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A 3.0MeV KOMAC/KTF RFQ Linac
J.M. Han, K.S. Kim, Y.J. Kim, M.Y. Park, Y.S. Cho, B.H. Choi, KAERI, Taejon, Korea
Y.S. Bae, K.Y. Shim, K.H. Kim, I.S. Ko, POSTECH, Pohang, Korea
S.J. Cheon, KAIST, Taejon, Korea
Y. Oguri, TIT, Japan
Abstract
The Radio-Frequency Quadrupole (RFQ) linac that
will accelerate a 20mA proton beam from 50keV to
3MeV has been designed and is being fabricated as the
first phase, KOMAC Test Facility (KTF), of the Korea
Multipurpose Accelerator Complex (KOMAC) project at
the Korea Atomic Energy Research Institute (KAERI).
The physical, engineering designs and fabrication status
of the RFQ are described.
1 INTRODUCTION
The linear accelerator for the KOMAC Project [1] will
include a 3MeV, 350MHz cw RFQ linac. The
KOMAC/KTF RFQ [2,3] concept is shown in Fig. 1 with
the main parameters given in table 1. The KTF RFQ
bunches, focuses, and accelerates the 50keV H+/H-
beams and derives a 3.0MeV beam at its exit. The RFQ
is a 324cm-long, 4-vanes type and consists of 56 tuners,
16 vacuum ports, 1 coupling plate, 4 rf drive couplers,
96 cooling passages, and 8 stabiliser rods. The RFQ is
machined into OFH-Copper, will be integrated from four
separate 81cm-long sections which are constructed by
using vacuum furnace brazing. RF power is supplied to
the RFQ which operates at 100% duty-factor by one
klystron of 1MW.
The physics and engineering design study of the KTF
RFQ linac are presented in section 2. Section 3 describes
the 450keV RFQ which is a test stand for the
development of the KTF RFQ linac. Section 4 presents
the present status of the KTF RFQ.Table 1. The KOMAC/KTF RFQ Linac Parameters.
PARAMETER VA LUE
Operating frequency 350 MHz
Particles H+ / H-
Input / Output Current 21 / 20 mA
Input / Output Energy 0.05 / 3.0 MeV
Input / Output Emittance,
Transverse/norm.0.02 /0.023
π-cm-mrad rms
Output Emittance,
Longitudinal 0.246 MeV-deg
Transmission 95 %
RFQ Structure Type 4-vanes
Duty Factor 100 %
Peak Surface Field 1.8 Kilpatrick
Structure Power 350 kW
Beam Power 68 kW
Total Power 418 kw
Length 324 cm
Low
Energ
yCoolin
g
PassagRF
Drive
PortCoupling
PlateTune
rVacuum
Port
High
Energ
y
Figure 1. 3MeV, 350MHz, cw KOMAC/KTF RFQ2 3.0MeV RFQ LINAC
2.1 Cavity Design
The design of the 3MeV RFQ has been completed. In
the KTF RFQ design, a main issue is to accelerate the
mixing H+/H- beam at the same time. The motion of the
mixing H+/H- beam into the RFQ has been studied by
using a time marching beam dynamics code, QLASSI[4].
Fig. 2 shows that the longitudinal beam loss increases
with the concentration of negative ions by the bunching
process which is distributed by attractive forces when the
mixing ratio of H- is more than 30%. The transverse
beam loss decreases with the mixing ratio of H- by the
space charge compensation in the low energy sections.
Figure 2. Dependency of the Beam Transmission Rate
and H- Mixing Ratio.
The average RFQ cavity structure power by rf thermal
loads is 0.35 MW and the peak surface heat flux on the
cavity wall is 0.13 MW/m2 at the high energy end. In
order to remove this heat, we consider 24 longitudinal
coolant passages in each of the sections. In the design of
the coolant passages, we considered the thermal
behaviour of the vane during CW operation, the
efficiency of cooling and fabricating cost. The thermal
and structure analysis was studied with the ANSYS code.
Because of the flow erosion of the coolant passages, we
considered the maximum allowable velocity of the
normal coolant as 4m/sec. From the thermal-structural
analysis of ANSYS, the peak temperature on the cavity
wall is 51.4 oC, the maximum displacement is 42µm and
the intensity stress is 13MPa. The temperature of the
coolants on the cavity wall varies to maintain the cavity
on the resonance frequency.
2.2 RF Power System
The total power simulated is 418kW, including beam
loading and power dissipation by a cavity wall, when an
additional 50% of the power is allowed as the differencebetween the theoretical model of the RFQ and the real
device built. This power is delivered by a single klystron,
capable of 1MW. The 350MHz klystron and RF
windows will be supplied by Thomson Co. Ltd. The
power is coupled in the cavity with a set of four coupling
loops. Each port will therefore carry an average rf power
of 120kW.
2.3 Cold Model
In order to test the fabrication accuracy and validate
the simulation studies which were performed by
PARMTEQM[5], VANES, SUPERFISH, MAFIA,
ANSYS codes, a full size 324cm-long RFQ cold model
was made of aluminium as shown in Fig. 3. Though this
model does not operate with rf power and under vacuum
pumping, it has rf power coupling ports, tuner ports,
stabilizer rod, coupling plate, end plates, and vacuum
ports which are given by the 3-dim drawing. By tuning
the undercut depth and end plates, we obtained the
optimum vane-end geometry and the required 350 MHz
resonant frequency.
Figure 3. The 3MeV RFQ Cold Model.
3 0.45MeV TEST RFQ
A 450keV test RFQ has been designed and fabricated
to understand the construction process, cooling, control,
rf drive system, and beam diagnostic techniques. Design
of the RFQ was done by KAERI and POSTECH, and
fabrication was done at Dae-Ung Engineering Company
and VITZRO TECH Co., Ltd.
Fig. 4 shows a 96.4cm long 0.45MeV RFQ which was
brazed in a vacuum furnace. The RFQ was brazed in a
vertical orientation with LUCAS BVag-8, AgCu alloy
with a liquid temperature of 780 oC. The four quadrants
of the RFQ have been fabricated separately and brazed.
Thus the RFQ is the completed monolithic structure and
the vanes are permanently aligned. This structure serves
to mitigate the cost and to simplify the mechanical
support system. Because of the leak of a brazing surface
and the strain of the RFQ structure by the furnace heat, it
is important to determine the appropriate shape of the
brazing area. To determine the appropriate shape, we
have performed two brazing tests. Testing of the brazed
RFQ showed it to be leak-tight. The coolant passages in
the cavity wall and vane area was the deep-hole drilled
and was brazed in a vacuum furnace.
Figure 4. A brazed 0.45MeV RFQ.
The frequency and unloaded Q was measured with a
network analyzer in atmosphere, i.e. not under a vacuum.
Without tuning, the measured frequency and Q were
349.63MHz and 5300 by observing the 1dB bandwidth,
respectively.
Fig. 5 shows the rf tuner which was fabricated to
tune the cavity to the operating frequency. By moving
four tuners to a 2.5cm inserted position, a total tuning
range of 2.5MHz was measured.
Figure 5. RF tuner.
The KTF RFQ has a rectangular undercut of the vanes.
The exact dimension of the undercut has been
determined empirically by cutting a vane which was
fabricated into the OFHC. Fig. 6 shows the variation of
the resonant frequency versus the depth of the undercut.
The resonant frequency of the RFQ cavity inversely
decreases with undercut depth. To maximise the effect
of the stabilizer rod, we determined that the undercut
depth and vane to end-plate length are 28mm and 35mm,
respectively. In this case, the quadrupole-dipole
separation was 10MHz.
Figure 6. Plot of the resonant frequency versus
the
depth of the undercut.
4 PRESENT STATUS
The design of the 3MeV RFQ has been completed. The
cold model with aluminium has been fabricated and
tested. The 3MeV RFQ with OFHC is being fabricated.
As a test bed for 3MeV RFQ, the design, construction,
electrical test, and vacuum test of the 0.45MeV RFQ
have been finished. The rf tuner has been fabricated and
was tested.
5 ACKNOWLEDGMENT
We are grateful to Dr. Kazuo Hasegawa at JAERI,
B.H. Lee and L.H. Whang at Vitzrotech Co. Ltd., and
D.S. Lim at DUE Co. This work has been supported by
the Korean Ministry of Science and Technology (MOST).
REFERENCES
[1] C.K. Park et al., “The KOMAC Project: Accelerator
and Transmutation Project in Korea”, Proceeding of
APAC98, Tsukuba, (1998).
[2] J.M. Han et al, “Design of the KOMAC H+/H- RFQ
Linac”, Proceeding of LINAC98, Chicago, (1998)
[3] J.M. Han et al., “Design and Fabrication of the
KOMAC RFQ”, Proceeding of PAC99, New York,
(1999).
[4] Y. Oguri et al., “Beam Tracking in an RFQ Linac
with Small Vane-Tip Curvature”, Nuclear Sci. Tech.,
30, 477 (1993).
[5] K.R. Crandall et al., “RFQ Design Codes”, LA-UR-
96-1836, (Revised February, 1997).
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DEVELOPMENT OF THE LOW ENERGY ACCELERATOR FOR KOMAC
B.H. Choi, J.M. Han, Y.S. Cho, K.S. Kim, Y.J. Kim, KAERI, Taejon, Korea
I.S. Ko, Postech, Pohang, Korea
K.S Chung Hanyang Univ, Seoul, Korea
K.H. Chung, Seoul Nat. Univ., Seoul, Korea
Abstract
KAERI (Korea Atomic Energy Research Institute) has
been performing the project named KOMAC (Korea
Multi-purpose Accelerator Complex) within the frame
work of national mid and long term nuclear research
plan. The final objectives of KOMAC is to build a 20-
MW (1 GeV and 20 mA) cw (100% duty factor) proton
linear accelerator. As the first stage, the low energy
accelerator up to 20 MeV is being developed in KTF
(KOMAC Test Facility). The low energy accelerator
consists of an injector, RFQ, CCDTL, and RF systems.
The proton injector with Duoplasmatron ion source has
been developed, and the LEBT with solenoid lens is
under development. The RFQ linac that will accelerate
a 20mA proton beam from 50keV to 3MeV has been
designed and is being fabricated. The RF system for
RFQ is being developed, and the CCDTL up to 20MeV
is being designed. The status of the low energy
accelerator will be presented.
1 INTRODUCTION
The KOMAC accelerator has been designed to
accelerate a 20 mA cw proton/H- with the final energy
1GeV cw super-conducting linac [1]. In the first stage of
the project, we are developing cw accelerating structure
up to 20MeV, and operate the accelerator in 10% duty
pulse mode. After the first stage, we will challenge the
cw operation of the accelerator. The 20MeV proton
accelerator is constructing in the KTF (KOMAC Test
Facility), and will be commissioned in 2003. After the
commissioning, KTF will provide the proton beam for
the many industrial applications.
In the KTF, we are developing the proton injector,
LEBT, 3MeV RFQ, 20MeV CCDTL, and RF system.
The proton injector is already developed, and the 3MeV
RFQ will be constructed in this fiscal year. Also we
have a plan to develop the basic Super-Conducting
cavity technology in the KTF for the second stage super-
conducting accelerator of the KOMAC. Fig. 1 shows the
plan of the KTF and Fig. 2 shows the status of the KTF.
The status of the low energy accelerator developments
in KTF will be introduced in this paper.
Figure 1: Plan of KTF 20MeV Accelerator
Figure 2: Status of KTF 20MeV Accelerator
2 PROTON INJECTOR [2]
For 20 mA proton beam at the final stage, KOMAC
requires the ion source with the proton beam current of
30 mA at the extraction voltage of 50 kV. Normalized
rms emittance of less than 0.3 π mm /GA2mrad is also
required for good matching of ion beam into RFQ.
The proton injector with a duoplasmatron ion source is
shown in Fig. 2 (left side). The system is composed of
an accelerating high voltage power supply, ion source
power supplies in a high voltage deck, gas feeding
system, and vacuum system.
The injector has reached beam currents of up to 50
mA at 50 kV extraction voltage with 150 V, 10 A arc
power. The extracted beam has a normalized emittance
of 0.2 π mm /GA2mrad from 90 % beam current and proton
fraction of over 80 %. The proton fraction is measured
with deflection magnet and scanning wire.
The beam can be extracted without any fluctuation in
beam current, with a high voltage arcing in 4 hours. The
cathode lifetime is about 40hr. To increase the filament
lifetime, it is necessary to lower the arc current or to
change the tungsten filament to other cathode such as
oxide cathode.
3 LEBT
Low-energy beam transport (LEBT) consists of two
solenoids, two steering magnets, diagnostic system,
beam control system, and funnelling system to transports
and matches the H+ for 20mA and H- for 3mA, beams
from the ion source into the RFQ. The main goal of the
LEBT design is to minimise beam losses. The design
codes used are TRACE 3D and PARMTEQM. The
PARMTEQM-simulated solenoid settings are B=2800G
and B=3900G, the RFQ transmission rate is more than
90%. Two solenoid magnets constructed are 20.7cm-
long, 16cm-i.d., are surrounded by a low carbon steel
and provide dc fields ≤5000G on the axis. During the
winter of 2000, we will test the LEBT to obtain a proper
matching condition with the RFQ.
4 RFQ [3]
The KTF RFQ bunches, focuses, and accelerates the
50keV H+/H- beams, and derives a 3.0MeV beam at its
exit, bunched with a 350MHz. The RFQ is a 324cm-long,
4-vanes type, and consists of 56 tuners, 16 vacuum ports,
1 coupling plate, 4 rf drive couplers, 96 cooling passages,
and 8 stabiliser rods. The RFQ is machined of OFH-
Copper, integrate from separate four sections which are
constructed by using vacuum furnace brazing. The RF
system for the RFQ is operated with 350MHz at 100%
duty-factor by one klystron of 1MW.
Its design was completed. In the RFQ design, a main
issue is to accelerate the mixed H+/H- beam at the same
time. The motion of the mixed H+/H- beam into the RFQ
has been studied by using a time marching beam
dynamics code QLASSI. The longitudinal beam loss
increases with the concentration of negative ions by the
bunching process which is distributed by attractive
forces when the ratio of H- is more than 30%.. Because
of the space charge compensation in the low energy
sections, the transverse beam loss decreases with the
mixing ratio of H-.
The average RFQ cavity structure power by rf thermal
loads is 0.35 MW and the peak surface heat flux on the
cavity wall is 0.13 MW/m2 at the high energy end. In
order to remove this heat, we consider 24 longitudinal
coolant passages in each of the sections. In the design of
the coolant passages, we considered the thermal
behaviour of the vane during CW operation and
manufacturing costs. The thermal and structure analysis
is studied with SUPERFISH and ANSYS codes. Thetemperature of the coolant passages on the cavity wall is
varied to maintain the cavity on resonance frequency.
As a test bed for 3MeV RFQ, the design, construction,
electrical test, and vacuum test of the 0.45MeV RFQ
have been finished. Design of the RFQ was done by
KAERI and POSTECH, a fabrication was done at Dae-
Ung Engineering Company and VITZRO TECH Co.,
Ltd. A difficult process in the fabrication of the RFQ
was to braze. Because of the leak of the brazing surface
and the strain of the RFQ structure by the furnace heat, it
is important to determine an appropriate shape of the
brazing area. To determine it, we have performed two
brazing test. Fig. 3 shows a 96.4cm long 0.45MeV RFQ
which was brazed in a vacuum furnace. The RFQ was
brazed in a vertical orientation with LUCAS BVag-8,
AgCu alloy with a liquid temperature of 780 oC. Testing
of the brazed RFQ showed it to be leak-tight.
The coolant passages in the cavity wall and vane area
were drilled with a deep hole. The entrances of deep
holes at the vane end was brazed. The exact dimension
of the undercut was determined empirically by cutting a
vane of the hot model which was fabricated of the OFHC.
In the case of the RFQ with a modulated vane tip, the
resonant frequency of the RFQ cavity linearly decreases
with undercut depth. However, in the case of the RFQ
cold model with a constant vane tip, the resonant
frequency of the RFQ cavity non-linearly decreases with
undercut depth.
Figure 3: A brazed 0.45MeV RFQ
The 3MeV RFQ cold model of aluminium was
fabricated and tested. A low-level RF control system,
which maintains proper amplitude(within ±1%) and
phase(within ±1°), has been designed. A cold model of a
tuner has been fabricated and is being tested.
Assembly works of the 3MeV RFQ will be done in
March, 2001.
5 CCDTL [4]
CCDTL will accelerate the 3MeV 20mA proton beam
to the energy of 20MeV. The structure design of
CCDTL is based on the 100% duty factor.
Table 1: Major Design Parameters of CCDTL cavity
- Structure : 700MHz CCDTL
- Length : 25m
- Aperture Diameter : 10/15mm
- No of EMG : 130 (8 βλ FODO)
- Total Structure Power : 1.15MW
- Sturcture Power per length : 50kW/m avg.
- Surface E : <0.9 Kilpatrick
The CCDTL cold models are fabricated to check the
design, the tuning method, and the coupling coefficients
and the fabrication method. The measured resonant
frequency is 700.8 MHz without air and humidity
compensation. The measured Q value of the cavity
without brazing is 87% of the calculated Q by
SUPERFISH without any surface cleaning. The super-
drilled coolant path is well fabricated, and this type
cooling method will be used for the CCDTL construction.
The field profile is measured with a bead perturbation
method. The field measurement system is shown in Fig.
4. A 2mm diameter and 2mm long alumina cylinder is
used for the bead. The stepping motor drive system
controls the position of the bead with an accuracy of
0.2mm. The frequency shift is measured with a network
analyzer (HP4306A/85064A). Because the temperature
controlled room is not available, the measurement was
carried with the careful check of the unperturbed
resonance frequency before and after the experiment.
Figure 4: CCDTL Field Measurement
Fig. 5 shows the one measured field profile in one
cavity of the aluminium cold model. The measured field
profile in a cavity agrees with the calculated profile. But,the field uniformity in the multi-cavity is not good. It is
necessary to increase the field uniformity by the fine
tuning of the cavity. This will be done with the brazed
copper cold model that will be fabricated in this year.
The copper model will be fabricated with the study. As
a back-up of the CCDTL, the design study for
conventional DTL will be performed.
0510152025
0 50 100 150
Figure 5: Measured Field Profile in One Cavity
(x: Position(mm), y: Field(Arb.))
6 SUMMARY
The low energy proton accelerator for KTF is
designed. The proton injector can provide the proton
beam for RFQ. The RFQ is fabricating and will be
tested with 1MW RF system. The CCDTL is studied
with cold models, and the hot model will be fabricated.
ACKNOWLEDGEMENT
This work was supported by the Korea Ministry of
Science and Technology.
REFERENCES
[1]B.H.Choi et al, “Overview of the KOMAC Project”,
Proceedings Of the 3rd International Conference
ADTTA'99 (1999).
[2] Y.S.Cho, et al, “‘High-current ion source
development for the Korea Multipurpose
Accelerator Complex”, Rev. Sci. Instr., 71,
969(2000).
[3] J.M. Han et al., “A 3MeV KOMAC/KTF RFQ
Linac”, in this conference (2000).
[4] Y.S. Cho et al, “Cold Model Test of The KOMCA
CCDTL Cavities”, Proceeding of EPAC2000,
Vienna, (2000).
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A HIGH-INTENSITY H– LINAC AT CERN BASED ON LEP-2
CAVITIES
M. Vretenar for the SPL Study Group, CERN, Geneva, Switzerland
Abstract
In view of a possible evolution of the CERN
accelerator complex towards higher proton intensities, a
2.2 GeV H- linac with 4 MW beam power has been
designed, for use in connection with an accumulator and
compressor ring as proton driver of a muon-based
Neutrino Factory. The high-energy part of this linac can
use most of the RF equipment (superconducting cavities
and klystrons) from the LEP collider after its
decommissioning at the end of 2000. Recent results
concerning low-beta superconducting cavities are
presented, and the main characteristics of the linac design
are described. The complete linac-based proton driver
facility is outlined, and the impact on the linac design of
the requirements specific to a Neutrino Factory is
underlined.
1. THE LEP-2 RF SYSTEM
The decommissioning of the CERN LEP e+ e– collider
at the end of 2000 will pave the way to the construction
of the Large Hadron Collider (LHC), but will also present
the unprecedented challenge of the removal, storage or
disposal, and possible recycling of the huge amount of
valuable LEP equipment.
A particularly valuable item is the 352.2 MHz
superconducting RF system built for the Phase 2 of LEP,
consisting of 288 four-cell cavities (Figure 1) operating at
4.5 °K and powered by 36 1 .3 MW CW klystrons. It
delivers a total accelerating voltage of about 3 GV to the
electron beam. Eight more klystrons are used to power
the normal-conducting RF system of LEP, for a total of
44 klystrons installed in the machine. Most of the
superconducting cavities (272) were produced using the
technique developed at CERN of sputtering a thin film of
niobium onto copper [1]. The cavities were initially
designed for a gradient of 6 MV/m, and during the 1999
run they achieved an average gradient of 7.5 MV/m, with
up to 9 MV/m in some cavities [2]. In the basic LEP
configuration, each klystron feeds 8 cavities via an array
of magic tees, equipped with circulators and loads. Four
cavities are grouped in a cryostat. The cavities and the
cryostats are fully equipped with slow and fast tuners,
power couplers matched for a beam current of 10 mA,
high-order-mode couplers, superinsulation and insulation
vacuum tanks.
The present plans foresee to store most of the RF
material for possible future use. The cryogenic system of
LEP will be used for the LHC magnets.Figure 1: The LEP-2 accelerating cavity
2. A LINAC BASED ON LEP CAVITIES
Some proposals for re-using this expensive hardware
have been made, such as for a Free Electron Laser [3] or
to build the ELFE machine on the CERN site, a
recirculating electron linac for nuclear physics [4].
An early proposal already opened the perspective of
using the LEP cavities in a high beam power
superconducting linac driving a hybrid reactor [5-7].
The main limitation for using these cavities in proton
linacs comes from the fact that they are designed for β=1,
their transit time factor drastically decreasing for a proton
beam at low beta. Figure 2 shows a calculation of the
effective cavity gradient as function of energy that can be
reached by LEP cavities operating at a nominal gradient
of 7.5 MV/m. While in principle they can be used for
proton acceleration from about 500 MeV, they become
efficient and economically justified only from about
1 GeV, i.e. in an energy range beyond the usual
requirements of high-power linacs for spallation sources,
transmutation or hybrid reactors.
Figure 2: Effective gradient of the LEP-2 cavities as a
function of energy
As will be seen in the following, applications of linacs
for physics research at energy > 1 GeV exist, but they
require beam powers of only a few MW and a well-
defined time structure of the beam. This imposes a pulsed
operation mode that has to be optimised to achieve a
reasonable mains-to-RF efficiency. High duty cycles are
preferable because they reduce the impact of the static
cryogenic losses, and long pulses minimise the relative
effect of the RF power lost during the relatively long (1-
2 ms) pulse rise time, when all the power is reflected
from the couplers.
However, the LEP cavities are well suited for pulsed
operation because of the inherent rigidity of the copper
cavity structure and of the relatively low gradient that
make them less sensitive to Lorentz force detuning and
vibration problems. The large (241 mm) aperture is
particularly useful for machines sensitive to beam losses.
3 APPLICATIONS OF A 2 GEV LINAC
The first proposal to replace the present 50 MeV linac
and the 1.4 GeV Booster in the CERN proton injector
chain with a Superconducting Proton Linac (SPL) dates
from 1996 [8]. In the original scheme this machine was
intended to accelerate mainly protons, and although in the
following studies the advantages of a common H–
operation for all the users became clear, the title of SPL
has been maintained. A first feasibility study [9]
considered a 2 GeV H– SPL equipped with LEP cavities
from 1 GeV energy, injecting at 0.8 Hz repetition rate
into the Proton Synchrotron (PS) ring. This new injector
would have several benefits over the present injection
scheme for the LHC:
- a factor 3 increase in the brightness of the proton
beam (density in transverse phase space) delivered
by the PS injector complex, due to the lower space
charge tune shift at injection, which is an advantage
for LHC;
- the potential for improving the peak intensity in the
PS for experiments requiring a high proton flux;
- the reduction of injection losses with charge
exchange injection and a chopped linac beam;
- the replacement of the PS injectors by modern and
standard equipment.
However, such a machine would be fully justified only
when pulsed at a higher rate, and the original feasibility
study aimed somehow arbitrarily for 5% duty cycle.
On the basis of this preliminary study, some user
communities have shown their interest in a high-intensity
facility at CERN. Firstly there is the strong demand for
second generation radioactive nuclear beam facilities in
Europe. The SPL could easily become the driver of a
facility based at CERN that would profit from the
experience gained at ISOLDE. The mean current required
is about 100 µA, preferably distributed in many low
intensity pulses and at a variable energy.Secondly, strong interest has been recently shown by
the physics community for the high-intensity high-quality
neutrino beams that can be provided by a Neutrino
Factory based on a muon decay ring. CERN has recently
started a study on the technological challenges of such a
Neutrino Factory, that resulted in the CERN Reference
Scenario of Figure 3 [10].
Figure 3: Possible layout of a Neutrino Factory
The main challenges for this machine come from the
need for high neutrino fluxes. The interest for physics
starts from some 1021 neutrinos/year, that can be obtained
with about 4 MW beam power from a driver accelerator
delivering protons on a target, producing pions which
decay into muons. After cooling and acceleration, the
muons are stored in a decay ring where they generate two
intense neutrino beams.
Simulations of particle production in the target
indicate that the number of pions is approximately
proportional to beam power for energies ≥ 2 GeV. This
suggests that a low-energy linac-based driver constitutes a
viable alternative to conventional high-energy, fast-
cycling synchrotrons. The HARP experiment at CERN
[11] is intended to provide experimental data on pion
production at different energies and from different
targets, for a final confirmation of the low energy choice.
Muon collection, cooling system and decay ring
impose a well-defined time structure for the beam on
target. This requires two rings after the linac, an
Accumulator to produce a 3.3 µs burst of 140 bunches at
44 MHz (the frequency of the muon phase rotation
section) and a Compressor to reduce the bunch length to
3 ns [12]. The rings have been designed to fit in the
existing ISR tunnel. Space charge and beam stability are
their major design concerns. To reduce space charge tune
shift at injection into the accumulator, the linac bunch
length has to be stretched in the transfer line, from about
30 ps to 0.5 ns, by means of two bunch rotating cavities.
4 THE SPL H– LINAC DESIGN
4.1 Main Parameters
The parameters of this machine (Table 1) had to take
into account the optimum operating conditions of the
superconducting cavities discussed in Section 2, and
are mainly determined by the needs of the Neutrino
Factory, by far the most demanding user in terms of
particle flux and time structure of the beam pulses.
Table 1 Main linac design parameters
Particles H–
Kinetic Energy 2 .2GeV
Mean current during pulse 11 mA
Repetition frequency 75 Hz
Beam pulse duration 2 .2ms
Number of particles per pulse1.51×1014
Duty cycle 16.5%
Mean beam power 4MW
RF Frequency 352 .2MHz
Chopping factor 42 %
Mean bunch current 18 mA
Transv. emittance (rms, norm.)0.6µm
The mean current during the pulse of 11 mA has
been selected as a compromise between the number of
klystrons needed in the superconducting section, the
efficiency of the feedback loops, the number of turns
injected into the accumulator and the power efficiency
of the room temperature section. The input couplers of
the LEP cavities are already matched for this current.
The linac energy of 2.2 GeV, the repetition rate of
75 Hz and the corresponding pulse length of 2.2 ms are
a compromise between the optimum operatingconditions of the superconducting cavities and the need
to limit the number of turns injected into the
accumulator, 660 in the present scenario.
A chopper in the low energy section is used to
minimise losses at injection in the accumulator and at
the transfer between the rings. In the present design,
42% of the beam is taken out at the chopper position,
leading to a source current and a bunch current of
18 mA. This value is within reach of present H–
sources and is well below the limits of space-charge
dominated beam dynamics.
The layout of the linac is shown in Figure 4 and key
data are given in Table 2.
Table 2 Layout data of the SPL H– linac
SectionOutput
Energy
(MeV)RF power
(MW)Nb. of
kly-
stronsNb. of
tetrodesLength
(m)
Source0.045 - - - 3
RFQ1 2 0 .25 1 -2
Chopper2 - - - 3
RFQ2 7 0 .6 1 -5
DTL 120 8 .7 11 -78
SC-lowβ1080 10 .6 12 74 334
SC–LEP2200 12 .3 18 -357
Total 32.5 43 74 782
The LEP RF frequency of 352 MHz has been
maintained for the whole linac. The choice of this
frequency for the low-beta superconducting cavities
that have to be built for the SPL allows to apply the
sputtering fabrication technique to the cavity and to use
couplers and cut-off tubes recuperated from LEP units.
This frequency provides the additional flexibility
that klystron or tetrode amplifiers can be used in the
RF system. Klystrons can feed the high-power cavities
Figure 4 : Layout of the SPL H– linacH-RFQ1 chop. RFQ2 RFQ1 chop. RFQ2RFQ1 chop. RFQ2 DTL SCDTLRFQ1 chop. RFQ2 β 0.52 β 0.7 β 0.8 LEP-II dump
Source Low Energy section DTLSuperconducting low-β45 keV 7 MeV 120 MeV 1.08 GeV 2.2 GeV
2 MeV 18MeV 237MeV 389MeV10m 78m 334m 357m
PS / IsoldeStretching and
collimation line
Accumulator RingSuperconducting β=1(room temperature and high-beta superconducting), with
a limited number of cavities connected to the same
klystron. Individual 65 kW tetrode amplifiers can,
instead, feed the low-beta superconducting cavities, thus
avoiding the potential dangers at low beam energy of a
field stabilisation based on the vector sum of many cavity
signals.
Keeping beam losses below the limit for hands-on
maintenance (1 W/m) has been a design issue from the
beginning. The main principles were a design of the linac
optics without excessive jumps in the focusing
parameters, to avoid the formation of halo from mismatch
at the transitions, a particular care to avoid crossing
resonances, and finally, the preference for large apertures
in spite of some reductions in shunt impedance. Wherever
possible, losses will be concentrated on localised dumps
by means of collimators.
4.2 Room Temperature Section
The design source current, 18 mA, is well within the
range of existing H– sources, while the required pulse
length and duty cycle are more challenging as compared
to existing sources. Reliability is also an important
concern. The study of an H– ECR source that could meet
the SPL parameters has been started, and collaborations
are envisaged.
The fast chopper (2 ns rise time) at 2 MeV could be a
travelling-wave stripline structure similar to the LANL
design [13]. An analysis of the options for the 1 kV pulse
amplifier indicates that a combination of vacuum tubes
driven by fast Mosfets could provide the required rise and
fall times [14]; this will be tested on a prototype.
The DTL starts at 7 MeV, and is composed of two
standard Alvarez tanks up to an energy of 18 MeV,
followed by a section of Side-Coupled DTL (SCDTL)
[15], 2-gap tanks connected by off-axis coupling cavities,
with quadrupoles placed between tanks. The 352 MHz
structure going up to 120 MeV is made of 98 small tanks
grouped in 9 chains, each one powered by a klystron [16].
4.3 Superconducting Section
The superconducting part of the linac is composed of
four sections made of cavities designed for β of 0.52, 0.7,
0.8 and 1 respectively. The LEP-2 cavities are used at
energies above 1 GeV. The cavities at β=0.52 and β=0.7
contain 4 cells, whilst the beta 0.8 cavities are made of 5
cells, to re-use the existing LEP cryostats.
The main parameters of the superconducting section
are summarised in Table 3. It has been assumed that the
LEP cavities will operate at 7.5 MV/m, while for the
newly-built β=0.8 cavities, cleaning procedures to
achieve high gradients can be applied and a design
gradient of 9 MV/m can be foreseen. During tests, a
β=0.8 cavity has already reached gradients of 10 MV/m
[17]. The transition energies between sections are definedin order to have the maximum effective accelerating
gradient and to minimise phase slippage.
Table 3 Superconducting linac section
BetaWoutGradientCavitiesCryost.Length
(MeV) (MV/m) ( m)
0.52 237 3 .5 42 14 101
0.7 389 5 32 8 80
0.8 1080 9 48 12 153
1 2200 7 .5 108 27 357
The cavities at β=0.7 and β=0.8 can be built of
niobium sputtered on copper. This technique, developed
at CERN, has many advantages with respect to bulk
Niobium for large productions:
a) the cost of the raw material is much lower, giving the
possibility to go for low frequencies where the iris
aperture is large, relaxing the mechanical tolerances
and reducing the probability of beam losses;
b) Nb/Cu cavities can be operated at 4.5 °K with Q-
factors of more than 109, simplifying the design of
the cryostats and of the power coupler;
c) the excellent mechanical properties of copper ensure
a better thermal and mechanical stability.
A development programme was started at CERN in
1996 to investigate the feasibility of the production of
cavities in the β range 0.5–0.8. The main results are the
prototypes of a 5-cell β=0.8 cavity and of a 4-cell β=0.7
cavity (shown in Fig. 5), that have achieved satisfactory
Q-values at high gradient (Fig. 6) [18,19]. Attempts to
sputter cavities at β<0.7 were not successful, thus the 42
cavities at β=0.52 have to be made of bulk niobium.
Figure 5: The prototype 4-cell β=0.7 cavity
Particular attention has been given to the pulsed
operation of the superconducting cavities. Feedback loops
are foreseen to minimise the effect on the beam of
cavity vibrations and of Lorentz forces. In the β=0.8 and
β=1 sections, where one klystron feeds 4 and 6 cavities
respectively, the compensation has to be made on the
vector sum. Simulations show that random oscillations of
the cavity frequency of up to 40 Hz amplitude can be
tolerated, without increasing the energy spread of the
beam outside the ± 10 MeV corresponding to the
acceptance of the accumulator [20].
Figure 6: Q vs. gradient of the sputtered-Nb cavities
Multi-particle simulations of the beam dynamics in the
superconducting section show a stable behaviour in the
presence of errors and mismatch of the input beam [21].
4.4 Layout on the CERN site
After considering some possible locations for the SPL
around the CERN Meyrin site, the option shown in Fig. 7
has been retained. Placing the linac and the parallel
klystron gallery in an area immediately outside of the
CERN fence on the Swiss side offers the advantages of an
economic trench excavation, of minimum impact on the
environment (the site is presently an empty field), of a
simple connection to the ISR tunnel and to the PS through
existing tunnels, and of an easy access from the road
along the fence.
The infrastructure for electricity, water cooling and
cryogenics makes a maximum use of existing facilities on
the Meyrin site.
Figure 7: Layout of the SPL on the CERN site
5 CONCLUSIONS
About 40% of the LEP-2 cavities, 57% of the cryostats
and all the klystrons plus other RF and HV equipment can
be used to construct a 2.2 GeV H– linear accelerator on
the CERN site that would improve the beam brightnessand intensity of the PS ring, provide a flexible and
powerful beam source for a second generation radioactive
beam facility and constitute the first step towards a
powerful Neutrino Factory.
REFERENCES
[1] C. Benvenuti et al., “Films for superconducting
accelerating cavities”, Appl. Phys. Lett. 45, pp. 583-
584, 1984.
[2]P. Brown, O. Brunner, A. Butterworth, E. Ciapala, H.
Fritscholz, G. Geschonke, E. Peschardt, J. Sladen,
“Performance of the LEP200 Superconducting RF
System”, 9th Workshop on RF Superconductivity,
Santa Fe, November 1999, CERN-SL-RF-99-075.
[3]R. Corsini, A. Hoffmann, “Considerations on an FEL
based on LEP Cavities”, CERN/PS 96-04.
[4]H. Burkhardt (ed.), “ELFE at CERN”, CERN 99-10.
[5]C. Rubbia, J.Rubio, “A tentative programme towards
a full scale energy amplifier”, CERN/LHC 96-11.
[6]D. Boussard, E. Chiaveri, G. Geschonke, J.
Tückmantel, “Preliminary Parameters of a Proton
Linac using the LEP 2 RF System when
Decommissioned”, SL-RF Tech. Note 96-4.
[7]C. Pagani, G. Bellomo, P. Pierini, “A High Current
Linac with 352 MHz cavities”, Linac’96, Geneva,
1996.
[8]R. Garoby, M. Vretenar, “Proposal for a 2 GeV
Linac Injector for the CERN PS”, PS/RF/Note 96-27.
[9]A.M. Lombardi, M. Vretenar (eds.), “Report of the
Study Group on a Superconducting proton linac as
PS Injector”, CERN/PS 98-064 (RF/HP).
[10]H. Haseroth, “Status of studies for a Neutrino
Factory at Cern”, EPAC2000, Vienna, June 2000,
CERN/PS 2000-026 (PP).
[11]F. Dydak (spokes-person) “Proposal to study the
hadron production for the neutrino factory and for
the atmospheric neutrino flux”, CERN-SPSC / 99-35.
[12]B. Autin et al., “Design of a 2.2 GeV Accumulator
and Compressor for a Neutrino Factory”,
EPAC2000, Vienna, June 2000, CERN-PS/2000-11.
[13]S. Kurennoy, J. Rower, “Development of Meander-
Line Current Structure for SNS Fast 2.5-MeV Beam
Chopper”, EPAC2000, Vienna, June 2000.
[14]M. Paoluzzi, “Design of 1 kV Pulse Amplifier for the
2.2 GeV Linac Beam Chopper”, PS/RF Note 2000-
018.
[15]J. Billen, F. Krawczyk, R. Wood, L. Young, “A New
RF Structure for Intermediate-Velocity Particles”,
Linac’94, Tsukuba, August 1994.
[16]F. Gerigk, M. Vretenar, “Design of a 120 MeV Drift
Tube Linac for the SPL”, PS/RF Note 2000-019.
[17]O. Aberle et al., “Technical Developments on
Reduced β Superconducting Cavities at CERN”,
PAC’99, New York, 1999.
[18]C. Benvenuti et al., “Production and Test of 352
MHz Niobum-Sputtered Reduced-β Cavities”, 8th
Workshop on RF Superconductivity, Abano, 1997.
[19]R. Losito, “Design and test of a 4-cell β=0.7 cavity”,
CERN SL-Note-2000-047 CT.
[20] J. Tückmantel, private communication.
[21]F.Gerigk, “Beam Dynamics in the Superconducting
section of the SPL (120 MeV-2.2 GeV)”, PS/RF
Note 2000-009, NF Note 24.0.1110
0 2 4 6 8 10 12
Eacc [MV/m]Q/109
0.8 single cell
LEP
0.7 4-cells
0.8 5-cells |
arXiv:physics/0008143 18 Aug 2000An ECR hydrogen negative ion source at CEA/Saclay: preliminary results.
R. Gobin*, P -Y. Beauvais, O. Delferrière, R. Ferdinand, F. Harrault, J -M. Lagniel.
Commissariat à l'Energie Atomique, CEA -Saclay, DSM/DAPNIA/SEA
91191 Gif sur Yvette Cedex, France
*e-mail: rjgobin@cea.fr
Abstract:
The development of a high intensity negative ion
source is part of a considerably larger activity presently
undergoing at CEA Saclay in the field of high intensity
linear accelerat ors. Preliminary studies toward the
construction of a 2.45 GHz ECR H - ion source have been
performed for few months. This new test bench takes
advantage of our experience on the French high intensity
proton source SILHI. In the new source, the high -energy
electrons created in the ECR zone are trapped by a dipole
magnetic filter. A rectangular 200 mm long plasma
chamber and an intermediate iron shield are used to
minimize the magnetic field in the extraction region. A
second magnetic filter separates electro ns and negative
ions in a 10 kV extraction system. To reduce the
electron/H - ratio, the plasma electrode is slightly
polarized. The design allows future evolutions such as
cesium injection, higher energy extraction and plasma
diagnostics. The installation of the source is now in
progress. The first helium plasma has been produced for
few weeks to verify the electron separator behavior. The
design, computations and the first results of the source are
presented.
I - Introduction
Potential applications of h igh current accelerators
include the production of high flux neutron beams for
spalliation reactions (ESS), future reactors, nuclear waste
treatment, exotic ion facilities or neutrino and muon
production for high -energy particle physics. The high
intensity beams for these accelerators may reach an
energy as high as 1 GeV. In France, CEA and CNRS have
undertaken an important R&D program on very high
beam power (MW class) light -ion accelerators for several
years. Part of the R&D efforts are concentrated on th e
IPHI (High Intensity Proton Injector) [1] demonstrator
project. This 10 MeV prototype of linac front end will
accelerate CW beam currents up to 100 mA. It will
consist of an intense ion source, a radio frequency
quadrupole (RFQ) and a drift tube linac (D TL). The High
Intensity Light Ion Source (SILHI) development, based
on the 2.45 GHz ECR plasma production, has been
performed for several years leading to a great experience
in high current proton beam production. Taking into
account this advantage, CEA wh ich is involved in the
ESS studies, decided to develop a hydrogen negative ion
source also based on the ECR plasma production.
Section II gives an overview of the SILHI proton
source performance in CW and pulsed mode. The new hydrogen negative ion source is described in section III
which also includes magnetic and trajectory
computations. Electron separator efficiency and proton
density have been measured and preliminary results are
reported in section IV. Then the conclusion presents, in
section V, the fu ture experiments planned to improve the
negative ion source performance.
II - SILHI, High Intensity Proton Source
To summarize the high intensity proton source
efficiency, the most significant results are reported
hereinafter. The SILHI proton source [2] has been
designed to reach a long lifetime and a very high
reliability. It operates at 2.45 GHz. The magnetic field
BECR = 875 Gauss is produced by 2 coils tunable
independently. The quartz RF window has been installed
behind a water -cooled bend to escape the beam of
electrons produced and accelerated back to the plasma
chamber, in the HV extraction system. The RF window
works well since the production of the first beam in July
1996. Nevertheless the boron nitride (BN) disc located at
the RF entrance is aff ected by the backstreaming electrons
and must be systematically replaced. Its lifetime is
estimated to be higher than 1000 hours for ~ 100 mA CW
beams, then more than 40 days of continuous operation
for such beams.
The plasma is easily obtained when the RF power is
larger than 350 W with the standard magnetic field and
operating gas pressure (10-3 Torr in the plasma chamber).
The source is generally operated 5 days a week for
8 hours daily runs. Less than 10 min. are needed each
morning to restart the sourc e with a 100 mA CW beam at
95 keV. The tune up time is reduced to 2 min. after a shut
down using an automatic procedure. Less than 6 hours are
usually needed to obtain the nominal beam parameters
after an operation in the source or in the low energy beam
transport (LEBT). This recovery time for pumping, HV
column conditioning and tune up is mainly induced by the
BN disc outgassing under plasma warming.
The best performances are clearly obtained when
two ECR zones are simultaneously located at both plasma
chamber extremities. The source efficiency increases to
0.145 mA/W (250 mA/cm2) for 850 W RF forward power
in these conditions instead of 0.105 mA/W with a single
ECR zone at the RF entrance.
Three long runs have been performed to analyze the
reliability – availability of the source. In October 1999,
with a 75 mA – 95 keV continuous beam, the reliability
reached 99.96 % for a 104 H long operation. Only one beam trip occurs during the test. The beam stopped during
2'30'' just one hour after the beginning of the statistics
leading to a 103 H uninterrupted operation.
In pulsed mode, rather short plasma rise and fall
times have been achieved during some preliminary
experiments done using a modulation of the 2.45 GHz
magnetron power supply. Plasma pulses with 10 µs rise
time and 40 µs fall time have been observed.
For a 80 mA CW proton beam, the nominal r,r' rms
normalized emittance is lower than 0.3 pi mm mrad and
the proton fraction better than 85 % (12 % for H2+ and
3 % for H3+). Several measurements have show n strong
improvements of the emittance (0.11 pi mm mrad) when a
buffer gas (H 2, N2, Ar or Kr) is injected in the LEBT. The
space -charge compensation (SCC) factor has been
measured at several points along the LEBT. A strong
dependence on the number of free electrons in the LEBT
line was found. The SCC can be much lower than
expected without an increased electron production
induced by adding heavy gases for example. Low SSC
leads to a strong emittance increase in the LEBT.
III - Negative hydrogen ion source design
Taking into account this experiment on high
intensity beams, it has been recently decided to study a
new source for negative hydrogen ion production. The
aim is to obtain a long lifetime source with high
reliability. As demonstrated with SILHI, thes e conditions
could be reached with sources in which the plasma is
generated by ECR. In classical sources, filament or
antenna lifetime reduces considerably the reliability.
To design the source, several contacts have been
undertaken with different nationa l and foreign
laboratories (Ecole Polytechnique, CEA Cadarache, CEA
Grenoble, CERN, Frankfurt University) which are
involved in H - studies. A step by step work has been
decided before the final design.
First, the production of hydrogen negative ions has
to be demonstrated in volume production mode before to
discuss Cesium or Xenon injection or other improvement
like Tantalum surface [3].
Extracted negative ions and electrons have to be
separated by means of a magnetic dipole. To avoid H 2
excited molecule destruction, high energy electrons have
to be eliminated in the plasma close to the extraction area
by using a magnetic filter. To do that preliminary
magnetic calculations have been performed to design the
C shape magnetic electron separator and magnetic filter.
Otherwise, the axial magnetic field provided by the two
coils to reach B ECR has also been calculated as well as the
iron shielding. All computations including particle
extraction (Fig. 1) have been done with Axcel [4], Opera
2D and 3D codes [5].
This source also operates at 2.45 GHz
(BECR = 875 Gauss) with a water cooled copper plasma
chamber. The rectangular (standard WR 284 waveguide)
plasma chamber length has been chosen at 210 mm
instead 100 mm for SILHI to reach an axial magnetic field as low as possible close to the extraction area in
order to limit the amount of high energy electrons in this
zone and to insert the C shape electromagnetic filter.
Fig 1: Hydrogen negative ions and electrons separation
The electrons are collected on the ele ctrode
The RF signal is produced by a 1.2 kW magnetron
source and is fed to the source via standard rectangular
waveguides and an automatic tuning unit. A three section
ridged waveguide transition is placed just before the
aluminum nitride (AlN) window. T his window is located
at the RF entrance in plasma chamber. It is protected from
backstreaming ions or electrons by a 2 mm BN sheet.
The Mo plasma electrode is biased to a few volt power
supply.
Figure 2 shows a cross -sectional view of the
experimental se tup. Several ports have been managed in
the plasma chamber for future plasma diagnostics
(Langmuir probe, Laser detachment, …).
The source and its ancillaries (power supplies, RF
generator, gas injection, …) are grounded and the 10 kV
extraction system is installed inside the vacuum chamber.
The collector is also linked to an independent HV power
supply. The 80 mm aperture C shape tunable electron
separator is located inside the vacuum vessel. By using
positive or negative HV power supplies, negative and
positive extracted beams could be respectively observed.
Fig 2: Cross -sectional view of the source
and extraction system
IV- Preliminary results
To verify the source behavior, the first plasma has
been produced by injecting hydrogen gas in the plasma chamber. After some outgassing troubles, it has been
quite easily obtained when the ECR zone was located at
the RF entrance in the plasma chamber. The RF forward
power was 500 W with a reflected power lower than 10 %
and with an operating pressure of 3 10-3 Torr in the
plasma chamber. As the magnetron pulsation is not yet
available, it has been decided to pulse the extraction
voltage (8 ms/s) with a continuous plasma for the
experiments described hereinafter.
The electron separator efficiency has been verif ied
with an Helium plasma. The total extracted negative
charges (20 mA trough a 5 mm diameter plasma
electrode) are collected on the extraction electrode for the
nominal value of the magnetic dipole.
In Hydrogen, the positive charge density has been
also checked with a - 6 kV pulsed extraction voltage. A
10 mA/cm2 beam density has been obtained with the
magnetic filter switched off. The density has been
reduced by a factor 2 with a Bdl ? 500 Gauss.cm
magnetic filter. With a pulsed positive extraction volta ge
(+ 5 kV), the electron density also decreases when the
magnetic filter is switched on. It also decreases by tuning
the plasma electrode voltage.
V- Conclusion
Figure 3 shows a general view of the source test
stand. The first plasma has been produced f or few weeks
and any negative hydrogen ion beam has been observed
since then. Copper (from plasma chamber body) and
carbon pollution due to O -ring attack by the RF has been
seen, it is probably the main reason of the hydrogen
negative ion destruction. Impr ovements to avoid this RF
attack are in progress. However, the electron separator
and the magnetic filter behavior seems quite satisfactory.
Fig 3: General view of the source test stand
Compare to the SILHI results, the positive ion
density is very l ow because of the magnetic configuration.
High negative ion beam intensity could not be produced
in such conditions. In the near future, after resolving the
O-ring problem, different studies will be done to enhance the plasma confinement in the extraction zone.
Computations have been already undertaken by using a
multipole magnetic configuration.
For higher energy extraction and to enhance the AlN
window lifetime, RF magnetron source pulsation work
will be also performed.
When the first step (volume produ ction mode of
hydrogen negative ions) will be reached, the second step
will consist to improve the performance by Xenon and
Cesium injection, Tantalum surface. Then to characterize
the beam, the source will be installed close to the SILHI
HV platform. The beam will be accelerated through a
dedicated accelerator column and analyzed in a diagnostic
box located on the platform.
The source efficiency will be analyzed at higher RF
power. It is also planned to characterize it at higher RF
frequency (10 GHz for e xample).
Acknowledgments
Many thanks to the members of the IPHI team for
their contributions, especially to G. Charruau and
Y. Gauthier for their technical assistance. The authors
would also thank M. Bacal, J. Faure, A. Girard,
C. Jacquot, G. Melin, J Sh erman, K Volk and the CERN
source team for their fruitful collaboration and valuable
discussions.
References
[1] P-Y. Beauvais et al, "Status report on the Saclay
High -Intensity Proton Injector Project (IPHI)", EPAC
2000 Vienne (Austria)
[2] J-M. Lagni el et al., Rev. Sci. Instrum. Vol. 71 n° 2,
830 (2000).
[3] J. Peters, Rev. Sci. Instrum. Vol. 71 n° 2, 1069 (2000).
[4] P. Spädtke, "Axcel -V 3.42", INP, D-65205
Wiesbaden, Germany.
[5] Opera 2D and 3D, © Vector Fields Limited, Oxford,
England. |
arXiv:physics/0008144 18 Aug 2000DESIGN OF THE ESS RFQs AND CHOPPING LINE
R. Duperrier, R. Ferdinand, P. Gros, J-M. Lagniel, N. Pichoff, D. Uriot
CEA-Saclay, DSM-DAPNIA-SEA
Abstract
The chopping line is a critical part of the ESS lin ac in
term of technical realisation of the choppers and
preservation of the beam qualities. A new optimised
design of the ESS RFQs and chopping lines is report ed.
The beam dynamics has been optimised with H - beam
currents up to 100-mA to have safety margin with re spect
to the ESS goals. The first RFQ transmits almost 99 .7%
of the beam up to 2 MeV. The line with two choppers
allows a perfect chopping between 2 bunches. The se cond
RFQ accelerates the particles up to 5 MeV with a
transmission close to 100%.
1 INTRODUCTION
The European Spallation Source (ESS) reference
design is described in ref. [1]. The accelerator is designed
to provide a proton beam power of 5 MW at a repetit ion
rate of 50 Hz. It comprises a 1.334 GeV H - linac (~ 10%
duty cycle) and two accumulator rings. The proposed lay-
out of the linac has two front ends, each one made up of
an 70 mA peak H - ion source, a low energy beam
transport, a first RFQ, a Medium Energy Beam Transp ort
(MEBT) with the choppers and a second RFQ. The bunc h
funnelling is done at 5 MeV and a 350 MHz Drift Tub e
Linac (DTL) accelerates the beam up to 70 MeV. In t his
reference design a 700 MHz normal conducting Couple d
Cavity Linac (CCL) further accelerates the H- beam to the
final energy.
2 DESCRIPTION
In high power proton accelerators for projects such as
ESS or the multi-user facility (CONCERT project [2] ), a
chopped beam is needed to reduce particle losses at
injection in the accumulator rings. The chopping sy stem
must be also used to create gaps between the batche s sent
to the different users by fast switching magnets. T he
choice of 2 MeV as chopping energy results of a
compromise between space-charge induced debunching
and emittance growth at low energy, chopper feasibi lity
problems increasing with the energy and the necessi ty to
stay below the first radio-activation threshold of copper
(2.16 MeV).
The H - source is pulsed with a rise/falling time of about
10 µs [3]. The beam is then pre-chopped in the LEBT
with a rising/falling time close to 1 µs. The optim isation
has been done with the beam injected at 95 keV in t he
first RFQ. Experiences gained on high power linac [ 4,5]
show that safety margins on both beam current and beam emittance has to be taken to obtain a robust design with
nominal parameters. For that purpose, 100 mA H - beam
with 0.25 π.mm.mrad transverse rms norm emittances was
the reference for the study presented here. The goa l was to
preserve the beam quality while the chopping occurs
between 2 rf bunches in about 2 ns. Multiparticle c odes
have been used to optimise the whole line (RFQ1 –
MEBT – RFQ2) to ensure realistic calculations of bo th
transmission and emittance growth.
2.1 RFQ1
The first RFQ is designed to accelerate the beam fr om
95 keV to 2 MeV. The beam dynamics is computed usin g
both PARMTEQM (z-code) and TOUTATIS, a more
sophisticated t-code [6]. The optimisation procedur e is to
design an initial RFQ for the full energy range (95 keV up
to 5 MeV) and then to cut it in two parts. The maxi mum
electric field is maintained below 1.7 Kp (31.3 MV/ m) in
order to avoid sparks in the cavity. The main RFQ1
parameters are specified in Table 1.
Table 1: RFQs specifications and results
Parameter RFQ1 Value RFQ2 value
Input energy 95 keV 2.0 MeV
Output energy 2.0 MeV 5 MeV
Input current 100 mA 97.1 mA
Input emit. 0.25 π.mm.mrad 0.29 π.mm.mrad
Length 5 m 3 m
Number of cells 527 83
Min. aperture a 3.52-4.13 mm 3.71-3.73 mm
Modulation 1-1.59 1.59-1.75
Vane voltage 87.3-117.7 kV 117.7-122.8 kV
Output emit. 0.26 π.mm.mrad 0.3 π.mm.mrad
Transmission 99.7 % 99.97 %
A transition cell is included between the last
accelerating cell and the fringe field [7] and the length of
this fringe field is adjusted to simplify the MEBT design.
The length of the RFQ1 cavity is mainly imposed by the
slow adiabatic bunching process needed to reach a h igh
capture efficiency as in the IPHI project [4]. The RF
segmented RFQ concept is kept from previous project
[4,5] for the construction of the ESS RFQ cavities.
2.2 MEBT
The aim of the Medium Energy Beam Transport line is
to match the beam into the second RFQ minimising
emittance growth and halo formation. Less than 0.01 % of
the chopped beam must enter the second RFQ, and the transmission of the non-chopped beam must be higher
than 95 %. The line is made up of 10 quadrupoles, 3
bunchers and 2 choppers. The total length (2.1m) is kept
as short as possible to minimise the emittance grow th. The
2 choppers are installed inside the quadrupoles.
Figure 1: Medium energy beam line.
Top : normal transport in the X plane.
Bottom : Y plane with the choppers (blue).
The 2 diaphragms are shown in red.
The first diaphragm prevents damages on the chopper s.
The second one collects the chopped beam and cleans the
non-chopped beam before RFQ2 (about 7 kW of stopped
particles is expected from the chopped beam). They are
sectored diaphragm to help in the tuning process.
Chopper plate
voltage Line
Transmission
800 V 0.01 %
600 V 0.16%
400 V 11.0 %
0 V 97.2 %
Table 2: transmission vs the
dynamic plate voltage
Figure 2 : Chopped and
unchopped beam
Emittance growth X Y Z
100 mA + 11 % + 13 % + 6 %
70 mA + 6 % + 9 % + 1 %
Table 3: Emittance increase through the MEBT.
Due to the non-linearity of the buncher fields and the
space-charge induced radial–longitudinal coupling,
emittance growth are minimum when the bunch transve rse
and longitudinal sizes in the bunchers are small. T he line
has been optimised to avoid these effects leading t o the
emittance growth given in Table 3 and a transmissio n of
97.2 %. 2.3 Choppers
Chopper requirements - The chopper tricky task is to
clear off an intact number of rf bunches with drast ic
rise/fall times (Table 4).
Electric length 2 x 240 mm or 1 x 480 mm
Gap 16 mm Min
Pulser voltage +/- 950 V Min
Field efficiency 84 %
Chopping time 2 × 600 µs 100 µs spaced out
Chopping frequency 360 ns off, 240 ns on, 50 Hz
Duty factor 5-65 %
Rise/fall time < 2 ns 2-98 %
Table 4: chopper main requirements
Figure 3 : 4 tracks 3D sections of the MAFIA model
Technologic choices - A micro-strip line meander
structure (Figure 3), like the one chosen for the S NS
design [8] [9], seems to suit this type of fast bea m
chopping. The total strip length has to be limited to avoid
pulse distortion. The choice of a notched line allo ws large
strips width on thin laminate keeping a 50 ohms
characteristic impedance [10]. The rise and fall ti me
requirement in the ns range implies a quasi TEM mod e for
the signal wave propagation.
Simulations and calculations - 2D simulations [11] give:
• The capacitance per unit length of the micro-strip
line using a static field solver.
• The self inductance per unit length using a transie nt
solver which takes into account the skin effect.
The self inductance increase due to notches is
empirically calculated from previous slit studies i n micro-
strip lines [12]. The chamfered extremities can be
predetermined from charts [13]. The characteristic
impedance of the line and the signal phase velocity can be
deducted from these calculations.
Figure 4 shows a coloured contour density of the
electrical field and equipotential lines. The field ripple is
less than 5% near the beam axe. Beam dynamics
calculations done using multiparticle codes do not show
evident effects resulting from this ripple.
MAFIA code [14] allows full 3D simulations in the
temporal domain. It confirms the previous calculati ons
within a 5% error margin. New calculations with mes h
refinement will be performed soon.
Figure 4: 2D electric field density
(18 tracks cross-section)
Moreover Pspice electrical simulations [15] dealing
with “lossy” line fitted with appropriate coefficie nts let us
assert that we can reach the desired rise and fall times
with a single 50 cm long chopper. Nevertheless, two
24 cm long choppers were used for the simulations. Table
5 summarises the geometrical and electrical values to
reach the chopping requirements.
Microstrip width 8 mm
Tracks period 10 mm
Separator thickness 500 µm 250 µm overhanging
Meander line width 78.2 mm 60.2 mm straight
Laminate thickness 3.04 mm Rogers RT 6002
Notches period 3.6 mm
Notch depth / width 3 / 1 mm
Lineic inductance 335 nH/m
Lineic capacitance 134 pF/m
Characteristic
impedance 50 ohms
Phase velocity 150 mm/ns 0.5*c
Table 5 : calculated main electrical characteristic s.
Total line length 2.87 m
Plate length 388 mm
38 tracks
Plate width 91 mm
Gap stroke 10-35 mm
Line thickness 70 µm
Global tolerance 5/100 mm
Table 6 : prototype main
characteristics
Figure 5 : 3D prototype
realistic view
Prototype design - A prototype (Figure 5) has been
recently launched and will be ready for tests and
measurements for the end of summer 2000. It is a fu ll 2
plates structure with limited length and line thick ness
(Table 6), a limitation due to the use of a standar d
microwave laminate (two Rogers RT 6002 12”x18” plat es
1.52 mm thick pasted together). A micrometric slide
allows the stroke adjustment to measure the impedan ce
variation according to the gap range. A TNC connect or
terminates each line end. 2.4 RFQ2
The second RFQ brings the bunched beam from 2 MeV
to 5 MeV. An inverse transition cell is added betwe en the
short matching section and the beginning of the
modulated vanes in order to preserve the beam
emittances. The 5 MeV final output energy allows th e
construction of the first drift tubes of the DTL us ing EM
quadrupoles as demonstrated by the IPHI R&D
programme. Again, 100 000 particles have been
transported from RFQ1 to RFQ2 through the MEBT to
ensure realistic simulations. The main RFQ2 paramet ers
are shown in Table 1.
3 CONCLUSION
The present design of the two 352 MHz RFQs and the
chopping line allows a perfect chopping between two
bunches without beam characteristics degradations. A
prototype is in construction and will be tested soo n to
confirm the calculated performance of the chopper.
4 ACKNOWLEDGEMENTS
The authors would like to thanks Sergey S. Kurennoy
(LANL) for his helpful advises and information.
5 REFERENCES
[1] The ESS technical study, ESS-96-53-M, Nov 96
[2] J-M. Lagniel, "High-Power Proton Linac for a mu lti-user
facility", EPAC2000, Vienna.
[3] R. Gobin et al., “last results of the CW high-i ntensity light
ion source at CEA-Saclay”, Rev. Sci. Instr. 69 n°2, 1009
(1998)
[4] P-Y. Beauvais, “Status report on the Saclay Hig h-Intensity
Proton Injector Project”, EPAC2000, Vienna.
[5] J. D. Schneider, “Operation of the low-energy
demonstration Accelerator: the proton injector for apt”,
PAC99, New-York, IEEE page 503.
[6] R. Duperrier, "Dynamique de faisceaux intenses dans les
RFQs -Toutatis", Thèse Université Paris-Sud ORSAY.
[7] K. Crandall, "Ending RFQ vanetips with quadrupo le
symmetry", Linac 94, p 227.
[8] SNS Interface Definition Document, “Beam Choppi ng and
Chopper Requirements”, DM Chapter 11- Sect 14, Sept . 98
[9] SNS PARAMETER LIST on July 8, 1999
[10] S.S Kurennoy, J.F. Power and D. Schrage, “Mean der-Line
Current Structure for SNS Fast Beam Chopper”, PAC 9 9
[11] Opera-2d, finite element code from Vector Fiel ds Ltd
Kidlington, UK
[12] W.J.R. Hoefer, “Equivalent Series Inductivity of a Narrow
Transverse Slit in Microstrip” , IEEE MTT-25, 822 ( 1977)
[13] MAFIA release 4.023, “Maxwell’s equations by m eans of
the Finite Integration Algorithm code, from CST Gmb H
Darmstadt, D
[14] K.C. Gupta, Ramesh Garg, Inder Bahl, Prakash B hartia, «
Microstrip Lines and Slotlines », second edition, A rtech
House Publishers
[15] Pspice, electrical simulation code from Micros im Corp.
(USA) |
arXiv:physics/0008145 18 Aug 2000STATUS REPORT ON THE 5 MeV IPHI RFQ
R. Ferdinand, P-Y. Beauvais, R. Duperrier, A. France, J. Gaiffier,
J-M. Lagniel, M. Painchault, F. Simoens, CEA-Saclay, DSM-DAPNIA-SEA
P. Balleyguier, CEA-Bruyères le Châtel, DAM
Abstract
A 5-MeV RFQ designed for a proton current up to
100-mA CW is now under construction as part of the
High Intensity Proton Injector project (IPHI). Its
computed transmission is greater than 99 %. The mai n
goals of the project are to verify the accuracy of the
design codes, to gain the know-how on fabrication, tuning
procedures and operations, to measure the output be am
characteristics in order to optimise the higher ene rgy part
of the linac, and to reach a high availability with minimum
beam trips. A cold model has been built to develop the
tuning procedure. The present status of the IPHI RF Q is
presented.
1 INTRODUCTION
Over the last 10 years, in-depth studies have been
carried out on the feasibility of high-power proton
accelerators capable of producing beams of several tens of
MW. With heavy targets, such beams can produce
extremely intense spallation neutron flux. Several
applications could benefit from the performance of this
new generation of high-power proton accelerators [1 ]:
spallation neutron sources for condensed matter stu dies,
hybrid reactors for nuclear waste transmutation, ne utrino
and muon factories, technological irradiation tools ,
production of radioactive ion beams, production of
radioisotopes, etc.
IPHI (“Injector of Protons for High-Intensity beams ”)
is a 1 MW low energy prototype, which could be used as
front end for such high-power proton accelerators [ 2].
This demonstrator is made up of the SILHI ECR sourc e
able to deliver more than 100 mA CW at 95 keV, the
5 MeV RFQ and a 10 MeV DTL. IPHI is designed to
operate up to 100% duty factor (CW).
2 RFQ BEAM DYNAMICS
The input energy of 95 keV results of a compromise
between RFQ length, source reliability and space-ch arge
control. The 5 MeV output energy results of a
compromise between cavity length, feasibility of th e DTL
using EM quadrupoles, and high beam transmission. T he
use of existing klystron at 352.2 MHz leads to an
optimum size of the cavity. The design current of 1 00 mA
has been selected to reach a high reliability at th e lower
currents needed by the different applications. The
expected normalised rms emittance from the source is
0.2 π.mm.mrad. Nevertheless, a safety margin is taken
using 0.25 π.mm.mrad in beam dynamics calculations.
The maximum electric field has been limited to 1.7 Kp (31.34 MV/m) taking into account experiences with t he
CRITS Experiment at Los Alamos and RFQs operated at
Saclay in the past. The RFQ cavity length was set t o 8 m.
Great care was taken on lost particles in the RFQ c avity.
The final design has been selected to avoid localis ed and
high-energy losses (activation), and to provide the highest
transmission avoiding any bottle neck [3]. Many bea m
dynamics computations including error studies have been
done using several complementary codes (PARMTEQM,
TOUTATIS [4,6], and LIDOS.RFQ [5]). Table 1 gives
the main parameters of the RFQ.
Table 1 : IPHI RFQ parameters
Structure 4 vanes
Frequency 352.2 MHz
Total length 8 m (8 sections)
Resonant coupling sections 4
Input/output Energy 95 keV / 5 MeV
Input beam characteristics 100 mA/0.25 π.mm.mrad
Mean aperture (R 0) 3.7 - 5.3 mm
Modulation (m) 1 - 1.75
Vanes voltage 87 - 123 kV (1.7 Kp)
PARMTEQM transmission 99.2 % (accel. particles)
Beam Power 490 kW
Total expected power 1650 kW
Stored energy 5.3 J
The new TOUTATIS code [6] allows to take into
account the field errors in the coupling section as well as
the mechanical defaults (vane extremities displacem ent...).
The LIDOS.RFQ code allowed to establish the require d
machining precision. Lots of errors were simulated with
the coupling gaps to ensure a “faults tolerant” des ign.
80% 85% 90% 95% 100%
0 50 100 150 200
a) Input beam current (mA) Transmission Transmitted
Accelerated
97% 98% 99% 100%
1 1.2 1.4 1.6 1.8 2
b) Input emittance ( ππ ππ.mm.mrad) Transmission 0.17 0.20 0.23 0.27 0.30 0.33
Transmitted
Accelerated rms
Total
27.66□MV/m 28.58□MV/m 29.04□MV/m 29.50□MV/m 29.96□MV/m 30.42□MV/m 31.34□MV/m 32.26□MV/m 33.19□MV/m 34.11□MV/m 35.03□MV/m 75% 80% 85% 90% 95% 100%
1.5 1.6 1.7 1.8 1.9
c) Kilpatrick Transmission Transmitted
Accelerated
0246810
0 1 2 3 4 5 6 7 8
d) RFQ length (cm) Losses (W/cm) Ideal case
Misaligned
input beam Gap 3
Figure 1: Transmission versus input beam current (a ),
input beam emittance (b), and rf field (c). Power
deposition due to beam losses along RFQ (d) 3 RFQ CAVITY
The structure is made up of 8 one-meter long sectio ns
accurately machined and brazed then assembled with a
resonant coupling every 2 meters (similar to the LE DA
design [8]). Details of the "Main steps for fabrica tion of
the IPHI RFQ" are published in these proceedings
(THD03, M Painchault et al.).
Figure 2: IPHI RFQ Artistic view with the 8 RF port s
The coupling plates allow a damping of the rf
longitudinal parasitic modes and the introduction o f
fingers to push away the dipolar modes. The RF desi gn of
the cavity requires intensive 3D simulations and
developments [7] with cross-checking on the cold mo del.
The field will be tuned using 128 tuners equally
distributed along the RFQ. The 8 Thomson RF windows
are already on hand, similar windows have been
successfully tested up to 700 kW at LANL for LEDA [ 8].
The low level RF is still under definition. A pick- up will
be used for the fast phase and amplitude control us ing
DSP. The slow frequency tuning will be done using t he
cooling system of the cavity based on the LEDA desi gn
[8]. The inlet water temperature is 10°C/50°F with water
flows tuneable up to 6 m/s. An erosion/corrosion an alysis
is presently done.
Great care has been put into the optimization of th e
pumping system. Two of the 1-m long sections are
dedicated to the rf feed while all the remaining se ctions
are equipped with a total of 72 pumping ports caref ully
designed to maximize the pumping speed. The running
pressure is expected to be 8 10 -6 Pa.
Figure 3:View of the RFQ cold model.
4 COLD MODEL
3.1 Objectives
The fabrication of a RFQ cold model started at the
beginning of 1999. The aluminium cavities have been
designed close to those of the final RFQ with possi ble fast
adjustments of the geometry to test different end v ane configurations, consequences of vane position error s... It
is now mainly used to develop the RF tuning procedu res
and the associated hardware and software. Some of t he
development will be used to help on vane positionin g
before the brazing step with an expected precision better
than 10 -5 m.
3.2 Design
The 1:1 scale cold model has been designed using
SUPERFISH for the main region and MAFIA for the end
regions. The transverse section is composed of flat faces
only, the tips of the electrodes are circular ( ρ/r 0 = 0,85).
The design has been done for a resonance frequency of
350.7 MHz with all the tuners flush mounted and
352.2 MHz with the tuners 5 mm inside the cavity.
3.3 Modularity
Figure 4 : transverse view of the RFQ cold model.
The cold model consists of octagonal 1-meter long
sections in which the 4 electrodes are screwed. Thr ee
different kinds of end pieces may be screwed to the main
electrodes: input/output beam, coupling region and plain
pieces. This last type allows to form a 2 meters lo ng RFQ
with a continuous electrode. Three different kinds of
segments can be used :
1- “pumping” type with the same slug tuners distrib ution
as the final pumping segments,
2- “RF coupling” type with one rectangular hole per
quadrant to allow the study of RF power coupling
through irises,
3- “Tuners over-equipped” type with 8 tuners per
quadrant instead of 4 for an accurate study of the
voltage law tuning.
One “pumping” section and one “Tuners over-
equipped” section are tested since August 99. The
measured resonance frequencies with the tuners at t he
nominal position are respectively 350.054 MHz and
351.24 MHz, the relative error is less than 3.10 -3. The
lower 2 dipole modes are about 700 kHz apart. Four more
segments will be tested soon.
An elaborated pulley system guides the bead on
different path through the 4 quadrants allowing a
comparison of the magnetic and electric fields
measurement in several locations (see Figure 5).
Figure 5: pulley system of the cold model
3.4 RF diagnostics
The fields are measured using the common bead–pull
perturbation method. A DC motor drives the bead and a
vector network analyser (VNA) measures the phase of the
transmission coefficient (s21). Operation of the DC motor
and the VNA is fully automatic, both being driven b y a
LabView program on PC. Measurements are readily
available in data files.
3.5 Measurement analysis
The data are then treated through a Matlab code. Th e
first step is the conversion of the measured phases into
voltages versus position for each quadrant, all exp ressed
in arbitrary units since no attempt is made to deri ve the
polarisability of the bead or the phase versus freq uency
slope of the s21. Smoothing and windowing produce
direct usable data. The second step is the analysis of the
data from the 4-quadrants. The RFQ is modelled as a
4-wire line system. The spectral theory of differen tial
operators is used to relate the measured voltages a long the
line to the physical parameters describing the whol e RF
circuit (the parallel capacitances C i or inductances L i of
each quadrant i versus the position and the end loads).
Figure 6: Estimated capacitances
(order = 2 for Q, S and T eigen modes) Figure 6 shows the result of a 4-quadrant analysis. The
magnetic field of the “over-equipped” type segment has
been perturbed with a titanium bead guided close to flush
mounted tuners. The left hand side figures are plot s of
modal combinations of capacitances over the theoret ical
capacitance C versus the longitudinal position z [m ]. The
plots at right are the C i/C for each quadrant. All the curves
are well within plus and minus 1% indicating a very
accurate machining and positioning of the 4 quadran ts.
The measurement of the capacitances is reproducible
within 2.10 -3 .
5 CONCLUSION
The detailed design of the IPHI RFQ is now nearly
completed. Great progress has been done on the RF
tuning procedures using the cold model. The constru ction
of a one-meter long copper prototype will be finish ed in
September. The delivery of the first RFQ section is
expected for the end of 2000 and the 8 sections mus t be
available mid 2002. The assembly will start before the
reception of the last section as shown in the plann ing
below. The first beam is expected late 2002 / begin ning
2003.
1234123412341234
IPHI□site□availability
Cooling□system□overhaul□
Source/LEBT□settling□in
RFQ/rf/HEBT/BS□assembly
RFQ□conditioning
5□MeV□pulsed□operation
5□MeV□CW□operation 2003
Power□supply□distribution□overhaul2001 2000 2002
REFERENCES
[1] J-M. Lagniel, "High-Power Proton Linac for a
Multi-User facility", EPAC 2000, Vienna, Austria.
[2] J-M. Lagniel et al., "IPHI, the Saclay High-Int ensity
Proton Injector Project", PAC 1997, Vancouver,
Canada.
[3] R. Ferdinand et al, "Optimization of RFQ design ”,
EPAC 1998, 1106, Stockholm, Sweden.
[4] R. Duperrier, "Dynamique de faisceaux intenses dans
les RFQs – Toutatis", PhD thesis n°6194, Université
Paris-sud ORSAY, France, July 2000.
[5] B. Bondarev, A. Durkin, S. Vinogradov "Multilev el
Codes RFQ.3L for RFQ designing", Moscow
Radiotechnical Institute, Proc of Computational
Accelerator Physics Conference (Virginia, USA,
1996).
[6] R. Duperrier et al, "Toutatis, the Saclay RFQ c ode",
this conference.
[7] P. Balleyguier, "3D Design of the IPHI RFQ Cavi ty"
this conference.
[8] J. D. Schneider, "Operation of the Low-Energy
demonstration Accelerator: The proton injector for
APT", PAC99, New York, pp 503-507. |
Continued Monitoring of the Conditioning of the Fermilab Linac 805 MHz
Cavities*
E. McCrory, T. Kroc, A. Moretti, M. Popovic, Fermilab, Batavia, IL 60510, USA
Abstract
We have reported previously on the conditioning of the
high-gradient accelerating cavities in the Fermilab Linac
[1, 2, 3]. Automated measurements of the sparking rate
have been recorded since 1994 and are reported here.
The sparking rate has declined since the beginning, but
there are indications that this rate may have leveled off
now. The X-rays emitted by the cavities are continuing
to decrease.
1. INTRODUCTION
Fermilab commissioned the seven, high-gradient 805
MHz RF accelerating modules in 1993. In order to
achieve the desired acceleration, gradients of up to 8
MV/m were required, which led to maximum surface
gradients of nearly 40 MV/m. These high fields caused
some concern about RF breakdown leading to beam loss
and to excessive X-ray exposure.
After seven years, it seems that the change in the rate
of these breakdowns has stabilized at a level well below
the original specifications: a lost beam rate due to RF
breakdown/sparking of 0.1% or less.
2. OVERVIEW OF MEASUREMENTS
Automated measurements of the sparking rate of each
of the seven 805 MHz RF cavities in the 400 MeV Fer-
milab Linac have been collected since April 1, 1994.
Also, we have automatically recorded the number of
beam pulses lost each day, presumably due to RF break-
down in one or more of the cavities, beginning in January
1994.
We have measured the X-ray production rate as a func-
tion of the power levels in one cavity on several oc-
casions over these years.
2.1. Sparking Rate
The sparking rate has been measured continually at the
15 Hz repetition rate of our RF system. These data have
been recorded daily. We have accumulated 1893 days of
data (82% of the available days). We record the number
of RF pulses for each of the seven 805 MHz cavities and
the number of times an RF pulse at that cavity was ruined
by an RF breakdown/spark. We have experimented with
various ways of detecting sparks in the cavities, and have
determined that watching for abnormal reverse power
from the cavity is the most reliable. We tried for ap-
proximately five years to correlate this reverse power
signal with vacuum activity in the cavity, and this worked
reasonably well in the early part of this period when the pressure was relatively high and stable. But now, better
vacuum conditions, coupled with regular, small vacuum
bursts unrelated to spark activity make the spark-induced
vacuum activity harder to identify. The ratio of these two
methods of counting varies by about a factor of two from
day to day, with an average ratio of 2 reverse-power-only
count for every reverse-power-and-vacuum-activity
count. The data we present here are for the reverse-
power-only method.
2.1.1. The Overall Rate
Table 1 shows the median number of sparks per day
for each of the years we have been accumulating data.
Most days have about 1.296x106 RF pulses per cavity.
The “Days” column represents the number of days
counted, based on the total number of RF pulses recorded
that year. Note that 1998 had only 196 equivalent days—
this is due to a series of major shutdowns in the Linac that
year. The jumps in the numbers in this Table, particu-
larly between 1995 and 1996 in Modules 1 and 2, corre-
spond to increasing the length of time the RF is at full
value (the “pulse length”).
There is no indication that sparking is correlated
among the cavities. So, one would expect that the sum of
the values in each row would represent the median num-
ber of sparks in the entire Linac per day.
We currently expect about 86 ± 32 sparks in the Linac
per day. This is the median number of sparks, summed
over all cavities, ignoring possible correlations. The error
bars represent the quadrature sum of the standard devia-
tion on the number of sparks per cavity, per day. This is a
rate of (6.6 ± 2.5) x10-5 sparks per RF cycle, or about one
spark every 17 minutes of operation. This is well below
the original specification of 1 spark in the Linac for every
1000 RF cycles.
2.1.2. Rates Per Cavity
The sparking rate of a cavity depends on many things,
and cold, startup effects often dominate getting a clean YearDaysM1M2M3M4M5M6M7
1994262 8551123542194
1995324 578661326134
199631812440441427143
1997289 92242671961
1998196 68876861
1999295 1110176762
2000141 2918195771
Table 1. The median number of sparks per day.
* Work supported by the US Department of Energy, contract #
DE-AC02-76CH0-3000.
reading of the rate per day. (The RF systems are inter-
locked, so coming out of an enclosure access often causes
small problems, which are generally manifested by high
reverse power that are not necessarily associated with
sparking.) Module 3 seems to have been the most stable
over these years, so we present the sparking rate per pulse
per day in Figure 1 for Module 3. The other modules
show similar characteristics, but because we have done
more experimentation with the pulse length on them, the
data are not as clear.
The most striking feature of this graph is that the spark-
ing rate has steadily declined for the entire measurement
period, and is only now beginning to show signs of level-
ing off. (Note that one spark per day would be a sparking
rate of just below 1x10-6, or “-6” on this graph).
The fit to these data for Module 3 says that in 1700
days, the sparking rate has decreased by a factor of ten.
We have experimented with changing the RF pulse
length on many of the cavities. We changed the pulse
length on Module 3 in June of 1999 from 60 to 67 micro-
seconds. According to our previous paper [2], we would
expect the sparking rate to increase by a factor propor-
tional to the fourth power in the pulse length. (67/60)4 =
1.55, which is consistent with the data presented here.
Prior to the pulse length change, it appears that the
sparking rate on Module 3 may have begun to level off at
a rate of one spark every 105 RF pulses. The other mod-
ules have a similar behavior, although it is difficult to
factor out the effect of the lengthening of the RF pulse.
We will continue to monitor the sparking rate and report
again in a few years.
2.2. Lost Beam
We also began counting the number of lost beam
pulses per day in 1994. The algorithm for determining
this, while not ideal, is reasonable: At the repetition rate
of the RF systems (15 Hz), we look for a beam pulse by
watching the current on the beam toroid at the beginning
of the 805 MHz section (at 116 MeV). If the beam cur-
rent is above 20 mA, then this cycle is a beam pulse. If,
then, the beam current out of the end of the linac (400
MeV) is less than 20 mA, we record this as a lost beam pulse. This records all sparks that result in a loss of
beam, but it also captures the occasional beam pulse dur-
ing routine tuning where the input current is just over 20
mA and the output current is just under that level. We
estimate that on days with ten or more lost beam pulses,
one can reasonably expect that one or two are from this
effect.
The data for the number of lost beam pulses are shown
in Figure 2. The line represents the median number of
lost beam pulses per day for the year, calculated on the
last day of the year. The number of lost beam pulses per
day was significantly larger in 1994 than it is now (an
average of 64.4 and a median of 9 with a standard devia-
tion of 398 in 1994 versus 2.1 ± 3.8 (median = 1) now).
Zero is represented as 0.1 on this log graph. The median
number of lost pulses per day in 1998 was zero because
we were down for a large fraction of that year.
With 30000 b eam pulses per day, we would expect
1.98 ± 0.74 lost beam pulses per day due to the RF break-
down rate of 6.6 x 10-5. Since we measure between 1 and
2 lost beam pulses per day, we can conclude that the
presence of beam does not have an appreciable effect on
the sparking rate in our cavities. In [1], we reported that
there is a 20% increase in the sparking rate during beam.
The statistics do not justify this conclusion now.
2.3. X-Ray Measurements
We have measured the X-ray levels at each of the four
sections of Module 5 on several occasions: once when it
was first commissioned, once for the 1996 paper, and
once again now. The data are shown in Figure 3.
The 1992 data were taken with a single detector placed
approximately four feet transversely from the center of
the module, between sections 2 and 3. The rest of the data
were taken with four detectors placed approximately 1
foot transversely from the center of each of the four sec-
tions of the module. The 1992 data have been multiplied
by four (assuming a quasi-line source) to suggest the
proper relationship to the other data that have not been
transformed.
We fit the data from each detector to the Fowler-Nord-
heim equation for an RF field that describes enhanced
field emission [4]. Figure 1. LOG(Module 3 Sparking Rate) per day
y = -0.0006x + 15.259
R2 = 0.275
-7-6.5-6-5.5-5-4.5-4-3.5-3-2.5-2
1-Apr-941-Apr-9531-Mar-9631-Mar-9731-Mar-9831-Mar-9930-Mar-0060 usec67 usecFigure 2, Lost Beam Pulses per Day
0.11.010.0100.01000.010000.0
1-Jan-941-Jan-951-Jan-9631-Dec-9631-Dec-9731-Dec-9831-Dec-99jF,5.7310
|
THE FABRE PROJECT AT TRIESTE
G. D'Auria, C. Rossi - Sincrotrone Trieste.
M. Danailov - Laboratorio Fibre Ottiche-Sincrotrone Trieste.
M. Ferrario - INFN Laboratori Nazionali Frascati.
N. Piovella, L. Serafini - Universita' di Milano and INFN.
Abstract
A program to design a high brilliance electron source
suitable for a short wavelength Linac-based FEL is
presented. The goal of the project is to develop a multi-
cell integrated photoinjector capable of delivering 1 nC
bunches with emittance below 1 mm mrad.
This will be the first step toward a possible
development of a IV generation light source test facility
based on the existing Trieste Linac. For this purpose a
common program between Sincrotrone Trieste and INFN-
Milano has been undertaken. Here a brief description of
the program and the first results of the RF Gun
electromagnetic structure with the beam dynamics on the
ELETTRA Linac are presented.
1 INTRODUCTION
The growth in interest over the last few years in IV
generation light sources based on FELs and the SASE
process [1,2] requires further considerable efforts for the
production of intense low emittance beams. In fact, in
order to reach and operate these facilities at shorter
wavelengths, the electron beam emittance εn, and the
coherent radiation wavelength λ, must be very close to
each other, to guarantee the maximum overlap between
the two beams in phase space. Moreover the efficiency of
the SASE-FEL emission process, defined as the ratio
between the photon beam power and the electron beam
power, scales like IPK1/3 with IPK the electron beam peak
current. The previous requirements clearly show that the
real figure of merit for the electron beam is the normalized
brightness, defined as Bn=IPK/4πεn2: a FEL in the X-ray
band will require 1014 to 1015 A/m2 and is nowadays the
main challenge to meet. In order to reach these values,
that are at the limit of expected performances for the next
decade, all the laboratories pursuing a long term program
in IV generation radiation sources have begun developing
experiments and test facilities to study the physics of high
brightness electron beams. The FaBrE project can be
viewed within the same context: its main aim is the study
and construction of a high brightness photo-injector to be
installed on the ELETTRA Linac for a FEL-SASE test
facility in Trieste. This ambitious program will also take
advantage of the future planned upgrading of ELETTRA:
after the commissioning of the new ELETTRA full
energy injection system [3], the present 1.0 GeV injector
Linac will be available for the proposed test facility.2 AIMS AND TIME SCHEDULE
Despite the large progress in brightness seen in the 90's
by electron sources based on RF photo-injectors, only
recently has a theoretical understanding of the
phenomenon of emittance degradation, hence of the
achievable brightness by a photo-injector, been made
possible. The success of the "invariant envelope" model,
that concerns the prediction of a new equilibrium mode for
a beam in the laminar flow regime, [4,5], will allow
further progress in this strategic field enabling an
optimum control of the emittance growth.
On this basis it has been shown recently that the
"integrated" photo-injector, whose first accelerating
section is integrated into the gun itself, has the optimal
configuration to produce a beam satisfying the invariant
envelope conditions when compared to the more popular
"split" version, where the accelerating section is
physically separated from the gun.
Starting from these considerations the goal of the
FaBrE project is the study and the construction of an
integrated photo-injector whose RF structure complies, as
much as possible, with the requirements imposed by
theory:
i) high spectral purity of the accelerating field profile
on the axis;
ii) normalized amplitude of the accelerating field,
α=eE0/2kRFmec2=1.3, corresponding at S-band to a
peak field of 80 MV/m;
iii) shunt impedance as high as possible (compatible
with the first requirement);
iv) capability to host a photo-cathode in ultra high
vacuum (≤ 10-9 torr);
v) stability and large separation between contiguous
resonant modes in the operating band.
In order to distribute as much as possible costs and
commitments, the whole program will be divided into
three different phases.
Phase I, with a quite modest commitment and cost,
will be limited to theoretical studies for the choice of the
most suitable electromagnetic structure, and the
construction of a whole accelerating section with its
electro-magnetic characterization at low RF power. On the
basis of the first numerical simulations carried out on
both Coupled Cavity Linac (CCL) and Plane Wave
Transformer (PWT) accelerating structures, it seems that
the second solution has a better matching with the
theoretical requirements previously mentioned [6], even ifmore simulations are required. We plan to get the first
copper model of the accelerating section before the end of
this year. This phase, already funded, should be completed
at the beginning of 2001 with the RF characterization of
the accelerating structure.
Phase II , in which the first prototype of the final
accelerating structure will be assembled and tested at high
RF power levels. The same section will also be used for
preliminary beam tests using a commercial Q-switched
Nd:YAG laser (Quantel YG 585_10), not synchronized
with RF, but already available in our laboratory.
Preliminary estimations show that with a slight
modification of the laser cavity and utilizing a BBO
crystal for IV harmonic generation, this laser can deliver 7
nsec pulses with 10 mJ at 266 nm in a nearly gaussian
transversal mode. An additional shortening and smoothing
of the delivered pulses are also expected by further
optimization of the laser cavity and, if necessary, using a
slicing with fast Pockels cell (the latter may turn out to
be necessary in order to avoid cathode damage). The goalof the second phase, not yet funded, is to have within
2002 a 12 to 20 MeV electron beam, with 100 to 200 A
peak current, and less than 5 mm mrad normalized
emittance.
Phase III , the photo-injector will be upgraded with a
mode-locked laser system able to deliver very short UV
pulses (230 nm, 10 ps, 200 µJ, with rise time and jitter
less than 1 ps). Such a performance can nowadays be
obtained by using several different types of mode-locked
systems (i.e. based on Nd:glass, Cr:LiSAF or
Ti:Sapphire); the final choice will be made on the basis of
further numerical simulations as well as on the experience
gained from phase II. At the end of this phase we expect
to have a 20 MeV pulsed electron beam with roughly 150
A peak current, 1 nC charge, 1 mm mrad emittance and
energy spread better than 1%. The expected quality of the
electron beam will allow the start-up of the suggested
R&D program installing the photoinjector on the existing
Linac.
Figure 1: Machine layout
3 PRELIMINARY BEAM
DYNAMICS SIMULATIONS OF THE
ELETTRA LINAC
3.1 General Layout of the machine
A complete desciption of the Trieste Linac can be found
in [7,8]. Neglecting the electron source and the
bunching section, the whole machine consists of two
different parts: i) a 100 MeV preinjector, made up of
two 3.2 m long constant impedance accelerating
sections (S0A, S0B), equipped with focusing solenoids
(up to 2.5 KGauss) and presently operated at 18 MV/m;
ii) the second part of the Linac is made up of seven 6.2
m long BTW accelerating sections (S1 to S7) equipped
with a SLED pulse compressor system. Eight
Thomson TH2132 45 MW klystrons feed the whole
machine. The maximum operating gradient has been
reached at 28 MV/m, and the machine can provide a
maximum energy of 1.2 GeV. Keeping fixed the first
100 MeV supplied by the preinjector the remaining
sections on average can easily supply 150 (or 100)
MeV/section with (or without) SLED respectively.
However one of the 6.2 m sections will be used for the
new full energy injector and the operational energy of
the test facility will be between 0.7 and 1.0 GeV
(with/without SLED).3.2 Preliminary beam dynamics studies
Preliminary beam dynamics studies have been carried
out on the whole ELETTRA Linac to estimate the
beam parameters that could be obtained after
installation of the photoinjector (following phase III).
All of the simulations have been made using the semi-
analytical code HOMDYN and considering the
emittance compensation theory. Even if in the future,
to reach lower emittance, different machine layouts
could be considered, at present, in order to minimize
costs, we have implemented only a slight modification
of the present machine layout, shifting the third Linac
section, S3, and leaving a 10 m drift space between
sections S2 and S3 to install the magnetic bunch
compressor, see Fig. 1 for a simplified scheme of the
machine layout. Magnetic optics between the sections
is not considered in these preliminary studies.
The photoinjector parameters have been optimized in
order to get a laminar waist and a maximum of the
relative emittance at the entrance of S0A, that results to
be a suitable condition to damp the emittance
oscillations [9]. The gun energy has been fixed at 20
MeV and the matched accelerating field of the first two
structures results to be 21 MV/m exiting the second
structure at 150 MeV, see Fig. 2 and 3.Ph.inj. S0A S1 S3 to S620 MeV150 MeV 400 MeV 1.0 GeV
S2 S0B45 m drift space, already available in
the present machine tunnel, for
undulator, beam spectrometer, beam
dump, photon diagnostics .020040060080010001200
00.511.522.53
010 2030 4050 6070T_[MeV] Dg/g_[%]T [MeV]Dg/g [%]
Z [m]
Figure 2: Energy gain and energy spread along the
Linac.
00.511.522.53
010 20 30 40 50 60 70<L>_[mm] <X>_[mm] enx_[mmmrad]<L> [mm], <x> [mm], enx [mm mrad]
Z [m]
Figure 3: Bunch length, beam envelope and transverse
emittance along the Linac.
As shown in Fig. 3, in the drift downstream of S0B
the emittance approaches its absolute minimum. The
beam is then injected in the first two sections of the
second Linac, S1 and S2, 20 degrees off crest at 21
MV/m to provide the necessary energy spread for the
magnetic compression expected at 400 MeV. The
magnetic compressor is modeled by HOMDYN as a
"one wiggler period" according to the wiggler hard edge
model reported in [10]. The equivalent period results to
be λw=π2lbend where lbend is the dipole magnet length and
the equivalent field strength is Bw=4Bbend/π. In the
present design lbend=0.5 m and Bw=0.11 T; a focusing
gradient in the chicane has also been considered with a
suitable pole shaping. At the exit of the compressor the
average peak current is 300 A, but inside the bunch, as
shown in Fig. 4, the slice peak current reaches higher
values over approximately one quarter of the bunch
length.
In the remaining 4 structures the beam is driven up
to 1 GeV with a further energy spread reduction to 0.4
%. The slight emittance growth shown in Fig. 3 after
the beam compression, is mainly due to a lack in the
required optics to compensate for space charge effects
induced by bunch compression.0500100015002000250030003500
-2 -1 0 1 2 3Current distribution along the bunchI [A]
Z-Zc [mm]Before Compression
After Compression
Figure 4: Current distribution along the bunch before
(lower curve) and after the magnetic compression.
4 CONCLUSIONS
An ambitious program for a high brilliance electron
source based on a 20 MeV integrated photoinjector has
been recently initiated at Sincrotrone Trieste under the
FaBrE collaboration. The preliminary results are
encouraging and in the near future the beam dynamics
simulations will be extended to include the Linac optics
and the effects of the beam interaction with the
undulator.
REFERENCES
[1]R. Bonifacio et al., "Collective instabilities and
high-gain regime in a Free Electron Laser", Opt.
Commun . 50, 1984.
[2]J. Arthur et al., "Linac Coherent Light Source
(LCLS) Design Study Report", SLAC-R-521,
April 1988.
[3]C.J. Bocchetta et al., "A full energy injector for
Elettra", EPAC 2000, Vienna, June 2000.
[4]L. Serafini et al., "Envelope analysis of intense
relativistic quasilaminar beams in RF photo-
injector: a theory of emittance compensation",
Phys. Rev. E, Vol. 55, June 1997.
[5]L. Serafini et al., "New generation issues in the
beam physics of RF laser driven electron photo-
injectors", SPIE-LASER '99 Conf., San Jose',
CA, January 1999.
[6]G. D'Auria et al., "The FaBrE project: design and
construction of an integrated photo-injector for
bright electron beam production", EPAC 2000,
Vienna, June 2000.
[7]D. Tronc et al. "The ELETTRA 1.5 GeV electron
injector", PAC '91, S. Francisco, May 1991.
[8]G. D'Auria et al., "Operation and status of the
ELETTRA injection Linac", PAC ’97, Vancouver,
May 1997.
[9]M. Ferrario et al., "HOMDYN study for the LCLS
RF photo-injector", LNF-00/004 (P), SLAC-Pub
9400, March 2000.
[10]H. Wiedemann, "Particle Accelerator Physics",
Spring-Verlag, 1993. |
arXiv:physics/0008148v1 [physics.acc-ph] 18 Aug 2000DESIGNOF THE7 MEV/U, 217MHZ INJECTORLINACFOR THE
PROPOSED IONBEAM FACILITYFORCANCER THERAPY
ATTHE CLINICINHEIDELBERG
B. Schlitt,GSI, Planckstraße1, D-64291Darmstadt,German y
A. Bechtold,U. Ratzinger, A. Schempp,IAP, Frankfurt amMai n, Germany
Abstract
A dedicated clinical synchrotron facility for cancer ther-
apy using energetic proton and ion beams (C, He and O)
has been designed at GSI for the Radiologische Univer-
sit¨ atsklinik at Heidelberg, Germany. The design of the
injector linac is presented. Suitable ion sources are dis-
cussed and results of ion source test measurements are re-
ported. The LEBT allows for switching between two ion
sources. A short RFQ acceleratestheionsfrom8keV/u to
400 keV/u. It is followed by a very compact beam match-
ingsectionanda3.8mlongIH-typedrifttubelinacforthe
accelerationto7MeV/u. Bothrfstructuresaredesignedfor
a resonance frequency of 216.816 MHz and for ion mass-
to-chargeratios A/q≤3(12C4+, H3+,3He+,16O6+).
1 INTRODUCTION
Since December1997nearly70patientshavebeentreated
successfully with energetic carbon ion beams within the
GSI cancer treatment program. Advanced technologies
like the intensity-controlled rasterscan method for 3-di-
mensionally conformal tumor treatment using pencil-like
ion beams and an active control of the beam intensity, en-
ergy, position and width during the irradiation have been
developed[1,2]. Thedevelopmentsandexperiencesofthis
programled to a proposalfor a hospital-basedion acceler-
ator facility for the clinic in Heidelberg [3]. It consists o f
a 7 MeV/u injector linac and a 6.5 Tm synchrotron [4] to
accelerate the ions to final energies of 50 to 430 MeV/u.
Table1: Majorparametersoftheinjectorlinac.
Designion12C4+
Operatingfrequency 216.816 MHz
Final beamenergy 7 MeV/u
Pulse currentsafterstripper ≈100eµA C6+
≈0.7mA protons
Beam pulselength ≤200µs@≤5Hz
Dutycycle ≤0.1 %
Norm.transverseexit
beamemittances(95%)1≈0.8πmmmrad
Exit momentumspread1±0.15%
Total injectorlength2≈13 m
1Not including emittance growtheffects inthe stripper foil .
2Including the ionsources andup tothe foilstripper.Three treatment areas (two isocentric ion gantries and one
fixedhorizontalbeamline)areproposedtotreatabout1000
patients/year. To cover the specific medical requirements,
the accelerator facility is designed to deliver both beams
of low-LET (linear energy transfer) ions (p, He) and high-
LETions(C,O).Therequestedmaximumbeamintensities
at the irradiation point are 1×109carbon ions/spill and
4×1010protons/spill. Only active and no passive beam
deliverysystemsareplanned.
2 INJECTOR LAYOUT
A compact injector linac with a total length of about 13 m
has been designed (Fig. 1 and Table 1). To provide a fast
switching between low and high-LET ion beams, the ana-
lyzed beams from two ion sources running in parallel can
beselectedbyaswitchingmagnetbeforeinjectionintothe
rf linac. For the production of the high-LET ion beams
an ECR ion source (ECRIS) is proposed. For the produc-
tion of the low-LET ion beams the installation either of
an ECRIS of the same type or of a much more compact,
cheaperandsimplergasdischargeionsourceis discussed.
To form short beam pulses, a fast macropulse chopper
will be used in the common straight section of the LEBT
line. For the intensity-controlledrasterscan method diff er-
ent beam intensities within an intensity range of 1/1000
are requested for each individual synchrotron cycle. The
required controlled beam intensity variation will be per-
formedalready alongthe LEBT line by changingthe driv-
ing currents of the quadrupole triplet magnets following
eachspectrometersectionfrompulsetopulse.
The 21 MV rf linac [5] is designed for ion mass-to-
charge ratios A/q≤3and an operating frequency of
216.816 MHz. It has a total length of only about 5.5 m
and consists of two cavities — a short RFQ structure and
an efficient IH-type drift tube linac. For stripping off the
remaining electrons prior to injection of the ions into the
synchrotron, a thin foil stripper located about 1 m behind
oftheDTL isusedforall ionspecies.
To avoid contaminations of the helium ion beams with
ions from other elements having the same A/q, the use of
3Heinsteadof4Heisproposed. Toreducespace-chargeef-
fects alongthe completeinjectorlinac in case of hydrogen
ion beamsand to increase the extractionvoltage of the ion
source, the production and acceleration of molecular H 2+
or H3+ion beams is planned. The molecules are breaking
upintoprotonsat thestripperfoil.400 keV/u 7 MeV/u 8 keV/u
5 m
O16 6+C4+12ECR
Ion Source: H2+H+
3
3He+orIon Source:QT SOL QT
Stripper
FoilMacropulse
ChopperQT
SOLQSSpectrometer
MagnetSlitsQT
QD QTIH − Drift Tube Linac RFQ
MagnetSwitching
Figure1: Schematicdrawingoftheinjectorlinac. SOL ≡solenoidmagnet,QS,QD, QT ≡magneticquadrupolesinglet,
doublet,triplet.
3 IONSOURCES
Toachievethedemandedbeamintensitiesattheirradiation
pointwithonlymoderaterequirementsforthe ionsources,
a multiturn-injection procedure with an accumulation fac-
tor of 10 is proposed for the synchrotron. Considering
reasonable loss factors for the complete accelerator chain
and the beam lines, the ion currents required from the ion
sourcesrangefromroughly100e µA O6+to about650 µA
H2+(Table2).
3.1 ECR ionsource
A high-performance 14.5 GHz fully permanent magnet
ECRIS called SUPERNANOGAN has been developed at
GANIL [6] and is commercially available from PAN-
TECHNIK S.A., France. To check the suitability of the
source for the therapy injector, test measurements have
been performed at the ECRIS test bench at GANIL. The
required ion currents could be exceeded by at least 50%
(C4+, H2+) up to a factor of about 3 (O6+, He1+) in a sta-
ble DC operating mode. The rf power transmitted by the
rf generatorwas about100 W forthe extractionof1.1 mA
He1+up to about 420 W for a 200 e µA C4+beam. The
measurednormalized90%transversebeamemittancesare
Table2: Ioncurrents Iionrequiredfromtheionsourcesand
ion source potentials VISneeded for a beam energy in the
LEBT of8keV/u.
Ionspecies Iion VISIonspecies
fromsource [ µA] [kV] tosynchrotron
16O6+100 21.316O8+
12C4+130 2412C6+
3He1+320 243He2+
1H2+650 16 protons
1H3+440 24 protons<0.5πmm mrad for 280 e µA O6+, about 0.6 – 0.65
πmm mrad for C4+and He1+,2+and roughly 0.7 πmm
mrad for a 1.5 mA proton beam. In the latter case, the
measured values may be limited by the acceptance of the
spectrometer system at GANIL. During the tests, some
high-voltage problems occurred above an extraction volt-
ageofabout20kV. Meanwhile,theseproblemshavebeen
analyzed by PANTECHNIK and some improvements will
be tested soon. The solution of these problems is essen-
tial for the therapy injector since extraction voltages of u p
to 24 kV are required for a beam energy of 8 keV/u (Ta-
ble 2). However, alternative high-performance ECR ion
sources using electromagnets are available, which can be
operatedattherequiredsourcepotentials. Forinstance,t he
ECR4-MtypeECRISavailablefromPANTECHNIKorthe
10GHz NIRS-ECRoperatedat HIMAC.
3.2 Gasdischargeionsource
Besideseconomicalreasons,agasdischargeionsourceop-
timized for the production of singly charged ions has sev-
eral advantages. In contrastto ECR ionsources, wherethe
H3+fractionoftheextractedhydrogenionbeamsisonlya
few percent,it can be optimizedto morethan90% in case
of a gas discharge ion source at low arc currents [7, 8, 9].
The acceleration of H 3+ion beams has the important ad-
vantage that the rf power levels in the linac cavities can
be identical in case of12C4+and for hydrogen ion beams.
Hence,fasterswitchingbetweenbothbeamswouldbepos-
sible as well as a very stable and reliable operation of the
cavities. Furthermore,higherioncurrentscan be extracte d
easily and very high beam qualities are achieved. For ex-
ample, for a 9 mA He+beam extracted with an extraction
voltageof17kVa normalized80% transversebeamemit-
tance of 0.003 πmm mrad was measured using the high-
current high-brilliancegas discharge ion source develope d
at the Institut f¨ ur Angewandte Physik (IAP) at the Univer-
sityofFrankfurt[10]. Withthesamesource,currentdensi-
ties of more than 40 mA/cm2could be achieved easily forH3+beamswith H 3+fractionsofabout94% [8, 9].
4 RF LINAC
A compact four-rod like RFQ structure for the accelera-
tion from 8 keV/u to 400 keV/u has been designed at the
IAP. The electrode length is 1.35 m, the electrode voltage
is 70 kV and the expected rf peak power is about 100 kW
at a low duty cycle around0.1%. For matchingthe output
beamparameterstothevaluesrequiredatinjectionintothe
IH-DTL a very compact scheme is proposed. For bunch-
ing the beam in the longitudinal phase plane a drift tube
directlyfollowingtheRFQstructurewillbeintegratedint o
the RFQ tank. Results of PARMTEQ simulations of the
RFQ as well as first results of model measurements and
MAFIA simulations regarding the integration of the drift
tubearereportedinRef.[11].
For focusing the beam in both transverse phase planes
and for correction of small angular deviations of the beam
at the RFQ exit, a magnet unit consisting of an xy-steerer
and a magnetic quadrupole doublet is flanged to the RFQ
tank. The unit has a total length of only 15 cm. It is fol-
lowedbya diagnosticchamberof5 cmlength,whichcon-
tainsacapacitivephaseprobeandabeamtransformer. The
simulationoftheparticledynamicsalongthematchingsec-
tion is included in the simulations of the RFQ and of the
IH-DTL[5,11].
The IH-type drift tube linac for the acceleration from
0.4 MeV/u to 7 MeV/u consists of four KONUS [12] sec-
tions housed in the same cavity of about 3.8 m in length
and 30 cm in diameter [5, 13]. It consists of 56 accelerat-
ing gaps and three integrated magnetic quadrupole triplet
lenses. The expected rf peak power is about 1 MW. To
achieveanapproximatelyconstantmaximumon-axiselec-
tric field of about 18 MV/m along the whole structure, the
gap voltage distribution is ramped from about 200 kV at
thelow-energyendtoabout480kVatmaximum. Bycare-
fuloptimizationoftheindividualKONUSsections,theac-
ceptance of the structure was increased to about1.3 πmm
mrad (norm.) in the transverse phase planes, and to about
3.0πnskeV/uin thelongitudinalplane.
Behind the IH-DTL the beam is focused on the strip-
per foil by another magnetic quadrupole triplet. The par-
ticle distributions at the stripper foil resulting from par ti-
cletrackingsimulationsalongtheDTLusingtheLORASR
codearepresentedinFig.2. Theparticledistributionsuse d
at injection into the DTL have been matched to the results
of the RFQ simulations. The 95% emittance areasat DTL
injectionare1.3 πnskeV/uinthelongitudinalphaseplane
and0.7 πmmmrad(norm.) inbothtransverseplanes. The
transverse beam emittances are based on the values mea-
sured for the ECR ion sources under discussion. The rela-
tivegrowthofthe95%ellipseareasalongtheDTLisabout
22%inallthreephasespaceprojections,thermsemittance
growth amounts to about 10% in each plane. Beam en-
velopesalongtheDTLhavebeenpresentedalreadyinear-
lier publications[5, 13]. The current limit for the IH-DTL-0.500.5
-10 0 10
Δφ / degΔW / W / %ε rms= 0.36 π ns keV/u
ε 95%= 1.62 π ns keV/u
-808
-2 0 2
x / mmx© / mradε n,rms= 0.17 π mm mrad
ε n,95%= 0.84 π mm mrad
-808
-2 0 2
y / mmy© / mradε n,rms= 0.17 π mm mrad
ε n,95%= 0.83 π mm mrad
Figure 2: Particle distributions at the stripper foil. The e l-
lipsescontain95% oftheparticles.
forionswith A/q= 3resultingfrombeamdynamicssim-
ulationsislargerthan20e µA[13]. Themomentumspread
oftheionbeamatthestripperfoilisabout ±0.15%. Itwill
beincreasedduetoenergy-stragglingeffectsinthefoil. T o
enhance the injection efficiency into the synchrotron, the
momentum spread will be reduced to ≤ ±0.1% by a de-
buncher cavity installed in the synchrotron injection beam
line. An 1:2 scaled rf model of the IH-DTL structure is
designed at present at GSI. First model measurements are
scheduledforthesecondquarterofthenextyear.
5 ACKNOWLEDGEMENTS
We would like to thank C. Bieth, S. Kantas, O. Tasset and
E. Robert (PANTECHNIK) for performing the SUPER-
NANOGANtestmeasurementsatGANIL. Thefruitfulco-
operation of L. Dahl (GSI) in the LEBT design is greatly
acknowledged.
6 REFERENCES
[1] G. Kraftet al.,in: Proc.EPAC98, Stockholm, 1998 , p. 212.
[2] H.Eickhoff,Th.HabererandR.Steiner,in: Proc.EPAC98,
Stockholm, 1998 , p. 2348.
[3] H. Eickhoff, D. B¨ ohne, Th. Haberer, B. Schlitt, P. Spill er,
J. Debus and A. Dolinskii, in: Proc. EPAC 2000, Vienna,
2000, inprint,and references therein.
[4] A. Dolinskii, H. Eickhoff and B. Franczak, in: Proc. EPAC
2000, Vienna, 2000 , inprint.
[5] B. Schlitt and U. Ratzinger, in: Proc. EPAC 98, Stockholm,
1998, p. 2377.
[6] P.Sortaiset al.,Rev. Sci.Instrum. 69, 656(1998).
[7] R. Hollinger, P. Beller, K. Volk, M. Weber and H. Klein,
Rev. Sci.Instrum. 71, 836 (2000).
[8] R. Hollinger,PhD Thesis,Universit¨ at Frankfurt a.M., 2000.
[9] R. Hollinger,private communication, 2000.
[10] K. Volk, W. Barth, A. Lakatos, T. Ludwig, A. Maaser,
H.KleinandK.N.Leung,in: Proc.EPAC94,London,1994 ,
p. 1438.
[11] A. Bechtold, A. Schempp, U. Ratzinger and B. Schlitt, in :
Proc. EPAC2000, Vienna, 2000 , inprint.
[12] U. Ratzinger and R. Tiede, Nucl. Instr. and Meth. in Phys .
Res. A415(1998) 229.
[13] S. Minaev, U.Ratzinger and B.Schlitt,in: Proc.1999 Part.
Accel. Conf.,NewYork, 1999 , p.3555. |
*This work is supported by the US DOE under contract no. W -31-109-ENG -38. A REAL -TIME ENERGY M ONITOR SYSTEM FOR TH E IPNS LINAC*
J.C. Dooling, F. R. Brumwell, M.K. Lien, G. E. McMichael, ANL, Argonne, IL 60439, USA
Abstract
Injected beam energy and energy spread are critical
parameters affecting the performance of our rapid
cycling synchrotron (RCS). A real -time energy
monitoring system is being installed to examine the H-
beam out of the Intense Pulsed Neutron Source (IPNS)
50 MeV linac. The 200 MHz Alvarez linac serves as the
injector for the 450 MeV IPNS RCS. The linac pr ovides
an 80 µs macropulse of approximately 3x1012 H- ions 30
times per second for coasting -beam injection into the
RCS. The RCS delivers protons to a heavy -metal
spallation neutron target for material science studies.
Using a number of strip -line beam p osition monitors
(BPMs) distributed along the 50 MeV transport line from
the linac to the RCS, fast signals from the strip lines are
digitized and transferred to a computer which performs
an FFT. Corrections for cable attenuation and
oscilloscope bandwidt h are made in the frequency
domain. Rectangular pulse train phasing (RPTP) is
imposed on the spectra prior to obtaining the inverse
transform (IFFT). After the IFFT, the reconstructed
time-domain signal is analyzed for pulse width as it
progresses along the transport line. Time -of-flight
measurements of the BPM signals provide beam energy.
Finally, using the 3 -size measurement technique, the
longitudinal emittance and energy spread of the beam are
determined.
1 INTRODUCTION AND MOTIVATION
The Intense P ulsed Neutron Source (IPNS) accelerator
system is equipped with a number of strip -line, beam
position monitors (BPMs) along the 40 -m transport line
from the 50 MeV Linac to the Rapid Cycling
Synchrotron (RCS). Operating at 30 Hz, the RCS
delivers 450 MeV protons to a heavy metal target
generating spallation neutrons for material science
research. Here we describe how signals from the first
four (upstream) BPMs in the
50 MeV line are used to determine bunch width, energy,
and energy spread in the beam. In jected beam energy
spread plays an important role in determining the
stability of circulating charge within a synchrotron.
Advancements in the speed of sampling oscilloscopes
and the rapid increase in processing power available
from personal computers all ow for real -time
measurement of the output microbunch shape from the
linac. As the bunch travels along the transport line, its
longitudinal size grows due to energy spread within the bunch. The growth in bunch length can be monitored
with the stripline B PMs and the energy spread
determined.[1]
The IPNS Alvarez, drift tube linac (DTL) began
operation in 1961 as the injector for the Zero Gradient
Synchrotron (ZGS). In 1981 after the ZGS program
ended, the linac became the injector for IPNS RCS. The
linac typically delivers 3.5 -3.7x1012 H- ions to the RCS
during an 80 -µs macropulse. During the early days of
the linac, the energy spread was measured to be 0.37
MeV. From numerical modeling, the capture efficiency
of the RCS at injection is optimized near a momentum
spread of 0.3 percent or approximately 0.3 MeV.
Instabilities arise if the energy spread is too low, whereas
high losses occur if the spread is too high; in either case,
RCS efficiency is reduced. A shift in energy during the
macropulse effectiv ely acts to increase energy spread
during injection into the RCS.
2 EXPERIMENTAL ARRA NGEMENT
The Energy Spread and Energy Monitor (ESEM)
diagnostic is presented schematically in Figure 1. The
upstream electrode of the first BPM (BPM 1) is located
5.455 m from the output flange of the last DTL tank.
The set of four BPMs included in the ESEM cover a
distance of 16.627 m along the beam path. This distance
is sufficient to allow observable growth in the
longitudinal size of the bunch without interference fr om
the return signal generated at the downstream electrode
of the BPM.
Figure 1. The Energy Spread and Energy Monitor. 3 DESCRIPTION OF ME ASUREMENTS
3.1 Pulsewidth
Time signals from the BPMs are recorded on a
Tektronix model TDS694c oscilloscope. The TDS694c
has a 3 GHz bandwidth and samples 4 channels
independently at 1010 samples/sec (10 GS/s). The
oscilloscope is controlled and data transferred via a
GPIB -to-Ethernet interface. This allows an office PC on
the network to communicate with the oscil loscope. After
receiving the initial trigger pulse from the chopper, the
oscilloscope waits for a controlled amount of time before
triggering its four channels. The delay time is adjustable
to allow for temporal examination of the macropulse.
Once the d elay period has expired, oscilloscope
triggering is enabled for the next zero crossing detected
on channel one. When the zero crossing is detected, all
four channels are triggered simultaneously. The trigger
time establishes the reference time for energy
measurements. A sampled waveform from BPM 1 is
presented in Figure 2.
Figure 2. Sampled waveform from BPM1.
Once signals from the BPMs have been collected and
transferred to the PC, the data can be displayed and
analyzed. Data analysis is implemente d using Visual
Basic. Analysis of the signals begins with rebinning the
data to obtain data sets with 2N samples as required for
the FFT[2].
In order to examine the same bunch on all four BPMs
and account for cable delays, a data window of 100 ns or
longe r is required. It is also desirable have good
frequency response for parameters derived from the
entire FFT. A 100 ns window allows us to look at
parameters derived from the FFT up to 5 MHz (Nyquist).
A period of 100 ns is a good compromise between
resolution in the FFT and frequency response of the
derived parameters. Extending the sample window
requires more time but also provides at cleaner signal.
After carrying out the FFT, the spectra are converted
to dBm and corrected for cable attenuation and in sertion
losses. RG -213 coax cable is employed to carry the beam signals from the BPM to the oscilloscope. An
attenuation correction[3] that is a function of frequency
and cable length is applied to each spectrum. Finally,
when performing the inverse tra nsform, Fourier filtering
is performed to remove low -level signals lost in the
noise. The pulsewidth is determined by evaluating the
following time function,
∑=+ =max
1) cos(1)(N
nn n
btn F
Ntf fw
where F n is the amplitude of the nth harmonic of the
beam bunch determin ed from the FFT. Employing
rectangular pulse train phasing (RPTP), φn may be
expressed as,
)2(0 0b
nt ntw f +=
The original phase is also available from the FFT by
taking the inverse tangent of the ratio of imaginary to
real parts of the signal. Figure 3 presents the cable -
corrected FFT spectrum for the time data giv en in Figure
2. Also indicated in the figure is the threshold level.
Finally, the reconstructed time signals using both RPTP
and original phasing are given in Figure 4. The pulse
shape in either case is similar to the “bipolar doublet”
described by Shaf er[4] and shown by Kramer[1].
Figure 3. Corrected FFT spectrum for data given in Fig. 2.
Figure 4. The reconstructed time signals.
3.2 Energy
Energy is determined by tracking a single microbunch
through the four BPMs. Time -of-flight (TOF)
measurem ents require accurate knowledge of the cable
lengths between the BPMs and the oscilloscope, and the
distance travelled by the beam along the transport line.
Cable lengths are determined by TDR. The path length
of the beam is measured directly or taken fr om survey
data.
The energy is calculated from the measured velocity of
the bunch. Velocity is determined by measuring the TOF
and correcting for signal propagation delays along the
cables. The time required for the signals from a given
bunch to reach the oscilloscope may be expressed as,
cs
vltj
pcj
j
b1
+= .
where j=2, 3, and 4 and t 1=lc1/vp. Taking t 1 as the
reference time, the bunch velocity in terms of the time
difference between arrival on channel 1 and channel j is
given as,
1 11
1
c cj pjj p
j
llvtsvc+−=b
where, v p is the phase velocity in the cable or
transmission line, l cj is the length of cable j, s 1j is the
distance the bunch travels from BPM 1 to BPM j, and
tj1=tj-t1. The sensitivity of the measurement increases
with distance between channel 1 a nd channel j. When
calculating the energy, the analysis detects the leading
edge of the bunch at the half -height of the pulse. To
obtain a measure of energy from bunch center -to-center,
the pulsewidth of the bunch at each location must be
accounted for.
3.3 Energy Spread
Energy spread is obtained by the three -size
measurement technique[5] neglecting space charge. The
advantage of this method is that it provides the
longitudinal Twiss parameters and therefore gives an
indication of axial focussing (i.e. , rf fields) at the output
of the DTL. The section of beam line between BPMs 2
and 3 includes two π/6 sector dipole magnets, both with
0.8 m radius of bend. Though the dipoles cannot affect
the energy or energy spread of the beam, they do cause a
change in the longitudinal size of the beam through
dispersion and coupling with the horizontal plane. This
effect is introduced into the longitudinal transfer matrix
by adjusting the path length between BPM 2 and 3. The
dipole length factor, f d, can be include d in the drift -space
transfer matrix R d(s). Labeling the three drift spaces
between the four BPMs as s 1, s2, and s 3, the transfer
matrix between BPM 1 and 3 for the 123 measurement
can be written as, )( ))(2 ( ),( 1 1 2 2 2 1 3sR lf sR ssR d d dr −=
For the 124, 134, and 234 meas urements, the respective
expressions for R 3 become,
)( ))(2 ( ),,( 1 1 3 2 2 3 2 1 3sR lf ssR sssR d d dr −+=
))(2 ()( ),,( 2 1 1 3 2 3 2 1 3rlf ssRsR sssR d d d−+ =
))(2 ()( ),( 2 1 3 2 3 2 3rlf sRsR ssR d d d− =
In theory, all four permutations should generate the
same result; in reality, f d may be varied to provide the
least rms error. Init ial data indicate f d is not a fixed
parameter but varies significantly from sample to sample
within a range of ±4. Negative values generally correlate
with lower energy spread, whereas the highest positive
values tend to correspond to the largest energy s pread.
4 RESULTS AND DISCU SSION
The maximum average current that the RCS has
achieved is 15.5 µA, reached in May of this year prior to
the annual summer shutdown. Since restarting the
machine on August 1st, maximum current has been
limited to about 14.8 µA. The ESEM diagnostic was
first put into service in May and observed some of the
higher current operations. ESEM energy data recorded
in May and August are presented in Figure 5.
ESEM energy data suggest that a greater fluctuation in
linac energy durin g the August 2000 run may be limiting
the efficiency of the RCS. Further tests are being
planned to determine the validity of this supposition.
Figure 5. ESEM linac energy data.
REFERENCES
[1] S. L. Kramer, Third European Particle Accelerator
Conf. (EP AC), Berlin, p. 1064(1992).
[2] G. D. Bergland, IEEE Spectrum, 41(1969).
[3] D. Sumner, ed, The ARRL Handbook for Radio
Amateurs , 77th ed, p. 19.5(2000).
[4] R. E. Shafer, IEEE Trans. Nucl. Sci., 32,
1933(1985).
[5] K. R. Crandall and D. P. Rusthoi, LANL R eport No.
LA-UR-97-886, May 1997. |
arXiv:physics/0008150 18 Aug 2000OPTICAL TRANSITION RADIATION (OTR) MEASUREMENTS OF AN
INTENSE PULSED ELECTRON BEAM†
C. Vermare, D. C. Moir and G. J. Seitz
LANL, Los Alamos, N.M. 87454, USA
† LA–UR – 00 – 2064Abstract
We present the first time resolved OTR angular
distribution measurements of an intense pulsed electron
beam (1.7 kA, 60 ns). These initial experiments on the
first axis of the Dual Axis Radiographic Hydro-Testing
(DARHT) facility and subsequent analysis, demonstrate
the possibility to extract, from the data, the energy and
the divergence angle of a 3.8 and 20 MeV electrons.
1 INTRODUCTION
By using the OTR angular distribution property [1], it
is possible to extract the energy and the angular
dispersion of several kinds of beam [2]. Recently, Le
Sage et al. [3] succeeded in producing a transverse
phase-space mapping of a 100 MeV electron beam by
coupling an interferometric measurement and a ’’mask’’
technique. We present the first OTR time resolved
angular distribution measurements made on an intense
pulsed electron beam with energies of 3.8 and 20 MeV.
The DARHT accelerator produces an intense pulsed
electron beam (1.7 kA, 20 MeV, 60 ns) that impinges on
a high-Z target. The quality of the X-ray source is
determined by spot size and dose. The spot size is
effected by magnet focal length, emittance, energy
spread and beam motion. The dose is determined by
beam energy, total charge, target material, and
convergence angle. We present results of OTR angular
distribution experiments performed at 3.8 and 20 MeV
electron beam energies on the first axis of DARHT and
compare these results to a 3D Ray-Tracing program
which is able to calculate the effect of each electron
beam parameter. Comparison shows that the data is most
sensitive to the electron beam energy and the
divergence/convergence angle. The maximum collection
angle of the optical system limits results at low energy
by mixing angular and spatial information.
The layout of the paper is as follows. The first section
describes OTR angular distribution properties. This is
followed by a brief description of the 3D Ray-Tracing
program. The third section describes the experimental
set-up. Section 4 shows initial OTR observations.
Comparison between results and simulations are made in
the fifth section. Time resolved measurements are
described in section 6. Limitations and planned
improvements of this diagnostic are discussed in the last
section of this part.2 OTR ANGULAR MEASUREMENTS
2.1 OTR properties
OTR is produced when a charged particle passes
between media with different dielectric constants as a
aluminium foil in vacuum. This light is emitted with a
characteristic angular distribution that depends of the
particle energy and direction. The Fig. 1 shows the OTR
density versus angle on the incidence plane (plane
defined by the beam axis and the normal vector of the
target). The zero angle corresponds to the “specular”
direction.
-400 -200 0 200 4000.05.0x10-61.0x10-51.5x10-5
single
electron
OTR foilθ
θ
OTR density (photons/ster./Angstrom)
angle (mrad) 3.8 MeV
20 MeV
Figure 1: OTR density versus angle on the incidence
plane for two different energy (the tilt angle of the
foil is 45 degree).
It is important to note that the light produced by OTR
is distributed with two different polarizations.
2.2 Ray-Tracing program
The system composed of the beam, the OTR foil and
the detection system has 3D geometry. We developed a
Ray-Tracing program able to follow photons produced
by each electron of a phase-space-defined beam. The
code calculates the image received by the screen for
each polarisation. For a simulation, a spectrum range,
foil material, lens and screen size and position and the
tilt angle of the target are chosen. The results obtained
are absolute as shown in Fig. 1.2.3 Experimental set-up
The experimental set-up is shown in Fig. 2.
Figure 2: Experimental set-up for angular
distribution measurement.
The electron beam passes through a thin aluminised
Kapton foil (10 microns) and produces OTR photons
from the aluminium-vacuum interface. This target is
tilted 45 degrees so the center of the OTR distribution is
emitted at 90 degrees from the beam axis. The OTR
light is collected by an achromatic doublet (focal length
= 200 mm and effective diameter = 70 mm) which
produces an angular image of the source on a screen.
This geometry requires the distance between the lens and
the screen to be equal to the focal length. The distance
between the target and the lens determine the maximum
collection angle of the system. Limitations are the screen
size, the "Cherenkov" background created inside the lens
by secondary electrons or X-rays and discoloration of the
optics caused by radiation damage. A 200-mm focal
length achromatic lens that gives a maximum collection
angle of 170 mrad is used. A polarizer can be added
between the lens and the screen to separate each
polarisation. An 8-frame gated camera records the image
formed on the screen. This device splits the light up to 8
different micro-channel plate (MCP) gated CCD. The
gating system of each MCP makes it possible to record
each image with a 10 nanoseconds duration.
This optical system is to first order independent of the
beam size and position. However, with large angle of the
OTR lobe (specially at low energy), a large beam size
can affect the angular distribution. In this case, a precise
analysis with the Ray-Tracing program is required.
2.4 OTR confirmation
To confirm that OTR is the main part of the light
collected by the system as opposed to prompt Cherenkov
light generated by X-rays or secondary electrons, initial
measurements were made of the polarization of light
from the source. The Fig. 3 shows three images
corresponding to (a) both polarizations (no polarizer),
(b) polarization in the incidence plane and (c)
polarization in the observation plane.
Figure 3: Experimental picture of the OTR
polarisation. The beam energy is 3.8 MeV
These pictures demonstrate that the light observed is
OTR. A second test was to block the light coming from
the target by placing a thin aluminium foil between the
target and the lens. The result confirms the low ratio of
the “Cherenkov” light coming from the target. Tests
were made at both 3.8 MeV and 20 MeV.
2.5 Extraction of beam parameters
According to the 3D model of OTR, a horizontal
(vertical) cut of the picture recorded gives information
about the beam. Results indicate that the distance
between the maximum positions is determined by the
energy and the filling at the centre is most sensitive to
beam envelope divergence/convergence angle at the
detector.
By changing the current on the guiding solenoid
before the OTR foil, the beam size and divergence angle
are varied. The Fig. 4 shows the effect predicted by the
simulations. These curves correspond to the full beam
duration and they are normalised.
-40 -20 0 20 400.00.20.40.60.81.0 5 mrad
10 mrad
15 mrad
20 mrad
LSB (a.u.)
Incidence direction (mm)
Figure 4: Simulations of the effect of the divergence
angle on the OTR angular distribution (20 MeV).
Fig. 5 shows a comparison of experimental data with
convergence angle determined by the magnet setting
demonstrating the sensitivity observed in the
calculations. -40 -20 0 20 400.00.20.40.60.81.0
Experimental
results DT#1=off
DT#1=140A
DT#1=240A LSB (a.u)
Incidence direction (mm)
Figure 5: Experimental results of the effect of the
divergence angle on the OTR angular distribution
(20 MeV).
For the energy measurement, induction cells were de-
energized at the end of the accelerator. An energy range
between 15 MeV and 20 MeV was obtained. Fig. 6
shows the overlaid experimental results.
-40 -20 0 20 40 full
2 cells off
4 cells off
8 cells off
16 cells off
LSB (a.u)
Incidence direction (mm)
Figure 6: Experimental results of the effect of the
beam energy on the OTR angular distribution.
These results are in agreement with the simulations.
The precision on the energy measurement for these data
is 250 keV at 20 MeV and 150 keV at 15 MeV.
2.6 Time resolved measurement
Fig. 7 is an example of a time resolved result. The
beam energy is constant through the pulse. Therefore,
we can observe the beam divergence/convergence as a
function of time. For this measurement, the vacuum was
reduced to 3.10-5 Torr to initiate a time dependent
focusing of the beam due to background gas
neutralization. This effect is, also, observed in the spatia l
measurement. This effect disappears then the vacuum
reach 5.10-6 Torr. These results need to be improved.
Currently, they show a time variation of the divergence
angle about 5 mrad. -15 -10 -5 0 5 10 150.20.40.60.81.0
LSB (a.u)
Incidence direction (mm) 10-20 ns
20-30 ns
30-40 ns
40-50 ns
Figure 7: Time resolved measurement of the beam
parameter using the OTR angular distribution
(20 MeV).
2.7 Discussion and perspectives
If the system is carefully aligned and the maximum
angle collected is more than four times 1/ γ, each beam
parameter can be consider independent. If not, they are
linked and a direct comparison with different
simulations is necessary. We found this complication at
3.8 MeV with our set-up.
After these measurements, we plan different
amelioration. First of all, the CCD dynamic range must
be improved (256 to 64k levels). There are calibration
concerns associated with the 8 different CCD camera
that need to be addressed before we can extract
quantitative information from the images. Next, we will
use a streak camera image a slice of the OTR
distribution directly as a function of time. Also, the
magnification of the OTR angular distribution image
must be increase to obtain a better definition of the
maximum and the centre.
CONCLUSION
The results presented here prove that it is possible to
use the OTR angular distribution information to measure
the energy and divergence/convergence angle of an
intense pulsed electron beam. The light intensity is
sufficient for time-resolved measurement of these
parameters in the 3.8-20 MeV energy range.
REFERENCES
[1]L. Wartski , “Etude du rayonnement de transition
optique produit par des electrons de 30 a 70 MeV”,
Thesis, Orsay (1976).
[2]D. W. Rule , “Transition radiation diagnostics for
intense charged particle beams”, Nucl. Inst. Meth.,
B24/25 (1987) 901-904..
[3]G.P. Le Sage et al. , “Transverse phase space
mapping of a relativistic electron beam using
Optical Transition Radiation”, PRST-AB, 2 (1999),
122802. |
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arXiv:physics/0008152 18 Aug 2000RF CAVITIES FOR THE MUON AND NEUTRINO FACTORY
COLLABORATION STUDY *
A. Moretti, N. Holtkamp, T. Jurgens, Z. Qian and V. Wu, FNAL, Batavia, IL 60510, USA
Abstract
A multi-laboratory collaboration is studying the
feasibility of building a muon collider, the first phase of
which maybe a neutrino factory. The phase space
occupied by the muons is very large and needs to be
cooled several orders of magnitude for either machi ne,
100,000 to 1 million for the collider and ten to 10 0 for the
factory. Ionization cooling is the baseline method for
muon cooling. This scheme uses hydrogen absorbers a nd
rf re-acceleration in a long series of magnetic foc using
channels to cool the muons. At Fermilab two rf cavi ty
types are under study to provide the required cooli ng rf re-
acceleration. A 805 MHz high gradient cavity for th e
collider and a 201 MHz high gradient cavity for the
neutrino factory. The 805 MHz cavity currently unde r
going cold testing is a non-periodic pi-mode cavity with
the iris openings shaped to follow the contour of t he
beam. The 201 MHz cavity uses hollow thin metal tub es
over the beam aperture to terminate the field in a pillbox
type mode to increase its shunt impedance. This is
possible because muons have little interactions wit h thin
metal membranes. Details of these cavities and cold
measurement data will be presented.
1 INTRODUCTION
An international collaborative study of muon collid ers
and neutrino factories has been going on for a numb er of
years [1, 2, 3]. The lead laboratories for this stu dy are
BNL, CERN, Fermilab and LBL. A collider or neutrino
factory for high-energy research needs a large numb er of
muons to produce a high luminosity beam. Because of the
muons short lifetime, they need to be transported q uickly
through the accelerator complex. The muons are prod uced
from pions decays off of a proton beam hitting a hi gh-Z
target in a solenoidal magnetic decay channel. The muons,
thus, produced occupy a very large 6-demensional ph ase
space which must be reduced (cooled) quickly by sev eral
orders of magnitude to meet the luminosity requirem ents.
Ionization cooling has been chosen as the cooling
technique. In this technique muons lose transverse and
longitudinal momentum as they pass through a low-Z
absorbing material. The longitudinal momentum is th en
restored by rf re-acceleration in large aperture rf cavities.
The process is repeated numerous times to reduce th e 6-
demensional phase space of the muons sufficiently f or
acceptance by the accelerator complex and meet its
luminosity requirements.
*Work supported by the US Dept. of Energy, contract
DE-AC02-76CH0-3000. Currently, at Fermilab and LBL high gradient, h igh
shunt impedance large beam aperture rf cavities are being
studied at 201 and 805 MHz [4]. Accelerating gradie nts of
15 MV/m and 30 MV/m for 201 and 805 MHz
respectively are required for the most favored scen arios.
LBL is studying pill-box type rf cavities with 125 micron
beryllium windows over the aperture, due to their h igher
shunt impedance and low radio of peak surface field to
accelerating field. Fermilab is studying a 201 MHz cavity
with thin hollow beryllium or aluminum tubes over t he
aperture. The tubes terminate the aperture electric fields in
a pillbox type mode and increase its shunt impedanc e
towards that of a true pill-box cavity. At Fermilab , also, a
805 MHz open cell cavity has been designed and a co ld
model has been built and tested. To increase its sh unt
impedance the iris openings have been dimensioned t o
follow the beam’s contour as it passes through the cavity.
2 GRIDDED 201 MHZ CAVITY DESIGN
The 201 MHz gridded cavity is bellow shaped to
increase its shunt impedance and has a set of cross ed
(gridded) hollow thin walled low-Z metallic tubes
covering the bean aperture, Fig.1. The tubes can be easily
forced gassed cooled, a great advantage over 125 mi cron
Be window covering the beam aperture. In this desig n the
tubes are made of aluminum 4 cm in diameter, 125
microns thick in the middle and 500 microns at its ends.
The cavity has a 0.60 m major radius, a length of 0 .64 m
and beam aperture of 0.64 m. Current mechanical and
electrical designs limit the Be window aperture des ign to
0.38 m. When connected to neighboring cavities, the
cavities are separately driven with a phase advance of pi
per cavity. Other phase advances are possible becau se the
grids were designed to minimize the coupling betwee n
neighboring cavities.
The computer program MAFIA was used to
optimize the design of the cavity. The number of tu bes
and their diameters were varied to maximize the shu nt
impedance, reduce the peak surface electric field,
minimize material intercepting the beam and the cou pling
between neighboring cavities with the beam aperture set
at 0.64 m. Following the above criteria, four verti cal and
four horizontal tubes, 4 cm in diameter resulted in the
most satisfactory cavity design. The cavity, Fig. 1 , has a
Qo of 63,000, shunt impedance of 32.0 MOhm/m and
requires 4.5 MW to achieve a accelerating gradient of 15
MV/m. The peak surface electric field at this gradi ent is
25 MV/m, an acceptable 1.7 times the Kilpatrick Lim it.Figure 1: Aperture Tube Layout
Figure 2: 805 MHz iris loaded cavity with a beam en velope matched aperture.
3 OPEN CELL 805 MHZ CAVITY DESIGN
The 805 MHz cavity is an iris-loaded structure with the
aperture of the iris dimensioned to follow the five -sigma
contour of the beam, Fig. 2. This allows the design to
maximize the shunt impedance without material in th e
beam path. This may increase cooling channel effici ency.
However, cooling simulations have shown little if a ny
improvement when compared to 125 Be window design.The design does eliminate the difficult mechanical and rf
electrical heating problems of the Be window design . The
beam aperture in the middle of the cavity is 0.16 m and at
the ends 0.08 m.
The computer programs Mafia and Superfish were
used to optimize the design. The criteria of the de sign was
to maximize shunt impedance while obtaining nearly
equal and reasonable peak surface electric fields o n all the
cavity irises. The cavity, shown in Fig.2, has a Qo of
35,600, a shunt impedance of 33.5 Mohm/m and requir es
27.7 MW for a accelerating gradient of 30 MV/m. The
peak surface electric field is 77 MV/m, 2.9 times t he
Kilpatrick limit. This might be acceptable for the required
short cooling pulse length of 210 microseconds. A h igh
power copper vacuum cavity is currently under
construction and breakdown studies are planned in a
Fermilab test facility under construction.
A full-scale aluminum model of the cavity has been
built. The model was built to test the accuracy of the
computer calculation and the machining accuracy of the
parts. The machining accuracy called for was +/- 13
microns. Measurements on the model were very good.
The measurements agreed with calculations to within 5
microns. Bead-pull measurements of the field profil e were
in agreement with calculations to within 5 %. The m odel
was further used to determine the size of the criti cal
coupling slot.
4 HIGH POWER RF COUPLER DESIGN
Mafia 3D time domain and 2D eigenmode solvers are
used for the coupler simulations [5]. The model con sists
of the first two cells of the six cell cavity with a
rectangular waveguide (the height is one half of th e
standard WR975 waveguide height) attached to the ou ter
wall of the first cell (see Figure 2). Energy coupl ing
between the waveguide and the cavity is through a
rectangular slot. The height of the slot is chosen to be that
of the waveguide, in order to minimize the ratio of the
maximum coupler voltage to waveguide voltage, while
the depth is the outer wall thickness of the cell. The width
is varied to achieve critical coupling. All corners of the
coupling slot’s cross section are rounded to a radi us of 7
mm. To simulate the total wall loss in cell 3 throu gh 6 in
the actual cavity, the conductivity of the second c ell is
adjusted to produce the loss. The conductivity is
determined using the 2D eigenmode solver where the wall
loss of each cell can be calculated; hence the cond uctivity.
In the simulation, it is important that cell 1 and 2 are in
tune and have the correct relative energy distribut ions.
The coupling coefficient ( β) calculation employs the
energy method [5] in which two-time domain runs are
needed. In the first time domain run, each cell is tuned
separately to 805 MHz and to have the right energy
distribution. After tuning, the two-cell structure is excited
by a monochromatic dipole signal located inside the
second cell. The 3D electric and magnetic fields ar e
recorded at four carefully chosen time steps. From these
fields, the power loss at the cavity wall and the p ower
flow out of the cavity into the waveguide are calcu lated.
The coupling coefficient is computed as the ratio o f
external power over wall loss. A low power test is
performed on a full-scale (six-cell) aluminum model to
check the simulations. For a set of coupler dimensi ons
that is close to critical coupling ( β = 1), the measured β is
0.976. The simulation result is 1.000. Finally, the couplingslot dimensions for critical coupling are determine d to be
(height, depth, width) = (6.2, 2.2, 8.2) cm.
5 CAVITY RESEARCH STATUS
Measurements of the frequency, field profile and
coupling slot size on the aluminum model were in
excellent agreement with Mafia and Superfish
calculations. A 805 MHz copper high power test cavi ty is
under construction. Electric field breakdown and va cuum
conditioning studies are to take place in a high po wer test
facility currently under construction at Fermilab. Design
studies of high gradient 201 MHZ cavities are curre ntly in
progress at Fermilab and LBL. The goal of these stu dies is
to produce several high power prototype cavities in the
next two years. A test facility at Fermilab is curr ently in
the early design stage.
REFERENCES
[1] R. Palmer, A. Tollestrup, and A. Sessler, Proc. of the
1996 DPF. DPB Summer Study "New Directions for
High Energy Physics", Snowmass, Co (1996).
[2] Status of Muon Collider Research and Developmen ts
and Future Plans, Fermilab-PUB 98/179.
[3] Neutrino Factory Physics Study, Fermilab Report :
Fermilab-FN-692.
[4] J.N. Corlett, et.al., Proc. of 1999 PAC, "Rf
Accelerating Structures for the Muon Cooling
Experment",pp 3149-3151, New York, 1999.
[5] D. Li,et.al., Proc. of 1998 Linac Conf., "Calcu lations
of the External Coupling to a Single Cell Rf Cavity ",
pp 977-979, Chicago, 1998. |
*This work is supported by the US DOE under contract no. W -31-109-ENG -38. RELIABILITY HISTORY AND IMPROVEMENTS TO THE ANL 50 MEV H-
ACCELERATOR*
L.I. Donley, V.F. Stipp, F. R. Brumwell, G. E. McMichael, ANL, Argonne, IL 60439, USA
Abstract
The H- Accelerator consists of a 750 keV Cockcroft
Walton preaccelerator and an Alvarez type 50 MeV
linac. The accelerator has been in operation since 1961.
Since 1981, it has been used as the injector for the
Intense Pulsed Neutron Source (IPNS), a national user
facility for neutron scattering. The linac delivers about
3.5x1012 H- ions p er pulse, 30 times per second (30 Hz),
for multi -turn injection to a 450 MeV Rapid Cycling
Synchrotron (RCS). IPNS presently operates about 4,000
hours per year, and operating when scheduled is critical
to meeting the needs of the user community. For man y
years the IPNS injector/RCS has achieved an average
reliability of 95%, helped in large part by the
preaccelerator/linac which has averaged nearly 99%. To
maintain and improve system reliability, records need to
show what each subsystem contributes to t he total down
time. The history of source and linac subsystem
reliability, and improvements that have been made to
improve reliability, will be described. Plans to maintain
or enhance this reliability for at least another ten years of
operation, will als o be discussed.
1 INTRODUCTION
After operating with 200 µs pulses at 0.25 Hz for 15
years as an injector for the Zero Gradient Synchrotron
(ZGS), the linac repetition rate was increased to 30 Hz
with 70 µs pulses in the mid -1970's when it also became
an injector for the Rapid Cycling Synchrotron (RCS).
Direct injection to the ZGS ended in 1979 and the RCS,
which was originally built to be a booster for the ZGS,
became the source of high -energy protons for the Intense
Pulsed Neutron Source (IPNS), a national user facility
for neutron scattering. Since IPNS bega n operation in
1981, it has accumulated over 60,000 hours of 30 Hz
beam operations. The overall accelerator reliability
(beam hours versus scheduled hours) exceeds 93% and
for the past 10 years, that number is greater than 95%.
Thoughout, the preaccelera tor/linac reliability has
hovered around 99%, and this has been achieved while
operating at roughly 30 times its original duty factor.
The evolution and performance of the preaccelerator
and linac was described in a 35th anniversary paper
presented at the Linac96 Conference [1]. In this present
paper, we examine overall system and subsystem
reliability and the impact of changes in hardware and
software on reliability, and discuss improvements that will be necessary to maintain current high performance
for at least another ten years.
The linac rf system was the first linac amplifier built
using the 7835 triode. The rf amplifier system was
designed and built by Continental Electronics
Corporation in the late 1950's and comprises a single 4 -
stage amplifier sy stem with a rated output of 5 MW and
a maximum pulse length of about 200 µs. With the
exception of the power supplies, most of which had to be
upgraded during the late 1970's when the duty factor was
increased, the original configuration and in many cases,
original equipment, is still functioning well. In the
almost 40 years since startup, seventeen new or rebuilt
7835 tubes have accumulated a total of over 200,000
filament hours in this single station, equivalent to over
23 years of continuous operation.
2 PREACCELERATOR
The preaccelerator is a standard 4 -stage 750 kV
Cockcr oft-Walton power supply built by Haefely. There
have been no failures of the Haefely high -voltage
components in the life of the facility. The original motor
generator set supplying 400 Hz ac to the high -voltage
rectifiers was replaced by one made in the USA. The
original 400 Hz isolated generator used for supplying
input power to the ion source enclosure was also replaced
with a brushless 60 Hz unit to reduce maintenance. It
has been trouble -free for the past 10 years.
The first ion source was a proto n source. This was
changed to a modified duoplasmatron -type H- source in
1976, and then to the present magnetron -type H- source
in 1983. The present source uses cesium to enhance the
H- current and produces 45 -50 mA, 70 microsecond
beam pulses at the req uired 30 Hz rate. It has been very
reliable and typically is dismantled and cleaned every six
months, with new electrodes installed about every year.
Close control of the cesium flow from the ion source is
necessary to keep the spark rate of the 750 kV c olumn
low. However, proof of success is the fact that the
column has not been dismantled in over 14 years.
Detailed operating logs have been kept since the
beginning of operation and for the last five years, the
trouble log has been computerized. Acceler ator
downtime is tracked, rounded to the closest 5 minutes.
Over the last five years (20,000 hours of beam operation)
there were 143 interruptions of 5 minutes or more
charged to the preaccelerator and 120 to the linac.
Breakdown by subsystem is given in Tables 1 and 2.
Table 1: Lost beam time in last 20,000 hours of operation
due to preaccelerator faults (number of beam
interruptions, lost beam time, mean -time-between -failure
and mean -time-to-repair).
Subsystem # of
Faults Down
Time
MTBF
MTTR
(hours) (weeks) (hours)
Haefely Supply 27 7.2 4.4 0.3
Chopper 15 4.0 7.9 0.3
Extractor Supply 4 1.0 29.7 0.3
Controls 23 11.3 5.2 0.5
Bouncer Supply 18 10.6 6.6 0.6
H- Source 25 14.0 4.7 0.6
Vacuum/Water 22 7.3 5.4 0.3
Beam Transport 9 7.3 13.2 0.8
3 LINAC
The 50 MeV linac is an Alvarez structure, constructed
as eleven sections that are bolted together to form a
single rf cavity that is 0.94 m in diameter and 33.5 m
long. There are dc quadrupole magnets in each of the
124 drift tubes powered by twel ve mag -amp type dc
supplies. These units are original to the linac and are
nearly 40 years old. The change from tube to
semiconductor regulators in 1988 cleared up most of the
maintenance and failure problems with these systems.
The tank, drift tubes an d quadrupoles have been very
reliable. With the exception of a failed interlock and loss
of water flow in the first eight quadrupole magnets in
1988 (in which external soft -solder joints melted and
separated but no noticeable internal damage to the
magnet coils occurred), these systems have caused no
significant downtime. Occasional rf arcing in the linac
tank after vacuum work has generally been cured by mild
reconditioning, and there is sufficient redundancy of
vacuum pumps that failures can be left to the next
scheduled maintenance period for repair.
Over the life of the machine, there have been
significant changes in the vacuum pumps used on both
the preaccelerator and linac. Some of the original 2000
l/s ion pumps are still in use (after many rebuild s to
replace the titanium elements and bead -blast internal
surfaces). Others have been replaced with cryopumps or
turbomolecular pumps to increase pumping speed and
allow operation to continue despite small leaks. Vacuum
pump upgrades are expected to con tinue over the life of
the facility because improvements in vacuum have
positive effects on many of the subsystems.
As is evident from Table 2, the rf transmitter accounts
for most of the downtime, and over the past five years
almost half the downtime for the preaccelerator/linac
combined is caused by two components, the dc blocking
capacitors in the triode cavity. These capacitors, which
are custom made by Continental Electronics and unique to our transmitter, only became a problem after the
change to 30 Hz operation. However, in the past twenty
years, the upper capacitor has failed 16 times and the
lower 10 times. Although average lifetimes are
approximately 25 and 40 operating weeks for the upper
and lower capacitors respectively, the range is broad.
