDoping nozzle gas valveskimmer9 cm2.6 cm3 cm 1-2 cm600 800 1000 1200 140002468 x=1 cm x=1 cm x=2 cm x=2 cmne [x1014 cm-3] Time [ns]1020304050 a) Experimental 3p/3d ratio Collisional-radiative modelingMgII 3p/3d Ratio 400 500 600 700 800 9000.00.51.01.5b) Stark broadening MgII line ratios Lower limit from the absolute intensity of ArIne[x1014 cm-3] Time [ns]1E14 2E14 3E143E14 3E144E144E14 4E145E146E147E147E14 -2-1012345678910110.00.51.01.52.02.5 Load sideGenerator side CathodeAnodeX [cm] Z [cm]100 200 300 400 500 60002468 2.8(7)cm/s 2.8(7)cm/s2.0(7)cm/s 2.3(7)cm/s X=5 cm X=2.5 cm (anode) X=0 cm (cathode)HeI 1s3d 1D level population [x108 cm-3] Time [ns]0.00.51.01.52.02.53.0 (a)Experiments HeI HeI HeII (x300) CR Modeling HeI HeII (x300)HeI 1s3d 1D, HeII n=4 populations [x108 cm-3] 200 400 600 800 1000 120005101520253035 (b)Te[eV] Time [ns]0.00.51.01.52.0 (a)Experiments Ar I Ar II Ar III CR Modeling Ar I Ar II Ar IIIAr I, Ar II, Ar III level populations [x109 cm-3] 400 600 800 1000 120005101520253035 (b)Te[eV] Time [ns]400 600 800 1000 12000510152025 Argon CR modeling Helium CR modeling CI ionization Helium absolute intensity Argon absolute intensity Helium, no electrodesTe[eV] Time [ns] 1 Spectroscopic investigations of a dielectric-surface-discharge plasma source R. Arad∗, K. Tsigutkin, Yu.V. Ralchenko, and Y. Maron Faculty of Physics, Weizmann Institute of Science, 76100 Rehovot, Israel Spectroscopic investigations of the properties of a plasma produced by a flashboard plasma source, commonly used in pulsed plasma experiments, are presented. The plasma is used to prefill a planar 0.4-µs-conduction time plasma opening switch (POS). A novel gas-doping technique and a secondary surface flashover plasma source are used to locally dope the plasma with gaseous and solid materials, respectively, allowing for spatially- resolved measurements. The electron density, temperature, and plasma composition are determined from spectral line intensities and line profiles. Detailed collisional-radiative modeling is used to analyze the observed line intensities. The propagation velocity and divergence angle of various ions are determined from time-of-flight measurements and Doppler broadening of spectral lines, respectively. This allows for distinguishing the secondary plasma ejected from the POS electrodes from the plasma of the flashboard source. PACS numbers: 52.50.Dg, 52.70.-m, 52.70.Kz, 52.75.Kq ∗Electronic mail: fnarad@plasma-gate.weizmann.ac.il 2 I. Introduction The need for reproducible plasma sources with various properties (electron density, electron temperature, plasma composition and uniformity) for different pulsed-power applications such as ion diodes [1] and plasma opening switches (POS) [2] encouraged the development of a variety of plasma sources. Among the most popular schemes are plasma guns [3], flashboards [4,5], gaseous plasma sources [6,7], and explosive emission sources [8]. In plasma (or cable) guns, a flashover occurs across the insulator at the end of a high-voltage cable. Flashboards are printed circuit boards consisting of a number of chains, each with a set of flashover gaps in series. A high-voltage pulse is applied to the ends of each chain and a surface breakdown is initiated across the gaps. The primary goal of this work is to study the plasma properties of a flashboard plasma source as a first step towards a thorough investigation of the physics of the interaction between plasmas and magnetic fields in general, and in particular that of POS’s. In our setup the flashboard plasma source is used to prefill a planar POS (0.4-µs- conduction time and 160 kA peak current) coupled to an inductive load. Previous experimental studies of flashboards include a detailed investigation of the plasma flow velocity and uniformity for various driving circuits and flashboard configurations [5] using electrical probes. A recent experiment [9], in which the electron density was determined both by spectroscopic measurements and by electrical probes, showed that electrical probes underestimate the electron density of a flashboard plasma by up to an order of magnitude. A spectroscopic investigation of the electron density, temperature, and plasma composition was carried out on a flashboard driven by a relatively low current through the discharge chains [10]. In that work, the peak electron density ne was determined to be 2×1013 cm-3 and the plasma was found to mainly consist of CII and CIII. However, it should be noted that the density of CIV and CV could not be determined in those experiments due to the low electron temperature. Plasma guns being closely related to flashboards have been studied [3,10,11,12] using various techniques that showed the plasma mainly consists of carbon ions with electron densities up to ≈1014 cm-3. Interferometry was implemented to determine the line-integrated electron density and its spatial distribution in a few POS experiments [13,14,15] where either plasma guns or flashboards were used. As will be shown below, determining the electron density, temperature and plasma composition in such plasma sources is a complex problem involving various processes such as plasma flow, ionization, and secondary plasma creation. To the best of our knowledge, these characteristics have as yet not been determined satisfactorily in flashboard and plasma gun plasmas. In this work we determine these plasma properties using different spectroscopic observations together with atomic-physics modeling. High resolution spectroscopy is employed to measure line profiles and intensities of a variety of spectral lines. Spatial resolution along the line of sight is obtained by doping the plasma with different elements and observing the characteristic emission from the doped species. To this end, we employ our newly developed gas injection [16] and surface flashover techniques to locally dope the plasma with gaseous and ionic material, respectively. The versatility of the doping methods also allows us to perform measurements using a wide variety of dopant elements which enables us to study many important plasma parameters. Furthermore, by measuring and analyzing spectral lines 3 belonging to a few different elements we are able to reduce the errors introduced by the uncertainty of the different atomic rates. The plasma electron density is obtained from Stark broadening of the Hα and Hβ lines of hydrogen. A second complementary method, based on spectral line ratios of MgII, is used to study the electron density at early times when ne < 1014 cm-3 and the Stark broadening yields less reliable results. It was verified that for high electron temperatures (Te > 10 eV) and low electron densities, the ratio between the MgII 3p and 3d level populations depends primarily on n e. A third method compares the absolute intensity of doped ArI lines with collisional-radiative calculations and the known argon density. The use of a few methods provides us with a more reliable electron density and enables measurements in a wide range of densities (1×1013 -2×1015 cm-3). The electron temperature is obtained from the temporal evolution and absolute intensities of spectral line emissions of various doped gases. The observed evolution of spectral lines from different charge-state ions is compared to collisional-radiative (CR) calculations [17]. Other, more conventional methods for determining the electron temperature such as spectral line ratios, could not be fully implemented here. A typical procedure would be to compare the line intensities from levels with significantly different energies both of which should not be too high. Here, this requirement could not be fulfilled due to the rapid ionization of low charge-state species and the possible presence of non-thermal electrons. The plasma composition is obtained from the absolute intensities of spectral lines belonging to a variety of species that make up the plasma. To this end, CR calculations that are based on the determined electron density and temperature are used for calculating the ratio between the excited level populations and the total density of the specific ion. The effects of charge exchange between CV and hydrogen are examined and found to significantly affect some of the CIV excited level populations, allowing the CV density to be determined. It is shown that the plasma mainly consists of protons, hydrogen, and carbon ions. By comparing the plasma composition with and without the POS electrodes in place we are able to determine which ions originate from the flashboard and which are coming off the POS electrodes. The use of the gas doping also allows for studying the non-radiating proton plasma at the front of the expanding plasma cloud by ob serving the excitation of the doped gas by the electrons in the proton plasma. We believe that the diagnostic methods described here can be used for studying plasma sources and transient plasmas in general. II. Experimental Setup Two flashboards are used to generate the plasma. A single 2.8-µF capacitor charged to 35 kV drives the flashboards via sixteen 75-Ω cables, giving a peak current of 6 kA per chain at t = 1.2 µs. The flashboards are positioned next to each other in such a way that the peak plasma density is generated in the middle of the plasma volume. Each of the flashboards has eight chains and each chain consists of 8 copper islands on a G-10 insulator of a 0.07-cm thickness with a 1.5-cm gap between the chains. The islands are 4 0.5 cm in diameter and are separated by 0.05 -cm gaps (in our experiments the flashboards are not coated with carbon spray as is commonly done). The flashboards are placed 3 cm above the 8-cm-long region of the POS transparent anode, see Figure 1. The planar POS electrodes are 14 cm wide (along the y- direction) and are 2.6 cm apart. In the POS region, the electrodes are highly transparent and consist of 0.1 cm-diameter wires separated by 1.4 cm. This high transparency ensures free plasma flow through the anode and little plasma stagnation near the cathode. The flashboard is operated ≈1.2 µs before the high-current pulse of the POS is applied. The following coordinates are defined: x = 0 is the cathode, y = 0 is the center of the electrodes, and z = 0 is the generator-side edge of the anode transparent region. The gas doping arrangement, consisting of a fast gas valve, a nozzle and a skimmer, is mounted below the cathode on a moveable stand that allows for 2D movement. The dopant-gas beam was diagnosed using a specially-designed, high spatial- resolution ionization probe array [16] that allows the absolute gas density and the beam width to be determined. The gas density can be varied from approximately 1013 to 3×1014 cm-3 by changing the time delay between the operation of the gas valve and the flashboard. The full width half maximum (FWHM) of the gas beam perpendicular to its injection could be varied from 1 to 2 cm by changing the aperture of the skimmer. Alternatively, for doping solid materials such as magnesium, we used an electrical discharge over an epoxy resin mixed with the desired element. A small G-10 board was placed 2 cm below the cathode and a discharge was driven by a 2 µF capacitor charged to 6 kV. The hereby-produced plasma was diagnosed spectroscopically for various time delays between this discharge and the operation of the flashboard. An upper limit of the electron density of the doped material is determined from the absolute intensity of the MgII 3p-3s transition without the operation of the flashboard plasma. The emission of this line is found to be very weak. To assure that the weak emissions do not result from an exceedingly low electron temperature (<1 eV) with a relatively high electron density, we compare the emission with and without the application of the flashboard and make use of the fact that once the doped column mixes with the flashboard plasma the electron temperature of the combined plasma is at least 3 eV and the combined electron density is ≈1013 cm-3 at t=450±30 ns. Then, by performing measurements with different time delays between the doping discharge and the flashboard operation we obtain an upper bound on the electron density of the doping discharge as a function of time. For the short time delays used in the present experiments, the electron density of the plasma created by the doping discharge was found to be less than 1×1013 cm-3, ensuring no significant effect on the flashboard plasma parameters. For the short time delays used here, the neutral and electron densities of the doped column are small enough inside the A-K gap to assure they do not significantly perturb the flashboard plasma. The density of neutrals such as MgI and CI are negligible in the A-K gap due to the rapid ionization caused by the flashboard-plasma electrons. The density of neutral hydrogen produced by the solid doping discharge is significantly lower than the hydrogen density ejected off the POS electrodes. We did not measure the density outside the A-K gap, however, based on comparison of measurements with different time delays we believe both the electron and the neutral densities to be <1014 cm-3 even 1 cm beyond the cathode. 5 Mirrors and lenses are used to direct the light onto a 1-meter spectrometer equipped with a 2400 grooves/mm grating. Observations along all three lines of sight are possible, however, in this work we present only observations performed along the y-direction. Observations along the x-direction are performed through the flashboards. A cylindrical lens images the output of the spectrometer onto a rectangular optical-fiber array at different magnifications allowing for observations with different spectral dispersions. The optical-fiber array consists of 12 vertical fiber stacks that deliver the light to 12 photomultipliers whose temporal resolution is 7 ns. The spectral resolution of our optical system is 0.07 Å. The absolute sensitivity of the optical system was measured using a few absolutely- calibrated lamps. In order to reduce the error in the relative calibration at different wavelengths we measured the light intensities of different spectral lines that originate from the same excited level. We observed the CII 2837-Å and 6578-Å lines, the NII 6284-Å and 3919-Å, and the SiII 2072-Å and 4130-Å lines. An accuracy of ±10% for the relative sensitivity and ±30% for the absolute calibration was determined. III. Experimental Results A. Measurements and Data analysis We observe the time-dependent intensities and spectral profiles of lines from the plasma constituents (mainly hydrogen, carbon, and oxygen) and the doped materials (such as helium, argon, and magnesium), as given in Table I. The line profiles are fitted using a Gaussian, from which the instrumental broadening is deconvolved. The width of the various lines is analyzed assuming Stark and Doppler broadening (all spectral lines used in this study are optically thin). The measured line intensities yield populations of the transition upper levels by using the Einstein coefficients of the transitions, the observed light emitting volume and the absolute sensitivity of the optical system. The level populations are then compared to time-dependent CR calculations in order to determine the plasma parameters such as ne and Te. The level populations given throughout this paper are divided by statistical weight. The typical velocities of species doped using the gaseous doping are 1×105 cm/s so that within 1 µs their motion is negligible. However, for the plasma and solid doping constituent the ion velocities are a few times 106 cm/s so that their motion during the time of interest is significant. To study the ion fluxes we used the fact that the ratio between different level populations and the total ionic density equilibrate within times determined by the radiative decay of the level (typically < 10-20 ns). Thus, the observed evolution of the level populations reflects the variations of the ionic density, ne and Te. Since (as will be shown bellow) ne and Te are determined from line widths, intensity ratios and line intensities of gaseous elements (whose motion is negligible), we could use the evolution of the observed intensities to determine the fluxes of the ions whose motion is significant. In the CR modeling [17] the time-dependent atomic/ionic level populations are calculated from the following system of rate equations: 6 where N(t) is the vector of all the level populations and A(t) is the rate matrix. The treatment is time-dependent (the rate equation system (1) is integrated explicitly), as is required for such highly transient plasmas. The number of atomic levels used in the calculations varies from about 10 for Hydrogen up to about 200 for the oxygen ions (OI- IX). The matrix A(t) depends on time via the electron density ne(t) and the time- dependent EED function. The basic atomic processes taken into account in the rate matrix are electron impact excitation and deexcitation, electron impact ionization and 3- body recombination, spontaneous radiative decays, radiative and dielectronic recombination, and charge exchange. It must be emphasized that in the present modeling we use the energy-dependent cross sections rather than Maxwellian-averaged rate coefficients, which allows for CR calculations with an arbitrary EED function. This is essential for studying the effects of the non-Maxwellian EED in the flashboard plasma. The atomic data required for our collisional-radiative modeling are either taken from existing atomic databases or newly calculated using available computational tools. The National Institute of Standards Atomic Spectra Database [18] is the principal source of energy levels and oscillator strengths; the missing values are calculated with Cowan’s Hartree-Fock relativistic atomic structure package [19] including configuration interaction and intermediate coupling. Since the available data on the electron impact excitation are rather limited, most cross sections are calculated with the Coulomb-Born- exchange code ATOM [20]. The excitation cross sections are mainly calculated in LS- coupling although, when necessary, other types of couplings are utilized (for example, jK-coupling for ArI). To improve the accuracy of dipole-allowed excitation cross sections, they are rescaled by the ratio between a more accurate oscillator strength, calculated by other methods (e.g., multi-configuration Hartree-Fock), and that calculated by ATOM (corrections are typically <50%). For some transitions between the levels with close energies this correction exceeded a factor of 2 (which may result in lower accuracy); however, the absolute magnitude of the corresponding cross sections is less important due to fast equilibration of the relevant level populations. Comparison between the rescaled excitation cross sections obtained with ATOM and available experimental data shows agreement within 20-30% for ions and within 50% for neutral atoms. For H and HeI-II more accurate excitation cross sections, calculated with the Convergent Close- Coupling method [21], are used to obtain the highest possible reliability. The use of accurate cross sections is especially important when the electron temperature is much smaller than the excitation threshold, and therefore the corresponding rate is mainly determined by the near-threshold cross section. The dipole-allowed cross sections for strong two-electron transitions are computed in the Van Regemorter approximation. The recommended ionization cross sections for the ion ground states [22] are also used for ionization from the excited states, the classical scaling σion ~ I-2 being implemented for the latter (I is the ionization energy). The Milne relation [19] is used to produce the radiative recombination cross sections from the Opacity Project results [23] for photoionization. The dielectronic recombination rates of Hahn [24] are used for most ions while for neutral and low-charge ions of argon the Burgess-Merts-Cowan-Magee parameterization [19] is utilized. Finally, the cross sections for the inverse processes are ())1( ),()(tNtAdttdN= 7 obtained using the principle of detailed balance which, although formulated for equilibrium plasmas, establishes relations between purely atomic quantities, and thus these relations are independent on particular plasma conditions such as, for example, the electron energy distribution function. B. Electron density Stark broadening measurements of the Hα and Hβ profiles are analyzed self- consistently to give ne and the hydrogen velocity distribution along the line of sight. In this analysis, the Stark broadening is assumed to mainly result from the quasistatic Holtzmark distribution of the ionic microfields with a small correction due to the electron dynamics, treated under the impact approximation. The line width calculations are performed according to Ref. [25]. The effects of ion dynamics (negligible for Hβ) are unimportant for Hα in this experiment due to the dominance of the Doppler broadening. The Doppler broadening results from the combined effect of the thermal velocity and the directed velocity, integrated along the line of sight. The Hα profiles are fitted by a Gaussian which is in good agreement with the observed profiles. Because of the large Doppler width of the hydrogen lines the use of Stark broadening for the determination of the electron density is limited to ne > 1014 cm-3. The Stark broadening dependence on the effective charge of the ions in the plasma is accounted for, based on the determined plasma composition (see Sec. /G03 III.E). The electron density measurements are performed in the POS anode-cathode (A-K) gap which is located 3-5.5 cm from the flashboard surface. H2 and CH4 doping is used to determine ne locally in 3D by observing the line emission from hydrogen produced from the dissociation of these molecules. Due to the complex chemistry of the doped CH4 molecules, whose detailed modeling is beyond the scope of this work, we performed the measurements using different densities of CH4 and H2 doping and compared them to experiments without doping. We were careful to operate with gas densities that are significantly smaller than the original plasma density to minimize the effects of the unknown plasma chemistry. The rise in the plasma electron density due to the ionization of these doped gases is estimated to be less than 5×1013 cm-3 due to the low density of the injected gas (ngas < 3×1013 cm-3). However, at t > 1 µs the density of hydrogen ejected from the POS electrodes becomes comparable to that of the doped hydrogen, resulting in line-integrated (along the y-direction) electron density determination. The evolution of ne at z = 3.5, y = 0, and x = 1 and 2 cm is shown in figure 2. In these experiments CH4 is injected into the entire POS region by removing the skimmer. The spatial resolution in these measurements is ±0.2 cm along the x-direction, ±0.5 cm along the z-direction, and spatially averaged along the y-direction. At each position two traces from two different discharges are shown to indicate the reproducibility ne, found to be within ±10% at t > 1 µs. The ratio between the MgII 3p and 3d level populations, determined from the intensities of the 3s-3p and 3p-3d transitions, is used in order to study ne at times earlier than 800 ns (prior to the ionization of the MgII). This method relies on the fact that for the densities 5×1012 < ne < 5×1014 cm-3 the 3p-3d excitation channel, which scales approximately as ne2, becomes comparable to the 3s-3d direct excitation from the ground 8 state, making the ratio between the 3p and 3d level populations dependent on ne. Moreover, due to the low excitation energies of these two levels the ratio between their populations is insensitive to the electron temperature for Te > 10 eV. Also, these spectral lines could be observed in a single discharge due to their similar wavelengths, overcoming the shot-to-shot irreproducibility. Figure 3(a) shows the time-dependent population ratio for these two MgII levels. Also shown is the ratio predicted by the CR calculations using ne from Figure 3(b) and Te(t) determined in Sec. /G03 III.D. The error bars on ne are determined by the experimental error, the uncertainty in the electron temperature, and by the uncertainties in the atomic-physics modeling. The larger uncertainties at later times result from the lower electron temperature at these times. At t = 800 ns ne so determined matches ne determined from Stark broadening to within ±10%. Figure 3(b) also shows a lower limit for ne obtained from the observed intensity of the ArI 6965-Å line. Here, we make use of the knowledge of the absolute density of the doped argon obtained from the ionization-probe array measurements [16]. By varying the electron temperature in the CR calculations as a free parameter we searched for the minimum electron density that would match the observed line intensity evolution, based on the known total ArI density. The presented lower limit accounts for the uncertainties in the ArI gas density and the absolute calibration of the spectroscopic system. By determining ne at various positions in the x-z plane we constructed a 2D map of ne at various times. Such a map of the electron density in the x-z plane integrated over the y-direction at t = 1200 ns, which corresponds to the time of application of the POS high- current pulse, is shown in Figure 4. The density minimum at z = 3.5 cm is below the contact region of the two flashboards. The density peaks at z = 1.5 and z = 5.5 cm result from a faster plasma expansion at the ground-side edges of the flashboards. The lower density at the generator side (z = 1.5 cm) relative to the load side (z = 5.5 cm) is due to the lower current through the flashboard at the generator side, as a result of a larger inductance of the cables driving that flashboard. The distribution of ne along the width of the electrodes (y-direction) is measured using observations along the z-axis, and the axially averaged electron density is observed to have a symmetric profile to within ±10% with a FWHM ≈ 12 cm. The density at the edges of the electrodes (along the y-direction) is found to be (1-2)×1014 cm-3 at t = 1200 ns. C. Plasma Expansion velocity In order to study the properties of the proton-plasma flowing at the front of the injected plasma, we observe light emission from the doped elements as they encounter the proton-plasma. The velocity of the proton plasma is determined from time-of-flight data obtained by observing the excitations of the dopants caused by the co-moving plasma electrons. The velocity of the slower carbon-dominated plasma is obtained from the time-of-flight data of spectral lines of carbon ions without doping. Figure 5 shows the intensity of the doped HeI 6678-Å line at different x-positions. In these experiments, the POS electrodes are removed in order to avoid electrode-plasma effects on the measurement. The time delay between the helium injection and the application of the flashboard current is varied in the experiments in order to compensate for the different travel times of the gas to the various x-positions. The plasma front is 9 seen to flow at a velocity of (2.8±0.5)×107 cm/s, followed by a plasma with a higher density and a lower velocity of (2.2±0.5)×107 cm/s, reaching the POS region at t = 400- 500 ns. The slower rise of the helium signal at positions farther from the flashboard indicates that the velocity at which plasma leaves the vicinity of the flashboard drops in time. The plasma that reaches the A-K gap 300-550 ns after the initiation of the flashboard discharge is primarily composed of protons with a small fraction of fast hydrogen. It is verified spectroscopically that the density of heavier ions such as carbon is negligible in this plasma front. In order to determine the proton transverse velocity we observe the Doppler broadened Hα profile in experiments without doping and without the POS electrodes in order to only observe the hydrogen originating at the flashboard. Since only fast hydrogen atoms (those produced via proton charge-exchange near the flashboard surface) may reach the A-K gap so early, the Hα Doppler width should yield the proton transverse velocity. Measurements of the Hα profile at x = 1 cm at early times (t = 400-700 ns) without gas injection and without POS electrodes show a constant Doppler profile with an average width that corresponds to a velocity of (3.5±0.5)×106 cm/s in the y-direction. Comparing this velocity to the proton propagation velocity, obtained from time-of-flight, one can obtain the divergence angle of the plasma, found to vary from ±(7±1.5)o at t = 320±40 ns to ±(10±2)o at t = 450±50 ns and to ±(13±2.5)o at t = 700±50 ns. Time-of-flight measurements of the CIII 2297-Å line without the POS electrodes showed a front propagating at a velocity of (7±2)×106 cm/s, reaching the A-K gap at t ≈ 800 ns. This is followed by CIII with a higher density propagating at a velocity of (5±1)×106 cm/s. The profile of this line indicated an average transverse velocity dropping from (2.4±0.3)×106 cm/s for the front of the CIII plasma to (1.7±0.2)×106 cm/s for the peak CIII density. The propagation velocity together with the transverse velocity yield a divergence angle of ±(20±8)o for the CIII plasma. D. Electron Temperature The electron temperature is determined as a function of time from the temporal evolution of absolute light intensities. To this end, the different effects that can affect the observed line intensities, such as variations of the ion or atom densities due to particle flow and ionization and the temporal variation of ne, had to be considered. Here, by doping gaseous materials with injection velocities of the order of 105 cm/s, we are able to minimize the effect of the flow. Using dopants with different ionization rates, it is also possible to study the effects of ionization. The behavior of the HeI 1s3d 1D level and the HeII n=4 level are modeled with the aid of CR calculations using the measured electron density (see Sec. /G03 III.B). Figure 6(a) shows the temporal evolution of these level populations, determined from the 6678-Å and the 4686-Å line intensities. The time-dependent electron temperature is determined by fitting the CR-code predictions for these level populations to the data. The modeling shown in Figure 6(a) is obtained using the electron temperature shown in Figure 6(b). 10 The modeling shows that the drop of the HeI 1s3d 1D population seen at t > 1 µs cannot result from ionization, due to the relatively low electron temperature at that time, indicating that this drop must occur due to a drop in Te as shown in Figure 6(b). The electron temperature at t = 600-800 ns is also obtained from the absolute line intensity and knowledge of the density of the injected gas [16]. The absolute intensity of the HeI 6678-Å line shows that the electron temperature is 11±2 eV at t = 700 ns. Because the ionization of HeI is small, the obtained electron temperature time- dependence is unique, since the HeI 1s3d 1D level population is mainly affected by the variation of the excitation rates due to changes in the electron temperature (ne(t) is known). A similar analysis is carried out for argon by measuring the temporal evolution of the ArI 6965-Å, ArII 4348-Å, and the ArIII 3286-Å lines. For ArI significant ionization occurs, requiring measurements of a few consecutive charge states in order to obtain a unique time dependence for the electron temperature. Figure 7(a) shows the temporal evolution of the ArI 4p'[1/2]1, ArII 3p4(3P)4p 4Do, and the ArIII 3p3(4So)4p 5P level populations, together with the predictions of the CR modeling. The experimental traces are averaged over 3 different discharges and have a temporal resolution of 20 ns. Figure 7(b) shows the electron temperature evolution that is used in the CR modeling to obtain the predictions in (a). The error bars on Te mainly result from the uncertainty in ne. The absolute intensity of the ArI 6965-Å line together with knowledge of the injected argon density show that Te = 14±2 eV at t = 550 ns. We also determined Te from the observed ionization time of CI produced from the disassociation of doped CH4. We assume that the drop observed in the CI intensity at t > 900 ns results from ionization after the replenishing of neutral carbon through the various disassociation channels of CH4 is finished. This assumption does not depend on a detailed knowledge of the chemistry of the CH4 molecules and the resulting disassociation. This measurement is only performed at x = 1 cm, since in this position the line intensity of CI without the CH4 doping is negligible. The ionization of CI at t = 900-1100 ns yields Te = 8±1.5 eV. The density of the doped Ar, He, and CH4 is varied from ≈2×1013 to ≈1014 cm-3 in order to examine whether the gas doping caused cooling of the plasma electrons. For the low densities of doped He, the plasma electrons are found to cool by <<1 eV. However, for Ar and CH4 doping the plasma electrons are found to cool by up to 2 eV (for a doped density of 1014 cm-3) resulting in an underestimate of Te. A summary of the electron temperature obtained using the temporal evolution of the helium and argon line intensities, the absolute intensities of helium and argon and the ionization time of CI, together with the error bars for x = 1 cm and z = 3.5 cm, is shown in Figure 8. Te is thus found to be 9±1.5 eV until t ≈ 1.0 µs, followed by a nearly linear drop to 5.0±0.5 eV at t ≥ 1.6 µs. Furthermore, the temperature at t = 1.2 µs (the time of the application of the POS current) is found to be 651 05..+ −eV near the cathode and somewhat lower near the anode (Te = 571 05..+ −eV). The agreement between the different methods and species substantiates the reliability of the determination of Te. Close to the generator-side edge of the plasma at z = -0.3 cm the electron temperature at t = 1.2 µs is found to be lower than at z = 3.5 cm, x = 1 cm. This seems to occur due to enhanced electron cooling caused by the solid anode surface at z < 0. 11 Also shown in Figure 8 is the electron temperature evolution determined using the temporal behavior of the HeI 1s3d 1D and the HeII n=4 spectral lines without the POS electrodes in place. It can be seen that the electron temperature without the POS electrodes in place is higher by ≈1 eV until t = 0.9 µs and by ≈2.5 eV at later times. The electron density at x = 1 cm is determined to be unaffected by the removal of the POS electrodes to within the accuracy of the measurement. The determination of the electron temperature in present work was carried out assuming an equilibrium Maxwellian electron energy distribution. Nonetheless, it is known that in highly-transient plasmas of high-current discharges there may exist deviations from a Maxwellian EED [26], often in a form of an excess of high-energy electrons. The effect of the overpopulated tail of the electron energy distribution function, which is normally negligible for atomic processes with small energy thresholds ΔE (ΔE < Te), can manifest itself especially in anomalous behavior of the highly-excited levels. This seems to be the case for the measured CIV lines, which originate from high lying levels (see Table 1). We were not able to consistently describe the absolute intensities of the CIV spectral lines with only a Maxwellian EED function, whereas an addition of a few percent of hot electrons (50-100 eV) resolved the discrepancy between modeling and measurements. We plan to address the issue of non-Maxwellian effects in the flashboard and POS plasmas in a separate publication. Such a small deviation from a Maxwellian description of the plasma electrons does not devaluate the above-described quantitative determination of the electron temperature since it is obtained from emissions of low lying levels (11-23 eV) for which the contribution of the Maxwellian electrons overwhelms that of the deviation. Thus, Te serves as a measure of the mean electron energy in the flashboard plasma. E. Plasma composition The plasma composition is determined from the observed light intensities and the known electron density and temperature. The total densities of HI, CI, CII, CIII, OII, and OIII are thus obtained from the intensities of the 6563, 2479, 2837, 2297, 4349, and 3047-Å lines, respectively. The effects of charge-exchange with neutral hydrogen on the observed lines of CII, CIII, OII, and OIII are estimated and found to be negligible. This modeling in which the EED is assumed to be Maxwellian, yielded, however, unreasonably high densities for CIV. One possibility to explain these line intensities is to assume the presence of non-Maxwellian energetic electrons (as described in the previous section). It is estimated that a few percent of 50-100 eV electrons are enough for such an explanation. The other possibility is that the high CIV line intensities are caused by charge-exchange processes between CV and H resulting in selective population of some of the CIV levels. The CV density could not be determined using line emissions from CV since no emission from CV lines was detected because the upper levels of all observable transitions are >300 eV above the ground state. To examine the effect of charge-exchange on the CIV 5801 and 2530 Å lines we performed experiments with a hydrogen density varying from 3×1012 cm-3 (by removing the POS electrodes and using no doping) to 1.5×1014 cm-3 (keeping the POS electrodes in place and using H2 doping) and observed the change in the CIV line intensities. 12 In the first configuration, in which the hydrogen density is too low for charge- exchange to contribute significantly to the level populations, are used to determine the density of CIV from CR calculations. It is important to note that since the CIV and CV originate at the flashboard (as described in Sec. /G03 III.F) their density is not affected by the removal of the POS electrodes. In the experiments with H2 doping the relative velocity between the hydrogen and CV, required for the charge-exchange modeling, is obtained from the flow velocity of CV, found to be (5±1)×106 cm/s (the doped hydrogen is slow). The hydrogen density determined from the Hα intensity and the state-selective cross sections for the charge exchange [27,28] are then used to determine the density of CV. The large uncertainty in the CV density mainly results from the uncertainty in the impact velocity and the resulting error of the state-selective charge exchange into the 3p level, which drops very sharply at low impact velocities, as the dominant channel for charge exchange becomes the 3d level [27] (the charge exchange into the 3d level has nearly no effect on the 3p population since the 3d level radiatively decays to the 2p level). A second uncertainty in the CV density results from the uncertainty in the H2 density. The cross section for charge exchange with these molecules is higher than the cross section with hydrogen atoms, especially at low velocities where the 3p level remains the dominant channel of charge exchange with H2 [29]. A lower limit for the CIV density can be obtained from the level populations observed during the POS operation when the electron temperature is found to rise [30] and the CIV level populations become dominated by electron excitation rather than by charge exchange processes. By taking the optimum excitation rates in the CR calculations, this analysis results in a lower limit for the CIV density, giving for t = 1.2 µs (6±1.5)×1013 and (2±0.5)×1013 cm-3 at x = 2.2 and x = 1 cm, respectively. An upper bound for the CIV density is the measured electron density and the requirement for charge neutrality. Measurements also show that the density of other heavier elements such as iron, copper, and aluminum are negligibly small except at a distance less than 1 mm from the electrode surfaces. This results from the low flow velocity of these ions into the POS region. Near the POS electrodes, however, heavy impurities, desorbed from the surfaces due to the impact of the primary flashboard plasma, are found to make up a few percent of the plasma density. The determination of the densities of most of the plasma constituents allows the proton density to be obtained from the balance between ne and the total non-protonic ionic charge. However, due to the large uncertainty in the density of CIV and CV there is a significant uncertainty in the proton density. Table II summarizes the plasma composition at x = 1 (the middle of the A-K gap) and x = 2.2 cm (near the anode) for z = 3.5 cm (the axial center of the POS) integrated along the y-direction. The composition is for the time 1.2 µs after the start of the flashboard current pulse. F. Origin of the various plasma constituents Measurements with and without the electrodes are used to distinguish between plasmas produced by the flashboard and secondary plasmas originating from the POS 13 electrodes. The electron density and temperature, required for the CR modeling of the observed line intensities, are determined for both configurations. The electron density is found to be approximately the same while the electron temperature is found to be higher at t > 0.9 µs if the POS electrodes are removed (see Figure 8). In experiments with the POS electrodes, the observed intensity of Hα is found to increase by more than an order of magnitude as a result of hydrogen originating from the electrode surfaces. Comparison of the hydrogen density near the anode and cathode shows that the hydrogen density emanating from the anode is approximately two times higher than that from the cathode. In experiments with the POS electrodes (without doping) the Doppler-dominated Hα width at t = 600-900 ns is found to depend on the location of the observation. In the middle of the A-K gap (x = 1 cm) Hα has a broad profile corresponding to an average hydrogen velocity of (4.5±0.3)×106 cm/s in the y- direction. Experiments near the anode (x = 2 cm) show a narrow profile with an average velocity in the y-direction of only (2.5±0.3)×106 cm/s. This data can be interpreted by assuming an injection of fast and slow hydrogen from the electrodes with a large divergence angle (±45o). As a result, the fast hydrogen reaches the middle of the A-K gap at t = 600 – 900 ns, whereas the denser and slower hydrogen only reaches the x = 2 cm position and the x=0.5 cm for hydrogen ejected from the cathode. The density of CII and CIII at x = 2 cm is found to be approximately the same for the two configurations until t ≈ 1 µs, after which the density of these ions increases sharply in the configuration with the electrodes and becomes about 6 times higher than without electrodes at t = 1.2 µs. A similar increase is also seen near the cathode (x = 0.3 cm). This means that most of the high density carbon plasma near the anode and cathode originates from the electrode wires rather than from the flashboard. However, unlike CII, CIII, and hydrogen, we may conclude that CIV and CV originate at the flashboard, since the temperature in the anode-cathode gap is too low to ionize CIII originating from the electrodes within ≈1 µs. IV. Summary and Discussion Plasma doping with various gaseous materials allows for studying many properties of the flashboard-generated plasma. Both Stark broadening of hydrogen lines and spectral line ratios of MgII are used to determine the evolution of ne. The agreement between the two methods at t = 800 ns is within ±10%. The measurement of spectral line ratios of MgII also allows for accurate determination of the temporal evolution of ne for electron densities as low as 1×1013 cm-3. At the time at which the POS is operated ne is found to vary between 3×1014 cm-3 near the cathode to 7×1014 cm-3 near the anode. The electron temperature of the expanding flashboard plasma at x = 1 cm, determined from the temporal evolution of different doped-species spectral lines, is found to drop from 18±3 eV at t = 500 ns to 10.5±1.5 eV at t = 800 ns and to 651 05..+ −eV at t = 1.2 µs. In experiments in which the POS electrodes are removed, the drop of the electron temperature is slower and an electron temperature of 10±1 eV is found at x = 1 cm at t = 1.2 µs (see Figure 8). The difference between the two configurations is explained by the cooling of the plasma due to the interaction with the electrodes. 14 As mentioned in Sec. /G03 III.D, the EED in our plasma seems to deviate from a Maxwellian one, with an excess of high-energy (~ 50-100 eV) electrons. We estimate that electrons with an energy of 100 eV have a mean free path of 1-3 cm for electron- electron collisions for an electron density of (1-3)×1015 cm-3 (this density is taken as the average electron density in the region between the flashboard and the POS, i.e., x > 2.6 cm). Thus, high-energy electrons produced near the flashboard may reach the POS region without thermalization, giving rise to the deviation of the EED from a Maxwellian. The relatively high accuracy in the determination of ne and Te allows for a reasonably reliable determination of the plasma composition. To this end, the effects of charge exchange processes on the line intensities are accounted for. It is shown that the possible presence of non-thermal electrons with energies above 50 eV mainly affects the uncertainty in the determination of the CIV density. In the center of the A-K gap the plasma is shown to consist of CIV with lower densities of CII, CIII, and CV. Hydrogen is found to be the only neutral element with a significant density (<10% of ne). The proton density determined from the balance between ne and the total non-protonic ionic charge has a large uncertainty due to the relatively large uncertainty in the CIV and CV densities. The density of oxygen ions is 3 - 10 times lower than that of carbon ions and the density of heavier ions (such as SiII, FeII, and NiII) are estimated to be less than 1012 cm-3. This work demonstrates the need for detailed CR modeling in order to avoid large errors in the determination of the electron temperature and plasma composition. By combining the electron density measurements with time-of-flight measurements and with the knowledge of the plasma composition it is possible to study the velocity and origin of the various plasma constituents. Here, we make the distinction between four different plasmas. The first plasma consists of protons traveling at a velocity up to (2.8±0.3)×107 cm/s and with a density less than 1×1013 cm-3. This is followed by a slower and more dense proton plasma with a propagation velocity of (2.1±0.3)×107 cm/s and a density of 2±1×1013 cm-3. The protons in both plasmas are found to have a very low divergence angle ±(7-13)o. The third plasma is found to mainly consist of carbon ions and protons with smaller concentrations of hydrogen, oxygen and silicon ions. The electron density of this plasma increases from 1014 cm-3 to >1015 cm-3 in the A-K gap. The most abundant charge state of the carbon ions in this plasma is CIV. The fourth plasma originates at the POS electrodes as a result of the bombardment by the proton plasmas. This plasma mainly consists of hydrogen, CII, CIII, and smaller contributions of OII and OIII and contributes most of these species in the A-K gap as is found by comparison between experiments with and without the POS electrodes. The carbon ions originating at the electrodes have an expansion velocity of (2±0.5)×106 cm/s, thus they are confined to a region < 1 cm from the electrodes at the time the POS is operated. This is also correct for most of the hydrogen ejected from the electrodes that has an expansion velocity of (1.5±0.5)×106 cm/s. However, approximately 20% of the hydrogen travels at a higher expansion velocity, (3-5)×106 cm/s, thus filling the entire A- K gap at the time the POS is operated. In our configuration, the effect of plasma ejection from the electrodes on ne is small since the electrodes consist of wires with a 94% transparency and the time delay between the creation of the electrode plasmas and the POS operation is small. This is also the reason why in our configuration the most abundant carbon species is CIV, which originates at the flashboard. In another configuration used earlier, in which a solid 15 cathode was used (rather than highly transparent) and the time delay between the flashboard discharge and the POS operation was longer, the density of the secondary electrode plasma in the A-K gap was higher than that of the flashboard plasma. One may, therefore, suggest that in other experiments [31,32], in which low-transparency POS electrodes are used and the time delay between the flashboard discharge and the POS operation is longer the effect of electrode plasmas is more important. This may influence the plasma composition (more ions with lower charge states), the electron temperature (lower Te) and electron density (higher ne) in most of the A-K. These results demonstrate that the properties of the prefilled plasma used in pulsed- power experiments, besides being influenced by the plasma-source geometry and the discharge circuit, can be highly affected by material release from the electrodes. This effect probably depends on the flux of plasma that strikes the electrode surfaces, on the electrode-surface area and conditions, and on the time delay between the source discharge and the system operation. Generally, knowledge of the plasma properties, and in particular the plasma composition, is highly important for understanding the device operation. We believe that the diagnostic methods here described can be used to systematically study most of the important plasma properties in a variety of experimental conditions and geometries. ACKNOWLEDGMENTS The authors are grateful to Y. Krasik and A. Weingarten for highly illuminating discussions, to V. A. Bernstam for aiding with the CR modeling, to I. Bray for providing us with atomic data and to P. Meiri for his technical assistance. This work is supported by the Minerva foundation (Munich, Germany). 16 Tables: Table I: A list of the spectral lines used for the determination of ne, Te, and the composition of the flashboard plasma. Species Wavelength Upper level of Energy of the Plasma constituents H 6563 n=3 12.1 H 4861 n=4 12.7 CI 2479 2p3s 1Po 7.7 CII 2837 3p 2Po 16.3 CIII 2297 2p2 1D 18.1 CIII 4647 2s3p 3Po 32.3 CIV 5802 3p 2Po 39.7 CIV 2530 5g 2G 55.8 CV 2271 1s2p 3Po 304.4 OII 4349 2p2(3P) 3p 4Po 25.8 OIII 3047 2p3p 3P 37.3 OIV 3063 3p 2Po 48.3 SiIII 2542 3p2 1D 15.2 CuII 2545/2403 5s 3D 13.4 Doped elements HeI 5875 1s3d 3D 23.1 HeI 6678 1s3d 1D 23.1 HeII 4686 N=4 51.0 MgII 2795.5 3p 2Po 4.5 MgII 2798 3d 2D 8.9 ArI 6965 4p'[1/2]1 11.5 ArII 4348 3p4(3P) 4p 4Do 19.5 ArIII 3286 3p3(4So) 4p 5P 25.4 17 Table II: The plasma composition at z = 3.5 cm 1.2 µs after the start of the flashboard current. Plasma composition at Plasma composition at Species x=1.0 cm [cm-3] x=2.2 cm [cm-3] H (1.5±0.5)×1013 (8±3)×1013 CI (1.1±0.3)×1012 (2±0.7)×1013 CII (1.3±0.4)×1013 (6±2)×1013 CIII (2.2±0.8)×1013 (9±3)×1013 CIV (7±4.5)×1013 (1±0.4)×1014 CV (2±1)×1013 (2±1)×1013 OII (1±0.4)×1012 (1±0.4)×1013 OIII (4±2)×1012 (2±1)×1013 OIV (1±0.6)×1013 (3±2)×1013 Σnizi (not Protons) 3.9×1014 7.5×1014 Protons from ne-Σnizi (0-2.5)×1014 ≈0×1014 ne (4.5±1)×1014 (7±1)×1014 18 Figure Captions: Figure 1: Schematic of the POS, the flashboard plasma sources, the gas-doping arrangements, and the spectroscopic system. The inter-electrode region of the planar POS is prefilled with plasma from two flashboard plasma sources. A fast gas valve, a nozzle, and a skimmer are used to locally dope the plasma with various species. Lenses and mirrors are used to collect light from the doped column into the spectrometer. A cylindrical lens focuses the output of the spectrometer onto a fiber bundle array that passes the light to 12 photomultiplier tubes. Figure 2: The time dependent electron density of the flashboard plasma for z = 3.5 cm (the middle of the plasma in the z-direction), determined from Stark broadening of hydrogen lines. Two traces obtained in different discharges are shown for x = 1 (near the middle of the A-K gap) and x = 2 cm (near the anode). Figure 3: (a) The observed time-dependent ratio of the MgII 3p 2Po and 3d 2D level populations time-averaged using a 30 ns filter and averaged over three experiments. The measurement is at x = 1, y = 0, and z = 3.5 cm (the middle of the plasma in the z- direction). The oscillations visible prior to t < 500 ns are caused by the weak signal of the MgII 2798 Å line and the resulting poor photon statistics. Also shown is the level population ratio predicted by the collisional-radiative modeling. (b) The electron density used in the CR calculations to obtain the predicted ratio shown in (a). Also shown is ne obtained from Stark broadening at t ≥ 780 ns. The thick solid curve represents a lower limit for ne obtained from the absolute intensity of the ArI 6965-Å line. Figure 4: A two dimensional map of ne integrated over the y-direction 1.2 µs after the operation of the flashboard. The spatial resolution in these measurements is ±0.5 cm in the z-direction and ±0.2 cm in the x-direction, respectively. The errors are ±0.7×1014 cm- 3. Figure 5: Time-of-flight measurements of the proton-plasma obtained from the HeI 6678- Å line intensity at various distances from the flashboard surface in the middle of the plasma in the z-direction. The results are averaged along 8 cm in the y-direction. a(b) cm/s means a×10b cm/s. Note that larger x-positions are closer to the flashboards. Figure 6: (a) The observed temporal evolution of the HeI 1s3d 1D and the HeII n=4 level populations (symbols) at x = 1 and z = 3.5 cm. Two traces corresponding to different experiments are shown for HeI. Also shown (thin lines) are the best-fit level populations predicted by the collisional-radiative modeling using the electron temperature evolution shown in (b). Figure 7: (a) The observed temporal evolution of the ArI 4p'[1/2]1, the ArII 3p4(3P)4p 4Do, and the ArIII 3p3(4So) 4p 5P level populations (symbols). The results are averaged over 3 cm along the y-direction, at x = 1 and z = 3.5 cm. Also shown (thin lines) are the 19 level populations predicted by the CR modeling. (b) The electron temperature evolution used in the CR code to obtain the curves shown in (a). Figure 8: The temporal evolution of the electron temperature at x = 1, y = 0, z = 3.5 cm obtained using the various methods discussed in the text. Also shown is the evolution of Te with no POS electrodes. References [1] T. A. Mehlhorn, IEEE Trans. Plasma Science, 25, 1336 (1997). [2] C. W. Mendel, Jr. and S. A. Goldstein, J. Appl. Phys. 48, 1004 (1977). [3] C. W. Mendel, Jr., D. M. Zagar, G. S. Mills, S. Humphries, Jr., and S. A. Goldstein, Rev. Sci. Instr. 51, 1641 (1980). [4] D. G. Colombant and B. V. Weber, IEEE Trans. Plasma Sci. PS-15, 741 (1987); B.V. Weber, R,J. Commisso, G. Cooperstein, J. M. Grossmann, D. D. Hinshelwood, D. Mosher, J. M. Neri, P. F. Ottinger, and S. J. Stephanakis, IEEE Tran. Plasma Sci. PS- 15, 635 (1987); R. W. Stinnet, IEEE Tran. Plasma Sci. PS-15, 557 (1987). [5] T. J. Renk, J. Appl. Phys. 65, 2652 (1989). [6] M. Sarfaty, Y. Maron, Ya. E. Krasik, A. Weingarten, R. Arad, R. Shpitalnik, and A. Fruchtman, Phys. Plasmas 2, 2122, (1995). [7] P. S. Anan'in, V. B. Karpov, Ya. E. Krasik, I. V. Lisitsyn, A. V. Petrov, and V. G. Tolmacheva, Zh. Tekh. Fiz. 61, 84 (1991). [8] D. D. Hinshelwood, IEEE Trans. Plasma Sci., PS-11, 188 (1982); D. D. Hinshelwood, ‘Explosive emission cathode plasmas in intense relativistic electron beam diodes’, PhD. Thesis, NRL, 1985. [9] A. Weingarten, A. Fruchtman, and Y. Maron, IEEE Trans. on Plasma Sci. 27, 1596 (1999). [10] A. Ben-Amar Baranga, N. Qi, and D. A. Hammer, IEEE Trans. On Plasma Sci. 20, 562 (1992); L. K. Adler, ‘Spectroscopic measurements of plasma density and temperature in a plasma opening switch’, Ph.D. Thesis, Cornell University 1993. [11] T. J. Renk, J. Appl. Phys. 77, 2244 (1995). [12] S. Humphries, Jr., C. W. Mendel Jr., G. W. Juswa, and S. A. Goldstein, Rev. Sci. Instr. 50, 993 (1979). [13] B. V. Weber, D. D. Hinshelwood, and R. J. Commisso, IEEE Trans. Plasma Sci. 25, 189 (1997). [14] A. S. Chuvatin, Rev. Sci. Instr. 64, 2267 (1993). [15] S. Kohno, Y. Teramoto, I. V. Lisitsyn, S. Katsuki, H. Akiyama, IEEE Trans. Plasma Sci. 27, 778 (1999). [16] R. Arad, L. Ding, and Y. Maron, Rev. Sci. Instr. 69, 1529 (1998). [17] M. E. Foord, Y. Maron, and E. Sarid, J. Appl. Phys. 68, 5016 (1990). [18] G.R. Dalton et.al., Proceedings of the Int. Conf. on Atomic and Molecular Data and their Interpretation, Gaithersburg, MD, 1997 (US Government Printing Office, Washington DC, 1998 p. 12 ). [19] R. D. Cowan, The theory of atomic structure and spectra (University of California Press, Berkeley, 1981). 20 [20] V. P. Shevelko and L. A. Vainshtein, Atomic physics for hot plasmas (IOP Publishing, Bristol, 1993). [21] I. Bray, private communication, 1999. [22] K. L. Bell, H. B. Gilbody, J. G. Hughes, A. E. Kingston, and F. J. Smith, J. Phys. Chem. Ref. Data 12, 891 (1983); M. A. Lennon, K. L. Bell, H. B. Gilbody, J. G. Hughes, A. E. Kingston, M. J. Murray, F. J. Smith, J. Phys. Chem. Ref. Data 17, 1285 (1987). [23] C. Mendoza and C.J. Zeippen, Proceedings of the Int. Conf. on Atomic and Molecular Data and their Interpretation, Gaithersburg, MD, 1997 (US Government Printing Office, Washington DC, 1998 p. 53 ). [24] Y. Hahn, J. Quant. Spectr. Rad. Transfer 49, 83 (1993). [25] H. R. Griem, Spectral line broadening by plasmas (Academic Press, N.Y., 1974). [26] See, e.g., K. G. Whitney and P. E. Pulsifer, Phys. Rev. E 47, 1968 (1993). [27] H. C. Tseng and C. D. Lin, Phys. Rev. A 58, 1966 (1998). [28] R. Hoekstra, J. P. Beijers, A. R. Schlatmann, R. Morgenstern, and F. J. de Heer, Phys. Rev. A 41, 4800 (1990). [29] M. Gargaud and R. McCarroll, J. Phys. B 18, 463 (1985). [30] R. Arad, K. Tsigutkin, A. Fruchtman, and Y. Maron, Proc. of the 12th IEEE Pulsed- Power Conf., Monterey CA, 1999, (Institute of Electrical and Electronics Engineering Inc. Piscataway, NJ, 1999, p.925 ). [31] W. Rix, A. R. Miller, J. Thompson, E. Waisman, M. Wilkinson and I. Wilson, in Proceedings of the IXth International Conference on High-Power Particle Beams, Washington D.C., 1992, (National Technical Information Service, Springfield VA, 1992, p.402 ). [32] D. D. Hinshelwood, B. V. Weber, J. M. Grossmann, and R. J. Commisso, Phys. Rev. Lett. 68, 3567 (1992).
arXiv:physics/0008093 16 Aug 2000AUTOMATED OPERATION OF THE APS LINAC USING THE PROCEDURE EXECUTION MANAGER R. Soliday, S. Pasky, M. Borland, Argonne National Laboratory, Argonne, IL 60439, USA Abstract The Advanced Photon Source (APS) linear accelerator has two thermionic cathode rf guns and one photocat hode rf gun. The thermionic guns are used primarily for APS operations while the photocathode gun is used as a free- electron laser (FEL) driver. With each gun requiri ng a different lattice and timing configuration, the nee d to change quickly between guns puts great demands on t he accelerator operators. Using the Procedure Executi on Manager (PEM), a software environment for managing automated procedures, we have made start-up and switchover of the linac systems both easier and mor e reliable. The PEM is a graphical user interface wr itten in Tcl/Tk that permits the user to invoke ‘machine procedures’ and control their execution. It allows construction of procedures in a hierarchical, paral lel fashion, which makes for efficient execution and development. In this paper, we discuss the feature s and advantages of the PEM environment as well as the specifics of our procedures for the APS linac. 1 TCL/TK CODE 1.1 PEM The Procedure Execution Manager (PEM) is a graphical user interface tool that allows the user to execute Tcl/Tk machine procedures and monitor their progress (see Figure 1) [1]. At the Advanced Photo n Source (APS), PEM procedures are used routinely dur ing the operations of the different accelerators includ ing the linac. A key advantage of the PEM is that it can b e easily expanded by adding new machine procedures without changing the familiar user interface. The PEM allo ws the user to select an execution mode: Automatic, Semi- Automatic, or Manual. These levels signify the amo unt of interaction and monitoring that will occur. The ma chine procedures include ‘steps’ at which the PEM can pau se. Manual mode pauses at all steps. Semi-Automatic on ly pauses at the first step. Automatic runs without p ausing at any of the steps. As shown in the figure, a collec tion of routines can be grouped together and given a title. The machine procedures shown here are specifically desi gned for linac operations. Additional PEM screens exist for all of the accelerators at the APS. Since it was first written in 1996, the PEM has proven to be a very reliable a nd useful program. Figure 1: Procedure Execution Manager 1.2 Machine Procedures A machine procedure is a Tcl/Tk procedure that foll ows a particular format where certain utility functions must be called from within the procedure [2]. These proced ures are loaded into the PEM by using Tcl/Tk’s built-in auto loading feature. The PEM then accesses a configura tion file that lists the machine procedures that can be executed during a session. A simple example is shown here: proc APSMpWriteThis {args} { APSMpStep “Writing to file” set fd [open /tmp/stuff w] puts $fd “$args” close $fd APSMpReturn ok “data written” } This procedure writes the value of ‘args’ to a file . Note that the last statement is APSMpReturn. This must be used in place of the return statement in all machin e procedures. For each machine procedure there can also be a companion description procedure defined as shown below. The return value from the description proce dure is displayed in the ‘Description’ frame of the PEM too l when the corresponding procedure is selected. proc APSMpWriteThisInfo {} { return “Place description of procedure here.” } Machine procedures are designed to be executed primarily from within the PEM. The PEM takes full advantage of the format of a procedure to permit monitoring and controlled execution. These machine procedures may also be executed from any APS Tcl/Tk library, which allows them to be executed like any other Tcl/Tk procedure. 1.3 Installation In order to run the PEM, Tcl/Tk with the Tcl-DP extension must be installed. The latest versions a re recommended. The APS Tcl/Tk library and the PEM packages must also be installed. All of these are located on the OAG web site1. 2 LINAC APPLICATIONS The linear accelerator has three guns: two thermio nic rf guns mainly used to support APS injections and o ne photocathode rf gun for experimental projects. All of these guns are important to APS as well as to futur e user demands. Normally experimental projects are parasitically operated during user beam mode. If s tored beam is lost, it is important to the Operations Gro up to have a fast and reliable transfer from the experime ntal project to the APS injection configuration. This f ast and reliable method of switching to and from the thermi onic guns has been accomplished using the PEM tools. In the past, operators manually initiated and monit ored all systems involved in the switchover using Motif Editor and Display Manager (MEDM) screens. Since the lina c has a multitude of MEDM screens that control every aspect of operations, the switchover was not an eas y task. Operators normally had to switch back and forth bet ween many MEDM screens as they worked. As demands on the operators increased due to system changes, some small shell scripts where written to perform tasks automatically. Although the scripts worked well, t hey were not always reliable because changes to machine ry and operational procedures were being made without warning. Also most of these scripts were not regul ated and did not have much of an error checking ability. When configured properly, PEM procedures follow the same steps an operator would take during equipment start -up. The PEM tools not only repeat steps faster, they al so provide reproducibility. Using PEM tools for linac operations, the operator no longer has to open multiple windows and work on one task at a time. Instead, the PEM is able to effici ently use multitasking to alleviate the burden on the operato rs in what can often be a stressful situation. By making the interface of the PEM simple and consistent, new mac hine procedures can be added. Operators can read the corresponding description and view the steps of a procedure to become familiar with it. This is not intended 1 http://www.aps.anl.gov/asd/oag/oaghome.shtml to reduce operator training, however; it merely act s as an additional source of information that may be valuab le to operators. 3 PEM DEVELOPMENT There are several different types of machine proced ures ranging from simple low-level procedures to a colle ction of many submachine procedures. Often a machine procedure is a collection of smaller core procedure s linked together either in series or in parallel. In doing so, we were allowed to work in a simple progression. Core procedures are the lowest level procedures that do the actual work and are written first. Once these core procedures have been tested, larger parent procedur es are formed that call on all of the necessary machine procedures. This process is repeated many times, c reating many levels of parent procedures. Table 1 shows an example of the machine procedures called when the photocathode start-up procedure is used. There are more layers than this table shows, but they are too nume rous to display. Table 1: Machine procedures called from the photocathode start-up procedure 1. Start up photocathode gun with power supplies a. Start-up power supplies i. Turn off unused power supplies ii. Turn on needed power supplies iii. Condition power supplies iv. Wait for conditioning to finish b. Start up photocathode gun i. Shut down thermionic gun ii. Bring modulators to standby iii. Bring modulators back up Using a modular method, the PEM can decrease the execution time by taking two or more nonsequential procedures and running them in parallel. These par allel procedures can split off indefinitely into subparal lel procedures. All procedural steps report back to th e original PEM and communicate the steps as they occu r. This causes an interleaving of the steps displayed by the PEM as the execution progresses. Eventually, prior to exiting or executing additional steps, the parallel procedures must be joined. This joining ensures th at all steps have been completed successfully prior to continuing with the program. Another advantage of a modular method is the abilit y to make a change in a core procedure. If this machine procedure is called by multiple machine procedures, the change will affect all of them. This means that mu ltiple source codes do not need to be changed if a design or operation change is implemented that requires a cha nge in a core machine procedure. Figure 2: Dialog for photocathode gun start up with power supplies The dialog screen shown for the switchover to the photocathode gun (see Figure 2) allows the operator to select a snapshot file to be restored at the end of a switchover. A snapshot file is a database file tha t includes all of the lattice power supply settings n eeded to reproduce the same beam as when the snapshot was recorded. Once executed, the PEM procedure opens another display window that shows each step as it occurs. The particular procedure shown in Table 1 will start up and condition the power supplies and in pa rallel start up the photocathode gun. The subprocedures f or the power supply start up and photocathode gun star t up are run in series. Because this machine procedure is able to do many tasks at once as well as perform ma ny safety checks, it allows the operator to attend to many other tasks. When an unexpected condition occurs, the PEM displays a dialog box to the operator containing a description of the problem and requests that the operator attempt to fix it manually. Buttons for continuing and/or aborting are often displayed on t hese dialog boxes. An abort button is always displayed on the PEM screen during execution for those situation s when something may go wrong and continued operation of the PEM may be unnecessary or unwise. Along these same lines, a log daemon is used with t he PEM to log any and all error messages that may occu r during normal operations. This has been used to tr ack down some obscure problems that occur infrequently. 4 CONCLUSION Without the use of the PEM, the multiple operating modes of the linac at the APS would not be possible . The PEM has proven itself to be a useful tool capab le of handling a wide array of tasks. Switching betwe en the operating modes with the assistance of the PEM has been made almost trivial for the operator. Without the assistance of the PEM, the method for switching operating modes would require detailed knowledge of all the systems, execution of all the switchover st eps in the correct order, and a lot of time. 5 ACKNOWLEDGMENT PEM was implemented by C.W. Sanders, formally of APS, based on concepts developed by M. Borland and C.W. Sanders. This work is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38. 6 REFERENCES [1] M. Borland, “The Procedure Execution Manager and its Application to Advanced Photon Source Operation,” Proceedings of the 1997 Particle Accelerator Conference, May 12-16, 1997, Vancouver, Canada, pp. 2410-2412 (1998). [2] C.W. Saunders, “PEM-Procedure Execution Manager.” http://www.aps.anl.gov/asd/oag/ manuals/APSPEM/APSPEM4.html.