There was a period from 1987 through 1995 (170 weeks)
with no failures, whereas others have failed after only a
few weeks. Frequency of failures have been higher over
the past few years, possibly due to higher -order modes in
the cavity and to difficulties the manufacturer is
encountering in maintaining a rebuild capability for such
limited -demand items.
Table 2: As Table 1 but for linac faults.
Subsystem # of
Faults Down
Time
MTBF
MTTR
(hours) (weeks) (hours)
DT power supplies 14 9.6 8.5 0.7
RF (capacitors) 15 109.6 7.9 7.3
RF (triode) 1 4.5 118.7 4.5
RF (other) 64 34.4 1.9 0.5
Vacuum/Water 19 11.3 6.2 0.6
Controls 7 3.1 17.0 0.4
The 7835 triode amplifier tubes used in our linac and
at several other accelerator facilities (e.g., LANL, FNAL)
were originally developed and manufactured by RCA
Corporation. For many years, the only supplier for these
tubes has been Burle Industries, a small company set up
by some former employees of RCA. Lifetime of the tubes
is a significant operational conside ration because of the
high item cost (of order $150,000 for a new tube and half
that if the failed tube can be repaired) and delivery times
are 6 to 12 months. Since startup in 1961, we have
accumulated over 200,000 filament hours on 17 new or
rebuilt tub es. In the early years of the facility, most of
the tubes failed during operation because of grid -cathode
shorts. Lifetimes varied from 3,000 to 17,000 hours
(10,000 hours average) for the eight tubes that developed
shorts, and each resulted in downtime to install and
condition the new tube. A manufacturing change by
RCA about 1970 cured the grid -cathode short problem,
and now the tubes last until the cathode degrades.
Cathodes degrade gradually, giving several months
during which the heater current can be periodically
increased to maintain sufficient emission and providing
time to schedule the replacement during a maintenance
period. Nine tubes have been replaced because of low
emission with lifetimes ranging from 9,000 to 20,000
hours (15,000 hours av erage). The only downtime
charged to the tubes in the last five years was a 4.5 hour
interruption in 1997, a few days after a blocking -
capacitor failure, when the tube developed an external
arc (carbon track on the ceramic). The tube was removed, cleaned and returned to service and proceeded
to run trouble -free for another 14,000 hours.
4 DISCUSSION
IPNS operation increased about five years ago to its
present level of about 25 weeks per year. During
operating periods, typically two to three weeks on w ith
one to two weeks off between runs, the facility is
expected to run 24 hours a day, seven days a week.
External users come for periods of a couple days to a
week or more to make measurements on samples
requiring individual exposure times that vary from less
than an hour to several days. In general, beam
interruptions of half an hour or less are relatively
insignificant to users unless they are frequent. Even
then, the main effect is a lack of throughput proportional
to the lost beam (equivalent to ope rating continuously at
reduced proton current on target). Long interruptions,
particularly those exceeding eight hours, may mean that
the user’s samples don’t all get exposed or statistics are
poor and the user may get ambiguous or uninterpretable
data. There have been only five beam interruptions from
preaccelerator/linac faults over the last five years that
exceeded eight hours and the longest was 20 hours. All
were associated with blocking -capacitor failures. The
average preaccelerator/linac reliabil ity (percent of
scheduled time when protons were delivered to target)
was 98.8% and half of the total downtime was associated
with blocking -capacitor failures. Downtime of all IPNS
accelerator systems over the past 12 years (including the
preaccelerator/l inac) is shown in Figure 1. Overall, the
preaccelerator/linac accounted for about one quarter of
the total accelerator downtime.
Figure 1. IPNS Accelerator Equipment Trouble Chart
The IPNS accelerators, for over 10 years, have
delivered high -energy prot ons to the spallation target
with an average reliability of 95%. Experience has
shown that this is a level at which the neutron -scattering
users are seldom disappointed. However, on the
occasional two -to-three week runs when reliability slips
below 90%, the chances of some users not completing their experiments are much increased. Also, the short
nature of individual experiments means that one cannot
compensate for a lost day of beam by operating at 10%
higher current for the next 10 days. Thus for IPNS ,
mean -time-to-repair is at least as important as total
downtime.
The challenge posed five years ago, of increasing
operation from the previous level of 15 -17 weeks per year
to 25 or greater without impacting reliability, has been
met. Present plans call for a further increase to about 30
weeks per year. Additional pressures on reliability are
that several systems, specifically the high -voltage
regulator and pulsed bouncer for the preaccelerator, and
the driver stage for the 7835 power triode in the linac rf
system, will have to be replaced because the tubes used in
these units are no longer available. As it is probable that
the new systems will result in an initial increase in
downtime, it will be important to aggressively attack the
linac rf capacitor p roblem which now accounts for about
half of our total downtime. The pressurized cavities used
for the 7835 tubes at FNAL and LANL are a possible
solution, but could not be installed in the 8 -10 week
summer shutdown of our present operating schedule.
Therefore we are hoping that our present work with the
manufacturer to develop a more robust capacitor, coupled
with our work to decrease higher -order modes in the
cavity, will provide relief.
ACKNOWLEDGEMENTS
The authors strongly believe that the reliability of the
IPNS accelerator systems is primarily attributable to the
designers, subsystem managers, engineers, technicians
and operators who have systematically eliminated the
weak components and have developed maintenance and
operating techniques that provide rapid detection and
repair when failures occur. Their response to calls, at
any hour of the day or night, is what keeps our mean -
time-to-repair manageable.
REFERENCES
[1] V. Stipp, F. Brumwell, and Gerald McMichael, "The
ANL 50 MeV H- Injector - 35 Year Anniversary,
1996 Linear Accelerator Conference, Geneva,
Switzerland, Aug 26 -30, 1996, CERN Report 96 -07,
Nov. 1996, p74. IPNS Accel. Equipment Trouble
0.002.004.006.008.0010.0012.0014.00
H~ Source
Linac
Beam Lines
Ring RF
RMPS
Magnets
Kicker
Cool/Misc
Vacuum
Utilities
Computer
MCR Eqp't
EquipmentHours/1000 Hours of OperationFY89-91Avg FY92-94Avg
FY95-97Avg FY98-00AvgAugust 10, 2000 |
arXiv:physics/0008154 18 Aug 2000TESTING AND IMPLEMENTATION PROGRESS ON THE ADVANCED
PHOTON SOURCE (APS) LINEAR ACCELERATOR (LINAC) HIGH -
POWER S-BAND SWITCHING SYSTEM
A.E. Grelick, N. Arnold, S. Berg, D. Dohan, G. Goeppner, Y.W. Kang,
A. Nassiri, S. Pasky, G. Pile, T. Smith, S.J. Stein
Argonne National Laboratory, Argonne, Illinois 60439 USA
Abstract
An S-band linear accelerator is the source of parti cles
and the front end of the Advanced Photon Source [1]
injector. In addition, it supports a low-energy un dulator
test line (LEUTL) and drives a free-electron laser (FEL).
A waveguide-switching and distribution system is no w
under construction. The system configuration was r evised
to be consistent with the recent change to electron -only
operation. There are now six modulator-klystron
subsystems, two of which are being configured to ac t as
hot spares for two S-band transmitters each, so tha t no
single failure will prevent injector operation. Th e two
subsystems are also used to support additional LEUT L
capabilities and off-line testing. Design consider ations for
the waveguide-switching subsystem, topology selecti on,
control and protection provisions, high-power test results,
and current status 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 the rmionic
rf gun via the exhaust of one accelerating structur e. L2,
L4, and L5 are conventional sectors, each using a S LED
cavity assembly [3] to feed four accelerating struc tures.
L3 supplies rf power to the photocathode gun locate d at
the beginning of the linac. For normal storage rin g
injection operation, L1, L2, L4, and L5 are operate d, and
for the SASE-FEL operation, all five units are oper ated.
A sixth klystron-modulator system was installed in the
linac gallery. Design work is in progress on a wa veguide
distribution and switching system to allow the thir d and
sixth subsystems to serve as hot spares. This is a change
from the original version of the system [4], that w ould
have allowed the sixth subsystem to serve as a hot spare
for any of the others. The most critical design is sue for
this system is waveguide switch reliability at 35 M W-
peak power.
2 TOPOLOGY CHANGES
The change from positron to electron operation in t he
APS storage ring, together with LEUTL operating
requirements, changed the linac configuration by
eliminating the L3 accelerating structure. The L3 klystron
therefore became an obvious candidate to be used as a hot spare. The current switching system topology is sh own in
Figure 1. There are now two separate sections. Th e first,
which covers the guns and lower energy sectors is i n the
process of being installed. In this low-energy sec tion, the
L3 klystron serves as a hot spare for the L1 and L2
klystrons and powers either the photocathode rf gun , to
support LEUTL operation, or the gun test room. In the
second, or high-energy section, the L6 klystron ser ves as a
hot spare for the L4 and L5 klystrons and powers th e test
stand for switches and other high-power waveguide
components. Implementation of the high-power secti on
has been put on hold pending firm decisions on the likely
use of higher power klystrons and additional accele rating
structures in order to provide increased energy for LEUTL
operation.
3 HIGH-POWER COMPONENTS
3.1 Waveguide Switches
The waveguide switches must be highly reliable at a
peak power of 35 MW. Tests had already confirmed t hat
commercially available, sulfur hexafluoride (SF6)
pressurized, WR284 waveguide switches were subject to
damage due to breakdown at peak powers greater than 30
MW [5]. Scaling to the same field strength in the larger
WR340 waveguide yielded a prediction of operation t o 43
MW before having significant breakdown problems.
Tests of WR340 waveguide switches were set up an
additional time at Stanford Linear Accelerator Cent er
(SLAC) Klystron Microwave Laboratory. The
unsuccessful results of the original WR340 waveguid e
switch tests [5] were traced to the fact that, cont rary to our
expectations, the purchased WR284-to-WR340 transiti ons
were not tapered transitions. This time, electrofo rmed,
tapered transitions were used. WR340 switches, whi ch
had been reworked by electropolishing, were operate d
when pressurized with SF6 at 30 PSIG. The results were
consistent with the prediction. Three out of four switches
operated at a peak power of 43 MW or greater before
repetitive arcing occurred. The fourth switch suff ered a
severe arc during conditioning and showed a decreas ed
return loss as evidence of degradation. To further
maximize high-power reliability, an SF6 conditioner -
dehydrator system is being used to supply pressure in the
interconnecting waveguides and switches.
.Figure 1: Switching system configuration.
A similar system, which also provides constant
circulation of the SF6, has been used successfully used at
the Duke University Free-Electron Laser Laboratory at up
to 34-MW peak power [6]. Addition of this further
refinement has not been implemented to date but rem ains
an option for the future.
3.2 New WR340 Window
The WR284 waveguide system uses purchased
waveguide windows to provide isolation between vacu um
and pressurized sections and to keep to an irreduci ble
minimum any possibility of contamination reaching
accelerating structures or SLED cavities.
A window is under development in-house for use with
the WR340 waveguide. A return loss of greater than 40
dB on all units has been set as a design goal. The
prototype window has achieved a pre-braze measured
return loss of 52 dB. However, very strong sensiti vity of
return loss to assembly pressure has raised concern over
the ability to maintain tolerances during the brazi ng
process. Therefore, a modified design, which
incorporates tuning adjustments that can be set aft er
brazing, has been created. Figure 2 is a drawing o f the modified window design, showing the tuning adjustme nt
provisions.
Figure 2: Tunable window design.
4 CONTROL IMPLEMENTATION
Implementation of controls for the switching system is
being coordinated with the new, more flexible equip ment
interlock system. Programmable logic controllers ( PLCs)
were chosen for the new interlocks for their ease o f
configuration and diagnostic capabilities.
Each sector of the linac will eventually have its o wn
interlock chassis where all interlock cabling is ru n. The
PLC that resides in this chassis (see Figure 3) wat ches all
inputs and generates the proper permits based on th e
interlock logic. The status of all inputs and outp uts from
each interlock chassis will be available in the for m of
operator screens. Additionally, a time-stamped fau lt stack
was created to aid in diagnostics.
The interface to the high-power components of the
switching system will utilize the same type of PLC as the
new interlock system. This PLC will be responsible only
for actuating the switch itself and verifying that the switch
is in a valid position. The switching PLC will not ify the
interlock PLCs via either discreet outputs or high- level
messages of the switch positions. The user interf ace to
the switching system will be a rack-mounted touch p anel
with a key switch. Activating the key switch will disable
klystron drive, enable waveguide switch power, and place
the switching PLC in the active mode; all accomplis hed
via hard-wired connections.
The design of the interlock logic is complicated by the
fact that the cause and effect relationships are de pendent
on the current operating mode of the linac. For ex ample,
if a loss of vacuum in one waveguide section occurs , the
proper modulator-klystron must be notified. It is the
responsibility of the local PLC interlock system to grant
or deny permits based on the input conditions, as w ell as
the switch positions.
Figure 3: PLC in interlock chassis.
5 CURRENT STATUS AND PLANS
Waveguide switch rework and testing is close to
completion. The window design is not complete and is
the probable gating item in the schedule. Interloc k and
interface circuits have been tested and are ready t o install
in most cases. A complete compatibility review is
required before the system design can be considered final.
In any event, commissioning is expected to be under way
during the first half of fiscal year 2001.
6 ACKNOWLEDGMENTS
The authors wish to thank G. Caryotakis and S. Gold of
SLAC for providing access to their laboratory to pe rform
high-power tests, and J. Eichner and G. Sandoval, a lso
from SLAC, for their help in assembling and operati ng the
test set-up. We also thank D. Meyer for operation of the
Argonne high power tests; J. Crandall, H. Deleon, J . Hoyt,
T. Jonasson, and J. Warren for setting up the vario us test
configurations; and C. Eyberger for editorial assis tance.
This work is supported by the U.S. Department of
Energy, Office of Basic Energy Sciences, under Cont ract
No. W-31-109-ENG-38.
7 REFERENCES
[1] 7-GeV Advanced Photon Source Conceptual Design
Report, ANL-87-15, April 1987.
[2] M. White, N. Arnold, W. Berg, A. Cours, R. Fuja ,
A.E. Grelick, K. Ko, Y. L. Qian, T. Russell, N.
Sereno, and W. Wesolowski, “Construction,
Commissioning and Operational Experience of the
Advanced Photon Source (APS) Linear Accelerator,”
Proceedings of the XVIII International Linear
Accelerator Conference, Geneva, Switzerland, 26-30
August, 1996, pp. 315-319, (1996).
[3] Z. D. Farkas et al., “SLED: A Method of Doublin g
SLAC’s Energy,” SLAC-PUB-1453, June 1974.
[4] A.E. Grelick, N. Arnold, S. Berg, R. Fuja, Y.W.
Kang, R.L. Kustom, A. Nassiri, J. Noonan, M. White,
“A High Power S-band Switching System for the
Advanced Photon Source (APS) Linear Accelerator
(Linac),” Proceedings of the XIX International Lina c
Conference, August 23-28, 1998, Chicago, IL, pp.
914-916 (1998)
[5] A. Nassiri, A. E. Grelick, R. L. Kustom, M. Whi te,
“High Peak-Power Test of S-Band Waveguide
Switches,” Proceedings of the 1997 Particle
Accelerator Conference, Vancouver, BC, Canada
(1998)
[6] P. G. O’Shea et al., “Accelerator Archaeology-T he
Resurrection of the Stanford Mark-III Electron Lina c
at Duke,” Proceedings of the 1995 Particle
Accelerator Conference, May 1-5, 1995, Dallas, TX,
pp. 1090-1092 (1996)
|
arXiv:physics/0008155v1 [physics.chem-ph] 18 Aug 2000GRECP/MRD-CI calculations of the spin-orbit splitting in t he
ground state of Tl and of the spectroscopic properties of TlH .
A. V. Titov∗, N. S. Mosyagin
Petersburg Nuclear Physics Institute, Gatchina, St.-Pete rsburg district 188350, RUSSIA
A. B. Alekseyev, R. J. Buenker
Theoretische Chemie, Bergische Universit¨ at GH Wuppertal , Gaußstraße 20, D-42097 Wuppertal,
GERMANY
(January 21, 2014)
Abstract
The generalized relativistic effective core potential (GRE CP) approach is
employed in the framework of multireference single- and dou ble-excitation
configuration interaction (MRD-CI) method to calculate the spin-orbit (SO)
splitting in the2Poground state of the Tl atom and spectroscopic constants
for the 0+ground state of TlH. The 21-electron GRECP for Tl is used and
the outer core 5 sand 5ppseudospinors are frozen with the help of the level
shift technique. The spin-orbit selection scheme with resp ect to relativistic
multireference states and the corresponding code are devel oped and applied in
the calculations. In this procedure both correlation and sp in-orbit interactions
are taken into account. A [4,4,4,3,2] basis set is optimized for the Tl atom
and employed in the TlH calculations. Very good agreement is found for the
equilibrium distance, vibrational frequency, and dissoci ation energy of the TlH
ground state ( Re= 1.870˚A,ωe= 1420 cm−1,De= 2.049 eV) as compared
with the experimental data ( Re= 1.868˚A,ωe= 1391 cm−1,De= 2.06 eV).
SHORT NAME: GRECP/MRD-CI calculations on Tl and TlH
KEYWORDS FOR INDEXING: Relativistic Effective Core Potential
Configuration Interaction, Molecule with heavy atoms, Elec tronic structure
calculation.
31.15.+q, 31.20.Di, 71.10.+x
Typeset using REVT EX
1I. INTRODUCTION
During the last few years a large number of publications have dealt with calculations of
the2Po
1/2−2Po
3/2splitting in the ground state of the Tl atom and spectroscopi c constants for
the 0+ground state of TlH. Such interest to these systems arises be cause of their relatively
simple electronic structure in the valence region. This mak es them very convenient objects
for testing methods for the description of relativistic and correlation effects. We can mention
some recent papers [1–7] in which the electronic structure o f thallium was studied and papers
[8–11] in which the calculation of spectroscopic constants for TlH was carried out. With the
exception of the atomic RCC calculation by Eliav et al. [6,7] and the atomic CI/MBPT2
calculation by Dzuba et al. [2], the published results cannot be considered to be very ac curate
and reliable, however, primarily because of the rather smal l basis sets and the small numbers
of correlated electrons.
In calculations of Tl and TlH with the use of the relativistic effective core potential
(RECP) approximation [12], in which only 13 thallium electr ons are treated explicitly (13e-
RECPs), one more problem appears. The correlation of the out er core (OC) and valence
(V) electrons, occupying the 5 dandns, np, nd (n= 6,7, . . .) orbitals, respectively, cannot
be satisfactorily described, mainly because the smoothed V -pseudoorbitals (pseudospinors)
have the wrong behaviour in the OC region. One-electron func tionsφcorr
x,k(r), being some
linear combinations of virtual orbitals, correlate to occu pied orbitals φocc
x(where x=c, v
stands for the OC and V orbital indices) and are usually local ized in the same space region
asφocc
x. Therefore, the original “direct” Coulomb two-electron in tegrals describing the OC-V
correlation of φocc
candφocc
vcan be well reproduced by those with the pseudoorbitals, des pite
their localization in different space regions. However, a tw o-electron integral describing the
“exchange” part of the OC-V correlation,
/integraldisplay
rdrφcorr†
c,k′(r)φocc
v(r)/integraldisplay
r′dr′φcorr†
v,k(r′)φocc
c(r′)1
r−r′, (1)
cannot be well reproduced because the V-pseudoorbitals are smoothed in the OC region
where the OC-pseudoorbitals are localized (for more theore tical details, see Ref. [13]).
The first RECPs for Tl with the 5 s,5pshells treated explicitly (21e-RECPs) for which this
disadvantage of the earlier “semicore” RECPs was overcome w ere generated and tested in
single-configurational calculations by Mosyagin et al. [14,15]. Some other inherent problems
of the “nodeless” RECPs were also solved with the 21-electro n Generalized RECP (21e-
GRECP) version presented in Ref. [14,15]. In Ref. [15], for t he case of the 21e-GRECP it
was also shown that the 5 s,5ppseudospinors could be frozen while still providing signifi cantly
higher accuracy than 13e-RECPs because the valence and virt ualnsandnp(n= 6,7, . . .)
pseudoorbitals in the former case already have the proper no dal structure in the OC region.
II. THE GRECP OPERATOR IN THE SPIN-ORBIT REPRESENTATION
In most existing quantum-chemical codes for molecular calc ulations with RECPs (as well
as in the MRD-CI code used in the present work) spin-orbit bas is sets are used. In these
versions the number of the two-electron integrals is substa ntially smaller than in the case
2of spinor basis sets providing the same level of correlation treatment. Spin-orbit basis sets
are preferable in the calculations in which correlation effe cts give a higher contribution to
the properties of interest than those of a relativistic natu re. This is usually the case for
valence and outermost core electrons, which mainly determi ne chemical and spectroscopic
properties of molecules.
Together with the spin-orbit basis set, the GRECP for Tl shou ld also be employed in
the spin-orbit representation. Following Ref. [16,17], th e components of the spin-averaged
part of the GRECP operator called the averaged relativistic effective potentials (AREP) are
written in the form [13,14]:
UAREP
nvl(r) =l+ 1
2l+ 1Unvl+(r) +l
2l+ 1Unvl−(r), (2)
UAREP
ncl(r) =l+ 1
2l+ 1Vncnvl+(r) +l
2l+ 1Vncnvl−(r), (3)
Vncnvl±(r) = [Uncl±(r)−Unvl±(r)]/tildewidePncl±(r) +/tildewidePncl±(r)[Uncl±(r)−Unvl±(r)]
−/summationdisplay
n′c/tildewidePncl±(r)/bracketleftbiggUncl±(r) +Un′cl±(r)
2−Unvl±(r)/bracketrightbigg
/tildewidePn′cl±(r), (4)
where Unl±(r) are the potentials generated for the ˜ ϕnl±(r) pseudospinors by means of the
Goddard scheme [18]; nxis the principal quantum number of an outercore ( nc), valence ( nv)
or virtual ( na) pseudospinor; landjare angular and total electron momenta; ±stands for
j=l±1/2;/tildewidePncl±(r) is the radial projector on the OC pseudospinors:
/tildewidePncl±(r) =/summationdisplay
m/tildewidest|nc, l,±, m/a\}bracketri}ht/tildewidest/a\}bracketle{tnc, l,±, m|. (5)
Clearly, the AREP component of the GRECP may be used in calcul ations with nonrela-
tivistic quantum-chemical codes in order to take account of spin-independent relativistic
effects.
The operator of the effective spin-orbit interaction can be d erived following the expression
for the spin-angular projector Pl±from Ref. [17]:
Pl±(Ω, σ) =1
2l+ 1/bracketleftBig/parenleftBig
l+1
2±1
2/parenrightBig
Pl(Ω)±2Pl(Ω)/vectorl/vectorsPl(Ω)/bracketrightBig
. (6)
Its components, called the effective spin-orbit potentials (ESOP), can be written as [13,14]
∆Unvl(r) =Unvl+(r)−Unvl−(r), (7)
∆Uncl(r) =Vncnvl+(r)−Vncnvl−(r), (8)
UESOP
nl =2∆Unl(r)
2l+ 1Pl/vectorl/vectors, (9)
3Pl=l/summationdisplay
m=−l|lm/a\}bracketri}ht/a\}bracketle{tlm|, (10)
where |lm/a\}bracketri}ht/a\}bracketle{tlm|is the projector on the spherical function Ylm.
Neglecting the difference between UAREP
nvLandUnvLJfor virtual pseudospinors with l > L
(for theoretical details see Ref. [13]), one can write the GR ECP operator Uas
U=UAREP
nvL(r) +L−1/summationdisplay
l=0/bracketleftBig
UAREP
nvl(r)−UAREP
nvL(r)/bracketrightBig
Pl
+/summationdisplay
ncL/summationdisplay
l=0UAREP
ncl(r)Pl+L/summationdisplay
l=1/bracketleftBig
UESOP
nvl+/summationdisplay
ncUESOP
ncl/bracketrightBig
Pl. (11)
Note that the nonlocal terms with the projectors on the most i mportant correlation
functions ˜ ϕcorr
nxl±;nklk±(r) (1) (where x=c, v) localized mainly in the OC and V regions and
with the corresponding potentials Ucorr
nklk±(r) can be taken into account in the considered ex-
pressions for the GRECP operator additionally to those with the OC projectors. Obviously,
the non-local GRECP terms for the frozen OC pseudospinors ca n be omitted in the sum on
(ncl) in Eq. (11).
We should emphasize that in spite of the rather complicated f orm of the above GRECP
operator, the main computational effort in calculating matr ix elements with the GRECP is
caused by the standard radially-local operator, which is al so a part of conventional RECP
operators, and not by the non-local GRECP terms. Thus, the ad ditional complications in
calculations with GRECPs are negligible in comparison with treatments employing conven-
tional semi-local RECPs if comparable gaussian expansions are used for the partial poten-
tials. The more critical point is that the effort in the calcul ation and transformation of
two-electron integrals is always substantially higher than that in the computation of RECP
integrals for all known RECP versions (including GRECPs) wh en appropriately large basis
sets are employed in the precise calculations.
III. FROZEN-CORE APPROXIMATION FOR THE OUTER-CORE SHELLS
To perform precise calculations of chemical and spectrosco pic properties, correlations
should be taken into account not only within the valence regi ons of heavy atoms and heavy-
atom molecules but in the core regions and between the valenc e and core electrons as well.
In practice, the goal is to achieve a given level of accuracy b y correlating as small a number
of electrons as possible, thus reducing the computational e ffort. However, as discussed in
the Introduction, the accuracy of the RECPs generated for a g iven number of explicitly
treated electrons cannot always satisfy the accuracy requi rements expected from correlating
all these electrons in the corresponding all-electron calc ulation. This is true, in particular,
for calculations of Tl, having a 5 d106s26p1leading configuration in the ground state, and its
compounds.
To obtain an accuracy on the order of 400 cm−1for the2Po
1/2−2Po
3/2splitting in the
ground state and for excitation energies to low-lying state s of Tl and to take account of the
core polarization, one should correlate at least 13 electro ns, i.e. include the 5 dshell. This is
4achieved in the present MRD–CI calculations with fandgbasis functions describing mainly
polarization of the 5 dshell (for other recent results see, e.g., [1,3,4]). Some da ta from our
13e-CI calculations of the SO-splitting in the ground state of Tl are collected in Table I in
comparison with the 3e-CI results, which in our DF/CI and GRE CP/CI calculations have
errors of about 600 cm−1.
We also should mention the recent relativistic coupled-clu ster (RCC) results of Landau
et al. [7], in which 35 electrons are correlated and a decrease of cl ose to 90 cm−1in the above
mentioned SO splitting is due to the Breit interaction. Note that this interaction is not yet
taken into account in the RECPs considered in the present wor k.
Obviously, the 5 dshell should also be explicitly treated in calculations of m olecules
containing Tl to take into account core relaxation and polar ization effects with satisfactory
accuracy. For these calculations it would be optimal to use t he RECPs with 13 electrons
of Tl treated explicitly (13e-RECPs) such as the RECP of Ross et al. [19] or our valence
RECP version [15]. None of the known nodeless 13e-RECPs can p rovide the aforementioned
accuracy, however. Although single-configurational tests [14,15] give errors of 100 cm−1or
somewhat more for excitation energies to low-lying states, they are dramatically increased for
13e-RECPs if all 13 electrons are correlated. The reasons ar e discussed in the Introduction
(one can also see the results of the 13e-RECP/MRD-CI calcula tions in Ref. [5] and of the
13e-PP/MRCI calculations in Ref. [4]).
To overcome this disadvantage, one should use RECPs with at l east 21 electrons, e.g.
21e-GRECP [14,15] and 21e-PP [4] for Tl. The 5 sand 5 ppseudospinors can be treated
as frozen, however, while still providing the aforemention ed accuracy. The 5 porbitals have
energies about four times higher and their average radii are 1.4 times shorter than those
for the 5 dorbitals. Moreover, their angular correlation is supresse d as compared with the
5dshell because the most important polarization functions (5 dfor the 5 porbitals and 5 p
for the 5 sorbitals) are completely occupied in the lowest-lying stat es. Therefore, the 5 s,5p
orbitals are substantially less active in chemical process es.
In order to freeze the 5 sand 5 ppseudospinors, one can apply the energy level shift
technique [13]. Following Huzinaga et al. [20], one should add the matrix elements of the
SCF field operators (the Coulomb and spin-dependent exchang e terms) over these OC pseu-
dospinors to the one-electron part of the Hamiltonian toget her with the level shift terms
/summationdisplay
ncf,l,±Bncfl±/tildewidePncfl±(r), (12)
where Bncfl±is at least of order |2εncfl±|andεncfl±is the orbital energy of the OC pseu-
dospinor/tildewideφncfl±(r) to be frozen. Such nonlocal terms are needed in order preven t collapse
of the molecular orbitals to the frozen states (the 5 s1/2,5p1/2,3/2pseudospinors for Tl). All
terms with the frozen core pseudospinors described here (th e Coulomb and exchange inter-
actions, and the level shift operator) can easily be present ed in spin-orbit form with the help
of eq. (6), as was done above for the GRECP operator.
More importantly, these OC pseudo spinors can be frozen in calculations with spin-orbit
basis sets and they can already be frozen at the stage of calcu lation of the one-electron
matrix elements of the Hamiltonian, as implemented in the MO LGEP code [21]. Thus,
any integrals with indices of the frozen spinors are complet ely excluded after the integral
calculation step.
5In single-configurational calculations with the numerical HFJ code [15] we have seen
that the SO splitting of the 5 pshell increases the resulting SO splitting of the2Poground
state by about 400 cm−1, whereas the SO splitting of the 5 dshell decreases the final SO
splitting by almost the same value. Therefore, it is importa nt to freeze the 5 p1/2and 5p3/2
(pseudo)spinors and not some averaged 5 p(pseudo)orbitals if the SO interaction is to be
taken into account in the 5 dand valence shells.
In Ref. [4], the 21e-“energy-adjusted” Pseudopotential (P P) having the features which
have been emphasized [13–15,22] as inherent for GRECPs (diff erent potentials for the 5 p
and 6ppseudospinors in the case of Tl) is generated and applied to t he calculation of the SO
splitting in Tl, with the core correlations described by the core polarization potential (CPP).
Some average OC pseudoorbitals are frozen and the SO splitti ng of 7810 cm−1obtained in
their 21e-PP/MRCI calculation is quite different than our re sult.
After applying the projection operator of eq. (6) to the leve l shift (12), Coulomb and ex-
change terms with the frozen core pseudospinors, the AREP an d ESOP parts of the GRECP
operator are to be modified to include these new contribution s. This technique was success-
fully employed in our earlier calculations of the spin-rota tional Hamiltonian parameters in
the BaF and YbF molecules [23].
The freezing technique discussed above can be efficiently app lied to those OC shells for
which the spin-orbit interaction is clearly more important than the correlation and relaxation
effects. If the latter effects are neglected entirely or taken into account within “correlated”
GRECP versions [13], the corresponding OC pseudospinors ca n be frozen and the spin-orbit
basis sets can be successfully used for other explicitly tre ated shells. This is true for the
5p1/2,3/2subshells in Tl, contrary to the case of the 5 d3/2,5/2subshells. Freezing the OC
pseudospinors allows one to optimize an atomic basis set onl y for the orbitals which are
varied or correlated in subsequent calculations, thus avoi ding the basis set optimization
for the frozen states and reducing the number of the calculat ed and stored two-electron
integrals. Otherwise, if the 5 pshell should be correlated explicitly, a spinor basis set ca n be
more appropriate than the spin-orbit one.
IV. THE MRD-CI METHOD
In the multireference single- and double-excitation CI app roach [24], the Λ S-basis sets of
many-electron spin-adapted (and space symmetry-adapted) functions (SAFs) are employed.
This method makes use of configuration selection and perturb ative energy extrapolation
techniques [24] and employs the Table CI algorithm [25] for e fficient handling of the various
open-shell cases which arise in the Hamiltonian matrix elem ents. Some new features of the
selection scheme used in this work are considered below. The higher excitations in the CI
treatment has been assessed by applying the generalized mul tireference analogue [27] of the
Davidson correction [26] to the extrapolated T=0 energies of each root.
After selecting the Λ S-sets of SAFs for a chosen threshold Ti(i= 1,2), they are collected
together in accord with the relativistic double-group symm etry requirements and a spin-orbit
CI (SO-CI) calculation is performed with these SAFs to obtai n some SO-roots (ΨSO,T i
I)
and their energies ( ESO,T i
I) which are of interest in a considered double group irreduci ble
representation (irrep). Then the linear T=0 correction is evaluated in the basis of the
6calculations with the T1andT2thresholds. Finally, the generalized Davidson (or full CI)
correction is applied to each root of interest.
The stage of the molecular spectroscopic constants calcula tion begins with the fitting
of the relativistic CI potential curves to polynomials whic h are employed to construct ap-
propriate Born-Oppenheimer nuclear motion Schr¨ odinger e quations solved by the Dunham
method with the help of the DUNHAM-SPECTR code of Mitin [28].
A. Features of the spin-orbit selection procedure
Let us define a Hamiltonian Hfor a molecule as
H=H(0)+Vcorr+HSO, (13)
where H(0)is an unperturbed spin-independent Hamiltonian, Vcorris a two-electron oper-
ator describing correlations, and HSOis a one-electron spin-orbit operator (ESOP in our
case). Let us choose an orthonormal basis set of SAFs {Φ(n)ΛS
I}in the Λ S-coupling scheme
(or “spin-orbit” basis set). In particular, these SAFs can b e solutions of Hartree-Fock equa-
tions with a spin-averaged RECP for the molecule considered . The H(0)Hamiltonian is
constructed to be diagonal in the given many-electron basis set:
H(0)Φ(n)ΛS
I =E(n)ΛS
I Φ(n)ΛS
I, (14)
where n= 0,1, . . .(see below the description of the indices in more detail). Ad ditionally
define H(0)so that
<Φ(n)ΛS
I|H(0)|Φ(n)ΛS
I>≡<Φ(n)ΛS
I|H|Φ(n)ΛS
I> (15)
in order to exclude the first-order PT contributions to total energies of molecular states (this
corresponds to the Epstein-Nesbet PT form).
We will ignore the two-electron spin-dependent (Breit) int eractions which ordinarily can
be neglected when studying chemical and spectroscopic prop erties. Breit and other quantum
electrodynamic (QED) effects are relatively large for lanth anides and actinides, but for the
V and OC shells they can be efficiently represented by the one-e lectron j-dependent RECP
terms.
Let us distinguish the following types of many-electron fun ctions which are considered
in a double-group symmetry:
• {Φ(0)ΛS
I, E(0)ΛS
I}N(0)ΛS
I=0 are reference SAFs (“Mains”) and their energies
E(n)ΛS
I =<Φ(n)ΛS
I|H(0)|Φ(n)ΛS
I> (16)
atn= 0 for those Λ S-irreps which are of interest for the final spin-orbit CI (SO- CI)
calculation;
• {Ψ(0)ΛS
I,E(0)ΛS
I}N(0)ΛS
I=0 are some of the CI solutions (“Λ S-roots”) and their energies
E(0)ΛS
I =<Ψ(0)ΛS
I|H(0)+Vcorr|Ψ(0)ΛS
I> (17)
in the Λ S-irrep which diagonalize the ( H(0)+Vcorr) in the subspace of Mains only;
7• {Ψ(0)SO
I,E(0)SO
I}N(0)SO
I=0 are some of the SO-CI solutions (“SO-roots” which are of inte r-
est) and their energies
E(0)SO
I =<Ψ(0)SO
I|H(0)+Vcorr+HSO|Ψ(0)SO
I> (18)
which diagonalize the complete HHamiltonian in the subspace of all Mains collected
from all the Λ S-irreps considered;
• {Φ(1)ΛS
I, E(1)ΛS
I}N(1)ΛS
I=0 are the singly-excited SAFs (SE-SAFs) and their energies (1 6)
atn= 1, i.e.
Φ(1)ΛS
I∈ {PΛSa+
paqΦ(0)Λ′S′
J } \ {Φ(0)ΛS
K} ∀ (p, q;J, K), (19)
where PΛS=|ΛS >< ΛS|is a projector on the subspace of the Λ S-states, a+
p(aq)
are the creation (annihilation) operators of one-electron states (spin-orbitals) φp(φq).
The SE-SAFs can be automatically selected because of their r elatively small number;
• {Φ(2)ΛS
I, E(2)ΛS
I}N(2)ΛS
I=0 are the doubly-excited SAFs (DE-SAFs)
Φ(2)ΛS
I∈ {PΛSa+
pa+
qarasΦ(0)Λ′S′
J } \({Φ(1)ΛS
K} ∪ { Φ(0)ΛS
L})∀(p, q, r, s ;J, K, L ) (20)
and their energies (16) at n= 2; a SAF Φ(2)ΛS
I should be selected in accordance with
some selection criteria to be used in the final SO-CI calculat ion. In principle, triple
and higher excited sets of SAFs can be similarly defined.
The correlation operator, Vcorr, has the symmetry of the molecule and, therefore, can
be rewritten as
Vcorr≡/summationdisplay
ΛSPΛSVcorrPΛS. (21)
It normally gives the most important contribution through t he second-order Brillouin-
Wigner PT energy correction in the basis set of Φ(n)ΛS
J (after appropriate redefinition of
H(0)in the subspace of Mains, see Ref. [29,30]):
/summationdisplay
n=1,2/summationdisplay
J|<Φ(n)ΛS
J|Vcorr|Ψ(0)ΛS
0>|2
EΛS
0−E(n)ΛS
J(22)
for the non-degenerate ground state ΨΛS
0with the exact energy EΛS
0in the Λ S-irrep (ob-
viously, terms with n≥3 are automatically equal to zero because Vcorris a two-electron
operator). A similar expression with the replacements Ψ(0)ΛS
0→Ψ(0)ΛS
I andEΛS
0→ EΛS
Ican
be applied for excited states ΨΛS
I(some precautions should be taken concerning the degen-
erate states and the orthogonality constraints with respec t to the lower-lying states with
J < I ). As a result, the first rows, columns and energies on the diag onal of the Hamiltonian
matrix
<Φ(n)ΛS
J|Vcorr|Φ(0)ΛS
I> , < Φ(n)ΛS
J|H|Φ(n)ΛS
J> (23)
8forn= 1,2 are usually employed in the selection procedures for SAFs {Φ(1,2)ΛS
I }based on
the nonrelativistic AkandBkapproximations (when HSOis not taken into account) [29,30]
or on the multi-diagonalization scheme [24] for subsequent calculations of ΨΛS
I. In spite of
some differences between these selection schemes, they are n ot very essential for the final
CI results if a high quality reference set (set of Mains) and a suitably small threshold are
chosen.
For molecules with heavy and very heavy atoms, the HSOoperator can give large contri-
butions to the energy both in second and in higher PT orders if a non-optimal set of Mains,
{Φ(0)ΛS
I}, is chosen after an SCF calculation with the SO-averaged pot entials (AREPs). The
latter is the usual practice and the set of Mains generated in such a manner can be smaller
than optimal for the case of large SO interaction. Therefore , not only second but third and
maybe even higher PT order(s) can be important in the selecti on procedure for a “bad”
set of the starting roots Ψ(0)SO
I. This means that the off-diagonal matrix elements of Hbe-
tween secondary many-electron basis functions (SE-, DE-SA Fs) may be introduced into the
selection procedure because HSOis a substantially off-diagonal operator contrary to Vcorr:
<Φ(n)ΛS
I|HSO|Φ(n′)Λ′S′
J > (ΛS) and (Λ′S′) can be different, n′∈ {n,|n±1|}.(24)
In particular, HSOgives zero matrix elements between SAFs belonging to the sam e ΛS-irrep
in the D2horC2vsymmetry groups.
For simplicity, let us consider the selection scheme based o n the Akapproximation (22).
In the nonrelativistic-type selection scheme, a SAF Φ(1,2)ΛS
J is selected in a Λ S-irrep if
|<Φ(1,2)ΛS
J |Vcorr|Ψ(0)ΛS
I>|2
E(1,2)ΛS
J − E(0)ΛS
I≥δEΛS
T, (25)
where I≤NSOandδEΛS
Tis a threshold criterion for the energy selection scheme in t he ΛS-
irrep. In (25) we have replaced the exact EΛS
Ienergies by the approximate E(0)ΛS
I values (17)
that corresponds to the Rayleigh-Schr¨ odinger PT case. Suc h a simplification is justified for
small δEΛS
Tand good reference states.
In a SO-CI calculation within some relativistic double-gro up irrep, substitutions for the
reference state (Ψ(0)ΛS
I →Ψ(0)SO
I) and the perturbation ( Vcorr→Vcorr+HSO) should be
used in the previous expression, so that
|<Φ(1,2)ΛS
J |Vcorr+HSO|Ψ(0)SO
I>|2
E(1,2)ΛS
J − E(0)SO
I≥δESO
T, (26)
where δESO
Tis a selection threshold for Φ(1,2)ΛS
J to be used in the subsequent SO-CI calcu-
lation.
In more detail, the matrix element in the PT numerator of the a bove formula can be
rewritten as
|<Φ(1,2)ΛS
J |Vcorr|Ψ(0)SO
I>|2(27)
+|<Φ(1)ΛS
J|HSO|Ψ(0)SO
I>|2(28)
+ 2ℜ(<Ψ(0)SO
I|Vcorr|Φ(1)ΛS
J><Φ(1)ΛS
J|HSO|Ψ(0)SO
I>) (29)
9by taking into account eq. (21) in the calculation of the matr ix elements for Vcorr, contrary
to those for HSO. In spite of mixing different Λ S-states due to HSO, the number of non-zero
matrix elements with HSOin eq. (29) is usually relatively small because the SO intera ction
is a one-electron operator (see eq. (24)) which is very local ized compared with the long-range
Coulomb interaction. Thus, one can see that the nonrelativi stic-type selection due to Vcorr
with respect to {PΛSΨ(0)SO
I}in each considered Λ S-irrep and automatic selection of all SE-
SAFs {Φ(1)ΛS
J}(19) can be efficiently applied instead of eq. (26). It must be e mphasized that
contrary to the selection schemes in the nonrelativistic ca se, SE-SAFs should be generated
with respect to the Mains from all the used Λ′S′-irreps. In a more simplified treatment, the
automatic selection of SE-SAFs can be done with respect to a s ubset of the most important
Mains, e.g. having largest CI-coefficients in the Ψ(0)SO
I roots.
Next let us consider the terms from the third-order PT energy (PT-3) for SAFs {Φ(1,2)ΛS
J }
which can be essential for the SO selection procedure. Below we shall discuss only matrix
elements in the PT numerators of the corresponding PT-3 term s because specific expressions
for the energy denominators are not essential for our analys is and conclusions. For simplicity,
we shall omit the terms conjugate to those considered.
The first two types of the PT-3 matrix elements are:
<Ψ(0)SO
I|HSO|Φ(1)ΛS
J><Φ(1)ΛS
J|HSO|Φ(1)Λ′S′
K ><Φ(1)Λ′S′
K |HSO|Ψ(0)SO
I> , (30)
<Ψ(0)SO
I|HSO|Φ(1)ΛS
J><Φ(1)ΛS
J|Vcorr|Φ(1)ΛS
L><Φ(1)ΛS
L|HSO|Ψ(0)SO
I> . (31)
The first intermediate state, Φ(1)ΛS
J, is a test SE-SAF and the indices for other intermediate
SAFs run over all the allowed ones. The PT-3 terms summed over the indices of the second
intermediate state give contributions (together with the c onjugate terms) for the selection
of the test SE-SAF. However, the SE-SAFs can be selected auto matically and these terms
are out of our particular interest.
The following matrix element type
<Ψ(0)SO
I|Vcorr|Φ(1,2)ΛS
J ><Φ(1,2)ΛS
J |HSO|Φ(1)Λ′S′
K ><Φ(1)Λ′S′
K |HSO|Ψ(0)SO
I> . (32)
can be used for the selection of Φ(1,2)ΛS
J and Φ(1)Λ′S′
K when summing over another set of
intermediate states in the PT-3 expression. As one can see, t his term can be used for the
selection of both SE-SAFs and DE-SAFs. The above expression is quadratic in the (large)
HSOinteraction contrary to the remaining terms considered bel ow. The contribution of the
terms with matrix elements (32) can be essential and their us e for the selection of DE-SAFs
Φ(2)ΛS
J can be important for a subsequent SO-CI calculation.
The following matrix element types contain a second order pe rturbation in Vcorrand,
therefore, we can suggest that in general they are less impor tant for our consideration than
the above terms:
<Ψ(0)SO
I|Vcorr|Φ(1,2)ΛS
J ><Φ(1,2)ΛS
J |Vcorr|Φ(1)ΛS
L><Φ(1)ΛS
L|HSO|Ψ(0)SO
I> , (33)
<Ψ(0)SO
I|Vcorr|Φ(1,2)ΛS
J ><Φ(1,2)ΛS
J |HSO|Φ(1,2)Λ′S′
K ><Φ(1,2)Λ′S′
K |Vcorr|Ψ(0)SO
I> . (34)
These terms, together with the conjugate ones, can be used fo r the selection of Φ(1,2)ΛS
J and
Φ(1)ΛS
L. The term
10<Ψ(0)SO
I|Vcorr|Φ(2)ΛS
J><Φ(2)ΛS
J|HSO|Φ(2)Λ′S′
K ><Φ(2)Λ′S′
K |Vcorr|Ψ(0)SO
I> (35)
can be analyzed separately because it contains both the inte rmediate states as DE-SAFs.
In general, it is more difficult to take such terms into account in the selection procedure,
because of the large number of tested DE-SAFs. We should note , however, that when a
tested Φ(2)ΛS
J DE-SAF is fixed, the other intermediate states, {Φ(2)Λ′S′
K }, are those DE-SAFs
which are only singly excited with respect to the tested one. Therefore, the number of them
will not be very high.
For completeness, the matrix element type which is cubic in t heVcorrperturbation
should be listed:
<Ψ(0)SO
I|Vcorr|Φ(1,2)ΛS
J ><Φ(1,2)ΛS
J |Vcorr|Φ(1,2)ΛS
K ><Φ(1,2)ΛS
K |Vcorr|Ψ(0)SO
I> . (36)
This term is of nonrelativistic type and it is out of our parti cular interest because it does
not contain the HSOperturbation. Again, we can separate the term
<Ψ(0)SO
I|Vcorr|Φ(2)ΛS
J><Φ(2)ΛS
J|Vcorr|Φ(2)ΛS
K><Φ(2)ΛS
K|Vcorr|Ψ(0)SO
I> . (37)
from the previous one only because the latter contains both t he DE-SAF intermediate states.
We should emphasize that the terms containing SE- or DE-SAFs in the intermediate
states of the PT-3 expressions are not taken into account in t heBkand multi-diagonalization
selection procedures, although these schemes include, in f act, contributions of higher than
the second-order PT terms.
When analyzing the above PT-3 terms, it can be concluded that if one replaces the
reference SO roots, Ψ(0)SO
I, by new reference states, Ψ(0+1) SO
I , which diagonalize the complete
Hamiltonian Hfor the sets of both Mains and SE-SAFs taken together, and app lies the
selection criterion based on the second-order PT (26), then the main part of the above PT-3
terms will be taken into account in such a selection. An excep tion occurs for terms (35) and
(37), but in general they are thought to be less important tha n the other third-order PT
terms.
In a more sophisticated treatment, the reference Ψ(0+1′)SO
I SO states can be generated
when diagonalizing Hfor the sets of Mains and those SE-SAFs ( {Φ(1′)ΛS
J}), which are au-
tomatically generated with respect to the most important su bset of Mains ( {Φ(0′)ΛS
I}). The
latter subset can be selected from a preliminary CI calculat ion for the set of Mains, e.g.
in a basis of configurations with the highest CI coefficients in Ψ(0)SO
I, and so on. This is
worthwhile in order to reduce the number of SAFs in the result ing reference states Ψ(0+1′)SO
I
rather than in Ψ(0+1) SO
I , thus reducing the selection time which can otherwise be ver y large.
We should also note that the trial SE- and DE-SAFs, which are t ested in the above
selection procedure, are generated only for the set of Mains and not for the {Φ(0+1′)ΛS
I }set.
Therefore, the number of the tested configurations and the se lection time are reasonably
limited. If the number of configurations used in {Φ(0+1′)ΛS
I }is not high, one can extend the
set of Mains by including the above subset of SE-SAFs, thus ob viously enlarging the set of
the consequently generated and tested SE- and DE-SAFs.
Again we should emphasize that it is not necessary to use the t hird-order PT or the
suggested automatic selection of SE-SAFs in a selection pro cedure if a fairly good set of
11the reference roots Ψ(0)SO
I is used, i.e., if they provide good approximations to the req uired
solutions ΨSO
I. In particular, if the {Ψ(0)SO
I}set is obtained from a preliminary series of
SO-CI calculations of the studied states, this can be superfl uous.
As an alternative to the above selection schemes with respec t to the PT energy, the PT
expressions for the CI coefficient of a trial SE- or DE-SAF can b e also explored. Applying
the above PT analysis to the case of the Ψ(0+1′)SO
I >reference state, a Φ(1′′,2)ΛS
J SAF is
selected if its CI coefficient C(1′′,2)ΛS
J satisfies the inequality
|C(1′′,2)ΛS
J | ≥Cmin,
where Cminis the selection threshold for the CI coefficients and
C(1′′,2)ΛS
J =<Φ(1′′,2)ΛS
J |H|Ψ(0+1′)SO
I >
E(0+1′)SO
I −E(1′′,2)ΛS
J, (38)
is the first-order PT value for the CI coefficient of a tested SAF which is not included in the
subset of the Φ(1′)ΛS
J reference SE-SAFs.
Such a means of selection can be preferable if those properti es of primary interest cannot
be calculated from potential energy curves or surfaces. Mor eover, the PT selection with
respect to both the energy and the CI coefficients can be applie d simultaneously if the
properties are of different nature.
V. CALCULATIONS
In the CI calculations of Tl and TlH we used the MRD-CI package [24] combined with
the SO selection codes based on the scheme described above. O ur test calculations have
shown that spin-orbit selection is very helpful for prepara tion of appropriate sets of Mains
and for reducing effort in the final CI calculations with an opt imal set of selected SAFs.
A. Spin-orbit splitting in the ground state of Tl
Calculations for the Tl atom were performed to optimize the b asis set and the level shift
GRECP parameters for the 21e/8fs-GRECP, i.e. the 21 electro n GRECP with 8 electrons
occupying the frozen OC pseudospinors, 5 s1/2and 5p1/2,3/2. The quality of the generated
basis set is analyzed by calculating the2Po
1/2−2Po
3/2splitting for the ground state.
The optimal basis set was selected in a series of MRD-CI calcu lations for Tl (with different
sets of primitives and numbers of contracted s, p, d, f andgfunctions) to minimize the sum
of energies for the ground2P1/2and2P3/2states. In these calculations, the SAFs were
selected in the2B1u,2B2u, and2B3uirreps of the D2hgroup (nonrelativistic-type degenerate
2Pground states belong to these irreps) because these doublet s are strongly mixed by the
SO interaction, resulting in the splitting of the ground2Pstate. We have found that two g
functions should be added to the basis set, giving a contribu tion of about 9000 cm−1to the
2Poground state total energies. The resulting [4,4,4,3,2] bas is set and GRECP parameters
for Tl can be found on http://www.qchem.pnpi.spb.ru .
12For the [4,4,4,3,2] basis set we have also performed MRD-CI c alculations including SAFs
from the2Auirrep and SAFs with quartet multiplicity (4B1u,4B2u,4B3u, and4Au). In our
calculations with different basis sets, their contribution s have decreased the SO splitting by
about 170 cm−1and the total energy by about 2000 cm−1. One can see from Table I that this
decrease is mainly caused by the p- and d-components which arise from reexpansion of the
leading spinor configuration in terms of the spin-orbit confi gurations. For good accuracy we
can recommend the inclusion of Λ′|S±1|-irreps for the calculation of states having leading
configurations in Λ S-irreps.
In Table I some of our final MRD-CI results are collected toget her with the atomic
relativistic coupled-cluster (RCC) results [6] obtained w ith a very large basis set. In these
MRD-CI calculations altogether 627 Mains in three basic irr eps and about 100 Mains in
five additional irreps were involved and SE-SAFs were automa tically generated for three
Mains to prepare the reference {Ψ(0+1′)SO
I }3
I=1states. Relatively small thresholds, T1=0.03
andT2=0.01 µEh, are used in the final runs with the [4,4,4,3,2] basis set (for theT=0
threshold and full-CI extrapolations), thus selecting res pectively about 190000 and 450000
SAFs altogether.
One can see that the best SO splitting calculated in the prese nt work underestimates
the experimental result [31] by about 400 cm−1(recall that additionally about 90 cm−1
is due to the Breit interaction [7]). Analyzing our previous GRECP/RCC calculations of
Hg [37] it can be concluded that this occurs due to the neglect of the OC-V correlations
with the OC 5 pand 4fshells, and to a lesser extent with 5 srather than due to the atomic
basis set incompleteness, the GRECP errors or the restricte d CI approximation. The OC-
V correlation (contribution to the total energy) in Tl and Hg will have the same order of
magnitude for respective pairs of correlated electrons (sp inors).
We also studied the reliability of the linear T→0 extrapolation procedure currently used
in the MRD-CI code. In the final results of our MRD-CI calculat ions the corresponding
correction gives the highest contribution to the cumulativ e error. So this is a bottleneck of
the present Tl and TlH calculations with the large number of M ains.
B. Spectroscopic constants of the ground state in TlH
The explicit treatment of 5 delectrons in precise TlH (TlX) calculations is necessary no t
only due to the strong correlation between these and the vale nce electrons of Tl, but also
because of the substantial influence of relaxation-polariz ation effects in this shell on the bond
formation. This cannot be very accurately taken into accoun t by employing a polarization
potential [32,33] in combination with, e.g., 3e-RECPs [14, 15]. The influence of other atoms
(X) in a TlX molecule on the 5 p, 5sand 4fshells of Tl is significantly smaller and can
be neglected if an accuracy of a few hundreds of wavenumbers f or the excitation energies
of low-lying states is sufficient. We neglected their contrib utions in calculation of the TlH
spectroscopic constants.
In calculating spectroscopic properties for the TlH ground state (Table II) we used the
contracted [4,4,4,3,2] basis set for thallium discussed ab ove and the [4,3,1] set for hydrogen
(seehttp://www.qchem.pnpi.spb.ru ) contracted from the primitive (6,3,1) gaussian basis set
of Dunning [38]. The SAFs were selected in the1A1,3B1,3B2and3A2ΛS-irreps of the
13C2vgroup because the triplet states are most strongly admixed b y the SO interaction to the
nonrelativistic1A1(or1Σ+inC∞v) ground state producing the relativistic 0+ground state
in the double C∗
∞vgroup.
We have performed three series of TlH calculations for 16 int eratomic distances. In these
runs, the Ψ(0+1′)SO
0 reference SO states are generated with the MRD-CI code by dia gonalizing
Hfor the set of Mains and the SE-SAFs which are automatically s elected with respect to the
single configuration SCF ground state (calculated with the S O-averaged GRECP), giving a
contribution of more than 90 % to the final wave function.
The first run is used for preparing an optimal set of Mains for t he second series of SO-CI
calculations. Only one SCF configuration which has the lowes t energy in each Λ S-irrep is
included into the subspace of Λ SMains and, consequently, the SO reference state consists of
these SCF configurations and the automatically selected SE- SAFs with respect to the SCF
configuration from the1A1irrep.
Those SAFs were selected as Mains for the second run which had the highest CI coeffi-
cients in the first run. As a result, 37 Mains in all irreps toge ther are employed in the second
run. Relatively small thresholds, T1=1.0 and T2=0.1µEh, are used in the second run (for
theT→0 extrapolation [24]), thus causing about 20000 and 85000 SA Fs to be selected in
the ΛS-irreps altogether.
In the most computationally consuming third run (with the se t of Mains consisting of the
SAFs having the largest CI coefficients in the wave function fr om the second run), about 320
Mains are used altogether and the thresholds are set at T1=0.1 and T2=0.05 µEh. About
70000 and 130000 SAFs, respectively, were used in the Λ S-irreps altogether in the final
SO-CI calculations.
One can see from Table II that the basis set superposition err or (BSSE) (see [39] and
references) must be taken into account for an accurate compu tation of spectroscopic con-
stants. The BSSE was studied in the Tl+ion calculations for the same interatomic distances
as in TlH and estimated also in the Tl−calculations for three distances, i.e. with the ghost
H atom. The same molecular basis set as in TlH was used for both the Tl and H atoms.
The contribution from BSSE to the total energy is decisive fo r the 5 d10and 6s2shells con-
sidered in the case of Tl+, while its changing due to addition of the 6 pelectrons (which are
bonding in TlH) can be considered as relatively small, becau se the difference in BSSE for
Tl+and Tl−is not significant in comparison with other errors. In the cal culations of the
spectroscopic properties with the counterpoise correctio ns (CPC), the calculated TlH points
on the potential curve were corrected with the calculated BS SE for Tl+, i.e. for the 5 d,6s
shells taken into account.
One can see that after applying the T=0, FCI, and counterpois e corrections, the calcu-
lated properties are in very good agreement with the experim ental data both in the second
and third runs. The accuracy obtained is notably better than for other existing results for
TlH (and not only for those presented in Table II). We suggest , however, that the very good
agreement of the calculated Dewith the experimental value can be fortuitous and the “real”
(full CI) value can be notably different from the listed one be cause of the approximations
made.
14VI. RESUME
The SO splitting in the ground2Pstate of Tl is calculated by the MRD-CI method
with the 21e-GRECP when 5 d10,6s2and 6p1electrons are correlated and the 5 s2and 5p6
pseudospinors are frozen in the framework of the level shift technique. A [4,4,4,3,2] basis set
is optimized for Tl and an underestimation of about 400 cm−1is found for the SO splitting
as compared with the experimental data.
Further improvement of the accuracy can be attained when cor relations with the outer
core 4 f,5pand 5sshells of Tl and Breit effects are taken into account. We expec t that
this can be efficiently done in the framework of the “correlate d” 21e/8fs-GRECP version
in which 13 electrons are treated explicitly as in the presen t calculation. The inclusion of
h-type functions is also desirable, as has been demonstrated for Hg in Ref. [37].
Fourteen electrons are correlated in the calculation of spe ctroscopic constants for the 0+
ground state of TlH and very good agreement with the experime ntal data is found.
The developed spin-orbit selection scheme and code are demo nstrated to be efficient
when large sets of basis functions and reference configurati ons are required in high-precision
electronic-structure calculations.
ACKNOWLEDGMENTS
This work was supported by the DFG/RFBR grant N 96–03–00069 a nd the RFBR grant
N 99–03–33249. AVT is grateful to REHE program of the Europea n Science Foundation for
fellowship grants (NN 14–95 and 22–95) to visit the laborato ry of one of us (RJB), where
part of the work was done. We are thankful to K. Shulgina and T. Isaev (PNPI) for writing
some codes used for automatic generation of Mains.
We are grateful to Dipl.-Ing. H.-P. Liebermann for the help i n combining the MOLGEP
and MRD-CI codes. We are also grateful to Dr. G. Hirsch (decea sed) for his kind hospitality
and invaluable help during visits to Wuppertal by AVT and NSM .
The main part of the present calculations was carried out at t he computer center of the
Bergische Universit¨ at GH Wuppertal. JECS codes developed by PNPI quantum chemistry
group were used for remote control of the calculations.
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17TABLES
TABLE I. Calculations of the spin-orbit splitting of the2Poground state in Tl (the [ ...],5d
spinors are frozen from the SCF calculation of the nonrelati vistically averaged [ ...]6s26p1configu-
ration).
Method SO splitting in cm−1
Spinor basis sets: [7,7,5] [7,7,5,3] [7,7,5,3,1]
81e-DF/3e-CI [13] 7129 7182 7206
21e/18fs-GRECP/3e-CI [13] 7133 7187 7211
Spin-orbit basis sets: [4,4,4] [4,4,4,3] [4,4,4,3,2]
21e/18fs-GRECP/3e-MRD-CI (Full CI)
(2B1u,2B2u,2B3uirreps of D2h) 7305 7373 7398
(...+2Au,4B1u,4B2u,4B3u,4Au) 7133 7205 7230
21e/8fs-GRECP/13e-MRD-CI + T=0 + FCI
(2B1u,2B2u,2B3uirreps of D2h) 7332 7222 7517
(...+2Au,4B1u,4B2u,4B3u,4Au) 7146 7044 7380
Spinor basis set: [35,27,21,15,9,6,4]
81e-DF/35e-RCC [6] 7710
Experiment [31] 7793
18TABLE II. GRECP/MRDCI calculations of the spectroscopic co nstants for the ground state
of TlH.
Re ωe De
Method (˚A) ( cm−1) ( eV)
SOCIEX: Tl [8,8,5,2] + H [4,3,1]
(Rakowitz & Marian, 1997 [8]) 1.86 1386 2.13
13e-RECP/SOCI: Tl [4,4,4,1] + H [4,2]
(DiLabio & Christiansen, 1998 [9]) 1.912 1341 1.908
13e-REP/KRCCSD(T): Tl [4,5,5,1] + H [3,2]
(Leeet al., 1998 [10]) 1.910 1360 2.02
21e-REP/KRCCSD(T): Tl [4,5,5,1] + H [3,2]
(Hanet al., 2000 [11]) 1.877 2.00
21e/8fs-GRECP/14e-MRD-CI Tl [4,4,4,3,2] + H [4,3,1]
(Present calculations)
37 Mains, T=0.1 1.858 1481 2.03
———”——— + CPC 1.872 1446 1.984
———”——— + T=0 + FCI 1.858 1453 2.10
———”——— + T=0 + FCI + CPC 1.872 1410 2.026
320 Mains, T=0.05 1.866 1408 2.23
———–”———— + T=0 + FCI 1.858 1449 2.124
———–”———— + T=0 + FCI + CPC 1.870 1420 2.049
Experiment (Grundstr¨ om & Valberg, 1938 [34]) 1.866a1390.7 2.06
Experiment (Urban et al., 1989 [36]) 1.872b1391.3
aHuber & Herzberg (1979) [35] have published value 1.87 ˚Awhich can be obtained from the
rotational constant Be.
bThis value is calculated by us from Be.
19 |
THE DARHT PHASE 2 LINAC*
H. L. Rutkowski, L. L. Reginato, W.L. Waldron, K. P. Chow, M. C. Vella, W. M. Fawley,
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720
R. Briggs, Science Applications International Corp., 7041 Koll Center Parkway, Suite 260
Pleasanton, CA 94566
S. Nelson, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550
Z. Wolf, Stanford Linear Accelerator Center, 2575 Sand Hill Rd., Menlo Park, CA 94025
D. Birx, Science Research Laboratory, 15 Ward Street, Somerville, MA 02143
Abstract
The second phase accelerator for the Dual Axis
Hydrodynamic Test facility (DARHT) is designed toprovide an electron beam pulse that is 2µs long, 2kA, and20 MeV in particle energy. The injector provides 3.2MeV so that the linac need only provide 16.8 MeV. Thelinac is made with two types of induction acceleratorcells. The first block of 8 cells have a 14 in. beam pipecompared to 10 in. in the remaining 80 cells. The otherprincipal difference is that the first 8 cells have reducedvolt-sec in their induction cores as a result of a largerdiameter beam pipe. The cells are designed for veryreliable high voltage operation. The insulator is Mycalex.Results from prototype tests are given including resultsfrom solenoid measurements. Each cell contains asolenoid for beam transport and a set of x-y correctioncoils to reduce corkscrew motion. Details of tests todetermine RF mode impedances relevant to BBUgeneration are given. Blocks of cells are separated by“intercells” some of which contain transport solenoids.The intercells provide vacuum pumping stations as well.Issues of alignment and installation are discussed.
1 INTRODUCTION
The Dual-Axis Radiographic Hydrodynamic Test(DARHT) facility at Los Alamos National Laboratory(LANL) is a pulsed X-ray radiography machine for thenational Stockpile Stewardship Program. It consists of
two linear accelerators oriented at 90 ° to each other and
with both pointed at the same target. The first linac(Phase 1) is a short pulse single shot accelerator usingferrite induction cells. The Phase 2 linac is a long pulselinac using Metglas induction cells. The beam at the exitof the accelerator is required to be 2 kA of electrons at 20MeV and with a 2 µs long flat current pulse. Downstreamkicker systems being designed and built by LawrenceLivermore National Laboratory (LLNL) will select 4short pulses out of the 2 µs macropulse for delivery to theX-ray conversion target. The physics design issues thatdominate the design are transverse RF mode impedancesand Q's that can generate BBU, transport solenoid field
alignment and energy flatness both of which contribute tocorkscrew motion of the beam centroid, vacuum quality,and emittance growth. The engineering issues thatdominate the design are packing enough volt-seconds ofinduction core material into a cell within the pre-setbuilding size constraints, high voltage reliability of thesystems, mechanical stability and alignment, andproviding sufficient space for diagnostics, sufficientmaintenance capability, and adequate vacuum pumping.The injector provides a 3.2 MeV, 2kA beam to theaccelerator in a pulse with 2.1 µs flat current and a 500 nsrise time (0-99%). The linac consists of 88 acceleratorcells adding 16.8 MeV to the beam energy. The first 8cells are called "injector cells" and are designed totransport the entire injector pulse to a Beam-head CleanUp Zone (BCUZ), which chops off the pulse rise time.This beam element is designed and built by LANL. Therest of the linac consists of 80 "standard cells", which arein blocks of 6 except for a final block of 8. Thecellblocks are separated by removable elements calledintercells. The beam exiting the linac should havetransverse motion of the centroid from all causes less than10% of the beam radius (5 mm) and normalized
emittance of no more than 1000
π mm-mrad.
2 ACCELERATOR INDUCTION CELLS
A drawing of the standard accelerator cell is shown in
fig. 2. The beam tube has an inner diameter of 10 inchesas chosen from preliminary BBU and beam lossconsiderations. The induction cores are Honeywell(Allied Signal) 2605-SC Metglas with a specified v-s
Figure 1 - DARHT Phase 2 Linac
*Work supported by the US Department of Energy under contract
DE-AC03-76SF00098product of 0.48 v-s per cell. Actual delivered cores are
providing up to 0.51 v-s. Each cell contains 4 pancakesub-cores and the entire ensemble is driven as a singleunit by a single pulser. The cores are assembled into thealuminium cell housing and then vacuum pumped beforethe cell is filled with Shell Diala high voltage oil. Thebeam tube is 304 stainless steel and is hollow toaccommodate a transport solenoid immersed in a water-cooling jacket.
The solenoid is a wound copper coil with polyester-
amide-imide insulated conductor wet-wound and paintedin Castall 301 epoxy. Each coil has 2400 turns in 12layers and can be driven to provide 0.2T field on thebeam tube axis within the limits of the cooling systemprovided by LANL. The solenoids will be operated in apulsed mode except for the injector cell solenoids.
The insulator is Mycalex (dielectric constant 6.7) and is
a structural member of the cell. It is a mica-glasscomposite chosen for its high strength without brittlenessand its excellent vacuum and high voltage properties. Itshigh dielectric constant was initially thought to presentTM mode impedance problems and considerable designeffort using the AMOS code was necessary to find anoptimal structure and ferrite damping design. Outside thebeam tube and in the oil-filled section, flexible PC boardtype corrector coils, both vertical and horizontal, arewrapped around the tube. The entire cell is supported bya 6 strut mounting system developed at LawrenceBerkeley National Laboratory (LBNL) for the AdvancedLight Source. It uses differential screws to allowprecision movement of the heavy cell (7 tons) duringalignment. The cells are vacuum-sealed to each otherusing a clamp ring-bellows system that allows movementfor alignment. Each accelerator cell will be fiducializedwith respect to the magnetic axis of its solenoid using astandard stretched wire technique by LANL. In that way,each cell can be aligned with respect to the ideal beamaxis as it is installed in the linac. The magnetic axis ofeach solenoid is to be within 400 µ of the ideal beam axis
(3
σ) for offset and within 3 mrad (3 σ) for tilt in both
Euler angles. The injector cells are a scaled version of the
standard cells. The beam tube ID was enlarged to 14 in.while keeping the outer cell dimensions constant. The
main effects are to enlarge the beam tube, solenoid, andinsulator while sacrificing Metglas volume on the insideof the cores. Consequently, the injector cells operate at175 kV pulse voltage compared to 193 kV for thestandard cells to maintain constant pulse length. Thebeam tube enlargement allows full transmission of theinjected pulse to the BCUZ and takes advantage of thewell known scaling (1/b
2 where b is the tube radius) for
BBU in pillbox cavities(1). Three prototypes were
constructed as part of the design. The first, without asolenoid, was built in two forms for high voltage testingand RF transverse mode measurements at LBNL. Thesecond was an earlier version of the injector cell designwhile the third was the final standard cell design. Bothwere sent to the THOR facility at LANL. The cells haveshown excellent high voltage performance and eachstandard production cell is tested at 200 kV for 2000shots before shipping.
3 TRANSVERSE MODE MEASUREMENTS
Measurements using both the standard TSD(2) method
for highly damped cavities and a new two-wire excitation- loop pickup method invented by two of the authors(Briggs and Birx) were used. The optimal dampinggeometry and ferrite placement was chosen by acomputer design activity using AMOS. The standard cellmeasurements were carried out initially on the firstprototype cell and finalized on the first productionstandard cell. The dipole transverse mode frequencieswere found to be 170, 230, and 577 MHz with realimpedances of 182, 259, and 283 ohms per mrespectively. These values were found to be sufficientlylow by the integrated beam dynamics team. The twomethods were found to agree within experimental error sothat the two wire-loop technique was used by itself on theinjector cell. The injector cell frequencies were 152 and200 MHz shifted from the standard cell and theimpedances were 152 and 149 ohms per m, well withinthe limits established by the beam dynamics studies. Thedamping ferrite used was CMI Technology, tile material,N2300, distributed azimuthally around the beam tube onthe oil side in pie sections, which were separated to avoidsaturation by the driver pulse current.
4 MAGNETIC CHARACTERIZATION
MEASUREMENTS
The solenoids were designed to be fine wire magnets
using many turns. This design eliminates the need topackage an internally cooled bulky conductor in a veryconfined space, reduces power consumption, andprovides very high quality field. Several (10) solenoidshave been characterised at Stanford Linear AcceleratorCenter (SLAC) prior to their installation in acceleratorcells. The solenoids are first installed in their beam tube
Figure 2 - Standard Accelerator Cellhousings, which are welded shut. They are then sent to
SLAC where they are aligned using standard SLACmeasurement procedures and their fields are measured. Arotating coil probe method is used, which gives data onthe transverse field components as a function of distancealong the axis in the solenoid. This data is then processedin a fitting routine at LBNL to find the effective magneticaxis of the solenoid. A few solenoids have also beenchecked for sextupole fields, which have turned out to benegligible. If a large effective tilt (>1.0 mrad) is detectedin one run, the solenoid is realigned and rechecked. Thelargest effective tilt between the magnetic and mechanicalsolenoid axes in the set of measurements taken with bothstandard cell solenoids and injector cell solenoids is 0.70mrad. The range is 0.03-0.7 mrad. After fieldmeasurements, the solenoids are returned to LBNL forassembly into accelerator cells.
5 INTERCELLS
The cellblocks in the linac are separated by intercells,which provide a removable element that allows slidingthe interlocking accelerator cells apart for removal in caseof maintenance. They also allow for diagnostic ringremoval at each cellblock, insertion of interceptingdiagnostics, and vacuum pumping. A cross-sectiondrawing of an azimuthally symmetric intercell is shownin fig. 3. The diagnostic ring, designed by LANL, isshown between the downstream bellows and the housingof the downstream cell. The intercell body provides thepositive side of the accelerating gap for the upstream cellwhose beam pipe and insulator are shown. This figureshows an intercell with a transport solenoid. Only thefirst 4 intercells have these solenoids, which are neededfor beam transport matching to reduce emittance growthand BBU. In the other 5 intercells, the solenoid is absentand the extra axial space is used for additional pumpingspeed. Twelve current return bars complete the circuitacross the pumping throat opening to smooth the B fieldof the return current. Each opening is filled with 95%open area stainless steel mesh to shield the intercell cavityfrom beam generated RF. The intercells are mounted onthe upstream cell with turnbuckle struts and the weight issupported by a single point suspension "pogo stick". Inthis way, the intercell can be independently aligned to theideal abeam line. Vacuum modelling, backed up byempirically determined outgassing rates on actualproduction cells, yields an average background pressureon axis of < 1.5x10
-7 Torr between the BCUZ exit and the
exit of the first four cell blocks of standard cells. It isapproximately 0.8x10
-7 Torr after that due to the higher
pumping speed available after the intercell solenoids areeliminated. This should be adequate to eliminate ion hoseinstability.
6 CONCLUSION
A high quality design for DARHT Phase 2 linacaccelerator cells has been achieved and cells are inproduction. These cells have been shown to haveexcellent high voltage performance even with some beamloss, and measurements have shown the RF modeimpedances relevant to BBU are sufficiently low toensure meeting design requirements. The backgroundpressure has been reduced by good vacuum systemdesign to safe limits. Also, a very high quality solenoidfield has been achieved within operational and coolingsystem constraints. Finally, a design has been achievedwhich delivers the energy and pulse length required inspite of serious space restrictions.
REFERENCES
[1] R. J. Briggs, D. L. Birx, G. J. Caporaso, V. K. Neil,
and T. C. Genoni, Particle Accelerators, 18, 1985,
pp 41-62
[2] L. S. Walling, D. E. McMurry, D. V. Neuffer, and H.
A. Thiessen, Nuc. Inst. Meth., A281, 1989, pp433-
447Figure 3 - Intercell |
arXiv:physics/0008157v1 [physics.acc-ph] 18 Aug 2000OPTIMIZEDWAKEFIELDCOMPUTATIONS USING A NETWORK
MODEL∗
J.-F. Ostiguy,K.-Y. Ng,FNAL, Batavia, IL 60510,USA
Abstract
Duringthe courseof the last decade,travelingwave accel-
eratingstructuresforafutureLinearColliderhavebeenth e
objectofintenseR&Defforts. Animportantproblemisthe
efficient computation of the long range wakefield with the
ability to include small alignment and tuning errors. To
that end, SLAC has developed an RF circuit model with
a demonstrated ability to reproduce experimentally mea-
sured wakefields. The wakefield computation involves the
repeated solution of a deterministic system of equations
over a range of frequencies. By taking maximum advan-
tageofthesparsityoftheequations,wehaveachievedsig-
nificant performance improvements. These improvements
make it practical to consider simulations involving an en-
tire linac ( ∼103structures). One might also contemplate
assessing, in real time, the impact of fabrication errors on
the wakefieldasan integralpartofqualitycontrol.
1 INTRODUCTION
During the course of the last decade, SLAC has been con-
ductingR&Donnewgenerationsofacceleratingstructures
for a futuremachine,the Next LinearCollider (NLC).The
culmination of this work is the Damped Detuned Struc-
ture(DDS).Sinceitisdifficulttodissipatedeflectingmode
power without also dissipating accelerating mode power,
this structure achieves high efficiency (shunt impedance)
byrelyingprimarilyondetuningtoproducefavorablephas-
ingofthedipolemodestomitigatethedipolesumwake. To
prevent the partial re-coherence of the long range wake, a
small amount of damping is provided by extracting dipole
mode energy through four manifolds which also serve as
pumpingslots.
A linear collider is a complex system and detailed nu-
merical simulations are essential to understand the impact
of differentrandomand/orsystematic structurefabricati on
errors on beam quality. Assuming a (loaded) gradient of
50 MV/m anda length of 2 m, each of the two armsof a 1
TeVinthecenter-of-massNLCwouldbecomprisedofap-
proximately 1000structures. To simulate the effect offab-
rication errorson emittance growth, one needs to compute
onewakeperstructure;consequently,thereisconsiderabl e
interest in performing these computations as efficiently as
possible. AtypicalNLCstructurecomprises206cells. Be-
cause of the large number of nodes, it impractical to re-
sort to standard finite element or finite difference codes to
computethewake. Tomakecomputationsmanageable,the
SLAC group has developed an RF circuit model. Despite
itslimitations,predictionshaveproventobeinremarkabl e
∗Work supported by U.S.Department of Energyagreement with experimental results. However, until now,
thewakecomputationsremainedtooslowtomakethesim-
ulation of a full linac practical. In this paper, we describe
algorithmicmodificationsthathaveledtoacodeachieving
threeordersofmagnitudeimprovementoverpreviouslyre-
portedperformance.
2 CIRCUITMODELFORDDS
InanRFcircuitmodel,Maxwell’sequationsarediscretized
usingaloworderexpansionbasedonindividualclosedcell
modes. The result is a system of linear equations that can
convenientlyberepresentedbyacircuitwherevoltagesand
currents are associated with modal expansion coefficient
amplitudes. A model suitable for the computation of the
fields excited by the dipole excitation of a detuned struc-
ture was developedby Bane and Gluckstern [1]. The con-
ceptofmanifolddampingwaslaterintroducedbyKroll[2]
and the circuit model was extended by the SLAC group to
include this feature [3]. The result is shown in Figure 1.
Thecorrespondingequationscanbeputintheform
Figure 1: Circuit model for Damped Detuned Structures.
Thethickhorizontallinesrepresenta transmissionline.
H−1
f2I H x 0
HT
xˆH−1
f2I−G
0 −G R
a
ˆa
A
=1
f2
b
0
0
(1)
where fis the frequency and Iis a unit diagonal. The
submatrices H,Hx,GandRareN×Nwhere Nis the
number of cells ( N= 206for the SLAC structure). H
andˆHdescriberespectivelytheTM 110-likeandTE 111-like
cell modecoupling, HxrepresentstheTE - TM crosscou-
pling, Rdescribes the manifold mode propagation and G
describes the TE-to-manifold coupling. The vectors a,ˆaare the normalizedloopcurrents (a=i/√Cn)forthe TM
and TE chains and Vis the normalized manifold voltage
at each cell location. Finally, the right hand side brepre-
sentsthebeamexcitation. Sincetheboundaryconditionsat
the cell interfacesimposethat the TM andTE components
must propagate in opposite directions, only the TM cell
modesareexcitedbythebeam. Thedipolemodeenergyis
coupledoutelectricallytothemanifoldviasmall slots; th e
TEcomponentofthefieldisthereforecapacitivelycoupled
to the manifold. Note thatthe manifoldisrepresentedbya
periodicallyloaded transmissionline for which only nodal
equationsmakesense,resultingin amixedcurrent-voltage
formulation.
3 SPECTRAL FUNCTION
Computing the wake of DDS structures involves solv-
ing (1) over the structure’s dipole mode frequency band-
with. A longitudinal dipole impedance is first obtained
by summing the cell voltages (in the frequency domain)
with appropriate time delays. The transverse impedance
is subsequentlyderived by invokingthe Panofsky-Wentzel
theorem. The circuit approach to wake computation in-
troduces a small non-causal, non-physical component to
the wake w(t)which can be suppressed by considering
[w(t)−w(−t)]u−1(t)instead. The sine transform of
this function, proportional to the imaginary part of the
impedance, is known as the spectral function S(ω). In
the context of circuit-based wake computations, S(ω)is
a more convenient quantity to compute than the dipole
(beam)impedance.
4 SPARSELINEAR EQUATIONS
IntheDDScircuitmodel,eachcellcouplesonlytoitsnear-
est neighbors. The resulting matrix is sparse and com-
plex symmetric (a consequence of electromagnetic reci-
procity). Computing the spectral function involves solv-
ing a sequence systems of linear equations. At each step
in frequency, the coefficient matrix changes slightly while
its sparsity structure remainsidentical. In addition, a go od
starting approximation to the solution for any frequency
step isprovidedbythesolutionfromthepreviousstep.
4.1 IterativeMethods
An algorithm suitable for symmetric complex systems
is the so-called Quasi Minimal Residual (QMR) algo-
rithm [4]. This algorithm is a relative of the well-known
conjugate gradient method which seeks to minimize the
quadratic form (Ax−b)T·(Ax−b). The QMR algo-
rithmminizimizesadifferentquadraticform;inbothcases
the key to rapid convergenceis suitable “preconditioning”
of the system Ax=bwith an approximate and easy to
computeinverse. TestswereperformedwithRDDS circuit
matricesusingstandardincompletefactorizationprecond i-
tioners; but the results were somewhat disappointing. It
is believed that with a suitable preconditioner, the methodcan be competitive; however, efforts to identify one were
abandonedafter a direct techniqueprovedto be more than
satisfactory.
4.2 DirectMethods
Directalgorithmsareessentiallyallrelativesoftheelem en-
tary Gaussian elimination algorithm, where unknowns are
eliminatedsystematicallybylinearcombinationsofrows.
A crucial point is that the order in which the rowsof
the matrix are eliminated has a direct impact on com-
putationalefficiency since a differentorder implies dif-
ferent fill-in patterns1.In principle, there exists an elim-
ination order that minimizes fill-in, which is notthe same
as the most numerically stable ordering. In some cases, it
is even possible to find an ordering that produces no fill-
in at all. Although the determination of a truly optimal
ordering is an NP-complete problem, it is possible using
practical strategies to find orderings that result in signifi -
cant computational savings. The most successful class of
orderingstrategiesareso-called“local”strategiesthat seek
to minimize fill-in at each step in the elimination process
regardlessoftheirimpactata laterstage.
The Markowitz Algorithm A good local ordering
strategy is the Markowitz algorithm. Suppose Gaussian
elimination has proceeded through the first kstages. For
each row iin the active (n−k)×(n−k)submatrix, let
r(k)
idenotethenumberofentries. Similarly,let c(k)
jbethe
numberof entriesin column j. The Markowitz criterionis
toselectaspivottheentry a(k)
ijfromthe (n−k)×(n−k)
submatrixthatsatisfies
min
i,j(r(k)
i−1)(c(k)
j−1) (2)
Using this entry as the pivot causes (r(k)
i−1)(c(k)
j−1)
entry modifications at step k. Not all these modifications
will result in fill-in; therefore, the Markowitz criterion i s
actually an approximationto the choice of pivot which in-
troducesthe least fill-in.
5 CODEDESCRIPTION
Ourcodeisbasedonthespectralfunctionmethodanduses
Markowitzorderingtosolvethecircuitequationsinthefre -
quencydomain. Comparedtotheprocedureoutlinedin[3],
the following changes have been made: (1) The manifold
voltage Aisnotseparatelyeliminated,inordertopreserve
sparsity. (2) Once the system (1) is solved, the loop cur-
rents are knownand the cell voltagescan be obtainedby a
simple matrix multiplication. There is therefore no need
toforman inverse [5].
1The elimination process creates non-zero entries at positi ons which
correspond to zeros in the original coefficient matrix. Thefi ll-in is theset
of all entries which were originally zeros and took on non-ze ros value at
any step of the elimination process.Two additionalremarksare in order. The process of de-
termining the Markowitz ordering can by itself be time-
consuming; however, since the structure of the RDDS
matrixremainsthesameateverystepin frequency,the
ordering needstobe determined only once . The relative
magnitudes of the equivalent circuit matrix entries do not
changeverysignificantlyoverthefrequencybandoccupied
by the dipole modes. This insures that the Markowitz
ordering remains numerically stable for all frequency
steps.
Implementationsof theMarkowitzalgorithmarewidely
available. We used SPARSE [6], a C implementation that
takes advantage of pointers to store the coefficient matrix
as a two-dimensional linked list. To each non-zero entry
correspondsa list node. Each node in turn points to struc-
ture which comprises the numerical value of the entry, its
two-dimensionalindicesandapointertoanupdatingfunc-
tion. A linked list makes sequential traversal of a row or
a column of the matrix efficient; however, random access
is expensive. To update the matrix at each frequency step,
wesequentiallyscantheentirelistandcallanupdatefunc-
tionbyindirectionusinga pointerstoredwithineachentry
structure.
The RDDS circuit matrix is not only sparse, it is also
symmetric. The SPARSE package does not exploit this
structurebecausethestandardeliminationprocessdestro ys
symmetry. We note that the Markowitz scheme can be ex-
tentedina waythat preservessymmetry.
6 RESULTS
Our optimized wakefield code was used to compute the
wake envelope of the RDDS structure, using parameters
provided by SLAC. On a 550 MHz Pentium III (Linux,
GNU gcc compiler) a complete calculation of the wake
takes approximately14 seconds. This represents a gain of
roughly three orders of magnitude compared to the previ-
ously reported performance and allows the generation of
wakes for an entire linac in less than four hours. Output
from the code is presented in Figures 2 and 3. The results
are identicalto thoseobtainedbytheSLACgroup.
7 ACKNOWLEDGMENTS
TheauthorswouldliketoexpresstheirappreciationtoNor-
man Kroll, Roger Jones, Karl Bane, Roger Miller, Zhang-
Hai Li and Tor Raubenheimer for in depth technical dis-
cussions about various aspects of the RDDS technology.
They also would like to extend special thanks to Norman
Kroll and Roger Jones for generously sharing personal
notes, providingparametersfor the RDDS as well as sam-
ple wakefieldcomputations.
8 REFERENCES
[1] K. L.F. Bane and R.L. Gluckstern, Particle Accls., 42, p123
(1993)0102030405060708090
14 14.5 15 15.5 16 16.5Spectral Function [V/pC/mm/m/GHz]
Frequency [GHz]RDDS1 Spectral Function
Figure2: ComputedspectralfunctionfortheRDDS1struc-
ture.
0.0010.010.1110100
0 10 20 30 40 50 60 70 80Wake Function Magnitude [V/pC/mm/m]
Distance [m]RDDS1 Wake
Figure3: Computedwakeforthe RDDS1structure.
[2] N. Kroll, The SLAC Damped Detuned Structure: Concept
and Design , Proceedings of the 1997 PAC(1997)
[3] R.M.Jones etal., ASpectral FunctionMethodAppliedtothe
Calculation of the Wake Function for the NLCTA , Proceed-
ings of the XVIIILinac Conference (1996).
[4] R. W. Freund and N. M. Nachtigal, “QMRPACK a Package
of QMR Algorithms”, ACM Transactions on Mathematical
Software,Vol. 22, pp. 46–77, 1996.
[5] For a detailed discussion, consult J.-F.Ostiguy and K.- Y. Ng,
FermilabReport FN-698(inpreparation)
[6] Kenneth S.Kundert, A. Sangiovanni-Vincentelli, SPARSE, A
Sparse Equation Solver , Dept. of Electrical Engineering and
Computer Science,UC Berkeley(1988)
[7] I.S. Duff, A.M. Erisman, K.K. Reid, Direct Methods for
Sparse Matrices , Oxford UniversityPress(1986). |
HIGH POWER OPERATIONS OF LEDA*
L. M. Young, L. J. Rybarcyk, J. D. Schneider, M. E. Schulzea, and H. V. Smith, Los Alamos
National Laboratory, Los Alamos, NM 87544, USA
Abstract
The LEDA RFQ, a 350-MHz continuous wave (CW)
radio-frequency quadrupole (RFQ), successfullyaccelerated a 100-mA CW proton beam from 75 keV to6.7 MeV. We have accumulated 111 hr of beam on timewith at least 90 mA of CW output beam current. The 8-m-long RFQ accelerates a dc, 75–keV, ~106-mA H
+ beam
from the LEDA injector with ~94% transmission. Whenoperating the RFQ at the RF power level for which it wasdesigned, the peak electrical field on the vane tips is33 MV/m. However, to maintain the high transmissionquoted above with the CW beam, it was necessary tooperate the RFQ with field levels ~10% higher thandesign. The RFQ dissipates 1.5 MW of RF power whenoperating with this field. Three klystrons provide the2.2 MW of RF power required by the RFQ to acceleratethe 100-mA beam. The beam power is 670 kW. Some ofthe challenges that were met in accelerating a 100-mACW proton beam to 6.7 MeV, will be discussed.
1 INTRODUCTION
The LEDA RFQ [1,2] (see Figure 1) is the highest
energy operational RFQ in the world [3-8]. Some of theunique features implemented in this RFQ to meet thisgoal include:• It is over 9 wavelengths long, by far the longest
4-vane RFQ in the world.
• The transverse focusing at the RFQ entrance was
reduced for easier beam injection.
• An electron trap is placed between the final
focusing solenoid and the RFQ. The electron trapprevents the electrons in the beam plasma from
flowing into the RFQ. With the electron trap turned
off electrons flowing into the RFQ reduced themeasured current as much as 25% from the correctvalue.
• The aperture and the gap voltage in the
acceleration section are larger than in previous
RFQ designs.
• The transverse focusing at the RFQ exit is reduced
to match the transverse focusing strength in thecoupled-cavity drift-tube linac [9].
• It is the first RFQ to utilize resonant coupling
[10,11]. The RFQ is composed of four 2-m-long
RFQs resonantly coupled together. The RF fields
throughout its 8-m length are nearly as stable as thefields in the 2-m-long RFQs from which it iscomposed.
• RF power from 3 klystrons is coupled to the RFQ
through 6 waveguide irises.
To implement the reduced focusing strength at the
entrance of the RFQ and have adequate focusing in the
__________________________________________
*Work supported by the US Department of Energy.
a General Atomics, Los Alamos, NM 87544 USA.Figure 1: Line drawing of 8-m-long RFQ. This drawing shows the six RF-waveguide feeds used to power the RFQ;
two on Section B1 and four on section D1.interior of the RFQ, the transverse focusing parameter is
increased smoothly from 3.1 to 7.0 over the first 32 cm of
the RFQ. The focusing parameter is proportional to
2
0/rV where V is the voltage between adjacent vane tips
and r0 is the average aperture. The voltage is held
constant in this region and the aperture is reduced toincrease the focusing parameter. On entry, the beam is notyet bunched, allowing the use of weak transversefocusing. By the time the beam starts to bunch, thefocusing is strong enough to confine the bunched beam.The reduced focusing strength at the entrance means thematched beam size is larger than it would have beenwithout the reduced focusing strength. This allows thefinal focusing solenoid to be placed farther away from theRFQ. Without this feature proper placement of the finalfocusing solenoid is right at the RFQ entrance. With thefocusing solenoid 30-cm from the RFQ, both simulationsand experimental evidence indicate the beam becomesun-neutralized in the last 10 cm before the RFQ matchpoint. Moving the final focusing solenoid 15-cm from theRFQ counteracts the effect of the defocusing from thebeam’s space charge.
2 RFQ DESIGN
2.1 Acceleration Section
In a typical RFQ that has constant focusing strength
and constant gap voltage, as vane modulation increases toaccelerate the beam, the aperture shrinks and beam can belost on the vane tips. In an RFQ, as the energy rises thecell length increases and, for a given modulation, theaccelerating gradient decreases inversely with cell length.Since the maximum practical modulation is about 2, theRFQ would become very long if the gap voltage remainedconstant. To reduce beam loss and shorten the RFQ, wemaintain a large aperture, and increase the vane voltage.The increased gap voltage substantially increases the
accelerating field, thus shortening the RFQ. However,even with this increased gap voltage, eight meters oflength is required to accelerate the beam to 6.7 MeV.
2.2 Resonate Coupling
A conventional 8-m-long, 350-MHz RFQ would not be
stable. Small perturbations would distort the fielddistribution intolerably [10,11]. Therefore, four 2-m-longRFQs (labeled as segments A, B, C and D in Figure 1) areresonantly coupled to form the 8-m-long LEDA RFQ.The resonant coupling is implemented by separating thefour 2-m RFQs by coupling plates. An axial hole in thecoupling plate allows the vane tips to nearly touch. Thecapacitance between the vane tips of one RFQ and thenext provides the RF coupling between the 2-m-long RFQsegments. The gap between the vane tips at the couplingjoint is 0.32 cm. To minimize the effect of this gap on thebeam, the gap is placed so that as a bunched beam pulsepasses the gap, the RF electric field crosses zero. The RFfield is in phase in all four segments. The “couplingmode” has a strong electric field across the 0.32-cm gapand has one longitudinal node in each 2-m RFQ segment.The coupling mode’s longitudinal component of theelectric field transmits RF power, and it is this modewhich provides the stability to the fields. When thecoupling mode is strongly excited (by a perturbation forexample), a saw-tooth pattern can appear on the fielddistribution [11].
2.3 RFQ Fields
Figure 2 shows a measurement of the fields in the
RFQ. The fields were measured with the beadperturbation technique in the magnetic field region closeto the outer wall. In this measurement a bead is mountedon a plastic tape that is supported at the ends of the RFQand at the coupling plates. The plastic tape with the beadmoves on a pulley system and travels through all 4quadrants of the RFQ. The bead perturbs the frequency ofthe RFQ in proportion to the stored energy of themagnetic field displaced by the bead. The frequencyperturbation is measured versus bead position and therelative magnetic field strength is derived. In Figure 2, thebumps in the field are caused by local perturbations in themagnetic field near the slug tuners. A total of 128 tunersare used to “tune” the RFQ to the correct field distributionand frequency. The larger dips in the quadrupolemagnetic field that occur every 200-cm are caused by thecoupling plates. These dips and bumps do not appear inthe electric field on axis. The RFQ is tuned using a “leastsquares” fitting procedure that minimizes the differencebetween the measured fields and the design fields. Theslug tuner insertions are the parameters in this “leastsquares” fit.
The minimum aperture occurs about 1.4 meters into the
RFQ, at the end of the gentle buncher. This is also the
Figure 2: Bead perturbation measurement of the RF fields
in the RFQ. The Quadrupole fields are normalized to100%. The two residual dipole modes mixed with the
RFQ fields are typically less then 2%.location where the transverse current limit goes through a
minimum. Typically, the end of the gentle buncher is theRFQ choke point that determines the maximum currentthat can be accelerated (~200 mA for this RFQ) [7]. Thetheoretical current limit assumes ideal quadrupole fieldsand can only be used as a rough guide of the actualcurrent limit.
2.4 RFQ Design
The RFQ was designed with the code PARMTEQM
[12]. PARMTEQM is an acronym for “Phase and RadialMotion in a Transverse Electric Quadrupole; Multipoles”.This code includes the effect of higher-order multipoles inthe RFQ fields that are important in accurately predictingbeam loss. In addition, PARMTEQM requires a realisticdescription of the input beam to accurately simulate beamlosses in the RFQ when the input beam is not ideal.Simulations of the beam transport through the LEBT [3]with PARMELA [13,14] produces a more realisticdistribution of particles for input into the RFQ simulationcode than the ideal input distributions generated internallyby PARMTEQM. Simulations of the RFQ withPARMTEQM predict output beam emittances in therange from 0.16 to 0.22 mm-mrad depending on the inputdistribution. The simulations also predict the x and yemittances to be the same. The measured x and yemittance [15,16] are 0.25 and 0.31 mm-mradrespectively. There are also minor differences betweenthe predicted and measured Twiss parameters [15].
3 LEBT MODIFICATIONS
Until we added an electron trap described below to the
LEBT, our transmission measurements were inaccurate.The input current was less than the current out of theRFQ. Electrons (from the beam plasma) flowing into theRFQ reduced the positive proton current measured by thetoroid at the RFQ entrance. These electrons areresponsible for neutralizing the proton beam spacecharge. We used the computer code PARMELA to
perform a simulation of the beam traveling through theLEBT with 98% space charge neutralization, except forthe last 10 cm in front of the RFQ. There, we made thesimple assumption that the space charge neutralizationchanged linearly from 98% to 0 in 10 cm. The results ofthis simulation showed that the beam could not beproperly “matched” into the RFQ. Space charge causedthe beam to defocus so much in the last 10 cm that it nolonger converged as it entered the RFQ. This limited themaximum beam current out of the RFQ to only 89 mA,equal to the maximum pulsed current that we obtained byAugust 23, 1999. Simulations showed that if we installedan electron trap to prevent the electrons from flowing intothe RFQ and decreased the solenoid-to-RFQ distancefrom 30 to 15 cm, then the beam could be matchedproperly into the RFQ. The electron trap is a ring placedat the entrance of the RFQ. The potential on this ring,−1 kV, prevents low-energy plasma electrons from going
through it, but does not affect the 75-keV protons. Wemade these changes to the LEBT on August 28–29, 1999.
4 TRANSMISSION THROUGH RFQ
Using the calculated beam from the modified LEBT,
PARMTEQM predicts 93% transmission with the RFQoperating at the design field levels. This transmission isslightly less then the 95% transmission previouslypredicted with assumption that the space charge is 96%neutralized all the way to the RFQ match point [17].RFQTRAK [18], a code that calculates the 3D space andimage charge effects in an RFQ, agreed very well withPARMTEQM. The measured transmission has been ashigh as 94% at 100 mA when the RFQ fields are 10%above the design field strength. Figure 3 shows acomparison between the calculated and measuredtransmission as a function of field strength. This figurealso shows an anomalous drop in transmission at the endof a 300-µs long pulse when the RFQ RF field strength is
at or below the design field strength.
4.1 Ion Trapping in RFQ
Figure 4 shows the transmission in a 300- µs-long beam
pulse when the RFQ fields are near the design field level.
0.400.500.600.700.800.901.00
0.85 0.9 0.95 1 1.05 1.1 1.15
RFQ Cavity Field Amplitude (1.00 = design)RFQ TransmissionPARMTEQM 108 mA calculation
Total transmission (end of pulse)
Total transmission (first 100 µsec)
Figure 4: RFQ output beam current vs. time for a 300- µs-
long pulse at ~97% of the design RF-field level.Figure 3: RFQ transmission versus cavity field using a
300-µs-long-beam pulse. The anomalous transmission
drop occurs at slightly higher fields for longer pulses and
CW beams. The calculated transmission is for accelerated
beam onl y.0.020.040.060.080.0100.0
0 100 200 300 400
Time (µµµµs)
The transmission drops unexpectedly about 150 µs into
the pulse. As we raise the RFQ fields the transmissionstays high for longer times. With fields above ~105% ofdesign we no longer observe this drop, even for longpulses and CW operation.
We observe higher-than-expected activation near the
high-energy end of the RFQ, consistent with high-energybeam loss. If uncorrected, the frequency of the RFQ dropswhen it accelerates a high average beam current. The lostbeam impinges upon and heats the vane tips, causingthem to expand inward, reducing the gap. However, thewater-cooling system reacts by increasing the temperatureof the outer wall to increase the gap, thereby restoring theresonate frequency. Operating the RFQ with fields about10% above design greatly reduces the magnitude of thisbeam loss.
The total RF power does not appear to change when the
transmission drops. However, when the transmissiondrops, the RF fields appear to increase slightly in the lastmeter of the RFQ as though there was less beam loadingin that section [4].
We theorize that the RFQ fields are trapping low-
energy H
+ ions near the axis [20]. This extra charge
causes the beam size to increase reducing thetransmission. This is also consistent with the observationthat the beam would cause the beam stop collimator ringto glow whenever the vacuum in the RFQ exceeded about1-2 x10
-7 Torr. Our conjecture is the beam ionizes more
of the residual gas (mostly H 2) and the resulting H+ is
likely to be trapped in the RFQ bore. Preliminarysimulations with a modified version of PARMTEQM, inwhich an artificial space charge is introduced near theaxis, show beam distributions similar to that shown inFigure 5 (b).At design fields or lower enough beam may strike the
vane tips, creating H
+ ions that get trapped temporarily in
the beam channel. As this trapped charge accumulates thebeam becomes larger still until the transmission dropssuddenly. Following Ref. [20] to calculate the amount ofcharge that can be captured both transversely andlongitudinally cannot explain the large drop intransmission. However, because the RFQ is 8 m long, alarge amount of charge may be captured by the transversefocusing fields temporarily, provided there is a largeenough supply of ions. This charge will tend to flow outboth ends of the RFQ, but enough charge can accumulateto significantly affect the transmission.
5 THE RF POWER AND RESONATE
CONTROL SYSTEM
The low-level RF (LLRF) system [21] controls the
amplitude, phase, and frequency of the RF powersupplied to the RFQ. We used X-ray-endpointmeasurements [7,22] to calibrate the fields at 3 differentvalues of the setpoint in the field-control-module (FCM).The resonance control module (RCM) determines theresonant frequency of the RFQ by comparing the phase ofthe forward power with a sample of the RF in the RFQ.The RCM sends a frequency error signal to the RFQ’swater-cooling system. When the resonant frequency erroris greater than a specified value (~20 kHz) from 350MHz, this module switches to a frequency agile mode andsynthesizes the frequency required to drive the RF at theRFQ’s resonant frequency.
The resonant-control cooling system, [23] using the
frequency-error signal from the RCM, controls thetemperature of the water flowing to 4 cooling systems oneach of the 4 segments. Each of these 4 cooling systemsprovides water to the outer-wall-cooling channels tomaintain the resonant frequency of the RFQ near 350MHz. Each of these outer-wall-cooling systems has amanually set mixing valve that combines water exitingthe outer wall cooling channels with the water providedby the resonant-control system. These manually-setmixing valves provide the compensation for thedifferential heating of the 4 RFQ segments. Amultiplexed system that uses 64 RF pickup probes in theRFQ measures the field amplitude. Four probes in eachquadrant of each of the 4 segments provide the field data.After inspecting a plot of the field data the 4 mixingvalves are adjusted to make the field distribution at highpower nearly the same as the field distribution measuredin the RFQ at low power.
6 CW OPERATION
On December 17, 1999 we had the first long run with
CW beam current of ~100 mA. This run had a few shortinterruptions but averaged 98.7 mA over 3.3 hr [7]. Keyfactors that were instrumental in reaching this goal are:Figure 5: Two vertical wire-scan measurements [19] of
beam profile; (a) during the first half, and (b) during last
half of beam pulses similar to that shown in figure 4. TheHEBT setting was for a Y-emittance scan (near theminimum width in Y) [15] for (a) and (b). The curves are
Gaussian fits the data (*).
(a) (b)• Reducing the distance between the LEBT solenoid 2
and the RFQ from 30 cm to 15 cm and adding the
electron trap at the RFQ entrance.
• Increasing the RF field level in the RFQ to 10%
above design reduced the anomalous beam loss in thehigh-energy end of the RFQ.
• The general improvement in the level of conditioning
of the RFQ with operation time. Observations
suggest that the vacuum in the RFQ must be about
1.x 10
-7 Torr or lower for good operation.
The RFQ is designed for peak fields at 1.8 Kilpatrick
field [24]. When operating at 2 times the Kilpatrick field,the spark rate is not a problem. The estimated spark rate isonly 1 to at most 2 sparks per minute average. When aspark is detected, the RF power is turned off for 100 µs.
After the RFQ is fully conditioned, most of the beaminterruptions are caused by injector arcs, HPRF, or LLRFproblems. When these problems and a few others arefixed we see no fundamental reasons why the RFQ cannot run for very long periods of time with only short~100 µs interruptions in the beam.
7 SUMMARY
The LEDA RFQ performs as designed, provided the
RF field is raised about 10% above the design level toreduce beam loss in the high-energy end of the RFQ andto reach the design transmission. The present RFQsimulation codes do not have the capability of simulatinglow-energy ions trapped in the RFQ focusing fields. Theaddition of an electron trap at the entrance to the RFQ isessential to the measurement of the transmission throughthe RFQ. Simulation of the beam transport through theLEBT with PARMELA allowed understanding theinjection of the beam into the RFQ.
The HEBT and beam stop have been moved ~11 m to
make room for a 52 quadrupole beam transport line. Thisbeam line will be used to study beam-halo of bothmatched and unmatched high-current beams [25].
REFERENCES
[1] D. Schrage et al. , “CW RFQ Fabrication and
Engineering,” Proc. LINAC98 (Chicago, 24-28
August 1998) pp. 679-683.
[2] 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.
[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] L. J. Rybarcyk et al. , “LEDA Beam Operations
Milestone and Observed Beam Transmission
Characteristics,” This conference.
[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] H. V. Smith, Jr. et al. , “Update on the
Commissioning of the Low–Energy DemonstrationAccelerator (LEDA) Radio–Frequency Quadrupole
(RFQ),” Proc. 2nd ICFA Advanced Accelerator
Workshop on the Physics of High-Brightness Beams(Los Angeles, CA) (in press).
[7] L. M. Young et al. , “Low-Energy Demonstration
Accelerator (LEDA) Radio-Frequency Quadrupole(RFQ) Results,” ibid. (in press).
[8] K. F. Johnson et al. , “Commissioning of the Low-
Energy Demonstration Accelerator (LEDA) Radio-
Frequency Quadrupole (RFQ),” Proc. PAC99 (New
York, 29 March - 2 April 1999) pp. 3528-3530.
[9] J. H. Billen et al., “A New RF Structure for
Intermediate-Velocity Particles,” Proc. 1994 Int.
Linac Conf., (Tsukuba, 21-26 Aug. 1994) pp. 341-
345.
[10]M. J. Browman and L. M. Young, “Coupled Radio-
Frequency Quadrupoles as Compensated Structures,”
Proc. of the 1990 Linear Accelerator Conference,
(Albuquerque, 10-14 Sept. 1990) LA-12004-C, 70.
[11]L. M. Young, “An 8-meter-long Coupled Cavity
RFQ Linac,” Proc. 1994 Int. Linac Conf., (Tsukuba,
21-26 Aug. 1994) pp. 178-180.
[12]K. R. Crandall et al ., “RFQ Design Codes,” Los
Alamos National Laboratory report LA-UR-96-1836
(Revised August 21, 1998).
[13]L. M. Young, “PARMELA,” Los Alamos National
Laboratory report LA-UR-96-1835 (Revised January
8, 2000).
[14]L. M. Young, “Simulations of the LEDA LEBT With
H
+, H 2+, and E− Particles,” Proc. of the 1997 Particle
Accelerator Conference (Vancouver, 12-16 May
1997) pp. 2749-2751.
[15]M. E. Schulze et al ., “Beam Emittance
Measurements of the LEDA RFQ,” This conference.
[16]W. P. Lysenko et al ., “Determining Phase-Space
Properties of the LEDA RFQ Output Beam,” This
conference.
[17]L. M. Young, “Simulations of the LEDA RFQ 6.7
MeV Accelerator,” Proc. of the 1997 Particle
Accelerator Conference (Vancouver, 12-16 May
1997) pp. 2752-2753.
[18]J. D. Gilpatrick et al ., “Beam Diagnostic
Instumentation for the Low-Energy Demonstration
Accelerator (LEDA): Comminssiong and Operational
Experience,” Proc. EPA2000 (Vienna,26-30 June
2000) (in press).
[19]N. J. Diserens, “Progress in the Development of a 3D
Finite Element Computer Program to calculate Space
and Image Charge Effects in RF Quadrupoles,” IEEETrans. Nucl. Sci., NS-32 , (5), 2501 (1985).
[20] M.S. deJong, “Background Ion Trapping in RFQs,”
Proc. 1984 Linac Conf. (Seeheim, Germany, 7-11
May 1984), pp.88-90.
[21] A. H. Regan et al ., “LEDA LLRF Control System
Characterization,” Proc. LINAC98 (Chicago, 24–28
Aug. 1998) pp. 944-946.
[22]G. O. Bolme et al., “Measurement of RF Accelerator
Cavity Field Levels at High Power from X-ray
Emissions,” Proceedings of the 1990 Linear
Accelerator Conference, (Albuquerque, 10-14 Sept.1990) LA-12004-C, 219.
[23]R. Floersch, “Resonance Control Cooling System for
the APT/LEDA RFQ,” Proc. LINAC98 (Chicago,
24–28 Aug. 1998) pp. 992-994.
[24]W. D. Kilpatrick, “Criterion for Vacuum Sparking
Designed to Include Both rf and dc,” Rev. Sci,
Instrum., 28, 824 (1957).
[25]T. P. Wangler, “Beam Halo in Proton Linac Beams,”
This conference. |
arXiv:physics/0008159 18 Aug 2000The Beam Halo Experiment at LEDA
P. L. Colestock, T. Wangler, C. K. Allen, R. L. Sheffield, D. Gilpatrick and the Diagnostics Group,
M. Thuot and the Controls Group, the LEDA Operations Team, Los Alamos National Laboratory,
M. Schulze and A. Harvey, General Atomics, Los Alamos, NM 87545
Abstract
Due to the potentially adverse effects of the gener ation of
halo particles in intense proton beams, it is imper ative to
have a clear understanding of the mechanisms that c an lead
to halo formation for current and proposed high-int ensity
linacs. To this end a theoretical model has been
developed, which indicates that protons under the
combined influence of strong space charge forces an d
periodic focussing in a linear transport channel ca n be
kicked into halo orbits. However, no experimental
measurements of beam halo in proton beams have yet been
carried out. In this paper we report the progress of an
effort to carry out an experiment to measure beam-h alo
using the existing high-intensity proton beam of th e LEDA
facility. A linear transport channel has been asse mbled
with the appropriate diagnostics for measuring the
expected small beam component in the beam halo as a
function of beam parameters. The experiment is bas ed on
the use of an array of high-dynamic-range wire and beam
scrapers to determine the halo and core profiles al ong the
transport channel. Details of the experimental des ign, the
expected halo measurement properties will be presen ted.
1 INTRODUCTION
The interest in understanding the formation of a ha lo
distribution around an intense proton beam has incr eased in
recent years with the development of new applicatio ns
requiring such beams. In order to understand this process,
a theoretical model has been developed and extensiv e
computer simulation has been carried out, reviewed
elsewhere in these proceedings.1 Although an extensive
theoretical literature has evolved, there has been no
definitive test of the model to date, owing in part to the fact
that few intense proton beams of the required inten sity
exist.
At the Low Energy Demonstration Accelerator (LEDA)2
at Los Alamos National Laboratory, we have embarked
upon a program to carry out a first test of the hal o
formation model using the available intense proton beam
from LEDA. A 52 quad transport line is being const ructed
following the RFQ of the LEDA injector, a 100 mA, 6 .7
MeV proton beam with a capability for continuousoperation. We will use, however, a pulsed beam wit h a 20
µsec pulse length and a 10-4 duty factor in order to
facilitate the use of direct wire and scraper measu rements
of the beam profiles.
The purpose of this program is to make a detailed
comparison, for the first time, between the theoret ical
model of halo formation and beam profiles in a cont rolled
way.
2 EXPERIMENTAL SETUP
An overall view of the halo transport line is seen in Fig. 1.
The transport channel consists of 52 quadrupoles wi th a G-
l = 2.5 T, the first four of which have extra stren gth to
permit mismatching the beam from the RFQ as it pass es
into the transport channel. An array of up to nine scanners
will be used to monitor beam profiles in two dimens ions
over the length of the transport channel, and detai ls of the
scanner design are given elsewhere in these proceedings.3
A scanner consists of 33 µ diameter carbon filaments that
can safely intercept the entire beam current and ca n be
passed through the core of the beam. Beam current is
determined by secondary emission. On the same asse mbly
a scraper plate is mounted which permits measuremen t of
the diffuse halo region. Taken together, the beam profile
measurements should permit a dynamic range of 105 .
The philosophy for scanner placement is the followi ng: the
first scanner is used to determine the beam distrib ution
emerging from the RFQ, and this provides a critical initial
condition on the halo evolution. The second group of four
scanners is placed after quadrupole 13, which is wh ere
simulations show the beam is largely debunched. Si nce the
transverse space charge forces depend on the longit udinal
phase space density, it is reasoned that a fairly u niform
domain for the halo formation should occur downstre am
from this point. Four scanners are used to ensure complete
coverage of the phase space over a full betatron pe riod (68
degrees per cell). A final array of four scanners at the end
of the transport channel permits a similar measurem ent of
phase space in the region where the halo is expecte d to be
fully developed. Additional diagnostics include an array
of beam position monitors, resistive-wall current m onitors,
current toroids and beam lossmonitors. Figure 1 Overall Layout of the Beam H alo Experiment on LEDA.
The transport line consists of 52 quadrupoles betwe en the LEDA RFQ and
the HEBT/beam stop. The first four quadrupoles are used to create a
controlled mismatch from the RFQ. A series of nine dual-axis, high
dynamic-range wire/scraper scanners will be used to measure halo
properties.
Details of the wire scanner design and implementati on are
given elsewhere in these proceedings4 , as well as aspects
of the complex control system required for driving the
scanners and acquiring scanner data.5
Beam operation will be limited to < 20 µsec macropulses
to limit the power delivered to the scraper element s, and a
software algorithm has been devised to prevent inse rtion of
scrapers into the core of the beam. However, the
combined use of wire filaments and scraper elements will
permit a complete beam profile to be obtained. Pro file
data will rely on shot-to-shot repeatability of the RFQ,
which has been measured to be about 1%. Position j itter
was also measured to be a negligible effect.
Because of previously existing constraints on LEDA, the
quadrupole magnets permit only a 3 cm bore beampipe ,
which is expected to be approximately 50% larger th an the
largest halo orbit. Moreover, the core beam size i s about 1
mm RMS, which requires 5 mil alignment tolerances. This
was carried out using a taut-wire system for magnet center
fiducialization, and a precision alignment rail spa nning the
length of the transport line. Each magnet was indi vidually
mapped and aligned relative to the rail, along with beam
position monitors. Load stresses and thermal expan sion
were measured to be negligible alignment factors. 1 – 2
mil tolerances were achieved. An array of steering
magnets will be employed to correct for misalignmen t
errors in conjunction with the eight beam position
monitors. The magnets are powered in strings of ei ght,
each with individual shunts for trim control. Four singlet
supplies are used to individually power the
match/mismatch magnets at the exit of the RFQ. Onc e the
beam has exited the transport channel, it enters th e high-
energy beam transport line (HEBT) and terminates in the
beam stop, both commissioned in a previous LEDA run .3 EXPERIMENTAL OBJECTIVES
The experimental objective is to verify the halo fo rmation
model which requires a detailed comparison with
simulations. This can, in principle, be obtained by the array
of nine scanners and the data will be fitted to the results of
beam simulations. Such an endeavor requires a mass ive
data acquisition and simulation effort and is expec ted to be
completed some months after data-taking has come to a
conclusion. However, as noted in reference 2, a workable
single-parameter measure of the halo formation is
described based on the fourth moment of the distrib ution.
This will serve as a rough measure of the existence of a
halo. However, a complete confirmation of the mode l will
require as complete a phase space picture as possib le.
Another signature of halo formation according to th e
theoretical model is the maximum extent of the halo
particles. This value is achieved shortly after th e mismatch
quadrupoles, with only the number of particles with in the
halo distribution increasing along the transport ch annel. A
measurement of this maximum particle radius using t he
sensitive scraper diagnostic, in particular as a fu nction of
the mismatch strength, will provide an important t est of
the theory.
An alternative approach is to vary the tune of the transport
line effectively rotating the distribution past a s et of
scanners with fixed orientation. Since simulations show
that both the core and halo distributions can be ma de to
rotate rigidly over some phase range, it may be pos sible to
use the an inverse Radon transform to form a detailed
picture of phase space. Whether this elegant techn ique can
be made to work in the experimental environment of an
intense beam remains to be seen.
4 RUN PLAN
At the present time, all of the magnets and beamlin e have
been installed, as well as most of the diagnostics hardware.
Final checkout of the associated electronics and co ntrol
systems has begun. Moreover, the LEDA RFQ and
injector are being readied for operation.
Current run plans indicate that beam operation will
commence shortly following this conference, with a
commissioning and shakedown period for new hardware .
Full data-taking will begin early in FY 01 and the
experiment is expected to continue until Spring 200 1.
Because of the necessity to handle large amounts of beam
profile data, the initial emphasis will be placed o n fully
integrating the new diagnostic hardware into the EP ICS-
based control system.
Due to the potential for beam-induced damage at the full
current available from the RFQ (100 mA), it is plan ned to
implement a fast-protect system based on beam losse s that
will prevent any such damage. An extended period o f
operation at low currents (10 mA) is planned until the fast-
protect system is in place, and all diagnostics hav e been
checked out at low power densities.
5 SUMMARY
We have described the elements of an experiment to test
the theoretical model for halo formation that is so on to be
carried out on the LEDA facility. The purpose of t he
experiment is to verify the model with a detailed
measurement of phase space along a linear transport
channel, and by comparing this experimental informa tion
to analytical models and simulations. The combinat ion of
large dynamic-range profile diagnostics and large-s cale
simulations should provide a unique and comprehensi ve
test of the theory.
ACKNOWLEDGMENTS
The authors acknowledge support from the U.S.
Department of Energy, and wish to give a special th anks to
the many dedicated staff who have braved a fire and other
challenges to put together this experiment in a sho rt time.
REFERENCES
1 T. Wangler and K. R. Crandall, “Beam Halo in Proto n
Linac Beams,” these proceedings.
2 J.D.Schneider, “Operation of the Low-Energy
Demonstration Accelerator: The Proton Injector for
APT,”Proc. 1999 IEEE Particle Accelerator Conf. (IE EE
Catalog No.CH36366, 1999), pp.503-507.
3 R. Valdiviez, N. Patterson, J. Ledford, D. Bruhn, R.
LaFave, F. Martinez, A. Rendon, H. Haagenstad,
J. D. Gilpatrick, and J. O’Hara, “Intense Proton Co re and
Halo Beam Profile Measurement: Beam Line Component
Mechanical Design, “ these proceedings. .
4 J. D. Gilpatrick, D. Barr, D. Bruhn, L. Day, J. L edford,
M. Pieck, R. Shurter, M Stettler, R. Valdiviez, J.
Kamperschroer, D. Martinez, J. O'Hara, M. Gruchal la,
and D. Madsen, “Beam Diagnostics Instrumentation fo r a
6.7-MeV Proton Beam Halo Experiment,” these
proceedings.
5 L.Day et al, “Control system for the LEDA 6.7 MeV
proton beam halo experiment,” these proceedings. |
arXiv:physics/0008160v1 [physics.chem-ph] 18 Aug 2000Generalized Relativistic Effective Core Potential Method:
Theory and calculations
A. V. Titov∗and N. S. Mosyagin
St.-Petersburg Nuclear Physics Institute,
Gatchina, St.-Petersburg district 188350, RUSSIA
(February 2, 2008)
Abstract
In calculations of heavy-atom molecules with the shape-con sistent Relativistic
Effective Core Potential (RECP), only valence and some outer -core shells
are treated explicitly, the shapes of spinors are smoothed i n the atomic core
regions and the small components of four-component spinors are excluded
from calculations. Therefore, the computational efforts ca n be dramatically
reduced.
However, in the framework of the standard nodeless radially local RECP
versions, any attempt to extend the space of explicitly trea ted electrons more
than some limit does not improve the accuracy of the calculat ions. The errors
caused by these (nodeless) RECPs can range up to 2000 cm−1and more for the
dissociation and transition energies even for lowest-lyin g excitations that can
be unsatisfactory for many applications. Moreover, the dir ect calculation of
such properties as electronic densities near heavy nuclei, hyperfine structure,
and matrix elements of other operators singular on heavy nuc lei is impossible
as a result of the smoothing of the orbitals in the core region s.
In the present paper, ways to overcome these disadvantages o f the RECP
method are discussed. The developments of the RECP method su ggested
by the authors are studied in many precise calculations of at oms and of the
TlH, HgH molecules. The technique of nonvariational restor ation of elec-
tronic structure in cores of heavy atoms in molecules is appl ied to calculation
of the P,T-odd spin-rotational Hamiltonian parameters including th e weak
interaction terms which break the symmetry over the space in version ( P) and
time-reversal invariance ( T) in the PbF, HgF, BaF, and YbF molecules.
SHORT NAME: GRECP method: Theory and calculations.
KEYWORDS FOR INDEXING: Relativistic Effective Core Potential
(Pseudopotential), Ab initio relativistic method, Electr onic structure calcu-
lation, Molecules with heavy atoms.
31.15.+q, 31.20.Di, 71.10.+x
Typeset using REVT EX
1 |
arXiv:physics/0008161 18 Aug 2000SNS□□SUPERCONDUCTING□□CAVITY□□MODELING
-ITERATIVE□LEARNING□CONTROL
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□Sung-il□Kwon,□□Yi -Ming□Wang,□□Amy□Regan,□□Tony□□Rohlev,
LANL,□Los□Alamos,□NM87544,□USA
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□Mark□Prokop,□□Dav e□Thomson,□Honeywell□□FM&T
Abstract
□□□□□□The□SNS□SRF□system□is□□operated□with□a□pulsed□beam.
For□ the□ SRF□ system□ to□ track□ the□ repetitive□ reference
trajectory,□ a□ feedback□ and□ a□ feedforward□ controllers□ has
been□ proposed.□ □ The□ feedback□ controller□ is□ to□ guarantee
the□ closed□ loop□ system□ stability□ and□ the□ feedforward
controller□is□to□improve□the□tracking□performance□for□ the
repetitive□ reference□ trajectory□ and□ to□ suppress□ the
repetitive□disturbance.□□As□the□iteration□number□increases,
the□error□decreases.
1□INTRODUCTION
The□ Spallation□ Neutron□ Source□ (SNS)□ Linac□ to□ be
built□at□Oak□Ridge□National□Laboratory□(ORNL)□ consists
of□a□combination□of□low□energy□normal□ conducting□(NC)
accelerating□ structures□ as□ well□ as□ higher□ energy
superconducting□ RF□ (SRF)□ structures.□ In□ order□ to
efficiently□ provide□ a□ working□ control□ system,□ a□ lot□ of
modeling□□has□performed.□□The□modeling□is□used□as□a□way
to□ specify□ RF□ components;□ verify□ system□ design□ and
performance□objectives;□optimize□control□parameters;□and
to□ provide□ further□ insight□ into□ the□ RF□ control□ system
operation.
The□modeling□addressed□in□this□note□deals□with□the□PI
feedback□controller□and□the□plug-in□feedforward□controller
(the□iterative□learning□controller).□□The□purpose□of□the□PI
feedback□controller□is□to□guarantee□the□robustness□and□the
zero□ steady□ state□ error.□ However,□ the□ PI□ feedback
controller□ does□ not□ yield□ the□ satisfactory□ transient
performances□□for□□the□RF□filling□□and□the□beam□ loading.
The□ feedforward□ controller□ proposed□in□ this□ note□ takes□ a
simple□form□and□is□effective.□In□order□to□generate□the□one
step□ahead□feedforward□control,□the□feedforward□controller
makes□ use□ of□ current□ error,□ the□ derivative□ of□ the□ current
error□ and□ the□ integration□ of□ the□ current□ error.□ This□ PID-
type□feedforward□controller□is□the□natural□consequence□of
the□ PI□ feedback□ control□ system□ where□ the□ inverse□ of□ the
closed□ loop□system□transfer□ matrix□has□ the□ same□ form□as
the□ transfer□ matrix□ of□ the□ PID□ system.□ The□ proposed
feedforward□controller□achieves□the□better□performance□for the□ repetitive□ reference□ trajectory□ to□ be□ tracked□ by□ the
system□ output□ and□ achieves□ the□ suppression□ of□ the
repetitive□disturbance□such□as□the□Lorentz□force□detuning.
2□□SUPERCONDUCTING□CAVITY□MODEL
The□ modeling□ of□ a□ superconducting□ □ cavity□ is□ based
on□the□assumption□that□the□RF□generator□and□the□cavity□are
connected□ with□ a□ transformer.□ □ The□ equivalent□ circuit□ of
the□ cavity□ is□ transformed□ to□ the□ equivalent□ circuit□ of□ RF
generator□ with□ transmission□ □ line□ (wave□ guide)□ □ and□ the
model□ is□ obtained[2].□ A□ superconducting□ cavity□ is
represented□by□the□state□space□equation□.
□□□□□□ IL IBuLBxLAx ) ( ) ( ) ( ω ω ω Δ+Δ+Δ=/G26 □□□□□□□□□□□□□□□□□□(1)
□□□□□□ xLCy ) (ωΔ=
and□□the□Lorentz□force□detuning□□is
□□□□ 2
22 2
12 1
z K
mz K
mL
mLτπ
τπ
ω
τω − −Δ− =Δ/G26 □□□□□□□□□□□□□□□□□□(2)
where
/Gfa/Gfa/Gfa/Gfa
/Gfb/Gf9
/Gea/Gea/Gea/Gea
/Geb/Ge9
− Δ +ΔΔ +Δ − −
=Δ
LL mL m
L
LA
τωωωω
τ
ω1) () (1
) (,□□□□□□□□□ /Gfa/Gfb/Gf9
/Gea/Geb/Ge9=1001C ,
/Gfa/Gfa/Gfa/Gfa
/Gfb/Gf9
/Gea/Gea/Gea/Gea
/Geb/Ge9−
=Δ
12
3232
12
) (
c
oZc
oZc
oZc
oZ
LBω,□□□ /Gfa/Gfb/Gf9
/Gea/Geb/Ge9
−−−=Δζζζζω
12323212)(c cc c
LIB ,
□
τcu R
c=1,□□□□□
τoQcu R
c
23= ,□□□□□ 2
] [] / [
/Gfa/Gfa
/Gfb/Gf9
/Gea/Gea
/Geb/Ge9
=
Vgap Vm MV oE
K K ζ
ζ:□Transformation□ratio,□□□□□□□□□ oQ:□Unloaded□□ Q
cu R:□Resistance□of□the□cavity□□equivalent□circuit
mωΔ :□Detuning□frequency[rad/s]
oZ:Transmission□line□impedance Lτ:□Loaded□cavity□damping□constant
τ:□Unloaded□cavity□damping□constant
mτ:□Mechanical□time□constant
K:Lorentz□force□detuning□Constant
[]T
fQ VfI V u= :□forward□Voltage□in□□I/Q
T
QIII I/Gfa/Gfb/Gf9
/Gea/Geb/Ge9= :□Beam□current□in□I/Q
/Gfa/Gfb/Gf9
/Gea/Geb/Ge9=QVIV x:□Cavity□Field□in□I/Q
The□ modeling□ of□ the□ cavity□ is□ based□ on□ the
assumption□that□the□exact□characteristics,□paramete rs□of□a
cavity□ are□ known.□ When□ there□ are□ parameter
perturbations,□ unknown□ deterministic□ disturbances□ a nd
random□ noises□ in□ the□ input□ channels□ or□ measurement
channels,□ those□ uncertainties□ are□ added□ to□ the□ stat e
equation□ or□ the□ output□ equation.□ For□ the□ control□ of □ this
uncertain□system,□□modern□robust□controllers□such□a s□ ∞H
controller,□ loop-shaping□ controller□ are□ applied.□ On □ the
other□ hand,□ PI□ (PID)□ controllers□ are□ designed□ by□ us ing
∞Hcontroller,□loop-shaping□controller□design□techniqu es.
3□ITERATIVE□LEARNING□CONTROL
The□SNS□SRF□system□is□operated□with□a□pulsed□beam.
The□period□of□the□beam□pulse□is□16.67□ sec m (60 1Hz ).
The□ objective□ of□ the□ SRF□ controller□ is□ to□ generate□ a
periodic□ reference□ trajectory□ whose□ period□ is□ 16.67
sec m (60 1Hz )□ and□□is□to□achieve□ a□ stable□ cavity□ field
periodically□ so□ that□ the□ RF□ power□ is□ delivered□ to□ t he
periodic□ beam□ pulse□ safely[3].□ □ A□ control□ system□ th at□ is
suited□ for□ this□ type□ of□ applications□ is□ Iterative□ L earning
Control□(ILC)□[1],[3].
Consider□a□controller□at□□the□ kth □iteration,
k
Fuk
Cuku += □□ □□□□□□□□□□□□ □□(3)
where□ k
Cu□is□the□output□of□the□PI□feedback□controller□and
k
Fu□is□the□output□of□□the□feedforward□ILC□controller.□ The
error□dynamics□is□expressed□as
k
cxIBK kecAke −= /G26
□□□□□□□□□□□□□□□□□□□□□□□□ rrLABIIBk
FBu /G26+Δ− −− ) (ω □□□□□□(4)
□□□ kek
cx= /G26where□ PBK LAcA −Δ= ) (ω .□ □ Since□ 3 1c c>> ,□ with□ the
proper□diagonal□terms□and□zero□off-diagonal□terms□o f□the
gain□ matrices□ PK□ and□ □ IK□ of□ the□ PI□ controller,□ the
diagonal□ terms□ of□ the□ matrix□ PBK LA−Δ ) (ω □ and□ □ the
matrix□ IBK □ are□ sufficiently□ large□ and□ so□ the□ I□ channel
error□and□the□Q□channel□error□□(4)□are□almost□decou pled.
□□□□□□□The□Laplace□transform□of□the□error□equation□ (4)□yields
□□□□□□□□ ) (1) ( ) ( ) ( ) ( sBIIBB seSsk
FU seS skE−− − =
□□□□□□□ ( ) ) ( ) (1) ( s RLA sI B seS ωΔ−−+ □□□□□□ □□(5)
where
BIBK
scA sI seS1
1
) (−
+−= /Gf7/Gf8/Gf6/Ge7/Ge8/Ge6□□□□□□□□□□□□□□□□□□(6)
□□□□□□□□Define□the□learning□control□rule□as□follows .
( )kLE k
FUf Qk
FU ⋅+⋅=+α1□□□□□□□□□□□□□□□□(7)
where□ ,f□□ 1 0<<f ,□is□□called□the□forgetting□factor□and
α,□□ 1 0<<α ,□is□a□design□constant.□□The□forgetting□factor
f□ and□ □ the□ constant□ α□ are□ to□ guarantee□ the□ robust
stability□ against□ uncertainties□ in□ the□ plant□ model□ and□ the
nonlinearity□ of□ the□ klystron.□ They□ also□ allow□ for
elimination□of□the□influence□of□random□noise,□spike s□and
glitches.□□ k
FU□is□the□Laplace□transform□of□the□feedforward
signal□in□iteration□ k□and□□ kE□is□the□Laplace□transform□of
the□ corresponding□ tracking□ error.□ Learning□ converge s□ if
the□ feedback□ loop□ is□ stable□ and□ the□ following□ condi tion
holds.□□For□ ,ℜ∈∀ω
□□□
∞−+<
∞+−+)( )(1)(1)(2ω ω ω ω jk
FU jk
FU jk
FU jk
FU ,
which□results□in□learning□convergence□condition
( ) 1<∞⋅−⋅eLS I f Qα □ □□(8)
□□□□□□□The□ Q-filter□ is□ designed□ such□ that□ it□ suppresses□ the
high□ frequency□ components□ at□ which□ the□ plant□ model□ is
inaccurate□and□passes□low□frequency,□at□which□the□m odel
is□ accurate.□ The□ Q-filter□ is□ either□ placed□ before□ the
memory,□ □ or□ in□ the□ memory□ feedback□ loop.□ Thus,□ the
bandwidth□of□the□ Q-filter□should□ be□ chosen□ greater□ than
or□equal□to□the□desired□closed□ loop□ bandwidth.□ From □ the
∞H□controller□design□point□of□view,□□(8)□interprets□t he□-
Q-filter□ as□ a□ weighting□ function□ for□ □ learning
performance,□i.e., ∞−<∞⋅−⋅1QeLS I fα □ □□□□□□□□□□□□□□□□□(9)
It□seems□natural□that□the□ Q-filter□is□viewed□as□a□measure
of□□learning□performance□and□the□cut-off□frequency□cω□of
the□ Q-filter□ □ is□ chosen□ as□ large□ as□ possible□ in□ order□ to
guarantee□zero□tracking□error□□up□to□frequency□□ cω.
□□□□□□□To□design□a□ L-filter,□detailed□knowledge□of□the□plant
is□ required.□ For□ low□ frequency□ dynamics,□ a□ competen t
model□of□ the□ plant□ often□ exists.□ □ However,□ identifi cation
and□modeling□of□□high□frequency□dynamics□is□difficu lt□and
may□ lead□to□an□ inadequate□ model.□ This□ could□ result□ in□ a
learning□ filter□ L□ that□ compensates□ well□ for□ low
frequencies□but□does□not□compensate□appropriately□f or□all
high□ frequencies□ and□ therefore□ causes□ unstable□ beha vior.
This□unstable□behavior□□is□□prevented□by□the□ Q-filter□and
to□ determine□ cω,□ a□ trade-off□ between□ the□ performance
and□the□robust□stability□is□necessary.□An□intuitive □synthesis
of□the□learning□ L-filter□□for□given□□ Q-filter□□is□as□follow.
□□ 1 1
) (1) (−+−=−= /Gf7/Gf8/Gf6/Ge7/Ge8/Ge6BIBK
scA sI seS s L □□□□□□□□□□□(10)
When□ the□ feedback□ PI□ controller□ gain□ matrix□ IK□ is
defined□as□a□diagonal□matrix,□□then□(10)□is□reduced □to
□□□ IK
sPK BLA sB s L1
)1) ( (1) ( +−−Δ−−= ω □□□□□□□□□□□(11)
Equation□ (11)□ shows□ that□ the□ learning□ L-filter□ has□ the
characteristics□of□□□PID[3]□.
4□SIMULATION
The□ closed□ loop□ system□ with□ PI□ feedback□ controller
and□ iterative□ learning□ controller□ was□ simulated.□ Fi gure□ 1
and□ figure□ 2□ □ show□ the□ field□ amplitude□ and□ □ the□ fie ld
phase,□ □ where□ the□ great□ □ improvement□ of□ the□ transie nt
behaviors□ both□ in□ RF□ filling□ and□ □ in□ beam□ loading□ i s
observed□as□iteration□number□increases.□□Also,□two□ figures
show□ that□ the□ periodic□ Lorentz□ force□ detuning□ effec t□ on
the□ field□ amplitude□ and□ the□ field□ phase□ is□ suppress ed
gradually□ as□ the□ iteration□ number□ increases.□ Figure □ 3
shows□ the□ Lorentz□ force□ detuning.□ Note□ that□ □ the□ st atic
value□ of□ the□ Lorentz□ force□ detuning□ calculated□ with □ the
cavity□ data□ ( 0 . 2− = K2Hz/(MV/m) ,□ 9 . 11 =acc E MV/m )
is□ □ -283□ Hz .□ With□ the□ RF□ On□ period□ 1.3□ msec
(300 sec µ field□settling□period□+□ 1000 sec µ beam□period),
the□Lorentz□force□detuning□is□developed□up□to□–200□ Hz .
REFERENCES
□[1]□Z.□Bien□and□J.-X.□Xu,□□ Iterative□Learning□Control:□Analysis,
Design,□ Integration,□ and□ Application .□ Kluwer□ Academic
Publishers,□1998. [2]□ B.□ R.□ Cheo□ and□ Stephan□ P.□ Jachim,□ “Dynamic□ inte ractions
between□ RF□ sources□ and□ LINAC□ cavities□ with□ beam□ loa ding,”
IEEE□Trans.□Electron□Devices ,□□Vol.□38,□No.□10,□□pp.□2264-2274,
1991.
□[3]□ Sung-il□ Kwon,□ Amy□ Regan,□ and□ Yi-Ming□ Wang,□ □ SNS
SUPERCONDUCTING□ □ CAVITY□ □ MODELING-ITERATIVE
LEARNING□ CONTROL□ (ILC) ,□ Technical□ Report,□ LANSCE-5-
TN-00-014,□Los□Alamos□National□Laboratory,□July,□□2 000.
Figure□ 1 □ Field□ Amplitude□ with□ PI□ Controller□ plus□ Iterative
Learning□Controller□(PI+ILC)□.
Figure□2 □□Field□Phase□with□PI□Controller□plus□Iterative□Lea rning
Controller□(PI+ILC).
Figure□3□□ Lorentz□Force□Detuning .0 0.5 1 1.5 2 2.5
x□10 -3 4.6 4.8 55.2 5.4 5.6
tim e(sec) FLD_ AMP(Volts) Dotted□Line:□1st□Iteration
Dash-dotted□Line:□2nd□Iteration
Dashed□Line:□3rd□Iteration
Solid□Line:□4th□Iteration R F□O N B EA M □O N R F□O FF
B EA M □O FF
0 0.5 1 1.5 2 2.5
x□10 -3 -3 -2 -1 01234
time(sec) FLD_ PHS(Degrees) Dotted□Line:□1st□Iteration
Dash-dotted□Line:□2nd□Iteration
Dashed□Line:□3rd□Iteration
Solid□Line:□4th□Iteration
RF□ON BEAM□ON RF□OFF
BEAM□OFF
0 0.5 1 1.5 2 2.5
x□10 -3 -250 -200 -150 -100 -50 0
time(sec) Loren tz□Force□Detu n in g□□ ΔωL(H z )
RF□ON□ BEAM□ON□ RF□OFF
BEAM□OFF□ |
Energy Transfer Mechanisms and Equipartitioning in
non-Equilibrium Space -Charge -Dominated Beams
R. A. Kishek , P. G. O’Shea, and M. Reiser,
Institute for Plasma Research, U. Maryland, College Park, MD 20742, USA
ramiak@ebte.umd.edu Abstract
A process of energy transfer is demonstrated in non -
equilibrium charged particle beams with anisotropy and
space charge. Equipartitioning of energy between
available degrees of freedom occurs in just a few betatron
wavelengths, without halo formation. Collective space
charge modes simil ar to those observed in recent
experiments provide the underlying coupling
mechanism. Since laboratory beams are commonly far
from equilibrium, the traditional K -V stability analysis
does not necessarily apply, implying that selection of an
operating point based on theory does not necessarily
avoid equipartitioning. Furthermore, the rate of
equipartitioning is shown to depend on a single free
parameter related space charge content of the final
(equipartitioned) beam, and does not depend on how the
kinetic e nergy is initially distributed between the two
planes.
Modern designs for high intensity linear accelerators
are frequently based on the presence of large anisotropies
between the longitudinal and transverse directions [1 -2].
Since the time scale for Coul omb collisions (intrabeam
scattering) is long relative to the size of the machine,
thermodynamic equipartitioning based on particle
collisions is usually ignored. Of more significance to the
designer is the collective space charge interactions which
may un der some circumstances couple the degrees of
freedom and allow the energy transfer. The question
which has preoccupied beam physicists for the past two
decades [1 -6] is precisely under what conditions does a
collisionless beam equipartition?
The wealth of theoretical studies in the past 2 decades
has contributed much to our understanding, but the
overall picture remains far from complete.
Thermodynamic considerations [1, 6] have been used to
predict the final equilibrium state, but are unable to
address the detailed energy transfer mechanism or the
timescales involved. As evidenced by some computer
simulations, unstable space charge modes have been
advanced as a likely mechanism [3 -4]. Using the same
framework that Gluckstern has used for an isotropic
beam [7], Hofmann has analytically derived the stability
properties of anisotropic distributions to small
perturbations [8]. The derivation assumes a Kapchinskij -
Vladimirskij (KV) -like but anisotropic equilibrium distribution in a uniform focusing channel, so as to make
the mathematics manageable, and results in charts
delineating stable and unstable regions that can be useful
to the accelerator designer, if correct. The theory has
been tested with simulations having KV and waterbag
initial distributions [3]. T he problem is that beams in
real machines are usually quite far from equilibrium, and
certainly not a KV. We therefore need to ask to what
extent are the features of such stability charts a result of
the choice of KV distribution?
To address this issue, w e self-consistently simulate
anisotropic beams using the particle -in-cell code WARP
[9] and using non -equilibrium initial distributions to
model realistic beams. The lack of equilibrium in the
initial distributions makes comparison to KV stability
theory m ore difficult. It also implies, as we shall see,
that certain space charge modes are born from the initial
mismatch of the distribution. In addition to examining
the energy transfer mechanism, we explore its scaling
and demonstrate that the rate of energy transfer depends
on a single free parameter related to the ratio of space
charge forces to external focusing forces.
In many respects, the simulations presented here are
very similar to those of a recent experiment, albeit in an
isotropic system, by Ber nal, et. al. [10]. The space -
charge-dominated electron beam in that experiment
exhibited wave -like density modulations which were
traced to the lack of detailed equilibrium at the source.
Simulations with the WARP starting with a semi -
Gaussian (SG) distrib ution [uniform density and
Gaussian velocity distribution with a uniform
temperature across x or y] have accurately reproduced
the density modulations [10]. A KV initial distribution,
on the other hand, did not reproduce the experiment. In
this paper we in troduce anisotropy into the such
simulations.
We use the 2 -½ D slice version of WARP [9], which
advances particles in a transverse slice under the action
of external forces and the self -consistent self -fields. To
simplify the issue, the external focusing is chosen to be
uniform along z (and equal in x and y), resembling the
focusing obtained from a uniform distribution of
background ions [11]. Typically we use a 256 × 256 grid
for the Poisson solver, a step size of 4 mm along z, and
20,000 particles, with test simulations up to 400,000
particles. Extensive testing of the numerics have demonstrated that the simulations are very robust with
respect to the choice of numerical parameters.
In the simulation shown in Fig. 1, a 10 kV, 50 mA
electron beam having a 7.5 mm radius is launched inside
a 1” radius circular pipe. The anisotropy is introduced by
fixing the external focusing strength at k o = 3.972 m-1 in
both directions, and picking initially different emittances
in x and y ( εx = 100 µm, while εy = 50 µm,
unnormalized effective). This implies that the tune
depressions are different, namely (k /ko)x = λβο/λβx =
0.41; (k /ko)y = λβο/λβy = 0.26, λβ being the betatron
wavelength and the subscript ‘o’ denoting zero space
charge. The initial beam sizes in x and y are chosen to be
matched solutions of the rms envelope equations,
assuming the emittances will not change. Naturally, any
change in emittance will also induce an rms mismatch.
In subsequent simulations, we vary the ratio εx / εy and
also the beam current to explore the scaling of the energy
transfer.
Fig. 1: Emittance exchange due to equipartitioning of a
SG beam in a symmetric uniform focusing channel ( αo =
kyo/kxo=1) and starting with εx = 2εy; ξ = (k/k o)final =
0.353, T = 3.17.
Hofmann [8] h as found that three dimensionless
variables are needed to describe the parameter space of
anisotropic beams. In this paper we choose the following
combination: (i) the ratio of zero -current betatron tunes:
αo ≡ kyo/kxo, which is set by the external lattice and is
not a free parameter for practical purposes; (ii) the
square root of the ratio of the total transverse kinetic
energy to the external field energy:
()()
22222 2
bkakb a
yo xoy x
++
≡e e
x ,
also invariant, due to conservation of energy; and (iii) the
initial ra tio of kinetic energies in the two transverse
directions: T ≡ Tx/Ty = εxkx / εyky. We set αo = 1 for this
paper. The 2nd parameter can be shown to approximately
equal the tune depression of the final equipartitioned
beam [12]. Note that the 3rd parameter is the only
variable measuring the degree of anisotropy and the only
one that changes as the beam equipartitions.
For the simulation in Fig. 1, ξ= (k/ko)final = 0.35 and T
= 3.17, placing it in the space -charge-dominated regime
with a fairly strong energy anisotropy. This simulation is
typical of simulations started with a SG distribution in
that the beam is observed to transfer energy between the
two directions until the beam sizes and emittances in the
two transverse planes are equal [Fig. 1]. The final
emittance satisfies conservation of energy [1, 13]. Close
examination of the density profile of the beam [14]
reveals that the energy transfer mechanism is precisely
the density oscillations that appear in the symmetric case
[10]. Noting that the speed of propagation of the density
crests from the edge to the center depends on the tune
depression [15], the wave velocities in an anisotropic
beam will be different because the tune depressions are
initially different in x and y. What begins as a ring at the
beam edge transforms into an ellipse with a different
eccentricity, i.e ., an initial temperature anisotropy
translates into a density anisotropy downstream. As the
wave breaks at the center of the beam, it generates
higher-order modes that couple the two transverse
directions and facilitate the transfer of kinetic energy
which leads to equipartitioning.
Simulations started with a KV distribution evolve
differently, and the KV beam does not always
equipartition [14]. The KV beam remains stable unless
the operating point allowed one of the instabilities
predicted by the KV stabi lity theory [8] to be excited.
Simulations started with a SG beam are not as easy to
reconcile with the theory, and the differences are
discussed more fully in [14].
Fig. 2: Evolution of emittance for beams with different
degrees of anisotropy; ξ = 0.188 and αo = 1 for all
beams, T varies from 1 to 16.
In Fig. 2, we vary the parameter T and find that the
characteristic distance (or time) over which a beam
equipartitions is over a large range independent of the
degree of anisotropy, provided αo and ξ are the same.
The 2nd parameter is therefore the only one which
governs the rate of equipartitioning. In other words, the
rate of equipartitioning is unaffected by the way the
0 5 10 15
s (m) 100
50 4εεrms (µµm) εx
εy
s (m) 0 5 10 154 εrms (mm-mr)
30100
εx
εykinetic energy is distributed in the two directions and
hence depends only on the final isotropic state, as long as
the total kinetic energy is held the same.
To quantify the rate of this process, we define a
“characteristic distance”, s eq, over which the emittances
approach their final value for the first time. In Fig. 3 the
beam current is systematically varied to explore the
dependence on the parameter, ξ. The only caveat is that
the ratio of emittances is held constant, so T changes
slowly because of changes in the matched beam size.
Nevertheless, as just demonstrated, changes in T do not
affect the equipartitioning time.
Fig. 3: The rate of equiparti tioning, defined (see text)
as λβo / seq, as a function of ξ; Tx/Ty varies slowly
between 2 and 4, αo = 1.
Three conflicting factors affect the equipartitioning
rate: (a) the space charge content (1 - ξ2) since that
determines the strength of the coupling ; (b) the initial
amplitude, and (c) the propagation speed of the
perturbation, since it is the main vehicle that effects the
coupling. We have a tradeoff between the amplitude of
the initial perturbation (b) and the space charge content
(a) since an equil ibrium thermal distribution converges
to a semi -Gaussian in the space charge limit where the
temperature (emittance) is zero. Further, the propagation
speed (c) is found in the isotropic case to peak at
intermediate tune depressions [15]. The strongest
coupling therefore takes place at the intermediate ξ of ~
0.3. Note that the equipartitioning distance can be as
small as two zero -current betatron periods, and increases
to larger values at either extreme. Even at the weak tune
depression of 0.87, the emitta nce can change
significantly in about 12 betatron periods.
In conclusion we pose a few questions on the
implications of this work. While theoretical studies have
been limited to small perturbations from equilibrium, it
is obvious that the possibility of e quipartitioning because
of a large perturbation needs to be investigated, since
ultimately beams in real machines will experience such
perturbations. Machine designs employing unequipartitioned beams need to be carefully reexamined
if space charge plays a role. Our finding that the rate of
equipartitioning depends only on the tune depression of
the final isotropic beam leads us to propose that a general
anisotropic beam can be modeled by using an “equivalent
isotropic beam” having the same ratio of total ki netic to
total external energies. Work along these lines needs to
be continued to explore cases where the external focusing
is not symmetric ( αo ≠ 1), and to explore transverse -
longitudinal equipartitioning in bunched beams.
We are grateful to A. Friedman, D. Grote and S. M.
Lund for the WARP code; to S. Bernal, C. L. Bohn, I.
Haber, I. Hofmann, and M. Venturini for valuable
discussions; and to M. Holland for assistance with
simulations. This work is supported by the U.S.
Department of Energy grant numbers DE -FG02-
94ER40855 and DE -FG02-92ER54178. The WARP code
runs on DOE supercomputers provided by NERSC at
LBNL.
REFERENCES
[1] T. P. Wangler, F. W. Guy, and I. Hofmann, Proc.
1986 Linac Conf., Stanford, CA, (1986).
[2] R. A. Jameson, IEEE Trans. Nucl. Sci. NS-28, 2408
(1981).
[3] I. Hofmann, IEEE Trans. Nucl. Sci. NS-28, 2399
(1981).
[4] I. Haber, et. al., Nuclear Instruments and Methods,
A415, 405 (1998).
[5] J-M Lagniel and S. Nath, Proc. EPAC 98, p. 1118
(1998).
[6] M. Reiser, Theory and Design of Charged Particle
Beams, (Wiley, New York 1994), chapters 5 and 6.
[7] R. L. Gluckstern, Proc. Linac Conf., Batavia, IL,
Sep. 1970, p. 811.
[8] I. Hofm ann, Phys. Rev. E, 57 (4), 4713 (1998).
[9] D. P. Grote, et. al., Fus. Eng. & Des. 32-33, 193-200
(1996).
[10] S. Bernal, R. A. Kishek, M. Reiser, and I. Haber,
Phys. Rev. Lett., 82, 4002 (1999).
[11] Simulations with a FODO lattice show similar
results, s ee refs. [3] and [15].
[12] χ = (1 - ξ2) = K/(k oa)2 is the space charge intensity
parameter defined in M. Reiser, et. al., Proc. PAC99,
p. 234 (1999).
[13] J. J. Barnard, et. al., AIP Conf. Proc 448 (1998), p.
221; M. Venturini, R. A. Kishek, and M. Reis er,
ibid., p. 278; S. M. Lund, et. al., NIM -A, A415,
345-356 (1998).
[14] R. Kishek, P. G. O’Shea, and M. Reiser, “ Space
Charge Wave -Induced Energy Transfer in non -
Equilibrium Beams, ” (to be published).
[15] R. A. Kishek, et. al., “Transverse space -charge
modes in non -equilibrium beams,” NIM -A, Proc.
Heavy Ion Fusion Conf., San Diego, Mar. 2000 (to
be published).
00.20.40.60.81
0 0.25 0.5 0.75 1
ξξλλβοβο/seq |
arXiv:physics/0008163 18 Aug 2000ANALYSIS AND SYNTHESIS OF THE SNS SUPERCONDUCTING R F
CONTROL SYSTEM
Y.M. Wang, S.I. Kwon, and A.H. Regan, LANL, Los Ala mos, NM 87545, USA
Abstract
The RF system for the SNS superconducting linac
consists of a superconducting cavity, a klystron, a nd a
low-level RF (LLRF) control system. For a proton li nac
like SNS, the field in each individual cavity needs to be
controlled to meet the overall system requirements. The
purpose of the LLRF control system is to maintain t he RF
cavity field to a desired magnitude and phase by
controlling the klystron driver signal. The Lorentz force
detuning causes the shift of the resonant frequency during
the normal operation in the order of a few hundreds hertz.
In order to compensate the Lorentz force detuning e ffects,
the cavity is pre-tuned into the middle of the expe cted
frequency shift caused by the Lorentz force detunin g.
Meanwhile, to reduce the overshoot in the transient
response, a feed-forward algorithm, a linear parame ter
varying gain scheduling (LPV-GS) controller, is pro posed
to get away a repetitive noised caused by the pulse d
operation as well as the Lorentz force detuning eff ects.
1 INTRODUCTION
To analyse the performance of the RF control system
for the SNS superconducting linac, a MATLAB model i s
created for each functional blocks, which includes the
superconducting cavity model, klystron model, PID
feedback controller, and a feed-forward controller[ 1]. An
equivalent resonant circuit couple with a coupling
transformer is used for the superconduncting cavity model
in which the Lorentz force detuning of the cavity
resonance frequency is included. The klystron is mo delled
as a cascade of a pass filter, determined by the ba ndwidth
of the klystron, and a phase-magnitude saturation c urve,
which represents the saturation characteristics of the
klystron. The phase-magnitude saturation curve is
obtained from the measurement and is further analys ed
using the curve fitting to generate the final model . The
main feedback controller is a PI controller for an easy
implementation and robustness concern. In order to
implement the RF control system in a full digital c ontrol
system, the latency analysis is needed to satisfy t he
performance requirement of the system. Finally, wit h the
results obtained from the numerical simulation and the
performance requirements, a full digital control sy stem for
the LLRF system is proposed. In this system, a comb ined
CPLD and DSP technology is used to cope with differ ent
requirements. The CPLD is applied to the critical p ath inwhich the time delay needs to be minimized. While t he
DSP is used to perform the complex linear parameter
varying gain scheduling (LPV-GS) control which requ ires
the computation power but needs only be fed to the control
signal in the next pulse.
2 SYSTEM MODELLING AND CONTROL
ALGORITHMS
2.1 Superconducting Cavity Model
The state space equation of the superconducting mod el
is given by
IIBBuxLAx ++∆=)(ω /c38 (1)
Cxy=
where,
− ∆+∆∆+∆− −
=∆
LLmLm
L
LA
τωωωω
τ
ω1) () (1
)(
the dynamics of the Lorentz force detuning satisfie s the
following equation
2222 1
QVK
mVK
mL
mL Iτπ
τπ
ω
τω − −∆−=∆ /c38 (2)
where, mω∆ is the synchronous phase detuning
frequency, Lω∆is the Lorentz force detuning frequency,
Lτis the loaded cavity damping constant, Kis the
Lorentz force detuning constant,
=QVIVxis the
cavity field in I/Q components, whereas, the system
matrices B, BI, and C are given in [1].
In the model, the Lorentz force detuning frequency
appears on in the system matrix A and all other sys tem
matrices are constant. In observing Equation (2), t he
Lorentz force detuning is a nonlinear function of t he
cavity field, which renders the system equation (1) a
nonlinear equation of the cavity field.2.2 Linear Parameter Varying Gain Scheduling
Controller (LPV GS)
The principles of the linear parameter varying gain
scheduling can be explained as the followings. Firs t, due
to the nonlinearity of the system equation, which c omes
from both the saturation characteristic of the klys tron and
the nature of the Lorentz force detuning effect, th e
maximum performance of the RF control system can on ly
be achieved by implementing a variable gain-profile based
on the equilibrium point at which the system operat es.
Secondly, at the equilibrium point, the system need s to be
linearized for solving the system equation (1). Fin ally,
both the feedback controller and the feed-forward
controller need to be implemented to suppress the
repetitive noise due to the pulsed operation and a known
effect of the Lorentz force detuning effect.
The equilibrium manifold of a linear parameter vary ing
system is given by
wEuBxAx )()()( ρρρ ++= /c38 (3)
xCy)(ρ= .
The above equations are a linearized version of the
system equation (1) at a specific operation point g iven by
ρ. Let ry be the desired trajectory to be followed by the
system output y. Then, the parameterised equilibrium
manifold of the system is defined by the solution o f the
algebraic equation given
= −
euex
CBAwE
ry 0)()()(
0)(0
ρρρ ρ (4)
Now we consider the open loop system as given in ( 1)
and the Lorentz force detuning as given in (2). First, let
[]T
Q IvvV=
be the desired output trajectory to be tracked by t he cavity
field I and Q. Then, the equilibrium manifold ),(eeux
of the open loop system as given in (1) is the sol ution of
the following algebraic matrix equation.
Icccc
uxc
Zc
ZcZcZ
Veeo oo o
LL mL m
L
−−−−−−−−−−−
+
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
− ∆+∆∆+∆− −
=
−−
0 00 02222
0 00 0
|||
1 00 1|222 2
|||
1) () (1
0
1 33 1
1 33 1
ζζζζ
τωωωω
τ
(5)
Solving Equation (5), we obtain
Vxe= (6)
−−−+
− ∆+∆∆+∆− −
−+−= IccccVcccc
ccZu
LLmLm
L oeζζζζ
τωωωωτ
1 33 1
1331
2
32
12222
1) () (1
)( 2
(7)
Note that the equilibrium manifold ),(eeux is
parameterized by not only the desired trajectory V, the
Lorentz force detuning Lω∆ but also the beam current I.
From (2), the Lorentz force detuning on the equilib rium
manifold is
2
222
12exKexKLeπ π ω − −=∆ . (8)
Using the equilibrium points obtained from (6) and (7),
we can design a linear parameter varying gain-sched uling
controller as
))(,,(exxVILFeuu − ∆+= ω (9)
In the controller (9), ),,(VIFLω∆ is the parameter
varying feedback gain matrix such that the closed l oop
system matrix
),,()()( VILBFLALclA ω ω ω ∆+∆=∆ (10)
is stable.
There are many design techniques for ),,(VILFω∆ .
A ∞H controller-based parametric varying controller and
a velocity-based gain-scheduling controller are two of
them. In addition, we can design a constant feedba ck gain
matrix F such that for all variations of Lω∆, V, and I
within given bounded sets, the closed loop system m atrix
(10) is stable. An eigenstructure control design t echnique
can be applied. Let the constant stable matrix rA be the
desired closed loop system matrix. Then, the feedba ck
controller gain matrix ),,(VILFω∆ is determined by
solving
),,()()( VILBFLALrA ω ω ω ∆+∆=∆ (10)
The solution of Equation (11) is
)((1),,(LArABVILF ω ω ∆−−= ∆ (10)
Assume that the desired closed loop system matrix i s a
diagonal matrix given by
=
2001
rara
rA .
Then,
+= ∆
22211211
2
22
11
2),,(FFFF
ccoZVILFω
( 13)
where,) (3)1
2(122) (1)1
1(321) (1)1
2(312) (3)1
1(111
Lmc
LracFLmc
LracFLmc
LracFLmc
LracF
ωω
τωω
τωω
τωω
τ
∆+∆−+ =∆+∆−+−=∆+∆++ =∆+∆−+ =
The controller as given in (9) together with (6), (7), and
(12) is a parametrically dependent controller where the
Lorentz force detuning Lω∆, beam current I, and the
desired trajectory V are parameters defining the
controller [1].
3 SIMULATION RESULTS AND
CONCLUSIONS
Figure 1 is the block diagram of the RF control sys tem. As
we can see that the fast signal path is the impleme nted
using the CPLD while the error feed-forward is
implemented using the DSP. The total frequency resp onse
of the system is given is Figure 2 illustrates the effect of
the Lorentz force detuning on the pole locations.
Figure 1. The block diagram of the RF control syste m.
Figure 2. Root loci of the characteristic equation
Figure 3. Field amplitude response for a closed-loo p
system with a LPV-GS controller.
The system performance is given in Figure 3 in whic h the
steady state value is within the error limit. In Fi gure 4, the
performance of the feed-forward control is represen ted in
a way so that the reduction of the repetitive noise due to
the beam pulse can be observed.
Figure 4. Pulse to pulse responses of the cavity fi eld with a
LPV-GS controller.
From the analysis and the simulation results obtain ed
from our modelling, it is obviously that the perfor mance
requirements have been achieved with a full digital control
system in which the latency of the digital system h as been
take into account in the modelling. However, in the real
operation, other problems may arise, such as the ef fect of
the microphonics. The performance of the proposed R F
control system in the real operation will be report ed when
the data is available.
REFERENCES
[1] S.I. Kwon, Y.M. Wang, and A.H. Regan, “SNS
Superconducting cavity modelling and linear paramet er
varying gain scheduling controller (LPV-GSC) and PI
controller syntheses, Technical Report LANSCE-5-TN-
00-013, LANL, June, 2000.DIGITAL
PID
CONTROLLERCAVITY
KLYSTRON
AMPLIFIER
CONTROLLER
DSPBUFFER
+ -+
-
RESONANCE
CONTROLLERWATER CONTROL /
CAVITY TUNERLINEAR PARA.
VVARYING GN
SCHEDULING
-4500 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0-6000-4000-20000200040006000
RealImagPole Location change with the change of Lorentz force d etuning
: ∆wL=-3972[rad/sec]
Damping Coeff.=0.5604
Natural Freq.=7780.1[rad/sec]
: ∆wL=0.0[rad/sec]
Damping Coeff.=0.8661
Natural Freq.=5034.3[rad/sec] 0 0.5 1 1.5 2 2.5
x 10-30100020003000400050006000
time(sec)CAV_FLD_ AMP(Volts)BEAM ONRF OFF
BEAM OFF
Solid Line : Response
Dashed Line : Reference
0 0.5 1 1.5 2
x 10-30123456
time(sec)FLD_AMP(Volts)Green Pulse: RF ON/OFF
Red Pulse: BEAM ON/OFF
Red Line: 1st Iteration
Green Line: 2nd Iteration
Blue Line: 3rd Iteration
Magenta Line: 4th Iteration |
arXiv:physics/0008164v1 [physics.chem-ph] 18 Aug 2000SUBMITTED TO BRIAN HEAD SPECIAL ISSUE OF JPC
Electronic coherence in mixed-valence systems:
Spectral analysis
Younjoon Jung, Robert J. Silbey, and Jianshu Cao∗
Department of Chemistry, Massachusetts Institute of Techn ology
Cambridge, MA 01239
(January 5, 2014)
Abstract
The electron transfer kinetics of mixed-valence systems is studied via solv-
ing the eigen-structure of the two-state non-adiabatic diff usion operator for a
wide range of electronic coupling constants and energy bias constants. The
calculated spectral structure consists of three branches i n the eigen-diagram, a
real branch corresponding to exponential or multi-exponen tial decay and two
symmetric branches corresponding to population oscillati ons between donor
and acceptor states. The observed electronic coherence is s hown as a result of
underdamped Rabi oscillations in an overdamped solvent env ironment. The
time-evolution of electron population is calculated by app lying the propagator
constructed from the eigen-solution to the non-equilibriu m initial preparation,
and it agrees perfectly with the result of a direct numerical propagation of the
density matrix. The resulting population dynamics confirms that increasing
the energy bias destroys electronic coherence.
∗To whom correspondence should be addressed. Electronic mai l: jianshu@mit.edu
1I. INTRODUCTION
Quantum coherence in the dynamics of condensed phase system s has become a subject of
recent experimental and theoretical studies. A central iss ue is the observability of electronic
coherence in electron transfer systems given the fast depha sing time in many-body quantum
systems. Experimentally, with the advance in ultrafast las er technology, oscillations in elec-
tronic dynamics have been observed in photo-synthetic reac tion centers and other electron
transfer systems and are believed to arise from vibrational and/or electronic coherence.1–3
Accurate measurements on photo-induced electron transfer in mixed-valence compounds
have demonstrated oscillations in electronic populations on the femtosecond time-scale.1,4
Theoretically, detailed path-integral simulations sugge st that such oscillations take place in
electron transfer systems with large electronic coupling c onstants and are sensitive to the
initial preparation of the bath modes associated with the tr ansfer processes. Lucke et al.5
extended the non-interacting blip approximation to incorp orate the non-equilibrium initial
preparation and carried out extensive path-integral quant um dynamics simulations for elec-
tron transfer reactions. According to their findings, large -amplitude oscillations are most
likely to be observed in symmetric mixed-valence systems th at are nearly adiabatic and
with initial configurations that are centered in the Landau- Zener crossing region. Using the
transfer matrix technique,6Evans, Nitzan, and Ratner7calculated short-time evolution for
the photo-induced electron transfer reaction in (NH 3)5FeII(CN)RuIII(CN) 5. Their results
show fast oscillations in the electronic population on the s hort time-scale(20 fs) followed
by a slower population relaxation on the long time-scale(10 0 fs). They pointed out that
these fast oscillations arise as the wave-function oscilla tes coherently between the donor
and acceptor states. The calculated long-time decay rate is considerably smaller than the
prediction by the golden-rule formulae,8,9confirming the inadequacy of non-adiabatic rate
theory in studying mixed-valence systems.
2In fact, a simple classical argument helps understand the na ture of the observed os-
cillations. As a function of the ratio between λ(the bath reorganization energy) and V
(the electronic coupling constant), there is a thermodynam ic transition from the localized
electronic state in a double-well potential to the delocali zed electronic state in a single well
potential.10–14(i) In the localized regime ( λ≫V), the large reorganization energy destroys
electronic coherence; hence, electron transfer is an incoh erent rate process, which can be de-
scribed by the non-interacting blip approximation or golde n-rule rate in the non-adiabatic
limit and by transition state theory in the adiabatic limit.15–17(ii) In the delocalized regime
(λ≤V), the electronic wave function extends to both the donor and acceptor states and
electronic coherence persists over several oscillations.10For mixed-valence compounds, the
electronic coupling constant is estimated to be in the range of 103cm−1, which is in the same
order as the reorganization energy.1,7Therefore, the observed oscillations and relaxation in
mixed-valence systems are the consequence of a highly non-e quilibrium coherence transfer
process.
Due to the delocalization nature of electronic states, an ad iabatic picture18is more useful
than the diabatic representation for analyzing the short-t ime dynamics in strongly-coupled
systems. In this picture, electronic coherence arises from Rabi oscillations between two adi-
abatic surfaces and decays because of electronic dephasing . Further, initial preparation and
wave-packet dynamics can modulate Rabi oscillations and th e overall electronic dynamics.
Thus, the adiabatic representation provides a simple pictu re for mixed-valence systems as
well as a simple analytical method to model fast electron dyn amics initiated by laser pulses.
As a general approach to describe condensed phase dynamics, we recently proposed a
spectral analysis method,19which is based on eigen-structures of dissipative systems i nstead
of dynamic trajectories. An important application of the ap proach is to analyze a set of
two-state diffusion equations, which was first used by Zusman to treat solvent effects on
3electron transfer in the non-adiabatic limit. The analysis allows us to characterize multiple
time-scales in electron transfer processes including vibr ational relaxation, electronic coher-
ence, activated curve crossing or barrier crossing. With th is unified approach, the observed
rate behavior, bi-exponential and multi-exponential deca y, and population oscillations are
different components of the same kinetic spectrum. Thus, sev eral existing theoretical mod-
els, developed for limited cases of electron transfer, can b e analyzed, tested, and extended.
In particular, rate constants extracted from the analysis b ridge smoothly between the adia-
batic and non-adiabatic limits, and the kinetic spectrum in the large coupling regime reveals
the nature of the localization-delocalization transition as the consequence of two competing
mechanisms.
In this paper, the spectral analysis approach developed in R ef. 19 is employed to study the
electron transfer dynamics in mixed-valence systems. We in voke the non-adiabatic diffusion
equation proposed by Zusman to describe the electron transf er process in the over-damped
solvent regime. As discussed earlier, electron transfer in mixed-valence systems takes place
in a different kinetic regime from the thermal activated regi me described by Marcus theory.
Thus, the time-scale separation is not satisfied, and multi- exponential decay and oscillations
are intrinsic nature of electron transfer kinetics. As a res ult, the kinetic spectra exhibit bifur-
cation, coalescence, and other complicated patterns. Care ful examination of these patterns
reveals the underlying mechanisms in mixed-valence system s.
The rest of the paper is organized as follows: The spectral st ructure of the non-adiabatic
diffusion equation is formulated in Sec. II. Numerical examp les of the spectral structure
of strongly mixed electron transfer systems are presented a nd discussed in Sec. III and
concluding remarks are given in Sec. IV.
4II. THEORY
There have been extensive studies of the solvent effect on ele ctron transfer dynamics
in literature with various approaches.20–24One of the most extensively studied models for
quantum dissipation is the spin-boson Hamiltonian,14,23
HSB=ǫ
2σz+Vσx+/summationdisplay
αp2
α
2mα+/summationdisplay
α1
2mαω2
α/parenleftBigg
xα−σzcα
mαω2α/parenrightBigg2
, (1)
whereǫis the energy bias between the two electronic states, Vis the electronic coupling
constant,σzandσxare the usual Pauli matrices, and {xα,pα}represents the bath degree of
freedom with mass mα, frequency ωα, and the coupling constant cα. In this model effects
of the bath modes on the dynamics of the system can be describe d via the spectral density
defined by,
J(ω) =π
2/summationdisplay
αc2
α
mαωαδ(ω−ωα). (2)
Equivalently, the spin-boson Hamiltonian in Eq. (1) can be s eparated into the electronic
two-level part HTLSand the nuclear bath part HB,
HSB=HTLS+HB. (3)
The two-level part of the Hamiltonian can be explicitly writ ten as
HTLS(E) =U1(E)|1/an}bracketri}ht/an}bracketle{t1|+U2(E)|2/an}bracketri}ht/an}bracketle{t2|+V(|1/an}bracketri}ht/an}bracketle{t2|+|2/an}bracketri}ht/an}bracketle{t1|), (4)
where the diabatic energy surfaces U1(E) andU2(E) are functions of the stochastic variable
E, which represents the polarization energy for a given solve nt configuration.20The trans-
formation from the spin-boson Hamiltonian to the two-level system Hamiltonian has been
shown in the literature23,25by the identity,
E({xα}) =/summationdisplay
αcαxα. (5)
5It is worthwhile to mention that the polarization energy Ewas recognized as the reaction
coordinate by Marcus in formulating non-adiabatic electro n transfer theory.15Since the
electron transfer process involves the collective motion o f a large number of solvent degrees
of freedom and the two-level system is linearly coupled to th e harmonic bath modes in
the spin-boson Hamiltonian in Eq. (1), the functional form f or the free energy surface is
harmonic,26thus giving
U1(E) =(E+λ)2
4λ, (6)
U2(E) =(E−λ)2
4λ+ǫ, (7)
whereλis the reorganization energy, which is related to the parame ters in Eq. (1),
λ=/summationdisplay
αc2
α
2mαω2α=1
π/integraldisplay
dωJ(ω)
ω. (8)
Considering the fact that electron transfer processes are u sually probed at room temper-
ature in polar solvents, we can treat the bath degrees of free dom inHBclassically. Then,
the spin-boson Hamiltonian in Eq. (3) can be used to derive a t wo-level classical equation
of motion,
i∂
∂tρ(t) =Lρ(t) = (LB+LTLS)ρ(t), (9)
whereiLB={HB,}is the Poisson operator for the classical bath and LTLS= [HTLS,]/¯h
is the Liouville operator for the two level system. Explicit ly, we express Eq. (9) in terms of
the density matrix elements,
˙ρ1=L1ρ1+iV(ρ12−ρ21), (10a)
˙ρ2=L2ρ2−iV(ρ12−ρ21), (10b)
˙ρ12=L12ρ12−iω12ρ12+iV(ρ1−ρ2), (10c)
˙ρ21=L21ρ21+iω12ρ21−iV(ρ1−ρ2), (10d)
6where the Planck constant ¯ his set to unity for simplicity, ρiis the diagonal matrix element
for electronic population, and ρijis the off-diagonal matrix element for electronic coherence .
Here,Ldescribes the relaxation process of classical bath, with Lidefined on the free energy
surface for the ith electronic state, and with L12andL21defined on the averaged free
energy surface. This set of semi-classical two-state equat ions has been previously derived in
different context by several authors.20,23,27It should be mentioned that the mapping from the
spin-boson Hamiltonian into the Zusman model requires the L orentzian form of the spectral
density,
J(ω) = 2λωωc
ω2+ω2c. (11)
Furthermore, we note that many chemically and biologically important electron transfer
processes take place in the over-damped solvent environmen t. Therefore, to describe the
density matrix evolution in the electron transfer kinetics in the mixed-valence system, we
invoke the non-adiabatic diffusion equation proposed by Zus man.20Then, the bath relaxation
operators in Eq. (9) are one-dimensional Fokker-Planck ope rators Lij,
Li=DE∂
∂E/parenleftBigg∂
∂E+β∂Ui(E)
∂E/parenrightBigg
, (12)
L12=L21=L11+L22
2=DE∂
∂E/parenleftBigg∂
∂E+β∂¯U(E)
∂E/parenrightBigg
. (13)
whereβ= 1/kBT,¯Uandω12are the average and the difference of the two free energy
surfaces, respectively,
¯U(E) =U1(E) +U2(E)
2, (14)
ω12(E) =U1(E)−U2(E). (15)
The energy diffusion constant DEis defined as
DE= Ω∆2
E, (16)
7where ∆2
Eis the mean square fluctuation of the solvent polarization en ergy
∆2
E=/an}bracketle{tE2/an}bracketri}ht= 2λkBT,
andτD= 1/Ω is the the characteristic timescale of the Debye solvent. T he correlation
function of the solvent polarization energy is given by
C(t) =/an}bracketle{tE(t)E(0)/an}bracketri}ht= ∆2
Eexp(−Ωt). (17)
Note that since the nuclear dynamics is modeled by the Fokker -Planck operator, the possi-
bility of the vibrational coherence is excluded in this mode l of electron transfer dynamics. It
is worthwhile to mention that one can obtain the non-adiabat ic diffusion equation starting
from the spin-boson Hamiltonian, by first deriving the evolu tion equation for the quantum
dissipative dynamics, and then taking the semi-classical l imit using the Wigner distribution
functions, and finally assuming the over-damped diffusion li mit.23
We investigate the spectral structure of the non-adiabatic diffusion operator by calcu-
lating the eigenvalues {−Zν}and the corresponding eigen-functions {|ψν/an}bracketri}ht}. Hereafter we
use Greek indices to denote the eigenstates and Latin indice s to denote the basis states of
the non-adiabatic diffusion operator. Because the non-adia batic Liouville operator is non-
Hermitian, the eigenvalues are generally given by complex v alues, and the right and left
eigen-functions corresponding to the same eigenvalue are n ot simply the Hermitian conju-
gate to each other.28For a given eigen-value Zν, the right and left eigen-functions of the
non-adiabatic diffusion operator are obtained from
L|ψR
ν/an}bracketri}ht=−Zν|ψR
ν/an}bracketri}ht, (18)
/an}bracketle{tψL
ν|L=−Zν/an}bracketle{tψL
ν|. (19)
The method of eigenfunction solution is well known for the di ffusion process on the har-
monic potential energy surface.29For a single quadratic potential U(x) =1
2mω2x2, the
8one-dimensional Fokker-Planck operator LFP=D(∂2
∂x2+β∂
∂xU′) can be transformed into
the quantum mechanical Hamiltonian in imaginary time,
Hs=−eβU(x)/2LFPe−βU(x)/2=−1
2µ∂2
∂x2+Vs(x), (20)
whereµ−1= 2D, and the quadratic potential is
Vs(x) =D/bracketleftbigg1
4(βU′(x))2−1
2βU′′(x)/bracketrightbigg
=1
2µγ2x2−γ
2, (21)
withγ=Dmω2/kBT. Since the transformed potential in Eq. (21) is just the same form as
for a simple harmonic oscillator with zero point energy comp ensation, the eigenvalues and
the eigen-functions for the original Fokker-Planck operat or can be constructed immediately
from the eigen-solutions of the harmonic oscillator Hamilt onian. Unlike the diffusion problem
on the single potential energy surface, there have been limi ted studies on the non-adiabatic
diffusion problem involving more than one potential energy s urface. In this aspect, Cukier
and co-workers have calculated the electron transfer rate b y calculating the lowest eigenvalue
of the non-adiabatic diffusion equation; however, their cal culation was limited to the weak-
coupling regime where the Zusman rate is applicable.27
An important issue in solving the non-adiabatic diffusion eq uation for electron transfer
is the choice of the basis functions since three different fre e energy surfaces are involved in
Eq. (9): two diabatic surfaces for the population density ma trix elements and one averaged
surface for the coherence density matrix element. In this pa per, the eigen-functions of L12
are used as our basis set to represent the non-adiabatic diffu sion equation. In principle,
one could have chosen the eigen-functions of L1orL2as basis functions, however, in that
case one has to evaluate appropriate Franck-Condon factors when calculating the coupling
matrix elements even with the Condon approximation. The Fok ker-Planck operator L12is
defined on the averaged harmonic potential centered at E= 0, and its eigen-solutions are
L12|φR
n/an}bracketri}ht=−nΩ|φR
n/an}bracketri}ht, (22)
9/an}bracketle{tφL
n|L12=−nΩ/an}bracketle{tφL
n|, (23)
where the right and left eigen functions are
φR
n(E) =1
(2nn!)1
2(2π∆2
E)1
4exp/parenleftBigg
−E2
2∆2
E/parenrightBigg
Hn/parenleftBiggE√
2∆E/parenrightBigg
, (24)
and
φL
n(E) =1
(2nn!)1
2(2π∆2
E)1
4Hn/parenleftBiggE√
2∆E/parenrightBigg
, (25)
whereHnis thenth order Hermite polynomial. As shown below, this choice of t he basis set
is convenient for our purpose.
To be consistent with the L12basis set, we separate the real and imaginary parts of the
coherence density matrix, namely, u=Reρ12andv=Imρ12, and rewrite Eq. (9) as
˙ρ1= (L12+δL)ρ1−2Vv, (26a)
˙ρ2= (L12−δL)ρ2+ 2Vv, (26b)
˙u=L12u+ω12v, (26c)
˙v=L12v−ω12u+V(ρ1−ρ2), (26d)
where we have defined δLas
δL=L11− L22
2. (27)
Then, all the relevant operators in Eqs. (26a)-(26d) can be e valuated in terms of the right
and left eigen-functions of L12, giving
/an}bracketle{tφL
n|L12|φR
m/an}bracketri}ht=−nΩδnm, (28)
/an}bracketle{tφL
n|δL|φR
m/an}bracketri}ht=−Ω/radicalBigg
λ
2kBT√
m+ 1δn,m+1, (29)
/an}bracketle{tφL
n|ω12|φR
m/an}bracketri}ht=/radicalBig
2λkBT(√mδn,m−1+√
m+ 1δn,m+1)−ǫδnm, (30)
/an}bracketle{tφL
n|V|φR
m/an}bracketri}ht=Vδnm, (31)
10where we assume the Condon approximation, i.e., the electro nic coupling matrix element
is independent of the solvent degrees of freedom. With the ba sis set, we can expand the
density matrix elements as
ρ1(E,t) =∞/summationdisplay
n=0an(t)φR
n(E), (32a)
ρ2(E,t) =∞/summationdisplay
n=0bn(t)φR
n(E), (32b)
u(E,t) =∞/summationdisplay
n=0cn(t)φR
n(E), (32c)
v(E,t) =∞/summationdisplay
n=0dn(t)φR
n(E). (32d)
Substituting Eqs. (32a)-(32d) into the eigenvalue equatio n Eq. (18), we have the following
coupled linear equations
−Zνan=−nΩan−Ω/radicalBigg
λ
2kBT√nan−1−2Vdn, (33a)
−Zνbn=−nΩbn+ Ω/radicalBigg
λ
2kBT√nbn−1+ 2Vdn, (33b)
−Zνcn=−nΩcn+/radicalBig
2λkBT(√
n+ 1dn+1+√ndn−1)−ǫdn, (33c)
−Zνdn=−nΩdn−/radicalBig
2λkBT(√
n+ 1cn+1+√ncn−1) +ǫcn+V(an−bn), (33d)
which is an explicit basis set representation for the two-st ate diffusion operator in Eq. (9).
The linear equations for the left eigen-solution as defined b y Eq. (19) can be written by the
transpose of Eqs. (33a)-(33d). Diagonalizing the 4 N×4Nmatrix (N= number of basis
functions) defined in Eqs. (33a)-(33d), we obtain the eigenv alues−Zνand the corresponding
eigenvectors of the non-adiabatic diffusion operator,
|ψR
ν/an}bracketri}ht=/summationdisplay
nRnν|φR
n/an}bracketri}ht, (34)
/an}bracketle{tψL
ν|=/summationdisplay
nLνn/an}bracketle{tφL
n|, (35)
whereRnνandLνnare elements of the transformation matrices.
11In general, due to the non-Hermitian nature of the non-adiab atic diffusion operator, the
right and left eigen-functions do not form an orthogonal set by themselves. However, when
the eigenvalues are all non-degenerate, the left and right e igen-functions form an orthogonal
and complete set in dual Hilbert space.30Explicitly, we have
/summationdisplay
n=0LνnRnν′=δνν′, (36)
for the orthogonality and
/summationdisplay
νRnνLνm=δnm, (37)
for the completeness. Using these properties, we can constr uct the real time propagator for
the operator Las
G(t) =/summationdisplay
ν|ψR
ν/an}bracketri}ht/an}bracketle{tψL
ν|e−Zνt, (38)
and express the time evolution of the density matrix by proje cting a given initial distribution
onto the eigenstates, giving
|ρ(t)/an}bracketri}ht=G(t)|ρ(0)/an}bracketri}ht=/summationdisplay
ν|ψR
ν/an}bracketri}ht/an}bracketle{tψL
ν|ρ(0)/an}bracketri}hte−Zνt. (39)
Hence, the eigen-solution to the two-state non-adiabatic d iffusion equation leads to a com-
plete description of electron transfer dynamics.
III. RESULTS AND DISCUSSIONS
In the section, we present the spectral structure of the non- adiabatic diffusion operator
by diagonalizing its matrix representation in Eqs. (33a)-( 33d). In principle, we need infinite
number of basis functions to diagonalize the non-adiabatic diffusion operator, however, in
practice, we have to truncate our basis set at some finite numb er. In all the calculations
below, we have used N= 50−200 to diagonalize the 4 N×4Nmatrix and the effect of finite
number basis on the spectral structure has been carefully ex amined.
12A. Spectral Structure
1. Mixed-valence systems
In the mixed-valence compounds, the electronic coupling co nstant has the same order
of magnitude as the reorganization energy and the electron t ransfer dynamics is usually
probed experimentally at room temperature in polar solvent s. To study this process, Evans,
Nitzan, and Ratner7carried out real time path-integral simulations for the pho to-induced
electron transfer reaction in (NH 3)5FeII(CN)RuIII(CN) 5. Based on their model, we chose the
parameters for the calculation shown in Fig. 1 as βΩ = 0.6716,βλ= 18.225,βV= 11.99,
andβǫ= 18.705. As mentioned in the introduction the mapping between th e spin-boson
Hamiltonian and the semi-classical Zusman equation is not r igorously defined. For example,
for the non-adiabatic diffusion equation, the solvation ene rgy correlation function takes
an exponential form with the rate Ω, whereas, for the spin-bo son model Hamiltonian, it
depends on the functional form of the spectral density. It ca n be shown that the Ohmic
spectral density with an exponential cut-off ωc
J(ω) =ηωexp(−ω/ω c), (40)
used in the calculation of Evans et al., leads to an energy correlation function with a
Lorentzian form at high temperature,23
CSB(t)≈2ηωckBT
π1
1 + (ωct)2. (41)
Then, the relaxation rate Ω used in our calculation is taken a s the inverse of the mean
survival time of CSB(t), which is Ω = 2 ωc/π.
In Fig. 1 the spectral structure of the non-adiabatic operat or is shown in complex space.
We have used N= 200(4N= 800) basis functions to calculate the eigenvalues. To remo ve
the effect of finite basis set from the resulting spectral stru cture, we only show the first
13400 eigenvalues in the complex plane. Since the non-adiabat ic diffusion operator is non-
Hermitian, the resulting spectrum shows complex conjugate paired eigenvalues as well as
real eigenvalues, giving rise to the tree structure with thr ee major branches (which we will
call the eigen-tree ). In Fig .1, we separate the real and imaginary parts of eigen value by
−Zν=−kν−iων. (42)
Obviously, the real part, kν, is always negative as all non-equilibrium physical quanti ties
decay to zero at time infinity, and it scales linearly with the indexνsince the relaxation rate
corresponding to the nth basis state φnis proportional to n. In general, the relative magni-
tudes of real and imaginary parts of eigenvalues determine t he time-evolution of the density
matrix: the real eigenvalues correspond to the simple expon ential decay components and
the complex conjugate paired eigenvalues correspond to the damped oscillation components.
To classify the eigenvalues quantitatively according to th eir dynamic behavior, we intro-
duce the dimensionless quantity θν
θν≡2πkν
|ων|, (43)
wherekνis the decay rate and 2 π/ω νis the oscillation period. The time-evolution of the den-
sity matrix component associated with the eigenvalue Zνis an exponential decay if θν=∞,
an under-damped oscillation if θν>1, and a damped oscillation if θν≤1. The relative
amplitude of the each component depends on the overlap matri x element between the ini-
tial density matrix and the eigenstate. As an approximate cr iterion for the classification
of the eigenvalues, the slope corresponding to θν= 1 is shown in the eigen-tree diagram
in Fig. 1. There are a few eigenstates around and below the θν= 1 line, with a typical
rate ofβkν≈5. For the parameters used in the calculation, βcorresponds to ∼170 fs in
real time, and, therefore, these eigenstates exhibits damp ed oscillations with a period and
a decay time in the femtosecond regime. In their real-time pa th integral simulations, Evans
14et al. showed that the population in the acceptor state oscillates with a few femtosecond
period and these oscillation decays in within 20 femtosecon ds. Thus, qualitative features
of the electron transfer dynamics can be predicted and under stood from a careful exami-
nation of the spectral structure. Since the spectral analys is presented here is based on the
semi-classical diffusion equation while the path-integral study is based on the quantum me-
chanical spin-boson Hamiltonian, the comparison between t he two approaches is expected
to be qualitative. In the following subsection, further ana lysis reveals the nature of these
oscillations.
2. Dependence on the coupling constant V
To examine the underlying spectral structure in more detail s, eigenvalues of the non-
adiabatic diffusion operator are plotted as functions of the electronic coupling constant in
Fig. 2. All the parameters except for the electronic couplin g constant are the same as used
in Fig. 1.
In Fig. 2(a), the real parts of the first 20 eigenvalues are sho wn as functions of the
electronic coupling constant. Note that eigenvalues corre sponding to complex conjugate
pairs have the same real part, thus they coalesce in the real e igenvalue diagram. When
the coupling constant is very small ( βV≪1), the real part of the first non-zero eigenvalue
is very well separated from the eigenvalues of excited state s, so the dynamics of electron
transfer can be considered as a incoherent rate process with a well-defined rate constant,
k1. When the coupling constant is larger ( βV≈1), the first excited state becomes close
to the second excited state, and they start to merge into a com plex conjugate pair. If the
coupling constant increases further, eigen-values show a b ifurcation behavior at βV≈10.
Therefore, in this regime, the electron transfer kinetics s how multiple time-scale relaxation
as well as coherent oscillation. The complicated behavior o f coalescence and bifurcation in
15the real eigenvalue appears more frequently at higher state s.
Another interesting feature of the real eigenvalue diagram is that a set of real eigenval-
ues decreases consistently as the coupling constant increa ses from zero. It turns out that
these eigenstates take on large imaginary parts, which are r esponsible for the onset of the
imaginary branches of the eigen-tree. In Fig. 2(b), the imag inary parts of the lowest 30
eigenvalues are plotted as functions of the coupling consta nt. Interestingly, the imaginary
part of the eigenvalue increases approximately linearly wi th the coupling constant at large
coupling regime. In fact, the dependence on the coupling con stant is similar to that of the
Rabi frequency for the two-level system,
ΩR=√
ǫ2+ 4V2, (44)
which is shown in Fig. 2(b). As pointed out in a recent paper,18electronic coherence in
mixed-valence systems arises from Rabi oscillations betwe en two adiabatic surfaces and
decays because of dephasing.
To demonstrate the correlation of the real and imaginary par ts of the eigenvalues as
functions of the coupling constant, we present a three dimen sional plot of the spectral
structure in Fig. 2(c). For clarity, only the positive branc hes of the imaginary eigenvalues
are shown. If we compare Fig. 2(c) with Fig. 2(a), the very rap idly decaying states shown in
Fig. 2(a) take on large imaginary parts corresponding to the Rabi oscillations as the coupling
constant increases, and these states are responsible for th e onset of the imaginary branches
in the eigen-tree for the mixed-valence system shown in Fig. 1.
B. Density Matrix Propagation
To check the validity of the spectral analysis as a density ma trix propagation scheme,
we calculated the time-evolution of the density matrix by ap plying the propagator defined
16by Eq. (38) to the initial density matrix for various energy b iases. Although it may seem
straightforward to use the spectral method as a propagation scheme, the case for a non-
Hermitian operator is not trivial and has not been explored. The main reason is that
though the left and right eigen-functions of a non-Hermitia n operator can be shown to form
a bi-orthogonal set for the non-degenerate eigenvalue case , numerically these eigen-functions
may not be stable enough to be used as a complete orthonormal b asis for the density matrix
propagation, especially in the nearly degenerate eigenval ue case. We can understand the
situation as follows: When the two nearly degenerate eigenv aluesZ1andZ2are obtained
from a non-Hermitian operator, the orthogonality implies t hat/an}bracketle{tL2|and|R1/an}bracketri}htare orthogonal
to each other as well as /an}bracketle{tL1|and|R2/an}bracketri}ht. When two eigenvalues become very close to each
other, unlike the Hermitian operator case, /an}bracketle{tL1|and/an}bracketle{tL2|almost coincide and so do |R1/an}bracketri}ht
and|R2/an}bracketri}ht, so that /an}bracketle{tL1|and|R1/an}bracketri}htbecome almost orthogonal to each other. To still satisfy
the normalization condition /an}bracketle{tLn|Rn/an}bracketri}ht= 1 in this case, the eigenfunction should be scaled
up, thus making the spectral structure very sensitive to the numerical error involved in the
calculation of eigenfunctions. For an interesting discuss ion on this point, one may refer
to the work by Nelson and co-workers.30Due to this numerical instability, the use of the
spectral method as a density matrix propagation scheme is no t without limitation.
Figure 3a shows the spectral structure and the time-evoluti on of the density matrix
propagation for the case of βΩ = 1,βλ= 15,βV= 12, and βǫ= 5. Generally, when
the energy bias is small ( βǫ≤5), the left and right eigenfunctions can form a complete
orthonormal basis set, so the spectral method is stable and c an be used as a numerical
propagation method for the density matrix. With a large ener gy bias, however, the calculated
eigenfunctions may not form a complete orthonormal basis. T o model for the photo-induced
back electron transfer experiment in the mixed-valence com pounds the initial density matrix
is chosen as a thermal equilibrium distribution of the donor state(i.e. 1-state) pumped to
17the acceptor state(i.e. 2-state),4,5,7
ρi(E,0) =1√
2π∆Eexp/parenleftBigg
−(E+λ)2
2∆2
E/parenrightBigg
δi2, (45a)
ρ12(E,0) =ρ21(E,0) = 0. (45b)
It would be straightforward to calculate the spatial distri bution of the density matrix in time
ρ(E,t) by applying the propagator in Eq. (38) to the above initial d ensity matrix; however,
to demonstrate the overall temporal behavior only the time e volution of the total population
in the acceptor state is calculated,
P2(t) =/integraldisplay
dEρ 2(E,t). (46)
In order to check the validity of the spectral method as a prop agation scheme in this case,
we also calculated the time evolution of the density matrix b y directly solving the 4 N
differential equations for the expansion coefficients of the d ensity matrix using the Bulirsh-
Stoer algorithm,31and the comparison in Fig. 3(a) shows a perfect agreement. If only the
transient behavior is concerned with, the direct propagati on method would be preferred over
the spectral method, however, the spectral propagation has the advantage when calculating
the long time behavior once the complete spectrum is known. O verall, the computational
costs for two method are comparable to each other. As expecte d from the spectral structure
shown in the previous section the population in the acceptor state shows an underdamped
coherent oscillation behavior at initial times followed by a damped oscillation behavior at
later times.
Further, we have also studied the density matrix propagatio n for different energy biases
to examine the electronic dephasing effect. As seen from Fig. 4(a), the increase in energy
bias destroys the electronic coherence dramatically. Anot her interesting observation is the
phase shift in the population dynamics as the energy bias is v aried, and it is because the
Rabi oscillation frequency increases with energy bias. We c an confirm the temporal behavior
18of the density matrix propagation by examining the spectral structure shown in Fig. 4(b).
The period of the initial coherence is estimated to be τosc≈0.25βfrom Fig. 4(a). In
comparison, the Rabi frequency for the corresponding adiab atic two-level system is given by
ΩR=√
ǫ2+ 4V2≈25β−1, which can also be obtained from the onset of imaginary branc hes
in the eigen-tree shown in Fig. 4(b), and the estimation is co nsistent with the oscillation
period observed in the dynamics since τosc≈2π/ΩR. The real eigenvalues of the lowest
excited states in the the imaginary branches are estimated t o beβk≈1−2, and they agree
with the decay time of the oscillation amplitude in Fig. 4(a) , confirming the validity of the
spectral method as a density matrix propagation scheme. Eve n though it has been well
known in the literature that the damping of population is enh anced with increased energy
asymmetry,14we have also confirmed this through the spectral analysis met hod.
As an example of the eigenfunction responsible for the coher ent oscillation behavior ob-
served in Fig. 4 (b), we show the left and right eigenfunction s corresponding to a complex
eigenvalueβZ= 2.6228±i26.394 for a symmetric case ( βǫ= 0) andβZ= 2.8057±i26.466
for an asymmetric case( βǫ= 5) in Figs. 5 and 6. The eigenfunctions corresponding to
a complex conjugate pair of eigenvalues are also complex con jugate to each other; there-
fore, the frequency spectrum of the density matrix evolutio n is proportional to the norm of
wavefunction. We note that the left eigenfunction is more ex tended than the right eigenfunc-
tion. Although the population distribution in the donor and acceptor states corresponding
to coherent oscillation is inverted with respect to the Bolt zmann distribution, it does not
contribute to the steady-state population distribution du e to the transient nature.
IV. CONCLUDING REMARKS
In this paper we have applied the spectral analysis method to the non-adiabatic two-state
diffusion equation, that describes electron transfer dynam ics in Debye solvents. In particular,
19we have examined electronic coherence in mixed-valence com pounds, and demonstrated
that underdamped Rabi oscillations are observed in an overd amped solvent environment.
Detailed study of the spectral structure of the non-adiabat ic operator for various energy
biases and coupling constants allows us to determine the und erlying mechanisms of electron
transfer kinetics. Eigenvalues form three branches in the e igen-diagram: a single branch of
real eigenvalues and two symmetric branches of complex conj ugate eigenvalues. In strongly
coupled systems, all three branches have a similar order of m agnitude, indicating that both
multiple-exponential decay and coherent oscillations can be observed experimentally.
We have investigated the dependence of the spectral structu re on the coupling con-
stant. In the very weak coupling regime, the lowest excited s tate is well separated from
higher states, which makes the electron transfer dynamics a well-defined rate process. In
the strong coupling regime, however, the eigenvalue diagra m shows coalescence/bifurcation
behavior in the complex plane. We have used the spectral meth od to calculate the time-
evolution of the density matrix, and indeed, observed elect ronic coherence in the temporal
behavior of population in the acceptor state for non-equili brium initial distributions. We
also found a good agreement between results of the spectral p ropagation method and of
the numerical propagation method for small energy bias case s. Due to non-Hermitianity of
the non-adiabatic operator, the spectral propagation meth od was not numerically stable for
large energy bias cases.
For an isolated quantum system, the eigen-solution to the Sc hr¨ odinger equation com-
pletely determines its dynamics. In a similar fashion, the e igen-solution to the non-adiabatic
diffusion operator completely characterizes the dynamics o f a dissipative system and thus
provides a powerful tool to analyze dissipative dynamics. I t is well known that quantum dy-
namics comes from the underlying spectra, especially in gas -phase chemical systems;32how-
ever, the spectral aspect of condensed phase dissipative sy stems has not been well recognized
20yet and deserves further investigation. Though the analysi s presented here is restricted to
semi-classical dissipative systems, it may also be applied to quantum dissipative dynamics.
In principle, we can derive the evolution equation for quant um dissipative systems either
from first principles or through numerical reduction, and th en pose the quantum dissipative
equation of motion as an eigen-value problem. Along this lin e, the dissipative dynamics of
the spin-boson Hamiltonian, which has been studied mostly a s a dynamic problem,6,33can
also be explored as a spectral problem in the future.
ACKNOWLEDGMENTS
The authors would like to thank NSF for financial support. One of us (YJ) would like
to thank the Korean Foundations for Advanced Studies for fina ncial support.
21REFERENCES
1Vos, M. H. ; Rappaport, F. ; Lambry, J.-C. ; Breton, J. ; Martin , J.-L. Nature 1993363,
320.
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2609.
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Chem. Phys. 1997107, 8397.
6Makarov, D. ; Makri, N. Chem. Phys. Lett. 1994221, 482.
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8Coalson, R. D. ; Evans, D. G. ; Nitzan, A. J. Chem. Phys. 1994101, 486.
9Cho, M. ; Silbey, R. J. J. Chem. Phys. 1995103, 595.
10Harris, R. A. ; Silbey, R. J. Chem. Phys. 198378, 7330.
11Silbey, R. ; Harris, R. A. J. Chem. Phys. 198480, 2615.
12Carmeli, B. ; Chandler, D. J. Chem. Phys. 198582, 3401.
13Chandler, D. in Liquides, Cristallisation et Transition Vitreuse, Les Hou ches, Session LI ,
edited by Levesque, D. ; Hansen, J. ; Zinn-Justin, J. , Elsevi er, New York, 1991.
14Leggett, A. J. ; Chakravarty, S. ; Dorsey, A. T. ; Fisher, M. P. A. ; Garg, A. ; Zwerger,
W.Rev. Mod. Phys. 198759, 1.
15Marcus, R. A. ; Sutin, N. Biochim. Biophys. Acta. 1985811, 265.
2216Newton, M. D. ; Sutin, N. Annu. Rev. Phys. Chem. 198435, 437.
17Bader, J. S. ; Kuharski, R. A. ; Chandler, D. J. Chem. Phys. 199093, 230.
18Cao, J. Chem. Phys. Lett. 1999312, 606.
19Cao, J.; Jung, Y. J. Chem. Phys. 2000 in press.
20Zusman, L. D. Chem. Phys. 198049, 295.
21Calef, D. F. ; Wolynes, P. G. J. Phys. Chem. 198387, 3387.
22Hynes, J. T. J. Phys. Chem. 198690, 3701.
23Garg, A. ; Onuchic, J. N. ; Ambegaokar, V. J. Chem. Phys. 198583, 4491.
24Sparpaglione, M. ; Mukamel, S. J. Chem. Phys. 198888, 3263.
25Cao, J. ; Voth, G. A. J. Chem. Phys. 1997106, 1769.
26Onuchic, J. N. ; Wolynes, P. G. J. Chem. Phys. 199398, 2218.
27Yang, D. Y. ; Cukier, R. I. J. Chem. Phys. 198991, 281.
28Simons, J. Chem. Phys. 19732, 27.
29Risken, H. The Fokker-Planck Equation (Springer-Verlag New York, 1984).
30Dahmen, K. A. ; Nelson, D. R. ; Shnerb, N. M. cond-mat/9903276 1999 .
31Press, W. H. ; Teukolsky, S. A. ; Vetterling, W. T. ; Flannery, B. P.Numerical Recipes in
FORTRAN 2nd edition (Cambridge University Press, Cambridge, 1992) .
32Field, R. W. ; O’Brien, J. P. ; Jacobson, M. P. ; Solina, S. A. B. ; Pollik, W. F. ; Ishikawa,
H.Adv. Chem. Phys. 1997101, 463.
33Wang, H. ; Song, X. ; Chandler, D. ; Miller, W. H. J. Chem. Phys. 1999110, 4828.
23FIGURES
FIG. 1. A plot of the lowest 400 eigenvalues for the non-adiab atic operator in a mixed-valence
system. The parameters are : βΩ = 0 .6716, βλ= 18.225,βV= 11.99, and βǫ= 18.705. The
dot-dashed line is for the case k=ω/2π.
FIG. 2. Plots of (a) real and (b) imaginary parts of the lowest 30 eigenvalues as a function of
the coupling constant, V. Except for the coupling constant, all the other parameters are set equal to
those used in Fig. 1. In Fig. 2(b), open circles correspond to the Rabi frequency Ω R=√
ǫ2+ 4V2.
Figure 3(c) shows the three dimensional plot of eigenvalues as a function of the coupling constant.
FIG. 3. Comparison between the result of direct numerical pr opagation and spectral propa-
gation. The parameters are chosen as βΩ = 1, βλ= 15, βV= 12, and βǫ= 3.
FIG. 4. Comparison of (a) the dynamics and (b) the spectra in t he mixed-valence system for
three different energy biases. Except for the energy bias, al l the other parameters are set equal to
those used in Fig. 3. Agreements between the results of numer ical and spectral propagation have
been checked in these cases.
FIG. 5. (a) Right and (b) left eigenfunctions with an eigenva lueβZ= 2.6228±i26.394 for a
symmetric bias case.( βǫ= 0) All the other parameters are set equal to those used in Fig . 3 except
for the energy bias. Each line corresponds to ρ1(solid), ρ2(dashed), u(dot-dashed), and v(dotted),
respectively.
FIG. 6. (a) Right and (b) left eigenfunctions with an eigenva lueβZ= 2.8057±i26.466 for an
asymmetric bias case.( βǫ= 5) All the other parameters are set equal to those used in Fig . 3 except
for the energy bias. Each line corresponds to ρ1(solid), ρ2(dashed), u(dot-dashed), and v(dotted),
respectively.
24−150 −100 −50 0 50 100 150010203040506070
βωβk0 5 10 15 20 2501234567
βVβk0 5 10 15 20 25−60−40−200204060
βVβω0102030405060
0123450510152025
βωβkβV0 1 2 3 4 5
t/00.10.20.30.40.50.60.70.80.91P2(t)spectralpropagationdirectpropagation
β0 1 2 3 4 5
t/00.10.20.30.40.50.60.70.80.91P2(t)
ββε=0βε=5βε=3−80 −60 −40 −20 0 20 40 60 800102030405060708090100
βωβk
βε=0
βε=3
βε=5-40 -20 0 20 40
E00.050.10.15 |ρ|-40 -20 0 20 40
E00.511.5 |ρ|-40 -20 0 20 40
E00.10.20.3 |ρ|-40 -20 0 20 40
E01234|ρ| |
Faraday Cup Measurements of Ions Backstreaming into a
Electron Beam Impinging on a Plasma Plume
G. Guethlein, T. Houck, J. McCarrick, and S. Sampayan, LLNL, Livermore, CA 94550,USA
Abstract
The next generation of radiographic machines based on
induction accelerators is expected to generate multiple,
small diameter x-ray spots of high intensity. Experiments
to study the interaction of the electron beam with plasmasgenerated at the x-ray converter and at beamline septa are
being performed at the Lawrence Livermore National
Laboratory (LLNL) using the 6-MeV, 2-kA Experimental
Test Accelerator (ETA) electron beam. The physics issues
of concern can be separated into two categories. The
interaction of subsequent beam pulses with the expanding
plasma plume generated by earlier pulses striking the x-
ray converter or a septum, and the more subtle effect
involving the extraction of light ions from a plasma by
the head of the beam pulse. These light ions may be due
to contaminants on the surface of the beam pipe or
converter, or, for subsequent pulses, in the material of theconverter. The space charge depression of the beam could
accelerate the light ions to velocities of several mm/ns.
As the ions moved through the body of the incoming
pulse, the beam would be pinched resulting in a moving
focus prior to the converter and a time varying x-ray spot.
Studies of the beam-generated plasma at the x-ray
converter have been previously reported. In this paper we
describe Faraday cup measurements performed to detect
and quantify the flow of backstreaming ions as the ETA
beam pulse impinges on preformed plasma.
1 INTRODUCTION
The interaction of an intense electron beam with the
x–ray converter in radiographic machines is an active area
of research[1]. A small, stable (constant diameter and
position) electron beam spot size on the converter is
essential to achieving a high-quality radiograph. Beam
parameters such as emittance and energy variation have
been considered limiting factors for realizing the optimum
spot size. However, advancements in induction accelerator
technology have improved beam quality to a level where
the beam interaction with the converter may be the
limitation for the next generation of radiographic
machines. Two areas of concern are the emission of light
ions [2] that can “backstream” through the beam due to
space charge potential, and interaction between the beam
and the plasma generated by previous pulses during
multiple pulse operation.
Previously reported studies have described the use of
Faraday cups to characterize the plasma plume generated
by the beam at the x-ray converter[3]. We have now used
Faraday cups to detect ions that are extracted from a
plasma plume by the electron beam and “backstream”through the beam. The studies reported below were
performed on the ETA-II accelerator at LLNL using a
6–MeV, 2-kA, 70-ns electron beam. The plasma plume
was generated by either a laser or a flashboard.
2 EXPERIMENTAL LAYOUT
2.1 Faraday Cups
The Faraday cups were comprised of two, electrically
isolated, concentric cylinders as illustrated in Fig. 1. The
inner cylinder could be biased up to 1.2 kV with respect
to the grounded outer cylinder. The OD was 5 cm with an
aperture of 1.9 cm. The low ratio of aperture to cup
length was to minimize the escape of secondary electrons
generated by the impact of the positive ions with the
inner cylinder. As shown in Fig. 2, the cups were situated
at the entrance of a solenoid operating with an on-axis
peak field of approximately 5 kG.
The inner cup discharged to ground through the 50- Ω
input of an oscilloscope, permitting the rate of charge
interception (current) to be measured. The sensitivity of
the cups to ion density, assuming single ionization, is:
minminnI
Aev= , where (1)
nmin is the minimum density, Imin is the minimum
detectable current, A is the aperture area, and v is the ion
velocity. For a nominal v of 2 mm/ns, nmin is 2x105 cm-3
(Imin was 80 µA).
2.2 Target Chamber
The x-ray converter was comprised of a rotating wheel
that held several “targets” to permit multiple shots before
the x-ray converter had to be replaced. The majority of
data was taken for tantalum targets of three thicknesses: 1
mm, 0.25 mm, and 0.127 mm. A series of experiments
were also performed wherein a thin foil was placed from
5 mm to 15 mm in front of the target to prevent ions
produced at the target from backstreaming into the beam.
Two-micron thick nitrocellulose and five-micron thick
Mylar foils were used. For some runs, Al was sputtered
onto the Mylar film to produce a conducting surface.
insulating
support
SMA
Feedthrough5 cm
7 cm1.9 cm
Figure 1: Schematic of the Faraday cup.Beam (2 kA, 5.5 MeV)Focusing
Solenoid
Faraday CupGraphite Safety Collimator
Target 25 cmMirror
Laser Beam
Flashboard
Figure 2: Schematic showing relative positions of the
faraday cups with respect to the beam line and target.
Two different target chambers were used. The principal
differences between their configurations were the location
of the focusing solenoid with respect to the target, the
angle of the viewing ports with respect to axis, and
number of ports (six or eight) that contained the Faraday
cups. The cups were located about 25 cm from the
beam/target intersection for both configurations. In the
first configuration the cups were at an angle of 30 ° from
the beam axis and located partially under the solenoid. Thesecond configuration reduced the angle to 20 ° and the cups
were just outside the solenoid entrance. Data shown beloware for two cups located approximately on opposite sides
of the beamline; cup #1 was towards the bottom and cup
#2 was at the top.
The actual chambers did not have the perfect cylindrical
symmetry shown in Figure 2. Various diagnostics andaccess ports were located around the chambers. However,
the largest deviation from the axial symmetry was the
target wheel. The axis of this wheel was located several
centimeters below the beam axis to avoid being struck anddamaged by the beam. The OD of the wheel extended fromabout two cm above the axis to about 7 cm below. Somecombination of the physical geometry of the system ledto azimuthal asymmetries in the Faraday cup data.
2.3 Laser and Flashboard
An 0.8J, 10 ns FWHM, Nd:YAG laser was directed at
the target and timed to produce a plasma of sufficient
density that would simulate target debris such as multiple
pulse electron beams might encounter near the target. Thepulse energy given below is the energy measured at the
laser. The energy at the target was approximately half thatvalue and decreased as the window was covered with
debris.
A flashboard was added to the experiment to generate a
"cooler" plasma than was possible with the laser. The
flashboard was constructed from semi-rigid coax cable by
cutting an end so the inner conductor protruded about 3
mm. The inner conductor was flattened and covered with a
graphite solution (aerodag) to enhance breakdown to theouter conductor. A 5 kV pulse was applied to the cable
producing an arc at the tip that was sustained for a few
microseconds. Although relatively simple in construction,the flashboard produced plasma plumes that were more
consistent in density and velocity than those generated
using the laser. The tip of the flashboard was located
about 1.5 cm from the beam/target intersection.
3 RESULTS AND DISCUSSION
The first ion signature measured by the Faraday cups
occurred when a 250-mJ laser pulse impacted the converter
about 30 ns prior to beam time. The output of one of thecups is shown in Figure 3 superimposed on a signal when
there was no prebeam laser pulse. The small signal
labeled "laser pulse at converter" was due to UV radiation
(generated by the laser at the converter) knocking electronsoff the surface of the cup. The larger signal labeled "e-
Beam Signal" is due to beam electrons scattered off the
converter[4]. Both signals are detected at the Faraday cup
within 1 ns of the respective events and were used as
timing fiduciaries. The large positive signal following thebeam was caused by ions extracted from the laser produced
plasma plume. In the 30 ns before beam arrival, this
plume would have expanded only a few mm from the
converter's surface. Ions were detected for experiments
where the laser pulses arrived on target as late as midway
through the e-beam pulse.
The species and energy of the ions can be inferred if
some assumptions are made about the path the ions took
to the cup, the relative time the ions were accelerated into
the e-beam, and the magnitude of the e-beam space charge
depression. With a number of qualifications, we believe
that the ion signal was due primarily to H
+ ions with
energy on the order of 100 KeV. An order of magnitude
estimate of the ion density can be made by assuming that
the cup intersected a portion of a uniform 2 π-stereradian
distribution of ions expanding from the target. For Figure3, the ion density in the vicinity of the target is about
10
13 cm-3.
-300306090
0 50 100 150 200 250 300Current (mA)
Time (ns)With laser plume
No laserLaser pulse
at converter
e-Beam
Signal
Figure 3: Faraday cup signal for an e-beam striking a laser
generated plasma. There was no bias on the cup.-500-400-300-200-1000100
100 150 200 250 300 350 400Current (mA)
Time (ns)No Flashboard
Cup #2
Cup #1e-Beam
Signal
Cup #2
Figure 4: Faraday cup signals for an e-beam striking a
flashboard generated plasma.
Figure 4 shows Faraday Cup signals when the beam
impacted the plasma plume generated by the flashboard in
front of the target superimposed on the baseline Cup #2
signal for no flashboard. The figure also illustrates the
difference in signals received at different azimuthal
positions. The flashboard was fired about a microsecond
before beam arrival to ensure that the plasma plume had
reached the beam/target intersection. The later arrival of
the main ion signal infers that heavier ions were involved
than for the case shown in Figure 3. For this case the ion
signal could have been caused by C+ ions with energy on
the order of 100 KeV in addition to H+ ions.
A non-conducting, thin film was suspended in front of
the target to impede the flow of ions into the beam. We
did not believe that sufficient energy would be deposited
into the film by the beam to ionize material. The lack ofions generated by the beam on Ta targets supported this
scheme. However, as shown in Figure 5, the beam
striking the film could generate a large ion signal. This
occurred sporadically when the film was 1.5 cm in front
of the target, essentially every time at 1.0 cm, and never
at 0.5 cm. A conductive coating on the film prevented ion
generation. A similar effect was noted on experiments
performed on the PIVAIR accelerator at CESTA[5].
-100-80-60-40-200204060
100 150 200 250 300 350 400Current (mA)
Time (ns)Cup #2e-Beam
Signal
nitrocellulose film
1.5 cm in front of
0.1 mm thick Ta targetion signal
Figure 5: Faraday cup signal for an e-beam striking a non-
conducting film 1.5 cm in front of a Ta target.-40-200204060
100 150 200 250 300 350 400Current (mA)
Time (ns)Laser with FilmLaser with no Film
e-Beam
Signal
Figure 6: Faraday cup signals showing effect of film on
backstreaming ions.
The film was effective at blocking ions in some cases.
Figure 6 shows the difference between film and no film
when a 53 mJ laser pulse fired 30 ns before the e-beam
was used to generate a plasma source for ions. The film
was placed 1.5 cm in front of the target for this data.
4 SUMMARY
The Faraday cups proved effective at detecting and
identifying ions backstreaming into the e-beam..
Indications of these ions were also supported by a time
varying beam spot on target as observed by the x-ray spot
produced. No ions were detected for the e-beam impinging
on a metallic target without a preformed plasma.
5 ACKNOWLEDGEMENTS
Rene Neurath provided and operated the Nd:YAG laser.
Dave Trimble fabricated the flashboard and pulsing
system. Cliff Holmes was responsible for the fabrication
of the Faraday cups and target chambers. All experiments
were performed under the guidance of John Weir at the
ETA-II Accelerator Facility. This work was 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.
REFERENCES
[1] Sampayan, S., et al., "Beam-Target Interaction Ex-
periments for Bremsstrahlung Converter Applica-
tions," Proc. 1999 Part. Accel. Conf., p. 1303.
[2] G. Caporaso and Chen, Y-J, “Analytic Model of Ion
Emission From the Focus of an Intense Relativistic
Electron Beam on a Target," Proc. XIX Int'l LINAC
Conf., p. 830 (1998).
[3] T. Houck, et al., "Faraday Cup Measurements of the
Plasma Plume Produced at an X-Ray Converter,"
Proc. XIX Int'l LINAC Conf., p. 311 (1998).
[4] Falabella,S., et al., "Effect of Backscattered Electrons
on Electron Beam Focus," this conf., TUB11.
[5] C. Vermare, et al., IEEE Trans. Plasma Sci., Vol.
27, No. 6, pp. 1566 –1571 Dec. 1999. |
arXiv:physics/0008166 18 Aug 2000OVERVIEW OF THE HEAVY ION FUSION PROGRAM*
C.M. Celata, Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
for the U.S. Virtual National Laboratory for Heavy Ion Fusion
* This work supported by the Office of Energy Research, U.S. Department of Energy, under contract number DE-AC03-
76SF00098.Abstract
The world Heavy Ion Fusion (HIF) Program for inertial
fusion energy is looking toward the development and
commissioning of several new experiments. Recent and
planned upgrades of the facilities at GSI, in Russia, and in
Japan greatly enhance the ability to study energy
deposition in hot dense matter. Worldwide target design
developments have focused on non-ignition targets for
nearterm experiments and designs which, while lowering
the energy required for ignition, tighten accelerator
requirements. The U.S program is transitioning between
scaled beam dynamics experiments and high current
experiments with power-plant-driver-scale beams. Current
effort is aimed at preparation for the next-step large
facility, the Integrated Research Experiment (IRE)-- an
induction linac accelerating multiple beams to a few
hundred MeV, then focusing to deliver tens of kilojoules
to a target. The goal is to study heavy ion energy
deposition, and to test all of the components and physics
needed for an engineering test of a power plant driver and
target chamber. This paper will include an overview of
the Heavy Ion Fusion program abroad and a more in-depth
view of the progress and plans of the U.S. program.
1 INTRODUCTION
The international program in Heavy Ion Fusion is at
the threshold of planning and constructing experiments
which will test many accelerator and target issues in
Figure 1: Induction linac driver subsystems & parameters
parameter ranges relevant to an eventual powerplant. In
Europe, Russia, and Japan the emphasis is on
measurements of stopping of heavy ions in matter.
Accelerator studies there focus on the challenge of
delivering very high-current beams to the target. In the
U.S., small scaled experiments have been completed, anda series of accelerator experiments using driver-scale
beams are in the construction and design phases. This
paper will concentrate on the U.S. program, since it relies
on a linac approach and has an accelerator physics
program dedicated to energy production research. Note
that much of the work cited here, and much more detail on
HIF can be found in [1].
2 HIF RESEARCH IN THE U.S.
2.1 The Induction Linac Approach
Present Heavy Ion Fusion indirect drive targets require
1-7 MJ of heavy ions delivered to the target in about 10
ns, at a kinetic energy ~ 2-10 GeV. The total charge
implied by this requirement leads to significantly higher
line charge densities than can be stored in a single storage
ring. While European designs have concentrated on
combining and compressing pulses from several rings in
the last phase of transport to the target, the U.S. approach
is to make use of the efficiency of the induction linac for
transporting high current beams in a single multibeam
linac. A schematic view of one possible power plant
accelerator (driver) is shown in Fig. 1. An injector
provides multiple (~30-200) beams of heavy ions (e.g.,
Cs+) at ~ 1.5-2 MeV, with current per beam of
approximately 1 A. The beams are space-charge-
dominated, with tune depressed by space charge to ~1/10
of the single particle tune. They are accelerated in parallel
through induction cores which encircle the array of beams.
Each beam is individually focused by quadrupoles-- in the
example of Fig. 1, electrostatic quadrupoles at low
energy, with a transition to magnetic quadrupoles at 50-
100 MeV. Maximizing the transverse current density of
the beam array, thereby minimizing the induction core
radius, is important in controlling the cost of the
accelerator. But electrostatic quadrupoles optimize at a
smaller aperture than is optimal for high overall current
density in magnetic quadrupoles. Therefore a 4-to-1
transverse combining of beams is included at the
transition to magnetic focusing. After combining, the
beam is accelerated to its final energy, then compressed
longitudinally by a factor ~ 10 to obtain the short pulse
required by the target. It is of utmost importance to keep
the emittance growth in the accelerator low, in order that
the beams can be focused to a spot of a few millimeter
radius at the target. Desirable final normalized emittance
is ≤ about 20 π mm-mrad.
~2-3 MeV
~1 A / beam~ 100 Mev
~10 A / beam~ 10 Gev
~400 A / beam~ 10 Gev
~4000 A / beam |
arXiv:physics/0008167v1 [physics.acc-ph] 19 Aug 2000DIPOLEMODE DETUNING INTHENLCINJECTORLINACS∗
K.L.F. Bane, Z. Li,SLAC, Stanford University,Stanford, CA 94309,U.S.A.
1 INTRODUCTION
A major consideration in the design of the accelerator
structures in the injector linacs of the JLC/NLC[1] is to
keep the wakefield effects within tolerances for both the
nominal(2.8ns)andalternate(1.4ns)bunchspacings. One
important multi-bunch wakefield effect is beam break-up
(BBU), where a jitter in injection conditions of a bunch
train is amplified in the linac; another is static emittance
growthcausedbystructuremisalignments.
The injector linacs comprise the prelinac, the e+drive
linac, the e−booster, and the e+booster. The first three
willoperateatS-band,thelastone,atL-band. Comparedto
the main (X–band)linac, the wakeswill tendto be smaller
byafactor 1/64and1/512,respectively,fortheS–andL–
band linacs. This reduction, however,—especially for the
S-band machines—, by itself, is not sufficient. Two ways
of reducing the wake effects further are to detune the first
pass-band dipole modes and to damp them. In this report
our goal is to design the accelerator structures for the in-
jectorlinacsusingdetuningalone,anoptionthatissimple r
than including damping. We will consider only the effects
of modes in the first dipole pass-band, whose strengths
overwhelminglydominate. The effects of the higher pass-
band modes, however, will need to be addressed in the fu-
ture. For a more detailed versionof this worksee Ref. [2].
Note that the design of the e+booster structure, which is
straightforward,will notbediscussedhere.
Machine properties for the injector linacs are given in
Table 1. Shown are the initial and final energies E0,Ef,
the machine length L, the initial (vertical) beta function
averaged over a lattice cell ¯β0, and the parameter ζfor
a rough fitting of the beta function to ¯β∼Eζ. The rf
frequencies are sub–harmonics of 11.424 GHz. As for
beam properties, for the nominal bunch train configura-
tion (95 bunches spaced at 2.8 ns), the particles per bunch
N= 1.20, 1.45, 1.45, 1.60×1010and normalized emit-
tanceǫyn= 3×10−8,10−4,10−4,.06rm,fortheprelinac,
e+drive, e−booster, and e+booster,respectively. For the
alternateconfiguration(190bunchesspacedat1.4ns) Nis
reducedby 1/√
2.
Table1: Machinepropertiesoftheinjectorlinacs.
Name E0,Ef[GeV] L[m] ¯β0[m] ζ
Prelinac 1.98,10.0 5588.61/2
e+Drive 0.08,6.00 5082.41/2
e−Booster 0.08,2.00 1633.41/4
e+Booster 0.25,2.00 1841.51
∗Work supported by the U.S. Department of Energy under contra ct
DE-AC03-76SF00515.2 EMITTANCE GROWTH
2.1 BeamBreak-Up (BBU)
In analogyto single-bunch BBU in a linac[3], multi-bunch
BBUcanalsobecharacterizedbyastrengthparameter,but
onedependentonbunchnumber m:
Υm=e2NLS m¯β0
2E0g(Ef/E0, ζ) [ m= 1, . . ., M ],
(1)
withMthenumberofbunchesina train. Thesumwake
Sm=m−1/summationdisplay
i=1W[(m−i)∆t] [ m= 1, . . ., M ],(2)
withWthe transverse wakefield and ∆tthe time interval
between bunches in a train. The wake, in turn, is given by
a sumoverthedipolemodesinthe acceleratorstructures:
W(t) =Nm/summationdisplay
n2knsin(2πfnt/c)exp( −πfnt/Qn),(3)
withttime and Nmthe number of modes; fn,kn, and
Qnare,respectively,thefrequency,thekickfactor,andthe
quality factor of the nthmode. The function g(x)in Eq. 1
dependson the focusingprofile in the linac. Assuming the
betafunctionvariesas ¯β∼Eζ,
g(x, ζ) =1
ζ/parenleftbiggxζ−1
x−1/parenrightbigg
[¯β∼Eζ].(4)
IfΥm, for all m, is not large, the linear approximation
applies, and this parameter directly gives the (normalized )
growth in amplitude of bunch m. The projected (normal-
ized) emittance growth of the bunch train then becomes
(assuming,forsimplicity,that,inphasespace,thebeamel -
lipseisinitiallyupright) δǫ≈1
2Υ2
rms0y2
0/σ2
y0,withΥrms0
the rms with respect to 0 of the strength parameter, y0the
initial bunch offset, and σy0the initial beam size. As jitter
tolerance parameter, rt, we can take that ratio y0/σy0that
yieldsa tolerableemittancegrowth, δǫt.
2.2 Misalignments
Ifthestructuresinthelinacare(statically)misalignedw ith
respect to a straight line, the beam at the end will have
an increased projected emittance. If we have an ensem-
ble ofmisalignedlinacsthen,tofirst order,thedistributi on
in emittance growth at the end of these linacs is given by
anexponentialdistribution exp[−δǫ//angbracketleftδǫ/angbracketright]//angbracketleftδǫ/angbracketright,with[4]
/radicalbig
/angbracketleftδǫ/angbracketright=e2NLa(xa)rmsSrms
E0/radicalbigg
Na¯β0
2h(Ef/E0, ζ)
(5)withLathe structurelength, (xa)rmsthe rms of the struc-
ture misalignments, Srmsthe rms of the sum wake with
respect to the average , andNathe number of structures;
the function hisgivenby(againassuming ¯β∼Eζ):
h(x, ζ) =/radicalBigg
1
ζx/parenleftbiggxζ−1
x−1/parenrightbigg
[¯β∼Eζ].(6)
Eq. 5 is valid assuming the so-called betratron term in
the equation of motion is small compared to the mis-
alignment term. We can define a misalignment tolerance:
xat= (xa)rms/radicalbig
δǫt//angbracketleftδǫ/angbracketright, with δǫtthe tolerance in emit-
tancegrowth.
Wearealsointerestedinthetolerancetocell-to-cellmis-
alignmentscausedbyfabricationerrors. Astructureisbui lt
as a collection of cups, one for each cell, that is brazed to-
gether, and there will be errors, small comparedto the cell
dimensions,inthestraightnessofeachstructure. Togener -
ate a wake (for a beam on-axis)in a structure with cell-to-
cell misalignments we use a perturbation approach based
ontheeigenmodesoftheunperturbedstructure[5][2].
3 WAKEFIELD DESIGN
Reducing emittance growth requires reducing the sum
wake. In the main (X-band) linac of the NLC, the strat-
egytodothisistouseGaussiandetuningtogenerateafast
Gaussian fall-off in the wakefield envelope; in particular,
at the positionof the secondbunchthe wake is reducedby
roughly 2 orders of magnitude from its initial value. At
the lower frequencies of the injector linacs we have fewer
oscillationsbetweenbunchesand thisstrategyrequiresto o
muchdetuning. Instead,wewill followa strategythatputs
early bunches on zero crossings of the wake, by a proper
choice of the average frequency. As for the distribution of
mode frequencies, we will aim for a uniform distribution,
forwhichthewakeis(for π¯ft/Qsmall):
W≈2¯k
Nmsin(2π¯ft)sin(π¯ft∆δf)
sin(π¯ft∆δf/Nm),(7)
withNmthenumberofmodes, ¯ktheaveragekickfactor, ¯f
the averagefrequency,and ∆δfthefullwidthofthedistri-
bution. The wake envelope initially drops with tas a sinc
function, but eventually resurges again, to a maximum at
t=Nm/(¯f∆δf).
For the 2nd bunch to sit on the zero crossing requires
that¯f∆t=n/2, with nan integer. For S-band, given
our implementationof the SLED-I pulse compressionsys-
tem, the optimalrfefficiencyisobtainedwhenthe average
dipole mode frequency is 4.012 GHz. For this case, with
the alternate (1.4 ns) bunch spacing, ¯f∆t= 5.62. The
half-integer is achieved by changing ¯fby−2%, a change
which, however, results in a net loss of 7% in accelerating
gradient. One way of avoiding this loss is to reduce the
group velocity by increasing the phase advance per cell of
the fundamentalmode fromthe nominal 2π/3. In fact, we
findthatbygoingto 3π/4phaseadvancewecanrecapture
thislossingradient.For the resurgence in the wake to occur after the bunch
trainhaspassedrequiresthat ∆δfbesignificantlylessthan
Nm/(M¯f∆t), which, in our case, is about 10%. Another
possibility for pushing the resurgence to larger tis to use
two structuretypes,whichcaneffectivelydoublethe num-
ber of modes available for detuning. This idea has been
studied; it has been rejected in that it requires tight align -
menttolerancesbetweenpairsofsuchstructures.
3.1 Optimization
The cells in a structure are coupled to each other, and to
obtain the wakefield we need to solve for the eigenmodes
ofthesystem. Weobtainthesenumericallyusingadouble-
bandcircuitmodel[6]. Thecomputerprogramweusegen-
erates 2Nccoupled mode frequencies fnand kick factors
kn, with Ncthe number of cells in a structure. It assumes
the modesare trappedat the endsof the structure. We will
use only the first Ncmodes (those of the first pass-band)
forourwakefieldsincetheyoverwhelminglydominateand
since thoseofthesecondbandarenotobtainedaccurately.
The constants(circuitelements) forthe programare ob-
tainedbyfittingtoresultsofa2Delectromagneticprogram
OMEGA2[7]appliedtorepresentativecellgeometries,and
thenusinginterpolation. Hereweconsiderstructuresofth e
disk–loaded type, with rounded irises. The iris and cav-
ity radii are adjustedto give the correctfundamentalmode
frequency and the desired synchronous dipole mode fre-
quency. Therefore, cell mcan be specified by one free
parameter, the synchronous frequency (of the first dipole
mode pass-band). The 3π/4S-band structure consists of
102 cells with a cell period of 3.94 cm, iris thickness of
0.584 cm, and cavity radius ∼4.2cm; the Qdue to wall
losses (copper) ∼14,500. Fig. 1 shows the first two dis-
persion curves of representative cell geometries (for iris
radii from 1.30to 2.00cm). The plottingsymbolsgive the
OMEGA2results,thecurves,thoseofthe circuitprogram.
Figure 1: The dispersion curves of the first two dipole
bandsofrepresentativecellsina 3π/4structure.
We will consider a uniform input (synchronous) fre-
quency distribution, but with a slanting top. This leaves
us with 3 parameters to vary: the (relative) shift in aver-
age frequency (from a nominal 4.012 GHz) δ¯f, the (rela-
tive)widthofthedistribution ∆δf,andthetiltparameter α
(−1≤α≤1, with α= 1giving a right triangle distribu-
tionwithpositiveslope). Varyingthese parameterswe cal-
culate Srms0andSrmsforthecoupledmodes,andforbothbunchtrainconfigurations,andweoptimize. Wefindthata
fairly optimalcase consistsof δ¯f=−2.3%,∆δf= 5.8%,
andα=−0.20,where Srms0=Srms=.004MV/nC/m2.
InFig.2weshowthedependenceof Srms0onδ¯fand∆δf
neartheoptimum.
Figure 2: Srms0[MV/nC/m2]vs.δ¯fand∆δfnear
optimum,for ∆t= 2.8ns(solid)and1.4ns(dashes).
In Fig. 3 we display, for the optimal case, the frequency
distribution (a), the kick factors (b), and the envelope of
thewake(c). Thedashedcurvesin(a)and(b)givethesyn-
chronous (input) values. The plotting symbols in (c) give
|W|at the bunch positions for the alternate (1.4 ns) bunch
train configuration. In (b) we see no spikes, thanks to the
fact that the synchronous point is near pi, and, serendipi-
tously, f0< fπfor cell geometries near the beginning of
the structure, f0> fπforthoseneartheend[6]. (Notethat
for the optimized 2π/3structure, for which f0> fπfor
all cell geometries,there is such a spike, and consequently
Srms0is 5 times larger than here[2].) From (c) we note
that many of the earlier bunches have wakes with ampli-
tudessignificantlybelowthe wakeenvelope.
Figure3: Resultsfortheoptimal 3π/4structure.
3.2 FrequencyErrors
Errorsincellmanufacturingwillresultinfrequencyerror s.
In Fig. 4 we give Srms0andSrms, when a random error
componentisaddedtothe(input)synchronousfrequencies
of the optimal distribution (each plotting symbol, with its
error bars, represents 400 seeds). With a frequency spac-
ing of ∼8×10−4, an rms frequency error of 1×10−4
is a relatively small perturbation, and for the 1.4 ns bunch
spacing its effect is small, whereas for the 2.8 ns spacing
it is not. The reason is that in the former case the beam
sits on the half-integer resonance (which is benign), whilein the latter case it sits on the integer(whichis not)[2]. As
to the effect in a linac, let us distinguish two types of er-
rors: “systematic random”and “purelyrandom”errors; by
the former we mean errors, random in one structure, that
are repeated in all structures of the linac; by the latter we
mean random also from structure to structure. We expect
theeffectofapurelyrandomerror,ofsay, 10−4(whichwe
thinkisachievable)tobesimilartoasystematicrandomer-
ror of 10−4/√Na.Na= 140,127,41 in, respectively,the
prelinac, the e+drive linac, and the e−booster; therefore
the appropriate abscissas in the figure become .8, .9, and
1.6×10−5. At these points,forthe 2.8nsspacing,we see
thatSrms0is onlya factor 2±1,2±1,3±2timeslarger
thantheerror-freeresult.
Figure4: Theeffectofrandomfrequencyerrors.
4 TOLERANCES
To obtain tolerances we performed particle tracking using
LIAR[8] and compare the results with the analytical for-
mulas given in Sec. 2. We take δǫt= 10%as accept-
able. For BBU the tightest tolerance is for the e+booster,
where rtis 3.8(2.2)analytically,5.5(3.0)numerically,for
∆t= 2.8(1.4)ns. Formisalignmentsthetightesttolerance
is for the prelinac, where xatis 2.9 (4.6) mm analytically,
3.2 (4.8) mm numerically. (For the other machines these
tolerances are /greaterorsimilar10times looser.) Purely randommachin-
ingerrors,equivalentto 10−4frequencyerrors,willtighten
these resultsby50-100%,buttheyarestill veryloose.
Finally, what is the random, cell-to-cell misalignment
tolerance? Performing the perturbation calculation men-
tionedearlierfor1000differentrandomstructures,wefind
thatSrms=.27±.12(.032±.003) MV/nC/m2for
∆t= 2.8(1.4) ns. We again see the effect of the integer
resonance on the 2.8 ns option result. For the prelinac the
cell-to-cell misalignment tolerance becomes 40 (600) µm
forthe2.8(1.4)nsconfiguration.
We thank T. Raubenheimer and attendees of the NLC
linac meetingsat SLACforcommentsandsuggestions.
5 REFERENCES
[1] NLCZDRDesign Report,SLACReport 474, 589(1996).
[2] K.Bane andZ.Li,SLAC-LCC-043,July 2000.
[3] A. Chao, “Physics of Collective Instabilities in High-E nergy
Accelerators”, John Wiley& Sons, NewYork(1993).
[4] K.Bane, et al,EPAC94, London, England, 1994, p.1114.
[5] R.M. Jones, et al,PAC99, New York, NY,1999, p. 3474.
[6] K.Bane andR.Gluckstern, Part.Accel. ,42, 123 (1994).
[7] X.Zhan, PhD Thesis,StanfordUniversity, 1997.
[8] R.Assmann, etal, LIARManual, SLAC/AP-103,1997. |
COLLECTIVE ACCELERATION OF IONS BY MEANS OF
PLASMOIDS IN RF WELLS OF FREQUENCY-MODULATED LASER
FIELD.
A.I. Dzergatch and S.V. Vinogradov
Moscow Radiotechnical Institute, 113519 Moscow, Russia
1. INTRODUCTION
The proposed linear accelerator ("scanator") consists
of a terawatt table-top laser and a set of passive
elements - beam splitters, dispersion elements for
stretching of the laser pulse and chirping of the splitted
beams, and dispersion elements for angle scanning of
crossed frequency-modulated laser beams. Ions are
trapped and accelerated in RF wells by the electron
component of plasmoids in the intersection zone of the
scanning laser beams. Computational studies give
encouraging results. A proof-of-principle experiment
on the base of a table-top laser is outlined.
Several groups investigate accelerators based on
plasma waves, which are excited by powerful short
laser or electron pulses (look, e.g.,[1] and references
therein).. These schemes are based on free oscillations
of the plasma and hence they directly depend on the
plasma tolerances and instabilities.
The present variant of acceleration is based on
forced oscillations of the charged plasma in laser-
generated moving or standing RF wells (HF traps,
ponderomotive- or quasipotential wells, M.A. Mil-ler’s
force, light pressure. This way leads to several schemes
of regular acceleration , based on far fields. The
dependence on plasma parameters is decreased in this
variant. One of these schemes [2], namely MWA
(moving well accelerator), is detailed and discussed in
this report.
Certain vacuum modes of fast electromagnetic
waves (far field) trap charged particles , electrons
(positrons) in the 1-st turn, near the minimums of the
envelope or near the zeros of the carrier frequency [2].
Both types of RFwells (“envelope wells” and “carrier
wells”) may be distant from the radiating surfaces,
hence the electric breakdown problems are moved
aside and concentrated fields with very high amplitudes
may be used. The RF wells may be effective (gradient
of the quasi- potential ~tens % of the field amplitude
Em), if the amplitude is large, Em~ mc2le/, e.g., 1
TV/m in case of electrons and a 1- mm laser.
This effect may be treated as 3-dimensional alternating
gradient focussing of the electron component of
plasmoids. The computed dimensions of plasmoids in
case of carrier RF wells are ~ λ/6 or smaller, and their
den-sity is sub-critical, so they are not larger thanseveral Debye lengths. It simplifies the plasma stability
problems.
Motion and acceleration of an RF well takes place, if
the given structure of its field in the moving frames
(e.g., a cylindrical wave mnE0) ( zr,,j ) is generated
by corresponding laboratory sources (the moving and
laboratory fields are connected by Lorentz transforms).
2. THE STRUCTURE OF THE FIELD
AND THE SCHEME OF THE
ACCELERATOR
The field structure is based on A.M. Sessler’s idea
[3] to use crossed beams of a small laser instead of the
expensive system of oversized resonators with
kilojoules of stored optical energy. The RF wells exist
in many points in the zone of intersection of the
focused laser beams.
These beams are crossed and focused (Fig.1) at the
center P of an RF well, which is accelerated along the
z-axis, if the field parameters have the proper
variations. The programs of the frequencies and angles
variations are defined by Lorentz- trans-formed values
of the RF well parameters, ,wq, prescribed in the
moving frames. The frequency wmay be constant, but
the inevitable variation of the angle q, i.e., of the RF
well form, limits its values near 45o15±. The sources
of these beams (focused dispersion radiators at the
Fig.2) are centered at 8 points (1x±; 0), (0;1y±),
(2x± ; 0) and (0; 2y± ), symmetric in the planes xz
and yz. The number of these partial beams may vary, in
principle, between 6 and infinity (cylindrical waves).
During the acceleration the beams are scanned (from
left to right at the Fig.1). This process is realized by
linear transforms (filtering) of the primary short (wide-
band) pulse of the feeding laser. This pulse is split into
a pair of pulses, and each of them is stretched and
frequency-modulated (FM, chirped) by means of,
positive and negative dispersion elements ±D.
The lags of the ion center from the electron center
and of the latter – from the RF well center must be
small (say, 0.01 l), if the number of accelerated ions
must be large; its increase leads to a decrease of the
number of accelerated ions. Some excess of electrons
ensures the longitudinal autofocalization of ions.Fig 1. Scheme of the scanator.
The injector may be simply a gas jet similar to that
used in printers . Some additional radiators (not shown)
may be installed (and fed from the same laser) as
correctors, if needed.
Estimated parameters of a proof-of- principle model
proton accelerator (Fig.1) are given in the Table 1
below:
Table 1. Some parameters of the scanator model.
Laser pulse energy/peak power: 3 J/300 GW
Laser wavelength ~1 mkm
Diameter of the FDR: 7 mm
Length of the acceleration path: 5 cm
Distance radiator-acceleration zone: 15 cm
Maximal angle scans: ~1 grad
FM deviations : ±1 %
Number of RF wells in the focal region: ~ 500
Focal field density 200 GV/m
Neutralization factor ~0.8
The number of accelerated ions per plasmoid is
defined by the ion density and by the plasmoid volume,
and it is proportional to the ratio er/l. The
accelerated current does not depend on the wavelength
l(at a given relative density cnn/).
The state of the art of tera- to peta-watt
subpicosecond lasers gives hope on the realization of
the proposed scheme.The above variant of the “scanator” is based on the
“carrier RF wells”, which are disposed with z-intervals
equal to a half of the z-wavelength. These wells are
relatively small, which simplifies the plasma stability
problems.
3. METHODS AND RESULTS OF
NUMERICAL STUDIES
“Multi-particles” programs are used for finding the
tolerable densities of the electron and ion components
of plasmoids and for final checking of the acceleration
concept in various regimes, including long
computations (~50 000 RF periods).
The axially-symmetric relativistic motion of many
electromagnetically interacting electrons and protons
was modeled in the rz-plane by the PIC method (2.5
measurements, ,,,,zzrr ′′ and the full velocity v,
rectangular toroidal macroparticles). Full system of
Maxwell’s equations and the equations of
macroparticles motion were solved for electrons and
ions in the co-moving (with the accelerated plasmoid)
ideal cylindrical resonator tuned to the same wave
011E as in the 1-particle case. The code was written in
C++ language. The number of macroparticles in the
calculations was usually ~50 000, the grid sizes about
30×30. The use of moving frames leads to large
economy of computation time. Special checks
(longitudinal waves in tubular beams, transverse waves
in plasma columns, several modes in an empty
cylindrical resonator) have shown the precision better
than several %. So the use of this “computational”
resonator is justified for the present case, when the
plasmoid is relatively small, ~ l/6 or less. The
additional physical parameters in the multi-particle
case are the initial densities of electrons and protons in
the charged plasmoid and some computational
parameters (the numbers of computation cells and steps
per RF period, etc). Preliminary values of the densities
were chosen with the account of Kapchinski —
Vladimirski equilibrium and its stability studies, which
lead to the AG focusing depression by the space charge
up to ~30%.. So the initial conditions were uniform
density and zero velocity for both electrons and ions,
which lead to very non-uniform density and losses
~20% of the particles at the initial several hundreds of
periods.
The physical parameters were field amplitude,
Brillouin angle, initial acceleration, 2 densities. Some
typical shots of electrons (upper bunches) and ions
(lower bunches) are shown at the Fig-s 2-3 for the
times 79977 and 650007 time units 2.2 λ/33c. The
initial distribution of electrons and protons was chosen
(for the economy of cells) as a spheroid, corresponding
to the RF well dimensions found in the preliminary 1-
particle modeling. Fig.4 shows the numbers of
electrons and protons in the accelerated RF well as
functions of time (in units 2.2 λ/c): after an initial
relatively swift (~1000 field periods ) loss the self-
consistent evolution process leads to acceleration of the
particles during ~50 000 periods with a relatively small
loss. The form of both bunches, electron and proton, is
gradually normalized, and then a slow “evaporation” of
particles takes place. This process is similar to halo
formation in the case of RFQ linac.
Fig 2-3. r-z portraits og electrons and accelerating
ions in RF well
Fig. 4.
The computed cartoons show the alternating
focusing-defocusing rz-oscillations, and the lag ofaccelerated ions from electrons, and of the electrons –
from the RF well center. Optimal amplitude of the field
was found to be mE2mc≈ /Re, where R is the
radius of the resonator, =R 2.2l for the present case
of Brillouin angle q≈60°, e is the electron charge.
The number of accelerated particles per plasmoid
(which decreases with the increase of the acceleration)
was found to be ~3000 electrons and ~1000 protons.
The initial value of the acceleration (it decreases with
growth of mass of the ions) was chosen in one of the
computational runs to be 0.000 001 2c/R, which
corresponds to the acceleration gradient
≈=MadzdW/ 500 MeV/m.
CONCLUSION
A compact proof-of principle collective accelerator
(“scanator”) may be built on the base of a table-top
terawatt laser and a passive optical sys-tem, which
splits and transforms the primary laser beam into
several frequency-modulated crossed scanning light
beams. The ions are accelerated by the electron
component of plasmoids (short plasma bunches), which
are trapped by moving RF wells (HF traps) of the
electromagnetic field in the in-tersection region. This
region is periodically scan-ned along the line of
acceleration. An estimation of parameters shows the
possibility of acceleration of, say, protons, from a gas
jet to 300 MeV, using a table-top terawatt 1-mkm laser
and a set of usual optical elements (mirrors, prisms,
diffraction gratings etc).
The numerical modeling confirms the possibility of
collective acceleration by charged plasmoids in RF
wells.
The encouraging computational results of the present
studies show, amongst other, the desirability of
theoretical nonlinear analysis of the problem which
might lead to its better understanding.
ACKNOWLEGEMENTS
This work was supported by the International
Scientific and Technology Center. The authors are
thankful to V.S.Kabanov, V.A.Kuzmin, and
F.Amiranoff for the help and discussions.
REFERENCES
[1]. E.Esarey, et al, IEEE Trans. on Plasma Science,
14, 2, 252-288 (1996).
[2]. A.I.Dzergatch, Proc. 4-th Europ. Particle Acc.
Conf. EPAC-94, 1, 814-816 (1994).
[3]. A.M.Sessler, Proposal, LBNL, March 1998.
|
ANALYSIS OF THERMALLY INDUCED FREQUENCY SHIFT FOR THE
SPALLATION NEUTRON SOURCE RFQ*
S. Virostek, J. Staples,
LBNL, Berkeley, CA 94720, USA
* This work is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy u nder
Contract No. DE-AC03-76SF00098.Abstract
The Spallation Neutron Source (SNS) Front-End
Systems Group at Lawrence Berkeley National Lab(LBNL) is developing a Radio Frequency Quadrupole(RFQ) to accelerate an H
- beam from 65 keV to 2.5 MeV
at the operating frequency of 402.5 MHz. The 4 section,3.7 meter long RFQ is a 4 vane structure operating at 6%duty factor. The cavity walls are made from OFE Copperwith a GlidCop
® outer layer to add mechanical strength.
A set of 12 cooling channels in the RFQ cavity walls arefed and controlled separately from 4 channels embeddedin the vanes. An ANSYS
® finite-element model has been
developed to calculate the deformed shape of the cavityfor given RF heat loads and cooling water temperatures.By combining the FEA results with a SUPERFISH RFcavity simulation, the relative shift in frequency for agiven change in coolant temperature or heat load can bepredicted. The calculated cavity frequency sensitivity is -33 kHz per 1 °C change in vane water temperature with
constant-temperature wall water. The system start-uptransient was also studied using the previously mentionedFEA model. By controlling the RF power ramp rate andthe independent wall and vane cooling circuittemperatures, the system turn-on time can be minimizedwhile limiting the frequency shift.
1 INTRODUCTION
The SNS is an accelerator-based facility to be built for
the US Department of Energy at Oak Ridge NationalLaboratory (ORNL) through a collaboration of six USnational laboratories. The facility will produce pulsedbeams of neutrons for use in scattering experiments byresearchers from throughout the world. LBNL isresponsible for the design and construction of the SNSFront End [1], which will produce a 52 mA, 2.5 MeV,6% duty factor H
- beam for injection into a 1 GeV linac.
The Front End consists of several components: an ionsource [2] and low energy beam transport line (LEBT)[3], an RFQ [4,5] and a medium energy beam transportline (MEBT) [6].
The RFQ resonant frequency is a function of both the
cavity geometry and particularly the spacing of the vanetips. There is a frequency shift of approximately 1 MHzper 0.001 inch (25 microns) change in the tip-to-tip
spacing. This dependence results in very tight machiningtolerances on the individual vanes of ±0.0003 inch (8
microns) in order to achieve a final frequency which iswithin the range of the fixed slug tuners. Duringoperation, a combination of RF power dissipated in thecavity walls and heat removal through the coolingpassages will cause the cavity to distort and shift infrequency. By appropriately controlling the temperatureof the RFQ cooling water continuously during operation,the cavity design frequency of 402.5 MHz will bemaintained.
This paper will summarize the studies conducted to
determine the RFQ frequency sensitivity to cooling watertemperature changes. The predicted values will becompared to those obtained experimentally throughtesting of the completed prototype RFQ module (Figure1). The operating scenario required to rapidly bring thesystem up to full RF power while maintaining therequired frequency through cooling water control willalso be presented.
Figure 1: First completed SNS RFQ module.
2 RFQ DESCRIPTION
The SNS RFQ will accelerate an H- ion beam from 65
keV to 2.5 MeV over its 3.73 meter length. The 4modules are constructed of C10100 oxygen free copper(OFE) with an outer layer of Glidcop AL-15. The OFEcopper was selected due to its superior brazingcharacteristics and the Glidcop for its ability to maintainstrength after brazing. The GlidCop is brazed to the outer
surface of the OFE and covers the cooling passages
which are milled into the back side of the copper vanepiece. The vacuum seals for all penetrations (RF ports,tuners, vacuum ports and sensing loops) are r ecessed
beyond the outer layer of GlidCop such that the br aze is
not exposed to the cavity vacuum. Also, since thecooling channels do not penetrate the ends of themodules, there are no water-to-vacuum joints in the entiresystem.
For joining the vanes together, a zero-thickness
brazing process has been selected in order to maintain the±0.001 inch (25 microns) vane tip-to-vane tip tolerance.
With this method, the joint surf aces are br ought into
close contact with Cusil wire braze alloy having beenloaded into grooves in the joint surf aces prior to
assembly. The alloy is spread throughout the joint duringthe braze cycle by means of capillary action. Thistechnique permits the RFQ modules to be assembled andthe cavity frequency measured prior to the braze cycle toallow for dimensional adjustments, if necessary.
The RF-induced thermal load on the cavity walls is
removed by means of a dual temperature water coolingsystem. This setup allows fine tuning of the structurefrequency in operation as well as during the RF powertransient at start-up. A schematic of the RFQ crosssection showing the OFE copper, Glidcop and coolingchannel geometry is shown in Figure 2. The 12 outerwall channels are on a separately controlled water circuitfrom the 4 vane channels.
Figure 2: SNS RFQ cross section geometry.3 FINITE ELEMENT MODELING
A finite element model of the RFQ has been developed
using ANSYS and consists of a 3-D slice of one quadrantof the RFQ cross section. The surface nodes on either
side of the slice are constrained to remain coplanar suchthat the longitudinal stresses are correctly calculatedwhile allowing for overall thermal growth along the RFQaxis. This could not be achieved with 2-D plane strainelements which would over-constrain the modellongitudinally and result in artificially high z-componentcompressive stresses. The loads and constraints appliedto the model include RF heating on the cavity walls,external atmospheric pressure, convective heat transferand water pressure on the cooling passage walls andboundary conditions imposed by symmetry constraints.With 18 °C water in the vane channels and 24 °C water in
the cavity walls, the resulting temperature profile rangesbetween 25 and 29 °C at full RF gradient with an average
linear power density of 90 W/cm (Figure 3). The averagepower density on the outer wall is 1.7 W/cm
2. The model
has also been used to calculate stresses and cavity walldisplacements.
Figure 3: Predicted RFQ cavity wall temperature profile.
4 FREQUENCY SHIFT STUDIES
In order to predict the frequency shift of the RFQ
cavity under various thermal conditions, a computerprogram was developed which combines the ANSYSdisplacement results with SUPERFISH calculations offrequency sensitivity. It was determined that the RFQfrequency shifts by –33 kHz for every 1 °C rise in the
vane cooling water. Preliminary measurements on thecompleted first RFQ module have yielded a value of –32
kHz/ °C. This sensitivity to vane water temperature will
be used to fine tune the RFQ frequency during operationbased on sensing probe measurements. The calculatedfrequency shift for changes in the wall water temperatureis +26 kHz/ °C. For equal changes in the vane and wall
water temperatures, the shift is -7 kHz/ °C.
The calculations described above were based on
nominal input temperatures for the vane and wall coolingchannels. However, as the water flows from the inlet tooutlet end of each RFQ m odule, its temperature will rise
as it absorbs heat. Also, there will be a net heat transferfrom the higher temperature wall water to the lowertemperature vane water. The predicted values of 2.7 °C
rise in vane water temperature and 0.4 °C rise in wall
water temperature create a different cross sectiontemperature profile at the outlet end of the RFQ module.The calculated frequency error due to the higher watertemperatures is –80 kHz from the inlet to the outlet end
of a 93 cm long module. This error is considered minorand can be corrected by adjusting the position of thefixed slug tuners along the length of the RFQ modules ifnecessary.
5 START-UP TRANSIENT
A series of transient analyses were performed using the
same FEA model to determine the frequency performanceof the system during ramp-up of the RF power. With18°C water in the vanes and 24 °C water in the walls and
no heat on the cavity walls, the resonant frequency is 216kHz higher than the nominal 402.5 MHz, outside thepassband of the cavity. Setting the vane water at a highertemperature and the wall water at a lower temperaturewill bring the cavity frequency down. As the RF poweris applied, the water temperatures are correspondinglyadjusted towards their nominal operating values.However, the ramp-up rate for the RF power must belimited since the cooling systems cannot respond fastenough to keep the frequency error low. Figure 4 showsa comparison between the frequency shift caused by thewater temperatures and that due to the cavity wall heat.
Figure 4: RFQ transient frequency error.
The error for wall heat is actually negative, but is plotted
as positive for comparison purposes. The differencebetween these 2 curves is the net frequency error versustime. As shown in the inset, the power is ramped up from10% to 100% of full power in 150 seconds. Initially, thewater temperatures are adjusted at the highest ratepossible until the 2 curves meet. This operating scheme
results in an acceptable frequency error of 45 kHz at lowpower and drops off to less than 5 kHz within 60seconds.
6 CONCLUDING REMARKS
The SNS RFQ resonant frequency will be regulated by
dynamically adjusting the water temperature in the 4cooling channels embedded in the vanes while holdingthe water temperature in the 12 wall channels constant.The theoretical frequency sensitivity to vane watertemperature of –33 kHz/ °C was confirmed by a
measurement of –32 kHz/ °C on the completed first RFQ
module. Also, by adjusting both wall and vane watertemperatures during start-up, the RF power can beincreased to its full value in 150 seconds or less whileholding the frequency error low enough to allow RFpower transfer.
7 REFERENCES
[1] R. Keller for the Front-End Systems Team, “Status
of the SNS Front-End Systems ”, EPAC ’00, Vienna,
June 2000.
[2] M.A. Leitner, D.W. Cheng, S.K. Mukherjee, J.
Greer, P.K. Scott, M.D. Williams, K.N. Leung, R.Keller, and R.A. Gough, “High-Current, High-Duty-
Factor Experiments with the RF Driven H
- Ion
Source for the Spallation Neutron Source ”, PAC ’99,
New York, April 1999, 1911-1913.
[3] D.W. Cheng, M.D. Hoff, K.D. Kennedy, M.A.
Leitner, J.W. Staples, M.D. Williams, K.N. Leung,R. Keller, R.A. Gough, “Design of the Prototype
Low Energy Beam Transport Line for the SpallationNeutron Source ”, PAC ’99, New York, April 1999,
1958-1960.
[4] A. Ratti, R. DiGennaro, R.A. Gough, M. Hoff, R.
Keller, K. Kennedy, R. MacGill, J. Staples, S.Virostek, and R. Yourd, “The Design of a High
Current, High Duty Factor RFQ for the SNS ”, EPAC
’00, Vienna, June 2000.
[5] A. Ratti, R.A. Gough, M. Hoff, R. Keller, K.
Kennedy, R. MacGill, J. Staples, S. Virostek, and R.Yourd, “Fabrication and Testing of the First Module
of the SNS RFQ ”, Linac ’00, Monterey, August
2000.
[6] J. Staples, D. Oshatz, and T. Saleh, “Design of the
SNS MEBT ”, Linac ’00, Monterey, August 2000.
|
STATUS REPORT ON THE LOW-ENERGY DEMONSTRATION
ACCELERATOR (LEDA)*
H. Vernon Smith, Jr. and J. D. Schneider,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract
The 75-keV injector and 6.7-MeV RFQ that comprise
the first portion of the cw, 100-mA proton linac for theaccelerator production of tritium (APT) project have beenbuilt and operated. The LEDA RFQ has been extensivelytested for pulsed and cw output-beam currents
≤100 mA.
Up to 2.2 MW of cw rf power from the 350-MHz rfsystem is coupled into the RFQ, including 670 kW for thecw proton beam. The emittance for a 93-mA pulsed RFQoutput beam, as determined from quadrupole-magnet-scanmeasurements, is
εx × εy = 0.25 × 0.31 (π mm mrad)2 [rms
normalized]. A follow-on experiment, to intentionallyintroduce and measure beam halo on the RFQ outputbeam, is now being installed.
1 INTRODUCTION
The LEDA RFQ [1] is a 100% duty factor (cw) linac
that delivers >100 mA of H+ beam at 6.7 MeV [2-4]. The
8-m-long, 350-MHz RFQ structure [5] accelerates the dc,75-keV, 110-mA H
+ beam from the LEDA injector [6]
with ~94% transmission. The primary objectives ofLEDA are to verify the design codes, gain fabricationknowledge, understand beam operation, measure outputbeam characteristics, learn how to minimize the beam-tripfrequency, and improve prediction of costs andoperational availability for the full 1000- to 1700-MeVAPT accelerator. This paper summarizes the RFQcommissioning results given in [1, 3, 4, and 7-13].
2 LEDA CONFIGURATION
The accelerator configuration for beam commissioning
of the LEDA RFQ is shown in Fig. 1. Major subsystems
Figure 1. LEDA configuration for RFQ commissioning.
_________________
* Work supported by the US Department of Energy.
DC1
DC2 VD1VD2
Inside RFQ end wall
is “match point” AC toroid 3
Electron Trap
–1 kV
Collimator
(water cooled) AC1
Figure 2. The LEBT beamline with optics and diagnostics.
are the injector [6], ion source and low-energy beam
transport (LEBT); RFQ [1, 4, 5]; high-energy beam trans-port (HEBT) [14]; and the beamstop [15]. The injector(Fig. 2) matches the 75-keV, 110-mA dc proton beam intothe RFQ. Simulations of offline injector measurements[16] indicate the RFQ input beam rms normalizedemittance is
≤0.23 π mm mrad [6]. A current modulator
feeding the microwave magnetron provides beam pulsing[17] for commissioning and beam-tuning activities. Theon-line LEBT diagnostics include a pulsed-current toroid,located directly before the RFQ (AC toroid 3), that is usedin determining the RFQ transmission.
A complete description of the LEDA RFQ, including
the RFQ rf-field tuning procedure, resonance control, andoperation with the high-power rf (HPRF) and low-level rf(LLRF) systems, is given in [1, 4, 5, 9, and 11] and thereferences contained therein. A schematic of the LEDAHEBT showing the location of beamline optics anddiagnostics is given in Fig. 3. The function of the LEDAHEBT is to characterize the properties of the 6.7-MeV,100-mA RFQ output beam and transport the beam withlow losses to a water-shielded ogive beamstop [15]. Thebeamline optics consist of four quadrupole-singlet andtwo X-Y steering magnets.
Capacitive Probe
dual unitA/C & D/C Toroid6” BPMBeam Profile
wire scanner
BIF Profile
RGAY
Beam StopRFQ ZQ#1 Q#2 Q#3Q#4
SM#1 SM#2A/C & D/C Toroid
2” BPM
2” BPMCapacitive Probe
single unit
2” BPM
Figure 3. Layout of HEBT beamline optics and
diagnostics. Beam direction is from left to right.InjectorRFQ
Beamstop
WaveguideHEBT118 min
020406080100120
12/17/99
16:1912/17/99
16:4812/17/99
17:1612/17/99
17:4512/17/99
18:1412/17/99
18:4312/17/99
19:1212/17/99
19:4012/17/99
20:09TimeIpeak
Avg CurrentAvg Current = 98.7 mA Total charge = 322.5 mA*hr Total time = 3.3 hrBeam Current (mA)
Figure 4. Archived RFQ output beam current (30-s inter-
vals) for a 3.3 hr period on Dec. 17, 1999. Any beam
interruptions during the last 118 min are <30-s long.
The HEBT beam diagnostics [18] allow pulsed-beam-
current, dc-beam-current, and bunched-beam-current aswell as transverse centroid, longitudinal centroid (i.e.,beam energy from time-of-flight and beam phase), andtransverse beam profile (wire scanner and beam-inducedfluorescence) measurements.
3 BEAM COMMISSIONING RESULTS
AND DISCUSSION
We have accumulated 21 hr of LEDA RFQ operation
with ≥99 mA of cw output beam current and >110 hr with
≥90 mA of cw output beam current [11] since modifying
the injector and increasing the RFQ rf fields to 5-10%above the design values as described in [1], [4], and [11].For one run of 118 min (Fig. 4), most of the beaminterruptions were 1-6 s in duration (Fig. 5). Recoveryfrom these interruptions, most of them arising from short-duration injector and/or rf-system sparks, was automatic(no operator intervention).
We find that during pulsed beam operation for RFQ rf-
field levels at the design value, for pulse lengths >200
µs,
051015202530
048 12 16 20 24 28 32 36 40 44 48 52 56 60Trip Duration (sec)Number of Beam Trips12/17/99 1100-2000
Figure 5. The number of beam trips vs. trip duration (data
archived in 1 s intervals) for a 9-hr time period that
includes the 3.3-hr interval displayed in Fig. 4. 020406080100
0 100 200 300 400
Time (µs)Current (mA)
Figure 6. RFQ output beam current vs. time into a 300- µs-
long pulse for the design RFQ rf-field level.
and for RFQ output beam currents >90 mA, the RFQ
transmission drops abruptly about 100 µs into the beam
pulse [11]. The transmission then remains constant at thelower value for the duration of the pulse, including longpulses. The RFQ output beam current for a 300-
µs-long
beam pulse is shown in Fig. 6. The current abruptly dropsby ~10% about 125
µs into the pulse. Figure 7 shows the
measured values for the total beam transmission at thestart and end of a 500-
µs, 2-Hz, 90-mA beam pulse. At
the end of the pulse the total transmission deviates fromthe PARMTEQM prediction for 108-mA output beamcurrent over the field-level range 88-98% of the design(Fig.7). The total transmission at the start of the pulsefollows the PARMTEQM prediction for the range 0.91-1.1 of the design rf-field level. For output beam currents
>90 mA, e.g. 100 mA, the RFQ transmission over the
whole pulse is increased to the design value by increasingthe rf-field level to 105-110% of the design field. Both
the rf-power system and the RFQ-cooling system allowthis increase
− the only drawback is that the RFQ requires
10-20% more input power.
The LEDA RFQ output beam emittance is determined
[12,13] from quadrupole-magnet scan measurements [12].For a 93-mA pulsed beam, three x quad scans are given inFig. 8 and three y quad scans in Fig. 9: also shown are thequad-scan simulations obtained using the particle-optics
0.400.500.600.700.800.901.00
0.85 0.9 0.95 1 1.05 1.1 1.15
RFQ Cavity Field Amplitude (1.00 = design)RFQ Transmission PARMTEQM 108 mA calculation
Total transmission (end of pulse)
Total transmission (first 100 µsec)
Figure 7. RFQ total beam transmission vs. rf cavity field
level at the start (crosses) and at the end (dashes) of a
500-µs-long, 90-mA beam pulse.02468101214
4.5 5.5 6.5 7.5 8.5 9.5 10.5Q2 (T/m)Xrms (mm)
Measured rms width
LINACa)
b)c)
Figure 8. 93-mA x-scan data (diamonds) taken on three
different days. The LINAC calculation (triangles, line)
has Twiss parameters as described in the text.
code LINAC. The LINAC curves in Figs. 8 and 9 are
determined by starting with the RFQ output beam Twissparameters calculated using PARMTEQM, then adjustingthese Twiss parameters to give the “fits” to the datashown in Figs. 8 and 9 [12]. The resulting RFQ outputbeam Twiss parameters are
αx = 1.8, αy = -2.5, βx = 36
cm, βy = 89 cm, εx = 0.25 π mm mrad, and εy = 0.31 π mm
mrad (rms normalized) [12]. The beam-optics codeIMPACT, which, like LINAC, includes non-linear space-charge effects, is used to calculate the beam profiles foreach of the quadrupole magnet settings used in the quadscans [13]. LINAC has 2-D (r-z) space charge; IMPACT,3-D. The Twiss parameters given above are used in thesecalculations. In Fig. 10 samples for 3 points in the x quadscan are given as a) - c) [Fig. 8] and for 3 points in the yquad scan as d) - f) [Fig. 9]. A computer program thatadjusts the Twiss parameters to obtain the best global fitto the measured beam profiles is being considered.
4 SUMMARY
The LEDA RFQ has operated with ≥99-mA cw output
beam for 21 hr cumulative: it has operated >110 hrcumulative with
≥90-mA cw output beam. The RFQ
output beam emittance for a 93-mA pulsed beam, deter-mined from quadrupole-magnet-scan measurements, is
εx × εy = 0.25 × 0.31 (π mm mrad)2 [rms normalized]. We
are now preparing to intentionally introduce and measurethe beam halo in a 52-magnet FODO lattice [19, 20].
d)
0246810121416
6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5
Q1 (T/m)Yrms (mm)
Measured rms width
LINACe) f)
Figure 9. Same as Fig 8, except a y scan.00.0050.010.0150.020.0250.030.0350.040.045
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
x (mm)Point 1 (Q2 = 4.74 T/m)
Experiment
IMPACT
00.050.10.150.20.250.30.350.40.45
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
x (mm)Point 4-1/2 (Q2 = 7.43 T/m)
Experiment
IMPACT
00.010.020.030.040.050.06
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
x (mm)Point 9 (Q2 = 10.70 T/m)
Experiment
IMPACT
00.0050.010.0150.020.0250.030.035
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
y (mm)Point 1 (Q1 = 7.52 T/m)
Experiment
IMPACT
00.010.020.030.040.050.060.070.080.090.1
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
y (mm)Point 5 (Q1 = 9.48 T/m)
Experiment
IMPACT
00.010.020.030.040.050.06
-40 -30 -20 -10 0 10 20 30 40Beam fraction per mm
y (mm)Point 11 (Q1 = 12.0 T/m)
Experiment
IMPACTa) b) c)
d) e) f)
Figure 10. IMPACT calculation (dashed) of the meas-
ured (solid lines) 93-mA x- (top) and y-scan (bottom)
profiles using the Twiss parameters in the text [12,13].
REFERENCES
[1] L. M. Young, et al., "High Power Operations of LEDA,"
this conf.
[2]J. D. Schneider and R. L. Sheffield, “LEDA − A High-
Power Test Bed of Innovation and Opportunity, ” this conf.
[3] H. V. Smith, Jr., et al., “Update on the Commissioning of
the Low-Energy Demonstration Accelerator (LEDA) Radio-Frequency Quadrupole (RFQ), ” Proc. 2nd ICFA Advanced
Accelerator Workshop on the Physics of High-BrightnessBeams (Los Angeles, 9-12 November 1999) (in press).
[4] L. M. Young, et al., “Low-Energy Demonstration
Accelerator (LEDA) Radio-Frequency Quadrupole (RFQ)Results,” ibid. (in press).
[5] D. Schrage, et al., “CW RFQ Fabrication and Engineering, ”
Proc. LINAC98 (Chicago, 24-28 Aug. 1998), pp. 679-683.
[6] J. Sherman, et al., “Status Report on a dc 130-mA, 75-keV
Proton Injector, ” Rev. Sci. Instrum. 69 (1998) pp. 1003-8.
[7] K. F. Johnson, et al., “Commissioning of the Low-Energy
Demonstration Accelerator (LEDA) Radio-FrequencyQuadrupole (RFQ), ” Proc. PAC99 (New York, 29 March - 2
April 1999) pp. 3528-3530.
[8] J. D. Schneider, “Operation of the Low-Energy Demon-
stration Accelerator: the Proton Injector for APT, ” Proc.
PAC99 (New York, 29 March - 2 April 1999) pp. 503-507.
[9] H. V. Smith, Jr., et al., “Commissioning Results from the
Low-Energy Demonstration Accelerator (LEDA) Radio-Frequency Quadrupole (RFQ), ” Proc. EPAC2000 (Vienna,
26-30 June 2000) (in press).
[10] J. D. Schneider, “Overview of High-Power CW Proton
Accelerators, ” ibid. (in press).
[11] L. J. Rybarcyk, et al., "LEDA Beam Operations Milestone
& Observed Beam Transmission Characteristics," this conf.
[12] M. E. Schulze, et al., "Beam Emittance Measurements for
the Low-Energy Demonstration Accelerator Radio-Frequency Quadrupole, ” this conf.
[13] W. P. Lysenko, et al., "Determining Phase-Space
Properties of the LEDA RFQ Output Beam, ” this conf.
[14] W. P. Lysenko, et al., “High Energy Beam Transport
Beamline for LEDA," Proc. LINAC98 (Chicago, 24-28August 1998) pp. 496-498.
[15] T. H. Van Hagan, et al., “Design of an Ogive-Shaped
Beamstop, ” ibid., pp. 618-620.
[16] H. V. Smith, Jr., et al., “Comparison of Simulations with
Measurements for the LEDA LEBT H
+ Beam,” Proc.
PAC99 (New York, 29 March - 2 April 1999) pp. 1929-31.
[17] T. Zaugg, et al., “Operation of a Microwave Proton Source
in Pulsed Mode, ” Proc. LINAC98 (Chicago, 24-28 August
1998) pp. 893-895.
[18] J. D. Gilpatrick, et al., "Beam Diagnostic Instrumentation
for the Low-Energy Demonstration Accelerator (LEDA):Commissioning and Operational Experience," Proc.EPAC2000 (Vienna, 26-30 June 2000) (in press).
[19] P. L. Colestock, et al., "The Beam Halo Experiment at
LEDA,” this conf.
[20] T. P. Wangler and K. R. Crandall, “Beam Halo in Proton
Linac Beams, ” ibid. |
arXiv:physics/0008171 19 Aug 2000SLAC Linac RF Performance for LCLS*
R. Akre, V. Bharadwaj, P. Emma, P. Krejcik, SLAC, Stanford, CA 94025, USA
* Supported by the U.S. Department of Energy, contra ct DE-AC03-76SF00515: LINAC2000 THC11: SLAC-PUB-857 4
† Throughout this paper °S, °F, and °C stand for degrees at 2856MHz, S-Band, degrees Fah renheit, and Celsius respectivelyAbstract
The Linac Coherent Light Source (LCLS) project at
SLAC uses a dense 15 GeV electron beam passing
through a long undulator to generate extremely brig ht x-
rays at 1.5 angstroms. The project requires electro n
bunches with a nominal peak current of 3.5kA and bu nch
lengths of 0.020mm (70fs). The bunch compression
techniques used to achieve the high brightness impo se
challenging tolerances on the accelerator RF phase and
amplitude. The results of measurements on the exist ing
SLAC linac RF phase and amplitude stability are
summarised and improvements needed to meet the LCLS
tolerances are discussed.
1 LCLS RF REQUIREMENTS
LCLS requires the SLAC linac to perform with
tolerances on RF phase and amplitude stability whic h are
beyond all previous requirements. The LCLS is divi ded
into four linacs L0, L1, L2, and L3 [1]. The phase and
amplitude tolerances for the four linacs operated a t S-
Band, 2856MHz, are given in Table 1.
Table 1: LCLS RF stability requirements.
Klystrons Phase rms °S†Amp. % rms
L0 2 0.5 0.06
L1 1 0.1 0.06
L2 34 0.1 0.15
L3 45 2.0 0.05
L0 is a new section of accelerator for the off axis
injector. L1, L2, and L3 are made of structures in the
existing linac from sector 21 to sector 30.
2 LINAC RF SYSTEM
2.1 The RF Distribution System
The RF distribution and control systems for the lin ac,
after upgrades 15 years ago[2] for the SLAC Linear
Collider(SLC) are shown in figure 1. The RF distri bution
system consists of coaxial lines with varying degre es of
temperature stabilisation, figure 1. The 3.125 inc h rigid
coax Main Drive Line, MDL, carries 476MHz down the 2
miles of accelerator. At the beginning of each of the 30
sectors the 476MHz is picked off and multiplied by 6 to
get 2856MHz. There is a temperature-stabilised coa xialPhase Reference Line, PRL, that carries the referen ce
signal to the Phase and Amplitude Detector, PAD, of the
eight klystrons in the sector.
Figure 1: SLAC linac RF station
The critical parameters for the short term and long term
variations in the RF phase and amplitude can be rea d back
through the existing control system. The phase and
amplitude from the output of the SLED energy storag e
cavity are compared and recorded by the PAD. There are
three methods of acquiring and displaying the data:
• The fast time plot gives 64 consecutive data points .
At 30Hz this is 2.1 seconds of data.
• The correlation plot collects data with a maximum
frequency of about 1Hz and can collect up to 512
data points.
• The history buffers are updated with a data point
every six minutes for the past week and every four
hours for the past 7 years.
The bit resolution of the ADC in the PAD is 0.04 °S.
Phase and amplitude stability has been measured for the
different time scales.2.2 RF Phase Stability
Phase fast time plots have an rms variation of 0.05 °S
and meet LCLS requirements on a two second time sca le.
On a larger time scale drifts of well over 0.1 °S are
observed as temperature of the regulating water and
environment changes, figure 2. The phase correlati on,
6°S/°F, is likely due to the high Q SLED cavity.
Figure 2: Top: Klystron phase, 2.1-second time scal e.
Center Klystron phase 14-minute time scale. Bottom
SLED water temperature °F 14-minute time scale.
During normal linac operation each klystron’s phase is
adjusted by a high power phase shifter to keep the phase
as read by the PAD within a few degrees of the set value.
The phase shifter is a rotary drum type and typical ly
moves about a dozen times a day by a stepper motor. The
resolution of the phase shifter is 0.125 °S, which is much
coarser than the short term phase variation seen on the fast
time plots. The position of this phase shifter is recorded
in history buffers. The phase shifter movement ove r a
three-day period has been correlated to outside
temperature and the coefficients listed in Table 2. The
klystrons are grouped according to their position w ithin
the sector and averaged over the 29 sectors from se ctor 2to sector 30. Position 1 is closest to the sub-boo ster
klystron and position 8 is at the end of the sector . During
the course of a year the outside temperature varies from
35°F to 95°F and as much as 35 °F diurnally.
Table 2: Klystron phase shifter movement
Klystron
Position Average
°S/°F Standard
Deviation Range
°S
1 0.33 0.11 20
2 0.41 0.10 25
3 0.46 0.11 28
4 0.49 0.14 29
5 0.60 0.14 36
6 0.69 0.13 41
7 0.80 0.16 48
8 0.64 0.19 38
2.3 RF Phase Measurement Accuracy
The critical phase stability of the RF with respect to the
beam is influenced at three levels within the RF
distribution and control system. The first level is the
stability of the phase reference system. The secon d tier is
the noise level and drifts associated with the phas e
measurement electronics, and the third level consis ts of
the errors introduced in the beam phase measurement
system.
The two-mile MDL has been studied [3] and the lengt h
electronically measured by an interferometer. From
reference [3] the length varies with pressure and
temperature over the 2 miles as follows:
∆φ(°S) ~ -2.64( ∆P(mBar)) + 1.36( ∆T(°F))
History buffers show that the pressure range, ∆P, is about
30mBar, which gives a phase variation of 79 °S. The
temperature range, ∆T, of the MDL is about 30 °F, half the
outside ∆T due to some insulation and temperature
regulation. This ∆T gives a phase variation of 41 °S. The
predicted phase variations based on the above analy sis
only accounts for about half the observed phase tun ing in
the linac that is necessary to keep the beam at con stant
phase to meet the beam energy and energy spread
requirements [4]. These additional errors indicate the
system is in need of an upgrade.
About 95% of the PRL is temperature controlled with
an rms value of 0.05 °F. The other 5% varies by about
10% of the surrounding temperature, which gives a
temperature variation of about 1.0 °F. The ½ inch heliax
has a temperature coefficient of 4ppm/ °C, 0.9°S/°F/sector.
The phase error is spread linearly from a minimum a t the
first klystron in the sector to a maximum at the ei ghth
klystron in the sector. The average phase variatio n of the
sector is ½ the phase variation of the PRL, 0.5 °S.
The multipliers are temperature stabilised to about
0.1°Frms and have temperature coefficients which range
from –1.7 °S/°F to +2.2 °S/°F. The phase errors from the
multipliers are on the order of 0.2 °S rms.
Additional errors are introduced between the phase
reference system and the beam by the variations in lengthdue to temperature of the accelerating sections and the
waveguide feeding them. These variations are ignor ed by
the feedback system since the PAD only measures the
signal at the output of the SLED cavity. Table 3
summarises the phase errors due to temperature chan ges
in the system. The dominant non-corrected error is due to
the accelerator structure temperature change. Meas uring
the RF phase at the output or input of the structur e as an
estimate for the phase of the structure as seen by the beam
would have an error of 0.8 °S rms, or half the phase
slippage of the structure.
Table 3: Phase/Temperature coefficients
°S/°C ∆Trms
°C ∆φrms
°S
Accelerator 10’ [5] 16.0 0.1 1.6
WR284 Cu WG 10’ [5] 0.25 0.2 0.05
½” Heliax 40’ @ 4ppm/ °C 0.16 1.0 0.16
7/8” Heliax 40’ @ 3ppm/ °C 0.12 1.0 0.12
1-5/8” Rigid 40’ MDL data 0.01 0.1 0.001
SLED [6] 23.6 0.1 2.4
PAD <0.5 0.1 <0.05
The measurement resolution of the PAD is good enoug h
to meet the LCLS requirements. Initial testing sho w
measurement drifts of the PAD from temperature
variations to be close to LCLS requirements. Furth er
testing will be done to better estimate the PAD err ors.
2.4 RF Amplitude Stability
Fast time plots for the klystron amplitude also sho w that
on a two second time scale the LCLS stability crite rion
can be met. The rms amplitude jitter measured by t he
PAD at the output of he SLED cavity is less than 0. 04%
of the amplitude. Correlation plots over a 14-minu te time
scale show the amplitude varies by as much as 0.5% peak
to peak. This change is correlated to the water
temperature of the SLED cavity and the magnitude of
variations is greatly effected by the tune of the c avity[7].
Klystron K02 on the SLAC accelerator has a slow
amplitude feedback and no SLED cavity. Measurement
of the amplitude variation over days is held to 0.0 6% rms.
Further work needs to be completed to determine how
stable the measurement is with respect to temperatu re
changes.
3 RF SYSTEM IMPROVMENTS
Extremely tight phase and amplitude tolerances
throughout the linac are required to meet the LCLS
specifications. The LCLS requirements listed in Ta ble 1
may still change as the design of the bunch compres sion
system evolves. Measurement of the individual klys trons
show that they are capable of attaining the desired
specification up to a two second time scale. The
challenge is to link the many klystrons together th rough a
RF distribution system and preserve the stability o ver
extended periods of time.On longer time scales where temperature changes are
significant, a new RF reference and distribution sy stem
located in the tunnel, which has rms temperature
variations less than 0.1 °F, is under consideration. The
new system will distribute 2856MHz to the klystrons and
provide a reference for phase measurements of the
accelerator RF and beam phase cavity RF. This new
phase system is expected to reduce the phase drifts and
errors along the kilometer linac from about 10 °S down to
as little as 0.1 °S. Even with such a phase stable RF
reference system, measuring the phase of the RF at the
input or output of the accelerator will result in e rrors of
0.8°Srms compared to the RF phase as seen by the beam
in an accelerator structure which has temperature
variations of 0.1 °Crms. In order to hold the RF to beam
phase to 0.1 °S a feedback system using a beam-based
measurement is necessary.
Further measurements will determine if the existing
amplitude measurement and control system with added
feedback is sufficient to meet LCLS requirements.
In LCLS L2 and L3, where there is a large number of
klystrons, it is likely that the phase errors will be
correlated with water temperature which spans group s of
16 klystrons, or outside temperature and pressure, which
is common to all. The larger number of klystrons d oes
not increase the tolerance of an individual klystro n by √n.
Further testing of the existing RF system as well as
development and testing of new systems is ongoing, the
results of which will lead to the design of the LCL S RF
system.
REFERENCES
[1] The LCLS Design Study Group, “Linac Coherent
Light Source Design Study Report”, SLAC-R-521,
December 1998.
[2] H. D. Schwarz, “Computer Control of RF at SLAC”,
SLAC-PUB-3600, March 1985, also PAC,
Vancouver, B. C., Canada, May 1985
[3] R. K. Jobe, H. D. Schwarz, “RF Phase Distribut ion
Systems at the SLC”, PAC89, Chicago, Il, 1989
[4] F.-J.Decker, R. Akre, R. Assmann, K.L.F. Bane,
M.G. Minty, N. Phinney, W.L. Spence, “Beam-Based
Analysis of Day-Night Performance Variations at the
SLC Linac”, PAC97, Vancouver, B.C., Canada, May
1997.
[5] R.B.Neal, “The Stanford Two-Mile Accelerator”, W .
A. Benjamin, Inc. New York, NY, 1968
[6] Z.D. Farkas, G. A. Loew, “Effect of SLED Cavity
Temperature Changes on Effective Accelerating
Field”, SLAC CN-124, October 1981.
[7] F.-J. Decker, R. Akre, M. Byrne, Z.D. Farkas, H.
Jarvis, K. Jobe, R. Koontz, M. Mitchell, R.
Pennacchi, M. Ross, H. Smith, “Effects of
Temperature Variation on the SLC Linac RF
System”, Proc. of the 1995 Particle Accelerator
Conference, pp. 1821-1823, 1995 |
arXiv:physics/0008172v1 [physics.acc-ph] 19 Aug 2000JLCPROGRESS
N.Toge, KEK,Tsukuba,Ibaraki 305-0801,Japan
Abstract
TheJLCisa linearcolliderprojectpursuedinJapanbyre-
searchers centered around KEK. The R&D status for the
JLC project is presented, with emphasis on recent results
from ATF concerning studies of production of ultra-low
emittance beams and from manufacturing research on X-
bandacceleratorstructures.
1 INTRODUCTION
Major elements of the current R&D activities for the JLC
project[1] includes: development of polarized electron
sources[2], experimental studies of a damping ring[3], de-
velopmentof X-bandtechnologiesas the main scheme for
the main linacs, and C-band RF development as a backup
technologyforthemainlinacs[4].
Figure1 shows a schematic diagram of JLC. The target
center-of-massenergyis250 ∼500GeVinphase-I,and ∼1
TeV or higher in phase-II. Since 1998, through an R&D
collaboration(InternationalStudyGroup–ISG)whichwas
formalized between KEK and SLAC[5], development of
hardware elements for the X-band main linacs has been
pursued based on the basic parameters common to both
JLC and NLC[6]. Tables1 and 2 give the most up-to-date
basicmachineparametersthathavebeenchosenasaresult
ofoptimizationprocessinISGdiscussions.
Table 1: Partial list of representativeJLC parametersas of
April, 2000 [5], if the main linacs are built based on the
X-bandtechnology.
Item Value Unit
#Electrons/ bunch 9.5 ×109
#Bunches/ train 95
Bunchseparation 2.8 ns
Trainlength 263.2 ns
RF frequency 11.424 GHz
RF wavelength 26.242 mm
Klystronpeakpower 75 MW
Length/ cavityunit 1.8 m
a/λ average0.18
Filling time 103 ns
Shuntimpedance 90 M Ω/m
Eacc(no-load) 72 MV/m
Eacc(loaded) 56.7 MV/m
Normalizedemittance 3.0 ×0.03(Linac) 10−6m.rad
4.5×0.1(IP) 10−6m.rad
Bunchlength 120 µmTable 2: RepresentativeJLC parameters(continued),if the
main linacs are built based on the X-band technology. Pa-
rametersthatwouldvaryfor ECM=500GeVand1.0TeV
areshown.
ECM 500GeV 1TeV
#cav/linac 2484 4968
#klystrons/linac 1584 3312
Length/linac 4.3 8.9 km
P(wall-plug) 94 191 MW
Rep. rate 120 120 Hz
β∗
x×β∗
y 12×0.12 12 ×0.15 mm ×mm
σ∗
x×σ∗
y 330×4.9 235 ×3.9 nm ×nm
<−∆E/E > 4.0 10.3 %
duetoBSM
Lum. pinch 1.1 1.43
enhancement
Luminosity 7 ×103313×1033cm−2s−1
This paper presents the R&D status of the JLC project,
with strong focus on (i) the most recent results from ATF
concerning studies of production of ultra-low emittance
beams and (ii) manufacturing research on X-band accel-
eratorstructures.
2 ATF
The Accelerator Test Facility (ATF)[3] at KEK (see Fig-
ure 2) is a test bed for an upstream portion of JLC, which
has to produce a train of ultra-low emittance bunches
of electrons (and positrons). It includes a multi-bunch-
capable electron source, a 1.54GeV S-band linac, and a
1.54GeV dampingring(DR)prototype.
Avarietyofstudiesweresuccessfullyconductedin1994
through 1996 on acceleration of multi-bunch beam (up to
12 bunches, 2.8ns bunch separation). The multi-bunch
beam loading compensation scheme based on the RF fre-
quencymodulationofasmallsetofacceleratingstructures
wassuccessfullydemonstrated.
Commissioning work of the ATF DR began in early
1997 with many participants from both inside and out-
side KEK. The accelerator has been operated in a single-
bunchmodewiththetypicalstoredintensityof ∼1×1010
electron/bunch or less at a repetition rate up to 1.56 Hz.
Achieved and design parameters relevant to operation of
ATF are summarized in Table 3. By Summer 1998, the
horizontalbeamemittance ǫxof∼1.4×10−9m(i.e. γǫx≃
3.5×10−6m) was measured by using a group of wire
scanners in the beam diagnostics section of the extraction
line[7].Detector4 km 11 km 11 km
E-Gun10 GeV Linac
e+ Target 1.98GeV LinacPre-damping Ring8GeV Pre-linac BC1BC2 Main Linac
Collimator, FFSMain Linac BC2
8GeV Pre-linacBC1
1.98 GeV Damping Ring 1.98 GeV Damping Ring
1.98GeV LinacE-GunSpin Rotator
Spin RotatorFFS, CollimatorIP0.5 km 0.5 km
~ 26 km
Figure1: SchematiclayoutofJLC inits Ecm= 1TeV configuration.
50.4m
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 Lec2 L13 L14 L15 L16 Lec1
120m 1.54 GeV Damping ring
1.54GeV S-band LINACDamped cavity Wiggler magnetExtraction Line Beam Diagnostics
Wiggler magnetWater cooling & Air condition facility
Water cooling & Air condition facility
Control roomModulator
Klystron
DC power supply for modulator714MHz RF source 53.4m 27.6m
Thermionic Gun80MeV PreinjectorL0Magnet power supply
Figure2: LayoutofATF –AcceleratorTest Facility–at KEK.
Table3: Achievedanddesignparametersat ATF.
Item Achieved Design Unit
LinacStatus
Max. beamenergy 1.42 1.54 GeV
Max. gradientwithbeam 28.7 30 MeV/m
Singlebunchpopulation 1.7 ×10102×1010
Multi-bunchpopulation 7.6 ×101040×1010
Bunchspacing 2.8 2.8 ns
Repetitionrate 12.5 25 Hz
Energyspread(fullwidth) <2.0% (90%beam) <1.0% (90%beam)
DampingRingStatus
Max. beamenergy 1.28 1.54 GeV
Circumference 138.6 ±0.003 138.6 m
Momentumcompaction 0.00214 0.00214
Singlebunchpopulation 1.2 ×10102×1010
COD (peak-to-peak) x∼2,y∼1 1 mm
Bunchlength ∼6 5 mm
Energyspread 0.06% 0.08%
Horizontalemittance (1.4 ±0.3)×10−91.4×10−9m
Verticalemittance (1.5 ±0.25)×10−111.0×10−11mmThe most recent set of measured horizontal and vertical
emittance values[8], as of April, 2000, are shown in Fig-
ure3.
0.51.0 1.52.02.53.03.5
0.501.01.52.02.53.0Emittance ratio ey/ex [%]X emittance [10 rad.m]-9
02 4 6 8 10
DR Beam Intensity [ electrons / bunch]109
0 2 4 6 8 10
DR Beam Intensity [ electrons / bunch]109Y emittance [10 rad.m]-11(A)
(B)
Figure 3: Measured values of (A) horizontal and vertical
beamemittance(unnormalized)andtheirratio(B)asfunc-
tionofbunchintensityfromATF.
SomeofthedevelopmentatATFwhichareconsideredto
playimportantrolesintheprogressoftheachievedvertica l
emittancevaluesaresummarizedasfollows:
•Improved resolution ( ∼20µm) of single-shot BPM
readoutelectronics.
•Improved understanding of the first order optics in
the DR, and corrections introduced by using “fudge”
factors for the field strength of quadrupole magnets
and quadrupole field components of the combined-
functionbendmagnetsintheDR.
•Improved dispersion and orbit correction algorithm
whicharetunedtominimizetheverticaldispersion( ∼
5 mm)in theDRwithoutoverlyupsettingtheCOD.
•Skew quadrupole magnet fields were introduced in
the arc sections of the DR by using trim windings of
sextupole magnets. They were used to minimize the
cross-plane coupling by using the x-ycoupling sig-
nalsinthediagnosticsofthefirst-orderoptics,aswell
asbyusingthetunedifference νx-νynearthecoupling
resonance.•Improved algorithm for the correction procedure for
theverticaldispersionintheextractionline,wherethe
wire scanner beam diagnostics instruments are situ-
ated.
10000
1000
100
10Vertical Emittance [nrad-m]Normalized beam emittances in
Linear Colliders
0.1 1 10 100
Horizontal Emittance [ µrad-m]SLC
TESLA(500)
CLIC
500ATF
TESLA(800)CLIC
3000JLC/NLC(500-1000)
CLIC
1000
Figure4: Normalizedhorizontalandverticalemittanceval -
ues that have been achieved at SLAC and ATF, compared
to what are requiredfor injectors of next-generationlinea r
colliders.
ThisresultfromATFmaybeputinperspectiveasshown
in Figure 4. It is seen that the beam emittance that is re-
quired for typical next-generation linear colliders, incl ud-
ing JLC/NLC, is nearly achieved. However, a number of
issuesstill remaintobe investigatedatATF. Forinstance,
1. Thebeamemittancevaluessofarconsideredthemost
reliable have been obtained by using wire scanners
in the extraction line. These and measurements from
synchrotron radiation (SR) monitor in the DR, which
utilizes the interferometry, are not totally consistent.
This is most likely due to effects of mechanical vi-
brations of the optical stands that are used for the SR
monitorsystem,butit requiresmorestudies.
2. Theremaybefielderrorsinthemagneticcomponents
in the beam extraction line or at the beam extraction
point. They might introduce x-ycross plane cou-
pling and fictitious signals of the growth of the ver-
ticalbeamemittance,whichmaynotyetbeaccounted
forinwire scannermeasurements.
3. Observed growth of vertical emittance or the emit-
tance ratio as shown in Figure 3 is found to have too
strongadependenceonthebunchintensity,compared
to existing model calculations of intra-beam scatter-
ing effects. It has not yet been resolved whether this
is due to errors in measurements, inadequate set-up
assumptionsortruebeamdynamicseffects.
In addition, the reported emittance numbers from ATF are
so far based only on single-bunch beam operations. After
the Summer shutdown period, the beam operation of ATFis scheduled to resume in October, 2000. Preparation is
currently under way for addressing the single-bunch emit-
tance issues as well as multi-bunch operation of the ATF
DR.
3 X-BAND ACCELERATING
STRUCTURE
Development of X-band accelerating structure at KEK
has been conducted in close collaboration with a group
at SLAC. The accelerating structure studied is based on
the damped-detuned concept [5]. The recent research fo-
cus at KEK has been on (i) fabrication of copper disks
for the RDDS (Rounded Damped-Detuned)structure with
a diamond-turning technique with ultra-high precision
lathes, and (ii) their assembly into structure bodies by
meansofthediffusionbondingtechnique[5, 9]. Theyhave
been pursued in conjunction with development of better
control of transverse wakefield and improved RF-to-beam
efficiency.
ODf 612b2a
pw
Figure5: SchematicdrawingofanRDDS disk.
Figure 5 shows a schematic drawing of the typical cop-
per disk for the RDDS structure, whose first prototype
(RDDS1)wassuccessfullybuiltin1999andwastestedfor
wakefieldcharacteristicsattheASSETfacilityofSLACin
2000. The RDDS1 (1.8 m long) consists of 206 similar
disks, each61mmin diameterand8.737mminthickness.
Duringdisk fabrication,muchattentionshave beenpaid
to the temperature control of the lathe, positioning of the
diamondcuttingtool,anditsmotion. Theradiusofthecut-
ting tool are pre-determined within 0.3 µm by machining
an aluminumtest hemisphere( φ60mm)andbymeasuring
thesurfacefeatureswitharoundnesstester. Figure6shows
a contour profile plot of a test RDDS disk that was mea-
sured with a CMM (Coordinate Measurement Machine)
with contouring capability. The solid line shows the de-
sign shape, while the blackdotsshow the measuredshape,
with the deviation from the nominal shape magnified by
200 times. The machined surface matched the design to
within ±1µm. Other dimensional parameters such as the
disk outer diameter, aperture radius 2a, which are deter-
minedbythediamondturning,arefoundtohavebeendone
with similarprecision.
In addition to mechanical quality control, careful mea-
surements of the fundamental and first-order transverse4681012
-6 -4 -2 0 2 4 6R [mm]
Z [mm]MeasurementDesign Contour
+/– 1 mmDesign
Contour
Figure6: ContourprofileplotofatestRDDS disknearthe
apertureopeningpart.
mode resonant frequencies of cells were performed for
individual disks. As an example, Figure 7 shows the
fundamental-mode frequencies, measured on a disk-stack
setup (white circles). The disks are mostly fabricated in
the order of the disk number. Figure 7 also shows the in-
tegrated phase error (brokenline), expected fromthe mea-
sured deviation of the fundamental-mode frequencies. It
is seen that by fine-adjusting the cell cavity size (denoted
as2bin Figure 7) based on the past trend of frequency
errors, the total integrated phase error can be controlled
with an extreme precision. Also, the first-order transverse
mode frequencies of the fabricated disks have been found
tohaveasmoothdistributionwithin0.4to0.6MHz. Over-
all, the micron-level precision to which the disks are ma-
chinedhavebeenverysuccessfullydemonstrated.
-3-2-10123
0 50 100 150 2002b offset [micron]
Freq. meas. & estim. [MHz]
Integ. phase2b offset introduced [micron]
Meas. offset of the Acc. mode [MHz]
Integrated phase error [degree]
Disk number
Figure 7: Fundamental mode frequencies, measured in a
disk-stacksetup,ofRDDS1 disks.
ThedisksarethenstackedonaprecisionV-block,where
a disk-to-disk alignment of 1 µm or better is possible dur-
ing stacking. A growth of the so-called “bookshelf” stack
errors is preventedby monitoring the inclination angles of
the surfaceofstackeddiskswitha two-axisautocollimator
andbyapplyingcorrectionsduringstacking.
Formationofthecompleteacceleratingstructureismade
through (i) the diffusion bonding procedure of the copperdisks that is conducted in two steps (prebonding and final
bonding),and(ii)the brazingofexternalcomponentssuch
ascoolingwatertubing,fixturesforsupportframes,waveg-
uides and flanges. It has been found repeatedly that the
disk-to-diskalignmentandthe“bookshelving”error(ofit s
absense thereof) of the disks are well maintained through-
out the diffusion bonding process, which create vacuum-
tightandmechanicallystrongenoughdisk-to-diskbonding
junctions.