arXiv:physics/0008094v1 [physics.gen-ph] 16 Aug 2000the quantum vacuum, fractal geometry, and the quest for a new theory of gravity Edward F. Halerewicz, Jr.∗ Lincoln Land Community College† 5250 Shepherd Road P.O. Box 19256, Springﬁeld, IL 62794-9256 USA August 11, 2000 Abstract In this letter recent developments are shown in experimenta l and theoretical physics which brings into question the validity of General Relativity. This letter em phasizes the construction of a fractal 3+ φ3spacetime, in N-dimensions in order to formalize a physical and consist ent theory of ‘quantum gravity.’ It is then shown that a ‘quantum gravity’ eﬀect could arise by means of the Str ong Equivalence Principle. Which is made possible through a pressure of the form −κ(Rca b−1 2gcσ abRc) =κTcσ ab. Where it is seen that nuclear pressures can be added to the gravitational ﬁeld equations by means of twistor spac es. Keywords: Fractal Geometry, New Relativity, quantum vacuum, EPR, zer o-point ﬁeld, quantum gravity, Mach’s Principle, Holographic Principle, Fermat’s Last Theorem, alternative gravity, Bohmian Mechanics. PACS no. 4.50, 4.60 1 Introduction Recently the validity of General Relativity (GR) has been br ought to question by Yilmaz [1], et al. Although such interpretations allow for gravitation to be mathemati cally consistent and singularity free. Such revisions fail to describe the behavior of test particles as adequately as G R, elevating GR as the correct theory [2]. Today certain questions about GR remain relevant, such as how does it relate to vacuum energy and quantum mechanics in general. It has been shown in previous works that GR remain s self consistent when including the quantum vacuum, or zero-point ﬁeld [3]. However, the search for a sel f consistent theory of “quantum gravity,” remains a major theoretical challenge today. Among the theoretical a rguments against the standard interpretation of GR is the choice of mathematical coordinate systems [4]. Special Relativity (SR), is based upon the structure of a ﬂat Minkowski spacetime given in a four-dimensional coordinat e system. Recently attempts have been made describing coordinate systems with fractal spaces as opposed to natura l ones [5, 6, 7]. Such an adaptation as the case with the Yilmaz approach eliminates singularities within the ﬁe ld equations. Recent observational and experimental data have also put in to question the validity of GR. The National Aeronautics and Space Administration (NASA) has reported a n anomalous acceleration of ±8.5×10−8cm, on spacecraft on the outer edge of the solar system [8]. This dat a was obtained from information gathered by the Jet Propulsion Laboratories (JPL), and the Deep Space Netwo rk (DSN). Thus far, no satisfactory conclusion has ∗email: edward@springnet1.com †This address is given for mailing purposes only, and is not re sponsible for the production of work within. This work was ma de possible through my own personal research prior to becoming a ﬁrst year student at the above establishment Ibeen given to explain the so called “anomalous acceleration towards the sun.” Not only have spacecraft provided some fundamental ﬂaws with gravitation, but laboratory res ults as well. Dr. Eugene Podkletnov, has reported a “gravitational shielding” eﬀect with composite bulk YBa 2Cu3O7−xceramic plates [9]. In light of all of these developments it is hard to consider GR as the correct theory. It is the opinion of the author that GR is a theory that “works,” however it doesn’t necessarily make it the cor rect theory. The goal of this letter is to show that GR is not the correct the ory of gravitation, but just works exceptionally well. Just as previously Newton’s law of universal gravitat ion worked exceptionally well. This letter is not intended to be a replacement for GR, nor is it intended to present theor etical ﬂaws of the that theory. This letter is only presented as an introductory work for an alternative theory of gravitation. The general theme of this letter is given by the following postulates: Postulate 1 (virtual gravitation). Spacetime is not a null energy ﬁeld, it consist of asymptotic vacuum ﬂuctu- ations, and behaves as a virtual “energy-sheet.” Postulate 2 (planckian invariance). The Planck length is a gauge invariant function for all (inte racting) brane observers. [an adaptation to the postulate of New Relativit y.] This letter is presented in the following format in section I I a brief introduction into uniﬁed ﬁeld theories are given. In section II.1 a few quantum gravity approaches a re introduced. In section III fractal geometry is introduced and its relations to a complex system are given. I n section III.1 the meaning of fractal geometry for QED is discussed. In section III.2 the meaning of a fractal ge ometry is discussed for QCD. In section IV a new theoretical particle is introduced utilizing fractal geom etry. In section IV.1 a relationship between N-dimensional and two-dimensional systems are given. In section V a philos ophy of geometry is given. In section V.1 the eﬀects of the quantum vacuum are discussed. In section V.2 a relatio nship between fractal geometries and the quantum vacuum are discussed. In section V.3 the meaning of Feynman d iagrams are discussed. In section V.4 the validity of quantum mechanics is brought into question. In section V.5 a n alternative description of gravity is given which may explain the EPR paradox. In section VI an overview of a canoni cal non Riemannian gravitational ﬁeld is given. In section VI.1 the planck length results as a function in can onical quantum gravity. In section VII a possible alternative for the “anomalous acceleration” of spacecraf t is given. In section VII.2 pseudo geodesic equations are presented. In section VIII a general discussion of this work is presented. Finally in section VIII.1 a discussion of the meaning of the planck length is given. 2 Uniﬁcation a brief history “I am convinced that He [God] does not play dice.” –A. Einstei n “Einstein, quit telling God what to do.” –N. Bohr The uniﬁcation of gravitation with quantum mechanics began with Einstein’s objections to the newly developed quantum theory. Although acknowledging the successes of th e new theory he believed it to be incomplete. Einstein was convinced that there was a deeper theory involved, one wh ich would also include GR, a uniﬁed ﬁeld theory was christened. Soon came the work of Kaluza and Klien, giving a p seudo mathematical uniﬁcation of electromagnetism and gravitation. The theory would soon die out and loose inte rest, until quantum mechanics came around. And asked the simple question, how does gravity behave at the qua ntum level; the answer Kalzua-Klien gravity [10]. Research in this area soon exploded, extra dimensions were s oon added to the ﬁeld equations, superstring theory was born. Particle physics began unifying fundamental forc es as well, the weak force, the strong force, the elec- tromagnetic force. But no gravitational force, Cosmologis ts helped out, the big bang and nucelarsynthesis would help to explain the problem. Soon physics became littered wi th Grand Uniﬁed Theories (GUTs) and Theories of Everything (TOEs), they all have the approach of a “uniﬁed ﬁe ld theory.” However, they missed the simple point Einstein was trying to make, how do quantum mechanics and rel ativity relate? This is a hierarchical question, the relevant question is how do macroscopic and microscopic wor lds communicate? II2.1 quantum gravity Historically there have been two models formulated for the c onstruction of a consistent quantum gravity theory. They are the canonical and Hamiltonian approaches [11]. The se two approaches have had limited success, however more recently the theory of loop quantum gravity [12] has bee n introduced into the sea. Out of the three approaches presented loop quantum gravity is generally accepted as the correct approach. However, for a more accurate description of the historic developments of quantum gravit y see [13]. In this letter I will focus on the canonical approach as it relates to gauge invariance. 3 Fractal Geometry “.. . If we’re built from Spirals while living in a giant Spira l, then is it possible that everything we put our hands to is infused with the Spiral?” –Max Cohen in the motion pictu reπ The presence of matter within GR disturbs the ﬁeld equations by the existence of singularities or “point- particles.” How can one avoid this eye sore in the equations, simple fractal geometry. If matter is fractal it can not condense into points, however this allusion can still ta ke place above the planck energy scale. It is hard to believe that this simple approach has only been attempted in recent times, fractal sets are more common in nature than simple polygons. First let us begin with a simple constr uction of a fractal set with the simple equation Z(n)= (Z(n−1))2+C. One must also realize that a fractal is composed of a complex number system, i.e. a+ib. Using this form one may wish to construct a complex averaging of the mean, which results from the golden mean φ= (√5−1)//radicalbig (5−1). Thus we have constructed a complex mean which has two poss ible solutions as seen below: Cm=√a−1/radicalbig (a−1)+√ −b−1/radicalbig (b−1)=  √a+2√ (a+2);√−b√ (b)⇔a=c √a√ (a);√ −b+2√ (b+2)⇔a/negationslash=c(1) this complex mean thus has non communcating solutions. Whic h from the stand point of imaginary numbers yields the general statements:√−n/radicalbig (n)=i↔/radicalbig (n)√−n=−i (2) Thus these statements would appear to be in agreement of the t heory of quaternions. Which is interesting enough in itself, a four-dimensional version of the complex number system. With this preliminary work set we can now construct a Minkows ki spacetime which that takes advantage of fractal dimensions. First one can construct the generalize d three-dimensional manifold as a three-brane, and thus incorporating time as a fractal set. Thus the fractal constr uction on spacetime is presented in the form 3 + φ3 a similar approach was made in Ref. [4]. It is here postulated the origin for this three-dimensional brane arises from planck scaling. The reason for this postulate is seen wh en the golden mean is applied to N-dimensions φn= (√ N−1)//radicalbig (N−1), the higher n the closer the mean is to 1. Thus only when n app roaches inﬁnity will we see that 3+ φ3will yield the standard Minkowski space of 3+1 dimensions. T his is of course in agreement with any system that we apply c=¯ h=κ=K=1 more evidently this corresponds to a time dilation eﬀect i n terms of special relativity. Such that we have the following revision to the ﬂ at four-dimensional Minkowski space: (ω,z2) =ωc2−(φ)z2 1−(φ)z2 2−(φ)z2 3 (3) It is interesting to note that this pseudo metric appears to b e an inverse of the standard Minkowski spacetime, this importance is seen in section V.5. So far this method has only left intriguing consequences, however it diverges from the point made earlier in this section. In string theory the atomism view of matter is replaced with v ibrating strings, these vibrating strings correspond to a real geometry. However, fractal geometries are allowed to break of these strings into imaginary components. These imaginary components thus make the string a complex fu nction, yielding a pseudo point-like string. To IIIanalyze this premise allow us to view the Nambu-Goto action S=µ0/integraltext d2ξ√g. If strings can indeed be made to subside with fractal geometry, then they would break oﬀ into imaginary components by the empirical action: S=ℵ0/integraldisplay ω2φ√−g (4) whereg=λ2dz2⊗d¯z2≡x+iy. This pseudo string forms many more string components in N-d imensions, that is the string fragment continues in inﬁntium. However, in th e physical sense the string exist a pseudo point-like particle, do to the scaling nature of the planck length. Thes e fractal strings then interact within a ﬁeld, known as gravitational or zero-point ﬁelds. When the fractal string s converge with other fractals, a self organization takes place, i.e. the production of virtual particles. This produ ction is made possible through the non communacating mathematics associated with quantum mechanics. As the part icles are produced they destroy one another, such that their world-sheets reverberate in a complex form. This complex reverberations in N-dimensions is responsible for the vibration of the string, which we na¨ ıvely associate with mass. 3.1 QED fractality Electromagnetic waves are the result of four-dimensional i nteractions, and its real wave would correspond to the results found in Classical Mechanics. However, it would int ern have a fractal complex ﬁeld, which would cause the ﬁeld to break periodically yielding a string fragment, a qua nta. This quanta in non self regulating, i.e. it is the nature of the real wave which causes the string to reverberat e. The above consideration may carry some controversy in the well known theory of Quantum Electrodynamics (QED). I f a quanta is just a fractal string, then what is the proper approach for the exchange of energy between the tw o systems? Well, the result appears as classical approximation, the quanta hits the string as a solid body, ca using a change in geometry, which virtual particles oppose. This causes the string to “bounce” back to its origin al form, emitting a real wave, but not necessarily a quanta, remember a quanta is given by a complex ﬁeld. Since qu antum mechanics is sketched out onto a point particle-like environment, its consequences would agree w ith the QED model. In fact the fractal model yields a much physical picture for non communacating relationships than the quantum theory. 3.2 QCD fractality The above result would agree with QED, however, its deﬁnitio ns are quite weak, in fact one may expand these deﬁnitions to Quantum Chromodynamics (QCD). Thus a nucleon may be made to reﬂect electromagnetic radiation as well, however, this dose not defy the documented experime nts in any magnitude. It would be the interaction of the system, i.e. what string perimeters give way to the colored, and other gauged forces, that yields its “particles”. Each string fragment consist of its own local vibrations (gauge i nvariance), which attributes its mass, i.e. diﬀerentiates between a Higgs particle and a quark. These states then have t heir own local statistics, their real waves would then correspond diﬀerently than the electromagnetic ﬁeld. Whic h results in the production of the celebrated Yang-Mills ﬁeld, and thus yields the production of colored particles su ch as the gluon. 4 Cardinal Strings Cantor pioneered the study of inﬁnities with his new theory o f Cardinal numbers, however he faced opposition in his time for this new theory [14]. Cardinal numbers oﬀer th e best insight into to the study of fractal strings in N-dimensions, these interpretations in fact have direct physical consequences. Most notably it can explain the situation unleashed by the infamous EPR-Bell paradox, wher e Faster Than Light (FTL) communication appears possible (depending on planckian scaling). For recent theo retical implications and interpretation of the EPR-Bell paradox see [15]. In each scaling the laws of physics would be very diﬀerent, and hence superluminal velocities would seem to appear in lower branes. Thus the traditional in teraction of strings should not be limited on a speciﬁed dimension, but behave as a set of Cardinal numbers. Thus a slight revision of the Nambu-Goto action should be given which yields S=ℵ0/integraldisplay ω2φ/radicalbig −˜g (5) IVwhere ˜g=−dt′⊗dt′+ [dr′⊗dr′+1 4(sin(2r′))2Ω−2]. 4.1 Fermat’s Last Theorem “To divide a cube into two or other cubes, a fourth power, or, i n general, any power whatever into two powers of the same denomination above the second is impossible. .. ” –F ermat Fermat’s Last Theorem can be associated with fractal geomet ry in one respect, there is no general real solution to fractal geometry above dimension 2. This may be a simple coin cidence and may have no deeper meaning, however, this is contrary to the recently created Holographic Princi ple (HP) [16]. The HP relates that the Universe may exist in dimensions of inﬁnitum status. However, the laws of physi cs are best projected onto a pseudo two-dimensional screen, and our three-dimensional world is only a pseudo man ifestation of an N-dimensional continuum. In string theory we can view our universe as made up of two two-dimensio nal branes (described by type IIA D2 membranes). Thus any other dimension outside the holographic conjectur e yields no physical meaning and no solution. Just as what is suggested by Fermat’s Last Theorem, therefore our fo ur dimensional slice of the brane is a pseudo physical manifestation of the holographic screen. There is only one e xplanation for this result, there must exist a physical constant for speciﬁed energy scales, i.e. the planck scale. Here another coincidence appears to arrive, the two dimensional wave equation for string theory/parenleftBig ∂2 ∂σ2−∂2 ∂τ2/parenrightBig χµ(σ,τ) = 0. It seems that both mathematically and physically there is a special importance with dimension 2. T his discussion is largely philosophical, however it is interesting to note that cardinal strings are given by complex numbers. In fact a cardinal string in fou r-dimensions is remarkably similar to the two-dimensional form, and seem s to correspond to a torus: S2=ℵ0/integraldisplay d2ξ√−g+ℵ0/integraldisplay d2ξ√−g≡ ℵ0/integraldisplay d4(x1 λ2 IIAt2 IIAM) (6) From this complex structuring, and properties of cardinal n umbers it can be seen why quaternions were alluded to in section III. 5 a matter of geometry Einstein’s theory of GR transformed Newton’s theory of a gra vitational force, to a direct consequence of geometry. However, although the idea of a force was replaced with a geom etry, the geometry still yields a force when explained in Riemannian geometry. An even more radical approach to gra vitation as a geometry was produced by Roger Penrose in his theory of twistor spaces. The geometry itself is more important than physical masses, in fact masses only come important when one expands this theory. This is tru e for a fractal revision of string theory, it exist a pure geometry, the nature of the geometry in fact produces ma ss. This seems almost a radical stance from the point of view of GR, however geometry remains a key factor as t he ideal of a Force to GR. 5.1 the vacuum The vacuum exist from a state of virtual particles being prod uced via cardinal string fragments. Since virtual particles are a pure construction of “particles” in N-dimen sions, they are not true strings (i.e. they violate the HP). These particles thus carry no mass-energy equivalent in our universe, never the less they still posses a geometry. This situation only holds true when the system is localized, however, when interacting with non cardinal strings can induce an energy exchange. By the well known Casimir eﬀec t the energy of the vacuum should be given by [20]: ρZP(ω) =¯hω3 2π2c3 and when interacting with an inertial mass system we have: mi=V0 c2/integraldisplay η(ω)ρZP(ω)dω V. This relates the fact that as mass is accelerated it pushes t he quantum vacuum energy (which is analogous to the assumption made be postulate one). In other words it reacts i n the same fashion as air molecules do when inertial masses accelerate on earth (producing pressure on the syste m). Furthermore, it can be assumed that a material body increases its rest mass by absorbing this zero-point-e nergy (this assumption must be given in order to satisfy conservation laws). Moreover, since they are cardinal strings they are uniﬁed in a manner, thus the vacuum is a geometrical manifestation of string particles. That is the geometrical patterns formed through string interactions is what we call a gravitational ﬁeld, i.e. a virtual gravitational ﬁel d. Since these string interactions are only virtual there is no reason to modify the Einstein Field Equation, unless one w ishes to discuss quantum string eﬀects. Furthermore N-dimensional spacetime metrics have shown to be very simil ar to the structure of four-dimensional spacetimes [17]. Therefore the classical gravitational ﬁeld is remove d from quantum mechanics as it exist in a virtual sense, thus quantum mechanics is a property of matter. 5.2 the stage The vacuum however is not currently treated as the geometry o f a fractal spacetime system, and hence is incom- patible with other vacuum theories [18, 19]. However a fract al model for quantum mechanics appears to agree with at least one interpretation of the quantum vacuum [20]. In fact this interpretation goes right along with loop quantum gravity, and string theory see Ref. [21]. The Hausdo rﬀ (or fractal) dimension suggest that dimensions maybe conﬁned to a D=3 spacetime, with fractal string scalin g. This principle was postulated earlier in this letter as the consequence of the “planckian scaling,” and is the lea ding postulate in New Relativity. Here the importance now becomes what is the meaning of de-Broglie phenomenon. Th e Einstein de-Broglie equation ¯ hωC=m0c2can be seen as a representation of the relativistic wave equatio n, i.e. mass is a quantiﬁable measure of energy. Which can be applied directly to string theory, the vibrations of t he string are given in a fractal frequency comparable to the Compton wave length. 5.3 the Feynman stage From the above consideration it can be equally applied the th e origin of a bodies mass intern comes from the gravitational ﬁeld itself. This requires the use of Feynman diagrams, and believe it or not this approach is indeed correct, if strings are represented by cardinal numbers (an d if quantum mechanics is considered to be correct). Since this allows for FTL communication at the classical lev el it can be interpreted at least at the quantum level that mass originates from spacetime (when measured at the pl anckian scale). In fact at the quantum level the production of virtual particles may be responsible for a lig ht paths geodesic curvature, yielding a quantum gravity theory. More correctly it may be viewed that the quantum theo ry is in reality a classical approximation of string theory. Theorem (Brussels approximation). Quantum mechanics exist as a classical approximation of str ing interac- tions which possesses an apparent time reversed symmetry. 5.4 quantum mechanics? Deriving this theorem let us consider the following thought experiment: If mass is composed of vibrating stings, and intern these strings produce gravitational ﬁelds in N-d imensions then space is vibrating. However, such an approach would imply that geometrically speaking the two sy stems are unaware of their own vibrations under gauge invariance. On the other hand, the interaction of frac tal strings are given in a complex ﬁeld, which itself is anti-communcating. Thus spacetime, or string space is su bjected to Uncertainty Principles as shown in Ref. [22]. Since at the classical level, the fractal space can giv e way to real solutions I now make the assertion that the quantum particles are at ﬂux, and not the space itself (fo r argumentative purposes only). Therefore when a quantum is observed, it is the space which becomes “fuzzy,” n ot the particle, and when not observed the inverse follows. Thus the theorem leads to two possible out comes dur ing an observation sequence. i) the fuzzy quantum particle becomes a point particle, when time symmetries are reversed. ii) spacetime is fuzzy, however when collapsed by a point particle elucidates to a “natural” state. Thus it i s seen that only when time symmetries are reversed VIFigure 1: An over simpliﬁcation of the EPR statistics for a sp in1 2system given in a binary plane with time reversed symmetries. Where the ovals represent the probability of be ing observed by the apparatus. And where the positions “on” or “oﬀ” represent the outcome of events interpreted by c lassical mechanics. does one obtain the laws commonly associate with quantum mec hanics. Making the only valid approximation of quantum mechanics the Brussels Interpretation [23], this m ay also explain the EPR paradox. However our thought experiment does yield one solution which interpretation i a nd ii are consistent. When a particle is observed by a frame, it is in reality observ ed by the local states (brane) of the cardinal string, which may be in any number of states. As a point particle (non l ocal string) enters the system it begins to collapse the wave function of the local state. This makes it appear tha t a fuzzy quantum particle has entered the system, much as a star appears to twinkle in the night sky. This collap sing of the state I will call the observational’s frame “present,” before the action the observational state was fu zzy. However, after the event a self organization took place, an event occurred, producing a present state. After t he quanta is observed by the observational frame, its present state then becomes certain in the terminology of cla ssical quantum mechanics. Therefore local brane string interactions can not take place until a non local (cardinal) string collapses the wave function of the system. 5.5 Bohmian Mechanics, gravitation, and EPR From the Feynman interpretation of the time reversed symmet ries of the gravitational ﬁeld, new conclusions about the nature of spacetime can be made. It is here postulated tha t classical mechanics is in reality a description of a quantum system given under an approximation of a time revers ed symmetry. Such that the following statements become true: •Reversal of Bohmian Mechanics (BM) yields Classical Mechan ics •Reversal of Brussels Interpretation (BI) yields Standard Q uantum Mechanics (SQM) With the fractalization of spacetime given in section III, w e may conclude that (with the use of Bohmian Mechanics) that inertial mass yields an expansion of spacet ime. Thus as a body gains mass as it accelerates in classical spacetime it causes the fractalization of the Boh mian system to increase (which is analogous to a Lorentz transformation). Which thus gives the allusion that spacet ime is contracting in the classical real frame. The gravitational force, thus is an inertial acceleration whic h radiates a pseudo center of gravity vector in terms of Newtonian mechanics. However, this is how we interpret the e vents classically, in reality it is the expansion of the fractal Bohmian space (the φ3term in section III, e.g. the time dimension)which yields in ertial acceleration. Therefore, it is the Brussels interpretation of quantum mec hanics which yields the EPR paradoxes, the connec- tion of the particles is created by the (incorrect) approxim ation of time reversed symmetries. That is to say the EPR paradox only includes simple (non fractal) states, whic h by time reversal appears to yield FTL communication (see ﬁgure 1). This appearance of FTL communication shouldn’t be taken to s eriously since recent experiments (cfr. Wang, et al [33]) appear to yield FTL communication. However, it is th e string interactions which yield quantum mechanics, in fact it yields the same interpretations as Bohmian Mechan ics [24]. Thus Bohmian Mechanics adequately describes the behavior of “particles” while, the Brussels Interpreta tions yields standard quantum mechanics [meaning that this system is only an approximation]. VII6 quantum gravity? Since complex spaces have been presented as a solution to the singularity problem, it is natural for a formulation of a complex spacetime. To proceed in this manner one must neg lect the cherished Einstein-Hilbert action s=/integraltext d4x√−gRand replace it with the Tucker-Wang action: s=/integraldisplay λ2R⋆1 Therefore we can now discuss a complex gravitational ﬁeld, w ithout the traditional Riemannian geometry. The classical Einsteinian relativity gives the generic ﬁeld fo r a spacetime geometry as ∇µGµν= 0. Such that I now wish to make the generalized statement ∇ˆµGˆµˆν/negationslash= 0, or in canonical terms ∇aGab= 0. As such a generalization of a purely idealistic spacetime governed by perfect ﬂuid beco mes: Gab= 16πGT ab (7) where under ideal cases one can have the geometry Ra b+1 4gabR. The reason the ﬁeld takes on the term Ra b, as opposed to Rab, can be seen with the use Riemannian metrics. First, let us be gin with a Ricci symmetric tensor of the form:∂Λσ µ ∂xv=ηµνσΦvΛσ µ which within a constant ﬁeld becomes Rµν= Λσ µFσν. This ﬁeld can thus transpose to Fµν=∂µΦν−∂νΦµ, and couple to an opposing electromagnetic ﬁeld by the connectio n Γσ µν= Λσ µΦν Which therefore leads to the following antisymmetric Riema nnian ﬁeld Rµν/parenleftbigg∂Φv ∂xσ−∂Φσ ∂xν+Cαβ νσΦαΦβ/parenrightbigg Λσ µ=FσνΛσ µ Whence therefore means that there must be an equivocal ortho normal action taking place, such that one has Ra b=dΛa b+ Λa c∧Λc b. In which the generic geometry for a perfect ﬂuid becomes tha t of Gab= 16πGTσ ab (8) which translates to the ﬁeld equation: Ra b−1 4g(ˆe(a)),ˆe(b))R=−16πG c4Tσ ab (9) This equation must be modiﬁed when given in an N-dimensional system such that on has: Ra b−1 2(n)g(ˆe(a),ˆe(b))R=−8π(n)G c2(n)Tσ ab (10) The above equation is in essence a canonical gravitational ﬁ eld equation, which appears to be a good candidate for a quantum gravity theory. Where the geodesic equations beco me d dˆl/bracketleftbigg/parenleftbigg 1 +γab(n) 2(n)/parenrightbiggdxµ dˆl/bracketrightbigg −Γ/negationslashα [/negationslashβ/negationslashλ]dx/negationslashβ dˆldx/negationslashγ dˆl/parenleftbigg 1 +γab(n) 2(n)/parenrightbigg /negationslash= 0 (11) However, this interpretation suggest that spacetime is qua ntitized by a canonical action however, quantum particles are given as classical particles. Hence this interpretatio n would be an inverse of understood quantum mechanics. However, here a paradox opens up, when time is reversed parti cles remain in one quantum state, thus SQM is not retrieved. Thus, it is seen that there exist no true “quantum gravity” theory. In fact if one applies this formulation with the planck length it destroys the principle of planckian invariance and gives an allusion to the existence of an Æther. VIII6.1 canonical approach fails Therefore the planck length no longer remains a constant but becomes a dynamical function. First let us write the planck length in terms of N-dimensions and apply it to the abo ve ﬁeld equation such that we have: lpκ= (κ¯hn/mn pc3(n))1/2(n)·ψ (12) Momentum must be reevaluated from E=±(pncn+mn 0c2(n))1/n, such that mn p0=∓(pncn/c−2(n))1/n. Thereby the planck length, and mass are actually given by a particles rest momentum. Such that the planck length is in reality given by the function: lp=±(κ¯hn/γmn p0c3(n))1/2(n)·ψ (13) This thereby has major implications, that the planck length is not really a constant at all but a function of momentum. Such that as an object increase in speed with respe ct to its rest momentum, its planck energy becomes larger. That is as a material body is subjected to length cont raction, its planckian energy is modiﬁed to compensate for the eﬀect. Since the momentum is measured at rest m remain s a constant, it is the velocity of the system which changes. Thereby meaning that length contraction in specia l relativity is not given by Lorentz transformations, but by the local rest momentum of the planck barrier. This result s when we interpreted this action canonically however under BM it yields expected results. Therefore it is seen tha t a canonical formulation fails to keep “planckian invariance” which represents a failed attempt at a quantum g ravity theory. 7 anomalous acclerations? Finally I bring light to an alternative explanation for the a cceleration of spacecraft [8]. Since the ﬁndings of the “anomalous acceleration towards the sun,” there have been a number of possible explanations given [27, 28, 29]. With the construction of a fractal N-dimensional spacetime , I view this as a quantum gravity eﬀect. As an object accelerates its fractal geometry changes [by means of BM], t hus resulting in pressure on the system. Pressures as the source of a gravitational ﬁeld were pioneered long ago by Einstein [30]: Rµν=−κ(Tµν−1 2gµνT) (14) +2 a2γµν=κ(σ 2−p) (15) 0 =−κ(σ 2+p) This method is not ad hoc, gravitational pressures for atomi c gases and radiation can be given by [37]: pgas=κ µHqT and p rad=1 3aT4(16) which lends the general results T=/parenleftbiggκ µH3 q1−β β/parenrightbigg1/3 q1/3(17) and p =/bracketleftBigg/parenleftbiggκ µH/parenrightbigg43 a1−β β4/bracketrightBigg1/3 q4/3=c(β)q4/3(18) Thus after a slight modiﬁcation of eq.(10), one can obtain th e following gravitational pressure: Rca b=−κ(Tca b−1 2gcσ abT) +2 a2γc ab=κ(σ 2−p) (19) Therefore the ﬂat ﬁeld equations can be given by: Γc ab(z)/negationslash= 0, Rca b(z)/negationslash= 0 (20) IXthus a line elements trajectory would be given by (ω,z2) =gcaωzcωza (21) therefore a metric in a complex fractal spacetime can be give n by: c=i/summationdisplay ab=1,2δωzc aωzc b≡0 (22) Since this quantum gravity eﬀect originates from the planck length it is very unlikely that the Yukawa interaction [31]: V(r) =−/integraldisplay dr1/integraldisplay dr2Gρ1(r1Xr2) r12[r+αexp(−r12/λ)] will take place (unless special conditions arise). However , if such an eﬀect does arise, it may yield peculiar motion for an obejects geodesic path. 7.1 SQM pseudo geodesic paths First let us begin with the two-dimensional Lagrangian Hami ltonian, so that we have an equation of motion from the simple action dqi dt=∂H ∂pi;dpi dt=∂H qi(23) In canonical terms motion is given by qi−∂H ∂pi;pi=∂H qi(24) lending a four-vector of the form p= ˙mx+e ca(x). In such that a Hamiltonian wave within a gravitational ﬁel d would be in motion according to the geodesic path: ∂H ∂qi−Γ/negationslashα [/negationslashβ/negationslashγ]dx/negationslashα dsdx/negationslashγ ds/negationslash= 0 (25) This geodesic unlike the prior for a classical particle, wil l not diﬀerentiate and thus its motion need not transverse through classical Euclidean space. Therefore it can be seen that complex spaces could impose unseen forces which would eﬀect a geodesic path for a body (or wave) in motion. A pr oposal made in Ref. [35], made a like was case in the relativistic sense so that one would have Fµ= ˆm0W′ Wdy dˆλdxµ dˆλ. Although there is no direct physical evidence of this, it is s till however an intriguing explanation. After all the so called “anomalous acceleration” is only experienced by s mall bodies, not massive ones such as planets. Thus a quantum interpretation of this eﬀect seems to ﬁt the observe d data better than any other approach. Alternatively Modanese has also predicted a macroscopic quantum gravity e ﬀect [32], however it is limited to the Podkletnov experiment [9]. 8 Discussion of theory The formulation of this theory was based on a desire for a refo rmulation of GR in order to describe a singularity free theory; in which a fractal formulation of the ﬁeld equations were derived. The second desire for this theory was the formulation of a quantum construction of GR, however the end result is a gravity theory which describes quantum mechanics. Therefore the gravitational ﬁeld and matter can be considered to be molded into the following form. Matter exist as a pseudo point particle whos ﬁeld of movement is restricted onto a two-dimensional (complex) frame. This two-dimensional frame’s movement is governed by BM, an d in part by the HP. Matter, is thus in reality a fractal vibrating string fragment which continues on into N -dimensions. The fractalization of this “cardinal string” produces virtual particles which posses a geometry, it is th is (virtual) fractal geometry that is responsible for the gravitational ﬁeld. XIn light of future studies it is likely that an adequate formu lation for an alternative to GR be given in the following forms. One the acceptance of a fractal (even if onl y quasi fractal) structure of matter and space as an adequate formulation for the geometry of spacetime. Two the acceptance of complex systems into the equations, e.g. quaternions, octonions, C* algebras, etc. And ﬁnally t hree, the acceptance of physical conditions which may not be “popular,” but yield results that are not contradicto ry to known data. The mathematical conditions are the most intriguing to author because there seems to be a hidd en mechanism in the mathematics. However, my advanced mathematics skills are mediocre at best so these av enues are still left open in this letter. 8.1 what layith beyond the planck length? Several physical arguments against the existence of singul arities have been given by Loinger [25, 26], as well as Einstein’s classic objections. Thus one may inquire what ha ppens at the planck length, i.e. what are the laws of physics? Here I now quote Kip Thorne, on our current understa nding of singularites and ‘quantum foam.’ “How probable is that a black hole’s singularity will give bi rth to ‘new universes?’ We don’t know. It might well never happen, or it might be quite common—or we m ight be on completly the wrong track in believing that singularites are made of quantum foa m. –Thorne (1994)” This now leads a discussion to recent attempts to model gravi ty in terms of N-dimesnional spaces (Arkani- Hamed, et al 62-69), in which the planck length varies with th e number of dimensions. Of course the planck length could be inﬁnitely small in an inﬁnite system, clearly a chal lenge to the principle of planck invariance. However, we note that with Mach’s Principle (MP) the planck length mus t be observed by an external mechanism to remain invariant. Thus the planck length exist as a fractalization of BM, which becomes an observational frame in classical real mechanics. Furthermore, from this it may be seen that th e laws of physics as we understand them are in direct consequence of the planck length. We may also assume the chos en string ﬁeld is quantitized (i.e. given by BM), because its mass is attributed to a complex pseudo oscillati on (vibration). Where I now quote David Bohm (cfr. Bohm, 22): “We may conclude that all systems which oscillate are quanti tized withE=n¯hνwhether these systems be mathematical oscillators, sound waves, or electromagne tic waves.” So what does physics look like beyond the planck length, reme mber the (local) laws of physics are given by two complex D2-branes. When we interpret these interactions we receive the traditional GR eﬀects at the macroscopic level. However, at the planck scale singularities don’t exi st such that the frame interacts via “cardinal strings” and not classical GR. Thus interactions on local branes ceas e, and supersymmetry takes over. However, only strings which are connected to a form of the D2-brane will hav e observable physical manifestations, this deals with “planckian invariance”. In fact each dimension may have its own unique planck length which governs its own local laws (explaining the limitation of classical string theory to a set number of dimensions). Which leaves open several areas in N-dimensional black hole mechanics, and planck len gth physics. 9 Conclusion I have shown that there is enough evidence at present to chall enge GR as the correct theory for gravitation. I have also introduced the study of a complex fractal spacetim e system and its possible relationship to the planck length. The given formulation for a canonical gravitationa l ﬁeld resulted in contradictory conclusions, thus ruling out a canonical approach to “quantum gravity.” Finally if my hypothesizes hold valid then SQM will begin to make invalid predictions for the behavior of particles near the p lanck length. Thus a fractal correction for SQM will be needed under certain gravitational ﬁelds, which may be comp arable to BM. A Appendix A: the equivalence principle A new Weak Equivalence Principle (WEP) for the gravitationa l ﬁeld can be postulated utilizing a Complex Fractal Minkowski Spacetime (CFMS) system (see eq.(3)). Since ther e is no spatial acceleration for the gravitational ﬁeld XI(in respect to MP), it is the acceleration of the pseudo time d imension in the CFMS which produces a gravitational curvature. Therefore a material body would have the traditi onal Minkowski spacetime, acting as a Lorentz frame. Since gravitational ﬁelds extend indeﬁnitely, this should cause time to continually progress within an inertial acceleration frame. This therefore means that as an object e nters a gravitational ﬁeld it becomes less massive, in terms of a Lorentz transformation. Equivocally it can be sta ted that energy is lost in curved spacetime. A similar eﬀect is all ready known, known as a “gravitational time dela y,” i.e the Shapiro Eﬀect . “.. . according to the general theory, the speed of a light wav e depends on the strength of the gravitational potential along its path.” –Shapiro (1964) This is however contradictory to SR, because it fails to desc ribe inertial acceleration within a gravitational ﬁeld correctly. However, the principle of relativity is still preserved, because the curvature of spacetime corr ects for the CFMS. Therefore the reason the equivalence principle is fundamental in GR is because it is the only priori condition which satisﬁes the principle of relativity . Thus without an equivalence principle, there would be no relativistic theory for the gravitational ﬁeld. The logarithm gravitational time delay, may also be respons ible for the apparent “anomalous acceleration” of spacecraft. David Crawford has oﬀered a similar explanatio n, where the gravitational term arises from interplane- tary dust [27]. A Appendix B: Yang-Mills gravity Here I now hit upon a topic hinted upon in section III.2; conve rting fractal geometry in the terminology of QCD. Let us now rewrite eq.(19), so that we have an equation of the f orm: −κ(Rca b−1 2gcσ abRc) =8π√−˜gTcσ ab (26) with this equation a Yang-Mills gravitational pressure can arise under the following ﬁeld: SE=1 4g2/integraldisplay ω4zFσ µνFσ µν+1 αo/integraldisplay Kia bKib a/radicalbig −˜gω2φ (27) which must be given in a conformal ﬁeld, i.e. gab=pδab, thus we have: S=1 2α0/bracketleftbigg/integraldisplay ωc2φp−1(φ)(∂2z)2+λab(∂az∂bz−pδab)/bracketrightbigg +ℵ0/integraldisplay pω2φ (28) From this it is now seen that a “cardinal string,” is in fact an N-dimensional world line. Which can communacate with other world lines, where we have a self-organization of the system by v(˜s) =1 4π/integraldisplay ω2φ/radicalbig ˜ggabǫjklm∂atab∂jklm (29) It is these interactions which generate a spinor space, whic h attributes mass to the geometry. Therefore the Yang- Mills ﬁeld is added to the gravitational ﬁeld, by means of a gr avitational pressure. Here a less restrictive form of the Strong Equivalence Principle (SEP) can be applied: (spinorpressure )·(energydensity ) = (strengthof gravitationalfield )·(gravitationalpressure ) Thus the interaction of two or more “cardinal strings,” prod uces a twistor like action, represented by Z∞atP∈ M. This also means that certain gravitational anomalies may no t only arise at the planck length, and may result in experimental veriﬁcation. XIIReferences [1] Yilmaz H. Did the apple fall? In M. Barone and F. Selleri, e ditors, Frontiers of Fundamental Physics. 115-124 (1994) [2] Fackerell E. Remarks on the Yilmaz and Alley papers. Proc eedings of the ﬁrst australaian conference on general relativity and gravitation, ed. D. L. Wiltshire, Un iversity of Adelaide (1996), 117 available URL: http://www.physics.adelaide.edu.au/ASGRC/ACGRG1/fac kerell.html [3] Castro C. On M Theory, Quantum Paradoxes and the New Relat ivity physics/0002019 [4] Castro C. Why We Live In 3 Dimensions hep-th/0004152 [5] Ya.Kobelev L. Do Gravitational and Electromagnetic Fie lds Have Rest Masses in the Fractal Universe? physics/0006043 [6] Ya.Kobelev L. The Theory of Gravitation in the Space-Tim e with Fractal Dimensions and Modiﬁed Lorents Transformations physics/0006029 [7] Ansoldi S. Loop Quantum Mechanics and the Fractal Struct ure of Quantum Spacetime, Chaos Solitons Fractals 10 (1999) 197 hep-th/9803229 [8] Anderson J, et al. Indication from Pioneer 10/11, Galileo, and Ulysses Data, o f an Apparent Anomalous, Weak, Long-Range Acceleration, Phys.Rev.Lett. 81 (1998) 2 858-2861 gr-qc/9808081 [9] Podkletnov E. Weak gravitation shielding properites of composite bulk YBa 2Cu3O7−xsuperconductor below 70 K under e.m. ﬁeld cond-mat/9701074 [10] Overduin J. and Wesson P. Kalzua-Klien Gravity. Phys.R ept. 283 (1997) 303-380 gr-qc/9805018 [11] Wallace D. The Quantization of Gravity –an introductio n gr-qc/0004005 [12] Rovelli C. Loop Quantum Gravity. Living Reviews in Rela tivity Vol 1 (1998) Max-Planck-Gesellschaft. /1998- 1rovelli available URL http://www.livingreviews.org/Ar ticles/Volume1/1998-1rovelli [13] Rovelli C. Notes for a brief history of quantum gravity g r-qc/0006061 [14] Motz L. and Weaver J. The Story of Mathematics, 1993 (New York: Avon Books) [15] Szab´ o L. Removing the last obstacle to the Einstein-Fi ne resolution of the EPR-Bell problem quant-ph/0002030 [16] ‘t Hooft G. Dimensional Reduction in Quantum Gravity gr -qc/9310026 [17] Casadio R. and Harms B. Black Hole Evaporation and Large Extra Dimensions hep-th/0004004 [18] Krasnoholovets V. and Ivanosky D. Motion of a particle a nd the vacuum, Physics Essays, vol. 6, no. 4, pp. 554-563 (1993) quant-ph/9910023 [19] Jaekel M. and Reynand S. Movement and ﬂutuations of the v acuum, Rept.Prog.Phys. 60 (1997) 863-887 quant-ph/9706035 [20] Haisch B. On the relation between zero-point-ﬁeld-ind uced inertial eﬀect and the Einstein-de Broglie formula, Phys.Lett. A268 (2000) 224-227 gr-qc/9906084 [21] Ansoldi S. Aurilia A. and Spallucci E. Loop Quantum Mech anics and the Fractal Structure of Quantum Spacetime, Chaos Solitons Fractals 10 (1999) 197 hep-th/98 03229 [22] Sasakura N. An Uncertainty Relation of Space-Time, Pro g.Theor.Phys. 102 (1999) 169-179 hep-th/9903146 [23] Bost¨ om K. Regaining time symmetry in the generalized q uantum mechaincs of the Brussels School quant- ph/0005024 XIII[24] Berndl K, et al. A Survey of Bohmian Mechanics, Nuovo Cim . B110 (1995) 737-750 quant-ph/9504010 [25] Loinger A. On the concept of mass point in general relati vity gr-qc/0006033 [26] Loninger A. On the propagation speed of wavy metric tens ors gr-qc/0007048 [27] Crawford D. A possible explanation for the anomalous ac celeration of Pioneer 10 astro-ph/9904150 [28] Tureyshev S, et al. The Apparent Anomalous Long-Rang Ac celeration of Pioneer 10 and 11 gr-qc/9903024 [29] Østvang D. Assumption of Static Gravitational Field Re sulting in an Apparently Anomalous Force gr- qc/9910054 [30] Einstein A. The Meaning of Relativity, 1921 (New York: M JF Books) [31] Long J. Experimental Status of Gravitational-Strengt h Forces in the Sub-Centimeter, Regime Nucl.Phys. B539 (1999) 23-34 hep-ph/9805217 [32] Modanese G. Theoretical Analysis of a Reported Weak Gra vitational Shielding Eﬀect. hep-th/9505094 [33] Wang L. Kuxmich A. and Dogarlu A. ”Gain-assisted superl uminal light propagation.” Nature 406 (2000) 277 [34] Arkani-Hamed N. Dimopoulos S. and Dvali G. ”The Univers e’s Unseen Dimensions.” Scientiﬁc American 283 (2000) 62-69 [35] Youm D. Extra Force in Brane Worlds hep-th/0004144 [36] Bohm D. Quantum Theory, 1951 (Toronto: Dover Books) [37] Tuaber G. Albert Einstein’s theory of General Relativi ty: 60 years of its inﬂuence on Man and the Universe, 1979 (New York: Crown Publishers, Inc.) [38] Thorne K. Black Holes & Time Warps: Einstein’s Outrageo us Legacy, 1994 (New York: Norton) pg. 478 ——————————————————————————– XIV
BEAM POSITION-PHASE MONITORS FOR SNS LINAC S.S. Kurennoy, LANL, MS H824, Los Alamos, NM 87545, USA Abstract Electromagnetic modeling with MAFIA of the combined beam position-phase monitors (BPPMs) for the Spallation Neutron Source (SNS) linac has been performed. Time-domain 3-D simulations are used to compute the signal amplitudes and phases on the BPPM electrodes for a given processing frequency, 402.5 MHz or 805 MHz, as functions of the beam transverse position. Working with a summed signal from all the BPPM electrodes provides a good way to measure accurately the beam phase. While for an off-axis beam the signal phases on the individual electrodes can differ from those for a centered beam by a few degrees, the phase of the summed signal is found to be independent of the beam transverse position inside the device. Based on the analysis results, an optimal BPPM design with 4 one-end-shorted 60-degree electrodes has been chosen. It provides a good linearity and sufficient signal power for both position and phase measurements, while satisfying the linac geometrical constrains and mechanical requirements. 1 INTRODUCTION Beam position-phase monitors in the SNS linac will deliver information about both the transverse position of the beam and the beam phase. Typical values for the beam position accuracy are on the order of 0.1 mm within 1/3 of the bore radius r b from the axis ( rb is 12.5 mm to 20 mm for the normal conducting part of the linac). The BPPMs have a high signal processing frequency, equal to the microbunch repetition frequency in the linac, f b=402.5 MHz (or its 2nd harmonics, 805 MHz). The beam phase measurement within a fraction of an RF degree is also required from the SNS linac BPPMs. Various options for the transducers (pickups) of the SNS linac BPPMs have been studied using the EM code MAFIA [3] in [1,2]. Electrostatic 2-D computations are used to adjust the pickup cross-section parameters to form 50- Ω transmission lines. 3-D static and time-domain computations were applied to calculate the electrode coupling. Time-domain 3-D simulations with SNS beam microbunches passing through the BPPM at a varying offset from the axis were used to compute the induced voltages on the electrodes as functions of time. After that an FFT procedure extracted the amplitudes and phases of the signal harmonics at individual outputs, as well as the amplitude and phase of the combined (summed) signal, versus the beam transverse position. This information was used to choose an optimal BPPM design. Section 2 summarizes the results of this study. In the SNS linac, there is a rare opportunity to put BPPMs and steering magnets inside the drift tubes in the drift-tube linac (DTL) to provide a better quality beam. This is due to the fact that every third drift tube (DT) is empty. The DTL RF fields, however, will produce an additional signal in BPPMs inside DTs at the DTL RF frequency 402.5 MHz that can exclude the BPPM signal processing at this frequency. For the coupled-cavity linac (CCL) there is no such problem, since its RF frequency is 805 MHz. In Sect. 3 we study the feasibility of using BPPMs in the DTL. 2 BPPM MODELING 2.1 BPPM Design After considering a few possible pickup designs, we decided to choose a BPM design having 4 stripline electrodes with one end shorted. A MAFIA model of the BPM consists of a cylindrical enclosure (box) with 4 electrodes on a beam pipe, see Fig. 1. Each electrode covers a subtended angle of 60 °. They are flush with the beam pipe, shorted at one end, and have 50- Ω connectors on the other end. For the CCL beam pipe radius rb=20 mm, the electrode length along the beam is taken to be 40 mm. The 50- Ω electrode connectors are modeled by discrete elements, 50- Ω resistors in this case. This design is non-directional, provides a rigid mechanical structure, has all four connectors on one end, and therefore can be mounted close to quadrupoles or fit inside a DT. Figure 1: MAFIA model of BPPM (1/2-cutout) with cone- tapered box end and electrodes (dark) with ridged transitions to connectors (shown as red pins). 2.2 Position Measurements Direct 3D time-domain computations with an ultra relativistic ( β=1) bunch passing the structure at the axis or parallel to the axis have been performed. A Gaussian longitudinal charge distribution of the bunch with the total charge Q=0.14 nC and the rms length σ=5 mm, corresponding to the 56-mA current in the baseline SNS regime with 2-MW beam power at 60 Hz, was used in the simulations. Presently, the MAFIA time-domain code T3 cannot simulate the open (or waveguide) boundary conditions on the beam pipe ends for non-ultra relativistic ( β<1) beams. The ultra relativistic MAFIA results are used to fix parameters of an analytical model of the BPPM at β=1, and then to extrapolate results for β<1 analytically. To study the BPM linearity, we perform simulations with the beam bunch passing through the BPM with different transverse offsets. The amplitudes Ã P and the phases of the Fourier transforms of the induced voltages on all four ( P=R,T,L,B for the right, top, left and bottom) electrodes are calculated as the functions of the beam transverse position. The BPM position sensitivity was found to be equal to 20log 10(ÃR/ÃL)/x ≅1.4 dB/mm. At high beam energies the signal power at 402.5 MHz changes between +4.6 dBm and –12.3 dBm for the beam position within a rather wide range, { x,y ∈(-rb/2,rb/2)}, i.e. the signal dynamical range is 16.9 dB. The BPM linearity results are presented in Fig. 2. MAFIA data showing the horizontal signal log ratio ln( Ã R/ÃL)/2 or the difference- over-sum ( ÃR-ÃL)/(ÃR+ÃL) for different vertical beam positions overlap, so that it is difficult to distinguish between the five interpolating lines in each group. We can conclude that this BPM design is insensitive to the beam position in the direction orthogonal to the measured one, and has a good linearity. 0 0.1 0.2 0.3 0.4 0.500.20.40.60.8 x/rS←(R−L)/(R+L)Ln(R/L)/2 →y/r=0 y/r=1/8y/r=1/4y/r=3/8y/r=1/2 Figure 2: Signal ratio S at 402.5 MHz versus the beam horizontal displacement x/rb, for a few values of the beam vertical displacement y/rb. 2.3 Analytical Model of BPM Assuming an axial symmetry of the beam pipe, the signals on the BPM electrodes of inner radius rb and angle ϕ can be calculated by integrating induced currents within the electrode angular extent. For a pencil beam bunch passing the BPM at the transverse position x=rcosθ, y=rsinθ at velocity v=βc, the signals are (e.g., [4]): ()( ) ()()0 01() () 4 22 () ()(, ,) s i n c o sm m bbmIg r Ig r Ig r I g rEfr C m mϕφ ϕ πθµ θ ν∞ ==+ + −∑ (1) where E=R,T,L,B are the Fourier amplitudes at frequency f of the voltages on the electrodes, ( µ,ν) are (0,0) for R, (0,π/2) for T, (π,0) for L, (π,π/2) for B, and Im(z) are the modified Bessel functions. All dependence on frequency and energy is through g=2 πf/(βγc), and overall coefficient C depends on the beam current. The parameters rb and ϕ can be considered as “free” parameters of the model. To find their effective values, we fit with Eqs. (1) the MAFIA results for β=1 at 402.5 MHz for the ratio S/(x/rb), where S is either ln( ÃR/ÃL)/2 or (ÃR-ÃL)/(ÃR+ÃL). The best fit to the numerical data was obtained [1] with the effective parameters reff=1.17 rb, ϕeff=1.24 ϕ (=74.5 °), where rb=20 mm, ϕ=60° are the geometrical values. Matching the amplitude of 402.5-MHz harmonics from an on-axis relativistic SNS beam bunch with Eqs. (1) fixes the constant C=1.232 V. Then the model reproduces MAFIA-computed 402.5-MHz signal amplitudes for the displaced beams with accuracy 1-2%. Assuming the effective parameters of the model applicable also at lower beam velocities, we extrapolate β=1 results to β<1. The signal power level for the on-axis beam is reduced by about 9 dB at β=0.073 (2.5 MeV). For the strongest signal in the beam displacement range (- rb/2, rb/2) both vertically and horizontally, this reduction is 4.4 dB, and for the weakest one it is 12.9 dB. As a result, the dynamical range of the 402.5-MHz signal would increase from about 17 dB for β=1 to about 25 dB at β=0.073, if the same radius of BPM were assumed. 2.4 Phase Measurements Two candidates for the beam phase detectors for the SNS linac – the capacitive probes and BPMs, either with signals from individual electrodes or with summed signals – have been studied and compared in Ref. [2]. MAFIA simulations with an ultra-relativistic beam, as well as measurements (for the capacitive probes), have shown a strong dependence of the measured beam phase on the transverse beam position inside a probe, when signals are picked up from individual connectors. For an off-axis beam, the signal phases from individual electrodes can differ from those for a centered beam by a few degrees, while the phase of a summed signal remains the same within the computation errors (0.1-0.2 °), even for the beam offsets as large as the pipe half-aperture. It is illustrated by Fig. 3 (the error bars are shown only for the summed signal). In the capacitive probe, the phase deviations from the centered beam phase grow as the beam offset increases, approaching 1 degree difference for large (half-aperture) offsets at the frequency 402.5 MHz. Based on the results of the analysis [2], we have chosen the BPMs with summed signals from all electrodes as the beam phase detectors in the SNS linac. 0 0.1 0.2 0.3 0.4 0.5−3−2−101 y/r∆Φ, degR T L B Σ Figure 3: 402.5-MHz signal phases on BPM electrodes and for summed signal versus beam vertical displacement y/rb, for the beam horizontal offset x/rb=1/4. 3 BPPM IN DTL We consider the tightest spot, the third DT in the 2nd DTL tank. The DT length along the beam is about 8 cm and its beam-pipe inner radius is 12.5 mm. The pickup design with four 60 ° electrodes is similar to that in the CCL, we only reduce the transverse dimensions and take the electrode length to be 32 mm. The beam-induced signals at the pickup electrodes are computed using MAFIA simulations with an ultra relativistic beam. The Fourier harmonics amplitudes for the on-axis beam are Ã 1=0.190 V at 402.5 MHz and Ã2=0.356 V at 805 MHz. We extrapolate these β=1 results to the H--beam energy of 7.5 MeV ( β1=0.126) analytically as described in Sect. 2.3. For the first harmonics, the ratio S(β1)/S(1)=0.80 results in the beam-induced signal amplitude 0.152 V at 7.5 MeV. For 805 MHz, the ratio S( β1)/S(1)=0.455 gives the signal amplitude 0.162 V. We want to compare these numbers with the signal amplitudes induced by the RF field in the DTL BPPM. To calculate the signal power on the BPPM electrodes induced by the 402.5-MHz RF field in the DTL tank, we put the DT with the BPPM inside in a cylindrical pillbox having the length of 96 mm (twice that of DTL half-cell), and adjust the pill-pox radius to tune the frequency of its lowest axisymmetric mode to 402.5 MHz. Integrating the electric field of the computed eigenmode along the electrode connector gives V con, and along the beam axis Vax. We calculate the scaling factor as the ratio of the on- axis voltage given by SUPERFISH design computations, V ax-SF = 2.96 ·105 V, to Vax. Multiplying Vcon by this scaling factor gives the RF-induced voltage amplitude Vind. The results are listed in Table 1 for a few different pickup positions inside the DT. Here z c is the longitudinal coordinate of the BPPM electrode center relative to the DT middle point, and z g is the same for the BPM annular gap center. Since the electrode length is 32 mm and the gap is 4 mm wide, we have z g - zc = 32/2 + 4/2 = 18 mm. Table 1: RF-induced signals versus BPPM position. zc, mm zg, mm Vind, V P, dBm 8 24 15.30 33.69 0* 18 3.28 20.32 -8 10 0.72 7.15 -12 6 0.36 1.13 -16 2 0.23 -2.77 -18 0** 0.22 -3.15 -20 -2 0.24 -2.40 * The electrode center is at the DT center ** The gap center is at the DT center Obviously, placing the BPPM gap near the DT center reduces the RF-induced signal significantly. This is due to the axial symmetry of the RF field, which penetrates effectively only through the annular gap, but not through the longitudinal slots between the electrodes. For the optimal BPM position inside the DT, the RF-induced voltages have the same order of magnitude as the beam-induced ones: 0.22 V versus 0.15 V at 402.5 MHz, and versus 0.16 V at 805 MHz. While this prevents us from processing BPPM signals at 402.5 MHz, we can be sure that this BPPM inside the DT can operate with the RF power on without damage to the cables or electronics, and the filtering out the 402.5-MHz signal will present no problem for the BPM signal processing at 805 MHz. 4 SUMMARY Electromagnetic MAFIA modeling of the SNS linac BPPMs has been performed. Based on the analysis results, an optimal pickup design with 4 one-end-shorted 60-degree electrodes has been chosen. It provides a good linearity and sufficient signal power for both position and phase measurements, while satisfying the geometrical and mechanical requirements, see in [1,2]. The feasibility of using BPPMs in the SNS drift-tube linac is demonstrated. The author acknowledges useful discussions with J.H. Billen, J.F. O ’Hara, J.F. Power, and R.E. Shafer. REFERENCES [1] S.S. Kurennoy, “Electromagnetic Modeling of Beam Position Monitors for SNS Linac ”, Proc. EPAC 2000; http://accelconf.web.cern.ch/accelconf/e00/PAPERS/ WEP2A08.pdf [2] S.S. Kurennoy, “Beam Phase Detectors for Spallation Neutron Source Linac ”, Proceed. EPAC 2000; http://accelconf.web.cern.ch/accelconf/e00/PAPERS/ WEP2A07.pdf [3] MAFIA Release 4.20, CST GmbH, Darmstadt, 1999. [4] R.E. Shafer, in AIP Conf. Proc. 319, 1994, p. 303. [5] S.S. Kurennoy, “BPMs for DTL in SNS Linac ”, Tech memo SNS:00-54, Los Alamos, 2000.
BRIDGE COUPLER FOR APT* Paul T. Greninger, Henry J. Rodarte, General Atomics, San Diego, CA Abstract The Coupled Cavity Drift Tube Linac (CCDTL) used in the Accelerator for the Production of Tritium (APT) is fully described elsewhere [1 ]. The module s are composed of several machined and brazed segments that must account for the accumulation of dimensional tolerances in the build up of the stack. In addition, space requirements dictate that power fed to the accelerator cannot be through the accelerating cavities. As well, we would like to remove a single segment of the accelerator without removing additional segments. These requirements combined with phasing relationships of the design and space limitations have resulted in a different bridge coupling method used in the module comprising 3-gap segments. The coupling method, phasing relationships and other features that enhance the flexibility of the design will be discussed. 1 BRIDGE COUPLER DE SIGN The Bridge Coupler below addresses all of the above problems. A unique feature is the ability to take up tolerances, and employ the design over a wide range of particle velocities . The center cavity is bent into a U, where the distance between the legs may vary, while maintaining a total constant length . This accommodate s different spacing between Accelerator Cavities (AC) . The bridge coupler consists of an odd number of cavities in order to preserve the π/2 operating mode of the RF structure. Figure 1 shows the basic bridge coupler design. The concept uses TM 01 mode pillbox cavities, with axes parallel to the accelerator center-line. A unique aspect of this structure is that due to phasing requirements, the center cavity is no longer a pill box cavity , but has been replaced with a waveguide operating in a TE 013 mode. Figure 2 defines the relative phase differences, shown with arrows, for the segments being coupled. The two Coupling Cavities (CC) (see Fig. 1) are unexcited in the π/2 mode, while the center cavity is excited to a power level determined by the relative sizes of the coupling slots shown. Since the center cavity is excited, RF power can be fed directly into it through a slot coupled to a waveguide. Work supported under contract DE-AC04-96AL89607Slot Cavity (5)AcceleratingCoupling Vacuum FlangeCoupling Cavity (4)Cavity (3)Center LocationStemAccelerating Cavity (1)Cavity (2)CouplingNoses for Tuning Power Feed SlotCenter Cavity Figure 1. Drawing of Bridge Coupler Cross Section. 2 FIELD PATTERN ACCCCenter Cavity Figure 2. Field Pattern Inside Bridge Coupled Model. Figure 3. Field Pattern in 3/2 λg Length Waveguide. The fields from the AC couple in the same direction as the CC. The fields from the center cavity couple in opposite directions so there is zero energy in the CC’s. The waveguide in Fig. 3 has its ends folded down. This gives the required 180 ° phase change at the guide.3 MAIN FEATURES • Coupling cavities and the center cavity have noses used for tuning. • The coupling cavities are split along a plane parallel to the accelerator center-line. A vacuum flange is seated here. Once disassembled a single segment of the accelerator can be removed without removing additional segments. • The center cavity is a waveguide bent into a U shape. The distance between the ACs can vary , while keeping the waveguide len gth and the resonant frequency unchanged. • The center cavity noses nearest the coupling slots enhance the fields locally. This increases the coupling between the center cavity and CCs. • The center cavity resonates in the TE 013 mode in order to provide the correct relative 1800 RF phasing of the accelerating cavities. The tuning noses are located at electric field maxima. • The relative sizes of the center cavity tuning noses near the coupling slots can be adjusted to mix in a lower order mode (TE 012) to give different H fields at the slots while maintaining resonant frequency. • Designate the AC and CC as 1,2 in the left hand side of Fig. 1. Then the coupling (k 12) is proportional to the slot size , the product H 1H2, divided by √(U1U2). The slot size is calculated by a special routine within the program CCT [ 2], developed by General Atomics. For the π/2 mode the energy in cavity 1 is related to the energy in cavity 3 by the following relationship : k12√U1=k23√U3. • Slots leading from the end of CC (2) into the center cavity (3) are semi-circular. This is done to enhance the H fields near the slot. See Fig. 4 below. The slot protruding into the waveguide enhances the action of the fields. Figure 4. Cross-section of Model for Test .• Coupling to an RF feed waveguide is accomplished by means of the slot labeled Center Cavity Power Feed Slot (see Fig. 1). Two slots are incorporated. • A chamfered post on the CC reduces the field s at the edge of the post . Also by reducing the post diameter , the tuning sensitivity is reduced. See Fig. 5 below . Figure 5. Chamfered CC Posts for Reduced Sensitivity. 4 LOCATION OF COUPLING CAVITY The coupling cavity slot is placed in a high magnetic field area. Elevating the position of the CC relative to the center axis of the AC insures there is only magnetic coupling through the slot. If the coupling slot is located further down from the top of the AC both electric and magnetic coupling terms are present. These terms are of opposite sign and add complexity to the design calculations. Raising the elevation of the coupling cavities introduced some intricacies to the mechanical design. The CC has an unsymmetrical split that is no longer through the center line (see Fig. 1). If the CC had a low elevation, chamfering the slot would have cut into the second half of the CC. 5 GENERAL DESIGN PROCEDURES The 2D cavities, without slots, are designed so that after the coupling slots are cut, giving the proper coupling, the structure resonates at the proper π/2 mode frequency. This usually means that a series of iterations are performed to arrive at a self-consistent solution. This procedure is outlined in [ 2]. In addition for the Coupled Cavity Drift Tube Linac , posts connect the drift tube (see Fig. 4). These posts are used for support and act as cooling passages to the drift tube . Coupling slots lower the frequency, while posts raise the cavity frequency. Similarly this effect is included in the iterative procedure described in [ 2]. 6 AC TO CC COUPLING When apertures (slots) are cut in a wall common to adjacent RF cavities, they will be coupled magnetically,Round Slot From CC into Center Cavityelectrically, or a combination depending upon slot location. The magnitude of the coupling depends on the fields in the two cavities at the location of the slot and on their stored energies. These fields and stored energies are calculated for unperturbed cavities (without slots). If the cavities are axially symmetric, 2-D codes can be used or, if the geometries are simple (such as rectangular), hand calculations are possible. Most of the coupling is magnetic : ( )jiH ji mag UUt HH eE eKelk) exp( )( )(3 0 02 03 0⋅− −=a pm where: µ0 is the permeability of free space. ε0 is the permittivity of free space. Hi (Hj) is the magnetic field of the 1st (2nd) cavity Ei (Ej ) is the electric field of the 1st (2nd) cavity. K and E are elliptical integrals of the 1st kind. Ui (Uj) is the stored energy of the 1st (2nd) cavity. l is the half-length of the slot. αH is a damping factor for evanescent modes in elliptical waveguides . t is the slot depth. 2 01   −=lwe see Fig. 6 below. In Fig. 6 full lengths and widths vs. half lengths and widths are defined respectively with upper and lower case letters. In the coupling equations the length L is always taken in the direction of the magnetic field. In our geometry l is the major axis of the ellipse. LWl w Fig. 6. Definition of Full and Half, Lengths and Widths. A perturbation can be used to estimate the frequency shift of a single slot in a cavity [ 3]. The frequency shift is ( )Ut H eE eKelf fH slotno mag) exp( )( )( 122 0 02 03 1 0⋅− −=Δa pm. 7 CC TO CENTER CAVITY COUPLING From Reference [ 4] expressions can be derived to eliminate Hj, Uj from the above coupling equations.   =aL baa UHend g cavity centercavity center 2sin 38 00p le wmb. The field cavity centerH is integrated over the slot length Lend.8 CENTER CAVITY TO WAVEGUIDE IRIS FEED The waveguide coupling factor ( β) for a cavity coupled to a waveguide can be defined as the ratio of the power emitted from the cavity into the waveguide (through a slot) to the power dissipated in the cavity walls. This calculation again lends itself to perturbation analysis [ 5]. The waveguide coupling factor is also related to the voltage standing wave ratio (VSWR) in the waveguide. If the waveguide is over-coupled, β is equal to the VSWR , while , if the waveguide is under-coupled, β is equal to 1/VSWR. For a rectangular slot , the equivalent area is placed in an ellipse, with the major axis one-half the length of the rectangular slot. ()() ( )P eEeKabHt le kZ wg H wg 2 0 026 4 010002 )( ( 9exp −⋅− Γ=a pb where: Z0= 00 emΩ, and k 0=2π/λ P = Total power dissipated The power is related to the total power per feed. One feed supplies power for several AC cells and an associated number of bridge coupler center cells. The power for the AC cells comes from Superfish. The power for the center cavity cells may be calculated by using Ref. [ 4]. 9 CONCLUSION We have developed a bridge coupler employing a 3/2 λg waveguide -center cavity. The design incorporates a split in the upper half allowing small segments of the accelerator to be removed. The design has the ability to stretch the distance between AC cavities. Using the design code developed by General Atomics gives the designer greater freedom. REFERENCES [1]Billen, J.H. et al. , 1995 Particle Accelerator Conference, Dallas TX. [2]P. D. Smith, “A Code to Automate the Design of Coupled Cavities”, these proceedings. [3]J. Gao, “Analytical formula for the resonant frequency changes due to opening apertures on cavity walls”, Nuclear Instruments and Methods in Physics Research A311 (1992) 437-443 [4]Sarbacher and Edson, “Hyper and UltaHigh Frequency Engineering”, Eq. 10.4, Eq. 10.33 [5]J. Gao, “Analytical formula for the coupling coefficient β of a cavity-waveguide coupling system”, Nuclear Instrumentation and Methods in Physics Research A309 (1991) 5-10
InjectorRFQ WaveguideBeamstopHEBT Figure 1: LEDA configuration for RFQ commissioning. Center and lower-upstream waveguides no longer used.LEDA BEAM OPERATIONS MILESTONE AND OBSERVED BEAM TRANSMISSION CHARACTERISTICS* L. J. Rybarcyk, J. D. Schneider, H. V. Smith, and L. M. Young, M. E. Schulzea, Los Alamos National Laboratory, Los Alamos, NM 87544, USA. * Work supported by US Department of Energy a General Atomics, Los Alamos, NM 87544, USA.Abstract Recently, the Low-Energy Demonstration Accelerator (LEDA) portion of the Accelerator Production of Tritium(APT) project reached its 100-mA, 8-hr CW beamoperation milestone. LEDA consists of a 75-keV protoninjector, 6.7-MeV, 350-MHz CW radio-frequencyquadrupole (RFQ) with associated high-power and low-level rf systems, a short high-energy beam transport(HEBT) and high-power (670-kW CW) beam dump.During the commissioning phase it was discovered thatthe RFQ field level must to be approximately 5-10%higher than design in order to accelerate the full 100-mAbeam with low losses. Measurements of a low-duty-factor,100-mA beam show the beam transmission isunexpectedly low for RFQ field levels between ~90 and105% of design. This paper will describe some aspects ofLEDA operations critical to achieving the abovemilestone. Measurement and simulation results forreduced RFQ beam transmission near design operatingconditions are also presented. 1 INTRODUCTION The LEDA RFQ is an 8-m-long linac that delivers a 6.7-MeV, 100-mA CW proton beam. The RFQ and itsvarious ancillary systems (Fig. 1) are described in detailelsewhere [1-6]. We recently completed the beam-commissioning phase where we accomplished our goal of8-hr, 100-mA, CW beam operation. During the coarse ofcommissioning the RFQ, we observed an overallreduction in beam transmission for peak currents >70 mA and RFQ field levels between ~90 and 105% of design.This paper presents results obtained during the 100-mACW beam-commissioning phase. It also summarizesnumerous measurements and simulations performed in anattempt to understand the aforementioned loss oftransmission. 2 LEDA PERFORMANCE From mid-Nov ’99 through early Apr ’00, LEDA was operated with beam currents in excess of 90 mA and dutyfactors ≥99.7%. During this time, while operating at duty factors ≥99.7%, LEDA accumulated 9.0 hr of ≥99.7 mA, 20.7 hr of ≥99 mA and 111 hr of ≥90 mA beam. The beam-current monitors were sampled at 30-sec intervals.In the analysis, a run was defined as a contiguous set ofsamples with the beam current ≥5 mA and the duty factor ≥99.7%. To aid in monitoring beam transmission through the RFQ, the duty factor was reduced from CW to 99.7%to allow accurate measurement of the RFQ input andoutput beam using the AC current monitors at the entranceand exit of the RFQ. These data are included in the CWanalysis. The longest run at greater then 90 mA was118 min at 99.3 mA. We accumulated a total of 694 runswith an average duration of 9.6 min each. A histogram ofrun duration statistics is shown in Fig. 2. During the commissioning phase, the following aspects of either LEDA hardware configuration or operationswere critical to achieving the 100-mA milestone: • LEBT beam properly matched to the RFQ. Space- charge effects were overcome through addition ofan electron-trap at the RFQ entrance and byreduction of final LEBT solenoid to RFQdistance [1]. • RFQ field quality. Monitored at 64 locations along structure and optimized through adjustment ofcooling flows on the four 2-m RFQ segments. • Resonance Control Cooling System performance. PID control parameters were adjusted for fasttransient response during high-power beamoperation. • Low operating pressure in RFQ. Pressure ~1.x10 –7 Torr for stable operation.• HEBT tuned for 100-mA beam. Facilitated using “notched” RF in synch with injector pulse toproduce short, high-current pulses for tuning. • High (>90%, design:~93%) beam transmission. Below ~90% the losses were too great to sustainstable operation. • RFQ field levels at 5-10% above design. 3 RFQ BEAM TRANSMISSION 3.1 Initial Observation Early on in the commissioning phase, an RFQ transmission curve first revealed a discrepancy betweenactual and expected beam transmission at high peakcurrents. Subsequently a series of measurements weremade to examine the transmission for various peak currentbeams from 70-100 mA. The measurement results alongwith a PARMTEQM [7] prediction for the nominal100 mA beam are shown in Fig. 3. All transmissionmeasurements were performed with low duty factor beamto limit the total beam loss. A substantial peak-current-dependent reduction in transmission was observed. Thisreduction in transmission for the 100mA beam ultimatelydictated a higher operating field level for the RFQ, i.e.field levels ~5-10% above design. 3.2 Simulations In an attempt to understand the loss of transmission at higher beam currents, numerous PARMTEQMcalculations were performed to investigate the effects ofvarying degrees of RFQ field tilt, beam mismatch,position and angle offsets to the beam, and beam currentenhancement on RFQ beam transmission. None of theabove results were able to reproduce the observed loss oftransmission. Initially, simulations were performed withfield tilts up to 10%. These results did not reproduce themeasurements. Also, these large tilts were not consistentwith our observations. (Quadrupole and dipole fielddistributions were derived from cavity signals sampled at64 locations along the RFQ.) Introducing mismatchedbeam into the RFQ reduced the overall transmission but nothing more. The code was then modified to allow forsmall displacements to be applied to the particlestransverse coordinates at a given cell. This was an attemptto approximate small misalignments between segments ofthe RFQ and small dipole field contributions from theRFQ. These results did not reproduce the transmissioncurves. The code was further modified to allow the beamcurrent in a bunch to be enhanced. This was an attempt tomimic background charges that might possibly becometrapped within the RFQ acceleration channel. In all theabove studies, the character of the predicted transmissioncurves was basically unchanged. The precipitous drop intransmission has not been reproduced by any of thesecalculations. 3.3 Additional Observations Further measurements revealed several interesting features. Time dependence in the loss of transmission wasseen while making measurements using short beam pulsesat reduced RFQ field strengths. We observed a step-change reduction in the beam current out of the RFQ asshown in Fig. 4. The leading portion of the pulse exhibitstransmission characteristics in agreement with simulationfor the nominal RFQ, the trailing edge does not.Simultaneous measurements of the cavity field amplitudesampled along the downstream portion of the RFQrevealed a small but measurable increase in the RFQ fieldlevel correlated with the decrease in beam transmission,also shown in Fig. 4. During this transition, the low-levelRF system maintained a constant drive signal. The RFQfield amplitude at the end of the pulse was observed toincrease exponentially towards the end of the RFQ. Thiswould be consistent with a reduction in beam loading, i.e.increase in beam loss, which would result in net higherfields. The beam loss would also be consistent withobserved high radio-activation levels at the high-energy0.0010.0100.1001.000 0 1 02 0 3 04 05 06 07 08 09 0 1 0 0 1 1 0Duration (m in)Runs per bin / total runs Figure 2: Histogram showing likelihood of success for a given duration run. Data set includes all CW runs o f ≥90 mA. Data binned in 5 min intervals.0.500.550.600.650.700.750.800.850.900.951.00 0.85 0.90 0.95 1.00 1.05 1.10 1.15RFQ Cavity Field Amplitude (1.00=design)RFQ Transmission70mA data(2/15) 80ma data(2/15) 90ma data(2/15) 100mA data(2/15) 100mA PARMTEQM calc Figure 3: LEDA RFQ transmission data. Measurements performed at 70-100 mA peak current. PARMTEQMprediction for nominal 100-mA beam also shown.end of the RFQ when it is operated at or below design field levels. We also observed that the temporal locationof the transition depends upon the RFQ field level. As thefield is reduced the start of the transition shifts towardsthe beginning of the pulse. Also, no difference was seen intransmission curves obtained using 90-mA beam underCW and low duty factor(~25%) RF operation. Changes in the wire scanner profiles were also seen across a beam pulse containing the transition. Thecentroid and rms width of both the horizontal and verticalprofiles were constant during the leading edge of thepulse. However, during the trailing edge as much as a100% increase in the rms size of the vertical profile wasseen as the field level in the RFQ was reduced from 10%above to 10% below design. Over that same fieldamplitude range a small change was observed for thehorizontal rms size while no change was observed ineither centroid. 3.4 One possible explanation The observed reduction in transmission might well be due to ions trapped within the RFQ accelerating channel.The potential well established in the RFQ is capable oftrapping slow moving ions [8]. These ions would increasethe effective space-charge force seen by the beam andcould result in a larger overall beam that could be lost onthe RFQ vanes. The source of ions, e.g. protons, might bebeam collisions with either residual gas molecules or theRFQ vane tips. The observed time dependence in theoutput beam current would be related to the ion builduprate. Previous simulations with PARMTEQM may havebeen to simplistic. A further study was performed wherePARMTEQM was modified to pre-load the space-chargemesh with additional charge. Very preliminary resultsshowed a background charge distribution could produce alarger, somewhat hollow beam. Transmission calculationshave not yet been performed with this code. However, a steady state, single-bunch code like PARMTEQM mightnot be appropriate for modeling this time dependentphenomenon. Along this line, work has begun ondeveloping a simple model of the RFQ using time as theindependent variable. A time-based code likeTOUTATIS [9] might also be more appropriate for thisstudy. More work needs to be done in this area. 4 SUMMARY The LEDA RFQ has performed well: it operated for 21 hr with RFQ output currents ≥99 mA during the recent beam-commissioning period. An unexpected reduction inhigh-peak current (>70mA) beam transmission wasobserved when the RFQ field levels were operatedbetween ~90 and 105% of design. Further investigationrevealed a time dependent character to the beamtransmission and correlated effects in wire scannerprofiles and RFQ field levels. Trapped ions in the RFQchannel is one possible explanation for the effect. ACKNOWLEDGEMENTS The authors thank the LEDA operations and support personnel without whom this work would not have beenpossible. REFERENCES [1] L.M. Young et al. , "High Power LEDA Operations," Proc. LINAC2000 (Monterey, 21-25 A ugust 2000) (to be published). [2] H.V. Smith et al., “Update On The Commissioning Of The Low-Energy Demonstration Accelerator(LEDA) Radio-Frequency Quadrupole (RFQ),”Proceeding 1999 ICFA Workshop on The Physics ofHigh Brightness Beams, in press [3] J.D. Sherman et al., “Status Report on a dc 130-mA, 75-keV Proton Injector,” Rev. Sci. Instrum. 69(1998) 1003-8. [4] D. Schrage et al., “CW RFQ Fabrication and Engineering”, Proc. LINAC98 (Chicago, 24-28August 1998) pp. 679-683. [5] D.E. Rees et al., “Design, Operation, and Test Results of 350 MHz LEDA RF System,” Proc.LINAC98 (Chicago, 24-28 August 1998) pp. 564- 566. [6] A.H. Regan et al, “LEDA LLRF Control System Performance: Model and Operational Experience”,Proc. 1999 Particle Accelertor Conf. (New York, 29March- 2 April, 1999) pp. 1064-1066. [7] K.R. Crandall et al., “RFQ Design Codes”, Los Alamos National Laboratory report LA-UR-96-1835(revised February 12, 1997). [8] M.S. deJong, “Background Ion Trapping in RFQs”, Proc. 1984 Linac Conf. (Seeheim, Germany, 7-11May, 1984), pp.88-90. [9] R. Ferdinand, et al, “TOUTATIS, the CEA-Saclay RFQ code”, Proc. LINAC2000 (Monterey, 21-25August 2000) (to be published).0102030405060708090100 0 100 200 300 400 Time (us)Current (mA) 1.001.011.021.031.041.05 rel. RFQ Field Level detector output Beam Current Field Level Figure 4: RFQ output current and field level versus time. RFQ nominal field level at ~97% of design.
RF Control System for the NLC Linacs* P. Corredoura✝, C. Adolphsen Stanford Linear Accelerator Center, Stanford, Ca 94309, USA Abstract The proposed Next Linear Collider contains a large num- ber of linac RF systems with new requirements for wide- band klystron modulation and accurate RF vector detection. The system will be capable of automatically phasing each klystron and compensating for beam loading effects. Accelerator structure alignment is determined by detection of the beam induced dipole modes with a receiver similar to that used for measuring the accelerator RF and is incorporated into the RF system topology. This paper describes the proposed system design, signal pro- cessing techniques and includes preliminary test results. 1. INTRODUCTION The NLC is essentially a pair of opposing X-band linacs designed to collide 500 GeV electrons and positrons. The main linac RF system is a major driving cost for the project consisting of 1600 X-band klystron tubes and relatedhardware.PulsestackingRFtechniquesareusedto efﬁciently develop the 600MW 300ns RF pulses required at each girder. A group of 8 klystrons (an 8 pack) deliver RF power to 8 girders, each girder supports 3 accelerator structures. Pulse stacking and beam loading compensation requires fast modulation of the klystron drive signal. A programmable digital IF solution will be presented. The absolute phase of the accelerating RF with respect to the beam is a critical parameter which must be mea- sured and controlled to the 1 degree X-band level. Even with a state-of-the-art stabilized ﬁber optic RF refer- ence/distribution system, there will still be phase drifts presentinthesystemwhichwillrequiremeasuringtherel- ativephasebetweenthebeamandacceleratingRFatregu- lar intervals. The techniques for making this and other X-band measurements are described in this paper can be applied to any linac RF system. Transverse alignment must be achieved to extremely tight tolerances to prevent excitation of transverse modes. Each accelerator is a damped detuned structure which is designed to load down the undesirable high-order modes (HOMs) and allow their external detection to facilitate alignment of each girder [1]. The shape of each individual accelerating cell in a structure is altered slightly along the length of the structure making the HOM frequencies a function of the longitudinal position on the structure. The frequencyofthelowestordertransversemoderangesfrom 14 to 16 GHz corresponding to upstream and downstream respectively [2]. A receiver will be outlined which mea- sures the beam induced HOMs allowing automated align-ment of each girder via remote mechanical movers. 2. KLYSTRON DRIVE GENERATION For the main linac system the output of 8 klystrons will be combined to deliver X-band RF to 8 accelerator girders through a high power distribution and delay system (DLDS) (ﬁgure 1). Each ~3us klystron pulse will be time multiplexed to steer RF power successively to 8 accelera- torgirdersbyquicklymodulatingtherelativephasesofthe klystron drive during a pulse. Wideband klystrons (>100MHz) will allow rapid phase switching to improve overall efﬁciency. Fixed delays in the DLDS deliver RF power a ﬁll time in advance of the beam. Fig.1. Diagramshowing8klystrondriving8linacgirders. Tocompensateforbeamloading,klystronswillbepre- cisely phase ramped during the ﬁrst ~100ns of each sub-pulse to direct some RF power to a port which is out oftimewiththebeam.Thisallowstheklystronstooperate in saturation while minimizing the energy spread across the 95 bunch train. To compensate for unknown transmis- sion phase shifts the system must produce any drive phase and allow a smooth transition to any other phase. Speciﬁ- cations for the main linac LLRF drive system are listed below in table 1. Parameter Value carrier frequency 11.424 GHz pulse width 3.1 us bandwidth >100 MHz phase range arbitrary, continuous phase resolution <1 degree dynamic range > 20 dB Table 1: Main linac LLRF drive requirements.KKKKKKKK 600 MW Delay Line Distribution Systemgirder #10girder #19 girder #1girder #28girder #37girder #46girder #55girder #64 *Work supported by Department of Energy, contract DE-AC03-76SF00515 ✝plc@slac.stanford.eduSLAC-PUB-8571 August 2000 Presented at the 20th International Linac Conference (Lina c 2000), Monterey Conference Center, California, August 21 -25, 2000.An in-phase/quadrature (IQ) drive generation system was produced for the next linear collider test accelerator (NLCTA) [3]. Two high speed (250 MS/S) DACs were usedtodriveanX-bandbasebandIQmodulator(ﬁgure2). The IQ technique works but is sensitive to mixer offsets, quadrature errors, baseband noise and requires two DACs. Fig.2. BasebandIQtechniqueusedtoproducetheX-band klystron drive waveform in the NLCTA. To reduce system cost and improve accuracy a digital IF approach is being pursued (ﬁgure 3). A modulated IF tone burst is generated by a single programmable DAC channel and up-mixed with a locked local oscillator (LO) to drive the klystron preampliﬁer. The IF frequency must be high enough to meet the system bandwidth requirements and allow ﬁlters to be realized which reject the image and the LOleakage.Frequencymultiplierscanbeusedtoraisethe IF frequency without increasing the DAC clock rate at the expense of phase resolution. A single sideband modulator can be used to reduce the image amplitude. Fig. 3. Digital IF technique for driving pulsed klystrons. ItisimportanttonotethatthephaseoftheoutputRFis a function of the phase of the (multiplied) DAC produced tone burst and the phase of the LO when the DAC is trig- gered. By choosing the IF frequency to be an integer mul- tiple of the bunch separation frequency (357 MHz for NLC), the phase of the accelerating RF will repeat for each bunch time slot (equations 1-4). This eliminates the need to load a differently phased DAC waveform or to resynchronize the LO before each machine pulse. Equations1-4. DerivationshowingRFphaserepeatsevery bunch separation (T) interval. T is for NLC.To estimate the DAC resolution required to produce drive phase shifts refer to ﬁgure 4. If we use the full DAC range to synthesize the IF waveform then the mini- mum phase shift we can resolve corresponds to a one bit changeatazerocrossing.A7bitDACattheIFfrequency is required. If multiplication is used to raise the IF fre- quency more bits are needed. A x8 multiplier requires 3 additionalbits.Producingthe89.15MHzsub-IFwitha12 bit device will allow operating the DAC below full scale. Fig. 4. Bits required to achieve phase resolution. Applying the digital IF signal generation technique to otherNLClinacsystemsoperatingatdifferentfrequencies simply requires a different RF modulator and LO refer- ence. The system bandwidth required to support X-band linacpulseswitchingwilleasilysupportSLEDcavityPSK or compressor beam loading compensation requirements. Maintaining the systems will also become less specialized since the DAC/IF hardware and some software will be identical for all klystrons operating at L,C,S, or X-band. 3. DIGITAL RF VECTOR DETECTION To conﬁgure the NLC main linac RF systems, measure- ment techniques must be available to allow proper align- ment of the accelerating RF to the beam. Again, a digital IF technique is being planned. The unknown RF signal is mixed down to an IF frequency, ampliﬁed, dither added (optional), and sampled with a high speed ADC (ﬁgure 5). The choice of the receiver IF frequency is less constrained than for klystron drive generation. The 89.25 MHz IF sys- tem shown will have 4 possible phase offsets for measure- ments triggered on 4 consecutive 357 MHz clocks. If the sample phase offset for each measured IF pulse were known it could be corrected for during post processing. Alternatively, a 357 MHz IF could be used. Undersam- pling techniques would be used to keep sample rates and memory requirements reasonable while maintaining sufﬁ- cient channel bandwidth. Fig. 5. Digital receiver to accurately measure IF vectors.I DAC Q DACvoltage to currentmemory memoryvoltage to currentbaseband IQ modulator 11.424 GHzX-band drive waveform output to TWT preampliﬁer RF reference DAC memorysingle sideband modulatorfrequency multiplier 10.710 GHzX-band drive 89.25 MHz714 MHzbandpass ﬁlter 357 MHzIF sub-IF11.424 GHz lockedLO reference clockwaveform output to TWT preampliﬁer φIFφLO+ φRF= 1[ ] 8φsubIFφLO+ φRF= 2 [ ] 8π 2-- -TφsubIF+  30 2π ( )TφLO+ ( )+ φRF= 3 [ ] 2 2π ( )T 8φsubIF+ 30 2 π ( )TφLO+ ( )+ φRF= 4 [ ] 357 MHz( )1 –1° + full scale - full scaleΔφ1bit} } full scale( ) ω tΔφ+( )sin 1 count = 5 [ ] full scale1 Δφ ( )sin-------------------1 1° ( )sin------------------ 57 counts = = = 6 [ ] bits required log22 57⋅ ( )7 bits at 714 MHz≅ = 7 [ ]At zero crossing: 1° X 11.33475 GHzIF ﬁlterADCmemoryIF gain DSP dither (optional)+ 357 MHz sample11.424 GHz RF signal to measure locked LO89.25 MHz clockRF ﬁlter (optional)Deriving wideband amplitude and phase vectors from asampledIFwaveformcanbeachievedbyapplyingaHil- bert transform. The Hilbert technique involves taking the Fourier transform of the sampled IF data, nulling all nega- tive frequency bins, scaling positive bins by 2, and ﬁnally takingtheinverseFouriertransform.Thisproducesacom- plextimedomainvectorallowingcalculationofamplitude and phase vectors (ﬁgure 6). While in the frequency domain, ﬁltering may be applied by nulling any undesired spectral bins, potentially enhancing algorithm efﬁciency. Fig. 6. Driving a klystron and detecting the output using digitalIFtechniques.Thereceiverbandwidthwas20MHz. To allow automated conﬁguration of the RF systems severalmeasurementsmustbeavailable(ﬁgure7).TheRF output of each klystron output coupler (P1,P2) must be measured to monitor klystron performance. The loaded accelerating RF is measured to allow compensating for beam loading and properly phasing the RF to the beam. Structuretransversealignmentisdeterminedbymeasuring thebeaminduceddipolemodes(X,Y).Abeampickupcan provide a phase reference if the receiver LO is chosen not to be locked to the machine reference. Fig.7. Measurementsneededforautomatedconﬁguration. Themostdifﬁcultmeasurementtaskisdeterminingthe RF/beam phase during single bunch operation. The struc- ture output RF port allows direct measurement of the accelerating RF, loaded RF, or the beam induced RF (no input RF applied). Dynamic range is an issue since a sin- gle pilot bunch of 1e9 particles induces RF 40dB below the accelerating RF. By applying dither to the IF (ﬁgure 5) and averaging 10 pulses, the pilot bunch phase can be measured to accuracy (no klystron RF). Alternativelythe phase of the accelerating RF can be compared to the loadedRFphaseduringasinglepulse.Thevectordiagram (ﬁgure 8) indicates that the relative angle must be mea- sured to accuracy. Simulations show 200 averages of a dithered 89.25 MHz IF using a 7 bit ADC to sample the 285ns burst would be required. Both techniques are truly differential allowing absolute resolution of RF/beam phase and support totally automated linac phasing. Fig. 8. Vector diagram of RF, beam, and loaded RF. Mod- ern techniques can measure the true RF/beam phase. Measurementofthe14-16GHzstructuredipolemodes requires a variable LO source to mix the desired signal down to the IF frequency. A ﬁlter and a RF limiter before the mixer are required to limit peak power when the struc- ture is misaligned. A triple bandpass ﬁlter passes dipole modes harmonically related to bunch spacing and corre- sponding to the center and both ends of the structure. AdditionalIFgain(30dB)willbeappliedwhenthesignals become small to produce the 65 dB dynamic range required to measure 1 micron offsets with a single shot pilot bunch (no averaging). Recently a receiver has been tested on a structure and beam reference pickup installed in the SLAC linac (ﬁgure 9). Fig.9. IFwaveform/resultsfromstructurealignmenttest. IFtoneburstwasdigitizedusinga1GHz8bitVXIscope. 4. CONCLUSION The digital IF technique proposed to produce arbitrary drive waveforms and accurately detect accelerator RF sig- nals has produced encouraging initial tests results. The development of a high speed DAC/ADC module is planned. A full 8 pack test installation is planned. 5. REFERENCES [1] J.W. Wang et al, “Accelerator Structure R&D for Linear Colliders”, PAC 99. [2] M. Seidel, “Studies of Beam Induced Dipole-Mode Signals in Accelerating Structures at the SLC”, PAC 97. [3] S. Holmes et al, “Low Level RF Signal Processing for the NLCTA”, PAC 97.arXiv:physics/0008098 16 Aug 200000.20.4amplitude − linearMeasured DAC 89.25 MHz subIF −2000200phase − degrees22.5 degree step 00.050.1amplitude − linearMeasured Klystron Output Using 89.25 MHz IF 0 200 400 600 800 1000 1200 1400 1600 1800 2000−2000200 time − nsphase − degrees180 degree step K DLDS combinerRF input loadbeam referencestructure output RFX YP2P1accelerator structuretunnel RF measurementsgallery RF measurements K dipole modes outputs ... 1°0.01° magnitude = S realimaginaryaccelerating RF vectorbeamrelative beam phase =Φ vector sum magnitude = Ainduced vector relative loaded/unloaded RF phase =αΦα ( )sin A S--- -----------------      asin α+ ≅ 0 50 100 150 200 250−0.1−0.0500.050.1 time − nsamplitude − voltsMeasured IF Beam Reference Mixed from 14.994 GHz −400−300−200−100 0100200300400050100150200250 beam position − micronsStructure Receiver Output vs. Beam Positionreceived power/phase − linear/degrees
Abstract Electron beams of linear induction accelerators experi- ence deﬂective forces caused by RF ﬁelds building up as aresult of accelerating cavities of ﬁnite size. These forcescan signiﬁcantly effect the beam when a long linac com-posed of identical cells is assembled. Recent techniques incomputational modeling, simulation, and experiments for20 MeV DARHT-II (Dual Axis Radiographic Hydrody-namic Test) accelerator cells were found to reduce thewakeﬁeld impedance of the cells from 800 ohms/meter to350 ohms/meter and experimental results conﬁrm theresults of the modeling efforts. Increased performance ofthe cell was obtained through a parametric study of theaccelerator structure, materials, material tuning, andgeometry.Asaresultofthiseffort,itwasfoundthatthick-ness-tuned ferrite produced a 50% deduction in the wake-ﬁeld impedance in the low frequency band and was easilytunable based on the material thickness. It was also foundthat shaped metal sections allow for high-Q resonances tobe de-tuned, thus decreasing the amplitude of the reso-nance and increasing the cell’s performance. For thegeometries used for this cell, a roughly 45 degree anglehadthebestperformanceinaffectingthewakeﬁeldmodes. 1 INTRODUCTION The modeling, simulation, design, and experimental activities for the DARHT-II accelerator cell (Figure 1)consisted of several stages in order to lower the wakeﬁeldimpedanceofthecellwhilemaintainingthevoltagebreak-down hold off characteristics of the pulsed power design.For the computational simulation and RF design of thecell, the A MOS1 code was used. Figure 1. The DARHT-II [standard] accelerator cell (left) and the DARHT-II injector accelerator cell(right) are shown during high voltage and RF (radiofrequency) tests at LBL. The cells are 6' in diameter,with 10" and 14" beampipe bores, have 1" accelerat-ing gaps, and have less than 200kV across the gap. The inside of the accelerator cell consisted of an accel- erating gap, a Mycalex™ insulator, a metglas core, andwas oil ﬁlled. As part of the design and simulation activi- ties for the cell, the material properties of the ferrite andMycalex used in the cell were obtained and a quick com-parison between the simulation code and an ideal pillbox cavity was generated to study the gridding effects in theresults. This was especially important since the ferritetuning effort was very sensitive to thickness variations.The accelerator cell was fabricated to match the resultingsimulationresultsandboth[standard]acceleratorcellsandinjector accelerator cells were tested experimentally 2. 2 SIMULATION OF THE CELL The AMOS2.5D FDTD (ﬁnite difference time domain) RFcodewasusedtosimulatethecrosssectionofthecell. Figure 2. The A MOSmodel of the original DARHT-II standard accelerator cell includes a Mycalex insula-torε r=6.8 to restrict the oil. The insulator is angled to reduce ﬁeld stress along the vacuum-side surface.The model was longitudinally lengthened to reduceinteractions with the left/right boundary conditionsand solutions to the model include transverse modes. Initial RF experimental results from the cell had poor performance and the ﬁrst models had peaks at 800 Ω/m so designeffortscommencedtoaddresstheseissues(seeFig-ure 7.) The resonances were caused by standing waves inthe cell in high-Q regions of the cell. These high-Qregions in the model were then identiﬁed via temporal-spatial Fourier transforms. This identiﬁed several proper-ties of the original cell: the high-Q regions in the cell arein the area of the accelerating gap and in the vertical oilcolumn;verylittleenergytraversestheuppercornerofthestructure; and the performance is very sensitive to the gapgeometry. Due to fabrication issues, it was not possible toconsider changes to the Mycalex piece. So subsequentdesign changes to the cell concentrated only on modiﬁca-tions to metal pieces or the addition of metal inserts to thecell. This also allowed for easy cell modiﬁcation. oil beampipemycalex vacuumferriteRF CELL MODELING AND EXPERIMENTS FOR WAKEFIELD MINIMIZATION IN DARHT-II Scott D. Nelson† and Michael Vella‡ Lawrence Livermore National Laboratory, Livermore, California 94550 USA† Lawrence Berkeley National Laboratory, Berkeley California 94720 USA‡Figure 3. The temporal-spatial Fourier transform shows the locations of the high-Q resonances (referto Figure 1 for the geometric layout of the compo-nents). The image on the left is the low frequencyresonance (see Figure 7), the image in the middle isthe high frequency resonance, and the image on theright is the pipe mode (for conﬁrmation purposes). As part of the simulation activities it was desirable to reduce the high-Q resonance that occurs at the low fre-quency point near 200 MHz. The tuning of the thicknessof the ferrite tiles in the oil region were found to be a veryeffective way to reduce this resonance. But simulations ofthese tiles are very sensitive to the tile thickness since itwas instrumental in the tuning process. Since the perfor-manceofthesimulationcodewiththesetunedferriteswascritical, a study looked at the errors based on cell size. Thepillboxequationfrom[3]withtheappropriatescale factors to correspond to the A MOScalibrations in MKS unitshasa8.987 ×1011scalefactortoconvertinto Ω/m. A sourceoferrorwasfoundtobethemeshingprocesswhichchanged the geometry of the solved problem — i.e. as aresult of the meshing, the geometry that was solved wasdifferent than the initial geometry. Since the ferrite tuningeffort was very sensitive to ferrite thickness, it was impor-tant to select cell sizes such that the ferrite and the gapcontainanintegernumberofcells. Failuretodosocausesan almost 20X increase in the error of the calculation (seeFigure 4).where are 1/b ×the zeros of , J 1andY1are the Bessel functions of the ﬁrst and second kind, isthe imaginary operator, cis the speed of light, Z 0=4π/c, Zs=2*Z0,v=βc, andb,d,R are as shown in Figure 4. Figure 4. The upper plot shows the results of com- paring an analytic pillboxto the A MOScalculation. The error is 6% (middle plot) and is caused by thegridding effects and not by the physics code itself.The lower plot shows the results of comparing thesame A MOScalculation with a pillboxof equal dimensions to that of the A MOSmesh. The error in this case is only 0.35%. 3 MATERIAL TESTING As part of the construction of the simulation model, the properties of the Mycalex and ferrite were measured usingthe coaxial line. 4,5,6This involves acquiring the [s] parametersforthesampleinacoaxiallineandthenapply-ing the following relationships: the sign choice for is resolved by requiring . Relativepermeabilityandpermittivityarethenobtainedas , where Z⊥ω()2cdZ0 πωb2---------------–ℑωd v-------sin2 ωd v-------2-----------------------1 H1ω()----------------  8.987× 1011⋅ = H1ω()ωb c-------J1'ωbc⁄() J1ωbc⁄()-------------------------G1'ωb,() G1ωb,()----------------------- – 1 d---1e2dµsω()–– µsω() ρs2b21–()------------------------------------------1 d---ω c----21e2d–εsω()– εsω()3------------------------------∑+ ∑–= Y1'ωbc⁄() ∂=xY1x()ωbc⁄,J1'ωbc⁄()=∂xJ1x()ωbc⁄ G1ωb,() J1ωbc⁄() C1ω()Y1ωbc⁄()×+ G1'ωb,()J1'ωbc⁄() C1ω()Y1'ωbc⁄()×+ == µsω() ρs2ωc⁄()2– εsω() βs2ωc⁄()2–=, = C1ω()jZsZ0⁄()J1'ωRc⁄() J1ωRc⁄()– jZsZ0⁄()N1'ωRc⁄() N1ωRc⁄()–------------------------------------------------------------------------------------ - –=ρsβs, J1'J1, ℑx{} %Error caused by gridding effects R ~ 3', 2d ~ 1", b ~ 0.127m grid size [mm] %Error 0 1 3 2012345678(not to scale) R b2d Frequency [MHz] %Error 0 200 400 600 800 100000.10.20.30.4 Peak % error2´1084´1086´1088´1081´109100200300400500600700800 V1S21S11+V2S21S11– X1V1V2– V1V2–---------------------- ΓXX21–± ZV1Γ– 1V1Γ–-------------------=, =, =, =, = ΓΓ 1≤ µrc2c1εrc2c1⁄=,= c1µr εr-----1Γ+ 1Γ–-------------2 c2µrεrc ωd-------1 Z---ln2 – ====Using these relations, the Mycalex was measured to have a permittivity of and apermeability of up to 800MHz. The error bars were determined from a sample ofteﬂon of the same size; but it should also be noted that thetechnique is dependent on standing wave patterns formedin the line, thus different materials have different errors.Further reﬁnement of the technique is required. Figure5.Thecoaxialair-lineteststandusedfortest- ing the material samples is shown on the left. Thepipe is a 50 Ωline, has type-N connectors on each end, and has a removable center conductor. Thetorque of each bolt was set to 50 oz.-in. and wasrequired for calibration purposes. On the right areshown two Mycalex samples (top row) and three fer-rite samples (bottom row). Fit quality in the line isimportant so copper tape was used to insure the ﬁt. The ferrite was measured using the same experimental conﬁguration as the Mycalex and Teﬂon. To be useful tothe A MOScode, the permeability has to be expanded using a Lorentzian model: yielding these coefﬁcients, valid for 1MHz - 1 GHz 4 CELL DESIGN MODIFICATIONS The modiﬁcations to the cell include the tuning of the ferrite tile located in the oil region, and the ﬂat cornerreﬂector in the vacuum region as shown in Figure 6. Thecorner reﬂector was examined using different angles andprotrusions into the cell. This is a low ﬁeld stress regionof the cell and so a great deal of latitude was allowed.Figure 6. The ﬁnal design modiﬁcations contain sim- ilarstructuresasFigure2butwiththeadditionofthethickness tuned ferrite in the oil region and the ﬂatcorner reﬂector in the vacuum region. 5 RESULTS The redesign activities of the cell improved the wake- ﬁeld performance of the cell by 2X of the dipole mode bylowering the Q of the structures. The results for the injec-tor cell further beneﬁtted from the increase beampipe boresize (14" vs. 10") thus dramatically reducing the high fre-quency resonance since it is above cutoff in a 14" bore.The monopole mode for this structure is extremely smalland the quadrupole mode (6000 Ω/m 2) does not steer the beam and so is not detrimental to the cell’s performance. Figure 7. The dipole-mode simulation of the 10" cell before modiﬁcation is shown with black diamonds,the results after modiﬁcation are shown with redsquares, and the results for the 14" injector cell aftermodiﬁcation are shown with blue triangles. Figure 8. The experimental data 7from the injector cellhasaloweramplitudethanexpectedcausedbyasplit peak at the low frequency point. The high-Qmode at ~500MHz is the pipe resonance.l= 123 αl=2.178×10102.571×10105.574×1010 βl=1.275×1077.375×1072.437×108 γl=3.390×10116.190×10101.880×1010εr6.8j0.5–() 0.2j0.2+()± = µr1.0j0.5+() 0.1j0.2+()± = µrf()1χmf() 11 2---αl βlγl–() j2πf–------------------------------------- -αl βlγl+() j2πf–--------------------------------------– l1=m ∑+=+= tuned ferritecorner reﬂector Cell#3, 10 in. cell, BEFORE 10 in. cell AFTER 14 in. cell AFTERCell performance: before and after 0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+026.00E+027.00E+028.00E+029.00E+02 0 MHz 100 MHz 200 MHz 300 MHz 400 MHz 500 MHz 600 MHz 700 MHz Re{Z ⊥} [ Ω/m] Re{Z⊥} 0.00E+005.00E+011.00E+021.50E+022.00E+022.50E+023.00E+02 0.00E+00 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 Frequency, Hz Ω/m6 CONCLUSIONS The redesign of the accelerator cell gap region improved the characteristics from the original 800 Ω/m down to 350 Ω/m. As part of this analysis, the following relationshipswereobserved:tuningtheferritethicknessofthe ferrite in the oil region reduces Re{Z ⊥}b y6 0Ω/m / mm(lowfreq.)and140 Ω/m/mm(highfreq.). Ferrite5-6 mm thick was selected and to be fully tuned, the ferritewas placed at normal incidence to the wavefronts. A cor-nerreﬂectorinthevacuumregionnearthegapaffectsper-formanceby50-100 Ω/mper30 °. The45°pointis“close” to the optimum angle in this design. Ferrite extending upthe oil gap reduces resonances by 20 Ω/m / cm (low freq.) as long as the ﬁelds remain normal to the surface. 13-15cm was chosen as the best compromise between perfor-mance and assembly efforts. Changing the insulator anglereduces resonances by 100 Ω/m / 10°(high freq.) but the insulator also has high voltage constraints and so theauthors did not change the insulator conﬁguration. From the comparisons with the analytic pillboxmodels, it was determined that most of the error in the A MOScal- culationsiscausedbygriddingeffectswhenthediscretiza-tionofthemeshdoesn’tmatchthephysicalgeometry. Forthose cases where the pillboxgeometry was assigned to match the cell-generated geometry, the error was less than0.35%. Theseeffectsareespeciallypronouncedduetothesensitivity of the performance of the cell vs. the tuned fer-rite thickness. Fromthemodelingandexperimentsontheinjectorcell, it was observed that the injector cell resonances scale withpipe radius. Also, the high frequency resonance seen withthe standard cell is virtually eliminated with the injectorcell due to the larger beampipe bore size for the injectorcell thus supporting a pipe mode at the high frequencypoint. 7 ACKNOWLEDGMENTS Thanks go to Dick Briggs for his comments on the experimental results, to Bill Fawley for his insights in themodeling efforts, to David Mayhall for his work on thepillboxveriﬁcation, to Jerry Burke for his material param- eter extraction, to Jim Dunlap for his experimental mea-surements, and to Brian Poole for his input on temporal-spatial Fourier transform techniques for identifying reso-nances. The authors would also like to thank, acknowl-edge, and morn the passing of Dan Birx who substantiallycontributed to the experimental effort and reviewed themodelingandsimulationefforts.Thisworkwasperformedunder the auspices of the U.S. Department of Energy bythe Lawrence Livermore National Laboratory under con-tract No. W-7405-Eng-48.8 REFERENCES [1] J. DeFord, S. D. Nelson, ‘‘A MOSUser’s Manual,” Lawrence Livermore National Laboratory , April 13, 1997, LLNL UCRL-MA-127038. [2] S. D. Nelson, J. Dunlap, ‘‘Two-wire Wakeﬁeld Mea- surements of the DARHT Accelerator Cell,” Lawrence Livermore National Laboratory , 1999, LLNL UCRL-ID- 134164. [3]R.J.Briggs,D.L.Birx,G.J.Caporaso,V.K.Neil,and T. C. Genoni, “Theoretical and Experimental Investigationof the Interaction Impedances and Q Values of the Accel-erating Cells in the Advanced Test Accelerator,” ParticleAccelerators, Vol. 18, 1985, pp.41-62. [4] G. Burke, ‘‘Broad-band Measurements of Electrical Parameters for Ferrite and Dielectric Samples,” Lawrence Livermore National Laboratory , 2000, as yet unpublished. [5]J.Baker-Jarvis,M.D.Janezic,J.H.Grosvenor,Jr.,and R. G. Geyer, ‘‘Transmission/Reﬂection and Short-CircuitLine Methods for Measuring Permittivity and Permeabil-ity,” U. S. Dept. of Commerce, NIST Technical Note1355-R, Dec. 1993. [6] A. M. Nicolson and G. F. Ross, “Measurement of the Intrinsic Properties of materials by Time-Domain Tech-niques,”IEEE Trans. Instrumentation and Measurement, Vol. IM-19, No. 4, Nov. 1970. [7] R. J. Briggs, M. Vella, S. D. Nelson, to be published report on the magnetic ﬁeld wakeﬁeld measurements ofthe DARHT-II injector cell.