However,differentialexpansionoftheceramicsendplate
supports relative to the first and the last copper disks lead
to a flaring of the structure ends during diffusion bonding.
Similar deformation occurredon RDDS1 when a stainless
steel manifold was installed on a mid portion of the struc-
ture duringbrazing. While the formererror couldbe recti-
fied later, the bonding techniques used in the assembly of
X-band structures call for some improvements in the near
future, in addition to studies of mass-production issues.
Results from wakefield measurements of RDDS1 proto-
typearereportedinacontributionsubmittedtothisconfer -
ence[9]. Also,issuespertainingtoRFprocessingandhigh
power operation of X-band accelerating structures at field
gradientupto70 ∼80MV/marebeinginvestigated[10].
4 OTHER ACTIVITIESONTHE X-BAND
RF R&D
Development work is also under way[11] for: klystron
modulators with semiconductor switching devices, con-
struction of X-band high-power klystrons with periodic-
permanent magnet (PPM) focusing, testing of the
DLDS (Delay Line Distribution System) components
(see Figure 8), development of X-band high-power RF
windows[12]. Someoftheeffortsarecarriedoutincollab-
orationwithagroupfromProtvinobranchofBINP,Russia,
aswell aswithSLACintheframeworkoftheKEK-SLAC
ISG.
TE01-TE11mode launcher
TE11->TE12
mode converterLoad
TE11->TE01mode converterTE01 mode extractor
TE01 tap-off Accelerating structures
55 m 55 m 55 mWaveguide ( f4.75inch)Klystron
3dB couplersRF Pulse
Beam direction
Figure 8: Schematic diagram of a DLDS concept where
the RF power from 8 klystrons are divided and distributed
tofourclustersofacceleratingstructuressituatedalong the
linac.
A low-power testing of transmission of X-band RF
througha long ( ∼50 m) waveguideas a proof-of-principleexperiment of the DLDS concept was successfully con-
ducted at KEK by a KEK-SLAC-Protvinocollaborationin
1999. Another testing is planned for Fall, 2000. A high-
power testing of Protvino-KEK RF windows have been
successfullycarriedoutatSLACinlate1999throughearly
2000. As of August, 2000, intense testing is being con-
ducted on KEK site for a PPM klystron that was designed
at KEKandbuiltin collaborationwithJapaneseindustry.
Inafewyears,whenthebasicR&DoftheseRFcompo-
nents, including the accelerating structures, becomes suf -
ficiently mature, it is considered highly desirable to build
a small part of the X-band linac, for instance a complete
unit set of the RF system as shown in Figure 8. While its
successfuloperationwithoutanybeamaccelerationshould
already mark a major milestone, possible acceleration of
low-emittance beams which would be hopefully available
by that time fromATF wouldplay a decisiverolein show-
ingthefeasibilityofJLC/NLC.
5 REFERENCES
[1] “JLCDesign Study”, KEKReport 97-1, April,1997.
[2] T. Nakanishi, et al., and K.Togawa, et al., contributed p a-
pers (4d052, 4d054) at APAC 98, KEK Proceedings 98-10,
November, 1998.
[3] F. Hinode, et al., ed., KEK Internal 95-4, June, 1995. S.
Takeda, et al, Particle Accel. 30(1990) 153. S. Kashiwagi,
et al, KEK-Preprint-96-110. Collaborators at ATF include
scientists from SLAC, CERN, DESY, BINP, PAL (Pohan,
Korea), IHEP (Beijing, China) and Tshinghua U. (China),
Grad.Univ.ofAdvancedStudies,HiroshimaUniv.,Kogakuin
Univ., Kyoto Univ., Nagoya Univ., Tohoku Univ., Tohoku-
Gakuin Univ.,and Tokyo Metropolitan Univ.
[4] Publications on the C-band R&D are compiled and availabl e
athttp://c-band.kek.jp . Also, see: J.-S. Oh, et al., THA12 at
this conference (LINAC2000), Monterey, Aug. 2000.
[5] KEK Report 2000-7 (also SLAC-R-559), April 2000. Avail-
able athttp://lcdev.kek.jp/ISGR .
[6] C. Adolphsen, et al, SLAC Report 474 (also LBNL-PUB-
5424, UCRL-ID-124161), May, 1996. For the most up-to-
date progress, see T. Raubenheimer, MO203 (invited talk) at
this conference (LINAC2000), Monterey, Aug. 2000.
[7] T. Okugi, et al., Phys. Rev. ST – Accelerators and Beams, 2,
022801 (1999).
[8] J. Urakawa, and K. Kubo, et al. , presentations
at EPAC 2000, Vienna, June, 2000. Available at
http://lcdev.kek.jp/ATF/Conf/EPAC2000.html . H. Hayano,
MOC01 at this conference (LINAC2000), Monterey, Aug.
2000.
[9] Y. Higashi, et al.,T. Higo, etal., J.W.Wang, et al.,and Z .Li,
et al.: TUA01, TUA02, TUA03, TUA9, and TUE4 at this
conference (LINAC 2000), Monterey, Aug. 2000.
[10] C. Adolphsen, TUE01 and TUE02 at this conference
(LINAC2000), Monterey, Aug. 2000.
[11] Y.H. Chin, et al., presentation at EPAC 2000, Vienna, Ju ne,
2000. KEKPreprint2000-70, Aug. 2000.
[12] S. Tokumoto, et al., THA02 at this conference (LINAC
2000), Monterey, Aug. 2000. |
OBSERVATION OF A H- BEAM ION INSTABILITY
Milorad Popovic and Todd Sullivan
Fermi National Accelerator Laboratory1
Batavia, IL 60510, USA
1 This work is supported by U.S. Dept. Of Energy through the University Research Association under contract DE-AC35-89ER40486 Abstract
We report the results of observations of H- beam
instabilities at the Fermilab Linac. By intentionally
creating “high” background pressure with different gases
in the 750 keV transport line we observed coherent
transverse beam oscillations. The minimal pulse length
required to observe oscillations and the frequency of
oscillations are functions of pressure and mass of the
background gas. The oscillations are present in both
transverse planes and very quickly reach saturation in
amplitude growth. The observed characteristics of beam
oscillations are in quantitative agreement with “fast
beam-ion instability” described by Raubenheimer and
Zimmermann[1]. Effects described here are occur far
from the normal operating range of the Fermilab Linac
but may be important for many future high intensity
accelerators.
1 INTRODUCTION
In 1985 a BPM system was introduced in the Fermilab
Linac[2] and fast H- beam transverse oscillations were
noticed when the pressure of the 750 keV line was
degraded by turning off the large ion pump near the H-
source. Recently we have revisited this phenomena in the
light of renewed interest in this type of beam instability.
In many future rings, this transient instability is predicted
to have very fast growth rates, much faster than the
damping rates of existing and proposed transverse
feedback systems, and thus is a potential limitation. The
instability described in this paper is caused by residual
gas ions. Charged particle beams, traversing a beam line
or circulating in a storage ring, ionize the residual gas and
generate free electrons and ions. The instability
mechanism is the same in the beam line and storage rings
assuming that ions are not trapped turn by turn in the
rings. The ions generated by the head portion of the beam
pulse oscillate in the transverse direction causing a
growth of the initial perturbation of the beam. In our
case, ions of the background gas are trapped and focused
by H- beam. They start to oscillate and create transverse
deformation of the H- beam. The model employed by
Raubenheimer and Zimmermann is in quantitative
agreement with our observations. In this model all ions
oscillate with the same frequency, the frequency of
small-amplitude oscillations of the centroid in the
potential well of the beam. In our experiment we see a frequency spread of oscillations which increases with
pressure.
2 EXPERIMETAL SETUP
During the experiment, a 750keV H- DC beam was
transported along a 10 meter long transport line to the
buncher cavity and Drift Tube Linac, see figure 1. The
beam size along the line does not change significantly
and averages about 2 cm in diameter. The peak current of
the beam was 65 mA at a starting background pressure of
1.2e-6 Torr. The background gas was mostly hydrogen
gas from the Ion source. The pulse length of 35 us is
created using a chopper which is at the beginning of the
line. Vacuum in the transfer line was maintained using a
turbo pump near the Chopper and another by Tank#1.
The pressure in the line was measured using an Ion
Gauge near the gas bleeding valve that was used for
introducing different gases and creating different
pressures in the line.
Figure 1
Beam current was measured at the entrance to Tank#1
and at the exit of Tank#2. The beam signal on the BPM
after Tank#2 was used to observe beam centred position Gas Valve BPM Chopper
during the pulse. The signals were recorded using a
LeCroy scope. For measurements of the beam oscillation
frequency, FFT signals were averaged over many beam
pulses. To avoid any frequency signal not related with
beam oscillations only the last 30 us of the beam pulse
was Fourier analysed. There was no noticeable difference
between horizontal and vertical planes, so all data was
taken looking at the horizontal plane only.
2 RESULTS
Using a bleeding valve in the middle of the transfer
line measurements were repeated for several different
gases. We used hydrogen, helium, argon and krypton as
the background gas. With the bleeding valve we where
able to
have fine control of the pressure in the line.
Figure 2
Two toroids were used to record the beam current in the
transfer line to insure that the current was constant in the
line for the whole range of pressure and gases.
Figure 3
As a way of monitoring that focusing properties of the
line did not change, we recorded beam current at end of linac, see figure 2. It is known that transmission through
the linac is very dependent on the quad settings in the
line. We have not measured beam profiles in the line but
know that the Buncher is an aperture restriction in the
line, and the high transmission is achieved only if the
beam has a waist at the Buncher position. We can say
with some degree of confidence that the high background
gas pressure did not change the beam profile in the line.
Under normal operation, the pressure in the line is
2.4x10-6 and in the experiment the highest pressure was
1.0x10-4.
Figure 4
The lower trace in figure 3 is a beam position signal from
the BPM at the entrance to Tank#3 under normal
operation with pressure in the line of 2.4x10-6Torr. Small
fluctuation of the signal are result of the noise in the
beam and pick-up. Upper trace is the same signal with a
pressure in the line of 4.8x10-5 Torr. It is clear that
Figure 5 Current in line
TT
1 >2 >
1) CH2: 20 mVolt 2 us
2) CH2: 20 mVolt 2 us Hydrogen, normal operations 2.4x10-6Torr
Nitrogen 3x10-5
4.8x10-5Torr T
T
TT
T
1 >
2 >
3 >4↑
5 >
1) p5e-5: 50 mVolt 5 us
2) p8e-5: 50 mVolt 5 us LinCurr,TranLineCurr&Pressure
20253035404550556065
1 4 7 10
Pressure(10-5Torr)Current(mA)Helium
Nitrogen
Argon
Krypton
THelium
TNitrogen
TArgon
TKryptonCurr, in Line
Lin .Current 3x10-5
4x10-5
5x10-5
8x10-5
1x10-4 oscillations start about 1.2 µs after the start of beam,
develop very quickly and saturate after one or two full
oscillations. The time for oscillations to develop and the
frequency of oscillations depend on the pressure and type
of background gas. Figure 4 shows scope traces of the
beam position for different gas pressures in Torr, when
the background gas is argon. To measure the frequency
of oscillations we used an FFT option built into the
LeCroy scope.
Figure 5
Figure 5 shows four traces: beam position, beam
intensity, FFT of beam position signal and the average of
the last 23 FFT beam signals. To exclude no-beam related
signals as in Figure 5 that come from noise on the pick-
up, we used only the last 30 us of the beam signal. For
low gas pressure we see a relatively sharp frequency
signal in the range of 0.5 MHz. As pressure is increased
the frequency signal is broadened and moves toward
higher frequency with the peak at 1.1 MHz for all gases.
Figure 6 4 CONCLUSIONS
We have intentionally created coherent transverse beam
oscillations. The oscillations have the following
characteristics:
• The time to develop oscillations strongly
depends on the pressure but not much to the gas
species,
• It takes only a few oscillations for the instability
to fully develop and saturate.
• At low gas pressure, the oscillation frequency is
~0.5MHz .
• At higher gas pressure, the oscillation frequency
peaks at ~1.1MHz for all gas species.
REFERENCES
[1] E. McCrory, G. Lee and R. Webber, “Observation of
Transverse Instabilities in FNAL 200MeV Linac”,
1988 Linear Accelerator Conf. CEBAF-89-001, pp
182-184.
[2] T.O.Raubenheimer and F. Zimmermann, Phys. Rev.
B52, 5487(1995).
Nitrogen 8x10-5
Nitrogen 3x10-5 |
2GeV SUPERCONDUCTING MUON LINAC
Milorad Popovic
Fermi National Accelerator Laboratory 1
Batavia, IL 60510, USA
1This work is supported by the U.S. Dept. of Energy through the Univ. Research Association under contract DE-AC35-89ER40486 Abstract
A muon collider as well as a neutrino factory
requires a large number of muons with a kinetic energy
of 50GeV or more. Muon survival demands a high
gradient linac. The large transverse and longitudinal
emittance of the muon beam coming from a muon
cooling system implies the need for a large acceptance,
acceleration system. These two requirements point clearly
to a linac based on superconducting technology. The
design of a 2GeV Superconducting muon Linac based on
computer programs developed at LANL will be
presented. The design is based on the technology
available today or components that will be avaible in the
very near future. |
BEAM ENERGY STABILIZATION OF THE KEK 40MEV PROTON LINAC
Z.Igarashi, K.Nanmo, T.Takenaka and E.Takasaki
High Energy Accelerator Research Organization, KEK
Oho1-1, Tsukuba, Ibaraki, 305-0801, Japan
Abstract
The new method to stabilize the beam energy of the
KEK 40MeV proton linac, is developed now. In this method, the signal of the velocity monitor installed upstream the debuncher in the 40MeV beam line, is processed and then fed to the phase shifter of the debuncher rf system so as to cancel the fluctuation of the beam energy.
In this article, the beam tests to prove the validity of
this method and the system are described.
1 INTRODUCTION
Various improvements to increase the beam intensity of
the KEK 12GeV PS for neutrino oscillation experiments have been continued these several years[1]. Since one of the causes that limit the beam intensity is the beam loss in the accelerators or the beam lines, the accelerators should be tuned with scrupulous care and these conditions be kept constant during the long operation period.
One of the beam parameters that affect the next
accelerations and the beam extractions is the centre energy (momentum) of the beam. In order to keep the beam energy being constant in the proton linac, the rf
system should equip the feedback loop which stabilizes the accelerating field. Unfortunately, since the rf system of our linac is operated near the saturation, the effects of the feedback are not expected much. Hence, the investigation of the new method to stabilize the beam energy was started.
2 BEAM TESTS
2.1 The layout of the linac
The KEK 40MeV proton linac that consists of the
prebuncher, the 20MeV tank, the 40MeV tank, the debuncher, and the 40MeV beam line is shown in Figure 1. Two velocity ( β) monitors are installed in the 40MeV
beam line, one (40 β1) is for the detection of the output
beam of the linac and the other (40 β2) is for the beam to
inject the 500MeV booster synchrotron. The debuncher is installed between them[2],[3]. The typical waveforms of the 20β1, the 40β1 and the 40 β2 are shown in Figure 2.
Figure 3 shows the variations of the 40 β1 and the linac
beam current during the operation period of about a month. The fluctuations of the accelerating energy are within 0.9%.
G.V 2T 5T 4CM 5T 3
P.Bend
D3
CM 6
CM 7
M8SV 3
SH 3
PR 5
G.V.Q7
Q9
B1
B 2G.V
G.V
PR 6
SV 4
Q12
Q13SH 5
SV 5
Q14
HB40 MeV F.C.500MEV BOOSTER
SYNCHROTRON
S1
M1PR 7Q11
B. INJ G.VS8PR 4
Q6Q3
Q4
Q5PR 2LINAC G.V 2
PR 1
Q1
Q2SHV 2SHV 1
Tr.m ag 1
Tr.m ag 2
CT 2
Q1
Q3Q2
Q4G.V
20MeV LinePR 3
Q10LEBTBeam Shutter3
P.bend
20MeV tankV e lo c ity mo n ito r
(20β)
40M eV tankCT & Profile
Linac G . V 1CT 1
Q8
V e lo c ity mo n ito r
(40β1)Debuncher
Pulse bend
Sl it
Analyzer
; loss m onitorEmittance monitor
40MeV lineEmittance monitor
Pre buncher
Velocity monitor (40β2)CT 3Emi tt anc e mo ni to r
PR 7SV 82.124m
3.277m
2.414m0.523m
Figure 1: The layout of the KEK 40MeV proton linac and the beam lines.
Figure 2:20 β(left upper), 40 β1(left lower) and
40β2(right)
Figure 3: The variations of the 40 β1(upper) and the
linac beam current(lower).
2.2 Beam tests
Figure 4 shows the layout of the beam tests. The
velocity monitor signals are acquired by the VME system and simultaneously observed by the scope. The debuncher rf system, therefore, the phase shifter is controlled by the PLC(Programmable Logic Controller).
Figure 4:The layout of the beam tests.
The result of the energy variation measured by 40 β1
and 40β2 versus the debuncher phase is shown in Figure 5.
Though the variation of the momentum spread ( ∆P/P), it
is obvious from Figure 5 that it is possible to change the beam energy within
±300MeV.
Figure 5: Energy variations versus the debuncher phase. In order to estimate the new method, we studied
whether it cancels the energy fluctuations due to the accelerating field of the two tanks, namely, the 20MeV tank level, the 40MeV tank level and the phase between two tanks.
Figure 6, Figure 7 and Figure 8 are the results of these
tests. In these graphs, the 40beta1 is the plots for the linac output energy, the 40beta2@D.B ON for the constant phase of the debuncher and the 40beta2@P.ADJ for the
adjusted phase to cancel the energy fluctuation.
Figure 6: The effect of the fluctuation of the 20MeV
tank level
. 4040.140.240.340.440.5
7 7.05 7.1 7.15 7.2 7.25 7.3 7.3540beta1
40beta2 @D.B ON
40beta2 @P.ADJ
20MEV TANK LEVELVELOCITY
MONITOR
(40b1)DEBUNCHER
VELOCITY MONITOR (40b2)3.277 m
2.414m0.523mPHASE
SHIFTERPHASE
DETECTOR
x2
20kW
AM PLIFIERS.GS/H
S/Hto SCOPEto VMEto VME
to SCOPE
PLC
PHASE
DETECTOR39.94040.140.240.340.440.540.640.7
- 2 02468 1 0 1 240beta1
40beta2
PHASE SET(V)
Figure 7: The effect of the fluctuation of the 40MeV
tank level.
In all tests, the input rf power to the debuncher is
15.5kW.
From these results, the fluctuations due to the
accelerating fields of the linac are within 0.08%.
Figure 8: The effect of the fluctuation of the tank phase.
3 THE SYSTEM PLAN FOR THIS
METHOD
In this method, to stabilize the beam energy, the
debuncher phase should be set to the appropriate value that is caluculated from the signals of the velocity monitor and the phasing system between the beam and the debuncher field. Furthermore, for the stabilization during
the pulse duration of the beam, the WE 7000 SYSTEM made by YOKOGAWA ELECTRIC COMPANY will be introduced as the data acquisition and the control system.
3 SUMMARY
It is proved that the new method by using the velocity
monitors and the debuncher is effective to stabilize the beam energy. Especially, the energy fluctuation due to the 20MeV tank level, the 40MeV tank level and the phase between two tanks are reduced within 0.08%.
It is expected that the new system will be completed
immediately and then will be used for the normal operation of the KEK 12GeV PS.
REFERENCE
[1]I.Yamane, H.Sato “Accelerator Development for K2K
Long-Baseline Neutrino-Oscillation Experiment,” January, 2000
[2]Z.Igarashi, K.Nanmo, T.Takenaka and
E.Takasaki,"Velocity Monitor for the KEK 40MeV Proton Linac," Proc. 1992 Linac Conference., (1992)
[3]Z.Igarashi, K.Nanmo, T.Takenaka and E.Takasaki, ”A
New RF System for the Debuncher at the KEK 40-Mev Proton Linac,” Proc. 1998 Linac Conference., 929 (1998)
4040.140.240.340.440.540.6
15 20 25 30 35 40 45 50 5540beta1
40beta2 @D .B ON
40beta2 @P.ADJ
TANK PHASE(deg)40.340.3540.440.4540.5
8.75 8.8 8.85 8.9 8.95 9 9.05 9.1 9.1540beta1
40beta2 @D.B ON
40beta2 @P.ADJ
40MEV TANK LEVEL |
High Current Proton Tests of the Fermilab Linac
M. Popovic, L. Allen, A. Moretti, E. McCrory, C.W. Schmidt and T. Sullivan
Fermi National Accelerator Laboratory1
Batavia, Illinois, USA
1 This work is supported by U.S. Dept. Of Energy through the University Research Association under contract DE-AC35-89ER40486 Abstract
The peak current limit for the Fermilab Linac
was recently studied. The purpose was to learn what
components of the present Linac can be used for the first
stage of a proposed proton driver[1]. For this application
the Linac must provide a H- beam in excess of 5000 mA-
µsec per pulse. The original Fermilab Linac was designed
for protons with a peak current of 75 mA and a pulse
length of four Booster turns (~10 µsec). The high energy
replacement was designed for a peak current of 35 mA
and a beam pulse length of 50 µsec. The present H-
source cannot deliver more than ~80 mA which produces
55 mA in the Linac. Using a proton source allows the
system to be tested to currents of ~100 mA and pulse
lengths long enough to observe the effects of long pulses.
This test has shown that the present Linac can accelerate
beam having a peak current up to ~85 mA with beam loss
comparable to the present Linac operation (~45 mA). The
results of the test will be presented.
1 INTRODUCTION
During its lifetime the Fermilab Linac has gone
through two mayor modifications. In both cases these
improvements were motivated by the need for higher
beam intensity from the Booster synchrotron.
Construction of the Linac began in 1968 and a 200-MeV
proton beam was first produced on November 30, 1970.
The design goal[2] of 75 mA and 10 µsec (four Booster
turns) was achieved quickly and surpassed. Although the
design intensity was 75 mA, the Linac was built to
accelerate at least 100 mA with similar beam emittance.
Emittance preservation is essential for successful
horizontal injection and stacking of four turns in the
Booster. Eight years later the proton source was replaced
by a H- source to accelerate a long, low-intensity H- beam
of 25 mA and build intensity in the Booster using multi-
turn charge-exchange injection. In the summer[3] of 1993
the Linac was upgraded again. The last four drift-tube
tanks were removed and a side-coupled structure installed
to increase the final energy to 400 MeV. The higher
injection energy in the Booster increased the magnetic
guide field at injection, reduced the frequency range of
the RF accelerating system and increased the Booster’s
space-charge limit at injection and therefore its possible
intensity. Although the Linac energy upgrade was
designed for a peak beam current of 35 mA, typical current at the end of the Linac was between 28 and 36
mA. Since the energy upgrade there have been small
changes in operating parameters of the ion source, low
energy transport line and Linac that have resulted in a
steady increase in peak beam current extracted from the
Linac. Figure 1 shows the peak beam current in the Linac
and corresponding Booster beam intensity over the past
thirty years of operations. In the constant quest for higher
beam intensity and considering that the side-coupled
structure was design for a maximum beam current of 35
mA, the intensity at 400 MeV has been greatly increased.
Figure 1. Linac and Booster intensity with time
2 SYSTEM MODIFICATION
Obtaining the anticipated current using an H- source
would have required a significant source development
program which, in part, was the purpose of this study.
Therefore an old proton source, a duoplasmatron, was
reinstalled in one of the Cockcroft-Walton pre-
accelerators. This source once produced several hundred
milliamperes for short pulse (3-5 µsec) 200-MeV Linac
operation and could easily provide a proton beam with a
current for this test. Other modifications required
changing the polarity of the preaccelerator high voltage
and magnetic transport dipoles. Also the 750-keV input
and 400-MeV output transport lines had to be retuned for
protons. The 750-keV transfer line is short. It has only
three quad triplets and a Buncher cavity. The polarity of
the quads were kept as for H- operation. A Trace2D Linac&Booster Peak Intesity
020406080100120140
1972
1976
1980
1989
1994
1997
2000
YearLinCurrent(mA)
0123456
BooIntesity(10^12)LinCurrBooInt
Protons
H- Ions model was used to tune the line, see Figure 2. This model
was also used to show that the line has sufficient
flexibility to match to the Linac with no change in the
Linac polarities or settings.
Figure 2. Trace2D run of the 750-keV transport line for
protons
3 COURSE OF TEST
H- operating conditions for beam transmission and
capture in the low-energy linac is 74% and 95% in the
high-energy linac. Thus there is a loss of ~30% that must
be accommodated by the source.
T
T14↑15↑
16 >17 >
14) CH1: 1 Volt 10 us
15) CH1: 1 Volt 10 us
16) CH2: 100 mVolt 10 us
17) CH2: 100 mVolt 10 us
Figure 3. Loading of the High-Energy Klystron RF pulse.
It was assumed that all limits on beam current will be
visible for a beam pulse between 10 to 30 µsec. Sparking
rates in the side-couple modules were monitored for any
increase, as were the amplitudes and reflected power on
the each RF station. The beam loss monitors were
carefully watched at all times. All high peak current
related measurements were done with a minimal number
of pulses. Figure 3 shows the RF gradient envelope and
reflected power signals for Station 7 of the side-coupled
linac. Red traces are RF signals with beam. Yellow traces
are the same signals without beam. The bumps are a result of gain and feed-forward adjustments. The feed-
forward correction compensates for beam loading starting
at the head of the beam pulse and lasting through the
duration of the pulse.
TT
10↓11 >12 >
13↓10) CH2: 100 mVolt 50 us
11) CH1: 100 mVolt 50 us
12) CH1: 100 mVolt 50 us
13) CH2: 100 mVolt 50 us
Figure 4. Loading of Low-energy RF pulse.
These signal were watched for signs of RF saturation
during high peak current running. Similar RF signals for
the low-energy linac tanks were watched. Figure 4 shows
a low-energy gradient and reflected power signal with
and without beam. This portion of the Linac was
originally design for higher peak currents. The dip in the
signal is present only during beam time and is a result of
beam loading and a lack of full beam loading
compensation. Every attempt is made to keep the gradient
signal constant during beam time.
Figure 5. Horizontal beam motion at 400 MeV.
Energy variation during the pulse can be observed at the
end of the Linac following a spectrometer magnet. These
variations are believed due to the variation of the
gradients along the drift-tube linac. Figure 5 shows the
horizontal position of the beam during the pulse.To
insure that the emittance of the high current beam from
the source was not degraded, the beam emittance at the
entrance to the Linac was measured, see Table 1. Clearly,
the beam emittance is rather constant and not
significantly dependant on the beam current for values
between 78 and 92 mA. This is important because there was significant change in the beam loss going from 85 to
95 mA or higher. Figure 6 shows toroids and beam loss
monitors along the Linac for a current of ~85 mA.
LinacInputEmittanceForProtonStudy
3/18/99E.McCrory
EmitPrT1InLinacOut(mA)E(95%Nor)
Horizontal 78 2.4
82 2.4
Emittance in pi-mm-mr 88 2.8
92 2.4
BuncherOFF 2.2
Vertical 78 3.1
82 2.9
88 2.9
92 3
BuncherOFF 3.1
Table 1. High current beam emittance.
Figure 7 shows the same signals for a current of ~95 mA.
The loss along the Linac has increased. Wire profiles
along the high-energy linac have not shown any visible
increase in beam size for the transverse planes.
Figure 6. Current and losses through Linac at ~85 mA.
Figure 7. Current and losses through Linac at ~95 mA.
3 SUMMARY
A proton beam with a peak current up to 90 mA
was accelerated through the Linac with similar losses as a
lower intensity H- beam. For currents above 90 mA there
is additional loss with indications that this loss is related
to a large energy spread and lack of RF voltage to
properly accelerate the beam and keep it in the bucket. It
is believe that the present Fermilab Linac can accelerate
up to 90 mA of H- beam for future uses with a suitable
source.
4 ACKNOWLEDGEMENTS
The authors would like to acknowledge the
work of several people who were instrumental in carrying
out this test. Although the test was relatively simple the
preparation and restoration of the Linac was extensive.
From the Linac, James Wendt and Ray Hren prepared the
source, its installation and later removal; Lester Wahl
assisted with RF monitoring and control. From the
Mechanical group, Danny Douglas, Mike Ziomek and
Ben Ogert prepared and restored the Cockcroft-Walton.
5 REFERENCES
[1] “A Development Plan for the Fermilab Proton
Source”, ed. S. D. Holmes, September 1997,
Fermilab TM-2021.
[2] Design Report, National Accelerator Laboratory,
United States Atomic Energy Com., July, 1968.
[3] Fermilab Linac Upgrade Conceptual Design,
November 1989.
|
arXiv:physics/0008177v1 [physics.acc-ph] 20 Aug 20002DSIMULATIONOF HIGH-EFFICIENCYCROSS-FIELD RF POWER
SOURCES∗
Valery A. Dolgashev,SamiG. Tantawi†,SLAC, Stanford, CA 94309,USA
1 INTRODUCTION
In a cross field device[1] such as magnetron or cross field
amplifier electrons move in crossed magnetic and electric
fields. Due to synchronismbetween electron drift velocity
andphasevelocityofRFwave,thewavebunchesthebeam,
electron spokes are formed and the bunched electrons are
decelerated by the RF field. Such devices have high effi-
ciency (up to 90%), high output power and relatively low
cost. Electrical design of the cross-field devices is diffi-
cult. The problem is 2D (or 3D) and highly nonlinear. It
hascomplexgeometryandstrongspacechargeeffects. Re-
cently, increased performance of computers and availabil-
ity of Particle-In-Cell (PIC) codes[2, 3], have made possi-
ble the design of relatively low efficiency devices such as
relativistic magnetrons or cross field amplifiers [4]. Sim-
ulation of high efficiency ( ∼90%) devices is difficult due
to the long transient process of starting oscillations. Use
of PIC codesfordesign ofsuch devicesis not practical. In
thisreportwedescribeafrequencydomainmethodthatde-
veloped for simulating high efficiency cross-field devices.
In the method, we consider steady-state interactionof par-
ticles with the modes of RF cavity at dominant frequency.
Self-consistencyofthesolutionisreachedbyiterationsu n-
til powerbalanceisachieved.
2 PHYSICALMODEL
Cross-field devices consist of a cathode and a surround-
ing anode. The structure is a cavity with a set of resonant
eigenmodes. Macroparticles are emitted from the cathode
and moved by forces of electromagnetic fields. The elec-
tromagneticfieldsaredeterminedbyappliedexternalelec-
tric potential between anode and cathode, oscillating field
of cavity modes, and space charge fields. We use geom-
etry with arbitrary piece-wise planar boundaries. In order
to solve theelectrostatic andelectrodynamicproblems,we
apply methods that do not require mesh generation. Inter-
action with magnetic field is determined by uniform mag-
netic field Hzwhich is parallel to z-axis and orthogonal
to the plane of simulation. There are several assumptions
thatweusetosimplifytheproblem. Theseassumptionsare
basedontheworkingregimeofthedevicesthatwewantto
simulate. Devices will have low current density, are non-
relativistic,andhaveresonantsystemswitharelativelyl ow
density of the cavity modes. Hence, we can neglect mag-
netic fields due to space charge and cavity modes. We can
∗This work was supported by the U.S.Department of Energy cont ract
DE-AC03-76SF00515.
†Also with the Communications and Electronics Department, C airo
University, Giza, Egypt.also use cavity modes with eigen-frequencies close to the
workingfrequency.
2.1 Basicequations
We are solving a steady state problem of electron beam
flow in self-consistent electromagnetic fields. Total fields
are superposition of static electric /vectorE′and magnetic /vectorH′
fields, and “oscillating” electric /vectorE(ω)and magnetic /vectorH(ω)
fieldsas
/vectorE(t) =/vectorE′+ℜe{/vectorE(ω)ejωt},/vectorH(t) =/vectorH′+ℜe{/vectorH(ω)ejωt}.
Hereωisangularfrequency, tistime. Weseparatetheelec-
trodynamicproblemintotwoparts. Thefirstpart–electro-
staticpotential Φisgeneratedby“external”anode-cathode
potential and by the static component of the space-charge
electric fields. The second part – the dynamic electromag-
netic fieldshaveaharmonic ejωttime(t)dependence.
2.2 Staticfields
We findthestatic electricfieldfrom /vectorE′=−∇Φ,usingthe
Poissonequation :
∇2Φ =−ρ
ǫ0, (1)
where ∇is the gradientoperator, ρis volumechargeden-
sity averaged over oscillation period T= 2π/ω.ǫ0is the
electricpermittivityofthevacuum.
2.3 Oscillatingfields
To solve the secondpart of the problem,we write the time
harmonic Maxwellequations as
∇ ×/vectorE=−jωµ0/vectorH,∇ ×/vectorH=jωǫ0/vectorE+/vectorJω.(2)
Here/vectorJωis electric currentdensity, µ0is the magneticper-
meability of vacuum. Oscillating fields inside a cavity are
expanded in terms of the cavity eigenmodes ( /vectorEs,/vectorHs)and
thefastoscillating electricpotential ϕωas
/vectorE=/summationdisplay
sAs/vectorEs− ∇ϕω,/vectorH=/summationdisplay
sBs/vectorHs.(3)
Heresismodeindex, AsandBsaretheeigenmodeam-
plitudes. Usingtheexpansion(3)wegetthe Poissonequa-
tionforthepotential:
∇2ϕ=∇ ·/vectorJω
jωǫ0=−ρω
ǫ0, (4)
where ρωis the oscillating space-charge density. Ampli-
tudesofthe electricfieldexpansionaregivenby
As=ω
j(ω2−ω2s)/integraltext
V/vectorJω/vectorE∗
sdV
ǫ0/integraltext
V/vectorEs/vectorE∗sdV. (5)
Hereωsisthemodeeigen-frequencyofthemode, Visthe
cavityvolume.2.4 Equationofmotion
Equationofmotionforan electronincrossed-fieldsis
d/vector p
dt=qe/vectorE(t) +µ0/vector v×Hz, (6)
where /vector pisthe relativisticmomentum, qeisthe charge,and
/vector visthevelocityoftheelectron. Currentdensityinducedby
theelectronmotionis /vectorJ=qe/vector vδ(/vector r),where /vector ristheposition
vectoroftheelectron,and δistheKroneckerdeltafunction .
3 NUMERICALMETHODS
Wecreatedseveralseparateprogrammodulestosimulatea
cross-fielddevice. Firstisan RFfieldsolver thatcalculates
eigenmodes and eigen-frequenciesin the cavity; second is
thePoisson solver that finds electric fields due to external
potential,static spacecharge,andoscillatingspacechar ge;
and third, the tracking module that performs tracking of
electrons through electromagnetic fields. For simulation,
we consideran arbitrary,piecewisebounded2Dgeometry.
3.1 Planargeometry
/K5A
/K50/K6F/K72/K74/K20/K32/K2C
/K59/K27
/K32
Γ/K5A/K27
/K31
/K78/K79/K50/K6F/K72/K74/K20/K31/K2C
/K59/K27
/K31/K5A/K27
/K32
/K31/K37/K38/K20
/K31/K32/K20/K34/K20
/K33/K20/K35/K20/K36/K20/K37/K20
/K39/K20/K31/K30/K20
/K31 /K31/K20
/K31/K32/K20
/K31/K33/K20
/K31/K34/K20
/K31/K35/K20
/K31/K36/K20/K31/K38
/K31/K39
Figure1: Planar geometry.
The geometry is cylindrical (uniformin the z-direction)
as illustrated on Fig. 1. It consists of planar sidewalls and
apertures. Thegeometryin the x, yplanecan bedescribed
byasetofpoints zs= (xs, ys),where s= 1,2..., N′;here
N′isthetotal numberofsidewallsandapertures. Periodic
boundary conditions are applied to the apertures. The pe-
riodic boundary allows us to use only part of the structure
and significantly reduce simulation time. In the particular
case shown in Fig. 1 the geometryhas N′= 19sidewalls,
two apertures (ports) with starting points p= 1,15, and
the cathode and anode determinedby s= 16,17,18,19,1
ands= 2,3, ...,15respectively.
3.2 RFfield solver
ThedescriptionoftheRFsolverthatisusedinthismethod
is published in [5]. Here we briefly outline its properties.
We use the scattering matrix approach [6] to calculate the
dispersionparametersoftheperiodic2Dstructure,it’sre s-
onantfrequencies,andthe correspondingfields. Thefields
are described by functional expansion. Boundary contour
mode-matching is applied in a piecewise bounded 2D re-
gionisappliedtoobtainthescatteringmatrixandfieldam-
plitudes [7]. The Galerkin method is used for the mode-
matchingprocedure. Thegeometryisdividedintoregions,
and electromagnetic fields in each region are expanded inseries of planewaves or (for low frequencies) Bessel func-
tions. Scattering matrices from the regions are combined
using the generalized scattering matrix technique. Reso-
nant and periodic boundary conditions [6] are used to ob-
tain resonant frequencies, dispersion parameters, and cor -
respondingfields. Wecalculatetheelectricfieldsonapolar
grid (only in the region of field-particle interaction), in o r-
der to speed up calculation of fields for the macroparticle
tracking. To obtain field at the macroparticle position we
use 2Dsplineinterpolation.
3.3 Poissonsolver
We use an efficient method for solving the Poisson equa-
tionforelectricfieldsina2-D,arbitrarilyshapedgeometr y.
The solution is based on the method of moments. Point-
matching in a piecewise bounded 2D region is applied to
obtainthechargedensityontheboundary. Theboundary’s
chargedensitydeterminesthefieldsandpotentialsthrough -
out the interior region. We use a complex representation
of the fields and potentials in the solution [8]. We apply
periodic boundary conditions to simulate the fields in the
periodicstructure.
Formulation We solve equation (1) in 2D. In the 2D
case it is advantageous to represent the position and field
vector’s (x, y)components by a single complex represen-
tation. We will work with functions of a complex variable
z=x+jy. Thefieldstrength /tildewideEcanbewrittenintermsof
the scalarpotential Φ = Φ( z)as
/tildewideE(z) =−dΦ∗
dz. (7)
Here∗representsthecomplexconjugate. Aneffectiveline
charge q(point charge in 2D geometry) has the complex
potential Φ = (q/ǫ0)logz. Weapproximatethechargedis-
tribution on the boundary of the region as a sum of “step”
functions. We divide each element(sidewall and aperture)
of the boundary into Nbstraight pieces or “charged lines”
withuniformchargedensity σalongthepiece. A uniformly
charged straight wall with beginning and end coordinates
z1andz2,respectively,willproduceacomplexpotentialat
the point zw
Φ(zw) =/integraldisplay
Lσ
ǫ0log(z−zw)dz, (8)
where Lis the contouralongthe line. Equation(8)is inte-
gratedanalytically.
Field strength of the charged wall We obtain the
electricfieldofthe chargedline bysubstituting(8)into(7):
/tildewideE(zw)ǫ0
σ=/bracketleftbigg|z1−z2|
z1−z2log/parenleftbiggzw−z1
zw−z2/parenrightbigg/bracketrightbigg∗
.(9)
Thevalueofthefunctionisundefinedontheline’scontour.
However,forus, the fieldsinside the regionare ofinterest.
Therefore, the direction of the field (for positive charge)on the line’s contouris chosento be directed inward. Also
singularities at points z1andz2can affect the field’s cal-
culation. Macroparticleswithfinitedimensionsareusedto
avoidthissingularity.
Periodicboundarycondition Weassumethatthepo-
tential and field strength are repeated on the period’saper-
tures (Fig. 1). Let z′
1∈Y′
1andz′
2∈Y′
2. If we shift the
region to the right so it coincides with the next period, the
coordinate z′
1will be transformed into coordinate z′
2. The
periodicboundaryconditionbecomes
Φ(z′
1) = Φ( z′
2),∂Φ(z′
1)
∂n=−∂Φ(z′
2)
∂n.(10)
WeassumetheDirichletconditiononthesidewalls(except
fortheapertures)as
Φ(Γ′) =ζ(Γ′),Γ = Γ′+Y′
1+Y′
2.(11)
Integral equations For periodic boundary conditions
(10)and(11)surfacechargedensity σmustsatisfythecou-
pledintegralequations
/integraltext
Γlog(zw−z)σ(z)dz=ǫ0ζ(zw), zw∈Γ′,/integraltext
Γlog(z′
1−z)σ(z)dz=/integraltext
Γlog(z′
2−z)σ(z)dz,/integraltext
Γ/braceleftBig
∂log(z′
1−z)
∂np/bracerightBig∗
σ(z)dz+πσ(z′
1) =
=−/integraltext
Γ/braceleftBig
∂log(z′
2−z)
∂npdz/bracerightBig∗
σ(z)dz−πσ(z2),
z∈Γ, z′
1∈Y′
1, z′
2∈Y′
2,
,
(12)
in which∂log(zw−z)
∂npdenotes the normal derivative of
log(zw−z)at the point zwassuming zis fixed; Γ =
Γ′+Y′
1+Y′
2; coordinates z1andz2are the same as in
(10);and ζ(zw)isthe externalpotential.
Numerical approximation We solve the integral
equation numerically, by approximating the source densi-
tiesbystep-functions[9]. Thuswedividethegivenbound-
aryΓintoNΓintervals and assume that the simple source
density σhas a constant value within each interval. Then
denoting these constant values by σi,i= 1,2, ..., N Γ, we
approximate ΦandEby
/hatwideΦ(zw) =NΓ/summationdisplay
i=1σiǫ0/integraldisplay
ilog(zw−z)dz,and (13)
/hatwideE(zw) =NΓ/summationdisplay
i=1σi
ǫ0/integraldisplay
i/parenleftbiggdlog(zw−z)
dzw/parenrightbigg∗
dz, (14)
where/integraltext
idenotesintegrationoverthe i-thintervalof Γ. We
substitute (13) and (14) into (12) to obtain numerical ap-
proximation for periodic solution. The unknowns (in the
systemobtained)arethechargedensityontheintervals σi,
the potential andthe electric field on the periodicaperture .
All coefficients in the system are calculated analytically.
For practical geometries, the matrix of coefficients is welldefined and there is no difficulty in solving the system di-
rectly. For macroparticle tracking, the electric field calc u-
lated onpolargridand theninterpolatedat the macroparti-
cle position(sameasforRF fields).
3.4 Tracking
We find a macroparticle trajectory by using the 4th order
Runge-Kutta method for integrating the equation of mo-
tion (6) in polar coordinates. Then, we integrate the com-
plexelectricfieldofthecavitymodesalongthetrajectoryt o
find coefficientsfor the cavity’seigenmodes(5). We mon-
itor energyconservationin orderto verifyaccuracyof cal-
culation. Forthat purposewe usetotal energythatconsists
ofkineticenergyofthemacroparticleandintegralofstati c
(due to external potential and static space charge) and os-
cillating(duetocavitymodesandoscillatingspacecharge )
electric fields along the trajectory. Initial charge and ve-
locity/vector varedeterminedbyaspace-charge-limited-emission
modelanda relaxationscheme.
3.5 Algorithm
We start simulation by calculatingdispersion the curvefor
the spatial period of the device (using the RF field solver ).
Then,wecalculateelectricfieldsfortheeigenmodes. Next,
(usingthe Poissonsolver )we calculateelectricfielddueto
external potential. Next, we start iterations using Track-
ingmodule tofindthemacroparticletrajectories,fieldinte-
grals alongthe trajectories, and electric fields due to spac e
charge. Next,weupdatethestaticandoscillatingfieldsand
start newiteration.
4 SUMMARY
We have written a C++ computer code that uses meth-
ods, described above. Accuracy of resonant frequency
calculation by RF field solver for typical geometries is
∼0.1%. We tested performance of Poisson solver and
Tracking module on diode geometries (without magnetic
field). We calculated diode current with typical accuracy
2-3% in comparison with analytical solution. Testing of
the codeoncross-fielddevicesisunderway.
5 REFERENCES
[1] G. B. Collins, “Microwave Magnetrons,” Boston tech. pub ., Inc.,
1964.
[2] B.Goplen at al, “User-configurable MAGIC Code for Electromag-
netic PIC Calculations,” Comp. Phys. Comm. , vol.87, pp. 54-86,
1995.
[3] K. R. Eppley, “Numerical Simulation Of Cross Field
Amplifiers,”SLAC-PUB-5183, 1990.
[4] X.Chen,atal,“2D/3Dmagnetronmodeling,” 2ndInt.Conf.OnCross
Field Devices and Appl. ,Boston, MA,USA,17-19 June, 1998.
[5] V.A. Dolgashev, S.G. Tantawi, “Method for Efficient Anal ysis of
Waveguide Components and Cavities for RF Sources,” EPAC’20 00,
26-30 June2000, Austria Center, Vienna.
[6] V. A. Dolgashev, “Calculation of Impedance for Multiple Waveg-
uide Junction Using Scattering Matrix Formulation,” prese nted at
ICAP’98, Monterrey, CA, USA,14-18 Sept., 1998.
[7] J. M. Reiter and F. Arndt, IEEE Trans. Microwave Theory Te ch.,
vol. 43,pp. 796-801, Apr.1995.
[8] R. B. Beth, “Complex Representation and Computation of T wo-
Dimensional Magnetic Fields”, Journal of Applied Physics, Vol. 37,
Number 7, June, 1966.
[9] L. M. Delves and J. Walsh, “Numerical Solution of Integra l Equa-
tions,” Clarendon Press,Oxford, 1974. |
F. Naito, K. Yoshino, C. Kubota, T. Kato, Y. Saito, E. Takasaki, Y. Yamazaki, KEK,
1-1 Oho, Tsukuba-shi, Ibaraki-ken,305-0801 Japan
S. Kobayashi, K. Sekikawa, M. Shibusawa, Saitama University
Shimo-Okubo, Urawa, 338-8570 Japan
Z. Kabeya, K. Tajiri, T. Kawasumi, Mitsubishi Heavy Industry
10 Oye-cho, Minato-ku, Nagoya, 455 JapanDEVELOPMENT OF THE 50-MEV DTL FOR THE JAERI/KEK JOINT
PROJECT
Tank No. 1 No. 2 No. 3
Energy (MeV) 19.7 36.7 50.1
No. of Cell 76 43 27
Length (m) 9.92 9.44 7.32
Tank dia. (mm) 561.1 561.1 561.1
DT dia. (mm) 140 140 140
Stem dia. (mm) 34 34 34
Bore dia. (mm) 13, 18 22 26Table 1. DTL design parametersFigure 1. rf-contactors
(a) rf-contactor between the end-plate and the tank, (b) rf-
contactor between the stem of the drift tube and the tank(a) (b)End plate
Vac.Tank
Vac.Copper SUS spring
Stem
TankAbstract:
An Alvaretz-type DTL, to accelerate the H- ion beam from
3 to 50 MeV, is being constructed as the injector for the
JAERI/KEK Joint Project. The following components of
the DTL have been developed: (1) a cylindrical tank, made
by copper electroforming; (2) rf contactors; (3) a pulse-ex-
cited quadrupole magnet with a hollow coil made by cop-
per electroforming; (4) a switching-regulator-type pulsed-
power supply for the quadrupole magnet. High-power tests
of the components have been conducted using a short-model
tank. Moreover a breakdown experiment of the copper elec-
trodes has been carried out in order to study the properties
of several kinds of copper materials.
1. INTRODUCTION
Construction has started of an Alvaretz-type DTL, to ac-
celerates the H- ions from 3 to 50 MeV, as the injector as
part of the JAERI/KEK Joint Project at the high-intensity
proton accelerator facility in Japan. The DTL consists of
three long tanks (maximum 9.9 m in length), each of which
is comprised of three short unit tanks (approx. 3 m in length),
to overcome difficulties with constructing the tank and as-
sembling the drift tube. The resonance frequency of the DTL
is 324 MHz. The rf pulse length is 600 µsec and its repeti-
tion rate is 50 Hz. The main design parameters of the DTL
are summarized in Table-1 [1]. This report describes three
aspects of the DTL construction; rf contactors, the charac-
teristics of the copper surface of the cavity, and the power
supply for the quadrupole magnet in the drift tube.
2. RF CONTACTOR
Two types of rf contactor have been developed: (a) a con-
tactor between the end plate and the tank cylinder; and (b) a
contactor between the stem of the drift tube and the tank.Cross-sectional views of the contactors are shown in Figure
1. The structure is very simple: a thin copper layer (0.5 mm
in thickness) surrounds a stainless steel spring. There is a
vacuum seal outside the rf contactor. The performance of
these was checked initially by a small test cavity and then
by the large cavities that are described in the next section.
3. VACUUM AND RF PROPERTIES OF
THE TEST TANK
The cavity cylinder for the DTL is made of iron, with the
inner surface covered by a copper layer (0.5 mm in thick-
ness) that was built by the Periodic Reverse (PR)
electroforming using pure copper sulphate bath, and then
finished by electropolishing [2]. A cylindrical cavity was
made to test the vacuum and the rf properties of the PR
copper electroforming surface. The size of the cylinder (560
mm in diameter, 3320 mm in length) is almost identical to
that of the longest unit tank of the DTL. The rf contactor
described in the previous section is used for the end plates.
The measured unloaded Q-value of the TM010 mode of the
cavity is 77000, which represents approximately 97 % of
the value obtained by analytical calculations. The results
indicate that (a) the electrical quality of the copper surface
is sufficiently high and (b) that the rf contactor functions
satisfactorily at a low rf-power level.
Vacuum property was also measured, and as the results in
Figure 2 show, the pressure level of the tank became 10-5 Pa
after 100 hours of evacuation. The outgas rate from the
tank surface was also measured by an integration methodMaterials The 1st breakdown field
(MV/m)
EF (PR, Pure copper sulfate) 41
EF (Copper sulfate with brightener) 13
EF (Pyrophosphate) 10
OFC (Lathe finishing) 20
OFC (Electro polishing) 16
OFC (Diamond bite) 70
(EF: Electro-Forming, PR: Periodic-Reverse
OFC: Oxygen Free Cooper)Table-2. The first breakdown fieldFigure 2. Ultimate pressure for the 3m tank
Figure 3. Pressure variation in the tank with outgas.Figure 4. Time variation in the current from the power sup-
ply. (1000A)
(a) reference pulse (input), (b) output current,
(c) output voltage, (d) deviation of the output current10-510-410-310-210-1100
10-210-11001011021031st
2nd
3rdPressure
Time[Pa]
(hour)
(build-up test) and the data is presented in Figure 3. The
outgas rate for the 2nd measurement was 5.2x10-8 Pa m3/s/
m2 (3.9x10-11 Torr l/s/cm2 ), which is closely consistent with
the value for the OFC.
4. QUADRUPOLE MAGNET AND
PULSED-CURRENT SUPPLY
One of the most important devices for the DTL is the
quadrupole magnet in the drift tube. We have developed a
compact quadrupole electromagnet with a hollow coil made
by the PR copper electroforming. Because the magnet is
operated in pulse mode to decrease the heating due to ohmic
loss in the coil, the pole piece of the magnet is made from a
stack of silicon steel plate (0.35 mm and 0.5 mm in thick-
ness). Details of the magnet are reported in reference [3].
A pulsed-current supply with a 20 kHz switching regula-
tor circuit (IGBT elements are used) has been developed
for the magnet. The requirements for the current supply are
as follows: (1) current stability, with the flat top of the out-put pulse being less than 10-3; (2) the duration time of the
stabilized flat top should be greater than 1 msec; (3) the
maximum current is 1000 A; (4) the rise time for the current
pulse is 5 msec. A typical measured pattern of the output
current from the supply with a dummy coil is shown in Fig-
ure 4. This shows that the stability of the current is about
5x10-4. The other requirements have also been achieved.
5. BREAKDOWN TEST OF THE COPPER
ELECTRODE
An electrical breakdown test has been conducted to deter-
mine the electrical characteristics of the electroformed cop-
per by the PR process for the DTL [4,5]. Electrodes made
by other processes were also tested, in order to compare
their properties. The top of the electrode is hemispheric in
shape, with a radius of 30mm.
The results show that the first breakdown field for the elec-
trode made by the PR copper electroforming is significantly
higher than those of the other electrodes, except for an elec-
trode made of OFC finished by a diamond bite. Table 2 shows
typical results for the first breakdown field level. The data0 1002 10-34 10-36 10-38 10-31 10-2
0 1002 1034 1036 1038 1031st
2ndPressure
Time(Pa)
(sec)
Figure 5. Schematic representation of the DTL hot model
Figure 6. Electric field along beam axisFigure 7. Conditioning history of the model04080120160200240
0100020003000400050006000
0102030405060Peak
AveragePeak PowerAverage power
Conditioning time (hour) (kW) (W)
-100.010203040
0.00.20.40.60.81.01.21.4measuredEz (arbitrary unit)
beam axis (m)indicates that the surface of the electroformed copper by
PR process has the most suitable properties for the accel-
erator cavity.
6. HIGH-POWER TEST OF A MODEL
TANK
A short tank (1.4 m in length) has been made for a high-
power test of the DTL components ( the rf contactors for
the stems and the end plates, the electroformed copper sur-
face, a drift tube with the quadrupole magnet, tuners, and
the input coupler). A schematic representation of the tank is
shown in Figure 5. Only the shortest drift tube has a
quadrupole magnet inside. The tank consists of the first three
cells and the last four cells of the DTL. The right half of the
3rd drift tube is half of the 142nd drift tube. Thus, the posi-
tion of the stem of the tube is not ideal, as the stem is lo-
cated at center of the tube. As this is likely to lead to non-
uniformity in the accelerating field (Ez) distribution, which
is measured by a bead perturbation method, the non-uni-
formity around the 3rd drift tube, that can be seen in the
data shown in Figure 5, is as expected. The measured un-
loaded Q-value was 46200, which is about 93 % of the esti-
mated value, and includes the effect of all components. The
shunt impedance is 54.7 MΩ/m. Because the design value
of the Ez is 2.5 MV/m, the required input rf-power is about160 kW.
The first high-power conditioning was carried out at the
end of April this year. Figure 7 shows the conditioning his-
tory of the tank. The design value for the peak power was
easily achieved with the short-pulse (several 10 µsec in
duration) operation, requiring just two days to achieve full-
power operation. The high-power test was terminated when
a ceramic window of the input coupler broke; however, when
the end plate of the tank was opened to check the inside, no
trace of the discharge was observed on the inner surface of
the tank ,the surface of the drift tube, or on the rf contactor
for the end plate.
7. CONCLUSION
The construction of the Alvaretz DTL for the JAERI/KEK
joint project has been started. The components developed
for the DTL have been tested in a high-power test of the
DTL model. In particular, the PR electroformed copper was
found to have excellent properties. There are problems
with the input coupler that need to be solved. However, mass
production of the magnet and drift tube is already in progress,
and the production of the pulsed-power supplies for the first
DTL tank is completed.
REFERENCES
[1] JHF Accelerator Design Study Report,
KEK Report 97-16 , KEK, Japan, (1998)
[2] H. Ino et al., "Advanced Copper Lining for Accelerator
Components", at this conference.
[3] K. Yoshino et al., "Development of the DTL Quadrupole
magnet with New Electroformed Hollow Coil for the
JAERI/KEK Joint Project", at this conference.
[4] Y. Saito et al., Proc. of the 25th LINAC meeting in
Japan, Himeji, Japan, 343 (2000) (in Japanese)
[5] S. Kobayashi et al., XIX’th ISDEIV,
Xi’an, China, (2000), to be published |
30
233548
Ø 3.3
685.3
5.5
3.53.3
7.5An electroformed layer.[mm]
( thickness: 1.0 mm ) 50.8
DEVELOPMENT OF A DTL QUADRUPOLE MAGNET WITH A NEW
ELECTROFORMED HOLLOW COIL FOR THE JAERI/KEK JOINT PROJECT
K. Yoshino, E. Takasaki, F, Naito, T. Kato, Y. Yamazaki, KEK,
1-1 Oho, Tsukuba-shi, Ibaraki-ken, 305-0801, Japan
K. Tajiri, T. Kawasumi, Y. Imoto, Z. Kabeya, Mitsubishi Heavy Industries,
10 Oye-cho, Minato-ku, Nagoya, 455, Japan
Abstract
Quadrupole electromagnets have been developed
with a hollow coil produced using an improved periodic
reverse electroforming. These will be installed in each of
the drift tubes of the DTL (324 MHz) as part of the JAERI/
KEK Joint Project at the high-intensity proton accelerator
facility. Measurements of the magnets’ properties were
found to be consistent with computer-calculated estimated.
The details of the design, the fabrication process, and the
measurement results for the quadrupole magnet are de-
scribed.
1 INTRODUCTION
The research and development of focusing electro-
magnets for the 324-MHz DTL as part of the Japan HadronFacility (JHF) started at KEK in 1996[1-3]. Since the op-
erating frequency is much higher (324 MHz) than the con-
ventional frequency (200 MHz), the size of the drift tube
(DT) becomes smaller, resulting in many technical diffi-
culties in fabricating a set of DT and quadrupole magnet
(Q-magnet). Beam dynamics[2] require that the magnets
and the DTs for the low-energy part of the DTL conform to
the following specifications:
1. The magnetic field gradient must be variable.
2. The electromagnet must be installed within the compact
DT (outer diameter within 140 mm, length about 52 mm).
3. The magnet must have a sufficiently large bore diameter
(nearly 16 mm) and a high magnetic field (an integrated
magnetic field gradient is 4.1 Tesla).
4. Expansion of the DT in the beam-axis direction should
be less than 10 µm on one side during operation.
5. The deviation of a quadrupole field center from the me-
chanical center must be within 15 µm.
In order to satisfy the these requirements, the pulsed
electromagnets have been selected, instead of permanent
magnets. However, if we use the conventional hollow con-
ductor type coil, it is very difficult to make an electromag-
net which can be installed within a 324-MHz DT, since the
rather large bending-radius is necessary for the hollow con-Table 1: Design parameters of the Q-magnets and the DTs
for the low-energy part of the DTL.
Magnet aperture diameter (mm) 15.6
Core length : L (mm) 33.0
Integrated field: GL(GLe) (Tesla) 4.1
Effective length: Le (mm) 39.2
Core material: Silicon steel leaves 1)
Main thickness of leaf (mm) 0.5
Yoke outer diameter (mm) 115
Nnumber of turns per pole (turns/pole) 3.5
Maximum Ampere-Turns (AT/pole) 3500
Excitation current ( Pulse ) (A) 780
Pulse repetition rate (Hz) 50
Pulse operation rise time 5 ms, flat top duration 2 ms
Minimum coil size (mm) h 5.5, w 5.3, t 1
Voltage drop (V) 1.8
Resistance (mm Ω) 2.3
Inductance ( µH) 18
Water flow rate (liter/min) 1.0
Water temperature increase (˚C) 3.0
Water pressure drop (kg/cm2) 1.8
DT outer diameter (mm) 140
DT aperture diameter (mm) 13.0
DT length (mm) 52.5
Note: 1) Nippon Steel Corporation., type 50H400[4]Figure 1: A detail of a corner part of the electroformed
hollow coil.ductors. Consequently, we newly developed an
electroformed hollow coil. This new method makes use of
an advanced Periodic Reverse (PR) copper
electroforming[5,6] combined together with various kinds
of machining processes without welding (except for the
connection to the outside of DT), where the outer surfaces
of the coil consists of an electroformed copper layer.
2 THE CHARACTERISTICS OF THE
Q-MAGNETS AND THE DTS FOR THE
LOW-ENERGY PART OF THE DTL
The design parameters of the Q-magnets and the
DTs that we developed for the low-energy part of the DTL
are listed in Table 1.
2.1 Electroformed Hollow Coil
Figure 1 shows a detail of a corner part of the
electroformed hollow coil. The coil manufacturing proc-
ess is outlined next. After cutting grooves for the water-
cooling channel in a copper block (Fig. 2a), the grooves
are filled with a wax, which is coated with silver powder to
achieve electronic conductivity (Fig. 2b). A copper layer
of 0.5 mm thickness is formed at each end face by PR cop-
per electroforming (Fig. 2c). After machining the surfaces,
additional copper deposits are formed by 0.5 mm thick-
ness. After removing the wax and boring the pole-piece
part (Fig. 2d), the coils of the end faces are separated using
an end mill of 0.8 mm in diameter (Fig. 2e). The coils in
the beam-axis direction are then separated to 1.0 mm by a
wire-cutting machine (Fig. 1). In this way, the coil inside
DT is molded without welding. Finally, the magnet leads
are connected to the coils with silver brazing (Fig. 2f).
In order to reduce pressure drops and the effects of
erosion, the water velocity in the coil is limited to under 2
m/s. For the same reason, a bending corner inside the wa-
ter-cooling channel is partly cut. As a result, the measured
pressure drops are 1.8 and 6.3 kg/cm2 for the water flows
of 1 and 2 Liter/minute, respectively.
2.2 Q-Magnet and DT
Some important properties of the magnets were
measured before installing into the DT. Figure 3 shows the
excitation-current dependences of the field gradients. The
measured data are compared with those 3-D analyzed by
MAFIA. Both are in agreement within approximately 2 %.
Furthermore, the higher order multipole components in the
magnetic field center measured by a rotating coil were suf-
ficiently small, being less than 0.11% in comparison with
the quadrupole component (Fig. 4). Also, the field center
was deviated only by about 4 µm from the mechanicalcenter.
After installing the magnet into the DT, some prop-
erties were also measured during field excitation. Figure 5
shows the dependence of the water-temperature increase
upon the excitation-current. The temperature increase in
the coil for water-flow rate of 1 liter/minute and the design
excitation-current of 780 A was 3 ˚C, which is within the
specification range. The drift tube was also water-cooled.
However, variations in the flow rate of the DT has no meas-
urable influence on the water-temperature of the coil. This
is probably because the heat load on the coil is not so heavy.
The resonant frequency of the test tank[7] of 1.4 m
was reduced by approximately 220 Hz, when the magnet
was excited at the maximum current (for the design water-
flow rate). This corresponds to an approximately 0.4 µm
Figure 2: Outline of the manufacturing process for the
electroformed hollow coil.Fig. 2f: Silver brazing of
magnet leads.Fig. 2e: Separation of the
coils by end mill.Fig. 2d: Cutting of the pole-
piece part.
Fig. 2c: PR electroformed
surface.Fig. 2b: Filling with silver
powder coated wax.Fig. 2a: Groove processing
for the water-cooling chan-nel.
Silver
brazingcopper
pipe
020406080100120
0 200 400 600 800 1000(Analyzed data)
(Measured data)Gy(x=2) [T/m]
Excitation current [A]
10-510-410-310-210-1100
012345678mesured data
;(DC current=179.9A);(GL=0.60456 T)Amplitude (r=6.5 mm)
Harmonics number n
012345
0 200 400 600 800 1000(Qmag:1L/min, DT:6L/min)
(Qmag:1L/min, DT:2L/min)
(Qmag:1.3L/min, DT:6L/min)Water-temperature rise
[˚C]
Excitation current [A]
Table 2: Specifications of all the Q-magnets in the DTL.
(A unit is mm)
DTL tank No. 1 2 3
Number of the DTs 77 44 28 Qmag outside diameter 115 Qmag core length 33 35 50 76 80 80 90 125 Qmag bore diameter 15.6 16 16 21 21 25 25 29 Number of the magnets 6 17 33 1 20 1 44 28expansion of the first drift-tube length.
3 THE SPECIFICATIONS OF
ALL THE Q-MAGNETS
The specifications of all the Q-magnets in the DTL
are shown in Table 2. Seven kinds of core lengths and five
kinds of bore diameters are chosen in order to make trade-
off between the requirements determined from the beam
dynamics and the reduction in the fabrication cost. The cross
sections of the coils for all the magnets are equal.
4 CONCLUSION
A prototype of the quadrupole electromagnet for
324-MHz DTL has been successfully made with the full
specifications. The measured characteristics satisfied the
requirements. In conclusion,
1. Quadrupole electromagnets have been developed with a
hollow coil produced using an improved periodic reverse
electroforming.
2. Measured field gradient agreed with the calculated one
within approximately 2 %, and higher-order multipole com-
ponents in the magnetic field center were sufficiently small,
being less than 0.11% in comparison with the quadrupole
component.
3. Since the pressure drop of the prototype coil is only 2
kg/cm
2 at the design water-flow rate, the electroformed coil
can be adapted to those for the longer magnets.
4. The temperature increase in the coil at the design water-
flow rate and excitation-current was 3 ˚C, which is within
the specification.
5. The specifications for all the Q-magnets have been de-
termined.
REFERENCES
[1] K. Yoshino et al., Proceedings of the 25th Linear
Accelerator Meeting in Japan, 273 (2000), in Japanese.
[2] KEK Report 97-16 (1998) chapter 4.
[3] Y. Yamazaki et al., “The Construction of the Low-
Energy Front 60-MeV Linac for the JAERI/KEK Joint
Project”, TUD07, this conference.
[4] Flat rolled magnetic steel sheets and strip of Nip-
pon Steel Corporation, DE104, 1998.1 Edition, Nippon
Steel Corporation, in Japanese.
[5] K. Tajiri et al., AESF/SFSJ Advanced Surface Tech-
nology Forum Proceedings, 145 (1998).
[6] H. Ino et al., “Advanced Copper Lining for Accel-
erator Components”, THE20, this conference.
[7] F. Naito et al., “DEVELOPMENT OF THE 50-MEV
DTL FOR THE KEK/JAERI JOINT PROJECT”, TUD08,
this conference.Figure 3: Comparison of excitation-current dependences
on magnetic field gradient for measured data and 3-D
analyzed data using MAFIA.
Figure 4: The higher-order multipole components in the
center of the magnetic field.
Figure 5: Excitation-current dependences on increase in
the water-temperature of the electroformed coil. |
DEVELOPMENT OF SUPER CONDUCTING LINAC
FOR THE KEK/JAERI JO INT PROJECT
M.Mizumoto, N.Ouchi, J.Kusano, E.Chishiro, K.Hasegawa, N.Akaoka, JAERI, Tokai, Japan
K.Saito, S.Noguchi, E.Kako, H.Inoue, T.Shishido, M.Ono, KEK, Tsukuba, Japan
K.Mukugi, C.Tsukishima, MELCO, Kobe, Japan
O.Takeda, Toshiba Corporation, Kawasaki, Japan
M.Matsuoka, MHI, Kobe, Japan
Abstract
The JAERI/KEK Joint Project for the high -intensity
proton accelerator facility has been proposed with a
superconducting (SC) linac option from 400 MeV to
600MeV. System design of the SC linac has been carried
out based on the equipartitioning concept. The SC linac is
planned to use as an injector to a 3GeV rapid cycling
synchrotron (RCS) for spallation neutron source after it
meets requirement to moment um spread less than ±0.1%.
In the R&D work for SC cavities, vertical tests of
single-cell and 5 cell cavities were performed.
Experiment s on multi-cell (5 cell) cavities of β=0.50 and
β=0.89 at 2K were carried out with values of maximum
electric surface peak fields of 23MV/m and 31MV/m,
respectively. A model describing dynamic Lorentz
detuning for SC cavities has been developed for pulse
mode operation. Validity of the model was confirmed
experimentally to simulate the performance.
1 INTRODUCTION
The Japan Atomic Energy Research Institute (JAERI)
and the High Energy Accelerator Research Organization
(KEK) are proposing the Joint Project for High Intensity
Proton Accelerator [1] by merging their original Neutron
Science Project (NSP)[2] and Japan Hadron Facility
(JHF)[3] . The accelerator complex for the Joint Project
consists of a 600 -MeV linac, a 3 -GeV RCS and a 50 -GeV
synchrotron. The linac comprises a negative ion source, a
3-MeV RFQ, a 50 -MeV DTL, a 200 -MeV SDTL
(Separated type DTL), a 400 -MeV CCL and a 600 -MeV
SC linac. Frequency of RFQ, DTL and SDTL is 324 MHz.
Frequency of CCL and SC linac is 972 MHz. T he
400MeV beams are injected to the RCS in the first step of
the Proje ct. Small m omentum spread , Δp/p less than
±0.1%, is required to inject to the RCS. The 600MeV SC linac will be used to improve the beam intensity after
acceptable beams to the RCS be achieved in pulsed
operation.
The R&D studies for the SC linac have been carried out
at JAERI in co llaboration with KEK. Dynamic behavior
of the Lorentz detuning is important for stable RF control.
Lorentz vibration model was established to describe
detuning behavior.
2 SYSTEM DESIGN OF S C LINAC
2.1 Layout of SC linac
Reference design of the SC proto n linac system from
400MeV to 600MeV has been made. Figure 1 shows the
schematic view of lattice structure. The SC linac is
divided into two cavity groups because proton velocity
increases as accelerating . The number of cells per cavity
is 7. Each cryomodu le unit consists of two cavities. T he
maximum electric surface peak field (Esp) of the cavities
is 30MV/m which corresponds to the magnetic surface
peak field Hsp of 525Oe . This criteria for the fixed
maximum magnetic field limit is determined based on the
experiences with multi -pacting condition of other SC
cavity experiments. Average synchronous phase angle
was set to be –30deg. The phase slip of the beam bunch in
the 7-cell cavity was within ±16deg.
The lattice design has been performed by considering
semi-equipartitioning condition of proton beam to reduce
emittance growth. In this condition, the equipartitioning
factor, γεnxσx /εnzσz, (ratio between transverse and
longitudinal values of emittance times phase -advance)
was taken to be 0.8 rather than 1. Lengths of quadrupole
magnets were determined from the limitation of Lorentz
stripping of the negative hydrogen beams. Design criteria
of stripping rate less than 10-8/m at bore radius ( 3cm) is
adopted for the magnetic field gradient with 10% margin.
Table 1 summarizes the design parameters. Quadrupole
magnet length and distance between magnets are 45 cm.
The Esp values are adjusted to achieve smooth phase
advance between groups. Total number of the
cryomodules is 15 with a t otal length of 69m.
80cm 50cm 80cm 45cmCRYOMODULE
CAVITY CAVITYQUADRUPOLE
MAGNETS
Focus DefocusFocus
Focusing Period 45cm45cm
Fig. 1 Bloak diagram of lattice structure for SC linacTable 1 Design parameters for the SC linac, H sp=525 (O e)
β L Esp/Eacc Hsp/Eacc K Eacc Esp
cm Oe/(MV/m) % MV/m MV/m
0.73 5.62 3.05 53.0 2.9 9.91 30.2
0.77 5.94 2.85 50.1 2.6 10.47 29.9 2.2 Beam simulation
Beam simulation has been carried out with the
modified PARMILA code using the parameters based on
the semi - equipartitioning condition[4]. The RMS
emittance growth rates in transverse and longitudinal
direction are 5% and –2%, respectively . Effects of the RF
phase and amplitude control error to the momentum
spread were evaluated in the energy region between 400
to 600MeV. The phase and amplitude errors were
introduced independently in the simulation assuming
uniform distribution . Intrinsic e nergy spread of the
injected beam at 400MeV was assumed to be ±0.2MeV.
The 1000 cases of the calculations were carried out in the
given error condition and the calculated averaged output
energy was obtained as a histogram. The energy spread is
then estimate d using the standard deviation of the
histogram. The values of standard deviations due to the
±1deg. phase error and ±1% amplitude error are
±0.23MeV and ±0.19MeV, respectively. The total energy
spread was estimate to be ±0.36MeV which corresponds
to Δp/p=±0.3% by including the ±0.2MeV intrinsic error.
3 SC CAVITY DEVELOPMENT
The SC cavity development is continued on the basis of
the design parameters for the JAERI original project
(NSP)[5], of which accelerating frequency is 600MHz
and number of cel ls in each cavity is 5 . Essential
differences with respect to fabrication method,
electromagnetic performance and mechanical property to
the cavity are not expected between two frequency
schemes. In the development work, vertical tests of a
single-cell cav ity and 5-cell cavities (β=0.50 and 0.8 9)
have been carried out.
3.1 Fabrication of 5-cell cavities
A 5-cell cavit y of β=0.50 was fabricated in the KEK
workshop . The Esp /Eacc, Hsp/Eacc, R/Q of the cavity
are 4.67, 94.8Oe/(MV/m), and 77.1Ω, respectively.
Equator straight lengths at both end cells are adjusted to
achieve flat electric field distribution on the beam axis.
Unexpected troubles were encountered in the fabrication
process resulting in the cavity structure with different cell
lengths. Pretuning of this cavit y was carrie d out.
Maximum deviation of the peak field at each cell center
was 37.5% before the pretuning. After the pretuning, the
deviation was reduced within 0.7. Field flatness within
2.1% was finally achieved after the pretuning but cavity
length became longer by about 6cm and frequency
increased by about 16MHz.
A 5-cell cavity of β=0.89 w as fabricated in Toshiba
corporation. The Esp/Eacc, Hsp/Eacc, R/Q of the cavity
are 2.04, 47.4Oe/(MV/m) and 443Ω, respectively.
Pretuning of the cavity was carried out. Maximum
deviation of the peak field at each cell center was 23%
before the pretuning. Field flatness was improved to 2% in the pretuning. Figure 3 shows the field distribution of
the 5 cell cavity before and after pretuning.
3.2 Vertical test of 5 -cell cavit ies
Surface treatment s of the 5 -cell niobium cavities
(β=0.50 and β=0.89) were carried out with t he same
procedure for the single -cell cavity, i.e., barrel polishing
(BP) and electro -polishing (EP). Average removal
thickness es in BP were 97 µm and 89µm for β=0.50 and
β=0.89 cavities, respectively. The EP processes were
made to β=0.50 cavity twice and β=0.89 cavity three
times with about 60 µm and 90 µm total removal
thicknesses, respectively, before last vertical test s were
carried out . In addition,
the heat trea tment at
750 C for 3hours and
HPR (high pressure
rinsing) for 1.5hours
were done . In the
vertical test, two kinds
of curves between
residual resistance and
temperature of cooling
down (Rs vs 1/T
curve) and between
quality factor and
maximum surface
electric field of the
cavity (Q0 vs Esp
curve) were obtained
experimentally. Figure
3 shows the 5 -cell
cavity of β=0.89
mounted on the
experimental set -up.
The cavity was just
taken out from the
cryostat and covered
with the frost.
Figure 4 shows the
curve of the residual
resistance as a
function of 1/T. The
surface resistance
Fig. 3 A 5 -cell cavity of β=0.89 00.20.40.60.811.2
0 200 400 600 800 1000 1200 1400Initial
Final
SUPERFISH
Position [mm]
Fig. 2 Field distribution of the 5-cell cavity ( β=0.89)
before and after pretuning Relative fieldvalues at 16MV/m for the β=0.50 and β=0.89 cavities are
10nΩ and 5nΩ at 2K, respectively.
Figure 5 shows the results with the vertical test s of the
5-cell cavities both for β=0.50 and β=0.89 at 4K and 2K .
For the β=0.50 cavity experiments, m aximum field
strengths of 23 and 18.7MV/m were obtained in 2K and
4.2K measurements, respectively. The field w as quench ed
at 2K and limited by the capacity of the RF power supply
at 4K. Quality factors Q were reasonable at low field
strength (2x1010 and 1x109 at 2K and 4.2K, respectively),
but were degraded a s field increase d.
For the β=0.89 experiments, m aximum field strengths
of 23 and 31MV/m were obtained in 2.1K and 4.2K
measurements, respectively. The field s were limited by
thermal quench at 2K and the capacity of RF power
supply at 4K due to a field emission . Good quality factors
Q of 5x1011 and 2x109 were obtained at 2K and 4K,
respectively.
The field strengths exceeded design values of
16MV/m for original 600MHz cavity. These
performance s, however, were not good compared with the
single-cell cavities which reached constantly to the values
more than 40MV/m[5]. The reasons for these results are
considered due to the cavity deformation in the pretuning
for β=0.50 cavity and insufficient surface treatment both
for β=0.50 and β=0.89. Further studies will be performed
to improve the performances to meet the final
requirement with the Esp value of more than 30MV/m.
4 DYNAMIC ANALYSIS OF LORENTZ
DETUNING
In the Joint Project, pulsed operation of the SC cavities
is planned with repetition rate of 50Hz and beam pulse
width of 0.5ms. Dynamic behavior of the Lorentz
detuning due to pulsed operation is the most important
issue for stable RF control of the cavities [6].
The dynamic analysis of the Lorentz detuning was
performed with the finite element model code ABAQUS.
The Lorentz force on the cavity wall was obtained from
the electromagnetic field distribution which was
calculated by the SUPERFISH code. The Lorentz
detuning and the cavity field influence each other. To
solve the dynamic Lorentz detuning and the dynamic
cavity field simultaneously, Lorentz vibration model
which describes dynamic behavior of the Lorentz
detuning is established . A programming language of
MATLAB/Simulink was used to solve the double
differential equation. The model was applied to the
simulation of the RF control successfully [7].
5 SUMMARY
System design of the SC proton linac has been carried
out for the JAERI/KEK Joint Project. R&D work of the
SC cavities for the high intensity proton linac has been
progressing and promising results are accumulated . A
model which describes the dynamic Lorentz detuning in
the pulsed operation was established. Design of a
prototype cryomodule, which include s two 5-cell cavities
of β=0.60, is in progress. Cavity and cryomodule tests
will be made early in 2001 . Experiments of the RF
control is planned using the prototype cryomodule.
REFERENCES
[1] The Joint Project Team of JAERI and KEK, “The
Joint Project for High Intensity Proton Accelerator ”,
JAERI-Tech 99 -056/KEKReport 99 -4 JHF-99-3 (1999)
[2] M. Mizumoto et al. , “The development of the high
intensity proton accelerator for the N eutron Science
Project”, Proc. APAC98., Tsukuba, Japan, p314 (1998)
[3] Y. Yamazaki et al., “Accelerat or Complex for the
Joint Project of KEK/JHF and JAERI/NSP”, PAC99,
New York, USA, p513 (1999)
[4] K. Hasegawa and T. Kato, “The Proton Linac For the
Joint Project” , in this Proceedings, 2000
[5] N. Ouchi et al., “Development of Superconducting
cavities for High Intensity Proton Accelerator in
JAERI”, IEEE Trans. on Applied Superconductivity,
vol. 9, No. 2, p.1030 (1999)
[6] E. Chishiro et al., “Study of RF Control System for
Superconducting Cavity”, Proc. of 12th Symposium on
Accelerator Science and Technol ogy in Japan (1999 )
[7] N.Ouchi et al, “Pulsed Proton SC Linac ”, in this
Proceedings, 2000 10810910101011
0 5 10 15 20 25 30 35
Esp(MV/m)J5003#1,2KJ5003#1,2K
J5003#1,4KJ5003#2,2KJ8903#1,2K
J8903#1,4KJ8903#3,2K
J8903#3,4KDesign value (16MV/m) for 600MHz.Q0
Fig.5 Vertical test results of the 5-cell cavities at 2K and
4K. Tests were done twice and three times for
β=0.50 and β=0.89 cavity, respectively.10-910-810-710-6
0.2 0.3 0.4 0.5J5003
J8903Rs(T)=A/T exp[- Δ/kT]+Rres
1/T [1/K]Rs
Fig.4 Residual resistance as a function of 1/T
(inverse of cooling temperature) |
STUDY OF NONLINEARIT IES AND SMALL PARTIC LE LOSSES
IN HIGH POWER LINAC
A. Kolomiets, S. Yaramishev, ITEP, Moscow, Russia
Abstract
The conception of High Power Linac developed in
Russian accelerator centres is based on the use of
independently phased SC r esonators with quadrupole
lenses between them. The type and parameters of the
resonators as well as focusing structure are varied along
the linac to optimise beam dynamics and the
characteristics of the linac. The beam evolution in the
linac was studied by simulation in 3D accelerating and
focusing fields by co mputer code DYNAMION. The
simulation includes all nonlinearities of external fields
and space charge forces. Estimations of particle losses in
the beam based on analysis of the spectral properties of
particle trajectories were carried out.
1 INTRODUCTION
The accelerator driven electronuclear installation for
numerous purposes requires the proton beam with energy
about 1 GeV and current between one and several tens
milliamp [1]. The only linear resonant accelerator is
considered as a choice if beam current above 10 mA is
required.
In accordance with the conception proton linac will be
built using one channel scheme. It consists of 0.1 MeV
DC injector, room temperature 300 MHz RFQ,
intermediate part with low beta 300 MHz SC
independently phased cavities and main part with
multigap 600 MHz SC cavities.
The paper is devoted to study of beam dynamics in the
intermediate part of the linac. It is clear that perturb ation
of the beam in this part of the linac i s the most strongly
marked due to low particle velocity and cons equently
high influence of space charge as well as relatively high
defocusing in the cavities
The intermediate part of linac has the energy range
from β ≈ 0.15 to β ≈ 0.5. In this part the sho rt (1 - 2 gaps)
SC cavities with independent RF excitation of each
resonator are the best choice as accelerating structure.
The most powerful method of the beam dynamics
study is computer simulation. Many codes are used for
this purpose. The output of the codes is usually set of
particles coordinates and velocities stored at some
structure positions. Evolution of the beam parameters is
estimated by calculation of rms or total beam emittances.
However the condition corresponding to the harsh
emittance growt h are wittingly out of acceptable range
for high power linacs where the particle losses are the
most critical problem. It means that the development of more sensitive methods suitable for analysis of computer
simulation results is actual task.
Some new app roaches to beam analysis have been
proposed and studied in ITEP. In the paper the methods
are described. The results of the dynamics study in low
energy section of intermediate part of HPL are presented.
2 TRANSVERSE DYNAMIC S OF THE
PERTURBED BEAM
The stu dies initiated by interest to high power linac
development and widely carried out showed that the halo
formation is connected with appearance of stochastic
elements in the dynamics under the influence of wide
range of factors causing the perturbation of li near motion.
These factors are space charge forces, influence of
longitudinal motion, mismatching, etc. The appearance of
stochastic elements means that certain number of the
particle trajectories became similar some random
function. It o ccurs even in the system where no random
forces influence the particle motion.
It follows from general theory of non -linear
dynamics [2] that the appearance of such trajectories is
the result of local instabilities in the system. This process
leads to the mixing of the traj ectories in phase space and
to the emittance increasing. If the local instability is the
main reason of increasing of particle amplitudes it is
sufficient to dete rmine the conditions when it appears and
find appropriate quantitative characteristic.
It is known [3] that the charged particle motion in
periodic focusing and accelerating channel is described
by Matieu -Hill equation. The fundamental solutions of
the equation are Floquet function. The characteristic
parameter of the equation determines stability or
instability of the solutions.
The transformation of particle coordinate through
focusing period of the channel is described be matrix T
with elements built from Floquet functions,
)()1( t t Tz z =+
where z – vector of particle coordinates in ph ase space, t
- dimensionless time ( tSv=t , v – particle velocity and
S – length of focusing period). Characteristic parameter l
can be found from expression:
)(21cosh TSpl= (1)
Real or complex values of l = k+im, correspond to
regions of instability of particle motion, imagine one l=im
- stable regions. The particle trajectory in the stable region is:
))(cos()( )( 0 0Jttm tr t + =Ax (2)
where r(t) and m(t) are module of Floquet function and
phase advance, A0 and q0 – initial conditions. In linear
theory all particle trajectories determine by the same
Floquet functions and depend only of initial amplitude
and phase. It can be assumed that for small perturbation
of linear motion above mentioned expressions are valid,
with r(t) and m(t) are the functions of position of the
particle in phase space.
The transformation matrix and therefore Floquet
functions in non -linear case can be found using code for
simulation of particle motion. It calculates particle
coordinates and velocities by linear t ransformation along
integration step under influence of external and space
charge forces F n. The elementary transformation matrix,
for example in X plane is:
n nnnn
n ddxx
xF
ddxx
ΔΔ
=
+ ttt
t11
1,
where Dt is step of integration. Matrix of full focusing
period T can b e obtained by multiplication of the
elementary matrixes. The characteristic parameter,
module and phase of Floquet function can be easily
calculated from the elements of the matrix. The
distribution of these parameters can characterize the
degree of pertur bation of the motion. The total
perturbation of the system can be expressed by summing
up all instability increments kj>1 over phase space:
∑=jk h0.
It is shown in [2] that in assumption that average value
of increment does not change in t ime, the increasing of
the volume occupied by particles in phase space,
appeared due to trajectory mi xing under influence of
local instability, can be determined as
t
ete02
0)(h
e= . (3)
The second proposed method consists in that stored
with certa in step particle coordinates obtained as a result
of computer simulation are considered as some random
set of points. The corresponding trajectory can be
reconstructed using well -known correlation function
method [4]. The particle trajectory is represented in this
case by series
∑+=
kk jka ax )cos( 0m (4)
It allows study spectral properties of the initial ra ndom
process generating the points. The application of this
method to the study of spectral properties of particle
motion in periodic focusing structu res is described in [5].
It is shown there that in spectra of particle transverse
frequencies in the presence of space charge forces always
presents zero frequency peak due to appearance in certain
number of partic le trajectories with the term a0.≠0. This coefficient is the constant term of the trajectory. It
is clear, that it can appear as the result of increasing of
amplitude of particle oscillation due to local instabilities
experienced by the given particle at some part of the
structure. It is shown in [6] that the probability of
increasing amplitude on the value Dx can be described by
Maxwell distribution
()2)(
32
4)(x
xpx
exxfΔ−Δ⋅=Δ (5)
Taking into account that the particle distributed on
transverse coordinate as
2)(
21
22)(s
p sx
e xg−
= , (6)
the probability for particle to increase its amplitude up to
aperture value i.e. x+ DDx > a and, therefore to be lost, can
be expressed as
∫∫−=a a
xadxdzzfxgp
0)()( (7)
3. SIMULATION RESULT S
The described methods have been applied to analysis
of simulation results carried out for intermediate part of
high power linac. The code DYNAMION [7] with routine
added for matrix coefficients calculation for each particle
has been used for simulations. The parameters of the
studied structure are given in Table 1.
Table 1. Parameters of the focusing period of
studied structure
Focusing lattice FODO
Length of focusing period (cm) 64.0
Length of gap (cm) 5.0
Number of gap in cavity 2
Voltage in gap (kV) 500
Length of quadrupole (cm) 12.0
Aperture (cm) 1.0
Gradient (T/m) 19.0
To increase statistic, the beam passed through studied
period 100 times. The initial particle distribution was
Gaussian. Matched Twiss parameters were calculated in
smooth approximation for the envelopes at 2s level. To
avoid the influence of the possible mismatching of the
initial distribution, beam preliminary passed through
period 200 times. To keep the average particle velocity
constant the reference particle passed each cavity at phase
-900. The accelerating field wa s chosen that to have the
design value of longitudinal oscillation frequency. The
3D distributions of the accelerating and focusing fields
were used for simulations. Particle – particle interaction
algorithm was used for space charge forces calculation.
Table 2 shows some beam parameters calculated with
elements of matrices and correlation function for several Table 2. Some calculated beam parameters
h <<mm>> h0 10-5 ss xx 10-3
0.0 0.97 1.3 0.084 0.86
0.03 0.94 6.8 0.092 2.54
0.12 0.80 83.5 0.110 3.78
0.17 0.75 114.0 0.143 5.29
0.21 0.71 149.0 0.198 7.16
0.23 0.73 213.0 0.229 8.09
values of Coulomb parameter h ~ I/Vp (I is beam current
and Vp is input normalized emittance [3]). <m> is average
phase advance, h0 – average instabi lity increment, s and
x are parameters of the distributions (5) and (6)
correspondingly. Fig.1 shows phase advance histograms
(left column of plots) and instability increment
histograms (central column of plots). Right column
shows phase space plots x/A 0(τ), (dx/d τ)/x/A0(τ)
represented by module and phases of Floquet functions
from (2). The rows of plots correspond to the values of
Coulomb parameters 0.028, 0.175 and 0.226. It is clear
seen how the increasing the number of unstable particles
leads to redis tribution of phase space.
The Fig.2 shows emittance growth for 100 periods
(upper curve of plot 1) and probability of particle losses
per meter (plot 2) in studied structure calculated from
expressions (3) and (7). The lower line in Fig.2 (1)
represents rms emittance of simulated beam after passin g the studied period 100 times. It can be seen from given
figures that there is no threshold of local instability and,
therefore, of emittance growth and particle losses. The
last ones are linear function of Coulomb parameter.
5. CONCLUSION
The proposed methods of analysis allow obtaining
from results of computer simulations of beam dynamics
the quantitative estimations of beam parameters, which
can characterise non -linear effects. They can be useful for
fast estimations with visualisation of the processes in the
beam caused by non -linear motion.
The results of the work confirm that emittance growth
is connected with local instab ility and can be estimated
by its value averaged over phase space.
The study of intermediate part of conceptual proton
high power linac using the methods showed that design
parameters are feasible and relative particle losses at
design beam current can be estimated at level 2 ⋅10-6 per
meter what is acceptable level for this part of the linac.
REFERENCES
[1] O.V. Shvedov et al., "Concept of HPL", Preprint
ITEP 35 -99, Moscow, 1999
[2] G.M. Zaslavskii, R.Z. Sagdeev, “ Introduction to
Nonlinear Physics”, “Nauka ”, Moscow, 1988.
[3] I.M. Kapchinski, "Theory of linear resonance
accelerators", Moscow, Energoizdat, 1982.
[4] V.V.Beloshapkin< G.M. Zaslavskii, “On the spectral
properties of dynamical system in the transition region
from order to chaos”, Physics Letter s, v.97a, N4, p.121
[5] Kolomiets A., et al., “The Study of Nonlinear Effects
Influenced by Space Charge in High Intensity Linac”,
PAC-95, Dallas.
[6] A. Kolomiets, S. Yaramishev “Comparative Study of
Accelerating Structures Proposed for High Power Linac”,
PAC-97, Vancouver
[7] A.Kolomiets, et al. “DYNAMION – the Code for
Beam Dynamics Simulations in High Current Ion Linac”,
EPAC-98, Stockholm
Figure 1. Phase advances (left column), increment of
instability (centre column), normalised particle
coordinates (right column) for beam currents 0 mA (a),
10 mA (b), 30 mA (c).
Figure 2. Emittance growth calculated for 100 periods
(plot 1) and specific relative particle losses. a
b
c 1 2 |
arXiv:physics/0008182 19 Aug 2000Control System for the LEDA 6.7-MeV Proton Beam Hal o Experiment1
L. A. Day, M. Pieck, D. Barr, K. U. Kasemir, B. A. Quintana, G. A. Salazar, M. W. Stettler
Los Alamos, Los Alamos National Laboratory, NM 87545, USA
1 This work supported by the Department of Energy un der contract W-7405-ENG-36.Abstract
Measurement of high-power proton beam-halo
formation is the ongoing scientific experiment for the
Low Energy Demonstration Accelerator (LEDA) facilit y.
To attain this measurement goal, a 52-magnet beam l ine
containing several types of beam diagnostic
instrumentation is being installed. The Experimenta l
Physics and Industrial Control System (EPICS) and
commercial software applications are presently bein g
integrated to provide a real-time, synchronous data
acquisition and control system. This system is
comprised of magnet control, vacuum control, motor
control, data acquisition, and data analysis. Uniqu e
requirements led to the development and integration of
customized software and hardware. EPICS real-time
databases, Interactive Data Language (IDL) programs ,
LabVIEW Virtual Instruments (VI), and State Notatio n
Language (SNL) sequences are hosted on VXI, PC, and
UNIX-based platforms which interact using the EPICS
Channel Access (CA) communication protocol.
Acquisition and control hardware technology ranges from
DSP-based diagnostic instrumentation to the PLC-
controlled vacuum system. This paper describes the
control system hardware and software design, and
implementation.
1 INTRODUCTION
As part of the linac design for the accelerator
production of tritium (APT) project the first 10-Me V
portion of this 100-mA proton accelerator was assem bled
at the Los Alamos Neutron Science Center (LANSCE) i n
1999 and was in operation for over one year. Now, t his
Low-Energy Demonstration Accelerator (LEDA)
provides the platform for a new experiment: attaini ng
measurements of high-power proton beam-halo formati on.
For this purpose a 52-magnet beam line has been ins talled
into the LEDA beam line between the Radio Frequency
Quadrupoles (RFQ) and the High Energy Beam Transpor t
(HEBT).
LEDA is using the distributed control system based on
EPICS [1, 2]. Extensions to the existing control sy stem
were developed for controlling the devices listed i n Table
1 in section 3.2. This table also shows device loca tions as
defined by the space upstream of the numbered
quadrupole magnet.EPICS is a toolkit for building distributed control
systems that originated at Los Alamos and is now
developed jointly by a collaboration of over 100
institutions [3]. It is the basis for operator cont rols
interfacing. Specific extensions to EPICS CA
communication protocol have been developed to integ rate
additional controls configuration and visualization options
into EPICS. IDL, a commercial visualization tool, has
been integrated to provide more complex data proces sing
and visualization options. LabVIEW has been integra ted
to enable simple, cost effective PC solutions to se lective
instrumentation control.
Provided below is a summary of the important contro l
system components. Brief descriptions of computer
system architecture, hardware, software, and extern al
interfaces are presented. These designs have been, or are
being integrated into the control system for LEDA.
2 CONTROL SYSTEM STRUCTURE
AND NETWORK TOPOLOGY
The Halo Experiment extends the LEDA controls
hardware architecture by 3 PC-Input/Output Controll ers
(IOC)s and 2 VXI-IOCs. The extension is comprised o f
four principal systems covering Quadrupole Magnet
Control, Steering Magnet Control, Diagnostic, and
Vacuum System Control.
The control system’s communication service is built on
a TCP/IP-based network and uses EPICS CA as the
primary protocol. Access to the controls network is
limited by an Internet firewall for safety and traf fic
congestion reasons. A second independent local area
network has been created to isolate the distributed I/O
modules from the control system’s network.
3 QUADRUPOLE MAGNET CONTROL
The 52-quadrupole-magnet focus/defocus (FODO)
lattice provides a platform to create phase space h alo
formation. The first four quadrupole magnets are ea ch
independently powered by a 500A/15V EMI/Alpha-
Scientific power supply (Singlet). Depending on how
those magnets are adjusted, a match or mismatch of the
RFQ output beam to the lattice is created.
The next 48 magnets are powered in groups of 8. Eac h
set of magnets is powered by an Alpha-Scientific
500A/100V Bulk Power Supply with an 8 Channel Shunt
Regulator that allows an individual current trim. W ithcertain magnet settings, the development of specifi c halo
formations can be observed along the lattice.
3.1 Quadrupole Magnet Control Hardware
The quadrupole magnet control subsystem uses as PC-
IOC computer an Intel Pentium II 500Mhz equipped wi th
two network cards. The first card establishes the p hysical
connection to the LEDA controls network. The second
card creates a local area network for National Inst ruments
(NI) modular distributed I/O system called FieldPoi nt
(FP). It includes analog and digital modules, and
intelligent Ethernet network module that connects t he I/O
modules to the PC-IOC computer.
Using the robust Ethernet networking technology to
position intelligent, distributed I/O devices close r to the
sensors, or units under test, leads to a significan t cost
savings and performance improvement. The most obvio us
benefit to this solution is the savings in signal w iring.
Replacing long signal wires with a single, low cost
network cable saves significant time and money duri ng
installation and testing. Furthermore, distributed I/O
systems, such as NI FP, also include special capabi lities to
improve the reliability and maintainability by usin g the
built-in, onboard diagnostic capabilities.
Table 1. Halo lattice beam line component location s
Device Locations
(Quadrupole Magnet #)
Singlet PS 1, 2, 3, 4
Bulk PS /Shunts 5-12, 13-20, 21-28, 37-44, 45-52
Steerers 4, 6, 15, 17, 26, 28, 36, 38, 47, 49
Fast Valve 2
Beam Line Valves 17, 36, 52
Ion Pumps 6, 10, 15, 21, 28, 34, 40, 44
Ion Gauges 13, 31, 42
WS/HS 5, 21, 23, 25, 27, 46, 48, 50, 52
3.2 Quadrupole Magnet Control Software
The NI LabVIEW-based quadrupole magnet control
system that runs on the Windows NT operating system
consists of 60 subroutines combined into 10 process es (4
Singlets + 6 Bulk/Shunt Regulator). These independe nt
software processes share CPU cycles based on priori ty.
To ensure proper operation, the control processes f or the
set-points have highest priority, while the display related
processes have lower priorities.
The LabView uses an in-house built ActiveX
Automation Server allowing the integration of the
LabVIEW system in the EPICS environment by serving
values of general interest to Graphical User Interf aces
(GUI) that are clients to CA.
Using the EPICS GUI called Display Manager (DM),
the control processes drive the power supply values andset-points by interpreting mouse clicks and text en tries
into commands.
To meet the magnetic field setability specification s,
magnet hysteresis must be compensated by ramping th e
field past the set-point and then reducing the fiel d to the
set-point. This led to a control process solution f or the
Bulk and Singlet power supplies that operates in fo rm of a
State Machine where the states are idle, ramp up, s oak
and ramp down. According to the magnet specificatio n the
process control ramps up to a desired set-point,
overshoots, stays for a specific time at that overs hoot
value and ramps back down to the operator desired s et-
point. Ramp up rate (Amps/step), overshoot value (A mps
above desired set-point), soak time (sec), and ramp down
rate (Amps/step) are individual settable by the ope rator.
Basic binary operation like AC Voltage On/Off, DC
Voltage On/Off, and interlock reset are provided as well.
The display processes are limited to voltage and cu rrent
read backs, status indication for PS interlocks, an d
Local/Remote status.
4 STEERING MAGNET CONTROL
The task of the beam’s 10 steering magnets (horizon tal
and vertical) is to correct the position of the bea m at the
end of every set of BPM/steering magnet associated pair
that is separated by approximately 10 quadrupole ma gnets
(see Table 1). Since the FODO-lattice period has a phase
advance of approximately 80 degrees, each pair of B PMs
can detect and each pair of steering magnets can co rrect
the beam’s position and angle that might be caused by
misaligned quadrupoles.
4.1 Steering Magnet Control Hardware
The steering magnet control system uses an Intel
Pentium II 450Mhz PC-IOC equipped with one network
card and two NI Plug & Play PCI General-Purpose
Interface Bus (GPIB/IEEE 488.2) controller cards. This
so called two-bus GPIB controller system interfaces 20
KEPCO 20V/5A power supplies, 10 for horizontal and 10
for vertical. All 10 power supplies on each bus are linked
together in a linear configuration (daisy-chain). T his
hardware architecture combines the cost-effectivene ss of
general-purpose PCs with the standardized and widel y
used GPIB solution. Due to the limitation of GPIB ( max
cable length 20m) the PC-IOC is located next to the 2
racks of steering power supplies.
4.2 Steering Magnet Control Software
The NI LabVIEW-based steering magnet control
system that runs on a Windows NT operating system
consists of 80 subroutines incorporated into 1 proc ess.
This all-in-one design follows a iterative control sequence
between the individual power supplies on both buses : 1)
writing to the first power supply on the first bus then
writing to the first power supply on the second bus . 2)reading from the first power supply on the first bu s and
then reading from the first power supply on the sec ond
bus. This continues until the 10th power supply has been
iterated and then starts over again. This strategy was
chosen for possible closed-loop beam correction whe re
the amount of beam spill is critical. Having the de scribed
procedure in place reduces possible beam spill by
reducing the reaction time of two corresponding ste ering
power supplies to a minimum of ~4ms.
The developed GUI shows the change of current and
voltage read backs as the operator changes the curr ent set-
points via text entry fields and slider controls.
Furthermore, the most important information about t he
status of the power supplies is read out from the 1 6-bit
status register.
5 VACUUM SYSTEM
The devices for the vacuum system for the Halo
Experiment beam-line comprise of one Fast Beam-line
Isolation Valve, three Beam Line Valves, three Ion
Gauges, three Convectron Gauges, and 8 Ion Pumps.
5.1 Vacuum System Control Hardware
The vacuum control subsystem operates as a standalo ne
system. That is, all hardware components are local to the
Halo Beam-line, hardwired together, network isolate d and
fully functional, i.e., the subsystem, contains all interlocks
and requires no input from remote computer control
equipment during normal operation.
Remote accessibility is established through the NI
distributed I/O modules called FP (see section 3.2) .
5.2 Vacuum System Control Software
The DirectSoft PLC is the heart of the vacuum contr ol
system and has incorporated all functionality. Ther e is no
direct access to the PLC’s CPU/memory components
during normal operation. Thus, all signals are prop agated
through the associated PLC input/output modules. A
custom ladder logic program containing equipment
interlocks, resides and runs (continuous loop) in t he
volatile memory of the CPU module. Though, the
program is lost during a power shut down, the ladde r logic
program is loaded from the flash memory and started
shortly after power is resumed. Its initial state a fter reboot
is a safe mode state in which all devices are turne d off.
6 WIRE SCANNER/HALO SCRAPER
There are nine Wire Scanner / Halo Scraper (WS/HS)
assemblies installed into the LEDA Halo Beam-line [ 5].
These assemblies contain three measurement devices for
each horizontal and vertical axis. Data acquired fr om
these devices is used to provide projected beam
distribution information. One wire scanner device
measures beam core distribution within +/- 3 rms wi dths
while two halo scraper devices measure the edges of beamdistribution outside 2.5 rms widths. These devices are
attached to an actuator driven by a stepper motor [ 6],
which drives either the WS or one of the HS into th e beam
in incremental steps. At each WS step, the amount o f
secondary electrons (SE) generated is measured and
normalized against synchronous beam current data [7 ]. At
each HS step, the proton beam charge is measured an d
normalized against the overlapping WS data. Togethe r,
this provides a complete beam distribution profile.
6.1 Control Solutions
Measurement control uses CA Server to communicate
between EPICS modules running on Kinetic Systems’
HKBaja60 in VXI IOCs, LabVIEW running on a PC IOC,
and IDL running on Sun Workstations. The ability of CA
Server to provide a communication means between
software applications running on different platform s
allows the flexibility to choose tools that best su it the
specific requirements.
EPICS real-time processing meets the requirement to
acquire synchronous beam pulse data at the rates of 1-6
Hz. Data is continuously acquired from beam current
monitors, wires, and scrapers by DSPs mounted on th e
VXI boards into local circular buffers. Data is ext racted
from these buffers by the EPICS database when trigg ered.
LabVIEW offers timely and cost effective methods fo r
controlling actuator motion. VIs were implemented t o
interface with off-the-shelf motor drivers. CA Clie nt runs
on the PC to enable communications between LabVIEW
and the other control system modules.
IDL provides flexible data processing and visualiza tion
options. Operators have the flexibility to process data
independently and fine-tune the processing on-line
without interrupting beam operations. The results a re
displayed graphically in plots as they are calculat ed.
The EPICS’ display manager (DM) serves as the
interface for operator control. Operators control
measurement parameters and view measurement status on
a DM GUI.
6.2 Implementation
The operator sets parameters and starts the
measurement from the WS/HS controls DM GUI. EPICS
database process variables are used to pass data an d
commands between EPICS WS/HS data acquisition,
LabVIEW motor control, and IDL data
processing/visualization modules. A SNL sequence
manages the execution of the separate control modul es by
setting process variables to known command values.
These variables are monitored through CA by associa ted
modules. When the expected command is received by a
module, execution is initiated. This flow is illust rated in
Figure 1.
For each step in a measurement, the SNL sequence
sends motor parameters to LabVIEW via a process
variable specifying Z location, horizontal or verti cal axis,and stepper motor position. LabVIEW sets a response
variable to notify the sequence when the stepper mo tor
has reached the required destination.
Meanwhile, each WS and HS pair has its SE charge
waveform signal acquired by its DSP during every be am
pulse. When LabVIEW notifies the SNL sequence that the
stepper motor is in position, the sequence sets a p rocess
variable to trigger the database to upload the acti ve wire
or scraper’s data along with the associated beam cu rrent
monitor’s data, for normalizing, into waveforms.
This synchronous data is locally processed within t he
EPICS database and also processed in IDL routines. The
local processing completes automatically within the
database. These functions produce results of global
control system interest such as unit conversion and
averaging. Because the EPICS database is loaded at boot
time, these functions are fixed, i.e, the majority of
parameters cannot be conveniently modified. However ,
the results are time-stamped, therefore, they can b e
archived and made available for synchronous data
retrieval.
IDL is utilised for specialized and flexible proces sing
and visualization. Functions can be created and/or
modified on-line from the control room without affe cting
beam operations. Furthermore, IDL provides more
complex options for viewing data. After data acquis ition
is complete, the SNL sequence notifies IDL at which time
IDL gets all necessary data for processing. The WS or HS
data is normalized and plotted against the motor’s position
producing the beam’s profile visually in real time.
Operators are able to view the measurement’s progre ss
and abort the scan if undesirable results are prese nted.
Processing parameters can be adjusted if desired, a nd a
new measurement started. REFERENCES
[1] L.R. Dalesio et all., Nucl. Instrum. Meth. In P hys
Research A352 (1994), 179-184
[2] D. Moore and L.R. Dalesio, “A Development and
Integration Analysis of Commercial and In-House
Control Subsystems” Conf. Proc. LINAC98, Chicago
August 1998
[3] M. Thuot, et al., ”The Success and the Future of
EPICS” LINAC96, Geneva, Aug. 1996.
[4] D. Moore and L.R. Dalesio, ”A Development and
Integration Analysis of Commercial and In-House
Control Subsystems”, LINIAC98, Chicago, Aug.
1998.
[5] J. D. Gilpatrick, et all, “Beam Diagnostics
Instrumentation for a 6.7-MeV Proton Beam Halo
Experiment,” these proceedings.
[6] R. Valdiviez, et al., “Intense Proton Core and H alo
Beam Profile Measurement: Beam Line Component
Mechanical Design,” these proceedings.
[7] Power, J., et al, “Beam Current Measurements for
LEDA,” Proceedings of the 1999 Particle Accelerator
Conference, New York, 1999Kinetic Systems
HKBaja60
VXI Module
DSP AcquisitionVXI
NI Motion ControlPCActuator
Wire & scraper
sensors
SUN
IDL
EPICS DMEPICS Database
SNL SequenceLab View
Ethernet (CA)
SNL Communication
Database CommunicationFigure 1: Wire Scan/Halo Scraper Control |
arXiv:physics/0008183 19 Aug 2000MULTIPLE-CHARGE BEAM DYNAMICS IN AN ION LINAC
P.N. Ostroumov, J.A. Nolen, K.W. Shepard
Physics Division, Argonne National Laboratory,
9700 S. Cass Avenue, Argonne, IL, 60439
Abstract
There is demand for the construction of a medium-
energy ion linear accelerator based on superconducting
rf (SRF) technology. It must be capable of producing
several hundred kilowatts of CW beams ranging from
protons to uranium. A considerable amount of power is
required in order to generate intense beams of rare
isotopes for subsequent acceleration. At present,
however, the beam power available for the heavier ions
would be limited by ion source performance. To
overcome this limit, we have studied the possibility of
accelerating multiple-charge-state (multi-Q) beams
through a linac. We show that such operation is made
feasible by the large transverse and longitudinal
acceptance which can be obtained in a linac using
superconducting cavities. Multi-Q operation provides
not only a substantial increase in beam current, but also
enables the use of two strippers, thus reducing the size of
linac required. Since the superconducting (SC) linac
operates in CW mode, space charge effects are
essentially eliminated except in the ECR/RFQ region.
Therefore an effective emittance growth due to the
multi-charge beam acceleration can be minimized.
1 INTRODUCTION
A preliminary design and beam dynamics study has
been performed for the rare isotope accelerator (RIA)
driver linac structure and is discussed elsewhere [1,2]. A
schematic view of the linac is shown in Fig.1.
Figure 1: Simplified layout of the Driver Linac.
The linac contains three main sections: a “pre-
stripper” section up to the first stripping target at 12.3
MeV/u, a medium energy section defined and separated
by the stripper targets and a high energy section with a
maximum uranium energy of 400 MeV/u. Total voltage
of the linac is 1.36 GV. The pre-stripper section consists of an ECR ion source followed by mass and charge
selection, an initial linac section consisting of an RFQ
and 96 low-beta independently-phased SRF cavities. The
middle section is based on 168 intermediate-beta SRF
cavities. The high-energy section consists of 172
elliptical cavities designed for three different velocities.
The heaviest ions, which are not fully stripped at the first
stripper, will be stripped a second time at ~85 MeV/u.
The charge state distribution of uranium ions is centered
at the charge state q0=+75 at 12.3 MeV/u from the first
stripper. Five charges encompassing ~80% of the
incident beam after the first stripper will be accelerated
simultaneously in the medium- /G45 section. After the
second stripper, 98% of the beam is in five charge states
neighbouring q0 = 89, all of which can be accelerated to
the end of the linac.
The accelerating field taking into account the cryostat
filling varies from 1.6 MeV/m to 5 MeV/m. Transverse
beam focusing over all of the driver linac is provided by
SC solenoids. The length of the focusing period depends
on the resonator type.
The behaviour of the uranium multi-Q beam has been
studied both by analytical and numerical methods. The
effects of various factors, such as beam mismatch,
misalignments, accelerating field errors and other factors
affecting the emittance growth of a multi-Q beam are
discussed.
2 BEAM DYNAMICS
2.1 Longitudinal beam dynamics
When a particle with a charge state, q, and mass
number, A, traverses an accelerating cavity of length, Lc,
and electric field E=Eg(z)cos /G5At, the energy gain per
nucleon /G27Ws,n is determined by the expression
scG 0 n,s cosL),(TeEAqW /G4D /G45 /G45 /G27 /G20 , where T( /G45, /G45G) is the
transit time factor, E0 is the average accelerating field of
the cavity and /G4Ds is the synchronous phase. /G45G is the
geometrical beta of the cavity. The transit time factor
(TTF) is a complicated function of both the field
distribution and the particle velocity. At low energy, the
particle velocity may change appreciably during the
passage through a multiple gap cavity. For this reason,
the TTF is most conveniently calculated numerically.
We define the synchronous phase for a given particle
traversing a given field with respect to that rf phase ECR
RFQ Low /G45 SRF St. 1
Medium- /G45 SRF St. 2
400 MeV/u Beam
High /G45 SRF
/G45=0.81 /G45=0.61 /G45=0.49 12 keV/u 160 keV/u
12.3 MeV/u
85.5 MeV/u 0,s
0q,s
icosAqcosAq/G4D /G4D /GB8
/GB9/GB7/GA8
/GA9/GA7/G20 /GB8
/GB9/GB7/GA8
/GA9/GA7producing maximum energy gain. The synchronous
phase, as with the TTF, is generally most conveniently
determined numerically. The synchronous motion of an
ion with charge state q can be considered as motion in an
equivalent traveling wave with the amplitude Em=
E0T( /G45, /G45G). For beam energies higher than several MeV/u
the accelerating field Em can be considered as changing
adiabatically along the linac. At lower energies the
adiabatic conditions of ion motion are not valid due to
the large increment of beam velocity in the cavity.
A heavy-ion linac is usually designed for the
acceleration of many ion species. In a SC linac the
cavities, fed by individual rf power sources, can be
independently phased. The phase setting can be changed
to vary the velocity profile for synchronous motion
along the linac. For a given, fixed phase setting, the
synchronous velocity profile, and the TTF profile are
fixed along the accelerator. To accelerate ions with a
charge-to-mass ratio (q/A)i different from the design
value, the following relation must be satisfied
.cosEAqcosEAq
0 i q,s 0,m
0q,si,m
i/G4D /G4D /GB8
/GB9/GB7/GA8
/GA9/GA7/G20 /GB8
/GB9/GB7/GA8
/GA9/GA7 Thus, the
velocity and the accelerated beam energy per nucleon do
not depend on the ion species.
In an independently phased cavity array such as an
SRF ion linac, beams of different charge-to-mass ratio
can be accommodated by changing either or both the
phase and amplitude of the electric field. Allowing both
parameters to vary permits the option of varying the
velocity profile. This can provide higher energies per
nucleon for ions with a higher charge-to-mass ratio.
The RIA driver linac will accelerate uranium ions at
charge state q0=75 after the first stripper and at q0=89
after the second stripper. The simultaneous acceleration
of neighbouring charge states becomes possible because
the high charge-to-mass ratio makes the required phase
offsets small for a limited states of charge states. We
note that different charge states of equal mass will have
the same synchronous velocity profile along the linac if
the condition
(1)
is fulfilled. The simultaneous acceleration of ions with
different charge states requires an injection of the beam
with each charge state q at a synchronous phase which is
determined from (1) /GBB/GBC/GBA
/GAB/GAC/GAA/G10 /G200,s0
q,s cosqqcosArc /G4D /G4D .
Figure 2 shows the synchronous phase as a function of
charge states when the synchronous phase for q0=75 is
.300,s/G24/G10 /G20 /G4D This particular example shows that if the
linac phase is set for charge state q0=75, a wide range of
charge states can be accelerated. As seen even for charge
state 70 only a small change from 30 to 23° in
synchronous phase is required. The longitudinal focussing of the linac being considered is sufficient to
accept the predicted beam emittance.
The separatrices for charge states q=73,75 and 77
calculated in a conservative approximation are shown in
Fig. 3. Each particle with different charge state q
oscillates around its own synchronous phase with
slightly different amplitude [2]. It would be entirely
feasible to eliminate the relative oscillations. If the linac
has been tuned for the acceleration of some charge state
q0, then the particle bunches of different, neighbouring
charge states could be injected into the linac at different,
neighbouring rf phases in order for each charge state to
be matched precisely to its own phase trajectory. The
higher the charge state, the sooner it must arrive at a
given point to be matched. One possible method of
adjusting the phase of multiple charge states would be a
magnetic system combined with rf cavities.
-60-50-40-30-20-100
65 70 75 80 85 90
Charge statePhase (deg)
Figure 2: Synchronous phase as a function of uranium
ion charge state. /G4Ds,0=-30° for q0=75.
Figure 3: Separatrices in the longitudinal phase space for
charge states 73, 75 and 77 of uranium beam.
For most applications, however, such a system is not
necessary since the acceleration of a multi-Q beam is
possible even without matching the different charge
states to their proper synchronous phase. If all charge
states are injected at the same time (at the same rf
phase), then, as described above, each charge state
bunch will perform coherent oscillations with respect to
the tuned charge state q0. One can view this as an
increase in the total (effective) longitudinal emittance of
the multi-Q beam, relative to the (partial) longitudinal
emittance of the individual charge state bunches. For
the heavy-ion SRF linac being considered, the
longitudinal emittance is predominantly determined by
the injector RFQ, and can be made as small as ~2.0
keV/u /G98nsec for a single charge-state beam [3]. For q=77q=75q=73
-80 -40 0 40/G27p/p (%)2
-2
Phase (deg)comparison, the linac acceptance, given by the area of
the separatrix shown in Fig. 3, for q0=75 is ~77
keV/u /G98nsec. As will be shown below, this provides
ample headroom for the effective emittance growth
introduced by the acceleration of multiple charge states.
It should be noted that if no phase-matching is done
for different charge states, additional emittance growth
will occur at frequency transitions in the linac. Heavy-
ion linacs typically have several such transitions to
permit efficient operation over the large velocity range
required. In the RIA driver linac the strippers will be
placed in the region of frequency transitions thereby
eliminating effective emittance growth of multi-Q
beams.
2.2 Transverse beam dynamics
We now consider the transverse phase space for this
same uranium beam through the medium- /G45 section of
the linac. The focussing period is defined by a lattice of
a SC solenoid following each pair of SRF cavities. The
present linac design calls for solenoid focussing
elements because SC solenoids are cost effective for this
application, but the following analysis is not particularly
restricted by this choice.
Table I shows the Twiss parameters for 5 different
charge states calculated for the focusing period at the 12
MeV/u region of the linac. The difference in Twiss
parameters for five charge states is sufficiently small
that all the charge states can be injected into the linac
with the same transverse parameters producing an
effective emittance growth of 6.5%.
The transverse beam emittance is determined by the
ECR source. Present-day ECR sources can produce
beam intensities up to ~1 p /G50A for a single charge state
of uranium, with an normalized emittance (containing
90% of the particles) equal to ~0.2 /G53/G98mm /G98mrad [4]. We
compare this emittance with the transverse acceptance of
the solenoidal focussing channel of the driver linac
assuming /G50x=60°, /G45x,max = 3.17 mm/mrad. The maximum
value of the /G45x -function occurs at the center of
solenoids, which has a bore radius of 15 mm. This
implies a normalized acceptance An = 11.6 /G53/G98mm /G98mrad.
The acceptance of this section is ~50 times larger than
the beam emittance at the entrance. It should be noted
that the acceptance of the next linac section, the high
energy part, is even larger, ~100 /G53/G98mm /G98mrad.
Transverse emittance can grow due to several error
effects. We discuss below two type of errors which can
most seriously impact the multi-Q beam dynamics. The
first type of error, mismatch, is caused by errors in
tuning or matching the beam into the linac and arises
because of errors in measurement of the input beam
parameters. The second type of error is transverse
displacements of the focussing lenses.
For a low-intensity single charge state beam,
mismatched betatron motion and coherent transverse Table I: Twiss parameters of the matched beam after the
first stripper for 5 different charge states.
q /G44x /G45x /G4Ax
73 0.428 1.536 0.770
74 0.435 1.518 0.783
75 0.441 1.500 0.783
76 0.448 1.483 0.809
77 0.455 1.467 0.823
Figure 4: Transverse beam envelopes for one charge
state beam (dots) and 5 charge state beam (solid line).
Input beam is mismatched by a factor 1.4.
oscillations will not increase effective transverse
emittance. For this case, the errors discussed above are
often of little consequence and can be easily corrected.
In the case of a multi-Q beam, however, the different
charge states have different betatron frequencies. As the
beam proceeds along the linac, the transverse
oscillations of the various charge states eventually
become uncorrelated and the effective total emittance,
summed over all charge states, increases.
One aspect of this behaviour can be illustrated by
considering a mismatched beam through the 58
focussing periods of the linac between the two strippers.
While the actual linac lattice will be slightly more
complex, it is sufficient for us to consider the periodic
focussing structure as having constant length. We
assume the solenoids to be tuned for a phase advance
over one period of /G50x = 60°, for charge state 75.
Although the phase advance per period does not depend
strongly on the charge state, over 58 periods the phase
differences between different charge states become
appreciable. If the input beam is mismatched, the phase
space ellipse begins to rotate, at twice the betatron
frequency, tracing out a (matched) ellipse of larger area.
Fig. 4 shows beam envelopes both for a single charge
state beam and also for a five charge state beam. The
oscillations of the mismatched beam remain coherent for
the single charge state, but not for the multi-Q case. To
summarize, the main difference between one- and multi-
Q beams is that mismatch of a single charge state beam
is generally correctable, and does not lead to transverse
emittance growth. For multiple charge states, correction
is more difficult, and will generally induce growth in the
transverse emittance. 00.10.20.30.40.5
0102030405060
Period numberBeam size (cm)One charge
Five chargesMulti-Q beams are also more severely affected by
misalignment errors. Misalignments produce a
transverse magnetic field on the linac axis and
coherently deflect the beam. For a single charge state
beam, misalignment causes lateral displacement of the
beam, but no emittance growth so long as the beam
remains in the linear region of the focussing elements.
With a beam containing multiple charge states, the
differing betatron periods, as well as the differing
displacements, cause growth in the transverse emittance.
We have performed Monte Carlo simulations of the
dynamics of multi-Q beams in the presence of alignment
errors. We considered a five charge-state uranium beam
in that portion of the linac between the first and second
strippers. To make the simulation more realistic, we
assumed a mismatch factor of 1.2 for the beam out of the
first stripper. We introduced alignment errors by
displacing separately both ends of each of the 58
focussing solenoids in both x and y by an amount
randomly varying over the range ±300 /G50m. The bar
graph in Fig. 5 is a histogram of the simulation results.
Note that for some sets of alignment errors, the
emittance growth factor can be as high as 8.5.
Figure 5: Probability of emittance growth in the
misaligned focussing channel. The emittance growth
factor is the ratio of (normalized) transverse emittance of
the beam at exit to that at entrance.
Even for the multi-Q beams, however, emittance
growth can be substantially reduced by simple corrective
steering procedures. We have modelled this by
assuming a measurement of the beam centroid position
and corrective steering to be performed once every four
focussing periods. This interval would correspond to the
space between cryomodules in the benchmark linac
design.
The transverse tune has important effects. We
consider the case of a 60° phase advance per focussing
period. For this case, the phase advance between the
points at which monitoring and steering is performed is
/G29x = 240°. This rotation in transverse phase space
transforms a centroid displacement at one corrective
station to a deflection at the succeeding station, which
can be directly corrected by simple one-element steering
at that point. In our simulations, the beam center was calculated as the center of gravity of a five charge state
beam with q=73…77. The results are shown in Fig. 5. It
can be seen that the emittance growth factor has, in all
cases, been reduced to less than 3. For the entire set of
cases of random errors, the most probable value for the
effective emittance growth factor is 1.4. The steering
procedure effectively reduces the divergence in
transverse phase space thereby reducing the possible
effective emittance growth.
3 EXAMPLES OF MULTI-Q BEAM
ACCELERTATION
3.1 Two charge state beam in the prestripper
linac
We have shown [3] that it is possible to accept two
charge states from an ECR and accelerate both to the
first stripping target at 12 MeV/u. The ECR is followed
by an achromatic bend with a charge selector which
transports a two-charge-state uranium beam to the
entrance of a multi-harmonic buncher upstream of the
RFQ. The fundamental frequency of both bunchers is
one half of the RFQ frequency. A combination of the
multi-harmonic buncher and RFQ bunches more than
80% of each charge state to an extremely low
longitudinal emittance (total emittance is lower than
~2.0 /G53 /G98keV/u /G98nsec) beam at the output of the RFQ. The
second buncher is located directly before the RFQ
entrance and changes the average velocities of each
charge state to the design input velocity of the RFQ.
Over the distance between the first, multi-harmonic
buncher, and second buncher the bunched beams of each
charge state are formed with a separation by 360 /G71 at the
RFQ frequency due to the different average velocities of
each charge state. Electrostatic quadrupoles provide
focussing and transverse matching to the RFQ
acceptance. The results of the design and beam
dynamics studies are presented at this conference (see
ref. [3]).
3.2 Five charge state beam in the medium- /G45
linac
We have carried out a Monte Carlo simulation of
multi-Q beam acceleration from the first stripper through
the second stripper and continuing to the end of the
linac. The simulation starts with a 12.3 MeV/u uranium
beam equally distributed over 5 charge states, all at the
same rf phase, and with a longitudinal emittance ~1.0
/G53 /G98keV/u /G98nsec. We consider in detail the behaviour of this
beam between the two strippers, a section of linac
consisting of 3-gap SRF cavities operating at 172.5 MHz
and 345 MHz. The rf phase throughout this section has
been set for acceleration of uranium with charge state
q0=75 at synchronous phase /G4Ds,75=-30°. The phase is
calculated using values of the electric field numerically 08162432
1 2 3 4 5 6 7 8
Emittance growth factorFractional probability (%)Empty bars: No corrective steering
Solid bars: One-element corrective steeringgenerated using realistic cavity geometries. The beam
tracking simulation was done with a modified version of
the LANA code [2]. This code completely simulates
beam dynamics in the six-dimensional phase space
including alignment errors. The phase space plot of all 5
charge state bunches at Wn=85.5 MeV/u, just before the
second stripper, is shown in Fig. 6 together with the
acceptance of 805 MHz elliptical SRF structure. The
acceptance is obtained by Monte Carlo simulation. After
the second stripper, the effective longitudinal emittance
of the multi-Q beam is increased by a factor of ~6. We
note, however, that this longitudinal emittance is still
substantially less than the acceptance of the remaining
portion of the SC linac.
Figure 6: Phase space plots of the multi-Q beam at the
location of the second stripper. Large dotted area
represents the acceptance of high-energy section.
The effect of rf field errors on longitudinal beam
dynamics in a multi-cavity linac becomes significant in
the present case mainly because of the large number of
individual cavities. These errors are caused by
fluctuations, which we assume to be random, in the rf
phase and amplitude of the electromagnetic fields in the
cavities. We have performed numerical simulation to
estimate the effects of this latter class of error, for both
single-charge-state and also for multi-Q beams. The
error effects are included by introducing phase and
amplitude errors for each of the cavities, randomly
distributed over the indicated range. We have found [2]
that even including both these effects, the total increase
in longitudinal emittance is still well below the
acceptance of the high energy part of the driver linac,
~77 /G53 /G98keV/u /G98nsec.
The construction of a high-intensity heavy-ion linac
requires at least two stripping foils. In order to avoid
beam losses in the high-energy part of the linac the low-
intensity unwanted charge states must be carefully
separated and dumped. It will require a system
containing dipole magnets and a rebuncher in order to
provide a unit transformation of the 6-dimensional beam
phase space (see Fig.1). We have designed such systems
for several cases: a 180 /G71 bend, a parallel translation of
~4.5 m and a chicane-like system for the straight line
transformation. After the stripping target, all charge states have the same velocity, therefore, such a matching
system is isopath for different charge states.
3.3 Multi-Q beam test at ATLAS
A test of the acceleration of multi-Q beams was
performed at the ATLAS accelerator. A 238U+26 beam
from an ECR ion source was accelerated to 286 MeV
(~1.2 MeV/u) and stripped just before the ‘Booster’
section of ATLAS. All charge states near q0=38 were
simultaneously accelerated in the Booster. The
parameters of each selected charge state were carefully
measured. Tuning of the focusing fields to get 100%
transmission was accomplished with the 58Ni+9 guide
beam prior to switching to the uranium mixed beam.
About 94% transmission of the multi-Q uranium beam
was detected. Six charge states were accelerated through
the Booster with an average energy spread within 1.5%.
Detailed experimental results are given in ref. [5].
4 CONCLUSIONS
The large longitudinal and transverse acceptance
characteristic of superconducting heavy-ion linacs makes
possible the acceleration of multiple charge state beams.
Our studies indicate that it is quite feasible to accelerate
2 charge states of uranium from an ECR to the first
stripper, 5 charge states of the same beams after the first
stripper and 5 charge states after the second stripper in
this linac. Such operation could provide ~120 kW of
uranium beam using a demonstrated performance of an
ECR ion sources.
5 ACKNOWLEDGEMENTS
The authors wish to thank our collaborators from
several laboratories. R. Pardo, M. Portillo (ANL), V.N.
Aseev (INR, Moscow), A.A. Kolomiets (ITEP,
Moscow), J. Staples (LBNL) participated in developing
of several aspects of multi-Q beam studies.
Work supported by the U. S. Department of Energy
under contract W-31-109-ENG-38.
6 REFERENCES
[1] K. W. Shepard, et al., in the Proceedings of the 9th
International Workshop on RF Superconductivity,
Santa Fe, New Mexico, 1999, to be published.
[2] P.N. Ostroumov and K.W. Shepard, Phys. Rev. ST
Accel. Beams 3, 030101 (2000).
[3] P.N. Ostroumov, et al, “Heavy Ion Beam
Acceleration of Two-Charge States from an ECR
Ion Source”, paper MOD01 in these Proceedings.
[4] C.M. Lyneis et al., Rev. Sci. Instrum., 69, 682
(1998).
[5] P.N. Ostroumov, et al, “Multiple Charge State Beam
Acceleration at ATLAS”, paper MOD02 in these
Proceedings.
-90 Phase, deg 805 MHz 90
/G27W/W (%) 1.6
-1.673
74
75
76
77 |
arXiv:physics/0008184 19 Aug 2000HEAVY-ION BEAM ACCELERATION OF TWO-CHARGE STATES FROM
AN ECR ION SOURCE
P.N. Ostroumov, K.W. Shepard, Physics Division, ANL, 9700 S. Cass Av., Argonne, IL, 60439
V.N. Aseev, Institute for Nuclear Research, Moscow 117312, Russia
A.A. Kolomiets, Institute of Theoretical and Experimental Physics, Moscow 117259, Russia
Abstract
This paper describes a design for the front end of a
superconducting (SC) ion linac which can accept and
simultaneously accelerate two charge states of uranium
from an ECR ion source. This mode of operation
increases the beam current available for the heaviest ions
by a factor of two. We discuss the 12 MeV/u prestripper
section of the Rare Isotope Accelerator (RIA) driver linac
including the LEBT, RFQ, MEBT and SC sections, with a
total voltage of 112 MV.
The LEBT consists of two bunchers and electrostatic
quadrupoles. The fundamental frequency of both
bunchers is half of the RFQ frequency. The first buncher
is a multiharmonic buncher, designed to accept more than
80% of each charge state and to form bunches of
extremely low longitudinal emittance (rms emittance is
lower than 0.2 /G53/G98keV/u /G98nsec) at the output of the RFQ.
The second buncher is located directly in front of the
RFQ and matches the velocity of each charge-state bunch
to the design input velocity of the RFQ. We present full
3D simulations of a two-charge-state uranium beam
including space charge forces in the LEBT and RFQ,
realistic distributions of all electric and magnetic fields
along the whole prestripper linac, and the effects of
errors, evaluated for several design options for the
prestripper linac. The results indicate that it is possible to
accelerate two charge states while keeping emittance
growth within tolerable limits.
1 INTRODUCTION
The Rare Isotope Accelerator (RIA) Facility requires a
1.3 GeV linac which would accelerate the full mass range
of ions and would deliver ~400 kW of uranium beam at
an energy of 400 MeV per nucleon [1,2]. The driver
would consist of an ECR ion source and a short,
normally-conducting RFQ injector section which would
feed beams of virtually any ion into the major portion of
the accelerator: an array of more than 400
superconducting (SC) cavities of seven different types,
ranging in frequency from 57.5 to 805 MHz. The linac
contains two stripping targets, at 12 MeV/u and 85
MeV/u, for the uranium beam. A novel feature of the
linac is the acceleration of beams containing more than
one charge state [3,4]. The front end of the RIA driver
linac consists of a ECR ion source, a LEBT, a 57.5 MHz
RFQ, a MEBT and a section of SRF drift-tube linac.
The present-day performance of ECR ion sources, and
considerations based on fundamental limiting processes in
the formation of high-charge state uranium ions in such sources, indicate that uranium beam intensities as high as
7 p /G50A in a single charge state of 29+ or 30+ are unlikely
to be obtained in the near future. Such a high current is
required in order to produce the RIA driver linac design
goal of a 400 kW uranium beam, even if we assume
multiple charge state beam acceleration following the first
stripper.
This paper discusses in detail a solution to this
limitation. It doubles the heavy-ion beam intensity by
accepting two charge states from the ion source.
2 DESIGN OF THE FRONT END
2.1 LEBT
The LEBT is designed for the selection and separation
of the required ion species and the acceptance of single-
or two-charge states by the following RFQ structure. The
first portion of the LEBT is an achromatic bending
magnet section for charge to mass analysis and selection.
For the heaviest ions, such as uranium ions, the transport
system must deliver to the entrance of the first buncher a
two-charge-state beam with similar Twiss parameters for
both charge states. The design features of the two-charge
selector will be discussed elsewhere. The ECR is placed
on a high voltage platform. The voltage V0=100 kV is
adequate to avoid space charge effects in the LEBT and
RFQ and to keep the RFQ length to less than 4 m.
A simplified layout of the second part of the LEBT is
shown in Fig. 1. This part of the LEBT solves the
following tasks: a) Beam bunching by a four-harmonic
external buncher B1 (the fundamental frequency is 28.75
MHz); b) Velocity equalization of two different charge
states by the buncher B2, operating at 28.75 MHz; c)
Charge-insensitive transverse focusing of the 2-charge
state beam and matching to the RFQ acceptance by the
electrostatic quadrupoles Q1-Q8.
The reference charge state for the design of the LEBT,
RFQ and MEBT is 29.5. The RFQ injector is designed to
accelerate any beam from protons to uranium to a
velocity v/c = 0.01893 at the exit of the RFQ.
The computer code COSY [5] was used in order to
design and optimize the LEBT by taking into account
terms through third order. The final beam dynamics
simulation has been performed by the DYNAMION code
[6] where the equations of motion are solved in a general
approximation using realistic 3D electrostatic fields of the
quadrupoles and rf bunchers, including space charge
forces for the multi-charged ion beams. Realistic 3D
fields for the electrostatic quadrupoles were calculated Figure 1: Layout of the LEBT.
Figure 2: Beam envelopes in the LEBT.
2930
dcb a
-400-2000200400600800-4-2024
-200-1000100200300400-4-2024-200-1000100200300400-4-2024
-200-1000100200-4-2024
Figure 3: Longitudinal phase space plots of a two-charge
state beam along the LEBT: a) after B1, b) before B2, c)
after B2, d) RFQ entrance with scale changed to RFQ
frequency.
by the SIMION code [7]. The 3D field distributions have
been used with both the COSY and DYNAMION codes.
Figure 2 shows beam envelopes along the LEBT
optimized by COSY. The total normalized emittance at
the exit assumes an ion source emittance of 0.5
/G53/G98mm /G98mrad. After careful optimization, including third
order terms, the rms emittance growth is less than 7% in
the horizontal plane. In the vertical plane there is no
observable rms emittance growth. Figure 3 shows the
transformation of the beam image in longitudinal phase
space. The first multi-harmonic buncher modulates the
beam velocity of the two charge states as shown in Fig.
3a. The drift space between the two bunchers (see Fig. 3)
is chosen from the expression
1qq)1q(q
AmeV2L
0 000
u0
12/G10 /G10/G10/G98 /G20 /G4F ,
where /G4F is the wavelength of the RFQ frequency, mu is
the atomic unit mass. This drift space separates in time the bunches of different charge states q0 and q0-1 by 360 /G71
at the RFQ frequency. The voltages of the multi-harmonic
buncher have been optimized, together with the RFQ
parameters, in order to obtain a total efficiency above
80%, while minimizing the longitudinal emittance for
each charge state. The second buncher is used to equalize
the velocities of the 2 charge states (see Fig. 3c).
2.2 RFQ
The formation of heavy-ion beams of low longitudinal
emittance has been discussed in ref. [8], in which the
lowest beam emittance was obtained by prebunching and
using a drift space inside the RFQ. This procedure works
well for a single charge state beam, but cannot
accommodate two charge states because of the different
velocities of different charge states coming from the same
ion source.
We describe below a new design for a low current RFQ
and injector system which can provide very low
longitudinal emittance for operation with both single
charge state and two-charge state beams. Low rms and
total longitudinal emittance are achieved by using an
external multi-harmonic buncher and an RFQ of modified
design. The RFQ has three main sections: 1) the standard
radial matcher, 2) the transition section and 3) the
acceleration section.
]The radial matcher transforms the RFQ acceptance to a
set of Twiss parameters that avoids large beam sizes in
the LEBT quadrupoles.
The transition section is a part of the RFQ with a linear
variation of the synchronous phase. The RFQ parameters
A, m, a and /G4DS are calculated self-consistently in order to
filter the longitudinal emittance in such a way that the low
populated area will be lost either inside the RFQ or in the
MEBT. It is performed by an iterative procedure of the
whole RFQ design and by observing the emittance
formation in the longitudinal phase space. The desired
result is achieved if the separatrix size is slightly larger
than the densely populated area of beam emittance. The
design goal is an rms emittance below 0.2 /G53/G98keV/u /G98nsec,
required for multiple charge state operation through the
rest of the linac.
The acceleration section is a portion of the RFQ with
constant synchronous phase which is equal to –24°. The
RFQ forms a beam crossover at the exit in both transverse
planes. This results in better matching of the two-charge
state beam in the MEBT.
In this way, the RFQ was designed for acceleration of
one- or two-charge state heavy-ion beams from 12.4
keV/u to 167 keV/u over a length of 4 m. The phase space
plots from numerical simulations of a two-charge state
uranium beam exiting the RFQ are shown in Fig. 4.
2.3 Beam simulation in the MEBT and SRF
linac
Between the RFQ and SRF Linac there is a matching
section – the MEBT. We found that SC solenoids placed
in individual cryostats are the best option to focus a two- Figure 4: Phase space plots of beams with charge state
29+ and 30+ exiting the RFQ. In the transverse planes the
charge states 29 and 30 occupy the same area, in the
longitudinal plane the bunches are separated by 360 /G71.
Figure 5: Two-charge state beam envelopes (rms and
total) along the MEBT and SRF Linac.
Figure 6: Phase space plots of a two-charge state beam
just prior to the first stripper.
charge state beam.
The portion of the linac prior to the first stripper
contains 96 cavities of four different types. Prior to ray
tracing a multiple charge state beam through the
prestripper linac, the transverse beam motion was
matched carefully using fitting codes for a trial beam of
charge state q=29.5. A particularly critical aspect of
fitting was to avoid beam mismatch at the transitions
between focusing periods of differing lengths. The
focusing lattice length is different for each of the four
types of SRF cavities. The phase advance per focusing
period was set at 60°. As mentioned in ref. [3], the correct
choice of phase advance is crucial for effective steering of
the multiple charge state beam. The design and simulation
of 3D beam dynamics in the SRF linac was performed by
the LANA code [9]. The pre-processor code generates the
phase setting for a uranium beam with average charge
state q=29.5. The rf phase is set to –30° with respect to
the maximum energy gain in each SRF cavity. Realistic field distributions for the SRF cavities were generated
using an axially-symmetric approximation of the actual
drift tube cavities. These fields are used both for the
design procedure and for the beam dynamics simulation.
The initial phase space distribution used for each charge
state was the beam at the exit of the RFQ, as simulated
with the DYNAMION code. The particle coordinates
were then tracked through the SRF linac. Figure 5 shows
the transverse beam envelopes (rms and total) along the
MEBT and SRF linac. Despite the slight mismatch of the
two-charge state beam along the linac, there is no rms
emittance growth in the transverse plane for an ideal linac
without any errors.
In longitudinal phase space the emittance of the two
charge state beam is always larger than for a single charge
state beam. Growth in effective emittance occurs due to
the oscillations caused by the slightly different, off-tune
synchronous phases for charge states 29+ and 30+.
However, the very low longitudinal emittance achieved
by the RFQ injector ensures that the total emittance of the
two-charge state beam remains well inside the stable area
in longitudinal phase space. Figure 6 shows the
longitudinal phase space just prior to the first stripper.
Note that the energy and phase acceptance of the
reminder of the SRF linac are /G723% and /G7230 /G71,
respectively.
3 CONCLUSION
The problem of acceptance and acceleration of two
charge states of a heavy-ion beam from a single ECR ion
source was successfully solved. A front end has been
designed for a driver linac for RIA that accelerates a two-
charge state uranium beam. The use of a two-charge state
beam is a powerful tool to double the total beam power
produced by the heavy ion driver linac.
This work was supported by the U. S. Department of
Energy under contract W-31-109-ENG-38
4 REFERENCES
[1] K. W. Shepard, et al., in the Proceedings of the 9th
International Workshop on RF Superconductivity,
Santa Fe, New Mexico, 1999, to be published.
[2] C. Leemann, Paper TU103, these Proceedings.
[3] P.N. Ostroumov and K. W. Shepard, Phys. Rew. ST
Accel. Beams, 3, 030101 (2000).
[4] P.N. Ostroumov, K. W. Shepard and J.A. Nolen,
Paper FR101, these Proceedings.
[5] M. Berz. COSY INFINITY. Version 8. User’s Guide
and Reference Manual, MSU, 1999.
[6] A.A. Kolomiets, et al, in the Proc. of the Sixth Eur.
PAC, Stockholm, Sweden, June 22-26, 1998.
[7] SIMION 3D. Version 6.0, User’s Manual, INEL-
95/0403 (Idaho Nat. Eng. Laboratory, 1995).
[8] J. Staples, in Proceedings of the XVIII Inter. Linac
Conf., V. 2, Tsukuba, Japan, 1994.
[9] D.V. Gorelov and P.N. Ostroumov, Proc. of the Fifth
Eur. PAC, p. 1271. Sitges, Spain, June 10-14, 1996. 00.30.60.91.21.5
0 10 20 30 40 50 60 70
Distance (m)Beam size (cm)q=29
q=30Total RMS . . . . .
.
-0.5-0.2500.250.5
-10 -5 0 5 10
Phase, deg 115 MHz/G27W/W (%)q=29q=30-20020320340360380-2-1012 dW/W(%)
Phase (degree)-0.3 0.0 0.3-40-2002040 Y' (mrad)
Y (cm)-0.3 0.0 0.3-40-2002040 X' (mrad)
X (cm) |
arXiv:physics/0008185 19 Aug 2000MULTIPLE CHARGE STATE BEAM ACCELERATION AT ATLAS
P.N. Ostroumov, R.C. Pardo, G.P. Zinkann, K.W. Shepard, J.A. Nolen, Physics Division, ANL,
9700 S. Cass Avenue, Argonne, IL60439, USA
Abstract
A test of the acceleration of multiple charge-state
uranium beams was performed at the ATLAS accelerator.
A 238U+26 beam was accelerated in the ATLAS PII linac to
286 MeV (~1.2 MeV/u) and stripped in a carbon foil
located 0.5 m from the entrance of the ATLAS Booster
section. A 58Ni9+ 'guide' beam from the tandem injector
was used to tune the Booster for 238U+38. All charge states
from the stripping were injected into the booster and
accelerated. Up to 94% of the beam was accelerated
through the Booster linac, with losses mostly in the lower
charge states. The measured beam properties of each
charge state and a comparison to numerical simulations
are reported in this paper.
1 INTRODUCTION
Simultaneous acceleration of multiple charge-state
beams has been proposed as a method of substantially
increasing the available beam current for the heaviest ions
from a RIA (Rare Isotope Accelerator) driver linac [1].
There is presently no facility where multiple charge-state
beam acceleration is used to increase the beam current.
Therefore, in order to demonstrate the concept, we have
accelerated a multiple charge-state uranium beam in the
existing ATLAS heavy-ion linac, and performed careful
measurements of the accelerated beam parameters for
comparison with the results of numerical simulations.
The acceleration of multiple charge-state uranium
beams has been observed at the ATLAS ‘booster’ as part
of the ‘normal’ uranium beam configuration. However,
the multiple charge states have been considered parasitic.
Therefore systematic studies of all the accelerated charge
states were not performed and accelerator parameters
were not chosen to optimize the acceleration of the other
charge states.
In this test, a 238U+26 beam was accelerated to 286 MeV
(~1.2 MeV/u) and stripped. All charge states near q0=38
were then simultaneously accelerated in the ATLAS
‘Booster’ linac. The parameters of each selected charge
state were carefully measured.
2 BEAM DYNAMICS SIMULATIONS
For a better understanding of the beam test results, a
multiple charge-state beam dynamics simulation in the
Booster was performed with the modified LANA code
[2]. At the position of the stripping target, input beam parameters were assumed to be the same for all charge
states. The beam longitudinal and transverse emittances
were taken to be equal to /G48L=2 /G53/G98keV/u /G98nsec and /G48T=0.25
/G53/G98mm /G98mrad. The ray-tracing code incorporated actual
resonator field profiles and field levels of the
superconducting cavities. The synchronous phase for
238U+38 was set to -30 /G71. Fig. 1 and 2 show the calculated
longitudinal phase space at the booster exit and the
transverse beam envelopes along the Booster.
Figure 1: Longitudinal phase space plots of the
accelerated multiple charge-state uranium beam exiting
the booster.
Figure 2: Transverse envelopes of the multiple charge-
state uranium beam along the booster.
As is seen one can expect acceleration of most charge
states produced after the stripping foil. The simulation of
the transverse motion does not include the misalignment
errors; and therefore the total effective emittance growth
is negligible.
3 DESCRIPTION OF THE EXPERIMENT
The 238U+26 beam from the ATLAS ECR-II ion source
was accelerated to 286 MeV (~1.2 MeV/u) in the Injector
Linac, and stripped in a 75 /G50g/cm2 carbon foil 0.5 m
before the ‘Booster’ linac as shown in Fig. 3. The beam 00.250.50.7511.25
0 4 00 8 00 1 200 1 600
Distance, cmBeam size, cm 40+39+
41+
42+
43+35+ 36+ 37+ 38+ -3-2-10123
-30 -20-10 0 1 0 20 30
Phase, deg/G27W/W, %
35+36+37+ 40+39+
41+
42+
43+38+energy was carefully measured by a resonant time-of-
flight (TOF) system [3]. The ATLAS Booster was tuned
using a 58Ni+9 ‘guide’ beam from the ATLAS tandem
injector whose velocity was matched to that of the
stripped 238U+38 and which has a similar charge-to-mass
ratio. The synchronous phase for 238U+38 was chosen to be
–30 /G71. Therefore the synchronous phase /G4DG required for
the guide beam is given by [1]
/G71 /G10 /G20/GBB/GBC/GBA
/GAB/GAC/GAA/G71 /G10/G98/G98/G10 /G20 27)30cos(23893858arccosG /G4D .
The synchronous phase in all 24 cavities of the booster is
set by an auto-scan procedure using a silicon detector for
beam energy measurements. Tuning of the focusing fields
to get 100% transmission was accomplished with the
guide beam prior to switching to the uranium mixed
beam.
Figure 3: Layout of ATLAS linac.
After optimising the Booster linac and 40 /G71-bend tune
with the 58Ni+9 guide beam, the stripped uranium beam
was injected into the Booster. Magnet slits were used to
cleanly select only the 38+ beam after the bending
magnets and the uranium injection phase was matched to
the guide beam’s phase empirically based on maximum
transmission through the system. Further tuning of the
bunching system and last PII resonator made small
adjustments to the uranium beam energy to better match
the guide beam’s velocity. After this tuning process, a
91% transmission of the multiple charge-state uranium
beam was achieved. The transmission improved to 94%
when a 10 mm aperture was inserted upstream of the
stripping target. Fig. 4 compares the intensity distribution
of the mixture of multiple charge-state uranium beams
accelerated in the booster to the measured stripping
distribution for the unaccelerated uranium. The difference
in the distributions is caused mainly by poo rer
transmission of lower charge states through the booster.
Also, some discrepancy is expected due to slightly
different tuning of unaccelerated and accelerated beams
and collimator slits in the 40 /G71 bend region.
The individual charge states then were analysed in the
40 /G71 bend region and sent to the ATLAS beam diagnostics
area (see Fig. 3). The parameters of each selected charge
state were carefully measured. Particularly the following
beam parameters were measured: /G78 Transverse emittance (the value and ellipse
orientation in phase space) by the help of
quadrupole triplet gradient variation [4] and a wire
scanner located 3.1 m apart.
/G78 Average beam energy using the ATLAS TOF energy
measurement system.
/G78 Beam energy spread with the silicon detector
measuring the bunch time width after a long drift
space to the ATLAS diagnostics area.
Figure 4: Comparison of intensity distributions for
accelerated and unaccelerated multiple charge uranium
beams.
Finally, the multi-charged uranium beam was stripped
for the second time at the exit of the Booster and 238U+51
was selected. The same beam parameters measurements
were performed and the beam was further accelerated in
the last section of ATLAS. As expected, the use of multi-
charged uranium beam on the second stripper increased
the intensity of double-stripped 238U+51 beam. The double-
stripped 238U+51 was accelerated up to 1400 MeV and used
for a scheduled experiment at ATLAS.
Basic results of these beam measurements are shown in
Fig. 5-8. Figure 5 presents transverse beam profiles at the
exit of the booster. The multiple charge-state uranium
beam has a larger size compared to the guide beam. As
was shown in ref. [1] misalignments of the focusing
elements and effective emittance growth of a multi-
charged beam compared to a single charge-state beam are
the main source of the larger beam size. Such errors must
be minimized in machines designed for the utilization of
multi-charge beams. The results of individual transverse
emittance measurements are presented in Table 1 and
Figure 6. The horizontal
60 mmH-planeV-plane
60 mmH-planeV-plane
Figure 5: Transverse profiles at the exit of the booster for
guide beam (left) and multi-Q uranium beam (right).
0.000.050.100.150.200.25
35 36 37 38 39 40 41 42 43 44
Charge stateNormalized current (rel. value)Unaccelerated
Acceleratedemittance is less than the vertical due to the charge
selection by the slits downstream of the bending magnet.
Therefore only emittances in the vertical plane are shown.
The double-stripped uranium beam 238U+51 contains all
information about the effective emittance of multi-
charged beam output of the Booster. The effective
emittance increases by a factor of 2 due to the
misalignment errors of the solenoids in the Booster.
Average energy and the FWHM energy spread of
individual charge states are shown in Fig. 7. For the
simulation of energy spread, the input longitudinal
emittance /G48L=2 /G53/G98keV/u /G98nsec was assumed. The graphs
show consistent behaviour of the energy spread as a
function of charge state. There is some discrepancy with
the average energy for the remote charge state which is,
probably, caused by the longitudinal tuning errors of the
SRF cavities. Certainly such tuning in high intensity
machines should be done with high precision, but for
Table 1: Twiss parameters of single charge state
beams at the exit of ATLAS for the vertical plane
Uranium
charge state /G44y /G45y,
mm/mrad /G48y, normalized,
/G53/G98mm /G98mrad
36+ 0.72 12.66 0.94
37+ 0.48 8.08 1.24
38+ 0.06 10.17 1.11
39+ 0.45 7.60 1.34
40+ 0.54 9.22 1.03
41+ -0.18 9.20 0.89
51+ 0.60 9.00 2.69
36
37
39
40
51
38
x', mrad
4
0
-4
-3 x, cm 3
Figure 6: Vertical phase space ellipses of single charge-
state beams. The black ellipse corresponds to double-
stripped U51+.
0.00.40.81.2
35 37 39 41
Charge stateEnergy spread (%)Measurement
Simulation
680690700710
35 37 39 41
Charge stateBeam energy (MeV)Si Detector
TOF
Simulation
Figure 7: Beam energy (left) and the FWHM energy
spread (right) of individual charge states. ADC ChannelIntensity (rel. unit)
Figure 8: Bunch time width for a single charge state
238U+38 beam (left) and a double-stripped 238U+51 beam
(right).
routine operation of ATLAS it is not required. Even with
these tuning errors, the average energy spread for three
neighbouring charge states 37-39, similar to the beam
energy distribution proposed for the RIA driver linac, is
only 0.7%.
Figure 8 presents bunch time width measurements of
beams transported to the ATLAS diagnostic area. The left
spectrum corresponds to charge state 38+ (before the
second stripping) and the right spectrum belongs to
double-stripped uranium beam 238U+51. The low intensity
background events are mostly due to detector system
background. The energy spectra at FWHM obtained from
these measurements are 0.4% for charge state 38+ and
1.3% for charge state 51+. So, the second stripping of the
multi-charged beam produced a 3 times larger energy
spread.
CONCLUSION
The results of this test are consistent with the
simulation and show that multi-charged beam
acceleration can substantially increase the intensity of
heavy-ion beams. A medium-energy high-power
machine, such as the RIA driver linac, can be designed to
utilize multi-Q beams if unwanted charge states after each
stripping target are cleaned by a corresponding magnetic
system. For low-intensity and low-energy linacs, such as
ATLAS, the double-stripped heavy-ion beam can be used
to obtain higher beam energy while providing more
intensity than with single charge-state acceleration.
Work supported by the U. S. Department of Energy
under contract W-31-109-ENG-38
REFERENCES
[1] P.N. Ostroumov and K.W. Shepard, Phys. Rev. ST
Accel. Beams 3, 030101 (2000).
[2] D.V. Gorelov and P.N. Ostroumov, Proc. of the Fifth
Eur. Part. Accel. Conf., p. 1271. Sitges, Spain, June
10-14, 1996.
[3] R. Pardo, et al, Nucl. Inst. Meth. A270, 226 (1988)
[4] P.N. Ostroumov, et al, Proc. of the Third Eur. Part.
Accel. Conf., p. 1109. Berlin, March 24-28, 1992.
|
arXiv:physics/0008186v1 [physics.acc-ph] 19 Aug 2000Determining Phase-Space Properties of the LEDARFQ Output B eam∗
W.P. Lysenko,J.D.Gilpatrick,L.J.Rybarcyk, J.D.Schneid er, H.V. Smith,Jr.,and L.M.Young,
LANL,Los Alamos,NM 87545,USA
M.E.Schulze, General Atomics,LosAlamos,NM 87544,USA
Abstract
Quadrupolescanswereusedtocharacterizethe LEDA RFQ
beam. Experimental data were fit to computer simulation
models for the rms beam size. The codes were found to
be inadequateinaccuratelyreproducingdetailsofthe wire
scanner data. When this discrepancy is resolved, we plan
to fit usingallthe datainwirescannerprofiles,notjust the
rmsvalues,usinga 3-Dnonlinearcode.
1 INTRODUCTION
During commissioning of the LEDA RFQ [1, 2], we found
that the beam behaved in the high energy beam transport
(HEBT)muchaspredicted. Thustheactual RFQbeammust
havebeenclose tothatcomputedbythe PARMTEQM code.
TheHEBTincluded only limited diagnostics[3] but we
were able to get additional information on the RFQbeam
distribution using quadrupole scans[4]. An good under-
standing of the RFQbeam and beam behavior in the HEBT
will be helpful for the upcoming beam halo experiment.
The problems with the quad scan measurements were the
strongspaceeffectsandthealmostcompletelackofknowl-
edge of the longitudinalphase space. Also, our simulation
codes, which served as the models for the data fitting, did
notaccuratelyreproducethemeasuredbeamprofilesatthe
wire scanner.
2 HEBTDESIGN
TheHEBT[5]transportsthe RFQbeamtothebeamstopand
provides space for beam diagnostics. Here, we discuss
HEBTpropertiesrelevanttobeamcharacterization.
•Design has Weak Focusing. Ideally, the HEBTwould
have closely-space quadrupoles at the upstream end
until the beam is significantly debunched, i.e., for
about one meter. After this point, we could use any
kind of matching scheme with no fear of spoiling the
beamdistributionwithspace-chargenonlinearities.
OurHEBTdesign uses four quadrupoles, which is
the minimum that provides adequate focusing for the
given length. Any fewer than four quadrupoles re-
sults in the generation of long Gaussian-like tails in
the beam,whichwouldbescrapedoffinthe HEBT.
•Good Tune is Important. If a tune has a small waist
in the upstream part of the HEBT, the beam will also
acquire Gaussian-like tails. Simulations showed that
∗Work supported by US Department of Energygood tunes existed for our four-quadrupole beamline
and were stable (slight changes in magnet settings or
inputbeamdidnotleadto beamdegradation).
•Beam Size Control. In our design, increasing the
strength of the last quadrupole (Q4) increases the
beam size in both xandyby aboutthe same amount.
This is because there is a crossover in xjust down-
stream of Q4 and a (virtual) crossover just upstream
of Q4 in y. If the beam turns out to not be circular,
thiscanbeadjustedbyQ3,whichmovestheupstream
crossoverpoint.
•Emittance Growth in HEBT. Simulationsshowed that
the transverse emittances grew by about 30% in the
HEBT. However,thisdidnotaffectfinalbeamsize. At
thedownstreamendofthe HEBTandinthebeamstop,
the beam is in the zero-emittance regime (very nar-
row phase-space ellipses). Simulations with TRACE
3-D, which has no nonlinear effects, and a 3-D par-
ticle code that included nonlinear space-charge pre-
dictedalmostidenticalfinalbeamsizes.
3 OBSERVED HEBT PERFORMANCE
Near the beamstop entrance, there is a collimator with a
sizelessthan3timesthermsbeamsize. Initialrunsshowed
beam hitting the top and bottom of the the collimator, in-
dicating the beam was too large in y. This was fixed by
readjustingQ3andslightlyreducingQ4toreducethebeam
size. Aftertheseadjustments,beamlosseswerenegligible .
Thisindicatedthe HEBTwasoperatingaspredictedandthe
RFQbeamwasaboutaspredicted. Therewerenolongtails
generated in the HEBTthat were being scraped off. Thus
our somewhat risky design, having only four quadrupoles,
workedasdesigned.
4 QUADRUPOLESCANS
4.1 Procedure
Only the first two quadrupoles were used. For character-
izing the beam in y, Q1, which focuses in y, was varied
andthe beamwas observedat the wirescanner,whichwas
about2.5mdownstream. ThevalueoftheQ2gradientwas
chosen so that the beam was contained in the xdirection
forall valuesofQ1. Forcharacterizing x, Q2wasvaried.
As the quadrupole strength is increased, the beam size
at the wire scanner goes through a minimum. At the min-
imum, there is a waist at approximately the wire-scanner
position. For larger quadrupolestrengths, the waist movesupstreaminthebeamline.
4.2 Measurements
Quadrupolescans were done a numberof times for a vari-
ety of beam currents for both the xandydirections. The
minimum beam size at the wire scanner was near 2 mm,
which was almost equal to the size of the steering jitter.
Approximatelyten quadrupolesettingswere used for each
scan. Data wererecordedandanalyzedoffline.
4.3 Fittingto Data
To determinethephase-spacepropertiesofthebeamat the
exit of the RFQ, we needed a model that could predict the
beamprofileatthewirescanner,giventhebeamatthe RFQ
exit. We parameterized the RFQbeam with the Courant-
Snyder parameters α,β, andǫin the three directions. We
usedthesimulationcodes TRACE3-DandLINACasmodels
for computing rms beam sizes in our fitting. The TRACE
3-Dcode is a sigma-matrix (second moments) code that
includes only linear effects but is 3-D. The LINACcode is
a particle in cell ( PIC) code that has a nonlinear r-zspace
chargealgorithm.
Figure 1 shows the rms beam size in the ydirection as
a function of Q1 gradient. The experimental numbers are
averagesfroma set ofquadscanruns[4]. Theothercurves
are simulationsusing the TRACE3-D,LINAC, andIMPACT
codes. The IMPACTcode is a 3-D PICcode with nonlinear
space charge. The initial beam (at the RFQexit) for all
simulationsis the beam determinedby the fit to the LINAC
model[4]. (Thisiswhythereislittledifferencebetweenth e
experimental points and the LINACsimulation.) There are
significant differences among the codes in the predictions
of the the rms beam size. Table 1 shows emittances we
246810121416
78910111213rms beam size in y (mm)
Q1 (T/m)TRACE 3-D
LINAC
IMPACT
Experiment
Figure 1: Rms beam size at wire scanner as function of
quad strength. All simulations used the fit to the LINAC
modelforthe inputbeam.
obtainedwhenfittingtothe TRACE3-DandLINACmodels.
Table 1: Rmsnormalizedemittances(mm ·mrad)
ǫx ǫy
Prediction( PARMTEQM )0.245 0.244
Measured( TRACE3-Dfit)0.400 0.401
Measured( LINACfit) 0.253 0.3145 QUAD SCAN SIMULATIONS
5.1 ProfilesatWireScanner
Since only the IMPACTcode has nonlinear 3-D space
charge, we would expect that this code would be the most
accurateandshouldbe usedto fit tothe data. Bothnonlin-
ear and 3-D effects are large in the quad scans. However,
we found that the IMPACTcode (as well as LINAC) could
not predict well the beam profile at the wire scanner. Fig-
ure 2 shows the projections onto the yaxis for two points
ofthe yquadscan,correspondingtoaQ1gradientsof7.52
and 11.0 T/m. The agreement for 11 T/m, which is to the
right of the minimum of the quad scan curve, is especially
poor. We see that the experimentalcurve (solid) hasa nar-
rower peak, with more beam in the tail than the IMPACT
simulationpredicts.
00.0050.010.0150.020.0250.030.035
-40-30-20-10010203040Beam fraction per mm
y (mm)Q1=7.52 T/m
Experiment
IMPACT
00.020.040.060.080.10.120.14
-40-30-20-10010203040Beam fraction per mm
y (mm)Q1=10.98 T/m
Experiment
IMPACT
Figure 2: Profile at wire scanner for yscan with Q1=7.5
T/m(left)andQ1=11T/m(right). Solidcurveistheexper-
imental measurement and the dashed curve is the IMPACT
simulationusingthe LINAC-fit beamasinput.
Figure 3 shows the yphase space just after Q2 for two
pointsin the yquadscan. AfterQ2, space chargehaslittle
effect and the beam mostly just drifts to the end (there is
little change in the maximum value of |y′|). The graph on
the left is for a Q1 value to the left of the quad scan mini-
mum (9.5 T/m). The graph at the right showsthe situation
to the right of the minimum (10.9 T/m). The distribution
in the left graph is diverging, while the one on the right is
converging. It is this convergencethat apparently leads to
the strange tails we seen in the experimentalprofilesat the
wire scanner. Figure 4 shows similar graphsa little before
-8-4048
-6-4-20246y' (mrad)
y (mm)-16-12-8-40481216
-4 -2 0 2 4y' (mrad)
y (mm)
Figure 3: Phase space after Q2 in ydirection for yscan
with Q1=9.5T/m(left)andQ1=11T/m(right).
the wire scanner, 2.35 m downstream of the RFQ. We see
how the tails in the yprojection form for the case of the
quad scan points to the right of the minimum, which cor-
respondto largerquadgradients. While thisappearstoex-
plainthenarrow-peak-with-enhanced-tailsseeninthewir e
scans, theeffectismuchsmallerthanintheexperiment.-10010
-20 -10 0 10 20y' (mrad)
y (mm)-20-1001020
-20 -10 0 10 20y' (mrad)
y (mm)
Figure4: SameasFig. 3butata pointjustupstreamofthe
wire scanner.
We studied various effects looking to better reproduce
the profiles seen at the wire scanner, all with negative re-
sults.
5.2 CodePhysics
We studiedthe effectsof meshsizes, boundaryconditions,
particle number, and time step sizes with no significant
changein results.
We investigatedthe possibilitythat there wereerrorsas-
sociated with using normalized variables ( px) in azcode,
whichIMPACTis. Forhigh-eccentricityellipses, thiscould
be problem. However,transformingdistributionsto unnor-
malizedcoordinates,whichareappropriatetoa zcode,did
notnoticeablychangetheresults.
5.3 EffectsofInput Beam
We used for input the beam generated by the RFQsimu-
lation code PARMTEQM . We also used generated beams,
which were specified by the Courant-Snyder parameters.
Using the Courant-Snyder parameters of the PARMTEQM
beam yielded similar results. Varying these parameters in
variouswaysdidnot makethe beamlookanycloser to the
experimentallyobservedone.
We tried various distortions of the input beam such as
enhancing the core or tail and distorting the phase space
by giving each particle a kick in y′direction proportional
toy2ory3. These changes had little effect, even for very
severe distortions. Kicks proportional to y1/3were more
effective. These are more like space-charge effects in that
the distortion is larger near the origin and smaller near the
tails. In general, we found that any structure we put into
the input beam tended to disappear because of the strong
nonlinearspace-chargeforcesat the HEBTfrontend.
5.4 EffectsofQuad Errors
Multipole errors were investigate using a version of
MARYLIE with 3-D space charge. We could generate tails
that lookedlike the experimentallyobservedones, but this
tookmultipolesthat wereabout500timesaslargeaswere
measuredwhenthequadrupolesweremapped.
Quadrupole rotation studies also yielded negative re-
sults.5.5 SpaceCharge
We investigated various currents and variations in space
charge effects along the beamline, as could be generated
byneutralizationorunknowneffects.
5.6 LongitudinalMotion
We hadpracticallynoknowledgeof thebeaminthe longi-
tudinal direction except that practically all of the beam is
very near the 6.7 MeV designenergy. Since the transverse
beam seems to be reasonably predicted by the RFQsimu-
lation code, we do not expect the longitudinalphase space
to be much different from the prediction. We tried various
longitudinalphase-spacevariationsandnoneledtoprofile s
at the wire scanner that looked similar to the experimental
ones.
6 DISCUSSION
In the upstream part of the HEBTthe beam size profiles
(xrmsandyrmsasfunctionsof z)forthequadscantuneare
not much different from those of the normal HEBTtune.
The differencesoccursquite a way downstream. But here,
space charge effects are small and are unlikely to explain
thedifferencesweseeinthebeamprofilesatthewirescan-
ner. Thisisamysterythatisstill unresolved.
If we succeedin simulatingprofilesat thewire scanners
that lookmorelikethe onesseenin themeasurement,then
it will bereasonableto fit the data to the 3-D IMPACTsim-
ulations. Inthatcase,wewilluseallthewire-scannerdata ,
taking into account the detailed shape of the profile and
not just the rms value of the beam width, as we did for the
TRACE3-DandLINACfits. While we were able to use a
personal computer to run the HPFversion of IMPACTfor
most of the work described here, the fitting to the IMPACT
modelwill haveto bedoneonasupercomputer.
7 ACKNOWLEDGEMENTS
We thank Robert Ryne and Ji Qiang for providingthe IM-
PACTcodeandforhelpassociatedwithits use.
8 REFERENCES
[1] H.V. Smith, Jr. and J.D. Schneider, “Status Update on the
Low-Energy Demonstration Accelerator ( LEDA),” this con-
ference.
[2] L.M. Young, et al., “High Power Operations of LEDA,” this
conference.
[3] J.D. Gilpatrick, et al., “ LEDABeam Diagnostics Instrumen-
tation: Measurement Comparisons and Operational Expe-
rience,” submitted to the Beam Instrumentation Workshop
2000, Cambridge, MA, May 8-11, 2000.
[4] M.E. Schulze, et al., “Beam Emittance Measurements of th e
LEDA RFQ ,”this conference.
[5] W.P. Lysenko, J.D. Gilpatrick, and M.E. Schulze, “High E n-
ergy Beam Transport Beamline for LEDA,”1998 Linear Ac-
celerator Conference. |
arXiv:physics/0008187 19 Aug 2000CONCEPT OF STAGED APPROACH
FOR INTERNATIONAL FUSION MATERIALS IRRADIATION FACI LITY
M. Sugimoto, M. Kinsho, H. Takeuchi, JAERI, Tokai, Ibaraki, Japan
Abstract
The intense neutron source for development of fusio n
materials planned by international collaboration ma kes a
new step to clarify the technical issues for realiz ing the
40 MeV, 250 mA deuteron beam facility. The baseline
concept employs two identical 125 mA linac modules
whose beams are combined at the flowing lithium tar get.
Recent work for reducing the cost loading concerns the
staged deployment of the full irradiation capabilit y in
three steps. The Japanese activity about the design and
development study about IFMIF accelerator in this y ear is
presented and the schedule of next several years is
overviewed.
1 INTRODUCTION
The International Fusion Materials Irradiation Faci lity
(IFMIF) is an IEA collaboration to construct an int ense
neutron source for development of fusion materials [1].
The 250-mA, 40-MeV deuteron beam is required to
satisfy the neutron flux level (wall load equivalen t to 2
MW/m2 ~ 9x1013 neutrons/cm2/s ~ 19 dpa/y for Fe) with
enough irradiation volume (>500 cm3). As the basic
concept discussed during these five years of CDA
(Conceptual Design Activity), a set of two identica l
175 MHz, 125 mA linacs is employed to achieve the
beam current requirement [2]. After the request fro m the
Fusion Program Coordination Committee (FPCC) in
January 1999, a plan with the reduction of the faci lity
construction cost (estimated at 1996) and the proje ct
schedule with a staged approach to match to the fus ion
reactor development plan is proposed at 2000 FPCC
meeting. It consists of three stages and each stage
achieves 20%, 50% and 100% of the full irradiation
capability shown above, respectively. The prospects for
materials development are recognized though the ser ies of
research items: the selection of materials for ITER test
blanket module as a near term milestone, the acquis ition
of engineering data for reactor prototype (like DEM O),
and the evaluation of lifetime of candidate materia ls.
From the accelerator technology viewpoints, some
essential key issues need to be solved before start ing the
construction, i.e. extremely stable 155 mA deuteron
injector, 175 MHz coupled cavity cw-RFQ, precise be am
dynamics simulation to realize the beam loss contro l, etc.
The most problem should be addressed by prototyping ,
however, some prior verification about the componen t
technology is necessary to initiate it. In the next several years, we concentrate on the restricted area of key
component technologies to proceed to the next comin g
Engineering Validation Phase (EVP) as a preparation of
construction phase.
2 STAGING CONCEPT
2.1 Overview
In the staged facility design, the layout of two 4 0 MeV
deuteron linac modules becomes simple coplanar form to
be upgraded easily. The major parameters of linac m odule
are summarized in Table 1 and the layout of one
accelerator module is shown in Fig. 1.
Table 1: Principal Parameters of Accelerator System
Item Specification Description
Particle D+ H2+ for tests
No. of Modules 1 or 2 1@ 1st/2nd stage
Beam Current 50/125/250mA 1st/2nd/3rd stage
Beam Energy 32 and 40MeV Selectable
Duty 100% CW Pulse for tests
Beam Size 20cmWx5cmH Uniform 1
Energy Spread □}0.2MeV Natural spread
RF Frequency 175MHz RFQ & DTL
RF Power 9MW 1MW unit x11
Availability > 88% Scheduled op.
Maintainability Hands-on HEBT ends at
target I/F valve
/c1
RFQ DTL
Injector High Energy
Beam Transport
Li Target D beam RF System
0.1MeV
5MeV
40MeV
125mA
Figure 1: Layout of IFMIF accelerator module.
1 Narrower width may be requested at 1st and 2nd stages to keep
charge density of full current beam. The number of irradiation test cells is reduced fro m two
to one that would be possible to rearrange the sche dule of
irradiation tests, so that the High Energy Beam Tra nsport
has only one beam line. The electromagnetic pump is
redesigned to minimize the volume of the loop for h igh-
speed lithium flow used as the neutron-generating t arget.
The resulting lithium inventory becomes 9 m3 from prior
value 21 m3. The newly estimated cost indicates that the
1st phase of 50mA operation can be started by 38% of t he
total cost (~$800M) formerly obtained at CDA phase and
the integral cost of all stages can be compressed t o 60%
of the former cost [3].
The construction/operation is divided into three st ages:
(1) 50 mA operation of a full performance linac for ~5
years, (2) full power 125 mA operation of the first linac
for ~5 years, and (3) 250-mA operation with an addi tion
of the second linac for more than 20 years. Other m ajor
parameters are not greatly changed from the CDA des ign
but the cut of redundancy of the reduced cost desig n
might influence to the overall availability, especi ally at
the initial stage operation.
2.2 Injector
The ion source for 155mA deuteron beam with require d
quality is almost available at the present technolo gy. Only
the verification of long-term stability and long li fetime
should be addressed, and these tasks will be perfor med in
a couple of year. As the actual operation starts fr om
50mA in the staged approach, the lifetime issue is also
relaxed.
On the other hand, LEBT is still problematic becaus e of
the less controllability of the space charge neutra lization.
The pulsing method to apply at the start up procedu re is
another unresolved issue. The use of H2+ beam at th e
prototype or commissioning phase brings the extra t ask to
calibrate and correlate the measurements with D bea m
case.
2.3 RFQ
As shown in Fig. 1, the output energy of IFMIF RFQ is
5MeV (CDA design employed 8MeV output) and the
final decision of the transition energy is made jus t before
the construction probably. In any case, the length of RFQ
exceeds 8 m and the coupled cavity technique develo ped
by LANL [4] is needed to maintain the field uniform ity
along the structure. The beam loss in RFQ usually o ccurs
at initial bunching section mainly and along the
acceleration section in a small part, as shown in F ig. 2.
The loss at low energy part will generate neutron d ue to
D(d,n) reaction for the self-impinged deuteron at t he vane
surface. It may helpful to be coated by high-Z mate rial at
vane tip and to use a method of surface cleaning to
remove deuteron gas periodically. For the loss at h igher
energy part high-Z material coating may also useful but
the better solution is stop of RFQ with a small ape rture
size. This might push the lowering of transition en ergy because RFQ with large bore is inefficient accelera ting
structure. Again the final decision requires the ac quisition
of many precise calculations and accurate measureme nts.
Figure 2: Typical result of particle tracking in IF MIF RFQ
using PARMTEQ (top:x, middle: φ-φs, bottom:W-Ws).
2.4 DTL
In the baseline design, Alvarez DTL is employed as the
main accelerator, with single stem and post coupler . CDA
design uses 3cm bore size for all drift tubes so th at
minimum incident energy is around 8 MeV if the
conventional electromagnetic quadrupole using FoDo
structure. The reduced cost design prefers the lowe r
transition energy so that either the focusing schem e
change like FoFoDoDo or bore radius change is neces sary.
Figure 3 indicates the PARMILA run of the former ca se.
The resultant emittance growth is larger than that for
FoDo case and we need to seek the best compromise o n
bore size. The gradient ramping at the beginning of DTL
is another issue to be addressed at prototyping and
extensive electromagnetic calculation is scheduled.
0.01.02.03.04.05.06.07.08.0
0 20 40 60 80 100 120 140
Cell NumberEmittance (100%, pi-cm-mrad)
Figure 3:Emittance profile of IFMIF DTL by PARMILA.
2.5 RF System
The most of accelerator tanks is configured as mult i-
drive form using 2 independent 1MW RF amplifier uni ts
shown in Fig. 4. At the first stage 50 mA operation is
achieved by removing one of two units and it is ins talled at the later stage. The circulator at the final out put is not
used in the current design because it might be erro r-prone
component from the experience at ICRF heating. The
serious analysis of such RF system control and resp onse is
necessary.
/G1Freq. /c1
Source & /c1
□□/A Cont. /c1Solid
Amp. /c1TH561 /c1
Tetrode TH628 /c1
Diacrode /c1100W /c1
Power /c1
Supply 1MW /c1TH781 /c1
Tetrode /c1
Power /c1
Supply Power /c1
Supply State
Figure 4: Layout of IFMIF RF power source unit.
2.6 HEBT
The design of new HEBT line is relaxed due to its
simplicity of the beam transport line, however, it still
requires the beam redistribution at target (20cm wi dth and
5cm height with uniform distribution except ramping at
both vertical ends). The resulting line consists of an
achromatic parallel translation with two dipoles an d static
multipole magnet and imager qudrupoles as redistrib ution
system and the last dipole bend, after that there i s a 14m
long drift space only till Li target.
The beam calibration dump placed at the middle of t wo
Li target stations in CDA design is disappeared now , and
alternative beam stop is desired for start up tunin g
purpose, which accepts several 100 kW power. The be st
place is straight end of the last dipole and it sho uld be
checked against the neutron back streaming from bea m
dump.
2.7 Superconducting Linac
From the beginning of the IFMIF design study, the
superconducting linac (SCL) was considered as the
promising alternative to DTL and the progress of ge neral
technology has been tracked. For the possible use i n the
future upgrade, the compatibility with DTL and SCL are
always concerned. Fig. 5 shows the one of the low β
structure for IFMIF purpose.
Figure 5: IFMIF SCAL quarter-wave structure [5].
3 DEVELOPMENT
The items covered in KEP include the long lifetime
injector of accelerator system, the lithium flow st ability
test of target system, the temperature control of s pecimens
of test cell facility, etc. The results of these te sts
contribute to realize the detailed design of the eq uipment
for the next coming EVP to achieve the stable syste m
operation. The items, injector test, RFQ cold mode l, DT
packaging test are proposed as KEP tasks to be carr ied
out in Japan with the possible international collab oration
and the cooperative sharing between JAERI and the
Japanese universities groups [6].
4 ADVANCED CONCEPT
The new scheme to realize the intense neutron sourc e is
a continuing task and a variation using Li flow wit hout
backwall us given in Fig.6, which is mixed with a p artial
energy recovery of deuteron beam to save electrical power.
2 beam ports for cwoperation
or Time sharing pulse modeSuperconducting
LinacInjector
Beamdump
NeutronsEd inj ~ 5-8MeV Ed exit ~ 40MeV
Ed ret ~ 30MeV
Li Taget
(Backwall-less)2 beam ports for cwoperation
or Time sharing pulse modeSuperconducting
LinacInjector
Beamdump
NeutronsEd inj ~ 5-8MeV Ed exit ~ 40MeV
Ed ret ~ 30MeV
Li Taget
(Backwall-less)
Figure 6:D-Li neutron source without backwall with
partial deuteron beam energy recovery
5 SUMMARY
The materials development is one of the most import ant
issues related to fusion programs, and it results i n a new
step to verify the key element technology, which ne eds to
be carried out by using all possible international and
domestic resources.
REFERENCES
[1] T.Kondo, H.Ohno, R.A.Jameson and J.A.Hassberger ,
Fusion Eng. Design, 22, 117 (1993) .
[2] IFMIF–CDA Team, IFMIF □^ International Fusion
Materials Irradiation Facility Conceptual Design
Activity Final Report, ENEA Frascati Report,
RT/ERG/FUS/96/11 (1996); IFMIF Conceptual
Design Evaluation Report, Ed. A.Moeslang, FZKA
6199, Jan. 1999.
[3] Fusion Neutronics Laboratory, Ed. M. Ida, IFMIF □^
International Fusion Materials Irradiation Facility
Conceptual Design Activity – Reduced Cost Report –
A Supplement to the CDA by the IFMIF Team,
JAERI-Tech 2000-014, Feb. 2000.
[4] L.M.Young, Proc. 1994 Int. Linac Conf., Tsukuba ,
1994, p.178.
[5] Y.Tanabe, et al.,Fusion Eng. Design, 36, 179 (1 997).
[6] Fusion Neutronics Laboratory, Ed. M. Ida, IFMIF □^
International Fusion Materials Irradiation Facility –
Key Element Technology Phase Task Description,
JAERI-Tech 2000-052, Aug. 2000. |
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.INDUCTIVE-ADDER KICKER MODULATOR FOR DARHT-2
E. G. Cook, B.S. Lee, S.A. Hawkins, F.V. Allen, B.C. Hickman, H.C. Kirbie
Lawrence Livermore National Laboratory
C.A. Brooksby – Bechtel Nevada
Abstract
An all solid-state kicker modulator for the Dual-Axis
Radiographic Hydrodynamic Test facility (DARHT-2) has
been designed and tested. This kicker modulator uses
multiple solid-state modules stacked in an inductive-adder
configuration where the energy is switched into each
section of the adder by a parallel array of MOSFETs. The
modulator features very fast rise and fall times, pulse
width agility and a high pulse-repetition rate in burst
mode. The modulator can drive a 50 Ω cable with voltages
up to 20 kV and can be easily configured for either
positive or negative polarity. The presentation will
include test data collected from both the ETA II accelerator
kicker and resistive dummy loads.
1 BACKGROUND
The DARHT-2 accelerator facility is designed to
generate 1 kA electron beam pulses of 2µs duration. The
LLNL designed fast kicker, based on cylindrical
electromagnetic stripline structures, cleaves four short
pulses out of this long pulse. The requirements for the
modulator that drives this kicker are listed in Table 1.
A ±10kV modulator design based on planar triodes was
originally used for this application [1]. While the hard-
tube performance was very good, concerns regarding future
availability and reliability of these devices led to
consideration of a solid-state replacement. Personnel
within this program had developed considerable expertise
with parallel and series arrays of power MOSFETs during
the successful design and testing of the Advanced
Radiograph Machine (ARM) modulator, a high power
pulser developed to show feasibility of solid-state
modulators for driving induction accelerators. While ARM
was designed for higher voltages and currents than required
by the kicker, its requirements for rise and fall times werealso significantly slower. After consideration of various
circuit topologies, the adder configuration used by ARM
was selected as the baseline for the kicker modulator;
MOSFETs were selected as the switching device.
The key parameter in the performance requirement is
the minimum pulsewidth of 16ns. As a class of devices,
1kV rated MOSFETs have demonstrated the required rise
and falltime; however, the critical information needed was
to determine whether MOSFETs are capable of switching
significant current while simultaneously achieving the
required minimum pulsewidth. Device datasheets do not
necessarily provide all the information required to make a
definitive decision: testing is essential.
2 DEVICE EVALUATION AND
SELECTION
In order to use a reasonable number of devices, only
MOSFETs capable of operation at voltages of ≥ 800
volts were evaluated. The evaluation circuit is a series
circuit consisting of a low inductance DC capacitor bank,
a resistive load, and the MOSFET. Devices were evaluated
on the basis of switching speed at various peak currents,
waveshapes, minimum output pulsewidth, and ease of
triggering. Extensive testing of many devices from several
vendors produced several that were acceptable and led to
the selection of the APT1001RBVR. During testing, this
device exhibited the cleanest rise and fall waveshapes and
met the pulsewidth, risetime, and falltime requirements.
We were also able to measure a peak current of ~35
amperes before seeing an unacceptable drain-source
voltage drop (we arbitrarily chose a voltage drop of < 20
volts during conduction of the current pulse as our
acceptance criteria). The APT1001RBVR has a 1000V
maximum drain to source rating, an average current rating
of 10A, and a pulsed current rating of 40A.
During the early testing of MOSFETs, it became
apparent that the MOSFET gate drive circuit was also an
essential element in achieving the best performance from
the individual devices. The coupling between the drive
circuit and the MOSFET had to have very low loop
inductance as the peak drive current required to achieve fast
switching performance was on the order of tens of
amperes. Even the devices within the gate drive circuit had
to be very fast and have short turn-on and turn-off delay
times. An early decision was that each MOSFET would
require its own dedicated gate drive. A simplifiedTable 1. Performance Requirements
Parameter Requirement
Output Voltage ±20kV into 50 Ω
Voltage Rise/Falltime ≤10ns (10-90%)
Flattop Pulsewidth 16ns–200ns (continuously
adjustable)
Burst Rate 4 pulses @1.6MHz(~600ns
between leading edges)schematic of the drive circuit is shown in Fig. 1. The
input device of the gate drive has a level-shifting TTL
input circuit internally coupled to a MOSFET totem pole
output. This circuit drives a fast, high current MOSFET
(peak current ±20 amperes) totem pole device which
drives the gate of the power MOSFET (capacitive load) to
turn it on and sinks current from the MOSFET to turn it
off. The gate drive circuit components require a dc voltage
of ~ 15 volts.
3 CIRCUIT TOPOLOGY
In the adder configuration shown in Fig. 2, the
secondary windings of a number of 1:1 pulse transformers
are connected in series. Typically for fast pulse
applications, both the primary and secondary winding
consists of a single turn to keep the leakage inductance
small. In this configuration, the output voltage on the
secondary winding is the sum of all the voltages
appearing on the primary windings. The source impedance
of the MOSFET array and the DC capacitor bank must be
very low (<<1 Ω) to be able to provide the total secondary
current, any additional current loads in the primary circuit,
plus the magnetization current for the transformer core.The layout for this circuit is important as it is necessary
to mimimize total loop inductance – it doesn’t take much
inductance to affect performance when the switched di/dt is
greater than 40kA/µs.
The MOSFETs shown in Fig. 2 have their source lead
connected to ground. This is chosen so that all the gate
drive circuits are also ground referenced, thereby
eliminating the need for floating and isolated power
supplies. The pulse power ground and the drive circuit
ground have a common point at the MOSFET source but
otherwise do not share common current paths thereby
reducing switching transients being coupled into the low
level gate drive circuits.
Overvoltage transients can be generated by energy
stored in the stray loop inductance, energy stored in the
transformer primary, and/or voltage coupled into the
primary circuit from the secondary (usually due to trigger
timing differences in stages of the adder). Transient
protection for the MOSFETs is provided by the series
combination of snubber capacitor and diode tightly
coupled to the MOSFET. The capacitor is initially
charged to the same voltage as the DC capacitor bank.
When the MOSFET is turning on, the diode prevents the
snubber capacitor from discharging through the
MOSFET. As the MOSFET turns off, transient voltages
that may exceed the voltage on the snubber capacitor turns
the diode on so that the capacitor can absorb the energy.
The parallel resistor allows the excess capacitor voltage to
discharge into the DC capacitor between bursts. Good
performance of the overvoltage circuit requires a low
inductance capacitor and a diode with a low forward
recovery voltage.
Not shown in the simplified circuit layout is the reset
circuit for the magnetic cores. The cores require reset so
that they do not saturate during a voltage pulse. As this
circuit operates in a well defined pulse format, it is not
necessary to actively reset the core between pulses.
Consequently, a DC reset circuit is used and is
implemented by connecting a DC power supply through a
large isolation inductor to the ungrounded end of the
secondary winding of the adder stack. In the interval
between bursts, the reset current will reset and bias the
magnetic cores. This approach is simple to incorporate
and requires few additional components but has the
disadvantage of requiring more magnetic core material in
the transformers.
4 COMPONENT LAYOUT
The overall circuit is chosen to have 24 MOSFETs per
primary circuit (12 per board). This gives a comfortable
margin in peak current capability that allows for extra
loading in the primary circuit, a reasonable magnetization
current, and total load current. The adder transformer is
designed to look very much like an accelerator cell of a
linear induction accelerator with the primary windingFig. 1 Simplified Schematic of MOSFET Drive Circuit
Drive
CircuitMOSFET
ArrayDC Capacitor+-
0
-V chgV chg
Vout
Transformer
Secondary+-
+-( ~ 4* V chg) Vpk0
VpkVout
TransformersTransformer
Core
Transformer
PrimaryRL
Drive
Circuit
Drive
Circuit
Drive
Circuit0
-V chg
0
-V chg
0
-V chgTransient
Voltage
Protection
Circuit
Fig. 2 Simplified Schematic of Adder CircuitTTL
Input
Level Shifting Logic
and DriverHigh Current
Totem-pole Driverto Power
MOSFET
GateVcc+totally enclosing the magnetic core (an annealed and
Namlite insulated Metlgas® 2605 S1A tapewound toroid
purchased from National/Arnold). A photograph of a
MOSFET carrier board connected to a transformer
assembly is shown in Fig. 4. The gate drive circuit boards
receive their trigger pulses from a single trigger circuit
which is connected to the pulse generator by either optical
fiber or coaxial cables. A complete adder assembly is
stack of transformer assemblies bolted together as shown
in Fig. 5. The secondary winding is usually a metal rod
that is positioned on the axial centerline of the adder
stack. The rod may be grounded at either end of the adder
stack to generate an output voltage of either polarity.5 TEST RESULTS
The modulator is undergoing extensive testing into
both resistive loads and into the kicker structure used on
the ETA II accelerator at LLNL. The modulator has been
operated at variable pulsewidths and at burst frequencies
exceeding 15 MHz. Fig. 6 is an oscillograph depicting
operation on ETA II (with ~500A electron beam current)
at ~18kV into 50 Ω (Ch1 is the drain voltage on a single
MOSFET and Ch4 is the output current at 100A/div).
The four pulse burst in Fig. 7 demonstrates the
pulsewidth agility of the modulator at variable burst
frequency at an output voltage of ~10kV also into 50 Ω.
6 CONCLUSIONS
A fast kicker modulator based on MOSFET switched
adder technology has been designed and tested. MOSFET
arrays in an adder configuration have demonstrated the
ability to generate short duration and very fast risetime
and falltime high-voltage pulses.
REFERENCES
[1] W.J. DeHope, et al, "Recent Advances in Kicker Pulser
Technology for Linear Induction Accelerators", 12th IEEE
Intl. Pulsed Power Conf., Monterey, CA, June 27-30, 1999
[2] Yong-Ho Chung, Craig P Burkhart, et al, "All Solid-state
Switched Pulser for Air Pollution Control System",12th IEEE
Intl. Pulsed Power Conf., Monterey, CA, June 27-30, 1999
Fig. 5 Complete Kicker Modulator Assembly
Fig. 4 Transformer Assy. with a MOSFET Carrier Bd.
Fig. 6 Operation of Kicker Pulser on ETA II
Fig. 7 Four Pulse Burst at 10 kV into 50 Ω Load |
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.REVIEW OF SOLID-STATE MODULATORS
E. G. Cook, Lawrence Livermore National Laboratory, USA
Abstract
Solid-state modulators for pulsed power applications
have been a goal since the first fast high-power
semiconductor devices became available. Recent
improvements in both the speed and peak power
capabilities of semiconductor devices developed for the
power conditioning and traction industries have led to a
new generation of solid-state switched high power
modulators with performance rivaling that of hard tube
modulators and thyratron switched line-type modulators.
These new solid-state devices offer the promise of higher
efficiency and longer lifetimes at a time when availability
of standard technologies is becoming questionable.
A brief discussion of circuit topologies and solid-state
devices is followed by examples of modulators currently
in use or in test. This presentation is intended to give an
overview of the current capabilities of solid-state
modulators in various applications.
1 BACKGROUND
Many of the high-voltage power conditioning
requirements of the accelerator community have been
satisfied by the use of conventional thyratron, ignitron,
sparkgap or, when more waveform control is required,
hardtube switches. Modulators using these switching
devices have limitations with regard to various
combinations of repetition rate, lifetime, efficiency, pulse
width agility, average power, cost, and sometimes switch
availability.
As accelerator requirements become more demanding,
particularly with regard to average power, lifetime,
pulsewidth agility, and repetition rate, some of these
conventional switching devices are inadequate. However
use of solid-state devices in these applications has been
held back by device limitations usually in either voltage
rating, peak switching power, or switching speed.
2 SOLID-STATE DEVICES
The most commonly used fast high-power
semiconductor device is the inverter grade thyristor which
is available with voltage ratings > 1kV at kA average
currents. These thyristors are closing switches that require
a current reversal through the device to commutate off,
and most high-voltage pulsed-power applications have
limited the use of thyristors as replacements for otherclosing switches. The semiconductor industry’s continued
development of devices for traction applications and high
frequency switching power supplies have created entire
families of devices that have a unique combination of
switching capabilities. For the first time we are seeing
production quantities of devices that may be considered to
be close to "ideal switches". These ideal switches are
devices that have a fast turn-off capability as well as fast
turn-on characteristics; devices that have minimal trigger
power requirements and are capable of efficiently
switching large amounts of energy in very short periods
of time and at high repetition rates if so required.
There are two devices that come close to meeting these
criteria for ideal switches, Metal Oxide Semiconductor
Field Effect Transistors (MOSFETs) and Insulated Gate
Bipolar Transistors (IGBTs). These devices and
applications utilizing these devices are the focus of this
paper. As shown in Table 1, both MOSFETs and IGBTs
are devices that can switch large amounts of power with
modest levels of trigger power. MOSFETs have
substantially faster switching speeds while IGBTs
generally are more efficient, handle more power, and are
capable of being manufactured at higher voltage ratings. A
very brief explanation of how these devices function
follows this section – more detailed information is readily
available from vendors and manufacturers.
Table1. MOSFET and IGBT Capabilities
Parameter MOSFET IGBT
Max. Peak Operating Voltage (V) 1200 3300
Peak Pulsed Current Rating (A) 100 3000
Derated Peak Power (kW/Device)* >80 >7000
Switching Speed – ON & OFF (ns) < 20 < 200
Gate Controlled Pulsewidth 20ns-DC 600ns-DC
Control Power (µJ/Pulse) < 5 < 30
Device Cost ($/kW switched) 0.30 0.15
*MOSFET – Voltage Derating ~ 80% of Max. Vpeak
*IGBT-Voltage Derating ~60% of Max. Vpeak
2.1 MOSFETs
As seen in the simplified schematic in Fig. 1,
MOSFETs are three terminal devices with the terminals
labeled drain, source and gate. MOSFETs are enhancement
mode devices that rely upon majority carriers for their
operation. Electrically the gate to source connection lookslike a capacitor and with zero gate-source voltage the
channel (the region between drain and source) is very
resistive. As the gate-source voltage is increased, the
electric field pulls electrons into the channel and increases
the flow of current, i.e. ,the flow of drain-source current is
enhanced by the gate-source voltage. Once the gate-source
capacitance is charged, no additional energy is required to
keep the device on.
For pulsed power applications where the goal is to turn
the device on very quickly, a fast, large gate current (10’s
of amperes) is required during turn-on but little power is
needed thereafter. Likewise, during turn-off, a large current
must be pulled out of the gate-source capacitance. The
gate drive circuit must be capable of sourcing and sinking
these currents at the required repetition rate.
2.2 IGBTs
As depicted in Fig. 2, an IGBT is also a three-terminal
device that combines the high input impedance and gate
characteristics of a MOSFET and the low saturation
voltage of a bipolar transistor. As the MOSFET is turned
on, base current flows in the pnp bipolar transistor,
injects carriers into the transistor and turns on the device.
While gating off the MOSFET initiates the turn-off
process for the transistor, the time required for fast turn-
off of the transistor also depends on other factors such as
carrier recombination time. Typically both the turn-on and
turn-off times for an IGBT are slower than those of a
MOSFET, but the peak current density in an IGBT is
approximately five times higher than that of a MOSFET
having the same die area.
3 SWITCH CIRCUIT TOPOLOGY
Single solid-state devices generally don't have the peak
voltage rating required for most accelerator applications
and, consequently, many devices are usually required for
their use in fast high-voltage circuits. Two circuit
topologies currently using MOSFETs and IGBTs to
achieve these high voltage levels offer significant
advantages over other circuit topologies including PFNs,
Blumlein lines, and, high step-up ratio transformers. The
first circuit approach is to connect as many switching
devices in series as is needed to meet the application's
requirements. The second approach uses what is
commonly referred to as an induction adder where the
switching devices drive the primary winding of multiple
transformers and the secondary windings of each of the
transformers are connected in series to generate therequired high voltage. In general, regardless of the switch
circuit topology, the total cost of the solid-state devices
required to switch a given peak power is determined by the
peak power capability of each switch - it matters not
whether the devices are arranged in series, parallel, or a
combination of both.
When MOSFETs or IGBTs are used, the capabilities of
a high-voltage pulsed circuit are greatly enhanced. Since
these devices can be gated off as well as gated on, the
circuit now has the capability of variable pulsewidth even
on a pulse-to-pulse basis, and the circuits may also be
operated at high repetition rates. Within the limits of their
current rating (which can be increased by paralleling
devices), these switches give the circuit topologies a low
source impedance, thereby allowing the load to vary over
a substantial impedance range. As with all solid-state
devices, the expected lifetimes of properly designed
circuits are very long.
3.1 Series Switch Topology
A common circuit topology for series stack approach is
shown in Fig. 3. In this topology a high voltage power
supply charges a DC capacitor bank. The series stack of
solid-state devices is connected between the capacitor bank
and the load. Gating the stack on and off applies the full
bank voltage across the load with the pulsewidth and
repetition rate being controlled by the gate trigger pulse.
The rise and fall times of the load voltage are determined
by the switching characteristics of the specific solid-state
devices used in the stack. Implementation of series stack
approach requires very careful attention to proper DC and
transient voltage grading of the switches in the series
stack to force a uniform distribution of voltage across all
the series elements under all conditions. All devices must
be triggered simultaneously - isolated trigger signals and
isolated power sources are usually required. Careful
attention to stray capacitance is very important.
Controls for the series stack are also critical, as the
control circuits must sense load conditions so faults can
be quickly detected and the stack gated off. Stacks
assembled with IGBTs can normally sustain short circuit
conditions for a short period of time (usually specified to
be ~10µs for high power devices) which gives more than
adequate time to turn the stack off.
An important operational advantage of the series stack
approach is the ability to obtain any desired pulsewidth
from the minimum pulsewidth as determined by the
capabilities of the switching devices out to and includingGateCollector
EmitterIGBT
Fig. 2 Simplified Schematic for an IGBTGateDrain
Source
Fig.1 MOSFET Device Symbol (N-type shown)a DC output. The series circuit topology can yield lower
hardware costs but this is partially offset by the additional
costs for the voltage grading components and costs
associated with achieving the appropriate isolation and/or
clearance voltages for the control system, gate circuits,
enclosures, etc.
3.2 Adder Topology
In the adder configuration shown in Fig. 4, the
secondary windings of a number of 1:1 pulse transformers
are connected in series. Typically, for fast pulse
applications, both the primary and secondary winding
consists of a single turn to keep the leakage inductance
small. In this configuration, the voltage on the secondary
winding is the sum of all the voltages appearing on the
primary windings. An essential criteria is that each
primary drive circuit must be able to provide the total
secondary current, any additional current loads in the
primary circuit, plus the magnetization current for the
transformer. This drive current criteria is easily met with
the circuit shown in Fig. 4 - the source impedance of a
low inductance DC capacitor bank switched by high
current IGBTs or a parallel array of MOSFETs or smaller
IGBTs is very low (<<1 Ω). The physical layout for this
circuit is important as it is necessary to maintain a small
total loop inductance – it doesn’t require much inductance
to affect performance when the switched di/dt is measured
in kA/µs.
In this layout, the solid-state devices are usually ground
referenced to take advantage of standard trigger circuits and
reduce coupled noise by taking advantage of ground
planes. The need for floating and isolated power supplies
is also eliminated. The pulse power ground and the drive
circuit ground have a common point at the switch source
lead but otherwise do not share common current paths
thereby reducing switching transients being coupled into
the low level gate drive circuits. The lower switch
voltages and the corresponding compact conductor loops,
i.e., low inductance, enables very fast switching times on
the order of tens of nanoseconds.
Voltage grading for individual devices is not a major
concern. The transformer provides the isolation between
the primary and secondary circuit and if the transformer is
designed as a coaxial structure, the high voltage isconfined to within the structure. It may help to think of
an adder as an induction accelerator with the beam replaced
with a conductor.
An adder circuit has a definite maximum pulse width
limitation as determined by the available volt-seconds of
the transformer magnetic cores. In comparison with the
series stack topology, the adder has the additional expense
of the transformer mechanical structure including the
magnetic cores, and a circuit to reset the magnetic cores
between pulses. The gate controls, being ground
referenced are simpler and less expensive.
4 MOSFETS AND IGBTS IN
ACCELERATOR APPLICATIONS
Including commercial ventures and government funded
projects, MOSFETs and IGBTs are being used in several
applications that impact accelerator technology. A few
examples are discussed.
4.1 Next Linear Collider (NLC) Klystron
Modulator
For the past several years, researchers at SLAC have
been developing a solid-state modulator based on an IGBT
switched adder topology. This aggressive engineering
project has made significant progress in demonstrating the
capabilities of high-power solid-state modulators. The
performance requirements are listed in Table 2.
The IGBTs currently used in the NLC modulator are
manufactured by EUPEC and are rated at 3.3kV peak and
800A average current. They have been successfully tested
and operated at 3kA peak current at 2.2kV. EUPEC and
other manufacturers have devices with higher voltage
ratings (>4.5 kV) that are also being evaluated.#1 #2 #nGate Drive Circuits and Controls
DC
Power
SupplyStorage
CapacitorLoad
Impedance
Fig. 3 Typical Series Switch Topology
Drive
CircuitSwitch
ArrayCapacitor+-
0 v
-V chgV chg
Vout
Transformer
Secondary+-
0 v
-V chg
0 v
-V chg+-( ~ 4* V chg) Vpk0
VVout
TransformersTransformer
Core
Transformer
Primary0 v
-V chgRL
Drive
Circuit
Drive
Circuit
Drive
Circuit
Fig.4 Simplified Schematic of Adder CircuitThe NLC modulator circuit has demonstrated combined
risetime and falltime that meet requirements. A 10-cell
prototype capable of generating a 22 kV pulse at 6 kA has
been operated as a PFN replacement (~1/10 the volume of
the thyrtron switched PFN) to drive a 5045 klystron in
the SLAC linac. Fig. 5 is a photo of the 10-cell
prototype.
4.2 Diversified Technologies Inc. (DTI)
DTI manufacturers and markets a broad range of series-
switched IGBT modulators that cover a voltage range of
up to 150 kV and peak power of 70 megawatts. They
have demonstrated switching times of <100ns and pulse
repetition rates from DC to 400 kHz. Their hardware has
applications in accelerator systems including klystron
modulators, ion sources, kicker modulator, and crowbar
replacements. These solid-state modulators are specifically
designed to compete with and replace vacuum tube based
systems. A photograph of one of their PowerMod™
systems is shown in Fig. 6.
4.3 Advanced Radiographic Machine (ARM) and
Fast Kicker Development at LLNL
The ARM II modulator was one of the first high-power
applications of power MOSFETs used in an adderconfiguration. Its specifications are listed in Table 3. A
photo of a single adder module is shown in Fig. 7.
Fast kicker pulser development at LLNL has been based
on the previous ARM modulator development. As listed
in Table 4, the major differences in requirements are
related to the faster risetime, falltime, and minimum
pulsewidth. To control the stray inductance, each parallel
array of MOSFETs drives a single pulse transformer. A
photo of the kicker assembly is shown in Fig. 8.
Fig. 5 IGBT Switched Induction Adder - 10 Cell
PrototypeTable 2. NLC Klystron Modulator Requirements
Number of NLC Klystrons 8 each
Operating Pulsed Voltage 500kV
Operating Pulsed Current 2000 amperes
Repetition Rate 120 Hz.
Risetime/Falltime <200ns 10-90%
Flattop Pulse Duration 3.0µs
Energy Efficiency >75%
Table 3. ARM II Modulator Specifications
Design Voltage 45 kV (15kV/Adder Module)
Maximum Current 4.8-6 kA
Pulsewidth 200ns-1.5µs
Maximum Burst PRF 1 Mhz
Number of MOSFETs 4032
Fig. 6 DTI's 125kV, 400A Solid-State Switch
Fig.7 ARM II Module
Table 4. Fast Kicker Specifications
Output Voltage 20 kV into 50 Ω
Voltage Risetime/Falltime <10 ns 10-90%
Pulsewidth 16ns-200ns variable within
burst
Burst Frequency >1.6 MHz - 4 pulses4.4 First Point Scientific
First Point Scientific has developed a variety of
MOSFET switched adder systems. Based on their
Miniature Induction Adder (MIA - see Fig. 9), they have
demonstrated high repetition rate, high-voltage systems
for air pollution control. First Point Scientific has also
developed a prototype for an economical, fast, high
repetition rate modulator featuring pulse width agility and
waveform control for a small recirculator to be used in ion
accelerator experiments.5 CONCLUSIONS
Faster and higher power solid-state devices are
constantly being introduced that offer significant
advantages for pulse power applications. These devices are
being incorporated into a significant number of modulator
designs and used in various projects for specific accelerator
applications. As the performance of these devices
continues to improve, they will replace more of the
conventional switch technologies.
REFERENCES
[1]H. Kirbie, et al, "MHz Repetition Rate Solid-State
Driver for High Current Induction Accelerators",
1999 Part. Accel. Conf ., New York City, Mar.29-
April 2, 1999, http :ftp.pac99.bnl.gov/Papers/
[2] R. Cassel, "Solid State Induction Modulator
Replacement for the Conventional SLAC 5045
Klystrons Modulator", LINAC 2000 - XX
International Linac Conf., Monterey, CA, August
21-25, 2000
[3] W. J. DeHope, et al, "Recent Advances in Kicker
Pulser Technology for Linear Induction Accelerators",
12th IEEE Intl. Pulsed Power Conf ., Monterey, CA,
June 27-30, 1999
[4] Yong-Ho Chung, Craig P. Burkhart, et al, "All Solid-
state Switched Pulser for Air Pollution Control
System", 12th IEEE Intl. Pulsed Power Conf.,
Monterey, CA, June 27-30, 1999
[5] M. Gaudreau, et al, "Solid State Modulators for
Klystron/Gyrotron Conditioning, Testing, and
Operation", 12th IEEE Intl. Pulsed Power Conf.,
Monterey, CA, June 27-30, 1999
[6] E. Cook, et al, "Inductive Adder Kicker Modulator for
DARHT-2", LINAC 2000 - XX International Linac
Conf., Monterey, CA, August 21-25, 2000
Fig. 9 First Point Scientific - Miniature Induction
Adder
Fig. 8 Photo of Complete Fast Kicker Assembly |
arXiv:physics/0008190v1 [physics.acc-ph] 20 Aug 2000ANALYSIS AND SIMULATIONOF
THE ENHANCEMENT OF THECSR EFFECTS
R. Li, JeffersonLab, 12000JeffersonAve.,NewportNews,VA 23606,USA
Abstract
Recent measurements of the coherent synchrotron radia-
tion (CSR) effects indicated that the observed beam emit-
tance growth and energy modulation are often bigger than
previouspredictionsbasedonGaussianlongitudinalcharg e
distributions. In this paper, by performing a model study,
we show both analytically and numerically that when the
longitudinal bunch charge distribution involves concentr a-
tion of charges in a small fraction of the bunch length,
enhancementoftheCSRself-interactionbeyondtheGaus-
sian prediction may occur. The level of this enhancement
is sensitivetothelevelofthelocalchargeconcentration.
1 INTRODUCTION
Whenashortbunchwithhighchargeistransportedthrough
amagneticbendingsystem,orbit-curvature-inducedbunch
self-interaction via CSR and space charge can potentially
induce energy modulation in the bunch and cause emit-
tancegrowth. Eventhoughtheearlieranalyticalresultsfo r
CSR self-interaction [1, 2] based on the rigid-line-charge
model can be applied for general longitudinal charge dis-
tributions, since the analytical results for a Gaussian bea m
are explicitly given, one usually applies the Gaussian re-
sultsto predicttheCSR effectsusingthemeasuredorsim-
ulated rms bunchlength. Similarly, a self-consistentsimu -
lation [3] was developedealier to study the CSR effect on
bunch dynamics for general bunch distributions; however,
the simulation is usually carried out using an assumed ini-
tial Gaussian longitudinalphasespacedistribution. Rece nt
CSRexperiments[4,5]indicatedthatthemeasuredenergy
spread and emittance growth are sometimes bigger than
previous Gaussian predictions, especially when a bunch is
fullycompressedorover-compressed.Inthispaper,weex-
plorethepossibleenhancementoftheCSRself-interaction
force due to extra longitudinalconcentration of chargesas
opposed to a Gaussian distribution. This study reveals a
generalfeatureoftheCSRself-interaction: wheneverther e
is longitudinal charge concentration in a small fraction of
a bunchlength,enhancementoftheCSR effectbeyondthe
Gaussian prediction can occur; moreover, the level of this
enhancement is sensitive to the level of the local charge
concentration within a bunch. This sensitivity should be
givenseriousconsidertationin designsoffuturemachines .
2 BUNCH COMPRESSIONOPTICS
Whenanelectronbunchisfullycompressedbyamagnetic
chicane, the final bunch length and the inner structure of
the final longitudinal phase space are determinedby many
details ofthe machinedesign. Inthis paper,we investigate
only the RF curvature effect, which serves as a model toillustrate the possible sensitivity of the CSR interaction to
the longitudinalchargedistribution.
In order to study the CSR self-interaction for a com-
pressed bunch, let us first find the longitudinal charge dis-
tribution for our model bunch when it is fully compressed
byachicane. Consideranelectronbunchwith Ntotalelec-
trons. Thelongitudinalchargedensityofthe bunchat time
tisρ(s, t) =Nen(s, t)(/integraltextn(s, t)ds= 1), where sis the
distancefromthereferenceelectron,and n(s, t)isthelon-
gitudinal density distribution of the bunch. At t=t0, let
the bunch be aligned on the design orbit at the entrance of
abunchcompressionchicane,withaGaussianlongitudinal
densitydistributionandrmsbunchlength σs0
n(s0, t0) =n0(µ) =1√
2πσs0e−µ2/2σ2
s0.(1)
Here welet eachelectronbeidentifiedbythe parameter µ,
whichisitsinitial longitudinalposition
s(µ, t0) =s0=µ(s >0forbunchhead ).(2)
In order to compress the bunch using the chicane, a linear
energy correlation was imposed on the bunch by an up-
stream RF cavity, along with a slight second-order energy
correlation due to the curvatureof the RF wave form. The
relativeenergydeviationfromthedesignenergyisthen
δ(µ, t0) =−δ1µ
σs0−δ2/parenleftbiggµ
σs0/parenrightbigg2
(δ1, δ2>0, δ2≪δ1),
(3)
where we assume no uncorrelated energy spread. When
thebeampropagatestotheendofthechicaneat t=tf,the
final longitudinalcoordinatesoftheelectronsare
s(µ, tf) =s(µ, t0) +R56δ(µ, t0) +T566[δ(µ, t0)]2(4)
=s(µ, t0)(1−R56δ1
σs0)−α[s(µ, t0)]2(5)
withα≡(R56δ2−T566δ2
1)/σ2
s0. One can obtain the
maximum compression of the bunch by choosing the ini-
tial bunchlengthandtheinitial energyspreadtosatisfy
1−R56δ1/σs0= 0, s(µ, tf) =sf=−α[s(µ, t0)]2.
(6)
For typical bunch compression chicane, one has R56>0
andT566<0. Therefore α >0, which implies sf≤0
fromEq.(6). UsingEqs.(6)and(2),we have
µ(sf) =/radicalBig
−sf/α (α >0, sf≤0).(7)
The final longitudinal density distribution can be obtained
fromchargeconservation n0(µ)dµ=n(sf, tf)dsf:
n(sf, tf) =1√
2πσsfesf/√
2σsf
/radicalBig
−sf/√
2σsfH(−sf),(8)σsf=/radicalBig
∝angbracketlefts2
f∝angbracketright − ∝angbracketleftsf∝angbracketright2=√
2ασ2
s0. (9)
where H(−sf)is the Heaviside step function, and σsfis
the rmsofthefinallongitudinaldistribution. Thefinal lon-
gitudinalphasespacedistributioncanbeobtainedas
sf≃ −(σsf/√
2δ2
1)δ2(10)
For example,when σs0= 1.26mm,R56= 45mm,and
δ1= 0.028,the compressionconditionEq. (6) is satisfied.
Withα= 0.08mm−1, Eq. (9) gives the final compressed
bunchlength σsf= 0.18mm. Fora realistic beam,uncor-
related energy spread δunshould be added to Eq. (3) (here
we assume δunhas a Gaussian distribution with ∝angbracketleftδun∝angbracketright= 0,
and rms width δrms
un). As a result, one finds the final rms
bunchlengthsatisfies
σeff
s=/radicalBig
∝angbracketlefts2
f∝angbracketright − ∝angbracketleftsf∝angbracketright2=σsf/radicalbig
1 +a2,(11)
withσsfgivenbyEq.(9),and a=R56δun/σsf. Anexam-
ple of the longitudinal phase space distribution described
by Eq. (10), with an additional width due to δun∝negationslash= 0as
givenbyEq.(11),isshownin Fig.1.
...................................................................................................................................................................................................................................................................................................................................................
-6 -4 -2 0 2
/-3-2-10123/1
un=0.un=0
Figure 1: Example of the longitudinal phase space distri-
butionfora compressedbeamwith RF curvatureeffect.
3 CSR FORA COMPRESSED BEAM
Next,westudytheCSRself-interactionofarigid-linecom-
pressedbunchinthesteady-statecircularmotion. Thelon-
gitudinaldensitydistributionfunctionofthe bunchis λ(φ)
forφ=s/R, with the rms angular width σφ=σs/Rfor
the rmsbunchlength σsandtheorbitradius R.
3.1 GeneralFormulas
The longitudinal collective force on the bunch via space-
chargeandCSRself-interactionis[1,2]:
Fθ(φ) =e∂(Φ−β·A)
βc∂t
=−Ne2
R2∂
∂φ/integraldisplay∞
01−β2cosθ
2 sin(θ/2)λ(φ−ϕ)dϕ(12)
where β=v/c,β=|β|,γ= 1//radicalbig
1−β2, andθis an
implicit functionof ϕvia the retardationrelation ϕ=θ−
2βsin(θ/2). In this paper, we treat only the high-energycase when γ≫θ−1andθ≃2(3ϕ)1/3. Inthis case Fθ(φ)
isdominatedbytheradiationinteraction:
Fθ(φ)≃−2Ne2
31/3R2/integraldisplay∞
0ϕ−1/3∂
∂φλ(φ−ϕ)dϕ.(13)
TheCSR powerduetotheradiationinteractionis
P=−N/integraldisplay
Fθ(φ)λ(φ)dφ. (14)
Results for the longitudinal collective force and the CSR
powerfora rigid-lineGaussian bunchare[1,2]:
λGauss(φ) =1√
2πσφe−φ2/2σ2
φ(σφ≫1
γ3),(15)
FGauss
θ(φ)≃Fgg(φ), Fg=2Ne2
31/3√
2πR2σ4/3
φ,(16)
PGauss≃N2e2
R2σ4/3
φ31/6Γ2(2/3)
2π, (17)
where Γ(x)istheGammafunction,and
g(φ) =/integraldisplay∞
0(φ/σφ−φ1)
φ1/3
1e−(φ/σ φ−φ1)2/2dφ1.(18)
3.2 CSR Interactionfora CompressedBunch
The angular distribution for a compressed bunch λcmpr(φ)
with intrinsic width due to δun∝negationslash= 0is the convolution of
the angular density distribution λcmpr
0(φ)withδun= 0and
a Gaussiandistribution λm(φ):
λcmpr(φ) =/integraldisplay∞
−∞λcmpr
0(φ−ϕ)λm(ϕ)dϕ, (19)
λcmpr
0(φ) =1√
2πσφeφ/√
2σφ
/radicalBig
−φ/√
2σφH(−φ), (20)
λm(φ) =1√
2πσmφe−φ2/2σ2
mφ, σmφ=R56δrms
un
R,(21)
where λcmpr
0(φ)is obtained from Eq. (8). We then pro-
ceedto analyzethelongitudinalCSR self-interactionforc e
for a rigid-line bunch with the density function given in
Eq. (19) under the condition σφ> σ mφ≫γ−3. Com-
bining Eq. (19) with Eq. (13), and denoting aas the in-
trinsic width of the bunch relative to the rms bunch length
(0< a < 1):
a=σw
σs(σw=R56δrms
un), (22)
one finds the steady-state CSR longitudinal force for a
compressedbunch:
Fcmpr
θ(φ) =/integraldisplay∞
−∞Fcmpr
θ0(ϕ)λm(φ−ϕ)dϕ. (23)It canbeshownthat Fcmpr
θ0(ϕ)inEq.(23)is
Fcmpr
θ0(φ)≃−2Ne2
31/3R2/integraldisplay∞
0ϕ−1/3∂
∂φλcmpr
0(φ−ϕ)dϕ
=−21/4FgdG(y)/dy (y=φ/σφ),(24)
withFggivenin Eq.(16),and
G(y) =H(−y)e−|y|/√
2|y|1/6Γ/parenleftbigg2
3/parenrightbigg
Ψ/parenleftbigg2
3,7
6;|y|√
2/parenrightbigg
+H(y)y1/6Γ/parenleftbigg1
2/parenrightbigg
Ψ/parenleftbigg1
2,7
6;y√
2/parenrightbigg
, (25)
where Ψ(a, γ;z)isthedegeneratehypergeometricfunction
Ψ(α, γ;z) =1
Γ(α)/integraldisplay∞
0e−zttα−1(1 +t)γ−α−1dt.(26)
Asa result,we have
Fcmpr
θ(φ) =21/4Fg√
2π a5/6f/parenleftbiggφ
a σφ;a/parenrightbigg
, (27)
f(y;a) =/integraldisplay∞
−∞G(a x)(y−x)e−(y−x)2/2dx.(28)
Similarly, the radiation power can also be obtained for the
compressedbunchusingEq.(14)with λcmpr(φ)inEq.(19)
andFcmpr
θ(φ)in Eq.(27),whichgives
Pcmpr
PGauss≃0.75I(a)
a5/6, (29)
I(a) =−/integraldisplay∞
−∞f/parenleftbiggφ
a σφ;a/parenrightbigg
λcmpr(φ)dφ.(30)
Numerical integration shows that |f(y;a)|max— the
maximum of |f(y;a)|for fixed a— is insensitive to afor
0< a < 1. As a result, for a compressedbunchwith fixed
σφ,wefoundfromEq.(27)theamplitudeoftheCSRforce
Fcmpr
θ(φ)varies with a−5/6. Therefore in contrast to the
well-known scaling law R−2/3σ−4/3
sfor the amplitude of
the longitudinal CSR force for a Gaussian beam, a bunch
describedbyEq.(19)has |Fcmpr
θ|max∝R−2/3σ−1/2
sσ−5/6
w
withσwin Eq. (22) denoting the intrinsic width of the
bunch. Likewise, for a=0.1, 0.2, and 0.5, we found from
numerical integration that I(a)≃0.76, 0.90 and 1.02 re-
spectively, and correspondingly Pcmpr/PGauss≃3.9, 2.6
and 1.4. This dependence of the amplitude of the CSR
force and power on the intrinsic width of the bunch for a
fixed rms bunch length manifests the sensitivity of the en-
hancement of the CSR effect on the local charge concen-
trationina longitudinalchargedistribution.
InFigs.2and3,weplotthelongitudinaldensityfunction
for various charge distributions with the same rms bunch
lengths (except the√
1 +a2factor in Eq. (11)), and the
longitudinalCSR collectiveforcesassociatedwith thevar -
ious distributions. The amplitudeof Fcmpr
θin Fig. 3 agrees
with the a−5/6dependence in Eq. (27). Good agreementof the analytical result in Eq. (27) with the simulation re-
sult[3]fortheCSRforcealongtheexampledistributionin
Fig. 1isshownin Fig.4.
ThisworkwasinspiredbytheCSRmeasurementledby
H. H. Braun at CERN, and by discussions with the team
duringthemeasurement. Theauthorisgratefulforthedis-
cussions with J. J. Bisognano, and with P. Piot, C. Bohn,
D. Douglas, G. Krafft and B. Yunn for the CSR measure-
ment at Jefferson Lab. This work was supported by the
U.S. DOEContractNo. DE-AC05-84ER40150.
-4 -2 0 2
/0.00.51.01.52.0()Gauss()0cmpr()fora=0cmpr()fora=0.5cmpr()fora=0.2cmpr()fora=0.1
Figure 2: Longitudinal charge distribution for a com-
pressed bunch with intrinsic width described by a, com-
pared with a Gaussian distribution. All the distributions
herehavethesameangularrmssize σφ.
-4 -2 0 2
/-8-6-4-2024F()/Fg
FGauss()Fcmpr() for a=0.5Fcmpr() for a=0.2Fcmpr() for a=0.1
Figure3: LongitudinalCSR forcealongthe bunchforvar-
iouschargedistributionsillustratedinFig. 2.
-6 -4 -2 0 2
/-2-101F()/Fg
FGauss()Fcmpr()fora=0.36simulation
Figure 4: Comparison of the analytical and numerical re-
sults of the longitudinal CSR force along the example
bunchillustratedin Fig.1. Herewehave σx≃3σs.
4 REFERENCES[1] Y. S.Derbenev, etal., DESY-TESLA-FEL-95-05,1995.
[2] B.Murphy, et al.,Part.Accel. 57, 9(1997).
[3] R.Li,Nucl. Instrum.Meth. Phys.Res.A 429,310 (1998).
[4] L.Groening, etal.,Proc.of2000EPACConf.,Vienna,2000‘
[5] P.Piot, et al.,Proc. of 2000 EPACConf., Vienna, 2000. |
arXiv:physics/0008191v1 [physics.acc-ph] 19 Aug 2000Ground MotionStudies andModeling for the InteractionRegi on
of aLinear Collider∗
A.Seryi, M.Breidenbach, J. Frisch
StanfordLinear AcceleratorCenter, StanfordUniversity, Stanford,California94309USA
Abstract
Groundmotionmaybealimitingfactorintheperformance
offuturelinearcolliders. Culturalnoisesources,animpo r-
tantcomponentofgroundmotion,arediscussedhere,with
data fromtheSLDregionat SLAC.
1 INTRODUCTION
Groundmotionmaybealimitingfactorintheperformance
of futurelinearcollidersbecauseit causescontinuousmis -
alignment of the focusing and accelerating elements. Un-
derstanding the ground motion, including finding driving
mechanisms for the motion, studying the dependence on
geology, local engineering, etc., and creating ground mo-
tion models that permit evaluation of the collider perfor-
mance, are essential for the optimization of the linear col-
lider.
There has been a lot of progress in understanding mo-
tion of the groundand its modeling in recent years, which
has allowed us to build both general and specific ground
motion models for a particular location. For example, the
model presented in [1] includes systematic, diffusive, and
fast motion based on various measurements performed at
the SLACsite.
However, several important features are not sufficiently
well studied and consequently are not yet adequately rep-
resented in this(orother)modelsor in the underlyingana-
lytical approach. Proper representation of cultural noise is
a major concern. The model mentioned above is based on
measurementsofthefastmotionperformedatnightinsec-
tor 10 of the SLAC linac [2], one of the quietest locations
at SLAC. The corresponding model of the correlation is
suitable forthecase whenthenoise sourcesarelocatedre-
motely from the pointsof observation. Cultural noise may
not only increase the fast frequency power spectrum, for
example as shown in Fig.2, but also the correlation model
may have to be changed if the noise sources are located in
the vicinity of or between the points of interest. Cultural
noise sources, located above or inside the tunnel, can lo-
cally increase the amplitudes of motion. The model, and
the analytical framework, however, assume that the spec-
trum of motion or the correlation do not depend on loca-
tion, which is natural for the spectral approach based on
theuseofthe2-Dspectrum P(ω,k),whichcannotdepend
on position. This issue should be handled by use of a lo-
cal addition p(ω,s)to the spectrum which would describe
(togetherwithcorrespondingcorrelationinformation)ea ch
∗Work supported by the U.S. Department of Energy, Contact Num ber
DE-AC03-76SF00515.noisesourcelocatedinthevicinity. Here ω= 2πf,f–fre-
quency,k–wavenumber, s–position. See[4]formorede-
tailed definitions. In some cases, a function ψ(ω,s)which
wouldcharacterizelocalamplificationofvibrations,fore x-
ampleduetotheresonantpropertiesofgirders,shouldalso
beused.
Cultural noise in the detector area of a linear collider is
ofspecialconcern. Themostseverepositiontolerancesare
for the final quadrupoles. Various systems of the detec-
tor and the detector hall will unavoidably alter the natural
“quietness”ofthearea.
Studies of vibrationnoise haverecently beenperformed
intheHERAHallEast[7]. TheobservedmotionatHERA
was found to be quite large, for example the rms motion
above 1 Hz reachs 100–200 nm. This high level of vibra-
tions at HERA appears to be caused by the high urbaniza-
tionofthearea.
In the studies presented below noise in the SLAC Large
Detectorhasbeeninvestigated.
2 NOISEINSLD DETECTOR AREA
Vibrations studies are currently being performed in the
SLD pit at SLAC. The SLD detector is shutdownand rep-
resents an ideal test bench for such studies. Eight seis-
moprobes have been installed in the detector area. Two
broadbandStreckeisenSTS-2seismometersareplacedun-
der the detectoronthe concretefloorwith 14 m separation
betweenthemasshowninFig.1. FourMarkL4geophones
are placed in the final focus tunnels, and two piezosensors
on the superconducting triplet and on the detector itself.
The complete results of these studies in the SLD hall will
bereportedelsewhere[3]. We presenthereonlytheresults
forthefloorvibrationunderthedetector.
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1SLD
seismometers 14m
20m
Figure1: SchematicsofSLDareashowinglocationofseis-
moprobes installed on the floor of the pit and in final fo-
custunnels. Thedoorsofthedetector,thesuperconducting
tripletsandprobesinstalled onthemarenotshown.The power spectra measured by the STS-2 probes dur-
ing day and night is shown in Fig.2. The high frequency
part of the spectra ( F>∼10Hz) is clearly much noisier
than that measured in sector 10. However, the day-night
variationisabsent,whichmeansthatthisnoiseisproduced
mostly by local (in the SLD building and nearby vicinity)
sources, while the contribution from more remote sources
(traffic, etc.) is much less pronounced. On the other hand,
the low frequencycontributionof the traffic and other cul-
turalnoisesproducedinsideandoutsideofSLAChasclear
day-nightvariations,asseenin Fig.2,3and4.
Figure 2: Power spectra measured at 2a.m in SLAC sector
10[2]comparedwithspectrameasuredbytheSTS-2probe
placed on the concrete floor under the South door of the
SLD detector.
The integrated amplitude shown in Fig.3,4 and 5 is
defined as the integral over the power spectrum from a
specific frequency Ftoa maximalfrequency:
rms =
Fmax/integraldisplay
Fp(f)df
1/2
.
Oneshouldnotethatthermsamplitudeofthedifferenceof
displacements of two points, in the case where these mo-
tions are uncorrelated, can exceed each of the individual
rmsvalues(forexamplebyafactorof√
2ifthesetwo rms
values are equal). The motion under the South and North
part of the SLD detector, shown in Fig.5, is mostly uncor-
related for frequencies higher than about 4 Hz, as seen in
Fig.6, though at some particular frequencies the correla-
tion is noticably nonzeroeven for f>∼4Hz. For identical
probes the imaginary part of the correlation must be zero
if the power spectra in the two places are equal [4]. One
can see in Fig.5 that these power spectra are, in fact, quite
differentandsotheimaginarypartofthecorrelationshown
in Fig.6isessentiallynonzero.
The measurementspresentedinFig.2,3 and4 wereper-
formed when most of the SLD electronics was on (with
itslocalventilation)andthebuildingventilationoperat ing.
ThewaterflowintheSLDconventionalsolenoidwassetto
one third of the nominal level, approximately 300 gallonsFigure 3: Rms amplitudes in different frequency bands
measured by the STS-2 probe placed on the concrete floor
undertheNorthdooroftheSLDdetectorinJuly2000.
Figure 4: Rms amplitudes in different frequency bands
of the difference of displacement measured by two STS-
2 probesplacedwith 14mseparationontheconcretefloor
undertheSouthandtheNorthdoorsofthe SLDdetector.
perminute. Thefloormotionwasfoundtobegreatlyinflu-
enced by the ventilation system of the building (located in
the North part of the SLD hall) and, to a lesser extent, by
the SLC and SLD water pumps located about 20 m North
ofthebuilding.
TheNLC,operatingatarepetitionfrequencyof120Hz,
will be sensitive to the jitter of its final focusingdoubleta t
frequencies above approximately 6 Hz (beam-based feed-
back can presumably take care of the beam offsets below
this frequency). As we see from the Fig.4, even without
any significant precautions to reduce the noise, the differ-
enceofthefloormotionmeasuredbythetwoSTS-2probes
separated by 14 m is about 8 nm for F>∼6Hz, which is
roughlytwice thetypicalNLCverticalbeamsize.
By turning off most (but still not all) of the equipment,
including the SLD and SLC water pumps, the building
ventilation and most of electronics (which would require
proper engineering of these subsystems for NLC) this dif-Figure 5: Integrated spectrum (amplitude for F > F 0)
corresponded to measurements in SLAC sector 10 at 2:00
compared with the spectra measured by probesplaced un-
dertheSLDdetectorwith14mseparationat15:00onJuly
21andat 15:00onAugust11;most ofthenoisesourcesin
the buildingturnedoffat thislater date.
Figure6: Correlation(realandimaginaryparts)ofthemo-
tion measured by the STS-2 probes placed under the SLD
detectorwith14mseparationonAugust11. Averagedover
123fileswith30secondsrecordlength.
ference can be decreased to about 2 nm [3]. As we see in
Fig.5, even in this case, the North probe, located closer to
the noisier North part of the building, shows larger vibra-
tions. Therefore, further reduction of the difference valu e
wouldseemto still bepossible.
Of course the motion of the final quadrupolescannot be
as low asthe motionof thefloor becausethe supportscan-
not be made ideally rigid. The strategy we consider in-
volves active stabilization of the final quadrupoles by us-
inginertialsensorspossiblyincombinationwithanoptica l
referencetotheground. Inoneoftheproposals[5] theop-
tical path would pass from the final quadrupoles through
the detectorto a commonlocationunderthedetector. This
has the disadvantage of putting significant constraints onthe detectordesign. Such a configurationof the detectoris
nowconsideredunlikelyto benecessary.
However, if the optical reference is desired in addition
to the inertial sensors to improve the performance of the
inertial stabilization, the optical reference can be made t o
the floor (possibly to local pits) under each of the final
quadrupoles (approximately at the same positions where
the STS-2 probes were placed in our measurements). The
necessary correction of the differential motion of the floor
could thenbe doneby usingseismometerslocated at these
referencelocations. Onecanseethatintheconditionssim-
ilar to those of the SLD area, where the spectrum of mo-
tiondropsquiterapidlywithfrequency,thisstrategywoul d
work even without significant additional engineering for
noisereduction.
The optical path for the reference to the ground could
be located outside of the detector, greatly simplifying its
design and operation. The newly designed final focussys-
tem[6],whichallowsadoublingof L∗,wouldsimplifythe
detectordesignevenfurther.
OnecanseethattheSLDarea,afterproperengineering,
or a site with similar characteristics, would be compatible
with alinearcolliderhavingnanometerscale beamsizes.
3 CONCLUSION
Several aspects of ground motion require particular atten-
tion, namely studies and modeling of cultural noises and
in particular those generated in the detector area of a lin-
earcollider. StudiesoftheculturalnoiseintheSLACSLD
areapresentedinthispaperwillhelptodeterminetheengi-
neeringrequirementsofvarioussubsystemsofthedetector
to becompatiblewithNLCrequirements.
We wouldlike to thank G.Bowdenand T.Raubenheimer
forvarioususefuldiscussions.
4 REFERENCES
[1] A.Seryi,inthis proceedings.
[2] NLCZDRDesign Group, SLACReport-474 (1996).
[3] A.Seryi,M. Breidenbach, et al.,tobe published.
[4] A.Sery, O.Napoly, Phys.Rev.E 53, 5323, (1996).
[5] G.Bowden, inProceed. of Snowmass 1996.
[6] P.Raimondi, A.Seryi,SLAC-PUB-8460;inthis proceed.
[7] C.Montag, inProceed. of EPAC2000. |
arXiv:physics/0008192v1 [physics.acc-ph] 19 Aug 2000Ground MotionModelofthe SLAC Site∗
AndreiSeryi, TorRaubenheimer
StanfordLinear AcceleratorCenter, StanfordUniversity, Stanford,California94309USA
Abstract
WepresentagroundmotionmodelfortheSLACsite. This
modelisbasedonrecentgroundmotionstudiesperformed
at SLAC as well as on historical data. The model includes
wave-like, diffusive and systematic types of motion. An
attempt is made to relate measurable secondary properties
of the ground motion with more basic characteristics such
asthelayeredgeologicalstructureofthesurroundingeart h,
depth of the tunnel, etc. This model is an essential step in
evaluatingsitesforafuturelinearcollider.
1 INTRODUCTION
In order to accurately characterize the influence of ground
motion on a linear collider, an adequate mathematical
model of ground motion has to be created. An adequate
modelwouldrequireanunderstandingofthetemporaland
spatial properties of the motion and identification of the
driving mechanisms of the motion. Eventually these must
be linked to more general properties of a site like geology
and urbanization density. In this paper, we consider one
particular model based on measurements performed at the
SLACsite[1,2,3,4,5]. Weusethismodeltoillustrateex-
isting methods of modeling, as well as potential problems
and oversimplificationsin the modelingtechniques. In our
particular case, the representationof the cultural noise, es-
pecially that generated inside the tunnel, is difficult to in -
corporate. However, the model provides a foundation to
whichmanyadditionalfeaturescanbe added.
In general, the ground motion can be divided into ‘fast’
and ‘slow’ motion. Fast motion ( f>∼a few Hz) cannot
be adequately corrected by a pulse-to-pulse feedback op-
erating at the repetition rate of the collider and therefore
results primarily in beam offsets at the IP. On the other
hand, the beam offset due to slow motion can be compen-
sated by feedback and thus slow motion ( f<∼0.1) results
only in beam emittance growth. Another reason to divide
groundmotionintofastandslowregimesisthemechanism
by which relative displacementsare producedthat appears
to bedifferentwith aboundaryoccuringaround0.1Hz. In
the following, we will first describe the ‘fast’ motion and
thenwewillpresentthe‘slow’motionwhichincludesboth
diffusiveandsystematiccomponents.
2 FASTGROUNDMOTION
Modeling of the groundmotion requiresknowledge of the
2-D power spectrum P(ω, k). The fast motion is usually
representedbyquantitiesthatcanbemeasureddirectly: th e
spectraofabsolutemotion p(ω)andthecorrelation c(ω, L)
which shows the normalized difference in motion of two
points separated by distance L. The spectrum of relative
∗Work supported by the U.S. Department of Energy, Contact Num ber
DE-AC03-76SF00515.motion p(ω, L)can be written as p(ω, L) =p(ω)2(1−
c(ω, L))whichinturncanbetransformedinto P(ω, k)[9].
Measurements [2, 6] show that the fast motion in a rea-
sonablyquiet site consistsprimarilyofelastic wavesprop -
agating with a high velocity v(of the order of km/s).
The correlationis then completelydefined by this velocity
(which may be a function of frequency) and by the distri-
bution of the noise sources. In the case where the waves
propagate on the surface and are distributed uniformly in
azimuthal angle, the correlation is given by c(ω, L) =
∝angbracketleftcos(ωL/v cos(θ))∝angbracketrightθ=J0(ωL/v)and the correspond-
ing 2-D spectrum of the ground motion is P(ω, k) =
2p(ω)//radicalbig
(ω/v(f))2−k2,|k| ≤ω/v(f).
The absolute power spectrum of the fast motion, as-
sumed for the SLAC model, correspondsto measurements
performed at 2 AM in one of the quietest locations at
SLAC, sector 10 of the linac [2], (see Fig.1). The spa-
tialpropertiesaredefinedbythephasevelocityfoundfrom
correlationmeasurements v(f) = 450 + 1900 exp( −f/2)
(withvin m/s, finHz)[2].
Figure 1: Measured [2] (symbols) and modeling spectra
p(ω)of absolute motion and p(ω, L)/2spectra of relative
motionforthe2AM SLACsite groundmotionmodel.
We believe that the frequency dependence of the mea-
sured phase velocity v(f)is explained by the geological
structure of the SLAC site where, as is typical, the ground
rigidity and the density increase with depth. The surface
motion primarily consists of transverse waves whose
phase velocity is given by vs≈/radicalbig
E/(2ρ)and which are
localized within one wavelength of the surface. If one
plots the quantity v2/λversus wavelength λ, we see that
this value is almost constant, varying from 3000m/s2at
λ= 100m to2000m/s2atλ= 1000m. This is consistent
with a ground density at the SLAC site that ranges from
1.6·103within the upper 100 m to 2.5·103kg/m3at a
kilometerdepthandaYoung’smodulus Ewhichincreases
from109Pa at 100 m to 1010Pa at 1000 m. These results
seem to be quite reasonableforthe SLACgeology,and,as
we will see below,theyalsoagreewithexplanationsofthe
observedslowmotion.3 SLOW GROUNDMOTION
Based on the argumentsabove, the wavelength at frequen-
cies below 0.1 Hz quickly becomes much larger than the
accelerator and eventually exceed the earth’s size. In this
case, the motion has little effect on the accelerator and at
some point the notion of waves is not really applicable.
Causes other than the wave mechanism must be responsi-
bleforproducingrelativemisalignmentsthatareimportan t
at low frequencies. Such sources include the variation of
temperature in the tunnel, undergroundwater flow, spatial
variation of ground properties combined with some exter-
nal driving force, etc. These causes can producemisalign-
mentswithrathershortwavelengthinspiteoftheirlowfre-
quencies.
The ATL model of diffusive groundmotion [7] is an at-
tempt to describe all these complex effects with a simple
rule which states that the varianceof the relative misalign -
ment∆X2is proportional to a coefficient A, the time T
and the separation L:∆X2=ATL. Inthe spectral repre-
sentation this rule can be written as P(ω, k) =A/(ω2k2).
It has been shown [10] that this rule adequately describes
available measured data in many cases, however, typically
only spatial or temporal information, but not both, was
taken for a particular data set. Measurements where good
statistics were collected, both in time and space and in a
relevant regionof parameterspace, are sparse and difficult
to perform. Thus, detailed investigation of slow motion is
an urgentissue forfuturestudies.
The diffusive component of the ground motion model
considered is based on measurements of slow motion per-
formed at SLAC. First, measurements performed in the
FFTB tunnel using the stretched wire alignment system
over a baselength of 30 m give the value of A≈3·
10−7µm2/(m·s) on a time scale of hours [3]. Second, a
48hourmeasurementofthelinactunnelmotionperformed
with the linac laser alignment system over a baselength of
1500 m gave A≈2·10−6µm2/(m·s) [4]. Finally, re-
cent measurements using a similar technique were made
over a period of one month and show that A≈10−7–
2·10−6µm2/(m·s) for a wide frequency band of 0.01–
10−6Hz [5]. In the latter case, the major source of the
slow1/ω2motion was identified to be the temporal vari-
ationsofatmosphericpressurecoupledtospatialvariatio ns
of ground properties [5]. The atmospheric pressure was
also thought to be responsible for a slow variation of the
parameter A.
The clear correlation of atmospheric pressure variation
with deformation of the linac tunnel, observed in [5], can
only be explained if one assumes some variation of the
ground properties along the linac. This variation can be
due to changes in the Young’s modulus E, changes in
the topology such that the normal angle to the surface
changes by ∆α, or changes in the characteristic depth h
of the softer surfacelayers. A roughestimate of the tunnel
deformation due to variation of atmospheric pressure ∆P
canbe expressedas0 5 10 15 20
time (year)0.00.51.0dy/dymax
tau=30 years, t0=−2 years, 25%@0, 81%@17 yearssymbols − SLC data for 1966−83
dashed lines − models with:
Figure2: DisplacementofsomepointsofSLAClinactun-
nel from 1966 through 1983 versus time and the approxi-
mationinEq.(2)with τ= 30andt0= 2years.
∆X, Y ∼h∆P
E·/parenleftbigg∆E
Eor ∆ αor∆h
h/parenrightbigg
(1)
The observeddeformationof the tunnel ∆Y= 50µm cor-
responding to ∆P= 1000Pa is consistent with this es-
timation if ∆E/E∼0.5,∆α∼0.5or∆h/h∼0.5and if
one assumes E/h ∼107Pa/m. The former assumption
is consistent with the heterogeneous landscape and geol-
ogyatSLACwhilethelatterappearstoagreewellwiththe
propertiesof the grounddeterminedin the previousSLAC
correlationmeasurements,if oneassumesthat h∼λ.
No direct conclusions can be drawn from the measure-
ments[5]todeterminethespatialbehavioroftheobserved
slow motion because the relative motion was only mea-
sured for one separation distance. However, the topology
of many natural surfaces (including landscapes) exhibits a
1/k2behavior of the power spectra [11]. Thus, it seems
reasonable to expect that temporal pressure variation can
also be a drivingterm of the spatial ATL-likemotion. Fur-
thermore,themeasuredparameter Acanbeextendedfrom
1500 m to a shorter scale, without contradicting the very
short baseline measurements[3] which produceda similar
valueof A.
Itisalsoworthnotingthatthecontributiontotheparam-
eterAdriven by the atmosphere scales as 1/E2or asv4
s
and therefore strongly depends on geology. Thus, the pa-
rameter A, at a site with a much higher vs, would not be
dominated by atmospheric contributions, while a site with
softer ground and a vshalf that at SLAC, may have a pa-
rameter Aashighas 3·10−5µm2/(m·s).
Finally, very slow motion, observed on a year-to-year
time scale at SLAC, LEP, and other places, appears to be
0 1000 2000 3000
s (m)−20.0−10.00.010.0y (mm)
SLC tunnel, vertical
1983−1966
Figure3: 17yearmotionofthe SLAClinactunnel[1].Figure4: Spatialpowerspectrumofverticaldisplacements
ofthe SLACtunnelfor1966to1983.
Figure 5: Rms relative motion versus time for L= 30m
forthe2a.m. SLACsite groundmotionmodel.
systematic in time, i.e. ∆X2∝T2[12]. For example,
measurementsoftheSLAClinactunnelbetween1966and
1983 [1] show roughlylinear motion in time with rates up
to1mm/yearinafewlocationsalongthelinac. Subsequent
measurements indicate that the rate of this motion has de-
creased over time although the direction of motion is still
similar as is illustrated in Fig. 2. In the case of SLAC, the
motionmayhavebeencausedprimarilybysettlingeffects,
while inLEP,the causemaymorelikelybe somethingdif-
ferentsuchasundergroundwater[12].
The temporal dependence of earth settlement problems
typicallyareapproximatedas:
∆y
∆ymax≈1−/parenleftBigg
1−/radicalbig
t/τ
(1 + 2/radicalbig
t/τ)/parenrightBigg
exp(−2.36t/τ)(2)
where the typical value of τis years. This type of solu-
tion exhibits√
tmotionat the beginningwhich then slows
andexponentiallyapproaches ∆ymax. Anexampleofsuch
a dependence is compared with the motion observed at
SLAC in Fig. 2. One can see that the early SLAC sys-
tematic motion can be also described reasonably well by
a linear in time motion, though nowadays the rate of the
motionshouldbealreadymuchlower.
Thespatialcharacteristicsofthissystematicmotionalso
seem to follow the 1/k2(or∆X2∝L) behavior. This
is evident in the displacements of the SLAC linac [1] af-
ter 17 years which is shown in Fig. 3. The correspond-
ing spatial spectrum is shown in Fig. 4 and it follows
1/k2in the range of λfrom 20–500m. Although there
is deviation from the 1/k2behavior at long wavelengthswhere there is limited data, this spectrum can be charac-
terized as Psyst(t, k)≈Asystt2/k2with the parameter
Asyst≈4·10−12µm2/(m·s2)forearlySLAC.Anestimate
of the rms misalignment due to this systematic motion is
then∆X2=AsystT2L. One can see that the transition
between diffusive and systematic motion would occur at
Ttrans=A/Asystwhich in our case, assuming the value
A= 5·10−7µm2/(m·s) for the diffusive component of
the SLAC ground motion model, would happen at about
Ttrans≈105s.
The SLACgroundmotionmodelincludesall of thefea-
turesthatwe havedescribed. Thetransitionfromthe ‘fast’
to the ‘slow’ motion is handled in a manner described in
Ref. [9]. The absolute spectrum p(ω)and the spectrum
of relative motion p(ω, L)are shown in Fig. 1. The sys-
tematic motion is not seen in this figure as it corresponds
to much lower frequencies. However, it is seen in Fig. 5
where the rms ∆Xis calculated for L= 30m by di-
rect modeling of the ground motion using harmonic sum-
mation [15]. One can see that this curve can be divided
into three regions: wave dominated ( T<∼10s), ATL-
dominated( 10<∼T<∼105s)andsystematicmotiondomi-
nated( T>∼Ttrans∼105s).
ThisgroundmotionmodelisincludedinthePWKmod-
uleofthefinalfocusdesignandanalysiscodeFFADA[13]
which can perform analytical evaluations using the model
spectra. The modelis also includedin the linac simulation
codeLIAR[14]wherethesummationofharmonicsisused
fordirectsimulationsofthegroundmotion.
4 CONCLUSION
WehavepresentedamodelofgroundmotionfortheSLAC
site. Thismodelincludesfast,diffusiveandsystematicmo -
tionwithparametersthatareconsistentwiththeknownge-
ologicalstructureoftheSLACsite. Itisbeingnowusedto
study the performance of the various systems in the Next
LinearCollider.
We would like to thank C.Adolphsen, G.Bowden,
M.Mayoud,R.Pitthan, R.Ruland, V.Shiltsev, and S.Takeda
forvariousdiscussionsofgroundmotionissues.
5 REFERENCES
[1] G.Fischer,M.Mayoud, CERN-LEP-RF-SU-88-07,1988.
[2] NLCZDRDesign Group, SLACReport-474 (1996).
[3] R. Assmann, C. Salsberg, C. Montag, SLAC-PUB-7303, in
Proceed. of Linac 96, Geneva, (1996).
[4] C.Adolphsen, G. Bowden, G.Mazaheri, inProc.of LC97.
[5] A.Seryi,EPAC2000, alsointhis proceedings.
[6] V.M. Juravlevet al.CERN-SL-93-53.
[7] B.Baklakov, etal.Tech. Phys. 38, 894 (1993).
[8] C.Montag, V. Shiltsev,et al.,DESYHERA95-06, 1995.
[9] A.Sery, O.Napoly, Phys.Rev.E 53, 5323, (1996).
[10] V. Shiltsev,inProc. IWAA95,KEK-95-12, 1995.
[11] R. Sayles,T.Thomas, Nature, 271, February2, (1978).
[12] R.Pitthan,SLAC-PUB-7043,8286, 1995,1999.
[13] O. Napoly, B.Dunham, inProceed. of EPAC94,1994.
[14] R. Assmann, et al.,inProceed. of PAC97, 1997.
[15] A. Sery,inProceed. of Linac 1996. |
arXiv:physics/0008193v1 [physics.acc-ph] 19 Aug 2000Simulation Studies ofthe NLCwithImprovedGround MotionMo dels∗
A. Seryi, L. Hendrickson,P. Raimondi,T.Raubenheimer, P. T enenbaum
StanfordLinear AcceleratorCenter, StanfordUniversity, Stanford,California94309USA
Abstract
The performance of various systems of the Next Linear
Collider (NLC) have been studied in terms of ground mo-
tionusingrecentlydevelopedmodels. Inparticular,thepe r-
formance of the beam delivery system is discussed. Plans
to evaluate the operation of the main linac beam-based
alignmentandfeedbacksystemsarealso outlined.
1 INTRODUCTION
Groundmotionisalimitingfactorintheperformanceoffu-
ture linear colliders because it continuously misaligns th e
focusing and accelerating elements. An adequate mathe-
matical model of ground motion would allow prediction
andoptimizationoftheperformanceofvarioussubsystems
ofthe linearcollider.
The ground motion model presented in [9] is based on
measurements performed at the SLAC site and incorpo-
rates fast wave-like motion, and diffusive and systematic
slow motion. The studies presented in this paper include,
inaddition,severalrepresentativeconditionswithdiffe rent
cultural noise contributions. These modelswere then used
in simulationsofthe NLCfinal focusandthemainlinac.
2 GROUNDMOTION MODELS
The ground motion model for the SLAC site [9] is based
onmeasurementsoffastmotiontakenatnightinoneofthe
quietestlocationsinthe SLAC,sector10ofthelinac[5].
Toevaluatedifferentlevelsofculturalnoise,weaugment
this model to represent two other cases with significantly
higher and lower contributions of cultural noise. The cor-
respondingmeasured spectra and the approximationsused
in themodelsareshownin Fig.1.
The“HERAmodel”isbasedonmeasurementsinDESY
[3] and correspondsto a verynoisy shallow tunnellocated
inahighlypopulatedareawherenoprecautionsweremade
to reduce the contribution of various noise sources in the
lab and in the tunnel. The “LEP model” corresponds to a
deeptunnelwherethenoiselevelisveryclosetothenatural
seismiclevel,withoutadditionalculturalsourcesoutsid eor
inside ofthe tunnel. The“SLACmodel”representsa shal-
low tunnel located in a moderately populated area with a
dead zone around the tunnel to allow damping of cultural
noise and with some effort towards proper engineering of
thein-tunnelequipment. (Note: thenamesofthesemodels
were used for convenience, and not to indicate the accept-
abilityofeachparticularlocation.)
∗Work supported by the U.S. Department of Energy, Contact Num ber
DE-AC03-76SF00515.10−1100101102
Frequency, Hz10−1310−1010−710−410−1102micron**2/HzUNK tunnel
LEP tunnel
Hiidenvesi cave
HERA tunnel
SLAC tunnel
SLAC 2am model
HERA model
LEP model
1/w4
Figure 1: Power spectra measured in several places in dif-
ferentconditions[1,3,5, 2]andtheapproximationcurves.
The correlation properties of the “LEP model” corre-
spond to a phase velosity v= 3000 m/s [1]. Both
the “SLAC model” and the “HERA model” use a phase
velosity correspondingto v(f) = 450 + 1900 exp( −f/2)
(withvin m/s, fin Hz) which was determined approxi-
mately in the SLAC correlation measurements [5]. This
approximation was found to be suitable for representing
the DESY correlation measurements [3], at least for fre-
quencies greater than a few Hz, which contain most of the
effectsoftheculturalnoise.
3 APPLICATIONSTO FFS
The groundmotion modelsdevelopedwere applied to two
versions of the NLC Final Focus, to the one described in
Ref. [5] as well as the current FFS described in Ref. [10].
The FF performance is usually evaluated using the 2-D
spectrum P(ω, k)given by the ground motion model plus
spectralresponsefunctionswhichshowthecontributionto
the beam distortionat the IP of differentspatial harmonics
ofmisalignment.
We summarize below the basics of the approach de-
veloped in [2, 4] and [5]. Considering a beamline with
misaligned elements, as in Fig.2, the beam offset at the
exit of the beamline and the dispersion (for example) can
beevaluatedusing
x∗(t) =N/summationdisplay
i=1cixi(t)−xfinandη(t) =N/summationdisplay
i=1dixi(t)
where ci=dx∗/dxianddi=dη/dx iare the coef-
ficients found using the parameters of the focusing ele-
ments and the optical properties of the channel. In a thin
lens approximation to linear order, ci=−Kiri
12and
di=Ki(ri
12−ti
126). Here Kiisr21of the quad ma-
trix, and ri
12andti
126arethe matrixelementsfromthe i-th
quadrupoleto the exit. Fig.3 shows the cicoefficients cal-
culatedforthe newNLCFinal Focus[10].entrancebeam
ref. linepp-dp
rel. offset
x*quadrupole
exitdispersionη
xi
xfin
si
Figure 2: Schematic showinghow quadmisalignmentsre-
sult in thebeamoffsetanddispersion.
Figure 3: Coefficients ci=dxIP/dxifor the new NLC
Final Focus. ComputedusingFFADAprogram[6].
It is straightforward then to combine these coefficients
into the spectral response functions which show the
contribution of misalignment spatial harmonics to the
relative beam offset or to the beam distortionat the IP. For
example,forthedispersion:
Gη(k) =/parenleftBiggN/summationdisplay
i=1di(cos(ksi)−1)/parenrightBigg2
+/parenleftBiggN/summationdisplay
i=1disin(ksi)/parenrightBigg2
Thespectralfunctionsfortherelativebeamoffset,longit u-
dinalbeamwaistshiftorcouplingcanbefoundinasimilar
mannerand examplesof the spectral functionsfor the new
NLCFF areshowninFig.4.
The time evolution of the beam dispersion, without the
effectoffeedbacks,canthenbeevaluatedusing
/angbracketleftη2(t)/angbracketright=/integraldisplay∞
−∞P(t, k)Gη(k)dk
2π
where P(t, k)representsa (t, k)incarnationof the ground
motionspectrum P(ω, k):
P(t, k) =/integraldisplay∞
−∞P(ω, k) 2 [1 −cos(ωt)]dω
2π
In the case where a feedback with a gain of F(ω)is
applied,the equilibriumbeamoffsetcanbeevaluatedas
/angbracketleft∆x∗2/angbracketright ≈/integraldisplay∞
−∞/integraldisplay∞
−∞P(ω, k)F(ω)G(k)dω
2πdk
2π
though more realistic simulations would be necessary to
produce a reliable result. In the examples given below,
we used an idealized approximation of the feedback gain
function F(ω) = min(( f/f0)2,1)withf0= 6Hz; this
isagoodrepresentationoftheSLCfeedbackalgorithmfor
120Hzoperation.
Such analytical evaluation of ground motion, using the
P(ω, k)spectrum and the spectral response functions forFigure4: SpectralresponcefunctionsofNewNLCFF.
Figure 5: Integrated spectral contribution to the rms equi-
librium IP beam offset for the traditional and new Final
Focus for the SLAC 2AM ground motion model. Ideal-
ized rigid supports of the final doublets are assumed to be
connected to the ground at ±SFDfrom the IP. The rela-
tive motion of the final doublets is completely eliminated
inthecase“ON”.Redarrowshowstheregionoffrequency
givingthelargestcontributiontothe rmsoffset.
the transport lines is included in the PWK module of the
final focusdesignandanalysiscodeFFADA [6].
Evaluation of the traditional and new Final Focus in
terms of the rms beam offset for the “SLAC model” is
shown in Fig.5. One can see that in terms of generalized
tolerancesthese two systems are verysimilar. However,in
the new system which has longer L∗, more rigid support
can be used for the final doublet which makes the perfor-
manceclosertotheideal. Onecanalsoseethatifonecould
eliminate the contribution from the final doublet by active
stabilization,it wouldremoveabout80%ofthe effect.
The free IP beam distortion evolution for the traditional
andnewNLCFFisshowninFig.6forthe“SLACmodel”.
Notethatanorbitcorrectionwhichcouldkeeptheorbitsta-
ble through the sextupoles would drastically decrease this
beam distortion. The picture presented is therefore useful
onlyforcomparisonoftheperformanceofthetwoFF sys-
tems. One can see, that the new FF, having longer L∗and
correspondinglyhigherchromaticity,hassomewhattighte r
tolerances. The orbit feedback, however, may be much
simpler sincethereare fewersensitiveelementsinthe new
system.
The analytical results presented in Fig.6 are in good
agreement with the tracking. One should note here thatFigure 6: Beam distortion at the IP for the traditional and
newNLCFF versustimeforthe“SLACmodel”ofground
motion, free evolution. Note that orbit feedback would
drastically decrease this beam distortion. Results were
computedusingthe FFADAprogram[6].
the tracking was done with an energy spread which is 3
timessmallerthannominal(see[10]forthesebeamparam-
eters) because otherwise the second order tracking routine
of the MONCHOU program used for misalignment simu-
lationdidnotproducereliableresultswhencomparedwith
otherprograms.
Comparison of the performance of the new FF in terms
of different groundmotion models is shown in Fig.7. One
canseethatasitelocatedinahighlypopulatedareawithout
proper vibration sensitive engineering would present sig-
nificantdifficultiesforalinearcolliderwiththeparamete rs
considered. Stabilization of only the final doublet would
not be sufficient in this case. A site with noise similar to
Figure 7: Integrated spectral contribution to the rms equi-
librium IP beam offset for the new Final Focus with FD
supports at SFD=±8m for different models of ground
motion. Dashed curves correspond to the complete elimi-
nationofrelativemotionofthefinal quads.Figure 8: LIAR generated misalignments of a linac for
“SLACmodel”and ∆T= 8hoursbetweencurves.
the “SLAC model” would certainly be suitable, while the
“LEPmodel”wouldbesuitableevenformuchmoreambi-
tiousbeamparameters. Theseresultsshouldnotbeconsid-
eredasanattempttoevaluateanyparticularsite,oreventh e
models, because for a fully consistent assessment, various
in-tunnel noise sources as well as vibration compensation
methodsmust beconsideredtogether.
4 APPLICATIONSTO LINAC
The models now developed, which more adequately de-
scribe the various components of ground motion, can
also be applied to simulations of the beam based align-
ment proceduresand cascaded feedback in the main linac.
Suchsimulationsrequiredirectmodelingofmisalignments
which is done by summing harmonics whose amplitudes
aregivenbythe2-Dspectrumofthecorrespondingground
motion model. In this case, since a large rangeof TandL
must be covered in a single simulation run, the harmonics
are distributed over the relevant (ω, k)range equidistantly
inalogarithmicsense[8]. Suchamethodofgroundmotion
modelingisnowincludedinthelinearacceleratorresearch
code LIAR [7] in addition to the previously implemented
ATL model. An example of the misalignments generated
byLIARisshownin Fig.8.
5 CONCLUSION
New ground motion models now incorporate various
sourcesof groundmotionsuch as wave-likemotion,diffu-
siveandsystematicmotion. Thesemodelsarebeingusedto
evaluate and optimize performance of various subsystems
oftheNLC.
6 REFERENCES
[1] V.M. Juravlevet al.CERN-SL-93-53.
[2] V.M. Juravlevet al.HU-SEFTR 1995-01.
[3] C.Montag, V. Shiltsev,et al.,DESYHERA95-06, 1995.
[4] A.Sery, O.Napoly, Phys.Rev.E 53, 5323, (1996).
[5] NLCZDRDesign Group, SLACReport-474 (1996).
[6] O.Napoly, B.Dunham, inProceed. ofEPAC94, 1994.
[7] R.Assmann, etal.,inProceed. of PAC97, 1997.
[8] A.Sery, inProceed. of Linac1996.
[9] A.Seryi,inthis proceedings.
[10] P.Raimondi, A.Seryi,SLAC-PUB-8460;inthis proceed. |
MEASUREMENT AND CORRECTION OF CROSS-PLANE COUPLING IN
TRANSPORT LINES*
M. Woodley, P. Emma, SLAC, Stanford, CA 94309, USA
* Work supported by the U.S. Department of Energy under Contract DE-AC02-76SF00515. Abstract
In future linear colliders the luminosity will depend on
maintaining the small emittance aspect ratio delivered by damping rings. Correction of cross-plane coupling can be important in preventing dilution of the beam emittance. In order to minimize the vertical emittance, especially for a flat beam, it is necessary to remove all cross-plane ( x-y)
correlations. This paper studies emittance measurement and correction for coupled beams in the presence of realistic measurement errors. The results of simulations show that reconstruction of the full 4
×4 beam matrix can
be misleading in the presence of errors. We suggest more robust tuning procedures for minimizing linear coupling.
1 INTRINSIC EMITTANCE
A four-dimensional (4D) symmetric beam matrix, σ,
contains ten unique elements, four of which describe coupling. The projected (2D) beam emittances,
εx and εy,
are defined as the square roots of the determinants of the on-diagonal 2
×2 submatrices. If one or more of the
elements of the off-diagonal submatrix is non-zero, the
beam is x-y coupled. Diagonalization of the beam matrix
yields the intrinsic beam emittances, ε1 and ε2.
2
12
1
2
2
22000
00 0,00 0
000Txx xx y x y
xx x xy xyRR
xy xy y yy
xy xy yy yε
εσσ σε
ε′′ <> <> <> <>
′′ ′ ′ ′<> <> <> < >== =′′ <> <> <> < >
′′ ′ ′′<> <> <> <>
The coupling correction process involves measuring the
ten elements of the beam matrix and finding a set of skew quadrupole strengths which block diagonalize the beam matrix, setting the projected emittances, for linear coupling, equal to the intrinsic emittances.
2 SKEW CORRECTION SECTION
The ideal skew correction section (SCS) contains four
skew quadrupoles separated by appropriate betatron phase advance in each plane such that the skew quadrupoles are orthonormal (orthogonal and equally scaled). A simple realization of such a system is possible if the skew quadrupoles each correct just one of the four x-y beam
correlations and if, in addition, the product
βxβy is equal at
each of the skew quadrupoles. Figure 1 shows such a system for the 250 GeV NLC beam, followed by a 4D emittance measurement section (described below). Skew quadrupoles at locations 1-4 (indicated at top of figure by diamond symbols) are used to correct the < xy>, <x
′y′>,
<x′y>, and < xy′> beam correlations, respectively, at
location 4. The horizontal and vertical betatron phase advances between the skew quadrupoles are also indicated on the figure. This scheme allows total correction of any arbitrary linearly coupled beam with correction range limited only by the available skew quadrupole strength.
0 50 100 150 200 250010203040506070β (m)
S (m)12 345 67 8 9
90°
90°180°
90°90°
90°90°
90°180°
90°90°
90°45°
45°45°
45°
Figure 1: SCS ( S=0-120 m) plus 4D emittance measurement
section (S=120-270 m): βx (solid), βy (dash). Diamond symbols
indicate skew quadrupoles; circles indicate wire scanners. The betatron phase advances between devices are shown in 2 rows above the plotted
β–functions ( x on top and y below).
3 4D EMITTANCE MEASUREMENT
The ideal 4D emittance measurement section contains
six beam size measurement devices (e.g. wire scanners) separated by appropriate betatron phase advance in each plane such that the four x-y beam correlations may be
measured independently. Figure 1 illustrates such a system. The wire scanners at locations 4-7 (circle symbols) are used to measure the < xy>, <x
′y′>, <x′y>, and
<xy′> beam correlations, respectively. Each wire scanner
has three independent angle filaments — an x-wire, a
y-wire, and an “off-axis”, or u-wire whose optimal
orientation is given by the inverse tangent of the uncoupled beam aspect ratio,
σy/σx [1]. At each of these
wire scanners σx, σy, and σxy are measured. An additional
two wire scanners (locations 8 and 9 in Figure 1) are required to determine the remaining in-plane correlations of the beam. There are a total of 10 beam parameters to determine (
εx,y, βx,y, αx,y, and the four x-y correlations) and
up to 18 beam size measurements, leaving 8 degrees of freedom in the analysis. The analysis consists of expressing the beam sizes at each wire in terms of the unknown beam parameters at the first wire, using the
wire-to-wire R-matrices, and solving the linear system.
Figures 2 and 3 each show the results of 5000 Monte
Carlo simulations of the 4D analysis and intrinsic vertical emittance calculation using this setup. The input beam is the nominal NLC beam at 250 GeV (
γε1=3×10−6 m,
γε2=3×10−8 m). For these emittances, the ideal rms beam
sizes at the wires range from 1.5-10 µm. In each
simulation, the real beam size on each wire is given a gaussian distributed multiplicative random error of rms f
err
()1sim err ideal f σσ=+
and the ensemble of simulated measurements is analyzed.
0500100015002000Nferr=1%
γε2=2.99±0.04
0100200300400Nferr=5%
γε2=2.80±0.34
012340 50 100150200
γε2 [10−8 m]Nferr=10%
γε2=2.63±0.69
012340 50 100150
γε2 [10−8 m]Nferr=20%
γε2=1.89±0.84
Figure 2: Results of simulations of 4D emittance measurement
and reconstruction of γε2 (coupled input beam). Vertical dotted
lines show the actual value γε20 used in the simulations.
0100020003000Nferr=1%
γε2=3.00±0.03
0200400600800Nferr=5%
γε2=2.90±0.16
012340 100200
γε2 [10−8 m]Nferr=10%
γε2=2.65±0.47
012340 50 100150
γε2 [10−8 m]Nferr=20%
γε2=1.83±0.72
Figure 3: Results of simulations of 4D emittance measurement
and reconstruction of γε2 (uncoupled input beam).
Figure 2 shows the results for four values of ferr when
the simulated input beam is coupled ( εy/ε2 = 1.5), while
Figure 3 shows the results for an uncoupled input beam (
εy/ε2 = 1). Figures 2 and 3 show that when the beam size
measurement errors are more than a few percent, the measurements become imprecise, and more importantly, the most probable computed value for the intrinsic vertical emittance becomes erroneously small. This bias may lead one to attempt to correct the implied coupling, which will actually introduce coupling rather than correct it. An additional problem, in the presence of errors, is that the 4D analysis can generate solutions for which the beam matrix is nonpositive, yielding imaginary emittances. As
f
err becomes larger, the fraction of simulations which yield
nonpositive beam matrices, the ‘rejection fraction ’,
increases to the point where 3 out of 4 measurements yield non-physical results when f
err reaches 20 %. Table 1
summarizes the results of the 4D measurement simulations for a coupled input beam; Table 2 summarizes the results for an uncoupled input beam. In each case, the most probable relative value of
ε2/ε20 is given, along with
the statistical rms width of the distribution (where ε20 is
the ‘real’ intrinsic emittance used in the simulations).
Table 1: 4D Simulation Results (coupled beam).
ferr ε2 /ε20 rejection fraction
1 % 1.00 ± 0.01 <0.1 %
5 % 0.93 ± 0.10 0.2 %
10 % 0.88 ± 0.23 22 %
20 % 0.63 ± 0.28 78 %
Table 2: 4D Simulation Results (uncoupled beam).
ferr ε2 /ε20 rejection fraction
1 % 1.00 ± 0.01 <0.1 %
5 % 0.97 ± 0.05 <0.1 %
10 % 0.88 ± 0.16 1.9 %
20 % 0.61 ± 0.24 59 %
4 2D EMITTANCE MEASUREMENT
An optimized 2D emittance measurement section
contains four wire scanners separated by 45 ° of betatron
phase advance in both planes. Figure 4 shows such a system preceded by an SCS. Each wire scanner has two independent angle filaments —an x-wire and a y-wire. At
each wire scanner
σx and σy are measured. There are a
total of three beam parameters to determine ( ε, β and α)
and four beam size measurements in each plane, leaving one degree of freedom in the analysis for each plane.
020406080100120140160180010203040506070β (m)
S (m)12 34 5 6 7
90°
90°180°
90°90°
90°45°
45°45°
45°45°
45°
Figure 4: SCS ( S=0-120 m) plus 2D emittance measurement
section (S=120-190 m): βx (solid), βy (dash).
Figures 5 and 6 each show simulations of the 2D
analysis and projected vertical emittance calculation using this setup. Figure 5 is for a coupled input beam, while Figure 6 is for an uncoupled input beam. 0100020003000Nferr=1%
γεy=4.50±0.05
05001000Nferr=5%
γεy=4.46±0.24
024680 100200
γεy [10−8 m]Nferr=10%
γεy=4.34±0.48
024680 50 100
γεy [10−8 m]Nferr=20%
γεy=3.95±1.02
Figure 5: Results of simulations of 2D emittance measurement
and reconstruction of γεy (coupled input beam). Vertical dotted
lines show the actual value γεy0 used in the simulations.
0100020003000Nferr=1%
γεy=3.00±0.03
05001000Nferr=5%
γεy=2.98±0.15
0 2 4 60 200400
γεy [10−8 m]Nferr=10%
γεy=2.89±0.31
0 2 4 60 100200
γεy [10−8 m]Nferr=20%
γεy=2.63±0.66
Figure 6: Results of simulations of 2D emittance measurement
and reconstruction of γεy (uncoupled input beam).
These figures show that the 2D projected emittance
measurement is far less sensitive to beam size measurement errors than the 4D intrinsic emittance measurement In addition, the 2D analysis does not generate non-physical solutions. Table 3 summarizes the 2D measurement simulations for a coupled input beam; Table 4 summarizes the results for an uncoupled input
beam (
εy0 is the ‘real’ projected emittance).
Table 3: 2D Simulation Results (coupled beam).
ferr εy /εy0 rejection fraction
1 % 1.00 ± 0.01 0
5 % 0.99 ± 0.05 0
10 % 0.96 ± 0.11 0
20 % 0.88 ± 0.23 0
Table 4: 2D Simulation Results (uncoupled beam).
ferr εy /εy0 rejection fraction
1 % 1.00 ± 0.01 0
5 % 0.99 ± 0.05 0
10 % 0.96 ± 0.10 0
20 % 0.88 ± 0.22 0
5 COUPLING CORRECTION
Given the unreliability of the 4D emittance
measurement, we propose, for the NLC, the coupling correction and 2D emittance measurement system shown in Figure 4. Coupling correction will be achieved by sequentially minimizing the measured projected vertical emittance with each of the four orthonormal skew quadrupoles. Figure 7 shows the Monte Carlo simulation of this process, assuming a coupled input beam (
εy/ε2 > 3)
and 10% beam size measurement errors. Because the optics of the SCS has been designed to make the skew quadrupoles orthonormal, a single pass through the set is sufficient to bring the projected vertical emittance down to its intrinsic value to within measurement errors.
01234567800.511.522.533.54
Skew Quad Scan Numberεy/εy0
SQ1(1)
SQ2(1)
SQ3(1)
SQ4(1)
SQ1(2)
SQ2(2)
SQ3(2)
SQ4(2)
Figure 7: Results of simulations of two full iterations of
coupling correction. Each circle gives the minimized value of
εy/ε2 after scanning the indicated skew quad.
Alternatively, the system shown in Figure 1 can be used
to remove the coupling more directly. Each skew quadrupole can be used to remove the measured < xy>
correlation at its associated wire scanner (skew
quadrupoles 1-4 correct < xy> at wire scanners 4-7,
respectively).
6 CONCLUSIONS
Although it may seem that the 4D emittance
measurement is the most direct way to compute skew corrections for a coupled beam, simulations show that realistic beam size measurement errors degrade the analysis to the point where it becomes counter-productive. The 2D emittance measurement is far more reliable, and when combined with an orthonormal skew correction system, provides the most robust method for correcting linear betatron coupling.
REFERENCES
[1] P. Emma, M. Woodley, Cross-Plane Coupling
Measurements with Odd-Angle Wire Scanners ,
ATF-99-04, KEK, Japan, March 1999. |
arXiv:physics/0008195v1 [physics.acc-ph] 20 Aug 2000Investigationsof SlowMotionsof the SLAC Linac Tunnel∗
Andrei Seryi
StanfordLinear AcceleratorCenter, StanfordUniversity, Stanford,California94309USA
Abstract
Investigations of slow transverse motion of the linac tun-
nel of the Stanford Linear Collider have been performed
over period of about one month in December 1999 – Jan-
uary 2000. The linac laser alignment system, equipped
with a quadrant photodetector, allowed submicron resolu-
tion measurement of the motion of the middle of the linac
tunnelwithrespecttoitsends. Measurementsrevealedtwo
majorsourcesresponsiblefortheobservedrelativemotion .
Variationoftheexternalatmosphericpressurewasfoundto
bethemostsignificantcauseofshortwavelengthtransverse
motion of the tunnel. The long wavelength component of
themotionhasbeenalsoobservedtohavealargecontribu-
tion fromtidal effects. Themeasureddata are essential for
determinationofparametersfortheNextLinearCollider.
1 INTRODUCTION
Theelectron-positronlinearcollidersenvisionedforthe fu-
ture must focus the beams to nanometer beam size in or-
dertoachievedesignluminosity. Smallbeamsizesimpose
strict tolerances on the positional stability of the collid er
components, but ground motion will continuously change
the componentpositions.
For linear colliders, the ground motion can be specifi-
cally categorized into fast and slow motion. Fast ground
motion (roughly f>∼0.1Hz) causes the beam position to
change from pulse to pulse. In contrast, slow ground mo-
tion (f<∼0.1Hz) doesnot result in an offset of the beams
at the interactionpoint since it is correctedbyfeedbackon
a pulse to pulse basis. However slow motion causes emit-
tancedilutionsinceitcausesthebeamtrajectorytodeviat e
from the ideal line. Investigations of slow ground motion
areessentialtodeterminetherequirementsforthefeedbac k
systemsandtoevaluatetheresidualemittancedilutiondue
to imperfectionsinthe feedbacksystems.
Investigationsof slow motion of the SLAClinac tunnel,
described in this paper, were performed in the framework
of the Next Linear Collider [1]. The measurements were
taken from December 8, 1999 to January 7, 2000. Earlier
measurementsusingthesametechniquewereperformedat
SLAC in November 1995 for a period of about 48 hours
[2]. The goal of the measurements was to systematically
studytheslowmotionandtofindcorrelationswithvarious
externalparametersin ordertoidentifythedrivingcauses .
∗Work supported by the U.S. Department of Energy, Contact Num ber
DE-AC03-76SF00515.2 RESULTS ANDDISCUSSION
Themeasurementsofslowgroundmotionwereperformed
usingtheSLAClinaclaseralignmentsystem[4]. Thissys-
tem consists of a light source, a detector, and about 300
targets, one of which is located at each point to be aligned
over a total length of 3050 m. The targets are installed in
a 2-foot diameter aluminum pipe which is the basic sup-
port girder for the accelerator. The target is a rectangular
Fresnellenswhichhaspneumaticactuatorsthatalloweach
lens to be flipped in or out. The light source is a He-Ne
laser shining through a pinhole diaphragm. The beam di-
vergence is large enough to cover even nearby targets and
only transverse position of the laser, but not angle, influ-
ences the image position. The lightpipe is evacuated to
about 15 microns of Mercury to prevent deflection of the
alignment image due to refraction in air. Sections of the
lightpipe, which are about 12 meters long, are connected
via bellowsthatallowindependentmotionoradjustment.
AschematicofthemeasurementsetupisshowninFig.1.
The measurements were done with a single lens inserted
whichwasnotmoveduntilthemeasurementswerefinished
inordertoensuremaximalaccuracy. (Inmultitargetmode
the repeatability of the target positioning limits the accu -
racy). We used the lens 14-9located at the end of the 14th
sectorof30total,almostexactlyinthemiddleofthelinac.
For these measurements, we replaced the standard de-
tector for this system with a quadrant photodetector (pro-
ducedbyHamamatsu)whichhasaquadraticsensitivearea
(∼10×10mm2)dividedintofoursectors. Bycombining
preamplified signals uifrom these quadrants, the quantity
to be measured X=x1+x3−2x2(see Fig.1) can be
determinedas X∝[(u1+u2)−(u3+u4)]/Σuiforboth
the horizontal and vertical ( Y) planes. Calibration of the
system was donebymovingthe detectortransversely. The
sensitivityis linearin therangeof ±1mm.
The measured data are shown in Fig.2. Two particular
characteristics are clearly seen: the tidal componentof th e
motionisverypronouncedandthereisastrongcorrelation
ofthemotionwithexternalatmosphericpressure.
Thelinactunnelwasclosed,withtemperaturestabilized
water through the RF structures during the entire period
SLC Laser Lightpipe FresnelLens
detectorQuadrant
~1500m~1500mLaser
xx
23
x1
Figure1: Schematicofthemeasurementsetup.342 346 350 354 358 362 366 370
Time□(day)-80-4004080X,□Y□(micron)
101210161020102410281032
Pressure□(mB)P
XY
19992000
Figure 2: Measured horizontal Xand vertical Ydisplace-
mentsplottedalongwith externalatmosphericpressure.
of the measurements. The girder temperature was stable
within 0.1oC over a day and within a few 0.1oC over a
week. TheRFpowerwasswitchedoffstartingDec.24and
turned on again Jan. 3. This resulted in a slow (weekly)
changeofthegirdertemperatureby0.5oCinthemiddleof
the linac and 1.5oC at the beginning. The average exter-
nal temperature varied by about 10oC over the month. No
significant correlationof the measureddata with these and
otherparameterswasobserved.
Figure3: Subsetofdatawheretidesareseenmost clearly.
Figure 4: Normalized 1/χ2showing quality of fit of the
measured data by sum of 37 tidal harmonics. Behavior of
1/χ2if the speed of one harmonic would vary. Example
fortheharmonicsM2(principallunar),N2andJ1. Vertical
linesshowtheoreticalspeedoftheseharmonics.
The tidal component of the motion has a surprisingly
large amplitude ( ∼10µm) (see Fig.3). The most pro-
nouncedharmonicsinthemeasureddataareM2(principal
lunar), N2 and J1. The primaryeffect of tidal deformation
istochangetheslopeoftheearth’ssurface( ∼100µm/1km
assumingtotaldeformation ∼0.5m). Thesecondaryeffect
is to change the curvatureof the surface ( ∼0.01µm/1km2
if one assumes uniform earth deformation). The laser sys-
temisnotsensitivetotheslopechange,butonlytothecur-Figure 5: Correlation (real and imaginary parts) of dis-
placementwith atmosphericpressure.
Figure 6: Spectra of displacement (multiplied by f2) and
thenoiseofelectronics. Peaksaround 10−5Hzcorrespond
to tides. Horizontallinescorrespondtothe ATLspectra.
vaturechange,whichisanadvantage. Theobserved 10µm
changeofthecurvaturecanonlybeexplainedifa localef-
fectofthetides,with Reffective ∼500km,isassumed. This
local anomaly at SLAC is caused by loading on the coast-
line as the ocean water level varies due to the tides. This
phenomenonhas been knownfor many yearsand is called
ocean loading. This effect is also responsible for an en-
hancement of the tidal variation of the earth surface slope
observed in the San Francisco Bay Area [5]. The ocean
loading effect vanishes away from the coastline. Regard-
less, these tidal effects are harmless for a linear collider ,
becausethe motionisslow, verypredictableand,most im-
portantly,hasa wavelengthmuchlongerthanthelengthof
the accelerator.
Correlation of the tunnel deformation with changes of
external atmospheric pressure, clearly seen in Fig.2 and
5, is significant from the lowest observed frequency up to
∼0.003Hz. Above this frequency the characteristic size
over which the pressure changes, which is ∼vw/f,where
vwis the wind velocity (typically 5m/s), becomes shorter
than the linac length and the correlations vanish. In this
frequencyrange, the ratio of deformationto pressure is al-
most constant at about 6 µm/mbar in Y and 2 µm/mbar in
X. Theinfluenceofsuchglobalchangesofpressureonthe
grounddeformationcanbeexplainedifthelandscapeorthe
ground properties vary along the linac. One should note
that deformations of the lightpipe itself or motion of the
targets caused by external pressure variation appear to be
eliminatedbydesign[4].
Thespectraofthetunneldeformationsexhibits 1/f2be-
havior over a large frequency band (see Fig.6). The 1/f2behaviorvanishesat f>∼0.01Hzwherethesignaltonoise
ratio becomes poor due to noise in the detector and elec-
tronics. Evaluation of this noise, also shown in Fig.6, has
been done by means of a light source attached directly to
the photodetector. The spot size and intensity of this light
source were very similar to those of the laser. Influences
of other sources of error (vacuum and temperature varia-
tion in the lightpipe, temperature in the tunnel, etc.) were
analyzedbutwere foundtobeinsignificant.
Above 0.1 Hz the signal to noice ratio again becomes
good as seen in the Fig.6. This is also confirmed by com-
parison of the measured lightpipe displacement with mea-
surements from a vertical broadband (0.01-100Hz) seis-
mometer STS-2 installed at the beginning of the linac,
whichmeasurestheabsolutemotionoftheground. Thesi-
multaneous measurementsof the tunnel motion and of the
absolute motion by STS-2 were performed during 3 days
fromJanuary4toJanuary7. ThecoherencebetweenSTS-
2andverticaldisplacementmeasuredbyphotodetectorwas
foundtobeabout0.5at F>∼0.2Hz.
During the 3-day period when the tunnel motion was
measured simultaneously with STS-2, only two remote
earthquakeswere detected by the seismometer. One of the
earthquakesdidnotproduceanynoticableeffectonthemo-
tion measured by the photodetector, probably because of
the specific orientation of the waves. The second earth-
quake,however,wasclearlyseeninbothsignals,asshown
inFig.7. Theratioofthemeasuredabsolutemotionandthe
relativedeformationofthetunnelisconsistentwithaphas e
velocity of about 2.5 km/s, consistent with earlier correla -
tionmeasurementsperformedatSLAC[1].
Figure 7: Displacement of the tunnel and displacement
measuredbySTS-2seismometerduringremoteearthquake
started January6, 2000at 02:49:00local time (supposedly
corresponds to 5.8MS earthquake at Alaska happened at
10:42:27 UTC). A passband filter 0.02–0.08Hz has been
appliedtothedata.
One model of slow ground motion is described by the
ATL-law [3]. For our 3 point motion, the ATL spec-
trum corresponding to the measured XorYisP(ω) =
4AL
ω2withL= 1500m. Fig.6 shows that the measured
spectrum corresponds to a parameter Aof about 10−7–
2·10−6µm/(m·s),somewhatchangingwithfrequency.
Spectral analysis of subsets of the data, however, shows
that this parameter actually varies in time (see Fig.8). The
variation of atmospheric activity is again responsible for342 346 350 354 358 362 366 370
Time□(day)1E-71E-6A□(micron**2/(m*s)□)
20001999Ax
Ay
Figure 8: Parameter Adefined from fit to spectra in the
band2.44E-4to 1.53E-2Hzforall data.
Figure 9: Parameter Aydefined from all vertical motion
datainthefrequencyband 3·10−5–10−3Hzversusampli-
tudeApofthe atmosphericpressurespectrum.
the variationof parameter A. The spectra of pressure fluc-
tuationswas foundto behavealso as Ap/ω2andits ampli-
tudeApcorrelates with the parameter A, as seen in Fig.9.
The temporal pressure variation can therefore be a major
drivingtermof the A/ω2-like motion. Thiseffectstrongly
dependsongeology[7].
3 CONCLUSION
Atmospheric pressure changes were found to be a major
causeofslowmotionoftheshallowSLAClinactunnel. In
deep tunnels or in tunnels built in more solid ground, this
mechanismwouldvanish,asitandlocation. Othersources
couldthendominate.
I would like to thank G.Bowden, T.King, G.Mazaheri,
M.Ross, M.Rogers, L.Griffin, R.Ruland, R.Erickson,
T.Graul, B.Herrmannsfeldt, R.Pitthan, C.Adolphsen,
N.Phinney and T.Raubenheimer for help, technical
assistance andvaluablediscussions.
4 REFERENCES
[1] NLCZDRDesign Group, SLACReport-474 (1996).
[2] C.Adolphsen, G. Bowden, G.Mazaheri, inProc.of LC97.
[3] B.Baklakov, P.Lebedev, V.Parkhomchuk, A.Seryi,
A.Sleptsov,V.Shiltsev, Tech.Phys. 38, 894 (1993).
[4] W.B.Herrmannsfeldt, IEEETrans. Nucl. Sci. 12, 9 (1965).
[5] MiltonD.Wood, Ph.D.thesis, Stanford, May1969.
[6] R. Assmann, C. Salsberg, C. Montag, SLAC-PUB-7303, in
Proceed. of Linac 96, Geneva, (1996).
[7] A.Seryi,EPAC2000, alsointhese Proceed. |
arXiv:physics/0008196v1 [physics.acc-ph] 20 Aug 2000ASECOND-ORDER STOCHASTICLEAP-FROG ALGORITHMFOR
LANGEVINSIMULATION∗
JiQiang and SalmanHabib, LANL,LosAlamos,NM 87545,USA
Abstract
Langevinsimulationprovidesaneffectivewaytostudycol-
lisional effects in beams by reducing the six-dimensional
Fokker-Planck equation to a group of stochastic ordinary
differential equations. These resulting equations usuall y
havemultiplicativenoisesince thediffusioncoefficients in
these equations are functions of position and time. Con-
ventional algorithms, e.g. Euler and Heun, give only first
order convergenceof moments in a finite time interval. In
this paper,a stochastic leap-frogalgorithmforthe numeri -
calintegrationofLangevinstochasticdifferentialequat ions
with multiplicative noise is proposed and tested. The al-
gorithm has a second-order convergence of moments in a
finite time interval and requires the sampling of only one
uniformly distributed random variable per time step. As
an example, we apply the new algorithm to the study of a
mechanicaloscillatorwithmultiplicativenoise.
1 INTRODUCTION
Multiple Coulomb scattering of charged particles, also
called intra-beam scattering, has important applications in
accelerator operation. It causes a diffusion process of par -
ticles and leads to an increase of beam size and emittance.
This results in a fast decay of the quality of beam and re-
ducesthe beam lifetime when the size of the beam is large
enoughtohitthe aperture[1].
AnappropriatewaytostudythemultipleCoulombscat-
tering is to solve the Fokker-Planck equations for the dis-
tribution function in six-dimensional phase space. Never-
theless, the Fokker-Planckequationsare very expensiveto
solve numerically even for dynamical systems possessing
onlyaverymodestnumberofdegreesoffreedom. Trunca-
tionschemesorclosureshavehadsomesuccessinextract-
ingthebehavioroflow-ordermoments,butthesystematics
of these approximations remains to be elucidated. On the
other hand, the Fokker-Planckequationscan be solved us-
ing an equivalent Langevin simulation, which reduces the
six-dimensional partial differential equations into a gro up
of stochastic ordinary differential equations. Compared t o
the Fokker-Planck equation, stochastic differential equa -
tionsarenotdifficultto solve,andwith theadventofmod-
ern supercomputers, it is possible to run very large num-
bersofrealizationsinordertocomputelow-ordermoments
accurately. In general, the noise in these stochastic ordi-
narydifferentialequationsaremultiplicativeinsteadof ad-
∗WorksupportedbyDOEGrandChallengeinComputational Acce ler-
ator Physics, Advanced Computing for 21st Century Accelera tor Science
and Technology Project, and Los Alamos Accelerator Code Gro up using
resources at the Advanced Computing Laboratory and the Nati onal En-
ergy Research Scientific Computing Center.ditive since the dynamic friction coefficient and diffusion
coefficient in the Fokker-Planck equations depend on the
spatial position. An effective numerical algorithm to inte -
gratethestochasticdifferentialequationwithmultiplic ative
noisewillsignificantlyimprovetheefficiencyoflargescal e
Langevinsimulation.
Thestochasticleap-frogalgorithmsintheLangevinsim-
ulationaregiveninSectionII.Numericaltestsofthisalgo -
rithms is presented in Section III. A physical application
of the algorithm to the multiplicative-noise mechanic os-
cillator is given in Section IV. The conclusions are drawn
in SectionV.
2 STOCHASTIC LEAP-FROG
ALGORITHM
In the Langevin simulation, the stochastic particle equa-
tions of motion that follow from the Fokker-Planck equa-
tionare(Cf. Ref. [2])
r′=v, (1)
v′=F
m−νv+√
DΓ(t), (2)
whereFis the force including both the external force and
the self-generated mean field space charge force, mis the
massofparticle, νisfrictioncoefficient, Disthediffusion
coefficient,and Γ(t)areGaussianrandomvariableswith
/an}bracketle{tΓi(t)/an}bracketri}ht= 0, (3)
/an}bracketle{tΓi(t)Γi(t′)/an}bracketri}ht=δ(t−t′). (4)
In the case not too far from thermodynamic equilibrium,
the frictioncoefficientisgivenas
ν=4√πn(r)Z4e4ln (Λ)
3m2(T(r)/m)3/2(5)
and the diffusion coefficient DisD=νkT/m[3]. Here,
n(r)isthedensityofparticle, T(r)isthetemperatureofof
beam, Zisthechargenumberofparticle, eisthechargeof
electron, Λis the Coulomb logarithm, and kis the Boltz-
mann constant. For the above case, noise terms enter only
in the dynamical equations for the particle momenta. In
Eqn. (6) below, the indices are single-particle phase-spac e
coordinateindices;theconventionusedhereisthattheodd
indicescorrespondtomomenta,andtheevenindicestothe
spatial coordinate. In the case of three dimensions,the dy-
namicalequationsthentakethegeneralform:
˙x1=F1(x1, x2, x3, x4, x5, x6) +σ11(x2, x4, x6)ξ1(t)
˙x2=F2(x1)˙x3=F3(x1, x2, x3, x4, x5, x6) +σ33(x2, x4, x6)ξ3(t)
˙x4=F4(x3)
˙x5=F5(x1, x2, x3, x4, x5, x6) +σ55(x2, x4, x6)ξ5(t)
˙x6=F6(x5) (6)
In the dynamical equationsfor the momenta,the first term
on the right hand side is a systematic drift term which in-
cludesthe effectsdue to externalforcesanddamping. The
second term is stochastic in nature and describes a noise
forcewhich,ingeneral,isafunctionofposition. Thenoise
ξ(t)is first assumed to be Gaussian and white as defined
by Eqns. (3)-(4). The stochastic leap-frog algorithm for
Eqns.(6)iswrittenas
¯xi(h) = ¯Di(h) +¯Si(h) (7)
The deterministic contribution ¯Di(h)can be obtained us-
ing the deterministic leap-frog algorithm. Here, the deter -
ministiccontribution ¯Di(h)andthestochasticcontribution
¯Si(h)of the above recursion formula for one-step integra-
tionare foundtobe
¯Di(h) = ¯ xi(0) +hFi(¯x∗
1,¯x∗
2,¯x∗
3,¯x∗
4,¯x∗
5,¯x∗
6);
{i= 1,3,5}
¯Di(h) = ¯ x∗
i
+1
2hFi[xi−1+hFi−1(¯x∗
1,¯x∗
2,¯x∗
3,¯x∗
4,¯x∗
5,¯x∗
6)] ;
{i= 2,4,6}
¯Si(h) = σii√
hWi(h) +1
2Fi,kσkkh3/2˜Wi(h)
+1
2σii,jFjh3/2˜Wi(h)
+1
4Fi,klσkkσllh2˜Wi(h)˜Wi(h);
{i= 1,3,5;j= 2,4,6;k, l= 1,3,5}
¯Si(h) =1√
3Fi,jσjjh3/2˜Wj(h)
+1
4Fi,jjσ2
jjh2˜Wj(h)˜Wj(h)
{i= 2,4,6;j= 1,3,5}
¯x∗
i= ¯xi(0) +1
2hFi(¯x1,¯x2,¯x3,¯x4,¯x5,¯x6)
{i= 1,2,3,4,5,6} (8)
where ˜Wi(h)is a series of random numbers with the mo-
ments
/an}bracketle{t˜Wi(h)/an}bracketri}ht=/an}bracketle{t(˜Wi(h))3/an}bracketri}ht=/an}bracketle{t(˜Wi(h))5/an}bracketri}ht= 0(9)
/an}bracketle{t(˜Wi(h))2/an}bracketri}ht= 1,/an}bracketle{t(˜Wi(h))4/an}bracketri}ht= 3 (10)
This can not only be achieved by choosing true Gaussian
randomnumbers,butalsobyusingthesequenceofrandom
numbersfollowing:
˜Wi(h) =
−√
3, R < 1/6
0, 1/6≤R <5/6√
3, 5/6≤R(11)2.092.12.112.122.132.142.152.162.172.182.19
0 0.1 0.2 0.3 0.4 0.5 0.6<x*x>
hwhite noise without damping
Figure1: Zerodampingconvergencetest. /an}bracketle{tx2(t)/an}bracketri}htatt= 6
as a functionof step size with white Gaussian noise. Solid
linesrepresentquadraticfitstothedata points(diamonds) .
where Ris a uniformlydistributed random number on the
interval(0,1). Thistricksignificantlyreducesthecomput a-
tionalcost ingeneratingrandomnumbers.
3 NUMERICAL TESTS
The above algorithm was tested on a one-dimensional
stochastic harmonic oscillator with a simple form of the
multiplicativenoise. Theequationsofmotionwere
˙p=F1(p, x) +σ(x)ξ(t)
˙x=p (12)
where F1(p, x) =−γp−η2xandσ(x) =−αx. The
stochastic leapfrog integrator for this case is given by
Eqns. (8) (white noise) with the substitutions x1=p,
x2=x.
As a first test, we computed /an}bracketle{tx2/an}bracketri}htas a function of time-
step size. Tobegin,we tookthecase ofzerodampingcon-
stant ( γ= 0), where /an}bracketle{tx2/an}bracketri}htcan be determined analytically.
The curve in Fig. 1 shows /an}bracketle{tx2/an}bracketri}htatt= 6.0as a function
of time-step size with white Gaussian noise. Here, the pa-
rameters ηandαare set to 1.0and0.1. The analytically
determined value of /an}bracketle{tx2/an}bracketri}htatt= 6.0is2.095222. The
quadraticconvergenceofthestochasticleap-frogalgorit hm
is clearly seen in the numerical results. We also verified
thatthequadraticconvergenceispresentfornonzerodamp-
ing (γ= 0.1). Att= 12.0, and with all other parameters
asabove,theconvergenceof /an}bracketle{tx2/an}bracketri}htasafunctionoftimestep
is shown by the curve in Fig. 2. As a comparison against
the conventional Heun’s algorithm [5], we computed /an}bracketle{tx2/an}bracketri}ht
as a functionof tusing100,000numericalrealizationsfor
a particlestartingfrom (0.0,1.5)inthe (x, p)phasespace.
Theresultsalongwiththeanalyticalsolutionandanumer-
ical solution using Heun’s algorithm are given in Fig. 3.
Parametersused were h= 0.1,η= 1.0,andα= 0.1. The
advantageinaccuracyofthestochasticleap-frogalgorith m
over Heun’s algorithm is clearly displayed, both in terms
oferroramplitudeandlackofasystematic drift.0.460.4650.470.4750.480.4850.490.4950.50.5050.510.515
0 0.1 0.2 0.3 0.4 0.5 0.6<x*x>
hwhite noise with damping
Figure 2: Finite damping ( γ= 0.1) convergence test.
/an}bracketle{tx2(t)/an}bracketri}htatt= 12as a function of step size with white
Gaussian noise. Solid lines represent quadratic fits to the
data points(diamonds).
-204812
0 100 200 300 400 500tExact
Error: Heun
Error: Leapfrog<<X2
Figure3: Comparingstochasticleap-frogandthe Heunal-
gorithm: /an}bracketle{tx2(t)/an}bracketri}htasafunctionof t. Errorsaregivenrelative
to theexactsolution.