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| q&$IWHUWKHKHDWLQJDQG5)WUDLQLQJWKHYDFXXPRI a7RUUZDVDFKLHYHG 5)32:(5 6833/< 7KHFRQWLQXRXV 5)VLJQDOI 0+] J\|P: IURPWKHPDVWHURVFLOODWRU LVPRYHGWRWKHIRUPLQJ DPSOLILHU DVVHPEO\ 8YLD$SKDVHVKLIWHU7KH IRUPLQJDPSOLILHURSHUDWHVDWWKHGRXEOHGIUHTXHQF\ 0+]DQGIRUPVWKH5)SXOVHRIaVZLGWKDQGRXWSXW SXOVHGSRZHUy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qyq&DFFXUDF\RIWHPSHUDWXUH VWDELOL]DWLRQ LVrq& 5(68/76 2)0($685 (0(176 :,7+ 7+(6+257% 81&+ 'XULQJWKHH[SHULPHQWV WKHPDLQJXQSDUDPHWHUV ZHUH x(OHFWURQJXQY ROWDJHN9 xXQSXOVHGFXUUHQW $ x&XUUHQWSXOVHZLGWK QV x5HSHWLWLRQ UDWH y+] )RUWKLVSXOVHZLGWKWKHEHDPUDGLDWLRQILHOGLVORZDQG LWGRHVQ WLQIOXHQFHH VVHQWLDOO\ RQWKHDFFHOHUDWLRQ7KH DFFHOHUDWLQJ ILHOGLQWKH$6LVGHILQHGE\WKHNO\VWURQ LQSXWSRZHURQO\$W WKDWRSHUDWLRQDO FRQGLWLRQV LW LV SRVVLEOHWRUHDFKWKHPD[LPXP DFFHOHUDWLQJJ UDGLHQWDQG KHQFHWKH PD[LPXP HQHUJ\RIWKHDFFHOHUDWHGEHDP 7KHRSHUDWLRQDO SDUDPHWHUV RISRZHUFRPSUHVVLRQ V\VWHPFDYLWLHVDUH FDOFXODWHGXVLQJWKHPHWKRG R I UHJUHVVLYHD QDO\VLVDQGWKH5)SXOVHVKDSHDIWHUWKHSRZHU FRPSUHVVLRQ V\VWHP x8QORDGHGT XDOLW\IDFWRU 4 x&RXSOLQJUDWLR E x7LPHFRQVWDQW W V DQGDOVRW\SLFDOWLPHVRILQSXWVLJQDOWUDQVLHQWSURFHVVHV x W VW\SLFDOWLPHRIJURZWKRILQFLGHQW x ZDYHDPSOLWXGHOHDGLQJSXOVHHGJH x W VW\SLFDOWLPHRISKDVHLQYHUVLRQ $VDQH[DPSOH)LJVKRZVWKHSXOVHVKDSHDIWHUWKH SRZHUFRPSUHVVLRQ V\VWHPDWWKHSXOVHGLQSXWSRZHU DPSOLWXGH 0: 0:0:0:0: 36/('5 HDO36/(' ,GHDO3,13 0:W2 PVHF36/('>0:@ W>PVHF@ )LJXUH3XOVHVKDSHDIWHUSXOVHFRPUHVVLRQ V\VWHP 2QH FDQFDOFXODWH WKHGLVWULEXWLRQ RIDFFHOHUDWLQJ HOHFWULFILHOGDPSOLWXGHD ORQJWKH$6DWGLIIHUHQWWL PHVE\ WKHVKDSHRI5)SXOVHDWWKH$6LQSXWDQG$6SDUDPHWHUV )RUFRQVWDQWLPSHGDQFH VWUXFWXUH D]HY]W( W]( JU ¸¸ ¹· ¨¨ ©§ ZKHUH(W (WLVHOHFWULFILHOGDPSOLWXGHD WWKH$6 LQSXWDW] 8VLQJWKHDFFHOHUDWLQJILHOGGLVWULEXWLRQ RQH FDQFDOFXODWHWKH HQHUJ\JDLQRIWKH DFFHOHUDWHG SDUWLFOHVWKDWHQWHUWKH$6DWYDULRXVWLPHV XQSXOVHGFXUUHQW LVPHDVXUHG E \WKHUHVLVWDQFH PRQLWRUORFDWHGDWWKHJXQRXWSXW0HDVXUHPHQWV RIEHDP HQHUJ\DUHPDGHE\PHDQVRIq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r 5(68/76 2)0($685 (0(176 :,7+ 7+(/ 21* %81&+ 7KHJRDORIWKHVHH[SHULPHQWV ZDVWKH$6WHVWDWWKH FRQGLWLRQV RSHUDWLQJRIWKHDFFHOHUDWRUIRUSURMHFW,5(1 >@7HVWFRQGLWLRQVZHUH x5HSHWLWLRQ UDWH y+] x(OHFWURQJXQY ROWDJH N9 xXQSXOVHGFXUUHQW $ x&XUUHQWSXOVHZLGWK QV x2XWSXWNO\VWURQSRZHU \|0: x5)SXOVHGXUDWLRQ PV x7LPHEHIRUHSKDVHLQYHUVLRQPV ,QWKDWH[SHULPHQW 5)EXQFKHUZDVQ WIHGE\WKHSRZHU DQGXQEXQFKHGE HDPZDVVWHHUHGWRWKH$6LQSXWWKDWLV ZK\WKHDFFHOHUDWHGEHDPKDGODUJHHQHUJ\VSUHDG7KH SKRWRJUDSK RIOXPLQHVFHQW VFUHHQDIWHUq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x0D[LPXP HOHFWURQHQHUJ\ 0H9 x0LQLPXP HOHFWURQHQHUJ\ 0H9 x7KHWRWDOFKDUJHRIWKHEHDP& xWRWDOQXPEHURISDUWLFOHV x7KHEHDPHQHUJ\FRQWHQW -RLOH 7KHGHWDLOVRIWKHH[SHULPHQWV ZLWKORQJEXQFKDUHLQ MRLQWUHSRUWSUHVHQWHGR QWKLVFRQIHUHQFH >@ 5()(5(1&(6 >@$9$OH[DQGURY HWDO(OHFWURQSRVLWURQ SUHLQMHFWRU RI9(33FRPSOH[ 3URFHHGLQJV RIWKH;9,,, ,QWHUQDWLRQDO /LQHDU$FFHOHUDWRU &RQIHUHQFH $XJXVWHQHYD6ZLW]HUODQG &(51 1RYHPEHU 9ROSS >@9$QWURSRY HWDO³,5(1WHVWIDFLOLW\DW-,15´ 3URFHHGLQJV RIWKH;9,,, , QWHUQDWLRQDO /LQHDU $FFHOHUDWRU&RQIHUHQFH $XJXVW*HQHYD 6ZLW]HUODQG9ROSS >@61'RO\D:,)XUPDQ 99.REHWV(0/DVLHY <X$0HWHONLQ 9$6KYHWV$36RXPEDHY WKH 9(337HDP³/LQDF/8()LUVWWHVWLQJUHVXOWV´ WKLVFRQIHUHQFH
arXiv:physics/0008101 17 Aug 2000THE CLIC MAIN LINAC ACC ELERATING STRUCTURE I. Wilson, W. Wuensch, CERN, Geneva, Switzerland Abstract This paper outlines the RF design of the CLIC (Compact Linear Collider) 30 GHz main linac accelerating structure and gives the resulting longitudinal and transverse mode properties. The critical requirement for multibunch operation, that transverse wakefields be suppressed by two orders of magnitude within 0.7 ns (twenty fundamental mode cycles), has been demonstrated in a recent ASSET experiment. The feasibility of operating the structure at an accelerating gradient of 150 MV/m for 130 ns has yet to be demonstrated. Damage of the internal copper surfaces due to high electric fields or resulting from metal fatigue induced by cyclic surface heating effects are a major concern requiring further study. 1 INTRODUCTION A major effort has gone into developing an accelerating structure for the CLIC main linac that can operate with an average gradient of 150 MV/m, that maintains an acceptable short-range wakefield, and that has highly suppressed long-range transverse wake fields. The latter design criterion is a result of multibunching - a common feature among linear collider designs in which multiple-bunch trains are accelerated during each RF pulse in order to reach design luminosities (1035 cm-2 sec-1 at 3 TeV for CLIC [1]) in a power-efficient way. For CLIC, beam dynamics simulations indicate that the amplitude of the single-bunch transverse wakefield must decrease from its short-range value by a factor of 100 during the first 0.67 ns (which is the time between bunches in the train and which corresponds to twenty 30 GHz accelerating mode cycles) and that it must continue to decrease at least linearly for longer times [2]. A structure, called the TDS (Tapered Damped Structure), capable of such transverse wakefield suppression, has been developed [3]. The suppression is achieved primarily through damping. Each cell of the 150-cell TDS is damped by its own set of four individually terminated waveguides. This produces a Q of below 20 for the lowest dipole band. The waveguides have a cutoff frequency of 33 GHz which is above the 30 GHz fundamental but below all other higher-order modes. Cell and iris diameters are tapered along the length of the structure in order to induce a frequency spread in the dipole bands which is called detuning. Iris dimensions range from 4.5 to 3.5 mm. Detuning de- coheres the wakefield kicks further suppressing the transverse wakefield.A first pass through the RF design of the TDS, for both fundamental mode performance and transverse wakefield suppression, has now been made. Certain weaknesses in the design, with regard to operation at very high gradients, have become apparent and are being addressed in a complete re-optimisation of the structure. 2. GEOMETRY AND FUNDAMENTAL MODE CHARACTERISTICS The geometry of a TDS cell is shown in Figure 1. A full structure consists of 150 cells. Iris diameters (2a) vary linearly from 4.5 mm at the head of the structure to 3.5 mm at the tail. The relationship between iris diameter (2a) and cell diameter (2b) necessary to maintain a fundamental mode frequency of 29.985 GHz was obtained using HFSS. Q, R'/Q, and group velocity (vg) as a function of b were then calculated. The relationship between a and b is plotted in Figure 2 and Q, R'/Q, and vg are plotted in figure 3. Figure1: Geometry of TDS cell and damping waveguides. The nominal CLIC beam consists of a train of 0.6 nC bunches spaced by 20 cm. Operating the structure at an average accelerating gradient of 150 MV/m results in the loaded and unloaded accelerating gradient profiles plotted in Figure 4. The corresponding power flows are plotted in Figure 5.Figure 2: Relationship between 2b (y axis) and 2a (x axis) for correct fundamental mode frequency. Units are mm. Figure 3: Fundamental mode characteristics as a function of cell number. R'/Q is in k /G3A/m, vg/c is a percentage. Q is nearly constant and has a value of 3600. Figure 4: Loaded and unloaded accelerating gradient [MV/m] as function of cell number.Figure 5: Loaded and unloaded power flows in units of MW as a function of cell number. The highest surface field occurs in the first cell of the structure and has a value of 420 MV/m. For a 150 ns long pulse, a pulsed temperature rise of about 250 /G71C occurs in the cell wall between the waveguides. 3. DIPOLE MODE CHARACTERISTICS. The damping waveguide terminating load is made from silicon carbide. The geometry of the load is shown in Figure 6. The principle behind this compact load, and the method by which the complex permittivity was obtained is described in reference [5]. The amplitude of the reflection coefficient, /G7ES11 /G7E, as a function of frequency was calculated using HFSS - see Figure 7. The transverse wakefield was then calculated using the double-band circuit model described in reference [4]. The result is plotted in Figure 8. The transverse wakefield suppression of the TDS and the validity of the tools used to model it was verified in an experiment in the ASSET facility [6] at SLAC. The calculated and measured wakefields are plotted in Figure 9. Figure 6: Geometry of the 30 GHz load.Figure 7: Amplitude of the reflection coefficient, /G7ES11 /G7E, as a function of frequency in units of GHz. Figure 8: Computed transverse wakefield. Figure 9: Results from the ASSET test. The amplitude of the transverse wake in units of V/(pC /G98mm /G98m) is plotted against time in ns. The test was made using a scaled 15 GHz structure.4. CONCLUSIONS A complete first pass has been made through the design of the TDS, for both the accelerating and the transverse modes. The transverse mode suppression is well within the specifications for CLIC and has been demonstrated experimentally. The situation for the fundamental mode for an accelerating gradient of 150 MV/m is less favourable. The 420 MV/m peak surface electric field at the input of the structure is quite high, the feasibility of which must clearly be demonstrated. The 250 /G71 pulsed surface heating of the cell walls is of even greater concern, since it is probably well above an acceptable value. The temperature rise may be reduced by optimising the cell geometry, in particular by decreasing both the thickness and width of the coupling iris. In this way the coupling of the damping waveguides to the dipole mode can be maintained while decreasing the perturbation of the fundamental mode currents. An overall shift to smaller irises (dimension 2a) and fewer cells - thus maintaining the same RF to beam efficiency - would reduce both the peak surface electric field and the pulsed surface heating in the first cell. Beam dynamics simulations are needed to determine how much increase in the transverse wakefield due to the smaller irises can be tolerated. ACKNOWLEDGEMENTS The authors would like to acknowledge their sincere and deep appreciation to Micha Dehler and Michel Luong, who have both made substantial contributions to the development of this structure. Special thanks are also extended to Erk Jensen for maintaining the circuit model. REFERENCES [1]J.P.Delahaye, I. Wilson, “CLIC, a Multi TeV e/G72 Linear Collider”, CERN/PS 99-062 (LP), November 1999. [2]D.Schulte, “Emittance Preservation in the Main Linac of CLIC”, EPAC’98, Stockholm, June 1996. [3]M.Dehler, I.Wilson, W.Wuensch, “A Tapered Damped Accelerating Structure for CLIC”, LINAC'98, Chicago, August 1998. [4]M.Dehler, “Modeling a 30GHz Waveguide Loaded Detuned Structure for the Compact Linear Collider (CLIC)”, CERN/PS RF 98-09. [5]M.Luong, I.Wilson, W.Wuensch, “RF Loads for the CLIC Multibunch Structure”, PAC’99, New York, March 1999. [6] C. Adolphsen et. al. “Wakefield and Beam Centering Measurements of a Damped and Detuned X-Band Accelerator Structure”, PAC’99, New York, March 1999.
TESTS OF THE CERN PROTON LINAC PERFORMANCE FOR LHC-TYPE BEAMS C.E. Hill, A. Lombardi, R. Scrivens, M. Vretenar CERN, CH-1211 Geneva 23, Switzerland A. Feschenko, A. Liou Institute for Nuclear Research, Moscow, 117312 Russia Abstract As the pre-injector of the LHC injector chain, the proton linac at CERN is required to provide a high-intensity (180mA) beam to the Proton Synchrotron Booster. The results of measurements at this intensity will be presented. Furthermore, the linac is now equipped with bunch shape monitors from INR, Moscow, which have allowed the comparison of the Alvarez tank RF settings with simulations. 1 INTRODUCTION Linac 2 has been in operation since September 1978 and routinely supplies protons during 6700 hours of operation per year. The machine consists of a Duoplasmatron proton source at 90 kV, a 750 keV RFQ, and three Alvarez tanks accelerating the beam to 50 MeV. In normal operation a 170 mA proton beam is injected into the Proton Synchrotron Booster (PSB) with a pulse length up to 150 µs, at a repetition rate of 0.8 Hz [1]. From 2005 onwards, the linac will function as the pre-injector for the Large Hadron Collider (LHC), for which 180 mA is desired [2]. This is difficult to achieve because the longitudinal beam dynamics are strongly space-charge limited at the low energy end of tank 1. Within the framework of upgrading towards this intensity the 750 kV Cockcroft-Walton was replaced with an RFQ in 1993. More recently, three Bunch Shape Monitors ( BSMs) have been installed, to allow the study of the beam dynamics of the Alvarez tanks. 2 HIGH CURRENT TEST S The optimisation of the linac to produce higher output currents was performed during 1999. A comparison of the readings of the current transformers along the length of the linac, for the high current case and during normal operation, is given in Figure 1. It is clear that the principle gain in this case was the 32% higher current from the source. From this higher current, the improvement of 10% was possible at the end of the linac, and the greater losses in the transfer line (after the linac) led to a final improvement at the entrance of the PSB injection line of 6%. The large losses from TRA02 to TRA06 are mostly due to the loss of the H 2+ beam at the entrance of the RFQ. The beam parameters were verified with the single-shot emittance measurement and the spectrometer at the beginning of the PSB injection line. No difference in the emittance or energy spread was seen between the high current case and the normal operation beam.Figure 1. Beam current measured along the proton linac, for the normal operation beam, and during tests of high current. TRA02 :After Source, TRA06:after RFQ; TRA10: after accelerator; TRA20-60; transfer line to the PSB. 3 BUNCH SHAPE MONIT ORS The principle of the Bunch Shape Monitor (constructed by INR, Troitsk) has been fully described in [3]. In short, a bunched ion beam impinges on a wire target held at high voltage, releasing secondary electrons that retain the initial bunch structure of the ion beam. The electrons are swept by a RF deflecting field, which allows the relative ion beam intensity at a given phase to be measured. By re-phasing the RF deflecting field with respect to the linac RF, the ion density distribution can be reconstituted. On Linac 2, three BSMs are now installed (see Figure 2). The first two are standard devices placed in the inter-tank sections between tanks 1-2 and 2-3. At the output of the linac, the 3D- BSM allows selection of a transverse portion of the beam and the measurement of the bunch shape in a single-shot, by an array of charge collectors. The 3D-BSM was installed in 1996 [4], and the results of its first measurements on the proton beam are given in [5]. Figure 2. Scheme of the CERN 202 MHz Linac 2 Alvarez tanks, with the locations of the three BSMs. The settings of the Alvarez tanks were studied by measuring the longitudinal movement of the bunch as a function of the RF phase and amplitude. By treating only the bunch centre, the effects of space charge in the calculation of the motion of the bunch are avoided. The bunch motion was calculated using the simulation tool DELTAT, based on the procedures given in [6].TRA02 TRA06 TRA07 TRA10 TRA20 TRA30 TRA40 TRA50 TRA60050100150200250300350Beam current (mA) Current Transformer During High Current Tests During Normal OperationThe measurements were performed on the linac with the normal operational beam, in a time window from 25-50 µs after the start of the proton pulse from the source. Measurements have shown that before this time, the proton beam is not well stabilised (due to current variation and the time constant of the RF feedback loops. 2.1 Tank 1 Measurements at the output of tank 1 have not been completed. The wire target of the IT-BSM is held at high voltage, but the current being drawn is too high for the HT supply (probably due to small discharges in vacuum). The rectification of this problem requires opening the vacuum of the Alvarez structure, which cannot be performed until the winter 2000-01 shutdown. The comparison of the measured bunch position in phase and the simulation is complicated by the degrees of freedom (the unknown RF amplitude and the unknown offset in the bunch phase). The measured data are shown in Figure 3 along with curves from a simulation, for different RF levels of the tank. Note that the measured data can be arbitrarily offset vertically. The simulation results in the same gradient as the measurement, for RF tank levels of 0.95A 0 , 1.10A 0 and 1.15A 0 (where A 0 is the nominal RF level). The power requirements to run the tank at 10% higher than the nominal values could not be fulfilled by the RF system, so the 0.95A 0 line would be the most likely. It is then estimated that the beam enters the tank with a phase of -30o during normal operation, whereas the initial synchronous phase of the tank is -35o (where 0o is the crest of the RF wave). With no second measurement of the beam of the bunch position as a function of the RF level, the results are not yet conclusive. 2.2 Tank 2 With measurements of the bunch position in phase as a function of the tank 2 RF phase and amplitude, the comparison of the simulated and measured data is much easier. In Figure 4a the measured data are compared to the simulations using DELTAT. The gradient of the measured data is very similar to that of the results obtained for an input phase of –10o, compared to the nominal synchronous phase of –25o. The bunch centre as a function of the tank 2 RF phase is given in Figure 4b, and is shifted along the x-axis such that the nominal phase corresponds to –10o. The simulation fits the measured data well.are Figure 3. Measured data for the phase position of the bunch at the output of tank 1 as a function of the RF phase of tank 1. Curves show simulated bunch position as a function of the RF phase, for different RF amplitudes. Figure 4. Measured data for the phase position of the bunch at the output of tank 2 as a function of a) the RF amplitude and b) the RF phase. Curves show simulated bunch phase position. 2.3 Tank 3 The measurements with the 3D-BSM located after tank 3 have concentrated on the longitudinal dynamics, and the transverse distribution of the beam is not considered here. As the electron bunch is measured with an array of transverse charge collectors, the resolution during the measurements reported here is approximately 4.5o, which is 2 to 4 times lower than the resolution of the BSMs located between the tanks. Improving the resolution results in a narrower total range of phases that can be measured. The measurements of the phase position of the bunch as a function of RF amplitude and phase are given in Figure 5. The simulations are in good agreement with the measured data based on an input phase of –40o compared to a nominal synchronous phase of –25o, and with a RF field level 6% higher than the nominal value.-80 -60 -40 -20 0 20 40 60-230-220-210-200-190-180-170 Nominal Phase Measured Data Sim 0.95A0 Sim 1.0A0 Sim 1.05A0 Sim 1.1A0 Sim 1.15A0Bunch centre (degrees) RF Phase of Tank 1 (degrees) 0.9000.9250.9500.9751.0001.0251.0501.0751.1001.1251.150-230-220-210-200Measured Data φ=-35o φ=-30o φ=-25o φ=-20o φ=-15o φ=-10oBunch centre (degrees) RF Level of Tank 2 (normalised) -40-35-30-25-20-15-10-50510-220-210-200-190 Nominal operating phase Measured Data 0.96A0 0.98A0 1.00A0 1.02A0 1.04A0Bunch centre (degrees) RF Phase of Tank 2 (degrees)Figure 5. Measured data for the phase position of the bunch at the output of tank 3 as a function of a) the RF amplitude and b) the RF phase of Tank 3. Curves show simulated bunch position. 4 CONCLUSIONS It has been demonstrated that the CERN lLinac 2 can provide the 180 mA beam required for the LHC injector chain. The larger resulting losses in this case mean that the beam is not at present used in routine operation at the PS complex.The BSM data compiled for tank 1 are incomplete and require the repair of the BSM and the measurement of the bunch phase position as a function of RF amplitude before final conclusions can be drawn. Tank 2 data are in excellent agreement with simulations of the bunch centre. Tank 3 data shows good agreement with simulations but with a RF level much higher than the nominal value. This exercise provides an excellent starting point for further simulation with a macro-particle code (e.g. PARMILA) to provide more complete understanding of the dynamics and limitations of the structure with higher currents. This should allow the losses at higher currents to be reduced for routine operation as the LHC preinjector . REFERENCES [1] C.E. Hill, A.M. Lombardi, E. Tanke, M. Vretenar, Present Performance of the CERN Proton Linac, Procs. of the 1998 Linear Accelerator Conf., Chicago, IL, p. 427 (1998). [2] K.H. Schindl, The Injector Chain for the LHC, Workshop on LEP-SPS Performance, Chamonix, Jan. 1999, CERN/PS 99- 018(DI). [3] Feschenko A.V., Ostroumov P.N., Bunch Shape Measuring Technique and Its Application for an Ion Linac Tuning, Proc. of the 1986 Linear Accelerator Conference, Stanford, 2-6 June, 1986, pp.323-327. [4] S.K. Esin, V.A. Gaidash, A.V. Feschenko, A.V. Liiou, A.N. Mirzojan, A.A. Menshov, A.V. Novikov, P.N. Ostroumov, O. Dubois, H. Kugler, L. Soby, D.J. Williams, A Three Dimensional Bunch Shape Monitor for the CERN Proton Linac, Proc. of the 1996 Linear Accelerator Conference, Geneva, Switzerland. [5] A.V. Feschenko, A.V. Liiou, P.N. Ostroumov, O. Dubois, H. Haseroth, C. Hill, H. Kugler, A. Lombardi, F. Naito, E. Tanke, M. Vretenar, Study of Beam Parameters of the CERN Proton Linac Using a Three Dimensional Bunch Shape Monitor, Proc. of the 1996 Linear Accelerator Conference, Geneva, Switzerland. [6] K. Crandall, RF Adjustment Techniques Using Relative Measurements, CERN PS/LR/Note 79-3 (1979).0.940.960.981.001.021.041.061.081.101.121.14-220-210-200-190-180-170 Measured Data φ=-200 φ=-250 φ=-300 φ=-350 φ=-400Bunch centre (degrees) RF Level of Tank 3 (normalised) -80 -70 -60 -50 -40 -30 -20 -10 0-220-210-200-190-180-170Nominal operating phase Measured Data 1.00A0 1.03A0 1.04A0 1.05A0 1.06A0Bunch centre (degrees) RF Phase of Tank 3 (degrees)
arXiv:physics/0008103v1 [physics.acc-ph] 17 Aug 2000Measurementof the six DimensionalPhase Space atthe NewGSI HighCurrent Linac P. Forck,F. Heymach,T.Hoffmann,A. Peters, P. Strehl Gesellschaftf¨ urSchwerionenforschung GSI, Planck Stras se1, 64291Darmstadt,Germany e-mail: p.forck@gsi.de Abstract For the characterization of the 10 mA ion beam deliv- ered by the new High Current Linac at GSI, sophisti- cated,mainlynon-interseptingdiagnosticdeviceswerede - veloped. Besides the general set-up of a versatile test bench, we discuss in particularbunchshape and emittance measurements. A directtime-of-ﬂighttechniquewith a di- amondparticledetectorisusedtoobservethemicro-bunch distribution with a resolution of ∼25ps equals 0.3oin phase. For the determination of the energy spread a co- incidence technique is applied, using secondary electrons emitted by the ion passing through an aluminum foil 80 cmupstreamofthediamonddetector. Thetransverseemit- tanceismeasuredwithinonemacropulsewithapepper-pot system equippedwitha highperformanceCCD camera. 1 BEAM DIAGNOSTICSFORTHE LINACCOMMISSIONING At GSI upgradedion sources[1] as well as a new 36 MHz RFQ- and IH-Linac designed for high current operation was commissioned in 1999 [2]. New beam diagnostic de- velopments were necessary due to the high beam power up to 1.3 MW at an energy of 1.4 MeV/u within a macro pulse length of maximal 1 ms. A beam diagnostics bench was installed behind each Linac-structure during the step- wise commissioning, the scheme is shown in Fig.1. Non- destructivedevicesareusedforfollowingtasks: Thetotal current is measured using beam transformer [3] madeofVitrovac6065corehavinga 2×10differentialsec- ondary winding. The resolution is 100 nA at a bandwidth of100kHz. Duetoafeedbackcircuitthedroopislessthan 1 % for5msmacro-pulses. Thebeam energy is determined by a time-of-ﬂight tech- nique using two 50 Ωmatched capacitive pick-ups [4] with 1 GHz bandwidth, separated by 2 m. A precision of ∆W/W = 0.1% isachieved. Thebeamposition is monitoredby digitizingthe powerof the6thharmonics of the rf frequency (216 MHz) of the 4 segmentsofthesepick-ups. Thebeam proﬁle is determined by a residual gas moni- tor [4], where residual gas ions are detected on a 23 strip printed board. For typical beam parameters no signiﬁcant broadeningof the proﬁles due to space chargeinﬂuence is expected. For lower current or shorter macro pulses con- ventionalproﬁlegridsareused. The instruments are now installed behind the last IH2 cavity as well as behind the gas stripper. In the following Figure1: Schemeofthetestbenchasarrangedforthecom- missioningofthe RFQ. wediscussthemeasurementinthelongitudinalplaneusing particle detectorsandofthe transverseemittanceusingth e pepper-potsystem. 2 MEASUREMENT OF BUNCH STRUCTURE Forthecomparisontocalculations,aswellasformatching of different Linac structures the knowledge of the bunch shape is important, but measurements are not as common as for the transverse case. At velocities much below the speed of light the signal on a transverse pick-up does not represent the details of the bunch shape due to the large longitudinal electric ﬁeld component. A comparison of the pick-up signal to the bunch shape measured with the methoddescribedbelowisshowninFig.2foravelocityof β= 5.5% (1.4 MeV/u) to visualize the broadeningof the pick-upsignaldetectingbuncheswithlessthan1nswidth. Wedevelopedadevicewherethearrivaloftheioninapar- ticle detector is measured with respect to the accelerating rf,seeFig.3. Themethoddemandsforlessthanoneionhit perrfperiod. ThisreductionisdonebyRutherfordscatter- ing in a 210 µg/cm2tantalumfoil( ∼130nmthickness)at a scattering angle of 2.5odeﬁned by a collimator with 16 cm distance and Ø0.5 mm apertures to give a solid angle of∆Ωlab= 2.5·10−4. The parameters are chosen to get an attenuation of ∼10−8of the beam current. A high tar- getmassispreferred,sotheenergyspreadforaﬁnitesolid angleislowerthantherequiredresolution. Forourparame- tersthelargestcontributiontotheenergyspreadarisesfr om the electronic stopping in the foil, which amounts e.g. for Arprojectilesto ∼0.25%andforUto ∼0.15%(FWHM)2 ns/div 100 mV/divpick-up FWHM=0.71 nsdiamond Figure 2: Comparison of a pick-up signal to the bunch shape determination using the particle detector setup for a 1.4 MeV/u Ar1+beam 3 m behind the IH2 output. (The 50Ωterminationofthe pick-upleadsto adifferentiation) Labcollimatoramp.start 1stopdisc.TDC disc.delay MCP diamondstart 2disc. accelerator rf thin Ta-targetsec. e from Al-foil θ beam Figure3: SketchofthedesignedTOFmethodforthebunch shape (Sec.2) and phase space distribution (Sec.3) mea- surementwithparticledetectors. at 1.4MeV/u. Moredetailscanbe foundin[5]. A drawbackof this method is the high sensitivity of the tantalum foil due to the heating by the ions energy loss. Therefore, the beam has to be attenuated, which can be done by defocusing. The device is now installed behind the gas stripper and the ﬁrst charge separating magnet so thatanothertypeofattenuationcanbeappliedbychanging thegaspressureorbyselectingadifferentchargestate. By thismeansalso spacechargeeffectscanbestudied. Anotherapproachwouldbetheuseofasupersonichigh density Xenon gas target instead of the Tantalum foil; es- timations of the effect of the larger elongation have to be done. After a drift of ∼1m the scattered ions are detected by a CVD diamond detector [5, 6]. Besides the very low ra- diationdamage,we gainmainlyfromtheveryfast signals, having a rise time below 1 ns. The conversion to logical pulsesisdonebyadoublethresholddiscriminator[7]. The logicalpulsesserveasastartofaVMEtime-to-digitalcon- verter (CAEN V488), where the stop is derived from the 36MHzusedforaccelerating. Thetimingresolutionofthe system is about 25 ps corresponding to a phase width of 0.3o. Asanexample,thebunchstructureofa120keV/uAr1+ beam at the output of the Super Lens (and an additional Figure 4: Example of the bunch shape measurements 2.4 m behindthe Super Lens. On the left bottomthe result for a low current0.1 mA Ar1+beam is shown, on top the cal- culatedemittanceis plotted. On the rightthe measurement andsimulationareshownforahighcurrentof5mAAr1+. drift of 2.4 m) is shown in Fig.4. The two measurements for low (left) and high (right) current show a quite differ- ent bunch shape having a larger, ﬁlamented emittance for the low current case. The particle tracking calculation [8] showsastrongioncurrentdependenceforthelongitudinal emittance. The applied rf power in the cavity counteracts the space charge force for a high current beam. For a low currentﬁlamentationoccursduetothemissingdampingby the space charge. Other experimental results can be found in [2]. 3 MEASUREMENT OFLONGITUDINAL EMITTANCE The main advantage of using particle detectors is the pos- sibility to measure the longitudinal emittance using a co- incidence technique. As shown in Fig.3, a second detector can be moved in the path of the scattered ions. It consists of a 15 µg/cm2Aluminum foil ( ∼50nm) where several secondary electrons per ion passage are generated. These electronsareacceleratedbyanelectricﬁeldof1kV/cmto- wards a micro channel plate equipped with a 50 Ωanode (Hamamatsu F4655-10). The time relative to the acceler- ating rf is measuredas well as the arrival time of the same particle at the diamond detector located 80 cm behind the MCP.Fromthedifferenceintimeofthe individual particles onecangeneratea phasespace plot. An example of such a measurement is given in Fig.5 (left) for a low current Ar beam 2.5 m downstream of the gas stripper. The arrival times at the diamond detector are used as the phase (or time) axis having a width of 1.4 ns equals 18ophase width. The time difference between diamond and MCP is plotted at the y-axis, the width is about0.4ns(FWHM)correspondingtoanenergyspreadof ∆W/W = 1.7%. For a high current beam (5 mA before stripping) a double structure is visible in the bunch pro- ﬁle and an energy broadening to ∆W/W = 2.8% with a clearcorrelationinthephasespace. Heretheattenuationi sFigure 5: Measured longitudinal phase space distribution for a low current Ar-beam (left) and a high current beam (right) 2.5 m behind the stripper. Note that the measured energyspreadmightbetoolarge. done by selecting a high charge state (Ar15+) far from the maximum of the stripping efﬁciency curve (Ar10+). The measured values are larger by a factor of 2 as expected by tracking calculation. Thereforeit is believed that some er - rorscontributetothemeasurement: Havingadriftlengthof only 80 cm between the two detectors and an ion velocity of 5.5% ofthe ions(correspondingto 48nstime ofﬂight) the accuracy in time has to be 25 ps to have a precision of ∆W/Wof0.1%. Theimperfectionsof thedevice,inpar- ticularthelackofhomogeneityoftheacceleratingﬁeldfor theelectronstowardstheMCPeffecttheresolutionintime. Anoptimizationhastobedone. Alargedistance(e.g. 3m) betweenthetwodetectorswouldlowertherequirementfor thetimeresolutionofthedetectors. Recentlyitwasdiscov - ered that there might be some problemsinside the stripper [2]duetoinhomogeneityofthegasjetresultinginawider energyspreadasthedesignvalue. Itisshown,thatthistypeofsetupcanbeusedforthede- terminationof the longitudinalemittance at low ion veloc- ities, but acarefuldesignofthecomponentsisnecessary. 4 MEASUREMENT OFTRANSVERSE EMITTANCE For the measurement of the transverse emittance two de- vices were installed at the diagnostic bench. A conven- tionalslit-gridsystem[4]havingacoordinateresolution of 0.05 mm and an angular resolution of 0.3 mrad. Due to the high beam power, this device can only be used for the lower energypart of the Linac. For high current operation we developed a pepper-pot system capable to measure the emittancewithinonemacro-pulse,see[9]formoredetails. Here the coordinates are ﬁxed by a 45×45mm2copper plate equippedwith 15×15holes with Ø0.1 mm. After a driftof25cmthebeam-letsarestoppedonaAl 2O3screen. Thedivergenceofthebeamiscalculatedwithrespecttothe image of the pepper-pot pattern. This image is created on the screenwith a HeNe laser,whichilluminatethepepper- pot via a pneumatic driven mirror. This calibration elimi- nate systematic errors due to mechanical uncertainties. A Figure6: Screenshotfromthepepper-potdeviceforanAr beam and, as an insert, the projection onto the horizontal plane. Figure 7: Phase space plot of the data shown above. highresolution12bitCCDcamera(PCOSensiCam)trans- mits the digital data via ﬁber optics. A typical image of such a measurement is shown in Fig.6, together with the projectionontothe horizontalorverticalaxis. Thisproje c- tionisusedfortheemittancecalculationwithanalgorithm like forthe slit-griddevice. For a precise measurement the beam width should be large enough to illuminate several holes in the pepper-pot plate (spacing 3 mm). This also avoids overheating of the pepper-pot,aswellassaturationofthelightintensityemi t- tedfromthe screen. Inaddition,a backgroundlevel(about 5 %) has to be subtracted from the data, probably caused by scattered light in the screen. Thereforethe beam optics have to be changed in some cases to use this modern and reliable system for a fast measurement in a high current operation. 5 REFERENCES [1] H.Reich, P.Sp¨ atke andL. Dahl,P.Sp¨ atke , thisproceed ings [2] W.BarthandW.Barth, P.Forckthis proceedings [3] N.Scheider, AIPproceedings 451, p. 502 (1998). [4] P. Forck, A. Peters, P. Strehl, AIP proceedings of the Bea m Instrumentation Workshop, Boston(2000). [5] P.Forcketal.,proceedings ofthe 4thDIPAC,Chester,p.176 (1999). [6] E. Berdermann et al., Proc. XXXVI Int. Winter Meeting of Nucl. Phys.,Bormio (1998). [7] C. Neyer, 3rdWorkshop on Electronics for LHC Experi- ments, London, CERN/LHCC/97-60(1997). [8] A.Schempp, LINAC96, procedings, Geneva (1996.) [9] T. Hoffmann et al., AIPproceedings of the Beam Instrumen - tationWorkshop, Boston(2000).
arXiv:physics/0008104v1 [physics.acc-ph] 17 Aug 2000Superconducting Superstructure forthe TESLACollider: Ne wResults N.Baboi, M.Liepe, J. Sekutowicz Deutsches Elektronen-SynchrotronDESY, D-22603Hamburg, Germany, M.Ferrario, INFN, Frascati, Italy Abstract A new cavity-chainlayout has been proposedfor the main linacoftheTESLAlinearcollider[1]. Thissuperstructure - layoutisbaseduponfour7-cellsuperconductingstanding- wave cavities, coupled by short beam pipes. The main advantages of the superstructure are an increase in the ac- tive accelerating length in TESLA and a saving in rf com- ponents, especially power couplers, as compared to the present 9-cell cavities. The proposed scheme allows to handle the ﬁeld-ﬂatness tuning and the HOM damping at sub-unit level, in contrast to standard multi-cell cavitie s. The superstructure-layout is extensively studied at DESY since 1999. Computations have been performed for the rf properties of the cavity-chain, the bunch-to-bunch energy spread and multibunch dynamics. A copper model of the superstructure has been built in order to compare with the simulations and for testing the ﬁeld-proﬁle tuning and the HOM damping scheme. A ”proof of principle” niobium prototype of the superstructure is now under construction andwill betestedwithbeamat theTESLATest Facilityin 2001. In thispaperwe presentlatest resultsof these inves- tigations. 1 INTRODUCTION The cost for a superconducting linear collider can be sig- niﬁcantlyreducedbyminimizingthenumberofmicrowave components, and increasing the ﬁll factor in a machine. Here the ﬁll factor is meant as a ratio of the active cavity lengthtothetotalcavitylength(activelengthplusinterc on- nection). These two conditions become partially fulﬁlled when the number of cells ( N) in a structure -fed by one fundamentalmode(FM)coupler-increases. Unfortunately therearetwolimitationsonthecell’snumberin oneaccel- eratingstructure: ﬁrstly the ﬁeld ﬂatness-the sensitivit yof theﬁeldpatternincreasesproportionalto N2-andsecondly trapped higher order modes (HOM). In order to overcome these limitations on N, the concept of the superstructure has been proposed for the TESLA main linac [1]. In this conceptfour7-cellcavities(sub-units)arecoupledbysho rt beamtubes. ThewholechaincanbefedbyoneFMcoupler attached at one endbeamtube. The lengthof the intercon- nections between the cavities is chosen to be half of the wave length. Thereforethe π-0 mode ( πcell-to-cell phase advanceand0cavity-to-cavityphaseadvance)canbeused for acceleration. In the proposed scheme HOM couplers canbeattachedto interconnectionsandtoendbeamtubes. All sub-units are equipped with a tuner. Accordingly the ﬁeld ﬂatnessandtheHOMdampingcanbestill handledat the 7-cellsub-unitlevel.2 REFILLING OFCELLS AND BUNCH-TO-BUNCHENERGY SPREAD The energy ﬂow through cell-interconnections and the re- sultingbunch-to-bunchenergyspreadhasbeenextensively studied forthe superstructurewith two independentcodes: HOMDYN [2] and MAFIA [3][4]. Negligible spread in the energy gain, smaller than 6·10−5for the whole train of 2820 bunches, proofs that energy ﬂow is big enough to re-ﬁllcellsinthetimebetweentwosequentialbunches;see Fig. 1. The energy spread results from the interference of theacceleratingmodewithothermodesfromtheFMpass- band. Thedifferencein energybecomessmallerat theend ofthepulse duetothe decayoftheinterferingmodes. Figure 1: Calculated energy gain for 2820 bunches accel- eratedbytheproposedsuperstructure. 3 FIELD FLATNESS TUNING Theπ-0 mode will be used for the accelerationof beam in thesuperstructure. Beforeassembly,eachofthefour7-cel l cavities will be pre-tuned for ﬂat ﬁeld proﬁle and the cho- sen frequency of the π-0 mode. The pre-tuning procedure is basedon measurementsof all modesofthe fundamental modepassband. Itallowstoadjusttheproﬁlewithaccuracy of better than 2-3 %for a 9-cell TESLA cavity. This error correspondstoa frequencyaccuracyoftheindividualcells of±30kHz. Afterthecavitychainofasuperstructurehas been assembled and is operated in the linac at 2K, the fre- quencyofeachsub-unitcanbecorrectedinordertoequal- ize the mean value of the ﬁeld amplitude in all sub-units (not between cells within one sub-unit). This ﬁeld proﬁle correctionispossibleduringthelinacoperation,sinceea ch 7-cell structure is equipped with its own frequency tuner. The method proposed to equalize the average accelerating ﬁeld of sub-units during operation is based on perturba- tion theory, similar to the standard bead-pull method ofL. Maier and J. Slater [5]. At ﬁrst, present ﬁelds of all sub-units are measured. For that, successively, the volume of each sub-unit is changed by the same amount (stepping motor of each tuner will be moved by the same numberof steps) to measure the frequency change of the π-0 mode. The changeis proportionalto the stored energyin the sub- unitofthesuperstructure. Foreachsub-unitrelativevalu es can be deﬁned and used to calculate frequencycorrections needed to equalize the ﬁeld. This method has been tested on a room temperature Cu model of the superstructure - see Fig. 2- and by computer simulations, see Fig. 3. One shouldnotethatthemethodrequiresonlyonepickupprobe for all 28 cells, and therefore effectively reduces the num- bersofcables,feedthroughsandelectronicsneededforthe control. cell 1 cell 8 cell 15 cell 21 cell 2800.511.5 field amplitude [arb. unit] cavity 1 cavity 2 cavity 3 cavity 4 bead pull measurement calculated from cavity−frequency perturbation measurement Figure 2: Field proﬁle before ﬁeld ﬂatness tuning. Shown is a comparison between the measured ﬁeld proﬁle (bead pulling on a Cu model of the superstructure) and the ﬁeld proﬁle calculated from the measured frequency perturba- tionsoftheindividualcavities. 0.940.950.960.970.980.991amplitude (arb. unit) cavity #with tuning of cavities without tuning 1 4 2 3 Figure3: Exampleofﬁeldﬂatnesstuningbytuningthein- dividual cavities (computersimulation). For the frequenc y ofthe individualcellsa variationof ±30kHzis assumed. 4 STATISTICS OF FIELD FLATNESS As discussed above the ﬁeld ﬂatness in a cold superstruc- turecanbehandledatthe7-cellsub-unitlevelbyadjustingthe frequency of each sub-unit. In order to verify this, the ﬁeld ﬂatnessina superstructurehasbeencalculatedbefore andafter tuningofthe individualcavities. The frequencie s of the cavities have been corrected accordinglyto the pro- posed tuning method. For the frequency of the individual cells a variationof ±30 kHz is assumed, based on the ex- perience with the TESLA 9-cell cavities. The statistics of 10000 calculated ﬁeld proﬁles is shown in Fig. 4. By ad- justing the frequencies of the individual cavities the ﬁeld unﬂatnessissigniﬁcantlyreduced. 0.8 0.85 0.9 0.95 10100200300400500600700800 Emin/Emax#0.8 0.85 0.9 0.95 1100200300400500600700800 # Emin/Emax(a) (b)0 Figure 4: Calculated ﬁeld ﬂatness statistics of 10000 su- perstructures before (a) and after ﬁeld ﬂatness tuning by adjustingthefrequenciesoftheindividualcavities(b). T he frequenciesoftheindividualcellsvariesby ±30kHz. 5 HOMDAMPINGANDMULTIBUNCH EMITTANCE The vertical normalizedmultibunchemittance at the inter- actionpointoftheTESLAcolliderisdesiredtobe 3·10−8 m·rad. Simulations of the emittance growth along the TESLA linac showed, that the dipole modes with domi- nating impedance (R/Q) should for that be damped to the level of Qext<2·105[6]. The interconnecting tubes of the superstructureallow to putHOM couplersbetweenthe 7-cell cavities. Measurementson a Cu modelof the super- structure at room temperature have demonstrated, that the requireddampingcanbeachievedwithﬁveHOMcouplers: three attached at the interconnectionsand one at bothends [7];seeFig.5. Notethatthesumofalllisteddipolemodes impedancesisalmosttentimessmallerthantheBBUlimit.Figure 5: Measured impedance values for dipole modes with hi gher R/Q. The impedanceshave been measured on a Cu modelofthesuperstructure. Forcomparisonalsothelimit i sshown,basedonbeamdynamicssimulation. 6 NB PROTOTYPE Aﬁrst”proofofprinciple”niobiumprototypeofthesuper- structureisunderconstruction[8]. Thesub-unitsareunde r fabricationand will be vertical tested similar to TESLA 9- cell cavities. The beam test for the prototype is scheduled forSpring2001. Itwillallowtoverifyenergyspreadcom- putations and RF measurements on the room temperature models. This will include the test of the HOM damping, the performance of the HOM couplers at higher magnetic ﬁeld andthe tuningmethodduringoperationat 2K. 7 CONCLUSIONS Thepresentedmeasurementsandcalculationsdemonstrate, thatintheproposedsuperstructurethereﬁllingofcells,t he HOM damping,the ﬁeld ﬂatnessand the ﬁeld ﬂatness tun- ingcanbehandled. Fortheﬁnalprove,thatthesuperstruc- ture layout can be used for acceleration, a niobium proto- type will be tested with beam at the TESLA Test Facility linac. 8 ACKNOWLEDGEMENTS This work has beneﬁted greatly from discussions with the membersofthe TESLAcollaboration. 9 REFERENCES [1] J. Sekutowicz, M. Ferrario and Ch. Tang, Phys. Rev. ST Ac- cel. Beams,vol. 2,No. 6(1999) [2] M. Ferrario,. A. Mosnier, L. Seraﬁni, F. Tazzioli, J. M. Tessier,ParticleAccelerators, Vol. 52(1996) [3] R. Klatt et al., Proc. of Linear Accelerator Conference, Stan- ford, June 1986 [4] M. Dohlus, private communication[5] L. Maier and J. Slater, Journal of Applied Physics, Vol. 2 3, No. 1,January 1952, page 68-77 [6] N. Baboi, R. Brinkmann, M. Liepe, J. Sekutowicz, EPAC2000, Viena,tobe published [7] H. Chen, G. Kreps, M. Liepe, V. Puntus and J. Sekutowicz, Proc.of the 9thWorkshopon rfSuperconductivity, Santa Fe, 1999 [8] R.Bandelmann etal.,Proc.ofthe9thWorkshoponrfSuper - conductivity, Santa Fe,1999
Beam Dynamics Studies for a High Current Ion Injector1 A. Sauer, H. Deitinghoff, H. Klein, Institut für Angewandte Physik, Universität Frankfurt am Main, Germany 1 Work supported by the European Commission. Abstract Recent ion source developments resulted in the generation of high brilliance, high current beams of protons and light ions. After extraction and transport the beams with large internal space charge forces have to be captured, bunched and preaccelerated for the injection into the following driver part of a new generation of high intensity beam facilities for neutron sources, energy production, transmutation e.g. A combination of RFQ and DTL is considered to be a good solution for such a high current ion injector. Some preliminary beam dynamics layouts have been investigated by multiparticle simu- lations. Basic parameters like frequency, ion energy and sparking have been varied for the International Fusion Material Irradiation Facility (IFMIF) scenario as an example. The main interest was directed to high transmis- sion, low losses and emittance conservation. The beam matching to the RFQ is shortly discussed as well as the matching between RFQ and DTL. 1 INTRODUCTION During the last years the feasibility of a new generation of high intensity ion accelerator facilities has been studied. They consist of an rf linac, in some cases followed by rings for postacceleration, beam accumulation or pulse compression. Well known examples are drivers for spal- lation neutron sources, transmutation of radioactive waste, energy amplifier e.g. A smaller scale project is a material test machine (IFMIF) [1]. In all proposed schemes the linac starts with high current ion sources followed by RFQ accelerators and drift tube linacs as initial part, in high energy beam facilities coupled cavity linacs or synchrotrons take over for further acceleration. The requested beam peak currents are in the range of some 10 mA to more than 100 mA of light and heavy ions. A high quality of the beam formation in the injector part is essential for the whole complex, because small emittances and well confined bunches are needed to avoid losses and activation along the accelerator. One critical aspect is the matching of the beam between the different parts of the driver, which will be discussed in the following for IFMIF. 2 THE IFMIF INJECTOR PART Fig. 1 gives a scheme of the proposed IFMIF facilities: Currents of up to 0.5 A D+ ions with variable final energies between 16 and 20 MeV/u are used to produce very high fluxes of neutrons for testing the wall materials for magnetic fusion reactors. Figure 1: Basic layout of the IFMIF facility. While in high energy machines the required high beam current at the injector end is generated by combining beams from 2 or more ion sources in funnelling steps, in the IFMIF case 2 to 4 accelerators in parallel are planned due to its comparatively small size. Each of them has to capture, focus and accelerate a beam of 125 mA in cw operation. Beam losses must be kept as small as possible to avoid activation, which becomes serious for beam energies higher than 1.0 MeV/u. A base line design has been worked out for the accelerator, parameters are listed in Table 1 [1]. Table 1: Base line parameters of the IFMIF injector Source RFQ DTL Iout [mA] 150 140 125 Wout [MeV/u] 0.05 4.0 20.0 εεRMS,N out [cm××mrad] 0.02 0.04 0.04 Length [m] 4.2 11.7 30.4 Frequency [MHz] 175 175 175 Beam Power [MW] 5.0 each A first beam dynamics layout of the RFQ employing equipartitioning rules has been carried out successfully already in 1996 [2]. Also a design of a DTL for 4 MeV/u was made. Recently two parameters of this base line design have been reconsidered: 1) The electrode voltage assumed for the RFQ was rather high: applying the Kilpatrick criterion for rf breakdown in the form , KE K ecmMVE MHzf085.0 24]/[10643.1][− ⋅ ⋅×= , [1] a Kilpatrick factor of 2.1 came out, which is reasonable for pulsed operation but considered as too high for cw operation of a structure. Therefore new designs have been attempted with a lower value of 1.7 Kilpatrick. 2) The RFQ is highly efficient in capturing and focusing high intensity beams, but the acceleration rate is rather low. Therefore the end energy of the RFQ of 4 MeV/u should be lowered to 2.5 MeV/u. 3 BEAM DYNAMICS LAYOUT OF RFQ AND DTL AT 175 MHZ In the baseline design the RFQ has to accelerate a 125 mA Deuteron beam from 50 keV/u to 4 MeV/u. One possible layout was found at 2.1 Kilpatrick introducing some equipartitioning rules into the design. The calculated beam behaviour was very satisfying: high transmission, small emittance growth and a well compressed bunch of small phase width, which is well prepared for direct injection into the DTL. Table 2: Design parameters of the different IFMIF 175 MHz RFQs 175 MHz RFQ design studies Ion D+ D+ D+ Design EP (Deshan) EP (Jameson) Conserv. (Sauer) N 5000 5000 5000 Input dist. 4d Wbag 4d Wbag 4d Wbag F [MHz] 175 175 175 Win /Wout [MeV/u] 0.05 / 2.5 0.05 / 2.5 0.125 / 2.5 V [MV] 0.100 - 0.164 0.111 - 0.151 0.126 b 2.13 1.7 1.7 RMSin 4 4 6 RMSout Crandall cell Crandall cell Crandall cell C / L [m] 382 / 8.21 660 / 12.32 565 / 14.52 φφsyn [°] -90. - -31. -90. - -35.3 -90. - -37. Iin [mA] 140 140 140 Iout [mA] 133.2 129.7 127.50 Tr 95,2% 92,7% 91,0% m 1.00 – 1.78 1.00 – 1.74 1.00 – 1.62 εεN,RMS trans [cm××mrad] 0.020 / 0.030 0.020 / 0.0226 0.021 / 0.0255 εεN,RMS long [cm××mrad] 0 / 0.057 0 / 0.046 0 / 0.054 Lowering the electrode voltage corresponding to a Kilpatrick factor of 1.7 made new designs necessary, at the same time the output energy of the RFQ was fixed to 2.5 MeV/u. Up to now the most promising design was found by R. Jameson employing equipartitioning again [3]. The third solution of Table 2 (Sauer) with simpler design methods following the rules of CURLI and RFQUICK have been obtained at higher input energies, nevertheless showing, that an RFQ design with a constant and modest voltage level is feasible. All examples from Table 2 show low losses and low emittance growth but a larger phase width of the bunch, the same is true for the 2.1 Kilpatrick design, if cut at 2.5 MeV/u. The matching of the RFQ input emittances has been done with the help of PTEQ [4] which delivers optimised Twiss parameters of the ellipses and a smoothed beam behaviour along the RFQ. This is an idealized assumption: Former inves- tigations on the matching to an RFQ with the help of a magnetic LEBT have shown [5], that small changes in current, solenoid fields and emittances in the order of a view percent do not cause losses but generate a large emittance growth in the RFQ. Therefore quite larger emittances than the ideal ones may be expected at the RFQ end. A provisional layout of an DTL has been done with SUPERFISH and PARMILA, to check the matching possibilities at the lower input energy of 2.5 MeV/u. The design was made for constant transverse focusing, constant rf field amplitude and some initial phase ramping. In the very first part of the DTL the filling factor had to be risen to 0.59 at pole tip fields below 1.0 T. Afterwards it was kept to 0.5 and lower. After some initial checks of the beam dynamics with matched input emittances generated by the program TRACE3D which showed a good beam behaviour, RFQ output emittances calculated with PARMTEQM were taken as an input to PARMILA. In all cases a short drift of βλ/2 (6.2 cm) was assumed between RFQ and DTL. Emittance growth was unavoidable in that case. Fig. 2 shows the calculated output emittances of the DTL at 20 MeV/u, with RFQ 3 of Table 2 as the previous accelerator. Figure 2: Output emittance of the 175 MHz DTL with direct input from the RFQ 3 from Table 2 and a 6.2 cm drift between RFQ and DTL. 4 MEBT DESIGN In the following a compact matching section (MEBT) with a length below 1.0 m is discussed. The design was made with TRACE3D which calculates the optimised field strength of the four magnetic quadrupoles (for transverse matching) and the two rebuncher cavities (for longitudinal matching). The result is a mismatch factor between RFQ and DTL smaller than 1 from first 2.5 without MEBT. Transfer of this MEBT to PARMILA and injection of the RFQ output emittances via matching line into the DTL showed a large improvement in the beam behaviour as can be seen in comparing Fig. 2 and Fig. 3. Figure 3: Output emittances of the 175 MHz DTL with a MEBT of 0.686 m between RFQ and DTL. The beam is quite nicely matched along the DTL with an rms emittance growth of 10 %, no losses and well shaped output emittances. Moreover such a design allows the installation of steerers as well as beam diagnostics and of a valve and bellows, which is important in the practical view of an accelerator facility [7]. 5 BEAM DYNAMICS LAYOUT OF RFQ AND DTL AT 120 MHZ Layout of RFQ and DTL have been repeated for a lower frequency of 120 MHz, which gives for the 4-Rod- RFQ still small geometrical dimensions [6]. The ad- vantage is a lower rf defocusing term and higher focusing efficiency, therefore RFQ layouts for 1.7 Kilpatrick are more easily reached for the reference energy also with the conventional design method. A transmission of more than 90 % with 40 % rms emittance growth were achieved. At 2.5 MeV/u the relation between longitudinal output emittance of the RFQ and the acceptance of the DTL is nearly the same as in the case at 175 MHz. Again the length of the RFQ output bunch of about 70° is too large for direct injection into the DTL. With the same matching procedure as before using TRACE3D and βλ/2 (which is now 9.1 cm) as basic length for drift, rebuncher cavities and quads a MEBT of 1.0 m has been designed for a low mismatch factor smaller than 1 again. Fig 5. shows the output emittances after the DTL, when the calculated output emittances of the 120 MHz RFQ were taken as input to MEBT and DTL for PARMILA. Again 100% transmission and an rms emittances growth of about 10% only were obtained for MEBT and DTL. Figure 4: Output emittances of the 120 MHz DTL with a MEBT of 1.001 m between RFQ and DTL. 6 CONCLUSIONS In the IFMIF scenario a beam dynamics layout for 125 mA D+ RFQ at 1.7 Kilpatrick. was successfully carried out for both frequencies 175 MHz and 120 MHz resp. While for a RFQ with high end energy and a very well bunched output beam with a small phase width of about ±20° a direct injection into the DTL shows smooth beam behaviour, low emittance growth and no losses. A matching line seems to be favourable in the case of a shorter RFQ with lower end energy and larger phase width of the output bucket to prevent filamentation and large emittance growth in the beam. In addition a matching line provides the possibility for beam diagnostics and adjusting devices. The advantages and disadvantages of both cases have to be studied carefully, especially the sensitivity against errors and initial mismatch, which may strongly influence the size and shape of the RFQ output beam. REFERENCE S [1] IFMIF Final Report, December 1996 (1997). [2] D. Li, R.A. Jameson, “Particle Dynamics Design Aspects for an IFMIF D+ RFQ”, EPAC’96, Sitges, June 1996. [3] R.A. Jameson, “Some characteristics of the IFMIF RFQ KP1.7 designs”, IFMIF Memo, RAJ-24-May- 2000. [4] R.A. Jameson, “A discussion of RFQ LINAC Simulation”, LA-CP-97-54, September 1997. [5] A. Sauer, Dipl. Thesis, Universität Frankfurt, 1998. [6] A. Schempp, “Advances of Accelerator Physics and Technologies”, Ed. H. Schopper, World Sci., 1993. [7] U. Ratzinger, private communications, August 2000. This figure "fig1.png" is available in "png" format from: http://arxiv.org/ps/physics/0008105v1This figure "fig2.png" is available in "png" format from: http://arxiv.org/ps/physics/0008105v1This figure "fig3.png" is available in "png" format from: http://arxiv.org/ps/physics/0008105v1This figure "fig4.png" is available in "png" format from: http://arxiv.org/ps/physics/0008105v1
arXiv:physics/0008106v1 [physics.acc-ph] 17 Aug 2000HIGH CURRENT BEAMDYNAMICS INAN ESSSC LINAC M.Pabst,K. Bongardt,ForschungszentrumJ¨ ulichGmbH,Ger many A. Letchford, RAL, Didcot,U.K. Abstract Three alternative designs of the European Spallation Source (ESS) high energy linac are described. The most promisingonesareeitheranormalconducting(nc)coupled cavity linac (CCL) up to ﬁnal energy or a change at 407 MeV to only one group of 6 cell superconducting (sc) el- liptical cavities. Fully3dMonteCarlosimulationsarepresentedforboth options, optimized for reduced halo formation at the linac end. For the error free matched case, especially the halo formation in the longitudinal plane is more pronounced for the hybrid solution with its superconducting cavities, caused by the unavoidable phase slippage, but still quite well acceptable for loss free ring injection. Simulations however for a 30% mismatched dense core, surrounded in addition by 1.5% halo particles are showing few parti- cles with very large amplitudes even in real space. This case represents halo formation in front to end simulations, causedby currentﬂuctuations,ﬁlamentedRFQ outputdis- tributionandenhancedbyaccumulatedﬁelderrors. 1 OPTIONSFORTHEHIGHENERGY PARTOF THE6%D.C. ESS LINAC The current reference design of the ESS contains a 1.334 GeVH−linacwitha5%dutycycle,a50Hzrepetitionrate andapeakcurrentof114mA.Thebeamcurrentischopped witha70%dutycycle[1]. Theradiofrequencyis280MHz for the two front end RFQs and DTLs. After funneling at 20 MeV ﬁnal acceleration to 1.334 GeV is accomplished in a nc CCDTL andCCL operatingat 560MHz. A sc ver- sion of the high energy linac is also being studied. The 1 msec long linac pulse is injected into 2 compressor rings, to producea ﬁnalbeampulselengthof1 µsec. Any design of the ESS high energy linac must ensure loss free ring injection. This demands an unﬁlamented 6d phasespacedistributionforthelinacbeam. Table 1, lists 3 high energy,6% duty cycle, linac design optionsfora 107mA, 60%choppedbeamusing700MHz structures. These were the linac parameters from the ESS study[2]. The700MHzlinacfrequencyisalsothesameas considered for the CONCERT [3] multi-user facility. The followingconclusionsarevalidforboth560and700MHz frequencies, but they are limited to linacs with about 6% dutycycle. For all three options, a doublet focusing system with warm quadrupolesis assumed either after 2 nc cavities [2] or after ( 2, 3, 4 ) 6-cell elliptical sc cavities. The acceler - ating gradient is kept constant at EoT= 2.8 MV/m for the nc cavities resp. Eo= ( 5 MV/m, 8.50MV/m, 13.7MV/mTable 1: Options for the high energy part of the ESS 6% d.c.,64mApulsecurrent,700MHzlinac Normal conducting (nc)linacSuper- conducting (sc)linacHybrid solu- tion Energy range105 - 1334 MeV: CCL120-1334 MeV: β=0.52, 0.65,0.8105-407 MeV: CCL >407 MeV: β= 0.8,sc Total length631m 493m 148m(nc) +267m(sc)= 415m # of cavi- ties232 212 62(nc) + 116 (sc) = 178 # of klystrons116 212 31(nc) + 116 (sc) = 147 Peak RF power per klystron2MW 0.4MW, 0.75MW2 MW(nc), 0.75MW(sc) Total peak RF Power232MW 101MW 137MW # of circu- latorsNONE 212 116 Cryogenic powerNONE 4MW 3 MW ) for the 3 sc cavities. Two power coupler/ cavity are as- sumed for the β= 0.8sc cavities. The synchronousphase iskeptconstantat −25ointhenccells,whereasonlythesc cavity midphase can be kept constant at −25oas a conse- quenceofphaseslippage. Allsccavitiesareassumedtobe madeoutof6identicalcells. Therelative βdependenceof the transit time factoris the same as forthe SNS 805MHz sc high β= 0.766-cell cavities [4], obtained from super- ﬁsh calculations where end ﬁeld effects are included. The average transit time factor is smaller by at least 10% than theπ/4 = 0 .79value of β= 1sc elliptical cavites. The averagesynchronousphasepercavityissmallerthan −37o at beginningresp. endofeachsc section. It is obvious from table 1, that a pure nc ESS linac ver- sion from 105 MeV on is the cheapest in capital cost, but not in operating cost. A sc ESS linac from 120 MeV on requires much less peak RF power, but it is in capital cost quite expensiveand substantial R & D is necessary for the 50Hzpulsedmodebehaviourof β= 0.52ellipticalsccav- ities [5, 6], including the ESS 2.7 msec long pulse option[2]. The hybridsolution with its two couplers/cavityis the shortest and the cheapest one for capital plus 20 years op- erating time cost. By having only one coupler/cavity the ESS hybrid linac will be longer than the correspondingnc one. Detailedpulsedpowertestswith2couplers/cavityare foreseen for the 500 MHz, β= 0.75sc cavity teststand at FZ J¨ ulich [7]. The open questionsare halo formationat the end of the ESS hybrid linac, resulting from the phase slippageandenhancedbymismatch. 2 MULTIPARTICLERESULTS FORTHE ERRORFREE MATCHED ESS LINAC In Fig. 1, 2 results from Monte Carlo simulations are shown for the ESS nc linac at 105 MeV injection and at the 1334 MeV ﬁnal energy. All simulations are done with 10000fullyin3dinteractingparticles. The700MHzbunch currentis107mA,thenormalizedrmsemittancesare 0.3π mmmradresp. 0.4πoMeV. Theratiobetweenthefulland zero current tune is greater than ( 0.6, 0.5 ) transversely resp. longitudinally. The ratio between the transverse and longitudinaltemperatureintherest systemis0.66atinjec - tion and about 1.3 at the linac end. All zero current tunes are below 90o. The rms radii are about 3 mm at injection resp. 2 mmat theﬁnal energy. The results in Fig. 1, 2 are obtained for the error free matched case. The upper row corresponds to an 6d wa- terbag input distribution limiting each particle coordina te to√ 8ofitsrmsvalue. AstheESShighintensitynclinacis designed to avoid all kind of instabilities, driven either b y high space charge, temperature anisoptropy or resonance crossing[8],almostnohaloformationisvisibleatthelina c end: there are no particles outside 20ǫrmsat the linac end. Thermsemittancesare changingbyless than10%. -10 0 10 x (mm)-404x' (mrad) -20 0 20 Phase (deg)-101Energy (MeV) -10 0 10 y (mm)-404y' (mrad) -10 0 10 x (mm)-404x' (mrad) -10 0 10 y (mm)-404y' (mrad) -20 0 20 Phase (deg)-101Energy (MeV) Figure1: InputdistributionfortheESSlinacwithmatched input. Upperrowwithout,lowerrowwith1.5%initialhalo In the lower row 1.5% halo particles are placed ini- tially outside the dense core at the surface of a 6d phase space boundary with 16ǫrms. Each particle coordinate is now limited to 4 times its rms value and there are in each 2d phase space projection less than 10−2particles outside-10 0 10 x (mm)-101x' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) -10 0 10 y (mm)-101y' (mrad) -10 0 10 x (mm)-101x' (mrad) -10 0 10 y (mm)-101y' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) Figure 2: Output distribution for the nc linac for matched input. Upperrowwithout,lowerrowwith1.5%initialhalo 10ǫrms. About 1% halo particles above 10ǫrmsare found in simulations of the 2.5 MeV ESS RFQ [9]. Phase space correlationsbetweenhaloparticlesofa bunchedbeamina periodic focusing channel are reported before [8, 10]. At the ESS nc linac end there are now about 10−3particles outside 20ǫrms, going up in phase space to about 40ǫrms. But still all particles are limited to ±10mm at the linac endwhich isless thanhalfof theassumed 22mmaperture radius. Therearelessthan 10−3particlesoutside ±1MeV. In Fig. 3 the output distributions are shown for the er- ror free matchedESS hybridlinac. Againthe upperrow is assuming an initial 6d waterbag distribution without halo, whereas the lower row assumes a distribution with 1.5% halo. For a constant transverse full current tune of 45oin the sc cavity section the ratio between the full and zero current tune is greater than ( 0.6 , 0.67 ) transversely resp. longitudinally. Theratio betweenthe transverseandlongi - tudinal temperature in the rest system is 0.36 at 407 MeV and about 1.1 at the linac end. All zero current tunes are below 90o. Thermsradiiareabout2mmat407MeVresp. 1 mmattheﬁnal energy. -10 0 10 x (mm)-101x' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) -10 0 10 y (mm)-101y' (mrad) -10 0 10 x (mm)-101x' (mrad) -10 0 10 y (mm)-101y' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) Figure 3: Output distribution for the hybrid linac. Upper rowwithout,lowerrowwith1.5%initialhaloBy comparing the nc output distribution in Fig. 2 with the hybrid one in Fig. 3 much more halo formation espe- cially in the longitudinal plane is visible at the end of the ESS linac. Less than 10−3particles are outside ±2MeV. Theinputdistributionwithinitially1.5%haloparticlesw ill lead to single particle amplitudes up to 9 mm at the ESS hybridlinac end,well outsidethe 6 mm boundaryvalueof twicethebeamcoresize,predictedbyparticle-coremodels [11]. The reason is the phase slippage especially at begin- ningandendofthescsection,wherethebeamvelocitydif- fersby±15%fromthecavitydesignvelocity. IntheMonte Carlosimulationsallparticlesareexperiencethechangeo f thesynchronousphasefromcelltocellinthe6-cellcavity. As a consequence, even the rms beam radii are oscillating along the ESS hybrid linac, as the nc to sc 6d phase space matchingisdonefortheaveragecavitysynchronousphase. 3 MONTE CARLOSIMULATIONSOF THEMISMATCHED ESS LINAC In Fig. 4 phase space distributions are shown at the ﬁnal 1334MeVenergyfortheESSncresp. hybridlinac,assum- ing a mismatched input distribution with 1.5% initial halo particles. The upper row is showing the nc and the lower row the hybridlinac outputdistributions. Excitedis a pure high mode with 30 % radial and 20% axial mismatch of the 3 bunch radii. The high mode oscillation frequencyof about 160o/period [8] causes halo formation in all 3 phase space planes as single particle have initial frequencies of halfthe highmodeoscillationfrequency. -10 0 10 x (mm)-101x' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) -10 0 10 y (mm)-101y' (mrad) -10 0 10 x (mm)-101x' (mrad) -10 0 10 y (mm)-101y' (mrad) -10 0 10 Phase (deg)-404Energy (MeV) Figure 4: Output distribution for the nc linac (upper row) and the hybrid linac (lower row). The mismatched input distributionis surroundedby1.5%initial halo. By assuming the same 30% pure high mode excitation, but foran initialdistributionwithouthaloparticles,the not shown resulting phase space distributions for both, the nc andthe hybridversionofthe ESS highintensitylinac look quite similiar to the shown distributionin Fig. 4. The only difference are somewhat less particles nearby the bunch core. But all requirements for loss free ring injection are fullﬁlled: the ﬁnal normalized transverse rms emittance issmaller than 0.4πmm mrad and there are about 10−3par- ticles outside 20ǫrms. Longitudinally there are less than 10−3particles outside ±2MeV , which is acceptable for energyspreadreductionbythebunchrotationsystem. Adding initially 1.5% halo particles to the 30% mis- matched dense core, the linac output distributions in Fig. 4 show a few particles withe quite large amplitudes even in real space. Studies are going on for the motion of these particles in the linac to compressor ring transfer line , sti ll effected by space charge forces [2]. Unconstrained hands on maintenance requires less than 1 W/m uncollected de- posited beam power at linac end and at ring injection.This valuecorrespondstolessthan 10−7/muncollectedparticle lossfortheESS acceleratorfacilitywithits5MWaverage beampower. In a periodic focusing system correlated ﬁeld errors of ±1%even for same limited periodes can cause noticable mismatch later on [12]. In front to end simulations of the complete ESS linear accelerator, including the chop- pingandfunnellines,thehaloformationatthelinacendis causedby currentﬂuctuations,ﬁlamentedRFQ outputdis- tributionandenhancedbyaccumulatedﬁelderrors. Theso resultinghaloformationcanbeestimatedbyassuming30% mismatch of an unﬁlamented beam, surrounded by 1.5% halo particles, at the entrance of the error free high energy linac section. 4 REFERENCES [1] I. S. K. Gardner et al., ’Revised Design for the ESSLinac’ , ESSrep.,ESS99-94-A, Aug 1999 [2] ’The European Spallation Source ESS Study’, Vol 1-3, March 1997; I. S. K. Gardner et al, Proc. PAC 97, Vancou- ver, Canada, p. 988 [3] J. M. Lagniel for the CONCERT Project team, Proc. EPAC 2000, Vienna, Austria [4] Y.Cho,’PrelimanaryDesignReport: SCRFLinacforSNS’, SNSreport SNS-SRF-99-101,Dec 99 [5] D.L.Schrage,’StructuralAnalysisofSuperconducting Ac- celerator Cavities’, Los Alamos report LA-UR-# 99-5826, Jan 2000 [6] N. Akaoka et al, ’Superconducting CavityDevelopment fo r High Intensity Proton Linacs in JAERI’, 9 th Workshop on RF Superconductivity, Santa Fe, USA, Nov 99; E. Chishiro et al., 12 th Symposium on Accelerator Science in Japan, Wako 99, p. 236 [7] E.Zaplatinetal.,’SuperconductingRFCavityDevelopm ent for ESS’,Proc.EPAC2000, Vienna, Austria [8] A. Letchfordet al.,Proc.PAC99, New York, USA,p. 1767 [9] A. Letchford, ’The ESS 280 MHz RFQ Design’, in ESS rep., ESS99-99-M-2, Dec 1999 [10] M. Pabst etal.,Proc. PAC97, Vancouver, Canada, p. 1846 [11] M. Ikegami, Phys.Rev. E 59, p. 2330, 1999 [12] K. Bongardt et al.,Proc.EPAC2000, Vienna,Austria
STUDIES OF COUPLED CAVITY LINAC (CCL) ACCELERATING STRUCTURES WITH 3-D CODES* G. Spalek, D.W. Christiansen, P.D.Smith, P.T. Greninger, C.M. Charman, General Atomics, San Diego, CA 92186, USA R.A. DeHaven, Techsource, Inc., Santa Fe, NM 87501, USA L.S.Walling, Ansoft Corp., Albuquerque, NM 87113 * Work supported by U.S. DOE contract DE-AC04-96AL89607Abstract The cw CCL being designed for the Accelerator Production of Tritium (APT) project accelerates protons from 96MeV to 211MeV. It consists of 99 segments each containing up to seven accelerating cavities. Segments are coupled by intersegment coupling cavities and grouped into supermodules. The design method needs to address not only basic cavity sizing for a given coupling and pi/2 mode frequency, but also the effects of high power densities on the cavity frequency, mechanical stresses, and the structure’s stop band during operation. On the APT project, 3-D RF (Ansoft Corp.’s HFSS) and coupled RF/structural (Ansys Inc.’s ANSYS) codes are being used to develop tools to address the above issues and guide cooling channel design. The code’s predictions are being checked against available low power Aluminum models. Stop band behavior under power will be checked once the tools are extended to CCDTL structures that have been tested at high power. A summary of calculations made to date and agreement with measured results will be presented. 1 INTRODUCTION To design a CCL structure, the RF design codes need to predict the pi/2 mode frequency (f pi/2) and nearest (k) and next-nearest-neighbor coupling constant (kk) of the CCL. Calculation accuracy has to be such that final frequency adjustment of fabricated structures can be accomplished by the usual methods of machining of reasonably sized tuning features (rings) or by small mechanical deformation of thinned portions of cavity walls. The pi/2 mode frequency is arrived at by calculating the frequency of the accelerating cavities (ac) without coupling slots and correcting this frequency by frequency shifts caused by coupling slots and next- nearest-neighbor coupling [1]. To obtain as high an accuracy as possible, the approach used in this work was to calculate the no slot absolute frequencies in 2-D using SUPERFISH and only calculate frequency differences in 3-D using HFSS.High RF power operation of the accelerator causes thermally induced stresses that can lead to: • Yielding of the cavity metal components leading to permanent frequency shifts. • Excessive frequency shifts of the accelerating mode. • Different frequency shifts for the coupling and accelerating cavities (i.e. change in stop band width leading to lower field stability). To predict high power effects, a code like ANSYS that couples electromagnetic, thermal, and structural calculations can be used to calculate the mechanical distortions and stresses caused by a particular cooling channel configuration and power level. The frequencies of the accelerating and coupling modes of the distorted structure can then be calculated in two ways: • By recalculating the modes of the distorted structures using the deformed geometry (“morphed” mesh), the coupled thermal- mechanical problem. • By calculating the frequency shifts caused by the cavity deformations using a technique such as the Slater [2] perturbation method. Since the expected frequency shifts are small (e.g. 100kHz for a 700MHz cavity), the perturbation method could be expected to yield more accurate results than the recalculation of the entire problem with a deformed and re-meshed geometry. This article reports on results of the HFSS and ANSYS calculations performed to date. 2 RF CALCULATIONS_RESULTS 2.1 Calculation Procedure The dimensions of a half scale (~1,400MHz) Aluminum model of segment 243 of the APT CCL were used for the calculations. The calculation procedure was: • Calculate the frequency f 0sf of the accelerating cavity using SUPERFISH [3]. • Construct an HFSS model of one period of the CCL segment consisting of ½ accelerating cavity and two ¼ sections of coupling cavities.• Use periodic boundary conditions with coupling cavity field phase shifts of 00, 360, 720, 1080, 1440, and 1800 to calculate the mode spectrum of the structure with HFSS. • Fit the mode spectrum to obtain the cavity frequencies (f hfss) including slots, and k hfss and kkhfss. • Generate a model of ½ of the accelerating cavity in HFSS using the same segmentation as in the periodic model (to guide the mesh and control cavity effective size ) and calculate its frequency f0hfss . • Calculate the (HFSS) accelerating cavity frequency shift ohfss hfss hfssf f df −= • Calculate the pi/2 mode frequency: hfsshfss sf hfss pi kkdf ff −+= 10 2/ Calculations were performed for several cases that included knife edged coupling slots with different coupling cavity depth (case 1 and case 2 below) and the enlargement of coupling slots by chamfering (case 3 and case 4 below). Frequency shifts were calculated for ac cavities only, future work will address the coupling cavities. For comparison with the calculations, Aluminum model no-slot ac frequencies and mode frequencies were measured. The mode spectra of the cavity model (11 cavities) were fit to extract the ac frequencies and coupling constants. 2.2 Results/Comparison With Measurements The calculated values and quantities extracted from fits of calculated and measured mode spectra are compared in Table 1. Table 1: Comparison with measured values, frequenci es are in MHz Case 1 Case 2 Case 3 Case 4 f0sf 1,432.352 1,432.352 1,432.352 1,432.352 f0hfss 1,436.960 1,436.960 1,436.960 1,436.960 fhfss 1,419.489 1,412.12 1,414.235 1,412.989 dfhfss -17.471 -19.047 -22.725 -23.972 fpi/2hfss 1,411.209 1,409.541 1,404.776 1,403.117 fpi/2mea 1,409.886 1,408.318 1,403.871 1,401.608 diff 1.32 1.22 0.91 1.5 khfss 0.03712 0.03981 0.04489 0.04671 kmeas 0.03764 0.04005 0.0444 0.0464 dif(%) 1.4 0.6 -1.1 0.67 kkhfss -0.00521 -0.00535 -0.00692 -0.00752kkmeas -0.00556 -0.00554 -0.00705 -0.00785 dif(%) 6.7 3.5 1.9 4.4 The differences between the calculated and measured pi/2 mode frequencies are well within the correction range (+/-3MHz) for a reasonably sized tuning ring. The nearest-neighbor coupling constants are very accurately calculated and the accuracy of the next-nearest-neighbor coupling calculation is also reasonable. 2 ANSYS CALCULATIONS Post-processor ANSYS routines implementing the Slater perturbation calculations in 3-D have been written. To test the implementation, frequency shifts for simple geometry cavities were calculated and checked by comparison to theory and to the fully coupled calculations. The calculated cases included: • Copper 1GHz pillbox cavity (TM 010 mode), frequency shift due to a uniform –100F temperature change. • Copper pillbox cavity, outside surface of metal held at constant temperature, inside heated by radio frequency (RF) fields. • Copper CCL cavity with cooling channels heated by RF fields. In addition, work is proceeding on modelling and calculations of modes and frequency shifts for a periodic APT CCL structure. 3.1Pillbox Cavity Results The radius of a 1GHz pillbox is: r=11.4742541cm Using an expansion coefficient for copper of 10-5 /0F, a 10 degree lowering of temperature gives a new frequency: ()Hzdrrcrf 011, 1000,000,1201=+=p Where r o1 is the root of J 0, c is the speed of light, and dr is the change in radius. So the frequency shift is: dfcalc=100,011Hz Using the same radius change and the surface fields calculated by ANSYS, the Slater perturbation method gives a frequency shift of: dfslater =100,083HzThe corresponding calculation results of the absolute cavity frequency using ANSYS linear and quadratic basis functions are: flinear =1,001,028,656Hz fquadratic =1,000,687,772Hz The error in calculating the absolute frequency of the pillbox is larger (~688kHz) than the frequency shifts of interest in the design of accelerator cavities. This error is due to mesh size, numerical accuracy, shape function order, etc. This indicates that errors can be expected to grow as the geometry of the cavity becomes more complicated by coupling slots, noses, and other features that increase the curvature of surfaces, distort the mesh, etc. Coupled calculations of a Copper pillbox were used to calculate the frequency shifts of a thick wall cavity heated internally by RF fields while the outside of the Copper was held at constant temperature. Two cases were calculated, one without including the effects of the motion of the cavity circular end walls and the other including these effects. The motion of the end walls should result in zero frequency shift and is thus a good test of the accuracy of calculation of electric and magnetic contributions to the frequency shift in the Slater method. The results are compared to the Slater calculations and theory below. The heating-caused radius change calculated by ANSYS was 4.8551*10-2cm. This gives a calculated frequency shift of : dfcalc=-4,213,469Hz The corresponding results of the coupled and Slater calculations using quadratic shape functions for the end- wall and no-end wall cases are listed in Table 2. Table2: Comparison of Copper Pillbox Results w/o end wall w/ end wall dfcoupled -4,248,333Hz -4,181,683Hz dfslater -4,234,887Hz -4,232,266Hz dfcalc -4,213,469Hz -4,213,469Hz The 0.06% difference in the Slater end-wall/no-end- wall results shows the degree of non-cancellation of the electric and magnetic field contributions to the frequency shifts on the end walls of the cavity. This error is quite small and the Slater agreement with the calculated value is excellent (0.4%). The larger 1.5% difference in the coupled results is probably due to the morphing of the mesh near the end-walls of the cavity.3.2 CCL Cavity Results Only preliminary calculations were made for the axi- symmetric CCL cavity with cooling channels. For a uniform -100F change in the temperature of a Copper CCL Cavity the resultant frequency shift results are: dfcalc=71,299Hz dfslater =73,100Hz The above 2.5% difference is larger than expected and needs to be investigated to make sure that displacements are calculated correctly in areas of small radii of curvature. The results for the cavity cooled by cooling channels are: dfcoupled =-198,202Hz dfslater =-202,368Hz The above frequency shifts differ by only 2.1%. They will be compared to 2-D calculations in the future. 2 CONCLUSIONS The comparisons made above indicate that: • 3-D codes such as HFSS can be used to calculate coupling constants of periodic CCL structures very accurately. • Frequency shifts caused by coupling slots can be calculated by 3-D codes and used to predict pi/2 mode frequencies with sufficient accuracy to be useful for design. If further comparisons between calculations and experiment are as successful as those above, 3-D accelerator structures can be designed with 3-D codes without the construction and testing of numerous Aluminum models. The preliminary work above also indicates that a 3- D Slater perturbation method can be used to predict thermally caused frequency shifts in accelerator structures while avoiding possible problems associated with mesh morphing in the ANSYS code. This method also shortens calculation times since a recalculation of the resonant mode frequencies is not necessary after the thermal/mechanical calculations are completed.REFERENCES [1] D. E. Nagle, E. A. Knapp, and B. C. Knapp, “Coupled Resonator Model of Standing Wave Accelerator Tanks”, Rev. Sci. Instr. 38, 1583 (1967).[2] J. C. Slater, “Microwave Electronics”,pp. 81-83, D. Van Nostrand Co., Inc. Princeton, N.J., 1950. [3] J. H. Billen and L. M. Young, “POISSON SUPERFISH”, Los Alamos National Laboratory report LA-UR-96-1834 (rev. March 14, 2000).