4 APPLICATION
In this section, we apply our algorithm to studying the ap-
proach to thermal equilibrium of an oscillator with multi-
plicativenoise. Thegoverningequationsare:
˙p=−ω2
0x−λx2p−√
2Dxξ2(t)
˙x=p (13)
where the diffusion coefficients D=λkT,λis the cou-
pling constant, and ω0is the oscillator angular frequency
without damping. In Fig. 4, we display the time evolu-
tion of the average energy with multiplicative noise from
the simulations and the approximate analytical calcula-
tions [6]. The analytic approximation resulting from the
application of the energy-envelopemethod is seen to be in
reasonable agreement with the numerical simulations for
kT= 4.5. The slightly higher equilibrium rate from the11.522.533.544.55
0 100 200 300 400 500 600 700 800<E(t)>
tnumerical simulation
analytical approximation
Figure 4: Temporalevolutionof the scaled averageenergy
/an}bracketle{tE(t)/an}bracketri}htwithmultiplicativenoisefromnumericalsimulation
andanalyticalapproximation.
analytical calculationis due to the truncationin the energ y
envelope equation using the /an}bracketle{tE2(t)/an}bracketri}ht ≈2/an}bracketle{tE(t)/an}bracketri}ht2relation
whichyieldsanupperboundontherateofequilibrationof
the averageenergy[6].
5 CONCLUSIONS
We have presented a stochastic leap-frog algorithm for
Langevin simulation with multiplicative noise. This
method has the advantages of retaining the symplectic
property in the deterministic limit, ease of implementa-
tion, and second-order convergence of moments for mul-
tiplicative noise. Sampling a uniform distribution instea d
of a Gaussian distribution helps to significantly reduce the
computational cost. A comparison with the conventional
Heun’salgorithmhighlightsthegaininaccuracyduetothe
new method. Finally, we have applied the stochastic leap-
frogalgorithmtoanonlinearmechanic-oscillatorsystemt o
investigatethethenatureofthe relaxationprocess.
6 ACKNOWLEDGMENTS
We acknowledgehelpfuldiscussionswith GrantLytheand
RobertRyne.
7 REFERENCES
[1] A. Piwinski, Proc. 9th Int. Conf. on High Energy Accelera -
tors,Standord, 1974 (SLAC,Stanford, 1974) p.405.
[2] H.Risken, TheFokker-PlanckEquation: MethodsofSolution
and Applications (Springer, New York,1989).
[3] M. E. Jones, D. S. Lemons, R. J. Mason, V. A. Thomas, and
D.Winske, J. Comput. Phys. 123, 169(1996).
[4] R.Zwanzig, J. Stat.Phys. 9,215 (1973).
[5] A. Greiner, W. Strittmatter, and J. Honerkamp, J. Stat. P hys.
51, 94(1988).
[6] K.Lindenberg and V.Seshadri, Physica 109A,483 (1981). |
RF PROCESSING OF X-BAND ACCELERATOR STRUCTURES
AT THE NLCTA*
C. Adolphsen, W. Baumgartner, K. Jobe, R. Loewen, D. McCormick,
M. Ross, T. Smith, J.W. Wang, SLAC, Stanford, CA 94309 USA
T. Higo, KEK, Tskuba, Ibaraki, Japan
Abstract
During the initial phase of operation, the linacs of the
Next Linear Collider (NLC) will contain roughly 5000 X-Band accelerator structures that will accelerate beams of electrons and positrons to 250 GeV. These structures will nominally operate at an unloaded gradient of 72 MV/m. As part of the NLC R&D program, several prototype structures have been built and operated at the Next Linear Collider Test Accelerator (NLCTA) at SLAC. Here, the effect of high gradient operation on the structure performance has been studied. Significant progress was made during the past year after the NLCTA power sources were upgraded to reliably produce the required NLC power levels and beyond. This paper describes the structures, the processing methodology and the observed effects of high gradient operation.
1 INTRODUCTION
Over the past four years, four NLC prototype X-Band
(11.4 GHz) accelerator structures have been processed to gradients of 50 MV/m and higher at the NLCTA [1]. The structures are traveling wave (2
π/3 phase advance per
cell), nearly constant gradient (the group velocity varies from 12% to 3% c) and 1.8 m long (206 cells) with a fill time of 100 ns. They were built in part to test methods of long-range transverse wakefield suppression. In two of the structures (DS1 and DS2), the deflecting modes are detuned, and in the other two (DDS1 and DDS2), they are damped as well [2]. The changes made for these purposes should not affect their performance as high gradient accelerators, which is the focus here. For completeness, results are included from a prototype JLC structure (M2) that was processed at the Accelerator Structure Test Area (ASTA) at SLAC [3]. This detuned structure is shorter (1.3 m, 150 cells) and has a somewhat lower group velocity (10% to 2% c) than the NLC structures.
2 FABRICATION AND HANDLING
The cells of four of the five structures where single-
diamond turned on a lathe, which produces better than 50 nm rms surface roughness: those of DS1 where turned with poly-crystalline diamond tools which yields a surface roughness of about 200 nm rms. Before assembly, the cells were chemically cleaned and lightly etched in a
several step process interleaved with water rinses. For the NLC structures, the steps include degreasing, alkaline soak, acid etch and finally an alcohol bath followed by blow drying with N
2. The M2 cells were cleaned in a
similar manner but with a weaker acid etch and the final cleaning was done with acetone. The bonding and brazing of the cells into structures were done in a hydrogen furnace for the NLC structures and in a vacuum furnace for M2. Before installation in the NLCTA, the NLC structures were vacuum baked at 450-550
°C for 4-6 days
and filled with N 2. The M2 structure was filled with N 2
after assembly and baked in situ at 250 °C for 2 days after
installation in ASTA. Based on the structure geometries and the vacuum pump configuration, the maximum pres-sure levels in the NLC structures after several days of pumping were estimated to be in the low 10 -
8 Torr scale
while the M2 pressure was likely in the mid 10 -8 Torr
scale.
3 RF PROCESSING
Of the five structures, only two (DDS1 and M2) were
systematically processed to gradients that were not limited by available power (about 200 MW was needed to produce NLC-like gradients). The other three structures were processed in the NLCTA to gradients of about 50 MV/m and used for beam operation. The bulk of processing was done at 60 Hz with 250 ns pulses (100 ns ramp, 150 ns flat top) at NLCTA and 150 ns square pulses at ASTA. The processing rate was paced by rf breakdown in the structures that reflected power toward the sources (klystrons) and increased the structure vacuum pressures. To protect the klystron windows, the rf power was shut off when more than 5 MW of reflected power was detected just downstream of them. The rf was then kept off for a period of time (30 seconds to many minutes) to allow the structures to pump down. Without this interlock, it is likely that the gas pressure would have built up in the structures over many pulses, causing breakdown on every pulse (the pressure threshold to cause breakdown appears to be in the high 10 -
7 Torr scale). Thus, the reflected
power limit did not greatly hinder the processing, and in fact it suppressed continuous breakdown (as added protection, the rf was automatically shut off if specific pump pressures were exceeded). ________________
*
Work Supported by DOE Contract DE-AC03-76F00515. The initial processing of the structures was done manu-
ally. That is, the power was slowly increased by an operator who also decided when to reset the power after a trip, for example, by monitoring vacuum pump readings. Such operator oversight was important since breakdown was generally accompanied by large pressure increases in the structures (10 to 100 times higher). Above gradients of about 60 MV/m, however, the pressure increases were typically below a factor of ten and sometimes too small to be detectable. With the smaller pressure increases, auto-mated control of the processing was practical. It was developed for both M2 and DDS1 processing, allowing unattended, around-the-clock operation, which improved the processing efficiency.
For DDS1, the processing control algorithm was typi-
cally setup to increase the structure input power by 1% if a reflected power trip did not occur within 2 minutes, and to decrease the power by 2% if a trip occurred within 10 seconds. These times were measured relative to the resetting of the rf power, which was ramped-up over a 30 second period starting immediately after a trip. If a trip occurred between 10 seconds and 2 minutes, the power level was not changed. Also, it would not be increased if any pump pressure readings were above tolerance levels. The algorithm for processing M2 used roughly the same logic.
The automated processing yielded smoothly varying
peak power levels when viewed over several hour time intervals. The mean power level depended on the algorithm parameters and on the reflected power trip threshold, which as noted above, was set to protect the klystron windows. When processing DDS1 near its maximum power level, roughly half of the trips were immediately proceeded by one or more consecutive pulses with significant reflected power. The first pulse in the sequence likely initiated a pulse-by-pulse gas build-up that eventually produced reflected power large enough to cause the trip (in many of these cases, the location of the breakdown appears to move upstream in the structure during this sequence of pulses). The distribution of reflected power per pulse was broad and peaked at low values. Above the minimum threshold that could be set, the rate of reflected power pulses was roughly ten times that above the nominal threshold (about 100/hour compared to 10/hour). If the trip threshold were lowered during processing, fewer multi-pulse trips would occur but the increase in the trip rate decreased the steady-state power level. So the nominal threshold was used to expe-dite processing.
The M2 structure was processed in ASTA during a
several week period dedicated for this purpose. An average gradient of 50 MV/m was reached in about 20 hours. Achieving higher gradients, however, took exponentially longer. The processing was stopped after 440 hours at which time a maximum gradient of 85 MV/m had been attained. Prior to its use in NLCTA, DS1 was also processed in ASTA [5]. Again, 50 MV/m was achieved fairly quickly, in about 30 hours. After 200
hours, the maximum source power was reached, which produced a 68 MV/m structure gradient.
In contrast, DDS1 was processed in a piecewise manner
over a three year period because of rf power generation and transport limitations (which were eventually over-come), and because it was used for beam operation. Thus, its processing history is harder to quantify. After 55 MV/m was reached using the nominal NLCTA rf pulse (100 ns ramp, 150 ns flat top), the pulse length was shortened (100 ns ramp, 50 ns flat top) in an attempt to speed up processing. Over the course of several hundred hours, the gradient was increased to 73 MV/m, after which it remained nearly unchanged during 300 hours of processing. A 250 ns square pulse was then used to better simulate NLC operation, which immediately reduced the maximum gradient to 70 MV/m. During the last 600 last hours of processing, the gradient has not increased above this level.
4 EFFECT OF PROCESSING
Past experience has shown that rf breakdown causes
surface damage to the tips of cell irises where the fields are highest (about twice the accelerator gradient). To quantify any changes in the NLC/JLC structures, both visual inspections and rf measurements were made. For the latter, a bead-pull technique was used as the primary means to measure the rf phase profile along the structures relative to the nominal phase advance [4]. Since a nylon string has to be pulled through the structure in this procedure, a noninvasive method of determining the phase profile using beam induced rf was developed. However, the dispersion that occurs during the rf propagation through the structure introduces systematic phase changes, so this technique is best used for measuring relative phase changes.
Table 1 summarizes the processing results from the five
structures including the net phase change and the number of cells with discernable phase shifts. The phase changes for all but DDS1 were determined from bead-pull meas-urement comparisons: the DDS1 value is based on a com-parison of the initial bead-pull profile with the latest beam-based measurement. As an example, Figure 1 shows
Table 1: Structure Processing Summary
Structure Hours
Operated Max
Grad.
(MV/m) Phase
Change
(deg.) # Cells
Affected
M2 440 85 25 70
DS1 550 54 7 80
DDS2 550 54 8 100
DS2 1000 50 20 150
DDS1 2700 73 60 120 bead-pull measurements of DS2 before and after high
power operation. One sees that a phase shift occurs at the upstream end of the structure, which is also true in the other four. In Figure 2, beam-based phase measurements of DDS1 are shown at three different times separated by about 300 hours of processing with the short pulse (the gradient was 60-70 MV/m in the first interval, and 68-73 MV/m in the second). Each curve is the phase profile of the rf induced by the passage of a 20 ns bunch train (11.4 GHz bunch spacing) through the structure. The rf is coupled out at the end of the structure so the rf induced in the first cell comes out last. The large phase increase after 100 ns is due to dispersion. The dispersion also smoothes out any fast phase variations as does the finite length of the bunch train (a 2 ns train is now being used). However, a progressive increase in the phase can be seen. The direction corresponds to an increase in the cell frequen-cies, which would occur if copper were removed from the tips of the cell irises (removing a 10
µm layer around the
curved portion of the irises yields a 1 ° phase shift per cell
in the front half of DDS1 where the irises are 1.0 to 1.5 mm thick). The phase shifts seen in the bead-pull comparisons also correspond to higher frequencies.
In Table 1, one sees that the phase change increases
with both gradient and operation time. The DS1 and DDS2 results are interesting to compare, especially since these structures were powered from the same rf source. While DS1 had been processed earlier (see above) and has conventionally machined cells, its phase change during operation in the NLCTA is nearly the same as DDS2, which was installed new and has diamond-turned cells. (DS1 was retuned after its processing at ASTA: the phase shift incurred there is not known).
The visual inspections of the structures were done using
a boroscope that could access the first and last 30 cells. All structures except DDS1 were examined. Essentially no damage was observed at the downstream end of M2 while the NLC structures showed a small amount of pitting on the tips of the irises in this region. The pits are generally less than 30
µm wide and cover less than a few
percent of the surface area. The depths of the pits are hard to estimate, but are probably less than their widths. With
this aspect ratio, it takes little rf energy to create them. For example, just 10
-5 of the energy in an rf pulse would
vaporize (30 µm)3 of copper if it were converted to heat.
For the upstream cells in both M2 and the NLC structures, it looks as if pitting has completely eroded off a layer of the iris surfaces, leaving them covered with 50-100
µm
wide dimples. Also, the surface color is a dull silver in contrast to the shiny copper color of the downstream cells.
5 CONCLUSION
Five prototype NLC/JLC structures have been proc-
essed to high power and show upstream iris damage and phase advance changes at gradients as low as 50 MV/m. One explanation is that the lower rf propagation imped-ance at the upstream ends of the structures (due to the higher group velocity) leads to more energy being absorbed in breakdown arcs, which act as low impedance loads [6]. This model may also explain why early proto-type cavities and structures that had long fill times or low group velocities performed well at high gradients (for example, a 75 cm long NLC structure with 5% group velocity was processed to 90 MV/m without any apparent phase change). A series of low group velocity structures are being built to verify this model as a first step to developing a high gradient version for NLC/JLC. Also, the characteristics of structure breakdown are being exten-sively studied [7,8].
6 REFERENCES
[1] R. D. Ruth et al., SLAC-PUB-7288 (June 1997). [2] J. Wang et al., PAC 99 Proc., p. 3423 (April 1999). [3] R. Loewen et al., SLAC-PUB-8399 (June 1997). [4] S. Hanna et al., SLAC-PUB-6811 (June 1995). [5] J. Wang et al., SLAC-PUB-7243 (August 1996). [6] C. Adolphsen, SLAC-PUB-8572 (in progress). [7] C. Adolphsen et al., SLAC-PUB-8573 (Sept. 2000),
which is an expanded version of this paper.
[8] Joe Frisch et.al., TUE03, these proceedings. -25-20-15-10-50510
0 50 100 150 200Net Phase Error (degrees)
Cell Number
Fig. 1: Bead-pull measurement of the DS2 phase profile
before (solid) and after (dotted) 1000 hours of high power
operation. -20020406080
0 20 40 60 80 100RF Phase (degrees)
Time (ns)
Fig. 2: Phase of beam induced rf measured at different
times separated by about 300 hours of processing (first
measurement = solid, second = dashed and third = dotted). |
FABRICATION AND TOLERANCE ISSUES AND THEIR INFLUENCE ON
MULTI-BUNCH BBU AND EMITTANCE DILUTION IN THE
CONSTRUCTION OF X-BAND RDDS LINACS FOR THE NLC1
R.M. Jones, R.H. Miller, T.O. Raubenheimer, and G.V. Stupakov; SLAC, Stanford, CA, USA
_____________
1 Supported under U.S. DOE contract DE-AC03-76SF00515.Abstract
The main linacs of the Next Linear Collider (NLC) will
contain several thousand X-band RDDS (RoundedDamped Detuned Structures). The transverse wakefieldin the structures is reduced by detuning the modalfrequencies such that they destructively interfere and byfour damping manifolds per structure which provide weakdamping. Errors in the fabrication of the individual cellsand in the alignment of the cells will reduce thecancellation of the modes. Here, we calculate thetolerances on random errors in the synchronousfrequencies of the cells and the cell-to-cell alignment.
Figure 1: Machined RDDS1 Cells
1. INTRODUCTION
In order to answer fundamental questions posed by
particle physics a high-energy e+-e- linear collider is being
designed at SLAC and KEK with an initial center-of-massenergy of 500 GeV and the possibility of a later upgradesto 1.0 TeV or 1.5 TeV. The heart of the collider consistsof two linear accelerators constructed from approximately10,000 X-band accelerating structures. These linacs willaccelerate a multi-bunch particle beam from 8GeV to500GeV. Each accelerating structure consists of 206 cells(two of which are shown in Fig. 1) which are bondedtogether. A displacement of the beam in the structuregives rise to a transverse deflecting force, or wakefield.There are two effects that are of concern: first, thetransverse wakefield can cause a multi-bunch beambreakup instability (BBU) which would make the colliderinoperable and, second, the wakefields caused by
misalignments of the cells and the structures will causemulti-bunch emittance dilution which will reduce thecollider luminosity.
The long-range transverse wakefield is reduced by
forcing the dipole modes to destructively interfere anddamping the modes with four manifolds per structure.However, errors in fabricating and aligning the cells cansignificantly increase the wakefield and thus it isimportant to carefully analyse each error component. Thefollowing section will discuss the effect of errors in thecell synchronous frequencies and the subsequent sectionfocuses on transverse cell-to-cell and structure-to-structure misalignment errors and the resulting toleranceimposed on the fabrication of the structures for aprescribed multi-bunch emittance dilution.
2. MACHINING ERRORS AND
EMITTANCE DILUTION
Small dimensional errors, generated when fabricating
the irises and cavities of an accelerator structure, give riseto errors in the synchronous frequencies [1]. Presently, itis possible to machine the cells to an accuracy of betterthan 1 µm [2,3], however, when fabricating several
thousand such structures, looser tolerances may reduce thefabrication costs.
The linacs consist of roughly 5000, nominally identical,
structures, each of which contains 206 slightly differentcells. The nomenclature that we adopt is an error typewhich is repeated in every cell of a structure but differs inevery structure is referred to as: a systematic-randomerror. Whereas, an error that is repeated in everystructure, but varies from cell-to-cell, we refer to as arandom-systematic error. We also consider random-random and systematic-systematic (potentially the mostdamaging) error types making a total of 4 error types.The random errors we consider have an RMS deviation of3MHz about the mean dipole frequency of the cells. Infabricating RDDS1, the RMS error in the synchronousfrequency prior to bonding the cells was 0.5MHz [2,3]and thus simulation of larger errors is pursued with a goalof understanding how much the cell-to-cell fabricationtolerances can be relaxed.
Cell-to-cell frequency errors within an individual
structure reduce the effect of the detuning cause a largerwakefield. Although BBU is a complicated effect, anindicator for the onset of BBU is provided by thewakefield at a particular bunch which is formed by
summing all wakefields left behind by earlier buncheswhich is denoted as the “sum wakefield” [4]. BBU willlikely arise when the RMS of the sum wake is the order of1 V/pC/mm/m or larger. When not in the BBU regime,the sum wakefield also provides an accurate method ofcalculating the multi-bunch emittance dilution and will beused in the following section.
An example of the sum wakefield for a structure with
3MHz RMS errors in the cell synchronous frequencies isplotted in Fig. 2 versus a change in the bunch spacing.Changing the bunch spacing is equivalent to changing allthe synchronous frequencies systematically. Thewakefield with the random errors is an order of magnitudelarger than in a perfect structure and if these cell errorsare reproduced in every structure it would be expected tocause significant BBU.
-0.75-0.5-0.2500.250.50.751
% Increase in Spacing0.250.50.7511.251.51.752RMS Dev. of Sum Wake
Figure 2. RMS sum wakefield for 3MHz RMS errors
This is confirmed by particle tracking simulations using
the code LIAR [5] in which the all structures are assumedto be perfectly aligned and the beam is initially offset by1µm. When all structures have identical random errors
(this is the case of random-systematic errors) and S
σ is of
the order of unity, the beam clearly undergoes BBU asillustrated in Fig. 3 and the emittance grows by roughly250%. This is supported by looking at the phase space atthe end of the linac which which is plotted in Fig. 4 (a). Incontrast, if the cell errors in every structure are different,the random-random case, BBU does not occur and theemittance growth is negligible as is also seen in Fig. 3.
Another important case, is that of an identical
systematic error in the synchronous frequencies of thecells and this is investigated by varying the spacing of thebunches in the train of particles. The case of asystematic-systematic error, corresponding to an error inall of the cell frequencies that is repeated in all of thestructures, is studied by choosing a particular bunchspacing that results in a peak in the sum wakefield. Suchan error also leads to BBU. However, imposing a smallrandom error (3MHz was utilised) from structure-to-structure prevents the resonant growth from occurring; thephase space at the end of the linac corresponding to this
random-systematic error is plotted in Fig. 4(b).
The results of relaxing the tolerance are documented in
[6] and it is found that even for the very relaxed case of a5MHz error in the synchronous frequencies BBU does notoccur and little emittance growth arises provided this cell-to-cell error is not repeatable over all structures.
0246810
BPM Position (km)050100150200250Percentage Emittance Growth
Figure 3. Emittance growth due to 3MHz RMS errors thatare (a) reproduced in every structure and (b) random fromstructure-to-structure.
-7.5-5-2.502.557.510
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(a) (b)
Figure 4. Phase Space (3MHz RMS error). The phasespace to the left (a) is for a linac composed of 4720structures assumed to have identical random errors in eachstructure. The phase space to the right (b) has beencomputed from a linac composed of structures with adifferent random error in the synchronous frequency (non-identical structures).
3. TOLERANCES IMPOSED ON
STRUCTURE ALIGNMENT
Next, assuming that BBU is not an issue, let us consider
the effect of misalignments of the cells and the structureson the multi-bunch beam emittance. In order to estimatethe growth of the projected emittance De of a train of
bunches caused by misaligned structure cells we uses thefollowing formula for the expectation value of De[5]
()1/2
0f 22 2 2
e0 s sk 1/2 3/2
0f1/rN LN S−γ γΔε = β Δγγ(3.1)
where re is the classical electron radius, N is the number
of particles in the bunch, b0is the average value of theIdentical Structure-to-
Structure Errors
Random Structure-to-
Structure Errorsbeta function at the beginning of the linac, Ns is the
number of structures in the linac, Ls is the length of the
structure, g0 and gf are the initial and final relativistic
factors of the beam, and Sk is the sum wake. The quantity
Sk is defined as a sum of the transverse wakes wi
generated by all bunches preceding the bunch number k,
Swkkik=/c61˚1 and DSkis the the difference between Sk
and the average value S, with SN SbkkNb=/c45
/c61˚1
1, where
Nb is the number of bunches. Also, SSk /c115=D212/. Eq.
(3.1) is derived assuming a lattice with the beta function
smoothly increasing along the linac as bE12/.
For small misalignments, wi is a linear function of cell
offsets, wW yii s skNc=
/c61˚1 which can be found from the
solution of Maxwell's equations for the structure. The
matrix Wis for the NLC structure RDDS1 with 206 cells is
based on the method described in [7]. It has a dimensionof N
b x 206. In our calculation we used Nb=95 for bunch
spacing 2.8 ns.
0 2 4 6 8 10
BPM Location Hkm L0246810D<e>
Figure 5: Percentage emittance growth down the linac
calculated with the tracking code LIAR for completestructures which are individually offset in a random
manner. The RMS offset of the structures is 40 µm.
In order to verify Eq. (3.1), we tracked a multi-bunch
beam through the complete linac (approx 11km) using thecomputer code LIAR [4] for RDDS1 with the followinglinac parameters: beam final energy - E
f=500 GeV,
number of structures in the linac – Ns = 4720, and number
of particles in the bunch – N =1.1 x1010. The multi-bunch
emittance growth which arises with structures rigidlymisaligned is shown in Fig. 5. It is evident that structuresrandomly misaligned with an RMS value of 40 µm gives
rise to an overall emittance growth of 10% which growslinearly along the linac as predicted by Eq. (3.1).
The result of many simulations in which each structure
is divided up into groups of cells and each individualgroup is moved randomly transverse to the axis of thelinac is illustrated in Fig. 6. It is seen that the analyticalformula based on the sum wakefield (line) generallyagrees well with the LIAR simulation (points). It should
be noted that the single bunch emittance growth due torigid structure misalignments imposes a much moresevere tolerance than that due to the multi-bunchemittance growth [8] however the multi-bunch effects setsthat tolerances on the alignment of the individual cells andshort pieces of the structure. The tolerance for the cellalignment is about 6 µm in this piecewise model.
Alternately, assuming a random walk model for the cell-to-cell alignment [9], each cell must be aligned withrespect to its neighbour with an RMS of 2 ∼3µm.
0 0.2 0.4 0.6 0.8 1
AlignmentLength StructureLength25102050100Tolerance HmmL
Figure 6: Tolerance vs. misalignment length in units of
the structure length Ls for 10% multi-bunch emittance
dilution. The solid curve shows the result of the analyticalcalculation based on Eq. (3.1); dots are the tolerancescalculated using LIAR
4. CONCLUSIONS
We have discussed four distributions of frequency
errors. BBU will arise in the NLC from cell frequencyerrors of many MHz which are repeated in everystructure. However, in practise it is expected thatfabrication errors will occur randomly from cell-to-celland from structure-to-structure and hence BBU is
unlikely to occur. Furthermore, to meet a prescribedmulti-bunch emittance growth of 10%, the cells in thepresent RRDS structure design structure must be alignedto better than 6 µm and the average alignment of the
structure must be better than 40 µm. Of course, the
average alignment tolerance is dominated by single bunchtolerances and must be closer to 10 µm[8].
5. REFERENCES
[1] R.M. Jones et al, EPAC96, (also SLAC-PUB 7187)
[2] T. Higo, et al., these proceedings (2000).[3] J.W. Wang, et al, these proceedings (2000).[4] K.L.F. Bane et al, EPAC94 (also SLAC-PUB 6581)[5] R. Assman et al,”LIAR”, SLAC-PUB AP-103, 1997[6] R.M. Jones, et al, EPAC2000, (also SLAC-PUB 8484)[7] R.M. Jones et al, PAC97 (also SLAC-PUB 7538)[8] NLC Zeroth Oder Design Report, SLAC-474 (1996).
[9] G. Stupakov, T.O. Raubenheimer, PAC99, p. 3444 |
PULSED SC PROTON LINAC
N. Ouchi, E. Chishiro, JAERI, Tokai, Japan
C. Tsukishima, K. Mukugi, MELCO, Kobe, Japan
Abstract
The superconducting (SC) proton linac is proposed in
the JAERI/KEK Joint Project for a high-intensity protonaccelerator in the energy region from 400 to 600 MeV.Highly stable fields in the SC cavities are required underthe dynamic Lorentz force detuning. A new modeldescribing the dynamic Lorentz detuning has beendeveloped and the validity has been confirmedexperimentally. The model has been applied successfullyto the rf control simulation of the SC proton linac.
1 INTRODUCTION
The Japan Atomic Energy Research Institute (JAERI)
and the High Energy Accelerator Research Organization(KEK) are proposing the Joint Project for High IntensityProton Accelerator[1,2]. The accelerator consists of 600MeV linac, 3 GeV RCS (Rapid Cycling Synchrotron)and 50 GeV synchrotron. SC structures are applied in thehigh energy part of the linac from 400 to 600 MeV.Momentum spread of the linac beams less than ±0.1% is
required for the injection to the RCS. At thecommissioning of the accelerator, 400 MeV beams willbe injected into the RCS. In this period, the SC linac willprovide the beams to the R&D for the ADS (AcceleratorDriven System) and the machine study will be carried outto obtain acceptable beam quality for the RCS. In order toincrease the beam intensity, the 600 MeV beams will beinjected into the RCS after the machine study.
The linac accelerates H
- beams in a pulsed operation;
repetition rate of 50 Hz, beam duration of 0.5 ms, peakcurrent of 50 mA and intermediate duty factor of 54 % bychopping. To meet the requirement of the RCS, rfamplitude and phase errors of the accelerating cavitiesshould be less than ±1% and ±1deg, respectively. In the
case of the SC cavities, the Lorentz force of the pulsed rffield induces dynamic deformation and detuning of thecavity, which disturb the accelerating field stability.
A new model which describes the dynamic Lorentz
force detuning has been established for the rf controlsimulation in the pulsed SC linac. The validity of themodel has been confirmed experimentally. The model hasbeen applied to the rf control simulation.
A new model, comparison between calculated and
experimental results, dynamic Lorentz detuning for themulti-cell cavity and the rf control simulation arepresented in this paper.2 MODEL FOR DYNAMIC LORENTZ
DETUNING
2.1 Stationary Lorentz Detuning
SC cavities are deformed by the Lorentz force of their
own electromagnetic field. The Lorentz pressure (P) onthe cavity wall is presented by the equation[3],
where H and E are magnetic and electric field strength on
the cavity surface. Since the cavity deformation isproportional to the Lorentz pressure, the detuning ( ∆f) is
proportional to the square of the accelerating field (E
acc)
by assuming linearity between the deformation and thedetuning. In our Joint Project, two kind of 972 MHzcavities, β
g (geometrical β of the cavity) = 0.729 and
0.771, are designed between 400 and 600 MeV region[4].The detuning constants k (= - ∆f/E
acc2) of these 7-cell
cavities are 1.61 and 1.42 Hz/(MV/m)2, respectively.
2.2 Lorentz Vibration Model
To simulate the rf control, the time-dependent cavity
field and detuning have to be solved simultaneously,because these affect each other. For this purpose, a newmodel which describes dynamic Lorentz detuning, namedLorentz Vibration Model, has been developed.
The basic idea of the model is listed below.
• The dynamic motion of the cavity is expanded in
terms of the mechanical modes. This method isknown as “Modal Analysis”.
• Cavity deformation for each mechanical mode is
converted to the partial detuning for each modeusing frequency sensitivity data.
• Total detuning is obtained by summing up the
partial detuning for each mode.
According to this basic idea, we have obtained the
Lorentz Vibration Model as the following equations.() P HE =−1
402
02µ ε ,
()() ()2
222
0
01d f
dt Qdf
dtf KV
V
K
mdf
dua F a
f f
where
Vkm k
mkk
mk k kC
k
kkk
k
k
c∆∆∆
∆ ∆++ =
=
•
•
=∑→
→→ →ωω ,
,
,
: cavity voltage (V) The inner products of (df/du) (ak) and (F0)(ak) mean the
detuning sensitivity of k-th mechanical mode and the
Lorentz force contribution to the k-th mechanical mode,
respectively. Parameters of ωmk, mk and (ak) are obtained
from the structural analysis code, ABAQUS. (F0) and
(df/du) are obtained from the SUPERFISH results.
2.3 Pulsed Operation in the Vertical Test
In order to observe the dynamic Lorentz detuning
experimentally, a pulsed operation was carried out in thevertical test of a single-cell 600MHz cavity of β
g=0.886.
In the test, one side of the cavity flange was fixed to thecryostat and the other side was free. The measurementwas made at 4.2 K. Unloaded and loaded quality factorsof the cavity were ~9 ×10
8 and ~9 ×107, respectively.
Figure 1 shows the rf power control signal, which is
proportional to the amplifier output power (max. 300W),and the surface peak field of the cavity (E
peak) in a pulse.
Rise time, flat top and repetition rate were 60 ms, 100 msand 0.76 Hz, respectively. The cavity was excited evenbetween the pulses with very low field (E
peak~0.7 MV/m)
in order to keep lock of a PLL (Phase Locked Loop)circuit. Dynamic Lorentz detuning was measured bytaking an FM control signal of the PLL circuit through alow path filter of 1 kHz. The signal was accumulated forabout 40 pulses and averaged to eliminate random noises.
Figure 2 shows the dynamic Lorentz detuning obtained
in the test. Vibration of the detuning was observed at theflat top and decay of the pulse. Impulses at the beginningof the rise and both ends of the flat top were due to theresponses of the PLL circuit. The frequency and the
quality factor of the vibration were estimated to be 122Hz and ~60, respectively, by analysing the waveform atthe decay.
To prepare the parameters for the Lorentz Vibration
Model, the SUPERFISH and the ABAQUS calculationswere performed and then we found that only the firstmode dominates the deformation. The frequency of themode was calculated to be 111 Hz, which agreed wellwith the experimental results. In the Lorentz VibrationModel calculation, V
C and Qm obtained experimentally
were used. Figure 3 shows the calculated result comparedwith the experimental results. The average detuning at theflat top for the calculated and experimental data agreedwithin 10 %. The calculation also reproduces thebehaviour of the vibration at the flat top and the decay.Since the geometry and the boundary conditions are verysimple in this calculation, the agreement between theModel calculation and the experiment indicates thevalidity of the Lorentz Vibration Model.
Small disagreement between the measurement and the
calculation shown in Fig. 3 is considered due to the errorsof the parameters used in the calculation as well asmeasurement error. In applying the Lorentz VibrationModel, ω
m1 and K1 (only the first mode dominates the
detuning in this case) were modified so as to reproducethe experimental data. Figure 4 shows the comparison ofthe modified calculated results and the experimental dataat the flat top region. In the figure, the agreementsbetween those data are very good.
Fig. 1 Rf power control signal and Epeak in the pulsed
operationFig. 2 Dynamic Lorentz Detuning obtained in the test
Fig. 3 Comparison between the calculated and the
measured datak
f
f
k
Q k
m kt h
F V
a kt h
df duk
mk
k
k : mechanical vibration mode number
: partial detuning for k - th mode (Hz)
: total detuning (Hz)
: angular frequency for - th mechanical mode
(rad / s)
: quality factor for - th mechanical mode
: generalized mass for mechanical mode (kg)
: Lorentz force vector at cavity voltage of (N)
: eigenvector for mechanical mode
: frequency sensitivity vector for displacement
mk∆
∆
ω
−
−→
→
→0 0
/
(Hz / m)
: cavity wall displacement (m)u
0.0 0.1 0.2 0.3 0.4 0.5024681012
Epeak=16MV/m
100ms 60ms ControlRF Power Control Voltage (V)
Time (s)024681012141618
Epeak (MV/m) Epeak0.0 0.1 0.2 0.3 0.4 0.5-1200-1000-800-600-400-2000200
Responce of feedback controllerResponce of feedback controller
Vibration
122 Hz
Qm~60Detuning (Hz)
Time (s)
0.0 0.1 0.2 0.3 0.4 0.5-1200-1000-800-600-400-2000200
Lorentz vibration modelMeasurementDetuning (Hz)
Time (sec) 3 DYNAMIC LORENTZ DETUNING FOR
MULTI-CELL CAVITY
The Lorentz Vibration Model has been applied to the
analysis of the dynamic detuning in the pulsed operationfor the 972 MHz 7-cell cavity of β
g=0.729. The thickness
of the cavity was set to be 2.8 mm.
3.1 Mechanical Modes of the 7-cell cavity
At the first step of the analysis, 150 mechanical modes
were calculated by the ABAQUS code. Figure 5 showsthe typical modes as well as the stationary deformation bythe Lorentz force. In this calculation, the left side of thecavity was fixed and the other side was supported by aspring as a tuner support. We found three kinds ofmechanical modes; (a) multi-cell modes, in which modes,cell position moves with lower frequency, (b) tuner andbeampipe modes, in which modes, only either end cell isdeformed, and (c) single-cell mode, in which modes, cell
position is fixed but each cell shape is deformed withhigher frequency. Some of the single-cell modes havedominant influences to the detuning. Multi-cell modeshave much less influences to the stationary detuning butare excited by the pulsed operation when the frequenciesmeet the multiple of the repetition rate.
Quality factors for the mechanical modes were set to be
250, 100 and 1000 for the multi-cell modes, tuner &beampipe modes and single-cell modes, respectively.Those values are based on our experimental experiences.
3.2 Choice of Mechanical Modes
According to the Lorentz Vibration Model, we can
consider the stationary condition by applying (d2∆fk/dt2) =
(d∆fk/dt) = 0. Then we obtain the stationary detuning for
each mode, ∆fk=Kk(VC/V0)2/ωmk2. From these data, we
chose dominant 21 modes out of the 150 modes for theLorentz Vibration Model calculation; 9 multi-cell modes,2 tuner & beampipe modes and 10 single-cell modes.
3.3 Conventional Pulsed Operation
The Lorentz Vibration Model was applied to the
calculation of the dynamic detuning for the conventionalpulsed operation, in which the cavity voltage increasesexponentially for 0.6 ms, holds for 0.6 ms and decreasesexponentially. Figure 6 shows the dynamic detuning, totaldetuning and some of the partial detuning, as well as thecavity voltage (V
C). In this calculation, some of the
single-cell mode vibrations are excited by a pulsedvoltage and the total detuning sways in the flat top andafter the pulse. The vibration in the flat top causes cavityfield error and the vibration after the pulse affects thenext pulse.
3.4 Cosine-shaped Cavity Excitation
In order to reduce the single-cell mode vibrations, we
applied the cosine-shaped cavity excitation, in whichcavity voltage increases and decreases in a cosine-shape.Figure 7 shows the cavity voltage and the dynamicdetuning for the cosine-shaped cavity excitation. The riseFig. 4 Comparison between the modified calculation
and the measured data at the flat top
Fig. 5 Typical mechanical modes for the 972 MHz 7-
cell cavity of β
g=0.729Fig. 6 Dynamic Lorentz detuning for the conventional
pulsed operation0.10 0.12 0.14 0.16 0.18 0.20 0.22-1050-1000-950
Measurement
Lorentz vibration model
(parameters of ωm1 and K1 were modified)
Detuning (Hz)
Time (sec)
Stationary Deformation by the Lorentz Force
Multi-cell Modes (Qm=250)
90.9 Hz
180 Hz
Tuner & Beampipe Modes (Qm=100)
922 Hz
3335 Hz
Single-cell Modes (Qm=1000)
1790 Hz
3377 Hz tuner (60,000 N/mm)Original Shape
Deformed Shape
0.0 0.5 1.0 1.5 2.0 2.5 3.0-200-150-100-50050Vc
Single-cell ModesBeampipe Mode
Total DetuningTuner Mode Lorentz Detuning (Hz)
Time (ms)
time and the flat top duration were the same as the
conventional pulsed operation. The cosine-shape decaywas connected smoothly to the exponential decay. Thevibration of the single-cell modes are much reduced asshown in Fig. 7.
Since this method is considered to be suitable to obtain
highly stable cavity field, it was applied to the rf controlsimulations described in the next section.
4 RF CONTROL SIMULATION
In the case that an rf source feeds the rf power to a
single cavity, the good stability of the cavity field isexpected because the influence of the dynamic Lorentzdetuning can be compensated by an rf low level controller.In the system design of the JAERI/KEK Joint Project,two cavities in a cryomodule are controlled in one rfsystem. In this work, the rf control system of two cavitieswith individual mechanical properties, which is caused byfabrication errors, has been simulated. We assumed thedifferent cavity wall thickness for providing individualmechanical properties and 2 simulations were performed;cavities of 2.8 mm and 3.2 mm thick, and 2.8mm and 3.0mm thick. Loaded quality factor of the cavities are set tobe 3.5 ×10
5 which is a half of the optimum one. This over-
coupled condition moderates the influences of thedynamic detuning to the cavity field stability, even
though additional rf power of about 20 % is required.
4.1 RF Control System
Figure 8 shows the schematic block diagram of the rf
control system. In this simulation, vector sum control oftwo cavities were applied. The feed forward (FF)controller provides the cosine-shaped waveform and thewaveform for the beam loading compensation. The cavityfield stabilization against the dynamic detuning isperformed by the feed back (FB) controller. The detuneoffset against the Lorentz detuning is optimized for eachcavity because of the individual mechanical property.
4.2 Simulation with cavities of 2.8 mm and 3.2
mm thick
The rf control simulation with 2 cavities was carried
out for 800 ms (40 pulses), where the wall thicknesses ofcavity #1 and cavity #2 are 2.8 mm and 3.2 mm,respectively. The dynamic Lorentz detuning including thedetune offset is plotted in Fig. 9 at every 0.1 second. Inthe figure, detuning of 0 degree means the optimumfrequency. The proper offset for each cavity provides thegood detuning in the beam period, ±~10Hz and ±~20Hz
for the cavity #1 and #2, respectively. Figure 10 showsthe amplitude and phase errors for those cavities. Theamplitude errors up to ±~0.1 % and phase errors up to
±~0.2 deg were obtained for both cavities, while only the
first pulse has slightly larger errors. Adopting the cosine-shaped cavity excitation and the proper detune offsetsprovide very good stability, which satisfies therequirement of ±1% and ±1deg.
4.3 Simulation with cavities of 2.8 mm and 3.0
mm thick
The rf control simulation with cavities of wall
thicknesses of 2.8 mm (cavity #1) and 3.0 mm (cavity #3)were also carried out for 1000 ms (50 pulses).Unfortunately, the cavity #3 has a multi-cell mode of349.5 Hz, which is very close to the multiple of theFig. 7 Dynamic Lorentz detuning for the cosine-shaped
cavity excitation0.0 0.5 1.0 1.5 2.0 2.5 3.0-200-150-100-50050Vc
Single-cell ModesBeampipe Mode
Total DetuningTuner Mode Lorentz Detuning (Hz)
Time (ms)
KlystronFB Ref.
FB
ControlFF
Control972MHz
C a v ity # 1 Cavity #2
Vector SumBeam
LoadingFF Ref.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-50050100150200
Beam Period
(0.5 ms)Cavity #2Cavity #1
Cavity #2 Detune Offset (149 Hz)Cavity #1 Detune Offset (174 Hz)
Detuning (Hz)
Time (ms)
Fig. 8 Schematic block diagram of the rf control systemFig. 9 Dynamic Lorentz detuning including the detune
offset at every 0.1 s in the simulation for 2.8 mm and 3.2
mm thick cavitiesrepetition rate, 350 Hz. Figure 11 shows the typical
dynamic detuning for cavity #3. The oscillation of themulti-cell mode is emphasized significantly as shown inFig. 11. The amplitude of the multi-cell mode is about 60Hz. Figure 12 shows the dynamic Lorentz detuningincluding the detune offset. The detune offset for thecavity #3 (116 Hz) was provided with consideration ofthe excited multi-cell amplitude. Therefore, the detuningof the cavity #3 in the first pulse is far from the optimumposition of 0 Hz in the beam period, but after severalhundreds ms it becomes closer value as emphasizing themulti-cell mode. Figure 13 shows the amplitude andphase errors for cavity #1 and #3. The errors are largewithin several hundreds ms, however, they becomesmaller after that because of the stationary vibration ofthe multi-cell mode. At the time of 900 ms, we obtainedthe errors of ±0.15% and ±0.6deg for amplitude and
phase, respectively. Even in this case, the stability of thecavity field satisfies the requirement.
5 CONCLUSION
In order to simulate the rf control and to estimate thefield stability of the SC proton linac, the Lorentz
Vibration Model describing the dynamic Lorentzdetuning has been developed. The validity of the newmodel has been confirmed experimentally. The modelwas applied successfully to the rf control simulation for972 MHz 7-cell cavity of β
g=0.729. Here, we have
obtained good cavity field stability which satisfied therequirement.
REFERENCES
[1] “The Joint Project for High-Intensity Proton
Accelerators ”, JAERI-Tech 99-056, KEK Report 99-
4, and, and JHF-99-3 (19969)
[2] K. Hasegawa et al., “The KEK/JAERI Joint Project;
Status of Design Report and Development ”, these
proceedings
[3] D. A. Edwards Ed., “TESLA TEST FACILITY
LINAC – Design Report ”, TESLA 95-01 (1995)
[4] M. Mizumoto et al., “Development of
Superconducting Linac for the KEK/JAERI JointProject ”, these proceedings-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0
Beam Period
(0.5 ms)t=0 1st pulse(Amplitude)
t=0 1st pulse(Phase)
Amplitude
ErrorPhase ErrorCavity #1 (2.8mmt)
Amplitude & Phase Error (%, deg)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0Cavity #2 (3.2mmt)t=0 1st pulse
(Phase)
t=0 1st pulse
(Amplitude)Phase Error
Amplitude
Error
Time (ms)Amplitude & Phase Error (%, deg)
Fig. 10 Amplitude and phase errors at every 0.1 s in the
simulation for 2.8 mm and 3.2 mm thick cavities0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-50050100150200Cavity #3 Detune Offst (116 Hz)Cavity #1 Detune Offst (174 Hz)
t=0.5ms 26th pulse t=0.4ms 21st pulset=0.3ms 16th pulset=0.2s 11th pulset=0.1ms 6th pulset=0 1st pulseCavity #3Cavity #1
Beam Period
(0.5ms)
Detuning (Hz)
Time (ms)
Fig. 12 Dynamic Lorentz detuning including the detune
offset at every 0.1 s in the simulation for 2.8 mm and 3.0
mm thick cavities
-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0t=0.5s 26th pulse
t=0.4s 21st pulse
t=0.3s 16th pulset=0.2s 11th pulse
t=0.1s 6th pulset=0 1st pulse
t=0.4s 21st pulse
t=0.3s 16th pulse t=0.2s 11th pulset=0.1s 6th pulset=0 1st pulse
Beam Period
(0.5ms)Amplitude ErrorPhase ErrorCavity #1 (2.8mmt)
Amplitude & Phase Error (%, deg)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0
t=0.5s 26th pulse
t=0.4s 21st pulse
t=0.3s 16th pulse
t=0.2s 11th pulse
t=0.1s 6th pulse t=0 1st pulset=0.4s 21st pulset=0.3s 16th pulse t=0.2s 11th pulset=0.1s 6th pulset=0 1st pulse
Phase Error
Amplitude
ErrorCavity #3 (3.0mmt)
Amplitude & Phase Error (%, deg)
815.0 820.0 825.0 830.0 835.0 840.0 845.0-150-100-50050100
349.5 Hz multi-cell mode
RF pulse RF pulseTotal Detuning
Detuning (Hz)
Time (ms)
Fig. 11 Typical dynamic Lorentz detuning for the cavity
#3 (3.0 mm thick)Fig. 13 Amplitude and phase errors at 0.1 s in the
simulation for 2.8 mm and 3.0 mm thick cavities |
null |
arXiv:physics/0008201v1 [physics.acc-ph] 20 Aug 2000APPLICATIONS OF TIME DOMAIN SIMULATION TO COUPLER
DESIGN FOR PERIODIC STRUCTURES∗
N. M. Kroll1,2, C.-K. Ng1and D. C. Vier2
1Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA
2University of California, San Diego, La Jolla, CA 92093, USA
Abstract
We present numerical procedures for analyzing the prop-
erties of periodic structures and associated couplers base d
upon time domain simulation. Simple post processing pro-
cedures are given for determining Brillouin diagrams and
complex field distributions of the traveling wave solutions ,
and the reflection coefficient of the traveling waves by the
input and output. The availability of the reflection coeffi-
cient information facilitates a systematic and efficient pro -
cedure for matching the input and output. The method
has been extensively applied to coupler design for a wide
variety of structures and to a study directed towards elim-
ination of the surface field enhancement commonly experi-
enced in coupler cells.
I. Introduction
Numerical simulation procedures for designing waveg-
uide couplers to accelerator structures are described in [1 ]
and an example of its application to the design of the in-
put coupler for the NLC linac is given in [2]. A coupler
cavity is designed with the intent of providing a matched
connection between a waveguide and a uniform accelerator
structure with dimensions corresponding to those of the
cell adjacent to the coupler cavity. A symmetric structure
consisting of two coupler cavities (with associated waveg-
uides) connected by a short section of accelerator structur e
(typically two cells worth) is modeled and subjected to a
(let us assume single frequency) time domain simulation.
The entire assembly is treated as a single structure with
two wave guide ports. The coupler cell dimensions are ad-
justed until an apparent match is achieved, that is, until no
reflection is experienced at the ports (the external match-
ing condition). To eliminate the possibility that the match
arises from a fortuitous cancellation between forward and
backward waves within the accelerator structure, both the
amplitude and phase of the accelerating field on the beam
axis are observed and required to have the periodicity and
phase advance properties appropriate to a pure traveling
wave (the internal matching condition). As a check one
may add a cell to the accelerator structure and see whether
all these conditions are still satisfied. It is often the case
that accelerator structures are slowly varying rather than
uniform, in which case the input and output couplers are
matched separately.
In practice the procedure (we refer to it as the standard
procedure) has been quite time consuming, involving trial
∗Work supported by the Department of Energy, contract DE-AC0 3-
76SF00515, grants DE-FG03-93ER40759 and DE-FG03-93ER407 93.and error rather than a systematic procedure to simultane-
ously satisfy both the internal and external matching con-
ditions. Another limitation arises from the fact that the
method assumes that evanescent bands can be neglected
but provides no procedure for demonstrating their absence.
In the next section we describe a new simulation pro-
cedure which has been found to be much more efficient,
and also which provides information about the presence of
evanescent bands. The basic elements of the method were
briefly described in [3] in connection with the design of a
coupler for the zipper structure. Because it has since re-
placed the old method for all of our coupler design work,
a more complete presentation together with examples will
be presented in the following sections.
II. The New Simulation Procedure
As in the case of the old standard procedure one applies
a single frequency time domain simulation by driving the
input port of a two port structure consisting of an input
cavity, anNcell periodic structure with period P, and an
output cavity. Instead, however, of focusing attention on
the S parameters of the structure as a whole, we direct our
attention to the simulated accelerating field Ez(z,t) eval-
uated along the beam axis. We assume a steady state has
been reached, so that the subsequent time dependence can
be expressed in terms of the complex Ez(z) (Ec(z) hence-
forth), obtained in the standard way by combining the sim-
ulated real fields at two times separated by a quarter pe-
riod. Then from Floquet’s theorem (neglecting evanescent
bands, losses, and an irrelevant overall phase factor)
Ec(z) =E(z)[exp(−jφ(z)) +Rexp(jφ(z))].(1)
HereE(z) is a real positive amplitude function with period
P, andφ(z) is a real phase function, periodic except for a
cell to cell phase advance ψ. Thus
E(z±P) =E(z),andφ(z±P) =φ(z)±ψ. (2)
Ris azindependent complex reflection coefficient. Note
that one is free to shift φby an arbitrary constant with a
compensating phase shift in R, since the overall phase of
Ecis irrelevant. This freedom corresponds to the choice of
reference plane through some point z0where we take φ= 0.
We now consider the quantities
Σ(z) =F+(z) +F−(z),and ∆(z) =F+(z)−F−(z),(3)
where
F±(z) =Ec(z±P)/Ec(z). (4)Elementary algebraic manipulation leads to the relations:
2Cos(ψ) = Σ(z), (5)
Rexp(2jφ) = [2Sin(ψ)−j∆(z)]/[2Sin(ψ) +j∆(z)].(6)
We note that while the RHS of (5) is formed of zdependent
complex quantities, it nevertheless turns out to be real and
zindependent. Similarly the absolute value of the RHS of
(6) is alsozindependent. Both these results should hold
for all “allowed” zvalues, i.e., values such that the three
pointsz+P,z, andz−Pall lie within the periodic portion
of the structure simulated, and together they constitute a
powerful constraint on the validity of the Floquet represen -
tation Eq. (1). Their failure beyond small numerical fluc-
tuations or small deviation from steady state is evidence
for the presence of evanescent bands. An example will be
presented in the section on the Zipper structure.
It is noteworthy that these relations allow one to de-
termine all the properties of the traveling wave solutions,
including the functions φ(z) andE(z) from a simulation
which contains an arbitrary mixture of forward and back-
ward waves. Of particular importance is the fact that it
gives the magnitude and phase of the reflection coefficient.
In contrast to the old standard method, there is here only
one matching condition to be satisfied, namely |R|= 0.
Typically match is achieved by varying two parameters in
the coupler design. Once one has determined how the real
and imaginary parts of Rvary with the parameters, one
can choose linear combinations of changes which acceler-
ate the process of converging to the origin in the complex
Rplane [4]. Because the phase of Rdoes depend upon
the position of the reference point relative to the couplers ,
one naturally keeps it fixed while carrying out this process.
Note that it is the output cavity that is matched by this
procedure. While not necessary, it is usually convenient to
construct a symmetric mesh. The input and output cav-
ities are then the same, and the structure as a whole is
matched when Rvanishes.
III. Applications and Examples
(a) The NLC four port output coupler
As an example of the principal features of the new
method we use the new NLC four port output coupler cav-
ity [5]. The purpose of the four port design was to provide
damping for those dipole modes that reach the end of the
structure while also providing an output for the accelerat-
ing mode. These dipole modes are typically those which
had been poorly damped because of decoupling of the last
cells from the manifolds. The four port symmetry provides
damping for both dipole mode polarizations and has the
added advantage of eliminating quadrupole distortion of
the coupler fields.
The design simulation was carried out with a three cell
periodic structure, and results are illustrated in Fig. (1) .
Two cases are shown, one matched, the other not. The re-
flection coefficients |R|as computed from Eq. (5) for the
two cases are shown as functions of z. The allowed z values
are those lying within the central cell, and one sees that forboth cases |R|is constant over that range. The real part
ofCos(ψ) is also plotted as a function of z. One sees that
the two values are indeed constant over the allowed range,
but contrary to expectations they differ somewhat from
each other and from the expected value of one half. This is
due to the fact that a different and coarser mesh than that
used to determine the phase advance parameter was used
for the time domain simulations. The two cases differ from
one another because the parameter variations in the cou-
pler associated with the matching procedure induce small
but global changes in the meshing. It has been confirmed
in a number of cases that there is good agreement be-
tween the phase frequency relation determined from single
cell periodic boundary condition frequency domain calcu-
lations and that determined from the time domain method
described here so long as the same mesh is used for both
simulations.
0.005 0.015 0.025 0.035
z (m)−1.0−0.50.00.51.0|R| and Cos( ψ)Matched
Unmatched
|R|
Cos( )ψ
Figure 1
|R|andCos(ψ) along the axis of the NLC four-port
output coupler.
(b) A Photonic Band Gap (PBG) structure
Figure 2
A snapshot of electric field in the PBG structure.
A coupler cell very similar to those of the SLAC struc-
tures has been designed for a PBG structure, that is, a
cylindrical cell with a pair of symmetrically placed waveg-
uide ports, a conventional beam pipe, and conventional
beam iris coupling to the periodic PBG structure. The
PBG cell structure [6] is a seven by seven square arrayof metallic posts aligned in the beam direction and termi-
nated by metallic end plates, the cell cavity being formed
by removing the central post. A circular aperture in the
end plates, identical to that between the coupler cell and
the adjacent PBG cell, provides cell to cell coupling and
a path for the beam. A perspective representation of the
four cell quarter structure used for the simulations is show n
in Fig. (2). Also shown is the simulated electric field dis-
tribution, scaled logarithmically to enhance the visibili ty
of weak field strengths. The figure illustrates the effec-
tiveness of the PBG structure in confining the acceleration
fields to the interior of the structure. The matching proce-
dure worked well, and, as in the four port coupler above,
there was no evidence for evanescent band contamination.
Fabrication of an experimental model with 5 coupled PBG
cells and complete with couplers is in progress at SLAC.
(c) The Zipper structure
The zipper is a planar accelerator structure described in
[3]. A 25 (counting the coupler cavities) cell W band model
has been built, cold tested, and subjected to bead pull
measurements as reported in [7]. The design was governed
by a decision to avoid bonded joints involving tiny structur e
elements such as the vanes which serve as cell boundaries
and also form the beam iris. The coupler cell is a quarter
wave transformer terminating in WR10 waveguide.
Early attempts at matching the coupler using the old
standard method failed, and it was this failure which led
to the development reported here. Matching using this
method was accomplished by making use of a time domain
simulation of a structure with 22 periodic cells. Fig. 3
shows the resultant ReCos (ψ),ImCos (ψ), and |R|plots
as computed from Eqs. (5) and (6). One sees large de-
viations from the expected zindependent behavior as one
moves away from the center of the structure. This effect in-
dicates a clear violation of Eq. (1). From the fact that the
violation fades away as one moves away from the couplers
indicates that the effect is due to the couplers generating an
evanescent band, the nearby monopole band pointed out in
[3]. This example demonstrates how the method indicates
the presence of evanescent band interference, and also how
one can carry out the matching procedure even when it is
present.
Figure 3
|R|andCos(ψ) along the axis of the zipper structure.IV. The Coupler Field Enhancement Problem
Electrical discharge damage has been commonly ob-
served in the coupler cells of accelerator structures and ha s
been attributed to the field enhancement noted in simula-
tions. We have taken advantage of our enhanced matching
capability to initiate a study of this long standing problem .
Exploration of the situation for the NLC coupler [2] showed
that the largest enhancement occurred on the coupler side
of the aperture of the iris separating the coupler from the
adjacent cell with azimuthal maxima opposite the coupler
waveguides and azimuthal minima 90 degrees away. This
observation was consistent with the pattern of discharge
damage [9]. It is pointed out in [8] that the azimuthal vari-
ation is due to the quadrupole component introduced by
the coupler waveguides and that the enhancement can be
reduced by introducing a racetrack like modification of the
coupler cell shape designed to eliminate it. This effect and
its cure have been confirmed in our own studies of the NLC
coupler. Two other modifications have also been explored.
The simplest and most effective was simply to reduce the
radius of the cell adjacent to the coupler. The effect for a
2% reduction is illustrated in Fig. 4 where it is seen that
the field on the coupler cell iris is significantly less than
that on the interior coupling irises. An undesirable conse-
quence is a 10 degree phase advance deficiency in the mod-
ified cell. An even larger field reduction would be obtained
by removing the quadrupole enhancement. We attribute
the reduction to an increase in group velocity. The other
modification consisted of enlarging the coupler iris com-
bined with an increase in the adjacent cell radius chosen so
as to preserve the cell phase advance, but the exploration
of this effect is incomplete. Experimental investigation to
determine whether such changes actually do reduce electri-
cal discharge damage in the coupler is clearly needed.
−0.005 0.005 0.015 0.025 0.035 0.045 0.055 0.065
z (m)0.00.51.01.52.02.5Electric Field (arbritray units)x direction
y direction
Figure 4
zvariation of electric field magnitude at radial positions
of the beam irises.
References
[1] C.K. Ng and K. Ko, Proc. CAP93 p243 1993
[2] G.B. Bowden et al, Proc. PAC99 p3426 1999
[3] N.M. Kroll et al Proc. PAC99 p3612 1999
[4] We are indebted to Roger H. Miller for suggesting this pro cedure
and emphasizing the advantage of tracking the complex Rrather
than |R|.[5] J.W. Wang, et al, Oral Poster TUA3, This Conference; R.M.
Jones, et al, Poster TUA8, This Conference
[6] D.R. Smith, et al, AAC94, AIP Conf. Proc. 335, p761 (1995)
[7] D.T. Palmer, et al, The Design, Fabrication, and RF Measu re-
ments of the First 25 cell W-Band Constant Impedance Accel-
erating Structure, AAC2000, to appear in AIP Conf. Proc. for
AAC2000
[8] J. Haimson, B. Mecklenberg, and E.L. Wright, AAC96, AIP
Conf. Proc. 398, p898 (1997)
[9] Juwen Wang, private communication |
AN INVESTIGATION OF OPTIMISED FREQUENCY DISTRIBUTIONS
FOR DAMPING WAKEFIELDS IN X-BAND LINACS FOR THE NLC
R.M. Jones1, SLAC; N.M. Kroll2, UCSD & SLAC; R.H. Miller1,
T.O. Raubenheimer1 and G.V. Stupakov1; SLAC
_____________
1 Supported under U.S. DOE contract DE-AC03-76SF00515
2 Supported under U.S. DOE grant DE-FG03-93ER407Abstract
In the NLC (Next Linear Collider) small misalignments
in each of the individual accelerator structures (or theaccelerator cells) will give rise to wakefields which kickthe beam from its electrical axis. This wakefield canresonantly drive the beam into a BBU (Beam Break Up)instability or at the very least it will dilute the emittanceof the beam. A Gaussian detuned structure has beendesigned and tested [1] at SLAC and in this paper weexplore new distributions with possibly better dampingproperties. The progress of the beam throughapproximately 5,000 structures is monitored in phasespace and results on this are presented.
1. INTRODUCTION
In all of our previous accelerating structures the celldimensions have been designed such that they follow anErf function profile and the uncoupled cells have aGaussian Kdn/df, kick-factor weighed density function,
-2 -1 0 1 2
Frequency00.20.40.60.81Kdn df
Figure 1: Optimisation with the idealised frequency
distributions illustrated
[2] profile. The normalised Gaussian is shown in Fig 1
together with the convolution of a number of “top hat”functions. A Gaussian distribution leads to a wakefieldwhich does not continue to fall rapidly because insampling the Gaussian (with a finite number of cells and aspecified frequency bandwidth) one is forced to truncatethe function and the resulting wakefield is the convolutionof a Gaussian function and a sinc function: sinc(f) =
sin(πf)/(πf). In this case, the spacing of the minima is not
uniform and thus a uniformly spaced multi-bunch train isunable to be precisely located at local minima. The
wakefield for a truncated Gaussian function (shown in Fig2) only follows a Gaussian decay for the initial part of thedecay (the first few bunches) and thereafter aconsiderable ripple occurs. Additional moderatedamping (Q~1000) is employed with four manifolds thatlie along the outer wall of the accelerator and this onlytakes effect after several meters down a bunch train of80m. Thus, these ripples can have serious consequenceson the wakefield.
0 1 2 3 4 5
s0.0010.010.11W‘
HsL
Figure 2: Envelope of wake function corresponding to
idealised distributions
In order to reduce the large ripple we have considered
various distributions to replace the Gaussian prescription.In this paper we will concentrate on a number ofconvolutions of the top hat function. A top hatdistribution has a sinc function as its Fourier tranform andthis falls of as 1/s. Each additional convolution leads to a1/s
k fall-off in the wakefield. Here, we consider k=2 (a
triangular distribution, g2) and k=3 (the convolution of a
triangular function with a top hat function, g3) and k=4 the
self-convolution of the triangular function and these areshown in Fig 1. The Fourier transform of the k=4 case isgiven by sinc
4 function and this is compared with the
truncated Gaussian in Fig 2. together with the k=2 andk=3 cases. The function described by the k=4 case isidentically zero at frequency units
±2 and thus enforced
truncation is not necessary. The peak values in theripples of the wakefield of the truncated Gaussian liebelow the sinc
2 but not below the sinc4 function. For this
reason we choose a g4 (sinc4 in wake space) design for a
new RDDS based upon a mapping function [3] re-parameterisation of RDDS1.14 14.5 15 15.5 16 16.5
Frequency HGHz L20406080100GHfLHVpCmm mGHz L
Figure 3: G(f), Spectral function, for a sinc4 variation
20 40 60 80
sHmL0.010.1110100WHVpCmm mL
Figure 4: Wakefield for Sinc4 distribution
-1 -0.5 0 0.5 1
Dsb0.20.40.60.81<S>HVpCmm mL
Figure 5: Sum wake function for optimised distribution
2. WAKE ENVELOPE FUNCTION FOR A
SINC4 DISTRIBUTION
We compute the wake envelope function using the
spectral function method [4] and this method has provenquite accurate in predicting the wakefield of a realisticstructure [1]. The spectral function for sinc
4 is shown in
fig. 3. and the main difference from the spectral functionof RDDS1 (fig 7) lies in the upper frequency end of thedistribution. In the case of RDDS1 the kick factorsincrease almost linearly with synchronous frequency andtowards the end of the high frequency end of the firstdipole band [2] the mode density (dn/df) has to increase inorder that Kdn/df be a symmetric function that falls with a14 14.5 15 15.5 16 16.5
Frequency HGHz L20406080100GHfLHVpCmm mGHz L
Figure 6: Spectral function RDDS1
20 40 60 80
sHmL0.010.1110100W‘
HVpCmm mL
Figure 7: Envelope of wakefunction for RDDS1
-1 -0.5 0 0.5 1
Dsb0.20.40.60.81<S>HVpCmm mL
Figure 8: Sum wake function for RDDS1
Gaussian profile. However, as dn/df increases then, of
course, the modal separation (approximately 50 MHz ormore compared to 7MHz in the center of the band)increases and hence the modes are not particularly welldamped by the manifold in the high frequency region.However, the sinc
4 possesses the useful property that the
modes are much more well damped in this region (15.8GHz and beyond) and this we attribute to the modes beingmore closely spaced in frequency. The wakefield corresponding to the spectral function ofFig 3, is shown in Fig 4 and the main improvement overthe wakefield of our present structure, RDDS1 (shown inFig. 7) lies in the region 0 to 10 m in which the wakefieldis improved by a factor of approximately 2 or more. Alsoshown in Figs 5 and 8 is S σ, the standard deviation of the
sum wakefield from the mean value, [Bane Ref] for thesinc
4 distribution and RDDS1 respectively, as a function
of bSΔ, the percentage variation in the bunch spacing.
The sum wakefield is useful in that it provides an
indicator as to whether or not BBU (Beam Break Up) will
occur. The abscissa in these curves is bSΔ and this
provides a convenient means of shifting all the cell
frequencies by a fixed amount and it corresponds to asystematic error in the synchronous frequencies [*Ref*]. From previous simulations, peaks in the standarddeviation of the sum wakefield close to unity have provedto be a symptom of BBU. However, BBU is indeed acomplex phenomena and, in order to be sure that BBUwill actually take place many particle tracking simulationswith the code LIAR [5] need to be undertaken. In thenext section the results on particle tracking simulations at
peak values in S
σ are presented.
3. BEAM DYNAMICS: TRACKING
THROUGH COMPLETE LINAC
In all of the tracking simulations we performed the bunch
train is offset by 1 µm and its progress down the linac is
monitored. Additional details regarding the simulationparameters are given in [6]. At the nominal bunch spacing
(84 cm) S
σ is approximately 0.15 V/pC/mm/m and 0.3
V/pC/mm/m for the new distribution and for RDDS1,respectively. Tracking through the complete linac forboth distributions indicates that that no significantemittance dilution occurs Also, in both cases there are
0 2 4 6 8 10
BPM Position Hkm L0123456De
Figure 9: Emittance growth for the sinc4 distribution and
RDDS1 at a bunch spacing which maximises S σ
peaks in S σ are very close (less than .05%) to the nominal
bunch spacing, however simulations show that these alsogive rise to no more than 1 or 2 percent dilution of the
beam emittance. The largest peak in S
σ for RDDS1 and
the new distribution are located at -0.35% and –0.48%away from the nominal bunch spacing, respectively. Theemittance growth after tracking through the linac at thesemodified bunch spacings is shown Fig. 9. For the sinc
4
distribution there is no emittance dilution arising fromlong range wakes. However approximately 6 % emittance
growth occurs for RDDS1. The phase space, shown inFig 10, indicates that for the sinc
4 distribution the particles
are well contained but for RDDS1 the bunch train isstarting to break up. Nonetheless, emittance growth isunlikely to be a problem for RDDS1 because: firstly thethe systematic shift is unlikely to be so large (-0.48% inthe bunch spacing corresponds to a shift in the dipolemode frequency of 72 MHz) and secondly, the shift is notexpected to be identical from structure-to-structure andthis has been shown [6] to significantly reduce anyemittance growth.
-1 -0.5 0 0.5 1
YHnorm. L-1-0.500.51Y'Hnorm. L
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495
-1 -0.5 0 0.5 1
YHnorm. L-1-0.500.51Y'Hnorm. L
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495
(a) (b)
Figure 10: Phase space for sinc4 distribution (a) and
RDDS1 (b) at a bunch spacing which maximises S σ
4. CONCLUSIONS
A sinc4 distribution for the uncoupled leads to improved
damping of the transverse wakefield. The mean value ofS
σ is approximately 2 times smaller than that of our
present structure, RDDS1 and, we have found that nosignificant emittance growth occurs over a broad range ofsystematic shifts in the synchronous frequencies of thecells. However, additional optimisation of the frequencydistribution and in the coupling of the wakefield to themanifold, should lead to even better damping of thewakefield. In the near future, we plan to embark on aprogram of iterative optimisation of the wakefield.
5. REFERENCES
[1] J.W. Wang et al, TUA03, LINAC200 (this conf.)
[2] R.M. Jones et al, EPAC96, (also SLAC-PUB 7187)[3] R.M. Jones et al, LINAC98 (also SLAC-PUB 7933)[4] R.M. Jones et al, LINAC96 (also SLAC-PUB 7287)[5] R. Assman et al,”LIAR”, SLAC-PUB AP-103, 1997
[6] R.M. Jones at al, TUA08, LINAC2000 (this conf.) |
COMPARISONS OF EQUIVALENT CIRCUIT PREDICTIONS WITH
MEASUREMENTS FOR SHORT STACKS OF RDDS1 DISCS, AND THEIR
POTENTIAL APPLICATION TO IMPROVED WAKEFIELD PREDICTION
R.M. Jones1, SLAC; T. Higo, Y. Higashi, N. Toge, KEK;
N.M. Kroll2, SLAC & UCSD; R.J. Loewen1, R.H. Miller1, and J.W. Wang1; SLAC
________________
1 Supported under U.S. DOE contractDE-AC03-76SF00515.
2Supported under U.S. DOE grant DE-FG03-93ER407.Abstract
In fabricating the first X-Band RDDS (Rounded
Damped Detuned Structure) accelerator structure,microwave measurements are made on short groups ofdiscs prior to bonding the discs of the entire structure.The design dispersion curves are compared with thefrequency measurements. The theory utilised is based ona circuit model adapted to a short stack of slowly varyingnon-uniform discs. The model reveals the nature of themodes in the structure and may also be used to refit theexperimental data to the parameters in a model of thewakefield given earlier [1]. This method allows a morefaithful determination of the wakefield that a beam willexperience as it traverses the structure. Results obtainedon the frequencies are compared to the original design.
1. INTRODUCTION
The design and fabrication of the first RDDSaccelerator structure (RDDS1) is described in [2]. Thebasic fabrication units of the structure are discs consistingof a rounded beam iris and a pair of identical half cellsone on each side of the iris. The disc also includes thefour circular waveguide sections which form the fourmanifolds together with the coupling slots which connectthem to the cells. Further details are found in [2] and [3].Microwave measurements were performed during thefabrication procedure to provide quality assurance. Thesewere of two sorts. The first consisted of resonancefrequency measurements on a single disc terminated withflat conducting plates in pressure contact with the disc [3].The second consisted of similar measurements on stacksof six successive discs (eg disc n to disc n+5) carefullyaligned and pressed together between two flat conductingplates. Each end plate was provided with an off centerprobe, similarly placed within the cell (rather than themanifold) region, and at an azimuth such that they facedone another. Both monopole and dipole resonantfrequencies were determined by means of of networkanalyser measurements of scattering matrix parameters,(primarily S
12) between the end plate probes.
The single disc measurements provide resonant
frequencies for the zero and π modes of the lowest
monopole band, the π mode of the first dipole band, and
the zero mode of the second dipole band. The primaryinformation that one might hope to aquire from the single
disc measurements would be the accelerating modefrequency (ie the 2 π/3 monopole), which is supposed to be
the same for each disc, and the tuning profile of thesynchronous frequency of the lowest dipole mode.Although this information is not provided directly, theaccelerating mode frequency is expected to be given toadequate accuracy from the monopole mode frequencies f
0
and fπ by the formula
()1/222
acc 0 0f2 f f / 3 f fππ=+ (1.1)
The lower dipole pi mode frequency is very close to the
synchronous frequency, so that its profile can provide anadequate surrogate for the synchronous mode profile.Also, these two frequencies uniquely determine two of theequivalent circuit parameters (independently of the valuesof the other parameters), which may then be compared toprofiles for these two parameters obtained by interpolationformulas from the five cells actually simulated. Furtherdetails regarding the application of the single discmeasurements may be found in [3], especially on thedevelopment of a rapid routine quality assuranceprocedure that could be integrated into a manufacturingprocess.
With the six disc stacks, more resonances can be
observed, thereby providing more complete informationregarding the accuracy of machining, simulation, andequivalent circuit representation. Of particular interest isthe fact that each of the stacks should, despite the fact thatthey are made up of discs dimensioned to fit the detuningprofile, should have a 2 π/3 phase advance monopole mode
all with the same 11.424 GHz frequency of theacceleration mode. Since the completed structure isdriven at 11.424GHz, a frequency error in the accelerationmode of a particular stack will translate into a phaseadvance error. Because these measurements wereperformed while fabrication was in progress,compensating dimensional changes could be made insubsequently fabricated cells so as to prevent theaccumulation of phase errors [2]. The stacks also providemore detailed information about the Brillouin diagram ofthe dipole modes. As discussed below, this informationwill be used to obtain more accurate information on thesynchronous frequency profile and on the couplingbetween the lower dipole modes and the manifold.2. FEATURES OF N DISC STACKS
We first consider the case of a uniform N disc stack.
Our attention will be focussed on the modes associatedwith the first monopole band, the first two dipole bands,and the first bands associated with the four manifolds.These are the low lying modes found in single discsimulations with periodic boundary conditions, their bandsbeing traced out by varying the phase advance.(Historically, one quarter of a single cell is simulated,with symmetry-symmetry boundary conditions for themonopole and a pair of nearly degenerate manifold bands,and metallic-symmetry boundary conditions for the lowertwo dipole bands and a manifold band. The results wouldbe identical for a disc.) The modes of an N disc stackterminated at each end by a conducting plate are equalamplitude superpositions of the two oppositely directedtravelling wave solutions. One may think of oneboundary as determining their relative phase, and then therelative phase required by the other boundary determineswhich phase advances correspond to modes. The "band
edge" solutions (ie 0 and π phase advance) are standing
rather than running waves, and are missing or included inthe mode spectrum accordingly as they do or do not fit theimposed boundaries. For the N disc stack one expects
modes corresponding to phase advances (n/N) π, with n =
0,1...N for the monopole band modes, n = 1,...N-1 for themodes of the manifold bands. The differences arise fromthe fact that the former are TM and the latter TE. Thedipole bands are TE-TM hybrids, but at the band edgesone or the other predominates. The consequence is that n= 0 is missing from the lower dipole band and n = pi ismissing from the upper dipole band. The four modes ofthe N = 1 case discussed above are illustrative of theserules. The total number of modes of an N disc stackassociated with the bands specified is 9N - 3 (countingboth dipole orientations).Experimental study of a uniform stack would provide acheck on the Brillouin diagram obtained from simulation.Also for N a multiple of three the sequence of phase
advances represented includes 2 π/3. Thus one gets a
better check on the accelerating mode frequency bystudying such stacks. The RDDS1 disc fabrication waschecked for the entire structure by measuring 34 six discstacks. The measured stacks belonged to the sequence ofdetuned cells and were therefore not uniform, but asmentioned above this does not affect the frequency of theaccelerating mode. For the monopole band observations,only the accelerating mode frequency was recorded. Thefrequencies of all dipole modes which could be seen fromthe probes were also recorded. These included the sixlower and upper dipole modes with probe coupledorientation, and those dipole phased manifold modesstrongly enough coupled to the cells to be seen from theprobes3. EQUIVALENT CIRCUIT ANALYSIS OF
DETUNED N DISC STACKS.
The equivalent circuit of [1] was designed to represent the
first two dipole bands, and the first manifold band of theDDS and RDDS structures. (Because the manifoldsinclude a transmission line in their representation, thereare solutions to the circuit equations which refer to higherorder manifold bands, but they have not been tailored torepresent the actual higher order manifold bands with anyreliability.) The actual structure contains a degeneratepair for each of these bands, but the circuit models only
Figure 1: Circuit diagram of 3-cell stack
one of them. The circuit for an N = 3 disc stack is
illustrated in Fig. (1). Here, to conform to earliernotations, we number the discs 1/2,..N-1/2. The full LCcircuits between discs n-1/2 and n+1/2 (n =1,...,N-1)represent the hybrid TE-TM modes of the cells betweenthe discs. Their loop currents are represented by the
amplitudes a
n, nˆa respectively for the TE and TM circuits
respectively. The shunted transmission line sections n
correspond to the portion of the manifold adjacent to then'th cell and are represented by the amplitude variables A
n
proportional to the voltages across the shunt. The halfcells at the ends of the TM chain with doubled C andhalved L correspond to the region between the end discsand the shorting plates, with amplitudes represented by
0ˆa and Nˆa, and the manifold transmission lines are
shorted at a distance one period away from the adjacent
shunts. Analogous to [1] the homogeneous circuit equations maybe written in the form:
RA-Ga = 0 (1.2)
-2
xˆ (H - f )a + H a - GA = 0 (1.3)
-2 t'
xˆˆˆ (H' - f )a + H a = 0 (1.4)
Here A, a, and ˆa are N-1, N-1, N+1 component column
vectors respectively. R, G, and H are N-1 x N-1 matrices,
ˆH' is N+1 x N+1, while the matrices Hx and t'
xˆH are N-1
x N+1 and N+1 x N-1 respectively. The matrix elements
of these matrices are the same as those given in [1] exceptManifold
TE C1 C2 V1 V2
L1 L2
C1 C2
L1
2 L2
2 L1
2 L2
2
TM L1
2 L1
2 L2
2 L2
2
1/2ˆ1C1 1/2ˆ2C2 1/42 C1
2 C3that the primes on ˆH and t'
xˆH indicate that their (0 1) and
(N+1 N) matrix elements are doubled. This asymmetry in
the equations could be removed by rescaling 0ˆa and Nˆa,
but the form of the eigenvectors is simpler with the
equations as they are. In particular, in the case of auniform stack direct substitution in (2), (3), and (4)verifies that the eigenvectors take the form
n1aK c o s ( n ) ; n 0 , . . N= ψ = (1.5)
n2 na = K A = sin(n ); n = 1,...,N-1 ψ (1.6)
with = (m/N) ; m = 0,...,N.ψ π (1.7)
The modal frequencies are determined by the phase (psi)-
frequency dispersion relation of Eq. (10) in [1]. Asindicated there, its three lowest roots provide the Brillouindiagram of the lower two dipole bands and the lowestmanifold band. An example appears later as Fig. (2).
Here the modal frequencies are determined by
substituting the ψ values of Eq. (7). The specification of
the eigenvectors is completed by computing K
2 from Eqs.
(2) and (6) and then K1 from (5) and (4). For two of the
three roots at ψ = 0 and at ψ = π, K1 = 0. All amplitude
variables then vanish, and hence
50 100 150
Phase (Deg.)12151821Frequency (GHz)0 50 100 150
121518
Figure 2: Brillouin diagram corresponding to RDDS1 cellstack 98 to 103 (average cell 100.5). The points areobtained from an experimental measurement and the linesare obtained from the circuit model in which the originaldesign was prescribed prior to the experiment.
these frequencies do not represent modes of the stack The
remaining frequencies at these phase values representpure TM modes for which K
1 can be assigned an arbitrary
non-zero value. For the detuned case numerical methodsmust be used to determine both the eigenvectors and themodal frequencies. Equations (2, 3, and 4) are a set of3N-1 linear homogeneous equations in the 3N-1amplitudes represented by a,
ˆa, and A. Modal
frequencies were determined by finding the frequencies atwhich the determinant of the coefficients vanishes. Thisroot search process was greatly facilitated by starting fromthe modal frequencies of a uniform stack with parameterscorresponding to those of the average cell in the stack.These may be obtained from the interpolation proceduredescribed in [4]. Once the modal frequencies are knowndetermination of the eigenvectors is a well known andnumericlly efficient linear algebraic procedure. The shiftof the modal frequencies from those of the average cell
uniform stack is usually quite small, in which case theindividual modes can still be designated by band andphase advance. Ambiguities can arise when there is neardegeneracy. Although distorted by detuning, equivalentcircuit eigenvectors can help resolve them. Likewise, thephase and magnitude of S_12 at the resonant peaks can dothe same for the stack experimental measurements We conclude with an example based on the stackformed by the six discs 98 to 103. The space between disc100 and 101 constitutes the average cell, which wedesignate as cell 100.5. Its Brillouin diagram is shown inFig. (2), and experimental points are also plotted. Table 1provides a numerical comparison between experimentalpoints, detuned stack equivalent circuit computed points,and points for a uniform 6 disc stack with parameters forcell 100.5. The particular frequencies selected for displayin the table are thought to be the most relevant for wakeprediction because of their bearing on cell to manifoldcoupling and synchronous frequency identification. Theequivalent circuit computed shifts of the detuned stackfrequencies from the uniform average stack frequenciesare indeed quite small in this case, so that the associationof the detuned stack frequencies with the uniform stackphas advances and bands seems quite unambiguous. Also
the eigenvectors do provide support for the identification,
although some of the patterns do show considerabledistortion. The identification of the experimentalfrequencies with phase advances and bands is so far basedprimarily on the pattern. The association of the lower 60degree and higher 120 degree frequencies with themanifold is, however, supported by the small amplitude oftheir S
12 peaks. We note that associated with each stack
we will have typically 27 measured frequencies, 12 fromthe single disc and usually 15 from the six cell stack. It isour intention to use this data to refine our parameterinterpolation curves and our synchronous frequencyprofile, and to thereby improve our wake predictions, butwe do not yet have a tested methodology for doing so.
ψ
60 90 120 150 180
fexp13.7
14.785614.8048
15.029215.110
16.75915.1388 15.1923
fmode13.6358
14.768614.7803
15.154415.0399
16.770115.1256 15.1556
fav13.6356
14.775914.7759
15.153915.0404
16.691815.1271 15.1537
Table 1: Experimentally measured stack frequencies, fexp,
modally determined frequencies, fmode and, average cell
frequencies.
4. REFERENCES
[1] R.M.Jones, et al, EPAC96 (also SLAC-PUB-7187)[2] J.W. Wang et al, TUAO3, LINAC2000 (this conf.)[3] R.H. Miller et al, TUA20, LINAC2000 (this conf.)
[4] R.M. Jones et al, LINAC98, SLAC-PUB-7934 |
New Development in RF Pulse Compression
Sami G. Tantawi*, SLAC, Menlo Park, CA94025, USA
Abstract
Several pulse compression systems have been proposed
for future linear colliders. Most of these systems requirehundreds of kilometers of low-loss waveguide runs. Toreduce the waveguide length and improve the efficiencyof these systems, components for multi-moding, activeswitches and non-reciprocal elements are beingdeveloped. In the multi-moded systems a waveguide isutilized several times by sending different signals overdifferent modes. The multi-moded components needed forthese systems have to be able to handle hundreds ofmegawatts of rf power at the X-band frequency andabove. Consequently, most of these components areovermoded. We present the development of multi-modedcomponents required for such systems. We also presentthe development efforts towards overmoded activecomponent such as switches and overmoded non-reciprocal components such as circulators and isolators.
1 INTRODUCTION
Rf pulse compression systems enhance the peak power
capabilities of rf sources. Indeed, it have been used as atool for matching the short filling time of an acceleratorstructure to the long pulse length generated by most rfsources such as klystrons. All rf pulse compression systemstore the rf energy for a long period of time and thenrelease it in a short time. For linac application associatedwith future linear colliders, the storage media is awaveguide transmission line. The energy required, tosupply a linac section or a set of linac sections, are storedin these lines. The length of these waveguidetransmission lines has the same order as
cτwhere τ is
the pulse length required by the linac and cis the speed
of light. For colliders based on X-band linacs such as theNLC [1] and JLC [2] these lengths are tens of meterslong. Since the collider usually contains several thousand-accelerator sections, the total waveguide system for thecollider is usually hundreds of kilometers long.
These long runs of waveguides have to be extremely
low-loss. At the same time it should be able to handlepower levels in the hundreds of Megawatts. Hence, thesewaveguides are usually highly over-moded circularwaveguide operating under vacuum. Because of vacuum,and tolerance requirements, these hundreds of kilometersof waveguide runs are expensive, hard to install andmaintain.
*Also with the electronics and communication department,
Cairo University, Giza, Egypt.To reduce these waveguide runs, several innovations
have been made both on the system and component levels:
1- RF pulse compression systems that have high intrinsic
efficiencies have been suggested. These systems areBinary Pulse Compression (BPC) [3], Delay LineDistribution System (DLDS) [4], and active pulsecompression system using resonant delay lines[5-6].
2- Enhancing the system power handling capabilities can
ultimately reduce the number of systems required. Onecan use a single system that services several rf sourcesand several accelerator sections. Hence, low-lossovermoded components have been developed for thesesystem, see for example [7-9]
3- Since these waveguide runs are over-mode one can
utilize these waveguides several times by sendingsignals over different modes. Such multi-modedsystems have been suggested [10] and conceptual testsfor components and concepts have been performed[11].
4- To implement active pulse compression systems
inexpensive supper-high-power semiconductorswitching arrays have been suggested [12], and tested[13]
In this paper we devote section 2 to an accurate
formulation for the length of waveguide runs required byseveral pulse compression systems. We then describe thework done to provide a supper high power test setup forthe components required by these systems in section 3. Insection 4 we describe the multi-moded planer componentsand associated tapers. Finally, in section 5, we show someattempts to provide a semiconductor microwave switch.
2 COMPARISON BETWEEN RF PULSE
COMPRESSION SYSTEMS
2.1 General Layout
To achieve pulse compression a storage system is
employed to store the rf power until it is needed. Differentportions of the input rf pulse T are stored for different
amounts of time. The initial portion of the rf pulse isstored for a time period t
m, the maximum amount of
storage time for any part of T. It is given by,
)1 (−=r m C tτ . (1)
where τis the accelerator structure pulse width and is
given by
rCT=τ (2)and rCis the compression ratio. The realization of the
storage system is usually achieved using low-loss
waveguide delay lines. These lines are usually guides thatpropagate the rf signal at nearly the speed of light. Themaximum length required for these guides, per
compression system, is
2max r
gmCvt l= ,( 3 )
where gv is the group velocity of the wave in the delay
line. The total number of rf pulse compression systems
required for the accelerator system is given by
crkkaa
cCnPPNNη= ;( 4 )
where aNis the total number of accelerator structure in
the linac,kP is the klystron (or the rf power source) peak
power, aPis the accelerator structure required peak
power, knis the number of klystrons combined in one
pulse compression system, and cηis the efficiency of the
pulse compression system.
Single-
Moded
Delay Lines3 dB 90 Degree Hybrid
Accelerator StructureTwo banks of power sources each has an nk/2 klystrons
3 dB 90 Degree Hybrid
Accelerator StructureTwo banks of power sources each has an nk/2 klystrons a) Single-moded Binary Pulse Compression
Single or
Multi-Moded
Delay LinesCirculatorShort Circuit
b) Binary pulse compression can have several improvements including
the use of a circulator and several modes to reduce the delay line
length.
Fig. 1 Binary Pulse Compression system
Thus the maximum total length of waveguide storage
line for the entire linac is given by
gr
kaa
ckc vC
PPN
nNl L τη 2)1 ( 1 max max −== . (5)
In general the total length L is given by
lRLLmax= ;( 6 )
where lR is a length reduction factor which varies from
one system to another and, in general, is a function of the
compression ratio. Finally, the total number of klystrons
in the system kN is given by,
kaa
crkPPN
CNη1= .( 7 )
2.2 Binary Pulse Compression system
For details of the original single moded system the
reader is referred to [3]. The system is shown in Fig. 1.The single moded BPC, in its original form, has a length
reduction factor
lR of rC/2 . It becomes more
economical at higher compression ratios. However, the
power being handled by the waveguides and rfcomponents is doubled at every stage of the BPC system.Naturally, the peak power depends on the number of
klystrons that one might use in one system, i.e.,
kn. The
length reduction factor is given by
r mlCncR−=2;( 8 )
where mn is the number of modes used in the system.
The parameter c determines the length reduction if a
circulator is used and is 1 if a circulator is used and 0
otherwise.
The efficiency of the system is given by
/Gf7/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7/Ge7
/Ge8/Ge6/Ge5
−/Ge5
−=
/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7
/Ge8/Ge6
−/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7
/Ge8/Ge6
−
==
mn
i mirmn
i mi
n
rCn
com cir c
C11
1010
101101
τατα
ηηη ;( 9 )
where iα is the attenuation constant in dB/unit time for
mode i, and cirη and comη are the circulator efficiency
and component efficiency respectively.
2.3 Delay Line Distribution System (DLDS)
The original description of the DLDS is found in [4]. A
modification to that system with multi-moded delay linesis discussed in [14]. However, accurate accounts for theefficiency and waveguide length are introduced here. Thesystem is shown in Fig. 2. To give an expression for thelength reduction factor in terms of the number of modes
mnwe first define the number of pipes per unit rf system
as
;5.01
/Gfa/Gfb/Gf9
/Gea/Geb/Ge9+−=
mr
pnCn (10)
where [.] means the integer-value function. The length
reduction factor is, then, given by
( )
)1 ()1 )(2/(1
−− −−=
r rp m r p
lCCn n CnR (11)
The efficiency of the system is given by:()()
/Gf7/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7/Ge7
/Ge8/Ge6
+
−−+= /Ge5 /Ge5−−−
=−−−−
=−−−−−
m p r
pmr jm
m jm p j r j nn C
jj nnCn
jnnjn jC
rcom
tC)1(1
11
20
120) (
20)(
20
10
1 10)1 10( 101τα
ταταταηη (12)
where jαis the attenuation of mode jin dB/unit time.
Delay Lines
Accelerator
StructuresBank of nk of klystrons
A set of hybrids that switches the combined rf
to different outputsNot all the output need to be used. The unused outputs areterminated by an rf load
a) A Unit of a Single-Moded DLDS
Multi-Moded Delay Lines. The total number of these lines is np
A mode launcher which takes nm inputs and produces nm modes into a single waveguide delay line
b) A Unit of a Multi-Moded DLDS
Single-Moded
Delay LinesAccelerator
Structures
A combiner
c) A Unit of an Active DLDSA High Power Microwave Switch
Fig. 2 Delay Line Distribution System
If a switch is used only one pipe is used and the length
reduction factor becomes 1/C r. The efficiency in that case
becomes
()()/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7
/Ge8/Ge6
+−−=−
rr
Coff
son
s off
sCoff
s
rCτ
ττηηηηηηηη1111
; (13)
where on
sηis the efficiency of the switch at the on state,
while off
sη is the efficiency of the switch at the off state.
The quantity τηis the efficiency of the waveguide due to
the attenuation of that waveguide for a period of time
2/τ .
2.4 Resonant Delay Lines
The original description of the resonant delay lines can be
found in [15]. An extensive analysis of the system and itsvariations using active switching are presented in [5].High power experimental results and techniques aredescribed in the next section of this article and detailed inRef. [7]. The system and its variations are shown in Fig. 3. Thelength reduction factor is given by
)1 (2
−−=
r r mlCCncR ; (14)
where cdetermine the length reduction if a circulator is
used and is 1 if a circulator is used and 0 otherwise.
The Efficiency of the system is given by
()2
10/
10/
01 10/
0 2
0 0 1010 1) 10(11/Gf7/Gf7
/Gf8/Gf6
/Ge7/Ge7
/Ge8/Ge6
−−−+=−
−− −
τα
ταταηηRRR RCrC
rcir; (45)
where αis the attenuation /unit-time in dB and is given
by /Ge5
==mn
ii
mn11α α ; and iαis the attenuation/unit-time
for mode i. The parameter 0R is a function of thecompression ratio [5] and is, approximately, given by
24 , 514.0 871.0)(164.0
0 ≤ −≈−
rC
r C e CRr.
Single or Multi-
Moded Delay LinesSingle or
Multi-Moded
Delay LinesShort Circuitklystrons
Coupling Irises
Accelerator Structures3 dB 90 Degree
Hybrid
a) Sled-II Pulse compression system
Short Circuitklystrons
Coupling Irises (can be
actively switched)
b) Sled-II pulse compression system with a circulator and active switchCirculator
Fig . 3 Resonant Delay Line pulse compression system
If one can design and implement a super high powerswitch, the intrinsic efficiency of the SLED-II system canbe enhanced. Intrinsic efficiency of this system isapproximately 82% [5], and the total efficiency is slightlyreduced from that number. The efficiency in this case hasa weak dependence on the compression ration.
2.4 Comparison
Table 1 shows the parameters of different single-modedpulse compression systems if used with the current designof the Next Linear Collider [1]. Clearly, these systemscomprise very long runs of low-loss vacuum waveguide.Several innovations are required to reduce the length andto make these systems operate at these high power levels.These are discussed in the following sections.
System Cr Waveguide
Lengthη
(%)Peak
PowerNumber
Of
Klystrons
4 131 km 85 600 MW 3168 DLDS
8 305 km 85 600 MW 1584
4 523 km 85 600 MW 3168 BPC
8 698 km 85 1200 MW 1584
4 180 km 82 493 MW 3277 (SLED-II)
8 124 km 59 716 MW 2258
Table 1: Parameters of single-moded different pulse
compression systems
3 HIGH POWER IMPLEMENTAION OF
THE RESONANT DELAY LINE SYSTEM
(SLED-II)
More technical details for the high power SLED-II
system can be found in [7]. Here we summarize the designand the obtained results.
To separate the input signal from the reflected signal,
one might use two delay lines fed by a 3-dB hybrid asshown in Fig. 4. The reflected signal from both lines canbe made to add at the forth port of the hybrid. Fig.4 showsthe pulse-compression system. For delay lines, it uses two22.48-meter long cylindrical copper waveguides, each is
12.065 cm in diameter and operating in the TE 01 mode. In
theory, these over-moded delay lines can form a storagecavity with a quality factor Q > 1x10
6. A shorting plate,
whose axial position is controllable to within ±4 µm by astepper motor, terminates each of the delay lines. Theinput of the line is tapered down to a 4.737 cm diameterwaveguide at which the TE
02 mode is cut-off; hence, the
circular irises which determine the coupling to the linesdo not excite higher order modes provided that they areperfectly concentric with the waveguide axis.
Fig. 4 The high power SLED-II system
A compact low-loss mode converter excites the TE
01
mode just before each iris [7]. These mode transducers,known as wrap-around mode converters, were developedspecifically for this application. The mode converters areconnected to two uncoupled arms of a high-power, over-moded, planar 3-dB hybrid. This hybrid is also designedspecially for this application so that it can handle thesuper high power produced by this system [9]. Thedistance from the irises to the center of the hybrid hasbeen adjusted to within ±13 µm to minimize reflections tothe input port. The iris reflection coefficient is optimizedfor a compression ratio of 8.
The system is designed to operate under vacuum. All
the components are designed to handle the peak fieldsrequired by the high power operating conditions of thesystem, at 11.424 GHz and 600 MW peak power themaximum field level is less than 40 MV/m.
The input and output pulse shapes of that system are
shown in Fig. 4. The output pulse reached levels close to500 MW. It was limited only by the power available fromthe klystrons.
4 MULTI-MODED STRUCTURES
Multi-moded system was first suggested for the DLDS
system [14]. Several designs for multi-moded componentshave recently been developed [16]. However, the mostpromising set of components are those based on planermicrowave structures [17]. These were an extension to theplanner hybrid designs developed for the High powerSLED-II Pulse compression system (see section 3 of thisarticle). These planner structures have the advantage of adesign that is insensitive to its height. Hence one can
increase the components height to any desired value toreduce the peak rf fields at the walls.
TE01TE20
TE10
TE11
Simulated electric fields of the multi-
moded circular to rectangular taper
139.8 mm
Taper Geometry (OperatingFrequency=11.424 GHz)40.64 mm
36.45 mm 36.63 mm
Fig. 5 Multi-moded circular to rectangular taper
To transfer the rectangular waveguide cross-section of
these components into a circular waveguide cross-section,thus making them compatible with the circular waveguidedelay lines, one need a special type of tapers. Tapers thattransforms waveguide modes from circular to rectangularhave been reported in [8]. These tapers could be extendedto operate with several modes at once. An example ofsuch a taper is shown in Fig. 5. The tapers takes the inputof a near square waveguide carrying the TE
10 and the TE 20
modes and transferring them into the circular waveguidemodes TE
11, and TE 01 respectively. These tapers perfectly
the planer multi-moded launcher and extractors presentedin [17].
5 ACTIVE SYSTSEMS
Supper-High-power microwave switches can reduce thecost of the DLDS while increasing its capabilities forhigher compression ratios. When used with DLDS onecan use one single pipe as shown in Fig. 2.
PIN diode array Active Window
• All doping profile and metallic terminals on the window are
radi al , i .e. perpendi cul ar to el ectri c f i el d of the TE 01 mode. /Ge0
Effect of doping and metal lines on RF signal is small when thedi ode i s reverse bi ased.
• Wi th f orward bi as, carri ers are i nj ected i nto I regi on and I regi on
becomes conductor /Ge0 RF signal is reflected.
P
N N side view (not to scale) metal line
(1.5um thick)
I
~10um
• Ba s e ma te ria l: high re s is tivity (pure ) s ilicon, <5000ohm-cm, n-
type
• Di ameter of acti ve regi on: 1.3 i nch
• Thickness: 220um
• Coverage (metal/doping line on the surface): ~10%Me ta l te rmina l
Radi al -l i ne PI N di ode array
structure (400 lines)220um2 inch
A B
Secti on A --B
Fig. 6 Implementation of a PIN diode active window
With resonant delay line systems active switches candramatically improve their efficiencies making it possible
Input
Output
Wrap-around Mode
ConverterH-Plane Over-moded
Hybrid
Iris
Delay Lines
Sled-II Configuration
0100200300400500
0 0.5 1 1.5 2Pulse compressor input
Pulse compressor outputPower (MW)
Time (micro-seconds)
Simulated Electric Field
of the Planer Hybrid
The wrap-around mode converter and
simulated electric field at its outputto utilize these compact systems for linear collider
applications. Indeed, these active switches have attracted the attentionof numerous researchers. However, most of the conceptsthat were suggested are either very expensive orimpractical. A promising concepts which combines botheconomical aspects and practical designs were suggestedrecently [13]. Also, the use of a several elements of such aswitch was explored [12]. The switch is shown in Fig. 6.The window shown operates in a waveguide carrying theTE
01 mode. Hence all the electric field lines are normal to
the doping and metalization lines. Because the TE 01 mode
does not carry any axial currents the separation of thewaveguide to supply the diodes with the required bias waspossible. These switches operated at power levels around10 MW at 11.424 GHz. This exceeds by orders ofmagnitude the capabilities of any known semiconductor rfswitch.
6 SUMMERY
Several pulse compression systems have developed for
use with the rf system of future linear colliders. Thesesystems suffer from very long waveguide runs. Some ofthe systems that have a compact nature also suffer fromefficiency degradation. To improve these systems severalinnovations were introduced. These innovations increasepower handling capabilities, make the system morecompact by utilizing several modes within a singlewaveguide, and finally improve the systems layout andperformance by turning them into active systems.
7 ACKNOLOWGMENT
This work reported in this paper is due to thecollaboration of several people, a partial list of them ismentioned here: C. Nantisat, N. Kroll, P. Wilson, F.Tamura, R. Ruth, G. Bowden, R. Lowoen, V. Dolgashev,K. Fant, A. Vlieks, R. Fowkes, C pearson, A. manegat,and the klystron mechanicalwork shop personel at SLAC.This work is supported by Department of Energy contractDE–AC03–76SF00515.
REFERENCES
[1]The NLC Design Group, Zeroth-Order Design Report
for the Next Linear Collider, LBNL-PUB-5424, SLAC
Report 474, and UCRL-ID 124161, May 1996
[2]The JLC Design Group, JLC Design Study, KEK-
REPORT-97-1, KEK, Tsukuba, Japan, April 1997.
[3]Z.D. Farkas, “Binary Peak Power Multiplier and its
Application to Linear Accelerator Design,” IEEETrans. MTT-34, 1986, pp. 1036-1043.
[4]H. Mizuno and Y. Otake, “A New Rf Power
Distribution System for X Band Linac Equivalent toan Rf Pulse Compression Scheme of Factor 2
N,” 17th
International Linac Conference (LINAC 94), Tsukuba,Japan, Aug. 21-26, 1994[5]S. G. Tantawi, et al. "Active radio frequency pulse
compression using switched resonant delay lines"Nuclear Instruments & Methods in Physic Research,Section A (Accelerators, Spectrometers, Detectors andAssociated Equipment) Elsevier, 21 Feb. 1996.Vol.370, No.2-3, pp. 297-302.
[6] Sami G. Tantawi et al: ‘Active High-Power RF Pulse
Compression Using Optically Switched ResonantDelay Lines’, IEEE Trans. on Microwave Theory andTechniques, Vol. 45, No 8, pp. 1486, A ugust, 1997
[7] Sami G. Tantawi, et al. , “The Generation of 400-MW
RF Pulses at X Band Using Resonant Delay Lines,”IEEE Trans. MTT, vol. 47, no. 12, December 1999;SLAC-PUB-8074.
[8] S.G. Tantawi, et al., “RF Components Using Over-
Moded Rectangular Waveguides for the Next LinearCollider Multi-Moded Delay Line RF DistributionSystem,” presented at the 18
th Particle Accelerator
Conference, New York, NY, March 29-April 2,1999.
[9] C.D. Nantista, et al ., "Planar Waveguide Hybrids for
Very High Power RF," presented at the 1999 ParticleAccelerator Conference, New York, NY, March 29-April 2, 1999; SLAC-PUB-8142.
[10] S.G. Tantawi, et al. , “A Multi-Moded RF Delay Line
Distribution System for the Next Linear Collider,”proc. of the Advanced Accelerator ConceptsWorkshop, Baltimore, MD, July 5-11, 1998, pp. 967-974.
[11] Sami G. Tantawi, et al., "Evaluation of the TE
12
Mode in Circular Waveguide for Low-Loss, High-Power RF Transmission," Phys. Rev. ST Accel.Beams, vol.3, 2000.
[12] Sami G. Tantawi and Mikhail I. Petelin: ‘The Design
and Analysis and Multi-Megawatt Distributed SinglePole Double Throw (SPDT) Microwave Switches’,IEEE MTT-S Digest, p1153-1156, 1998
[13]Fumihiko Tamura and Sami G. Tantawi, ”Multi-
Megawatt X-Band Semiconductor MicrowaveSwitches,” IEEE MTT-S Digest, 1999
[14]S. G. Tantawi, et al. “A Multi-Moded RF Delay Line
Distribution System for the Next Linear Collider ,”
Proce of the Advanced Accelerator ConceptsWorkshop, Baltimore, Maryland, July 5-11, 1998, p.967-974
[15]P.B. Wilson, Z.D. Farkas, and R.D. Ruth, “SLED II:
A New Method of RF Pulse Compression,” LinearAccel. Conf., Albuquerque, NM, Sept. 1990; SLAC-PUB-5330.
[16]Z. H. Li et al, “ Mode Launcher Design for the Multi-
moded DLDS ,” Proc. of the 6th European Particle
Accelerator Conference (EPAC 98), Stockholm,Sweden, 22-26 Jun 1998, p. 1900-1903.
[17]C. Nantista and Sami G. Tantawi, “ A Planar
Rectangular Waveguide Launcher and Extractor for aDual-Moded RF Power Distribution System, ” Thisproceedings. |
arXiv:physics/0008205v1 [physics.acc-ph] 20 Aug 2000IMPEDANCE OF ABEAMTUBE WITHSMALL CORRUGATIONS∗
K.L.F. Bane, G. Stupakov,SLAC, Stanford University,Stanf ord, CA 94309,U.S.A.
1 INTRODUCTION
In accelerators with very short bunches, such as is envi-
sioned in the undulatorregionofthe Linac CoherentLight
Source (LCLS)[1], the wakefield due to the roughness of
thebeam-tubewallscanhaveimportantimplicationsonthe
required smoothness and minimum radius allowed for the
beam tube. Of two theories of roughness impedance, one
yieldsanalmostpurelyinductiveimpedance[2],theothera
singleresonatorimpedance[3];forsmoothbunches,whose
lengthislargecomparedtothewallperturbationsize,thes e
two modelsgivecomparableresults[4].
Using very detailed, time-domain simulations it was
found in Ref. [3] that a beam tube with a random, rough
surfacehasanimpedancethatissimilartothatofonewith
small,periodiccorrugations. It was further found that the
wake was similar to that of a thin dielectric layer (with
dielectric constant ǫ≈2) on a metallic tube: Wz(s)≈
2K0cosk0s,with wavenumberandlossfactor
k0=2√
aδandK0=Z0c
2πa2; (1)
withathe tube radius, δdepth of corrugation, and Z0=
377 Ω. For the periodic corrugation problem this result
was inferred fromsimulations for which the period p∼δ.
On the other hand, at the extreme of a tube with shallow
oscillations, with p≫δ, the impedance was found, by a
perturbation calculation of Papiernik, to be composed of
many weak, closely spaced modes beginningjust above pi
phaseadvance[5].
In this report we find the impedance for two geometries
of periodic, shallow corrugations: one, with rectangular
corrugations using a field matching approach, the other,
with smoothly varying oscillations using a more classical
perturbation approach. In addition, we explore how these
results change character as the period-to-depthof the wall
undulation increases, and then compare the results of the
two methods.
2 RECTANGULAR CORRUGATIONS
Let us consider a cylindrically-symmetric beam tube with
the geometryshown in Fig. 1. We limit considerationhere
to the case δ/asmall; for the moment, in addition, let δ/p
notbesmall. Wefollowtheformalismofthefieldmatching
program TRANSVRS[6]: In the two regions, r≤a(the
tuberegion,RegionI)and r≥a(thecavityregion,Region
II) the Hertz vectors are expandedin a complete, orthogo-
nalset; EzandHφarematchedat r=a;usingorthogonal-
itypropertiesaninfinitedimensional,homogeneousmatrix
equation is generated; this matrix is truncated; and finally ,
∗Work supported by the U.S. Department of Energy under contra ct
DE-AC03-76SF00515.Figure1: Thegeometryconsidered.
the eigenfrequencies are found by setting its determinant
to zero. We demonstrate below that, for our parameter
regime, the system matrix can be reduced to dimension 1,
andtheresultsbecomequitesimple.
Inthe tuberegion,the z-componentofthe Hertzvector
ΠI
z=−∞/summationdisplay
n=−∞An
χ2nI0(χnr)
I0(χna)e−jβnz,(2)
withI0themodifiedBessel functionofthe firstkind,and
βn=β0+2πn
p, χ2
n=β2
n−k2,(3)
withβ0the phase advance and kthe wave number of the
mode. Inthecavityregion,
ΠII
z=−∞/summationdisplay
s=0Cs
Γ2sR0(Γsr)
R0(Γsa)cos[αs(z+g/2)],(4)
αs=πs
g,Γ2
s=α2
s−k2, (5)
R0(Γsr) =K0(Γs[a+δ])I0(Γsr)−I0(Γs[a+δ])K0(Γsr),
(6)
withK0themodifiedBessel Functionofthesecondkind.
EzandHφaregivenby
Ez=/parenleftbigg∂2
∂z2+k2/parenrightbigg
Πz, Z 0Hφ=−jk∂Πz
∂r.(7)
Matching these fields at r=a, and using the orthogo-
nality of e−βnzon[−p/2, p/2], andcos[αs(z+g/2)]on
[−g/2, g/2]weobtainahomogeneousmatrixequation. To
findthe frequencies,thedeterminantisset to zero; i.e.
det/bracketleftbigg
R −/parenleftbigg2g
p/parenrightbigg
NTIN/bracketrightbigg
= 0, (8)
with thematrix Ngivenby
Nns=2βn
(β2n−α2s)g/braceleftbigg
sin(βng/2) : seven
cos(βng/2) : sodd,(9)
andthediagonalmatrices RandIby
Rs= (1+ δs0)ka/parenleftbiggR′
0
xR0/parenrightbigg
Γsa,In=ka/parenleftbiggI′
0
xI0/parenrightbigg
χna.
(10)For the beam, on average, to interact with a mode, one
spaceharmonicofthemodemustbesynchronous. Wewill
pick the n= 0space harmonicto be the synchronousone;
i.e.letβ0=k(we take the particle velocity to be v=c).
Let us truncate the system matrix to dimension 1, keeping
only the n= 0ands= 0terms in the calculation. Now
ifkδis small, then the s= 0term in Rbecomes R0=
2/(kδ), then= 0term in IisI0=ka/2, andN00≈1.
Eq.8thenyields
k=/radicalbigg2p
aδg, (11)
which,for p= 2g,equals k0ofEq.1.
Thelossfactorisgivenby K=|V|2/[4Up(1−βg)][7],
withVthe voltage lost by the beam to the mode, Uthe
energy stored in the mode, and βgthe group velocity
overc. The voltage lost in one cell is given by the syn-
chronous (n= 0)space harmonic: V=A0p, and the
energy stored in one cell, U= 1/(2Z0c)/integraltext
E·E∗dv,
is approximately that which is in the n= 0space har-
monic: U=πA2
0a2p(1 +k2a2/8)/(2Z0c)(for details,
see Ref. [6]). For βg, we take Eq. 8 truncated to dimen-
sion 1, and expandnear the synchronouspoint. Taking the
derivative with respect to β0and then setting β0=kwe
obtain:
(1−βg) =4δg
ap. (12)
Thelossfactorbecomes K=K0.
The above method can be extended to modes of higher
multipole moment m, in which case the beam will excite
hybrid modesrather than the pure TM modes of above[6].
Again the system matrix can be reduced to the n= 0and
s= 0terms, and the lowest mode wave number and loss
factorhaveasimpleform(for 1≤m≪a/δ):
k=/radicalBigg
(m+ 1)p
aδgandK=Z0c
πa2(m+1),(13)
and(1−βg) =m(m+ 2)δg/(ap). In particular,we note
thatthedipole (m= 1)frequencyisequaltothemonopole
(m= 0)frequency.Also,thewakeattheoriginisthesame
asfortheresistive-wallwakeofacylindricaltube[8],asw e
expect.
Fig. 2 shows a typical dispersion curve obtained by
TRANSVRS. Here k/k0= 1.07,K/K0=.94. Note
that even when δ/ais not so small, e.g.for bellows with
δ/a≈.2[9], the analytical formulasare still useful. Fig. 3
showshow the strengthand frequencyofthe modechange
as the period of undulation is increased. The scale over
whichKdropstozerois p0≈π/radicalbig
aδg/2p. Byp∼p0,the
one dominant mode has disappeared, and we are left with
the manyweak,closelyspacedmodesofPapiernik.
3 SINUSOIDALCORRUGATIONS
Let usassume nowthat thepipesurfaceisgivenby
r=a−hsinκz, (14)Figure2: Dispersioncurveexample.
Figure3: Anexampleshowingtheeffectofvarying p.
where 2π/κis the period of corrugation, and his its am-
plitude. We assume that both the amplitude and the wave-
length are small, h≪aandκa≫1. This allows us
to neglect the curvature effects and to consider the surface
locally as a plane one. We will also assume a shallow cor-
rugation hκ≪1,i.e.the amplitudeof oscillation is much
smaller thantheperiod.
Introducing a local Cartesian coordinate system x,y,z
withy=a−r(directed from the wall toward the beam
axis),and xdirectedalong θ,thesurfaceequationbecomes
y=y0(z)≡hsinκz. The magneticfield near the surface
Hx(y, z)doesnotdependon x(thatis θ)duetotheaxisym-
metryoftheproblem. It satisfiesthe Helmholtzequation
∂2Hx
∂y2+∂2Hx
∂z2+k2Hx= 0 (15)
with theboundarycondition
(/vector n∇H)|y=y0= 0, (16)
where /vector nis the normal vector to the surface, /vector n=
(0,1,−hκcosκz).
Note that the longitudinal electric field Ezcan be ex-
pressedintermsof Hx,
Ez=−i
k∂Hx
∂y. (17)Usingthesmallparameter h/a,wewilldevelopaperturba-
tion theory for calculation of Hxnear the surface and find
howEzis relatedto Hx.
In the zeroth approximation,the zdependenceof Hxis
dictatedbythebeamcurrentperiodicity,
Hx(y, z) =H(y)eikz. (18)
Putting Eq. (18) into Eq. (15) we find that d2H/dy2= 0,
henceH(y) =H0+Ay, where the constant Acan be
related,throughEq. (17),totheelectricfieldonthesurfac e,
A=ikEz. We will see belowthat Aissecondorderin h.
For a flat surface, for which /vector n= (0,1,0), from the
boundary condition (16), we would conclude that A= 0,
however,thecorrugationsresultina nonzero A,andhence
Ez. Substitutingthemagneticfield(18)intotherighthand
side ofEq. (16)onefinds
/vector n∇H=−1
2ihkκH 0/bracketleftBig
ei(k+κ)z−ei(k−κ)z/bracketrightBig
−ikζH 0eikx.
(19)
Clearly, the boundary condition is not satisfied in this ap-
proximation. Tocorrectthis,wehavetoaddsatellitemodes
to thefundamentalsolution(18)
Hx(y, z) =H(y)eikz+H1(y, z),(20)
where
H1(y, z) =B+(y)ei(k+κ)z+B−(y)ei(k−κ)z.(21)
The dependence of B±versus ycan be found from the
Helmholtzequation,
B=B±
0e−y√
κ2±2κk, (22)
where B±
0are constants. In order for B±to exponentially
decayin y,we haveto assumeherethat k < κ/ 2.
Substituting H1terms into the boundary condition
(16) generates first order terms that have x-dependence
expi(k±κ)x, and second order terms proportional to
exp(ikx). Fromtheformeronefindsthat
B±
0=−ikκH 0h
2√
κ2±2kκ, (23)
andthelattergivesanexpressionforthetangentialelectr ic
field onthesurface,
Ez=1
4ikh2κHx√
κ2+ 2kκ+√
κ2−2kκ√
κ2−4k2.(24)
OnecannowsolveMaxwell’sequationswiththebound-
aryconditiongivenbyEq. (24)(seedetailsin[10]). Itturn s
out, that in the region of frequencies k < κ/ 2there exist
a solution corresponding to a wave propagating with the
phasefrequencyequalto thespeedoflight. Thefrequency
and the loss factor of the mode are shown in Fig. 4 (solid
lines). We seethatdecreasingtheheightofthecorrugation
resultsinsmallerwakes,andhenceleadstothesuppressionFigure4: Frequencyandlossfactorasfunctionofheight.
of the interaction of the synchronouswave with the beam.
Inthelimit ofsmall frequencies, k≪κthe frequencyis
k1=2
h√aκ. (25)
We have to mention here that the perturbation theory
breaks down for very small values of h. Indeed, we im-
plicitlyassumedthatthesatelliteharmonicsinEq. (22)ar e
localized near the surface, otherwise our approximationof
plain surface becomes invalid. Hence, we have to require
thatκ−2k≫a−1, which gives the following condition
of applicability: h > a−1/4κ−5/4. Thisconditionexplains
why this mode was not found by Papiernik: being pertur-
bative in parameter hthe approach developedin his paper
isapplicableonlywhen hcanbemadearbitrarilysmall.
Finally,inFig.4weincludealsotheresultsofFig.3,ob-
tained by field matching for δ/a=.003(the dashes). For
the comparison we make the correspondences p= 2π/κ
andδ= 2h. Wenotethateventhoughthegeometryforthe
field matching results violate our requirement for smooth-
ness, theresultsforthetwo methodsareverysimilar.
4 ACKNOWLEDGEMENTS
We thank A. Novokhatskii for his contribution to our un-
derstandingofthe problemofroughnessimpedance.
5 REFERENCES
[1] Linac Coherent Light Source (LCLS) Design Study Report.
SLAC-R-521,Apr 1998.
[2] K. Bane, et al, PAC97, p. 1738 (1997); G.V. Stupakov, Phys.
Rev. AB1,64401 (1998).
[3] A.Mosnier and A.Novokhatskii, PAC97, p. 1661 (1997).
[4] K.Bane andA.Novokhatskii, SLAC-AP-177,March1999.
[5] M. Chatard-Moulin and A.Papiernik, IEEETrans. Nucl.Sci.
26, 3523 (1979).
[6] K. Bane and B. Zotter, Proceedings of the 11thInt. Conf. on
HighEnergyAccelerators, CERN,p. 581(1980).
[7] See,e.g., E. Chojnacki, et al, PAC93, p. 815, 1993; A. Mil-
lich,L.Thorndahl, CERN-CLIC-NOTE-366,Jan. 1999.
[8] A. Chao, “Physics of Collective Instabilities in High-E nergy
Accelerators”, John Wiley& Sons, NewYork(1993).
[9] K.Bane andR.Ruth, SLAC-PUB-3862,January 1986.
[10] G.V. Stupakov in T. Roser and S. Y. Zhang, eds., AIP Con-
ference Proceedings 496, 1999, p.341. |
arXiv:physics/0008206v1 [physics.acc-ph] 21 Aug 2000COMPRESSION OF HIGH-CHARGE ELECTRONBUNCHES∗
M. J.Fitch, A. C. Melissinos;UniversityofRochester, Roch esterNY 14627,USA
N.Barov, J.-P. Carneiro, H. T.Edwards, W.H. Hartung;FNAL, BataviaIL 60510,USA
Abstract
TheAØPhotoinjectoratFermilabcanproducehighcharge
(10-14 nC) electron bunches of low emittance ( 20πmm-
mrad for 12 nC). We have undertaken a study of the opti-
mal compression conditions. Off-crest acceleration in the
9-cell capture cavity induces an energy-time correlation,
which is rotated by the compressor chicane (4 dipoles).
The bunch length is measured using streak camera images
of optical transition radiation. We present measurements
under various conditions, including the effect of the laser
pulse length (2 ps sigma Gaussian vs. 10 ps FWHM flat
top). The best compression to date is for a 13.2 nC bunch
withσz= 0.63mm(1.89ps),whichcorrespondstoapeak
currentof2.8kA.
1 INTRODUCTION
Electron beams with short bunch lengths are desirable for
high energy physics, free-electron lasers, and other appli -
cations.
In this paper we report on studies of compression at the
AØ Photoinjectorof Fermilab[1,2]with a chicaneof four
dipoles as measured by a picosecond streak camera. The
photoinjector was prototyped for the TeSLA Test Facility
[3], and in that context, there are three stages of acceler-
ation and compression, and the chicane is the first com-
pressor. The gun is a 1.625-cell π-mode normal conduct-
ing copper structure at 1.3 GHz whose backplane accepts
a molybdenum plug coated with a Cs 2Te photocathode.
Solenoidsforemittancecompensationsurroundthegun. A
superconducting Nb nine-cell cavity accelerates the beam
to 16–18 MeV. After the dipole chicane are experimental
anddiagnosticbeamlines.
Thesestreakcamerameasurementssupportelectro-optic
sampling measurements reported in a companion paper in
these proceedings. Emittance measurements are reported
by J.-P. Carneiro et al. in these proceedings,andfor 12nC
the normalized emittance is ǫn= 20πmm-mrad without
compression. The issue of emittance growth during bunch
compression[4–8]isunderstudy,thoughpreliminarystud-
iessuggestanemittanceincreaseofapproximatelyafactor
oftwo.
∗Work supported in part by Fermilab which is operated by
URA, Inc. for the U.S. DoE under contract DE-AC02-76CH03000 .
e-mail:mjfitch@pas.rochester.edu2 EXPERIMENT
ThestreakcameraisaHamamatsuC5680-21Sstreakcam-
era with M5676 fast sweep module1read out by a Pulnix
progessivescandigitalCCDcamera. Calibrationwasdone
using a short UV laser pulse and a thick fused silica delay
block. We find a calibration of 3.9 pixels/ps at the fastest
sweepspeedwithalimitingresolutionofabout1ps. After
oneyear,thecalibrationwasrepeated,andthehigh-voltag e
sweepofthestreaktubehaddegradedsomewhatto3.6pix-
els/ps, so we report at most a 10% systematic uncertainty
in thebunchlengthmeasurements.
Optical transition radiation (OTR) light [9–11] was im-
aged by all-reflective optics to the slit of the streak cam-
era. OTR is prompt, and has a characteristic opening an-
gle of 1/γ, and in our case E=γmc2∼16MeV, so
1/γ∼32 mrad or 1.8◦. An out-of-plane(periscope) bend
rotates the image so that the vertical direction of the beam
falls on the horizontal slit. This is desirable to diagnose
aberrationssincethe chicanebendsin theverticalplane.
The photocathodedrive laser (built by the University of
Rochester) is a lamp-pumped Nd:glass system frequency-
quadrupled to the UV ( λ= 263nm) [12]. The UV laser
pulse is a Gaussianwith σt= 1.9ps. Then,theUV pulses
are temporally shaped to an approximate flat-top distribu-
tion with 10.7 ps FWHM. We have measured the bunch
lengthwithboththe longandtheshortlaserpulse.
Anumberofstreakimageswasacquiredateachsetting.
Aftersubtractingaconstantbackgroundtracefromthepro-
jectedimage,(unsubtractednoisewhichvariedfromimage
to image was within ±0.5pixel.) each streak image trace
is fit to a Gaussian. The mean value from the ensemble
at each setting is reported, with error bars assigned from
thestatistical spreadofvaluesfromthisensemble. Theex-
ception to this is in Figure 1, where each streak image is
correlated with the charge measured on that shot, and the
errorbaristhe errorintheGaussianfit.
3 RESULTS: LONGLASER PULSE
In Figure 1 we give the uncompressed bunch length ver-
sus charge. At low charge,the bunchlength is the same as
thatoftheUVlaserpulseonthecathode,butincreasesdra-
matically at highercharges. At highcharge,( ∼11–13nC),
the beam was compressed and measured as a function of
the accelerating phases (Figure 2). We set the middle pair
of dipole chicane magnets to the nominal values (current
I= +2 .0A or 680 Gauss), and reduce the outer pair
1Wethank A.Hahn, FNALBeams Division, for the loan of the stre ak
camera.0510152025303540
0 2 4 6 8 10 12Uncompressed Bunch Length vs ChargeFWHM of Gaussian fit [ps]
Charge [nC]
Figure 1: Uncompressed bunch length vs. charge for the
long(10psFWHM)laser pulselength.
slightly for vertical steering (typically −1.9to−1.95A).
ThephasesofthegunRFand9-cellcavityRFarerecorded
asthe“setphase”fromthecontrolsystem( UNIX). Inaddi-
tiontothesetphase,wegivethe9-cellphaseformaximum
energy (crest). The gun phase is referenced to PARMELA
bythecurveofchargetransmissionvs. gunphase.
Thepointofbestcompression(Figure2)isnotsensitive
to thegunphase,howeverthebunchlengthensifthephase
is too early. Even for high charge, the measured bunch
lengthiseasily compressedtoless than1mm σz(or3ps),
andtheoptimumis0.63mm σz(1.89ps).
05101520253035
-130 -120 -110 -100 -90 -80 -70High Charge (11-13nC) Bunch Length vs. Phase
Gun φG=-40o, Parmela φP=+45o
Gun φG=-30o, Parmela φP=+55o
Gun φG=-50o, Parmela φP=+35oFWHM from Gaussian fit [ps]
9-cell phase [deg]max energy
Figure 2: Compression vs. Phases of the 9-cell cavity and
gunforthe10pslaserpulselength. Thewidthofthefocus
mode image was 2.77 ps FWHM, and the data were cor-
rected assuming this broadening adds in quadrature to the
real width.
Repeating this experiment as a function of charge, we
find that the minimum bunch length is shorter at lower
charge(Figure3), because non-linearspace chargegrowth
is uncompensated. The phase of optimal compression isonly weakly dependent on the charge, shifting by 4◦from
1 nCto 10nC.
0510152025
-95 -90 -85 -80 -75 -70 -65Compression vs. Charge: Long laser pulse
10nC
8nC
4nC
1nCFWHM of Gaussian fit [ps]
9-cell phase [deg]max energy -56
Figure 3: Compression vs. chargefor the 10 ps laser pulse
length.
4 RESULTS: SHORTLASER PULSE
The compression experiments were repeated with a short
Gaussian laser pulse ( σt= 2ps) on the cathode, with
the expectation that the space charge growth of the bunch
lengthwouldbemoresevere.
With the chicane dipole magnets off and degaussed, we
measured the uncompressed bunch length with a streak
cameralookingatOTRradiationasbefore(Figure4)from
1nCto5nC.Evenatlowcharge,thebunchlengthismore
thanafactoroftwolongerthantheinitiallaserpulselengt h
on the cathode, and increases linearly with charge. The
compressedbunchlengthfortheshortlaserpulseisshown
in Figure 5, and the minimum is slightly larger, and at a
largerangleoff-crest.
5 PEAK CURRENT
One figure of merit for (sub)picosecond electron bunches
is the peak current, which depends on both the charge and
the bunchlength. Ifthebeamis Gaussianintime,
I(t) =Q√
2π σtexp(−t2/(2σ2
t)) (1)
Thenthepeakcurrentisbydefinitionthepeakvalueofthe
currentprofile:
Ip=Q√
2π σt=2√
2 ln 2Q√
2π τ(2)
for the rms bunch length σzor the full width at half maxi-
mum(FWHM) τ.
We know of two other facilities which report peak cur-
rent at or above 2 kA which are 1.97 kA at the AWA [13]0246810
0 1 2 3 4 5 6Uncompressed Bunch Length vs. Charge
for short (2 ps σ) laser pulseσT [ps]
Charge [nC]
Figure 4: Uncompressed bunch length vs. charge for the
2 pslaser pulselength.
012345678
-85 -80 -75 -70 -65Compressed Bunch Length vs Charge
for short laser pulse (2 ps σ)
σT [ps] (4nC)
σT [ps] (6nC)
σT [ps] (8nC)σT [ps]
9-cell phase [deg]max energy
-40 deg
Figure 5: Compressedbunchlengthvs. chargefor the2 ps
laser pulselength.
and2.3kAattheCLICTestFacility(CTF-II)atCERN[8].
Ourreportedbestpeakcurrentof2.8kAisasignificantim-
provement.
6 REFERENCES
[1] Eric Ralph Colby. Design, Construction, and Testing of a
Radiofrequency Electron Photoinjector for the Next Gener-
ation Linear Collider . PhD thesis, University of California
Los Angeles, 1997.
[2] J.-P. Carneiro et al. First Results of the Fermilab High-
Brightness RFPhotoinjector. InA.LuccioandW.MacKay,
editors,Proceedings of the 1999 Particle Accelerator Con-
ference, pages 2027–2029, 1999.
[3] D.A. Edwards. TTF Conceptual Design Report. Technical
Report TESLA95-01, DESY,1995.[4] Bruce E. Carlsten and Tor O. Raubenheimer. Emittance
growth of bunched beams in bends. Physical Review E ,
51(2):1453–1470, 1995.
[5] Bruce E. Carlsten. Calculation of the noninertial space -
charge force and the coherent synchrotron radiation force
for short electron bunches in circular motion using the re-
tarded Green’s function technique. Physical Review E ,
54(1):838–845, 1996.
[6] Bruce E. Carlsten and Steven J. Russell. Subpicosecond
compression of 0.1–1nC electron bunches with a magnetic
chicaneat8MeV. PhysicalReviewE ,53(3):R2072–R2075,
1996.
[7] M. Dohlus and T. Limberg. Emittance Growth due to
Wake Fields on Curved Bunch Trajectories. Proceedings
ofthe1996FreeElectronLaserConference (FEL96) ,1996.
TESLA-FEL96-13.
[8] H.Braun, F.Chautard, R.Corsini,T.O.Raubenheimer, an d
P. Tenenbaum. Emittance Growth during Bunch Compres-
sionintheCTF-II. PhysicalReviewLetters ,84(4):658–661,
2000.
[9] U. Happek, A. J. Sievers, and E. B. Blum. Observation
of Coherent Transition Radiation. Physical Review Letters ,
67(21):2962–2965, 1991.
[10] Yukio Shibata, Toshiharu Takahashi, Toshinobu Kanai,
Kimihiro Ishi, Mikihiko Ikezawa, Juzo Ohkuma, Shuichi
Okuda, and Toichi Okada. Diagnostics of an electron beam
of a linear accelerator using coherent transition radiatio n.
Physical Review E ,50(2):1479–1484, 1994.
[11] R. Lai, U. Happek, and A. J. Sievers. Measurement of the
longitudinal asymmetry of a charged particle bunch from
the coherent synchrotron or transition radiation spectrum .
Physical Review E ,50:R4294–R4297, 1994.
[12] A.R. Fry, M.J. Fitch, A.C. Melissinos, and B.D. Taylor.
Laser system for a high duty cycle photoinjector. Nuclear
Instruments & Methods in Physics Research A , 430:180–
188, 1999.
[13] M. E. Conde, W. Gai, R. Konecny, X. Li, J. Power,
P.Schoessow, andN.Barov. Generationandaccelerationof
high-charge short-electron bunches. Physical Review Spe-
cial Topics - Accelerators and Beams , 1:041302, 1998. |
arXiv:physics/0008207v1 [physics.acc-ph] 21 Aug 2000ELECTRO-OPTICSAMPLING OFTRANSIENT FIELDSFROM THE
PASSAGE OF HIGH-CHARGEELECTRON BUNCHES∗
M.J. Fitch,A. C. Melissinos,UniversityofRochester, Roch esterNY 14627,USA
P. L. Colestock,J.-P. Carneiro, H.T. Edwards,W. H.Hartung , FNAL,BataviaIL 60510,USA
Abstract
When a relativistic electron bunch traverses a structure,
strong electric fields are induced in its wake. We present
measurements of the electric field as a function of time as
measured at a fixed location in the beam line. For a 12 nC
bunchofduration4.2psFWHM,thepeakfieldismeasured
>0.5MV/m. Time resolution of ∼5 ps is achieved using
electro-optic sampling with a lithium tantalate (LiTaO 3)
crystal and a short-pulseinfraredlaser synchronizedto th e
beam. We present measurements under several different
experimentalconditionsanddiscuss the influenceof mode
excitationin thestructure.
1 INTRODUCTION
Sincethepioneeringexperiments[1–3],electro-opticsam -
pling(EOS)hasbeenshowntobeapowerfultechniquefor
fast time-domainmeasurementsofelectricfields[4,5].
The use of electro-optic sampling for accelerator appli-
cationshasbeenpreviouslysuggestedby[6–8]andothers.
Detectionofthebeamcurrentbymagneto-opticeffectshas
beendemonstratedby[7]withatimeresolutionthatissub-
nanosecond.
Recently, at Brookhaven, electro-optic detection of a
charged particle beam was reported by detecting a faint
light pulse through crossed polarizers as the beam passed
by an electro-optic crystal [9]. The time resolution possi-
ble here is limited by the speed of the photodetectors and
amplifiers, which similar to that available with capacitive
beam pickups ( ∼100 ps). Earlier at Brookhaven, an RF
phasemeasurementusingtheelectro-opticeffectandphase
stabilizationbyfeedbackwasdemonstrated[10].
We have used electro-optic sampling to measure the
electricfieldwaveformsinvacuuminducedbythepassage
of electron bunches with an estimated time resolution of
∼5ps, limitedbythe laserpulselength[11,12].
Independentlyof our work, a group at FOM Rijnhuizen
(Nieuwegein,TheNetherlands)hasusedelectro-opticsam-
pling in ZnTe to resolve the sinusoidal electric field of
the free electron laser FELIX at the optical frequency
(λ= 150 µm) [13]. Of note is the rapid-scanning cross-
correlation technique (a fast data-acquisition trick). Th e
same group has sampled the electric field of the transition
radiation from the electron beam exiting a beryllium win-
dow [14] and the electric field in vacuum [15] from which
∗Work supported in part by Fermilab which is operated by URA, I nc.
for the U.S. DoE under contract DE-AC02-76CH03000. Current address
of P.L.Colestock is LANL.e-mail: mjfitch@pas.rochester.e duthe bunchlengthismeasured.
We have thus far been unable to reproduce their results
with ZnTe;wesuspecta problemwith ourcrystal.
2 EXPERIMENT
The linear electro-optic effect (or Pockels effect) is one o f
severalnonlinearopticaleffectsthatarisefromthesecon d-
order susceptibility tensor χ(2), and is described in many
standard texts, such as [16]. For our purposes, it suf-
fices that the polarization of light is altered by an electric
field applied to the crystal. By analyzing the polarization
change, the electric field can be measured. Using a short
laser pulse and a thin crystal, the electric field is sampled
at a particular time T iwhen the laser pulse arrives at the
crystal. By changing the delay of the probe laser arrival
time, and repeatedly measuring the electric field, the elec-
tric field waveformisrecoveredbyelectro-opticsampling.
The data acquistion is handled by LabVIEW and a digital
oscilloscope.
PD b
PD a
Polarizer
10 nC e-
3-10 ps
UV to gun
10 ps
Photocathode
Drive Laser2 ps IRPolarizer
DelayCompensatorDifferential Detection
(a-b) / (a+b)Balanced Photodiode (PD) Pair
Electro-Optic
Crystal LiTaO3
zx
yy
z3 1
x,2Coordinate
System
Crystal
OrientationGlass Viewports
Figure1: EOS configuration,sensitiveto (Ez+Eθ)/√
2.
ExperimentswereperformedattheAØPhotoinjectorof
Fermilab [17,18]. A lamp-pumped Nd:glass laser system
built by the University of Rochester is quadrupled to UV
(λ= 263nm)forphotocathodeexcitation. The UV pulses
are temporally shaped to an approximate flat-top distribu-
tion with a 10.7 ps FWHM. Unconverted infrared light isthe probe laser for the electro-optic sampling, so that jit-
ter between the beam and the probe laser vanishes to first
order. The photoinjector produces 12 nC bunches with
normalizedemittanceof 20 πmm-mrad(uncompressed)in
pulse trains up to 200 pulses long with interpulse spacing
1µsec. A chicane of four dipoles was used for magnetic
compression. In a companion paper in these proceedings
we present some compression studies. The best compres-
sion to date is σz= 0.63mm (1.89 ps) for a charge of
13.2nC, whichgivesa peakcurrentof2.8kA.
-0.1-0.0500.050.10.15
-500 0 500 1000 1500 2000 2500 3000EOS signal: 3mm LiTaO3 7nC, Chicane offΓ [rad]
time [ps]
Figure2: EOS waveform,sensitiveto (Ez+Eθ)/√
2.
0246810
0 5 10 15 20 25 30Fourier Transform (FFT): 3mm LiTaO3 7nC Chicane offMagnitude
Frequency [GHz]2.7
3.4
7.2
9.511.8
18
20
Figure3: FFT ofwaveformin Figure2
We have taken data using several different configura-
tions. The elements common to all of the setups are a po-
larizer, the crystal, the compensator, and another polariz er
(analyzer). The ellipsometry can be simplified for perfect
polarizersandsmallpolarizationchangesinthecrystal. F or
two detectors AandB(silicon photodiodes)after the ana-
lyzer,theintensitymeasuredat IA≡Ais:
A=Iosin2(δΓ +φ) (1)
where the intensity incident on the analyzer is Io, andφ
is a constant which represents the compensator and/or the
static birefringence of the crystal ( φs=ω(no−ne)L/c).
Thetermproportionaltotheelectricfieldis δΓ =ωδn L/c,
andputtingintheelectro-opticcoefficientforLiTaO 3withthe electricfieldalongthe3-axis,wefind
δΓ =ω
c(n3
or13−n3
er33)E3L. (2)
For the electric field along the 2-axis of LiTaO 3, the
electro-opticcoefficientis δΓ =ωn3
or22E2L/c. Itisclear
from Equation 1 that if φ= 0, then for small signals,
A∝Io(δΓ)2.
The second detector Bmeasures the orthogonal polar-
izationcomponent,so B=Iocos2(δΓ +φ). Itisseenthat
fora choiceof φ=π/4,
A−B
A+B= sinδΓ∼δΓ∝E (3)
independentof Io. Thecompensatorthenisusedtobalance
the detectors in the absence of electro-optic modulation.
However, the static birefringenceis a function of tempera-
ture, so we make one furthersubtractionto cancel driftsto
formtheexperimental Γ.
Γ =/parenleftbiggA−B
A+B/parenrightbigg
signal−/parenleftbiggA−B
A+B/parenrightbigg
background(4)
Forthebackgroundpoints,ashutterisclosedwhichblocks
the UV for the photocathodebut allows the infrared probe
laser to go to the crystal. The field magnitudeis estimated
by calibrations on a duplicate crystal on the bench. A
fieldE3= 100kV/m induces a rotation Γ = 0 .046rad,
while E2= 100kV/m induces Γ = 0 .003rad, all for the
7×8×1.5mmLiTaO 3crystal(thickness L= 1.5mm).
3 RESULTS
With the sensitive axisof the crystalorientedso that E3=
(Ez+Eθ)/√
2,usingtheconventionthattheelectronbeam
velocity defines the +zdirection, the measured waveform
in shownin Figure2. Theinitially surprisingfeatureis the
presence of strong oscillations that persist beyond the end
of the delay stage (3 ns). These are attributedto excitation
of modes in the structure, and an FFT of the waveform is
shown in Figure 3. We can, for example, identify the fre-
quencies near 3 GHz as trapped modes in the 6-way cross
[19].
With the sensitive axis of the crystal oriented so that
E3=Er, the measuredwaveformis quite different,being
nearly sinusoidal (Figure 4). In the cylindrical beam pipe
(radius b= 2.2cm), there is a propagating (waveguide)
TM1,1modewithfrequency ν= (3.83)c/2πb= 8.4GHz,
and it may be the origin of the observed 8.8 Ghz compo-
nent. The slow build-up (and beat near 1900 ps) in the
envelope could be explained by a small splitting of this
modeintotwofrequencies,whichareinitiallyoutofphase.
The FFT (figure 5) suggests a splitting, but the resolution
(limited by the length of the scan) is poor. More will be
presented and discussed in a future publication. A second
roundof experimentsis plannedwith the goal of detecting
the directCoulombfield ofthe bunch.-0.4-0.3-0.2-0.100.10.20.30.4
-500 0 500 1000 1500 2000 2500 3000EOS scan: Radial E, Q=12nC, FWHM=6psΓ [rad]
time [ps]
Figure 4: Electro-optic sampling waveform, sensitive to
Er.
01020304050607080
0 10 20 30 40 50FFT of scan: Radial E, Q=12nC, FWHM=6psMagnitude
Frequency [GHz]7.2 GHz
8.2 GHz8.8 GHz
11.8 GHz
Figure5: FFT ofwaveformin Figure4
The direct Coulomb field of the bunch, if detected, is
simply connected with the charge distribution ρ(z)with
sensitivity to head-tail asymmetries. As the electro-opti c
effect has a physical response at the femtosecond level,
thetechniqueofelectro-opticsamplingcouldbeavaluable
method for bunch length measurements at the <100fs
level. The transient (wake) fields we measured off-axis
could be applied to on-axis measurements of the wake
function and beam impedance. Higher-order mode cou-
plinganddampinginstructuresmayalso beofinterest.
4 REFERENCES
[1] J. A. Valdmanis, G. Mourou, and C. W. Gabel. Picosec-
ondelectro-opticsamplingsystem. AppliedPhysicsLetters ,
41:211–212, 1982.
[2] J. A.Valdmanis, G.Mourou, andC.W.Gabel. Subpicosec-
ond Electrical Sampling. IEEE Journal of Quantum Elec-
tronics, QE-19:664–667, 1983.
[3] David H. Auston and Martin C. Nuss. Electrooptic Gen-
eration and Detection of Femtosecond Electrical Tran-
sients.IEEE Journal of Quantum Electronics , 24(2):184–
197, 1988.
[4] Q.WuandX.-C.Zhang. Free-spaceelectro-opticssampli ng
of mid-infrared pulses. Applied Physics Letters , 71:1285–
1286, 1997.[5] A.Leitenstorfer,S.Hunsche,J.Shah,M.C.Nuss,andW.H .
Knox. Detectors and sources for ultrabroadband electro-
optic sampling: Experiment and theory. Applied Physics
Letters, 74(11):1516–1518, 1999.
[6] P.J.Channell. UseofKerrcellstomeasureRFfields. Tech -
nicalreport,LosAlamosNationalLaboratory,1982. Accel-
erator Theory Note AT-6:ATN-82-1.
[7] Yu. S. Pavlovand N. G. Solov’ev. Formation and Measure-
ment of Picosecond Beams of Charged Particles. In Proc.
VIIIAll-UnionCharged Part.Accel.Conf. ,volume 2,pages
63–67, Protvino1982.
[8] M. Geitz, K. Hanke, and A. C. Melissinos. Bunch Length
Measurements at TTFL using Optical Techniques. Techni-
cal report, DESY, 1997. Internal report TESLA collabora-
tion.
[9] Y. K. Semertzidis et al. Electro-Optical Detection of
Charged Particle Beams. Proceedings of the 1999 Particle
Accelerator Conference (PAC'99) ,pages 490–491, 1999.
[10] K. P. Leung, L. H. Yu, and I. Ben-Zvi. RF Phase Stabi-
lization of RF Photocathode Gun Through Electro-Optical
Monitoring. Proc. SPIE - Int. Soc. Opt. Eng. , 2013:147–
151, 1993. BNL-49276.
[11] M. J. Fitch et al. Electro-optic Measurement of the Wake
Fields of 16MeV ElectronBunches. Technicalreport, Uni-
versityofRochester,1999. UR-1585andFERMILAB-TM-
2096.
[12] M. J. Fitch, A.C. Melissinos, and P.L. Colestock. Picos ec-
ondelectronbunchlengthmeasurementbyelectro-opticde-
tection of the wakefields. Proceedings of the 1999 Particle
Accelerator Conference (PAC'99) ,pages 2181–2183, 1999.
[13] G.M.H.Knippels etal. GenerationandComplete Electri c-
Field Characterization of Intense Ultrashort Tunable Far-
Infrared Laser Pulses. Physical Review Letters , 83:1578–
1581, 1999.
[14] D. Oepts et al. Picosecond electron-bunch length measu re-
ment using an electro-optic sensor. Proceedings of the 21st
International FEL Conference (FEL99) , 1999. 23–28 Au-
gust 1999, DESY.
[15] X. Yan et al. Sub-picosecond electro-optic measuremen t of
relativistic electron pulses. submitted to Physical Review
Letters, 2000.
[16] Amnon Yariv. Optical Electronics . Holt, Rinehart & Win-
ston, Inc., 3rdedition, 1985.
[17] Eric Ralph Colby. Design, Construction, and Testing of a
Radiofrequency Electron Photoinjector for the Next Gener-
ation Linear Collider . PhD thesis, University of California
Los Angeles, 1997.
[18] J.-P. Carneiro et al. First Results of the Fermilab High -
Brightness RF Photoinjector. Proceedings of the 1999 Par-
ticle Accelerator Conference (PAC'99) , pages 2027–2029,
1999.
[19] Ch. X. Tang and J. Ng. Wakefields in the Beamline of TTF
Injector II. Technical report, DESYTESLA97-11, 1997. |
arXiv:physics/0008208 21 Aug 2000Efficient Design Scheme of Superconducting Cavity
Sang-ho Kim, Marc Doleans, SNS/ORNL, USA
Yoon Kang, APS/ANL, USA
Abstract
For many next-generation high intensity proton
accelerator applications including the Spallation N eutron
Source (SNS), superconducting (SC) RF provides the
technology of choice for the linac. In designing th e
superconducting cavity, several features, such as p eak
fields, inter-cell coupling, mechanical stiffness, field
flatness, external Q, manufacturability, shunt impe dance,
higher order mode (HOM), etc., should be considered
together. A systematic approach to determine the op timum
cavity shape by exploring the entire geometric spac e of
the cavity has been found. The most efficient use o f RF
energy can be accomplished by adjusting the cell sh ape. A
small region in parameter space satisfying all reas onable
design criteria has been found. With this design pr ocedure,
choosing the optimum shape is simplified. In this p aper,
the whole design procedure of this optimisation sch eme is
explained and applied to the SNS cavity design.
1 INTRODUCTION
In many recently initiated or proposed projects for high
intensity proton acceleration, SCRF technology has been
selected for the main part of the linac, which uses
elliptical shape SC cavities. SNS will be the first high
intensity proton accelerator with a SC linac. The b asic
parameters of the SNS SC linac are shown in Table 1 .
Table 1: Basic parameters of the SNS SC linac
RF frequency 805 MHz
Energy range 185-1000 MeV
Average beam current 2 mA
Number of beta sections 2 (0.61 and 0.81)
Transition energy between sections ~380 MeV
Cavity shape elliptical (6 cells)
In designing the cavity, RF and mechanical prope rties
are considered together, especially for the cavity whose
beta is less than one. The general design bases and issues
for the SNS cavity are summarized in terms of cavit y
parameters.
For the inner cell design;
• Minimise the peak surface fields
• Provide a reasonable mechanical stiffness
• Maximize the R/Q
• Achieve a reasonable inter-cell coupling coefficien t
For the end cell and full cavity design;
• Obtain a good field flatness• Obtain a lower (or same) surface fields at end cell s
than (or with) those of inner cell
• Achieve a reasonable external Q, Qex
All the issues listed above are directly linked to the
shape, and the effects of shape on these issues are
different. In some aspects, the effects compete, an d
optimization among tradeoffs becomes necessary. A
systematic scheme is introduced here for choosing t he
optimum cavity shape.
2 INNER-CELL DESIGN
Figure 1 shows the geometric parameters of the
elliptical cell. Adjusting four of these five param eters
(Req, α, Rc, a/b, Ri) determines a cell shape that satisfi es
required beta and frequency. Usually the equator ra dius is
used for tuning, since its effect on the resonance
frequencies is large and its influence on the other cavity
parameters is negligible.
2a2b
Aspect Ratio
(a/b)Slope Angle
R Dome
(Rc)R Equator (Req)
R Iris (Ri)(α)
Figure 1: Geometric parameters of the cell.
In order to understand the influences of cell param eters
on the cavity performance, the entire geometric spa ce was
explored. The following procedures were established from
this understanding. Peak fields (Ep & Bp), inter-ce ll
coupling coefficient (k), R/Q, and Lorentz force de tuning
coefficient (K) are used as cavity parameters. The first
step is to determine relations between the dome rad ius and
the iris ellipse aspect ratio at fixed iris radius and slope
angle. At any dome radius, the surface electric fie ld
profile can be changed by adjusting a/b. In this ad justment
variations of other cavity parameters are negligibl e.
Figures 2 (a) and (b) are comparisons of surface el ectric
fields for given a peak surface electric field and
accelerating field, respectively. The line 2 in Fig ure 2 (a)
has higher accelerating field than the others. The line 2 in
Figure 2 (b) has lower peak electric field than the others.
The field profile of line 2 provides the efficient use of RF
energy.12
31
2
3E
zE
zz z
(a) (b)
Figure 2: Surface electric fields of cells with dif ferent
values of a/b; (a) at same peak surface electric fi eld, (b) at
same acceleration gradient.
The best values of a/b are automatically determined by
fixing the other geometric cell parameters. In this
procedure, a/b’s are found as a function of Rc’s at given
Ri and α. Slope angles above 6 degree are required for the
rinsing process. Due to its small cell length, only a small
angle region from 6 to 8 degrees leads to a good ce ll
performance for the medium beta case. Finally the c ell
geometry can be defined with the remaining two cell
parameters, Ri and Rc, at fixed slope angle.
0.30.40.50.60.70.80.9
30 32 34 36 38 40
Dome Radius (mm)Iris Aspect Ratio (a/b) Ri=50 mm
Ri=45 mm
Ri=40 mm
Figure 3: “Efficient-set” lines of cell geometry fo r SNS
b=0.61 cavity at the slope angle of 7 degree.
Figure 3 is an example of SNS medium beta ( β=0.61)
inner cell at the slope angle of 7 degrees. SUPERFI SH
was used for the analysis [1]. These lines all sati sfy the
condition of a flat field around the iris. In Figur e 4,
relative values of cavity parameters are plotted fo r the cell
geometry on the solid line in Figure 3. The cell wi th
efficient-set having Rc=30 mm is used as normalisat ion
reference. The Lorentz force detuning coefficient K is
calculated with fixed boundary condition and stiffe ner
location of 70 mm from the cavity axis. This coeffi cient is
sensitive to the shape of the iris ellipse, especia lly in low
beta case. Similar graphs can be done for other Ri values
to cover all the geometric parameter space.
An optimum cell shape that satisfies all the design
criteria can be found by combining the results from
different Ri’s.
Figure 5 is an example of SNS medium beta cavity at
the slope angle of 7 degree. The SNS design criteri a are;
Ep=27.5 MV/m, Bp<60 mT, k>1.5 %, K<3 Hz/(MV/m)2,
and Eo>11.9 MV/m for the reference geometry.0.20.40.60.81.01.21.41.6
30 32 34 36 38 40
Dome radius (mm)Relative Valuesk
Ep
R/Q
Bp
K
Figure 4: Relative cavity parameter behaviors for R i=50
mm and a=7 degree versus Rc. “Efficient-set” geomet ry is
represented by dome radius, Rc.
The Eo value used here pertains to the inner-cell o nly.
All the design criteria are marked with bold lines in Figure
5. There is a small region where all the design cri teria are
satisfied. Selecting the final cell geometry is a m atter of
the strategy. SNS chose the cell geometry for the h igh
accelerating gradient within the design criteria.
The results of inner cell design for the high beta cavity
show similar behaviours except a few aspects. The
Lorentz force detuning is not sensitive to cell sha pe, so
this is not an issue in the high beta cavity. The s lope angle
can be chosen from 6 to 12 degrees, for specified c avity
parameters. The larger slope angle is better for en d cell
tuning.
1.51.82.12.42.73.0
30 32 34 36 38 40
Dome Radius (mm)Ep/EoBp=65 mTBp=60 mTBp=55 mT
(at Ep=27.5 MV/m)K=2 Lorentz detuning
K=3 Hz/(MV/m)2
K=4
k=2.5 %
k=2.0 %
k=1.5 %SNSEp/Eo
(at Ri=50 mm)
Ep/Eo
(at Ri=45 mm)
Ep/Eo
(at Ri=40 mm)
Figure 5: Overall comparisons of cavity parameters on the
cell geometric parameter.
3 END CELL DESIGN
End-cells should be tuned separately due to the att ached
beam pipes. Changing the shape of end-cells must le ad to
a reasonable axial electrical field flatness below ~2 %.
Peak surface fields must be equal or lower than inn er cells
value. Many different end-cell shapes can satisfy t hese
criteria. The Figure 6 shows the axial electric fie ld profile
for different acceptable cavity geometries.
Each end-cell designs necessitate a different appro ach
because one is connected to the power coupler. A co axial
type power coupler will be used in SNS. 0.0E+005.0E+061.0E+071.5E+072.0E+072.5E+07
0 20 40 60 80 100 120
Axial Distance (cm)Electric Field on the Axis (V/m)
SNS81 (I)
SNS81 (II)
SNS81 (III)
Figure 6: Axial electric field profiles for three d ifferent
SNS high beta ( β=0.81) cavities at Ep=27.5 MV/m.
Required Qex’s are 7.3 ×105 and 7.0×105 for medium
and high beta cavities, respectively. A computer st udy of
Qex has been done following a scheme introduced in [2].
Four-parameter space has been explored (Figure 7). The
geometry of the power coupler is not used as a para meter
in the study.
The inner conductor tip position has a strong effec t on
the coupling between the cavity and the power coupl er. As
shown in Figure 8, about 25 mm displacement results in
one order variation on the Qex value.
LPC
ITP BPRGOCREG1) GOC
(Geometry of Coupler)
2) BPR
(Beam Pipe Radius)
3) REG
(Right End-cell Geometry)
4) LPC
(Longitudinal Position of Coupler)
5) ITP
(Inner conductor Tip Position)
Figure 7: Five parameters that can affect Qex.
1.E+041.E+051.E+061.E+07
-40 -30 -20 -10 0 10 20 30
Inner Conductor Tip Position (mm)QexSNS61
SNS81 (I)
SNS81 (II)
7E+05
Figure 8: Variations of Qex’s as a function of ITP for
three different cavities. Same GOC and 7 cm of LPC are
used for each calculation.
Since keeping the same iris radius as the inner cel l leads
to a high Qex value, enlarging the beam pipe size c an
provide a solution. This option is efficient below a certain
diameter. Figure 9 shows that over 62 mm, increasin g the
beam pipe size has a weak influence on the high bet a case.
The points marked with triangle are not on the line . This
results from the change in field profile after end celltuning. These points still satisfy all requirements , that
means the Qex can be also controlled by changing en d cell
shape only.
01234567
5.0 5.5 6.0 6.5 7.0
Beam Pipe R adius (cm)Qex (*105)
SNS81 (I)
(7 cm)Da1
(6.6 cm)
Da3 (6.4 cm)SNS81 (II)
(6.2 cm)Da5 (6.0 cm)Da6 (5.2 cm)
Field profiles are
almost same.
Field profiles are
different from above
2-Die 3-Die 4-Die
Figure 9: Variation of Qex’s as a function of BPR f or SNS
high beta cavity. Same GOC, 0 cm of ITP and 7 cm of
LPC are used for each calculation.
The effect of LPC is also examined from 8.5 cm to 7
cm, the lowest possible distance. The Qex decreases
linearly by a factor of three.
Many end cell shapes could satisfy the requirements .
The final decision will depend on the amount of
engineering margin.
4 HIGHER ORDER MODE (HOM)
HOM analysis for the reference geometry has been
done. Many trapped modes are found even in referenc e
geometry. Beam dynamics issues related with HOM are
under study and the intermediate results suggest th at the
cumulative beam break-up is not an issue in SNS [3] . In
order to investigate the effects of mechanical impe rfection
of the cavity on the trapped modes, Monte-Carlo ana lysis
is in progress.
5 SUMMARY
The cavity performance is visualised in the geometr ic
parameter space by the systematic scheme introduced .
6 ACKNOWLEDGEMENT
We are grateful to R. Sundelin, P. Kneisel at Jlab, and
James Billen at LANL for giving many useful advises and
comments.
This work is sponsored by the Division of Materials
Science, U.S.Department of Energy, under contractio n
number DE-AC05-96OR22464 with UT-Bettelle
Corporation for Oak Ridge National Laboratory.
REFERENCES
[1] J. Billen and L. Young, “POISSON SUPERFISH,”
LA-UR-96-1834 (2000)
[2] Pascal Balleyguier, “External Q Studies for APT SC-
Cavity Couplers,” Proc. of LINAC98, pp133 (1999)
[3] Dong-O Jeon, SNS.ORNL.AP.TN019 (2000) |
STRUCTURAL ANALYSIS OF
SUPERCONDUCTING ACCELERATOR CAVITIES *
D. Schrage, LANL, Los Alamos, N.M., USA
* Work sponsored by he Japan Atomic Energy Research Institute
under contract DE-FI04-91AL73477-A012Abstract
The static and dynamic structural behavior of
superconducting cavities for various projects wasdetermined by finite element structural analysis. The
β =
0.61 cavity shape for the Neutron Science Project wasstudied in detail and found to meet all design requirementsif fabricated from five millimeter thick material with asingle annular stiffener. This 600 MHz cavity will have aLorentz coefficient of –1.8 Hz/(Mv/meter)
2 and a lowest
structural resonance of more than 100 Hz.
Cavities at β = 0.48, 0.61, and 0.77 were analyzed for a
Neutron Science Project concept which would incorporate7-cell cavities. The medium and high beta cavities werefound to meet all criteria but it was not possible togenerate a
β = 0.48 cavity with a Lorentz coefficient of
less than –3 Hz/(Mv/meter) 2.
1 INTRODUCTION
There are quite a few accelerator projects underway for
which elliptical superconducting cavities are planned.This paper documents structural analysis of
β < 1
superconducting cavities for the Neutron Science Projectof JAERI [1] and includes consideration of Lorentz forcedetuning, cavity fabrication, vacuum loading, tuningforces, and mechanical resonant frequencies.
2 TECHNICAL CONSIDERATIONS
While each accelerator has specific technical
requirements with regard to the values of β, the number of
cells, and the bore sizes of the cavities, there are someother physics and engineering considerations that must beincluded in the design of the cavities. Some of theseparameters, the peak electric and magnetic surface fieldsalong with the bore radius, affect the performance of thecavities. Other parameters, such as the material thicknessand the wall slope, are related to the practical matter ofmanufacture of the cavities. Lastly, the presence orabsence of annular stiffeners has a significant effect uponthe Lorentz for RF detuning, the mechanical resonantfrequencies, and the tuning forces. A detailed discussionof this is given in References 2 and 3.
The parameters are listed in Table 1. The value selected
for B
peak/Epeak is arguable and some organisations would
suggest that a higher value would be more suitable if itresulted in a lower peak electric field. Indeed, all of the
values are to some extent arbitrary; they are certainly notabsolute. However, they do serve as guidelines forpreliminary design of cavities.
Table 1: Cavity Design Parameters
PARAMETER ALLOWABLE
VALUES
Peak Electric Field Epeak/Ea = minimum
Peak Magnetic Field Bpeak/Epeak~ 1.71
mT/(Mv/meter)
Fabrication Rmin > 2*thickness
BCP Cleaning Slope > 6o
Mech. Resonances ωi > 60 Hz
Radiation Pressure k < 2.0 Hz/(Mv/meter)2
Tuning Sensitivity < 5.0 #/kHz
Vacuum Loading σvon-Mises < 3,500 #/in2
3 STATIC ANALYSIS OF CAVITIES
Three cavity mid-cell shapes were analysed: β = 0.48,
0.61, and 0.77. These were obtained from Reference 4 andare shown on Figure 1. A
β = 1 cross-section is shown for
reference.
The structural analysis was carried out using
COSMOS/M [5]. Two-dimensional axi-symmetric
elements were used for the analysis of half-cells todetermine the tuning forces plus the deflections, stressesand frequency shifts under vacuum load and Lorentzpressure. The frequency shifts were determined from theoutput of SUPERFISH [6].
The main consideration was the Lorentz force de-
tuning. The analyses were performed for various stiffenerring radii. The results for the
β = 0.61 cavity are shown on
Figure 2 for material thicknesses of 3, 4, and 5 mm. Theresults are similar for the other two cavities. Withoutannular stiffeners none of the cavity shapes will satisfy therequirement that the Lorentz detuning coefficient of thecavity be less than –3 Hz/(Mv/meter)
2. However, for the β
= 0.61 with the 4 mm thickness, the curve is quite flat sothe selection of the 7 inch stiffener radius is not rigid.
Some cases were run with two stiffener rings but these
resulted in unacceptably high tuning forces. Use of twostiffener rings would also increase fabrication costs.-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.00 1. 00 2. 00 3. 00 4. 00 5. 00 6. 00 7. 00 8. 00 9. 00
STIFFENER RADIUS inchesLORENTZ COEFFICIENT3 mm Thick
4 mm Thick
5 mm Thick
Figure 2: Effect of Stiffener Radius and Material
Thickness for β = 0.61 Cavity
Figure 1: Cavity Cross-Sections
The results for the three cavities are listed on Table 2.
The β = 0.48 cavity does not meet the fabrication criteria
(Rmin > 2.0*t) and has a Lorentz coefficient that is greater
than the specified value. However, with the lower- β
cavities operated at lower gradient (the requirement is thatE
peak < 16.0 Mvolt/meter [1]) this may be acceptable. At
this peak electric field, the accelerating field is only 3.7Mvolt/meter and the Lorentz detuning is reduced to 1/8ththe value at E
a = 10 Mvolt/meter.Table 2: Static Analysis Results for Stiffened Cavities
β = 0.48 β = 0.61 β = 0.77
Thickness, mm 5.0 4.0 4.0
Rmin 1.4*t 4.0*t 3.5*t
k Mvolt/m2-3.3 -1.8 -0.9
Tuning #/kHz 1.06 0.97 1.78
Vac. Stress #/in23496 3811 2896
The deformation of the β = 0.48 cavity under Lorentz
pressure resulting from an accelerating gradient of 10Mvolt/meter is shown on Figure 3. The Lorentz pressuresare quite low with the maximum being 0.48 #/in
2. The
axial deformations are similarly low; the maximum is 6.7X 10
-6 inch. This corresponds to a frequency shift of –330
Hz.
Figure 3: Lorentz Pressure Deformation for
β = 0.48 Cavity
Three-dimensional finite element models were used to
determine the gravity deformations of the complete 5-celland 7-cell cavities. These analyses were run usingCOSMOS/M with three-node shell elements. The weightsand mid-length transverse deflections of the cavities arelisted on Tables 3 and 4. The presence of the stiffenersproduces a significant reduction of the deflection.
Table 3: Static Deflections of 7-Cell Cavities
β = 0.48
5 mm Thickβ = 0.61
4 mm Thickβ = 0.77
4 mm Thick
Un-Stiffened
Wt (#) 237. 195. 211.
Disp. (in) 0.01474 0.01537 0.02854
Stiffened
Wt (#) 278. 232. 251.
Disp. (in) 0.00059 0.00047 0.00061
Table 4: Static Deflections of 5-Cell Cavities
β = 0.48
5 mm Thickβ = 0.61
4 mm Thickβ = 0.77
4 mm Thick
Un-Stiffened
Wt (#) 169. 139. 151.
Disp. (in) 0.00395 0.00622 0.00778
Stiffened
Wt (#) 199. 166. 179.
Disp. (in) 0.00017 0.00020 0.000194 DYNAMIC ANALYSIS OF CAVITIES
The three-dimensional finite element models described
in the previous paragraph were used to determine themechanical resonant frequencies. A cross-section of a 5-cell, un-stiffened
β = 0.61 cavity is shown on Figure 5 and
the results for 5-cell and 7-cell cavities are listed onTables 5 and 6 respectively. For these cases, the irises ofthe end-cells were held rigidly fixed in all coordinates.Use of other boundary conditions would have resulted inlower frequencies.
Figure 5: Lowest Mode of β = 0.61 5-Cell Cavity
Table 5: Cavity Structural Frequencies of 5-Cell Cavities
CAVITY WALL
THICK
mmUN-
STIFFENED
CAVITY
LOWEST
FREQUENCY
HzSTIFFENED
CAVITY
LOWEST
FREQUENCY
Hz
β = 0.48 5.0 47. 181.
β = 0.61 4.0 40. 217.
β = 0.77 4.0 37. 251.
Table 6: Cavity Structural Frequencies of 7-Cell Cavities
CAVITY WALL
THICK
mmUN-
STIFFENED
CAVITY
LOWEST
FREQUENCY
HzSTIFFENED
CAVITY
LOWEST
FREQUENCY
Hz
β = 0.48 5.0 27. 130.
β = 0.61 4.0 22. 130.
β = 0.77 4.0 20. 142.
Past experiments [7] have shown good agreement of
measured mechanical resonant frequencies with thepredicted values. It is important to note that the analyseswere run for simple cavities; there were no beam tubes,power couplers, HOM couplers, etc. included. In addition,there is no consideration of the stiffness of the cavitysupport structure. Inclusion of any or all of these itemswill reduce the mechanical resonant frequencies. Thus,the frequencies listed in Tables 5 and 6 must be regardedas ideal maximums. As in the case of a similar study of
the cavities for the APT linac [8], it was found that theannular stiffeners would be required to meet the dynamicrequirements, in particular when the effects of the beamtubes, etc. are included.
5 CONCLUSIONS
There are many variables to consider in the design of
superconducting cavities. However, in meeting therequirements listed in Table 1, the options diminishrapidly. It is clear that for values of
β < 0.5, the structural
design of these cavities is a challenge at 600 MHz.Minimization of the Lorentz force detuning will likelyrequire operation of
β < 0.5 cavities at Ea < 10
Mvolt/meter. It is also clear that stiffeners will be requiredto meet the mechanical resonant frequency requirement.
6 ACKNOWLEDGEMENT
Rick Wood provided the software support for the
calculation of the frequency shifts. Jim Billen and FrankKrawczyk provided cavity designs and SUPERFISH runsfor these analyses.
REFERENCES
[1] M. Mizumoto et al., “A Proton Accelerator for the
Neutron Science Project at JAERI, ” 9th Workshop on
RF Superconductivity, Santa Fe, November 1999.
[2] J. Billen, “Superconducting Cavity Design for SNS, ”
LANL Memo LANSCE-1:99-149, August 1999.
[3] D. Schrage, “Structural Analysis of Superconducting
Accelerator Cavities, ” LANL Technical Report LA-
UR:99-5826, November 1999.
[4] T. Wangler, “A Superconducting Design for the
JAERI Neutron Science Project, ” LANSCE-1:99-
193(TN), November 1999.
[5] Structural Research & Analysis Corporation,
http://www.cosmosm.com/
[6] J. Billen & L. Young, “POISSON SUPERFISH, ” Los
Alamos National Laboratory report, LA-UR-96-1834,Revised April 22, 1997.
[7] G. Ellis & B. Smith, “Modal Survey of Medium
Energy Superconducting Radio Frequency Cavity forAccelerator Production of Tritium Project, ” 9th
Workshop on RF Superconductivity, Santa Fe,November 1999.
[8] D. Schrage et al., “Static & Dynamic Analysis of
Superconducting Cavities for a High PerformanceProton Linac, ” 7th Workshop on RF
Superconductivity, Saclay, October 1995. |
CW PERFORMANCE OF THE TRIUMF 8 METER LONG RFQ FOR EXOTIC
IONS
R. L. Poirier, R. Baartman, P. Bricault, K. Fong, S. Koscielniak, R. Laxdal, A. K. Mitra, L. Root, G.
Stanford, D. Pearce, TRIUMF, Vancouver, B. C. Canada
Abstract
The ISAC 35 MHz RFQ is designed to accelerate ions of
A/q up to 30 from 2keV/u to 150keV/u in cw mode. TheRFQ structure is 8 meters long and the vane-shaped rodsare supported by 19 rings spaced 40 cm apart. An unusual
feature of the design is the constant synchronous phase of
-25°; the buncher and shaper sections are eliminated in
favor of an external multi-harmonic buncher. All 19 ringsare installed with quadrature positioning of the four rodelectrodes aligned to +/- 0.08 mm. Relative field variationand quadruple asymmetry along the 8 meters of the RFQwas measured to be within +/- 1%. Early operation at peakinter-electrode voltage (75kV) was restricted by the rapidgrowth of dark currents due to field emission; the nominaloperating power of 75 kW increased to 100 kW in a fewhours. A program of high power pulsing, followed by cwoperation have all but eliminated the problem leading to asuccessful 150 hour test at full power. Successful beamtest results confirm beam dynamics and rf designs.
1 INTRODUCTION
The accelerator chain of the ISAC radioactive ion beamfacility includes a 35.3 MHz split ring RFQ, operating incw mode, to accelerate unstable nuclei from 2 keV/u to150 keV/u. The RFQ structure is 8 meters long and thevane-shaped rods, comprised of 40cm long cells, aresupported by 19 rings. Full power tests on a single module[1] and on a three-module assembly [2] enabled us tocomplete the basic electrical and mechanical design forthe RFQ accelerator. An initial 2.8m section [3] of theaccelerator (7 of 19 rings) was installed and aligned toallow preliminary rf and beam tests to be carried out. Thefull complement of 19 rings shown in Fig. 1 has now been
tested to full rf power with beam.
2 DESIGN CONSIDERATIONS
The design of the RFQ is dominated by threeconsiderations. Firstly, the low charge-to-mass ratio of theions dictate a low operating frequency to achieve adequatetransverse focusing. Secondly, continuous wave (cw)operation is required to preserve beam intensity. Thirdly,the desire to minimize the length of the structure and itscost. No single feature of this RFQ gives it exceptionalstatus, but the combination of novel features and unusualdesign parameters adopted to address these considerationscan be argued to give it “landmark” status.The relative tuning difficulty of an RFQ scales roughly asthe vane-length, L, divided by the free-space wavelength
λ; for ISAC this ratio is ≈ 1 which is typical of RFQs.However, the alignment difficulty scales as L/r
0 where r0
is the bore radius; and the structure length L=8 m of ISAC
makes this aspect unusually challenging. The vane voltageof 75 kV is moderate, and the electric field limitationcomes not from consideration of the Kilpatrick factor, butrather from the c.w. requirement and cooling limitations.Both these considerations feed into the challenge andcomplexity of the mechanical design regarding stiffness,stability and tolerances. Though there are several c.w.proton RFQs and one light-ion RFQ, ISAC is unique inc.w. operation for heavier ions. There is no other RFQoperating in cw mode in this frequency range with acharge to mass ratio of 1/30.
Figure 1. Full compliment of 19 rings installed and
aligned
2.1 Beam Dynamics [4]
Though it is now well accepted that a design strategy
different from that for high current proton linacs, be usedfor low current, light and heavy ion RFQs, this was not soat the design time six years ago. The Kilpatrick factor at1.15 is rather modest. However, because of the power vs.cooling requirement one cannot increase the accelerationrate by merely raising the voltage. In order to reduce thestructure length, the buncher and shaper sections werecompletely eliminated in favour of a discrete four-harmonic saw-tooth pre-buncher located 5m upstream.This has also the benefits of reduced longitudinalemittance at the RFQ exit and of allowingexperimentalists to do “time of flight” work with an 86 nstime structure. These gains are made at the expense of aslightly lower beam capture of 80%. Acceleration startsimmediately after the radial matching section (RMS) andthe vane modulation index ( m) ramps quickly from 1.124
to 2.6, while the bore shrinks from 0.71 to 0.37 cm in theremaining booster and accelerator sections. A
conventional LANL-type design, as for protons, would
have resulted in a 12 meter long linac.To maintain reasonable acceptance, the vane design hascharacteristic radius to pole tip r
0 = 0.741cm. Tho ugh one
could single out m and φs as unusual, it is the combination
of parameters, chosen so as to hasten acceleration(particularly in the early cells), which is remarkable. Thefocusing parameter B=3.5, which is “low to typical ” of
ion-RFQs, is carefully balanced against a comparatively
large peak RF-defocusing parameter ∆= –0. 0408. For
RFQs in general, φ
s rarely exceeds -30o and m is rarely
above 2 while in ISAC the synchronous phase φs = -25o is
large and constant which maximizes the acceleration andm=2.6 which is a record for operational RFQs. Here we
adopt the definition φ
s =-90o/0o gives min/max
acceleration.There are several other features of the beam dynamics andvane shapes, which at the time of design were consideredquite novel. An exit taper was substituted by a muchshorter transition cell, and a transition cell was introducedbetween the RMS ( m=0) and the booster. Both entrance
and exit region fields of the RFQ vanes were modeledwith an electrostatic solver. To minimize machining costs,
vanes with constant transverse radius of curvature ρ = r
0
were adopted; this leads to significant departure (up to
35% for ISAC) from the two term potential either where
ka ~ unity and/or where m is large. Here k=2π/(βλ) and a
is the local minimum bore radius. The cell parameters a,m
were systematically corrected to compensate for thiseffect.The RFQ is also unusual in that the vanes are rotated 45degrees from the usual horizontal/vertical orientation.Matching into and out of the RFQ therefore requires around beam. The matching into the RFQ is achieved byfour electrostatic quadrupoles. They are the same designas the other quadrupoles in the beam transport line exceptfor the last one. In order to retain an acceptance of greater
than 100 π mm-mrad through the matching region this
quadrupole is very small (1 inch long by 1 inch insidediameter).
2.2 RF
From a structural point of view, the low frequency of theRFQ dictates that a semi-lumped resonant structure beused to generate the required rf voltage between theelectrodes. Various RFQ models were built [5] and thestructure proposed for the ISAC-I accelerator, is a variantof the 4-rod structure developed at the University ofFrankfurt [6]. The unique design feature of this 4-rodstructure is the single split ring rather than a separate ringfor each pair of electrodes resulting in only 8% of thepower being dissipated in the tank walls, negating theneed for water cooling the tank walls. The choice of the
split ring design along with the choice of making ρ = r
0
has negated the requirement for individual stem tuners.
This type of structure was chosen for its relatively highspecific shunt impedance, its mechanical stability and the
elimination of an unwanted even-type transmission linemode in favour of the desired odd-mode. The parasiticeven-mode was identified as cause of a serious loss oftransmission in the HIS RFQ [7] at GSI.
2.3 Mechanical
The mechanical design of the RFQ [8] has two majorunique features: (1) The vacuum tank is square in cross-section and split diagonally by an “O” ring flange into
two parts, the tank base and the tank lid. Each part wasplated separately using the triangular enclosure as thecontainer for the cyanide bath solution. In thisconfiguration, the copper plating is easier and it givesfull unobstructed access to the RFQ modules for ease ofinstallation and alignment. (2) The basic design of theRFQ structure is different from other RFQ structures inthat the maximum rf current carrying surface (rf skin)has been de-coupled from the mechanical supportstructure (strong-back). The rf skin encloses the strong-back but is attached only where it meets the electrodesupports, thus allowing for thermal expansion of the skinwithout loading the structure and causing electrodemisalignment.
3 ALIGNMENT
Figure 2. Mechanical alignment of RFQ rings
The alignment goal was to achieve quadrature positioning
of the rod electrodes to within 0.08 mm. Beam dynamicscalculations indicate that at this level, emittance growth issmaller than 1%. The alignment philosophy [8] was basedon manufacturing 19 identical rings and mounting them onprecision ground platens, which are accurately aligned inthe vacuum tank prior to ring installation. Each platen isspecial steel 63.5 mm thick with an offset axial rail boltedand doweled in place. The platen and rail are accuratelyground in one set- up, thus providing an accurate datumfor mounting and locating the ring bases. Each platen has5 adjustable mounting points, 3 vertical and 2 lateral, andspecial alignment targets. The platens are adjusted andaligned in the tank using the theodolite intersectionmethod. Once the platens are aligned they are locked inposition and ready for installation of the rings. The-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1
024681 0 1 2 1 4 1 6 1 8 2 0Ring NumberDeviation in mmVertical
Horizontalalignment of the ring assemblies on the platens was
accomplished by the same method. Because of themanufacturing procedures and alignment philosophyadopted, when the electrodes were installed on theirmounting surfaces they were aligned by definition,assuming that the fabrication tolerances were met.The three dimensional theodolite technique involveslocating two theodolites within a known grid thenmeasuring the angles to monument targets to computetheir coordinates. The theodolites and grid lie along oneside of the RFQ, and so the horizontals off axismeasurements are less accurate because they are close tothe theodolite sight line. The alignment results are shownin Fig. 2.
4 SIGNAL LEVEL TESTS
Figure 3. RFQ tank ready for signal level tests
4.1 Frequency, Q and Impedance Measurements
Following the mechanical alignment the lid was installedas shown in Fig. 3 ready for signal level tests. Thefrequency and Q are measured with a network analyzerand the shunt impedance is derived from two independent
R/Q measurements; ∆C method and input admittance
method [9]. Results are compared to MAFIA calculationsin Table 1.Table 1. Comparison of measured values with calculatedMAFIA values.
Parameter MAFIA Measured
Frequency (MHz) 34.7 35.7
Q 14816 8400
Rshunt (k-ohms) 61.9 36.75
Rshunt (k-ohms-m) 470.4 279.3
R/Q 4.18 4.375
4.2 Bead-pull measurements
Since the electrodes have no shoulder upon which to rest
the dielectric bead on when measuring the lower gap, thesagging of the bead was overcome by fabricating a beadcarriage from teflon that traveled down the center bore ofthe RFQ and was guided by the straight edges of the
electrodes. Nine bead pull runs were made for each set ofmeasurements; carriage only, four separate runs with thedielectric bead in each of the four quadrants and fourseparate runs by rotating the carriage and the beadtogether. The carriage only run was used as the averageperturbation reference, the run by rotating the carriagewith the bead was used to correct any asymmetry in thecarriage and the four separate runs were correctedaccordingly.Both the average peak field variation and the quadrupolefield asymmetry were deduced from the measurements andare shown in Fig 4. The results are within the target of +/-
1% field strength variation.
Figure 4. Bead pull measurement results
4.3 Transmission line mode measurement .
The decision to install the rf power amplifiers for the ISAC
accelerator RF systems several wavelengths away from theRF structures, was based on our experience with matchinghigh Q loads to power sources via a long length oftransmission line on other RF systems at TRIUMF [10]. Inorder to minimize the possibility of a parasitic oscillationat a transmission line mode, it is best for the transmissionline electrical length to be a multiple of λ/2. The
parameters of the amplifier tuned circuit and the resonantcavity coupled at each end of the transmission line have aneffect on the equivalent electrical length of thetransmission line. For the RFQ system, the transmissionline resonances were measured to be 2.45 MHz apart. Theelectrical length of the line was adjusted via a trombone tonλ/2 indicated by the cavity resonance f
o being centrally
located between the two resonances making the differencefrom f
o to the first transmission line resonance 1.225 MHz.
This is a necessary adjustment for the stable operation of
the rf system.
5 FULL POWER TESTS
In preparation for full power tests the RFQ tank was
baked out for three days at 60 ° C by uniformly powering
eighty-four 500W heaters on the tank walls, covered witha glass fiber blanket to contain the heat (Fig 5). At thesame time 60 degree water was circulated through thestructure cooling system. A base pressure of 1.4 *10-7 torrwas achieved, which increased to 4.0*10-7 torr with fullRF power applied.
Figure 5. RFQ system ready for bake-out.
Careful cleaning procedures and high power pulsing
drastically reduced the growth rate of dark currentsassociated with field emission. The pulses were 128 uslong at a rate of 500 Hz at a peak amplitude of ~100 kVpeak. The gradual reduction of dark currents is indicatedin Fig. 6 by the reduction of the slope of the sequentialgraphs, which are in chronological order from top tobottom. Each graph plotted is for a constant voltage aftertwo hours of high power pulsing.
Figure 6. Increase in power level due to dark currents
Initially at the nominal voltage of 74 kV, the dark currents
caused an increase of power from 75 kW to 100 kW in 2hours. Now the power due to dark currents increases byonly 5 kW and then levels off in 2 days. A successful 150
hour test at full power was achieved. With no darkcurrents present, the power requirement to reach an inter-electrode voltage of 75 kV is 75 kW. The amplifier iscapable of 150 kW for peak power pulsing.
6 BEAM TESTS
In 1998 an interim beam test was completed with the first7 ring section (2.8m) accelerating beams to 55keV/u. In1999 the final 12 rings were added. Beam commissioningof the complete 19 rings was finally completed this year.The RFQ was operated in cw mode for all beam tests.Beams of
4He1+, 14N1+, 20Ne1+ and 14N21+ all have been
accelerated to test the RFQ at various power levels.In the initial 7-ring test a dedicated test facility was placeddownstream of the RFQ. For the final 19-ringconfiguration the test station was placed downstream ofthe MEBT and the first DTL section. The test facility
includes a transverse emittance scanner, and a 90 ° bending
magnet and Fast Faraday cup for longitudinal emittanceestimations.
Figure 7. (a) RFQ beam test results showing capture
efficiency for beams of N
+ as a function of relative vane
voltage. The beam capture for both bunched andunbunched initial beams are recorded (squares) and arecompared with PARMTEQ calculations (dashed lines). In(b) the results for both N
+ and N+
2 are plotted with respect
to absolute vane voltage.
Beam capture has been measured as a function of RFQ
vane voltage for each ion and for both unbunched and
6065707580859095100105
0 5 10 15 20 25 30Time in HoursPower (kW)26 AUG 99
13 OCT 99bunched input beams. The MEBT quadrupoles were used
as a velocity filter to remove the unaccelerated beam. Theresults for atomic and molecular Nitrogen are given in Fig.7 along with predicted efficiencies based on PARMTEQcalculations. The RFQ capture efficiency at the nominalvoltage is 80% in the bunched case (three harmonics) and25% for the unbunched case in reasonable agreement withpredictions. A separate measure of the timing pulse trainobtained from scattering the beam in a gold foil shows that5% of the accelerated beam is distributed in the two 35.4MHz side-bands. This means that after chopping theoverall capture efficiency in the 11.8 MHz bunches willbe 75%. This will eventually be increased to 80% byadding a fourth harmonic to the pre-buncher.By varying the MEBT rebuncher while measuring theproduct of energy spread and time width at an energy ortime focus gives an estimate of the longitudinal emittance.The results are shown in Fig. 8 for a
4He1+ beam and give
an emittance of 0.5 π keV/u-ns in agreement with
simulations. The measured energy of 153keV/u also is inagreement with design.Transverse emittances were measured before and after theRFQ. The results show that, when the matching isoptimized, the emittance growth in both planes isconsistent with zero for the 7-ring configuration for an
initial beam of 15 π-mm-mrad. In the 19-ring test the
emittance scanner was moved after the 90 ° bend in
MEBT. In this case the emittance growth was non-zerobut less than a factor of two. It has not been determinedwhat part of the emittance growth is in the RFQ and whatpart is contributed by the optics.The transverse and longitudinal acceptances wereexplored with a so-called ‘pencil beam ’ defined by two
circular apertures of 2mm each separated by 0.7m placedin the RFQ injection line. One steering plate was availabledownstream of the collimators to steer the ‘pencil beam ’
around the RFQ aperture. In the case of the longitudinalacceptance the energy and phase of the incident beam wasvaried while recording the beam transmission.
Figure 8. Energy spectrum and corresponding pulse width
for an accelerated beam of
4He1+.
Based on the steering/transmission data the transverse
acceptance was estimated to be ≤ 140 π mm-mrad. The
longitudinal acceptance was measured for both a centeredand an off-centered beam (Ac=2.7 mm) at the nominalRFQ voltage using the pencil beam. The energy and phasesettings where the acceptance dropped to 50% of the peakvalue were used to define the longitudinal acceptancecontour. The acceptance of the centered beam was
estimated to be 180 π %-deg at 35MHz or 0.3 π keV/u-ns.The acceptance opens up for off-centered beams with
values of 400 π %-deg at 35 MHz or 0.7 π keV/u-ns. The
expected longitudinal acceptance based on PARMTEQ
simulations is 0.5 π keV/u-ns.
In general the beam test results demonstrate a strongconfirmation of both the beam dynamics design and theengineering concept and realization.
7 ACKNOWLEDGMENT
We would like to thank Gerardo Dutto, and Paul Schmor,for their valuable technical and managerial discussions.We are especially grateful to Roland Roper (machineshop) who took on the responsibility of the fabrication andmanufacturing details of rings, jigs and fixtures. A specialthanks to Bhalwinder Waraich for the mechanicalassembly and installation of the rings, and to Peter Harmerfor the organization and integration of the RFQ with allthe ancillary systems.
8 REFERENCES
[1] R.L. Poirier, P.J. Bricault, K. Jensen and A. K. Mitra, “The
RFQ Prototype for the Radioactive Ion Beams Facility atTRIUMF ”, LINAC96, Geneva, Switzerland
[2] R.L. Poirier, P. Bricault, G. Dutto, K. Fong, K. Jensen, R.
Laxdal, A.K. Mitra, G. Stanford, “Construction Criteria and
Prototyping for the ISAC RFQ Accelerator at TRIUMF ”, Proc.
1997 Particle A ccelerator Conference.
[3] R. L. Poirier, et al, “RF Tests on the Initial 2.8m Section of
the 8m Long ISAC RFQ at TRIUMF ”, LINAC98, Chicago,
USA.
[4] S. Koscielniak et al, “Beam Dynamics of the TRIUMF ISAC
RFQ”, LINAC96, Geneva, Switzerland.
[5] P. J. Bricault, et al, “RFQ Cold Model Studies ”, PAC95,
Dallas Texas, USA.
[6] A. Schempp, et al., Nucl. Instr. & Meth. B10/11 (1985) p.
831.
[7] J.Klabunde, et al, “Beam dynamics simulations in a four-rod
RFQ”, LINAC94, Tsukuba, Japan, pg.710.
[8]G. Stanford, D. Pearce, R. L. Poirier, “Mechanical Design,
Construction and Alignment of the ISAC RFQ Accelerator atTRIUMF ”, LINAC98, Chicago, USA.
[9] P. Bourquin, W. Pirkl and H.-H. Umstatter, “RF and
Construction Issues in the RFQ for the CERN Laser IonSource”, Proc. XVIII LINAC Conference, CERN, p381(1996)
[10] R. L. Poirier et al, “Stabilizing a Power Amplifier Feeding
a High Q Resonant Load ”, PAC95, Dallas Texas, USA.
[11] R. Laxdal, et al, “First Beam Tests with the ISAC
RFQ”
LINAC98, Chicago, USA.
|
LOCAL AND FUNDAMENTAL MODE COUPLER DAMPING OF THE
TRANSVERSE WAKEFIELD IN THE RDDS1 LINACS
R.M. Jones1; SLAC, N.M. Kroll2; UCSD & SLAC, R.H. Miller1, C.-K. Ng1 and J.W. Wang1; SLAC
_____________
1Supported under U.S. DOE contractDE-AC03-76SF00515.
2Supported under U.S. DOE grant DE-FG03-93ER407.Abstract
In damping the wakefield generated by an electron beamtraversing several thousand X-band linacs in the NLC weutilise a Gaussian frequency distribution of dipole modesto force the modes to deconstructively interfere,supplemented with moderate damping achieved bycoupling these modes to four attached manifolds. Most ofthese modes are adequately damped by the manifolds.However, the modes towards the high frequency end ofthe lower dipole band are not adequately damped becausethe last few cells are, due to mechanical fabricationrequirements, not coupled to the manifolds. To mitigatethis problem in the present RDDS1 design, the outputcoupler for the accelerating mode has been designed so asto also couple out those dipole modes which reach theoutput coupler cell. In order to couple out both dipolemode polarizations, the output coupler has four ports. Wealso report on the results of a study of the benefits whichcan be achieved by supplementing manifold damping withlocal damping for a limited number of cells at thedownstream end of the structure.
1. INTRODUCTION
The transverse wakefield in an accelerator structure is dueto dipole modes which are excited in the structure whenthe beam traverses it off center. In the manifold dampeddetuned structures [1] two measures are taken whichmitigate the deflecting effect of these modes on trailingbunches. The primary measure taken is to modify theindividual cells so as to achieve a smooth (currently a kickfactor weighted truncated Gaussian) distribution of modefrequencies which via destructive interference betweenthe modes leads to a large decrease in deflecting forces onbunches arriving some 1.4 to 7 nanoseconds later. Due tothe discreteness of the modes this effect begins to breakdown at subsequent time delays, earliest for the mostwidely spaced modes, latest for the most narrowly spaced(ie where the mode density in frequency is highest). Thislatter point, typically a maximum, is often referred to asthe recoherance peak because it occurs where the mostclosely spaced modes are again in phase (~100 ns). Thedamping manifolds, four tapered circular waveguides withelectric field coupling to all cells except a few at the ends,which serve to drain amplitude from these modes with atime constant ~20 ns thereby limiting the magnitude of therecoherance effect and ultimately leading to a steady
reduction in wake amplitude. This paper is devoted to adiscussion of two aspects of this scenario which havelimited the effectiveness of this approach. As mentionedabove a few cells at the ends are not coupled to themanifolds. This has been done to avoid mechanicalinterference problems. The consequence has been thatone to a few modes at the high frequency end and withsignificant kick factors are very poorly damped. Theyshow up as sharp peaks in the high frequency end of thespectral function and elevate the wake amplitude. Thiseffect has been more prominant in the RDDS design thanin the earlier DDS designs due to the increase in thedetuning frequency span (11.25%) and in the ratio offrequency span to Gaussian width (4.75). Nevertheless itwas manifold radiation observed in the DDS3 ASSETexperiment which suggested a cure. Sharp high frequencypeaks were observed in the manifold radiation when thebeam was displaced vertically but not when it wasdisplaced horizontally suggesting that these modes werebeing damped by the output coupler for the acceleratingmode which then had two horizontal waveguide ports.This led to the design of a four port output couplerintended to damp both polarizations of the dipole modes.The design of the coupler, the equivalent circuit analysisof its effect and confirmation by simulation, by RFmeasurements, and by observation of beam inducedradiation from the output coupler will be discussed infollowing section. The other aspect has to do with the risein the wake amplitude which occurs at the earliest timesand is due to the early onset of recoherance for the mostwidely spaced modes. The effect of manifold dampingdoes not set in soon enough to mitigate this effect. Thisled to an investigation of the effect of locally damping alimited number of cells at the downstream end of thestructure. The results of this study will follow thediscussion of the four port output coupler
2. THE FOUR PORT OUTPUT COUPLER
(1) Design
The standard output coupler for the DDS structures has
been fitted with a pair of WR90 rectangular outputwaveguides polarized in the electric field directionemerging on opposite sides of the coupler cell in thehorizontal direction. In order to provide an output for both
dipole mode polarizations these output waveguides havebeen replaced by four WR62 waveguides oriented so as toform an "X" with respect to the horizontal and verticalaxes [1]. The reduced width was chosen to convenientlyfit the space limitations dictated by the output celldimensions, and the orientation was chosen to avoidmechanical interference with the manifold couplers,which are located in the vertical and horizontal planes.The accelerating mode and the lower dipole modes, whilethe TE
01 polarisation is cut-off for both of them. The
coupler was matched for the accelerator mode by varyingthe cell radius and wavguide iris width as described in [2]using the mesh shown in Fig. (1).
Fig 1: Fundamental four-port coupler
(2) Assessment of the damping effect
The first step in the assessment was the determination of
the loading effect on the output cell. A symmetry reduced
section of the output cell with dipole π mode boundary
conditions was probe driven in a broad band time domainsimulation with outgoing wave boundary conditions onthe output waveguides to determine the resonantfrequency and Q
ext. The KY [3] frequency domain method
was also applied for corroboration, and a Qext of 36
established. This information was used in a number of
Fig. 2: Circuit model of S11 of output port
ways in the equivalent circuit calculations to investigate
the damping effect on the structure modes. An effect isexpected of course only for those modes which reach the
output cell, and all of the Q's are expected to be muchhigher than the 36 referenced above. The most directequivalent circuit information was obtained by computingthe dipole mode S
11 as seen from the output coupler,
parameterised by making use of the simulationdetermined Q
ext (Fig. (2)). The Qext of the broadest peak
shown was of the order of 300, a figure which agreeswith the result obtained from a time domain simulation ofa ten-cell model (without manifolds) terminated with theoutput coupler. One notes that the peaks narrow as onemoves to the lower frequency peaks due to an increase inQ
ext. This effect combined with the onset of manifold
damping leads to the total disappearance of resonanceeffects below 16 GHz. The spectral function wasrecomputed with an appropriate series resistance insertedin the TM circuit of the last cell [4]. The sharp peakswhich had been observed previously
Fig. 3 Spectral function RDDS1 for 4 cells decoupledeither end of structure and no external loading of cells
14 14.5 15 15.5 16 16.5
Freq. (GHz)20406080100Spectral Fn. (V/pC/mm/m/GHz)
Fig. 4: Spectral function for 4 cells decoupled either endof structure and Q ~36 for the last cell.
(Fig. (3) were no longer observed (Fig. (4)) and Q's
estimated by Lorentzian based fits to spectral functionamplitudes were in qualitative agreement with thoseobserved as described above. The associated wakeenvelope functions are constructed in Figs (5) and (6), anda striking improvement may be notedOutgoing portsFigure 5: Envelope of wake function for RDDS1
excluding external loading.
20 40 60 80
s (m)0.010.1110100Wake Function (V/pC/mm/m)
Fig. 6: Envelope of wakefield for 4 cells decoupled eitherend of structure and Q~36 for the last cell
III) The Effect of Damping a Limited Number
of Cells
The spectral function plotted in Fig. (4) shows a number
of peaks on the downward sloping high frequency endwhich are more widely spaced and somewhat strongerthan those found at the lower frequencies. As noted above,because of their wide spacing their contribution todeconstructive interference disappears at quite early timesand is responsible for the sharp rise in the wake whichfollows the first minimum. This observation suggested anexploration of the effect of adding moderate localdamping to a limited number of cells towards the outputend of the structure. The investigation has so far beencompletely phenomenological, that is, based on anassumed distribution of Q values rather than a design ofthe damped cells. We have explored tapered distributionsof Q's confined to the cells with cell numbers above 187and values no less than 500. The best results wereobtained with distributions which smoothed out the peakswhile leaving the mean value of the spectral functionmore or less intact. Figure (7) shows a spectral functionresulting from a linear taper of 1/Q from 1/6500 at cell14 14.5 15 15.5 16 16.5
Freq. (GHz)20406080100Spectral Fn. (V/pC/mm/m/GHz)
Fig 7: Spectral function for RDDS with the last 10 Cells
Q =500and 1/x taper up to 6500 from Cells 197 to 187.
187 to 1/500 at cell 197 and thence constant. The ripples
are indeed smoothed out and the Gaussian like form isintact. A strong suppression of the early rise in the wakemay be noted Fir. (8). These results suggest that thisapproach warrants further investigation.
20 40 60 80
s (m)0.010.1110100Wake Function (V/pC/mm/m)
Fig 8: Wake envelope function for RDDS with the last 10
Cells Q =500and 1/x taper up to 6500 from Cells 197 to187.
4. CONCLUSIONS
We conclude that the four port output coupler undoes the
harmful effects of decoupling the last few cells from thedamping manifolds. Alternatively this might be done bylocally damping a limited number of cells at the outputend of the structure with additional benefits, especially inthe early time portion of the wake.
5. REFERENCES
[1] J.W. Wang et al, TUA03, LINAC200 (this conf.)
[2] N.M. Kroll et al, TUE04, LINAC2000, (this conf.)
[3] N.M. Kroll and D.U.L. Yu, Part. Acc.,34, 231 (1990)[4] R.M. Jones, et al, LINAC96, (also SLAC-PUB-7287) |
AN OVERVIEW OF THE
SPALLATION NEUTRON SOURCE PROJECT
Robert L. Kustom, SNS/ORNL, Oak Ridge, TN
Abstract
The Spallation Neutron Source (SNS) is being designed,
constructed, installed, and commissioned by the staff of
six national laboratories, Argonne National Laboratory,
Brookhaven National Laboratory, Jefferson National
Accelerator Laboratory, Lawrence Berkeley National
Laboratory, Los Alamos National Laboratory, and Oak
Ridge National Laboratory. The accelerator systems are
designed to deliver a 695 ns proton-pulse onto a
mercury target at a 60-Hz repetition rate and an average
power of 2-MW. Neutron moderators that will convert
the spallation neutrons into slow neutrons for material
science research will surround the target. Eighteen
neutron beam lines will be available for users, although
initially, only 10 instruments are planned. The Front-
End Systems are designed to generate a 52 mA, H-
beam of minipulses, 68% beam on, 32% beam off,
every 945 ns, at 2.5 MeV for 1 ms, 60 times a second.
The Front-End systems include a RF driven, volume-
production ion source, beam chopping system, RFQ,
and beam transport. The linac consists of a drift tube
linac up to 86.8 MeV, a coupled-cell linac to 185.7
MeV, and a superconducting RF linac to the nominal
energy of 1 GeV. The design of the superconducting
section includes 11 cryomodules with three, 0.61-beta
cavities per cryomodule and 15 cryomodules with four,
0.81-beta cavities per cryomodule, with space to install
six more 0.81-beta cryomodules. The accumulator ring
is designed for charge exchange injection at full energy
and will reach 2.08x10E+14 protons/pulse at 2-MW
operation. The goal is to reduce uncontrolled beam
losses to less than 1x10E-4. A detailed overview of the
accelerator systems and progress at the various
laboratories will be presented.
1 INTRODUCTION
The Spallation Neutron Source (SNS) facility under
construction at Oak Ridge National Laboratory is
designed to generate pulses of neutrons at intensities
well beyond any of the world’s existing spallation
neutron sources. The accelerator systems are designed
to deliver a 695 ns proton-pulse onto a liquid mercury
target at a 60-Hz repetition rate with an average proton
beam power of 2-MW. The target station will have 18
shutters that ultimately will be able to support 24
neutron instruments. An initial complement of ten
instruments is planned at the start of operation in 2006.
A site master plan is shown in Figure 1.
Figure 1: Site Master Plan.
The SNS is being designed and built as a partnership of
six DOE national laboratories: Lawrence Berkeley
(LBL) in California, Los Alamos (LANL) in New
Mexico, Argonne (ANL) in Illinois, Oak Ridge (ORNL)
in Tennessee, Brookhaven (BNL) in New York, and
Thomas Jefferson (JLAB) in Virginia. The Front-End
Systems (FES) are the responsibility of LBL. The drift
tube linac (DTL), coupled-cell linac (CCL), and warm
parts of the linac, including the end-to-end physics
design and RF system design are the responsibility of
LANL. The superconducting RF cavities, cryomodules,
and cryogenic equipment are the responsibility of
JLAB. The accumulator ring and high-energy transport
lines between the linac and the ring (HEBT) and the
ring and the target (RTBT) are the responsibility of
BNL. The target station and conventional facilities are
the responsibility of ORNL. The neutron instruments
are the responsibility of ANL. Project integration,
direction, and planning for operation are the
responsibilities of the SNS office at ORNL This article
describes the combined effort on the part of staff at
these laboratories.
Considerably more detail is provided in a number of
excellent papers being presented at this conference.
A summary of key design parameters for the SNS
facility is presented in Table 1.Table 1. Summary of key design parameters
for the SNS Facility
Proton beam power on target, Mw 2
Average proton beam current on target, mA 2
Pulse repetition rate, Hz 60
Chopper beam on duty factor, % 68
Front-end and linac length, m 335
DTL output energy, MeV 87
DTL frequency, MHz 402.5
CCL output energy, MeV 185
Number of SRF cavities 92
Linac output energy, GeV ≈1
CCL and SRF frequency, MHz 805
Linac beam duty factor, % 6
High Energy Beam Transport (HEBT) length,
m170
Accumulator ring (AR) circumference, m 248
Ring orbit revolution time, ns 945
Number of turns injected into AR during fill 1060
AR fill time, ms 1
Gap in AR circulating beam for extraction, ns 250
Length from AR to production target (RTBT),
m150
Peak number of accumulated protons per fill 2.08E+14
Proton pulse width on target, ns 695
Target material Hg
Number of neutron beam shutters 18
Initial number of instruments 10
Number of instruments for complete suite 24
2 TECHNICAL DESIGN OF THE
ACCELRATOR SYSTEMS
2.1Front-End Systems
The Front-end Systems (FES) are designed to generate
an H- beam of mini-pulses with 68% on time, every 945
nanoseconds for a period of 1 millisecond at a 60 Hz
repetition rate. The FES include a RF driven, volume-
production ion source, beam chopping system, RFQ, a
low-energy beam transport (LEBT) system, and a
medium-energy beam transport (MEBT) system. The
FES must deliver 52 ma at 2.5 MeV at the input to the
drift tube linac. The key FES parameters are listed in
Table 2[1].
The H- ion source utilizes a 2-MHz, RF driven
discharge to generate the plasma. The plasma is
confined by a multi-cusp magnet configuration. A
magnetic dipole filter separates the main plasma from
the region where low-energy electrons generate the
negative ions. A heated cesium collar surrounds the
production chamber. Electrons are removed from the
ion beam by a deflecting field from a dipole magnet
arrangement in the outlet plate of the plasma generator.
The ion source is tilted with respect to the LEBT to
compensate for the effect the electron-clearing field has
on the ion beam.The LEBT structure is based on an earlier design [2]
that proved the viability of purely electrostatic
matching. There are two einzel-lens in the LEBT. The
second is split into quadrants that can be biased with
D.C. and pulsed voltages to provide angular steering
and pre-chopping. Chopping voltages of +- 2.5 kV and
300 ns are rotated around the quadrants. Corrections in
transverse beam displacement are achieved by moving
the ion source and LEBT with respect to the RFQ [3]. A
schematic of the ion source and LEBT that will be used
for startup of the facility is shown in Fig. 1. Its
performance goal is 35 ma, and it will be a significant
step towards developing the full 65 mA estimated for 2
MW operation.
Table 2. FES Key Performance Parameters
Ion Species H-
Output Energy, MeV 2.5
H- current @ MEBT output, mA 52
Nominal H- current @ ion-source output,
mA65
Output normalized transverse rms emittance,
π mm mrad0.27
Output normalized longitudinal rms
emittance, π MeV deg0.13
Macro pulse length, ms 1
Duty factor, % 6
Repetition rate, Hz 60
Chopper rise & fall time, ns 10
Beam off/beam on current ratio 10E-4
The RFQ will accelerate beam from 65 keV to 2.5 MeV
with an expected transmission efficiency of better than
80%. It is built in four modules using composite
structures with a GlidCop shell and four oxygen-free-
copper vanes. The length of the RFQ is 3.72 m.
Figure 2: Schematic of the startup ion.Figure 3: End-on view of the assembled RFQ source
for the SNS module.
The design frequency is 402.5 MHz. Peak surface fields
reach 1.85 Kirkpatrick and require 800 kW during the
pulse. The output of RFQ is directed into the MEBT [4].
Matching from the RFQ to DTL is performed in the
MEBT. Final chopping of the bunches is also performed
in the MEBT.
2.2Linac Systems
The linac consists of a drift tube linac up to 86.8 MeV, a
coupled-cell linac (CCL) up to 185.7 MeV, and a
superconducting linac up to a nominal energy of 1 GeV.
The superconducting linac is divided into a medium-
beta cavities and high-beta cavity sections [5]. The
medium-beta cavity is designed for a geometric β of
0.61, and the high-beta cavity is designed for a
geometric β of 0.81. The nominal transition energy
between the medium and high beta sections is 378.8
MeV.
The DTL consists of six separate tanks each driven by a
402.5 MHz, 2.5 Mw klystron. The focusing lattice is
FFODDO with a six βλ period. The focusing magnets
are permanent magnet quadrupoles with constant GL of
3.7 kG and a bore radius of 1.25 cm. There are one- βλ
inter-tank gaps for diagnostics. Empty drift tubes
contain BPMs and steering dipoles. There are 144
quadrupoles and 216 drift tubes in the DTL. The energy
gain per real estate meter is 2.3 MeV/m in the DTL.
Key parameters for the DTL are listed in Table 3.
The CCL operates at 805 MHz. There are eight
accelerating cells brazed together to form a segment.
Six segments are mounted and powered together as a
single module using 2.5- βλ coupling cells, one of which
is powered. A 3-D schematic of Module 1 is shown inFig. 4 and a cutaway view of the segments and the
powered coupler is shown in Fig 5.
Table 3: DTL Parameters
Tank
#Final
Energy
(MeV)Power
(Mw)Length
(m)# of
cells
1 7.46 0.52 4.15 60
2 22.83 1.6 6.13 48
3 39.78 1.93 6.48 34
4 56.57 1.93 6.62 28
5 72.49 1.87 6.54 24
6 86.82 1.88 6.61 22
Figure 4: 3-D Schematic of CCL module 1.
Figure 5: Cutaway view of CCL through segments 1 &
2 and the powered coupler.
There are a total of eight modules. Four, 5-megawatt
klystrons drive the CCL. Each klystron drives two
modules. The peak power is 11.4 Mw and the maximum
accelerating field on axis is 3.37 MV/m (E οT). Theenergy gain per real estate meter is 1.7 MeV/m. The
transverse focusing system is a FODO lattice in the
CCL. The bore radius goes from 1.5 cm to 2.0 cm. The
total length of the CCL is 55.12 m.
The high-energy end of the linac, above 185.7 MeV,
uses superconducting cavities. The design is based on a
conceptual design study completed by scientists from
many institutions and lead by Yanglai Cho [6].
Two different superconducting cavity designs are used
in the SNS linac, one with a geometric β of 0.61,
defined as the medium- β cavity, and the other a
geometric β of 0.81, defined as the high- β cavity. There
are six cells per cavity in the medium and high- β
sections. More than six cells per cavity results in
excessive phase slip for a particular beta and fewer than
six cells per cavity results in inefficient use of real estate
and higher cost due to increased parts count. The
cavities will be fabricated using 4 mm-thick Nb with
stiffening or reinforcement plates. The initial design
assumes a peak field of 27.5 MV/m, +-2.5 MV/m,
however, with conditioning and future processing,
higher gradients are expected. The design value for Q ο
is 5x10E+9, and the loaded Q design value is 5x10E+5.
The effective accelerating gradients are 10.5 MV/m in
the 0.61- β section and 12.8 MV/m in the 0.81- β section.
The design values for Lorentz detuning, referenced to
the geometric accelerating field, are 2.9 Hz/(MV/m)^2
in the medium- β cavities and 1.2 Mz/(MV/M)^2 in the
high- β cavities. The 6- σ design value for microphonics
is +-100 Hz. Cold tuning will allow the cavities to be
taken off resonance by 100 kHz. Each cavity is driven
by a single, 550 kW klystron operating at 805 MHz [7].
There are three cavities per cryomodule in the medium-
β section, and a total of eleven medium- β cryomodules
in the linac. There are four cavities per cryomodule in
the high- β section of the linac. Initially, fifteen high- β
cryomodules will be installed. There are, however,
Figure 6: Schematic view of a medium- β cryomodule
and superconducting cavities.additional straight-section spaces to install as many as
twenty-one high- β cryomodules in the future. A
schematic sectional view of the medium- β cryomodule
through the superconducting cavities is shown in Fig. 6
A summary of the key superconducting linac
dimensions is listed in Table 4 and key cryogenic
parameters are listed in Table 5.
Table 4. Key superconducting RF cavity dimensions
Nb thickness, mm 4.0
Minimum bore radius, medium- β, cm 4.3
Cryomodule length, medium- β, m 4.239
Cryomodule length, high- β, m 6.291
# of medium- β cryomodules 11
# of high- β cryomodules (initial) 15
Warm space between cryomodule cells,
m1.6
Total length of SRF linac with extra 6
cryomodules, m235.92
Table 5. Cryogenic requirements for
superconducting linac
Operating temperature, K 2.1
Primary circuit static load, w 785
Primary circuit dynamic load, w 500
Primary circuit capacity, w 2500
Secondary circuit temperature, K 5.0
Secondary circuit static load, g/s 5
Secondary circuit dynamic load, g/s 2.5
Shield circuit temperature, K 35-55
Shield circuit load, w 5530
Shield circuit capacity, w 8300
2.3 Accumulator Ring
The accumulator ring for SNS is a FODO arc with
doublet straight sections [8]. This lattice has four-fold
symmetry with zero dispersion in the straight sections.
A plan view of the ring and transport lines is shown in
Fig. 7. The ring circumference is 248 m. The zero
dispersion regions include two-6.85 meter sections and
one long 12.5 meter section. Each of the four straight
sections has a dedicated function. The injection straight
includes the injection septum magnet, eight bump
magnets for horizontal and vertical injection painting,
the stripper foil, and dump septum. The collimator
section includes moveable scattering foils and three
fixed collimators. The extraction section includes
fourteen full-aperture-ferrite, extraction kicker magnets
and a Lambertson extraction septum magnet. The rise
time of the extraction kickers is 200 ns. The RF sectionhas three first-harmonic cavities operating at 1.058 MHz
and a second harmonic cavity. The total voltage
generated at the first harmonic is 40 kV and at the
second harmonic is 20 kV.
At 2 MW operation, 2.08x10E14 protons are
accumulated in a 650-700 ns bunch in 1060 turns. The
injection process is direct charge exchange using a
painting scheme to achieve uniform transverse charge
distribution and a second harmonic RF system to spread
the beam more uniformly in the longitudinal plane. The
expected fractional space-charge tune-shift is 0.14. The
goal for gap cleanliness is 10E-4 beam-in-gap/total
beam.
Achieving low uncontrolled beam loss, less than 10E-4,
is a key element of the accumulator ring design. The
design of the injection process, collimation scheme, RF
system design, emittance and acceptance ratio, and
extraction system are all designed to achieve this low
level of beam loss.
Figure 7: Plan view of the SNS accumulator ring.
3 PROJECT STATUS
Major construction has started on the conventional
facilities and the technical components. Excavation on
the site is currently about 40% complete. Much of the
building and utility detailed design has started and
major civil procurements, such as bulk concrete and
structural steel, are well along in the procurement cycle.
The start-up ion source and all electrostatic LEBT have
been successfully operated at LBL at 42 mA, greater
than the initial 35 mA needed for the start of
commissioning. The first of the RFQ modules has been
fabricated and tested at full field and pulse length. A
cold model of the DTL is in fabrication and a coldmodel of the CCL has been successfully tested at
LANL. A significant number of major linac
procurements, such as the 402.5 MHz, 5 MW klystrons,
402.5 MHz circulators, and transmitters for klystron
control, have been awarded. The copper model for the
0.61- β single cell has been brazed and is being tested at
JLAB. Six-cell Nb cavities are being fabricated. The
procurement of Nb for construction of all the cavities
and much of the hardware for the cryogenic facility and
cryomodule production has been awarded. Procurement
of ferrite for the ring RF systems has been awarded and
sample is being tested. Ring dipole, quadrupole, and
corrector magnets have tested, and procurement of these
magnets has started at BNL.
In summary, major construction has started and the
project expects to meet the goal of first beam injected
into the accumulator ring by July 2004, and first beam
on target by January 2005.
REFERENCES
[1]R. Keller, “Status of the SNS Front-End Systems,”
EPAC 2000, Vienna, Austria, July 2000.
[2]J. W. Staples, M. D. Hoff, and C. F. Chan, “All-
electrostatic Split LEBT Test Results,” Linac ’96,
1996.
[3]J. Reijonen, R. Thomae, and R. Keller, “Evolution of
the LEBT Layout for SNS,” Linac2000,
Monterey, CA, August 2000.
[4]J. Staples, D. Oshatz, and T. Saleh, “Design of the
SNS MEBT,” Linac2000, Monterey, CA, August
2000
[5]J. Stovall, et al., ”Superconducting-Linac for the
SNS,” Linac2000, Monterey, CA, August 2000.
[6]“Superconducting Radio Frequency Linac for the
Spallation Neutron Source,” Preliminary Design
Report, SNS Project, Report #SNS-SRF-99-101,
Oak Ridge, TN
[7]M. Lynch, “The Spallation Neutron Source (SNS)
Linac RF System,” Linac2000, Monterey, CA,
August 2000.
[8]J. Wei, et al., “Low-Loss Design for the High-
Intensity Accumulator Ring of the Spallation
Neutron Source,” Physical Review ST, To be
published, 2000. |
DESIGN OF 11.8 MHZ BUNCHER FOR ISAC AT TRIUMF
A.K. MITRA, R.L. POIRIER, R.E. LAXDAL, TRIUMF
Abstract
The high energy beam transport (HEBT) line for the
ISAC radioactive beam facility at TRIUMF requires an
11.8 MHz buncher. The main requirements of the buncher
are to operate in cw mode with a velocity acceptance of2.2% and an effective voltage of 100 kV, which for athree gap buncher gives a drift tube voltage of 30 kV. Alumped element circuit is more suitable than a distributedrf structure for this low frequency of operation. The
resonant frequency of 11.8 MHz is obtained by an
inductive coil in parallel with the capacitance of the drifttube. The coil is housed in a dust free box at atmosphericpressure whereas the drift tube is placed in a vacuumchamber and an rf feedthrough connects them. Two
design of this feedthrough, one using disk and one using
tubular ceramics, operating at 30 kV rf, are described inthis paper. MAFIA and SUPERFISH codes are used tosimulate the fields in the feedthroughs, particularlyaround the ceramic metal interfaces. Test results of theprototype feedthroughs are presented and the choice of
the proposed final solution is outlined.
1 INTRODUCTION
The beam from the DTL of the ISAC radioactive beam
facility goes thru the high energy beam transport (HEBT)and is delivered to various target stations. The DTL
produces beams fully variable in energy from 0.15-1.5
MeV/u with mass to charge values of 3 ≤ A/q ≤ 6. A low-
β 11.78 MHz buncher placed approximately 12 m down
stream from the DTL can provide efficient initial
bunching for beams from 0.15-0.4 MeV/u [1]. The basic
parameters of the HEBT low- β buncher are given in Table
1.
Table 1: Basic parameters of the HEBT low- β buncher
Resonant frequency, f 11.78 MHz
Velocity ( βc) 0.022
Charge to mass ratio 1/3 ≥q/A≥1/6
Energy range 0.15 0.4 MeV/u
Veffective, maximum 100 kV
Vtube 30 kV
Number of gaps 3
βλ/2 29.28 cm
Beam aperture, diameter 2.0 cm
Cavity length 70.0 cm
Voltage stability ± 1.0%
Phase stability ± 0.3 %
operation cwThe maximum required effective voltage from the
buncher is 100 kV. For a 3 gap structure, the effectivevoltage , Veff is given by V
eff = 4 V t To , where V t is the
drift tube voltage and T o is the transit time factor.
2 DESIGN
A prototype of a two gap structure is designed to
produce 30 kV tube voltage at the HEBT buncherfrequency. Since the resonant frequency is low, a lumped
element circuit is found to be more suitable than a
distributed structure. An inductive coil in parallel withthe capacitance of the drift tube and circuit capacitancesproduces the desired resonant frequency. The coil,shorted at one end, is placed in a dust free box. Two
designs of feedthroughs are tested. An rf feedthrough
connects the open end of the coil to the drift tube, whichis in a vacuum box. Two nose cones are also attached inthe vacuum box to simulate the gap capacitance of theHEBT buncher. The prototype buncher is shown in Fig.1. Since the power dissipated is estimated to be
approximately 700 watts, the coil is water cooled.
Insulators support the coil to reduce vibration due to waterflow.
Figure 1: Prototype of HEBT buncher
2.1 The coil
A 6 turn coil is made of a hollow copper tube of ½
diameter. Inductance of this coil is 6.1 µH with turn-to-
turn spacing of 1.5, coil diameter of 9 and a coil lengthof 15. A ¼ hollow tube is inserted in this ½ tubebefore the coil is made. Water flows through this ¼ tubeand flows out from the ½ tube. The coil is installed in analuminum box 20x20x24 and the water inlet, outlet
and coupling loop are located on top of this box.
2.2 Disk ceramic feedthrough
The disk feedthrough [2] uses a ceramic disk of 4.5 outer
diameter with inner hole of 1.75 diameter and athickness of 0.375. The ceramic is not metalized and it is
held in position by bolting two halves of both inner and
outer conductors as shown in Fig. 2. Helicoflex rings areused for vacuum seal between ceramic and the metalparts. In case of window failure, only the ceramic need bereplaced and the metal parts can be reused. The MAFIA
static solver is used to design the contour of the metal
around the ceramic and is shown in Fig. 3.
Figure 2: Sectional view of disk ceramic feedthrough
Figure 3:MAFIA plot of e-fields2.3 Tubular ceramic feedthrough
Figure 4: Tubular ceramic in a coaxial housing
The tubular feedthrough uses an Alumina ceramic from
Jennings, which has an outer diameter of 2.38 and alength of 3.5 and is metalized at the edges. The ceramicassembly in a coaxial housing is shown in Fig. 4. Coronashields are incorporated in the design near the ceramic tometal joints. A SUPERFISH simulation is used to
optimize the shape of the electrodes around the ceramic.
Fig. 5 shows the electric field distribution in the ceramicand on the corona shields.
Figure 5: SUPERFISH simulation of e-fields in the
tubular feedthrough
2.3 Insulating support for the coil
Since the coil is water cooled, it needs to be supported in
order to reduce the vibration induced by the water flow. Apolycarbonate material known as Lexan, is used as an
insulator to support the coil. Tests shows that it can
withstand high dc voltage and has low rf losses at the
design frequency of the buncher. Unfortunately, it breaks
down when rf is maintained for a while and catches fireand carbonizes. The mechanism of failure of Lexan underrf operation is not fully understood. This material isabandoned and a 2 diameter Teflon rod is used instead.
3 RF MEASUREMENTS
3.1 Signal level
The feedthroughs are assembled and connected to the coil,
which is housed in the dust free box. The tubularfeedthrough with the coil connected shows a Q of 1920
and a shunt impedance of 685 k Ω at 11.975 MHz. The
capacitance of the feedthrough is measured to beapproximately 27 pF. The disk feedthrough in parallel
with the same coil shows a rather low Q value of 50. Thisimplies that the ceramic is contaminated and further test isabandoned until the cause of such contamination isunderstood.
3.2 Power
The power test is done with the tubular feedthrough and
the coil. The water cooling of the coil is extended to theceramic-metal joint and the flow is 16 liters/minute. Thevacuum of the test box is 2.10
-7 Torr without rf applied. A
1 kW solid state amplifier is used for the test. A 4.5
diameter loop installed inside the coil at the short circuit
end, is used to couple power. The loop can be turned toprovide 50 Ω matching of the power amplifier and the
resonant circuit. Under cw operation, 30 kV at the drifttube at 11.975 MHz is maintained for 6 hours without any
breakdown
or interruption. The maximum temperature on
the ceramic is measured to be 38 degrees C. Measuredvalues are shown in Table 2.
Table 2: Measured rf parameters of the prototype buncher
Resonant Frequency 11.975 MHz
Q, unloaded 1920
Rshunt 685 kΩ
R/Q 357
Vtube 30 kV
Pmeasured 750 watts
Ptheoretical 657 watts
Ceramic temperature 38oC
The drift tube voltage is calibrated by measuring emitted
x-rays. A glass window is provided in the test box for this
purpose. Also, rf pick up probes are calibrated with themeasured shunt impedance. Fig. 6 shows the measuredtube voltage with input rf power varying from 800 wattsto 1000 watts. This shows excellent agreement of themeasured voltage with x-ray and rf pick up probes.29303132333435
700 800 900 1000 1100 Pinput, W attsVolta ge, kV
Vgap
X -ray
Figure 6: X-ray measurement of drift tube voltage
4 CONCLUSION
Since no commercial rf feedthrough is available which
can withstand the 30 kV rf voltage at 11.8 MHz under cwmode, it has been decided to develop such a feedthroughat TRIUMF. The prototype tests have shown excellenthigh voltage performance of the tubular feedthrough. Thiswill be used to design the final HEBT low- β buncher. The
3 gap HEBT buncher requires two parallel circuits,
consisting of two rf feedthroughs and two coils tuned tothe same frequency. Hence, two fine tuners will berequired for the operation of the buncher. A singlecoupling loop driven by a 2 kW power amplifier will beadequate.
5 ACKNOWLEDGMENTS
The authors like to thank Erk Jensen, CERN, Switzerlandfor providing the design of the CERN disk ceramicfeedthrough. Thanks are due to Joseph Lu for making the
coil, the box assembly and rf measurements, Al Wilson
for the detail drawings of disk and tubular feedthroughs.Thanks are also due to Balwinder Waraich and PeterHarmer for providing technical assistance and MindyHapke for the photographs. We also wish to thank S.Arai, KEK, Japan and R.A Rimmer, LBL, USA for many
helpful discussions.
6 REFERENCES
[1] R.E Laxdal, Design Specification for ISAC HEBT,
TRIUMF Design Note, TRI-DN-99-23[2] R. Hohbach, Discharge on ceramic windows and
gaps in CERN PS cavities for 114 and 200 MHz,
CERN/PS 93-60 (RF) |
arXiv:physics/0008214v1 [physics.ins-det] 23 Aug 2000Predictions about the behaviour of diamond,
silicon, SiC and some AIIIBVsemiconductor
materials in hadron fields
I.Lazanuaand S. Lazanub
aUniversity of Bucharest, Faculty of Physics, P.O.Box MG-11 ,
Bucharest-Magurele, Romania, electronic address: ilaz@s cut.fizica.unibuc.ro
bNational Institute for Materials Physics, P.O.Box MG-7, Bu charest-Magurele,
Romania, electronic address: lazanu@alpha1.infim.ro
Abstract
The utilisation of crystalline semiconductor materials as detectors and devices op-
erating in high radiation environments, at the future parti cle colliders, in space
applications, in medicine and industry, makes necessary to obtain radiation harder
materials. Diamond, SiC and different AIIIBVcompounds (GaAs, GaP, InP, InAs,
InSb) are possible competitors for silicon to different elec tronic devices for the up-
mentioned applications. The main goal of this paper is to giv e theoretical predictions
about the behaviour of these semiconductors in hadron fields (pions, protons). The
effects of the interaction between the incident particle and the semiconductor are
characterised in the present paper both from the point of vie w of the projectile, the
relevant quantity being the energy loss by nuclear interact ions, and of the target,
using the concentration of primary radiation induced defec ts on unit particle flu-
ence. Some predictions about the damage induced by hadrons i n these materials in
possible applications in particle physics and space experi ments are done.
PACS :
61.80.Az: Theory and models of radiation effects 61.82.-d: R adiation effects on
specific materials
Key words: Diamond, Silicon, SiC, AIIIBVsemiconductors, Hadrons, Radiation
damage properties
1 Introduction
The crystalline materials for semiconductor devices used i n high fluences of
particles are strongly affected by the effects of radiation. A fter the interaction
Preprint submitted to Elsevier Preprint 13 December 2013between the incoming particle and the target, mainly two cla sses of degrada-
tion effects are observed: surface and bulk material damage, the last due to the
displacement of atoms from their sites in the lattice. After lepton irradiation,
the effects are dominantly at the surface, while heavy partic les (hadrons and
ions) produce both types of damages.
Up to now, in spite of the experimental and theoretical effort s, the under-
standing of the behaviour of semiconductor materials in rad iation fields, the
identification of the induced defects and their characteris ation, as well as the
explanation of the degradation mechanisms are still open pr oblems.
The utilisation of semiconductor materials as detectors an d devices operating
in high radiation environments, at the future particle coll iders, in space appli-
cations, in medicine and industry, makes necessary to obtai n radiation harder
materials.
Diamond, SiC and different AIIIBVcompounds (GaAs, GaP, InP, InAs, InSb)
are in principle, possible competitors for silicon in the re alisation of different
electronic devices.
All analysed materials have a zinc-blend crystalline struc ture, with the excep-
tion of SiC, that presents the property of polytypism [1]. Th e polytypism refers
to one-dimensional polymorphism, i.e. the existence of diff erent stackings of
the basic structural elements along one direction. More tha n 200 polytypes
have been reported in literature [2], but only few of them hav e practical im-
portance. These include the cubic form 3 C(β), and the 4 Hand 6Hhexagonal
forms. For the cubic polytype, the symmetry group is T2
d, while for the hexag-
onal ones this is C4
6v. Silicon is at the base of electronic industry, diamond
and the AIIIBVcompounds present attractive electrical and/or luminesce nce
properties, of interest for different applications, while t he utilisation of SiC
as a radiation detector, both in high energy physics and in th e field of X-ray
astronomy is now under extensive investigation more confer ences in the field
having sections dedicated to SiC.
The diamond has the reputation of being a radiation hard mate rial and it
is considered as a good competitor to silicon, but non all its properties as a
radiation hard material have been proved experimentally.
The main goal of this paper is to give some theoretical predic tions about the
behaviour of different semiconductors in hadron fields (pion s, protons), these
materials representing potential candidates for detector s and electronic devices
working in hostile environments.
The treatment of the interaction between the incident parti cle and the solid
can be performed from the point of view of the projectile or of the target.
In the first case, the relevant quantity is the energy loss (or equivalently the
2stopping power) and in the second situation the effects of the interactions are
described by different physical quantities characterising material degradation.
There is no a physical quantity dedicated to the global chara cterisation of
the effects of radiation in the semiconductor material. A pos sible choice is the
concentration of primary radiation induced defects on the u nit particle fluence
(CPD), introduced by [3]. It permits the correlation of dama ges produced in
different materials at the same kinetic energy of the inciden t hadron. For
the comparison of the effects of different particles in the sam e semiconductor
material, the non ionising energy loss (NIEL) is useful.
As a measure of the degradation to radiation, in the present p aper the energy
lost by the incident particle in the nuclear interaction and the concentration
of primary defects induced in semiconductor bulk are calcul ated. If the energy
loss of the incident projectile is, in principle, a measurab le physical quantity,
the concentration of primary defects is not directly observ able and measurable
and can be put in evidence only indirectly, e.g. from the vari ation of macro-
scopical parameters of the material or/and of electronic de vices. It is to be
noted that there exists also a kinetics of the defects induce d by irradiation
giving rise to the annealing process. A general treatment of the evolution pro-
cesses is not possible, so, only some particular models exis t in literature, see
for example [4] and the references cited therein in the case o f silicon. In these
circumstances, in the present paper only the primary proces s of defect genera-
tion is modelled .This way, the energy range of incident hadr ons for which the
concentration of primary defects do not affect irreversibly the device properties
could be established (theoretically predicted).
2 Model of the degradation
2.1 Energy loss
At the passage of the incident charged particle in the semico nductor material
same of its energy is deposited into the target. The charged p articles interact
with both atomic and electronic systems in a solid. The total rate of energy
loss, could, in general, be divided artificially into two com ponents, the nuclear
and the electronic part.
The energy lost due to interactions with the electrons of the target gives rise
to material ionisation, while the energy lost in interactio ns with nuclei is at
the origin of defect creation.
A comprehensive theoretical treatment of electronic stopp ing, which covers all
energies of interest, cannot be formulated simply because o f different approxi-
3mations concerning both the scattering and contribution of different electrons
in the solid. For fast particles with velocities higher then the orbital velocities
of electrons, the Bethe-Bloch formula is to be used [5]. At lo wer velocities,
inner electrons have velocities greater than particle velo city, and therefore do
not contribute to the energy loss. This regime has been model led for the gen-
eral case by Lindhard and Scharff[6] and particular cases hav e been treated,
e.g. in reference [7]. If the particle has a positive charge, and a velocity close to
the orbital velocity of its outer electrons, it has a high pro bability of capturing
an electron from one of the atoms of the medium through which i t passes. This
process contributes to the total inelastic energy loss sinc e the moving ion has
to expend energy in the removal of the electrons which it capt ures.
The nuclear stopping depends on the detailed nature of the at omic scattering,
and this in turn depends intimately on the form of the interac tion potential. At
low energies, a realistic potential based on the Thomas-Fer mi approximation
has been used in the literature [6] and at higher energies, wh ere scattering
results from the interaction of unscreened nuclei, a Ruther ford collision model
is to be used.
2.2 Bulk defect production
The mechanism considered in the study of the interaction bet ween the incom-
ing particle and the solid, by which bulk defects are produce d, is the following:
the particle, heavier than the electron, with electrical ch arge or not, interacts
with the electrons and with the nuclei of the crystalline lat tice. The nuclear
interaction produces bulk defects. As a result of the intera ction, depending on
the energy and on the nature of the incident particle, one or m ore light par-
ticles are produced, and usually one or more heavy recoil nuc lei. These nuclei
have charge and mass numbers lower or at least equal to those o f the medium.
After the interaction process, the recoil nucleus or nuclei , if they have suffi-
cient energy, are displaced from the lattice positions into interstitials. Then,
the primary knock-on nucleus, if its energy is large enough, can produce the
displacement of a new nucleus, and the process could continu e as a cascade,
until the energy of the nucleus becomes lower than the thresh old for atomic
displacements.
The concentration of the primary radiation induced defects on unit fluence
has been calculated starting from the following equation:
CPD (E) =1
2Ed/integraldisplay/summationdisplay
idσi
dΩL(ERi)dΩ (1)
where Eis the kinetic energy of the incident particle, Edthe threshold energy
4for displacements in the lattice, ERithe recoil energy of the residual nucleus,
L(ERi) the Lindhard factor describing the partition between ioni sation and
displacements and dσi/dΩ the differential cross section for the process respon-
sible in defect production. In the concrete calculations, a ll nuclear processes,
and all mechanisms inside each process are included in the su mmation over in-
dexi. Because of the regular nature of the crystalline lattice, t he displacement
energy is anisotropic.
In the concrete evaluation of defect production, the nuclea r interactions must
be modelled, see for example references [3,8–10]. The prima ry interaction be-
tween the hadron and the nucleus of the lattice presents char acteristics re-
flecting the peculiarities of the hadron, especially at rela tively low energies.
If the inelastic process is initiated by nucleons, the ident ity of the incoming
projectile is lost, and the creation of secondary particles is associated with
energy exchanges which are of the order of MeV or larger. For p ion nucleus
processes, the absorption, the process by which the pion dis appears as a real
particle, is also possible.
The energy dependence of cross sections, for proton and pion interaction with
the nucleus, presents very different behaviours: the proton -nucleus cross sec-
tions decrease with the increase of the projectile energy, t hen have a minimum
at relatively low energies, followed by a smooth increase, w hile the pion nu-
cleus cross sections present for all processes a large maxim um, at about 160
MeV, reflecting the resonant structure of interaction (the ∆ 33resonance pro-
duction), followed by other resonances, at higher energies , but with much less
importance. Due to the multitude of open channels in these pr ocesses, some
simplifying hypothesis have been done [10].
The process of partitioning the energy of the recoil nuclei ( produced due the
interaction of the incident particle with the nucleus, plac ed in its lattice site) by
new interaction processes, between electrons (ionisation ) and atomic motion
(displacements) is considered in the frame of the Lindhard t heory [11].
The factor characterising recoil energy partition between ionisation and dis-
placements has been calculated analytically, solving the g eneral equations of
the Lindhard theory in some physical approximations. Detai ls about the hy-
pothesis used could be found in reference [12]. All curves st art, at low energies,
from the same curve; they have at low energies identical valu es of the energy
spent into displacements, independent on the charge and mas s number of the
recoil. At higher energies, the curves start to detach from t his main branch.
This happens at lower energies if their charge and mass numbe rs are smaller.
The maximum energy transferred into displacements corresp onds to recoils
of maximum possible charge and mass numbers. The curves pres ent then a
smooth increase with the energy. For the energy range consid ered here, the
asymptotic limit of the displacement energy is not reached.
5For binary compounds, the Lindhard curves have been calcula ted separately
for each component of the material, and the average weight Br agg additivity
has been used. In this case, a threshold for atomic displacem ents must be
considered for each atomic species and for each direction in the crystal. In
the concrete calculations, a weighted value, independent o n the crystalline
direction has been used.
3 Results, discussions and some possible applications
The nuclear stopping power presents an energy dependence wi th a pronounced
maximum. It is greater for heavier incident particles: prot ons compared to
pions. In a given medium, the position of the maximum is the sa me for all
particles with the same charge. In figure 1, the nuclear energ y loss in diamond,
silicon, silicon carbide, GaP, GaAs, InP, InAs and InSb is re presented for
protons and pions respectively, as a function of their kinet ic energy. In the
same medium, the position of the maximum is the same for pions and protons.
The behaviour of these materials in proton and pion fields is c haracterised by
the CPD. In Figure 2, the dependence of the CPD as a function of the protons
kinetic energy and medium mass number is presented for diamo nd, silicon, SiC
GaAs and InP - see reference [13] and references cited therei n. The values for
diamond degradation are from reference [9], the correspond ing ones for silicon
are averaged values from references [14] and [15], SiC - from reference [16] and
those for GaAs and InP are from reference [15]. Low kinetic en ergy protons
produce higher degradation in all materials. The discontin uity in the surface
is related to differences in the behaviour of the CPD for monoa tomic materials
(or binary ones with close elements), and binary ones with re mote elements
in the periodic table.
For pion induced degradation, the energy dependence of CPD ( as well as of the
NIEL) presents two maxima, the relative importance of which depends on the
target mass number: one in the region of the ∆ 33resonance, more pronounced
for light elements and compounds containing light elements , and another one
around 1 GeV kinetic energy, more pronounced for heavy eleme nts. At higher
energies, an weak energy dependence is observed, and a gener alA3/2
average de-
pendence of the NIEL can be approximated [10,12]. In Figure 3 , the CPD
for all analysed materials (diamond, Si, SiC, GaP, GaAs, InP , InAs, InSb) is
represented as a function of the pion kinetic energy and of ma terial average
mass number. The differences in the behaviour of these materi als are clearly
suggested by the discontinuity in the mesh surfaces.
In the energy range considered in the paper, it could be obser ved that the
CPD produced by pions and protons, and characterising the bu lk degrada-
6tion, are very different and reflect the peculiarities of the i nteractions of the
two particles with the semiconductors. For pions, there are two maxima, one
in the region of the ∆ 33resonance, corresponding to about 140 - 160 MeV
kinetic energy, and the other at higher energies, around 1 Ge V. The relative
importance of these maxima depends on the mass number of the m aterial.
In comparison with this behaviour, the CPD produced by proto ns decreases
abruptly with the increase of energy at low energies, follow ed by a smooth and
slow increase at higher energies.
In relation to their behaviour in pion fields, these material s could be separated
into two classes, the first with monoatomic materials or mate rials with rela-
tively close mass numbers (diamond, silicon, GaAs and InSb) , and the second
comprising binary materials with remote mass numbers of the elements (SiC,
GaP, InP, InAs) with similar behaviours inside each group. T he diamond is
the hardest material from all considered here. A slow variat ion of the primary
defect concentration has been found for pion irradiation of diamond, silicon,
SiC, GaP and GaAs, in the whole energy range of interest, with less than
2 displacements/cm/unit of fluence. In contrast to these mat erials, there are
others, characterised by a low CPD in the energy range up to 20 0 MeV, (which
represents this way the upper limit of the energy range where their utilisation
in pion field is recommended), followed by a pronounced incre ase of displace-
ment concentration with energy to more than 8 displacements /cm/unit fluence
for InSb.
It is to be mentioned that, in the present model hypothesis, f or SiC, negligible
differences have been found between different polytypes in wh at regards the
effects of pion and proton degradation [16], conclusions in a ccord with the
experimental results [17]. The behaviour of SiC in radiatio n fields is between
the corresponding one of diamond and silicon.
As it is well known, the analysed semiconductors are possibl e materials for de-
tectors and electronic devices which have to work long time i n particle physics
experiments, space applications, etc., in intense fields of hadrons, and in ex-
perimental configurations which impose high reliability of devices, and must
present a controlled degradation of their parameters. As po ssible applications
we will analyse two hypothetical cases: the utilisation of d iamond, silicon, SiC
or GaAs as detectors at the Large Hadron Collider (LHC) at CER N, and the
long time exposure of the electronic devices in the field prod uced by cosmic
rays.
For the LHC, the standard physics programme is based on the st udy of proton
- proton interactions, at about 7 TeV beam energy, on an integ rated luminos-
ity of 5 x105pb−1which corresponds to 9 year of operation, for an annual
operation time of 1.9x107s. The irradiation background is continuous. The
charged hadrons are produced in the primary interactions, w hile the neutrons
7are albedo particles. The charged pions are the dominant par ticles, followed
by protons, antiprotons and kaons. As an illustration of the above calculations
for the degradation of different semiconductors in proton an d pion fields, the
results of the simulation of Gorfine and Taylor [18] have been chosen, for pion
and proton fluxes in the inner detector assembly region, para llel to the beam
axis. Both protons and pions are transported down to thermal energies in
the detector, by nuclear interactions. The particle flux ene rgy spectra have
been simulated for a first layer of Si detectors (situated at 1 1.5 cm), and with
complete moderator. The obtained spectra have been found to be slowly de-
pendent on the material of the inner detector and so, in the pr esent paper the
same hadron spectra for diamond, silicon carbide, silicon a nd gallium arsenide
have been utilised.
The convolution of the pion and proton spectra with the energ y dependence
of the CPD has been done in the energy range 50 MeV - 10 GeV, and 1 0
MeV - 10 GeV for pions and protons respectively. Below 50 MeV, a realistic
estimation of materials degradation to pions is very difficul t due to the lack
of experimental data on pion - nucleus interaction and also t o the increase of
the weight of Coulomb interaction.
The results of these calculations are summarised in Fig. 4, f or the diamond,
silicon carbide, silicon and GaAs options, both for pion and proton degra-
dation. In the analysed case, diamond and SiC are the hardest materials in
both pion and proton fields; the diamond is harder to pions tha n to protons.
The behaviour of silicon is similar in both particle fields. T he GaAs option
is not recommended because of an order of magnitude higher de gradation in
comparation with all other considered materials.
Another possible utilisation of semiconductor devices in r adiation fields is
related to space applications. In the primary cosmic radiat ion, the most abun-
dant particles are the protons [5]. Other charged particles (for example π+/−,
e+/−,µ+/−,νmu, etc.) are produced in the interaction of the primary cosmic
rays in air. The damage induced in diamond, Si, SiC, GaAs and I nP by the
primary cosmic field has been estimated for protons in the ene rgy range 10
MeV - 10 GeV, and the results are presented in Figure 5. The dev ices have
been supposed to be exposed directly to the cosmic field. In th is case too,
diamond has been found to be the hardest material. GaAs and In P suffer a
degradation of a factor of about 50 times higher in comparati on with diamond
and this behaviour can affect irreversibly the properties of these materials for
long time operation.
The degradation produced by the particle field at LHC (pions a nd protons)
and by the free protons from the primary cosmic rays respecti vely are of the
same order of magnitude for each of the materials investigat ed.
84 Summary
A systematic theoretical study has been performed, investi gating the interac-
tion of charged hadrons with semiconductor materials and th e mechanisms of
defect creation by irradiation.
The nuclear stopping power has been found to be greater for he avier incident
particles (protons compared to pions), and for lighter medi a. The position of
its maximum is the same for protons and pions in the same mediu m.
The mechanisms of the primary interaction of the hadron with the nucleus
(nuclei) of the semiconductor lattice have been explicitly modelled and the
Lindhard theory of the partition between ionisation and dis placements has
been applied.
For protons, the low kinetic energy particles produce highe r degradation in all
materials.
For pions, the energy dependence of CPD presents two maxima, the relative
importance of which depends on the target mass number: one in the region of
the ∆ 33resonance, more pronounced for light elements and compound s con-
taining light elements, and another one around 1 GeV kinetic energy, more
pronounced for heavy elements. At higher energies, an weak e nergy depen-
dence is observed. A slow variation of the primary defect con centration has
been found for pion irradiation of diamond, silicon, GaP and GaAs, in the
whole energy range of interest, with less than 2 displacemen ts/cm/unit of
fluence. In contrast to this situation, for the other semicon ductor materials
analysed, a low CPD is estimated in the energy range up to 200 M eV (which
represent the energy range up to their utilisation in pion fie ld is recommended),
followed by a pronounced increase of displacement concentr ation to more than
to more than 8 displacements/cm/unit of fluence at high energ ies.
The behaviour of this semiconductor materials has been anal ysed compara-
tively both in relation to particle physics experiments (in ner part of the detec-
tion system at LHC) and to space applications (the devices be ing considered
to be exposed directly to the cosmic ray field.
References
[1] A. P. Verma and P. Krishina, ”Polymorphism and Polytypis m in Crystals”,
Wiley, New York, 1966.
[2] W. R. L. Lambrecht, S. Limpijummong, S. N. Rashkeev, and B . Segall, Phys.
St. Sol. (b) 202 , n.5 (1997) 5.
9[3] I. Lazanu, S. Lazanu, E. Borchi, M. Bruzzi, Nucl. Instr. a nd Meth. Phys.
Research, A 406 (1998) 259.
[4] S. Lazanu and I. Lazanu, ”Annealing of radiation induced defects in silicon
in a simplified phenomenological model”, e-preprint LANL ph ysics/0008077,
submitted to Nucl. Instr. and Meth. in Phys. Research A.
[5] C. Caso et al., Review of Particle Properties, Eur. J. Phy s.C3(1998).
[6] J. Lindhard and M. Scharff, Phys. Rev. 124(1961) 128.
[7] E. Morvan, P. Godignon, S. Berberich, M. Vellvehi and J. M illan, Nucl. Instr.
and Meth. Phys. Research B 147 (1999) 68.
[8] S. Lazanu, I. Lazanu, U. Biggeri, E. Borchi, and M. Bruzzi , in Conf. Proc. Vol.
59, ”Nuclear Data for Science and Technology”, Eds. G. Reffo, A. Ventura and
C. Gradi. (SIF, Bologna, 1997) 1528.
[9] I. Lazanu, and S. Lazanu, Nucl. Instr. and Meth. Phys. Res earchA 432 (1999)
374.
[10] S. Lazanu, I. Lazanu, U. Biggeri and S. Sciortino, Nucl. Instr. and Meth. Phys.
Research A 413 (1998) 242.
[11] J. Lindhard, V. Nielsen, M. Scharff and P. V. Thomsen, Mat . Phys. Medd. Dan
Vid. Sesk. 33(1963) 1.
[12] S. Lazanu and I. Lazanu, ”Analytical approximations of the Lindhard equations
describing radiation effects”, e-preprint LANL hep-ph/991 0317, submitted to
Nucl. Instr. and Meth. in Phys. Research A.
[13] I. Lazanu, S. Lazanu and M. Bruzzi, ”Expected behaviour of different
semiconductor materials in hadron fields”, Proc. Conf. ENDE ASD, Stockholm,
June 2000, e-preprint LANL physics/0006054.
[14] A. van Ginneken, preprint Fermilab, FN-522 , 1989.
[15] G. P. Summers, E. A. Burke, P. Shapiro, S. R. Messenger an d R. J. Walters,
IEEE Trans. Nucl. Sci. NS-40 (1990) 1372.
[16] I. Lazanu, S. Lazanu, E. Borchi, M. Bruzzi, ”A comparati ve study of the
radiation properties of SiC in respect to silicon and diamon d”, e-preprint LANL
physics/0007011, submitted to Nucl. Instr. and Meth. in Phy s. Research A.
[17] E. Wendler, A. Heft and W. Welsch, Nucl. Instr. and Meth. in Phys. Res. B
141(1998) 105.
[18] G. Gorfine and G. Taylor, preprint INDET-NO-030, UM-P-9 3/103 (1993).
10Figure captions
Figure 1: The nuclear energy loss in diamond, silicon, silic on carbide, GaP,
GaAs, InP, InAs and InSb as a function of the kinetic energy of the incident
particle: protons (up) and pions (down) respectively.
Figure 2a: The concentration of primary defects on unit fluen ce (CPD) in
diamond, silicon, SiC, GaAs and InP induced by protons, as a f unction of the
kinetic energy and average mass number of the semiconductor material. The
mesh surfaces are drawn only to guide the eye.
Figure 2b: The dependence of CPD as a function of proton kinet ic energy, for
the same semiconductors.
Figure 3a: The energy and material dependence of the CPD on un it pion
fluence for diamond, Si, SiC, GaP, GaAs, InP, InAs and InSb. Th e mesh
surfaces are drawn to guide the eyes.
Figure 3b: The CPD as a function of the kinetic energy of incid ent pions, for
the same semiconductors.
Figure 4: Estimated CPD on unit fluence induced in diamond, si licon, SiC and
GaAs, by the simulated flux energy spectra of pions and proton s in the inner
detector at LHC.
Figure 5: Estimated CPD on unit fluence induced by the primary cosmic ray
flux energy spectra in diamond, Si, SiC, GaAs and InP (only the effects pro-
duced by protons are considered), in the hypothesis that the se semiconductor
materials are exposed directly in the radiation field.
11/G31 /G30/G20 /G2d/G34/G31 /G30/G20 /G2d/G33/G31 /G30/G20 /G2d/G32/G31 /G30/G20 /G2d/G31/G31 /G30/G20 /G30
/G50/G72 /G6f/G74/G6f/G6e /G20/G4b/G69/G6e /G65/G74/G69 /G63/G20/G45/G6e /G65/G72/G67/G79 /G20/G5b/G4d/G65 /G56/G5d/G30/G31 /G30/G30/G32 /G30/G30/G33 /G30/G30/G34 /G30/G30/G35 /G30/G30/G36 /G30/G30
/G4e/G75/G63/G6c/G65/G61/G72/G20/G53/G74/G6f/G70/G70/G69/G6e/G67/G20/G50/G6f/G77/G65/G72/G20/G5b/G4d/G65/G56/G20/G63/G6d/G32/G2f/G67/G5d/G43 /G53 /G69/G43 /G53/G69 /G49 /G6e/G50
/G47 /G61/G50 /G47/G61 /G41/G73 /G49/G6e /G41/G73 /G49/G6e /G53/G62
/G31 /G30/G20 /G2d/G34/G31 /G30/G20 /G2d/G33/G31 /G30/G20 /G2d/G32/G31 /G30/G20 /G2d/G31/G31 /G30/G20 /G30
/G50/G69 /G6f/G6e/G20/G4b /G69/G6e/G65 /G74/G69/G63 /G20/G45/G6e/G65 /G72/G67/G79 /G20/G5b/G4d/G65 /G56/G5d/G30/G31 /G30/G32 /G30/G33 /G30/G34 /G30/G35 /G30/G36 /G30/G37 /G30/G38 /G30/G39 /G30
/G4e/G75/G63/G6c/G65/G61/G72/G20/G53/G74/G6f/G70/G70/G69/G6e/G67/G20/G50/G6f/G77/G65/G72/G20/G5b/G4d/G65/G56/G20/G63/G6d/G32/G2f/G67/G5d/G43 /G53 /G69/G43 /G53/G69 /G49 /G6e/G50
/G47 /G61/G50 /G47 /G61/G41 /G73 /G49/G6e /G41/G73 /G49/G6e /G53/G6200.511.522.53
020406080-2-10123
lg(E [MeV])Alg(CPD [1/cm])10 010 110 210 3
Proton Kinetic Energy [MeV]10 -210 -110 010 110 2CPD [1/cm]C
SiC
Si
GaAs
InP12345
050100150-2-1.5-1-0.500.51
lg(E [MeV])Alg(CPD [1/cm])10 210 310 410 5
Pion Kinetic Energy [MeV]10 -210 -110 010 110 2CPD (1/cm)C
SiC
Si
GaP
GaAs
InP
InAs
InSbCSiCSiGaAs10 210 310 4CPD [1/cm]
pions
protonsCSiCSiGaAsInP10 210 310 4CPD [1/cm] |
arXiv:physics/0008215v1 [physics.atom-ph] 23 Aug 2000Laser spectroscopy of simple atoms and
precision tests of bound state QED
Savely G. Karshenboim1
D.I. Mendeleev Institute for Metrology, 198005 St. Petersb urg, Russia
Max-Planck-Institut f¨ ur Quantenoptik, 85748 Garching, G ermany2
Abstract. We present a brief overview of precision tests of bound state QED and
mainly pay our attention to laser spectroscopy as an appropr iate tool for these tests.
We particularly consider different precision tests of bound state QED theory based on
the laser spectroscopy of optical transitions in hydrogen, muonium and positronium
and related experiments.
I INTRODUCTION
Precision laser spectroscopy of simple atoms (hydrogen, de uterium, muonium,
positronium etc.) provides an opportunity to precisely test Quantum Electro dy-
namics (QED) for bound states and to determine some fundamen tal constants with
a high accuracy. The talk is devoted to a comparison of theory and experiment
for bound state QED. Experimental progress during the last t en years has been
mainly due to laser spectroscopy and, thus, the tests of boun d state QED are an
important problem associated with modern laser physics.
The QED of free particles (electrons and muons) is a well-est ablished theory
designed to perform various calculations of particle prope rties (like e. g. anomalous
magnetic moment) and of scattering cross sections. In contr ast, the theory of bound
states is not so well developed and it needs further precisio n tests. The QED theory
of bound states contains three small parameters, which play a key role: the QED
constant α, the strength of the Coulomb interaction Zαand the mass ratio m/M
of an orbiting particle (mainly—an electron) and the nucleu s. It is not possible to
do any exact calculation and one has to use some expansions ov er some of these
three parameters.
The crucial theoretical problems are:
•The development of an effective approach to calculate higher -order corrections
to the energy levels.
1)E-mail: sek@mpq.mpg.de
2)Summer address•Finding an effective approach to estimate the size of uncalcu lated higher-order
corrections to the energy levels.
The difference between these two problems is very important: any particular evalu-
ation can include only a part of contributions and we must lea rn how to determine
the uncertainty of the theoretical calculation, i. e. how to estimate corrections
that cannot be calculated. We discuss below some important h igher-order QED
corrections, our knowledge on which determines the accurac y of the bound state
QED calculations. Doing ab initio QED calculations, none can present any theo-
retical prediction to compare with the measurements. With t he help of QED one
can only express some measurable quantities in terms of fund amental constants
(like e. g. the Rydberg constant R∞, the fine structure constant α, the electron-
to-proton ( me/mp) and electron-to-muon ( me/mµ) mass ratio). The latter have to
be determined somehow, however, essential part of the exper iments to obtain the
values of the contstans involves measurements with simple a toms and calculations
within bound state QED.
The study of the Lamb shift in the hydrogen atom and the helium ion about
fifty years ago led to a great development of Quantum electrod ynamics. Now, in-
vestigations of the lamb shift are still of interest. After r ecent calculations of the
one-loop, two-loop and three-loop corrections to the Lamb s hift in the hydrogen
atom, the main uncertainty comes from higher order two-loop contributions of the
orderα2(Zα)6m. They contain logarithms (ln( Zα)) which enhance the correction3,
and the leading term with the cube of the logarithm is known [1 ]. The uncer-
tainty due to the uncalculated next-to-leading terms is est imated as 2 ppm. It is
competible with an experimental uncertainty from laser exp eriments (3 ppm) and
essentially smaller than the 10-ppm inaccuracy of computat ions because of the lack
of an appropriate knowledge of the proton charge radius [2]. This 10-ppm level of
uncertainty due to the proton structure is an obvious eviden ce that the QED is
an incomplete theory which deals with photons and leptons (e lectrons and muons)
and cannot describe the protons (or deutrons) from first prin ciples. To do any
calculations with hydrogen and deuterium one has to get some appropriate data
on their structure from expreriment.
We particularly discuss here precise tests of the bound stat e QED theory due
measurements of the 1 s−2sand other optical transitions in hydrogen, muonium
and positronium and related experiments. We consider a numb er of different two-
photon Doppler-free experiments in hydrogen and deuterium in Sect. II. In Sect.
III we present the status of the Lamb shift study and discuss s ome running auxilary
experiments which can help us in the understanding of the hig her-order corrections
and the proton structure. Since the problem of the proton str ucture limits use-
fulness of extremely precise hydrogen experiments, the stu dy of unstable leptonic
atoms can provide some tests QED which are efficient and compet itive with the
study of hydrogen [3]. We discuss these in Sect. IV.
3)In hydrogen, one can find: ln3(Zα)−1∼120, ln2(Zα)−1∼24 and ln( Zα)−1∼5.The paper contains also a brief summary and a list of referenc es. The latter is
far incomplete. However, most problems concerning the prec ision study of simple
atoms were discussed at the recent Hydrogen atom, 2 meeting, which took place
June, 1-3, 2000, in Italy and we hope that one finds more refere nces on the subject
therein [4].
II TWO-PHOTON DOPPLER-FREE SPECTROSCOPY
The effect of Doppler broadening used to be a limiting factor i n the measurement
of any transition frequency. A way to avoid it is to apply two- photon transitions,
which are not sensitive to the linear Doppler shift. First of all, a success in pre-
cision spectroscopy is to be expected from 1 s−2smeasurements, because of the
metastability of the 2 sstate and, hence, of its narrow natural radiative width.
Since a value of the Rydberg constant can only be determined f rom measurements
with hydrogen and deuterium transitions, one has to measure at least two different
transitions for any applications to QED tests ( i. e. one measurement is for the
Rydberg constant, while the other is to test QED using the val ue of the Rydberg
constant). An appropriate option is to study the 1 s−ns, 2s−nsor 2s−nd
transitions in hydrogen, or the 1 s−2stransition in other atoms.
A Studying hydrogen and deuterium atoms
To precisely test bound state QED theory of the Lamb shift one has to measure
two different transitions in hydrogen and/or deuterium. Com bining them prop-
erly one can exclude the contribution of the Rydberg constan t to the transition
frequency
E(nl) =−c h R∞
n2(1)
and find a value determined by some known relativisitc correc tions ( ∼α2R∞) and
by the Lamb shifts of the involved levels. Since a number of di fferent states is
involved, a number of Lamb shifts have to be determined.
A specific combination of the Lamb shifts
∆L(n) =EL(1s)−n3EL(ns) (2)
is important [5] for the evaluation of the data from optical m easurements in Garch-
ing [6] and Paris [7], obtained by means of two-photon Dopple r-free laser spec-
troscopy. The use of this difference allows to present all unk nown Lamb shifts of
nsstates in terms of only one of them (usually—either EL(1s) orEL(2s)). The
uncertainty in this difference is also determined by unknown higher-order two-loop
terms, but the leading term which includes a squared logarit hm is known [10]. It is
important to underline that the status of this difference [8, 2] differs from the status
of the 1s Lamb shift [9]. It is free of most theoretical proble ms and it is in some
sense not a theoretical value, but a mathematical one.1057 900 kHz 1057 850 kHz 1057 800 kHzLS
FS
OBF
SY / K
theory
0.8620.8050.847 grand average
averages
comparison Garching - Paris
Garching 1s 2s
2s 8s, d 12d
Paris
FIGURE 1. 2s−2pLamb shift: compari-
son of theory and experiment. Grand aver-
age denotes the average of all data from the
Lamb shift (LS), fine structure (FS) and op-
tical beat frequnecy (OBF) measurements.FIGURE 2. Level schemes of absolute fre-
quency measurements at MPQ (Garching)
and LKB (Paris)
Absolute measurements of two-photon transitions
Now let us consider some recent experiments. The most accura te result for the
Lamb shift by optical means (Fig. 1, see [2,9] for references )) can be achieved from
a comparison of the Garching data and the Paris data (see Fig. 2, the references
can be found in Refs. [2,6,7,4]). These are results after abs olutely measuring some
transitions frequencies ( i. e. by measuring with respect to the cesium standard).
In the case of the Garching experiment, a small electric field allows a single-photon
E1 transition from the 2s state to the 1s level and a resonance in the intensity of
this decay was used as a signal when tuning the laser frequenc y. The measurement
accuracy of the 1s-2s transition is high and it can be used for other applications,
like e. g. a search for variation of constants (see e. g. [11]) .
Relative measurements of two-photon transitions
A comparison of two absolute frequencies involves cesium st andards and a lot
of metrology. It is possible to avoid comparing two frequenc ies within the same
experiment. Level schemes of three experiments (see for det ail Refs. [2,6,7,4]) are
presented in Fig. 3. Most of them used only two-photon transi tions. In all three
experiments, the pair of measured frequencies consists of t wo values that differ by
a factor either about 4 (Garching, Paris) or 2. These factor a ppear within the
leading non-relativistic approximation ( i. e. from the Schr¨ odinger equation with
the Coulomb potential), in which the energy levels are deter mined by Eq. (1).
Multiplying the smaller frequencies by the proper factor (4 or 2) and comparing
them with the larger frequencies experimentally, the beat f requency signals wereextracted. For some rather historical reasons, the final res ults are less accurate
than in the case of the comparison of two absolute measuremen ts (see Fig. 1 which
contains an average value over all three experiments).
Garching 1s 2s 4s, d
Yale 1s 2s 4p
Paris 1s 3s
2s 6s, d
FIGURE 3. Level schemes of relative frequency measurements at MPQ (Ga rching), Yale Uni-
versity and LKB (Paris)
III THE LAMB SHIFT IN THE HYDROGEN ATOM
Only a part of the experiments mentioned above were performe d for both hy-
drogen and deuterium, and essentially more experimental da ta are available for
hydrogen. Below we discuss only the Lamb shift in the hydroge n atom.
A Present status
The current situation of the comparison of theory and experi ment is summarized
in Fig. 3 (see Refs. [2,9] for details). A result marked with SY/K (Sokolov-and-
Yakovlev value, corrected by us) is not included neither in t he average over the
Lamb shift measurements ( LS) nor into the grand average . This corrected result of
Sokolov and Yakovlev is claimed to be the most precise, howev er its real accuracy is
an open question. The theory is presented with three differen t values of the proton
size published some time ago. The uncertainty of these theor etical results is about
4 ppm. In our opinion, a reasonable theory is not so accurate a nd its margins
are presented as a filled area (10-ppm uncertainty). All expe rimental results but
one by Sokolov and Yakovlev are consistent with our concerva tive estimate of the
theoretical value.
The present status can be briefly described as following:
•The experiments are mainly consistent to each other and with theory.
•In particular, the optical data evaluated with the help of th e difference in
Eq. (2) are consistent with the microwave data ( LSandFSin Fig. 1) found
without the use of Eq. (2).•The uncertainty of the grand average value for the 2 s−2p1/2Lamb shift is
about 3 ppm.
•The computational uncertainty is about 2 ppm and it originat es from the
unknown higher-order two-loop corrections of the order α2(Zα)6mwhich are
known only in part [1].
•The uncertainty due to the finite size of the proton is about 10 ppm and it is
due to the inaccuracy in the determination of the proton char ge radius [2].
Below we discuss some current laser experiments which offer s ome solutions for the
problems with the theory, both: of the proton size and of the h igher-order two-loop
contributions.
B Proton structure
There are a few ways to study the proton chagre distribution. One of them is
to look for elastic scattering of electrons by protons at low momentum transfer q.
One can determine the proton electric form factor from the sc attering data and
extrapolate in to zero momentum transfer
GE(q2) = 1 −R2
p
6q2+... (3)
Unfortunately, the scattering data were not evaluated prop erly and a comprehensive
description cannot be available. The claimed uncertainty l eads to a 3.5-ppm error
bar for the Lamb shift, but we expect it to be rather 10-ppm.
A promising project to determine the proton charge radius is now in progress
at Paul Scherrer Institut (Villigen). It deals with muonic h ydrogen. The muon is
about 200 times heavier than the electron and hence the Bohr o rbit of the muon
lies much lower than the one of the electron and the level ener gies are more affected
by the proton structure. The used atomic level scheme is pres ented in Fig. 4. It is
similar to the one applied for muonic helium some time ago. A m ain advantage is a
slow-muon beam at PSI. The use of slow muons allows to make use of a low-density
gas target which reduces the collisional decay rate of the me tastable 2 sstate. It
has been checked experimentally that, under the conditions of the PSI experiment,
the 2sstate is metastable enough and not destroyed by collisions. That allows
one to go to the next step: to excite the atoms in the 2 sstate to the 2 pstate by
a laser and to look for the intensity of the X-ray Lyman- αas a function of the
laser frequency. In case of successful measurement the resu lt will be the Lamb
shift in muonic hydrogen with an essential contribution due to the proton size, and
eventually with a value of the proton charge radius more accu rate by an order of
magnitude than the current scattering values.1s2s2p
X - raylaser
FIGURE 4. The level scheme used in the PSI experiment on the Lamb shift i n muonic hydrogen
C Higher-order two-loop corrections
The other problem of theory of the Lamb shift in hydrogen is th e unknown
higher-order two-loop corrections. They are proportional toZ6, while the leading
contribution to the Lamb shift is ∼Z4. That means that some less accurate
measurements at higher Zcan nevertheless give some efficient results for these
higher-order terms. There are three projects for low- Zions [4]:
•Lamb shift measurement in the4He+ion (Z= 2) at the University of Windsor
(recently completed);
•two-photon 2 s−3stransition in the4He+ion (Z= 2) at the University of
Sussex (in progress);
•fine-structure (2 p3/2−2s) measurement in hydrogen-like nitrogen14N6+(Z=
7) and14N6+at the Florida State University.
The scheme of the last experiment is presented in Fig. 5. It is similar to the
previous one with the muonic hydrogen. It is expected (see My ers’ paper in [4]) to
be sensitive to the higher-order two-loop corrections.
1s2s2p3/2
X - raylaser
2p1/2 Lamb splitting Fine structure
FIGURE 5. The level scheme of the FSU experiment on the fine structure in hydrogen-like
nitrogen
23S1
1644THz
4110
13S1 Theory Stanford (& Bell)1233607210 MHz 1233607230
FIGURE 6. Three-photon ionisation in
positronium at Stanford UniversityFIGURE 7. Positronium 1 s−2s: theory
and experiment
IV LEPTONIC ATOMS
Since the theory of the energy levels in the hydrogen atom is l imited by nuclear
structure effects, one can try to study protonless hydrogen- like atoms: muonium
and positronium. In both of them the 1 s−2sinterval was measured.
A 1s-2s transition in positronium
In the positronium spectrum there are a number of values whic h were or are under
precise experimental study. In all cases the uncertainty of the positronium energy
(n= 1,2) is known up to α6m. The only double logarithm ( α7mln2α) is known to
the next order [1,12]. The inaccuracy of the theory originat es from the non-leading
terms (single logarithm and constant) of radiative and radi ative-recoil corrections
of order α7m. For the majority of measurable quantities (hfs of 1 s, 1s−2sinterval,
fine structure of n= 2, orthopositronium and parapositronium decay rate), the
theory is competitive with the experiment and actually slig htly more accurate. We
present a level scheme of a measurement of the 1 s−2sinterval in positronium in
Fig. 6, while in Fig. 7 we compare the theory with the experime nt.
The positronium 1 s−2sexperiment is quite different from the hydrogen one,
because of the short lifetime of the orthopositronium tripl et 13S1state which the
experiment starts from. The three-photon annihilation lea ds to a lifetime of 13S1
state of 1 .4·10−7s. The method applied was three-photon ionization which has a
resonance due to the two-photon transition.
B 1s-2s transition in muonium atom
The muonium nucleus, the muon, lives about 2.2 µs and an idea used to measure
1s−2sinterval in muonium is similar to the positronium experimen t (Fig. 6).
However, the application is very different. In contrast to po sitronium (in which, e.g. the 1 s−2sexperiment is really competitive with the study of 1 shfs), a much
more efficient test of QED can be provided by the hyperfine struc ture of the ground
state which was measured very precisely. The uncertainty of the calculation of the
1shfs interval originates from some unknown corrections of th e fourth order. Some
of these, including the large logarithms (ln( Zα)−1∼5 or ln( M/m)∼5), are known
in the double logarithmic approximation [1,13] and non-lea ding terms lead to an
uncertainty of the theoretical expression as large as 0.05 p pm. The uncertainty
arises from the unknown next-to-leading radiative-recoil (α(Zα)2(m/M)EF) and
pure recoil (( Zα)3(m/M)EF)4terms (which are essentially the same as the α7m
terms in positronium). However, the budget for the theoreti cal uncertainty contains
not only the computational items. Actually, the largest con tribution to the budget
comes from a calculation of the Fermi energy because of the la ck of a precise
knowledge of the muon-to-electron mass ratio. We summarize in Fig. 8 a few
of the most accurate values for the mass ratio (see Refs. [13, 3] for references).
The most accurate result there is from study of the Zeeman effe ct of the 1 sstate in
muonium. Another way to determine this ratio is the 1 s−2smuonium experiment.
Two other values are extracted from the study of muon spin pre cession in different
media.
100 200 300 400 500 600 700 800 900
muon-to-electron mass ratio: mµ/me = 206.768 xxxµ+ in waterµ+ in Br2Zeeman effect in muonium 1s1s-2s in muonium
FIGURE 8. Some determinations of the muon-to-electron mass ratio
4)EFstands here for the Fermi energy, which is a leading order con tribution to the hyperfine
structure which is a result of the non-relativistic interac tion of the magnetic moments of the
electron and the muon.V SUMMARY
In our talk we briefly discuss several precision tests of boun d state QED. The
short overview shows that theory and experiment are consist ent within their un-
certainty and the crucial corrections in bound state QED in p resent are:
•higher order two-loop corrections (hydrogen Lamb shift);
•radiative-recoil and pure recoil terms of order α7m, the calculation of which
involves an essential part of the QED, binding and two-body e ffects (positro-
nium and muonium).
The study of these corrections is necessary to develop an effic ient theory competitive
with experiment.
AKNOWLEDGEMENTS
I would like to thank T. W. H¨ ansch and S. N. Bagayev for suppor t, hospitality
and stimulating discussions. The stimulating discussions with K. Jungmann, D.
Gidley, R. Conti and G. Werth are also gratefully acknowledg ed. I am grateful to
J. Reichert for useful remarks. The work was supported in par t by RFBR (grant
# 00-02-16718), NATO (CRG 960003) and Russian State Program Fundamental
Metrology .
REFERENCES
1. S. G. Karshenboim, JETP 76, 541 (1993).
2. S. G. Karshenboim, Can. J. Phys. 76(1998) 168.
3. K. Jungmann, this conference .
4. Hydrogen Atom II: Precision Physics of Simple Atomic Syst ems. Book of abstracts
(ed. by S. G. Karshenboim and F. S. Pavone), Castiglione dell a Pescaia, 2000. The
Proceedings will be published by Springer in 2001.
5. S. G. Karshenboim, JETP 79, 230 (1994).
6. R. Holzwarth et al.,this conference .
7. F. Biraben, this conference .
8. S. G. Karshenboim, Z. Phys. D 39, 109 (1997).
9. S. G. Karshenboim, invited talk at ICAP 2000, to be publish ed, e-print hep-
ph/0007278.
10. S. G. Karshenboim, JETP 82, 403 (1996); J. Phys. B 29, L21 (1996).
11. S. G. Karshenboim, Can. J. Phys. to be published, e-print physics/0008051.
12. K. Pachucki and S. G. Karshenboim, Phys. Rev.A 602792 (1999);
K. Melnikov and A. Yelkhovsky, Phys. Lett. B 458, 143 (1999).
13. S. G. Karshenboim, Z. Phys. D 36, 11 (1996). |
arXiv:physics/0008216v1 [physics.acc-ph] 23 Aug 2000NEW METHODOF DISPERSION CORRECTIONINTHEPEP-IILOW
ENERGYRING∗
I.Reichel, Y. Cai, SLAC, Stanford, California
SLAC-PUB- physics/0008216
Abstract
The sextupole magnets in the Low Energy Ring (LER) of
PEP-IIaregroupedinpairswithaphaseadvanceof180de-
grees. Displacing the magnets or moving the orbit to dis-
placethebeaminthemagnetsinanantisymmetricwaycre-
atesadispersionwavearoundthering. Thiscanbeusedto
correcttheverticaldispersioninLERwithoutchangingthe
local coupling. Resultsfromsimulationsareshown.
1 INTRODUCTION
The luminosity of PEP-II is currently mainly determined
bytheverticalbeamsizeattheinteractionpoint(IP).Inth e
LowEnergyRing(LER)theverticalbeamsizeattheIPis,
tosomeextent,causedbytheresidualverticaldispersioni n
the ring. It is hoped that by lowering the dispersion in the
ring, the vertical beam size at the IP can be decreased and
the luminositythereforeincreased.
The sextupoles to correct the chromaticity in LER are
grouped in non-interleaved pairs of the same strength.
Therefore moving one of the sextupoles of a pair up (or
moving the beam in the sextupole using a closed orbit
bump) and the other one of the pair down createsa disper-
sionwavearoundtheringwithoutaffectingthecouplingor
theorbitoutsidetheregion. Wewanttouseoneormoreof
these dispersion waves to try to cancel some of the resid-
ual vertical dispersion in the ring in order to minimize the
verticalbeamsizeat theIP.
2 SIMULATION
2.1 LEGO
The simulations are done using LEGO [1]. Five different
seeds are used for the misalignment. All five seeds give
RMS dispersions and orbits of the size that is observed in
the real machine. Orbit and dispersion are corrected using
the same algorithmsthat areused inthe controlroom. The
couplingis minimizedusing theclosest tuneapproachina
way similar to the procedure used in the control room on
the realring.
2.2 EffectsofSextupoleAlignment
For a verticalalignmenterrorof 0.5mmforthe sextupoles
(thisisassumedtobetheerrorinthemachine),theresidual
verticaldispersionsfortheseseedsarebetween5and 6cm.
∗Work supported by the Department of Energy under Contract No .
DE-AC03-76SF00515Using an error of 1mm, the dispersions are only slightly
larger for four seeds and grow from 6 to almost 7.2cmfor
oneseed. Figure1showsthedependenceofthedispersion
ontheaverageerror. average.
sextupole alignment error in mmvertical RMS dispersion in cm
5.566.577.58
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Figure1: Dependenceoftheresidualverticaldispersionon
the verticalsextupolealignment(averageoverfiveseeds).
2.3 MovingSextupolePairs
In the simulations we currently move the sextupoles by
changingtheiralignment. Thisiseasierand”cleaner”than
a closedorbitbump.
Asextupolepairischosen. Thesimulationprogramthen
loopsoverpositionchangesfrom −5to+5mminstepsof
one mm. One of the two magnets is moved up by the ap-
propriate amount, the other one down (taking into account
the original(mis-)alignmentofthemagnet).
At each step the dispersionand the verticalbeam size at
the IP are calculated. The vertical beam size is calculated
usingthealgorithmdescribedin[2].
The correction algorithm was studied for vertical sex-
tupolemisalignmentsof 0.5and1.0mm,wheretheaverage
dispersionforthe five seedsis 5.5and5.9cmrespectively.
TheaveragebeamsizesattheIPare 3.52and4.11µm. For
eachcalculationtwodifferentsextupolepairsandthesame
fivedifferentseedsasbeforeareused.
3 RESULTS
Figure 2 showsthe results of a typical simulationrun. The
dispersion and vertical beam size at the IP are plotted ver-
sus the alignment change of the sextupole pair. One cansextupole movement in mmvertical dispersion in cm
sextupole movement in mmvertical beamsize in mum66.26.46.66.877.27.47.6
-4 -2 0 2 4
3.944.14.24.34.4
-4 -2 0 2 4
Figure 2: Simulation result for one sextupole pair and one
seed.
see that the two parameters have their minimum not at the
same misalignment.
From eachthe dispersionand the beamsize one can ob-
tain an optimal position for the respective sextupole pair
(see Fig. 2). However, the results of the two don’t always
agreewitheachother.
Moving one sextupole pair such that the minimum dis-
persion is obtained, the vertical beam size on average
shrinks slightly to 3.51and4.07µm respectively (to be
comparedto 3.52and4.11µm). Inthiscasethesextupoles
have to be moved on average by 0.8and1.2mm respec-
tively.
Usingtheminimumoftheverticalbeamsizeonecanob-
tain slightlybetterresults: 3.50and4.06µmrespectively.
4 CONCLUSION
Unfortunately for this method, the general steering algo-
rithminPEP-IIisgoodenoughto obtainsmall dispersions
and small vertical beam sizes. Moving sextupole pairs
can decrease the vertical beam size only by about 0.1µm,
which is of the order of two to three percent. Using this
method on the real machine is made complicated by two
things: In the machine closed orbit bumpshave to be used
insteadofmovingthemagnetsthemselvesanditisnotvery
easy to optimize on the vertical beam size at the IP itself
which seemsto bethe methodto beused givingthe differ-
encein locationoftheminima.
Neverthelessthemethodmightbeusefulontherealma-
chine as the dispersion correction in the steering package
currentlyworksnotverywell. Thismethodmightbefaster
andmoreefficient. Thiswill bestudied.
5 REFERENCES
[1] Y. Cai et al.: LEGO: A Modular Accelerator Design Code,
Proceedings of the 17th IEEE Particle Accelerator Confer-
ence, Vancouver, Canada, 1997.[2] Y. Cai: Simulation of Synchrotron Radiation in an Electr on
StorageRing,Proceedingsofthe15thAdvancedICFABeam
Dynamics Workshop on Quantum Aspects in Beam Physics,
1998. SLAC-PUB-7793 |
arXiv:physics/0008217v1 [physics.acc-ph] 23 Aug 2000SNS FRONT ENDDIAGNOSTICS∗
L.Doolittle,T.Goulding,D. Oshatz,A. Ratti, J.Staples,
E. O.Lawrence Berkeley NationalLaboratory,Berkeley,CA 9 4720,USA
Abstract
TheFrontEndoftheSpallationNeutronSource(SNS)ex-
tends from the Ion Source (IS), through a 65keV LEBT, a
402.5MHzRFQ,a2.5MeVMEBT,endingattheentrance
to the DTL. The diagnostics suite in this space includes
stripline beam position and phase monitors (BPM), toroid
beam current monitors (BCM), and an emittance scanner.
Provisionisincludedforbeamprofilemeasurement,either
gasfluorescence,laser-basedphotodissociation,oracraw l-
ingwire. Mechanicalandelectricaldesignandprototyping
of BPM and BCM subsystems are proceeding. Significant
efforthasbeendevotedtopackagingthediagnosticdevices
in minimal space. Close ties are maintained to the rest of
the SNS effort, to ensure long term compatibility of inter-
facesandinfactsharesomedesignworkandconstruction.
Thedataacquisition,digitalprocessing,andcontrolsyst em
interface needs for the BPM, BCM, and LEBT diagnostic
are similar, and we are committed to using an architecture
commonwiththerest oftheSNS collaboration.
1 INTRODUCTION
The SNS Front End consists of an H−Ion Source,
Low Energy Beam Transport (LEBT), a Radio Frequency
Quadrupole (RFQ) with 65keV injection energy and
2.5MeV output energy, and a 3.6m long Medium En-
ergyBeam Transport(MEBT),that matchesandchopsthe
2.5MeV H−beam before injection into the remainder of
the SNS linac[1].
The extremely compact 65keV LEBT leaves no room
for conventional diagnostics. Only one measurement of
beampropertiesremains,asplit-collectorcurrentmeasur e-
ment, that goes under the name “LEBT Diagnostic.” No
beamdiagnosticdevicesatall areincludedinthe RFQ.
Table 1 shows the instruments that will be assembled
on the 2.5MeV, 3.6m long MEBT. Figure 1 shows their
placement along the beam line. This paper will discuss
eachofthese instrumentsinturn.
2 LEBT DIAGNOSTIC
Beam current will be monitored on a four-way split elec-
trode (LEBT chopper target), placed at the exit of the
LEBT. Thecurrentbalancebetweenelectrodesatdifferent
timesduringthechoppercyclecanbeusedtoqualitatively
determineoffsetsfromthe RFQ axis[2]. With appropriate
manipulationofthebeamsteering,someinformationmight
be gainedaboutbeamsize.
∗Work supported by the Director, Office of Science, Office of Ba -
sic Energy Sciences, of the U.S. Department of Energy under C ontract
No. DE-AC03-76SF00098Table1: MEBT instrumentationsummary
Device Qty. zextent Measures
LEBT 1 0 mm centering
BPM 6106 mm* position,phase
BCM 2 59 mm current
Profile 5 51 mm xandyprofile
Emittance 12×51 mm x-x′andy-y′
*all but23mmoverlapswithquadrupolemagnet
Figure2: StriplineBPM assembly
3 BEAM POSITION MONITORS
BPMs will be installed in six locations in the MEBT,
spaced roughly every 90◦of betatron phase advance [3].
The BPMs will primarily be used as a secondary standard
for restoring the beam, where the primary standard is the
nullpointforquadrupolesteering. TheBPMsalsoserveto
measure the trajectory of systematically deflected bunches
(thispatternisrelatedtothebetatronoscillationofpart icles
inthebunch,butdiffersduetospacechargeeffects)andto
providebeamphaseinformationfortuningthelongitudinal
optics by way of the rebuncher cavities. Thus, reliability,
repeatability and linearity are more important than initia l
zeroset.
To minimizethe amountof beamlinespace dedicatedto
BPMs, the stripsare relativelynarrow(22◦) so as to fit be-
tweenquadrupolepoletips.
The electrical processing will use the 805MHz signal
component, since the fundamental 402.5MHz signal will
be contaminated by fringe fields from nearby 402.5MHz
rebunchercavities.
Sincethisisalowvelocitybeam( β= 0.073)wire-based
calibrationwill not givea propercalibrationcurve. A sim-
plenumericalmodelwillconvertelectricalsignalstrengt hs
to linearizedposition.
Measurementsofa prototypeshowtheexpectedshorted
50Ωstriplinebehavior,withnospuriousresonancesbelow
8GHz. Construction of all required BPMs is nearly com-
plete.BPM
BPM
BPM
BPM
BPM
BPMProfile
Profile
Profile
ProfileToroid
ToroidProfile
3.6 m
Figure1: OverviewofMEBT.
Electronics to measure longitudinal bunch information
now uses the signals coming from the BPM pickup, to
avoid the need for separate beamline hardware (see sec-
tion 6 below). For relative phase measurement with a sin-
gle BPM, thisisfairlyeasy. Forabsolutemeasurementbe-
tweenpairsofBPM’s,thisrequiresextraattentiontocable s
and calibration. All BPMs are installed in the same direc-
tional orientation, so those phase signals can be compared
with noadditionalsensorcalibrationterm.
Figure3: Currenttransformerassembly
4 BEAM CURRENTMONITORS
The MEBT beamline includes two current transformersto
measurebeamcurrent,onebeforeandoneaftertheMEBT
chopper target. These will be used to measure the cur-
rentwaveformsthataregeneratedbytheLEBTandMEBT
chopping processes. They also provide the first calibrated
measureofbeamcurrentandintegratedbeamcharge.
The toroidal transformer is nearly a standard Bergoz
FCT-082-50:1 [6], using a high permeability core to keep
droop to a minimum during the 0.65 µs chopped beam
pulse. These transformers have a measured droop of
0.06%/ µs.
These devices are mounted 37mm from the main fo-cussing magnet pole tips (1.16 diameter), leading to con-
cerns that the DC magnet fringe field would saturate por-
tions of the toroid core. The result would be a increased
droop rate, and sensitivity of the measurement results on
quadrupole drive current. Tests have shown this is indeed
the case: with the quadrupole running near its design gra-
dient (38T/m), the current transformer’s droop approxi-
mately doubles. The design shown above, however, in-
cludes a 3.2mm thick shield made from mild steel. With
thisfieldclampinserted,thedroopofthetransformerisnot
measurablyaffectedbyquadrupoleoperation.
Thetransformershavebeendelivered,theremainingme-
chanicalbeamlinepartshavebeenfabricated,andassembly
isunderway.
Figure 4: Wire Scanner concept, with provisions for RGF
orLP device.
5 BEAM PROFILEMONITORS
MeasurementsofbeamprofileintheMEBTareconsidered
essential tocheckthat thetransversebeamopticsisbehav-
ing as intended. In final operation of SNS, these measure-
ments should be made without disturbing the operation of
the machine. The two leading contenders to provide such
functionalityareResidualGasFluorescence(RGF)[4]and
Laser Photodissociation (LP) [5]. Unfortunately, both of
these techniques are considered experimental at this time,
andcannotbecountedontodeliverreliableprofiledatafor
beamlinecommissioningin2002.The current plan is to provide conventional crawling
wire scanners, with co-located optical ports, for eventual
addition of an RGF or LP monitor. The wire scannerswill
be used first to commission the beamline, and then to test
andcommissionanopticaltechnique.
Brookhaven Nat. Lab. will provide the flange-mounted
wire scanners for the whole SNS project, including the
MEBT. Thatdesignwillbecustomizedtofitthetightspace
allotment. Unlike the final optical devices, the wire scan-
nerisintendedtoworkonlywhenthebeamrunsatreduced
dutyfactor. Ratherthanthefull1mspulseat60Hz,weex-
pect 100 µm wire to survive 100 µs pulses at 6Hz (1% of
the nominal 6% duty factor). This is adequate to commis-
sion,but notoperate,theaccelerator.
The beamboxis designedto accepta wire scanner,plus
two pairs (in and out, xandy) off/2.8 windows on the
beam, and a gas jet that could be part of the fluorescence
experiments. Spaceissufficientlytightthatthebeamboxes
will likely be manufacturedas part of neighboringcompo-
nents (chopper electrodes, chopper target, and emittance
scanner). Wehopethatthisopticalaccesswillbesufficient
to deployafinal non-invasiveprofilemeasurement.
6 EMITTANCE
The 1999 SNS Beam Instrumentation Workshop [7]
strongly recommended that a way be found to measure
the emittance of the beam as it leaves the MEBT on its
way to the DTL. By subsuming phase measurement into
the BPM pickup system, a slit and multisegment collector
assembly (at 51mm each) could be fit into the beamline.
Note that the drift space between these devices contains
onefocussingquadrupole,oneBPM, andoneprofilemon-
itor. The engineeringdesign of this subsystem has started.
Theslit cannotabsorbthefullbeampower.
For each position of the movable slit, all the beam di-
vergenceinformationis recordedsimultaneouslyby a seg-
mented collector assembly. Each segment has its own
front-endelectronicsequipment,consistingofachargeam -
plifier andsample-and-hold.
Table 2 shows a plausible parameter set for the MEBT
emittance device. With these parameters, the error in re-
constructed emittance and the error of the reconstructed
Twissbetafunctionaretypicallyontheorderof2%orless.
Table 2: MEBTEmittanceDeviceparameterset
Slit width 0.2mm(7.9mils)
Totalslit movementrange 5.0mm
Slit positionsformeasurement 50
Collectorsegments 64
Collectorsize 30mm(square)
Collectorcenter-centerspacing 0.5mm
Slit-collectorspacing 205mm7 SIGNALPROCESSING ELECTRONICS
The signal processing needs for the BPM, LEBT diag-
nostic, and BCM are similar, both among themselves and
with their cousins in the larger SNS project. Collaborative
andcompetitivedevelopmentofelectronicsisunderwayat
LBNL, LANL,andBNL.
Most of the relevant information can be collected with
a moderate rate (34-68 MHz), moderate resolution (12-14
bit) digitization of a suitably conditioned signal. We are
investigating digitization and signal processing platfor ms
that can reliablyandcost-effectivelydealwith this volum e
of data, and interact with the Global Controls (EPICS).
That platform would then be used for all BPM, BCM, and
LEBTsignalhandling,andpossiblyotheruseswithinSNS.
Each instrument has unique analog signal conditioning
requirements. ForBPM processing,at least onechannelof
“vectorvoltmeter”isrequiredtoprocessbeamphaseinfor-
mation. Thisisexpectedtofunctionwithamixeranddirect
IFsampling. Suchprocessingcanalsobeusedforposition
readout. Log amp circuitry is also under considerationfor
the actual signal strength measurements: it has a dynamic
rangeadvantageoverordinarylinearanalogprocessing.
TheBCMsignalconditioningrequirementsforthefront
end are actually quite modest, essentially a 40dB ampli-
fier and filter. This simplicity has to be balanced against
compatibility with the future BCM signal conditioningfor
the SNS ring, where the signal has an additional 60 dB of
dynamicrange,andturn-turndifferencesareimportant[8] .
8 ACKNOWLEDGEMENTS
The authors would like to thank all our collaborators at
LANL, BNL, andORNL fortheir variousrolesin keeping
this project moving. Contributions from Tom Shea, John
Power, Marty Kesselman, Pete Cameron, and Bob Shafer
havebeenparticularlyhelpful.
9 REFERENCES
[1] R. Keller et al., “Status of the SNS Front-End Systems,” 7 th
European ParticleAccelerator Conference, Vienna,2000.
[2] J.W. Staples et al., “The SNS Four-Phase LEBT Chopper,”
Proceedings of the 1999 Particle Accelerator Conference,
New York, 1999
[3] J. Staples et al., “Design of the SNS MEBT,” paper MOD18
at thisconference.
[4] J. Kamperschroer, “Initial Operation of the LEDA Beam-
InducedFluorescenceDiagnostic,”9thBeamInstrumentati on
Workshop, Cambridge, 2000
[5] R. Shafer, “Laser Diagnostic for High Current H−Beams,”
19thInternational LinacConference, Chicago, Illinois,1 998.
[6] Bergoz Instrumentation, Crozet, France,http://www.b ergoz.-
com/dfct.htm
[7] SNSBeam Instrumentation Workshop, Berkeley, 1999
[8] M.Kesselmanetal.,“SNSProject-WideBeamCurrentMon-
itors,”7thEuropeanParticleAcceleratorConference,Vie nna,
2000. |
arXiv:physics/0008218v1 [physics.acc-ph] 23 Aug 2000HIGHPOWERMODELFABRICATIONOF BIPERIODICL-SUPPORT
DISK-AND-WASHERSTRUCTURE
H. Ao∗, Y. Iwashita,T.Shirai and A. Noda,
Accelerator Laboratory,NSRF, ICR, KyotoUniv.,
M. Inoue,Research ReactorInstitute,KyotoUniv.,
T.Kawakita,K. Ohkuboand K. Nakanishi,MitsubishiHeavy In dustries,Ltd.
Abstract
The high power test model of biperiodic L-support disk-
and-washer was fabricated. Among some trouble in the
fabrication, the main one was a vacuum leak in a brazing
process. The repair test of the leak showed a good result;
fourunitswererecoveredoutoffiveleakunits(recoverrate
80%). While an acceleratingmodefrequencywastunedat
anoperatingfrequencyof2857MHzbysqueezingmethod,
acouplingmodefrequencyof2847MHzandthe3.4%field
flatness(peakto peakratio)wereachieved.
1 INTRODUCTION
A disk-and-washer(DAW)structureisdevelopedasanad-
vanced structure of a coupled cell cavity. The DAW struc-
ture has a high shunt impedance in the high βregion and
good vacuumproperties. A couplingconstant of the DAW
is much larger than that of a side-couple cavity, which
bringseasyfrequencytuningandlargetoleranceinfabrica -
tion. The large coupling does not require a frequencytun-
ingforeachcell. Onlytheaveragefrequencyofentirecells
must be controlled. Electric field distribution can be ad-
justed by slight movementof the washer position, because
the distribution depends on the coupling constant balance
betweencells.
Highermodeaccelerationcomplicatesmodeanalysisfor
cavitydesign. TheDAWrequiresafewsupportswhichdis-
turbanaxialsymmetry,sothattheelectromagneticfieldof
the DAW is more complicated than that of a side couple
structure. Nevertheless, the feature of the DAW is attrac-
tiveforahigh βregionaccelerator. ThisstudyoftheDAW
introducedthebiperiodicL-supportstructurewhoseadvan -
tagewasproposedbyacalculationstudy.[1]Thisstudyin-
vestigates its feasibility through the test model fabricat ion
andthemeasurement.
A coaxialbridgecouplerconnectstwo acceleratortubes
of 1.2m length. The total lengthis about 2.8m. The oper-
atingfrequencyofthistestmodelis2857MHzwhichisthe
same as that of the disc-loaded linac in our facility. These
specifications were designed so that a high-powertest can
be carried out with the existing beam line and RF sources
in future.
This paper describes the fabrication process of two ac-
celerationtubes(No.1andNo.2)andthetuningoperation.
∗Present address: High Energy Accelerator Research Organiz ation
(KEK),1-1 Oho,Tsukuba, Ibaraki, 305-0801, Japan2 FABRICATION
2.1 Fabricationprocessandstructure
The DAW structure is fabricated by three brazing
steps.(Seefig.1)
Figure 1: Fabrication steps of the biperiodic L-support
DAW
The wall loss on the metal surface requires water paths
in the washer and support. Cooling water enters from one
side of the supports and goes out from another side. Two
roughprocessedhalfwasherswerebrazedtogether(step1),
andthenit wasmachinedto thefinaldimensions.
The frequency and the field flatness of each unit were
measured at the stage of step2 with aluminum units and
terminating plates, which are used for the cold model test.
These propertieswere optimized by fine correctionsbased
on the measurement[ ?]. A detail of the measurement and
optimizationaredescribesin §3.
Theacceleratortubeisinstalledintoawaterjacketmade
ofSUS, whichkeepsthestrength.
2.2 Brazing
InthisDAWstructure,somebrazedareasseparatevacuum
region from water. The reliability of the brazing is impor-
tant for the DAW. The test model was fabricated in five
times for the physical and technical R&D (from 1st to 5th
model generations). Following sections describe the prob-
lemsandactionstakenthroughoutthisstudy.
LeakThefabricatedmodelwasinspectedagainstvac-
uum leak in every brazing step. Although only the wash-
ers that passed the leak-test at step1 were used in step2,
vacuum leaks arose in washer parts. Rates of the leak are
summarizedinTable1.
Althoughthe 1st and5thmodelshadnoleak, the 2ndto
4th models exhibited bad yield (50 to 67%). The reason
is considered as follows. When fifty washers were brazed
whichweremainlyusedfromthe2ndto4thmodels,avac-
uum leak arose on half of them. These washers were re-
paired by putting an additional brazing filler metal on the
washer surface and brazing again. Although the leak was
repaired at the time, the fine machining, as mentioned in
§2.1, removed the surface and the brazed area became toModelgeneration Leak/Total %
1st 0/6 0
2nd 5/8 63
3rd 6/12 50
4th 8/12 67
5th 0/12 0
Table1: Leak-testresultsafterbrazingSTEP2
thin. Eventhesewasherpartspassedtheleak-test. Heating
in step2,however,causedthevacuumleakagain.
The original reason for the washer repair is considered
that theywaited aboutone year afterthe machiningtill the
step1brazing. Itcausedoxidationonthesurfaceofbrazing
area,whichdegradedthebrazingquality.
The 1st and 5th model had no idle time, hence there is
novacuumleak.
Leak repairing The five leak unitswere tested for re-
pair, and the four of them were successful at the stage of
step2. The way of repairing was as follows. After paste
mixed with a brazing metal powder was injected into a
cooling path, these units were heated up again. Because
the paste metal must not fill up the cooling path, viscos-
ity of the paste was adjusted by additional acetone, and
compressed air was blown after inserting the paste. The
repaired washer was set at lower side in a heat up process
forthebrazingmetalnottoflow out.
Figure2: Repairingavacuumleakbypasteofbrazingfiller
metal.
Although three units had no vacuum leak after an en-
tire brazing (step3), one unit caused a small vacuum leak
again. The leak rate was 5.0×10−9[torrl/s]. Because the
brazing metal for the repair has melting temperatureat the
middlebetweenstep2 andstep3,the marginofthe melting
temperature was ensured. Endurance of the repaired area,
however,seemsnotenough.
This structure requiresthree steps brazing, so that some
improvements are required in the washer fabrication. It is
necessarytoreducethewaitingtimefromthefinemachin-
ingtothebrazingprocess. Asforabasicimprovement,the
innerstructurehavemorestrongbrazedjunction.
3 MEASUREMENTS OFPROPERTIES
Regularunits
Frequency The accelerating and coupling mode fre-
quenciesweremeasuredatstep2. Figure3showsthemea-
surementresultsasa scatterdiagram.
Figure3: Singleunitmeasurementsoffrequencies.The frequency errors are small within the same model
generations. It means that the fabrication process keeps
good reproducibility. It is important to examine the de-
pendence of the frequency on the number of units for the
designofamulti-cell-DAW.Manyaluminummodelswere
used to optimize dimensions[3]. Figure 4 shows the fre-
quency convergence up to 12 units of the optimized test
model.
Figure4: Unitnumberdependenceontheacceleratingand
couplingmodefrequencies.
The tolerance of the accelerating mode frequency is
about1MHzatthisstage. Becausethesqueezemethodcan
raisetheacceleratingmodefrequencyuptoabout1.5MHz.
The couplingmode frequencyis about 10MHz lower than
the operating frequency 2857MHz from this result. This
10MHz erroris consideredastolerable,because the DAW
hasalargecouplingconstant,andthusthemodeseparation
isabout30MHz.
Electric field distribution The electric field distribu-
tiondependsonthedisplacementofthewasher. Thisstudy
chosethefollowingwaytokeeptheprecisionofthewasher
position.
1. A coordinate measuring machine measures the sup-
port(fixedina flame)positionsafterstep1brazing.
2. Thesupportsocketismachinedonthewashersurface
so thatthewasher centerfitsonthebeamline.
3. Acarbonrodholdsthecenterlinethroughouttheheat-
upprocessinthestep2brazing.
The tolerance of the washer position is ±0.1mm from a
concentriccenteranda parallelposition.
Two sources of assembling error are considered. One is
the displacement in the heat-up process. The other is the
miss handling after brazing. The carbon rod must be re-
movedwithoutwasherdisplacement. Thisworkwassome-
timesnoteasy. Thisisbecauseathermalexpansionofcop-
per might tighten the clearance between the beam-hole of
the washerandthecarbonrod.
Theholdingschemeofcarbonsupportsshouldbeeasyto
removeafterbrazing. These assemblingerrorinthe beam-
axis direction can be easily corrected. It is not easy on
the transverse direction. The assembling tolerance and the
loosecontactshouldbecompromised.
In the DAW, the flatness of the electric field distribution
can be corrected without changingthe frequencies. Figure
5 shows the example of the field correction. The field dis-
tributionwascheckedbeforewholebrazing(step3)assem-
bled temporarily. A coordinate measuring machine mea-
sured the washer positions, and then they were corrected
by hand. The final field distribution is also shown in Fig.
5.Figure 5: Correction of the electric field distribution and
the finaldistributionsofNo.1andNo.2acceleratingtubes.
4 CONCLUSION
The feasibility of the biperiodic L-support DAW was con-
firmedatS-bandastheresultofthisstudy. Thefabrication
procedurewas established;machining,brazingandassem-
bling. We took care of following points. The unit num-
ber dependence on the frequency must be considered dur-
ing the frequency optimization. This optimization would
need some test model measurements. The brazing pro-
cess is most important in the fabrication. The reliability
of the washer should be kept for the three brazing steps.