arXiv:physics/0008108v1 [physics.acc-ph] 17 Aug 2000EstimatesofDispersive Effectsin aBent NLC MainLinac∗ M.Syphers and L.Michelotti,Fermilab, Batavia, IL 60510,U SA Abstract An alternative being considered for the Next Linear Col- lider (NLC) is not to tunnel in a straight line but to bend theMainLinacintoanarcsoastofollowanequipotential. Webeginhereanexaminationoftheeffectsthatthiswould haveonverticaldispersion,withitsattendantconsequenc es on synchrotronradiation and emittance growth by looking at two scenarios: a gentle continuousbendingof the beam to follow an equipotential surface, and an introduction of sharp bends at a few sites in the linac so as to reduce the maximumsagittaproduced. 1 CONTINUALGENTLE BENDS In our ﬁrst scenario, the Main Linac remains as close as possible to an equipotential surface. Minimalism suggests that we try bending the beam by vertically translating al- ready existing NLC quadrupoles,without introducingnew elements or additional magnetic ﬁelds. We thus propose that steering be accomplished by precisely aligning all the quads “level” along the equipotential and then raising the vertically defocusing (D) quadrupoles to steer the beam through the centers of the vertically focusing (F) quads.1 Bending at the D quad locations will minimize the gener- ated dispersion. To estimate the order of magnitude of dispersion pro- duced by such an arrangement, we calculate (a) assuming aperiodicsequenceofmagnetswhile(b)neglectingtheef- fectsofacceleration[1]and(c)keepingonlyleadingterms in the bend angle. Our results will be reasonably correct providedthat upstreaminjectionintothe Main Linacis re- designedto match the new arrangement. Furtherdetails of thecalculationandothersdiscussedinthispaperaredocu- mentedelsewhere. [2] Figure 1 shows the physical layout of quadrupoles and identiﬁes the geometric parameters. It is practical to write the vertical offset of the quadrupole relative to the equipotential, d−ysag. In terms of the distance between quadrupoles, L, the local betatron phase advance per cell, µ, andtheradiusofthe earth, R,thisoffsetis d−ysag=L2 R/parenleftbigg1 2+1 sin(µ/2)/parenrightbigg (1) To make a numerical estimate of this offset at the high en- ergy end of the linac, we take L≈19 m, µ≈π/2,and R≈6400 km .Eq.(1)thenyields d−ysag≈108µm.The dispersion can be estimated easily using two observations: ∗Work supported by the U.S.Department of Energy 1The usual convention is for “F” (“D”) to indicate a horizonta lly fo- cusing(defocusing )quadrupole. Wedotheoppositehere,be causeweare considering dynamics only in the vertical plane.Lθ dyo ysag CL Figure1: DescriptionofparametersfordescribingtheCR thinquadcalculations. (1) in passing through a thin bending magnet, the slope of thedispersionfunction, D′,changesbyanamountapprox- imately equal to the bend angle; (2) by symmetry, the dis- persionattainsitsmaximum(minimum)valueatthecenter of the focussing (defocussing) quadrupole. The values of thedispersionfunctionat thethin-lensquadrupoleswill b e Dmin=L2 Rsin2(µ/2)(1−sin(µ/2)), Dmax =L2 Rsin2(µ/2). Using the same parameters as before, this provides the numericalestimate,at thehighenergyendofthe linac, Dmin= 0.032 mm , Dmax= 0.11 mm . If we take a large ∆p/p≈∆E/E= 0.02,because of BNS damping, and assume that the “invariant emittance” γǫy/π≈100 nmandβy≈40matapointwheretheelec- tron’s energy is E= 100GeV, then Dmax·∆p p= 2.2µm comparedto σy=/radicalbig βyǫy/π= 4.6µm. Vertical bending will produce synchrotron radiation, which, in its turn, will add to the vertical emittance of the beam. Athighenergy,thetotalenergyradiatedbyoneelec- tronisgivenbytheexpression,2 U=/integraldisplay (cdt)1 6πǫo/parenleftbigge ρ/parenrightbigg2 γ4, where, ρis the bend radius, γis the relativistic 1//radicalbig 1−(v/c)2,and the other variablesneed no introduc- tion. Intermsoftheelectronenergy, E,andthebendangle, θ, producedbythequadoveritslength, ℓ, U= (1.41×10−5mGeV−3)·E4θ2/ℓ . (2) 2For example, see Equations 8.6 and 8.10 of Edwards and Sypher s. [3]Using the same parameters as before, θ≈5.6µrad; at the high energy end of the linac, E≈500GeV, and ℓ≈1m;ourestimateofthetotalradiatedenergy(perelec- tron per bend) is about 28 keV. Put another way, the ra- tio,U/E≈6×10−8.It isinconceivablethat suchasmall fractional change in beam energy could seriously damage the emittance, but we will estimate its effect anyway. The additionalinvariantemittancedueto synchrotronradiati on is approximatedas ∆(γǫy/π)≈55 6√ 3re¯hc mc2/angbracketleftH/angbracketright/angbracketleftbigg1 ρ2/angbracketrightbigg θγ6,(3) where ǫy=πσ2 y/βyandreisthe classical electronradius. Thequantity /angbracketleftH/angbracketright ≈ D2 y/βyat thequadrupolelocation. Plugging in the same numbers as before, estimating βy≈60m, and using our previous estimates for Uand Dmax, we obtain, ∆(γǫy/π)≈1.8×10−7nm – as ex- pected,averysmall number. 2 LOCALIZED SHARPBENDS Althoughthe2 µmoffsetofanoff-momentumparticlepre- dicted inthe previoussection isnot catastrophic,neither is itcompletelynegligible. Wewillnowconsidereliminating it by employing the second scenario: constructing a Main Linacthatislaserstraightexceptforhighlylocalizedben ds at a few, widely separatedlocations. These bendsites then allowustofollowtheequipotentialinacoarser,piece-wis e fashion. Ifwethinkofbendingeverykilometer,orso,then thebendangleshouldbeabout160 µrad. We’lltakethisas the “canonical”valueforcalculationsinthissection. We will proceed again in a minimalist way. In order to minimize the modiﬁcation of existing lattice hardware and optics, we adopt the use of combined function mag- nets to both bend and focus the electron beam. Other pos- sibilities could be considered at a later date. If we only “bend” at each of the local sites, a dispersion wave of am- plitude ∼4 mm would be generated at each bend center. To match the trajectories at the end of each local bending region for particles with various momenta, the total bend angle of each region is distributed across four neighboring (combined function) dipoles. The strategy is akin to that of an 18thcentury optician designing a simple focussing achromat. The results at the lowest energy bending location are shown in Figure 2. The dashed line follows the residual dispersion, now completely contained within the ≈40 m long bending region, with maximum amplitude of about 0.6 mm. The maximum orbit distortion of 1 mm is too large an offset from the central (curved) axis of the local bendingmagnets. Theywouldhavetobedisplacedsoasto follow the new orbit. A few iterations of these manipula- tions should then convergeon an acceptable design. How- ever, the ﬁnal orbit and its local residual dispersion shoul d not be much different from what we have calculated here. For now, we simply display these results as indicating the orderofmagnitudeofthe effects.900 1000 1100 1200 Azimuth [m]−2−1012Orbit and residual dispersion [mm]Orbit Dispersion Figure2: Orbitdeviationrequiredtozerotheresidualdis- persion. The distortions in the trajectory at ten locations located ∼1 km apart in the NLC are of the order 1-2 mm, with the higher displacements occuring at the higher energies. The corresponding dispersions generated along the linac are shown in Figure 3. A residual dispersion of 1 mm re- mains in the neighborhood of the bends. Again assuming that∆p/p≈0.02,wehave Dmax·∆p p= 20µmcompared toσy=/radicalbig βmaxǫy/π= 4.6µm.This is a large increase, butitexistsonlynearthebendsites. Awayfromthesesites, the dispersion is (essentially) zero, and its contribution to emittance is negligible. We note in passing that an advan- tage of this calculation is that one can envision making it operational. Finally, we estimate the synchrotronradiationand emit- tance growth incurred by our second scenario, once again using Eqs.(2) and (3). The valuesof ∆(γǫy/π)at all bend locations are plotted in Figure 4. Each site contains one 0 5000 10000 Azimuth [m]−2−1012Dispersion [mm] Figure 3: (a) Residual dispersion generated by the partial achromatsat all tenlocationsalongtheNLC.dominant,verysharpbend. Itseffectismostapparentnear thehighenergyendofthelinac,wherethe E6dependence becomesoverwhelming. Evenso,theadditional ≈1nmin invariant emittance is less than 1% of the 140 nm vertical emittanceexpectedwithintheinteractionregion. Notice that although the synchrotron radiation is rather high at the end of the linac, the ratio U/E = 49 MeV /473 GeV ≈10−4isstill asmall number. In passing, we note that some attention should be given to coherent synchrotron radiation (CSR) from individual bunches. A quick look[2] shows that the current NLC de- sign has an aperture a≈7mm, so CSR is forbidden,even at the high energy end of the linac, because of “shielding” from the walls of the beam pipe. However, the margin of safety is not comfortablylarge. It may be necessary to re- examinethisissue. 0 5000 10000 Azimuth [m]0.00.51.0[nm] Δ(γε/π) Figure 4: Emittance growth due to synchrotron radiation in sharpbends. 3 ULTIMATE ENERGY OF “CURVED”LINAC While present NLC designs with beam energies in the range of a few hundred GeV to 1 TeV may not be very sensitive to the curvatureof the earth, there will be a prac- ticallimittotheupgradedenergyofsuchadevice. Suppose the schemeofsteeringthe beamwith offsetquadrupolesis adopted. Then,athighenergies,eventuallytheenergygain within a FODO cell will be equal to the energy lost due to synchrotron radiation as the beam is bent by the offset quadrupole. Thislimitcanbe easilywrittenas Elim=/parenleftbiggπ Cγℓq LEcvR2/parenrightbigg1/4 where Ecvistheenergygainpermeterofthelinac. Asthe energy is increased, both the quadrupole length (strength) andthehalf-celllengthincreaseroughlyproportionally. As an example, using Ecv= 50 MeV/m, and ℓq/L= 0.025, thenwe get Elim≈6.5TeV.Energy, emittance vs. Linac length 012345678910 0 20 40 60 80 100 120 140 160 180 200 Path Length (km)Energy (TeV) 00.10.20.30.40.50.60.70.80.91 Growth in Normalized Emittance ( µm) Energy (150 MeV/m) (50 MeV/m) Δ(γεy/π) (150 MeV/m) (50 MeV/m) Figure 5: Beam energy and emittance growth vs. path length of linac which follows the curvature of the earth. Results with accelerating gradients of 50 MeV/m and 150MeV/mareshown. Thisconclusionis illustratedin Figure5. Theemittance growth due to synchrotron radiation is also plotted, show- ingthatagrowthof ∆(γǫy/π)= 100nm(roughlyequalto theNLCnominaldesignvalueatcollision)occurswellbe- fore the ﬁnal energyis reached, with the emittance growth being a steep function of energy ( ∼γ6). For 50 MeV/m, the practical limit of a curved linac may be only about 2- 3 TeV per beam, which would have a length of about 50 km. For150MeV/m,thelimitsare3-4TeVperbeamover about 20 km. Note that a “laser straight” linear collider with20kmperlinac,andalengthyinteractionregion,with its two ends near the surface of the earth would have its collisionpointlocatedroughly50mbelowthesurface. Acknowledgements We aregratefultoCourtlandtBohnforsuggestingthepos- sible importanceofcoherentsynchrotronradiation. 4 REFERENCES [1] Leo Michelotti. A two-parameter accelerating FODO cell . FERMILAB-FN-688,January 2000. [2] L. Michelotti and M. Syphers. Estimates of dispersive ef - fects in a bent NLC main linac. FERMILAB-FN-690, May 2000. [3] D. A. Edwards and M. J. Syphers. An Introduction to the Physics of High Energy Accelerators . John Wiley & Sons, New York, 1993.
ANALYSES OF KLYSTRON MODULATOR APPROACHES FOR NLC ∗ Anatoly Krasnykh, SLAC, Box 4349 Stanford, CA 94309 ∗ Work supported by Department of Energy contract DE-AC03-76SF00515Abstract Major changes to the Next Linear Collider (NLC) design were facilitated by the experimental testing of the 75 MW X-band klystron at a 3.0 µsec pulse width 1 and new component development allowing the delay line distribution system (DLDS) to operate with eight bins instead of four. This change has a direct effect on the design of the klystron modulator. The general approaches,which are being studied intensively, are: the conventional base line modulator with two klystrons 2, a Hybrid version of the baseline with a solid-state on/off switch, a solid- state induction type modulator that drives eight klystrons 3, and a solid-state direct switch modulator 4. Some form of pulse transformer is the matching element between the klystron beam and the energy store in the all of theseapproaches except the direct switch. The volume and costof the transformer is proportional to the peak pulse powerand the output pulse width. The recent change in the NLCdesign requires double the transformer effective core area, and increase both the size and cost of modulator. In the direct switch model there is no pulse transformer. Theklystron beam potential is practically equal to the potentialof the energy storage element. Here the solid-state switchblocks the 500 kV DC voltage of the storage element. In this paper transformerless modulator approaches are presented based upon a Marx method of voltagemultiplication using on/off Insulated Gate BipolarTransistors (IGBT’s) instead of on switches. DC voltagepower supply system is much simpler as compared to thepower system of the direct switch approach. INTRODUCTION The current design of the future linear colliders consists oftwo X-band linacs powered by over 1,600 klystrons. Thefacility will consume in excess of 200 MW of averagepower. The total number of NLC klystrons and modulators can be reduced if RF pulse duration is longer. The pulse width for NLC klystrons is not precisely defined for thetime being. The present klystron modulator for NLCrequires a 500 kV, 530 A, 3 microsecond flat top pulsewith 120 PPS to drive a pair of 75 MW klystrons. These parameters are used in planning for the NLC, but R&D is continuing on klystron improvements. The design ofmultiple beam klystrons with lower beam potential is anexample of a possible improvement. The modulator forsuch a klystron or group of klystrons will be simpler,reliable, and less expensive.ENERGY TRANSFER EFFICIENCY The major design requirements for the modulator are cost, efficiency, reliability. The efficiency is one importantparameter due to large number of modulators for NLC. It is useful to identify losses into two separate efficiencies. The first component is a ratio of the energy delivered intoklystron beam and PFN storage energy. It is the dischargeenergy efficiency. A second component is a ratio of theflat top portion of the klystron pulse energy and the energyof pulse width. It is energy transfer efficiency. The efficiency of modulator is a product of these components. It was shown 2 that the discharge energy efficiency of ~92-93% can be obtained for PFN using oil filledcapacitors, two gap thyratron and conventional pulse transformer for the 150 MW and 1.5 µsec output levels. The energy transfer efficiency is still close to ~70%. The pulse transformer is one element of modulator, which limits the energy transfer efficiency due to a limitation of the transmission band of the pulse power. It is well knownthat the rise time of pulse transformer is limited by the period which is determined by the leakage inductance L and the distributed capacitance C, as following LCt~1. The geometry of high voltage pulse transformer fixes L and C. It is evident also that a transformer with a small number of turns ought to have a wider bandwidth. On the other hand the reduction of the number of turns produces larger pulse droop and core size for the given pulse width. Moreover, for transformers with a low L and C the inductance of the current loop between the PFN and a primary winding through switch (Lc) and the load capacitance (Cg) can play a noticeable role on the rise/fall time as well as energy transfer efficiency. For optimal case it has been found that the energy transfer efficiency can be evaluated by 2 11 11141 3211 211 kkpdpw tt rfte +⋅+⋅+⋅⋅+≅ +≅ λπη where trf is the flat-top duration, w1 and p are the number turns and the circumference of primary winding, d is the average distance between primary and secondarywindings. Coefficients k 1=Lc/L and k2=Cg/C are the additional parts of the primary inductance and the secondary load capacitance, which have been normalized by L and C of the transformer. The effective length of rf- pulse is λ=ctrf /(n-1)ε0.5, where c is the velocity of light, n is the turn ratio, and ε is the dielectric constant of the media between windings. The coefficient w1p/λ<1 is responsible for the transmission of the energy and has to be minimized (i.e. to use a media with lower ε, and lowerturn ratio) for the given amount of w1p. For example, this coefficient is ~0.035 for the conventional pulse transformer. The coefficient (1+4d/p) 0.5 is responsible for the degree of the high voltage isolation between windings and it is ~1.24. The contribution of k2 is higher as compared to k1 for the high power modulators. High core dynamic capacity effect can take noticeable place during transmission of the high pulse power through transformer. Actually equations, which describe theprocesses in the pulse transformer, assume that a velocityof field propagation in the core material is instantaneous.However the core magnetizing velocity is a function of conductivity of the ribbon, effective pulse permeability, etc. The step of the power starts to propagate in the corematerial from periphery to the center. The velocity of themagnetization has to be extremely fast in the beginning ofprocess. This can happen when an extremely large spike ofthe magnetizing current amplitude takes place. The quality of the core material, ribbon thickness, and the core geometry are responsible for this effect. The charge of theload capacity, of the transformer distributed capacity, andof the effective core dynamic capacity goes practicallysimultaneously. The value of this capacity depends on dB/dt and amplitude of H (t) for the simple case. In spite of described effects the demonstrated efficiency of modulator is rather high. For instance, the energytransfer is ~70% for 1.5 microsecond rf-pulse duration 2. It is expected that the total efficiency can be improved on~12% for 3 microsecond pulse width in the conventionalmodulator configuration. The use of features of coupled transmission lines lies in the base of an idea how the energy transfer efficiency canbe improved. The first approach is related to the use ofcoupled transmission lines, which support true TEMmodes. The approach consists of a bunch of small transformers, which are connected as shown in Fig. 1a). Fig.1a) Both lines are uniform and linear for a simple case. The power step of the modulator is launched at the input of thecoupled line. The number of cells and their parameterswill determinate the transmission band of the pulsetransformation. If the group velocity of the exciting TEMwave of the first line is equal to the group velocity of the second line, the energy of the pulse will be launched on the load with high efficiency. Two coupled pieces of transmission lines are added to the primary and secondary winding of PT 0 as shown in Fig. 1b). The initial part of the pulse power is launched into the load though the fast coupling lines. The coupling is due to fast PT 1. The rest of power is transmitted through PT0. Modulators based on schemes of Fig. 1a) and 1b) consist of cells, which are not used in the pulse formation.They are used as transmission line elements. The buffer Fig. 1b) elements always will additionally dissipate some amountof energy. Fig 1c) Fig. 1: a) scheme of distributed line transformer, b) scheme of a conventional transformer with additional fast cells, and c) scheme of modulator based on coupled pulseforming lines. Approach presented in Fig 1c) has no buffer elements. One port of primary transmission line can be open. Otherport of this line has the switch for discharge of PFN. Here PFN is created by the lumped C 0 capacitors and the primary winding of pulse transformers PT. The energy is stored in capacitors of the first transmission line. Primary windings of pulse transformers have a floating potential.The charge of the second line is equal to zero. The outputpulse is started when the switch is on. The TEM wave starts to propagate toward the end of line. The storage energy is launched partly in the load and partly on thecharge of the second line capacitors C 1. After the double propagating time, the energy is extracted at the load. Thisapproach was evaluated experimentally as well as by computer simulation. The aim of these experiments was ‘proof of principle ’ i.e. to show a possibility of the pulse formation on the load. The leakage inductors anddistributed capacitances here are not parasitic elements.They are parts of pulse forming network. The outputwaveform of the voltage for two-coupled PFN with a step up voltage transformation is shown in Fig. 2. -1000100200300400500600 0 200 400 600 800 1000 t, nsecVout, V Fig. 2: Output waveform for two-coupled PFN with a step up voltage transformation. The rise and fall times for this approach can be better as compared to the conventional modulator. The energy transfer efficiency will be higher if a ferrite material isused as a core in this design. The process of the dynamicmagnetization of a ferrite material in nanosecond rangegoes with much lower loss as compared to the iron ribbon.Evaluation of the total efficiency for a distributed line transformer approach gives a value of ~83% for 1.5 usec pulse duration. TRANSFORMERLESS APPROACH This modulator has no problems, which relate to the use of a ferromagnetic media for transformation of the energy (such as a limitation of the volt-second product, dynamic losses, etc.). Presented approach is based upon a modifiedMarx method. The modified Marx method chargescapacitors in parallel and partially discharges them inseries through IGBT assembly 5. The possible circuit schematics are shown in Fig. 6a) and b). a) b) Figure 3: A Marx type modulator concepts. Because the switch elements are on/off devices, pulse duration is variable, and the efficiency is high. The voltagemultiplication of a Marx circuit is equal to the number ofcells used. Pulse droop is low depending only on energystorage capacity and load impedance. Compensation for small droop is possible via a bouncer at the low potential side. Extrapolating this transformerless approach for theNLC seems to be attractive especially for time being whenthe pulse width is not fixed. The circuit schematic shown in Fig. 3a) is a topology with active recharge switches. They allow a reduction of the charging loss. The topology of Marx type modulatorwith recharging through isolation transformers ispresented in Fig. 3b). Here the NiZn ferrite coretransformers are used for charging process. The isolationdistance d between primary windings is varied according to the applying pulse voltage. The main problem of the Marx type modulator is parasitic capacitance. The amplitude of the transient inrush current and its derivative di/dt, which flows thought the switches, can destroy solid-state assembly. To get some experience with the transformerless approach a 10 cell solid state Marx type modulator was designed, manufactured and tested 5. Methods of reduction of inrush current were experimentally evaluated. It was shown thatamplitude of transient current can be reduced by factor of~10 to keep the solid state switch working point within Reverse Biased Safe Operating Area (RBSOA). Thevolume of solid state Marx type modulators is larger as compared to the transformer concepts due to the absence of ferromagnetic media with a high permeability. Thedesign of multiple beam klystrons is presently under studyat SLAC 6. Authors offer the possibility of considerably lower cathode voltage. The transformerless approach canbe effective and low cost solution for multiple beam klystron modulator design. CONCLUSION The discharge energy efficiency can be ~92-93% for the 150 MW and 1.5 µsec output level. However, the energy transfer efficiency is only ~70%. This result was received on Test Bed modulator 2. For high power modulators the dynamic capacitive effect of the core magnetizing can play a noticeable role. Effect is inherent for all high powerpulse transformers. The bandwidth of the PFN-KLYSTRON energy transmission is limited by this effectand by the well-known geometrical effects. Ways of therise of energy transfer efficiency had been discussed. The use of the inductively coupled distributed lines can overcome the limitation on the pulse transformationbandwidth. Evaluation of total efficiency for a distributedline transformer concept gives a value of >80%.Computer simulations and low voltage experiments suggest the possibility to get a faster rise time on the load with a higher efficiency. A transformerless modulator approach is presented on a Marx method of the voltage multiplication using on/of IGBT’s. Approach has no problems, which relate to the use of a ferromagnetic media for transformation of the energy. The transient process in Marx modulator was analyzed and effective methods of reduction of inrushcurrent was experimentally obtained and kept the solid stage switch working point within of RBSOA. A new semiconductors based on materials with a wide band gap (for example, Silicon Carbide) will push up thevoltage hold-off and make solid state modulator designs athigher voltages simpler and more efficient in the future. ACKNOWLEDGEMENTS This material would not be prepared without the help ofR. Koontz, S. Gold, and R. Akre. REFERENCES 1 D. Sprehn et al, SLAC-PUB-8346, March 2000. 2 R. Koontz et al, PAC 1997, Vancouver, B. C., Canada. 3 R. Cassel et al, 24th International Power Modulator Symposium, 2000, Norfolk, VA, USA.4 M. Gaudreau et al, 23th International Power ModulatorSymposium, Rancho Mirage, CA, 1998 5 R. Akre et al, A Solid State Marx Type Modulator for driving a TWT, 24th International Power Modulator Symposium, 2000,Norfolk, VA, USA.6 G. Caryotakis et al, The Next Linear Collider Klystron Development Program, this Conference.
OPERATIONAL ASPECTS OF THE HIGH CURRENT UPGRADE AT THE UNILAC J. Glatz, J. Klabunde, U. Scheeler, D. Wilms, GSI, Darmstadt, Germany Abstract With the new GSI High Current Injector, the beam pulse intensity will be increased by more than two orders of magnitude. The high beam power and the short stopping range at particle energies below 12 MeV/u can destroy accelerator components even during a single beam pulse. Therefore, the operation of the whole accelerator facility has required major changes in hardware, software and operating strategy. A sophisticated beam diagnostic system is indispensable for a safe operation. Preferably non-destructive devices were installed. Destructive elements, e.g. beam stoppers, slits, apertures, were improved in order to withstand the high beam power. Automatic damage prevention was realised by beam loss monitors comparing and evaluating very fast beam current transformer signals. Additionally, the component status will be controlled permanently. For foil stripping at 11.4 MeV/u, a magnetic beam sweeping system was installed Thereby, the hit area will be increased during one 100 µs pulse. During operation, manual variation of parameters has to be reduced. Set-up and automatic beam adjustment procedures have to exclude uncontrolled beam loss. The versatility of the UNILAC is enhanced by the possible three-beam operation on a pulse-to-pulse basis. Since November 1999 the upgraded UNILAC is serving the experiments.1 INTRODUCTION With the installation of the High Current Injector HSI, the beam pulse power has been increased considerably. The schematic layout of the UNILAC is shown in Fig. 1. where key parameters for the high current acceleration are listed, uranium being the reference ion. Considering the high beam pulse power ( maximum of 1250 kW at the gas stripper section ) and especially the short stopping range at the UNILAC energies, accelerator components could be destroyed. Even one pulse with a length of 100 µs can melt metal surfaces. Necessary consequences for a safe operation will be discussed in the following sections, the topics are: control of the high intensity beam, safety of accelerator components, foil stripping at high beam power, operating strategy including the time sharing operation. The stepwise installation and commissioning of the new linac sections were carried out from January to November 1999. There are separate reports concerning the commissioning of the high current ion sources, LEBT, HSI, and the new stripper section at 1.4 MeV/u. The summary report on the commissioning contains an extensive list of references. (Ref. [1]). Fig. 1: Schematic layout of the UNILAC with some key parameters, uranium as reference ion.
arXiv:physics/0008111v1 [physics.gen-ph] 17 Aug 2000Looking Forward to Pricing Options from Binomial Trees Dario Villaniaand Andrei E. Ruckensteinb (a) 10 Brookside Drive, Greenwich, CT 06830 (b) Department of Physics and Astronomy, Rutgers Universit y, 136 Frelinghuysen Road, Piscataway, NJ 08854 (September 14, 2013) Abstract We reconsider the valuation of barrier options by means of bi nomial trees from a “forward looking” prospective rather than the more conven tional “backward induction” one used by standard approaches. Our reformulat ion allows us to write closed-form expressions for the value of European and American put barrier-options on a non-dividend-paying stock. 1I. INTRODUCTION Closed-form valuation within the Black-Scholes-Merton eq uilibrium pricing theory [1,2] is only possible for a small subset of ﬁnancial derivatives. In the majority of cases one must appeal to numerical techniques such as Monte Carlo simulati ons, or ﬁnite diﬀerence methods and much of the eﬀort in the ﬁeld has been in developing eﬃcien t algorithms for numerically solving the Black-Scholes equation [3]. An alternative dir ection has been the evaluation of discrete-time, discrete-state stochastic models of the market on binomial and trinomial trees [4,5]. Not only is this discrete approach intuitive an d easily accessible to a less mathe- matically sophisticated audience; but it also seems to us to be a more accurate description of market dynamics and better suited for evaluating more inv olved ﬁnancial instruments. Moreover, the few exact Black-Scholes results available ca n be recovered in the appropriate continuous-time trading limit. The main diﬃculty in pricin g with binomial trees has been the non-monotonic numerical convergence and the dramatic i ncrease in computational eﬀort with increasing number of time steps [6,7]. For example, the state of the art calculations involve memory storage scaling linearly (quadratically) w ith the number of time steps, N, for European (American) options, while the computation tim e increases like N2in both cases [8]. In this paper we reconsider valuation on binomial trees from what we call a “forward look- ing” prospective: we imagine acting as well-educated consu mers who attempt to eliminate risk and estimate the value of an option by looking at its futu re expected values according to some reasonable dynamical model. As will be described in g reat detail below the main intuitive idea of our computations is to regard the movement of the price on the tree as a random walk (with statistical properties consistent with a risk-neutral world) with “walls” imposed by the nature of the option, such as the possibility o f early exercise (American op- tions) or the presence of barriers. Our mathematical formul ation then has two conceptually distinct components: the ﬁrst ingredient is an explicit des cription of the possible “walls”. For example, in the case of barrier American options both the barrier and the “early ex- ercise” surface need to be speciﬁed. The second step will be t o compute the probability that the price reaches particular values at every accessibl e point on the tree. This involves counting the number of paths reaching that point in the prese nce of “walls”, a somewhat involved but exactly solvable combinatorics problem. Once these two steps (specifying the walls and computing the probabilities) are accomplished th e value of both European and American options, with and without barriers, can be written down explicitly. In an attempt to be pedagogical, we will limit ourselves to the simplest ca ses: we will only treat Euro- pean, simple American and European with a straight “up-and- out” barrier. Although the calculation can be simply extended to the barrier American o ption that discussion merits a separate publication. We note that, as far as we know, in the case of trees explicit fo rmulas like the ones we are proposing exist in the literature only in the simplest case of conventional European options [5,9]. For the more complicated case of American opt ions, the main issues are best summarized in the last chapter of Neil Chriss’ book [10]: “Th e true diﬃculty in pricing American options is determining exactly what the early exer cise boundary looks like. If we could know this a priori for any option (e.g., by some sort of formula), we could produ ce pricing formulas for American options.” Below we propose a s olution to this problem in the 2context of binomial trees. Our formulation complements the earlier studies of American options in the limit of continuous-time trading [11–13] whi ch also focus on the presence of an early exercise boundary for the valuation of path-depend ent instruments. The study of the continuum limit of our formulas is instructive and will b e left for a future publication. II. BINOMIAL TREES To establish notation we begin by dividing the life of an opti on,T, into Ntime intervals of equal length, τ=T/N. We assume that at each discrete time ti=iτ(i= 0,1,2, ..., N ) the stock price moves from its initial value, S0=S(t0= 0), to one of two new values: either up to S0u(u >1) or down to S0d(d <1) [8]. This process deﬁnes a tree with nodes labeled by a two dimensional vector, ( i, j) (i= 0,1,2, ..., N ;j= 0,1, ..., i) and characterized by a stock price S(i, j) =S0ujdi−j, the price reached at time ti=iτafterjup and i−jdown movements, starting from the original price S0. The probability of an up (down) movement will be denoted by pu(pd= 1−pu); and thus each point on the tree is also characterized by the probability, pj u(1−pu)i−j, which represents the probability associated with a single path ofitime steps, j(i−j) of which involve an increase (decrease) in the stock price. Computing the probability of connecting the origin with point ( i, j) requires, in addition to the single path probability, a factor counting the number of such possible paths in the presence of a barrier and/or the possibility of early exercise. The calcu lation of this degeneracy factor involves the details of each ﬁnancial derivative and it will be discussed in turn for each of our examples. The binomial tree model introduces three free parameters, u, dandpu. Two of these are usually ﬁxed by requiring that the important statistical pr operties of the random process deﬁned above, such as the mean and variance, coincide with th ose of the continuum Black- Scholes-Merton theory [3]. In particular, puu+ (1−pu)d=erτ(1) erτ(u+d)−ud−e2rτ=σ2τ, (2) where ris the risk-free interest rate, and the volatility, σ, is a measure of the variance of the stock price. We are left with one free parameter which can be chosen to simplify the theoretical analysis; one might choose, for example, u= 1/d[4], which simpliﬁes the tree geometry by arranging that an up motion followed by a down mot ion leads to no change in the stock price. This condition together with (1) and (2) imp ly: u=eσ√τ(3) d=e−σ√τ(4) pu=erτ−d u−d. (5) We stress that Equations (1-5) are to be regarded as short-ti me approximations where terms higher order in τwere ignored. With these deﬁnitions out of the way we can begin discussing t he valuation of put options with strike price Xand expiration time T. 3A. European Put Options The simple European put option is a good illustration of our “ forward looking” approach. We are interested in all those paths on the tree which, at expi ration time i=N, reach a price, S(N, j) =S0ujdN−j< X, for which the option should be exercised. That implies that j≤j∗= Int [ln( X/S 0dN)/ln(u/d)], where Int refers to the integer part of the quantity in square brackets. The mean value of the option at expiration c an then be written as a sum over all values of j≤j∗of the payoﬀ at j,X−S0ujdN−j, multiplied by the probability of realizing the price S(N, j) =S0ujdN−jafterNtime steps, P[N, j]. As already mentioned above, P[N, j] =ℵE[N, j]pj u(1−pu)N−j, where ℵE[N, j] counts the number of paths starting at the origin and reaching the price S(N, j) inNtime steps. For the case of conventional European options this is just the number of paths of Ntime steps, with jup and N−j down movements of the price, and is thus given by the binomial coeﬃcient, ℵE[N, j] =/parenleftBigg N j/parenrightBigg =N! j!(N−j)!. (6) The resulting expression for the mean value of the option at m aturity is then discounted to the time of contract by the risk-free interest rate factor ,e−rT, to determine the current expected value of the option: ¯VE=e−rTj∗/summationdisplay j=0/parenleftBigg N j/parenrightBigg pj u(1−pu)N−j/parenleftBig X−S0ujdN−j/parenrightBig . (7) This expression is not new: it was ﬁrst discussed by Cox and Ru binstein [5] who also showed that in the appropriate continuous trading-time lim it (τ→0) (7) reduces to the Black-Scholes result [1]. B. European Put Barrier Options We are now ready to extend (7) into an exact formula for the mea n value of an European put option with a barrier. Although our approach can be used f or other barrier instruments, we consider the simplest case of an “up-and-out” put option w hich ceases to exist when some barrier price, H > S 0, higher than the current stock is reached. With the choice u= 1/dan explicit equation for the nodes of the tree which constitute the barrier can be written down: S(jB+ 1 + 2 h, jB+ 1 + h) =S0ujB+1+hdh, h = 0,1, ..., h B (8) Here, jB= Int [ln ( H/S 0)/ln(u)] deﬁnes the ﬁrst point just above the barrier, ( jB+1, jB+1), andhBlabels the last relevant point on the barrier corresponding to the time closest to the maturity of the option, i.e., hB= Int t[(N−jB−1)/2]. Since the probability that any allowed path starting with th e present stock price, S0, reaches an exercise price at maturity, S(N, j)< X, is still pj u(1−pu)N−j(with j≤j∗) the average value of the European barrier option can be written i n a form similar to (7): ¯VEB=e−rTj∗/summationdisplay j=0ℵEB[N, j]pj u(1−pu)N−j/parenleftBig X−S0ujdN−j/parenrightBig , (9) 4whereℵEB[N, j] is the number of paths Ntime-steps long involving jup and N−jdown movements of the price excluding those paths reaching any of the points on or above the barrier (8). As we will explain below, ℵEB[N, j] is given by ℵEB[N, j] =/parenleftBigg N j/parenrightBigg −hM/summationdisplay h=0ℵres EB[jB+ 1 + 2 h, jB+ 1 + h]/parenleftBigg N−jB−1−2h j−jB−1−h/parenrightBigg ,(10) where the second term on the right-hand side represents the c ontribution from the unwanted paths which hit the barrier (8) before reaching an exercise p oint (N, j). To understand the form of the excluded contribution in (10) w e ﬁrst note that reaching the excluded region requires that the path hits the barrier a t least once. Thus in counting the excluded paths we can simply focus on those paths reachin g the barrier. One might think that the number of unwanted paths can then be calculate d by (i) counting the number of paths connecting the origin to a given point on the barrier ; (ii) multiplying this by the number of paths connecting that point on the barrier with the exercise point ( N, j) [14]; and ﬁnally (iii) summing over all points of the barrier (8). T his seems correct except that all paths connecting a barrier point with ( N, j) already include those paths which have already hit the barrier once or more before reaching the give n barrier point. To eliminate overcounting we must make sure that in (i) we only include pat hs reaching the particular point on the barrier without having previously visited any o ther point on the barrier. The number of such restricted paths (reaching the point ( jB+ 1 + 2 h, jB+ 1 + h)) is denoted by ℵres EB[jB+ 1 + 2 h, jB+ 1 + h]. Finally note that the ﬁnal sum over the length of the barrie r is restricted to h≤hM=min(hB, j−jB−1) with j≥jB+ 1, corresponding to the fact that, in general, the exercise point ( N, j) cannot be reached from all points on the barrier. This completes our explanation of (10). We are then left with computing ℵres EB. From its very deﬁnition it is not hard to see that ℵres EB[h]≡ ℵres E[jB+ 1 + 2 h, jB+ 1 + h] satisﬁes the following recursion relation: ℵres EB[0] = 1 (11) ℵres EB[h] =/parenleftBigg jB+ 1 + 2 h jB+ 1 + h/parenrightBigg −h−1/summationdisplay l=0ℵres EB[l]/parenleftBigg 2(h−l) h−l/parenrightBigg , h≥1, (12) with the sum in (12) removing contributions from previously visited barrier points. Obvi- ouslyℵres EB[0] = 1 as there is a single path involving jB+ 1 up moves connecting the origin with the point ( jB+ 1, jB+ 1) on the tree. To solve Equations (11) and (12) we ﬁrst combine the sum on the right-hand side of (12) with the term on the left and rewrite the resulting equat ion in the form of a discrete convolution: h/summationdisplay l=0ℵres EB[l]/parenleftBigg 2(h−l) h−l/parenrightBigg =/parenleftBigg jB+ 1 + 2 h jB+ 1 + h/parenrightBigg , (13) where the boundary condition, ℵEB[0] = 1, is already included as the h= 0 contribution to (13). Note that (13) can be solved by standard Laplace tran sform (or Z-transform) techniques [15]. Since in applying these ideas to the more co mplicated American options we will lose the convolution form of (13) – the kernel will depen d onhandlseparately and not 5only through the diﬀerence, h−l– we prefer to proceed in a more general way and stay in “conﬁguration space” until the very end. We prefer to regard (13) as a matrix equation to be solved by matrix inversion. We proceed by reformulating (13) in the following matrix for m: LEBΠres EB=DEB, (14) where ΠEBandDEBarehM+ 1 dimensional vectors, with components ΠEB,h=ℵres EB[h] andDEB,h=/parenleftBigg jB+ 1 + 2 h jB+ 1 + h/parenrightBigg ,h= 0,1,2, ..., h M, and the ( hM+ 1)×(hM+ 1) dimensional matrix, LEB, can be written as, [LEB]h,l=/parenleftBigg 2(h−l) h−l/parenrightBigg θ(h−l). (15) Note that in (15) we have explicitly added a θfunction ( θ(x) = 1 for x≥0 and vanishes otherwise) to stress that LEBis a lower triangular matrix with unity along and zeros above the diagonal. This simple observation allows us to rewrite ( 14) in the convenient form, LEB=1(hM+1)×(hM+1)+QEB, (16) where QEis a nilpotent matrix of order hM,QEBy= 0 for y≥hM+ 1; and [ QEB]h.l=/parenleftBigg 2(h−l) h−l/parenrightBigg θ(h−l−1). The nilpotent property of QEBallows us to write down the explicit solution for (14), ΠEB= [1+QEB]−1DEB=hM/summationdisplay R=0(−1)RQR EBDEB, (17) which in turn leads to the following formula for the value of t he option, ¯VEB=¯VE−¯Vres EB (18) ¯Vres EB=e−rTj∗/summationdisplay j=jB+1hM(j)/summationdisplay h,l,R=0(−1)R/parenleftBigg N−jB−1−2h j−jB−1−h/parenrightBigg/bracketleftBig QR EB/bracketrightBig h,l/parenleftBigg jB+ 1 + 2 l jB+ 1 + l/parenrightBigg (19) ×pj u(1−pu)N−j/parenleftBig X−S0ujdN−j/parenrightBig . Note that the lower limit on the external sum, j=jB+ 1, excludes all paths unaﬀected by the presence of the barrier; also we have explicitly indicat ed the Nand/or jdependence of the various quantities involved; and have separated out the contribution to ℵEB[N, j] from unrestricted paths (the ﬁrst term on the right-hand side of ( 10)) which simply leads to the value of the European put option given in (7). We expect that, since we have an analytical formula, we shoul d be able to recover the exact solution of the continuum Black-Scholes theory for th is simplest of barrier options [16] as was already done for conventional European puts [5]. Figu re 1 shows the numerical convergence of the binomial value of a representative “up-a nd-out” European put option to its analytic value [16]. The same general idea used in the case of European barrier opt ions will now be used to write down an exact formula for the price of a simple American option, regarding the latter as an option with an early-exercise barrier. 6C. Conventional American Put Options Using this view to valuate American options requires the kno wledge of those points on the tree where it ﬁrst becomes proﬁtable to exercise the op tion. This set of points, parameterized as ( i, jx[i]), constitute the “early exercise barrier” (EXB). Determi ning the explicit form of the surface, jx[i], seems very diﬃcult (if at all possible) as it already impli es a knowledge of the mean value of the option at some ﬁnite numbe r of points on the tree. In this section we show that there is a self-consistent exact fo rmulation of the problem which proceeds in the following three steps: (i) we assume that the early exercise surface, jx[i] is given and compute an explicit formula for the value of the opt ion at each point on the tree, f(i, j;jx[i]), which depends parametrically on jx[i]; (ii) the fact that early exercise at ( i, j) only occurs when X−S0ujdi−j≥f(i, j;jx[i]) gives us an explicit formula for the EXB which corresponds to the strict equality, /parenleftBig X−S0u˜j[i;jx[i]]di−˜j[i;jx[i]]/parenrightBig =f(i, jx[i]); jx[i] = Int/braceleftBig˜j[i;jx[i]/bracerightBig . (20) [Note that, on the right-hand side of (20) we have not used f(i,˜j[i]) which might appear at ﬁrst sight as a more natural choice for deﬁning the EXB. As w ill become clear below, (20) is the simplest and most natural choice which resolves t he ambiguity of deﬁning f(i, j) away from points on the tree.] Finally, (iii) substituting t he solution (20) into the formally exact valuation expression gives us the value of the option. Although this strategy leads to an exact solution of the price of an American option, explici t numbers require rather heavy numerical computations except in the simplest example of a s traight EXB. Let us proceed in carrying out the program outlined above by a ssuming that the EXB, i.e.,jx[i], is explicitly given. To begin our calculation we will need some very general properties of the barrier. These follow from two simple char acteristics of early exercise: (i) if the point ( i, j) is an early exercise point, then so are all points “deeper in -the-money”, (i, j′), j′= 0,1, ..., j−1; and (ii) if two adjacent points at the same time step, ( i+ 1, j+ 1) and (i+ 1, j), are both early exercise points so is the point ( i, j). (The latter property follows from a conventional “backwardation” argument [3] w hich indicates that the average expected payoﬀ at ( i, j), discounted at the risk-free interest rate, is smaller tha n the actual payoﬀ, thus making ( i, j) itself an early exercise point.) It is not hard to see that (i ) and (ii) guarantee that the inner part of the early exercise region ca nnot be reached without crossing the EXB. Thus, if we deﬁne iAto be the ﬁrst time for which early exercise becomes possible and parametrize the points on the EXB as ( i=iA+h, jx[iA+h]) with h= 0,1,2, ..., N−iA, it then follows that jx[iA] = 0. Moreover, the structure of the tree ensures that jx[i] is a nondecreasing function of i; more precisely, for each time step, jx[i] either increases by one or remains the same. The formal expression for the price of an American option can be written down once one recognizes that once a path hits the EXB the option expire s and thus any point on the barrier can be reached at most once. As a result, the value of the option is a sum of (appropriately discounted) payoﬀs along the barrier, weig hted by the probability of reaching each point on the barrier without having visited the barrier at previous times. We can then write the expected value of an American option as: 7¯VA=N−iA/summationdisplay h=0e−r(iA+h)τℵres A[h]pjx[iA+h] u (1−pu)iA+h−jx[iA+h]/parenleftBig X−S0ujx[iA+h]diA+h−jx[iA+h]/parenrightBig , (21) whereℵres Adenotes the number of paths reaching the EXB in iA+htime steps without having previously visited any points on the barrier. The counting problem can be solved along similar lines to tho se followed in the case of European options: ℵres A[h] satisﬁes an equation analogous to (12), namely, ℵres A[0] = 1 (22) ℵres A[h] =/parenleftBigg iA+h jx[iA+h]/parenrightBigg −h−1/summationdisplay l=0ℵres A[l]/parenleftBigg (h−l) jx[iA+h]−jx[iA+l]/parenrightBigg , h≥1, (23) where the ﬁrst term on the right-hand side counts the total nu mber of unrestricted paths from the origin to the point ( iA+h, jx[iA+h]) on the barrier, while the second term excludes those paths which, before reaching ( iA+h, jx[iA+h]) visited any of the previous barrier points, ( iA+l, jx[iA+l]), l= 0,1,2, ..., h−1 [14]. As in the case of the European barrier option (23) is rewritte n as a matrix equation: LAΠres A=DA. (24) HereΠAandDAareN−iA+ 1 dimensional vectors, with components ΠA,h=ℵres A[h] and DA,h=/parenleftBigg iA+h jx[iA+h]/parenrightBigg ,h= 0,1,2, ..., N−iA, and the ( N−iA+1)×(N−iA+1) dimensional matrix, LA, takes the form, [LA]h,l=/parenleftBigg h−l jx[iA+h]−jx[iA+l]/parenrightBigg , l≤h= 0,1,2, ..., N−iA. (25) Note that, in contrast to (15) and (16), LAdepends on the indices handlseparately; also, we have used the identities jx[iA] = 0 and/parenleftBigg iA jx[iA]/parenrightBigg = 1, to incorporate the boundary condition, ℵres A[0] = 1, in (24) in a symmetric way. As in (16), we can decompose LAas, [LA]h,l=δh,l+ [QA]h,l (26) [QA]h,l=/parenleftBigg h−l jx[iA+h]−jx[iA+l]/parenrightBigg θ(h−l−1), (27) where Qhas nonzero elements starting just below the diagonal and it is thus a nilpotent matrix of degree N−iA+ 1. Thus, ΠA= [1+QA]−1DA=N−iA/summationdisplay m=0(−1)mQm ADA, (28) leading in turn to the ﬁnal formula for the value of the option , 8¯VA=/summationtextN−iA h,l,m=0e−r(iA+h)τ(−1)m[Qm A]h,l/parenleftBigg iA+l jx[iA+l]/parenrightBigg ×pjx[iA+h] u (1−pu)iA+h−jx[iA+h]/parenleftBig X−S0ujx[iA+h]diA+h−jx[iA+h]/parenrightBig . (29) One last step is the determination of f(i, j), the value of the American put at every point (i, j) on the tree which, in turn, will allow us to derive the equati on for the EXB. This is easily done by simply translating the origin in (29): f(i, j) =N−iA/summationdisplay h,l,m=i−iAe−r(iA+h−i)τ(−1)m[Qm A]h,l/parenleftBigg iA+l−i jx[iA+l]−j/parenrightBigg ×pjx[iA+h]−j u (1−pu)iA+h−jx[iA+h]−i+j/parenleftBig X−S0ujx[iA+h]diA+h−jx[iA+h]/parenrightBig . (30) Together with (20) this then leads to the rather formidable- looking equation for the barrier height ˜j[iA+k] at the ( iA+k)-th time step ( k= 0,1, ..., N−iA), as a functional of the barrier position at all future time steps before expiration: /parenleftBig X−S0u˜j[iA+k]diA+k−˜j[iA+k]/parenrightBig =N−iA/summationdisplay h,l,m=ke−r(h−k)τ(−1)m[Qm A]h,l/parenleftBigg l−k jx[iA+l]−jx[iA+k]/parenrightBigg ×pjx[iA+h]−jx[iA+k] u (1−pu)h−k−jx[iA+h]+jx[iA+k](31) ×/parenleftBig X−S0ujx[iA+h]diA+h−jx[iA+h]/parenrightBig jx[iA+k] = Int/braceleftBig˜j[iA+k]/bracerightBig . (32) [It should now be clear that in (30) jmust be restricted to points on the tree as the binomial coeﬃcient/parenleftBigg 0 jx[xA+l]−˜j[xA+l]/parenrightBigg would be ill-deﬁned – hence the choice (20).] Equations (31) and (32) for the boundary together with the formula for t he value of the option, (29), constitute an exact pricing strategy for a conventional Ame rican put. A similar formula for an American put with an “up-and-out” barrier will be discuss ed in a future publication. It is instructive to consider Equations (29), (31) and (32) i n the explicitly solvable case of a straight barrier. We begin with the observation that at exp iration, k=N−iA, (31) reduces to the deﬁnition of j∗= Int[ln( X/S 0dN)/ln(u/d)], deﬁned in the case of the European option, and thus, the barrier goes through the point ( N, j∗) at expiration. Moreover, starting from the exact point ( N, j∗) on the barrier and decreasing jx[i] by one with each backward time step we reach iA=N−j∗along the straight line, jx[i] =i−N+j∗. Recall that, since with each increasing time step, jx[i] either increases by one or remains the same, this straight l ine represents a lower bound for the early exercise barrier. For this straight barrier (28) and (29) reduce to, Pstraight A =e−r(N−j∗)τ(1−pu)N−j∗/parenleftBig X−S0dN−j∗/parenrightBig +j∗/summationdisplay h=1e−r(N−j∗+h)τℵstraight A [h]ph u(1−pu)N−j∗/parenleftBig X−S0uhdN−j∗/parenrightBig (33) with 9ℵstraight A =/parenleftBigg N−j∗+h h/parenrightBigg −/parenleftBigg N−j∗+h−1 h−1/parenrightBigg (34) We expect that the result for the true barrier should approac h the straight line formula for coarse enough time steps, τ > N−j∗−iA, (where this iAis the ﬁrst time of early exercise in the limit of continuous-time trading). III. CONCLUSION We have presented a scheme for pricing options with and witho ut barriers on binomial trees. To the best of our knowledge ours is the ﬁrst explicit d erivation of exact formulas treating barriers on binomial trees. It is our expectation t hat in the limit of continuous- time trading we should be able to recover the few exact result s available in the literature, especially for American options [12,13]. We also hope that o ur explicit formulas may provide a framework for improving the eﬃciency of numerical computa tions. IV. ACKNOWLEDGEMENTS The authors dedicate this paper to Professor Ferdinando Man cini, a remarkable teacher, colleague and friend, on the occasion of his 60th birthday. W e are grateful to Stanko Barle for reading the manuscript and bringing the work of referenc es [12] and [13] to our attention. 10REFERENCES [1] F. Black and M. Scholes, J. Finance 27, 399 (1972); J. Pol. Econ. 81, 637 (1973). [2] R. Merton, Bell J. Econ. Manag. Sci. 4, 141 (1973). [3] J. C. Hull, Options, Futures and Other Derivatives , Prentice-Hall (1999). [4] J. Cox, S. Ross, and M. Rubinstein, J. Fin. Econ. 7, 229 (1979). [5] J. Cox and M. Rubinstein, Options Markets , Prentice-Hall (1985). [6] E. Derman, I. Kani, D. Ergener, and I. Bardhan, Enhanced Numerical Methods for Options with Barriers , Goldman, Sachs & Co. (1995). [7] S. Figlewski and B. Gao, J. Fin. Econ. 53, 313 (1999). [8] P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives , Cambridge Univ. Press (1999). [9] S. R. Pliska, Introduction to Mathematical Finance , Blackwell Publishers (1997). [10] N. A. Chriss, Black-Scholes and Beyond: Option Pricing Models , McGraw-Hill (1997). [11] R. Geske and H. E. Johnson, J. Finance 39, 1511 (1984). [12] I. J. Kim, Rev. Fin. Studies 3, 547 (1990). [13] B. Gao, J-Z Huang and M. G. Subrahmanyam, An Analytical Approach to the Valuation of American Path-Dependent Options , working paper (1996). [14] We recall that the number of paths between two arbitrary points on the tree, say ( i, j) and (i′, j′) (with i′> i, j′> j) is given by the binomial coeﬃcient, ( i′−i)!/[(j′−j)!(i′− i−j′+j)!]. [15] See, for example, K. S. Miller, Linear Diﬀerence Equations , W. A. Benjamin, Inc. (1968). [16] M. Rubinstein and E. Reiner, RISK 4, 28 (1991). 11FIGURES 0 10 20 30 40 50 60 N2.02.53.03.54.04.55.05.56.0 V FIG. 1. Convergence to analytic value [16] of a three-month E uropean “up-and-out” put option on a non-dividend-paying stock as the number of binom ial intervals Nincreases. The solid line joining the squares is a guide to the eye. The stock price S0is 60, the risk-free interest rate r is 10%, and the volatility σis 45%. The barrier level His at 64. The analytic value (continuous line) is 2 .524. As a point of reference, we give the European put option v alue (i.e., H→ ∞) 4.6 after the Black-Scholes solution (dashed line). 12
arXiv:physics/0008112v1 [physics.acc-ph] 17 Aug 2000DESIGNAND PERFORMANCE SIMULATIONSOF THE BUNCH COMPRESSOR FOR THEAPS LEUTLFEL∗ M.Borland, ANL,Argonne, IL 60439,USA Abstract A magnetic bunch compressor was designed and is be- ing commissioned to provide higher peak current for the Advanced Photon Source’s (APS) Low-Energy Undulator Test Line (LEUTL) free-electronlaser (FEL) [1]. Of great concern is limiting emittance growth due to coherent syn- chrotronradiation(CSR).Tolerancesmustalsobecarefull y evaluatedtoﬁndstableoperatingconditionsandensuretha t the system can meet operational goals. Automated match- ing andtolerancesimulationsallowedconsiderationof nu- merous conﬁgurations, pinpointing those with reduced er- ror sensitivity. Simulations indicate tolerable emittanc e growth up to 600 A peak current, for which the normal- ized emittance will increase from 5 to about 6.8 µm. The simulationsalso providepredictionsofemittancevariati on with chicane parameters, which we hope to verify experi- mentally. 1 INTRODUCTION A companion paper [2] reviews magnetic bunch compres- sion and shows a schematic of our system. I assume the reader is familiar with this paper. The APS bunch com- pressor design is an outgrowth of studies [3] by P. Emma and V. Bharadwaj of Stanford Linear Accelerator Center (SLAC). They explored a number of designs, including symmetric and asymmetric four-dipole chicanes. Starting from this work, I investigated a large number of conﬁgu- rations with various values of R56, asymmetry, and ﬁnal current. For each conﬁguration, detailed longitudinal and transverse matching was performed, followed by tracking with CSR and wakeﬁelds. Then, sensitivity analysis was performedforallconﬁgurations,followedbyjittersimula - tionsfortheleast sensitiveconﬁgurations. Thisworkreliedon elegant [4],a6-Dcodewith afast simulationofCSReffects,pluslongitudinalandtransvers e wakeﬁelds. elegant also performsoptimization of actual tracking results, such as bunch length, energy spread, and emittance. Simulationofthelinacusesthe RFCAelement,a matrix- based rf cavity element with exact phase dependence. Our linac has quadrupoles around the accelerating structures. Hence, I split each 3-m section into about 20 pieces, be- tween which are inserted thin-lens, 2nd-order quadrupole elements. A series of such elements is used for each quadrupole. A Green’s function technique is used to model wake- ﬁelds,usingatabulationoftheSLAC-structurewakefunc- tions provided by P. Emma [5]. To reduce running time, ∗Work supported by the U.S. Department of Energy, Ofﬁce of Bas ic Energy Sciences, under Contract No. W-31-109-ENG-38.one longitudinal wake element is used per 3-m section, whichisagoodapproximationforrelativisticparticles. F or transversewakes,Iusedonewakeelementperrfcavityel- ement(about20persection). The CSR model used by elegant is based on an equa- tion [6] for the energy change of an arbitrary line charge distribution as a function of the position in the bunch and in a bending magnet. Details of this model will be pre- sentedbytheauthoratanupcomingconference. Effectsof changesin the longitudinaldistributionwithin a dipole ar e included. CSR in drift spaces is included by propagating the terminal CSR “wake” in each bend through the drifts with thebeam. 2 MATCHING Longitudinal and transverse matching has the goal of pro- vidingconﬁgurationsforthe300-Aand600-ALEUTLop- erating points[2]. The starting point for the simulations i s macro particle data generated [7] with PARMELA, giving the6-Ddistributionafterthephotoinjector(PI).SeeFigu re 1 in[2]. Longitudinalmatching involvesadjusting the phase and voltage of L2 (see [2] for nomenclature) to obtain the de- sired current and energy after the chicane. Then, L4 and L5 are adjusted to minimize the energy spread and obtain the desired ﬁnal energy. Longitudinal matching includes longitudinal wakeﬁelds, rf curvature, and higher-order ef - fectsinthebeamtransport,bymatching trackedproperties ofthesimulatedbeam. Figure1showsthelongitudinalphasespaceforthe300- Acasewith R56=−65mm,whichexhibitsacurrentspike of nearly 1200 A. The matching ignores this spike (which isshorterthanaslippagelengthfor530nm)becauseofthe way “current”isdeﬁned[2]. Figure1: Typicallongitudinalphasespace (300-Acase)Figure2: Typicaltwissparametersinthe chicaneregion. Followinglongitudinalmatching,transversematchingis done for each conﬁguration. Initial Twiss parameters are obtained from the rms propertiesof the PARMELA beam. Starting values for the quadrupoles were obtained from matching “by hand” for one conﬁguration. Four sequen- tialelegant runs work the beta functions down the linac. The most important constraints maintain small beta func- tions in the linac (for transverse wakeﬁeld control), small horizontal beta in dipole B4 (to reduce CSR effects), and matching for the emittance measurement sections. Figure 2 showssampleTwissparametersin thechicaneregion. The matching is highly automated, so that only the de- sired beamcurrentand energyneedsto be speciﬁed. Eval- uation of tolerances and randomized simulations are also automated, being set up by scripts from the corresponding matchingruns. Transferof data between simulationstages ishandledusingSDDSﬁlesandscripts[8],reducingerrors andincreasingthenumberofconﬁgurationsthatcanbeex- amined. For example, a script is used to scan all conﬁgu- rationsandgivepowersupplyspeciﬁcations. A distributed queue utilizing 50 workstations is used to run the simula- tions. Figure 3 shows emittance vs. R56for the symmetric (A=1) and asymmetric (A=2) cases at 300 A and 600 A. For 300 A, the symmetric and asymmetric cases are very similar. For 600 A, the difference is 10% or more, which shouldbemeasurable. One surprise in Figure 3 is that the emittance does not uniformly increase as \|R56\|increases, even though elegant shows the expected monotonic increase (due to CSR) vs bending angle for a single dipole with a constant inputbeamdistribution. Thisisapparentlyduetovariatio n in the compressed bunch distribution between cases with the same “current” but different R56. For smaller \|R56\|, there are higher current spikes at the head of the bunch, leading to a larger and more rapidly changing CSR wake, which in turn leads to larger emittance growth. The effect isevenmorepronouncedin the1200-Acases(notshown). Insertion of the scraper between B2 and B3 to remove the low-energy part of the beam can reduce the height and width of the current spike, resulting in lower emittance. Unfortunately, this also reduces the current in the rest ofthe bunchconsiderably. Earlier simulations showed that emittance trendscan be changed signiﬁcantly by inconsistent values of the hori- zontal beta function at the exit of B4. All of these sub- tleties will make for difﬁcult interpretation of experimen ts in which R56is varied. However, because compression to different currents for ﬁxed R56involves only adjustment of the rf phases and voltages, comparision of the emit- tance growth for different amounts of compression should bemorestraightforward. Figure3: Horizontalnormalizedemittancevs. R56. 3 TOLERANCE DETERMINATION Tolerances are driven by the FEL gain length, trajectory, and wavelength stability requirements [9]. The 10% rms gain length variation limit is easy to use in elegant as it computes FEL performance directly using Xie’s parame- terization [10]. Beam trajectory limits ( ∼50µm,∼50µr) are included separately as they are not incorporated into Xie’s formula. The 1-nm rms wavelength variation limit is a challenging goal at 530 nm as it puts a 0.1% limit on energyvariation. The analysis begins by running single-parameter “sweeps” to assess the effect on the constrained quanti- ties (gain length, trajectory, and wavelength) of accelera - tor parameters (e.g., rf phase). Sweeps included rf phase and voltage; photoinjectortiming, charge,andenergy;and chicanedipolestrength. Fromthese sweeps, a scriptdeter- mines the limit on each parameter change due to the var- ious speciﬁcations, showing that conﬁgurations with the largest R56are least sensitive to difﬁcult-to-control tim- ing and phase errors. These conﬁgurations experience the mostemittancedegradationfromCSR,buttendtoyieldthe shortestgainlengthastheyhavethesmallestenergyspread (L2beingcloserto crest). The limits, shown in Table 1, are larger for the 600-A case because the 1-nm wavelength constraint is easier at 120 nm than 530 nm. Nine parametersare limited primar- ily by the wavelength constraint and four others by hori- zontal trajectory constraints. Hence, to determine the rms tolerance, one simply divideseach sweep limit by√ N,NTable 1: Selectedsweeplimitsfor R56=−65mm quantity 300-Alimit 600-Alimit L2phase 0.17◦0.49◦ L4/L5phase 0.77◦1.45◦ L2voltage 0.11% 0.31% L4/L5voltage 0.52% 1.4% PItiming 0.29ps 0.88ps PIenergy 0.26% 1.1% PIcharge 12% >20% beingthenumberofparameterslimitedbyaparticularcon- straint. For the horizontal trajectory, Nwas doubled to eight to allocate half the budget to nonswept parameters (e.g., corrector magnets). Some of these phase and timing tolerancesarebeyondthestate oftheart. 4 RANDOMIZED SIMULATIONS Randomized simulations were used to conﬁrm the toler- ances and examine errors not covered by the sweeps (e.g., corrector jitter, quadrupole jitter, and alignment). Thes e were done for the most stable conﬁgurations (i.e., R56= −65mm). Because some tolerances are beyond the state of the art, I used randomized simulations to determine the impact of “relaxed” tolerances, assuming these rms levels [11]: 1◦rf phase jitter, 0.1% rf voltage jitter, 1 ps timing jitter,5% chargejitter,and2%PI energyjitter. Tables 2 and 3 show the results, respectively, for the sweep-derivedtolerance levels and the relaxed levels. The sweep-derivedtolerancelevels result in meeting the speci - ﬁcations for the FEL, while the relaxedlevels, not surpris- ingly, do not. One surprise in the relaxed case is the large jitter in the vertical plane. This results from uncorrected nonlinear dispersionin a vertical dogleg between the linac and the LEUTL, a problem which can be readily remedi- ated usingtwo sextupoles[5]. Table 2: Results of 300 randomized simulations with sweep-determinedtolerancelevelsfor R56=−65mm 300A 600A quantity rms%rms % jitterinside jitterinside /angbracketleftx/angbracketright(µm)71835791 /angbracketleftx′/angbracketright(µr)29932496 /angbracketlefty/angbracketright(µm)13100 11100 /angbracketlefty′/angbracketright(µr)19981799 Lgain(m)0.01990.016 100 λ(nm)0.83720.29100Table 3: Results of 300 randomized simulations with re- laxedtolerancelevelsfor R56=−65mm 300A 600A quantity rms %rms% jitterinsidejitterinside /angbracketleftx/angbracketright(µm)89728981 /angbracketleftx′/angbracketright(µr)59646858 /angbracketlefty/angbracketright(µm)638812779 /angbracketlefty′/angbracketright(µr)138 6224539 Lgain(m)0.048 6831.3 λ(nm)9.6 92.827 5 ACKNOWLEDGEMENTS The technical note [3] by P. Emma and V. Bharadwaj pro- vided a valuablestarting point. I acknowledgehelpful dis- cussions and assistance from H. Friedsam, E. Lessner, J. Lewellen, S. Milton, and G. Travish. J. Lewellenprovided the PIbeamdistributiondata. 6 REFERENCES [1] S.V. Milton et al., ”Observation of Self-Ampliﬁed Spont a- neous Emission and Exponential Growth at 530 nm,” Phys. Rev. Lett.,tobe published. [2] M. Borland et al., “A HighlyFlexible Bunch Compressor fo r the APSLEUTLFEL,”these proceedings. [3] P.Emma,V. Bharadwaj, private communication. [4] M. Borland, unpublished program. See www.aps.anl.gov/asd/oag/manuals/elegant ver14.1 /ele- gant.html. [5] P.Emma,private communication. [6] E. L. Saldin et al., “On the coherent radiation of an elect ron bunch moving inanarc of acircle,” NIMA398 (1997) 392. [7] J.Lewellen, private communication. [8] M. Borland, “A Universal Postprocessing Toolkit for Acc el- eratorSimulationandDataAnalysis,”Proc.1998ICAPCon- ference, Monterey, tobe published. [9] S.Milton, private communication. [10] M. Xie, “Design Optimization for an X-Ray Free Electron LaserDrivenbySLACLinac,”Proc.1995 PAC,Dallas,May 1-5, 183. [11] G. Travish,private communication.
DESIGN OF THE SNS NORMAL CONDUCTING LINAC RF CONTROL SYSTEM Amy Regan, Sung-il Kwon, Tony S. Rohlev, Yi-Ming Wang, LANL, Los Alamos, NM 87544, USA Mark S. Prokop, David W. Thomson; Honeywell FM&T Abstract The Spallation Neutron Source (SNS) is being designed for operation in 2004. The SNS is a 1 GeV machineconsisting of a combination normal-conducting and super-conducting linac as well as a ring and target area. The linac front end is a 402.5 MHz RFQ being developed by Lawrence Berkeley Lab. The DTL (at 402.5MHz) and the CCL (at 805 MHz) stages are beingdeveloped by Los Alamos National Laboratory. Theexpected output energy of the DTL is 87 MeV and that ofthe CCL is 185 MeV. The RF control system underdevelopment for the linac is based on the Low EnergyDemonstration Accelerator’s (LEDA) control system withsome new features. This paper will discuss the new designapproach and its benefits. Block diagrams and circuitspecifics will be addressed. The normal conducting RFcontrol system will be described in detail with referencesto the super-conducting control system where appropriate. 1 RF CONTROL FUNCTION OVERVIEW The RF system for the SNS linac is well described in M. Lynch ’s paper in these proceedings [Ref. 1]. Specifically of interest to the RF Control System(RFCS) is the fact that one control system is required foreach klystron. The RF control system must supportoperation of 402.5 MHz and 805 MHz normal conducting(NC) cavities, as well as 805 MHz superconducting(SRF) cavities. The intent of the RF Control systemdesign is to provide a system which requires minimalhardware changes to support all three cavity types. Foreach cavity type, the governing specification is to providecavity field control within ±0.5% amplitude and ±0.5 ° phase. The functions required of the RFCS are: Cavity Field Control, Cavity Resonance Control, HPRF Protection,and Reference generation and distribution. Figure 1shows a block diagram of the RFCS. We have selected aVXIbus architecture for the RFCS. The present design combines cavity field control and resonance control into a single double-wide VXIbusmodule. The HPRF Protect function will be performed byanother VXIbus module. Both are supported by a ClockDistribution Module. Physically this design differs fromits predecessor (the Accelerator Production of Tritium RFcontrol system) where Resonance Control and FieldControl were individual modules and the HPRF Protectfunction required a VXI module plus multiple outboard chassis. Experience with LEDA has showed us we canreduce the number of channels supported by the HPRFProtect circuitry in such a way as to perform all requiredfunctions in a single VXI module only. We have alsoseen that combining the Field and Resonance Controlfunctions into a single VXIbus module reduces theamount of backplane cross-communication and simplifiesmodule-to-module interconnections. Linac CavityDigital ControllerKlystron Resonance Control LOLO50 MHz ReferenceHigh Power Protect 50 MHz Cavity FeedbackUpconverter Downconverter Figure 1: RF Controls for SNS. A conceptual VXIbus crate layout is shown in Figure 2. Due to the physical separation of the klystrons for theNC cavities, we are only putting one RF Control systemin a VXI crate. The SRF cavities ’ klystrons are located close enough together to encourage savings in crate andrack cost by co-locating two control systems in a singlecrate. We cannot do this for the NC systems because thedistances between adjacent klystrons will detrimentallyaffect our control margin due to increased signal groupdelay. Figure 2. VXIbus RF Control System Crate Layout.2 VXIBUS MODULE DESIGNS 2.1 Clock Distribution Module The Clock Distribution Module (CDM) is quite similar to that of LEDA. It receives a phase stable referencesignal from the Reference Distribution system andgenerates a 40 MHz ADC (analog-to-digital converter)clock (digital) and a 50 MHz IF (analog) for use by theField/Resonance Control Module. A block diagram of theCDM is presented in Figure 3. The CDM also receivespulse timing information from the Brookhaven NationalLaboratory Timing Module and distributes it to the rest ofthe RFCS to synchronize the RFCS with the rest of theaccelerator. 0/4 VCO 50 MHz IF REF40 MHz ADC CLK10 MHz SYNCH 40CO 0/20CO2.5 MHz REF LO2.5 MHz LO2.5 MHz locked to, and with phase stability of, the LOD FLIP-FLOP PHASE/ FREQ DETECTOR T0 RF GATELOOP FILTERPHASE/ FREQ DETECTOR PHASE/ FREQ DETECTORLOOP FILTER LOOP FILTER TIMING LOGIC/16 Figure 3. Clock Distribution Module Block Diagram. 2.2 Field/Resonance Control Module The Field/Resonance Control Module has two primary functions. 1) It determines the current resonance conditionof the cavity and sends a correction signal to the CavityResonance Control System (water cooling for the NCcavities) which brings the cavity back to resonance andmaintains it. It also generates a frequency-shifted drive forconditions when the cavity is far off-resonance [Ref 2]). 2)The module also samples the cavity field and outputs thecorrect control signals for the klystron in order to keep thecavity field phase and amplitude within specification. Ituses both PID (proportional-integral-derivative) controland an adaptive feedforward algorithm. The adaptivefeedforward algorithm we refer to as an Iterative LearningController and is covered in a separate paper at theseproceedings [Ref.3]. Figure 4 is a simplified block diagram of the Field/Resonance Control Module. This module makesextensive use of modern high-speed digital circuitry.Downconversion of the RF signals for the resonancecontrol function and upconversion of the controlled RFdrive signal for the klystron are the only analog circuitson the board. Significant digital components are twodigital signal processors (DSPs) and four ComplexProgrammable Logic Devices (CPLDs). The two TI C60family DSPs are used for the Resonance Control and Field Control functions (one each). IF To Cavity Resonance Control SystemTo KlystronD/AD/ADSP1 DSP1 LO A/D VCO IQ Mod U/C LOCPLD I/Q Det. & FIR Fld & Kly. PID ControllerD/A D/ACLK40 MHzCav IFCPLD I/Q Det. & FIRLO A/DFwd RFLO A/DCPLD I/Q Det. & FIRRefl RFLO A/DCPLD I/Q Det. & FIRBeam RF Figure 4. Simplified Control Module Block Diagram. The Resonance Control algorithm is the same as used in LEDA [Ref. 2]. In the fast field control signal path,multi-rate digital processing is performed in CPLDs foroptimized throughput. The Field Control DSP performsthe slower, pulse-to-pulse, adaptive feedforward and gainscheduling features, while the Field Control CPLD doesthe actual fast feedback PID algorithm. We will use the Altera EP20KE series CPLD family. Three of the four CPLDs are identical, containing amultiplexer/ multiplier (I/Q detector), digital filter, 2x2rotation matrix and a PID controller). The I and Q outputdata rate is 20 MHz, and the expected delay/latencythrough the CPLD is 23.5 cycles (1174 ns). Figure 5shows a block diagram of the CPLD signal flow and theassociated delay/latency at each step. Note that theprimary source of delay is the group delay through theFIR filter. The fourth CPLD in the Control Moduleincludes this basic structure as well as the functionality toperform klystron phase control. ADC Data (40 MHz)Mux/Multiplier FIR Filter2x2 RotationPID ControllerRotationMatrixWeights (I) Weights (Q) Set Point (Q) PID Gains (Q)Set Point (I) PID Gains (I) I Data (20 MHz) Q Data(20 MHz)FIR Filter PIDController Delay Latency11.517 1101120 cycles3.5 cycles Figure 5. CPLD Signal Flow and Expected Delay.2.3 HPRF Protect Module The HPRF Protect function is fulfilled by a single VXIbus module. It is based on the HPRF Protect systemfor LEDA, which consists of a VXIbus module and fourdifferent chassis. Based on lessons learned on LEDA, wehave been able to dramatically simplify this function.The module ’s purpose is to turn off the RF drive to the klystron should a fault occur within the HPRF transportline, be it arcing in the waveguide or unexpectedly highreflected power. There are six RF channels per module formonitoring RF power and ten inputs from the fiber opticarc subsystem for monitoring waveguide arcs. Instead ofsimply turning off the drive to the klystron on any givenfault, logic is built into each channel to allow for acertain number of faults within a certain period of time(fault frequency) before declaring an RF-off state. Besides waveguide arc monitoring via fiber optic detectors, logic is built into the module to interpret whena cavity arc occurs based on RF power signal fromdirectional couplers at the cavity itself and the internalcavity field. Like the Field/Resonance Control Module, the HPRF Protect Module is primarily digital. The analog front endconsists of a 20 MHz bandwidth input filter at the RFfrequency (402.5 or 805 MHz) and a true RMS powerdetector (AD8361). The input filter is the only part thatis frequency-dependent. A six Ms/s analog-to-digitalconverter (Zilog XRD 6418 with 6 channels used) with10 bits of resolution is used to digitize the power withinthe pulse. After that, all the comparators are digital, andwith a simple Altera PLD for decode/actions, the expectedtotal response time is 10 µs. A block diagram of a single RF channel (which is duplicated six times on the board) isgiven in Figure 6. BPFRMS Detector ADC10 bit latch 10 bit latch10 bit comparator Control / Readout Logic10 bit data A>B MUX control 10 10RF inANALOG FRONT END 10 Figure 6. Single RF Channel Block Diagram. 3 REFERENCE DISTRIBUTION The phase stable RF Reference required for the linac will be distributed via an insulated 3 1/8 ” coaxial line along the tunnel of the accelerator. The local oscillatorfrequency is distributed (352.5 MHz, or 755 MHz) and ismixed with the cavity field from individual pickup loopsfrom each cavity in order to send a 50 MHz cavity fieldsignal up to the klystron gallery where the RF Controlelectronics are located. The phase stability of the referenceline is maintained to 0.1 ° at 755 MHz. [Ref. 4] A mockup of this system will be built at LANL later thisyear. 4 SUMMARY The design of the RF Control System for the Normal Conducting SNS linac is well underway. Individualmodules have been identified, specified, and are mostlythrough the initial design phase. In the next few monthswe will begin bread-boarding these modules and puttingtogether an initial test system. REFERENCES [1] M. Lynch, et al “The Spallation Neutron Source (SNS) Linac RF System ”, Linac 2000, August 2000. [2] Y. Wang, et al, “Algorithms and Implementations of APT Resonant Control System ”, PAC 1997, May 1997. [3] S. Kwon, et al, “An Iterative Learning Control of a Pulsed Linear Accelerator ”, Linac 2000, August 2000. [4] T. S. Rohlev, “Phase Stability Requirements for the SNS Reference Distribution System ”, Internal LANL Technical Note LANSCE-5-TN-017, 8/3/00.
arXiv:physics/0008114v1 [physics.acc-ph] 17 Aug 2000AHIGHLYFLEXIBLEBUNCH COMPRESSOR FORTHEAPS LEUTLFEL∗ M. Borland,J. Lewellen,S. Milton,ANL, Argonne,IL 60439,U SA Abstract The Low-Energy Undulator Test Line (LEUTL) free- electron laser (FEL) [1] at the Advanced Photon Source (APS) has achieved gain at 530 nm with an electron beam current of about 100 A [2, 3]. In order to push to 120 nm and beyond, we have designed and are commissioning a bunch compressor using a four-dipole chicane at 100-210 MeV to increase the currentto 600 A or more. To provide options for control of emittance growth due to coherent synchrotronradiation(CSR),thechicanehasvariable R56. The symmetry of the chicane is also variable via longitu- dinal motionofthe ﬁnal dipole,whichis predictedto have aneffectonemittancegrowth[4]. Followingthechicane,a three-screenemittancemeasurementsystem shouldpermit resolution of the difference in emittance growth between variouschicaneconﬁgurations. A vertical bendingmagnet analysis line will permit imaging of correlations between transverseandenergycoordinates[5]. A companionpaper discussesthephysicsdesignin detail[4]. 1 APSLINACOVERVIEW The APS injector consists of a linac, an accumulator ring, anda7-GeVboostersynchrotron. Inadditiontodelivering beamtotheaccumulator,thelinaccanbeconﬁgured[6]to deliver beam to the LEUTL experimenthall [1]. The linac consists of 13 Stanford Linear Accelerator Center (SLAC) type accelerating sections powered by four klystrons, two thermionicrfguns(TRFG)[7,8,9]powered(oneatatime) byasingleklystron,andonephotocathodegun(PCG)[10] powered by a single klystron. Figure 1 shows a schematic ofthesystemandthelocationofthenewly-installedbunch compressor. The original purpose of the linac was to create positron beams and deliver them to the accumulator ring for injec- tion into the APS. The positron target was subsequently removed when the APS switched to electron operation. In both situations, the requirements on the linac were mod- est intermsofemittance,energyspread,bunchlength,and stability. However, the requirements for reliability were andareveryhigh,whichwasonereasonforeliminationof positron operation. The FEL project requires much higher beam quality and beam stability. The required beam qual- ity is typically only achieved using a photocathode gun; however,thereliabilityof suchguns(particularlythe dri ve laser) is insufﬁcient to act as an injector for the APS. The dualthermionicgunshaveadistinctadvantagehere,having proven themselves as components of the injector at SSRL ∗Work supported by the U.S. Department of Energy, Ofﬁce of Bas ic Energy Sciences, under Contract No. W-31-109-ENG-38.[11]. The use of alpha magnets [7] for magnetic bunch compression in these guns allows the guns to be placed off-axis, leaving the on-axis position for the PCG. This is an important consideration in preserving the PCG beam brightness. 2 MAGNETICBUNCH COMPRESSION The principle of magnetic bunch compression is well- known, so we only review the basic idea here. In a mag- netic chicane (see Figure 1) the path length traveled by a particleis s=so+R56δ,where soisthecentralpathlength andδ= (p−po)/poisthefractionalmomentumdeviation. Forsimplechicanes, R56<0sothathigh-energyparticles take ashorterpath. Phasingthebeamaheadofthecrestintheprecompressor linacintroducesan“energychirp”intothebeam,sothatthe tail of the beam has higher energy than the head. As a re- sult,thetailwillcatchuptotheheadinthechicane,giving a shorter bunch. If the beam is undercompressed,then the energy spread imparted in the precompressor linac can be removedbyphasingbehindthecrestinthepostcompressor linac. It ispossibletoderiveformulaeforthephasingrequired to obtain a desired bunch length and minimized energy spread. However, accurate calculation requires including wakeﬁeld effectsand dependson the detailedinitial bunch shape. Hence, we used simulation to ﬁnd the optimal val- ues[4]. 3 LEUTL BEAM REQUIREMENTS The primary goal of the bunch compressor is to provide highercurrentbeamtotheLEUTLFEL.Asecondarygoal is characterization of CSR effects. The bunch compres- sor was designed with two LEUTL operating points in mind. These operating points, distinguished primarily by the beam current of 300 or 600 A, are summarized in Ta- ble 1. The requirements for charge and emittance are not difﬁcult compared to the state-of-the-art for photoinject or systems. We hope that these parameters can be achieved repeatably and easily to provide for routine and stable op- eration. Because of the very non-Gaussian longitudinal phase- spacedistributionsonetypicallyseesinthecompressor,w e use the following deﬁnition for the beam current: I80= 0.8∗Qtotal ∆t80where Qtotalis the total chargein the beam and ∆t80is the length in time of the central 80% of the beam. The value of 80% was used because this includes most of theparticlesbuttypicallyexcludeshigh-currentspikest hat tend to occur at the head and tail. Also, when we refer toTC RF guns L1 L2 (SLED) to FELB1B2 B3 B4Q1 Q2emittance measurementmatching quads BPMα magnetsPC gun L4 (SLED)emittance measurementCTR diagnostic vertical bend diagnostic scraper flagL3 L4 (SLED) Figure1: SchematicoftheAPS linacwith thebunchcompresso r. bunchlength,wemean ∆t80. Because the initial emittance is relatively large, it is de- sirable that compression not make it larger. For the 600-A case, however,simulations[4]predictanemittancegrowth of up to 40% due to CSR. Hence, this part of the LEUTL requirementmaynotbemet. Table1: DesiredLEUTLOperatingPoints Nominal 300A600A Current(A) 100 300600 Energy(MeV) 217 217457 RMS en. spread(%) <0.1<0.1<0.15 Initialcharge(nC) 0.5 0.50.5 Finalcharge(nC) 0.5 0.420.42 ∆t80(ps) 4 1.10.55 Norm. emittance( µm) 5 55 Lightwavelength(nm) 530 530120 4 BUNCHCOMPRESSOR FEATURES Figure 1 provides a detailed schematic of the compressor chicane. One sees that most of the beam energy at the en- trance to the bunch compressor is due to the “L2” sector of the linac, which consists of a single SLEDed klystron driving four SLAC-type 3-m structures, delivering a beam energy of up to 210 MeV. The bunch compressor was de- signedwiththe rangefrom100-210MeV inmind. Table 2 shows some of the principle parameters of the bunch compressor. A noteworthy feature of the APS de- sign is that the R56is designed to be variable, which will beaccommodatedthroughtransversemotionofthecentral dipole pair (B2 and B3 in Figure 1). This permits varia-tionofthebendinganglewithouthavingtodesignmagnets with largegoodﬁeldregions. Asa result,we canvary R56 between 0 and -65 mm. Presently, due to delivery prob- lems with the ﬂexible chambers, the chicane is installed withﬁxedchambers. Laterthisyearwewillinstallﬂexible curved chambers in all the dipoles and telescoping cham- bers between the dipoles. The hardware required for mo- tionofthemagnetsisalreadyinplace. Table 2: BunchCompressorParameters Maximumbendangle 13.5◦ Maximumbendﬁeld 0.86T Effectivebendlength 192mm Maximum R56 -65mm Maximumtransversemotion 184mm Maximumlongitudinalmotion 602mm The symmetry of the chicane will also be variable through longitudinal motion of the ﬁnal dipole, B4. The ratio of the B3-B4 distance to the B1-B2 distance will be variable from1.0 to 2.0, correspondingto variationsin the ratio of the angle of B1 to the angle of B4 from 1 to 1.8. Two “tweaker” quads are required within the chicane to allow matching the dispersion for asymmetric conﬁgura- tions. Variable R56and symmetry is thought to be interesting inthattheeffectofCSRshouldchangewiththeseparame- ters(or,morefundamentally,withthebendingangles). The asymmetricconﬁgurationshaveweakerbendingin B3and B4,wherethebeamisshortest,whichshoulddecreaseCSR effects. However, these conﬁgurations also have a larger driftbetweenB3 andB4,whichallowsCSR moreroomto act. Simulations show a slight beneﬁt to the asymmetricconﬁguration for 600 A, and greater beneﬁt beyond that. We hope to test these predictions once the ﬂexible cham- bersandemittancediagnosticsarefullyimplemented. At present, no attempt has been made to shield CSR by placing small-gap chambers in the dipoles. Our intention is to add this feature if we ﬁnd it necessary and to thus measuretheeffect. 5 DIAGNOSTICS Because of concerns about CSR and jitter effects in the bunchcompressor,wehaveplannedforextensivediagnos- tics for the system. Although not all diagnostics are com- pleted at this time, we expect completionthis year. Figure 1 showsmostoftheplanneddiagnostics. There are BPMs upstream and downstream of the chi- cane, plus one in the center of the chicane that will give information on the energy centroid. This new design is monopulse-receiver-basedandshouldhavesingle-shotres - olutionand reproducibilityof 15 µm forchargeof 0.1 to 2 nC. Thecompressorwillhaveatotalofsevenbeam-imaging ﬂags. One ﬂag is in the chicane center, downstream of the two-blade beam scraper. Another is at the exit of B4, where a small horizontalbeamsize is requiredto minimize CSReffects. Threeﬂagswith1-mspacingprovideathree- screen emittance measurement system. Several of these ﬂagsuseanewdesignincorporatingtwocameras—onefor low magniﬁcationandanotherforhighmagniﬁcation. The highmagniﬁcationcamerasshouldachieveabeamsizeres- olutionof7 to15 µm,dependingonchargesensitivities. The chicane bends the beam in the horizontal plane. A verticalspectrometermagnetisinstalleddownstreamofth e chicane with two ﬂags. The ﬁrst ﬂag allows imaging the x−δcorrelationsinthebeam[5],whichshouldgiveinfor- mation on the effects of CSR and wakes. The second ﬂag is usedforenergyspreadandcentroidresolution. For bunch length measurements, we will initially use a coherenttransitionradiation(CTR)diagnostic[3]. Thisd i- agnostichasbeensuccessfullyusedwithoneoftheTRFGs and showedfeatureson the 100-fsscale. We havealso left space for synchrotron light ports on all of the dipoles and may use this radiation for bunch-length measurements in the frequencydomain[12]. 6 FUTURE DEVELOPMENTS We arealso interestedinuseofthebunchcompressorwith the TRFGs. Bunch lengths of 350 fs have been obtained with one of these guns, using alpha-magnet-based com- pression [3]. Simulations predict that by also using the bunch compressor, bunch lengths of 5-10 fs are possible with currents on the order of 500 A. While this is not use- ful for FEL work, it may be useful to those interested in ultrashortpulses. WearealsoplanninganenergyupgradetotheAPSlinac to allow energyof up to 1 GeV. The present limit with thebunchcompressorisabout600MeV. 7 ACKNOWLEDGEMENTS We would like to acknowledge valuable suggestions from and calculations done by Paul Emma and Vinod Bharad- waj, both of SLAC. Their technical note [13] provided a valuablestartingpointforourdesign. 8 REFERENCES [1] S.V. Milton et al., “The FEL Development at the Advanced Photon Source,” Proc. FEL Challenges II, SPIE, January 1999, tobe published. [2] S.V. Milton et al., ”Observation of Self-Ampliﬁed Spont a- neous Emission and Exponential Growth at 530 nm,” (sub- mittedtoPhys. Rev. Lett.). [3] N.S. Sereno et al., “Use of Coherent Transition Radiatio n to Set Up the APS RF Thermionic Gun to Produce High- Brightness Beams for SASE FEL Experiments,” these pro- ceedings. [4] M. Borland, “Design and Performance Simulations of the BunchCompressorfortheAPSLEUTLFEL,”theseproceed- ings. [5] D.Dowelletal.,ICFAWorkshoponHighBrightnessBeams, Nov. 9-12, 1999, UCLA,tobe published. [6] R. Soliday et al., “Automated Operation of the APS Linac UsingtheProcedureExecutionManager,” theseproceedings . [7] M. Borland, “A High-Brightness Thermionic Microwave ElectronGun,”SLAC-Report-402,1991.StanfordUniversit y Ph.D. Thesis. [8] M. Borland, “An Improved Thermionic Microwave Gun and Emittance-PreservingTransportLine,”Proc.1993PAC,May 17-20, 1993, New York, 3015-3017. [9] J.W. Lewellen et al., “A Hot-Spare Injector for the APS Linac,”Proc.1999PAC,March29-April2,NewYork,1979- 1981. [10] S.G. Biedron et al., “The Operation of the BNL/ATF Gun- IV Photocathode RF Gun at the Advanced Photon Source,” ibid.,2024-2026. [11] J.N.Weaveretal.,“ExperiencewithaRadioFrequencyG un onthe SSRLInjector Linac,”Proc. 1993 PAC,op. cit.,3018- 3020. [12] B.X.Yang, private communication. [13] P.Emma,V. Bharadwaj, private communication.
arXiv:physics/0008115 17 Aug 2000SPACE CHARGE EFFECTS IN BUNCH SHAPE MONITORS A.V.Feschenko, V.A.Moiseev Institute for Nuclear Research, Moscow 117312, Russia Abstract The operation and parameters of Bunch Shape Monitors using coherent transformation of time structure of an analyzed beam into a spatial one of low energy secondary electrons emitted from a wire target is influenced by the characteristics of a beam under study. The electromagnetic field of a bunch disturbs the trajectories of secondary electrons, thus resulting in a degradation of phase resolution and in errors of phase position reading. Another effect is the perturbation of the target potential due to the current in the wire induced by a bunch as well as due to current compensating emission of the secondary electrons. The methods, the models and the results of simulations are presented. 1 INTRODUCTION Bunch Shape Monitors (BSM) are used to measure longitudinal microstructure of the accelerated beam in a number of accelerators [1-4]. The principle of operation of the BSMs is based on the coherent transformation of a longitudinal distribution of charge of the analyzed beam into a spatial distribution of low energy secondary electrons through transverse RF modulation. Typically the phase resolution of the detectors is about 1° at the frequencies of hundreds MHz. The resolution is determined by a number of parameters. The most complicated effects are due to the influence of electromagnetic fields of the analyzed beam. The fields disturb the trajectories of the electrons thus resulting in degradation of accuracy of the measurements. This effect was estimated earlier [5-7] but for extremely simplified model and detector geometry. We studied the effect for the typical geometry (fig.1) of the existing detectors and analyzed the motion of the electrons through the whole electron line from target 1 to the plane of electron collector 2. Another effect is the perturbation of the potential of the target due to the current in the wire induced by a bunch as well as due to the current compensating emission of the secondary electrons. The model of a transmission line is used for estimating the effect. 2 DESCRIPTION OF THE MODEL The motion of the electrons inside chamber 3 is analyzed for the 3D geometry. Downstream of collimator 4 a 2D model is used. Target 1, target holders 5 and theplatform 6 are at the HV negative potential argtU. Chamber 3 is at zero potential. Figure 1: Geometry of the Bunch Shape Monitor. The field inside the chamber satisfies the Poisson equation: 0),(),(ερtrtrEdiv = , ),(tfΓ=Γφ , (1) where ),(trρ is a charge density in the bunch of the analyzed beam at the moment of time t, Γφ is a boundary potential generally depending on time. One can split the problem (1) into three independent problems to find the fields 21,EE and 3E ( 321EEEE ++= ). Problem 1: 0)(1=rEdiv , )(01Γ=fφ (2) The field 1E can be found from a solution of the Laplace equation for the potential )(1rφ without a beam: )( )(11 rgradrE φ−= Problem 2: 0),(2=trEdiv , ),(),(2 2 tftΓ=Γφ (3) We assume this process to be a quasi stationary one and 1 2EE<<. In this case the potential satisfies the Laplace equation and ),( ),(22 trgradtrE φ−≈ . Formulation of this problem is an attempt to take into account the effects of distortion of the boundary potential. The distortions of the target voltage are estimated below but are not taken into account in the simulation process. Problem 3: Generally the bunch generates both electric and magnetic fields and a complete system of Maxwell equations must be solved. To simplify the problem we consider it to be electrostatic in the reference frame moving with the bunch. The magnetic field due to the charges located on the moving boundaries in this frame is neglected. With this assumptions for each fixed moment1 cmIon Beam Electrons yxz z315647 2of time 0t the electric field in the beam frame 03E can be found from the Poisson equation 000 0003),(),(ερtrtrEdiv = , 0),(0003 =Γtf (4) The subscript “0” indicates that the beam frame is considered. The electric and magnetic fields in the laboratory frame can be found by Lorenz transformations. The equations (2) and (4) were solved numerically for the mesh 0.5mm×0.5mm×0.5mm (in the laboratory frame). Near the target in the region where the motion of the electrons is of interest the radial component of the electric field is approximated by the function ryKzyxEr)(),,(1≈ for the problem 1 and 2 4)( 3)(),,(ryK r eyKzyxE−≈ for the problem 3 [8]. Here r is the distance from the target center. The functions )(1yK and )(4yK were selected to satisfy the condition argteUW−= for the fixed position of the bunch, where Wis the energy of electrons passing through slit 4 and emitted from the target with the zero energy. Phase resolution of the detector is defined as maxXXLΔ=Δϕ . To avoid mixture of the effects LXΔ in our analysis is considered to be the size of the focused electron beam in the plane of electron collector 2 with the RF deflecting field in deflector 7 off. maxX is the amplitude of the displacement of the electrons. For simplicity we assume σ2≈ΔLX , where σ is rms size of the focused electron beam. Due to space charge the size LXΔ increases thus resulting in a degradation of ϕΔ. Another effect is the changing of the average position of the focused electron beam LXδ. This effect is the reason for the error of phase reading maxXXLδδϕ=. Modulation of the electron velocity in y direction also can result in distortion of the results of the measurement because of possible displacement of the beam outside the active area of the electron detector. To estimate the distortion of the potential of the target it was considered as a transmission line (fig. 2). Figure 2: Equivalent circuit of the target. Charge density on the conducting surface can be written as 0εσ⋅=nE, where nE is the component ofelectric field perpendicular to the surface. Linear density of the charge in the wire: ∫=π ασ2 0),( drtyqt, where tr is a target radius. The relation of the changing of the charge distribution due to motion of the bunch and the current in the wire can be written as: ytyi ttyq ∂∂=∂∂ ),(),(. For the transmission line model the voltage ),(tyU due to the current ),(tyi satisfies the equation ),(),(),(tyRittyiLytyU+∂∂=∂∂ (5) Taking into account that the typical transverse dimensions of the target are much smaller than those of the holders 5 the boundary conditions for 2argtly±= , where argtl - is the length of the target wire, can be set to zero. Zero moment of time can be selected to correspond to the position of the bunch at the entrance of the chamber and the initial condition also can be set to zero. The emission current is represented by the current source ),(tyI. One can show that the voltage Uin the transmission line due to this current satisfies the equation tILRItURC tULC yU ∂∂+=∂∂− ∂∂− ∂∂ 22 22 (6) The boundary and initial conditions can be the same as previously with the zero moment of time corresponding to the middle point between bunches. The model of a transmission line cannot be considered as a rigorous one. We did not use it for problem 2. However we believe it to be useful for estimationg the perturbations of the target voltage and for making a conclusion about the applicability of problems 1 and 3. 3 RESULTS OF SIMULATIONS The simulations have been done for the 3D Bunch Shape monitor [3] installed in the CERN Linac-2 (I=150 mA, ϕσ=15°, yσ=3.5 mm, zσ=2 mm, f=200 MHz, argtU=-10 kV, maxX=27 mm). The trajectories of the electrons, emitted at different phases, in the space between the target and collimator 4 are presented in fig. 3. Figure 3: Trajectories of electrons in the beam frame emitted at different moments of time. The difference of electron energy with and without space charge along the trajectory for the electronsyΔ ),(tyi),(tyi ),(tyi),(tyU ),(tyI),(tyyiΔ+ ),(tyyiΔ+),(tyyUΔ+ ),(tyi /G10 /G18/G13/G18/G14 /G13/G14 /G18/G15 /G13/G15 /G18 /G10 /G17/G13 /G10 /G15/G13 /G13 /G15 /G13 /G17 /G13 /G19 /G13 /G1B /G13 /G33 /G4B/G44 /G56 /G48/G0F/G03 /G47/G48/G4A/G3D/G0F/G03 /G50 /G50 /G56 /G4C /G4A /G50 /G44 /G15 /G0D /G56 /G4C /G4A /G50 /G44/G03corresponding to the head and the center of the bunch is given in fig. 4. Modulation of energy of the electrons at the collimator 4 and time of flight from the target to RF deflector 7 is shown in fig. 5. Figure 4: Difference of energies of the electrons with and without space charge. Figure 5: Modulation of electron energy and time of flight. The behavior of phase resolution and error of phase reading is shown in fig. 6. Figure 6: Phase resolution and error of phase reading. The displacement of the electrons in the plane of electron collector 2 in the y direction, due to space charge with respect to their position without space charge, is presented in fig. 7. Figure 7: Displacement of the electrons due to space charge for different initial coordinates y. Perturbation of the target voltage due to the current induced by the bunch and calculated for 61024.1−⋅=L H/m, 121093.8−⋅=C F/m and R=61 Ohm/m (the parameters of the equivalent coaxial transmission line with the diameter of the inner and outer conductors of 0.1 mm and 50 mm correspondingly at 1 GHz) is shown in fig. 8. The solution of equation (6) also gives a bipolar shape of perturbation. The magnitude of the perturbation for the experimentally estimated emission current of 250 µA does not exceed ± 5 µV.Figure 8: Perturbation of the target voltage due to the current induced by the bunch. 4 CONCLUSION The limits of the report do not enable a thorough study of the effects to be presented. Although each particular set of parameters requires special calculations, nevertheless some general rules can be mentioned. Thus increasing the target potential evidently decreases all the space charge effects. Although decreasing the transverse dimensions of the beam increases the space charge forces, nevertheless it leads to the shrinking of the region of the intensive interaction of the electrons with the bunch and as a result decreases the influence of space charge. Decreasing the longitudinal bunch size initially gives rise to the space charge effects. However further decreasing results in decreasing the time of interaction of the electrons with the bunch and hence in decreasing the final effects. The effects of voltage perturbation are essential for short bunches and for some practical parameters are dominating. REFERENCES [1]P.N.Ostroumov et al. Proc of the XIX Int. Linear Acc. Conf., Chicago, 1998 pp.905-907.A [2]A.V.Feschenko et al. Proc. of the 1997 PAC, Vancouver, 1997, pp.2078-2080. [3]S.K.Esin et al. Proc. of the XVIII Int. Linear Acc. Conf., Geneva, 1996, pp.193-195. [4]S.K.Esin et al. Proc. of the 1995 PAC and International Conf. on High-Energy Accelerators, Dallas, 1995, pp.2408-2410. [5]V.Vorontsov, A. Tron. Proc of the 10th All-Union Workshop on Part. Acc. Dubna 1986, V.1, pp.452- 455 (in Russian). [6]A.Tron, I.Merinov. Proc. of the 1997 PAC, Vancouver, 1997, V.2, pp.2247-2049. [7]E.McCrory et al Proc. of the XVIII Int. Linear Acc. Conf., Geneva, 1996, pp.332-334. [8]C.Badsel, A.Lengdon. Phisika Plazmy I chislennoe Modelirovanie, Energoatomizdat, Moscow, 1989 (in Russian). /G10 /G19/G17 /G10 /G18/G13 /G10 /G16/G19 /G10 /G15/G14 /G10 /G1A /G1A /G15/G14 /G16/G19 /G18/G13 /G19/G17/G10 /G15/G16/G10 /G15/G14/G1C /G10 /G14 /G11 /G18/G10 /G14 /G11 /G13/G10 /G13 /G11 /G18/G13 /G11 /G13/G13 /G11 /G18/G14 /G11 /G13/G14 /G11 /G18/G15 /G11 /G13/G15 /G11 /G18 /G39 /G52 /G4F /G57 /G44 /G4A /G48 /G0F/G03 /G39 /G33/G4B /G44 /G56 /G48 /G0F/G03/G47 /G48 /G4A/G3C /G0F/G03/G50 /G50 /G10 /G14 /G18/G13/G10 /G14 /G13/G13/G10 /G18 /G13/G13/G18/G13/G14/G13/G13/G14/G18/G13 /G10 /G16 /G13 /G10 /G15 /G13 /G10 /G14 /G13 /G13 /G14/G13 /G15/G13 /G16/G13 /G33 /G4B /G44 /G56 /G48 /G0F/G03 /G47/G48/G4A/G28 /G51 /G48 /G55 /G4A /G5C/G03 /G50 /G52 /G47 /G58/G4F/G44 /G57 /G4C /G52 /G51/G0F/G03 /G48/G39 /G10 /G16/G10 /G15/G10 /G14/G13/G14/G15/G16/G17 /G10 /G16/G13 /G10 /G15/G13 /G10 /G14/G13 /G13 /G14/G13 /G15/G13 /G16/G13 /G33 /G4B/G44 /G56 /G48/G0F/G03/G47 /G48 /G4A/G37/G4C /G50 /G48 /G0F/G03 /G53 /G56 /G13/G13 /G11 /G15/G13 /G11 /G17/G13 /G11 /G19/G13 /G11 /G1B/G14/G14 /G11 /G15/G14 /G11 /G17/G14 /G11 /G19/G14 /G11 /G1B /G10 /G16 /G13 /G10 /G15 /G13 /G10 /G14 /G13 /G13 /G14 /G13 /G15 /G13 /G16 /G13 /G33 /G4B /G44 /G56 /G48 /G0F/G03 /G47/G48/G4A/G35 /G48 /G56 /G52 /G4F /G58 /G57 /G4C /G52/G51 /G0F/G03 /G47 /G48 /G4A /G10/G14 /G11 /G15/G10/G14/G10/G13 /G11 /G1B/G10/G13 /G11 /G19/G10/G13 /G11 /G17/G10/G13 /G11 /G15/G13/G13/G11/G15/G13/G11/G17/G13/G11/G19 /G10/G16 /G13 /G10/G15 /G13 /G10/G14 /G13 /G13 /G14/G13 /G15/G13 /G16/G13 /G33 /G4B /G44 /G56 /G48 /G0F/G03 /G47/G48 /G4A/G33 /G4B /G44 /G56 /G48/G03/G48 /G55/G55 /G52 /G55 /G0F/G03 /G47 /G48/G4A /G10 /G16/G10 /G15/G10 /G14/G13/G14/G15/G16 /G10 /G16/G13 /G10 /G15/G13 /G10 /G14/G13 /G13 /G14/G13 /G15/G13 /G16/G13 /G33 /G4B/G44 /G56 /G48 /G0F/G03 /G47/G48 /G4A/G27/G4C /G56 /G53 /G4F/G44 /G46 /G48 /G50 /G48/G51 /G57 /G0F/G03 /G50 /G50/G10 /G16/G11/G1C /G19 /G15 /G10 /G16/G11/G13 /G1B /G14/G18 /G10 /G15/G11/G15 /G13 /G14/G14 /G10 /G14/G11/G16 /G15 /G13/G1A /G10 /G13/G11/G17 /G17 /G13/G15/G15 /G13/G11/G17 /G17 /G13/G15/G15 /G14/G11/G16 /G15 /G13/G1A /G15/G11/G15 /G13 /G14/G14 /G16/G11/G13 /G1B /G14/G18 /G16/G11/G1C /G19 /G15/G03/G10 /G14 /G18 /G13/G10 /G14 /G13 /G13/G10 /G18 /G13/G13/G18/G13/G14/G13/G13/G14/G18/G13/G15/G13/G13/G15/G18/G13/G16/G13/G13 /G13 /G18 /G14/G13 /G14/G18 /G15/G13 /G15/G18 /G3D/G0F/G03 /G50/G50/G28 /G51 /G48/G55 /G4A /G5C/G03 /G47 /G4C/G49/G49/G48 /G55 /G48/G51 /G46 /G48 /G0F/G03 /G48 /G39 /G4B /G48/G44 /G47 /G46 /G48 /G51 /G57 /G48 /G55
arXiv:physics/0008116v1 [physics.acc-ph] 17 Aug 2000USE OFCOHERENT TRANSITIONRADIATIONTOSETUP THEAPS RF THERMIONICGUNTO PRODUCE HIGH-BRIGHTNESSBEAMSFOR SASE FEL EXPERIMENTS∗ N. S. Sereno, M.Borland, A. H.Lumpkin, Argonne/APS, Argonne, IL, 60439-4800,USA Abstract We describe use of the Advanced Photon Source (APS) rf thermionic gun [1], alpha-magnet beamline, and linac [2] to produce a stable high-brightnessbeam in excess of 100 amperes peak current with normalized emittance of 10 πmm-mrad. To obtain peak currents greater than 100 amperes, the rf gun system must be tuned to produce a FWHMbunchlengthontheorderof350fs. Bunchlengths this short are measured using coherent transition radiatio n (CTR) produced when the rf gun beam, accelerated to 40 MeV, strikes a metal foil. The CTR is detected using a Golay detector attached to one arm of a Michelson inter- ferometer. The alpha-magnet current and gun rf phase are adjusted so as to maximize the CTR signal at the Golay detector,whichcorrespondstotheminimumbunchlength. The interferometer is used to measure the autocorrelation oftheCTR.Theminimumphaseapproximation[3]isused to derive the bunch proﬁle from the autocorrelation. The high-brightnessbeam is accelerated to 217 MeV and used to produceself-ampliﬁedspontaneousemission(SASE) in ﬁve APS undulatorsinstalled in the Low- Energy Undula- tor Test Line (LEUTL) experiment hall [4]. Initial optical measurementsshoweda gainlengthof1.3mat 530nm. 1 INTRODUCTION TheAPSrfthermionicgunservesbothasaninjectorforthe APS[2]storageringaswellasahigh-brightnesssourcefor SASEFELexperimentsaspartoftheAPSLEUTLproject. Tuning of the gun as a high-brightnesssource was accom- plished using CTR from the rf gun beam accelerated to 40 MeV. The gun, linac, and CTR setup are shown in Fig- ure 1. The beam emerges from the 1.6 cell πmode rf gun and proceeds to the alpha magnet via a beamline contain- ing focusing, steering, a kicker, and an entrance slit. The alpha-magnet vacuum chamber contains a scraper that is usedtoremovethelowenergy/highemittancetailfromthe beam. Afterthealpha-magnet,thebeamtraversessomefo- cussing and correction elements, then proceeds through a 3-mSLACs-bandacceleratingwaveguidetotheCTRfoil. The CTRfoil ismountedonan actuatoralongwith a YAG crystal, which is used to focus the beam to a small beam spotatthefoilposition. TheCTRiscollectedbyalensand sent to a Michelson interferometer with a Golay detector mounted on one arm. The autocorrelation of the CTR is ∗Work supported by U.S. Department of Energy, Ofﬁce of Basic E n- ergy Sciences, under Contract No. W-31-109-ENG-38.Alpha Magnet1.6 cell rf thermionic gun. SLAC Accelerating WaveguideScraper SlitCTR Foil InterferometerGolay Detector Figure 1: Layout of rf gun, alpha magnet, CTR apparatus, andlinacbeamlinecomponents. performed by moving one arm of the interferometer while recording the Golay detector output. The Golay detector outputcanbemaximizedat thepeakoftheautocorrelation scan andused to adjust rf gunpowerand phase, beamcur- rent,andalpha-magnetcurrentsoastominimizethebunch lengthoutofthealphamagnet. Oncethisisdone,theCTR signalis agoodrelativemeasureofthebunchlength. 2 BEAM OPTIMIZATION To prepare the rf gun to produce a high-brightness beam one must ﬁrst scan the alpha-magnet current to ﬁnd the minimum bunch length. Typically the rf gun is powered anywherefrom1.5to1.7MW,andtheheatercurrentisad- justedtoproduce1to2nCinatrainof23s-bandbunches. The gun power and beam current are kept constant during the scans. Prior to the scan, the beam is focused on the YAG using quads before and after the alpha magnet, with the alpha magnet “close” to the setting required for mini- mumbunchlength. Duringthescan,therfgunphasemust be adjustedlinearlyto compensateforpathlengthchanges in the alpha magnet. To maximize scan resolution, the in- terferometer is set to maximize the Golay detector signal. Figure2showsatypicalalpha-magnetscanshowingapeak at175amperes. ThecurverepresentstheoutputoftheGo- lay detectorfroma gatedintegratorampliﬁer. Once the minimum bunch length has been found, an alpha-magnetscraperscanisperformed. Simulationsshow amicrobunchproﬁlethathasalow-emittance,high-energy core beam and a high-emittance, low-energy tail. The scraper scan is performed to optimize removal of the low- energy tail. Figure 3 shows a typical scraper scan where the CTR signal is plotted vs scraper position. The edge of the core beam is at approximately9.5 cm. Figure 4 showsFigure2: CTRgatedintegratorsignalvsalphamagnetcur- rent. a plot of CTR signal vs beam current, as measured by a beam position monitor (BPM) adjacent to the CTR foil, taken during the scraper scan. Included with the data is aquadraticﬁt,showingtheexpectedquadraticdependence ofthe coherentradiationonthenumberofparticles. Figure 3: CTR gated integrator signal vs alpha-magnet scraperposition. 3 BUNCH PROFILEMEASUREMENT Once the scraper position is determined, the interferome- ter is used to measure the autocorrelation of the digitized gated integrator CTR signal. Figure 5 shows the auto- correlation measured for a beam of 1 nC in 23 S-band micropulses. Autocorrelation processing begins with tak- ing the fast Fourier transform (FFT) of the autocorrela- tion, which gives the square of the bunch spectrum. The method of Lai and Sievers is then used to reconstruct the phase spectrum from the amplitude spectrum by comput- ing a principal value integral. Once the phase spectrum is obtained, an inverse FFT is performed to derive the mi- crobunch proﬁle. Additional processing is performed toFigure4: Plot ofCTR gatedintegratorsignal vsbeamcur- rent as measured by a BPM. The plot shows a quadratic ﬁt along with the data indicatinga strong quadraticdepen- denceoftheCTR. Figure 5: Autocorrelationof the gatedintegratorCTR sig- nal. correct for the reduced response of the Golay detector at lowfrequencies(longwavelengths). Sinceanybunchspec- trumapproacheslowfrequenciesquadratically,aquadrati c ﬁt is performedfor frequenciesfrom the Golay detector3- dBpointtoauser-selectablehigherfrequency,typicallyi n- cluding 3 to 5 frequency points [3]. The ﬁt is then used to extrapolatequadraticallyto DCfromthe Golaydetector 3-dB point. Figure 6 shows the amplitude spectrum de- rived from the measured autocorrelationand the corrected spectrumforlowfrequencies. Themaineffectsofthislow- frequency correction is to broaden the derived bunch pro- ﬁleandﬂattenthedipsintheautocorrelationadjacenttoth e peak. Thesedipsareunphysicalsincetheautocorrelationi s always positive. Figure 7 shows the derived bunch proﬁle fromthethecorrectedautocorrelationspectrum. Theover- all proﬁle contains a high-current peak ( >100 amperes), a lower current shoulder, and is overall about 400 fs wide. ThisbeamwasusedforSASE measurements.Figure 6: Amplitudespectrumderivedfromthe autocorre- lationandcorrectedspectrumatlongwavelengths. Figure 7: Bunch proﬁle derived from corrected autocorre- lationamplitudespectrum. 4 MEASUREMENT OFSASE GAIN The beam prepared as described above was accelerated to 217MeV.Theemittancewasmeasuredinthetransportline usingthestandardthree-screentechnique,givinganormal - ized emittance of approximately 10 πmm. The energy spreadisestimatedtobe 0.1%. Thebeamwastransported to the undulator hall and passed through ﬁve APS undu- lators with diagnostics stations between them. Figure 8 shows the measured photon intensity (corrected for spon- taneous background) at each undulator diagnostic station. The solid line is an exponential ﬁt to the data showing a gainlengthof1.3mforbothundulatorradiationandcoher- enttransitionradiationdata[5],in agreementwithacalcu - lation using the previously listed peak current, emittance , andenergyspread. The rf thermionic gun beam was quite stable once tun- ing was completed. One limitation of the beam is that the microbunch length is on the order of the electron slippage length. The ﬁnal saturated power is therefore expected to be lowerforthisbeam.Figure 8: SASE gain measured at undulator diagnostics stations. 5 ACKNOWLEDGEMENTS The authors thank J. Lewellen, S. Milton, and J. Galayda forusefulcommentsandsuggestions. 6 REFERENCES [1] M. Borland, “An Improved Thermionic Microwave Gun and Emittance-PreservingTransportLine,”Proc.1993PAC,May 17-20, 1993, New York, 3015-3017. [2] J. Lewellen et al., “Operation of the APS RF Gun,” Pro- ceedingsofthe1998LinacConference,ANL-98/28,863-865 (1999). [3] R. Lai and J. Sievers, “Determination of Bunch Asymmetry fromCoherentRadiationintheFrequencyDomain,”AIPVol. 367, 312-326 (1996). [4] S.V. Milton et al., “Observation of Self-Ampliﬁed Spont a- neous Emission and Exponential Growth at 530 nm,” (sub- mittedtoPhys. Rev. Lett.). [5] A. H. Lumpkin et al., “First Observation of Z-Dependent Electron Beam Microbunching Using Coherent Transition Radiation,” (submittedtoPhysical Review).
COHERENT SYNCHROTRON RADIATION MEASUREMENTS IN THE CLIC TEST FACILITY (CTF II) H.H. Braun, R. Corsini, L. Groening, F. Zhou, CERN, Geneva, Switzerland A. Kabel, T. Raubenheimer, SLAC, Menlo Park, CA 94025, USA R. Li, TJNAF, Newport News, VA 23606, USA T. Limberg, DESY, Hamburg, Germany Abstract Bunches of high charge (up to 10 nC) are compressed in length in the CTF II magnetic chicane to less than0.2 mm rms. The short bunches radiate coherently in thechicane magnetic field, and the horizontal andlongitudinal phase space density distributions areaffected. This paper reports the results of beam emittanceand momentum measurements. Horizontal and verticalemittances and momentum spectra were measured fordifferent bunch compression factors and bunch charges.In particular, for 10 nC bunches, the mean beammomentum decreased by about 5% while the FWHMmomentum spread increased from 5% to 19%. Theexperimental results are compared with simulations madewith the code TraFiC 4. 1 INTRODUCTION Short electron bunches traversing a dipole with bending radius /G55 can emit coherent synchrotron radiation (CSR) at wavelengths longer than the bunch length. Theenhancement of the radiated power, with respect to theclassical synchrotron radiation, can be expressed as [1]: 3432 02 2028.0 ZcohecN P /G56 /G55 /G48/G7C /G27 where a longitudinal Gaussian distribution of the N particles with constant rms bunch length /G56Z along the curved trajectory are assumed.The CSR induces an average momentum loss and a momentum spread on the bunch. For relativistic beamsthe momentum loss is independent of the beam energyand the effect can be treated like a wake-field. Since ittakes place in a dispersive region, the transverse phasespace distribution is also affected, and the beam emittancein the bending plane increases. These effects are a concern in all accelerator applications in which high-charge short bunches areneeded, e.g., free electron lasers, linear colliders and two-beam accelerators. Analytical treatments have beendeveloped for idealised conditions [1-4] but for manypractical cases the effect must be treated numerically.Some codes have been developed for this purpose [5,6],but a complete benchmark of the codes withmeasurements is so far lacking. The parameters of thedrive beam in the second Compact Linear Collider TestFacility, CTF II [7] are well suited to study the CSReffect experimentally. CTF II (see Fig. 1) was built to demonstrate the feasibility of two-beam acceleration at 30 GHz. The high-charge drive beam is generated in a 3 GHz RF gun, andaccelerated in two Travelling Wave Structures (TWS).The drive beam bunches are then compressed in amagnetic chicane, and generate 30 GHz RF power in aseries of Power Extraction and Transfer Structures(PETS). This power is used to accelerate a low-chargeprobe beam, in a parallel beam line. Figure 1: The CLIC Test Facility CTF II. The CSR experiments were performed in the drive beam line (upper part).Only the drive beam line of CTF II has been used to perform the CSR experiment. Single bunches withcharges of 5 and 10 nC have been used. For both charges,the horizontal and vertical beam emittances and themomentum spectra of the drive beam have beenmeasured as a function of the deflection angle in thechicane. Similar measurements have been performedalready in CTF II [8], but limited to the emittances. Theresults of simulations made using the code TraFiC 4 [5] are reported for comparison. 2 EXPERIMENTAL SETUP Bunch compression is used in CTF II to enhance the 30 GHz power production, and is achieved by accelerationoff-crest (such that particles at the tail of the bunch havehigher energies than those at the head), in combinationwith a magnetic chicane composed of three rectangulardipoles. The deflection angle in the first and the lastdipole can be varied from 3.7 /G71 to 14 /G71 (twice these values in the central dipole). The corresponding energydependence of the path length R 56 = ds / (dE/E) ranges from 6 mm to 90 mm. Due to the construction of thevacuum chamber, the chicane cannot be switched offcompletely. Using this scheme, rms bunch lengths of lessthan 0.4 mm can by achieved for bunch charges of 10 nC. The beam momentum spectra were measured at the entrance to the chicane by switching off the last twochicane magnets and using the first one as a spectrometer.A second spectrometer at the end of the line permits themeasurement of the beam spectra after the passage in thechicane. The PETS were deactivated by shielding themwith metallic tubes, in order not to perturb the momentumdistribution. Two wall-current monitors placed in front ofthe chicane and in front of the second spectrometermeasured the beam intensities, in order to monitor thelosses. The bunch lengths were measured behind the chicane by analysing the frequency spectrum of the mm-waveradiation excited by the beam passing an RF waveguideconnected to the vacuum chamber [9]. Transverse beamprofiles were recorded after the chicane using an opticaltransition radiation (OTR) screen and a camera. Thehorizontal and vertical emittances were measuredsimultaneously by using the quadrupole scanningtechnique. In order to cover the full range of phaseadvances from 0 /G71 to 180 /G71 in both transverse planes, the quadrupole strengths of a bipolar triplet were variedindependently. For accurate determination of the beamwidths, only those profiles were selected which fitted onthe OTR screen. The off-crest RF phase and the range of chicane settings were selected in order to achieve over-compression of the initial bunch length, thus covering asufficient range in bunch lengths.3 MOMENTUM SPECTRA AND BUNCH LENGTH The beam momentum spectra measured for different bunch compressor settings are shown in Fig. 2 (5 nCcase) and Fig. 3 (10 nC case). The spectra from TraFiC 4 simulations are also shown. Since the spectrometer at theend of the beam line consists of a deflecting dipole it alsorepresents a source of CSR, and the spectra result fromthe sum of the CSR effects in the compressor chicane andin the spectrometer. 40 45 40 458.1 mm 0.69 mm13 mm 0.46 mm18 mm 0.36 mmdN/dp [arb. units]25 mm0.29 mmR 56 = 32 mm σbunch = 0.14 mm42 mm 0.13 mm 51 mm 0.13 mm Momentum [MeV/c] 62 mm0.31 mm75 mm 0.46 mm 40 45 Figure 2: Measured (solid line) and calculated (dotted line) momentum spectra at the end of the beam line fordifferent compressor settings R 56, with a bunch charge of 5 nC. The measured bunch lengths are also shown (lowervalues). 30 40 30 409.6 mm 0.82 mm15 mm 0.52 mm21 mm 0.36 mm 25 mm 0.34 mmR56 = 29 mm σbunch = 0.36 mm34 mm 0.36 mm 39 mm 0.37 mm44 mm 0.48 mm49 mm 0.65 mm 61 mm 0.54 mm75 mm 0.87 mm Exp CTF II TraFiC4 30 40dN/dp [arb. units] Momentum [MeV/c] Figure 3: Measured (solid line) and calculated (dotted line) momentum spectra at the end of the beam line fordifferent compressor settings R 56, with a bunch charge of 10 nC.38404244 123 0.511.5 20 40 60p0 [MeV/c ] qbunch = 5 nCqbunch = 10 nC∆prms [MeV/c ] Exp CTF II TraFiC4 R56 [mm]σl [mm]Exp CTF II TraFiC4 20 40 60 Figure 4: Measured and calculated rms bunch lengths, mean momenta and momentum spreads as functions ofR 56, for bunch charges of 5 nC (left) and 10 nC (right). As the bunches are compressed, the spectrum broadens, and in the 10 nC case it develops two maximabeyond maximum compression. During over-compression the spectrum narrows again. The measured bunch lengths and the mean momenta and rms momentum spreads calculated from themeasured spectra are plotted in Fig. 4, as function of R 56. They are compared with the results of the TraFiC4 simulations. For 5 nC, no significant changes of the mean beam momentum was observed, although the momentumspread increased by a factor four at full compression withrespect to the initial spread, and decreased at over-compression. For charges of 10 nC, a decrease of themean momentum by 2 MeV/c, i.e. 5% of the initial beammomentum, was measured at full compression. Themomentum spread shows a behaviour similar to the onedescribed for 5 nC. In the 5 nC case the dependence ofthe measured bunch length on R 56 shows a symmetric shape as expected from linear longitudinal dynamics. Onthe other hand, the asymmetric shape of thecorresponding curve for 10 nC indicates a strong impactof CSR on the longitudinal properties for R 56 > 20 mm. Using the program TraFiC4, the bunch, consisting of 500 macro particles, was tracked from the entrance to thechicane to the end of line spectrometer dipole. At eachstep, the sum of fields resulting from magnetic fields,space charge and CSR wake-fields was applied to themacro particles. The resulting bunch lengths, emittancesand momentum spectra have been calculated from the final distributions. In order to determine the initial conditions to be used in the simulations, the bunch lengths measured after thechicane have been used together with the momentumspectra recorded at the entrance to the chicane. The initialphase space distributions in the longitudinal plane havebeen reconstructed taking into account the contribution tothe intra-bunch momentum correlation of the off-crest RFphase and of the short-range longitudinal wake-fields inthe accelerating structures. The initial bunch length hasbeen chosen to fit the momentum spectra at the chicaneentrance and the measured bunch lengths, assuming thatthe CSR does not influence the bunch length. As alreadymentioned above, this assumption seems to be justifiedonly for R 56 < 20 mm in the 10 nC case. Indeed, while a linear longitudinal model describes well the measuredbunch length dependence from R 56 for 5 nC, no set of initial conditions could be found that can describe themeasured values for 10 nC. Therefore only the bunchlengths for small R 56 have been used in this case. Since only a few measurement points are available for the fit,the input parameters for the 10 nC measurements areknown with less accuracy. In the case of 5 nC the dependence of the observables on R 56 was well reproduced. For 10 nC the calculated bunch lengths agreed with the measured ones until fullcompression, but the asymmetric shape of the bunchlength curve was not reproduced, although the TraFiC 4 simulation looks more accurate than the simple linearcalculation. The amount of momentum loss for 10 nC wascalculated correctly, but the experimental and theoreticalcurves are shifted with respect to each other, possiblyindicating different initial longitudinal beam parametersin the experiments and the simulations. Another possibleexplanation is a fluctuation in the initial beam parameterswhile the measurement was taken. Especially, variationsin RF power and phase could be invoked, since the sameinitial conditions have been used to simulate all of themeasurement points and a phase variation of a fewdegrees would be enough to explain the discrepanciesfound. On the other hand, a very good matching of the measured and calculated momentum spreads was foundfor both cases. In the case of 5 nC, the simulations revealno splitting of the maximum and the shapes of the spectrashow a good agreement. In the 10 nC case, although therms widths are in good agreement, the measured andsimulated momentum spectra have somewhat differentshapes. The formation of two maxima occurs in thesimulations for smaller values of R 56. Since the CSR wake depends on the exact longitudinal charge distributionwithin the bunch, the details of the momentum spectrumshape should depend on that as well.24 246CompressorSpectrometerploss/po [%] ∆prms/po [%] s [m]σbunch,rms [mm] 0.51 2 4 6 8 10 12 14 Figure 5: Evolution of the bunch length, the rms momentum spread and the momentum loss (from bottomto top) along the CTF II beam line as simulated withTraFiC 4, for 10 nC. The results with two different setting of the chicane are shown, corresponding to R56 = 34 mm (solid line) and R56 = 15 mm (dashed line). In the simulation an initial Gaussian charge distribution has been assumed, but previous bunch lengthmeasurements with a streak camera indicate a different,asymmetric charge distribution. The simulations showthat the splitting of the maxima in the spectra for 10 nC isdue to CSR in the chicane, since such a feature is presentin the simulated spectrum right after the chicane. On the other hand, the broadening of the spectra seems to be caused by space charge during the drift to thespectrometer. This is apparent in Fig. 5, where theevolution of the bunch length, the momentum spread andthe momentum loss along the beam line, simulated withTraFiC 4, are shown. In spite of the fact that some features of the simulated momentum loss curve are not yet understood, fromFig. 5, it can be seen that the momentum loss is mainlyconcentrated in the chicane region where the bunches areshort, and is a direct indication of CSR emission. On theother hand, for R 56 = 34 mm, the momentum spread increases along the whole beam line after the chicane,showing that such growth is not directly caused by a CSReffect.4 EMITTANCES The measured horizontal and vertical emittances after the chicane are plotted in Fig. 6 as function of R56, together with the bunch lengths. The measured vertical emittances for 5 nC are constant within the error bars, and are consistent with thesimulation results. The measured horizontal emittancesfor 5 nC increase until full compression, then a saturationoccurs. The four highest values seem to be shifted withrespect to the lower values. This shift might be due todispersion at the location of the OTR screen, caused by amismatch of the relative magnetic strength of the chicanedipoles. It must be noted that these four points wererecorded after switching off the last two magnets of thechicane in order to check the beam momentum, and thenswitching on the magnets again. For 10 nC the measured vertical emittance shows a maximum at full compression. This qualitative behaviouris found also in the simulations. The measured data pointsfor the horizontal emittance scatter after over-compression, thus not allowing meaningful conclusions.The huge growth of the horizontal emittance predicted bythe simulations was not found experimentally. It can benoted that the increase of the momentum spread alsoaffects the accuracy of the emittance measurements andcontributes to the error bars and the scattering of the datapoints. 100200300 100200300 0.511.5 20 40 60γεx [mm mrad ] γεy [mm mrad ]qbunch = 5 nC qbunch = 10 nC Exp CTF II TraFiC4 R56 [mm]σl [mm]Exp CTF II TraFiC4 20 40 60 Figure 6: Measured and calculated rms bunch lengths and transverse beam emittances as function of R56 for bunch charges of 5 nC (left) and 10 nC (right).5 CONCLUSIONS AND OUTLOOK The measurements made in the CTF II drive beam line showed clear signs of CSR emission. In particular themomentum loss and bunch length as a function of thebunch compressor setting in the 10 nC case are wellexplained by the CSR effect, such as the two-peakmomentum spectrum shape at 10 nC. A good agreementwas found between measurements and simulations madewith the TraFiC 4 code for all observables, except the emittance growth for bunches of 10 nC, which is not yetunderstood. Future experiments will aim in particular at an experimental investigation of the shielding effect of thebeam pipe on the CSR emission. The extension of theCSR spectrum is limited at the high-frequency end by thebunch length, and at the low-frequency end by the cut-offof the beam pipe. When the beam pipe dimensions aresmall with respect to the bunch length, CSR emission issuppressed. While this is indeed the case in manyaccelerators, in the CTFII chicane the free spaceapproximation is valid. It would be interesting to explorethe intermediate regime since often only a partialshielding can be used to reduce the unwanted CSR effectin the case of very small bunches. Several approaches tothe calculation of the shielding effect exist [1-4, 10],using different approximations. The TraFiC 4 code is also capable of treating the shielding, and could bebenchmarked against the measurements. Short bunches will be passed through a new four- magnet chicane installed downstream of the bunchcompressor. Three vacuum chambers of different heightwill be used in order to provide different shieldingenvironments. The new chicane has been designed forlarge deflection angles and small dispersion functions, i.e.small R 56. Therefore the bunch length can be made short and relatively constant in the new chicane, while CSRemission in this region will be high. A further advantagewill be the smaller disturbance from possible dispersionerrors.REFERENCES [1] J.B. Murphy, S. Krinsky, R.L. Gluckstern, “Longitudinal Wakefield for an Electron Moving ona Circular Orbit”, BNL-63090, (1996). [2] L.I. Schiff, “Production of Particle Energies beyond 200 MeV”, Rev. Sci. Instr., 17, 1, 6-14, (1946). [3] J.S. Nodvick, D.S. Saxon, “Suppression of Coherent Radiation by Electrons in a Synchrotron”, Phys. Rev. 96, 1, 180-184, (1954). [4] S.A. Kheifets, B. Zotter, “Coherent Synchrotron Radiation, Wake Field and Impedance”, CERN SL95-43 (AP), (1995). [5] A. Kabel, M. Dohlus, T. Limberg, “Using TraFiC 4 to Calculate and Minimize Emittance Growth due toCoherent Synchrotron Radiation”, to be published inNucl. Inst. and Meth. A. [6] R. Li, “Self-Consistent Simulation of the CSR Effect”, Proceedings of the 6 th European Particle Accelerators Conference, Stockholm, (1998). [7] H.H. Braun, “Experimental Results and Technical Development at CTF II”, Proceedings of the 7th European Particle Accelerators Conference, Wien,June 2000. [8] H.H. Braun, F. Chautard, R. Corsini, T.O. Raubenheimer, P. Tenenbaum, “Emittance GrowthDuring Bunch Compression in the CTF II”, Phys.Rev. Lett. 84, 658 (2000). [9] H.H. Braun, C. Martinez, “Non-Intercepting Bunch Length Monitor for Picosecond Electron Bunches”,Proceedings of the 6 th European Particle Accelerators Conference, Stockholm, June 1998. [10]R. L. Warnock, K. Bane, “Coherent Synchrotron Radiation and Stability of a Short Bunch in aCompact Storage Ring”, SLAC-PUB-95-6837,(1995).
Physics Design Considerations of Diagnostic X Beam Transport System* Yu-Jiuan Chen and Arthur C. Paul, LLNL, Livermore, CA 94550, USA Abstract Diagnostic X (D-X) transport system would extract the beam from the downstream transport line of the second- axis of the Dual Axis Radiographic Hydrodynamic Test facility (DARHT-II[1]) and transport this beam to the D-X firing point via four branches of the beamline in order to provide four lines of sight for x-ray radiography. The design goal is to generate four DARHT-II-like x-ray pulses on each line of sight. In this paper, we discuss several potential beam quality degradation processes in thepassive magnet lattice beamline and indicate how they constrain the D-X beamline design parameters, such as thebackground pressure, the pipe size, and the pipe material. 1 INTRODUCTION The D-X beamlines would transport several sections of the 20 MeV, 2 kA, 2 µs long beam from the DARHT-II accelerator exit to the D-X firing point to provide four lines of sight for x-ray radiography. The DARHT-II-like x-ray pulses will be generated for each line of sight. Therefore, maintaining the beam quality in the D-X system is essential. Since the DARHT-II accelerator operates at 1 pulse per minute or less, it is also crucial to design the D-X beamline configuration requiring a minimum number of tuning shots. Modification to the DARHT-II building and the downstream transport should be minimized. Lastly, the D-X system should not preclude the future upgrade for additional lines of sight. In this report, we discuss several beam degradation processes in the passive beamline (excluding the active components such as the kicker systems) and how they constrain the beamline design. We establish thespecifications for beam emittance in Sec. 2, pipe material and size in Sec. 3, the vacuum requirement in Sec. 4. We discuss the emittance growth in a bend and the alignment specifications in Secs. 5 and 6. In Secs. 7 and 8, we discuss the design objectives on how to extract beams from the DARHT-II beamline and to minimize the tuning shots. A conclusion will be presented in Sec. 9. 2 EMITTANCE AND X-RAY DOSE The internal divergence angle (θ = ε /af ) of the beam hitting the Bremsstrahlung target increase the radiation cone angle, and hence lower the forward x-ray dose. Here ε is the un-normalized Lapostolle emittance, and af is the beam radius on the target. The scaling law for the forward x-ray dose D created by an electron current I with a pulse length τp hitting a 1-mm thick tungsten plate is given by the functional form[2] of _______________________ * This work was performed under the auspices of the U.S. Department of Energy by University of CaliforniaLawrence Livermore National Laboratory under contract No. W-7405-Eng-48.DIp ≅× − − + []−2 10 0 07 0 0103 0 0008742 8 2 3τγ θ θ θ.exp . . . , (1) with I in kA, τp in ns, θ in degrees, and D in Roentgen at 1 m. The DARHT-II final emittance specification is 1500π mm-mr. To achieve the DARHT-II x-ray dose within ±5 % with a 2.1 mm diameter spot size, the final emittance for the D-X should be less than 1875 π mm-mr. 3 CONDUCTIVITY AND PIPE SIZE There are several transport issues regarding the choice of beam pipe size and material, especially, conductivity of the pipe material. The finite conductivity of the pipe wallcan cause both transverse[3] and longitudinal[4] resistive wall instabilities and beam energy loss. 3.1 Transverse Resistive Wall Instability The transverse resistive wall instability arises from the diffusion of the return current into the wall due to its finite conductivity. The reduction in the magnetic forces of the dissipating return current acts as frictional forces on the particles. The friction forces enhance the slow wave of the transverse beam displacement over the length of the beam path and lead to beam loss and large time integrated spot sizes on the targets. For a continuous focusing system, the instability grows approximately as exp[1.5(z/Ltr)2/3], where Ltr is the instability’s characteristic growth length[3] given as LI Ikb ctro=23γβ πσ τβ, (2) where kβ is the space charge suppressed betatron wavenumber, Io is the Aflvén current (~ 17 kA), σ and b are the beampipe’s conductivity and radius. The characteristic growth length ( ltr) in a drift region is given as lI Ib ctro=γβ πσ τ243 . (3) Comparing the equations above, we observe that ltr = (Ltr/ kβ )1/2/2. Chopping the 2 µs long pulse to shorter pulses soon, using a large, highly conductive beampipe, and transporting beam with relatively strong focusing fields can reduce the instability growth. Two pipe materials are examined for the instability growth: 304 stainless steel at 68 Fo, and 6061-TO Al with at 68 Fo. Based on the growth lengths given in Table 1, we have chosen the beampipe to be 8 cm radius, 6061-TO Al. Note that the beams in most of the D-X beamline are short pulses (~100 ns) extending over 2 µs instead of the full 2 µs long pulses, and hence, the true growth lengths are expected to be longer than those in Table 1. We have also limited the maximum length for each beamline branch to 200 m, and that for individual drift sections to 10 m. Since the nominal D-X beam radius is 1 cm, the chance of having the beam melt the Al pipe wall should be small.Pipe materialConductivity (sec-1)Overall System’s Growth LengthGrowth Length in Drift Region 304 SS 1.25 x 101651.2 m 6.4 m 6061-TO Al 2.368 x 1017222.8 m 13.3 m Table 1 The resistive wall instability’s growth length for a 2 µs long D-X beam. The pipe radius is 8 cm. Using a 13 cm radius stainless steel pipe also gives the same growth length. However, the transport system would be 60 % longer. Furthermore, the accuracy of the beam position monitors needed for a larger pipe would lead to a larger uncorrectable centroid displacement which provides the free energy for emittance growth. 3.2 Parasitic Energy Loss When the beam is traveling down a pipe, the return current flows in the resistive pipe wall in the opposite direction and creates a longitudinal voltage drop along the wall. For a 2 kA beam pulse with a 5 ns rise time traveling for 100 m in a 8 cm radius Al pipe, the energy loss at 50% of the peak current is only about 1 keV, i.e., 0.005 % of the beam energy. 3.3 Longitudinal Resistive Wall Instability The ohmic losses due to the image currents flowing in the resistive wall coupled with perturbations in the current’s line density lead to longitudinal resistive wall instability. Since the ohmic losses are negligible for the D-X transport, the beam needs to travel a long distance before accumulating enough perturbation in the line density from bunching or debunching. Therefore, the longitudinal resistive wall instability should not aconcern. Without any Landau damping, the growth lengthis 98 km for the instability at 1 GHz and 219 km for the instability at 200 MHz. 4 VACUUM REQUIREMENT Collisional ionization of the background gas may affect the beam propagation in a substantial way. Secondary electrons created in the collisions can be expelled by the beam electrons’ space charge fields, while the ions remain trapped in the potential well of the electron beam. Those ions forming an ion channel can provide the background gas focusing forces and cause the ion hose instability [5- 9]. The background gas can also cause an emittance growth by scattering the beam electrons. 4.1 Background Gas Effects The accumulated difference in the phase advance between the beam head and the beam tail over a distance L due to the ion focusing effects of the background gas can be approximated as ∆φγβτ ≈I IPL aop3[ns] [torr], (4) A large difference in the phase advance causes beam envelope oscillations within a pulse, and hence, increase the time integrated spot size. Therefore, to achieve a small time integrated spot size, ∆φ should be much less than2π. The pressure in the D-X needs to be much less than 5 x 10-8 torr for a 2 µs beam in a 120m long beamline. However, the electron beam is 2 µs long only while it is being extracted from the DARHT-II beamline. It then will be chopped into four shorter pulses with the longest pulsebeing about 500 ns wide. The subsequent kickers will chop these four pulses further into even shorter pulses. Most of the ions trapped by the electron beam’s space charge potential well would escape to the wall quickly during the pulse separation time. Therefore, a vacuum of ~10 -8 torr should be sufficient. 4.2 Ion Hose Instability The ion-hose instability occurs when the electron beam is not collinear with the ion channel. The instability grows with time and distance. It grows the fastest when the beam travels in a preformed offset, constant-strength ion channel with all the ions oscillating at the same frequency. The maximum number of e-folding growth[6] (Γ) for an emittance dominated Bennett beam at its equilibrium travel a distance L is given as Γ≅3 3IfL Ie oγβ ε. (5) Let d be the initial separation between the beam and the ion channel, and λ be the instability’s transverse beam displacement normalized to the beam radius. We now have λ = (d/a) eΓ. Rearranging Eq. (5) yields to the pressure requirement as PI Ia dpo[torr][ns][cm-rad] ≅  γβ τελ3 3ln. (6) Assume that the initial beam-channel separation is 0.5 mm, for a 2 µs long, 1 cm radius beam. To achieve a transverse beam motion at a x-ray target less than 10% of the beam size, P ≤ 1 x 10-8 torr. A spread in ion’s mass or oscillation frequency damps the instability growth[7]. The detuning of the ion oscillating frequency due to the beam envelope changes either by design or by mismatch along the beamline also suppress the growth[8]. Furthermore, the recent studies[8, 9] suggest that its saturation amplitude is always small compared with the beam radius. All these may lead to a more relaxed vacuum requirement. 4.3 Background Gas Scattering Scattering in a background gas can cause emittance growth. For a 20 MeV beam traveling 120 m in 10-8 torr of oxygen, the total emittance growth caused by background gas scattering is 1.38 x 10-3 π mm-mr which is insignificant compared with the nominal 1000 π mm- mr emittance exiting the DARHT-II accelerator. 5 EMITTANCE GROWTH IN BENDS A typical D-X beamline consists of 4 bending systems and two active kickers. To achieve the emittance specification at the D-X targets, the relative emittance growth for each bending system is limited to 5 %. According to Ref. 10, the emittance growth arising from nonlinear fields in the bend geometry is negligible for theD-X. Most of the emittance growth would come from the nonuniform image charge and current distributions due to the geometric difference between the cross-sections of thebeam and the pipe[11]. To minimize the emittance growth, the shape of the beam pipe needs to be the same as the beam. Typically, the shape of the beam cross- section in a bending system, consisting of several dipole magnets and quadrupoles, varies from magnet to magnet. A nominal D-X bending system is about 10 m in length.Making the pipe cross-section similar to the beam cross- section along the bend will be difficult and costly. For practical reasons, the pipe cross-section is round through out the D-X beamline except where the beampipe is split. The D-X transport tune is chosen to make the beam as round as possible and its size much smaller than the pipe size so that the nonlinear image force effects are small. 6 MISALIGNMENT Coupling of chromatic aberration of a misaligned transport system and the time dependent energy variation within a current pulse leads to a time varying transverse motion called corkscrew motion[12, 13]. When the difference in the total phase advance ( φtot ) of the beam transport through the entire system is small compared with unity, the corkscrew amplitude for this transport system with randomly misaligned magnets is given as ηργ γφ ργ γφ≈   ≈  n nrms rms tot32/∆ ∆ , (7) where φ is the phase advance after traveling through a single misaligned magnet. The quantity ρrms is the r.m.s. value of the electron gyro-radii of the reference beam slice resulting from n randomly misaligned magnets. Assume \| ρrms \| ~ ∆θrms l ~ ∆rms , where ∆θrms is the r.m.s. value of the magnet tilts, ∆rms is the r.m.s. value of the magnet offsets, and l is the magnet length (nominally 20 cm). The amplitude of the time varying transverse beam motion on the DARHT-II target should be less than 10% of the beam radius. The beam typically has to travel through approximately 100 more magnets to reach its D- X converter target than to reach the DARHT-II target. Let us assume that the corkscrew amplitude arising from misalignment of those 100 magnets alone is also onetenth of the beam radius. The corkscrew amplitudes caused by random tilts and offsets generally add in quadrature. Thenet corkscrew amplitude, including the contribution from the misaligned DARHT-II magnets, at a D-X target will be about 14 % of the beam radius. For the worst case, thetime integrated D-X spot size is now 14 % larger versus 10 % larger for the time integrated DARHT-II spot size if the same focusing field is used for both systems. We can easily compensate for this difference by increasing the final focusing field at the D-X target area by 4 %. The energy variation within the pulse is less than or equal to ±0.5 %. The total phase advance on a D-X beamline is nominally ~14 π. We assume that the random tilts and the random offsets contribute to the corkscrew motion equally. According to Eq. (7), the r.m.s. tilt has to be lessthan 0.8 mrad, and the r.m.s. offset has to be less than 0.16 mm if no dynamic steering would be implemented tocontrol the corkscrew motion. These specifications are very similar to the DARHT-II alignment specifications:1.95 mrad for the 3 σ in tilts and 0.45 mm for the 3 σ in offsets. Our experience on dynamically controlling the corkscrew motion on ETA-II[13] and FXR[14] and the simulations of corkscrew control on DARHT-II indicate that corkscrew motion on D-X should not be a concern. 7 BEAM EXTRACTION AND MODULARITY OF THE BEAMLINE The first design task in the Diagnostic X beamline is the beam extraction from DARHTII. Due to the space constraint, the beam will be extracted before it enters the DARHT-II kicker. The beam will be 2 µs long with an unknown amount of beam head current. To prevent the beam from drilling a hole through the D-X Al beampipe, the extraction line should coexist with the DARHT-II shuttle dump so that the dump can be used to characterize the beam and to tune the DARHT-II accelerator before bending the beam into the D-X transport system. In order to minimize the number of shots needed for tuning the D-X beamline, the beamline will consist of several “identical” modules, such as 10o, 45o and 90o achromatic bend systems, 90o kicker achromatic bend systems, etc. A matching section precedes each module. Operationally, we would tune each type of module once, and then set all the identical module’s magnets at the same tune. We then would only tune individual matching sections to match beams into the following modules. 8 CONCLUSIONS Many transport concerns have been discussed in this paper. Based on our current knowledge of those potentiallydegrading mechanisms, we believe that they can be suppressed. 9 REFERENCES [1] M. Burns, et. al., Proc. of PAC 99, New York, N. Y. March 27 – April 2, 1999, p. 617. [2] Y.-J. Chen and E. A. Rose, memo, April 2000. [3] G. J. Caporaso, W. A. Barletta, and V. K. Neil, Particle Accelerator, 11, 71, (1980). [4] M. Reiser, “Theory and Design of Charge Particle Beams”, John Wiley& Sons (1994). [5] H. L. Buchanan, Phys. Fluid, 30, 221, (1987). [6] R. Briggs, LBNL Engineering Note #M7848, Jan. 25, 2000. [7] G. V. Stupakow, Phys. Rev. Special Topics-Accel. And Beams, 3, 19401 (2000). [8] G. J. Caporaso and J. F. McCarrick, this proceedings. [9] T. C. Genoni and T. P. Hughes, DARHT-II Ion Hose Workshop, LBNL, Feb. 8, 2000. [10] Y.-J. Chen, Proc. of PAC 97, Vancouver, B. C. Canada, May 12-16, 1997, p. 1864. [11] B. Poole and Y.-J. Chen, this proceeding. [12] Y.-J. Chen, Nucl. Instr. and Meth. A292, 455 (1990). [13] Y.-J. Chen, Nucl. Instr. and Meth. A 398, 139 (1997). [14] R. Scarpetti, Private communications.
Physics Design of the ETA-II/Snowtron Double Pulse Target Experiment* Yu-Jiuan Chen, Darwin D.-M. Ho, James F. Mccarrick, Arthur C. Paul, Stephen Sampayan, Li-Fang Wang, John T. Weir, LLNL, Livermore, CA 94550, USA Abstract We have modified the single pulse target experimental facility[1] on the Experimental Test Accelerator II (ETA- II) to perform the double pulse target experiments to validate the DARHT-II[2, 3] multi-pulse target concept. The 1.15 MeV, 2 kA Snowtron injector will provide the first electron pulse. The 6 MeV, 2 kA ETA-II beam will be used as the probe beam. Our modeling indicates that the ETA-II/Snowtron experiment is a reasonable scaling experiment. 1 INTRODUCTION The DARHT-II facility will provide four 2.1 mm spot size, x-ray pulses within 2 µs with their x-ray doses in the range of several hundred rads at a meter for x-ray imaging. To achieve its performance specifications, the DARHT-II x-ray converter material is inertially confined after it turns into plasma by the heating of previous beam pulses[2]. Furthermore, the beam-target interactions, such as the focusing by backstreaming ions from desorbed gas from the target surface for the first pulse and by those from the target plasma for the subsequent pulses, and the instabilities of the beam propagating in dense plasma, should be mitigated. We will perform the ETA- II/Snowtron double pulse target experiments to validate the DARHT-II multi-pulse target concept. To simulate theDARHT-II beam-target interactions, a target plasma will be created first by striking the 1MeV, 2 kA Snowtron beam on one side of a 5 mil Ta foamed target, and then, the 6 MeV, 2 kA ETA-II beam will enter from another side to probe the target. The foam target density is 1/5 of the solid Ta density. To validate the DARHT-II target confinement concept and to ensure generating the required x-ray dose, the better characterized ETA-II beam will be used to hit the target first. We will then measure the x-ray dose created by the Snowtron beam to benchmark our hydrodynamics modeling and x-ray dose calculations. In this paper, we discuss the Snowtron beam parameters and how much target plasma is needed to do the scaled multi-pulse target experiment in Sec. 2. In Sec.3, the hydrodynamics modeling of the target plasma with the Snowtron beam parameters is presented. Both the 6MeV ETA-II beam and the 1 MeV Snowtron beam will _______________________ * This work was performed under the auspices of the U.S. Department of Energy by University of CaliforniaLawrence Livermore National Laboratory under contract No. W-7405-Eng-48.use the same final focus lens. The final focus configuration is presented in Sec. 4. The summary is in Sec. 5. 2 SNOWTRON PARAMETERS Since the fourth DARHT-II pulse has to travel through the longest plasma column. It will experience the worst beam-target interaction. The double pulse experiment is designed to simulate the beam-target interactions for the third and the fourth DARHT-II pulse. The DARHT-II target plasma discussed in this section is the plasma seen by the fourth pulse (the worst case). The Snowtron injector was a predecessor of the ETA-II injector. The 1 MeV, 2 kA Snowtron injector, using a velvet cathode, will deliver a 70 ns (FWHM) long beam with a 35 ns flattop. The energy variation during the flattop is ± 1 %. The injector voltage waveform is given in Fig. 1. The Snowtron beam’s normalized Lapostolle emittance is 1200 Fig.1 The Snowtron injector voltage waveform .π mm-mr. To simulate the DARHT-II beam-target interactions, the ETA-II beam needs to travel through an over-densed target plasma which fully charge-neutralizes the beam and has a magnetic diffusion time of the order of 1 ns. The plasma temperature should be similar to that ofthe DARHT-II plasma. The temperature is linearly proportional to the energy density deposited by the beam. Assume that only the flattop portion of the Snowtronbeam can deposit energy into a small spot due to chromatic aberration of the lens, and ignore the energy deposited by the rise and fall of the beam. The DARHT-II target is a 1 cm long Ta foamed target with 1/10 of the solid Ta density[2]. To achieve the temperature of theSnowtron Voltage Waveform -0.4-0.200.20.40.60.811.2 0 50 100 150 200 time (ns)Normalized Voltage target plasma seen by the fourth DARHT-II pulse, we need to focus the Snowtron beam on the ETA-II target to a 1 mm FWHM spot. The focusing effects of the target plasma and the backstreaming ions can over-focus the beam. The length of the ion channel or the plasma column needed to disrupt the final focus spot size R is proportional to R/(γ /I)1/2. The plasma column created by the Snowtron beam needs only 40 % of the DARHT-II plasma column length to disrupt the ETA-II beam. Since the backstreaming ion channel increases in time, its net focusing effects also increase in time. The time needed to observe the beam disruption, after the backstreaming ions are born, also follows the same scaling. For the case that the beam traveling through a pre-existing target plasma, thebackstreaming ions may appear as soon as the beam arrives at the target. The beam disruption time for the ETA-II beam is only 40 % of that for the DARHT-II beam. Therefore, it is easier to observe the backstreaming ions’ focusing effects on the ETA-II beam. 3 SNOWTRON TARGET PLASMA We have modeled the target plasma created by the Snowtron beam using the LASNEX hydrodynamics code. We assume that the Snowtron beam has a Gaussian distribution. The energy deposited by the non-flattop of the beam is ignored. The deposited energy is distributed inthe target according to a Monte Carlo calculation using the MCNP code. The third DARHT-II pulse will travel through a 1.1 cm long plasma column. According to the discussion in the previous section, the ETA-II beam has to travel through a 0.5 cm plasma column to experience similar beam-target interactions. The Snowtron plasma atthe ETA-II side will have expanded approximately 0.5 cm at 1 µs after the Snowtron beam hits the target. Therefore, firing the ETA-II beam at 1000 ns 1.0 3.4 2.6 1.8 z (cm)0.1.2.3.4R (cm).6 .5 Fig. 2 The plasma density contours at 1 µs after the Snowtron beam hits the 5 mils Ta foam target with 1/5 of the solid Ta densitythis time can simulate the beam-target interaction for the third DARHT-II pulse. The density contours of the simulated target plasma for the third pulse are shown in Fig. 2. The foamed target is located at z = 2.3 cm in the plot. Due to the low beam energy, the large scattering by the target makes the electrons in the Snowtron beam deposit more energy at the entrance side of the target than at the exit side. Therefore, there is less plasma expansion at the exit side. The corresponding plasma density and temperature along the z-axis are shown in Figs. 3(a) and (b). The ETA-II beam’s number density is 5.3 x 1013 c.c.-1 which is much less than that for the target plasma.Therefore, the beam is full charge neutralized by thetarget. The Snowtron plasma’s temperature is about half of the DARHT-II plasma. The magnet diffusion times varies from 0.01 ns to 1 ns depending on the beam radius along the axis. Therefore, the interactions between the ETA-II beam and the Snowtron target plasma would be similar to that between the DARHT-II beam and and its target plasma. Similarly, to simulate the fourth DARHT- II pulse’s beam-target interaction, the ETA-II beam shouldbe fired at 1.5 µs after the Snowtron beam. Z (cm)1.0 1.8 2.6 3.4 np (1018 c.c.-1) 024 13 Z (cm)1.0 1.8 2.6 3.4 T (1000 oC)12 8 4 0 Fig. 3 The plasma (a) density and (b) temperature along the z-axis at 1 µs after the Snowtron beam hits the 5 mil Ta foam target with 1/5 of the solid Tadensity. AP magnetSleeve : 5.35 cm thickness, 22 cm ID, 23.5 cm lengthSleeve 0.635 cm thickness, 22 cm ID, 30 cm length 11.80 Fig. 4 The Snowtron injector and beamline configuration. The target is in the center of the AP magnet. The ETA-II beam strikes the target from the right. 4 FINAL FOCUS The configuration for the Snowtron beamline is shown in Fig. 4. In order to provide enough space for diagnostics, the final focus magnet (AP magnet) for theETA-II target experiment has a large inner bore with an IDof 39.4 cm, and the ETA-II target is located inside a focusing magnet near the center. Therefore, the same magnet will also be used to focus the Snowtron beam. Since the Snowtron beam energy is much less than the ETA-II beam energy, two iron sleeves will be inserted from the Snowtron side of the magnet to reduce the excess magnetic field. A 23.5 cm long sleeve with a thickness of5.35 cm is located near the center of the final lens. The second sleeve, 17.5 cm away from the end of the first sleeve, is 30 cm long and 0.635 cm thick. Both sleeve’s ID are 22 cm. The excitation of the final focus lens will be set by focusing the ETA-II beam to a 1 mm FWHM spot. The nominal focusing field is 4800 Gauss. A solenoid (M80) is placed between the two sleeves to match the Snowtron beam from the injector exit into the final focus region for a 1 mm FWHM spot size also. TheM80 magnet is 15.4 cm in length and has an 18.2 cm ID and a 25.4 cm OD. These two sleeves and the matching lens can also be moved in and out as a unit to improve thetunability for the Snowtron beam. According the MCNP calculation, about half of the Snowtron beam electrons will be backscattered to the backward hemisphere[4]. We estimate that approximately half of these backscattered electrons would be confined by the final focus field to within the Snowtron beam radius in the final focus region. However, we do not anticipate any serious problems in focusing the Snowtron beam in the presence of these backscattered electrons. Since the sleeves and the matching solenoid are movable, we can adjust theirposition to allow a stronger focus field to compensate the backscattered electrons’ defocusing effects. 5 CONCLUSIONS We have modified the single pulse target experimental facility on the ETA-II to perform the double pulse target experiments to validate the DARHT-II multi-pulse target concept. By using the 1.15 MeV, 2 kA Snowtron beam togenerate a target plasma and then firing the 6 MeV, 2 kA ETA-II beam at an appropriate time, we can simulate the beam-target interactions for various DARHT-II pulses. 6 ACKNOWLEDGEMENTS We would like to thank G. Caporaso and G. Westenskow for many useful discussions. 7 REFERENCES [1] S. Sampayan, et. al., this conference proceeding. [2] M. Burns, et. al., Proc. of PAC 99, New York, N. Y. March 27 – April 2, 1999, p. 617. [3] Y.-J. Chen, et. al., Proc. Of PAC 99, New York, NY, March 27 – April 2, 1999, p. 1827. [4] S. Falabella, et. al., this conference proceeding.
A NEW LINEAR INDUCTIVE VOLTAGE ADDER DRIVER FOR THE SATURN ACCELERATOR M. G. Mazarakis, R. B. Spielman, K. W. Struve, F. W. Long Sandia National Laboratories, Albuquerque, NM 87185, USA Abstract Saturn is a dual-purpose accelerator. It can be operated as a large-area flash x-ray source for simulation testing or as a Z-pinch driver especially for K-line x-ray production. In the first mode, the accelerator is fitted with threeconcentric-ring 2-MV electron diodes, while in the Z-pinch mode the current of all the modules is combined viaa post-hole convolute arrangement and driven through acylindrical array of very fine wires. We present here a point design for a new Saturn class driver based on a number of linear inductive voltage adders connected inparallel. A technology recently implemented at theInstitute of High Current Electronics in Tomsk (Russia) isbeing utilized[1]. In the present design we eliminate Marx generators and pulse-forming networks. Each inductive voltage addercavity is directly fed by a number of fast 100-kV small-size capacitors arranged in a circular array around eachaccelerating gap. The number of capacitors connected inparallel to each cavity defines the total maximum current. By selecting low inductance switches, voltage pulses as short as 30-50-ns FWHM can be directly achieved. The voltage of each stage is low (100-200 kV). Many stages are required to achieve multi-megavolt acceleratoroutput. However, since the length of each stage is veryshort (4-10 cm), accelerating gradients of higher than 1 MV/m can easily be obtained. The proposed new driver will be capable of delivering pulses of 15-MA, 36-TW,1.2-MJ to the diode load, with a peak voltage of ~2.2 MVand FWHM of 40-ns. And although its performance willexceed the presently utilized driver, its size and cost couldbe much smaller (~1/3). In addition, no liquid dielectrics like oil or deionized water will be required. Even elimination of ferromagnetic material (by using air-corecavities) is a possibility. 1 INTRODUCTION Saturn is a pulsed power accelerator presently in operation at Sandia[2]. It represents a modification of the old PBFA I[3] accelerator used for ion fusion research. Itis named Saturn because of its unique multiple ring diodedesign which is utilized while operating as an x-raybremsstrahlung source. As an x-ray source Saturn candeliver to the three-ring e-beam diode a maximum energy of 750 kJ with a peak power of 32 TW providing an x-ray exposure capability of 5 x 10 12 rads/sec over a 500cm2 area. As such it is the highest power electrical driver forbremsstrahlung production in the world. As a z-pinchdriver Saturn can deliver up to 8 MA to a wire or gas-puff load. A 700-kJ total x-ray output was obtained fromaluminum or tungsten wire pinches, and a 60-70-kJ K-line radiation from aluminum 2 PRESENT ACCELERATOR CONFIGURATION Fig. 1 Schematic representation of the present Saturn configuration Figure 1 is a schematic representation of the Saturn configuration presently in effect. It utilizes conventional pulse power architecture. The driver starts with 36 Marx generators as the principal energy source. Then ensues acascade of pulse compression stages to convert themicrosecond FWHM Marx output to the 50-ns final pulsethat powers the electron diode or the z-pinch load. Thepower flow follows the following stages: The microsecond pulses are stored in 36 water intermediate store capacitors. From there through 36 triggered gasswitches they charge 36 pulse-forming lines whichthrough 36 self-breaking water switches launch the shortpulses into 36 triplate water transmission lines andimpedance matching transformers. Finally, the short (now ~50 ns) pulse is applied to the diode through a water-vacuum insulating ring interface. The 36 verticaltriplate transmission lines are connected to threehorizontal triplate disks in a water convolute section. Inthe vacuum section the power is split into three parts andfeeds three conical-triplate magnetically insulated transmission lines (MITL) that power the three separate rings of the e-diode. A final pulse compression of 25:1 isachieved. To accomplish all that, the device requiressubstantial dimensions. The overall diameter is 30 m andthe height 5 m. The 36 Marxes are immersed in a 250,000-gallon oil-filled annular tank while the rest of the device is in 250,000 gallons of deionized water. In our proposed design, utilizing the fast-pulser technology[5], the entire Saturn device will be smaller than 17 m in diameter and will be only 5 m in height. In addition, no water and oil tanks are required. Except for the 5-m high,4-m diameter cylindrical central section which is invacuum, the entire accelerator is in the air and readilyserviceable. The three diodes are concentric and, hence, have different power requirements. The innermost diode requires less power than the outside one to achieve thesame x-ray area average output. The diode design is suchthat a power partition with a ratio of 3:2:1 from outsidering to center ring is required. The bottom triplate MITLfeeds the largest outer diode ring while the top MITL is connected with the innermost diode ring. 3 NEW SATURN DESIGN BASED ON OUR FAST PULSER ARCHITECTURE In our new design we utilize the revolutionary technology of the fast pulser[4]. We select 24 modules of ~700 kA, 2.2 MV each to produce a total current of the order of 14 MA. The novelty of the design is that each ofthe 24 modules is a self-magnetically-insulated voltageadder. No liquid or solid insulator is utilized between theinner cathode electrode and the outer anode cylinder. Acoaxial geometry is adopted. The stages of each module voltage adder are of relatively low voltage (100 kV). Therefore, in principle a 20-stage structure would beenough to obtain a 2 MV output. However, because weadopted a design where the output impedance is equal tothe characteristic impedance of the pulser[4], the peakvoltage of each stage will not exceed 55 kV. Hence, 40 stages per module will be necessary. The prime power per stage is a circular array of 24, 30-kA, maximum-current fast capacitors. The inductance and capacitance ofeach capacitor is 25 nH and 11 nF respectively, which provide a very fast, LC=17 ns, time constant for the system. Hence, no further pulse power compression is required. The power-pulse FWHM from its onset has therequired width to be applied directly to the diode. The capacitor arrays are switched to the accelerating gaps through very low inductance, ~ 1 nH, externally triggered switches. Two candidates are presently considered and being evaluated. The first, developed inRussia[5], is a ring switch that has a fraction of 1-nHinductance switching large currents up to ~ 1 MA throughmultiple conducting channels. It can do that at a low 100-kV voltage. The second is being developed in Sandia[6]. It is a low inductance (~50 pH) high-gain photoconductive semiconductor switch (PCSS) that canswitch up to 250-kV, 7-kA currents in a parallel array ofsix, 2-inch wide GaAs wafers. We believe thephotoconducting switch can be further improved toincrease the current per unit width by at least one order of magnitude for the same size. The Russian switch has already reached maturity, and the design and hardware arereadily available. Those switches make it possible toswitch at low, 100-kV voltage and still produce very narrow 50-ns pulses directly from the capacitors without the need of cumbersome and expensive pulse compression. The fastest capacitors available to date are the compact Maxwell Laboratories S type capacitors model 31165 [7].They are very small (58x150x274 mm 2) and very fast (L=25 nH and C=40 nF). These capacitors could easily be modified to the faster ones (L=25 nH, C= 11 nF) considered in the present point design. Although thecapacitor dimensions could be further reduced or specialsemi-circular geometries could be developed, for thepurpose of our point design the dimensions of thepresently available off-the-shelf 31165 Maxwell capacitors were assumed. Fig. 2 Cross-sectional view of the new design The accelerator gaps are magnetically insulated with Metglas.TM A very small MetglasTM cross-sectional area (10 cm2) is needed because of the modest, 2.4x10-3 voltsecs of each gap. Therefore, the dimensions of eachmodule are as follows: The overall diameter is 2.2 m andthe length 2.4 m (Fig. 2). The cathode electrode is conicalstarting with a diameter of 1.17 m and terminating at the output end with a 1.08 m diameter cylinder. The anode electrode is a 1.20-m inner diameter cylinder. A 4-mcoaxial MITL vacuum transmission line connects eachmodule to the respective conical triplate MITL of eachdiode ring. The coaxial MITL is longer than theminimum required (2 m) in order to provide easy accessibility and servicing of the accelerator. Throughout the power flow from the voltage adders to the diode ringsself-magnetic insulation and impedance matching wasrigorously implemented. The power partition ratio of3:2:1 for the diode rings was retained since the same diode as the one presently operating was assumed to be utilized One of the advantages of the proposed design is that the diode insulating stack of the central section is eliminated.The only insulator left is the short plastic ring insulatingeach of the ~60-kV accelerating gaps of the modules. A total of 24 modules are required to provide the 15 MA current to the diode. To maintain the proper power ratio,4, 8, and 12 modules are connected respectively to thetop, middle and bottom conical triplate MITL’s. A minimum anode-cathode gap of 1 cm was maintained throughout the entire power flow transport. Table 1: Comparison of the two designs’ performance New Point Design Level Zsource OhmsZload OhmsVload MVI load MA Top 0.91 0.99 2.5 2.5 Middle 0.45 0.38 2.2 5.7 Bottom 0.30 0.33 2.4 7.4 Present Accelerator Performance Top 0.66 0.99 2.1 1.8 Middle 0.33 0.38 2.1 3.9 Bottom 0.22 0.33 2.1 6.2 Table 1 summarizes the peak pulse power parameter of the new point design and compares it with the presentaccelerator performance. The design was done analytically and verified numerically utilizing the circuit design code SCREAMER[8]. Figure 3 provides the voltage current and power for the outer-diode ring connected to the bottom MITL. -202468 -505101520 0 50 100 150 200Vload (MV) Iload (MA) Pload (TW)Pload (TW) time(ns) Fig. 3 Outer diode ring (bottom MITL) waveforms. Similar pulse waveforms are applied to the other two diode rings. Figure 4 is a three dimensional rendition ofthe entire accelerator. Fig. 4 A 3 D visualization of the new Saturn point designThe same design can deliver to a Z-pinch load a maximum total-current of 15 MA. In this case only 2 conical MITLs will be utilized with each connected to 12 voltage-adder modules. Details of this application anddesign optimization will be presented in futurepublication. 5 SUMMARY We have developed a point design for an alternative pulsed power driver for the Saturn accelerator. It has thesame or higher power and energy output as the onepresently in operation. However, its size is appreciablyreduced from the present 30-m diameter to 17 m. Theoverall height remains the same and of the order of 5 m. The expensive large amounts of oil and deionized water are eliminated together with the requirement for a centralwater-vacuum interface-insulating stack. A first cutestimate of the cost suggests a relatively modest value of~$10 M. A cheaper device of ~$5 M could be built if the Metglass TM cores are substituted with air core equivalents. Because of the very short pulse 40-ns FWHM, thedimensions of an air core device will not be appreciablylarger than our Metglass TM point design 6 REFERENCES [1] B.M. Kovalchuk, and A. Kim "High Power Driver for Z-pinch Loads." Private Communication, 1998. [2] D.D. Bloomquist, R.W. Stinnett, D.H. McDaniel, J.R.Lee, A.W.Sharpe, J.A.Halbleib, L.G. Schlitt. P.W. Spence, and P. Corcoran, "Saturn, A Large AreaX-Ray Simulation Accelerator," Proc. 6 th IEEE Pulsed Power Conference, Arlington, VA, 1987, p.310. [3] T.H. Martin, J.P. VanDevender, G.W.Barr, S.A. Golstein, R.A. White, and J.F. Seamen, "IEEE Transactions on Nuclear Science," NS-28, No. 3,3368 (1981). [4] M.G. Mazarakis, and R.B. Spielman, "A Compact, High-Voltage E-Beam Pulser," 12 th IEEE Pulsed Power Conference, Monterey, CA, July 1999. [5] B. M. Kovalchuk, "Multi-Gap Spark Switches," Proc. 11th IEEE Pulsed Power Conference, 1997, p. 59. [6] F.J. Zutavern, G.M. Loubriel, et al., "Properties of High Gain Gas Switches for Pulsed Power Applications," Proc. 11th IEEE Pulsed Power Conference, 1997, p. 959. [7] Maxwell Laboratories, Inc., "Capacitor Selection Guide" Maxwell Laboratories, Inc. San Diego, CA,Bulletin No. MLB-2112B. [8] M. L. Kiefer, K. L. Fugelso, K. W. Struve, M. M. Widner, "SCREAMER, A Pulsed Power DesignTool, User’s Guide for Version 2.0." Sandia National Laboratory, Albuquerque, N.M., August 1995 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy underContract DE-AC04-94AL85000.
arXiv:physics/0008121 17 Aug 2000TOUTATIS, THE CEA-SACLAY RFQ CODE R. Duperrier∗, R. Ferdinand, J-M. Lagniel, N. Pichoff, CEA-Saclay, 91191 Gif s/Yvette cedex, France ∗ Contact rduperrier@cea.frAbstract A CW high power linear accelerator can only work with very low particles losses and structure activation. At low energy, the RFQ is a very sensitive element to losses. To design the RFQ, a good understanding of the beam dynamics is requested. Generally, the reference code PARMTEQM is enough to design the accelerator. TOUTATIS has been written with goals of cross- checking results and obtaining a more reliable dynamics. This paper relates the different numerical methods used in the code. It is time-based, using multigrids methods and adaptive mesh for a fine description of the forces without being time consuming. The field is accurately calculated through a Poisson solver and the vanes are fully described, allowing to properly simulate the coupling gaps and RFQs extremities. Differences with PARMTEQM and LIDOS.RFQ are shown. 1 TOUTATIS ALGORITHM The scheme used by TOUTATIS to simulate the beam dynamics in RFQ is simple. The charge distribution, ρ, is discretized in a 3D mesh with a “cloud in cell” scheme. In the same grid, the vane geometry is embedded and likened to a Dirichlet boundary. The Poisson equation is solved with the obtained grid. The solver is detailed in the following sections. Finally, forces are extracted from the potential. This scheme allows to take into account external fields, space charge and image effects. Forces are applied to macro-particles via the following step to step scheme: [ ]   γ+γδ+β γ=β γ= γδ+δβ+ = + ++ ++ n1n n 1n1n 1nn2 nn 1n ) a() a(c 2t)( )()r ( Emq) a(a2ttcr r /c114 /c114 /c114 /c114/c114 /c114 /c114/c114 /c114 /c114 /c114 (Eq. 1) with δt, the time step; a, the acceleration; E, the electrical field; r, βc, γ, q, m, respectively the position, speed, relativistic factor, charge and mass of the partic le. The main advantage of this scheme is that its Jacobian is strictly equal to one. Then, the code is preserved from phoney damping of emittance which may occur with “leap frog” scheme [1]. This algorithm can be looped to reach any longitudinal position in the RFQ.2 FINITE DIFFERENCE METHOD In TOUTATIS, the Poisson equation is solved using the Finite Difference Method. The purpose of this section is not to describe in detail this well known method. The reader will find in literature many specialized books [2,3]. Only the main principles are presented. In the mesh (Fig.1), a particular node, labelled 0, is bind to its neighbours, labelled from 1 to 6, by a finite equation. This equation is a function of the electrical potential on each node, Ψi, the charge density on the considered node, ρ0, and some weighed coefficients, αi: ∑ =Ψαρ=Ψ6 1iii00 ),( f (Eq. 2) The coefficients are function of the distance between nodes, hi. Figure 1: Illustration of the Finite Difference Method. This kind of weighting allows to take into account the vane shape very accurately. The famous “stairs” discretization is then avoided. The principle is to compute each node of the grid with its associated equation taking into account the new values calculated for the previous nodes. Once all nodes of the mesh computed, the scheme can be looped to reach convergence, in other words, until the values of the electrical potential don’t change anymore. This particular way to use finite difference equation is calle d Gauss-Seidel relaxation. The accuracy of this method is only a function of h. When h tends towards zero, the solution becomes exact [2]. However, the convergence is slow enough to become prohibitive for the simulation of a whole RFQ with reasonable values of h and δt. For instance, one week of computation on a Pentium 450 MHz is necessary for the IPHI design [4]. Several methods have been developed to get acceleration of the0 1 2 34 56 h2h1h6 h5h4 h3relaxation process. We can quote the Chebyshev acceleration [5] and the Frankel-Young acceleration [2]. The next section describes the method used by TOUTATIS to reduce this computation time from one week to 5 hours. 3 MULTIGRID METHODS Practical multigrid methods were first introduced in the 1970s by Brandt [6]. Basically, we need to solve the following equation: ρ=∆Ψ (Eq. 3) with ρ, the source term; Ψ, the researched scalar potential; ∆, the Laplacian operator. The source term is discretized in a fine grid. Performing i Gauss-Seidel cycles on this fine grid, we obtain a rough estimation, Ψi, of Ψ. The Laplacian of Ψi is not equal to ρ, the difference: ρ−∆Ψ=ρii~(Eq. 4) is called the residual or defect. This residual is the solution of a second Poisson equation dealing with the error: ii~~ρ=Ψ∆ (Eq. 5) where i~Ψ is the scalar correction which allows to get Ψ via the relationship: ii~Ψ−Ψ=Ψ (Eq. 6) This is an important point in multigrid methods, we are going to estimate the error after a few relaxations rathe r than the final solution Ψ step by step. In order to get rapid estimation of this error, the equation (Eq. 5) is solved performing a relaxation process using a coarser grid, the residual having been previously discretized in this new mesh ( restriction ). This coarser grid is also marred by mistakes which can be estimated employing the same technique, and so on…To correct one fine grid with the coarser one result, an interpolation process, named prolongation , is performed. This is the main principle of the multigrid methods. The user has to combine the different stages in respect of his problem. This gives many possibilities of cycle architectures. We can quote the V cycle which is very common [7]. The cycle used by TOUTATIS is described in the figure 2. Figure 2: Representation of the TOUTATIS cycle (GS = 3 Gauss-Seidel relaxations, R = Restriction, P = Prolongation). 4 ADAPTIVE MESH REFINEMENT In order to take into account neighbour bunches, the longitudinal dimension of the grid is set to βλ and a longitudinal periodicity is imposed in the relaxation process. The main drawback of this technique occurs during acceleration of the bunch. As the phase spread decreases, the resolution on the bunch decreases also. To simply solve this problem, TOUTATIS uses a second mesh which is embedded in the main grid (Fig. 4). Its dimensions are function of bunch rms sizes while the big grid dimensions are function of the vane geometry. Figure 4: Scheme of the Adaptive Mesh Refinement 5 TESTS 5.1 Theoretical comparison The multigrid solver has been validated with a gaussian cylindrical beam. Figure 3 shows the radial component of the electrical field calculated with different resolutions for the finest grid (653, 333, 173, 93) compared to the theoretical value. Figure 3: Theoretical field and computed fields for different resolutions of the finest grid (653, 333, 173, 93). This test shows the good agreement achieved with this solver. The maximum discrepancy is less than 0.7 % for the 653 and 333 cases. It is also interesting to notice that the low resolution cases give a reasonable agreement which allows very fast calculations (15 minutes). 5.2 Experimental comparison The reference [1] describes in details an experimental confrontation between TOUTATIS and RFQ2 measurements performed in 1993 at CERN [8]. It is shown that the discrepancy is in the same region of measurements errors, around 5 %, while PARMULT discrepancy is around 15 %. 6 SIMULATION OF COUPLING GAPS The main advantage of the numerical approach of TOUTATIS is the possibility to simulate any vanes geometry. For example, the effect of discontinuity as the coupling gaps for segmented RFQs can be estimated. This is a very important point, especially when the geometry of these gaps (Fig. 5) is slightly complicated in order to reduce the sparking probability [4,9]. Figure 5: Vane profile with coupling gap. An elliptical curvature avoids a field enhancement without impairing the focusing forces significantly. To minimize the coupling gap perturbation, Lloyd Young, from LANL, has put into practice a new technique consisting in locating the gap at the longitudinal position crossed by the synchronous particle when the RF power is equal to zero [10]. Applying this concept in a particular cell, this gives the law: πφ=s cLz (Eq. 5) for the position gap center; with Lc, the cell length; φs, the synchronous phase. The figure 6 shows a typical TOUTATIS result for the electrical potential calculation in the horizontal plane without and with a coupling gap. Figure 6: Equipotentials in the horizontal plane without and with coupling gap. In favor of the IPHI project, several configurations for coupling gaps have been tested especially by varying gap width and location [11]. The table 1 compiles the significant results for the three gaps of the IPHI design. Table 1: Main results about gaps effects (* ≡ gaps @ exactly 2, 4, 6 m; + ≡ gaps @ Young’s location). Gap width (mm) ∅3.5* 3.5+2.2* 2.2+ in, tout, t~/~εε (%) 4 28 12 12 8 Transmission (%) 97 95 96 97 97 This study shows that: • The coupling gaps must be included in beam dynamics simulations to avoid too optimistic forecasts (emittances, losses, activation). • The gap width has to be set as small as possible and the center located at Young’s positions. 7 CONCLUSION A new RFQ code for beam simulation, TOUTATIS, has been written with goals of cross-checking the results of other codes and reaching a more reliable description of the electrical fields in the linac. Its numerical approach allows to simulate accurately, for any vanes geometry, the whole beam zone contrary to PARMTEQM which is limited by cylindrical harmonics [12,13]. The multigrid solver permits fast calculations compared to LIDOS which uses Chebyshev acceleration [5]. An adaptive mesh refinement is implemented in order to describe as well as possible the charges distribution without impairing the computation time. TOUTATIS has been also written to be a friendly user code (multiplatforms, PARMTEQM input file can be directly used as TOUTATIS input file). REFERENCES [1] R. Duperrier, “Intense beam dynamics in RFQs linacs”, PhD thesis n° 6194, University of Orsay, Orsay, July 2000. [2] E. Durand, “Electrostatique, Tome III, Méthodes de calcul, Diélectriques”, Masson & Cie, 1966. [3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, “Numerical Recipes, the Art of scientific computing”, Cambridge University Press, 1992. [4] R. Ferdinand & al., “Status report on the IPHI RFQ ”, this conference. [5] B. Bondarev, A. Durkin, S.V. Vinogradov, “Multilevel Codes RFQ.3L for RFQ designing”, CAP’96, Virginia, 1996. [6] A. Brandt, “Mathematics of Computation”, 1977, vol. 31, pp. 333-390. [7] P. Pierini, “A multigrid-based approach to modelling a high current superconducting linac for waste transmutation”, ICAP’98, Monterey, September 14-18 1998. [8] A. Lombardi, E. Tanke, T.P. Wangler, M. Weiss, “Beam dynamics of the CERN RFQ2 comparison of theory with beam measurements”, CERN report PS93-13 (HI), March 1993. [9] P. Balleyguier, “3D Design of the IPHI RFQ Cavity”, this conference. [10]L. Young, private communication. [11]R. Duperrier & al., “Study of coupling gaps effects on beam dynamics in favor of the RFQ IPHI design”, CEA report CEA/DSM/DAPNIA/SEA/IPHI 2000/07, Saclay, January 2000. [12]K.R. Crandall, “Effects of vane-tip geometry on the electric fields in Radio-Frequency Quadrupole linacs», LANL report LA-9695-MS, 1983. [13]R. Duperrier & al., “Field description in an RFQ and its effect on beam dynamics”, LINAC’98, Chicago, August 1998.
arXiv:physics/0008122 17 Aug 2000RF SYSTEM UPGRADES TO THE ADVANCED PHOTON SOURCE LINEAR ACCELERATOR IN SUPPORT OF THE FEL OPERATION* T.L. Smith/c250, M.H. Cho/c173, A.E. Grelick, G. Pile, A. Nassiri, N. Arnold Argonne National Laboratory, Argonne, IL60439 USA * Work supported by the U.S. Department of Energy, O ffice of Basic Sciences, under Contract No. W-31-10 9-ENG-38. /c250 Email; tls@aps.anl.gov /c173 Permanent address; Physics Department, POSTECH, Po hang, S. Korea, Email: mhcho@postech.ac.kr Abstract The S-band linear accelerator, which was built to b e the source of particles and the front end of the Advanc ed Photon Source [1] injector, is now also being used to support a low-energy undulator test line (LEUTL) an d to drive a free-electron laser (FEL). The more severe rf stability requirements of the FEL have resulted in an effort to identify sources of phase and amplitude instability and implement corresponding upgrades to the rf generation chain and the measurement system. Test d ata and improvements implemented and planned are described. 1 INTRODUCTION The rf power for the APS linear accelerator [2] is provided by five klystrons (L1 through L5), each of which feeds one linac sector; L1 feeds rf power to a ther mionic rf gun via the exhaust of one accelerating structur e (AS). The L2, L4, and L5 sectors are conventional sectors , each using a SLED cavity assembly [3] to feed four ASs. L3 supplies rf power to the photocathode gun located a t the beginning of the linac. For normal storage ring inj ection operation, L1, L2, L4, and L5 are operated; for sel f- amplified spontaneous emission (SASE)-FEL operation , all units are operated. A pulsed solid-state amplif ier is used to drive the klystrons. The pulsed amplifier i s preceded by a drive line that feeds all sectors, ph ase shifters, a PIN diode switch used for VSWR protecti on, and a preamplifier. The rf for the entire linac is derived from an ovenized, synthesized source and 10-Watt Ga As FET amplifier, which feeds both the drive line and a reference line. The VXI-based measuring system for each sector is housed in a separate cabinet, and each sy stem uses the reference line to derive phase measurement s. The overall APS linac rf system is schematically shown in Fig. 1. Figure 1: APS linac rf schematic diagram. 2 PHASE STABILITY 2.1 Phase Stability Test Unit A new phase measurement circuit (Fig. 2) was built that allowed two 2856-MHz signals to be input to a doubl e- balanced mixer with the resultant mixer output (IF ) showing a phase relationship between the two input signals. A network analyzer was used to measure the phase at 2856 MHz of a variable phase shifter for calibration over the variable phase shifter vernier scale. For calibration of the phase stability test unit, a 2856- MHz signal generator output was split and connected to the two inputs. Both variable attenuators were adju sted so the mixer LO and RF inputs = +10 dBm. The variable phase shifter was adjusted around the zero crossing to get a calibrated reading of 1 mV = 0.219o. To make accurate calibrated linac phase measurements, LO and RF port s were always set to +10 dBm and measurements were always taken at the mixer IF output zero crossing. Figure 2: Block diagram of phase stability test uni t. 2.2 Modulator Upgrades The modulator is a classic line design with a pulse forming network (PFN) and thyratron switch. The DC high-voltage power supply section charges the PFN t o a voltage of up to 35 kV before each pulse. When the modulator is triggered, the PFN is discharged by th e thyratron (EEV1836A) through the primary of a pulse transformer, resulting in a high-power pulse on the secondary (approx. –300 kV for 5 µs) being applied to the klystron cathode. Precise regulation of the PFN cha rging voltage is desired, since the klystron phase and am plitude are sensitive to this voltage. Original modulator p ower supplies had 3-phase variable transformers for the incoming AC voltage control, which are ineffective for high speed regulation. The electromechanical regula tor plus the series-connected command-charging switch combination also did not satisfy our goal of better than 99% availability. All modulators have been upgraded and now use a constant current mode, solid state, high frequency inverter type high-voltage power supply (EMI-303L). This pow er supply provides better than ±0.3% charging voltage regulation performance [4], improving klystron outp ut phase stability. The phase stability measured acros s the klystrons has improved by a factor of 2 to 2.5. 2.3 Adjustment of Phase Compensation Circuit The low-level pre-amp phase shifter assembly operat es on a 2856-MHz CW signal with the add-on function of phase compensation during the klystron drive pulse. The compensation is designed to cancel out rather slow phase settling of the 400-W amplifier output pulse that i s due to the thermal time constant of the pulsed output tran sistors within the amplifier. The leading edge of the rf ga te triggers an adjustable R-C circuit, providing an exponential waveform that is applied to a dedicated compensation phase shifter [5]. The practice has be en to adjust the compensation for the flattest drive phas e after settling of a leading edge transient. It has been d etermined that adjusting for best phase at the klystron outpu t can improve system performance. Figures 3 and 4 show th e L3 klystron phase before and after the compensation adjustment. Figure 3: L3 Klystron phase – before adjustment of phase compensator. Figure 4: L3 Klystron phase – after adjustment of p hase compensator. 2.4 Modulator Trigger Adjustments A typical “phase pushing” specification from the klystron vendor (Thomson) could be approx. 600 V = 1o of phase shift. This varies with the cathode voltag e used and the unique characteristics of each klystron. Th e modulator trigger could be adjusted so that the fla ttest section of the cathode voltage is present during th e time when the beam is present. This would produce a more stable phase. Figures 5 and 6 show data before and after modulator adjustments. Figure 5: L2 before modulator trigger adjustments. Figure 6: L2 after modulator trigger adjustments. 2.5 Trigger Jitter Each modulator-klystron subsystem receives three modulator triggers, an rf drive gate, a SLED PSK tr igger, and a klystron pulse trigger. Each trigger has uniq ue timing requirements that are functions of trigger distribution delays, internal delays, and the beam transmission time. A number of commercial precision delay generators (SRS:DG535) and VXI-based delayed trigger generators are the main hardware units for the trigger system. An approximately 10-ns trigger jitter has been observed, for example, between the rf drive and the SLED PSK trigger during normal injection operation to th e booster ring. However, the requirement of the phase stability for the current on-going SASE-FEL experim ent (LEUTL [6]) is less than 2o, which is equivalent to 2-ps in time. An experimental study is planned to explore t he possibility of satisfying the above requirement, wi thout replacing the existing trigger delay system, by imp roving the flat-top of the klystron output power character istics. 3 ON-LINE PHASE MEASUREMENT SYSTEM Phase is measured online using reference and sample signals that are synchronously downconverted to 20 MHz. A vector detector operating at 20-MHz produces I an d Q vector signals from which phase is calculated using software [7]. Each existing I/Q demodulator is implemented with analog circuits utilizing two double-balanced mixer s in an insulated, ovenized enclosure. Because 10 MHz bandwidth is required to follow SLED waveforms accurately, there is a 20 MHz component that cannot be completely filtered out and constitutes noise in th e measurements because it is asynchronous to the puls e timing and hence the sample in each pulse. The planned upgrade will use a 12-bit flash A-to-D converter sampling synchronously to the 20-MHz IF a t 80-MSa/s, to allow retrieval of data unaffected by carrier noise. Together with a reduction of LO noise, achie ved by upgrading to a newer IF reference oscillator design , it is expected that the repeatability of the relative pha se value in consecutive readings will be within a fraction o f one degree. This should allow the use of feedback to re duce phase perturbations. 4 FUTURE IMPROVEMENTS Linear accelerator operation for a SASE FEL, the 4th generation light source, requires state-of-the-art performance from the all linac subsystems, especial ly rf power (within ±0.1% shot-to-shot) and phase (less than 1o rms jitter shot-to-shot and ±5o rms long-term drift). To satisfy such tight criteria, both low-level and hig h-power rf systems are under evaluation together with devel opment of a precision on-line phase detector system. Along with the upgrade of relevant hardware components and subsystems, a beam-based optimization study of the operation parameters, such as trigger and phase compensation adjustment, is also planned. 5 REFERENCES [1] 7-GeV Advanced Photon Source Conceptual Design Report, ANL-87-15, April 1987. [2] M. White et al., “Construction, Commissioning a nd Operational Experience of the Advanced Photon Source (APS) Linear Accelerator,” Proc. XVIII Int’l Linear Accelerator Conf., Geneva, Switzerland, August 26-30, 1996, pp. 315-319, 1996. [3] Z.D. Farkas et al., “SLED: A method of doubling SLAC’s energy,” SLAC-PUB-1453, June 1974. [4] R.Fuja et al., “Constant-current charging suppl ies for the APS linear accelerator modulators,” Proc. PAC’97, Vancouver, Canada, May 1997, pp. 1269- 1271, 1998. [5] A.E. Grelick et al., “Phase control & intra-pul se phase compensation of the APS linear accelerator,” PAC’95, Dallas, May 1995, pp. 1082-1084, 1996. [6] S.V.Milton et al, “Observation of Self-Amplifie d Spontaneous Emission and Exponential Growth at 530 nm,” PRL 85(5), pp. 988-991, July 2000. [7] A.E. Grelick et al., “RF & beam diagnostic instrumentation at the APS linear accelerator,” Pro c. XVIII Int’l Linear Accelerator Conf., Geneva, Switzerland, August 26-30, 1996, pp. 851-853, 1996. S
Status of Superconducting RF Linac Development for APT∗ K. C. D. Chan, B. M. Campbell, D. C. Gautier, R. C. Gentzlinger, J. G. Gioia, W. B. Haynes, D. J. Katonak, J. P. Kelley, F. L. Krawczyk, M. A. Madrid, R. R. Mitchell, D. I. Montoya, E. N. Schmierer, D. L. Schrage, A. H. Shapiro, T. Tajima, J. A. Waynert, LANL, Los Alamos, NM 87544, USA J. Mammosser, TJNAF, Newport News, VA 23606, USA J. Kuzminski, General Atomics, San Diego, CA 92186, USA ∗ Work supported by US Department of EnergyAbstract This paper describes the development progress of high- current superconducting RF linacs in Los Alamos, performed to support a design of the linac for the APT (Accelerator Production of Tritium) Project. The APT linac design includes a CW superconducting RF high- energy section, spanning an energy range of 211–1030 MeV, and operating at a frequency of 700 MHz with two constant-β sections ( β= 0.64 and β=0.82). In the last two years, we have progressed towards building a cryomodule with β=0.64. We completed the designs of the 5-cell superconducting cavities and the 210-kW power couplers, and are currently testing the cavities and the couplers. We are scheduled to begin assembly of the cryomodule in September 2000. In this paper, we present an overview of the status of our development efforts and a report on the results of the cavity and coupler testing program. 1 INTRODUCTION Development of Superconducting (SC) RF Technology for high-current CW proton linacs, performed in support of the linac design for the APT Project [1], has been underway at Los Alamos since 1997 [2]. The goal is to design, build and test 5-cell cavities and power couplers to their specifications and integrate them into a prototype cryomodule. Since our report given at the 19th International Linac Conference [3] in August 1998, we finished the design of the cavities and couplers, which were built and tested. We also completed the design of the prototype cryomodule. The major components of the cryomodule are now being fabricated. In this paper, we will report the test results of the 5- cell cavities, the test results of the power couplers, and the progress of cryomodule fabrication. We will also report the results of a series of tests at cryogenic temperature that determined experimentally the heat leaks between the 2-K operating temperature of the cavities and room temperature. Because of space limitation, we will not describe the designs of the cavities, the couplers, or the cryomodule, as they can be found in Ref. 3.2 STATUS OF CAVITY DEVELOPMENT The APT 5-cell cavities consist of the bare cavities made of niobium (RRR=250), and the inner and outer helium vessels made of unalloyed titanium [4]. A completed bare cavity is shown in Fig. 1. These bare cavities were fabricated by CERCA in France. The half- cells were formed by spinning, and they were electron- beam welded to form bare cavities. The fabricated bare cavities are RF-tested before they are sent to Titanium Fabrication Corporation for installation of an inner helium vessel by an electron-beam-welding process. To date, inner helium vessels have been installed on two of the four cavities (Fig. 2). Following inner-helium-vessel installation, the cavities will be RF tested again to insure that cavity performance has not been degraded during the inner-helium-vessel installation. After the required performance is confirmed, outer helium vessels, instrumentation, and tuners will be installed to complete the cavities for integration into the cryomodule. Figure 1 Bare niobium 5-cell cavity Cavity testing is being shared between Los Alamos National Laboratory (LANL) and Thomas Jefferson National Accelerator Facility (TJNAF). To date, a total of three cavities have been tested successfully. Before testing, cavity processing includes the following: 1) removal of 150 µm with (1,1,2) buffered- chemistry polishing at acid temperatures below 15˚C; 2)high-pressure water rinsing; and 3) cavity baking at 150˚C. Figure 2 Cavity with inner helium vessel installed Caviy test results so far are summarized in Fig. 3. The Q-values achieved have been more than a factor of two higher than the required 5x109 at the operating gradient of 5 MV/m. The highest gradient reached of 10 MV/m is a factor of two higher than the operating gradient. For these cavities, we observed electron activities starting at 3 MV/m. The highest gradients were achieved with helium processing. After helium processing, cavity performance shown remained if the cavities were kept at LHe temperature. The highest gradient reached was limited by excess radiation and/or not enough RF power, and not by quenches. Figure 3 Performance of APT 5-cell cavities 3 STATUS OF COUPLER DEVELOPMENT The APT power couplers are required to transmit 210 kW of RF power [5]. They were tested on a test stand at room temperature (Fig. 4) [6]. Three types of tests were performed: transmitted power, totally reflected power, and condensed gas. The first test, transmitted-power capability, was performed simultaneously with two power couplers up to 1-MW. Power was transmitted from one coupler, through a copper pillbox cavity, to a second coupler, and finally to a water-cooled RF load. The couplers and copper cavity were matched for 100%transmission. The second test, totally-reflected power was performed by replacing the RF load with a “sliding short” and completely reflecting forward power, thus setting up a standing-wave pattern in the coupler-cavity system. By varying the location of the sliding short, we were able to locate the maximum of the standing-wave pattern at sixteen locations over one wavelength along the system. This test allowed us to simulate the situation when the CW beam is interrupted. The third test, the condensed-gas test, was performed by cooling the tapered end of the outer conductors to LN temperature, allowing observation of effects of residual-gas condensation. Condensed-gas effects were reported to be important in enhancing multipacting. Figure 4 Test stand for coupler at room temperature Couplers with fixed and adjustable coupling were tested. The adjustability of the coupler was accomplished with BeCu bellows (tip bellows) located close to the tip of the inner conductor (Fig. 5). RF power was fed through RF windows with two ceramic disks. These windows were fabricated by EEV, England, and were tested to 1 MW. Results of high-power testing are summarized here: 1. Both fixed and adjustable couplers achieved a power level of 1 MW CW in the transmitted-power tests; 2. In the reflected-power tests, the fixed couplers achieved 850 kW CW at the APT operating condition; achieved 550 kW CW and 850 kW at 50% duty cycle at all sliding short positions; 3. We did not observe any significant multipacting in any of the tests. With the couplers operating at 10-7 Torr, there was some vacuum activity that changed the residual gas pressures at levels of 10-9 Torr around 250 kW; 4. Data from the condensed-gas tests did not show any enhancement of multipacting1.0E+081.0E+091.0E+101.0E+11 0 2 4 6 8 10 12 14 Eacc (MV/m)Sylvia Germaine Eleanore APT spec.Figure 5 Adjustable inner conductor After high-power operations of the adjustable couplers, we observed failures of the tip bellows and electron etching. Two tip bellows failed after reaching 750 kW. Preliminary inspection indicated that the failures were caused by excess temperature at the tip bellows, which agreed with temperatures as calculated in thermal simulations. Thermal simulations also showed that the temperature of the tip-bellows could be effectively reduced by 400˚C by copper plating the BeCu bellows and cooling using gaseous helium (instead of air) as in the cryomodule design. The electron etching is marks on the inner conductors that look like electron tracks. The density of the etching and the depths of the marks increase with RF power levels. The coupler test results have been encouraging. The APT Project Office has initiated a change in the coupler specification from 210 kW to 420 kW. This change decreases the number of couplers by one-half, leading to a cost saving of more than $60M in the APT plant design. 4 STATUS OF CRYOMODULE DEVELOPMENT We have completed the final design of the cryomodule [7]. Fabrication of major components is in progress at Ability Engineering Technology in Chicago (Fig.6). All the components will be delivered to Los Alamos by the middle of September. 5 CRYOGENIC TEST RIG RESULTS We completed the experiments that measured the heat leaks from room temperature to 2 K via a power coupler [8]. In our design, this heat leak is minimized by a double-point heat-intercept approach. Without simulated RF heating, the heat leak to 2 K by one coupler is measured as one watt. The low-temperature and high- temperature heat intercept, respectively, removed 2.5 watts and 12 watts of heat. With RF heating simulated for 210 kW of coupler transmitted power, the heat leak to 2 K was 1.4 watt. The heat removed by the low- temperature and high-temperature heat exchangers was,respectively, 10 and 30 watts. This performance is as predicted by our thermal model. Figure 6 Milling the o-ring grove on the cryostat 6 SUMMARY The SCRF development has tested the 5-cell cavities and couplers experimentally. The measured RF and thermal performance greatly exceeds the APT required performance. We will be ready to assemble the cryomodule in September 2000. REFERENCES [1]P. W. Lisowski, The Accelerator Production of Tritium Project, Proc. 1997 Part. Accel. Conf., Vancouver, BC, Canada, May 1997, p.3780. [2]K. C. D. Chan, et al., Engineering Development of Superconducting RF Linac for High-Power Applications, Proc. 6th Euro. Pact. Accel. Conf., Stockholm, Sweden, June 1998, p.1843. [3]K. C. D. Chan, et al., Progress of APT Superconducting Linac Engineering Development, Proc. 19th Intern. Linac Conf., Chicago, USA, August 1998, p.986. [4]S. Atencio, et al., Design, Analysis, and Fabrication of the APT Cavities, Proc. 1999 Part. Accel. Conf., New York City, USA, May 1999, p.965. [5]E. N. Schmierer, et al., Development of the SCRF Power Coupler for the APT Accelerator, ibid., p.977. [6]E. N. Schmierer, et al., High-Power Testing of the APT Power Coupler, this Proceedings. [7]B. M. Campbell, et al., Engineering Analysis of the APT Cryomodules, Proc. 1999 Part. Accel. Conf., New York City, USA, May 1999, p.1327. [8]Joe Waynert, Los Alamos National Laboratory, private communication.
arXiv:physics/0008124v1 [physics.acc-ph] 17 Aug 2000Analysisof Longitudinal Bunching inan FELDrivenTwo-Beam Accelerator∗ S. Lidia,LBNL, Berkeley, CA USA J.Gardelle, T.Lefevre, CEA/CESTA, LeBarp, France J.T.Donohue,CENBG, Gradignan, France P. Gouard,J.L. Rullier,CEA/DIF, Bruyeres leChatel, Franc e C. Vermare, LANL,Los Alamos,NM USAAbstract Recent experiments [1] have explored the use of a free- electron laser (FEL) as a buncher for a microwave two- beamaccelerator,andthesubsequentdrivingofastanding- waverfoutputcavity. Herewepresentadeeperanalysisof the longitudinal dynamics of the electron bunches as they are transported from the end of the FEL and through the output cavity. In particular, we examine the effect of the transport region and cavity aperture to ﬁlter the bunched portionofthebeam. 1 INTRODUCTION Since 1995, free-electron laser (FEL) experiments at the CEA/CESTA facility have addressed the problem of the generationofasuitablebuncheddrivebeamforatwo-beam accelerator using linear induction accelerator technolog y. Inthesetrials,a32periodlongbiﬁlar-helixwiggleriscou - pledwitha35GHz,5kWmagnetrontoprovideaneffective FEL interaction with the beam. Early experiments[2], [3] demonstratedopticaldiagnostictechniquestoshowbunch- ingofthebeamat the35GHzFEL resonantfrequency. In the ﬁrst cavity experiments [1], the induction linac ’PIVAIR’ was utilized, since its design energy of 7.2MeV isnearoptimumforaKa-bandtwo-beamacceleratorbased upon the relativistic-klystron mechanism [4]. During op- eration, PIVAIR delivered a 6.7MeV, 3kA, 60ns (FWHM) electronbeam. Theemittanceoutoftheinjectorisapprox- imately 1000 πmm mrad, and the energy spread is less than 1% (rms) over the pulse length. The full current is collimated to 830 ±30A at the FEL entrance. Two 6- period adiabatic sections are used to inject the beam into the properhelical trajectoryinsidethe wiggler,andthen t o release thebeambackintothetransportlineafterwards. After the wiggler follows a short transport beamline to capture and focus the beam through a narrow-aperture(4- mmID),35GHz,single-cellrfoutputcavity[5]. Thebeam- line consists of a section of stainless steel pipe (39mm ID, 1.2m long) with a set of solenoid magnets to provide fo- cusing through the rf cavity. The rf power generated by the beam in the cavity is collected and analyzed, while the beamitself isdumped. Thisset-upisshowninFigure1. ∗TheworkatLBNLwasperformed undertheauspices oftheU.S.D e- partment of Energy by University of California Lawrence Ber keley Na- tional Laboratory under contract No. AC03-76SF00098. Figure 1: Schematic of downstream transport beamline, andrfdiagnostics. Figure 2: FEL power and bunching evolution in the wig- gler,measuredversusperiodnumber. 2 SIMULATIONCODES Twoseparatenumericalsimulationcodesareusedtomodel the system behavior. The ﬁrst is the steady-state, 3-D FEL code,SOLITUDE [6]. The evolution of the FEL mode power, both as measured and as calculated by SOLITUDE , is shown (circles and solid line, respectively) in Figure 2. Also shownis the calculatedvalueofthe bunchingparam- eter (dashed line) [7]. Not shown is the evolution of the beam current duringtransport throughthe wiggler. Exper- imentally, the current exiting the wiggler was observed to be∼250A.ThisvaluewasreproducedintheFELsimula- tions[8]. TheRKScode [9] is then used to propagate the beam02468101214161820 00.20.40.60.811.21.4 z [m]RMS Envelopes [mm] Solenoid Field [kG]Beam Pipe Radius [mm] RF Cavity Radius [mm] xy Figure 3: Simulation results of beam transport from the wigglerexitthroughtherfcavity. 00.10.20.30.40.50.60.7 00.20.40.60.811.21.4 z [m]Bunching Parameter Bunch Current [kA]CavityWiggler Exit Figure 4: Simulation results of current and bunching pa- rametertransport. from the end of the wiggler through the cavity, and to cal- culate the interaction of the beam with the rf output struc- ture. The6D particledistributionofthe beamat theendof thewiggler,ascalculatedby SOLITUDE ,isusedasinputto RKS. The evolution of the beam rms envelopes are shown in Figure 3. The cavity acts as a collimator, reducing the beam current, as can be seen in Figure 4. This degree of current loss was observed experimentally. The calculated power developedin the rf cavity is also comparableto that observedexperimentally[1]. 3 FILTERING AND BUNCH ENHANCEMENT TheinterestingfeaturetoobserveinFigure4,isthediscon - tinuous growth of the bunching parameter, and simultane- ous current loss, as the beam is partially focused through the cavity. This is preceded by the gradual loss of cur- rent, and the gradual increase in the bunching parameter in the transport region between the end of the wiggler and1213141516 -4-3-2-101234γ s = c * t [mm] Figure 5: Longitudinalphase space distribution at wiggler exit. the entrance to the rf cavity. As was pointed out in [1], the transport line and cavity appear to act as a ﬁlter that preferentially selects the bunched portion of the beam for transmission. We seek to analyze this behaviorin terms of thedynamicsofthebeaminthetransportlinefollowingthe wiggler. The longitudinal phase space of the bunches near the middle of the beam pulse at the exit of the wiggler are shown in Figure 5. As shown, there is signiﬁcant initial bunching (b ∼0.4) as well as ’tilt’ (energy-phasecorrela- tion). The presence of this tilt arises from the fortuitous extractionof the beamat an appropriatesynchrotronoscil- lation phase in the ’saturated’ regime of the FEL interac- tion. This tilt contributes to continued bunching in a bal- listic transportline. Theeffectofspacechargeforcesupo n debunchingarelimitedbythelowcurrent(250A)andhigh kinetic energy (6.7 MeV), and signiﬁcant debunching will only appearafterseveral meters[10]. Thistilt can account for a modest rise in the bunching parameter, from 0.4 to ∼0.5,asdiscussedbelow. Inadditiontotheenergytilt,thebunchesemergingfrom the wiggler exhibit nonuniform bunching over the trans- versedistribution. ThisisshowninFigure6. Displayedare contours of constant bunching parameter as a function of transverseposition. Thetransversepositioncoordinateh as been normalized by the appropriate rms transverse beam size (σxorσy). While the average value of the bunching parameteris ∼0.4,there is a largedegreeof variationwith the high-brightness, central core more strongly bunched than the outlyingedges. Collimation of the beam can then strip awaythe less-bunchedregions,resultingin anoveral l enhancementoftheaveragebunchingparameter. The origin of the transverse variation can be related to the variation of the electromagnetic signal co-propagatin g with the beam in the waveguide of the FEL. Optical guid- ing studies [11] show that both the beam density and the electromagnetic mode amplitude decrease with increasing0.7 0.4 0.1 -6-4-20246 x / σ-6-4-20246 y / σy x Figure 6: Transversedistributionof bunchingparameterat wigglerexit. 00.10.20.30.40.50.60.7 00.20.40.60.811.21.4Current [kA], Bunching z [m]Bunching Current 10mm10mm 19.5mm19.5mm 30mm30mm45mm 45mm Figure 7: Evolution of beam current and bunching param- eter withvaryingbeampipediameters. transverse distance from the beam axis. There is, then, a smaller coupling between the beam and the mode at dis- tancesfromthebeamaxis,withthesubsequentdecreasein the forcesresponsibleforbunching. A series of simulations were performed in which the beam pipe radiusof the transport line between the wiggler and rf cavity was varied. The purpose of this was to ex- plore the relative inﬂuenceof the two bunchingeffectsde- scribed above. The results are shownin Figure 7, showing evolution of beam current and bunching parameter along the beamline. As shown, the smaller pipesact as collimat- ing agents, while the larger pipes transmit nearly 100% of the beam current from the wiggler to the entrance of the cavity. In the simulations, the smaller beam pipes allowed the less-bunched portions of the beam to be intercepted, thereby increasing the average bunching parameter. How- ever,allsimulationsdemonstratedballisticbunchingdue to the energy-phase tilt. At the end of the transport line lies the cavity with a 2mm bore radius, which acts as a ﬁnal collimatorandlimitsdrasticallythepercentageoftransm it- ted current, while also stripping away the unbunched por-0.8 0.6 0.1 -6-4-20246 x / σ-6-4-20246 y / σ xy Figure 8: Transversedistributionof bunchingparameterat cavityexit. tions from the highly-bunched core. The ﬁnal transverse distribution of the bunching parameter is shown in Figure 8, taken at the exit plane of the cavity. This distribution exhibits a broader central plateau with a greater degree of bunchingthanseenin Figure6. 4 CONCLUSIONS We have presented results of simulations to assist in the analysis of experimental measurements of current and bunching transport in a high-frequency, two-beam accel- eratorprototypeexperiment. We haveshownthat theaver- agedegreeoflongitudinalbunchinginabeamthatexitsan FEL ampliﬁer can be improved by collimation. However, thismayalsobeaccompaniedbysigniﬁcantlossofcurrent. 5 REFERENCES [1]T.Lefevre,et. al., Phys. Rev. Lett. 84(2000),1188. [2]J. Gardelle,et. al., Phys. Rev. Lett. 76(1996),4532. [3]J. Gardelle,et. al., Phys. Rev. Lett. 79(1997),3905. [4]G.Westenskow,et. al. ,ProceedingsoftheVIIIInterna- tionalWorkshoponLinearColliders ,Frascati (1999). [5]S.M.Lidia,et. al., ProceedingsoftheXIXInternational LinearAcceleratorConference ,Chicago(1998),97. [6]J. Gardelle,et. al., Phys. Rev. E 50(1994),4973. [7]Thebunchingparameterisdeﬁnedas b=\|/angbracketleftexp (iψ)/angbracketright\|, whereψ=kwz−ω(z/c−t)is the usual phase variable ofanelectroninthebunch,andwhere /angbracketleft/angbracketrightdenotesthebunch ensembleaverage. [8]S.M.Lidia,et. al., Proceedingsofthe1999IEEE Particle AcceleratorConference ,NewYork(1999),1797. [9] S.M. Lidia, Proceedingsof the 1999IEEEParticle Ac- celeratorConference ,New York(1999),2870. [10]J. Gardelle,et. al., Proceedingsof theXIX InternationalLinearAcceleratorConference ,Chicago (1998),794. [11]A. Bhattacharjee,et. al., Phys. Rev. Lett. 60(1988), 1254.
PROGRESS ON THE 140 KV, 10 MEGAWATT PEAK, 1 MEGAWATT AVERAGE POLYPHASE QUASI -RESONANT BRIDGE, BOOST CONVERTER/MODULATOR FOR THE SPALLATION NEUTRON SOURCE (SNS) KLYSTRON POWER SYSTEM 1 William A. Reass, James D. Doss, Robert F. Gribble, Michael T. Lynch, and Paul J. Tallerico Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87544 Abstract This paper describes electrical design and operational characteristics of a zero -voltage -switching 20 kHz polyphase bridge, boost converter/modulator for klystron pulse application. The DC -DC converter derives the buss voltages from a standard 13.8 kV to 2300 Y substation cast -core transformer. Energy storage and filtering is provided by self -clearing metallized hazy polypropylene traction capacitors. Three “H -Bridge” IGBT switching networks are used to generate the polyphase 20 kHz transformers primary drive waveforms. The 20 kHz drive waveforms are chirped the appropriate duration to generate the desired klystron pulse width. PWM (pulse width modulati on) of the individual 20 kHz pulses is utilized to provide regulated output waveforms with adaptive feedforward and feedback techniques. The boost transformer design utilizes amorphous nanocrystalline material that provides the required low core loss at d esign flux levels and switching frequencies. Resonant shunt peaking is used on the transformer secondary to boost output voltage and resonate transformer leakage inductance. With the appropriate transformer leakage inductance and peaking capacitance, zer o-voltage -switching of the IGBT’s is attained, minimizing switching losses. A review of these design parameters and a comparison of computer calculations, scale model, and first article results will be performed. Figure 1: Simplified Block D iagram. _______________________________________ 1 Work supported by The U.S. Department of Energy 1 MAJOR COMPONENTS The simplified bock diagram of the converter/modulator system is given in Figure 1. This system economizes costs by the utilizati on of many standard industrial and utility components. The substation is a standard 13.8 kV to 2300 Y cast -core transformer with passive harmonic traps and input harmonic chokes. These components are located in an outdoor rated NEMA 3R enclosure that do es not require secondary oil containment or related fire suppression equipment. An SCR regulator is helpful to optimize the IGBT switching parameters. The SCR regulator also accommodates incoming line voltage variations and other voltage changes resultin g from transformer and trap impedances, from no -load to full - load. The SCR regulator also provides the soft -start function. The SCR regulator provides a nominal +/ - 1500 Volt output to the energy storage capacitors. The energy storage capacitors are self -clearing metallized hazy polypropylene traction motor capacitors. As in traction application, these capacitors are hard bussed parallel. These capacitors do not fail short, but fuse or “clear” any internal anomaly. At our capacitor voltage rating (2 kV) there has not been a recorded internal capacitor buss failure. In this application, as in traction motor application, the capacitor lifetime is calculated to be 1e9 hours. Insulated Gate Bipolar Transistors (IGBT’s) are configured into three “H” bridge c ircuits to generate a three phase 20 kHz square wave voltage drive waveform. The IGBT’s are “chirped” the appropriate duration to generate the high voltage klystron pulse, typically 1.3 mS. Due to the IGBT switching power levels, currents , and frequencies involved, low inductance busswork and bypassing is of paramount importance. The boost transformers utilize amorphous nanocrystalline material that has very low loss at the applied frequency and flux swing. Operating at 20 kHz and about 1.6 Tesla bi -directional, the core loss is about 1.2 watts per pound in our application, or 320 W per core. Each of the “C” cores (one for each phase) weigh 260 lbs. and has a 3.5” by 5” post. By appropriately spacing the secondary from the primary, the transformer leakage inductance can be resonated with secondary shunt peaking capacitors to maximize voltage output and tune the IGBT switch current to provide “zero -voltage -switching”. The zero -voltage - switching is provided when the IGBT gate drive is positive, but reverse transformer primary circulating current is being carried by the IGBT internal freewheel diode. We have tuned for about 4 uS of freewheel current before the IGBT conducts in the normal quadrant. This tuning provides for about 15% control range for IGBT pulse width modulation (PWM). The ability of IGBT PWM of the active klystron voltage pulse permits adaptive feedback and feedforward techniques with digital signal processors (DSP’s) to regulate and provide “flat” klystron voltage pulses, irrespective of capacitor bank “start” voltage and related droop. Line synchronization is not absolutely required as the adaptive DSP can read bank voltage parameters at the start of each pulse and calculate expected droop. A standard six -pulse rectification circuit is used with a “pi -R” type filter network. The diodes are fast recovery ion -implanted types that are series connected with the appropriate compensation networks. The diodes have the second highest power loss (after the IGBT’s) and are fo rced oil cooled. The filter network must attenuate the 120 kHz switching ripple and have a minimal stored energy. The stored energy is wasted energy that must be dissipated by the klystron at the end of each pulse. With the parameters we have chosen, th e ripple is very low (~300 volts) and the klystron fault energy (in an arc-down) is about 10 joules. Even if the IGBT’s are not turned off, the transformer resonance is out of tune because of the fault condition, and little difference in klystron fault en ergy is realized. If the IGBT’s fail short, through the transformer primary winding, the boost transformer will saturate in about 30 uS, also limiting any destructive faults to the klystron. In a faulted condition, the klystron peak fault current is about twice nominal, with low dI/dT’s. 2 MODELING The complete electrical system of the converter/modulator system has been modeled in extreme detail. This includes design studies of the utility characteristics, transformer and rectification methodology, I GBT switching losses, boost transformer parameters, failure modes, fault analysis, and system efficiencies. Various codes such as SCEPTRE, MicrocapIV, Flux2D, and Flux3D have been used to complete these tasks. SCEPTRE has been primarily used to examine I GBT and boost transformer performance in great detail to understand design parameters such as switching losses, IGBT commutation dI/dT, buss inductance, buss transients, core flux, core flux offset, and transformer Eigen frequencies. Flux2D and Flux3D h ave been used to examine transformer coupling coefficients, leakage inductance, core internal and external flux lines, winding electric field potentials, and winding field stresses. The Flux2D and Flux3D were particularly useful to examine transformer sec ondary winding profiles that gave the desired coupling coefficients with minimized electrical field stresses. MicrocapIV has been used to examine overall design performance of the system. This includes the utility grid parameters such as power factor, l ine harmonics, and flicker. We have optimized the design to accommodate the IEEE -519 and IEEE -141 standards. The MicrocapIV uses simplified switch models for the IGBT’s, which does not accurately predict their losses. However, the code has been very us eful to examine tradeoffs of circuit performance with the lumped elements such as the boost transformer, shunt peaking capacitance, the filter networks, and the input energy stores. MicrocapIV is also adept at making parametric scans to determine componen t sensitivities and tolerances. Comparisons between the SCEPTRE and MicrocapIV codes show no significant differences in the system operational performance such as switching currents, switching voltages, and output voltage parameters. A typical output of MicrocapIV, Figure2, shows the zero - voltage -switching of a pair of IGBT’s in the top waveform as compared to the transformer primary winding current in the middle waveform. Figure 2: Details of Waveforms. The negative IGBT current before turn -on is the freewheel diode current. This negative current is of paramount importance to provide a loss -less turn -on of the IGBT, excessive PWM can cause the IGBT to fall out of synchronization with the winding current. Limits of PWM are necessary to ensure this co ndition does not happen, causing over -heating of the IGBT’s. A detail of the klystron voltage rise time, about 75 uS, is shown at the bottom. With the appropriate optimization, good klystron pulse fidelity can be obtained as shown in Figure 3. The model shows the resulting droop from the capacitor energy stores. Figure 3 Klystron Cathode Voltage. 3 SCALE MODEL RESULTS A scale model has been built and has been used extensively to prove various design concepts. As electronic control subsystems become available, they can be tested and debugged with the scale model. Figure 4 shows the first results we obtained with our adaptive power supply feedback and feedforward system. The lower most trace shows the capacitor bank voltage droop of 20%, five times worse than the SNS bank design (~4% droop). The scale model shows a droop that follows the capacitor voltage (as can be expected) without any adaptive controls. With adaptive control, the flattop performance is considerably improved as well as having a b etter rise time. Further efforts to optimize the loop parameters should be able to provide a flat output voltage pulse. Figure 4: Adaptive Feedfoward/Feedback 4 PROJECT STATUS The status of the first article is moving along well . The construction efforts are nearing completion with only a few subassemblies needed before final integration. The oil tank basket is shown in Figure 5. The high -voltage rectifier assemblies are in the lower portion of the figure. The IGBT switch boa rds can be seen on top of the converter/modulator in their “edge” connectors. The oil tank and safety enclosure is depicted in Figure 6. Most prominent is the cooling panel for the IGBT’s and oil/water heat exchanger. The enclosure is an integral part o f a ground plane (also noted in the picture) to minimize any radiated noise. The converter/modulator is completely (electrically) sealed at all points. Unfortunately, vendor delays of the SCR phase control rack will not permit full average power operatio n until December 00 or January 01. We will therefore use a pair of 10 kW power supplies for initial debug operations and testing to full output voltage, current, and pulse length, but at a reduced pulse rate. Figure 5: Oil Tank Basket Assembly Figure 6: Oil Tank & Safety Enclosure 5 CONCLUSION The converter/modulator will be demonstrating many new technologies that will hopefully revolutionize long-pulse klystron modulator design. These items include self -clearing capacitors in a modulator application, la rge amorphous nanocrystalline cores, high-voltage and high -power polyphase quasi -resonant DC-DC conversion, and adaptive power supply control techniques. These design and construction efforts will hopefully come to closure in less than a year after concep tion. Design economies should be realized by the use of industrial traction motor components (IGBT’s and self -clearing capacitors) and standard utility cast -core power transformers. The modular design minimizes on -site construction and a simplified utility interconnection scheme further reduces installation costs. The design does not require capacitor rooms and related crowbars. Design flexibility is available to operate klystrons of different voltages by primarily changing the boost transformer turns r atio. All testing of individual full -scale components and scale model results all agree to date with modeling efforts that indicate the design technology will be imminently successful.
arXiv:physics/0008126v1 [physics.acc-ph] 17 Aug 2000The RTA Betatron-Node Experiment: Limiting Cumulative BBUGrowthInALinear Periodic System S. Lidia, LawrenceBerkeley NationalLaboratory T. Houck,G. Westenskow,LawrenceLivermoreNational Labor atory Abstract ThesuccessfuloperationofaTwo-Beamacceleratorbased on extended relativistic klystrons hinges upon decreasing thecumulativedipoleBBUgrowthfromanexponentialtoa moremanageablelineargrowthrate. Wedescribethetheo- reticalschemetoachievethis,andanewexperimenttotest this concept. The experiment utilizes a 1-MeV, 600-Amp, 200-nselectronbeamandashortbeamlineofperiodically- spaced rf dipole-mode pillbox cavities and solenoid mag- nets for transport. Descriptions of the beamline are pre- sented, followed by theoretical studies of the beam trans- portanddipole-modegrowth. 1 INTRODUCTION A Lawrence Livermore National Laboratory (LLNL) and Lawrence Berkeley National Laboratory (LBNL) collab- oration is studying the application of induction accelera- tor technology to the generation of microwave power. We refer to this scheme of power generation as the Relativis- tic Klystron Two-Beam Accelerator (RK-TBA) [1]. This scheme is considered a TBA approach as the extraction of microwavepowerisdistributedalongadrivebeamparallel to the high-energy rf linear accelerator. The RK designa- tion indicates that the power is generated by the interac- tion of the relativistic modulateddrivebeam with resonant structuressimilarto thoseusedina conventionalklystron . TheprimaryadvantageofTBA conceptsisthatthe con- version of drive beam power to microwave power can be highly efﬁcient ( >90%). This efﬁciency is realized by distributing the power extraction over an extended length. The interest in RK-TBA’s is that induction accelerators are efﬁcient at producingvery high power electron beams. Present induction accelerators operate at currents of sev- eral kilo-amperesand accelerate the beam to 10’s of MeV for beam power of 100’s GW [2]. The induction accelera- tor can realize improvedefﬁciency at convertingwall plug power into beam power by replacing the standard electro- magnetsolenoidsfor beamtransportwith permanentmag- nets. Evenhigherefﬁciencycanbeattainediftheinduction cells are used as high-voltage step up transformers driven by a relative low voltage ( ∼20 kV) pulsed power system. Present designs of a RK-TBA predict efﬁciency of about 40%inconversionofwallplugpowerintobeampower[3], oratotalwallplugtomicrowavepowerefﬁciencyofabout 36%. The main section of an RK where the microwave power is generated is comprised of many repeating mod- ules as illustrated in Figure 1. Within each module,the in-ductioncellsreplacetheenergyextractedfromtheelectro n beamby themicrowaveoutputstructure. The efﬁciencyof this process — extraction and reacceleration — is nearly 100%. Not shownin Figure1 are thebeam generationand modulation sections and the ﬁnal beam dump. Fixed en- ergylossesinthoseprocesseshavetobeincludedincalcu- lating the total beam energyto microwave conversionefﬁ- ciency. Thus,itisimperativethattheRK havemanyofthe efﬁcient extraction and reacceleration cycles to reduce th e relativevalueofﬁxedlosseswithrespecttothetotalenerg y transferredtothebeam. Several proof-of-concept experiments have been per- formed to demonstrate the viability of the RK-TBA con- cept. These experiments have shown the generation of collider-scaledrivebeamininductionlinacs,production of high-quality, high-power microwaves from standing- and traveling-wave structures driven by induction accelerato r beams, and multiple reacceleration and extraction cycles [4,5]. Aswillbedescribedbelow,wearecontinuingtoper- formexperimentsto studyspeciﬁc physicsandtechnology issueswhile constructinga prototyperelativistic klystr on. 2 BEAM DYNAMICSISSUES TheultimateefﬁciencyofaRKisdeterminedbytheinduc- tion beam dynamics i.e. the number of extraction struc- tures that the beam can transit. We have identiﬁed three critical areas of beam dynamics that must be understood. The ﬁrst involvesmaintainingthe longitudinalmodulation of the beam or ”rf bucket” structure. In the drifts between output structures, space charge forces will cause the beam to lengthen in phase space, i.e., ”debunch”. If this effect is not corrected, the rf current (Fourier component of the beam at the modulationfrequency)will decrease resulting inadecreaseinthemicrowavepowerthatcanbeextracted. Inductively detuning the output structures, similar to the penultimate cavity in conventional klystrons, can counter the space charge forces. The requirement for long-term longitudinal stability is reestablishing the initial long itudi- nal charge distribution at the end of a synchrotron period. Computer simulations have shown that with proper detun- ing, the rf current can be maintained over the 150 output structuresenvisionedforafullscale RK-TBA [6]. The other issues involve transverse instabilities. The beamwillexcitedipolemodesintheinductioncellacceler- ating gapsaswell asin the resonantoutputstructures. The induction cell accelerating gaps can be severely damped with rf absorbers for all resonant modes since the applied voltage pulse is quasi-static compared to the resonant fre-quencies. In addition, the natural energy spread over the rf bucket contributes to phase mixed, or Landau, damp- ing. The combination of rf absorbers and energy spread is expectedtomaintainthe transverseinstabilitydueto th e dipolemodesin theacceleratinggapsat acceptablelevels. The resonant output structures present a more difﬁcult transverse instability issue. The fundamental mode must couplesufﬁcientlywiththebeamtoextracttherequireden- ergy. Varioustechniquesexist todamphigherordermodes in both output and accelerating structures. However, the permanent magnet focusing system envisioned for an RK- TBA allows the applicationof a new techniquethat we re- ferto asthe BetatronNode Scheme. Transversebeaminstabilitytheoryiswelldevelopedand the exponential growth predicted is supported by exper- iment. However, the standard theoretical approach as- sumes that the discrete cavities interacting with the beam are closely spaced compared to the betatron wavelength due to the focusing system. Our design for an RK-TBA system requires strong focusing to maintain the required beam radius and a constant average energy over each ex- traction/reacceleration cycle. This combination leads to spacing between output structures of one betatron wave- lengthandthe basicassumptionofthestandardtheoretical approachdoesnothold. An alternative approach to studying the transverse in- stability uses transfer matrices [7]. Assuming a monoen- ergetic beam and a thin cavity, Equations (1) through (3) indicatethe salient partsofthistheory. Equation(1)repr e- sentsthetransversemomentumchangeanelectronreceives passing through the cavity. Ris an integral operator that accounts for the part of the beam that has already passed throughthecavity. TheﬁrstmatrixontheRHSofEquation (2) is then the transfer matrix for the beam going through thecavity. Forasufﬁcientlythincavity,thetransversepo si- tiondoesnotchange. Onlythemomentumisaffected. The second matrix represents the betatron motion of the elec- trons between cells where θis the phase advance. Thus, Equation(2)advancesthepositionandmomentumofelec- tronsfrom the exit of on cavity to the exit of the following cavity. Byrepeatedlymultiplyingthetwotransfermatrice s, thepositionandmomentumattheexitofanycavitycanbe related to the initial conditions. For the situation where θ is constant for all sections and θ≪1, the series of matrix multiplications can be shown to yield the same expected exponentialgrowthasthemorestandardapproach. ∆p⊥=R·x (1) /bracketleftbigg x p⊥/bracketrightbigg n+1exit= /bracketleftbigg1 0 R1/bracketrightbigg/bracketleftbigg cosθsinθ ω ωsinθcosθ/bracketrightbigg /bracketleftbiggx p⊥/bracketrightbigg nexit(2) For the case where θ= 2π(or any integral multiple of π), the matrix multiplication is greatly simpliﬁed. The be- tatronmotionreturnstheelectronstotheoriginaltransve rseSolenoidPumping Port Period ~ 60 cmRF Cavity RF BPMAperture 4.75 cm CL Figure1: Minimumbeamlineconﬁguration. positionandmomentum(oppositelydirectedforoddmulti- ples of π). The multiplicationinvolvesonly the matrix de- scribing the effectof the cavity, and,as shownin Equation (3),thisleadstoalineargrowthinthetransverseinstabil ity /bracketleftbiggx p⊥/bracketrightbigg n+1exit= =/bracketleftbigg 1 0 R1/bracketrightbiggn/bracketleftbigg x p⊥/bracketrightbigg (n=1) exit =/bracketleftbigg 1 0 nR1/bracketrightbigg/bracketleftbigg x p⊥/bracketrightbigg (n=1) exit(3) There are many non-ideal factors in a realistic acceler- ator including cavities of ﬁnite thickness and variation in phase advance due to energy and/or focusing errors. Pa- rameterstudiesthroughcomputersimulationsindicatetha t the transverse instability is signiﬁcantly reduced for sys - tems with reasonable variations in parameters. We intend toexperimentallytestthevalidityandrobustnessoftheBe - tatronNodeScheme. 3 BETATRON NODESCHEME EXPERIMENT The basic elementsinvolvedin a test of the BetatronNode Schemeare: asetofdevicesthatgeneratealocalizedtrans- verse impedance, a tunable focusing and transport system, and diagnostics to measure the BBU mode signal on the beamasafunctionoftimeanddistancealongthebeamline. A schematicfora possiblebeamlineisshownin Figure1. The localized impedances are generated in simple pill- box cavities, tuned so that the TM110mode frequency matches the modulation of the beam; a series of solenoid magnetsprovidetunablefocusing;andrfBPMsplacedbe- tween cavities provide a means of collecting the dipole mode signal carried by the beam. We have built several sections of this beamline, using off-the-shelf components wherever possible. Each section is one betatron wave- lengthlongandiscomprisedofonepillboxcavity,apump- ing port, a diagnostic, and three solenoids. The rf cavities have a simple pillbox design, with a dielectric insert (Alu- mina 99.5%, ǫ≈9) to adjust the mode frequency. The dipole mode resonates at ∼5.2 GHz, with a wall-loaded Q-value of ∼100 and a normalized transverse impedance/bracketleftBig Z⊥ Q/bracketrightBig ∼6.5Ω. Computer simulations of the increase in power mea- sured by the rf diagnostics at the dipole mode frequency100101102103104105106107 0.9 0.95 1 1.05 1.1Power (normalized) Bz (kG)After 10 Sections Figure 2: Dipole mode power vs. solenoidal ﬁeld (phase advance). 10-1100101102103104105106 0 2 4 6 8 10Power (normalized) Section #0.92 kG 0.98 kG 1.04 kG1.06 kG1.08 kG Figure 3: Dipole mode power vs. length for varying solenoid ﬁelds, displaying linear and exponential growth regimes. are showninFigures2and3. Variations of ±10% in the solenoidal ﬁeld (betatron phase advance) from the optimum should produce several orders of magnitude increase in measured mode power af- ter only a few sections. The graphs indicate the maxi- mum power expected during the main body of the beam (”ﬂat top”). Thetemporalpowervariationduringthepulse (not shown) is predicted to have different characteristics betweenunder-andover-focusedscenarios. 4 SUMMARY The long-termgoal of the RTA Facility is to build a proto- type relativistic klystronthat hasall the majorcomponent s required for a RK suitable for collider applications. The prototypewouldserve asa test bed forexaminingphysics, engineering, and cost issues. The ﬁrst major component, the1-MeV,600-A,inductionelectrongun,oftheprototypehasbeencompletedandcommissioned. Beforecontinuing with the next section of the prototype, we intend to per- formaseriesofbeamdynamicsexperiments. Inparticular, wewilldemonstratetheeffectivenessoftheBetatronNode Scheme. We arealsocontinuingtostudyandoptimizecol- liderdesignsbasedontheRK-TBA scheme. 5 ACKNOWLEDGMENTS We thank Swapan Chattopadhyay,George Caporaso, Kem Robinson, and Simon Yu for their support and guidance. Dave Vanecek and Wayne Greenway provided invaluable mechanical engineering and technical services. John Cor- lett and Bob Rimmer designed the rf diagnostics. This work was performed under the auspices of the U.S. De- partment of Energy by University of California Lawrence Berkeley Livermore National Laboratory under contract No AC03-76SF00098 and Lawrence Livermore National LaboratoryundercontractNo. W-7405-Eng-48. 6 REFERENCES 1. Sessler, A.M.andYu,S.S., Phys. Rev. Lett. 54,889 (1987). 2. Burns,M.J.,et al.,”DARHTAcceleratorUpdateAnd PlansForInitialOperation”,in ProceedingsofIEEE1999 Part. Accel. Conf. ,NY, 1999,pp. 617-621. 3. Houck,T.L.(ed.),etal., “AppendixA: A RF Power SourceUpgradeto theNLCBasedontheRelativistic KlystronTwo-BeamAcceleratorConcept”,SLAC-474, StanfordUniversity(1996). 4. Westenskow,G.A.andHouck,T.L., IEEETrans. PlasmaSci. 22,pp. 424-436(1994). 5. G.A. WestenskowandT.L.Houck,“Resultsofthe ReaccelerationExperiment: ExperimentalStudyofthe Relativistic KlystronTwo-BeamAccelerator”,in Proceedingsof the10thIntl. ConferenceonHighPower Particle Beams ,SanDiego,CA (1994). 6. Giordano,G., et. al.,“Beam DynamicsIssuesinan ExtendedRelativistic Klystron”, ProceedingsofIEEE 1995Part. Accel. Conf. ,Dallas, TX,1995,p. 740-742. 7. Neil, V.K.,Hall, L.S.,andCooper,R.K., Part. Accel. 9, pp. 213-222(1979).
SIMULATION OF RETURNING ELECTRONS FROM A KLYSTRON COLLECTOR Z. Fang, The Graduate University For Advanced Studies S. Fukuda, S. Yamaguchi, And S. Anami, High Energy Accelerator Research Organization (KEK) 1-1, Oho, Tsukuba-Shi, Ibaraki, 305-0801, Japan Abstract Spurious oscillations in klystrons due to returning electrons from the collector into the drift tube were observed and studied at KEK. Simulations of returning electrons using EGS4 Monte Carlo method have been performed. And the oscillation conditions are described in this paper. 1 INTRODUCTION A high-power 324MHz klystron (3MW output power, 650µs pulse width, and 110kV beam voltage) is being developed at KEK as a microwave source for the 200 MeV Linac of the KEK/JEARI Joint Project for the high- intensity proton accelerator facility. However during high-voltage processing of the klystron tube #1, strong spurious oscillations were observed when there was no input power. These oscillations were caused by the returning electrons from the collector into the drift tube of the klytron. As the returning electrons modulated by the gap voltage of the output cavity will induce a gap voltage in the input cavity, an input cavity voltage is possibly regenerated by the returning electrons to cause the oscillations. In order to examine the conditions of these oscillations, a simulation of returning electrons was conducted using EGS4[1]. The coefficients and energy distributions of the returning electrons were investigated at a range of beam voltages and with a variety of collector shapes, in order to examine the oscillation conditions thoroughly. 2 OSCILLATION PHENOMENA AND OSCILLATION SOURCE During the high-voltage processing of the 324MHz klystron tube #1, unexpected oscillations were observed from both the output and input cavities when there was no driving input power. These oscillations occurred when the beam voltage was either 63~71kV or larger than 90kV, and had a frequency close to 324MHz. When a magnetic field was applied at the collector region to deflect the electron beam, the oscillationsstopped. This indicates that the source of these oscillations was the collector region. However, as there was no resonance of frequency close to 324MHz corresponding to the dimensions of the collector, the oscillations were assumed to be the result of returning electrons. The collector size was accordingly changed to evaluate its effect on the returning electrons. A collector with a larger radius and longer in length was used in the tube #1A, where the results of the experiment indicated that the oscillations disappeared for the low beam voltage region and started from 95kV. When the length of the collector was increased further in the tube #2, again there were no oscillations at the low beam voltage region, and these did not occur until the beam voltage level was even higher. The shapes of the collectors are shown in Fig.1 and the oscillation regions for the three tubes are summarized in table 1. Figure 1: Collector shapes of tubes #1, #1A, and #2. Table 1: Collector shapes and experiment results of the beam voltage regions of the oscillations. TubeCollector radius(cm)Collector length(cm)Beam voltage regions of the oscillations (kV) #16.5 62.463<V<71,V>90 #1A11.5 92.4 V>95 #211.5 122.4V>104 or largerRadius (cm) 051015 020406080100120140 Length (cm)Tube #1 Tube #1A Tube #23 OSCILLATION STUDY 3.1 Oscillation Mechanism In the klystron, after the electrons of the main beam bombard the surface of the collector, back-scattered electrons are produced due to an inelastic process with atomic electrons, and some of these back-scattered electrons return into the drift tube of the klystron. As these returning electrons are modulated by the gap voltage of the output cavity(Vo), they induce an rf voltage in the input cavity(Vf). In this way, the returning electrons and the main beam of the klystron form a feedback loop(Fig.2) and it’s possible to cause spurious oscillations . In Fig.2, A is the voltage gain induced by the main beam of the klystron; A = Vo/Vd, where Vd is the input cavity voltage modulating the main beam. ββ is the voltage gain induced by the returning electron current; ββ = Vf/Vo. \|ββ\| is proportional to the returning electron current as the onset of the oscillations is in the small-signal linear region. Figure 2: Block diagram of the feedback loop of the klystron due to the returning electrons. 3.2 Simulations of Returning Electrons The EGS4(Electron Gamma Shower) Monte Carlo method was applied to the simulation of back-scattered and returning electrons in a klystron collector with an external magnetic field[2]. The trajectories for the incident electron beam were calculated initially with space-charge forces, relativistic effects, self-field and external magnetic fields. The cathode radius was 4.5cm and the magnetic field on the cathode surface was 130Gauss. The beam radius was 3.5cm at the entrance of collectors and the focusing magnetic field on the Z axis in the collector region is shown in Fig.3. The conditions at the start of the EGS4 Monte Carlo simulation were based on the above calculations for the incident electron positions, and their velocities and energies when bombarding the surface of the collector. The trajectories of the incident electron beam, and the back-scattered and returning electrons are shown in Fig.4 for tubes #1, #1A, and #2. Calculations of the coefficients and distributions of theZ-component of the energy(Ez) normalized by the beam voltage(Eo) for the returning electrons reveled that the coefficients and the peaks of the energy distributions for Ez/Eo were also identical under different beam voltages for a given collector. Simulations of the returning electrons were carried out with a variety of collector shapes. When the size of the collector was enlarged, the amount of returning electrons decreased. The returning coefficients were 0.68%, 0.17%, and 0.14% for the collectors of tubes #1, #1A, and #2, respectively. The energy distributions for Ez/Eo are shown in Fig.5. The shapes of the distributions differ slightly between the different collectors. Figure 3: Magnetic field on the Z axis in the collector region. Figure 4: Trajectories of the incident electron beam, and the back-scattered and returning electrons in the collector of tubes #1(a), #1A(b), and #2(c).A ββVi Vd Vo + Vf+Bz (G) 050100150200250 286529653065316532653365 Z(mm) Figure 5: Energy distribution of the returning electrons of tubes #1, #1A, and #2. 3.3 Oscillation Conditions Returning electrons modulated by the output cavity voltage will induce an rf voltage in the input cavity. If the phase of this induced voltage is the same as the phase of the voltage modulating the main beam, and the amplitude is larger than the latter, then an input cavity voltage is regenerated by the returning electrons, and spurious oscillations will occur. In other words, the oscillation conditions are: \|A/G9Aββ\|>1 (1) arg(A/G9Aββ) =arg(A)+arg(ββ) = 2nπ, n = 0, 1, 2… (2) where the complex variables A and ββ are the voltage gains induced by the main beam and returning electrons respectively. A = \|Vo/Vd\| ∠-θ0 ββ = \|Vf/Vo\| ∠-θ0 ′ where θ0 and θ0′ are the delayed angles corresponding to the DC transit angles, which are a function of frequency and beam voltage. A can be obtained using the klytron simulation code JPNDISK, the one-dimension disk-model code. ββ can be calculated from the rf current of the returning electrons. As the returning electron current is very small, and the onset of the spurious oscillations is in the small- signal-linear region, it is possible to apply the ballistic theory. As the contribution from the output cavity voltage is larger than the others, we just consider the output and input cavities. It is necessary to modify the formula for the rf current in the ballistic theory for the returning electrons, to take into consideration the energy distributions, η(x), of the returning electrons, which is a polynomial function fitted to the distribution shape for each of the collectors, as shown in Fig.5.where x = Ez/Eo, Ib is the total current for the returning electrons, J1(X′) is the Bessel function, and X′ is the bunching parameter for the returning electrons. Fig.6 shows the curve of A(ω)/G9Aββ(ω) for tube #1 as the frequency changes from 322 to 326MHz under beam voltages of 65kV, 70kV, and 75kV. We can see that, between 65 and 70kV, some frequency components exist that satisfy the oscillation conditions, \|A(ω)/G9Aββ(ω)\| >1 and arg[(A(ω)/G9Aββ(ω)] = 2nπ. The oscillation regions of tube #1 obtained from the calculations were 64~70kV and larger than 79kV. For tubes #1A and #2, the calculation indicate that the oscillation regions of beam voltage larger than 100kV and 105kV, respectively, due to decreases in the current for the returning electron. These calculations are close to values obtained for the oscillation regions in the experiments. Figure 6: Curves of A(ω)/G9Aββ(ω) as frequency of 322 ~326MHz under beam voltages of 65kV, 70kV, and 75kV. 4 CONCLUSION A Mechanism to produce returning electrons in a collector was successfully simulated using EGS4. A simple analysis was conducted, which took into account the returning electrons, and was in close agreement with the experiment results. Although reversing electrons are sometimes produced at the output cavity and cause instabilities, those were not analyzed in the present paper, which focused on the oscillations due to returning electrons from the collector. REFERENCES [1] W. R. Nelson, H. Hirayama and D. W. O. Rogers, SLAC-report-265, 1985. /G3E/G15/G40 Z. Fang, S. Fukuda, S. Yamaguchi, and S. Anami, Proceedings of the 25th Linear Accelerator Meeting in Japan, 216-218, July 2000.∫∫− = dxxdxt XJxI Ib rf)()' cos()'()(22 1 ηθω ηRelative No. 00.20.40.60.811.2 00.10.20.30.40.50.60.70.80.911.1 Ez/E0#1 #1A #2 0123 0306090 120 150 180 210 240 2703003300 1 2 3ωω 324MHzarg[A(ω)ββ(ω)] 324MHz324MHz ω\|(A(ω)ββ(ω)\| 65kV 70kV 75kV
arXiv:physics/0008128v1 [physics.acc-ph] 18 Aug 2000SIMULATIONOF ANINTRA-PULSE INTERACTION POINTFEEDBACK FORFUTURE LINEARCOLLIDERS D. Schulte,CERN, 1211Geneva,Switzerland Abstract In future normal-conducting linear colliders, the beams will be delivered in short bursts with a length of the or- der of 100 ns. The pulses will be separated by several ms. In order to maintain high luminosity, feedback is neces- saryonapulse-to-pulsebasis. Inaddition,intra-pulsefe ed- backthatcancorrectbeampositionsandangleswithinone pulse seem technically feasible. The likely performances of different feedback options are simulated for the NLC (NextLinearCollider[1])andCLIC(CompactLinearCol- lider[2]). 1 INTRODUCTION A verticalpositiondisplacementbetweenthebeamcentres at the interaction point (IP) will cause luminosity reduc- tion. Two main sources of beam jitter at the interaction point IP are expected. Firstly, the beam entering the ﬁnal focus system may jitter in angle and position. At the IP, theresultingverticalpositionerror,normalisedtothebe am size, and the resulting angle error, normalised to the beam divergence, are expected to have the same size. Secondly, transverse jitters of the ﬁnal focus magnets, especially of the ﬁnal doublet, will mainly change the position of the beams at the IP, not so much the angle. The jitter at the IP canthusbedescribedby /parenleftbigg/angbracketleft(∆y)2/angbracketright σ2y/parenrightbigg =/parenleftBigg /angbracketleft(∆y′)2/angbracketright σ2 y′/parenrightBigg +/parenleftbigg/angbracketleft(∆ffsy)2/angbracketright σ2y/parenrightbigg . Here,∆yand∆y′aretheoffsetandangleerrorofthebeam at the IP, σyandσy′are beam size and divergence,also at theIP. ∆ffsyisthecontributiontothepositionerrordueto the ﬁnal focus system. If it is large, the effect of the angle at theIP canbeneglected. 2 BEAM-BEAM INTERACTION Whenthe beamscollidewith a verticaloffset,theywill re- ceive a strong kick from the beam-beam interaction. The angle of the outgoing beam can therefore be used to mea- suretherelativepositionsofthebeams. Thedependenceof kick angle and luminosity on the position and initial angle have been simulated with the program G UINEA-PIG[3], varying both parameters. The luminosity Las a frac- tion of the nominal L0, is shown in Fig. 1, as a function on the relative beam position error and beam angle error. The kick angle is shown in Fig. 2 as a function of the off- set. If the beams collide without an offset but with an an-0.50.60.70.80.91 00.511.522.53L/L0 ∆y/σy , ∆y'/σy'NLC, ∆y CLIC, ∆y NLC, ∆y' CLIC, ∆y' Figure 1: The luminosity as a function of the beam offset and angle attheIP.CLICisnotverysensitiveto ∆y′because thever- tical beta-function atthe IP ismuch largerthan the bunchle ngth. 020406080100120140 00.511.522.53θ [µradian] ∆y/σyNLC CLIC Figure2: The kickangle θas a function of the beam offset. IPbeam 2 beam 1BPM kicker Figure 3: View of the feedback system from above. The beams collide with a ﬁxed horizontal angle θc. The BPM measures the verticalpositionofbeam1andthekickercorrectsbeam2acc ord- ingly. gle, their initial angle is roughly preserved in the beam- beam interaction. For comparison: the beam divergenceis σy′≈26µradianfor NLC and σy′≈11.7µradianfor CLIC. 3 POSITIONFEEDBACK MODEL In order to have a fast correction, corrector and beam- position monitor (BPM) need to be located close together. Here, they are located on the same side of the IP at a dis- tance of 1.5 m, see Fig. 3. Thus the correction is not ap-246810121416182022 00.020.040.060.080.10.12∆L [%] g Figure 4: The luminosity loss in NLC (with feedback) for a beam positionerror as a function of the gain g. plied to the measured beam but to the other one. This sig- niﬁcantlyreducesthetimenecessarytotransportthesigna l from the BPM to the kicker. The feedback response time τdisgivenby τd=τp+τk+τpf+τkf+τs (1) Here,τpis the time the BPM electronicsneedsto measure the beamoffsetsandto processthedata, τkisthe response time of the kickerand τsis the transporttime of the signal from BPM to kicker. τpfandτkfare the times of ﬂight from the IP to the BPM and from the kicker to the IP, re- spectively. In the following, a total of τd= 20 nsis as- sumed,halfofwhichisdueto τpf+τkf. Thepulselengths are100 nsinCLICand 266 nsin NLC. The hardware for this feedback has not yet been de- signed. With a solid state ampliﬁer it should be possible to correct an offset of 2σy[4], with an additional stage of tubeampliﬁcationthismayevenbeextendedto 20σy[5]. It is assumed that the feedback changes the beam posi- tionby δyaftereachmeasuredbunchaccordingto δy=gθ σy′σy fora measuredangle θ. Thegainfactor gischosento give optimal performance. The additional crossing angle, that results from the correction is orders of magnitude smaller thanthebeamdivergenceandcanbeneglected. 4 RESULTS OF POSITIONFEEDBACK Here,onlypositionerrorsareconsidered. FirstNLCisdis- cussed. In Fig. 4, the luminosity loss with a beam offset ∆y= 2σyis shown as a functionof the gain g. As can be seen,g= 0.06seemsagoodchoice. Verysmallgainslead to a slow correction,verylarge onesto an over-correction. Both result in a larger luminosityloss. With g= 0.06, the luminosity loss is reduced by a factor 6, compared to the casewithoutfeedback. Forasmalleroffsetof ∆y= 1/8σy aboutthesamefactorisfound. Twomainsourcesofnoisecanleadtoanincreasedlumi- nosity loss with feedback: a bunch-to-bunchpositionjitte r of the incoming beam, and the position resolution of theBPM. Forthechosengain g= 0.06,theadditionallossin- duced by the feedback is very small, comparedto the case without feedback. To estimate the required BPM resolu- tion, simulations are performed with perfect beams and a position error of the BPM of σBPM= 15µmfor a single bunch. The luminosity loss, averaged over 100 cases, is only∆L/L= 0.7×10−3. The limit on the BPM resolu- tion seems therefore not to be very stringent compared to the resolutions that must be achieved in other parts of the machine. For a very large offset of ∆y= 12σy, the luminosity withoutfeedback,isonly 3.5%ofthenominalvalue. Ifthe feedback has the required correction range, it can recover 73%of the full luminosity. For the experiment, this can make the difference between a complete failure and still acceptablerunningconditions. For CLIC, the machine with a centre-of-massenergy of 1 TeVis simulated. At higher energies, Ecm= 3 TeV or Ecm= 5 TeV, a large number of electrons and positrons willbeproducedduringthecollisionofthetwobeams,ina processcalledcoherentpaircreation[6];alreadyat Ecm= 3 TeV, the number of these particles is about 20 %of the number of beam particles. They induce a strong signal in the BPM, and due to their large angle could even hit it. Theirpropertiesneedtobestudiedindetailbeforeonecan suggesta feedbackforthehighenergymachines. In CLIC at Ecm= 1 TeV, the feedback response time is assumed to be the same as in NLC. With the optimum gaing= 0.005, the luminosity loss is reduced by a factor 3. This is not as good as in NLC, since the bunch trains areshorterinCLIC.ABPMresolutionof σBPM= 15µm leads to a luminosity loss of only ∆L/L= 1.2×10−4. This is better than in NLC because of the lower gain and the slightlylargerkickangleforanoffsetof ∆y=σy. 5 INFLUENCE OFANGLES If the beams at the IP have angle jitters, this reduces the luminosity. In addition, the BPM measures the additional angle and the feedback tries to correct a non-existing off- set. The latter problemcan be solved by measuringthe in- coming beam angle error and subtracting it from the value measured by the feedback. Two options are discussed in reference[7],onesuggestedbyM.Breidenbach. Bothhave some difﬁculties and neither correct the angle error, but only its effect on the position feedback. As shown below, this is not sufﬁcient, because the luminosity loss will stay large. If the angle jitter is signiﬁcant, an additional angl e feedbackisneededforeachbeam,asdescribedbelow. 6 ANGLEFEEDBACK MODEL Each angle feedback consists of a BPM and a strip-line kickerwhichareplacedinthebeamdeliverysectionbefore thedetector,seeFig.5. Thisassumesthattheanglejitteri s created before this system, as is to be expected. The BPM has to be at a phase advance of (n+k+1 2)πfrom the IP,IP Kicker BPMnπ(n+k+1/2) π Beam Figure5: Schematic layout of the angle feedback. 051015202530354045 -2-1.5-1-0.500.511.52∆L/L [%] ∆ y'/σy'no feedback uncorr. angle corr. angle angle feedback Figure 6: The total luminosity loss as a function of the initial angle of the measured beam. Thebeam-beam positionseparati on inthe interactionpoint is ∆y= 2σ∗ y. whereanangleerrorattheIPcanbemeasuredasaposition error. ThekickerhastobeclosertotheIP,at nπ,tobeable to transport the signal in the same direction as the beam. Here, the angle at the IP can be corrected by applying a kick. Oneneedslargebeta-functions,attheBPMtohavea goodsignal, and at the kicker to havea smaller divergence and thus correction angle. Possible positions exist in the beam delivery system [8]. The kick angles have to be sig- niﬁcantlylargerthanfortheoffsetfeedback[7],anditmay be difﬁcultto achievethis. This feedback is relatively simple, and uses a constant gainforeachbunch. Theresponsetime τdisgivenbyequa- tion(1). Inthepresentcase τpfisnegative,sincethebeam reachestheBPMbeforetheIP.Withsignaltransmissionat thespeedoflight,onewouldobtain τs+τpf+τkf= 0and consequently τd=τp+τk. Inthefollowing, τd= 15 nsis assumed. 7 RESULTS OFANGLEFEEDBACK The angle feedback is simulated for NLC. The optimum gainisdeterminedinthesamewayasforthepositionfeed- back. Ifonlyangleerrorswerepresent,theluminosityloss would be reduced by a factor 6, as for the position feed- back. TherequiredresolutionfortheBPMdependsonthever- tical beta-function at its position. It must correspond to a resolutionofthebeamanglein theIPof 0.2σy′,toachieve aluminositylossofonly ∆L/L= 10−3forperfectbeams.Finally, the combination of angle and position error is considered. Figure 6 shows the fractional luminosity loss for a constant beam position errorof ∆y= 2σyas a func- tionoftheangleerror. Ifnofeedbackisused,theluminos- ity loss is high. An additional angle error can increase it evenmore. Ifonlyapositionfeedbackisused,whichdoes not correct the angle error of the incoming beam, the lu- minositylossis smallas longasthe angleerrorsare small. If∆y′/σy′becomes comparable to ∆y/σy, the loss is al- most the same as without feedback. If one measures the incoming angle, and subtracts it from the measured value, the situation does not improve very much. If ﬁnally, a po- sition feedback at the IP and an angle feedback for each beamareused,theluminositylossissigniﬁcantlyreduced, independentoftheinitial angleerror. 8 CONCLUSION If the appropriate hardware can be built, the intra-pulse feedback at the interaction point offers a reduction of the luminosity loss due to pulse-to-pulse jitter by a factor of about 6 in NLC and 3 in CLIC. Even in case of a very large offset of 12 times the beam size, more than 70 %of the luminosity is recovered in NLC. Without feedback the luminositywouldbealmostzero. If the anglejitter is signiﬁcant,it is not sufﬁcient to cor- rect the measured kick angle accordingly. To reduce the luminositylossduetotheangleerrors,thedescribedangle feedback is necessary. Whether it is feasible needs to be studied. 9 ACKNOWLEDGEMENTS I would like to thank T. Raubenheimer for inviting me to SLAC, where I did most of the work presented. I am grateful to M. Breidenbach, P. Emma, J. Frisch, G. Haller, T. Raubenheimer and P. Tennenbaum for very helpful dis- cussions. 10 REFERENCES [1] NLCparameters canbe found under URL http://www-project.slac.stanford.edu/lc/local/ AccelPhysics/Accel Physics index.htm . [2] J.-P. Delahaye. The CLIC Study of a Multi-TeV e+e−Lin- ear Collider. CERN/PS99-005 (LP) (1999). [3] D. Schulte. Study of Electromagnetic and Hadronic Back- ground in the Interaction Region of the TESLA Collider. Phdthesis. TESLA-97-08(1996). [4] G. Haller,M. Breidenbach. Privatecommunication. [5] J. Frisch.Private communication. [6] D.Schulte.HighEnergyBeam-BeamEffectsinCLIC. PAC 1999, New York,USA andCERN/PS99-017 (LP) (1999). [7] D. Schulte. Simulation of an Intra-Pulse Interaction Po int Feedback for the NLC. CLIC-Note415 (1999). [8] T. Raubenheimer and P. Tennenbaum. Private communica- tion.
arXiv:physics/0008129v1 [physics.acc-ph] 18 Aug 2000TOWARDSRELIABLEACCELERATIONOF HIGH-ENERGYAND HIGH-INTENSITYELECTRONBEAMS K. Furukawa∗and LinacCommissioningGroup† HighEnergy Accelerator Research Organization(KEK) Oho 1-1,Tsukuba,Ibaraki, 305-0801,Japan Abstract KEK electron linac was upgraded to 8 GeV for the KEK B-Factory (KEKB) project. During the commissioning of the upgraded linac, even continuing SOR ring injec- tions,wehadachievedaprimaryelectronbeamwith10-nC (6.24×1010)perbunchupto3.7-GeVforpositrongenera- tion. ThiscouldbeclassiﬁedasoneofthebrightestS-band linac’s. Since the KEKB rings were completed in December 1998, those 3.5-GeV position and 8-GeV electron beams have been injected with an excellent performance. More- over, we have succeeded in switching among the high- intensity beams for KEKB and beams for two SOR rings with sufﬁcientreproducibility. After the commissioning of the KEKB ring started, we have launched a project to stabilize the intensity and qual- ityofthehigh-currentbeamsfurthermore,andhaveaccom- plishedit investigatingeveryconceivableaspect. 1 INTRODUCTION KEK B-factory (KEKB) project has started in 1994 to studyCP-violationinB-mesondecayswithanasymmetric electron-positron collider. The performance of the exper- iment depends on the integrated luminosity of KEKB and hencethebeaminjectionefﬁciencyfromtheinjectorlinac. Inordertoachievetheefﬁcientfull-energyinjection,the original2.5-GeVelectronlinacwasupgradedupto8GeV, with enforcingaccelerationgradient by a factor of 2.5 and with extending the length of the facility by about 40 %. Becauseofthesitelimit,twolinac’swith1.7-GeVand6.3- GeV were combined using a 180-degree bending magnet system to form a J-shape linac. And the primary electron beam was designed to be 10 nC per bunch to produce3.5- GeV positronwith0.64nC. The upgraded electron/positron linac has been commis- sioned since the end of 1997 even continuingthe injection to Photon Factory (PF). We had overcome many practical difﬁculties, andhad alreadyachievedmost ofthe designed beamparameters[1, 2]. However,topursuethecapabilityofthelinacandKEKB to its utmost limit, we still continueto improvethe quality ∗e-mail: <kazuro.furukawa@kek.jp> †LinacCommissioning Group: N.Akasaka, A.Enomoto,J. Flana gan, H. Fukuma, Y. Funakoshi, K. Furukawa, T. Ieiri, N. Iida, T.Ka mitani, M. Kikuchi, H. Koiso, T. Matsumoto, S. Michizono, T. Nakamur a, Y. Ogawa, S. Ohsawa, K. Oide, Y. Onishi, K. Satoh, M. Suetake a nd T.Suwadaofthebeams. 2 COMMISSIONING Thecommissioningstartedattheendof1997usingtheﬁrst partofthelinacjustbeforethecompletionoftheupgraded linac. In order to carry it a task force called a linac com- missioninggroupwasformed,inwhich7personsfromthe linacand12personsfromtheringperticipated. Thisgroup later became a part of the whole KEKB accelerator com- missioninggroup. The beam was operated at the linac local control room at the beginning. After the completion of the KEKB rings theoperationroomsforthelinacandtheringweremerged with some computer network and video switch prepara- tions. Part of the operation log-book has been recorded electronically to facilitate communication between local engineersandremoteoperators. 3 STABILITY AND RELIABILITY After the commissioning of the KEKB ring started, we have realizedthat it was necessaryto manipulatethe beam delicatelyandcontinuouslyinordertomaintainthequalit y ofthehigh-intensitybeamsforalongtermwithoutdegrad- ing the injection performance. Thus, we have launched a project to stabilize the intensity and quality of the high- currentbeams. 3.1 HighCurrentBeam At the beginningof the commissioningit wasnecessary to make much effort to transport a 10-nC electron beam on to the positron generation target. It was often difﬁcult to keep the beam more than an hour. Otherwise local bumps had to be made to cure the beam instabilities, which was caused by the ﬂuctuationof accelerator equipmentand the transversewake-ﬁelds. Suchdifﬁculties,however,wereresolvedgraduallyafter understanding the sources of the instabilities with carefu l beam studies as surveillance systems were installed for rf systems and other equipment[]. Since the commissioning hadstarted beforethecompletionofthe wholelinac,some part of the accelerator equipment was not operated at the optimum condition. The largest contributions to the in- stabilities came from many parameters in the pre-injector section[3].Thus we had realized that it was important to study the tolerancesofbeamstoeachparameter. Table1showssome ofthoseresults. Table1: Tolerancesofa10-nCbeam Parameter Tolerance Gunhighvoltage ±0.38% Guntiming ±45. ps SHB1(114MHz)phase ±1.1deg. SHB2(571MHz)phase ±1.3deg. Buncherphase ±1.7deg. Buncherpower ±0.47% Sub-booster-Aphase ±3.5deg. Sub-booster-Bphase ±4.0deg. These tolerance values were obtained to keep 90 % of the maximum beam current at the positron production tar- get by changing only one parameter around a good set of parameters. Software to ﬁnd correlation was used in order to acquirethesedata[5]. Foralongtermeachparametermaydriftindependently. If the room temperature changes, most parameters may correlate with it. Thus, while above tolerance values are good reference to consider the beam stability, parameters ofequipmenthaveto bekeptinmuchbetterlimits. In order to stabilize equipment parameters following above guidelines, stabilization software, which will be de - scribedlater,wasimplementedaswellasthehardwareim- provement. Aftersuchchallengingeffort,wehadachievedaprimary electron beam with 10-nC ( 6.24×1010) per bunch up to 3.7-GeV for positron generation, without any loss at the 180-degreebendingsystem. Thiscouldbeclassiﬁedasone ofthe brightestS-bandlinac’s. 3.2 FourBeam Modes It was anticipatedthatit mightdegradethe performanceof the linac to switch beams between four injection modes. After the high-current beam was achieved, we had some- times found that the beam parameters were not optimal. Actually, the beam parameters in four beam modes are quitedifferentasshowninTable 2.The major challenging issues here were reproducibility of the beams in one of fourmodes, reliability of switching andtheswitchingspeedtoimprovetheintegratedluminos- ity. In this area, software to switch beam modes had been developed since the beginning of the commissioning. In order to accomplish above tasks the software was reﬁned especially in the magnetinitialization for the reproducib il- ity and in recovery of the equipment failures for the relia- bility. Itcanbeevenre-conﬁguredeasilyinseveralaspect s byanoperator. Thedetailsaredescribedelsewhere[6]. 13.7-GeV primary electron beam.Table2: Beam Modesofthe Linac KEKB PF PF-AR HER LER Energy 8GeV 3.5GeV 2.5GeV 2.5GeV Particle e−e+e−e− Charge 1.28nC 0.64nC 0.2nC 0.2nC (10nC)1 Repetition 50Hz 50Hz 25 Hz 25Hz Reﬁll Time 1-2min. 5-10min. 3-5min. 3-5min. Interval 1- 2hr. 1-2hr. 24 hr. 2- 4hr. Usingthisenhancedsoftwarethelosstimecausedbythe beammodeswitchingwasmadenegligible,andthebeams became well reproducedover the frequent mode switches. Switching time for the KEKB modes became 90 to 120 seconds, which is acceptable. Thus it is not a major issue at linacanymore. There are several plans for experiments that use high- energy electrons in the linac. An example is the slow positron facility for solid-state and particle physics[7] . Whiletheprioritiesoftheseexperimentsarecurrentlylow , newbeammodesforthemmaybeaddedtotheroutineop- erationifit ispossibletosolvenewswitchingissues. 3.3 BeamFeedbackLoops Even with the efforts on beam stabilization and reliable beam mode switching, it was sometimes necessary to tune the equipment parameters delicately in order to maintain some beam parameters in a long term. Only some experts couldtunethe beamandit tooksometime. Simple feedback loops to limit energy ﬂuctuations of the beams had been installed since the beginning of the commissioning[8]. And the same software was applied to stabilizeequipmentparametersasalreadydescribedabove . Anditwasalsoappliedtostabilizebeamorbits. Morethan 30 feedbackloopshavebeeninstalledandare workingde- pending on the beam modes. The details are described elsewhere[6]. These feedback loops have improved short-term linac stability,andhavecuredlong-termdriftsaswell. 3.4 BeamOptics In orderto reproducethe beam well underdifferentcondi- tions the beam optics along the linac has to be understood well. We have investigatedseveral aspects to ﬁnd discrep- ancybetweendesignandrealoptics. In order to measure the beam emittance well, both the Q-magnet-scan method and wire scanners have been used dependingonthelocations. Theerrorsinenergygaineval- uations along the linac were not small unfortunately. We are tryingto reﬁneit using a gain derivedfromthe rf mea- surement, beam energymeasurement by an analyzer mag- net andlongitudinalwake-ﬁeldestimation.Usingsuchbeaminformationsoftwaresystemswerede- velopedtomatchthebeamopticsattheﬁxedenergy[9]and tore-matchtheopticsafterarf-powerre-conﬁguration[10 ]. Although it does not cover whole linac yet since we have severalmatchingpointsalongthelinac,theyareuseddaily . The effect of the transverse wake-ﬁeld is not small es- pecially with high-intensity beams, and it degrades the beamemittanceandthe stability. Evaluationandreduction of the wake-ﬁeld effects are tried with some success[11]. Quadrupole wake-ﬁeld effects were also observed for the ﬁrst time[12]. 4 OPERATIONSTATISTICS With the help of above improvement, the linac operation has become fairly reliable. The total operation time in FY 1999 was 7296 hours, which much increased because of the full KEKB operation[1]. The availability of the linac forinjectionwas99.0%,whichhavebeenmuchimproved. Theaverageintensityofthepositroninspring2000was 0.62nC,whichisjust lessthanthesafetylimitatthebeam transportline. 5 MORECHALLENGES 5.1 Dischargein AcceleratorStructures The discharge in the accelerator structures at sections A1 (buncher and the ﬁrst normal structure) and 21 (positron generator) became severe in March 2000, where beam charge (and loss) is high and is surrounded by solenoid coils. It was found that the discharge frequently occurred nearthetrailingedgeofthe rfpulses. The wave guides at these sections, hence, were re- arranged to shorten the pulse width and the rf-power was optimizedtotheimprovedbeamswithlowervoltage. Then suchdischargedecreasedtotherate lessthanoncea day. Since it is important to understand the phenomena deeply, a test stand for such stations was built for the in- vestigation of the discharge phenomena as well as for the conditioningofacceleratorstructures. 5.2 Two BunchAcceleration Inordertodoublethepositronbeamcharge,itisconsidered to have two bunches in a linac rf pulse. Because of the rf synchronization scheme between the linac and the ring thosebuncheshaveto beseparatedby96nsat minimum2. A preliminary study was made on this two-bunch scheme, and got promising results on the energy compen- sation of the second bunch with a carefull rf-pulse timing control. Energy difference was estimated to be 2.5 % for the 8-nCbeamcomparingthelongitudinalwake-ﬁeldwith the one for a low-intensity beam. Devices for this scheme is underpreparation. 2275th bucket in the linac and 49th bucket in the ring6 CONCLUSIONS In the commissioning of the KEKB injector linac we have overcome challenging issues and have accomplished the stabilizationprojectinvestigatingeveryconceivableas pect. The linac is providing fairly stable beams with very high availability. During normal operation operators rarely change the beamparameters. Instead,softwareforbeam-modeswitch- ingandfeedbackloopstakescareofthem. Sincethecharge limit at the beam transport line induced by the safety rea- sons will be removed soon, the performance of the linac maybemoreenhanced. During this improvement, we had valuable experiences on tolerancestudiesand stabilizationtechniqueof the tim - ing and rf systems especially at the buncher section. We alsogainedaknowledgeonthephysicalphenomenaofthe beamsparticularlyofanemittancegrowth. Theyareindis- pensableforthe designandconstructionofthe nextgener- ation accelerators such as a linear collider, an FEL and an injectorforsuper-high-luminositymachines. 7 REFERENCES [1] A. Enomoto et al., “Performance of the KEK 8-GeV Elec- tron Linac”, Proc. EPAC2000, Vienna, Austria, to be pub- lished. [2] Y. Ogawa et al., “Commissioning Status of the KEKB Linac”, Proc.PAC’99, New York,U.S.A.,1999. Y. Ogawa et al., “Commissioning of the KEKB Linac”, Proc. Linac’98, Chicago, U.S.A.,1998. [3] S. Ohsawa et al., “Pre-injector of the KEKB Linac”, Proc. EPAC2000, Vienna, Austria,tobe published. [4] H. Katagiri et al., “RF Monitoring System in the Injector Linac”, Proc.ICALEPCS’99,Trieste,Italy, 1999, p.69. [5] K. Furukawa et al., “Accelerator Controls in KEKB Linac Commissioning”, Proc.ICALEPCS’99,Trieste,Italy,1999, p.98. [6] K. Furukawa et al., “Beam Switching and Beam Feedback Systems at KEKBLinac”, these proceedings. [7] T. Shidara et al., “The KEK-PF Slow-Positron Facility at a New Site”,Proc.LINAC’98, Chicago, U.S.A.,1998. [8] K. Furukawa et al., “Energy Feedback Systems at KEKB Injector Linac”, Proc. ICALEPCS’99, Trieste, Italy, 1999, p.248. [9] N. Iida et al., “Recent Progress of Wire Scanner systems for the KEKB Injector LINAC and Beam Transport Lines”, Proc. EPAC2000, Vienna,Austria, tobe published. [10] T. Kamitani et al., “Optics Correction for Klystron Switch- ing atthe KEKBInjector Linac”, Proc.EPAC2000, Vienna, Austria, tobe published. [11] S. H. Wang et al., “Simulations of Wake Effects on a High- Current Electron Beam at the KEKB Injector Linac”, KEK Report 2000-4. [12] Y. Ogawa, “Quadrupole Wake-Field Effects in the KEKB Linac”, Proc.EPAC2000, Vienna, Austria,tobe published.
A LOW-CHARGE DEMONSTRATION OF ELECTRON PULSE COMPRESSION FOR THE CLIC RF POWER SOURCE R. Corsini, A. Ferrari, J.P. Potier, L. Rinolfi, T. Risselada, P. Royer, CERN, Geneva, Switzerland Abstract The CLIC (Compact Linear Collider) RF power source is based on a new scheme of electron pulse compressionand bunch frequency multiplication using injection bytransverse RF deflectors into an isochronous ring. In thispaper, we describe the modifications needed in thepresent LEP Pre-Injector (LPI) complex at CERN inorder to perform a low-charge test of the scheme. Thedesign of the injector (including the new thermionic gun),of the modified linac, of the matched injection line, andof the isochronous ring lattice, are presented. The resultsof preliminary isochronicity measurements made on thepresent installation are also discussed. 1 INTRODUCTION The time structure of the CLIC drive beam is obtained by the combination of electron bunch trains in rings usingRF deflectors [1]. The next CLIC Test Facility (CTF3) atCERN will be built in order to demonstrate the schemeand to provide a 30 GHz RF source with the nominalparameters [2]. CTF3 will be installed in the area of thepresent LPI complex. As a preliminary stage, the existinginstallation will be modified in order to perform a test ofthe combination scheme at low charge. The layout of theLPI complex after the modifications is shown in Figure 1.The first part of the LEP Injector Linac (LIL) will bedismantled and shielding blocks will be added, creatingan independent area that can be used for component testsand later for the commissioning of the new CTF3injector. The LIL bunching system will be moveddownstream and a new gun [3] will be installed. Eight ofthe sixteen accelerating structures of LIL, the extractionlines to the PS complex, and the positron injection line between LIL and EPA will be removed. 2 THE INJECTOR AND THE LINAC The new triode gun has a design voltage of 90 kV. Its control grid can be pulsed in order to provide a train of upto seven pulses with a repetition rate of 50 Hz. Thenominal pulse length is 6.6 ns FWHM, and the pulses arespaced by 420 ns, corresponding to the Electron PositronAccumulator (EPA) circumference. The nominal peakcurrent of the gun is 1 A. The present bunching system ofLIL, composed of a single-cell pre-buncher and astanding wave buncher, will be used. It will be poweredby a 30 MW klystron and will provide a 3 GHz bunchedbeam at 4 MeV, with a normalised rms emittance of50 /G53 mm mrad. Each pulse will then be composed of 20 bunches each with a charge of 0.1 nC and a length ofabout 10 ps FWHM. These values are extrapolated frommeasurements made on the present installation [4]. The pulse train will be accelerated to a maximum energy of 380 MeV, using eight travelling waveaccelerating structures, powered in groups of four by two40 MW klystrons. The beam parameters have been chosen to minimise the energy spread generated by beam-loading in the LILstructures, while still keeping a charge per bunch which ishigh enough to give a good resolution for the streakcamera measurements of the beam time structure. Thebeam-loading parameter in LIL is 0.2 MeV/nC perstructure. The resulting energy spread within each pulseis about 3 MeV. An additional energy difference ofroughly 3 MeV will occur between the first two pulses,i.e., before the steady state is reached. CTF IICTF3 Injector LIL 10 mGun and bunching systemInjection line EPA Figure 1: Layout of the LPI complex after the modifications planned for the CTF3 preliminary phase.The total energy spread of about 6 MeV is within the EPA acceptance ( /G72 1% total). It can nevertheless be reduced by a factor two by delayed RF filling of thestructure or by dumping two out of the seven pulses ofthe train. For the frequency multiplication test, one needsfive pulses maximum. The linac optics will be adapted to the new layout. Two quadrupole triplets, one located after the bunching systemand the other between the first two accelerationstructures, will provide matching to the modified LILFODO lattice. The last two LIL structures will beremoved and replaced by a matching section to theinjection line. The design of the linac optics and of thematching sections is presently in progress. Apart from thenew gun, no new equipment is needed and only a re-arrangement of the existing components suffices. 3 THE INJECTION LINE The present line from LIL to EPA is achromatic in both planes. In addition to the main horizontal bendingmagnets, the line contains two small vertical dipoles,since the levels of LIL and EPA differ by 15 cm in orderto allow injection from the inside of the ring. However,the line must also be made isochronous, in order topreserve the bunch length from the linac to the ring. Thisis essential for the combination process, for which shortbunches (< 20 ps FWHM) are needed. Furthermore, theline has to be re-matched to the new ring lattice. A newoptics of the line has been found which satisfies therequirements of the CTF3 preliminary phase. If R is the 6x6 dimensional transfer matrix of the injection line, the achromatic condition implies that thematrix elements R 16 , R26 , R36 and R46 are equal to zero. To preserve the bunch length, the elements R51 , R52 , R53 , R54 and R56 must be small. In particular this last condition is not satisfied in the present line, for which R56 = -1.05 m (against a requirement of \| R56\| < 0.1 m). R56 will be controlled in the new arrangement by three high-gradientquadrupoles, placed between the bending magnets andthe septa. The eight other coefficients, as well as the two transverse lattice functions /G45 X and /G45Y, can be controlled by five quadrupoles in the straight section of the injectionline (the gradients and the positions of these fivequadrupoles were chosen independently in order to fulfilall the requirements). The dispersion and the /G45-functions of the new injection line are displayed in Figure 2. The Twiss parameters atthe entrance of the line are the ones required for thetransverse matching at the injection point in the ring ( /G45 X = 31 m, /G44X = -2, /G45Y = 5 m, /G44Y = -1). The beam envelope is within the acceptance of the line. The R51 , R52 , R53 , R54 and R56 coefficients remain small enough for no bunch lengthening to occur.0.0 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. s (m)0.0500.1000. β(m) -6.-5.-4.-3.-2.-1.0.01. D(m) βxβy DxDy Figure 2: Optics functions of the new injection line. Two quadrupoles must be added to the present layout, and some of the existing quadrupoles must be moved. Allquadrupoles in the line will be fed with independentpower supplies. However, the geometry of the line ispreserved, such that no major hardware modifications areneeded. The quadrupoles and power supplies from thedismantled beam lines can be reused in the new injectionline. 4 THE RING LATTICE The lattice of the EPA ring will be modified to become isochronous and thus to preserve the bunch length andspacing during the combination process (three to fiveturns). A first isochronicity test has been performed in the present ring [5]. The isochronous lattice was obtained bychanging the strength of the quadrupoles without makingany hardware modification. Measurements of the beamtime structure were made using a streak camera. Thebunch length increased rapidly over a few turns in thenormal case, while no significant bunch lengthening wasobserved over 50 turns in the isochronous case. Anevaluation of the momentum compaction has beenobtained by measuring the bunch spacing, yielding valuesof /G44 as small as 2.3 /G75 10 -4, close to the goal of the future CTF3 (\| /G44 \| /G64 /G72 10-4). The dispersion in the modified isochronous lattice is shown in Figure 3, together with that of the normal latticeused for LEP operation. In the isochronous case, thedispersion varies strongly in the straight sections. Theinjection line was therefore badly matched to the ring,and the larger beam envelopes gave rise to beam losseson the first few turns. The situation will be worse whenthe RF deflectors are installed, since they will reduce theavailable aperture. To solve this problem, in the new isochronous lattice the dispersion is made zero in the straight sections [6].The dispersion and the /G45-functions are shown in Figure 4. This needs the displacement of four quadrupoles, and thedecoupling of one of the existing families.-10. 0.0 10. 20. 30. 40. 50. 60. 70. s (m)-4.-3.-2.-1.0.01.2.3.4.Dx(m)isochronous optics standard optics Figure 3: Dispersion in the present EPA ring, for different optics (half ring shown). -10. 0.0 10. 20. 30. 40. 50. 60. 70. s (m)0.010.20.30.40.50.60.70.80.β(m) -12.-10.-8.-6.-4.-2.0.02.4. Dx(m) βxβyDx Figure 4: Optics functions of the new EPA ring lattice (half ring shown). 5 THE COMBINATION TEST Two transverse RF deflectors will replace the present fast injection kickers. They will create a time-dependentclosed bump of the reference orbit, allowing interleavingof three to five bunch trains (see Figure 5). Thecombination test requires C = n ( /G4F /G72 /G4F/N) where n is an integer, C is the ring circumference, N is the combination factor and /G4F is the RF wavelength in the deflectors and the linac. Combination factors of 3, 4 and 5 will be testedin the preliminary phase of CTF3. One arc of the ring willbe displaced by 7.5 mm, in order to fulfil the conditionwith N = 4. The other combination factors can be tested by changing the RF frequency by /G72 150 kHz. The bandwidth of the klystrons is wide enough to cover thisrange. The accelerating structures will be tuned inoperation to follow the change of the RF frequency, byvarying their temperature /G72 3 /G71C. The RF deflectors are travelling wave iris-loaded structures, for which theresonant mode is a deflecting mode with a /G53/2 phase advance per cell and a negative group velocity. Theyhave been used in the past to measure the bunch length inLIL [7]. Each one is 27 cm long, with 6 cells and an irisdiameter of 2.3 cm.local orbitsinjection line Transverse deflector field /G4F = 10 cm deflector deflectorseptum /G4F/5 (2 cm)1st turn 5th turn Fig. 5: Principle of injection by transverse RF deflectors, for a combination factor of 5. After injection, thecirculating bunches follow orbits that are inside theseptum. After five turns the beam is extracted. Each will be powered by one of the existing 30 MW klystrons and a power of about 4 MW each is needed forthe nominal deflecting angle of 2 mrad at 380 MeV.While one of the fast injection kickers will be removed,the other will be displaced but kept in the ring to allowconventional single-turn injection. It will be used duringcommissioning, to check the ring optics prior to theinstallation of the RF deflectors. Also, one of the positroninjection kickers, located at the opposite side of the ring,will remain in place and will be used to extract thecirculating beam to a dump. REFERENCES [1] H. Braun and 16 co-authors, “The CLIC RF Power Source-A Novel Scheme of Two Beam Acceleration for e/G72 Linear Colliders ”, CERN 99-06 (1999). [2] CLIC Study Team, “Proposals for Future CLIC Studies and a New CLIC Test Facility (CTF3) ”, CERN/PS 99-047 (LP) and CLIC Note 402 (1999). [3] H. Braun, L. Rinolfi, "Technical description for the CLIO gun", CTF3 Note 2000-07, April 2000. [4] R. Corsini, A. Ferrari, L. Rinolfi, T. Risselada, P. Royer, F. Tecker, "New measurements of the LILbunch length and lattice parameters", PS/LP Note2000-02 and CTF3 Note 2000-13, July 2000. [5] R. Corsini, J.P. Potier, L. Rinolfi, T. Risselada, “Isochronous Optics and Related Measurements in EPA ”, Proceedings of the 7 th European Particle Accelerators Conference, Wien, June 2000 [6] T. Risselada, to be published.[7] M.A. Tordeux, "Etude des longueurs de paquets du LIL à 4 MeV", PS/LP Note 93-14 (MD).
arXiv:physics/0008131v1 [physics.acc-ph] 18 Aug 2000BEAM DYNAMICS SIMULATIONFOR THECTF3 DRIVE-BEAM ACCELERATOR D. Schulte,CERN, 1211Geneva,Switzerland Abstract A new CLIC Test Facility (CTF3) at CERN will serve to study the drive beam generation for the Compact Linear Collider (CLIC). CTF3 has to accelerate a 3.5 Aelectron beamin almostfully-loadedstructures. Thepulse contains more than 2000 bunches, one in every second RF bucket, and has a length of more than one µs. Different options for the lattice of the drive-beam accelerator are presented , basedonFODO-cellsandtripletsaswellassolenoids. The transverse stability is simulated, including the effects o f beamjitter,alignmentandbeam-basedcorrection. 1 INTRODUCTION In the nominal stage of CTF3, the drive-beam accelerator will haveeightklystrons,eachfeedingtwo1 m-longstruc- tures. The structures are almost fully loaded, transferrin g more than 90 %of their input power to the beam. The av- erageenergygainperstructureis ∆E≈9.1 MeV[1]. The beampulseconsistsoftenshorttrainsofabout210bunches each. The ﬁrst train ﬁlls odd buckets, the immediatelyfol- lowing second train ﬁlls even buckets; this pattern is then repeated. An RF-deﬂector at half the linac frequency is used to separate the trains after acceleration [2]. The ini- tial beam energy is E0≈26 MeV, the ﬁnal beam energy Ef≈170 MeV , the bunch charge q= 2.33 nC, its length σx≈1.5 mm[3]andthetransversenormalisedemittances areǫ∗ x=ǫ∗ y= 100 µm. 2 STRUCTURE MODEL The simulations below have been performed using PLACET[4]. Thelong-rangetransversewakeﬁeldisrepre- sented by the lowest two dipole modesof each cell. These havebeencalculatedneglectingthecouplingbetweencells and the effect of the damping waveguides [5]. The damp- ing of the lowest dipole mode has been found [6] to be in the range Q= 11toQ= 19for perfectloads. In the sim- ulation,themodesareconﬁnedtotheircells,whichallows one to take into account the angle of the beam trajectory in the structure. The lossfactorsused inthe simulationare 50 %larger than in [5]. This is to account for the effect of higher-order modes. Also, the damping is conservative in the simulation; Q= 30andQ= 400are used for the lowest and the second dipole band. The short-range lon- gitudinal [5] and transverse [7] wakeﬁelds have been cal- culated and are included in the simulation. Almost perfect compensation of the long-range longitudinal wakeﬁelds is predicted[1].Quadrupolewakeﬁeldsmaybeimportantandhavebeen implemented in PLACET. The corresponding modes have not yet been calculated but need to be includedin the sim- ulationassoonastheyareavailable. 3 LATTICES Three different lattices were investigated. One consists o f simple FODO-cells, with one structure between each pair ofquadrupoles. Theothertwolatticesarebasedontriplets . In one case (called T1 below), one structure was placed between two triplets; in the other case two structures(T2). The weaker triplet lattice (T1) and the FODO lattice are roughlycomparablein length and cost, whereasthe strong tripletlattice (T2)is signiﬁcantlylongerandmorecostly . In the FODO lattice, the phase advance is µ= 102◦ per cell, with a quadrupole spacing of 2 m. In T2 one has µx= 97◦andµy= 110◦,andadistanceof 4.2 mbetween triplets. The sum of the integrated strengths of the outer two magnets is slightly larger than that of the inner one. Withthisarrangement,thehorizontalandtheverticalbeta - functions are equal in the accelerating structures, and the energyacceptanceofthelatticeismarkedlyimproved. For T1 the phase advancesare µx= 84◦andµy= 108◦for a triplet spacing of 3 m. The transverse acceptance is 4.2σ fortheFODO lattice, 4.9σforT2and 5.8σforT1. Since the beams have to be compressed after the accel- eration,the RF-phase cannotbe used to optimisethe beam transport. It must be chosen to achieve the required com- pression and to limit the energyspread of the beam before the combinerringto the latter’senergyacceptance. An RF phase ΦRF= 6◦isusedinthefollowing. 4 TRANSVERSE BEAM JITTER No estimate of the transverse jitter of the incoming beam exists. Therefore,onlythejitterampliﬁcationiscalcula ted. Inthesimulation,eachbunchiscutintoslices; thebeamis set to an offset of ∆xand tracked through the linac. The normalisedampliﬁcationfactor Afora slice isdeﬁnedas A=σx,0 ∆x/radicalBigg/parenleftbiggxf σx,f/parenrightbigg2 +/parenleftbiggx′ f σx′,f/parenrightbigg2 Here, σx,0andσx,fare initial and ﬁnal beam size, σx′,0 andσx′,fare initial and ﬁnal beam divergence, ∆xis the initial beamoffsetand xfandx′ faretheﬁnal positionand angle of the centre of the slice. For a slice with nominal energyand without wakeﬁeld effects, one has A= 1. The-3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) -3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) Figure 1: The ampliﬁaction factor of the beam at the end of the drive-beam accelerator, using the FODO lattice, without a r amp (left) and with a ramp (right). A mono-energetic beam withou t wakeﬁelds should stayon the innermost circle. -3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) -3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) Figure2: Thebeamattheendofthedrive-beam acceleratorina tripletlattice. Ontheleft-handsideT2,ontheright-hand sideT1. maximum ampliﬁcation factor ˆAis the maximum over all slices. The left-hand side of Fig. 1 shows the bunches at the end of the accelerator using the FODO lattice. Differ- ent quadrupole strengths were used to ﬁnd the best phase advance. Somebunchesarekickedsigniﬁcantly;themaxi- mum ampliﬁcation is ˆA= 3.7. Without knowledgeof the acceptancedownstreamandthesize ofthebeamjitter,it is not possible to decide whether the ampliﬁcation is accept- able. Within the linac, evena largejitter of ∆x=σxdoes notleadto beamloss. The ﬁrst few bunches in each train are kicked particu-345678910111213 051015202530acceptance [ σ] s [m]FODO T2 T1 Figure 3: The minimum acceptance along the linac, with a gra- dient error. For each lattice, 100 machines have been simula ted and their minimum acceptance at each point isplotted. larly hard. This can be preventedby addingcharge ramps. Towardstheendofatrainthatﬁllsevenbuckets,thebunch chargeisslowly decreasedfromthe nominalbunchcharge to zero. At the same time one increases the charge in the odd buckets from zero to nominal, to keep the beam cur- rent constant. Thus the two consecutive trains practically overlap. On the right-handside of Fig. 1, one can see that inthiscaseallbunchesarewellconﬁned,withamaximum ampliﬁcationof ˆA= 2. In the triplet lattices, the horizontal plane has a larger jitterampliﬁcationthantheverticalone. Buteventhehori - zontal ampliﬁcations are signiﬁcantly smaller than in the FODO lattice. Figure 2 shows the examples of a pulse without charge ramps, the ampliﬁcation factors being 1.8 (T2) and 1.5 (T1). With charge ramps, they are reduced to 1.5 and 1.3. If the beam jitters signiﬁcantly, the triplet latticesaremarkedlybetterthanthe FODOlattice. 5 BEAM-BASED ALIGNMENT To keep operation as simple as possible, only one-to- one correction is considered. All elements are assumed to be scattered around a straight line following a normal distribution with σ= 200 µm. In the FODO lattice, corrector dipoles are located after each quadrupole and beam position monitors (BPM) are placed in front of each quadrupole. In the triplet lattices, the correctordipoles are positioned after the triplets and the BPMs are positioned in front and after the triplets. The correctors are used to bring the average beam position to zero in the BPMs. For eachcase,100differentmachinesaresimulated. Thesmall growthsofabout 0.5 %arealmostthesameforall lattices. 6 GRADIENTAND PHASEERRORS The limit on the variation of the bunch energy is 1 %[8], much smaller thanthe single-bunchenergyspread. In nor- mal operation, the additional dispersive effects are there - fore small. Static local energy errors are of more concern andarediscussedhere.0123456 -20-15-10-505101520min acceptance [ σ] ∆G/G [%]FODO T2 T1 T2, weak Figure4: Theminimumacceptance ofthelinac,asafunctionof the RF-gradient error;20 machines were simulatedfor eachc ase. The initial andﬁnal beamenergycan be well measured, and from this the average gradientcan be derived. A local variation of the gradient is more difﬁcult to detect. It will leadtoaquadrupolestrengththatisnotadaptedtothebeam energy. Theworstcaseistoolowagradientintheﬁrsttwo structures, which are fed by one klystron. In the simula- tion,100differentmachineswithagradientintheﬁrst two structuresthatistoolowby 10 %(20 %)arecorrectedwith the beam. The emittance growth found after correction is 1 %(5 %)intheFODOlatticeand 0.5 %(2 %)inT1,which seemstobesufﬁcientlylow. InT2,thevaluefora 10 %er- ror is small, 2 %, but for an error of 20 %it starts to be large: 14 %. The transverse acceptanceis reducedto 3.8σ (3.2σ)intheFODOlattice, 4.7σ(3.8σ)inT2andto 5.2σ (5.1σ)inT1. Figure3showstheacceptanceforagradient errorof 20 %. FortheFODO lattice, andtoa lesser degree also for T2, one starts to worry about beam losses. How- ever, an error of 10 %seems acceptable with all lattices. To be able to use the FODO lattice or T2, it necessary to measurethelocalgradienttobetterthan 10 %,tobeableto correct the lattice accordingly. For T1, a precisionof 20 % is sufﬁcient. The RF power produced by a klystron has a systematic phase variation duringthe pulse. One hopesto correctthis effect globally, but local variations will remain. To esti- matetheirimportance,alinearchangeinphaseof 20◦over the pulse is assumed for the two structures driven by one klystron. The next pair hasan exactly oppositephase vari- ation. The resulting bunch-to-bunchenergy spread is 2 %, full width, which is not acceptable in the combiner ring; so a bettercompensationwouldbeneeded. Incontrast,the emittance growth seems acceptable with about 1.5 %av- eraged over 100 machines for all lattices; the acceptance is hardly decreased. This phase variation does not cause signiﬁcanttransverseeffects. 7 ENERGY ACCEPTANCE During commissioning of the linac, large energy errors may occur. To study the sensitivity to this, 20 machines were simulated for each lattice in the following way: the linac is corrected with a nominal beam; then the incom--3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) -3-2-10123 -3-2-10123(x’f/σx’,f) (σx,0/∆x) (xf/σx,f) (σx,0/∆x) Figure 5: The ampliﬁcation of beam jitter with and without the charge rampfor the latticeT2including the injector soleno id. ing beam is assumed to be accelerated at a different gra- dient. Figure 4 shows the minimum transverse acceptance of the three lattices as a function of the RF-gradient error. The ﬁnalenergyerroris about 1.4timeslargerthanthat of the RF-gradient, since the beam loading does not change. TheFODOlatticeandT1haveacomparableenergyaccep- tance, whereas that of T2 is slightly smaller. By reducing thefocalstrength,theenergyacceptancecanbefurtherim- provedatthecostofhighertransversewakeﬁeldeffects. By reducingthestrengthofT2to µx= 83◦andµy= 94◦,the energy acceptance becomes larger than that of the FODO lattice. The maximum ampliﬁcation of an initial jitter in- creases from 1.8 to 2.6 but is still smaller than the factor 3.7intheFODOlattice. Witheachlattice,thelinacenergy acceptanceislargelysufﬁcientduringnormaloperation. 8 SOLENOID PLACET has been modiﬁed to also simulate the effects of solenoids with acceleration. This allows to include the last two structures of the injector which are placed inside a solenoid. Two triplets are used to match the end of the solenoid to the T2 version of the drive-beam accelerator. The ﬁeld of the solenoid is 0.2 Tand its length is chosen such that a horizontaljitter of the nominal beam leads to a ﬁnal horizontal offset. The end ﬁelds of the solenoid are modelled as thin lenses. Neither space charge nor the dif- ferenceoftheparticlevelocitiesfromthespeedoflightar e taken into account, but the wakeﬁelds are considered, in contrasttocalculationsdonewithPARMELA [9]. Figure 5 shows the ampliﬁcation factor. While there is some contribution from the structures in the solenoid, the overallampliﬁcationseemsstill acceptable.9 CONCLUSION The simulations show that the lattices considered here can be acceptable; the best is the strong triplet lattice T1. The tripletlatticeT2seemstobeabetterchoicethantheFODO lattice. The FODO lattice is less expensivethan T2,which is much cheaper than T1. To ﬁnd the best compromise, more information is needed. For the FODO lattice the ramps have to be studied in more detail. For all lattices, the matching from the injector to the linac and from the linac tothecombinerringneedstobeunderstood. 10 REFERENCES [1] I. Syratchev. Private communication. [2] D.Schulte.TheDrive-BeamAcceleratorofCLIC. Proceed- ings of Linac 1998, Chicago, USA andCERN/PS 98-042 (LP)(1998). [3] L.Rinolﬁ.Private communication. [4] D.Schulte. PLACET:A ProgramtoSimulateDriveBeams. Proceeding of EPAC 2000, Wien, Austria andCERN-PS- 2000-028 (AE) (2000). [5] L.Thorndahl.In: TheCLICRFPowerSource. CERN99-06 (1999). [6] E. Jensen, A. Millich and L. Thorndahl. Private communi- cation. [7] A. Millich.Private communication. [8] R. Corsini.Private communication. [9] F.Zhou. Tobe published as aCTF3-Note.
arXiv:physics/0008132v1 [physics.acc-ph] 18 Aug 2000BEAMLOADING COMPENSATIONINTHEMAINLINACOF CLIC D.Schulteand I. Syratchev,CERN, 1211Geneva,Switzerland Abstract Compensationofmulti-bunchbeamloadingisofgreatim- portance in the main linac of the Compact Linear Collider (CLIC). The bunch-to-bunch energy variation has to stay below 1 part in 103. In CLIC, the RF power is obtained by decelerating a drive beam which is formed by merging a number of short bunch trains. A promising scheme for tackling beam loading in the main linac is based on vary- ing the lengths of the bunch trains in the drive beam. The schemeanditsexpectedperformancearepresented. 1 INTRODUCTION Multi-bunch beam loading is a strong effect in the main linac ofCLIC. It needstobe compensatedwith helpofthe RF to avoid extreme variations of the beam energy along the pulse. Several approaches to solve the problem exist. All of them are based on manipulations of the drive beam whichgeneratesthe RF power. The ﬁrst possibility is to reduce the bunch charge in the ﬁrst part of the drive-beam pulse [1]. In this scheme, the ﬁrst bunch has about 70 %of the nominal charge. The chargeisthenslowlyincreasedfromonebunchtothe next until it reaches the nominal value. This charge ramp cre- ates a ramp in the RF voltage. By carefully shaping the chargeramp,onecanachievebeam-loadingcompensation. In principle,thiscompensationcan beperfect. However,it may be very difﬁcult to control the bunch charge with the requiredprecisionso as to achievethe requiredcompensa- tion of the gradient variation ∆G/G 0≤10−3(G0is the nominalgradient). Anothermethodisdescribedinreference[2]. Itachieves ∆G/G 0≈2×10−3. It requires additional hardware and may compromise the stability of the drive beam in the de- celerator. In a third option, presented in this paper, one creates a ramp in the current of the drive-beampulse comparableto the ﬁrst option. But instead of varying the bunch charge, one varies the number of bunches per unit length of the pulse. This can be achieved by modifying the drive beam inthedrive-beaminjector[3]. Tounderstandthis,itisnec - essary to understand the drive-beam generation, which is describedbelow. 2 THE DRIVE-BEAM GENERATION Thedrivebeamisproducedandacceleratedat a frequency of about 937 MHz . In the injector of the drive-beamlinac, one has a sub-harmonicbuncherwhich can be switched toFigure 1: Schematic layout of the delay loop after the drive- beam accelerator in CLIC. The two RF-deﬂectors are shown as rectangles. ﬁll either odd or even buckets. In the drive-beam accel- erator, the beam then consists of short trains of bunches that ﬁll every second bucket. The ﬁrst train ﬁlls the odd buckets, the immediately following second train ﬁlls the even ones, and this pattern is then repeated [4]. The cur- rent in the drive-beam accelerator, and consequently the beam loading, therefore remains constant. After acceler- ation, the trains are separated using an RF-deﬂector run- ning at half the linac frequency. The ﬁrst train is deﬂected into a delayloopandmergedwith thesecondonein a sec- ond RF-deﬂector, see Fig. 1. The newly created pulses are separated by gaps that allow conventional deﬂectors to be switchedonandoff. They are sent into two combiner rings [5]. These rings have a circumference equal to the distance between two pulses plus (or minus) a quarter wavelength. This allows to merge four pulses to form a single one, using an RF- deﬂector. The new pulse has four times as many bunches as each of the initial ones, with a distance between the bunchesthatisfourtimessmaller. Thebunchescomprising the four pulses have been inter-leaved by this operationso thattheﬁrstbunchofeachoftheinitialpulsesisoneofthe ﬁrst fouroftheﬁnal pulse. The ﬁrst ring is followed by a second one, four times larger, which merges four of the pulses of the ﬁrst ring. At the end, the bunch-to-bunchdistance has been reduced from the initial 64 cmto only 2 cm. In the following,each 64 cmlong section of the beam pulse is called a bin and it contains32bunches. Thebunchesthatwereintheﬁrst bin ofeachinitialtrainareintheﬁrstbinoftheﬁnalpulse. The bunches that were in the second bin of an initial pulse are in thesecondbinoftheﬁnal pulse,andso on.train binNominal Switching Delayed Switchingbefore first RF−deflector after second RF−deflector before first RF−deflector after second RF−deflector Figure 2: The scheme of delayed switching. In this example, each train contains 5 bins (an arbitray number chosen for bet ter visibility). In the upper case the phase is switched at the no minal time, creating a rectangular pulse. In the lower case the pha se switchis delayed tocreate aramp. 3 DELAYED SWITCHING In orderto create a currentrampin the ﬁnal pulse, the ﬁrst few bins of this pulse must contain a smaller number of bunches than nominal. This in turn requires that some of the pulses after the delay loop have less than the nomi- nal two bunches per bin. This can be achieved by delay- ing the switching of the sub-harmonicbuncher. The effect of the nominal switching is illustrated in the upper part of Fig. 2. The two trains before the delay loop and the pulse after this loop are shown. In the delay loop, the bunches of the ﬁrst train are delayed by one nominal train length. In the lower part of the ﬁgure, the sub-harmonic buncher is switched slightly later. The bunch that, in the nominal scheme,wouldhavebeentheﬁrstoneofthesecondtrainis thereforeappendedtotheﬁrsttrain. Thesecondtrainstart s onebunchlater thannominal. Asa consequence,thepulse afterthedelayloopcontainsonlyonebunchintheﬁrstbin. The last bunch of the ﬁrst train is appenended after at the endofthepulse. Theadditionaltailofthepulsecreatesnoprobleminthe combiner ring, as long as the distance to the ﬁrst bin of thenextpulseislongenoughtoswitchtheejectionkickers of the rings on and off. In the drive-beam decelerator, the additionaltail isnotimportant,sinceit will just adda lit tle tail to the RF-pulse produced in the power extraction and transferstructures(PETS). The switching time can be individually chosen for each train,soaratherﬁnerampintheﬁnalpulsecanbecreated. Thissolutiondoesnotrequireanyadditionalhardware;one must only be able to switch the sub-harmonic buncher at non-regularintervals. 4 NUMERICALRESULTS To achieve beam-loading compensation in the CLIC main linac, 11 of the 32 initial trains need to be delayed in the drive-beam linac. The maximum delay necessary is 11 bins. The gradient seen by the main-linac bunches can be00.20.40.60.81 020406080100120140160G/G0 t [ns] Figure 3: Shape of the RFpulse which is produced by the drive beam ifdelayed switching isapplied. -0.015-0.01-0.00500.0050.010.0150.020.0250.03 020406080100120140160∆G/G0 [%] bunch number Figure 4: The deviation from the nominal gradient as seen by each bunch inthe mainlinac. simulated with ASTPC [6]. In this program, the transient effects in the PETS of the drive-beam decelerator, as well as in the main-linac accelerating structures are taken into account. Eachstructureisrepresentedbyaseriesofreﬂec- tors that are located at the cell boundaries. This makes it possible to simulate precisely the beam acceleration in the time domain. The gradient errors depend on which of the trains are delayed. To ﬁnd a good choice, a number of different de- lay patterns was created randomly. These were evaluated with the program and the best case was accepted. For this case, Fig. 3 shows the RF-pulse as it is produced by the PETS. This pulse leads to a bunch-to-bunchgradient error inthemainlinacthatremainsbelow ∆G/G 0= 5×10−4, see Fig. 4. This is better than the required precision of ∆G/G 0≤10−3. The method described achieves a constant amplitude of the accelerating ﬁeld in the main linac. The main beam is, however, not accelerated on the crest of the RF wave, but at a small phase in the main part of the linac, ΦRF= 6◦. At the end of the acceleration, this phase is even larger, ΦRF= 30◦. Since the amplitude is increased in the RF phaseandthe beamloadingisin phasewiththe beam,this00.511.522.533.54 050100150200250300350Amplification quadrupole numbercase 1 case 2 Figure 5: The maximum ampliﬁcation of an initial beam jit- ter along the decelerator. The case of a rectangular current pulse (case1)iscomparedtotheonewitharampinbunchnumber(cas e 2). leads to an effective phase shift of the total accelerating ﬁeld during the ﬁrst part of the main-linac pulse. In order to prevent this, one can think of shifting the delayed trains beforetheyaremergedwiththeotherones. Theshifthasto be such that the bunches are in phase with the main beam. Inthiscase,notonlytheamplitudebutalsotheacceleratio n phaseismaintained. 5 SIMULATIONOFTHE DRIVEBEAM To estimate the impact of the beam-loading compensation on the stability of the drive beam in the decelerator, simu- lations are performedusing PLACET [7]. A lattice is cho- seninwhicheachsix-waveguidestructurefeedsthreemain linac structures. Asameasureofthestability,themaximumampliﬁcation of an initial jitter is used, which is determined as follows: inthesimulation,eachbunchiscutintoslices. Thebeamis offset and then tracked throughthe decelerator. The maxi- mum offset that the centre of any slice reaches, dividedby the initial offset, is the maximum ampliﬁcation. Figure 5 shows this ampliﬁcation of a transverse jitter along the drive-beam decelerator. If no transverse wakeﬁelds were present, the ﬁnal ampliﬁcation factor would be A=√ 10 from the adiabatic undamping of the motion. As can be seen, a rectangular current pulse (case 1) is close to this case. The bunch ramp increases the ampliﬁcation some- what (case 2). This seems tolerable. Most of the effect is duetothe trailingbunches. Ifthedelayedtrainsareshiftedinphase,soastoprevent phaseshiftoftheaccelerationﬁeld,thewakeﬁeldeffectsi n the drive-beam decelerator may become worse. The sim- ulation shows that also in this case, the jitter ampliﬁcatio n is almost the same as without the shift; they could not be distinguishedintheplot. Themethodthereforeseemstobe practical. But other methods, such as a slow phase change alongthetrain,mightachievethesame result.-0.0200.020.040.060.080.10.12 020406080100120140160∆G/G0 [%] bunch number Figure 6: Deviation from the nominal gradient, as seen by the main-beam bunches inCTF3. 6 APPLICATIONTO CTF3 Delayed switching could also be used in CTF3, the new CLIC Test Facility, which will be constructed at CERN. In this case, the switching time will be longer, about 4 ns. Since only ten pulses are merged to form the drive beam, one only delaysthree of them. Again,differentcases were searched for an optimum. The achieved compensation is verygood,about ∆G/G 0≈1.2×10−3, seeFig. 6. 7 CONCLUSION The method presented,to compensate the beam loading in the main linac, achieves the required precision of better than one part in 1000. It is very simple, can be adjusted to different switching times, and requiresno additional hard - ware. It seemsto bethemethodofchoiceforCLIC. 8 REFERENCES [1] L.Thorndahl. DriveBeam Bunchlet Trainsfor Multibunch - ing.CLIC-Note291 (1995). [2] R. Corsini, J.-P. Delahaye and I. Syratchev. CLIC Main Linac Beam-Loading Compensation by Drive Beam Phase Modulation. CLIC-Note408 (1999). [3] D. Schulte. Pulse Shaping and Beam-Loading Compensa- tionwiththe DelayLoop. CLIC-Note434 (2000). [4] D. Schulte. The Drive-Beam Acclerator of CLIC. Proceed- ings of LINAC 1998, Chicago, USA andCERN/PS 98-042 (LP)(1998). [5] R.CorsiniandJ.-P.Delahaye. TheCLICMulti-DriveBeam Scheme.CLIC-Note331 (1997). [6] I. Syrachev and T. Higo. Numerical Investigation of Tran - sient Beam Loading Compensation in JLC X-Band Main Linac.KEK-Preprint-96-8 (1996). [7] D.Schulte. PLACET:AProgram toSimulateDriveBeams. Proceeding of EPAC 2000, Wien, Austria andCERN/PS 2000-028 (AE) (2000).
arXiv:physics/0008133v1 [physics.acc-ph] 18 Aug 2000Double Kicker systemin ATF T.Imai,K.Nakai,Science UniversityofTokyo,Chiba, Japan H.Hayano,J.Urakawa, N.Terunuma,KEK,Ibaraki, Japan Abstract A double kicker system which extracts the ultra-low emit- tancemulti-bunchbeamstablyfromATFdampingringwas developed. The performance of the system was studied comparing an orbit jitter with single kicker extraction in single bunch mode. The position jitter reduction was es- timated from the analysis of the extraction orbits. The re- ductionwasconﬁrmedforthedoublekickersystemwithin a resolution of BPMs. More precise tuning of the system withawirescannerhasbeentriedbychanginga βfunction atthesecondkickertogetmorereductionofkickanglejit- ter. Theresultsofthese studiesaredescribedindetail. 1 INTRODUCTION KEK/ATF is an acceleratortest facility for an injector part of a future linear collider. It consists of an S-band injecto r linac, a beam-transport line, damping ring and extraction line [1]. Themain purposeofATF is to generateandmea- sure ultra-low emittance multi-bunch beam (2.8nsec spac- ing, 20bunch)and developtechnologythat can stably sup- plythe beamtothe mainlinac. Septum Magnet Extraction Line Damping Ring Injected BeamExtracted BeamExtraction kicker (2nd kicker) Damping Ring Extraction kicker (1st kicker) Injection kicker /BnZr/BnZrL1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 Lec2 L13 L14 L15 L16 Lec1 Damping Ring S-band injector LinacL0KEK/ATF (Accelerator Test Facility ) Beam Transport LineExtraction Line Figure1: Layoutofinjection/extractionregionofATF The stable beamextractionfromthe dampingring ises- sential for linear collider to achieve high luminosity, be- cause the position jitter would be magniﬁed by transverse wakeﬁelds in the linac and reduce the luminosity. There- fore,thejittertoleranceofextractionkickermagnetisve ry tightandestimatedtobe 5×10−4assuming βx=10m[2]. Itwillbeappliednotonlytoauniformityofpulsemagnetic ﬁeldforthetoleranceofmulti-bunchbuttoapulse-to-puls e stability. In ATF, double kicker system was developed for the stable beam extraction [3]. The system uses two iden-tical kicker magnets for beam extraction. The ﬁrst kicker is placed in the damping ring and the second one for jitter compensationin theextractionline. The ATF extractionkickerconsistsof 25electrodepairs with ferrite loaded in a vacuum chamber. A ceramic tube with TiN coated inside is used for beam channel in the kicker in order to reduce an beam impedance. The spec- iﬁcationofthekickerissummarizedin Table1. Table 1: ATFExtractionKickermagnet Kickangle 5 mrad Impedance 50 Ω Magnetlength 0.50 m Magneticﬁeld 513 Gauss Rise andFall time 60 nsec Flat top 60 nsec Maximumvoltage 40 kV Maximumcurrent 800 A 2 DOUBLE KICKERSYSTEM Thedoublekickersystemconsistsofonepulsepowersup- ply andtwo kickermagnetsseparatedbyphaseadvance π. It is, in principle, able to compensate kick angle variation oftheextractionkickerindampingring. Wheneachkicker has a kick angle variation ∆θ1and∆θ2,(x, x′)at the sec- ondkickercanbewrittenas /parenleftbiggx x′/parenrightbigg =M1→2/parenleftbigg0 ∆θ1/parenrightbigg +/parenleftbigg0 ∆θ2/parenrightbigg (1) Here,M1→2isatransfermatrixfromtheﬁrstkickertothe second one. Since a phase difference of the two kickers is π, /parenleftbiggx x′/parenrightbigg = −/radicalBig β2 β10 −α2−α1√ β2β1−/radicalBig β1 β2 /parenleftbigg0 ∆θ1/parenrightbigg +/parenleftbigg0 ∆θ2/parenrightbigg (2) then, x= 0, x′=−/radicalBigg β1 β2∆θ1+ ∆θ2 (3) are obtained. When the two kickers are identical, that is ∆θ1= ∆θ2,thevariationcouldbecanceledwiththesame βfunction. If the two kickers are not identical, compen- sation also can be done by adjusting βfunction. In case of multi-bunch extraction, only the similarity of the ﬂat- topwaveformbetweentwokickersisrequiredforthejitter reduction in each bunch. A tight ﬂatness requirement for everybunchin amulti-bunchisnotnecessary.DC Power supplyPFLThyratron Coaxial cable1st Kicker ( in damping ring ) 2nd Kicker ( in Ext. line )Coaxial cableLoad Load 1st kicker2nd kickerBeam design orbit extra kick Δθ1 Δθ2 = (Δθ1) Figure2: Doublekickersystem 3 ORBITJITTER REDUCTION We measured horizontal beam orbit jitter of the extraction line in single bunch operation and compared the perfor- manceofthedoublekickersystem withthecase ofextrac- tionwithoutthesecondkickerwhichwedeﬁnedasasingle kickermode. The operation condition at the measurement was single bunch mode, beam energy 1.3GeV, and repetition rate is 0.78Hz. The beam is extracted from damping ring to the extractionline by one kicker magnetand three DC septum magnets. Kickangleofthekickerisdesignedto 5mrad. In single kicker mode, a dipole magnet was installed instead the secondkicker. The beam orbit shot by shot was measured by strip-line BPMs in the extraction line [4]. Total 14 BPMs are used forthemeasurementandanalysis. 3.1 Analysismethod The horizontal beam position jitter in the extraction line came from a kick angle jitter of the extraction kickers and a momentum ﬂuctuation of the beam. The horizontal dis- placement at a pointafter the second kickercan be written as ∆xi= ∆ xkicker + ∆xmomentum =R12(1, i)∆θ1+R12(2, i)∆θ2+ηi∆p p(4) where R12(1, i)isatransfermatrixcomponentfromthe ﬁrst kickerto the pointi , R12(2, i)is fromthe secondone andηiisadispersion. Inthisanalysis, ∆θ1,∆θ2and∆p/p were obtained by ﬁtting the measured displacement from the average position at BPMs with eq(4). ∆xkickerwas calculated by subtraction ηi∆p pfrom the displacement at each BPM. In single kicker mode, the same analysis also has been done, but ∆θ2was set to zero because the dipole magnetinstalledasthesecondkickerisDCmagnet.3.2 Resultof jitterreduction Table 2 shows the comparison of the position jitter caused by kick angle variation in both modes at the BPM where hasthemaximum R12fromthekicker. σkickerwascalculatedbyusingmodelvalueof R12and ﬁtting value of kick angle variation of each shot. The ef- fect of jitter reduction was compared in the two kick an- gle region. As a result, the double kicker system reduced the jitter down to the resolution of BPM which is about 20µm in case of small kick variation, however the reduc- tion rate was not sufﬁcient in case of large kick variation. Inbothcases,thepositionjitterreductionwasobservedfo r the double kicker conﬁguration, however, a precise tuning of the system and a high resolution position monitor are still necessaryforafurtherreduction. Table 2: Comparisonofpositionjittermeasurement mode #of σkicker #of σkicker meas. [ µm]meas. [ µm] double 115 37 181 24 single 60 78 248 34 ∆θ1≥0.007mrad ∆θ1≤0.007mrad 4 OPTICSTUNING In order to get more jitter reduction, βfunction at the sec- ond kicker was surveyed using high resolution position monitor. One of the wire scanners was used as a position jitter detector[5]. 4.1 Jittermeasurementwith wirescanner Thewirescannerwhichhave10 µmdiametertungstenwire wasusedforthejittermeasurement. Scatteredgammarays from the wire are detected by air Cherenkov detector with photo multiplier. Before orbit measurement we measured horizontal beam proﬁle and set the wire at the position which is middle of the slope of the proﬁle. The distribu- tionofdetectedgammaraysisconvertedtothedistribution of the position with beam proﬁle. Horizontal beam size wasaround100 µmin thesemeasurements. 4.2 Opticstuningmethod Changing βfunction at the second kicker, a minimum po- sition jitter was surveyed. However, the condition of jitte r compensation was different in each optics setting because dispersionandorbitwascorrectedineachsettingindepen- dently. Then the position jitter caused by kick angle vari- ation was comparedby normalizingeach conditionwith β function2.5matthewireandphaseadvance1.5 πfromthe secondkicker. TheresultissummarizedinTable 3. With these values a βfunction of maximum jitter re- duction and kick angle jitter are estimated. Assuming model optics and kick angle ratio k, kick angle variationγ signal 2ΔγBeam profile horizontal position 2Δxwire setting position Figure3: Beamjitter measurementusingwirescanner Table3: Resultofopticstuning βfunction estimated βfunction phase 2ndKicker σkicker wire advance 5.0[m] 11.4[ µm] 2.47[m] 1.507[ π] 6.0 11.0 2.49 1.501 8.0 10.9 2.65 1.492 10.0 11.4 2.59 1.482 12.0 12.9 1.81 1.501 will be ∆θ1=k∆θ2, horizontal displacement is written by∆xkicker =R12(1, wire)∆θ1+R12(2, wire)∆θ2=√βwire∆θ1(√β1−k√β2). By ﬁtting σestimated =/radicalBig σ2 kicker+σ2 resolution at eachopticssetting, k= 0.833 ∆θ1= 6.7µrad σresolution = 10 .7µm were obtained. Measured βfunction at the ﬁrst kicker was 4.95m, so βfunction at the second kicker for maximum jitter reduc- tion was estimated to be 7.13m. The estimated resolution 10.7µmis includedmonitorresolutionand incomingposi- tionandanglejitter. Theresolutionofthispositionmonit or using wire is estimated about 1 µm, then it seemed that the beam orbit of damping ring had ﬂuctuation in these mea- surements. The kick angle ratio 0.833 is not explained by the difference of the cable length between the two kickers (9.4m). It seemed that the difference of ceramic coating betweenthe two kickerscausedthe muchdifferenceof the ﬁeld strength. 5 CONCLUSION The performance of the double kicker system was studied with measurementof horizontalorbit jitter in single bunch mode. The system has the jitter less than the resolution of1011121314 4 6 8 10 12 14Estimated Horizontal Position Jitter [ micron ] Calculated Beta function at 2nd Kicker [ m ] Figure 4: Position jitter dependence on βfunction at the secondkicker BPM in small kick variation. The survey of the maximum reduction optics changing βfunction at the second kicker was performed with wire scanner. The βfunction which gave a minimum position jitter was found. Estimated β functionforitwas7.13mandthiscorrespondedtothekick angleratioofthetwokickers0.833. 6 ACKNOWLEDGMENTS The authors would like to acknowledge Professors H.Sugawara,M.Kihara,S.IwataandK.Takatafortheirsup- port andencouragement. We also expressourthanksto all themembersoftheATFgroupfortheirsupporttothebeam experiments. 7 REFERENCES [1] Edited by F.Hinode et al.,”ATF Design and Study Re- port”,KEKInternal 95-4,1995 [2] T.O.Raubenheimer et al.,”Damping Ring Designs for a TeV Linear Collider”,SLAC-PUB-4808,1988 [3] H.Nakayama, KEK Proceedings 92-6,1992,p326-p334 [4] H.Hayano et al., “Submicron Beam Position Monitors for Japan Linear Collider”,LINAC’92,Ottawa,1992 [5] H.Hayano,”Wire Scanners for Small Emittance Beam Mea- surement inATF”,thisconference
arXiv:physics/0008134v1 [physics.acc-ph] 18 Aug 2000BEAMSWITCHINGANDBEAMFEEDBACK SYSTEMSATKEKBLINAC K. Furukawa∗, A. Enomoto,N. Kamikubota,T. Kamitani,T. Matsumoto, Y. Ogawa,S. Ohsawa, K.Oideand T.Suwada HighEnergy Accelerator Research Organization(KEK) Oho 1-1,Tsukuba,Ibaraki, 305-0801,Japan Abstract TheKEK8-GeVelectron/3.5-GeVpositronlinachasbeen operatedwith verydifferentbeamspeciﬁcationsfordown- stream rings,KEKB, PF andPF-AR. Forthereliableoper- ation amongthese beam modesintelligentbeamswitching andbeamfeedbacksystemshavebeendevelopedandused since itscommissioning. A software panel is used to choose one of four beam modes and a switching sequence is executed in about two minutes. Most items in a sequence are simple operations followed by failure recoveries. The magnet standardiza- tion part consumes most of the time. The sequence can bere-arrangedeasilybyacceleratoroperators. Linacbeam modes are switched about ﬁfty times a day using this soft- ware. Inordertostabilizelinacbeamenergyandorbits,aswell assomeacceleratorequipment,aboutthirtysoftwarebeam feedback loops have been installed. They have been uti- lized routinely in all beam modes, and have improved its beam quality. Since its software interfaces are standard- ized, it is easy to add new feedback loops simply deﬁning monitorsandactuators. 1 INTRODUCTION The KEK electron/positron linac had been upgraded for KEK (KEK B-factory) asymmetric electron-positron col- lider since 1994. Commissioning of the ﬁrst part of the linac has started at the end of 1997 and has already achieveddesignedbeamparametersafterits completionin 1998[1]. IthasbeenprovidingbeamsfortheB-physicsex- periment (Belle) of the CP-violation study at KEKB since 1999. Theperformanceoftheexperimentdependsontheinte- gratedluminosityattheKEKB,whichislargelydependent on stability and intensity of linac beams. Since the linac have to provide four beam modes which are very different (KEKB e+, KEKB e−, PF-Ring, PF-AR), it had been re- alized that it was important to achieve reproducibility and stability ofeachofthosefourbeammodes[2]. 2 LINAC CONTROLS The linac control system was also upgraded[3] to support the upgraded high-intensity linac based on the system re- juvenation in 1993[4]. It consisted of layered components ∗e-mail: kazuro.furukawa@kek.jpthat communicate each other, where hardware and lower- layer information were hidden from the upper layer and only useful features are exposed to the upper layers. New components were added to accommodate new accelerator equipmentandfeaturesfortheKEKBinjection. Especially software for beam position monitors was developed and databaseforequipmentandbeamlineswasmuchenriched. In the commissioning, many pieces of application soft- wareweredevelopedasclientstothecontrolsystem. Many of them were designed with user interface on X-Window employing SAD-Tk or tcl/tk scripting languages for rapid development and simple manipulation. They use common library routines to facilitate maintenance as well as devel - opment. The number of application programs exceed 100 includingonesforbeamstudies. 3 COMMISSIONING In the commissioning of the upgraded linac the quality of beams had gradually improved as the beam study ad- vanced, and design values were achieved for a short term. It was, however, realized that much effort was required to reproducethequalityandtomaintainitforalongerperiod. One of the main reasons was switching between quite differentfourbeammodes. Theotherwasshort-terminsta- bilitiesandlong-termdriftsofequipmentparameters[2]. In order to cure these, software for beam-mode switch- ing and feedback loops has been reﬁned, while they had beendevelopedsincethebeginningofthecommissioning. 4 SOFTWARE The software has been developed with the tcl/tk scripting language under the same environmentas other application software[3]. 4.1 BeamModeSwitch Inthelinacbeammodeswitching,asdescribedabove,it is important to select operation conditionsand parametersof acceleratorequipmentreliably,andtoachievereproducib le beam qualities. In order to meet the purpose, the software wasdevelopedtobeeasilyre-arranged,anditcurrentlyhas followingswitchingitems. •Suspension of beam feedback loops and other sub- systems.Figure 1: An example of linac beam mode switch panel. In this example, KEKB e+injection was selected. Check- buttons on the left are used to select items to go through. Pull-down-menus are used to choose parameter ﬁles. Boxesontherightshowexecutionstatus. •De-gauss of a bending magnet (only for PF injec- tion)1. •Simple standardizationofmagnets. •Selectionofa gun,magnets,andrfsystems. •Parametersformagnets(mostlymagneticﬁelds). •Parametersforrfsystems(mostlyphases). •Parametersfortimingsystems. •Parametersforguns. •Operationonpositrontargetsandchicane. •Operation mode of beam instrumentations and their dynamicranges. •Initial beamproﬁlemonitorselection. •Initial beamrepetitionrate. •Selectionofbeamtransportlines. •Informationtodownstreamringcontrolsystems. •Review ofequipmentparameters. •Displayandrecordofequipmentstatusandparameter differences. •Resumptionofcorrespondingbeamfeedbackloops. 1Thisshouldbereplaced byasimplestandardization afterso mebeam studies.•Informationtooperatorsviaa speechprocessor. Itemsrelatedtotheradiationsafetyarenotincluded,and arehandledbyaseparatesafety-interlocksystem. Fig. 1 shows an example of the software panel. Each itemonthepanelcanbeenabledordisabledbyanyopera- tor, and its status can be saved or restored. New items can be introduced by adding entries in the database. If some troublesoccurs,whichcannotberemovedthroughthecon- trolsystem,thateventisreportedtotheoperator,whomay retryit aftertheproblemwasremoved. Itemslistedas‘Parameters’arenormallytakenfromthe equipment parameters when the last time the same beam mode was used, while other parameter sets can be chosen fromthemenuifa operatorneedsone. Fortheinitializationofthemagnetsfollowingissuesare repeatedly tested: reproducibility of magnetic ﬁelds, tol - erance of the power supplies to the steep current changes and failure recoveries in control and application software . Since this part consumes most of the time in the switch, it isstill beingimproved. 4.2 BeamFeedbackLoops Figure 2: Energy feedback panel at the R sector as an ex- ample. Parameters and processing speciﬁcations can be modiﬁedanytime. Software feedback loops installed in the linac are cate- gorized into three groups: stabilization for equipment pa-rameters, the beam energy and the beam orbit. Their ba- sic software structure is the same and is built of following parts. •Check the conditions of beam modes, beam current, parameterlimit, etc. •Read the monitor value applying moving average, limit checkandotherspeciﬁcpost-processing. •Derive the feedbackamountapplyingconversionfac- tors, gainandlimit check. •Set the actuator value applying limit check and other speciﬁc pre-processing. •Flow control, graphics display, recording and inter- face toothersoftware. Fig. 2 shows an example panel for one of the energy feedback loops. Each parameterin the panel can be modi- ﬁed anytime. Energy feedback loops are composed of a monitor of a beam position at a large dispersion and an actuator of rf phases at two klystron stations, in order to maintain the energy spread small. This type of energy feedback is in- stalled at 4 locations at 180-degree arc and the end of the linac. Some parameters are different depending on the beammodes[5]. Orbitfeedbackloopsusebeampositionsasmonitorval- ues and steering magnets as actuators. A monitor value is actually a weighed average of beam position monitors (BPM’s) over a betatron wavelength according to the re- sponse function for the corresponding steering magnet. Normally two subsequent regions, which are apart by 90- degreebetatronphase,aregroupedasinFig.3. Somefeed- back loops read only one BPM and used to keep the beam positionatthe endofthelinac. Steerings Weighed Average of BPM'sBeam Orbits Figure 3: A group of orbit feedback loops. Weighed av- erage of BPM’s over a betatron wavelength as a response functionisfedintosteeringmagnetstrength. This type of orbit feedback loops are installed at many sectorsandthe currentnumberofgroupsreached15. SinceBPM’scanbereadat1Hz,mostofthebeamfeed- back loops are operated at this speed[6]. Feedback gains are chosento besmall,0.2to0.5,inorderto avoidoscilla- tion. The same feedback software has been applied to stabi- lize accelerator equipment. Although these loops may be gradually moved to local controllers or even to hardware, they are effective to suppress newly found instabilities in acceleratorequipment. Recently,itwasappliedtosuppres slong-termdriftofthetriggertimingoftheelectrongunand foundtobeeffective[7]. New feedbackloopscan beeasily built simplybydeﬁn- ing monitors, actuators and some parameters. Standard software libraries provide environment to tune those new loopsevenduringoperation. Inordertomanagelargenumberoffeedbackloops,sev- eral softwarepanelsweredevelopedsuchas a globalfeed- backstatusdisplayandfeedbackhistoryviewer. 5 CONCLUSIONS Software was developed to stabilize the linac beam, and successfully improved the beam reproducibility and relia- bility. Since it wasdesignedto be re-arrangedeasily, oper - atorscouldsolveproblemsmodifyingsoftwareparameters evenwhena beamoperationmodehadtobemodiﬁed. The beam-mode switching panel has become much re- liable and reproduced beams sufﬁciently with switching more than 50 times a day. Switching time, which is im- portant for integrated luminosity, was shorten to be 90 to 120seconds. Feedback loops cured both the short-term instabilities and long-term drifts of the beam energy, orbits and equip- ment parameters. Dependingonthe acceleratorstatusthey suppressedthebeaminstabilitytoahalfanddriftstoaﬁfth without any operator manipulations. It also was useful to keepbeamswhenbeamstudieswerecarriedunderunusual beamconditionsandtoﬁndsomeanomaliesintheacceler- ator. Those software systems were used in the routine opera- tion and contributed to enhance the KEKB experiment ef- ﬁciency. 6 REFERENCES [1] A.Enomoto etal.,“PerformanceoftheKEK8-GeVElectron Linac”, Proc.EPAC2000, Vienna, Austria,tobe published. [2] K.Furukawa etal.,“Towards Reliable Accelerationof High- Energy and High-Intensity Electron Beams”, these proceed- ings. [3] K. Furukawa et al., “Accelerator Controls in KEKB Linac Commissioning”, Proc. ICALEPCS’99, Trieste, Italy, 1999, p.98. [4] K. Furukawa et al., “Upgrade Plan for the Control System of the KEK e−/e+Linac”, Proc. ICALEPCS’91, Tsukuba, Japan, 1991, p.89. [5] K. Furukawa et al., “Energy Feedback Systems at KEKB Injector Linac”, Proc. ICALEPCS’99, Trieste, Italy, 1999, p.248. [6] N. Kamikubota et al., “Data Acquisition of Beam- Position Monitors for the KEKB Injector-Linac”, Proc. ICALEPCS’99,Trieste,Italy, 1999. T. Suwada et al., “New Data-Acquisition System of Beam- Position and Wall-Current Monitors for the KEKB”, Proc. APAC’98,Tsukuba, Japan, 1998. [7] S. Ohsawa et al., “Pre-injector of the KEKB Linac”, Proc. EPAC2000, Vienna, Austria,tobe published.
THE RF-SYSTEM OF THE NEW GSI HIGH CURRENT LINAC HSI G. Hutter, W. Gutowski, W. Hartmann, G. Kube, M. Pilz, W. Vinzenz, GSI, Darmstadt, Germany Abstract The RF part of the new high current injector-linac HSI consists of five cavities with the new operating frequency of 36 MHz instead of 27 MHz of the removed Wideroe type injector. The calculated power requirements of the cavities including beam load in three structures were between 110 kW for a rebuncher and 1.75 MW pulse-power for the two IH-cavities. The beam load is up to 150 kW for the RFQ and up to 750 kW for the two drift tube tanks. An additional 36 MHz debuncher in the transfer line to the Synchrotron (SIS) will need 120 kW pulse power. We decided to fulfil these demands with amplifiers of only two power classes, namely three amplifiers with 2 MW and six amplifiers with 200 kW pulse output power. The latter ones are also used as drivers for the 2 MW stages. The 200 kW amplifiers were specified in detail by GSI and ordered in the industry. The three 2 MW final amplifiers were designed, constructed and built by GSI. The paper gives an overview of the complete RF system and the operating performance of amplitude and phase control with beam load. It further describes some specialities of the new 2 MW amplifiers like the simplicity of the anode circuit, a very sophisticated socket for a cathode driven amplifier with cathode on dc ground, the parasitic mode- suppression, shielding and filtering of unallowable RF- radiation and operating experience since October 1999.1 INTRODUCTION Within the beam intensity upgrade program at GSI[1], the old Wideroe type injector with four tanks, working at 27 MHz, was replaced in 1999 by the high current injector HSI. During the last year of operation, one of the old 27 MHz am plifiers, originally built by Herfurth GmbH and redesigned by the Unilac-RF-group in 1984, was replaced by a 200 kW amplifier, manufactured by Hüttinger Elektronik GmbH, Freiburg[2]. In this way we could remove the old amplifier during beam-operation of the Unilac and gain a powerful plant for prototype activities of the future 2 MW amplifiers, as a connection to a 24 kV anode supply was available there. We changed the working frequency of a second 200 kW Hüttinger amplifier from 27 MHz to 36 MHz to get a prototype driver. In parallel, five 200 kW amplifiers where built at Thomcast AG Turgi, three of them with all the electronics needed for the final 2 MW stages except the anode supply and the filament transformers[3]. A prototype of the 2 MW amplifier was b rought to operation until the end of December 1998. After a redesign of the prototype three final amplifiers were built at GSI in 1999. The first of them delivered RF to the RFQ just in time, the others for the IH-structures were ready well before the tanks. Figure 1 shows the complete 36 MHz RF distribution. master oscillator 108.408 MHz 10 W cwf / 3 power splitter 1:N 36.136 MHz 0.25 W phase shifter0.25 W 0.25 W 0.25 W 0.25 W 0.25 W phase control control Transistor 2 kW 0,5 kW 200 kW 40 kW1ms 6ms 1ms 6ms 2 MW 1ms 400 kW 6ms tuning 450 kW40 kW 6ms2 kW 200 kW0,5 kW1ms 1ms6msTransistor 40 kW 6ms2 kW 200 kW0,5 kW1ms 1ms6msTransistor 40 kW 6ms2 kW 200 kW0,5 kW1ms 1ms6msTransistor 2 kW 6ms 40 kW 6ms2 kW 200 kW0,5 kW1ms 1ms6msTransistor RFQ 150 kWSuper Lens400 kW2 MW 1ms 6ms IH 1400 kW2 MW 1ms 6ms IH 2 110 kWRebuncher 120 kWTC Debuncher 1,69 MW 1,75 MW6ms 130 kWshifter controlcontrolphasephase shifter controlphase controlphase phase controlphase controlshifter shifter controlphase controlphase shifter controlphase controlphase cavity tuningcavity tuningcavity tuningcavity tuningcavity tuningcavityTransistorphase lock ampl. control Figure 1: 36 MHz High Current Injector RF-SystemFigure 2: Overview of the new installation 2 THE 200 KW AMPLIF IERS 2.1 New amplifiers from Thomcast AG, CH Most details of the five new 200 kW amplifiers have been reported in [3]. In 1999 all five amplifiers were installed at GSI (after the old Wideroe-amplifiers were removed) and brought to operation without remarkable problems after introducing additional damping for higher frequencies through GSI staff in the anode circuits. The only massive breakdowns occurred at the dc-power- supplies of the solid state 2 kW drivers, which had been built by a subcontractor of Thomcast. The original power-supplies have all been replaced meanwhile, but the new ones still have problems and only with the use of a spare-amplifier we could provide continuos operation. 2.2 Modified old amplifiers Two 27 MHz / 200 kW amplifiers built by Hüttinger Elektronik GmbH, Freiburg, Germany[2] were designed for 10 % duty cycle . As the anode transformers of these amplifiers have 75kVA, there was a very good safety margin to use one of these amplifiers for a buncher at the end of the new injector that needs up to 30 % duty cycle, but only about 100 kW. For this purpose we had to change the frequency of the amplifier to 36 MHz. The Π-filter anode circuit could be easily brought to the proper frequency by replacing its inductance by a strip transmission line and a different tuning of the two variable vacuum capacitors. The input circuit could be tuned without changing any hardware. To suppress self- oscillations near the fundamental frequency, a neutralisation had to be introduced to the amplifier, which additionally isolated the input from the output by about 17 dB. The second amplif ier was changed to 36 MHz, too, and will be used as a driver for the old 2 MW prototype and also as spare unit. 3 FINAL 2 MW AMPLIF IERS 3.1 Tube Selection As the Siemens tube RS2074HF is installed in the five amplifiers of the Alvarez-section of the Unilac and delivers reliable 1.6 MW at 108 MHz with 25% duty cycle, we decided to take its low frequency version, the RS2074SK for the new 2 MW amplifiers. This tube is identical to the HF-version, but misses a lower- lossy- material for the screen-grid contact area, where the pyrolitic graphite grid is mounted to its support. 3.2 Input Circuit From the old installations of the Wideroe-RF, the 1MVA / 24kV anode-power-supply was recuperated, as it fit all requirements with a few small changes. The reuse of the power-supply, however, implicated that the cathodes of the new amplifiers had to remain on dc- ground. On the other hand, we did not dare to build grid- driven amplifiers with the given higher frequency and power level, but chose a grid1 / grid2 based circuit with cathode drive. The RF-potential on the cathode necessitated to feed the heating-current over two parallel λ/4 strip-lines with a strong capacitive coupling against each other. On the RF-cold ends, one of them is connected to ground while the other one is connected to the heating transformer via a 1200 A filter. Figure 3 shows in a schematic diagram, how the input- impedance of the tube is transformed to 50 Ω. RF input290pF~~ vacuum capacitor1nF line 34cm40nH feedlinecoaxial 28cm525 pFLoad of cathode current ~~matching circuit tube equivalent strip-circuit diagram 304 Figure 3: Equivalent input circuit diagram 5 Figure 4 gives a comparison of a Supercompactå calculation and an analyzer measurement of the input impedance. Five parallel low inductive 20 Ω resistors simulated the cathode current in the measurement. 0 0.2 0.5 1 20.51 2 -0.5 -1-240MHz30MHz 8 0 0.2 0.5 1 20.51 2 -0.5 -1-240MHz30MHz 8 Smith charts calculated measured Figure 4: Input impedance from 30 MHz to 40 MHz.3.3 Tube Socket Controlgrid and screengrid are both twice capacitively blocked in the socket as close as possible to the rf- ground without any tunability. The capacitors used for this purpose were especially developed by the Swiss company Güller together with GSI. They use a copper- silver-plated polyester foil that is covered with two silver-plated brass plates, while all isolating foil parts are covered with silicone-rubber. By this treatment the capacitors are mechanically robust and absolutely waterproof. They were tested up to 8 kV successfully without partial discharges. Figure 5: Parts of the sophisticated socket When we started operation of the prototype, the output power was first limited to about 1 MW by excessive G1- and G2 current, caused by a high level of the third harmonic in the input circuit, as this frequency is near the λ/4 resonance of the G1-G2-circuit. Introducing a ring of ferrite between G1 and G2 inside the socket and changing the length of the coaxial line to the driver solved the problem and led to the data given in table 1. 3.4 Anode Circuit The anode circuit is a λ/4 resonator with a shrinked in house designed and built cylindrical capton capacitor as anode-dc-blocker, which was tested up to 50kV. Strong galvanic output coupling led to the needed low loaded-Q-value of 4, which means a 3 dB bandwidth of about 9 MHz. This makes tuning of the once adjusted anode circuit unnecessary, even after a tube change. 3.5 Filtering and Shielding Unwanted RF-radiation of the new amplifiers was about 60 dB lower than the radiation of the ol d Unilac amplifiers, due to an excessive RF-contact-design of all circuits (there are up to 7 contacts in series) and to sufficient filtering of all dc-power-lines, including especially developed ferrite filled high voltage cables. 3.6 Damping of Parasitic Oscillations When pulsing the G1 to lower values without RF- drive, the tube started parasitic oscillations around 900 MHz. As some tubes, especially an also tested RS2074HF, produced these modes even with RF, we placed 6 ferrite rods around the anode ceramic, as thishad shown sufficient damping for a RS2074SK in an earlier test which GSI-staff did at HIMAC. As some RS2074HF still oscillated and as the ferrite kept cold at full power, we introduced 30 rods to each amplifier. Now both tube types are usable without oscillations. 3.7 Operating Parameters Table 1:Reached Data of two operational modes Pulse Output Power 400 kW 2 MW Repetition Frequency 50 Hz 20 Hz Duty-Cycle 30 % 6.0 % Average Output Power 120 kW 120 kW Pulselength 6 ms 3 ms Pulse Drive Power 26 kW 85 kW Filament Voltage 13 V 13 V Filament Current 900 A 900 A G1 DC-Voltage pulsed from -700V to -600V G1 Pulse Current 0.0 A 4.6 A Anode DC Voltage 24.5 kV 24.5 kV Anode Pulse Current 45 A 120 A G2 DC Voltage 1450 V 1450 V G2 Pulse Current 0.0 A 4.0 A Efficiency 44 % 78 % 4 OPERATING PERFORM ANCE Six 200 kW and three 2 MW amplifiers were installed and brought to satisfying operation from January to August 1999. The implementation of a RF-feedback- system was not necessary because of the redesigned controls[3]. A hard to find malfunction of the control occurred at the IH1-amplifier: After some warm-up time a higher order resonance of the cavity drifted exactly to the 9th harmonic, which influenced the reference values. A low pass filter in both reference lines solved this problem. 5 OUTLOOK Some work will have to be done not to fire the crowbar if a cavity sparks which still leads to an averaged breakdown of a few minutes per month. We also intend to test the socket-compatible smaller RS2042SK of the old Wideroe amplifiers in the rebuilt prototype amplifier with additional vacuum capacitors. We still have four of those tubes and they are powerful enough to feed the RFQ. REFERENCES [1] U. Ratzinger; “The New GSI Prestripper LINAC for High Current Heavy Ion Beams” Proc. of the 1996 LINAC Conf., Geneva, 288 [2] G. Hutter et al; “New Power Amplifiers for 200 kW at 27 MHz”, Proc. of the 1992 EPAC Conf, 1203 [3] W.Vinzenz, W.Gutowski, G.Hutter, GSI, B.Rossa Thomcast AG, Proc. of the 1998 LINAC Conf., 219
COMMISSIONING OF IH-RFQ AND IH-DTL FOR THE GSI HIGH CURRENT LINAC W. Barth, P. Forck, J. Glatz, W. Gutowski, G. Hutter, J. Klabunde, R. Schwedhelm, P. Strehl, W. Vinzenz, D. Wilms, GSI Darmstadt, Germany U. Ratzinger, Institute for Applied Physics, University Frankfurt am Main, Germany Abstract The new 1.4 MeV/u front end HSI (HochStromInjektor) of the Unilac accelerates ions with A/q ratios of up to 65 and with beam intensities in emA of up to 0.25 A/q. The maximum beam pulse power is up to 1300 kW. During the stepwise linac commissioning from April to Septem- ber 1999 the beam behind of each cavity was analysed within two weeks. A very stable Ar1+ beam out of a vol- ume plasma source MUCIS was used mainly. The meas- ured norm. 80 % emittance areas around 0.45 π mm mrad are close to the results from beam simulations. Up to 80 % of the design intensity at the linac exit were achieved. In February 2000 an U4+ beam from the MEVVA source was accelerated for the first time. 1 INTRODUCTION H-type RFQ- and DTL-structures are well suited and frequently used meanwhile at the front end of proton and heavy ion linacs. The recent GSI heavy ion linac project is characterized by three quite ambitious parameters: The high A/q-value of up to 65, the high beam current – for example 15 emA of 238U4+, and the normalized horizontal emittance of less than 0.8 π mm mrad for the U73+ beam at synchrotron injection after having passed two stripping processes. Multiturn injection into the horizontal phase space of the synchrotron SIS 18 should then allow the accumulation and acceleration of up to 4·1010 U73-ions. This aim as well as the state of the art in ion source devel- opment dictated the parameter choice for the new linac [1]. The original Wideröe DTL section was replaced by H-type structures (Fig. 1). They allow to increase the voltage gain by a factor of 2.5 while keeping the total length of the installations. Additionally, the injection energy into the Unilac was reduced from 11.4 keV/u to 2.2 keV/u. These changes allow a convenient matching of high current beams from MUCIS or MEVVA ion sources to the Unilac [2,3,4]. Several aspects of the new 91 MV linac are described in ref. [5,6,7]. The new 36 MHz power amplifier is described in ref. [8]. The choice of this fre- quency, which is one third of the succeeding Alvarez- DTL frequency resulted from beam dynamics calculations as well as from corresponding cavity sizes [7]. RF- and beam results from the IH-RFQ and from the Super Lens SL, which were invented during this project, are reportedin this paper as well as the corresponding results of the IH-DTL. Fig. 1: General drawing of the HSI 2 TIME SCHEDULE The main injector of the GSI complex had to be replaced at the end of this project. A nine month-period was per- mitted to remove the Wideröe-DTL as well as to install and to commission the new injector, rf equipment and the 1.4 MeV/u charge separator. During that period all GSI experiments were supplied exclusively by the High Charge State Injector [9]. The linac assembly and com- missioning was done tank by tank and a beam diagnostics bench [10] was installed behind the component under investigation. The following time schedule shows the key events: 17. Dec.1998Last beam from the Wideröe structure March1999Beam injection into the 2.2 keV/u transport line 28. April1999Acceleration by the IH-RFQ 31. May1999Beam injection into the Super Lens 22. July1999Acceleration by the IH-DTL, tank 1 06. Sept.1999Acceleration by the IH-DTL, tank 2 and beam transport to the gasstripper 3 COMPONENT TESTS AND ALIGNMENT The rf tuning results from all cavities as well as rf power tests with the first and the last of 10 RFQ modules are described in ref. [11]. The flatness of the voltage distribu- tion along the mini vanes is within ± 1 %. The vane alignment within each module is given by the precision of the inidividual components. No adjustment during the final assembly was foreseen. The deviation in the trans- versal position of the vane-carrier rings was below 0.05 mm. The precision of each carrier ring is within± 0.01 mm. The RFQ modules were then bolted together and after the final installation in the Unilac tunnel the vertical module displacement was up to 0.1 mm along a sine-shaped half wave, while the horizontal displacement was up to ± 0.25 mm along an S-shaped curve. As the total RFQ length is 9.4 m, the related angular deviations from the beam axis are quite acceptable. The Super Lens [12], an 11-cell RFQ with enlarged aper- ture and synchronous phase – 90°, acts as a 3-dimensional focusing lens into the IH-DTL and is bolted on the low energy end plate of cavity IH1. The large tank modules of the DTL cavities were manufactured and copper plated in time as well as the bulk copper drift tubes. Some technical effort became necessary during the design and construc- tion of the cavity internal quadrupole triplets: the cores are laminated and the material is cobalt steel alloy (Vaco- flux50, Fa. Vacuumschmelze). The quadrupoles with core aperture diameters of 38 mm and 50 mm reach up to 1.27 T and 1.23 T, respectively, at the pole tips. These fields were demonstrated successfully in all cases. Due to the complex pulse structure of the Unilac the magnets have to allow pulsed operation with a pulse rise time of 16 ms only and pulse repetition rates of up to 20 Hz. This corresponds to voltages of up to 700 V applied on the coils. With the exception of one singlet, which is showing intolerable leak currents at repetition rates above 5 Hz, all lenses have reached the specifications. The present situa- tion causes no restriction with respect to synchrotron injection. Anyway the corresponding triplet will be re- placed at the first opportunity. The alignment results of the triplets are ± 0.1 mm for the magnetic axis of each singlet within the corresponding triplet. The deviations of all linac components from the beam axis are up to ± 0.25 mm. The triangulation measuring technique was applied. After final installation in the Unilac tunnel the vacuum pressure became better than 2·10-8 hPa for all structures within one month. All cavities came on rf power after about 2 days of preconditioning, where only some 10 W were accepted at vacuum pessures up to 7·10-6 hPa. The RFQ as well as the Super Lens show pronounced dark current contributions at voltage amplitudes above the 75 % design level [7]. IH1 shows modest dark current contributions while IH2 seems to become free of that effect after some more operation time. 238U4+-levels (that is 90 % of the A/q = 65 design level) were provided and used in beam times since 02/2000. 4 BEAM COMMISSIONING The mobile diagnostics instrumentation consisted of 4-segmented capacitive pick-up probes, beam transform- ers, slit-grid as well as single shot, pepper pot type emit- tance measurement devices and diamond detectors for bunch length detection [10]. Beam Energy: The time of flight technique was used to measure the beam energy [5-7]. In all cases the correct energy was received immediately after first beam injection and at the nominal rf parameters. Additionally, the exit energy dependence on rf amplitude and phase can be measured and compared to beam simu- lations. Fig. 2 shows as an example the energy profile of IH2, which was calculated by LORASR and verified quite well by TOF measurements. Fig. 2: Dependence of the IH2 exit energy on rf ampli- tude and phase of the same tank (LORASR simulation). Pulse Shaping by the LEBT Chopper: An important feature of the high current pulse shape is the rise time to full intensity: This defines the amount of beam energy to be handled, while machine parameters are changed or optimized. The linac beam pulse is defined by a chopper, which is located immediately in front of the RFQ. It has a rise time of about 0.5 µs [4]. After reaching the chopper voltage design levels pulse rise times of 5 µs are measured now as shown qualitatively by the dashed curves in Fig. 3, it is the time resolution of the current transformers. The curves measured during the commis- sioning show the quality of the flat top as well as the con- servation of the beam pulse shape along the linac. This demonstrates the performance of the rf control loops, which are acting very well though the beam current is close to the design level of 10 mA Ar+ in this example. In case of the IH-DTL, up to 44 % of the rf power arepumped into the beam. More detailed investigation on the pulse shape will be based on capacitive pick-up signals [5]. Transmission: At a given beam current from the ion source the injected beam current into the linac was varied by cutting the hori- zontal beam emittance with slits. Especially the beam transmission along the RFQ depends very much on the slit width and is ranging from 60 % at Iin = 18 mA (measured in front of the quadruplet lense) up to around 100 % at Iin< 4 mA. The transmission along IH1 (IH2) is above 90 % (95 %) for the whole current range and is approaching 100 % at current levels below 2 mA. So far the RFQ has reached the design current (10 mA), while a maximum of 8 mA was measured behind of IH2. Transverse Emittance: Down to the RFQ exit the slit-grid emittance device was used for all current levels. At higher beam energies, high current and full pulse length, only single shot measure- ment techniques could be applied. Fig. 5 shows maximum current emittance plots from different commissioning steps. The measured LEBT emittance shows, that the matching into the RFQ causes beam losses at high input beam currents: the normalized RFQ acceptance was cal- culated to be 0.3 π mm mrad only. At the exit of IH2 andat a beam current of 7.5 mA the measured norm. 80 % emittance area of. 0.45 π mm mrad is well contained within the 90 % area resulting from LORASR calcula- tions. More detailed studies about the RFQ transmission and comparisons with simulations are planned for the future. Bunch shape measurements with high resolution were done by using diamond detectors [10]. Fig. 3: Beam transformer signals along the HSI. Dashed lines correspond to an improved chopper per- formance (see Ref. 4) Fig. 4: Horizontal emittance measurements taken during different commissioning steps along the HSI REFERENCES [1]U. Ratzinger, Proc. 1996 Linac Conf. Geneva, CERN 96-07, p. 288 [2]P. Spädtke, H. Emig, K.D. Leible, C. Mühle, H. Reich, B.H. Wolf, Proc. 1998 Linac Conf., Chicago, ANL 98/28, p. 884 [3]H. Reich, P. Spädtke, Rev. Sci. Instr., Vol. 71 (2000), p. 707 [4]L. Dahl, P. Spaedtke, these Proceedings [5]W. Barth, these Proceedings [6]U. Ratzinger, Proc. EPAC 2000, Vienna, to be published[7]U. Ratzinger, Proc. HIF 2000, San Diego, accepted for publication in Nucl. Instr. Meth. A [8]G. Hutter, W. Gutowski, W. Hartmann, G. Kube, M. Pilz, W. Vinzenz, these Proceedings [9]J. Klabunde, Proc. 1992 Linac Conf. Ottawa, AECL-10728, p. 570 [10]P. Forck, F. Heymach, T.Hoffmann, A. Peters, P. Strehl, these Proceedings [11]H. Gaiser, K. Kaspar, U. Ratzinger, S. Minaev, proc. 1999 PAC Conf., New York, p. 3552 [12]U. Ratzinger, R. Tiede, Proc. 1996 LINAC Conf, Geneva, p. 128
arXiv:physics/0008137 18 Aug 2000MAIN STEPS FOR FABRICATION OF THE IPHI RFQ M Painchault, J Gaiffier, C Chauvin, J Martin. SEA / DAPNIA / CEA, France Abstract The RFQ of the project IPHI [1] is a 8 meter long, high power, very precise tolerances (0.01 mm on 1 meter long for example) device to accelerate protons. This RFQ is similar to the RFQ of the LEDA project. So, we real ize a thermal and mechanical studies followed by differen t tests for machining and brazing copper. We describe in this paper those different steps and the way we pro ceed to supply the fabrication itself by an independent com pany. 1 SUBMISSION OF THE PROBLEM The main difficulties to fabricate this RFQ and consequences on the design are summarized on table 1: Main characteristics Consequences /G143 1.2MWHF = 15 W/cm2 average with pick at 150 W/cm2. /G143 High Q; low ∆F /G143 Waved profile of the vane tip /G143 Good vacuum/G143 Structure in copper. /G143 Control of thermal expansion /G143 Drilling the cooling channels. /G143 High tolerances: + Minimum number of pieces + brazing in 2 steps + Precise machining /G143 Metallic linkage for helicoflex Table1: Characteristics So, we first do a thermal study to determine dimens ions and positions of the channels and optimize the cool ing characteristics. Second, we design the section. Third, we do a lot of tests to qualify the machinin g and brazing procedures. 2 THERMAL STUDY 2.1 Cooling the vane. 2.1.1 – Topics We search a correct frequency of the cavity (+/- 50 kHz), a constant frequency all along the cavity, va nes at the right position, a maximum stress below the yiel d point. On another hand, it would be more convenient during running operations to obtain small displacements on thevane tip in transition cold / hot period. So, we wa nt the lowest running temperature as possible. 2.1.2 Method of calculation. The geometry is given on figure 1: Figure1: temperature map Figure1: temperatures map First stage is the optimization of the initial 2D 8th section which receives the greatest power density. So we did the following steps: Second stage: 2D calculation of final section: We j ust verify that there is no important derive. Third stage: 3D calculation of the 8th sections. We obtain the temperatures map, stresses map. We have also the evolution of the vane tip position, t he Loop on v1 (Speed of channel in the vane tip) to obtain ∆F=0.Power density issued from Superfish. Lowest values for water temperature and 5 m/s for speeds except channel 1. Correction on power density with the real temperature Final map of temperatures, stresses and RF power. Repartition of power in the 5 channels, Increase of water temperature.Loop on power densityManual loop on position and number of channel.evolution of the bottom of the cavity, the evolutio n of the frequency all along the section: Fourth stage: 3D: Inlet temperature of channel 1 an d 2 are fixed at 10°C. We search a single inlet tempera ture of channel 3,4 and 5. In fact the RFQ will be regulate by channels of the bottom of the cavity. 2.1.3 Results: The temperatures map is given on figure 1. Maximum temperature is 41°C on the bottom. The vane tip are at 13°C. Von Mieses stresses were below 26 MPa, frequency varies between 0 and –50kHz without the extremities. Vane tip moves less than 5 µm. Stresses are greater in 3D than in 2D. So, we re- optimize the geometry. Inlet temperatures of channel 3,4,5 vary between 10 °C and 13°C from section2 (minimum) and section 8 (maximum). 2.2 Segment ends Local RF heat is underestimated by RF simulations. P Balleyguier shows at lease a factor 4 on the power displayed on the edges [2]. But, in fact, global va lues are reliable. They give a good idea of displacements. Displacements are given on figure 2 where the maximum is 0,07 mm at the end of the vane type (blu e color): By the way, an inclined end instead of a rectangula r cut like LEDA RFQ decreases displacements of the vane t ip. The maximum temperature is evaluated to 80°C. The maximum von Mieses stress is about 80 MPa. So, the conclusions for the design is to do smooth edges to spread the RF power and to reduce distance between the channel and the end face. This work is still in progress to be more confident on local values and r educe temperature and stresses. 3 COPPER We are chiefly interested by grains size and therma l expansion. For the thermal expansion coefficient an d the brazing operation, we need chiefly an homogenous copper in the three directions. Grains size is impo rtant to avoid leaks. Properties of copper are strongly dependent on its history. So, we specify hardness, easier to measure and equally dependent on the way to elaborate it. The copper specification precises this two paramete rs. Of course, we fabricate all copper in a single time . We control the forging at the supplier. 4 DESIGN OF THE RFQ The design lays upon the principles detailed in the first paragraph and upon results of the thermal studies. Further, to determine the way to cut the RFQ in segment, in section of 1 meter long, each with four vanes: 2 majors and 2 minors, we completely reproduce the LEDA schedule.[3] We add stainless steel flange for metallic linkage. Flanges for the tuner are also in stainless steel. The output for the cooling is reported on the copper structure . The two stages of brazing are one for stainless ste el components, one for copper pieces. Figure 3: section 1 To align, we will use taylor balls. The general operations of the fabrication procedure is chronologically: + Drilling the channels, + Rough machining of the vanes, + Final machining of the major vanes and control, + Final machining of the minor vanes and control, + Surface treatment, + Assembling minor and major vanes, + RF tests, + Last machining issued from the assemble, + Brazing all components together and control (dimensions and RF). 5 THE DIFFERENT STEPS OF DEVELOPMENT. We have two topics: first determine the best way fo r machining and brazing (it is the technical topic), second to find a supplier for these operations (the commer cial purpose).So, we plan the following schedule: Table2: schedule of development. To design the rough model, we first design the prototype and we keep the most sensible aspects. 6 ROUGH MODEL. It is 500 mm long, has a single vane, one assembled face, and reference faces to see deformations. Results are: Form tolerance of the tip vane: 0,02 m m, flatness of assembled faces: 0,01 mm, parallelism between the assembled face and the vane tip: 0,01 m m. The brazing operation add a 0,01 mm deformation. Th e drilling test gives an displacement below 0,5 mm bu t the operation stays risky. It was the hoped values which were held by SICN. So , it was the chosen supplier. 7 TEST ON 1 METER LONG. In fact, we identify the technical difficulties and imagine a test for each one. Those tests and their results are given on table 3. One of them is the assemblage of stainless steel an d copper. We chose a brazing solution because electro n beam welding has been tested to assemble copper wit h too greatest deformations. The brazed method needs precise machining tolerances but experience of prec ise values were obtained for vanes. To determine the desired tolerance, we use a therma l mechanical program called Castem, developed by CEA. This program can simulate the elastic and plastic b ehavior of materials during thermal cycles. 8 PRE - PROTOTYPE. The pre – prototype is mechanically the same than t he prototype but we simplify it to reduce costs. For e xample, there is only one vane tip machined with the correc t profile, holes for pumping ports are done but just one will be brazed, few cooling holes will be drilled. At the opposite, we could make cuts to visualize deformations and see the quality of the brazed junc tion. We yet have roughly machined the pieces. We still have to machine the waved profile on one major piec e and to braze the pre - prototype. Figure 4: major vane roughly machined 9 CONCLUSION. We plan to achieve the pre prototype for middle of October 2000, fabrication of the first section woul d be ended beginning 2001 and the RFQ in middle of 2002. REFERENCES [1] R Ferdinand, Status report on 5 MeV IPHI RFQ, l inac 2000. [2] P Balleyguier, 3D design of the IPHI RFQ cavity, this conference (linac 2000, Monterey). [3] D L Schrage, CW RFQ Fabrication and Engeenering , Linac 1998Tests Topics Results Machining faces on 1 m long block + thermal treatment./c55 /c82 /c3/c89 /c72 /c85 /c76 /c73 /c92 /c3/c83 /c82 /c86 /c86 /c76 /c69 /c76 /c79/c76 /c87/c76 /c72 /c86 /c3/c82 /c73 /c3/c87/c75 /c72 /c3 /c80 /c68 /c70 /c75 /c76 /c81 /c72 /c68 /c81 /c71 /c3 /c87/c75 /c72 /c3/c72 /c73/c73 /c72 /c70 /c87 /c3/c82 /c73 /c3 /c75 /c76 /c74 /c75 /c3/c87/c72 /c80 /c83 /c72 /c85 /c68 /c87/c88 /c85 /c72 /c86 /c17Flatness 0,01 mm. Variation 0,02 mm. Drilling 1 m long holes. /c41 /c72 /c68 /c86 /c76 /c69 /c76 /c79/c76 /c87/c92 Precision 0,4 mm Material control after thermal treatment/c38 /c75 /c68 /c85 /c68 /c70 /c87 /c72 /c85 /c76 /c93 /c72 /c3/c74 /c85 /c68 /c76 /c81 /c3/c86 /c76/c93 /c72 /c17 Done. Brazing stainless steel flange on copper cavity/c55 /c82 /c3/c80 /c76 /c81 /c76/c80 /c76 /c93 /c72 /c3/c71 /c72 /c73/c82 /c85 /c80 /c68 /c87/c76 /c82 /c81 /c3/c82 /c81 /c3/c87 /c75 /c72 /c3 /c89 /c68 /c81 /c72 /c87/c76/c83 /c17Variation of the position below 0,01 mm. Brazing on 500 mm long /c55 /c82 /c3/c89 /c72 /c85 /c76 /c73 /c92 /c3/c87/c75 /c72 /c3/c69 /c85 /c68 /c93 /c76 /c81 /c74 /c3/c70 /c82 /c81 /c71 /c76 /c87/c76 /c82 /c81 /c86Deformation: 0,01 mm ; Compactness with ultrasounds: 90% Brazing on 1m long /c55 /c82 /c3/c84 /c88 /c68 /c79/c76 /c73/c92 /c3 /c87/c75 /c72 /c3 /c90 /c75 /c82 /c79 /c72 /c3/c83 /c85 /c82 /c70 /c72 /c71 /c88 /c85 /c72 /c69 /c72 /c73/c82 /c85 /c72 /c3/c87 /c75 /c72 /c3 /c83 /c85 /c72 /c3/c83 /c85 /c82 /c87/c82 /c87 /c92 /c83 /c72 /c17In progress Obtrusion of holes /c55 /c82 /c3/c84 /c88 /c68 /c79/c76 /c73/c92 /c3 /c87/c75 /c72 /c3/c83 /c85 /c82 /c70 /c72 /c71 /c88 /c85 /c72 No leak Machining and brazing the pumping ports/c55 /c82 /c3/c84 /c88 /c68 /c79/c76 /c73/c92 /c3 /c87/c75 /c72 /c3/c83 /c85 /c82 /c70 /c72 /c71 /c88 /c85 /c72 /c3/c69 /c72 /c73/c82 /c85 /c72 /c3 /c87 /c75 /c72 /c83 /c85 /c72 /c3/c83 /c85 /c82 /c87 /c82 /c87/c92 /c83 /c72Tolerance within 0,01 mm Machining the definitive vane tip on 300 mm long and control it./c55 /c82 /c3/c89 /c68 /c79 /c76 /c71 /c68 /c87/c72 /c3/c87/c75 /c72 /c3/c80 /c68 /c70 /c75 /c76 /c81 /c76/c81 /c74 /c3 /c83 /c85 /c82 /c74 /c85 /c68 /c80 /c68 /c81 /c71 /c3 /c71 /c72 /c87 /c72 /c85 /c80 /c76 /c81 /c72 /c3/c83 /c85 /c72 /c70 /c76/c86 /c76 /c82 /c81 /c3/c82 /c73/c3/c87/c82 /c82 /c79 /c86 /c17 /c3 /c55 /c82 /c89 /c68 /c79 /c76 /c71 /c68 /c87/c72 /c3/c87/c75 /c72 /c3/c70 /c82 /c81 /c87/c85 /c82 /c79 /c3/c83 /c85 /c82 /c70 /c72 /c71 /c88 /c85 /c72 /c17Form factor < 0,02 mm. Machining the final vane tip on 1 m long./c55 /c82 /c3/c89 /c68 /c79 /c76 /c71 /c68 /c87/c72 /c3/c82 /c81 /c3/c87/c75 /c72 /c3 /c87/c85 /c88 /c72 /c3 /c71 /c76 /c80 /c72 /c81 /c86 /c76 /c82 /c81 /c86 In progressTechnical steps Topics Commercial steps Calendar 1 –Preliminary tests = on small pieces. Invitation to tender 3 months 2 –representative rough model. (500 mm long)+ analyse the manufacturing constraints + First procedureInvitation to tender 3 months 3 –Tests on 1 meter long To validate all parameters First step of the market: firm10 months 4 –pre – prototype First repetition First step of the market: firm3 months 5 –Prototype Second step of the market: firm3 months 6 –All sections.Optional: depends on results on the prototype 20 months
arXiv:physics/0008138v1 [physics.acc-ph] 18 Aug 2000RESULTS ONPLASMA FOCUSING OF HIGH ENERGYDENSITY ELECTRONAND POSITRONBEAMS∗ J.S.T. Ng,P. Chen, W. Craddock, F.J. Decker, R.C. Field, M.J .Hogan,R. Iverson, F. King,R.E. Kirby,T.Kotseroglou,P. Raimondi,D. Walz, SL AC, Stanford, CA. 94309,USA H.A.Baldis†,P. Bolton,LLNL,Livermore,CA. 94551,USA D. Cline, Y.Fukui, V. Kumar,UCLA, LosAngeles,CA. 90024,US A C. Crawford, R. Noble,FNAL,Batavia, IL.60510,USA K. Nakajima,KEK, Tsukuba,Ibaraki 305-0801,Japan A. Ogata,HiroshimaUniversity,Kagamiyama,Higashi-Hiro shima,739-8526Japan A.W.Weidemann,UniversityofTennessee, Knoxville,Tenne ssee37996,USA Abstract We present results from the SLAC E-150 experiment on plasma focusing of high energy density electron and, for the ﬁrst time, positron beams. We also discuss measure- mentsonplasmalens-inducedsynchrotronradiation,longi - tudinaldynamicsofplasmafocusing,andlaser-andbeam- plasmainteractions. 1 INTRODUCTION The plasma lens was proposed as a ﬁnal focusing mecha- nism toachievehighluminosityforfuturehighenergylin- ear colliders [1]. Previous experimentsto test this concep t were carried out with low energy density electron beams [2]. In this paper, we present preliminary results obtained recently by the E-150 collaborationon plasma focusing of highenergydensityelectronandpositronbeams. Table 1: FFTB electron and positron beam parameters for thisexperiment. Parameter Value Bunchintensity 1.5×1010particlesperpulse Beam size 5 to8 µm (X), 3 to5 µm (Y) Bunchlength 0.7mm Beam energy 29GeV Normalizedemittance 3 to5 ×10−5m-rad(X), 0.3to0.6 ×10−5m-rad(Y) Beam density ∼7×1016cm−3 2 EXPERIMENTAL SETUP The experiment was carried out at the SLAC Final Focus Test Beam facility (FFTB)[3]. The experiment operated ∗Work supported in part by the Department of Energy under cont racts DE-AC02-76CH03000, DE-AC03-76SF00515, DE-FG03-92ER406 95, and DE-FG05-91ER40627, and the Univ. of California Lawrenc e Liv- ermore National Laboratory, through the Institute for Lase r Science and Applications, under contract No. W-7405-Eng-48; and by the US-Japan Program for Cooperation in High Energy Physics. †Also at UCDavis, Dept. of Applied Science.parasitically with the PEP-II B-factory; the high energy electron and positron beams were delivered to the FFTB at 1 - 10 Hz from the SLAC linac. The beam parameters aresummarizedinTable 1. A layout of the beam line and a schematic drawing of the plasma chamberare shownin Figure 1. The beam size was measuredusinga wirescannersystem. A carbonﬁber 4µm or7µm in diameter was placed downstream of the plasmalens,adjustablealongthebeamaxisin arangeof8 to 30 mm fromthe centerof the lens. The Bremsstrahlung photonsweredetectedinaCherenkovtypedetectorlocated 35mdownstreamofthelens. Asetofionizationchambers interleaved with polyethylene blocks, located 33 m down- stream of the lens, was used to monitor the synchrotron radiation emitted as a result of the strong bending of the beam particles by the plasma lens. This detector provided an independent measure of the focusing strength. Also, a Cherenkov target was installed in the electron beam line downstreamtoenablestreakcameradiagnosticsofthelon- gitudinalplasmafocusingdynamics. To create the plasma lens, a short burst (800 µs dura- tion) of neutral nitrogen or hydrogengas, injected into the plasmachamberbyafast-pulsingnozzle,wasionizedbya laserand/orthehighenergybeam. Theneutraldensitywas determined by interferometryto be 4×1018cm−3for N 2 and5×1018cm−3forH 2ataplenumpressureof1000psi. The injected gas was evacuated by a differential pumping system which made operationof the gas jet possible while maintaining ultra-high vacuum in the beam lines on either side ofthechamber. 3 PLASMA FOCUSING For a bunched relativistic beam traveling in vacuum, the Lorentz force induced by the collective electric and mag- netic ﬁelds is nearly cancelled,makingit possible to prop- agate over kilometers without signiﬁcant increase in its emittance. In response to the intruding beam charge and current,theplasmaelectrondistributionisre-conﬁgured to neutralize the space charge of the beam and thereby can- cel itsradialelectricﬁeld. Fora positronbeam,theplasma electrons are attracted into the beam volume thus neutral-/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1 /0/0 /1/1 /0/0/0 /1/1/1scanner /0/0/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1/1/1/1 ~ 1 m Scale/0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0 /1/1/1/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 Wire Not to/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/1/1 /0/0/0/0/0 /1/1/1/1/1 /0/0/1/1Final Quads Wire scanner DetectorSynch. Radiation Monitor/ Photon beam line Beam dumpSteering dipolePlasma Lens e / e- e / e-+ + Gas-jet nozzleLaserblower portRoots e / e- + 2 mm dia. foilsTurbo pump portDipole targetCherenkov to streak camera Figure1: Layoutoftheplasmalensmeasurementsetupandsch ematicsoftheplasmachamber. izing it; for an electron beam, the plasma electrons are expelled from the beam volume, leaving behind the less mobile positive ions which neutralize the beam. When the beam radius is much smaller than the plasma wave- length, the neutralization of the intruding beam current by theplasmareturncurrentisineffectivebecauseofthesmal l skindepth. Thisleavestheazimuthalmagneticﬁeldunbal- anced which then “pinches” the beam. In this experiment, typical plasma densities were of the order of 1018cm−3, corresponding to a plasma wavelength of approximately 30µm which was indeed much larger than the incoming beamradius. The plasma was created by meansof beam self-induced ionization and laser avalanche ionization. For the case of beam self-ionization, a small fraction of the neutral gas molecules was ionized due to collisions with the high en- ergy beam particles. The secondary electrons from this impact ionization process were accelerated by the intense collective ﬁeld in the beam, transverse to the direction of propagation,to furtherionizethe gas[4]. That is, the head of the bunch was able to ionize the gas while the core and thetailofthebunchwerefocused. Amorequantitativeun- derstandingrequiresdetailedcalculationswhicharenoty et availableforthisexperimentalsetup. Theresultsonlaserpre-ionizationplasmafocusingwere obtained using a turn-key infrared ( λ= 1064nm) laser system. Itdelivered1.5Joulesofenergyperpulseof10ns FWHMat10Hz. Thelaserlightwasbroughttoalinefocus atthegasjet;theplasmathusproducedwasapproximately 0.5mmthickasseenbythe e+/e−beams.With the relatively long infrared laser pulse, the pulse front was able to ionize a small fraction of the gas by multiple-photon absorption; the resulting secondary elec - tronswereaccelerated,transversetothelaser’sincident di- rection, to further ionize the gas. This process led to an avalanche growth in plasma density, similar to the beam self-ionizationcase. 3.1 Resultson plasmafocusing The results for laser (and beam) ionization plasma focus- ing of electron and positron beams are shown in Figures 2 and 3, respectively. The measured transverse beam size is shown as a function of the distance (Z) between the wire scanner and the plasma lens. The axis of the gas jet is at Z = -10.5 mm. In the X-dimension, the beam enve- lopeisshownconvergingwithoutplasmafocusing(triangle points);whilewithlaser(andbeam)inducedplasmafocus- ing(ﬁlledcircles),thebeamenvelopeisshownconverging towards a reducedwaist and then divergingbecause of the strong focusing. In the Y-dimension, the waist is at a lo- cation close to the the plasma lens beyondthe reach of the wire scanner; the beam envelope is seen diverging due to the strong plasma focusing. Focusing is also observed for beam-inducedplasmawith thelaserturnedoff. 4 OTHERRESULTS Discussionsonadditionalresultsobtainedfromthisexper - imentcanbefoundin[6]. A briefsummaryis givenhere.Plasma Focusing of Electron Beams 012345678 -10 -5 0 5 10 15 20 25Z(mm)Beam size σx (µm) Laser (and beam) induced plasmaNo plasma Beam induced plasmaPlasma LensE150 Preliminary (Jul.2000) N2 gas 0246810 -10 -5 0 5 10 15 20 25Z(mm)Beam size σY (µm)Laser (and beam) induced plasma No plasmaBeam induced plasmaPlasma Lens E150 Preliminary (Jul.2000)N2 gas Figure2: PlasmafocusingforelectronbeamsintheX(top) andY (bottom)dimensions. Duringthe plasmafocusingmeasurements,thefocusing strength was also measured independently by monitoring the synchrotron radiation emitted by particles focused by the lens. The critical energy was estimated to be a few MeV, corresponding to a focusing gradient of 106T/m. The longitudinal focusing dynamics was diagnosed with a streak camera with pico-second time resolution and, as expected, the focusing was strongest at the longitudinal center of the bunch. The laser- and beam-plasma interac- tion was studied by varyingthe laser pre-ionizationtiming withrespecttothebeamarrivaltime;weobservedadelay- correlatedmodulationoftheplasmafocusinginthe“after- glow”regime. 5 SUMMARY AND OUTLOOK Results on plasma focusing of 29 GeV electron and, for the ﬁrst time, positron beams have been presented. Beam self-ionization turned out to be an economical method for producing a plasma lens. The infrared laser with a 10 ns longpulsealsoprovedtobeefﬁcientinplasmaproduction, resulting in the strong focusing of electron and positron beams. Dataonotheraspectsofplasmafocusingwerealso collected;detaileddiscussionispresentedelsewhere[6] .Plasma Focusing of Positron Beams 0246810 -10 -5 0 5 10 15 20 25Z(mm)Beam size σx (µm) Laser (and beam) induced plasmaNo plasma Beam induced plasmaPlasma LensE150 Preliminary (Apr.2000) N2 gas 0246810 -10 -5 0 5 10 15 20 25Z(mm)Beam size σY (µm)Laser (and beam) induced plasma No plasmaBeam induced plasmaPlasma Lens E150 Preliminary (Apr.2000)N2 gas Figure3: PlasmafocusingforpositronbeamsintheX(top) andY (bottom)dimensions. Design studies for linear collider applications are just starting. The ﬁrst issue to resolve is the effect of beam jit- ter on the achievableluminosityof plasma focusedbeams. Optimization of plasma lens parameters requires bench- markingofcomputercodesaswellasbetterunderstanding of the various plasma production processes. The experi- encegainedinthisexperimentwill serveasabasisforfur- therengineeringdesignstudiesforaneventualplasmalens application. 6 REFERENCES [1] P.Chen, Part.Accel., 20, 171(1987). [2] J.B. Rosenzweig et al., Phys. Fluids B 2, 1376(1990); H. Nakanishi et al., Phys. Rev. Lett, 66, 1870(1991); G. Hairapetian et al., Phys. Rev. Lett. 72, 2403(1994); R.Govilet al.,Phys.Rev. Lett. 83, 3202(1999). [3] V. Balakin etal., Phys.Rev. Lett. 74, 2479(1995). [4] R.J.BriggsandS.Yu,LLNLReportUCID-19399,May1982 (unpublished). [5] B.Chang et al.,Phys.Rev. A 47, No.5, 4193(1993). [6] J.S.T.Ng etal.,Proceedingsofthe9thAdvancedAccelerator Concepts Workshop, June 2000, Santa Fe,NM.
arXiv:physics/0008139v1 [physics.atom-ph] 18 Aug 2000Frequency dependent hyperpolarizabilities of atoms; calc ulations using density-functional theory Arup Banerjee and Manoj K. Harbola Laser Physics Division Center For Advanced Technology, Ind ore 452013, India Abstract Using the orbitals generated by the van Leeuwen-Baerends po tential [Phys. Rev. A 49, 2421 (1994)], we calculate frequency-dependent response proper- ties of the noble gas atoms of helium, neon and argon and the al kaline earth atoms of berrylium and magnisium, with particular emphasis on their non- linear polarizabilities. For this, we employ the time-depe ndent Kohn-Sham formalism with the adiabatic local-density approximation (ALDA) for the ex- change and correlation. We show that the results thus obtain ed for frequency- dependent polarizabilities (both linear and nonlinear) of the inert gas atoms are highly accurate. On the other hand, polarizabilities of the alkaline earths are not given with the same degeree of accuracy. In light of th is, we make an assessment of ALDA for obtaining linear and nonlinear respo nse properties by employing time-dependent density-functional theory. 1I. INTRODUCTION Over the past decade, the time dependent density-functiona l theory (TDDFT) [1,2] is be- ing used increasingly to study frequency dependent respons e properties [3–14] and excitation energies [2,11,15] of many-electron systems. The theory fo und its ﬁrst application in the cal- culation of frequency dependent polarizabilities of atoms and photo-absorption cross section of atoms and molecules [3,4]. However, the existence of TDDF T was not formally proved at that time. It was some years later that Deb and Ghosh [16], B artolotti [17] and Runge and Gross [18] laid rigorous foundations of TDDFT. Analogou s to its stationary counter- part TDDFT is exact in principle but its practical implement ation requires approximating the exchange-correlation (XC) energy functional. In most o f the applications the adiabatic local density approximation (ALDA) [1] is used. In this appr oximation the nonlocal time- dependence of the functional is ignored and the spatial depe ndence is also treated locally. As such the functional has the same form as the local-density approximation (LDA) of the stationary-state theory [19,20]. Results for the response properties using the LDA for the unperturbed system and ALDA for perturbation calculations show that with these approx- imations the polarizability at zero frequency (static pola rizability) as well as its frequency dependence are overestimated. Use of LDA+ALDA for the respo nse property calculations introduces errors at two diﬀerent levels. First, the use of t he LDA for calculation of the unperturbed orbitals gives potentials and densities which are asymptotically not correct. In particular the tail of the LDA potential is exponential rath er than its correct −1 rbehaviour [21]. Thus the physical propreties that depend on the asympt otic nature of the ground-state orbitals and densities, for example the response propertie s, are not determined accurately by the LDA orbitals. Secondly, errors in the response proper ty calculations arise beacause: (i) further change in the potential itself is being calculat ed approximately, and (ii) ALDA is valid only in the limit of zero frequency. There have been att empts to rectify the problem at the ﬁrst level by employing methods which reproduce the asym ptotic behaviour of the po- tential correctly. Thus Senatore and Subbaswamy [7] applie d the self-interaction correction 2[22] method to obtain accurate orbitals with proper asympto tic decay. Later Zong et. al [8] devised a scissors operator technique and Gisbergen et al. [ 9] employed a model potential with the desired asymptotic behaviour to improve the polari zabilities. Gisbergen et al. [11] also studied frequency dependent polarizabilities of He, B e and Ne by employing exact XC potential for the ground-state of these atoms coupled with A LDA for the XC kernels. Nearly exact ground-state wavefunction along with the ALDA for XC k ernels were employed by us [13] to calculate nonlinear optical coeﬃcients of helium, a nd to investigate the accuracy of ALDA in predicting the frequency dependence of nonlinear po larizabilities. All these studies show that the inaccuracy in the response pr operties arise mainly from the use of the LDA to obtain the unperturbed densities; For pe rturbative calculations, on the other hand, ALDA appears to be a reasonably accurate appr oximation, particularly at frequencies in the optical range. One may therefore conclud e that if the asymptotic nature of the potential is corrected, the response properties at op tical frequencies will come out to be copmparable to the ab-initio or experimental values even with the use of ALDA. However, the conclusions above cannot said to be general since most of the studies have been conﬁned to investigating the linear polarizabilities. Against this background it then becomes necessary to invest igate how does ALDA per- form in the calculation of frequency dependent nonlinear op tical coeﬃcients of heavier atoms when asymptotically accurate ground-state orbitals are us ed for this purpose. For the helium atom we could perform such a study with a near exact ground-st ate orbital [23] obtained from its Hylleraas wavefunction. However, this is not possi ble for systems with larger num- ber of electrons. For such systems we generate the ground-st ate orbitals and potentials by using the model potential introduced by van Leeuwen and Baer ends (LB) [24] as a cor- rection to the LDA XC potential. Recently we have employed th is potential to calculate the static nonlinear response properties of several atoms a nd ions [25]. We found that the orbitals given by this model potential, when used with the LD A for the higher derivatives of XC potential, give reasonably accurate static hyperpolari zabilities for the inert gas atoms. Thus to assess the accuracy of ALDA in predicting the frequen cy dependence of nonlinear 3polarizabilities, in this paper we ﬁrst study dynamic hyper polarizabilities of these atoms calculated by employing the orbitals generated by the LB pot ential. In our study [13] of the static properties, we also found that the hyperpolarizabil ities for the alkaline earths show some improvement with the inclusion of the LB correction alt hough their linear polarizabil- ities remain unaﬀected. In this paper we study the frequency -dependent polarizabilities of the alkaline earth atoms of Be and Mg also and show that the dyn amic polarizabilities of these systems follow the same trend as their static counterp arts. In the following, we ﬁrst brieﬂy describe the LB potential. We follow that with a short description of the variation- perturbation method of calculating the dynamic response pr operties. We then present our results and conclude the paper with a discussion. van Leeuwen and Baerends [24] proposed a correction to the LD A potential primarily to correct its asymptotic behaviour. For this they ﬁrst noted t hat the Becke’s correction [26] to the LDA functional gives the correct exchange-energy densi ty but fails to give the potential correctly. Thus they suggested that a Becke-like correctio n be added to the potential directly, and found that such a correction brings the approximate pote ntial very close to the exact one. Thus the highest occupied orbital eigenenergy as obtai ned from this potential is close to the ionization energy of a many-electron system. More rec ently, it has been shown that the corresponding total energy is also quite accurate [25] . The eﬀective potential proposed by LB has the form vxc(r) =vLDA xc(r) +vLB(r) (1) with vLB(r) =−βρ1 3x2 1 + 3βxsinh−1(x), (2) where x=\|∇ρ\| ρ4 3andβ= 0.05. Note that the extra term vLB(r) added to the LDA potential is like the Becke term. It therefore represents the correcti on to only the exchange component of the potential. Using the potential given by Eq.(1) and (2) we generate the ground-state orbitals and potentials for the above mentioned atoms and th en employ these orbitals for the calculation of frequency dependent linear and nonlinea r polarizabilities. 4We perform calculations of the optical response properties by employing the variation- perturbation method [10] within the time dependent Kohn-Sh am (TDKS) formalism of TDDFT. This method has been used in the past to calculate [13] polarizabilities corre- ponding to several nonlinear optical phenomena. The method and expressions for various hyperpolarizabilities have been discussed [10,13] in our e arlier works. Thus we do not de- scribe these in detail here. It is suﬃcient to mention that th e linear polarizability and the coeﬃcient corresponding to the degenerate-four-wave-mix ing (DFWM) are obtained directly by minimizing the second-order and fourth-order changes in the quasi-energy with respect to the ﬁrst and second-order orbitals, respectively. The ot her nonlinear coeﬃcients are not related directly with the fourth-order energy change. Howe ver, expressions for these coeﬃ- cients in terms of second-order orbitals have been derived [ 13,27] within TDKS theory and it is these expressions which we employ here. To perform our c alculations, we represent the radial part of the induced orbitals by a linear combination o f the Slater type orbitals (STO). Thus it is given as f(r) =/summationdisplay iCirnie−ηir, (3) where niandηiare the parameters of STO which are ﬁxed and Ciare the linear vari- ational parameters. Parameters niandηiare chosen in such a way that the functions correctly represent the excited states. We have optimized t hese parameters by calculat- ing the static poalrizabilities and hyperpolarizabilitie s from the LDA ground-state orbitals, and matching the results with the numbers obtained by Stott a nd Zaremba [4]. For the exchange-correlation energy we have used the Gunnarsson-L undquist parametrization [28], which is the same as used by Stott and Zaremba. Thus our result s for hyperpolarizabilities are around 10% less than those of Senatore and Subbaswamy [7] who use the Perdew-Zunger [22] parametrization. We now present the results of our calculation. Although our m ain focus in this paper is on the frequency dependent hyperpolarizabilities, for c ompleteness we ﬁrst discuss the results for frequency dependent polarizabilities. 5In Figs.1-3 we show the ratioα(ω) α(0)for the inert gas atoms He, Ne and Ar as a function of frequency ωand for comparison also display the corresponding ab-initi o results. For helium we compare our results with the results of Bishop and Lam [29] and for neon and argon the comparison is made with the MP2 results of Rice [30]. Since α(0) is already reproduced [25] quite accurately with the LB orbitals, it is clear from t hese ﬁgures that α(ω) is also accurate and matches quite well with the ab-initio results. For helium the ab-initio results are essentially reproduced by our calculations for frequen cies up to 0.5 a.u. (wavelength of about 914 ˚A). For neon the match with the MP2 results is good till about 0 .1 a.u. and for argon good match is obtained up to ω= 0.07 a.u.. For comparison we also show the corresponding LDA results. It is quite clear that the LDA res ults are highly inﬂated in comparison to both the LB corrected and the ab-initio number s. In Figs. 4 and 5, we present the results ofα(ω) α(0)for the Be and Mg atoms. The results for these atoms are quite diﬀerent from those of the inert gas atoms discussed above. Here the LDA and the LB corrected results are essentially the same . This is intriguing since the energies and the highest occupied eigenvalues obtained from the two schemes diﬀer signiﬁcantly. A possible reason for this could be that since the outer electrons in these systems are less tightly bound, the exchange and correlatio n eﬀects play a relatively more important role in determining the polarizabilities, and AL DA is not suﬃcient to represent their eﬀects accurately. Thus although the unperturbed orb itals are improved by inclusion of the LB correction, this alone is not suﬃcient to get over th e inaccuracy of ALDA for these systems. Having discussed linear polarizabilities, we now proceed o n to present the results for the nonlinear coeﬃcients corresponding to third harmonic g eneration (THG) and DFWM of these atoms. As is the case with polarizabilities, these r esults are also obtained from the orbitals generated by the LB potential with the use of ALDA fo r perturbative calculations. First, we discuss the results for helium. Exact results corr esponding to the above men- tioned nonlinear optical eﬀects for helium are available ov er a range of frequencies. In Figs. 66 and 7 we display our results in comparison with the ab-initi o results [31]. Plotted in Fig. 6 is the DFWM coeﬃcientγ(ω) γ(0), and in Fig. 7 we plot the THG coeﬃcientγ(3ω) γ(0)as a function ofω. It is evident from the ﬁgures that the results for hyperpola rizabilities obtained from the LB orbitals are also highly accurate. The LDA, on the othe r hand, overestimates the frequency dependence of these quantities by a large amount. A comparative study like this is not possible at all frequencies for neon and argon because of the lack of available data. However, some experimental results are available [32,33] a t few discrete frequencies for all three atoms. We now compare our results with these experimen tal numbers. In Table I we present our results for the THG coeﬃcients of the atoms considered in this paper at two distinct wavelengths, namely, λ= 10550 ˚A (ω≈0.0433 a.u.) and λ= 6943 ˚A (ω≈0.0658 a.u.). We do so because experimental results for THG by h elium, neon and argon exist at these wavelengths. For a complete picture, we give the results obtained from both the LDA and the LB orbitals. It is again clear that althou gh the LDA orbitals give a large error in the estimates of these quantities for the nobl e gas atoms, the LB corrections to the LDA eliminates almost all of this error. Further, the n umbers obtained from the LB orbitals lie within the experimental error bounds. For th e other two atoms also the LB corrected results for the THG coeﬃcients are less than the LD A numbers. However, there are no experimental numbers available for these systems. No netheless, on the basis of their zero frequency results, we expect the LB numbers to be closer to experiments than the LDA numbers. In Table II we present the results of DFWM coeeﬁcien ts at the same frequencies, although no experimental data exists for this eﬀect. Howeve r, we expect these results to be quite accurate for the noble gas atoms, and moderately acc urate for the alkaline earths. This is because the maximum deviation from the ab-initio or t he experimental results is observed for THG coeﬃcients which have already been shown to be accurate. To conclude, our study above indicates that when accurate or bitals are employed to calculate response properties, ALDA reproduces the freque ncy dependence of linear as well as nonlinear response properties of the noble gas atoms quit e accurately. On the other hand, for the alkaline earths its behaviour with respect to the lin ear and nonlinear polarizabilities 7is quite diﬀerent. Thus it appears that for systems, such as B e and Mg, where the electrons are loosely bound, ALDA is not a good approximation to calcul ate the eﬀects of exchange and correlation. In a recent study, it has been shown that ALD A is a major component [34]of the exchange-correlation functional for time-depe ndent hamiltonians. This has been done by considering two electrons moving in a time-dependen t potential. However, our study indicates that this is true only if the electrons are tightly bound. 8REFERENCES [1] E.K.U. Gross, J.F. Dobson and M. Petersilka, in Density Functional Theory , Edited by R.F. Nalewajski, Topics in Current Chemistry vol. 181 (Spri nger, Berlin, 1996). [2] M. Casida, in Recent Advances in Density Functional Methods Edited by D.P. Chong (World Scientiﬁc, Singapore, 1995). [3] A. Zangwill and P. Soven, Phys. Rev. A 21, 4274 (1980). [4] M.J. Stott and E. Zaremba, Phys. Rev. A 21, 12 (1980); Phys. Rev. A bf 22, E2293 (1980). [5] S.K. Ghosh and B.M. Deb, Chem. Phys. 71, 295 (1982); J. Mol. Struc. (Theochem) 103, 163 (1983). [6] L.J. Bartolotti, J. Chem. Phys. 80, 5187 (1984); J. Phys. Chem. 90, 5518 (1986). [7] G. Senatore and K.R. Subbaswamy, Phys. Rev. A 35, 2440 (1987). [8] H. Zong, Z. Levine and J.W. Wilkins, Phys. Rev. A 43, 4629 (1991). [9] S.J.A. Gisbergen, V.P. Osinga, O.V. Gritsenko, R. van Le euwen, J.G. Snijders and E.J. Baerends, J. Chem. Phys. 105, 3142 (1996). [10] A. Banerjee and M.K. Harbola, Phys. Lett. A 238, 525 (1997). [11] S.J.A. Gisbergen, F. Kootstra, P.R.T. Shipper, O.V. Gr itsenko, J.G. Snijders and E.J. Baerends, Phys. Rev. A 57, 2259 (1998). [12] S.J.A. Gisbergen, J.G. Snijders and E.J. Baerends, J. C hem. Phys. 109, 10644 (1998). [13] A. Banerjee and M.K. Harbola, Eur. Phys. J. D 5, 201 (1999). [14] F. Aiga, T. Tada and R. Yoshimura, J. Chem. Phys. 111, 2878 (1999). [15] M. Petersilka, U.J. Grossmann and E.K.U. Gross, Phys. R ev. Lett. 76, 1212 (1996). 9[16] B.M. Deb and S.K. Ghosh, J. Chem. Phys. 77, 342 (1982). [17] L.J. Bartolotti, Phys. Rev. A 24, 1661 (1981); Phys. Rev. A 26, 2243 (1982). [18] E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984). [19] R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford, New-York, 1989). [20] R.M. Dreizler and E.K.U. Gross, Demsity Functional Theory: An Approach to Many- Body Problem (Springer Verlag, Berlin, 1990). [21] C. O. Almbladh and U. von Barth, Phys. Rev. B 31, 3231 (1985). [22] J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [23] T. Koga, Y. Kasai and A. J. Thakkar, Int. J. Quantum Chem. 46, 689 (1993). [24] R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994). [25] A. Banerjee and M. K. Harbola, Phys. Rev. A 60, 3599 (1999). [26] A. D. Becke, Phys. Rev. A 38, 3098 (1988). [27] A. Banerjee, Ph. D Thesis (Devi Ahiliya Vishva Vidyalay a, Indore, India, 1999). [28] O. Gunnarsson and B. I. Lundquist, Phys. Rev. B 13, 4274 (1976). [29] D. M. Bishop and B. Lam, Phys. Rev. A 37, 464 (1988). [30] J. E. Rice, J. Chem. Phys. 96, 1992. [31] D. M. Bishop and J. Pipin, J. Chem. Phys. 91, 3549 (1989). [32] H. T. Lehemeier, W. Leupacher and A. Penzkofer, Opt.Com m.56, 67 (1985). [33] G. H. C. New and J. F. Ward, Phys. Rev. Lett. 19, 556 (1967). [34] I. D. Amico and G. Vignale, Phys.Rev. B 59, 7876 (1999). 10A. Table Captions Table I : THG coeﬃcients with and without the LB correction to the LDA along with their experimental values (in atomic units). Table II : DFWM coeﬃcients with and without the LB correction to the LD A (in atomic units). 11Table I Atom λ= 6943 ˚A λ= 10550 ˚A LDA LDA+LB Expt. [33] LDA LDA+LB Expt. [32] He 98.46 48.35 53.6±7 89.12 45.11 44.07±4.8 53.6±2.4 Ne 230.30 103.68 119±13 202.67 95.50 78.27±8.3 96.5±4.8 Ar 2532.8 1560.3 1691±167 1925 1275.1 1021.3±107 786±48 Be 6.42×1081.45×108- 1.41×1051.07×105- Mg no minimum no minimum - 1.88×10113.17×109- Table II Atom λ= 6943 ˚A λ= 10550 ˚A LDA LDA+LB LDA LDA+LB He 87.62 44.57 84.86 43.58 Ne 198.16 94.06 190.45 91.65 Ar 1863.0 1226.1 1725.5 1154.8 Be 9.83×1047.33×1046.17×1044.83×104 Mg 4.46×1051.98×1052.08×1051.14×105 12Figure Captions Fig. 1 : Plot of α(ω)/α(0) as a function of frequency ωfor helium. The squares, open circles and ﬁlled circles represent the LDA, the LDA+LB and ab-initio results [29], respectively. Fig. 2 : Plot of α(ω)/α(0) as a function of frequency ωfor neon. The squares, open circles and ﬁlled circles represent the LDA, the LDA+LB and ab-initi o results [30], respectively. Fig. 3 : Plot of α(ω)/α(0) as a function of frequency ωfor argon. The squares, open circles and ﬁlled circles represent the LDA, the LDA+LB and a b-initio results [30], respec- tively. Fig. 4 : Plot of α(ω)/α(0) as a function of frequency ωfor Be. The squares and ﬁlled circles represent the LDA and the LDA+LB results, respectiv ely. Fig. 5 : Plot of α(ω)/α(0) as a function of frequency ωfor Mg. The squares and ﬁlled circles represent the LDA and the LDA+LB results, respectiv ely. Fig. 6 : Plot of γ(3ω)/γ(0) (THG) as a function of frequency ωfor helium. The squares, open circles and ﬁlled circles represent the LDA, the LDA+LB and ab-initio results [31], respectively. Fig. 7 : Plot of γ(ω)/γ(0) (DFWM) as a function of frequency ωfor helium. The squares, open circles and ﬁlled circles represent the LDA, t he LDA+LB and ab-initio results [31], respectively. 130.000.020.040.060.080.101.01.11.21.31.41.5α(ω)/α(0) ω0.000.020.040.060.080.101.21.51.8 α(ω)/α(0) ω0.0 0.1 0.212 γ(ω;ω,−ω,ω) /γ(0) ω0.00 0.04 0.081.001.011.021.03 α(ω)/α(ο) ω0.000.020.040.060.080.101.0001.0051.0101.015 α(ω)/α(ο) ω0.00 0.02 0.04 0.06 0.08 0.101.0001.0051.0101.015 α(ω)/α(ο) ω0.0 0.1 0.2369γ(3ω;ω,ω,ω)/γ(0) ω

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.