More R&D’s are needed for managements of brazing and
improvements of the inner structure of the washer. A ma-
chining process is important for frequency control. The
finemachiningofNCturningcenterachievedthesufficient
precision for the reproduction. The electric field distribu -
tioncanbecorrectedbyhandafterassemblingbasedonthe
measurement data, so that it is not necessary to keep the
fine precision throughout the fabrication. This correction
has no influence on the frequencies. The squeeze has the
1.5MHz tuning range. This tolerance would be enough to
optimize the cavity dimensions considering the frequency
convergence.
5 REFERENCES
[1] Y. Iwashita, ”Disk-and-washer structure withbiperiod ic sup-
port”, Nucl. Instrm.and Meth. inPhys.Res. A348(1994)15-
33
[2] H. Ao, et al.,”FABRICATION OF DISK-AND-WASHER
CAVITY”, Proc. First Asian Particle Accelerator Conf.,
KEK,Japan (1999) p.187.
[3] H. Ao, et al., ”Model Test of Biperiodic L-support
Disk-and-Washer Linac Structure”, Jpn. J. Appl. Phys.
Vol.39(2000)651-656 Part1.No.2A |
arXiv:physics/0008219v1 [physics.flu-dyn] 24 Aug 2000Integral methods for shallow free-surface
flows with separation
Shinya Watanabe
Dept. of Mathematical Sciences, Ibaraki University, 310-8 512, Mito, Japan
Vachtang Putkaradze
Dept. of Mathematics & Statistics, University of New Mexico , Albuquerque, NM 87131-1141, USA
Tomas Bohr
Dept. of Physics, The Technical University of Denmark, Kgs. Lyngby, DK-2800, Denmark
Submitted on August 16, 2000.
Abstract
We study laminar thin film flows with large distortions in the f ree surface using the
method of averaging across the flow. Two concrete problems ar e studied: the circular
hydraulic jump and the flow down an inclined plane. For the cir cular hydraulic jump
our method is able to handle an internal eddy and separated flo w. Assuming a variable
radial velocity profile like in Karman-Pohlhausen’s method , we obtain a system of
two ordinary differential equations for stationary states t hat can smoothly go through
the jump where previous studies encountered a singularity. Solutions of the system
are in good agreement with experiments. For the flow down an in clined plane we
take a similar approach and derive a simple model in which the velocity profile is not
restricted to a parabolic or self-similar form. Two types of solutions with large surface
distortions are found: solitary, kink-like propagating fr onts, obtained when the flow
rate is suddenly changed, and stationary jumps, obtained, e .g., behind a sluice gate.
We then include time-dependence in the model to study stabil ity of these waves. This
allows us to distinguish between sub- and supercritical flow s by calculating dispersion
relations for wavelengths of the order of the width of the lay er.
11 Introduction
In this paper we develop a simple quantitative method to desc ribe flows with a free surface
which can undergo large distortions. Our method is capable o f handling flows whose velocity
profile may become far from parabolic — even including separa tion and regions of reverse
flow. We are concerned with the case when the fluid layer is thin . For low Reynolds number
flows the lubrication approximation can be used with great su ccess (see e.g. [15]). For
high Reynolds number flows without separation an inviscid ap proximation and the shallow
water equations [43] are widely used. For moderate Reynolds numbers where these limiting
approximations are invalid it is important to take both iner tial and viscous effects into
account in a consistent way, and yet one would like to keep the model simple enough to
be tractable. In this paper we show that integral methods, li ke the ones developed by von
Karman, can handle a class of such problems successfully. To be concrete we develop the
method in the context of two physical examples: the circular hydraulic jump and the flow
down an inclined plane . Both geometries support jump- or kink-like solutions with abrupt
changes in the surface shape and internal velocity profiles. Analytical solutions for such
flows are extremely difficult to obtain, and simple approximat e theories that capture the
phenomena are invaluable.
The two flows are studied in separate sections, and an introdu ction is provided in the
beginning of Secs. 2 and 3, respectively. In Sec. 2 we develop the theory for the circular
hydraulic jump. We first study the boundary layer approximat ion to the full Navier-Stokes
equations, and reduce it to a simple set of equations by avera ging over the thickness. Sta-
tionary solutions are obtained by solving a two-point bound ary value problem for a system of
only two ordinary differential equations. The solution is co mpared to previous experiments,
showing good agreement. Taking advantage of the simplicity of the reduced equations, it
is possible to obtain analytic approximations for the stati onary solution. Two “outer” so-
lutions connected by an “inner” transition region are studi ed separately and we obtain a
relationship analogous to the shock condition in the classi cal shock theory, but within our
viscous model.
The flow down an inclined plane is then studied in Sec. 3. We use the same strategy
as in Sec. 2 to derive a simple model for the two-dimensional fl ow. One family of solutions
found in this model is kink-like traveling wave solutions th at occur, e.g., when the flow rate
is suddenly changed. Their velocity profiles along the incli ned plane are found to stay close
to parabolic even when a variable profile is assumed. There is another family of solutions
with a sudden change in the surface that would correspond to t he circular hydraulic jump in
case of the radial geometry. These solutions can be interpre ted as the stationary hydraulic
jump, created behind a sluice gate in a river, even though tur bulence is not included in the
model. The flow downstream of the jump approaches a simple sta tionary flow, but the flow
upstream is an expanding flow with a linear growth in thicknes s. The velocity profile departs
considerably from parabolic near the jump.
It is not easy to analyze the stability of the solutions with j umps obtained in Secs. 2
and 3, even in the linear geometry. Instead, in Sec. 4, we incl ude time-dependence in the
2models and study the dispersion relation for the stationary flow with constant thickness. A
well-established concept in the inviscid theory is to class ify flows as super- and subcritical
when the thickness is small and large, respectively. They do not have obvious counterparts,
however, when viscosity is included. By looking carefully a t the dispersion relation in the
long and medium wave regime, we can classify the stationary fl ows into these two categories
in our viscous model. The model shows spurious divergencies in the short wavelength region
which we do not know how to overcome at present. This makes the model unsuited for
direct time-dependent simulations. A short paper describi ng some of the main results has
appeared earlier [8].
3Figure 1: (a) Schematic view of the circular hydraulic jump. (b) Snapshot of a nearly
perfect, stationary and circular hydraulic jump. Ethylene -glycol is used.
2 The circular hydraulic jump
2.1 Introduction to the problem
When a jet of fluid hits a flat horizontal surface, the fluid spre ads out radially in a thin,
rapidly flowing layer. At a certain distance from the jet a sud den thickening of the flow
takes place, which is called the circular hydraulic jump. Th is is commonly seen, e.g., in the
kitchen sink, but it is also important as a coating flow and in j et-cooling of a heated surface
[29]. In these practical flows with typically high Reynolds n umbers, disturbances often make
the jump non-stationary and distorted. In controlled labor atory experiments corresponding
to a more moderate Reynolds number, an apparently stationar y, radially-symmetric flow can
be achieved. Such experiments was carried out by C. Ellegaar det al. and the results have
been published elsewhere [6, 7, 16, 17, 30]. We thank them for providing us with data and
pictures. A schematic view and a video image of the circular j ump are shown in Fig. 1.
In these experiments the hydraulic jump is formed on a flat dis c with a circular rim.
The rim height dcan be varied, and is an important control parameter. Since t he rim is
located far from the impinging jet with the diameter of the di sc around 36cm, it does not
affect the jump except that it changes the height of the fluid la yerhextexterior to the jump.
The jump still forms even when d= 0, but a larger dmakeshextlarger and, therefore, the
jump stronger. Typically, hextexceedsdby 1-2mm. The surface profiles for varying dare
shown in Fig. 2. An interesting transition in the flow structu re has been observed [6, 7] as
dis varied. For d= 0, it was noticed before [40, 14, 23, 32, 33] that the jump con tains an
eddy on the bottom, called a separation bubble , whose inner edge is located very close to the
position of the abrupt change on the surface, as illustrated in Fig. 3(a). Such a hydraulic
jump is referred to as a type I jump. While dremains small, this jump is stable, but as d
is increased further, a wave-breaking transition occurs [6, 7] which results in a new state of
the flow. In this type II state, the flow has an additional eddy, called a roller or asurfing
4Figure 2: Height profiles h(r) for different values of the external height hext. (The rim
heightdis controlled but not shown.) The height h(r) approaches hextfor large values of r.
Parameters are: the flow rate Q= 27[mℓ/s] and viscosity ν= 7.6×10−6[m2/s], corresponding
to the characteristic scales: radius r∗= 2.8[cm], height h∗= 1.4[mm], and radial velocity
u∗= 12[cm/s]. Figure taken from [6].
Figure 3: A schematic picture showing two observed flow patte rns: (a) type I flow, with a
separation bubble, which occurs for small d, and (b) type II flow, with an additional roller
eddy, for large d. Transitions between these states occur at a certain d, with surprisingly
small hysteresis.
wave, just under the surface as shown in Fig. 3(b).1This state resembles a broken wave in
the ocean, but is still apparently laminar. On reducing d, the type I pattern reappears, and
there is almost no hysteresis associated with this transiti on. The transition from type I to
II often leads also to breaking of the radial symmetry. An int riguing set of polygonal jumps
[16, 17] are created rather than the circular one. In this pap er we shall concentrate on the
type I flow which already poses considerable difficulties. We h ope to be able to generalize
our approach in the future to be able to handle the transition to the type II flow.
Considering how simple and common the circular hydraulic ju mp appears to be, it is
surprising that a satisfactory systematic theory does not e xist. The approach considered as
“the standard” for the study of hydraulic jumps is to combine the inviscid shallow water
equation with Rayleigh’s shocks [13]. In the beginning of th e century Lord Rayleigh treated
[37] a discontinuity in a one-dimensional linear flow geomet ry. Such a structure is usually
called a river bore if it is moving and a hydraulic jump if it is stationary and is created due
to, e.g., variations in the river bed. His approach was based upon the analogy between the
1Ifdis increased even further, the jump “closes” as seen in Fig. 2 .
5shallow water theory and gas theory [43]. He assumed that, ac ross such a shock, the mass
and momentum flux are conserved but not the energy flux.
In a coordinate system moving with the shock, the flow velocit yv1and heighth1upstream
of the jump as well as v2andh2downstream of the jump are taken to be positive constant
values. Then, conservation of mass flux Qacross the jump is given by
v1h1=v2h2=Q. (1)
Conservation of momentum flux is
h1/parenleftbigg
v2
1+1
2gh1/parenrightbigg
=h2/parenleftbigg
v2
2+1
2gh2/parenrightbigg
. (2)
These shock conditions lead to the relation
h2
h1=1
2(−1 +/radicalBig
1 + 8F2
1) =2
−1 +/radicalBig
1 + 8F2
2(3)
whereF1=/radicalBig
v2
1/gh1= (hc/h1)3/2is the upstream Froude number ,F2=/radicalBig
v2
2/gh2=
(hc/h2)3/2thedownstream Froude number , and
hc= (Q2/g)1/3(4)
is called the critical height. It is easy to see that hcis always between h1andh2, and that
F1>1> F2ifh1< h c< h2, andF1<1< F2ifh1> h c> h2. In other words the jump
connects a supercritical flow withF >1 on the shallower side ( h<h c) to a subcritical flow
withF <1 on the deeper side ( h>h c). Since the Froude number measures the ratio of the
fluid velocity vand the velocity of linear surface waves√gh, it means that, in the moving
frame, the flow moves more rapidly than the surface waves on th e shallower side, but moves
slower on the deeper side — in a precise analogy with the gas th eory [43, 39]. Further, it is
found that the upstream h1must be supercritical by considering the change in the energ y
flux across the jump [43]:
Qe2−Qe1=−gQ(h2−h1)3
8πh1h2(5)
whereQedenotes the energy flux. Since the energy must be dissipated t hrough the jump,
i.e.Qe2−Qe1<0, rather than generated, it is required that h1< h2. The origin of the
dissipation is usually attributed to the turbulent motions at the discontinuity and surface
waves carrying energy away from it.
It is possible to apply this theory, combined with an assumpt ion of the potential flow,
for describing the circular hydraulic jump. However, it lea ds to incorrect estimates [42, 5]
of the radius of the jump Rj. Most notably, Rjis predicted to be sensitive to the radius of
the impinging jet which should be greatly influenced by radiu s and height of the inlet nozzle
where liquid comes out. In experiments [42, 5] such a strong t endency was not observed.
Instead, it has been found that Rjscales with the flow rate Qwith a certain power, and
6it supports a model in which viscosity plays an important rol e. Watson [42] constructed
a model of the flow consisting of the inviscid and viscid regim es, and solved the viscid
part assuming a similarity profile. By connecting to the spec ified external height hextvia a
Rayleigh shock, he obtained a prediction for the radius of th e jump which compares favorably
with the measurement [42, 5], as we explain in Sec. 2.5. In his model the viscous layer starts
from the stagnation point at r= 0 on the plate and quickly reaches the surface at a small r.
There is a fairly long stretch from this rtoRjin which the flow is fully viscous.2Thus, one
could neglect the inviscid region and assume a fully viscid fl ow everywhere in order to derive
a simpler model. This assumption was made by Kurihara [25] an d Tani [40] who started
from the boundary layer equations developed by Prandtl [34, 38]. They took an average of
the equations over the thickness, assuming also a similarit y velocity profile. It resulted in
a single ordinary differential equation for the stationary j ump. This theory was elaborated
in [5] who realized that the flow outside the jump would natura lly lead to a singularity at a
larger. By identifying this singularity with the outflow over the ri m of the plate, the flow
outside the jump could be uniquely specified. By introducing a Rayleigh shock, the jump
radius and its parameter dependence was calculated and comp ared to measurements. The
model predicted the observed Rjreasonably well, as we review in Secs. 2.2–2.5.
Obviously, treating the jump as a discontinuity provides us with no information on the
internal structure of the jump region such as the type I to II t ransition of the flow patterns.
It also seems inconsistent to assume a Rayleigh shock when vi scous loss occurs in the whole
domain. Why do we assume an extra energy loss at the “jump” whe re the flow is stationary
and apparently laminar? It seems possible to attribute the e nergy dissipation entirely to
laminar viscous forces, and to construct a viscous theory wh ich produces a smooth but kink-
like surface shape without the need for a discontinuity. Nev ertheless, such a description must
overcome a difficulty arising from the Goldstein-type singul arity [19, 26] of the boundary
layer equations in the vicinity of separation points. This s ingularity is thought to be an
artificial one created by truncation of higher derivatives f rom the Navier-Stokes equations.
It also arises in the “usual” boundary layer situation where a high Reynolds number flow
passes a body, e.g., a wing. In such cases inviscid-viscid in teraction is taken into account
in order to resolve the singularity in a technique called the inverse method [10]. In our
situation, however, there is no inviscid flow outside the lay er. In Secs. 2.6–2.7 we propose a
way to resolve the trouble in the following manner. We first in clude an additional degree of
freedom in the velocity profile to make it non-self-similar, just like in the Karman-Pohlhausen
method [20] for the usual boundary layer theory. To describe the evolution, in r, of this free
parameter, we couple the layer thickness to the pressure by a ssuming hydrostatic pressure.
This serves as an alternative to the inverse method in the abs ence of a potential external
flow. The resulting model for a stationary solution is two cou pled ordinary differential
equations, and reproduces the type I flow with a separation bu bble — the one shown in
Fig. 3(a). Comparison with the experiment is made in Sec. 2.7 . It is possible to approximate
2This assumption is confirmed by recent laser-doppler measur ements of the velocity profile before the
jump [30]. Thus, the assumption made by [18, 4, 9] that the jum p occurs at the point where the growing
viscous layer touches the surface and the flow becomes fully d eveloped, is incorrect.
7analytically the stationary solution found in the model. In Sec. 2.8 the analysis is presented
separately for the regions before and after the jump (i.e. tw o “outer” solutions) and the
“inner” solution inside the jump region. An interesting obs ervation on the inner solution is
that a formal parameter µcan be introduced so that Rayleigh’s shock condition is reco vered
in the limit µ→0.
2.2 The full model
We write down the complete model to describe the circular hyd raulic jump under the assump-
tion that the flow is laminar and radially symmetric without a ny angular velocity component.
We take the radial and vertical coordinates ˜ rand ˜z, and denote the velocity components by
˜uand ˜w, respectively.3The governing equations are the continuity equation
˜u˜r+˜u
˜r+ ˜w˜z= 0 (6)
and the Navier-Stokes equations:
˜u˜t+ ˜u˜u˜r+ ˜w˜u˜z=−1
ρ˜p˜r+ν/parenleftbigg
˜u˜r˜r+1
˜r˜u˜r−˜u
˜r2+ ˜u˜z˜z/parenrightbigg
˜w˜t+ ˜u˜w˜r+ ˜w˜w˜z=−1
ρ˜p˜z−g+ν/parenleftbigg
˜w˜r˜r+1
˜r˜w˜r+ ˜w˜z˜z/parenrightbigg (7)
where subscripts denote partial differentiations such as ˜ u˜t=∂˜u/∂˜t. For the boundary
conditions we impose no-slip on the bottom:
˜u(˜z= 0) = ˜w(˜z= 0) = 0. (8)
The dynamic boundary conditions on the free surface ˜ z=˜h(˜t,˜r) are
˜p−2ρν
1 +˜h2
˜r/bracketleftBig˜h2
˜r˜u˜r+ ˜w˜z−2˜h˜r( ˜w˜r+ ˜u˜z)/bracketrightBig/vextendsingle/vextendsingle/vextendsingle
˜z=˜h=σ˜k
ν/bracketleftBig/parenleftBig˜h2
˜r−1/parenrightBig
( ˜w˜r+ ˜u˜z)−2˜h˜r(˜u˜r−˜w˜z)/bracketrightBig/vextendsingle/vextendsingle/vextendsingle
˜z=˜h= 0(9)
whereσis the coefficient of surface tension and ˜kis the mean local curvature of the free
surface. We also need to satisfy the kinematic boundary cond ition on the free surface:
˜h˜t+ ˜u˜h˜r= ˜w on ˜z=˜h(˜t,˜r). (10)
We are mostly interested in stationary solutions in this sec tion. When the flow is stationary,
we may integrate (6) over ˜ zfrom 0 to ˜h, and use (10) to obtain
˜r/integraldisplay˜h(˜r)
0˜u(˜r,˜z)d˜z=q=Q
2π. (11)
This quantity, the total mass flux Qor the mass flux per angle q, is a constant, given as a
parameter in the experiment.
3We use tildes for the dimensional variables, dependent or in dependent. Dimensionless variables will be
expressed by the same symbols but without tildes. In figures, however, we do not use tildes for simplicity.
82.3 Boundary layer approximation
Since it is a formidable task to treat the full model as it stan ds, some simplifications need to
be made. As explained in Sec. 1, the Reynolds number for the flo w of the circular hydraulic
jump is too large to justify the lubrication approximation, but is not large enough to use the
inviscid approximation. Fortunately, the flow is “thin,” i. e. runs predominantly horizontally
along the plate. Truncation of the full model by the boundary layer approximation is quite
natural in such a situation, and has indeed been used in previ ous literature [25, 40, 5]. In the
boundary layer approximation pressure, viscous, and inert ial terms in (7) are all assumed to
be of the same order, but there are only a few dominant terms in each group. For instance,
a viscous term ν˜u˜r˜ris assumed to be negligible compared to ν˜u˜z˜z. The dominant terms in
the first equation in (7) are determined in the usual manner: ˜ u˜t(if time-dependent), inertia
terms ˜u˜u˜rand ˜w˜u˜z, the pressure term ˜ p˜r/ρ, and the dominant viscous term ν˜u˜z˜z. Similarly,
from the second equation in (7) we assume the dominant balanc e between ˜p˜z/ρandg. Here,
unlike the usual boundary layer theory, we have taken into ac count the effect of gravity.
This will couple the surface height hto the pressure, and will later turn out to be crucial for
removing the singularities of the boundary layer approxima tion.
If we denote the characteristic radius and height by r∗andz∗, respectively, then the
second dominant balance requires the characteristic press ure to beρgz∗. Then, the first
balance relation requires
u∗
t∗=u2
∗
r∗=u∗w∗
z∗=ρgz∗
ρr∗=νu∗
z2∗(12)
whereu∗andw∗are typical radial and vertical velocities, respectively, andt∗is the charac-
teristic time scale. The mass flux relation (11) requires tha t
u∗r∗z∗=q. (13)
while the continuity equation (6) requires
u∗
r∗=w∗
z∗. (14)
Solving (12), (13), and (14) uniquely determines the charac teristic scales:
r∗= (q5ν−3g−1)1/8≃2.7[cm],
z∗= (qνg−1)1/4≃1.5[mm],
u∗= (qνg3)1/8≃12[cm/s],
w∗= (q−1ν3g)1/4≃6.7[mm/s],
t∗= (qν−1g−1)1/2≃0.22[s](15)
where the estimated values correspond to a typical set of par ameters used in the experiments:
ν≃0.1 cm2/s (for mixture of ethylene-glycol and water) and Q≃30 cm3/s, i.e.,q≃5
cm3/s. The values for r∗andz∗correspond well to a typical jump radius and fluid thickness
9in the experiments. Also, the predicted scaling can be exper imentally tested by, for instance,
measuring the dependence of the jump radius by changing the p arameters such as q. In [5]
evidence of the scaling and validity of the underlying assum ption was given.
We now use the characteristic scales (15), together with the pressure scale p∗=ρu2
∗, to
non-dimensionalize the full equations. From (7), we obtain
ut+uur+wuz=−pr+uzz+ǫ2/parenleftbigg
urr+1
rur−u
r2/parenrightbigg
ǫ2(wt+uwr+wwz) =−pz−1 +ǫ2wzz+ǫ4/parenleftbigg
wrr+1
rwr/parenrightbigg
,(16)
where
ǫ=z∗/r∗=/parenleftBig
q−3ν5g−1/parenrightBig1/8. (17)
Sinceǫ=z∗/r∗= 0.05 for the typical parameter values above, the assumption th at the flow
is “thin” is well satisfied, and we shall drop the terms of orde rǫ2and higher in the equations
(16). We also focus on stationary solutions in the rest of the section, and thus we obtain the
simplified equations of motion:
uur+wuz=−pr+uzz
0 =−pz−1.(18)
Correspondingly, within the error of O(ǫ2), the dynamic boundary conditions (9) are just
p|z=h=Whrr
uz|z=h= 0.(19)
Here we have introduced the Weber number
W=σz∗
ρu2∗r2∗=σ
ρgr2∗=ℓ2
2r2∗=σρ−1(q−5ν3g−3)1/4. (20)
whereℓ= (2σ/(gρ))1/2is the capillary length. For the parameter values above toge ther with
σ∼70[dyn/cm] (maximum), we estimate that W∼0.01 andℓ∼3.8[mm]. Since Wis
small, we neglect it in the study of stationary states.4,5The second equation of (18) and the
first condition of (19) with Wset to zero yield hydrostatic pressure:
p(r,z) =h(r)−z. (21)
Combining (18) and (21), we obtain the stationary boundary l ayer equations:
uur+wuz=−h′+uzz, (22)
4However, the term influences dispersion of short waves, so sh ould be included in the stability analysis
of stationary states, possibly together with the neglected terms of O(ǫ2) and higher in (18).
5The Reynolds number, defined as R=u∗z∗/ν= (q3ν−5g)1/8≈18. The Reynolds number at the nozzle
outlet is much higher, but it becomes moderate near the jump.
10where the prime denotes the derivative with respect to r. This is supplemented by the
dimensionless continuity equation:
ur+u
r+wz= 0, (23)
and mass flux condition:
r/integraldisplayh(r)
0u(r,z)dz= 1. (24)
The boundary conditions have been reduced to:
u(r,0) =w(r,0) = 0
uz|z=h(r)= 0.(25)
In addition to these conditions, boundary conditions in the radial direction also need to be
specified. We do not elaborate on them, however, since the in- and outlet conditions arise
naturally without the need for prescription when we obtain a simplified system.
The boundary layer equations (22)–(25) form a closed system and can be solved numer-
ically, but pose a difficulty when separated regions exist. Su ppose that there is a separation
point atr=rsandz= 0 on a flat plate where the skin friction uzvanishes. In its vicin-
ity one finds [38] that generic solutions of (22) develop sing ularities of the Goldstein-type
u∼√rs−r,w∼1/√rs−r. On the other hand, experiments [6] show separation and
reversed flow just behind a jump, so it is necessary to overcom e this difficulty, which is well-
known in the “usual” boundary layers around a body immersed i n a high Reynolds number
external flow. No such singularities are observed in numeric s of the full Navier-Stokes equa-
tions in that case, and thus the trouble is thought to be due to truncation of the terms
involving higher derivatives in r, i.e. the terms in (16) of the order O(ǫ2) and higher. An
attempt to include those terms leads to intractable equatio ns, so the inverse method [10] is
often used. In this method the feedback from the boundary lay er into the external potential
flow is taken into account, and the coupled system is iterativ ely solved to remove the singu-
larity. Without such an external flow present for the circula r hydraulic jump, Higuera [22]
has still obtained the velocity and height profiles from the b oundary layer equations. His
method, called marginal separation, is to force the boundar y layer equations through the
point of separation by choosing a special non-divergent vel ocity profile at the point. The
physical reasoning for the choice of such a particular profil e is rather unclear. Since our aim
is also to obtain a simple tractable model, we have chosen a di fferent approach.
2.4 Averaged equations
Rather than solving the partial differential equation (22) i tself, we shall be content with
satisfying only the mass and momentum conservation laws, de rived from averaging (22) over
the transverse z-direction. To do this we make an ansatz for the radial veloci ty profileu. One
might expect that the singularities at separation points do not contribute to the averages
and do not cause any harm. Such an expectation is too naive as s hown in the next section,
11since the model still shows singular behavior near the jump i f the simplest velocity profile is
assumed. Nevertheless, we show in Sec. 2.7 that the model bec omes capable of going through
the jump smoothly once enough flexibility is introduced in th e assumed profile.
We first define the average velocity at rby
v=1
h/integraldisplayh
0u(r,z)dz. (26)
The total mass flux condition (24) can be written as
rhv= 1. (27)
Next, for each fixed r, we integrate the radial momentum equation (22) over zfrom 0 to
h(r), and use the continuity equation (23) with the surface boun dary conditions (25). We
obtain the averaged momentum equation
1
rhd
dr/bracketleftBigg
r/integraldisplayh
0u2dz/bracketrightBigg
=−h′−1
huz|z=0. (28)
Usingvand
G=1
h/integraldisplayh
0/parenleftbiggu
v/parenrightbigg2
dz, (29)
we obtain
v(Gv)′=−h′−1
huz|z=0. (30)
Equations (27) and (30) are the total mass and momentum equat ions.
2.5 Similarity profile for u
The simplest assumption for the radial velocity profile is a s elf-similar ansatz:
u(r,z)/v(r) =f(η) (31)
whereη=z/h(r) takes values between 0 (bottom) and 1 (surface). Using (23) , the
ansatz can be rewritten in the alternate form: w(r,z) =ηh′u(r,z). It is also equivalent
to the requirement that the local inclination of the streaml ines at (r,z) be proportional to
ηh′=zh′(r)/h(r). Clearly, such an ansatz is too simple and “rigid” to descri be a flow
with separation. However, this is the assumption used in the previous literature, and we
summarize its consequences. For more details, see [5].
The conditions (25) and (27) now imply
f(0) = 0,
f′(1) = 0,/integraldisplay1
0f(η)dη= 1.(32)
12They are not sufficient to uniquely determine f, and we choose one that is physically reason-
able. Thus, a parabolic profile f(η) = 3η−3/2η2is a simple candidate. Using this choice,
G= 6/5 is a constant from (29), and (30) becomes
6
5vv′=−h′−3v
h2. (33)
Other choices for flead to the same equation with different numerical coefficient s. Since all
such equations, corresponding to different choices of f(η), can be further transformed to
vv′=−h′−v
h2(34)
by suitably including numerical coefficients in the characte ristic scales (15), the choice of f
is not important in the study of qualitative behaviour and of parameter dependence.
Using (27), the equation reduces to a single ordinary differe ntial equation for v(r):
v′/parenleftbigg
v−1
v2r/parenrightbigg
=1
vr2−v3r2. (35)
This Kurihara-Tani equation was derived and studied in [40] , in its dimensional form, and
in [5]. The results can be summarized as follows. To find a solu tion corresponding to a
hydraulic jump, the velocity vshould be large for small r, and decrease smoothly as r
increases. However, the model does not have such a solution. The coefficient of v′on the left
hand side generically vanishes at some rwherev′diverges. If (35) is solved in a parametric
form on the ( r,v)-plane, all solutions spiral around and into the fixed point (r,v) = (1,1),
that is a stable focus in the plane. Therefore, one must still connect solutions in the interior
and the exterior by means of, e.g., a Rayleigh shock across wh ich mass and momentum flux
are conserved. When this is carried out, one finds that the sho ck occurs very close to r= 1
in the dimensionless coordinates, implying that the radius of the jump in the dimensional
coordinates scales roughly as r∗in (15), i.e.:
Rj∝/parenleftBig
q5ν−3g−1/parenrightBig1/8. (36)
This scaling relation (36) was compared to experiments [5, 2 1] by changing qfor several
differentν. The radius of the jump indeed scaled with the mass flux q, but the exponent
observed in the experiment was about 3/4 rather than 5/8 sugg ested by (36). To explain
the discrepancy, Rjwas calculated more accurately [5]. It was first proven that t here is
no solution for v(r) to the Kurihara-Tani equation that extends to r=∞. All solutions
were found to diverge at some r=rend(constant) like h∼ {log(rend/r)}1/4. By identifying
this singularity as the end of the plate where the water runs o ff, one may always find the
solution of (35) diverging at the end of the plate of a given ra diusr=rend. By following the
solution to smaller r, the solution before the jump and the position of the shock ar e uniquely
determined assuming a connection via a Rayleigh shock. The s hock location constructed in
this way showed a good agreement [5, 21] with the experiment.
132.6 Profile with a shape parameter
An ansatz more flexible than (31) must be used for resolving th e flow pattern in the vicinity
of the jump. We shall allow the function fin (31) to depend also on r. The simplest modifi-
cation we can make is to assume f=f(η,λ(r)) so that the velocity profile is characterized by
a single “shape parameter” λ(r). The approach follows the ideas developed by von Karman
and Pohlhausen [38] for the usual boundary layer flow around a body. There, separation of
the boundary layer can occur when the pressure gradient, imp osed by the external inviscid
flow, becomes adverse. In our case, there is no external flow, b ut there is a pressure gradient,
along the bottom z= 0, that is proportional to h′(r) due to the hydrostatic pressure (21).
Thus, the possibility arises that the flow separates on z= 0 near the jump where h′is large
and pressure is increasing in r, as in the usual boundary layer flow.
As an improvement over the parabolic profile, we approximate the velocity profile by the
cubic:
u(r,z)/v(r) =aη+bη2+cη3, (37)
wherea,b,care now functions of r. Due to the boundary condition (25) and mass flux
condition (27), the coefficients a,b, andccan be expressed in terms of one parameter λas,
for example:
a=λ+ 3, b =−(5λ+ 3)/2, c = 4λ/3. (38)
The separation condition
uz|z=0= 0 (39)
is now equivalent to a= 0, orλ=−3. Theu-profile is parabolic when c= 0, orλ= 0.
Now that we have two unknowns h(r) andλ(r), two equations are necessary. We use the
averaged momentum equation (30) as the first equation. Note t hatGis now not a constant,
but depends on the shape parameter λ. From (29), we obtain
G(λ) =6
5−λ
15+λ2
105. (40)
Following the Karman-Pohlhausen choice, we choose the seco nd equation to be the momen-
tum equation (22) evaluated at z= 0:
h′=uzz|z=0. (41)
This connects the pressure gradient on z= 0 withλ. Using (38) and (40), the two equations
(30) and (41) can be written as
vd
dr{G(λ)v}=−h′−v
h2(λ+ 3)
h′=−v
h2(5λ+ 3)(42)
14which can be simplified to
(G(λ)v)′=4λ
h2
h′=−v5λ+ 3
h2.(43)
Finally, eliminating vusing (27), we obtain a nonautonomous system of two ordinary differ-
ential equations for h(r) andλ(r):
h′=−5λ+ 3
rh3
dG
dλλ′=4rλ
h+G(λ)h4−(5λ+ 3)
rh4.(44)
This is the model for the stationary circular hydraulic jump . It does become singular, but
only on the lines h= 0 andλ= 7/2 which does not cause any trouble in describing a flow
with a separated zone ( λ <−3). We show in the next section that the highly simplified
model indeed contains solutions which describe the observe d circular hydraulic jumps. A
similar approach using momentum and energy conservation wa s used in [1], but they did not
succeed in finding continuous solutions through the jump.
2.7 Numerical solution of the integrated model
The model (44) can be solved as a boundary value problem by spe cifying two boundary
conditions for different values of r. Thus we impose
(r1,h1(r1)) and (r2,h2(r2)), r 1<r2 (45)
where the values are taken from the measured surface height d ata. There is no fitting
parameter once they are chosen, and the function h(r) and the shape parameter λ(r) are
determined. In particular, we do notneed to specify the shape parameter as a part of the
boundary conditions. This is an advantage of the simplified m odel since one no longer needs
to specify the velocity profile at the inlet and/or outlet bou ndaries, which is not easy to do.
In fact, we see that specifying both handλat oner, either inside the jump or outside, and
solving (44) as an initial value problem is unstable. The sys tem is extremely sensitive to the
initial condition if one integrates (44) in the direction of increasingrfrom a small ror in the
direction of decreasing rfrom a large r. Therefore, we choose r1andr2near 1, typically r1
around 0.4-0.8 and r2around 1.2-1.6. Then, a straightforward shooting method fr om either
boundary is sufficient to obtain a solution. After this is achi eved, the solution is extended to
r<r 1and tor>r 2by integrating (44) backward from r1and forward from r2, respectively.
Integrations in these directions are stable.
Figure 4(a) shows two solutions of such a boundary value prob lem. They correspond
to the two type I solutions in Fig. 2, reproduced here as dot-d ashed curves. From each
curve the boundary data are taken at ˜ r1= 11.8[mm] (corresponding to dimensionless value
r1= 0.42) and ˜r2= 30.0[mm] (tor2= 1.07). The computed solutions h(r) corresponding
15Figure 4: (a) Two surface height profiles of type I flow, taken f rom the experiment in (2)
are shown as the dot-dashed curves. Numerical solutions of t he model (43) are shown as
solid curves in both panels, and show reasonable agreement. To obtain each of the numerical
solutions,hvalues were read from the experimental data at r= 11.8[mm] and r= 30.0[mm],
then a boundary value problem was solved by the shooting meth od. The thick dashed
curve represents an analytical approximation of the soluti ons before the jump, described in
Sec.2.8.1. The formula (52) and (53) shows good agreement wi th one fitting parameter. (b)
The computed shape parameters λ(r), characterizing the velocity profiles, corresponding to
the two numerical solutions in (a). The flow is separated behi nd the jump where λ <−3,
and approaches the parabolic profile λ= 0 asrincreases. Again, the dashed curve is an
analytical approximation. (c) Two trajectories of (43) are shown in the ( h,λ)-plane. They
correspond to solid curves in (a) and (b).
to the data are shown in solid curves. Each curve shows a gradu al decrease for small ˜ ras
˜rincreases, reaches a minimum at some ˜ r≈15[mm], and then undergoes a sharp jump at
˜r≈22- 23[mm], and a slow decay after the jump. The location of th e jump is about 10% off
in each case, and the slope behind the jump is noticeably diffe rent. However, the qualitative
behavior is well captured by the simple model. Figure 4(b) sh ows the shape parameter λ.
The velocity profile changes suddenly almost simultaneousl y with the rapid increase of the
surface height, and a region where λ<−3, corresponding to separation, is observed in each
case.6The parameter λ(r) recovers and appears to converge to λ= 0 (the parabolic profile)
asrbecomes large.
The flow structure is more directly shown in Fig. 5, where the u-velocity profiles are
computed from λat equidistant locations in r. Since magnitudes of the velocity vary a lot
between small and large r, the profiles are scaled by the average velocity, so that the p rofiles
6If the downstream height is further reduced, however, the sh ape parameter λdoes not reach λ=−3,
and there is no separated region. Thus, our model predicts th at a (weaker) jump without an eddy is possible.
The flow near the bottom still decelerates just after the jump .
16Figure 5: Visualization of the type I flow pattern based on the computed shape parameter
λ(r) from the model. The velocity profiles at equidistant locati ons inrare the horizontal
component u, thus they are not tangential to the streamlines. Since magn itudes of the
velocity vary greatly between small and large r, the profiles of u(r,z)/v(r) are shown. The
streamlines separate zones which carry 10% of the flow rate. A separation bubble is present
in the range of rwhereλ<−3. Note the difference in the scales for the axes. The paramete rs
differ from those of Fig. 4. They are: Q= 33[mℓ/s] andν= 1.4×10−5[m2/s], corresponding
tor∗= 2.5[cm],z∗= 1.7[mm], and u∗= 16[cm/s].
ofu(r,z)/v(r) are shown. The stream function ψis computed from the definition
u=ψz/r , w =−ψr/r. (46)
The dimensionless stream function varies from ψ= 0 onz= 0 toψ= 1 onz=h. Inside
the separated region ψ <0. The contours at ψ=−0.1,0,0.1...,1 are shown in the figure.
That is, a region between two neighboring contour curves car ries 10% of the mass flux.
The surface velocity Upredicted from the model is shown in Fig. 6. The parameters ar e
the ones used in Fig. 5. The model again misses the location of the jump by about 20%,
so measurements and the curve from the model are offset, but qu alitative features are well
reproduced. The velocity outside the jump is small and decay s likeU∝1/r, as can be seen
from the log-log plot in the inset. This is consistent with an almost constant hand a nearly
parabolic velocity profile, which we analytically demonstr ate in the next section. On the
other hand, the surface velocity decreases almost linearly before the jump. This region is
harder to explain intuitively, but an analytical approxima tion is also obtained in the next
section. At the jump a rapid, cusp-like drop in the velocity i s noticed.
Finally, we discuss the dependence of the solutions on the ex ternal height hext. Both in
experiments and in the model the height inside the jump is lit tle affected by the change in
171075310
7
5
3
2U[cm/s]
r[cm]
r[cm]U[cm/s]
8 4 0100200
Figure 6: Comparison of the prediction from the model with a s urface velocity measurement
by C. Ellegaard, A.E. Hansen, and A. Haaning [6]. The paramet ers are the same as in Fig. 5.
Marker particles and a high-speed camera were used in order t o obtain the surface velocity U
shown as dots. The theoretical dotted curve was computed by fi nding a stationary solution
h(r) andλ(r) of a boundary value problem using two data points taken from the measured
surface profile (not shown). Although the location of the jum p is about 20% off, the model
reproduces qualitative feature of the measurement very wel l. At small r, the velocity drops
rapidly and almost linearly. It then shows a cusp-like drop a t the jump, and decays gradually
for larger. The final decay is proportional to 1 /ras can be seen from the slope of about −1
in the log-log plot of the exterior region (inset).
the external boundary condition h2(r2). The numerical solutions as well as the measured
surface profiles in Fig. 4(a,b) apparently overlap in the int erior to the jump. Of course, the
two solutions must correspond to different trajectories of t he model (43) and cannot collapse
exactly onto a single curve. However, the closeness of the so lution curves in the interior to
the jump is the cause of the difficulty of solving the initial va lue problem starting from a
smallr.
If the external height is further increased, a transition fr om type I to II is observed in
the experiment, as illustrated in Fig. 2 and Fig. 3. Unfortun ately, no such transition is
reproduced in the model when h2is increased. Instead, one finds a computed solution of
the model similar to the ones in Fig. 4 even for a much larger h2. A physical mechanism
to “break” the wave into a type II flow appears to be missing. In fact, a solution with a
roller is prohibited by the model (43). The surface velocity on a roller is negative (inward).
According to (38), the velocity at the surface is
U=v(a+b+c) =v9−λ
3, (47)
wherev >0 is the average velocity. Thus, U <0 iffλ >9. However, since we start with
λ≃0 and the line λ= 7/2 makes (43) singular, a solution with a roller is not possibl e. It
18seems likely that this behavior can be traced back to the assu med pressure distribution (21)
which does not provide any pressure gradient along the surfa cez=h. In a recent simulation
of the circular hydraulic jump by Yokoi et al.[41] pressure buildup just behind the jump
is observed and claimed to be crucial in breaking the jump. Th e non-hydrostatic pressure
arises partly due to the surface tension in (19.1), but also d ue to the truncated viscous terms
in (18) and (19). We do not know at present how best to extend ou r model to include the
type II flows.
2.8 Asymptotic analysis of the averaged system
In this section we approximate the solutions of (43) analyti cally using formal perturbation
expansions. We obtain explicit expressions for two “outer” regions: the region before the
jump and the one after the jump. Moreover, we derive a single o rdinary differential equation
for the “inner” region near the jump. Analysis in the inner re gion connects a previous model
using a Rayleigh shock with our model.
2.8.1 Outer solution 1 (before the jump)
First, we analyse the region before the jump where thickness of the fluid as well as the
radius are small, compared to the exterior region. We denote the typical thickness, in the
dimensionless coordinates, as θ, and treat it as a formal small parameter. We rescale the
variables into H,R, andVas
h=θH,
r=θαR,
v=θ−1−αV,(48)
and require consistent balance of the terms in (44) or, equiv alently, (43). The rescaling for v
in the third equation of (48), is chosen to ensure mass conser vation (27) for all θ. In terms
of the new variables, (43) can be written as
θ−2α−1d
dR(G(λ)V) =θ−24λ
H2,
θ1−αH′=−θ−α−3V5λ+ 3
H2.(49)
From the first equation the only consistent choice is to take α= 1/2. Then, in order to
balance the power of θon both sides of the second equation, we need
λ=−3/5 +θ4λ1+.... (50)
The form is also motivated by Fig. 4 in which λstays close to the value −0.6 before the
jump.
To findH(R) and the correction λ1, substitute (50) into the first equation of (49). To
the lowest order in θwe obtain
G(−0.6)/parenleftBigg1
H2RdH
dR+1
HR2/parenrightBigg
=12
5H2, (51)
19whereG(−0.6) = 1088/875≃1.243. Solving this equation yields
H=C1
R+4
5G(−0.6)R2, (52)
whereC1is an arbitrary integration constant. The functional form a grees with Watson’s
self-similar solutions [42]. We also compare the lowest ord er term ofθin the second equation
of (49), and find that
λ1=RH3
5dH
dR.
By substituting Hin (52) we obtain an approximate expression for λ:
λ=−3
5+θ4/bracketleftBiggH4
5−12
25G(−0.6)R2H3/bracketrightBigg
. (53)
We test the approximations (52) and (53) in Fig. (4). The dash ed curves are the theoretical
curves ofH(R) andλ(R), shown in the dimensional coordinates. They match the nume rical
solutions and the measurements well before the jump. Here, t he formal parameter θis taken
as unity, and the one free parameter C1was fitted to be 0 .25.
2.8.2 Outer solution 2 (after the jump)
Let us now consider the behavior of (43) for large r. We again introduce a formal small
parameterθ, but we now rescale r=θ−1R. If we moreover assume that the height is of
order 1, i.e., h=H, then the rescaling of the velocity is necessarily v=θVdue to (24).
Using these new variables, Eqs. (43) become:
θ2d
dR(G(λ)V) =4λ
H2
dH
dR=−V5λ+ 3
H2.(54)
In order to balance the terms in the first equation we choose
λ=θ2λ1+.... (55)
This is again consistent with the bottom panel of Fig. 4 where λapparently tends to 0,
corresponding to the parabolic profile. Then, the terms of or der unity in the second equation
are
dH
dR=−3
RH3(56)
whose solution is
H=/parenleftbigg
12 logRend
R/parenrightbigg1/4
(57)
20whereRendis an integration constant representing the radius where th e height goes to 0.
Thus, (43), as well as the simpler Kurihara-Tani model (33), becomes singular when r→ ∞.
This seems to be a general property of models based on the boun dary layer equations [5].
The absence of regular solutions for the system (22)-(25) wh enr→ ∞ was proved in [36].
We have attributed this lack of asymptotic solutions to the i nfluence of the finite size of the
plate. Indeed, a solution with vanishing height such as (57) reminds one very much of a flow
running off the edge of a circular plate.
The height H(R), given by equation (57), is a very slowly varying function o fR. There is
a long regime 1 ≪R≪Rendwhere the height appears almost constant. In this intermedi ate
regime the leading order of (54.1) becomes
G(0)d
dR/parenleftbigg1
RH/parenrightbigg
=4λ1
H2(58)
whereG(0) = 6/5. Therefore,
λ=θ2λ1=−θ2G(0)H2
4/parenleftBigg1
RH2dH
dR+1
R2H/parenrightBigg
≈3
10r2/parenleftbigg3
H3−H/parenrightbigg
. (59)
We conclude that λ(R)∝1/R2→0 which explains the observed approach to the parabolic
velocity profile for large r.
2.8.3 Inner solution near the jump: conservation of momentu m
Finally, we analyze the region around the hydraulic jump. Re call that in the Kurihara-Tani
theory (33) the jump was obtained by fitting a Rayleigh shock. In this section, we show that
our model (44) is a natural generalization of the equation.
To do this we return to (42), and introduce a formal parameter µin the left-hand side
of the second equation.
vd
dr{G(λ)v}=−h′−v
h2(λ+ 3)
µh′=−1
rh3(5λ+ 3).(60)
wherev= 1/(rh). The first equation describes the balance of inertia, hydro static pressure,
and viscous forces. The value µ= 1 corresponds to (42).
Settingµ= 0 givesλ=−0.6. Then the first equation becomes the Kurihara-Tani
equation (33), except that the coefficient 6 /5 = 1.2 is changed to G(−0.6)≈1.243 here, since
the profile is not parabolic. (As discussed before, the veloc ity profile is not so important in
their model as long as it is self-similar.) Since our model co rresponds to µ= 1, the parameter
µinterpolates between the two models, but the correspondenc e of the two is not obvious
because the limit µ= 0 is a singular limit. We treat µas a formal small parameter, and
carry out a singular perturbation analysis to investigate t he connection as well as to obtain
an approximation in the jump region.
21In Kurihara-Tani model a shock is needed to extend the soluti on from small to large
values ofr. Suppose the shock is situated at r=r0. Consider a small region of size µaround
r=r0, and rescale the coordinate as r=r0+µX. Then, in the inner coordinate X, Eq. (60)
becomes
1
r0hd
dX/braceleftBiggG(λ)
r0h/bracerightBigg
=−dh
dX+O(µ)
dh
dX=−5λ+ 3
r0h3+O(µ).(61)
We see that λ=−0.6 withhan arbitrary constant are the only possible fixed points of
(61). Thus the solutions must satisfy λ→ −0.6 forX→ ±∞ . This correctly matches the
external solution before the jump, but not after the jump, wh ereλ→0.7The first equation
can be integrated once, giving the momentum conservation.
G(λ)
r2
0h+h2
2=C3 (62)
with an integration constant C3. Now we solve the second equation of (61) for λ, and
substitute it into this equation. Using (40) in the form G(λ) =1
105/parenleftBig
λ−7
2/parenrightBig2+13
12, we obtain
an ordinary differential equation for honly:
1
105/parenleftBiggr0h3
5dh
dX+41
10/parenrightBigg2
+13
12+r2
0h3
2=C3r2
0h. (63)
We look for a solution h(X) withh→h1asX→ −∞ andh→h2asX→+∞where
h1andh2are constants. Then, Eq. (63) with the first boundary conditi on determines the
constantC3in terms of r0andh1. Eliminating C3we obtain
1
105
/parenleftBiggr0h3
5dh
dX+41
10/parenrightBigg2
h1−/parenleftbigg41
10/parenrightbigg2
h
+13
12(h1−h)−r2
0
2h1h(h2
1−h2) = 0. (64)
Plugging the second boundary condition into this equation y ields a relation between h1and
h2, givenr0.
h1h2
2+h2
1h2−2h3
c= 0. (65)
where
hc= (G(−0.6)/r2
0)1/3(66)
7Note that the singularity of the outer solution after the jum p (57)-(59) for r→0 does not allow correct
matching for X→+∞when µ→0. Nevertheless, our method reproduces the structure of the separation
zone quite well.
22is the critical height for the circular hydraulic jump.8Solving this equation, we obtain an
equation analogous to the shock condition (3):
h2
h1=1
2/parenleftbigg
−1 +/radicalBig
1 + 8(hc/h1)3/parenrightbigg
=2
−1 +/radicalBig
1 + 8(hc/h2)3. (67)
It is easy to see that hcis always between h1andh2, i.e.,h1<hc<h2orh2<hc<h1. The
Froude number in this case could naturally be defined as F(X)2= (hc/h(X))3.9
Whenh1is close tohc, the final height h2is close tohcas well. Then, the Froude number
is close to unity for all X, and the jump is weak, i.e. hc−h1=δ≪1. Then, we see from
the balance of the terms in (64) that h=hc+δY(δx). The leading balance reduces to
Y′=γ(1−Y2)
with
γ=196875
1312/parenleftbigg7
17/parenrightbigg2/3
r5/3
0≈83.1r5/3
0. (68)
Thus, in the weak jump limit, the height is given by
h(x) =hc+δtanh(δγx). (69)
It is interesting to note that we can connect from h1atX=−∞toh2atX= +∞if
h1< h2, but not if h1> h2, just like in the Rayleigh shock. This requirement comes fro m
the equation (64) self-consistently rather than making a hy pothesis on the energy loss like
we did in (5). To see this, consider the stability of the fixed p ointsh1andh2with respect to
the governing equation (64) for h.10Linearizing (64) around the uniform solutions hi(where
i= 1,2), we obtain an equation for the perturbation δhiin the height:
d
dX(δhi) =Kiδhi
where
Ki=2625
41r0{2h3
c+h1(h2
1−3h2
i)}
2h3
ih1. (70)
Ifh1< h c< h2, thenK1>0> K 2, showing that the fixed point h=h1is unstable and
h=h2stable. A trajectory departing from h1atX=−∞and arriving at h2atX= +∞
8In dimensional variables, the critical height is ˜hc= (G(−0.6)q2/g˜r2
0)1/3. This is identical to the critical
height (4) that appeared in the Rayleigh shock, apart from th e numerical factor and the influence of ˜ r0
reflecting the radial geometry. The viscosity νonly enters in the coefficient of dh/dX in the dimensional
version of (64), thus does not affect ˜hc.
9However, it is not clear whether Fdefined in this way can be a measure of super- and subcriticali ty
since the governing equations are not the shallow water equa tions and therefore propagation of disturbances
do not obey the well-known velocity√gh.
10Of course, this stability analysis is to study existence of s tationary solutions, and not to study the
stability of such solutions in the time-dependent theory.
23Figure 7: Comparison between the full numerical solution of (43), the same two solutions
as in Fig. 4 shown as solid curves, and solutions of the asympt otic equation (64), shown as
dashed curves. Even though the asymptotic analysis assumes µ→0, the solutions compare
fairly well with the full numerics corresponding to µ= 1. The asymptotic analysis connects
the model (43) with the Rayleigh shock condition. See text.
is not prohibited, and we can indeed find such a trajectory sho wn in Fig. 7. In contrast, if
h1> h c> h2, then the stability of the fixed points is reversed, and there is no trajectory
going from h1toh2.
Whenh1< hc< h2so that such a trajectory exists, the departure from h1is generally
rapid, giving an impression of a “sharp corner” at the beginn ing of the jump, and the arrival
ath2is much smoother just as shown in Fig. 7. This is because the ma gnitude of the stability
coefficientK1is large compared to that of K2. The feature is most pronounced when h1is
small (so,h2is large). It vanishes as ( h2−h1)→0 whenK1andK2both tend to zero.
In Fig. 7 we compare solutions of (64) with the two solutions o f the full numerical solution
of (43) shown in Fig. 4. The jump region is enlarged. Solution s of (64), shown as solid curves,
are computed by fitting the values for h1andh2, and solving the equation using r0obtained
from (65) and (66). We chose an initial condition to be somewh ere inside the jump, and
integrated (64) forward and backward from it. Since (64) has a translational invariance with
respect toX, the initial condition fixes the location of the jump without affecting the shapes
ofhorλ. The analysis assuming µ→0 performs surprisingly well against the numerical
solution for µ= 1. The size of the jump region is now of order µ, i.e., unity, and the internal
structure is non-trivial. The single ordinary equation (64 ) is capable of describing the eddy
formation in this region.
243 Flow down an inclined plane
3.1 Introduction to the problem
The properties of waves running down an inclined plane is a su bject of great theoretical and
practical importance, and has attracted the attention of ma ny researchers. Starting with
the pioneering work of Kapitsa & Kapitsa [24], some of the maj or contributions to this field
are found in [2, 3, 31, 35, 11, 12, 28, 27] . The physical pictur e is the following. A fixed flux
of fluid is constantly poured onto the inclined plane from abo ve. The fluid forms a stream
moving downwards under the action of gravity – an idealized m odel of a river. If the influx
of fluid upstream is suddenly increased, it causes the height upstream to increase, and the
extra mass of fluid to propagate downstream. In a river, this m ay be caused by the melting
of snow at regions neighbouring the river’s source, or by sud den rain. A river bore, on the
other hand, is introduced at the mouth of the river by a tidal w ave, for instance, and moves
upstream. In both cases, a solitary wave can be formed, movin g at a constant velocity c
without changing its shape.
We are particularly interested in kink-like solitary wave s olutions going from one constant
heighth1to anotherh2. One can identify such a solution with a heteroclinic orbit, connecting
two stationary states [35]. The speed cdepends on how much the fluid level is increased,
i.e., the heights h1andh2. Alternatively, we can consider cas a parameter, and study the
existence of the stationary solution h≡const. depending on c. It is rather straightforward to
see that two solutions with h≡h1andh≡h2exist ifcis sufficiently large. However, even if
cis in that regime, it is hard to judge whether there exists a sm ooth solution connecting the
two states. Based on the method of averaging in Sec. 2, we deve lop a simple model which
helps us to derive criteria for their existence and to comput e the wave form. The model also
enables us to ask whether they appear as “Rayleigh shocks” in the sense that the flow is
supercritical in front of the kink structure and subcritica l behind it. As we shall elaborate,
the distinction between super- and subcritical flows is a con cept inherent in inviscid shallow
water theory, and is not at all obvious for a viscous flow since now the waves will show
dispersion as well as damping. Indeed, we find that the wave ve locities corresponding to the
largest wave lengths will always propagate both forward and backwards, as in a subcritical
flow. Nevertheless, if we focus on wavelengths of the order of the depth of the fluid layer, a
clear distinction can be made.
There is another kind of flow in the linear geometry in which a s udden thickening of
height is observed. This solution is not only relevant for, e .g., the flow of water exiting from
a sluice but is also a direct analog of the circular hydraulic jump. The flow streams rapidly in
a region immediately after the sluice, and then abruptly slo ws down at a certain downstream
position. It is stationary (i.e. c= 0) with a constant discharge, and is notobtained as a
state connecting two “equilibrium” heights. In fact, the ra pid flow before the jump cannot
be extended arbitrarily far upstream. We shall show that our models provide physically
reasonable solutions in this case, too.
In Secs. 3.2 and 3.3 we write down the complete system for the i nclined plane problem,
25non-dimensionalize it, simplify it using the boundary laye r approximation, and average over
the thickness in two ways. These steps are in parallel with th ose in Sec. 2, but we go through
them briefly not only for completeness but also since the geom etry and the characteristic
scales are different. To seek stationary and traveling wave s olutions, we write the equations
in a coordinate frame moving at a constant speed in Sec. 3.4. T raveling waves are studied
in detail in Sec. 3.5, and the stationary jumps in Sec. 3.6.
3.2 The governing equations
We consider a viscous, incompressible, two-dimensional flo w. The coordinate system is
˜xin the downstream direction parallel to the inclined plane, and ˜yin the perpendicular
direction above the plate. Denote the velocities in these di rections by ˜ u(˜x,˜y,˜t) and ˜w(˜x,˜y,˜t),
respectively, the pressure by ˜ p(˜x,˜y,˜t), and the height by ˜h(˜x,˜t). The governing equations
for this problem are the continuity equation
˜u˜x+ ˜w˜y= 0 (71)
and the Navier-Stokes equations
˜u˜t+ ˜u˜u˜x+ ˜w˜u˜y=−1
ρ˜p˜x+gsinα+ν(˜u˜x˜x+ ˜u˜y˜y)
˜w˜t+ ˜u˜w˜x+ ˜w˜w˜y=−1
ρ˜p˜y−gcosα+ν( ˜w˜x˜x+ ˜w˜y˜y)(72)
Here,αis the angle of the inclined plane (between 0 and π/2) measured downward from the
horizontal line, and the subscripts denote the partial deri vatives as before. The boundary
conditions are identical to those of the radial geometry, i. e., (8)–(10), by reading ˜ ras ˜xand
˜zas ˜y. The local mass flux is:
˜q(˜x,˜t) =/integraldisplay˜h(˜x,˜t)
0˜ud˜y.
Integrating the continuity equation (71) in ˜ yover the thickness and using the boundary
conditions, we obtain the flux conservation equation:
˜h˜t+ ˜q˜x= 0. (73)
The equations above form a complete system apart from the inl et and outlet conditions. They
possess a trivial stationary solution (Nusselt solution) w ith a constant ˜hand the parabolic
velocity profile:
˜u(˜x,˜y,˜t)≡gsinα
ν/parenleftBigg
η−η2
2/parenrightBigg
, (74)
whereη= ˜y/˜h. Given this equilibrium flow, the local flux ˜ qis also uniform and steady, and
is a function of ˜h:
˜q=/integraldisplay˜h
0˜ud˜y=g˜h3sinα
3ν. (75)
26In a non-equilibrium flow we assume that the inclined plane is infinitely long, and the
flow sufficiently far downstream approaches this equilibrium flow. We then treat the flow
rate ˜qfor ˜x→ ∞ as the characteristic mass flux q∗. The corresponding height ˜husing
(75) is used as the length scale h∗, andv∗=q∗/h∗becomes the characteristic velocity.
We non-dimensionalize the governing equations by these sca les. The continuity equation is
unchanged in form:
ux+wy= 0, (76)
and the Navier-Stokes equations become
ut+uux+wuy=−px+3
R+1
R(uxx+uyy)
wt+uwx+wwy=−py−3
Rtanα+1
R(wxx+wyy)(77)
where the pressure is normalized to ρu2
∗, and the Reynolds number is
R=v∗h∗
ν=q∗
ν=gh∗3sinα
3ν2. (78)
The dimensionless mass flux is q(x,t) =hvin terms of the average velocity
v(x,t) =1
h/integraldisplayh
0udy (79)
whereby (75) becomes
q=hv=h3(80)
in an equilibrium flow of height h.
3.3 Boundary layer equations and averaged models
Since the flow on the inclined plane is expected to be predomin antly in the x-direction, the
boundary layer approximation should be applicable [11, 12] as long as separation does not
occur. In a similar manner as the radial case, the dominant te rms of (77) are:
ut+uux+wuy=−px+3
R+1
Ruyy
0 =−py−3
Rtanα.(81)
The dynamic boundary conditions on z=hreduce, as before, to:
p|y=h=Whxx
uy|y=h= 0(82)
27with the Weber number in this case being
W=σ
ρh∗v∗2=9σ
ρgh ∗2sin2α. (83)
From (81.2) and (82), the pressure is hydrostatic with contr ibution from the surface tension:
p(x,y,t) =3
Rtanα(h(x,t)−y) +Whxx (84)
so, (81.1) becomes
ut+uux+wuy=3
R−3
Rtanαhx+1
Ruyy+Whxxx. (85)
The mass conservation (73) is non- dimensionalized to
ht+ (hv)x= 0. (86)
Now, we make an ansatz for the u-profile, and average over the thickness in order to
obtain two simplified models. First, we use the self-similar velocity profile:
u(x,y,t)/v(x,t) =f(η) (87)
whereη=y/h(x,t) and the function f(η) satisfies
f(0) = 0
f′(1) = 0/integraldisplay1
0f(η)dη= 1.(88)
Plug this ansatz into (85), multiply it by h, and average over yto obtain
(hv)t+G(hv2)x=3h
R−3
Rtanαhhx−3v
Rh+Whh xxx (89)
together with the mass conservation (86). Here,
G=1
h/integraldisplayh
0(u/v)2dy=/integraldisplay1
0f2(η)dη
is a constant for a given profile in this model. We shall use G= 6/5 for concreteness,
corresponding to the parabolic profile f= 3(η−η2/2). Equation (89) is the Cartesian
analogue of the Kurihara-Tani equation (33), with time-dep endent and surface tension terms.
Next, we assume a variable one-parameter profile for u. As before, we use a third-order
polynomial
u(x,y,t) =v(x,t)(aη+bη2+cη3) (90)
witha=λ+ 3,b=−(5λ+ 3)/2, andc= 4λ/3 chosen to satisfy the conditions (88) for
f. The shape parameter λ(x,t) is the single variable characterizing the velocity profile . To
28describe the evolution of λ(x,t) andh(x,t) we choose the same set of equations as in the
circular hydraulic jump. The first equation is the mass flux eq uation (86). In addition, we
use the momentum equation (85) multiplied by hand averaged in y, and also (85) evaluated
aty= 0:
(hv)t+ (hv2G(λ))x=3h
R−3
Rtanαhhx−v
Rh(λ+ 3) +Whh xxx
0 =3
R−3
Rtanαhx−v
Rh2(5λ+ 3) +Whxxx(91)
whereG(λ) is given by (40) as before. This system can be cast into the mo re compact form:
(hv)t+ (hv2G(λ))x=4vλ
Rh
hxcotα= 1−v
3h2(5λ+ 3) +WR
3hxxx.(92)
In the following we call (89) with (86) the “similarity model ”11and (92) with (86) the
“one-parameter model”. Both models inherit the trivial uni form solution from the complete
Navier-Stokes model: h=v=q≡1, andλ≡0 (parabolic profile) for the one-parameter
model.
3.4 Stationary solutions in a moving coordinate frame
Here, we are concerned with either stationary solutions or t raveling waves whose surface
profiles may show abrupt changes. Both types of solutions can be sought as stationary
solutions in a moving coordinate system with a suitable cons tant velocity c, including the
possibilityc= 0. Thus, we use the traveling wave coordinate ξ=x−ct, and rewrite the
models within this frame.
Using the chain rule, the mass conservation (86) used in both models becomes
−chξ+ (hv)ξ= 0
which can be integrated to
−ch+hv≡Q(const.) (93)
whereQis the mass flux, viewed in the moving frame.12The flow must approach the uniform
equilibrium flow h= 1 in theξ→ ∞ limit. Suppose it also approaches another equilibrium
flowh=h2in theξ→ −∞ limit. Then, using (80), the condition becomes
−ch2+h3
2=Q=−c+ 1. (94)
Of course,h2= 1 is a solution of this equation. In this case we might still b e able to find a
non-trivial solution of a pulse-like solitary wave form. Su ch solutions have previously been
11The similarity model is the “Shkadov model” considered in [1 1, 12] when W∝ne}ationslash= 0.
12Note that the flux q(x, t) in the laboratory frame is, in general, not a constant. The d ischarge at the
inlet, e.g., at x=−∞must be varied in time accordingly.
29studied well [11, 12], and we do not further seek this type of s olutions. For a solution of (94)
other than h2= 1, we need
c=h2
2+h2+ 1. (95)
The solution that can be positive is
h2=−1 +√4c−3
2
which is positive if and only if c>1.
Whenc >1 two different equilibrium solutions exist, and we hope to fin d a kink-like
solution which connects the two limiting flows. However c>1 is only the necessary condition
for its existence. Sufficiency for the existence depends on th e models and the parameters: R,
α, andc. In Sec. 3.5 we shall clarify the parameter regime for finding such solutions. It turns
out that the velocity profiles in this type of solutions do not deviate much from parabolic
even in the one-parameter model. In this sense they correspo nd to somewhat “mild” jumps
in terms of the flow structure.
In Sec. 3.6 we find another family of solutions which approach esh= 1 asξ→ ∞ when
c<1. These solutions do notstart from an equilibrium state at ξ=−∞. Instead, they are
only valid for ξlarger than some value ξ0. In the similarity model they are not interesting
since they approach h= 1 smoothly. However, within the one-parameter model, an ab rupt
change is developed in both the surface and velocity profiles , sometimes with separation. We
interpret this solution, when c= 0, as the analogue of the circular hydraulic jump in the
Cartesian geometry.
The presence of surface tension makes the order of the equati ons higher and makes it
more difficult to compute the solutions even when they exist. W e assume that Wis small and
negligible, and set W= 0 in this section. Under this assumption we convert the aver aged
models into the moving coordinate frame at velocity c. Equation (89) in the similarity model
becomes:
−c(hv)ξ+6
5(hv2)ξ+3
Rtanαhhξ=−3v
Rh+3h
R. (96)
Using the condition (93), vcan be eliminated. We obtain a first order differential equati on
forh:
dh
dξ=15
R(h−1)(h2+h+ 1−c)
c2h2−6(1−c)2+ 15h3/(Rtanα). (97)
Similarly, (92) in the one-parameter model is converted to:
−c(hv)ξ+ (hv2G(λ))ξ=4vλ
Rh
hξcotα= 1−v
3h2(5λ+ 3)(98)
to be solved with (93). One variable, for instance v, can be eliminated so that the system
becomes two-dimensional for handλ.
30In the following sections we treat these averaged models as “ dynamical systems”, and
viewξas a time-like variable. Fixed points of these systems corre spond to the uniform,
equilibrium solutions of the original time-dependent equa tions. Note that stability in terms of
the variable ξis not equivalent to temporal stability of the original time -dependent equations.
3.5 Traveling wave solutions
Due to the relationship (95) which is a one-to-one map betwee ncandh2in the range c>1,
we may treat h2orcas the primary parameter interchangeably. Using h2as a parameter
corresponds physically to varying the height and discharge upstream and then observing the
corresponding change in the wave velocity. The condition c>1 is equivalent to h2>0, and
h2>1 ifc>3. The two regimes h2>1 andh2<1 are qualitatively different. For h2>1
the discharge at ξ→ −∞ is increased, and a forward-facing front travels downstrea m. As we
shall see in this section, this state exists for small enough R. In contrast, h2<1 corresponds
to a backward-facing front which is found to exist for large e noughRbut seems to us very
likely unstable. Thus, we concentrate on the case h2>1 in the following.13
3.5.1 The similarity model
Since (97) is a first order autonomous ordinary differential e quation, the necessary condition
for the existence of a heteroclinic orbit starting from h2(>1) and arriving at h= 1 is that
the fixed point h= 1 is stable and h2is unstable. By linearization, the fixed point h= 1 is
found to be stable if
c2−6(1−c)2+ 15/(Rtanα)>0 (99)
or,
Rtanα<15
6(1−c)2−c2=15
5h4
2+ 10h3
2+ 3h2
2−2h2−1≡f1(h2) (100)
where the denominator is positive for c>3. Similarly, h2is found to be unstable if
Rtanα<15h2
−h4
2−2h3
2+ 3h2
2+ 10h2+ 5≡f2(h2). (101)
The denominator of f2vanishes only at h2=hmax
2≈2.13 for the region h2>1. Ifh2>hmax
2,
thenf2<0 and (101) cannot be satisfied. We discard this region of h2. For 1<h2<hmax
2
one finds that f2(h2)>1>f1(h2). Thus, the necessary condition for the existence is simply
(100). Once the necessary condition is fulfilled, sufficiency is guaranteed. To see this, we
13If we used the geometric mean of the up- and downstream height s/radicalbig˜h1˜h2as the characteristic length,
we would obtain equations whose symmetric appearance makes it easy to study the forward- and backward-
facing fronts simultaneously. However, we have chosen to sc ale by the downstream height ˜h1in order to
treat the traveling waves as well as the stationary jumps.
31Figure 8: Computed examples of the traveling wave solutions connecting two equilibrium
states. Here, the angle of the plane α= 2[deg], and the height h→h2= 1.5 asξ→ −∞ ,
corresponding to the front velocity c= 4.75. Three solutions for R= 3.5, 4.5, and 5.5 are
shown. (a) Height hfrom solution of the similarity model (97). The front become s steeper
asRincreases. (b) Height hfrom solution of the one-parameter model (98). The curves
are quite similar to the ones in (a) except for the oscillatio n in the shallower side when R
becomes close to a critical value. (See text.) (c) Shape para meterλcorresponding to the
solutions in (b). They deviate from the parabolic profile λ= 0 and oscillate (for R= 5.5),
but only slightly. This explains the similarity between (a) and (b).
only need to ensure that the denominator on the right hand sid e of (97) does not vanish in
the region 1 <h<h 2. Suppose it vanished at hs, then we would have
c2h2
s−6(1−c)2+ 15h3
s/(Rtanα) = 0. (102)
Comparison with (99) gives us
c2(1−h2
s) + 15(1 −h3
s)/(Rtanα)>0.
It is clear that hs>1 is impossible. Thus, hs<1, and there is no vanishing denominator
in 1<h<h 2. In Fig. 8(a) we show computed solutions of (97) for three diff erent Reynolds
numbers. The parameters αandh2are fixed, such that (100) becomes R <6.95. Within
this range, a larger Rmakes the propagating front sharper.
323.5.2 The one-parameter model
We can eliminate vfrom (93) and (98), and think of trajectories on the phase por trait for
(h,λ). We look for a heteroclinic orbit starting from a fixed point (h2,0) and arriving at (1 ,0)
asξ→ ∞. It is necessary for its existence that the point ( h2,0) has at least one unstable
direction and (1 ,0) has at least one stable direction. Linearizing around the equilibrium
point ash=he+δhandλ= 0 +δλ, wherehe= 1 orh2, we obtain:
/parenleftBigg
δhξ
δλξ/parenrightBigg
=J/parenleftBigg
δh
δλ/parenrightBigg
.
It is straightforward to calculate the 2 ×2 Jacobian matrix J, and show that
detJ=60(c−3h2
e) tanα
Rh7e. (103)
For the point ( h2,0) we have c−3h2
e= 1 +h2−2h2
2<0 whenh2>1. This means that
detJ <0 forh2>1, and the fixed point is always a saddle, having exactly one un stable
direction.
For the point (1 ,0) we have det J >0 sincec−3h2
e=h2
2+h2−2>0 whenh2>1.
Thus, we must also compute the trace of Jforhe= 1 which can be shown to be
trJ=−60
R+ (33−61c+ 25c2) tanα.
For the stability of (1 ,0) we need tr J <0. Since 33 −61c+25c2>0 forc>3, this condition
becomes
Rtanα<60
33−61c+ 25c2=60
−3−11h2+ 14h2
2+ 50h3
2+ 25h4
2≡fs(h2). (104)
When this is satisfied, the fixed point is locally attracting, and a trajectory may reach it
from any direction. Indeed, we find numerically that the cond ition (104) also seems to be
sufficient. For any Randαwe have tried in the range (104), a heteroclinic solution was
found. Computed solutions for three different values of Rare shown in Fig. 8(b) and (c).
The parameters αandh2are identical to the ones used for the similarity model in Fig . 8(a).
The condition (104) yields R <5.59. The height profiles in (b) are essentially identical to
the ones in (a). This is because the shape parameter λshown in (c) does not deviate much
fromλ= 0, the parabolic profile.
In Fig. 8(b) and (c), the solution is oscillatory around h=h1andλ= 0 forR= 5.5.
This is a feature seen when Rbecomes close to the critical value given by (104). It happen s
when the type of the fixed point (1 ,0) changes from a stable node to a stable focus. The
point is a focus when det J >(trJ)2/4, which is equivalent to f+(h2)< Rtanα < f −(h2)
where
f±(h2) =60
−7−9h2+ 16h2
2+ 50h3
2+ 25h4
2±2√
5D(105)
33and
D= 2 + 3h2−9h2
2−19h3
2+ 3h4
2+ 15h5
2+ 5h6
2.
It can be seen that f+(h2)<fs(h2)<f −(h2) forh2>1. Therefore, a heteroclinic solution
can be found and exhibits oscillations in a small region f+(h2)< Rtanα < f s(h2). In
Fig. 8(b) and (c) this condition corresponds to 4 .81< R < 5.59, so only the solution for
R= 5.5 shows oscillations.
3.6 Stationary jumps
Ifc <1, the two averaged models have only one fixed point h= 1. Therefore, one might
imagine that it is too limited to show any jump-like structur es. Nevertheless, we look for
trajectories that approach to the fixed point as ξ→ ∞. Even though c= 0 is the physically
most interesting case, we treat the general case 0 ≤c<1. Since there is no h2, we usecas
the prime parameter in this section.
3.6.1 The similarity model
The sole fixed point h= 1 must be stable to be the limiting point of a trajectory as ξ→ ∞.
For 0≤c<1, the condition is similar to (99) but with reversed inequal ity
c2−6(1−c)2+ 15/(Rtanα)<0. (106)
The singular height hsof the governing equation is still given by (102), and, using a similar
argument as before, it is easy to see that 0 ≤hs<1 is impossible when c<1. Thus, there is
a trajectory which approaches h= 1 from below if (106) holds. When 1 >c> (6−√
6)/5≃
0.71,c2−6(1−c)2>0 and (106) cannot be satisfied. When c<(6−√
6)/5, the condition
is equivalent to
Rtanα>15
6(1−c)2−c2, (107)
which is satisfied in a range of Rtanαsince the denominator of the right hand side is positive.
Computed solutions for R= 50, 70, and 100 are shown in Fig. 9 as dashed curves using
α= 3[deg] and c= 0. The condition (106) becomes R >47.7, and is satisfied for all three.
Each solution simply approaches h= 1 smoothly, clearly reflecting the first order nature
of the model (97). As ξdecreases, the height vanishes at a finite ξand an inlet must be
placed before this happens. If his very small, (97) simplifies to dh/dξ = 5/{2R(1−c)}. The
solution is
h(ξ) =2.5
R(1−c)(ξ−ξ0) (108)
for someξ=ξ0whereh= 0. There is no abrupt change in the solutions that resembles
a stationary shock structure. If we use Rsmaller than the critical value, then there is no
solution converging to h= 1. Therefore, we view the similarity model as inadequate fo r
describing stationary jumps.
34Figure 9: Computed stationary solutions for α= 3[deg] and c= 0. Dashed curves are
solutions of the similarity model (97) for R= 50, 70, and 100. Solid curves are solutions of
the one-parameter model (98) for R= 30, 50, and 70. A larger Rcorresponds to a slower
convergence to the equilibrium flow h= 1. These solutions do not show any shock-like
structure.
3.6.2 The one-parameter model
The sole fixed point of this model when c <1 is (h,λ) = (1,0). The Jacobian and its
determinant is still given by (103), but now c−3h2
e=c−3<0 and, thus, det J <0.
Therefore, the fixed point is always a saddle in this range of c, and there is one direction
convergent to the fixed point as ξ→ ∞. It is easy to compute the corresponding trajectory
by integrating backward in ξfrom the vicinity of the fixed point. This solution seems to
exist for all values of R,αandc<1. We are interested in solutions which approach h= 1
from below, and tend to h= 0 at some ξ=ξ0asξdecreases. (To be physical, an inlet
condition must be specified at some ξ > ξ 0.) We can analyze the solutions asymptotically
nearξ0by assuming that h∼A(ξ−ξ0) asξ→ξ0+ 0. Then, using (93) and Q= 1−cin
(94), we obtain v∼(1−c)/{A(ξ−ξ0)}. Substituting these into (98.2) yields
λ∼ −0.6 +3A3
5(1−c)(1−Acotα)(ξ−ξ0)3.
Finally, comparing coefficients of the dominant terms in (98. 1) determines Aas
A=12
5RG(−0.6)(1−c)≈1.93
R(1−c).
We observe two qualitatively different types depending on th e parameter values. If λ
increases at the point ξ=ξ0, then the solution reaches the parabolic profile λ= 0 mono-
tonically. This occurs when Ris large, and three computed solutions are shown in Fig. 9
as solid curves. The height profile is qualitatively identic al to the ones from the similarity
model shown in dashed curves. They do not show any jump struct ure.
On the other hand, if λdecreases at ξ0, then the trajectory makes an excursion to smaller
λ, sometimes into the separation zone λ<−3, before recovering toward λ= 0. The condition
to obtain the second type is Acotα>1, or,
Rtanα<12
5G(−0.6)(1−c)≃1.94(1−c) (109)
35Figure 10: (a) Computed height hof the stationary solutions for the one-parameter model
(98) usingα= 3[deg],c= 0, andR= 5 and 10. A shock-like structure is visible, with a fast
shooting flow in front of it and a slow equilibrium flow behind. (b) The shape parameter
λcorresponding to the solutions in (a) shows separation, λ <−3, in both solutions. (c)
Corresponding trajectories on the phase portrait of hversusλ. In addition to the two
solutions for R= 5 and 10, three more solutions for R= 20, 30, and 50 are shown. An
excursion to small λbefore convergence to the fixed point at (0 ,1) is visible for trajectories
with small R.
withG(λ) given by (40). Two solutions satisfying this condition are shown in Fig. 10(a) and
(b). Both the height profile and the shape parameter vary in a s imilar manner to the one we
obtained in the circular hydraulic jump. The phase portrait in (c) demonstrates how rapid
and large the excursion can become for small R. This type of solution could be realized, for
instance, as a stationary flow ( c= 0) exiting a sluice gate placed at some ξ >ξ 0.14
14A full-scale channel flow such as a river certainly requires a turbulence modelling, but we have been able
to construct a miniature experimental model in which the flow remains laminar. However, our preliminary
observation is that a pair of edge waves are created from the e nds of the gate, which makes the flow three-
dimensional.
364 Linear stability of equilibrium states
It is quite difficult to carry out linear stability analysis ar ound the stationary solutions
and traveling wave solutions found so far. They have non-uni form profiles obtained only
numerically and some of the solutions have singular points b eyond which they cannot be
continued. Moreover, the inlet boundary condition can stro ngly affect the stability properties
of the solutions. We shall therefore focus on the linear geom etry, and only study stability
of the equilibrium flow h≡const. The results are, however, expected to be applicable
to the equilibrium flow sufficiently far downstream of the jump in the stationary solutions
and to flows sufficiently up- and downstream of the moving front in case of the traveling
wave solutions. Since the dispersion relation scales with t he chosen characteristic length, as
described in Sec. 4.4, we only need to consider the flow h≡1. Both the similarity model (89)
and the one-parameter model (92) are considered, including the surface tension term which
is expected to be relevant [35] for stability. One of our aims is, of course, to judge when
infinitesimal disturbances grow and whey they decay, but the ir propagation velocities are
also of our great interest. By comparing the velocities to a r eference velocity, which is zero
for the stationary jump and c(>3) for the traveling wave, we are able to classify different
parts of the solutions as either super- or subcritical.
4.1 Dispersion relations
The first step is to linearize the models around the fixed point h=v= 1 and, for the
one-parameter model, λ= 0. We assume infinitesimal disturbances δh,δv, andδλ, and
decompose them into Fourier modes:
δh,δv,δλ ∼ei(kx−ωt). (110)
Plugging them into the linearized equations for the similar ity model (86) and (89), we
obtain:
ω2+ω/parenleftbigg3i
R−12
5k/parenrightbigg
+/parenleftbigg
−9i
Rk+6
5k2−3
Rtanαk2−Wk4/parenrightbigg
= 0. (111)
Solving the equation the dispersion relation is found to be
ω±=−3i
2R+6
5k±/radicalBig
D0 (112)
where the discriminant is
D0=−9
4R2+27i
5Rk+ 3k2/parenleftbigg2
25+1
Rtanα/parenrightbigg
+Wk4. (113)
Similarly, from the one-parameter model (86) and (92), we ob tain the dispersion relation:
ω±=−6i
5R+61
50k±3
5/radicalBig
D1 (114)
37where
D1=−4
R2+178i
15Rk+/parenleftbigg421
900+20
3Rtanα/parenrightbigg
k2+i
9 tanαk3+20W
9k4+iRW
27k5.(115)
Note that this model also has only two dispersion relations, ω+(k) andω−(k) because the
second equation of (92) does not include time-derivatives.
4.2 Long wave limit
We first study the long wave limit k→0 by taking only the lowest order terms in k. For
the similarity model, the dispersion relation (112) become s
ω+= 3k+ik2(R−cotα) +O(k3)
ω−=−3i
R−3
5k−ik2(R−cotα) +O(k3).(116)
Ask→0, the group velocities dω+/dk→3 anddω−/dk→(−3/5). Therefore, waves
corresponding to ω−propagate upstream, and the flow is subcritical irrespective of R. By
studying the dominant imaginary components of ω±, we also find that the reverse propagating
branchω−is always stable, i.e. the disturbances decay, for small eno ughkwhereas the
forward propagating branch ω+is stable only for small enough Reynolds number satisfying
Rtanα<1. (117)
The limiting dispersion is identical in the one-parameter m odel apart from numerical
coefficients. For small k, (114) becomes
ω+= 3k+ik2(5
4R−cotα) +O(k3)
ω−=−12i
5R−14
25k−ik2(5
4R−cotα) +O(k3).(118)
Thus, the flow is always subcritical since the long waves in th eω−branch propagate upstream
with velocity −14/25. Again, this branch is stable for any Rwhile theω+branch is stable
only for small Reynolds numbers:
Rtanα<4/5. (119)
4.3 Intermediate range of k
It is quite unexpected that the flow is subcritical for any R. One would intuitively expect
that disturbances cannot propagate upstream for sufficientl y rapid flows. An explanation
can be made by a more careful study of the dispersion relation s (112) and (114), or, in
particular, the discriminants D0andD1.
38Figure 11: Real part of the dispersion relation showing the p ropagation of disturbances
on the equilibrium flow. (a) Similarity model using (112) for R= 25, 30, and 35. (b)
One-parameter model using (114) for R= 20, 25, and 30. In both models α= 5[deg]
andW= 0.01 are fixed. Three dashed and solid curves correspond to the ω+andω−
branches, respectively, of the dispersion relation. The ω+has a positive slope, or group
velocity, for all k, while theω−branch has positive slope only when Ris large. However, for
large enough R, the region of kin which both branches have positive slopes extends from
smallkcorresponding to wavelengths beyond the system size to larg ekwith wavelengths
smaller than the thickness of the flow. In this case the flow is e ssentially supercritical since
disturbances are all carried away downstream.
We first consider the similarity model. If the O(k2) term dominates in D0, then the
corresponding group velocities become
c±=dω±
dk≈6
5±/radicalBigg
6
25+3
Rtanα. (120)
Bothc+andc−become positive for
Rtanα>5/2. (121)
We attempt to estimate such a range of k. For brevity we assume Rtanα≪25/2 so that
the coefficient of k2inD0can be approximated by 3 /(Rtanα). If the magnitude of the
O(k2) dominates in D0, then we must have
3k2
Rtanα≫9
4R2,27k
5R, Wk4,
that is,
max
/radicalBigg
3 tanα
4R,9 tanα
5
≪k≪/radicalBigg
3
RWtanα. (122)
UsingR= 30,α= 5[deg], and W= 0.01, for instance, the condition (121) and (122) gives
a window 0 .16≪k≪10.7 in which we can hope that the O(k2) term dominates.
Rather than attempting a more accurate estimate of the zone, we demonstrate that such
an interval can be in fact quite long, by plotting the real par t ofω±(k) for (112) in Fig. 11(a).
39Three different values of Rare used while αandWare fixed. The ω+branch, shown as
dashed curves, has a positive slope for any k. Both phase and group velocities of this branch
are positive. On the other hand, the ω−branch, shown as solid curves, qualitatively changes
withR. ForR= 25 its slope appears to be negative for all k, indicating a subcritical flow.
However, for a larger Rthere is an interval of kin which the slope becomes positive. In
the limitk→0, the branch still has a negative slope in accordance with th e analysis of the
long wave limit in the previous section. However, the subcri tical region near k= 0 can be
very small. One sees in Fig. 11(a) that the curve has a positiv e slope already when k>0.05
andR= 35. The slope continues to be positive until k= 2, corresponding to a wavelength
of half the thickness of the equilibrium flow. Since the syste m length is finite in practice,
the subcritical flow in the k→0 limit cannot be achieved, and the flow becomes essentially
supercritical for all the wave numbers observed. This define s the super- and subcritical flows
within our viscous model, and confirms the intuitive picture of having a supercritical flow
when the flow is sufficiently rapid.
The situation is qualitatively identical in the one-parame ter model. We obtain
Rtanα>20/11 (123)
and
max
/radicalBigg
3 tanα
5R,50 tanα
89
≪k≪min
60 tanα
R,/radicalBigg
3
RWtanα,/parenleftbigg180
R2Wtanα/parenrightbigg
(124)
as the corresponding equations to (121) and (122), respecti vely. Again using R= 30,
α= 5[deg], and W= 0.01, the interval becomes 0 .05≪k≪0.18. The upper limit
comes from the O(k3) term inD1, and is estimated to be rather small since we have only
compared the magnitudes. In fact, when we plot the real part o f the dispersion relation
(114) in Fig. 11(b), we find that the ω−branch has a positive group velocity for a much
longer range of k. The supercritical flow near the k= 0 limit is very small once again if R
becomes as large as R= 25.
4.4 Super- and subcriticality for moving fronts
The intermediate- kbehavior enables us to decide whether a given equilibrium flo w is “inher-
ently” super- or subcritical. This distinction is made base d on wave velocities with respect
to the laboratory frame. A more classical distinction of the two types arises in the context
of the shock theory, as reviewed in Sec. 2.1. In this case velo cities are measured with respect
to a moving front; we call the flow “supercritical” if the grou p velocity of all the waves is
less than the front velocity c, and “subcritical” if there is a wave component whose group
velocity is larger than c. Here, we briefly note that the averaged equations can descri be this
traditional classification, too.
Take a moving front such as the one shown in Fig. 8. We concentr ate on the long wave
limitk→0. Forξ→ ∞ the flow approaches an equilibrium flow with h= 1. Linear waves
propagate forward and backward with the group velocities dω+/dk= 3 anddω−/dk=−3/5
40according to the dispersion relation for the similarity mod el (116). This is a subcritical
situation in the laboratory frame, but, since the front velo city isc= 1 +h2+h2
2>3, both
these waves propagate into the front. Therefore, the flow is s upercritical with respect to the
front.
To derive the dispersion relation of the equilibrium flow wit h heighth2forξ→ −∞ ,
consider rescaling the height by h2. That is, we use this height as the characteristic length
so that a wave number kmust be multiplied by h2. Since the flow rate is q2=h3
2from
(80), the velocity has to be scaled by q2/h2=h2
2. Thus, the group velocities for this flow in
the laboratory frame are dω+/dk= 3h2
2anddω−/dk=−(3/5)h2
2. It is easy to show that
3h2
2>c= 1+h2+h2
2forh2>1. Thus, one wave component propagates into the front while
the other moves away from it so that the flow behind the front is subcritical.
Therefore, the moving front has a supercritical flow on the sh allower side and a subcritical
flow on the deeper side, and can be regarded as a classical shoc k. Using the one-parameter
model instead of the similarity model is qualitatively iden tical.
4.5 Short wave limit
We now come back to the stationary equilibrium flow, and study the dispersion relation in
the short wave range. Since the derivation of the averaged eq uations relies on the assumption
of predominantly horizontal flow, it is not our aim to accurat ely resolve wave components
whenkis large. We only hope that the short waves decay so that they d o not interfere
with meaningful dynamics when we simulate the time-depende nt model. Unfortunately, the
one-parameter model performs poorly in this respect compar ed to the similarity model.
The dispersion relation of the similarity model (112) can be approximated in the large k
limit as
Reω±=±√
Wk2+O(k)
Imω±=−3
2R+O(1/k).(125)
Thus, short waves in (89) are damped out if W >0.
If we neglect the surface tension and set W= 0, the dispersion relation for large kis
ω±=c±k−3i
2Rc±−3
c±−6/5+O(k−1) (126)
wherec±is the velocity of the corresponding wave given by
c±=6
5±/radicalBigg
6
25+3
Rtanα. (127)
Sincec−<6/5 from (127), the branch ω−is always stable, as can be seen from (126). On
the other hand, since c+>6/5, the condition for the stability of the branch ω+isc+<3,
which is equivalent to
Rtanα<1. (128)
41For a large Rthe equilibrium state is no longer stable, but this is reason able in the absence
of surface tension.
Now, we turn into the dispersion relation of the one-paramet er model (114). For large k,
it behaves as
ω±∼ ±k5/2/radicalBig
iW/75 ifW >0, (129)
and as
ω±∼ ±k3/2/radicalBig
i/(25 tanα) ifW= 0. (130)
In either case one of the branches has an unstable component a sk→ ∞, irrespective of R
orα. We have been unable to find a natural modification to the one-p arameter model which
prevents this unphysical behavior. Its cause may well be tha t the evolution of short waves
is not well represented by the boundary layer approximation we started with. In fact, in
the boundary layer equations (81) the higher order derivati ves ofxthat are thought to be
crucial for stability of the high- kmodes are neglected. In this view the similarity model (89)
provides surprisingly reasonable behaviour for large k, even starting from (81).
425 Conclusions
In this article we have presented a simple but fairly quantit ative method of reducing flows
with strongly deformed free surfaces to a manageable system of equations. By assuming a
“flexible” velocity profile whose shape parameter is another dependent variable, flows with
an internal eddy can be described. In the radial geometry our results compare well with
experiments and we have obtained analytic expressions for t he circular hydraulic jump.
We have also studied the flow down an inclined plane. The reduc ed equations possess not
only the traveling wave solutions (heteroclinic orbits) st udied previously but also stationary
jump solutions. We have found that the stationary solutions show a stronger change in the
velocity profile than the traveling waves.
Finally, we have classified different parts of the flows into su per- and subcritical by study-
ing the dispersion relation around the equilibrium flow. Thi s classification is standard for
inviscid shallow water flow and in shock theory, but is is not o bvious in the context of vis-
cous flow. Indeed, for sufficiently long waves the averaged equ ations show that supercritical
flow is not possible. However, waves with intermediate lengt hs can make the flow essentially
supercritical.
The only but serious defect of our reduced model which we have been unable to overcome
is its short wavelength behavior. As it stands now, some arti ficial dissipation term to stabilize
the short waves is necessary before time-dependent simulat ions are attempted. To our dismay
a more natural treatment of this problem has so far eluded us.
Acknowledgements
The core part of this work was carried out while authors were a t the Center for Chaos &
Turbulence Studies (CATS) at the Niels Bohr Institute to whi ch SW and VP are grateful
for hospitality and an inspiring environment. SW thanks the Institute for Mathematics
& its Applications (IMA) of the University of Minnesota for p roviding him with a place
and atmosphere to continue the work. Research supported in p art under Grant-in-Aid for
Scientific Research of JSPS. VP acknowledges the hospitalit y of University of Chicago and
support through the NSF grant No. DMR 9415604 and MRSEC, NSF G rant No. DMR
9808595.
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46 |
arXiv:physics/0008220v1 [physics.optics] 24 Aug 2000Binary Representations of ABCD Matrices
S. Ba¸ skal∗
Department of Physics, Middle East Technical University, 0 6531 Ankara, Turkey
Y. S. Kim†
Department of Physics, University of Maryland, College Par k, Maryland 20742, U.S.A.
Abstract
TheABCD matrix is one of the essential mathematical instruments in o ptics.
It is the two-by-two representation of the group Sp(2), which is applicable to
many branches of physics, including squeezed states of ligh t, special relativity
and coupled oscillators. It is pointed out that the shear rep resentation is ori-
ented to binary logic which may be friendly to computer appli cations. While
this is a future possibility, it is known that para-axial len s optics is based on
the shear representation of the Sp(2) group. It is pointed out that the most
general form of the ABCD matrix can be written in terms of six shear ma-
trices, which correspond to lens and translation matrices. The parameter for
each shear matrix is computed in terms of the three independe nt parameters
of the ABCD matrix.
Typeset using REVT EX
∗electronic address: baskal@newton.physics.metu.edu.tr
†electronic address: yskim@physics.umd.edu
1I. INTRODUCTION
In a recent series of papers [1,2], Han et al. studied possible optical devices capable of
performing the matrix operations of the following types:
T=/parenleftbigg1a
0 1/parenrightbigg
, L =/parenleftbigg1 0
b1/parenrightbigg
. (1)
Since these matrices perform shear transformations in a two -dimensional space [3], we shall
call them “shear” matrices.
However, Han et al. were interested in computer applications of these shear mat rices
because they can convert multiplications into additions. I ndeed, the Tmatrix has the
property:
T1T2=/parenleftbigg1a1
0 1/parenrightbigg /parenleftbigg1a2
0 1/parenrightbigg
=/parenleftbigg1a1+a2
0 1/parenrightbigg
, (2)
and the Lmatrix has a similar “slide-rule” property. This property i s valid only if we restrict
computations to the T-type matrices or to the L-type matrices.
What happens if we use both LandTtypes? Then it will lead to a binary logic. In the
present paper, we study this binary property of the ABCD matrix, which takes the form
G=/parenleftbiggA B
C D/parenrightbigg
, (3)
where the elements A, B, C andDare real numbers satisfying AD−BC= 1. Because of
this condition, there are three independent parameters.
We are interested in constructing the most general form of th eABCD matrix in terms of
the two shear matrices given in Eq.(1). Two-by-two matrices with the above property form
the symplectic group Sp(2). Indeed, we are quite familiar with the conventional rep resenta-
tion of the two-by-two representation of the Sp(2) group. This group is like (isomorphic to)
SU(1,1) which is the basic scientific language for squeezed states of light [4]. This group
is also applicable to other branches of optics, including po larization optics, interferometers,
layer optics [5], and para-axial optics [6,7]. The Sp(2) symmetry can be found in many other
branches of physics, including canonical transformations [3], special relativity [4], Wigner
functions [4], and coupled harmonic oscillators [8].
Even though this group covers a wide spectrum of physics, the mathematical content
of the present paper is minimal because we are dealing only wi th three real numbers. We
use group theoretical theorems in order to manage our calcul ations in a judicious manner.
Specifically, we use group theory to represent the most gener al form of the ABCD matrix in
terms of the shear matrices given in Eq.(1), and to translate the group theoretical language
into a computer friendly binary logic.
With this point in mind, we propose to write the two-by-two ABCD matrices in the
form
TLTLT . . .. (4)
Since each matrix in this chain contains one parameter, ther e are N parameters for N matrices
in the chain. On the other hand, since both TandLare real unimodular matrices, the final
2expression is also real unimodular. This means that the expr ession contains only three
independent parameters.
Then we are led the question of whether there is a shortest cha in which can accommodate
the most general form of the two-by-two matrices. We shall co nclude in this paper that six
matrices are needed for the most general form, with three ind ependent parameters. While
we had in mind possible future computer applications of this binary logic, we are not the
first ones to study this problem from the point of view of ray op tics.
Indeed, in 1985, Sudarshan et al. raised essentially the same question in connection with
para-axial lens optics [7]. They observed that the lens and t ranslation matrices are in the
form of matrices given in Eq.(1). In fact, the notations LandTfor the shear matrices of
Eq.(1) are derived from the words “lens” and “translation” r espectively in para-axial lens
optics. Sudarshan et al. conclude that three lenses are needed for the most general fo rm
for the two-by-two matrices for the symplectic group. Of cou rse their lens matrices are
appropriately separated by translation matrices. However , Sudarshan et al. stated that the
calculation of each lens or translation parameter is “tedio us” in their paper.
In the present paper, we made this calculation less tedious b y using a decomposition of
theABCD matrix derivable from Bargmann’s paper [9]. As far as the num ber of lenses
is concerned, we reach the same conclusion as that of Sudarsh anet al.. In addition, we
complete the calculation of lens parameter for each lens and the translation parameter for
each translation matrix, in terms of the three independent p arameters of the ABCD matrix.
In Sec. II, it is noted that the Sp(2) matrices can be constructed from two different sets of
generators. We call one of them squeeze representation, and the other shear representation.
In Sec. III, it is shown that the most general form of the Sp(2) matrices or ABCD matrices
can be decomposed into one symmetric matrix and one orthogon al matrix. It is shown
that the symmetric matrix can be decomposed into four shear m atrices and the orthogonal
matrix into three. In Sec. IV, from the traditional point of v iew, we are discussing para-axial
lens optics. We shall present a new result in this well-estab lished subject. In Sec. V, we
discuss other areas of optical sciences where the binary rep resentation of the group Sp(2)
may serve useful purposes. We discuss also possible extensi on of the ABCD matrix to a
complex representation, which will enlarge the group Sp(2) to a larger group.
II. SQUEEZE AND SHEAR REPRESENTATIONS OF THE SP(2) GROUP
Since the ABCD matrix is a representation of the group Sp(2), we borrow mathematical
tools from this group. This group is generated by
B1=1
2/parenleftbiggi0
0−i/parenrightbigg
, B 2=1
2/parenleftbigg0i
i0/parenrightbigg
,
L=1
2/parenleftbigg0−i
i0/parenrightbigg
, (5)
when they are applied to a two-dimensional xyspace. The Lmatrix generates rotations
around the origin while B1, andB2generate squeezes along the xyaxes and along the axes
rotated by 45orespectively. This aspect of Sp(2) is well known. Let us consider a different
representation.
3The shear matrices of Eq.(1) can be written as
/parenleftbigg1s
0 1/parenrightbigg
= exp ( −isX1),
/parenleftbigg1 0
u1/parenrightbigg
= exp ( −iuX2), (6)
with
X1=/parenleftbigg0i
0 0/parenrightbigg
, X 2=/parenleftbigg0 0
i0/parenrightbigg
, (7)
which serve as the generators. If we introduce a third matrix
X3=/parenleftbiggi0
0−i/parenrightbigg
, (8)
it generates squeeze transformations:
exp (−iηX3) =/parenleftbiggeη0
0e−η/parenrightbigg
. (9)
The matrices X1, X2, and X3form the following closed set of commutation relations.
[X1, X2] =iX3,[X1, X3] =−2iX1,
[X2, X3] = 2iX2. (10)
As we noted in Eq.(6), the matrices X1andX2generate shear transformations [3,10,11].
The matrix X3generate squeeze transformations. Thus what is the group ge nerated by one
squeeze and two shear transformations?
The generators of Eq.(7) and Eq.(8) can be written as
X1=B2−L, X 2=B2+L, X 3= 2B1, (11)
where L, B 1andB2are given in Eq.(5). The Sp(2) group can now be generated by two
seemingly different sets of generators namely the squeeze-r otation generators of Eq.(5) and
the shear-squeeze generators of Eq.(11). We call the repres entations generated by them the
“squeeze” and “shear” representations respectively. It is quite clear that one representation
can be transformed into the other at the level of generators. Our experience in the conven-
tional squeeze representation tells us that an arbitrary Sp(2) matrix can be decomposed into
squeeze and rotation matrices. Likewise then, we should be a ble to decompose the arbitrary
matrix into shear and squeeze matrices.
We are quite familiar with Sp(2) matrices generated by the matrices given in Eq.(5). As
shown in Appendix A, the most general form can be written as
G=/parenleftbiggcosφ−sinφ
sinφcosφ/parenrightbigg /parenleftbiggeη0
0e−η/parenrightbigg /parenleftbiggcosλ−sinλ
sinλcosλ/parenrightbigg
, (12)
where the three free parameters are φ, ηandλ. The real numbers A, B, C andDin Eq.(3)
can be written in terms of these three parameters. Conversel y, the parameters φ, ηandλ
4can be written in terms of A, B, C andDwith the condition that AD−BC= 1. This
matrix is of course written in terms of squeeze and rotation m atrices.
Our next question is whether it is possible to write the same m atrix in the shear repre-
sentation. In the shear representation, the components sho uld be in the form of TandL
matrices given in Eq.(1) and a squeeze matrix of the form
/parenleftbiggeη0
0e−η/parenrightbigg
, (13)
because they are generated by the matrices given in Eq.(7) an d Eq.(8). But this mathemat-
ical problem is not our main concern. In the present paper, we are interested in whether it
is possible to decompose the ABCD matrix into shear matrices.
III. DECOMPOSITIONS AND RECOMPOSITIONS
We are interested in this paper to write the most general form of the matrix Gof Eq.(3)
as a chain of the shear matrices. Indeed, Sudarshan et al. attempted this problem in
connection with para-axial lens optics. Their approach is o f course correct. They concluded
however that the complete calculation is “tedious” in their paper.
We propose to complete this well-defined calculation by deco mposing the matrix Ginto
one symmetric matrix and one orthogonal matrix. For this pur pose, let us write the last
matrix of Eq.(12) as
/parenleftbiggcosφsinφ
−sinφcosφ/parenrightbigg /parenleftbiggcosθ−sinθ
sinθcosθ/parenrightbigg
, (14)
withλ=θ−φ. Instead of λ,θbecomes an independent parameter.
The matrix Gcan now be written as two matrices, one symmetric and the othe r orthog-
onal:
G=SR, (15)
with
R=/parenleftbiggcosθ−sinθ
sinθcosθ/parenrightbigg
. (16)
The symmetric matrix Stakes the form [2]
S=/parenleftbiggcoshη+ (sinh η) cos(2 φ) (sinh η) sin(2 φ)
(sinhη) sin(2 φ) cosh η−(sinhη) cos(2 φ)/parenrightbigg
. (17)
Our procedure is to write SandRseparately as shear chains. Let us consider first the
rotation matrix.
In terms of the shears, the rotation matrix Rcan be written as [10]:
R=/parenleftbigg1−tan(θ/2)
0 1/parenrightbigg /parenleftbigg1 0
sinθ1/parenrightbigg /parenleftbigg1−tan(θ/2)
0 1/parenrightbigg
. (18)
5This expression is in the form of TLT, but it can also be written in the form of LTL. If we
take the transpose and change the sign of θ,Rbecomes
R′=/parenleftbigg1 0
tan(θ/2) 1/parenrightbigg /parenleftbigg1−sinθ
0 1/parenrightbigg /parenleftbigg1 0
tan(θ/2) 1/parenrightbigg
. (19)
BothRandR′are the same matrix but are decomposed in different ways.
As for the two-parameter symmetric matrix of Eq.(17), we sta rt with a symmetric LTLT
form
S=/parenleftbigg1 0
b1/parenrightbigg /parenleftbigg1a
0 1/parenrightbigg /parenleftbigg1 0
a1/parenrightbigg /parenleftbigg1b
0 1/parenrightbigg
, (20)
which can be combined into one symmetric matrix:
S=/parenleftbigg1 +a2b(1 +a2) +a
b(1 +a2) +a1 + 2ab+b2(1 +a2)/parenrightbigg
. (21)
By comparing Eq.(17) and Eq.(21), we can compute the paramet ersaandbin terms of η
andφ. The result is
a=±/radicalBig
(coshη−1) + (sinh η) cos(2 φ),
b=(sinhη) sin(2 φ)∓/radicalBig
(coshη−1) + (sinh η) cos(2 φ)
coshη+ (sinh η) cos(2 φ). (22)
This matrix can also be written in a TLTL form:
S′=/parenleftbigg1b′
0 1/parenrightbigg /parenleftbigg1 0
a′1/parenrightbigg /parenleftbigg1a′
0 1/parenrightbigg /parenleftbigg1 0
b′1/parenrightbigg
. (23)
Then the parameters a′andb′are
a′=±/radicalBig
(coshη−1)−(sinhη) cos(2 φ),
b′=(sinhη) sin(2 φ)∓/radicalBig
(coshη−1)−(sinhη) cos(2 φ)
coshη−(sinhη) cos(2 φ). (24)
The difference between the two sets of parameters abanda′b′is the sign of the parameter
η. This sign change means that the squeeze operation is in the d irection perpendicular to
the original direction. In choosing abora′b′, we will also have to take care of the sign of
the quantity inside the square root to be positive. If cos(2 φ) is sufficiently small, both sets
are acceptable. On the other hand, if the absolute value of (s inhη) cos(2 φ) is greater than
(coshη−1), only one of the sets, abora′b′, is valid.
We can now combine the SandRmatrices in order to construct the ABCD matrix. In
so doing, we can reduce the number of matrices by one
SR=/parenleftbigg1 0
b1/parenrightbigg /parenleftbigg1a
0 1/parenrightbigg /parenleftbigg1 0
a1/parenrightbigg /parenleftbigg1b−tan(θ/2)
0 1/parenrightbigg
×/parenleftbigg1 0
sinθ1/parenrightbigg /parenleftbigg1−tan(θ/2)
0 1/parenrightbigg
. (25)
6We can also combine making the product S′R′. The result is
/parenleftbigg1b′
0 1/parenrightbigg /parenleftbigg1 0
a′1/parenrightbigg /parenleftbigg1a′
0 1/parenrightbigg /parenleftbigg1 0
b′+ tan( θ/2) 1/parenrightbigg
×/parenleftbigg1−sinθ
0 1/parenrightbigg /parenleftbigg1 0
tan(θ/2) 1/parenrightbigg
. (26)
For the combination SRof Eq.(25), two adjoining Tmatrices were combined into one T
matrix. Similarly, two Lmatrices were combined into one for the S′R′combination of
Eq.(26).
In both cases, there are six matrices, consisting of three Tand three Lmatrices. This
is indeed, the minimum number of shear matrices needed for th e most general form for the
ABCD matrix with three independent parameters.
IV. PARA-AXIAL LENS OPTICS
So far, we have been investigating the possibilities of repr esenting the ABCD matrices
in terms of the two shear matrices. It is an interesting propo sition because this binary
representation could lead to a computer algorithm for compu ting the ABCD matrix in
optics as well as in other areas of physics. Indeed, this ABCD matrix has a deep root in
ray optics [6].
In para-axial lens optics, the lens and translation matrice s take the form
L=/parenleftbigg1 0
−1/f1/parenrightbigg
, T =/parenleftbigg1s
0 1/parenrightbigg
, (27)
respectively. Indeed, in the Introduction, this was what we had in mind when we defined
the shear matrices of LandTtypes. These matrices are applicable to the two-dimensiona l
space of
/parenleftbiggy
m/parenrightbigg
, (28)
where ymeasures the height of the ray, while mis the slope of the ray.
The one-lens system consists of a TLT chain. The two-lens system can be written as
TLTLT . If we add more lenses, the chain becomes longer. However, th e net result is one
ABCD matrix with three independent parameters. In Sec. III, we as ked the question of
how many LandTmatrices are needed to represent the most general form of the ABCD
matrix. Our conclusion was that six matrices, with three len s matrices, are needed. The
chain can be either LTLTLT orTLTLTL . In either case, three lenses are required. This
conclusion was obtained earlier by Sudarshan et al. in 1985 [7]. In this paper, using the
decomposition technique derived from the Bargman decompos ition, we were able to compute
the parameter of each shear matrix in terms of the three param eters of the ABCD matrix.
In para-axial optics, we often encounter special forms of th eABCD matrix. For instance,
the matrix of the form of Eq.(13) is for pure magnification [12 ]. This is a special case of the
decomposition given for SandS′in Eq.(21) and Eq.(23) respectively, with φ= 0. However,
7ifηis positive, the set a′b′is not acceptable because the quantity in the square root in
Eq.(24) becomes negative. For the abset,
a=±(eη−1)1/2, b =∓e−η(eη−1)1/2. (29)
The decomposition of the LTLT type is given in Eq.(20).
We often encounter the triangular matrices of the form [13]
/parenleftbiggA B
0D/parenrightbigg
or/parenleftbiggA0
C D/parenrightbigg
. (30)
However, from the condition that their determinant be one, t hese matrices take the form
/parenleftbiggeηB
0e−η/parenrightbigg
or/parenleftbiggeη0
C e−η/parenrightbigg
. (31)
The first and second matrices are used for focal and telescope conditions respectively. We
call them the matrices of BandCtypes respectively. The question then is how many shear
matrices are needed to represent the most general form of the se matrices. The triangular
matrix of Eq.(30) is discussed frequently in the literature [12,13]. In the present paper, we
are interested in using only shear matrices as elements of de composition.
Let us consider the Btype. It can be constructed either in the form
/parenleftbiggeη0
0e−η/parenrightbigg /parenleftbigg1e−ηB
0 1/parenrightbigg
(32)
or
/parenleftbigg1eηB
0 1/parenrightbigg /parenleftbiggeη0
0e−η/parenrightbigg
. (33)
The number of matrices in the chain can be either four or five. W e can reach a similar
conclusion for the matrix of the Ctype.
V. OTHER AREAS OF OPTICAL SCIENCES
We write the ABCD matrix for the ray transfer matrix [12]. There are many ray tr ansfers
in optics other than para-axial lens optics. For instance, a laser resonator with spherical
mirrors is exactly like para-axial lens optics if the radius of the mirror is sufficiently large [14].
If wave fronts with phase is taken into account, or for Gaussi an beams, the elements of the
ABCD matrix becomes complex [15,16]. In this case, the matrix ope ration can sometimes
be written as
w′=Aw+B
Cw+D, (34)
where wis a complex number with two real parameters. This is precise ly the bilinear
representation of the six-parameter Lorentz group [9]. Thi s bilinear representation was
discussed in detail for polarization optics by Han et al. [17]. This form of representation is
useful also in laser mode-locking and optical pulse transmi ssion [16].
8The bilinear form of Eq.(34) is equivalent to the matrix tran sformation [17]
/parenleftbiggv′
1
v′
2/parenrightbigg
=/parenleftbiggA B
C D/parenrightbigg /parenleftbiggv1
v2/parenrightbigg
, (35)
with
w=v2
v1(36)
This bilinear representation deals only with the ratio of th e second component to the first in
the column vector to which ABCD matrix is applicable. In polarization optics, for instance ,
v1andv2correspond to the two orthogonal elements of polarization.
Indeed, this six-parameter group can accommodate a wide spe ctrum of optics and other
sciences. Recently, the two-by-two Jones matrix and four-b y-four Mueller matrix have been
shown to be two-by-two and four-by-four representations of the Lorentz group [1]. Also re-
cently, Monz´ on and S´ anchez showed that multilayer optics could serve as an analog computer
for special relativity [5]. More recently, two-beam interf erometers can also be formulated in
terms of the Lorentz group [18].
CONCLUDING REMARKS
The Lorentz group was introduced to physics as a mathematica l device to deal with
Lorentz transformations in special relativity. However, t his group is becoming the major
language in optical sciences. With the appearance of squeez ed states as two-photon coherent
states [19], the Lorentz group was recognized as the theoret ical backbone of coherent states
as well as generalized coherent states [4].
In their recent paper [2], Han et al. studied in detail possible optical devices which
produce the shear matrices of Eq.(1). This effect is due to the mathematical identity called
“Iwasawa decomposition” [20,21], and this mathematical te chnique is relatively new in op-
tics. The shear matrices of Eq.(1) are products of Iwasawa de compositions. Since we are
using those matrices to produce the most general form of ABCD , we are performing inverse
processes of the Iwasawa decomposition.
It should be noted that the decomposition we used in this pape r has a specific purpose.
If purposes are different, different forms of decomposition m ay be employed. For instance,
decomposition of the ABCD matrix into shear, squeeze, and rotation matrix could serve
useful purposes for canonical operator representations [1 3,22]. The amount of calculation
seems to depend on the choice of decomposition.
Group theory in the past was understood as an abstract mathem atics. In this paper, we
have seen that it can be used as a calculational tool. We have a lso noted that there is a
place in computer science for group theoretical tools.
9APPENDIX A: BARGMANN DECOMPOSITION
In his 1947 paper [9], Bargmann considered
W=/parenleftbiggα β
β∗α∗/parenrightbigg
, (A1)
withαα∗−ββ∗= 1. There are three independent parameters. Bargmann then o bserved
thatαandβcan be written as
α= (cosh η)e−i(φ+λ), β = (sinh η)e−i(φ−λ). (A2)
Then Wcan be decomposed into
W=/parenleftbigge−iφ0
0eiφ/parenrightbigg /parenleftbiggcoshηsinhη
sinhηcoshη/parenrightbigg /parenleftbigge−iλ0
0eiλ/parenrightbigg
. (A3)
In order to transform the above expression into the decompos ition of Eq.(12), we take the
conjugate of each of the matrices with
C1=1√
2/parenleftbigg1i
i1/parenrightbigg
. (A4)
Then C1WC−1
1leads to
/parenleftbiggcosφ−sinφ
sinφcosφ/parenrightbigg /parenleftbiggcoshηsinhη
sinhηcoshη/parenrightbigg /parenleftbiggcosλ−sinλ
sinλcosλ/parenrightbigg
. (A5)
We can then take another conjugate with
C2=1√
2/parenleftbigg1 1
−1 1/parenrightbigg
. (A6)
Then the conjugate C2C1WC−1
1C−1
2becomes
/parenleftbiggcosφ−sinφ
sinφcosφ/parenrightbigg /parenleftbiggeη0
0e−η/parenrightbigg /parenleftbiggcosλ−sinλ
sinλcosλ/parenrightbigg
. (A7)
This expression is the same as the decomposition given in Eq. (12).
The combined effect of C2C1is
C2C1=1√
2/parenleftbiggeiπ/4eiπ/4
−e−iπ/4e−iπ/4/parenrightbigg
. (A8)
If we take the conjugate of the matrix Wof Eq.(A1) using the above matrix, the elements
of the ABCD matrix become
A=α+α∗+β+β∗,
B=−i(α−α∗+β−β∗),
C=−i(α−α∗−β+β∗),
D=α+α∗−β−β∗. (A9)
10It is from this expression that all the elements in the ABCD matrix are real numbers.
Indeed, the representation αβis equivalent to the ABCD representation. In terms of the
parameters λ, ηandφ,
A= (cosh η) cos(φ+λ) + (sinh η) cos(φ−λ),
B= (cosh η) sin(φ+λ) + (sinh η) sin(φ−λ),
C= (cosh η) sin(φ+λ)−(sinhη) sin(φ−λ),
D= (cosh η) cos(φ+λ)−(sinhη) cos(φ−λ). (A10)
11REFERENCES
[1] D. Han, Y. S. Kim, and M. E. Noz, J. Opt. Soc. Am. A 14, 2290 (1997); D. Han, Y. S.
Kim, and M. E. Noz, Phys. Rev. E 56, 6065 (1997).
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[3] Y. S. Kim and E. P. Wigner, Am. J. Phys. 58, 439 (1990).
[4] Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific,
Singapore, 1991).
[5] J. J. Monz´ on and L. L. S´ anchez-Soto, Phys. Lett. A 262, 18 (1999).
[6] H. Kogelnik and T. Li, Applied Optics 5, 1550 (1966), and the references listed in this
review paper.
[7] E. C. G. Sudarshan, N. Mukunda, and R. Simon, Optica Acta 32, 855 (1985).
[8] D. Han, Y. S. Kim, and M. E. Noz, Am. J. Phys. 67, 61 (1999).
[9] V. Bargmann, Ann. Math. 48, 568 (1947).
[10] A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
[11] D. Onciul, Optik 96, 20 (1994).
[12] A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (John Wiley &
Sons, New York, 1975).
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[14] W. K. Kahn, Applied Optics 4, 758 (1965).
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Electronics 34, 1075 (1998).
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[18] D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. E 61, 5907 (2000).
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A. Benjamin, New York, 1966).
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[22] M. Nazarathy and J. Shamir, J. Opt. Soc. Am. 72, 356 (1982); H. Sasaki, K. Shinozaki,
and T. Kamijoh, Opt. Eng. 35, 2240 (1996).
12 |
arXiv:physics/0008221v1 [physics.chem-ph] 24 Aug 2000Investigations of Amplitude and Phase Excitation Profiles in
Femtosecond Coherence Spectroscopy
Anand T.N. Kumar, Florin Rosca, Allan Widom and Paul M. Champ ion
Department of Physics and Center for Interdisciplinary Res earch on Complex Systems,
Northeastern University, Boston MA 02115
Abstract
We present an effective linear response approach to pump-pro be femtosecond coherence spectroscopy in the well
separated pulse limit. The treatment presented here is base d on a displaced and squeezed state representation
for the non-stationary states induced by an ultrashort pump laser pulse or a chemical reaction. The subsequent
response of the system to a delayed probe pulse is modeled usi ng closed form non-stationary linear response
functions, valid for a multimode vibronically coupled syst em at arbitrary temperature. When pump-probe sig-
nals are simulated using the linear response functions, wit h the mean nuclear positions and momenta obtained
from a rigorous moment analysis of the pump induced (doorway ) state, the signals are found to be in excellent
agreement with the conventional third order response appro ach. The key advantages offered by the moment
analysis based linear response approach include a clear phy sical interpretation of the amplitude and phase of
oscillatory pump-probe signals, a dramatic improvement in computation times, a direct connection between
pump-probe signals and equilibrium absorption and dispers ion lineshapes, and the ability to incorporate co-
herence such as those created by rapid non-radiative surfac e crossing. We demonstrate these aspects using
numerical simulations, and also apply the present approach to the interpretation of experimental amplitude
and phase measurements on reactive and non-reactive sample s of the heme protein Myoglobin. The role played
by inhomogeneous broadening in the observed amplitude and p hase profiles is discussed in detail. We also
investigate overtone signals in the context of reaction dri ven coherent motion.
I. INTRODUCTION
Femtosecond coherence spectroscopy (FCS) is an ultrafast p ump-probe technique that allows the experimen-
talist to create and probe coherent vibrational motions and ultrafast chemical reactions in real time [1–20]. In
a typical pump-probe experiment, an ultrashort pump laser p ulse is used to excite the sample of interest. The
subsequent non-stationary response of the medium is monito red by an optically delayed probe pulse. Owing to
the large spectral bandwidth available in a short laser puls e, it can generate non-stationary vibrational states
in a molecular system as shown in Fig. 1(a). The subsequent nu clear dynamics modulates the optical response
as detected by the probe pulse. Coherent vibrational motion in the ground state has been observed in crys-
talline and liquid phase systems [2,12], and in biological s pecimens having short excited state lifetimes such
as bacteriorhodopsin [8,9], and myoglobin [14]. For molecu les that have long-lived excited states, however, the
excited state coherence is dominant and has been identified i n several dye molecules in solution [1,6,20], in small
molecules in the gas phase [5,7,10], and in photosynthetic r eaction centers [11,17].
Apart from “field driven” coherence directly prepared by the laser fields, vibrational coherence can also be
driven by rapid non-radiative processes. For example, if we consider a third electronic state |f∝an}bracketri}htthat is coupled
non-radiatively to the photo-excited state |e∝an}bracketri}htas in Fig. 1(b), the wave-packet created in the excited state by the
pump can cross over to |f∝an}bracketri}ht, leaving a vibrationally coherent product [15]. Fig. 1(b) s uggests the importance of
taking a multidimensional view of the problem, whereby the s urface crossing between the reactant excited state
|e∝an}bracketri}htand the product state, |f∝an}bracketri}htalong the reaction coordinate Ris accompanied by the creation of a vibrational
coherence along the Qcoordinate that is coupled to the non-radiative transition . In earlier work, we have
presented expressions for the time dependent population an d first moment evolution of vibrational dynamics
following a Landau-Zener surface crossing [21]. These expr essions are rigorously valid, provided the quantum
yield for the reaction is unity (a condition that holds for NO and CO photolysis from heme proteins [22–24]).
A common theoretical formulation for pump-probe spectrosc opy is based on the third order susceptibility χ(3)
formalism, which provides a unified view of four wave mixing s pectroscopies [25–28] with different combinations
of fields, irrespective of whether they are continuous wave o r pulsed. However, the separation of the pump and
probe events is not clear in this formalism, since the pump in duced density matrix is implicitly contained in
the third order response functions. Thus, it is also attract ive to treat the pump and probe processes separately
in the well separated pulse (WSP) limit. For example, the “do orway/window” picture has been developed [28]
1which can be used to represent the pump and probe events as Wig ner phase space wave packets. This readily
enables a semi-classical interpretation of pump-probe exp eriments [29–31]. Another view of the WSP limit is
based on the effective linear response approach. In this appr oach, the pump induced medium is modeled using a
time-dependent linear susceptibility [32–34,8,27]. This has the appealing aspect that a pump-probe experiment
is viewed as the non-stationary extension of steady state ab sorption spectroscopy. |
arXiv:physics/0008222v1 [physics.gen-ph] 24 Aug 2000P-Loop Oscillator on Clifford Manifolds and
Black Hole Entropy
Carlos Castro∗and Alex Granik†
Abstract
A new relativity theory, or more concretely an extended rela tivity
theory, actively developed by one of the authors incorporat ed 3 basic
concepts. They are the old Chu’s idea about bootstarpping, N ottale’s
scale relativity, and enlargement of the conventional time -space by in-
clusion of noncommutative Clifford manifolds where all p-br anes are
treated on equal footing. The latter allowed one to write a ma ster
action functional. The resulting functional equation is si mplified and
applied to the p-loop oscillator. Its respective solution i s a general-
ization of the conventional point oscillator. In addition , it exhibits
some novel features: an emergence of two explicit scales del ineating
the asymptotic regimes (Planck scale region and a smooth reg ion of
a conventional point oscillator). In the most interesting P lanck scale
regime, the solution reproduces in an elementary fashion th e basic
relations of string theory ( including string tension quant ization). In
addition, it is shown that comparing the massive ( super) str ing degen-
eracy with the p-loop degeneracy one is arriving at the propo rtionality
between the Shannon entropy of a p-loop oscillator in D-dime nsional
space and the Bekenstein-Hawking entropy of the black hole o f a size
comparable with a string scale. In conclusion the Regge beha vior
follows from the solution in an elementary fashion.
∗Center for Theoretical Studies of Physical Systems,Clark A tlanta University,Atlanta,
GA. 30314; E-mail:castro@ts.infn.it
†Department of Physics, University of the Pacific, Stockton, CA.95211; E-
mail:galois4@home.com
11 Introduction
Recently a new relativity was introduced [1] -[8] with a purp ose to develop a
viable physical theory describing the quantum ”reality” wi thout introducing
by hand a priori existing background. This theory is based upon 3 main
concepts:
1) Chew’s bootstrap idea about an evolving physical system g enerating its
own background in the process.
2) Nottale’s scale relativity [9]-[10] which adopts the Pla nck scale Λ = 1 .62×
10−35mas the minimum attainable scale in nature.
3) a generalization of the ordinary space-time ( the concept most important
for our analysis) by introduction of non-commutative C-spa ces leading to
full covariance of a quantum mechanical loop equation. This is achieved by
extending the concepts of ordinary space-time vectors and t ensors to non-
commutative Clifford manifolds (it explains the name C-spac e) where all
p-branes are unified on the basis of Clifford multivectors. As a result, there
exists a one-to-one correspondence between single lines in Clifford manifolds
and a nested hierarchy of 0-loop, 1-loop,..., p-loop histories in Ddimensions
(D=p−1) encoded in terms of hypermatrices.
The respective master action functional S{Ψ[X(Σ)]}of quantum field theory
in C-space [11, 4] is
S{Ψ[X(Σ)]}=/integraltext[1
2(δΨ
δX∗δΨ
δX+E2Ψ∗Ψ) +g3
3!Ψ∗Ψ∗Ψ+
g4
4!Ψ∗Ψ∗Ψ∗Ψ]DX(Σ).(1)
where Σ is an invariant evolution parameter (a generalizati on of the proper
time in special relativity) such that
(dΣ)2= (dΩp+1)2+ Λ2p(dxµdxµ) + Λ2(p−1)dσµνdσµν+...
+(dσµ1µ2...µp+1dσµ1µ2...µp+1)(2)
and
X(Σ) = Ω p+1I+ Λpxµγµ+ Λp−1σµνγµγν+... (3)
is a Clifford algebra-valued line ”living” on the Clifford man ifold outside
space-time. Multivector XEq.(3) incorporates both a point history given by
the ordinary ( vector) coordinates xµand the holographic projections of the
nested family of all p-loop histories onto the embedding coordinate spacetime
2hyperplanes σµ1µ2...µp+1. The scalar Ω p+1is the invariant proper p+ 1 = D-
volume associated with a motion of a maximum dimension p-loop across the
p+1 = D-dim target spacetime. Since a Cliffordian multivector with Dbasis
elements ( say, e1, e2, ..., e D) has 2Dcomponents our vector Xhas also 2D
components.
Generally speaking, action (1) generates a master Cantoria n (strongly frac-
tal) field theory with a braided Hopf quantum Clifford algebra . This action is
unique in a sense that the above algebra selects terms allowe d by the action.
In what follows we restrict our attention to a truncated vers ion of the theory
by applying it to a linear p-loop oscillator.
This truncation is characterized by the following 3 simplifi cations. First, we
dropped nonlinear terms in the action, that is the cubic term ( corresponding
to vertices) and the quartic (braided scattering) term. Sec ondly, we freeze
all the holographic modes and keep only the zero modes which w ould yield
conventional differential equations instead of functional ones. Thirdly, we
assume that the metric in C-space is flat.
2 Linear Non-Relativistic p-loop Oscillator
As a result of the postulated simplifications we obtain from t he action (1) a
C-space p-loop wave equation for a linear oscillator
{−1
21
Λp−1[∂2
∂xµ2+ Λ2∂2
(∂σµν)2+ Λ4∂2
(∂σµνρ)2+...+ Λ2p∂2
(∂Ωp+1)2]+
mp+1
21
L2[Λ2pxµ2+ Λ2p−2σµν2+...+ Ω p+1]}Ψ =TΨ(4)
where∂2
(∂xµ)2=gµν∂
∂xµ∂
∂xν,∂2
(∂σµνρ)2=gµαgνβ∂
∂σµα∂
∂σνβ, ..., etc. ,Tis tension of
the spacetime-filling p-brane, mp+1is the parameter of dimension ( mass)p+1
, parameter L(to be defined later) has dimension lengthp+1and we use units
¯h= 1, c= 1. A generalized correspondence principle1allows us to introduce
the following qualitative correspondence between the para meters mp+1, L,
and mass mand amplitude aof a point (particle) oscillator:
mp+1(”mass”)←→m,
L(”amplitude ”)←→a
1In the limit of Λ /a→0 volume Ω p+1, holographic coordinates σµν, σµνρ, ...→0, and
p-loop oscillator should become a point oscillator, that is p-loop histories collapse to a
point history
3We rewrite Eq.(4) in the dimensionless form as follows
{∂2
∂˜x2µ+∂2
∂˜σ2µν+...−(˜Ω2+ ˜x2
µ+ ˜σ2
µν+...) + 2T}Ψ = 0 (5)
whereT=T//radicalBig
Amp+1is the dimensionless tension,
˜xµ=A1/4Λp
Lxµ,˜σµν=A1/4σµνΛp−1
L, ...,˜Ωp+1=A1/4Ωp+1
L
are the dimensionless arguments, ˜ xµhasCD1≡Dcomponents, ˜ σµνhas
CD2≡D!
(D−2)!2!components, etc. and
A≡mp+1L2/Λp+1
Without any loss of generality we can set A= 1 by absorbing it into L.
This will give the following geometric mean relation betwee n the parameters
L, m p+1, and Λ
L2= Λp+1/mp+1
The dimensionless coordinates then become
˜xµ=/radicalBig
Λp+1mp+1xµ/Λ,˜σµν=/radicalBig
Λp+1mp+1σµν/Λ2, ...,
˜Ωp+1=/radicalBig
Λp+1mp+1Ωp+1/Λp+1
As a result we obtain a new dimensionless combination Λp+1mp+1indicating
existence of two separate scales : Λ and (1 /mp+1)1
p+1. It is easily seen that
this dimensionless combination obeys the following double inequality:
/radicalBig
mp+1Λp+1<1</radicalBigg
1
mp+1Λp+1(6)
Relations (6) define two asymptotic regions:
1)the ”fractal” region characterized by mp+1Λp+1∼1(area of Planck scales)
and
2)the ”smooth” region characterized by mp+1Λp+1>>1.
4Since equation (5)is diagonal in its arguments we represent its solution as a
product of separate functions of each of the dimensionless a rguments ˜ xµ,˜σµν, ...,
Ψ =/productdisplay
iFi(˜xi)/productdisplay
j<kFjk(˜σjk)... (7)
Inserting (7) into (5) we get for each of these functions the W hittaker equa-
tion:
Z′′−(2T−˜y2)Z= 0 (8)
where Zis any function Fi, Fij, ..., ˜yis the respective dimensionless variable
˜xµ,˜σµν, ..., and there are all in all 2Dsuch equations. The bounded solution
of (8) is expressed in terms of the Hermite polynomials Hn(˜y)
Z∼e−˜y2/2Hn(˜y) (9)
Therefore the solution to Eq.(5) is
Ψ∼exp[−(˜x2
µ+ ˜σ2
µν+...+˜Ωp+1)]/productdisplay
iHni(˜xi)/productdisplay
jkHnjk(˜σjk)... (10)
where there are Dterms corresponding to n1, n2, ..., n D,D(D−1)/2 terms
corresponding to n01, n02, ...,etc. Thus the total number of terms corre-
sponding to the N-th excited state ( N=nx1+nx2+...+nσ01+nσ02+...) is
2D.
The respective value of the tension of the N-th excited state is
TN= (N+1
22D)mp+1 (11)
yielding quantization of tension.
Expression (11) is the analog of the respective value of the N-th energy state
for a point oscillator. The analogy however is not complete. We point out
one substantial difference. Since according to a new relativ ity principle [1]
-[8] all the dimensions are treated on equal footing (there a re no preferred
dimensions) all the modes of the p-loop oscillator( center of mass xµ, holo-
graphic modes, p+ 1 volume) are to be excited collectively. This behavior is
in full compliance with the principle of polydimensional in variance by Pez-
zagalia [12]. As a result, the first excited state is not N= 1 ( as could be
naively expected) but rather N= 2D. Therefore
T1→T2D=3
2(2Dmp+1)
5instead of the familiar (3 /2)m.
Recalling that Lis analogous to the amplitude aand using the analogy
between energy E∼mω2a2and tension T, we get T=mp+1Ω2L2. Inserting
this expression into Eq.(11) we arrive at the definition of th e ”frequency” Ω
of the p-loop oscillator:
ΩN=/radicalbigg
(N+ 2D−1)mp+1
Λp+1(12)
where we use L=/radicalBig
Λp+1/mp+1.
Having obtained the solution to Eq.(5), we consider in more d etail the two
limiting cases corresponding to the above defined 1) fractal and 2) smooth
regions. The latter ( according to the correspondence princ iple) should be
described by the expressions for a point oscillator. In part icular, this means
that
˜xµ=xµ
Λ/radicalBig
mp+1Λp+1→xµ/a (13)
where parameter a >> Λ is a finite quantity ( amplitude) and we use R= 1.
Using Eq.(13) we find
mp+1→(MPlanck)p+1(Λ
a)2<<(MPlanck)p+1(14)
where the Planck mass MPlanck≡1/Λ. From Eqs.(11) and (12) follows that
in this region
TN∼(MPlanck)p+1(Λ
a)2<<(MPlanck)p+1
ΩN∼(ωPlanck)p+1Λ
a<<(ωPlanck)p+1
ωPlanck = 1/Λ(15)
in full agreement with this region’s scales as compared to th e Planck scales.
At the other end of the spectrum ( fractal region) where mp+1Λp+1∼1 we
would witness a collapse of all the scales to only one scale, n amely the Planck
scale Λ. In particular, this means that a∼Λ, and the oscillator parameters
become
˜xµ=xµ
Λ/radicalBig
Λp+1mp+1∼xµ
Λ, m p+1∼1
Λp+1≡(MPlanck)p+1, T∼mp+1∼1
Λp+1
(16)
These relations are the familiar relations of string theory . In particular, if we
setp= 1 we get the basic string relation
6T∼1
Λ2≡1
α′
Above we got two asymptotic expression for mp+1
mp+1=/braceleftBigg
Λ−(p+1)(Λ/a)2if Λ/a << 1
Λ−(p+1)ifmp+1/Λp+1∼1, a∼Λ
This indicates that we can represent mp+1Λp+1as power series in (Λ /a)2(e.g.,
cf. analogous procedure in hydrodynamics [13]):
mp+1Λp+1= (Λ
a)2[1 +α1(Λ
a)2+α2(Λ
a)4+...]
where the coefficients αiare such that the series is convergent for a∼Λ.
Ifp= 1 then using the fact that in this case energy Tground = 2m2(see
footnote2), returning to the units ¯ h, and introducing 1 /a=ω( where ωis
the characteristic frequency) we get (cf.ref [5])
¯heff= ¯h/radicalBig
1 +α1(Λ
a)2+α2(Λ
a)4+...
Truncating the series at the second term , we recover the stri ng uncertainty
relation
¯heff= ¯h[1 + (α1/2)(Λ/a)2]
Interestingly enough, the string uncertainty relation sti ll did not have ” a
proper theoretical framework for the extra term” [14]. On th e other hand,
this relation emerges as one of the results of our theoretica l model.
As a next step we find the degeneracy associated with the N-th excited level
of the p-loop oscillator. The degeneracy dg(N) is equal to the number of par-
titions of the number Ninto a set of 2Dnumbers{nx1, nx2, , ..., n xD, nσµν, nσµνρ, ...}:
dg(N) =Γ(2D+N)
Γ(N+ 1)Γ(2D)(17)
where Γ is the gamma function.
We compare dg(N) (17) with the asymptotic quantum degeneracy of a mas-
sive (super) string state given by Li and Yoneya [15]:
dg(n) =exp[2π/radicalBigg
nds−2
6] (18)
2that is for a point oscillator we get Eground = ¯hω/2 =/radicalbig
Tground /8
7where dsis the string dimension and n >> 1. To this end we equate (18)
and degeneracy (17) of the first excited state ( N= 2D) of the p-loop. This
could be justified on physical grounds as follows. One can con sider different
frames in a new relativity: one frame where an observer sees o nly strings (
with a given degeneracy) and another frame where the same obs erver sees a
collective excitations of points, strings, membranes,p-l oops, etc. The results
pertinent to the degeneracy (represented by a number) shoul d be invariant
in any frame.
Solving the resulting equation with respect to√nwe get
√n=1
2π/radicalBigg
6
ds−2Ln[Γ(2D+1)
Γ(2D+ 1)Γ(2D)] (19)
The condition n >> 1 implies that D >> 1 thus simplifying (19). If we
setds= 26 ( a bosonic string) and use the asymptotic representatio n of the
logarithm of the gamma function for large values of its argum ent
LnΓ(z) =Ln(√
2π) + (z−1/2)Ln(z)−z+O(1/z)
we get:
√n≈2Dln(2)
2π∼2D−1∼N (20)
From (Eq.18) follows that for n >> 1 Ln[dg(n)]∼√n. On the other hand,
Li and Yoneya [15] showed that in this case√n∼SBH,where SBHis the
Bekenstein-Hawking entropy of a Schwarzschild black hole. Taking into ac-
count Eq. (20) we obtain
SBH∼2D−1(21)
This is a rather remarkable fact: the Shannon entropy of a p-loop oscillator
inD-dimensional space ( for a sufficiently large D), that is a number N=
2D( the number of bits representing all the holographic coordi nates), is
proportional to the Bekenstein-Hawking entropy of a Schwar zschild black
hole.
Finally, Eq.(20) allows us to easily derive the Regge behavi or of a string
spectrum for large values of n >> 1. To this end we associate with each bit
of ap-loop oscillator fundamental Planck length Λ, area Λ2, mass 1 /Λ,etc.
As a result, l2
s∼Area s= N×Λ2, ms2= N×M2
Planck. On the other hand,
according to (20) N∼√nwhich yields
ls∼√nΛ2;m2
s∼√n M2
Planck
8Therefore the respective angular momentum Jis
J=m2×ls2∼nM2
PlanckΛ2=n
where we use MPlanckΛ≡1 by definition.
3 Conclusion
Application of a simplified linearized equation derived fro m the master action
functional of a new ( extended) relativity to a p-loop oscillator has allowed
us to elementary obtain rather interesting results. First o f all, the solution
explicitly indicates existence of 2 extreme regions charac terized by the values
of the dimensionless combination mp+1Λp+1:
1) the fractal region where mp+1Λp+1∼1 and 2 scales collapse to one, namely
Planck scale Λ
and
2) the smooth region where mp+1Λp+1<<1 and we we recover the description
of the conventional point oscillator. Here 2 scales are pres ent , a character-
istic ”length” aand the ubiquitous Planck scale Λ ( a << Λ)demonstrating
explicitly the implied validity of the quantum mechanical s olution in the
region where a/Λ<<1.
For a specific case of p= 1 ( a string) the solution yields ( one again in an
elementary fashion) one of the basic relation of string theo ryT= 1/α′). In
addition, it provides us with a derived string uncertainty r elation, which in
itself is a truncated version of a more general uncertainty r elation [5].
Comparing the degeneracy of the ground state of the p-loop fo r a very large
number of of dimensions Dwith the respective expressions for the massive (
super) string theory given by Li and Yoneya [15] we found that the Shannon
entropy of a p-loop oscillator in D-dimensional space ( for a sufficiently large
D), that is a number N= 2D( the number of bits representing all the
holographic coordinates), is proportional to the Bekenste in-Hawking entropy
of a Schwarzschild black hole.
The Regge behavior of the string spectrum for large n >> 1also follows from
the obtained solution thus indicating its, at least qualita tively correct, char-
acter. Thus a study of a simplified model ( or ”toy”) problem of a linearized
p-loop oscillator gave us ( with the help of elementary calcul ations)a wealth
of both the well-known relations of string theory ( usually o btained with the
9help of a much more complicated mathematical technique)and some addi-
tional relations ( the generalized uncertainty relation). This indicates that
the approach advocated by a new relativity might [4, 11] be ve ry fruitful,
especially if it will be possible to obtain analytic results on the basis of the
full master action functional leading to functional nonlin ear equations whose
study will involve braided Hopf groups.
Acknowledgements The authors would like to thank E.Spalucci and
S.Ansoldi for many valuable discussions and comments.
References
[1] C. Castro , ” Hints of a New Relativity Principle from p-Brane Quantum
Mechanics ” J. Chaos, Solitons and Fractals 11(11)(2000) 1721
[2] C. Castro , ” The Search for the Origins of MTheory : Loop Quantum
Mechanics and Bulk/Boundary Duality ” hep-th/9809102
[3] S. Ansoldi, C. Castro, E. Spallucci , ” String Representa tion of Quantum
Loops ” Class. Quant. Gravity 16(1999) 1833;hep-th/9809182
[4] C.Castro, ” Is Quantum Spacetime Infinite Dimensional?” J. Chaos,
Solitons and Fractals 11(11)(2000) 1663
[5] C. Castro, ” The String Uncertainty Relations follow fro m the New
Relativity Theory ” hep-th/0001023; Foundations of Physic s , to be
published
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Relativity ” physics/ 0002019;
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tum Paradoxes”, J.Chaos, Solitons, and Fractals 11(11) (2000) 2167.
[8] C. Castro, ”An Elementary Derivation of the Black-Hole A rea-Entropy
Relation in Any Dimension ” hep-th/0004018
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Scale Relativity ” World Scientific, 1993;
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11 |
arXiv:physics/0008223v1 [physics.flu-dyn] 25 Aug 2000New complex variables for equations
of ideal barotropic fluid
A.L. Sorokin
Institute of Thermophysics, 630090 Novosibirsk, Russia
February 2, 2008
Abstract
We propose new construction of dependent variables for equa tions
of an ideal barotropic fluid. This construction is based on a d irect gen-
eralization of the known connection between Schroedinger e quation
and a system of Euler-type equations. The system for two comp lex-
valued functions is derived that is equivalent to Euler equa tions. Pos-
sible advantages of the proposed formulation are discussed .
1 Introduction
When solving a partial problem of fluid dynamics or exploring general prop-
erties of governing equations one often use different choice of the dependent
variables. Introduction of a stream function is common prac tice for two-
dimensional problems. For a general case of a 3D time-depend ent flow one
can use a vector potential, a pair of stream functions (for in compressible
case), Clebsch potentials and etc. Clebsch potentials are m ainly used with
intention to exploit preferences of Lagrange description o f a fluid motion.
The new representation is based on the use of multi-valued po tentials and
Euler approach. The paper is composed as follows. In the seco nd section
we analyze Madelung transformation that connects a generic Schroedinger
equation with a system of Euler-type equations. Some genera lization will be
made for the case of potential flows of a barotropic fluid. In th e next sec-
tion the generalization of Madelung transformation for a ge neral vector field
1will be derived, that leads to the system of equations (8) for two complex-
valued functions with arbitrary potentials. In the fourth s ection we use this
arbitrariness and propose the choice of potentials, that ma ke the system
equivalent to Euler equations for an ideal barotropic fluid. To substantiate
this we will derive Euler equations from the system (8). In th e last section we
discuss possible preferences of new choice of dependent var iables and their
relation to vortices.
2 Madelung transformation
Since pioneer work by E.Madelung [1] physical literature co ntains many ex-
amples of connection between Schroedinger equation of quan tum mechanics
and fluid dynamics. Typical exposition of this connection is the substitution
ψ=√ρeiϕ
βinto
i∂ψ
∂t=−β
2∆ψ+Vψ (1)
that leads to∂ρ
∂t+∇ ·(ρ∇ϕ) = 0 (2)
∂ϕ
∂t+(∇ϕ)2
2=−V+β
2∆√ρ√ρ(3)
This trick looks slightly mystical for novice. Some histori cal notes and elu-
cidation can be found in [3]. More clear is back substitution . Following
Madelung [2], let’s linearize equation (2) using substitut ion
ρ=ψψ, ϕ =−iβ
2ln/parenleftBiggψ
ψ/parenrightBigg
(4)
wereβhas dimension of kinematical viscosity. After simple algeb ra one can
obtain
ψ/parenleftBigg∂ψ
∂t−iβ
2∆ψ/parenrightBigg
+ψ/parenleftBigg∂ψ
∂t+iβ
2∆ψ/parenrightBigg
= 0
Choice∂ψ
∂t−iβ
2∆ψ=iVψ
leads to Schroedinger equation. Here Vis a real-valued function of a time,
coordinates and/or ψ. We can conclude that this equation leads to conser-
vation of probability, but dynamics is completely defined by potentialV.
2Now from hydrodynamical viewpoint let’s summarize restric tions that
were implicitly used in this derivation. First, interpreti ngρas density of some
fluid with an arbitrary equation of state, we see that fluid flow is supposed
to be potential. Second, we use dimensional constant β.
To describe an ideal fluid, we can to overcome the second restr iction using
a non-dimensional form of equation (2) ( β= 1) and the potential
V= Π (ρ) +1
2∆√ρ√ρ
This choice give Cauche-Lagrange equation for barotropic fl uid
∂ϕ
∂t+(∇ϕ)2
2=−Π
but leads to
i∂ψ
∂t=−1
4/parenleftBigg
∆ψ−ψ
ψ∆ψ/parenrightBigg
+
−1
8/parenleftBigg
∇ln/parenleftBiggψ
ψ/parenrightBigg/parenrightBigg2
+ Π/parenleftBig
ψψ/parenrightBig
ψ
that differs from Schroedinger equation. This form of equati on of an ideal
barotropic fluid seems to be unknown.
3 Generalization of Madelung transformation
We consider a direct generalization of the previous scheme f or the case of
two complex-valued functions and introduce definitions
ρ=ρ1+ρ2,J=ρV=ρ1∇ϕ1+ρ2∇ϕ2 (5)
Obviously, permutation of indexes should not have any physi cal consequence.
For velocity and vorticity we obtain
V=ρ1
ρ∇ϕ1+ρ2
ρ∇ϕ2,∇ ×V=ρ1ρ2
ρ2∇ln/parenleftBiggρ1
ρ2/parenrightBigg
× ∇(ϕ1−ϕ2) (6)
The requirement of possibility to represent a vector field wi th a non-zero
total helicity
H=/integraldisplayρ1ρ2
ρ2ln/parenleftBiggρ1
ρ2/parenrightBigg
(∇ϕ1× ∇ϕ2)·dσ/negationslash= 0
3implies a multi-valuedness of potentials [4] (here integra l should be taken over
some closed surface). That is admissible due to usage of the c omplex-valued
variables.
Linearizing
∂ρ
∂t+∇ ·J= 0 (7)
after some algebra we obtain
ψ1/parenleftBigg∂ψ1
∂t−i
2∆ψ1/parenrightBigg
+ψ1/parenleftBigg∂ψ1
∂t+i
2∆ψ1/parenrightBigg
+ψ2/parenleftBigg∂ψ2
∂t−i
2∆ψ2/parenrightBigg
+ψ2/parenleftBigg∂ψ2
∂t+i
2∆ψ2/parenrightBigg
= 0
By inspection one can show that choice
∂ψk
∂t−i
2∆ψk=Ukψk
with
U1=ρ2
2ρI−iV1, ,U 2=−ρ1
2ρI−iV2
whereI,V1,V2are real-valued functions of time, coordinates and/or ψksolve
this equation. We obtain the following system of equations
i∂ψ1
∂t=−∆ψ1
2+/parenleftBiggρ2
2ρiI+V1/parenrightBigg
ψ1, i∂ψ2
∂t=−∆ψ2
2+/parenleftBigg
−ρ1
2ρiI+V2/parenrightBigg
ψ2(8)
Substitutions ψk=√ρkexp(iϕ) give the equivalent system
∂ρk
∂t+∇·(ρk∇ϕk) = (−1)k−1ρ1ρ2
ρI,∂ϕk
∂t+(∇ϕk)2
2=−Vk+1
2∆√ρk√ρk(9)
Equation (7) follows from the first two equations of this syst em.
4 New form of Euler equations
To apply the derived system to description of an ideal barotr opic flow we
need a proper choice of the potentials I,V1,V2. By inspection it was found
that
V1= Π (ρ)−ρ22
2ρ2w2+1
2∆√ρ1√ρ1(10)
4V2= Π (ρ)−ρ12
2ρ2w2+1
2∆√ρ2√ρ2(11)
I=∇ ·w+w
ρ·/parenleftBigg
ρ2∇ρ1
ρ1+ +ρ1∇ρ2
ρ2/parenrightBigg
(12)
make system equivalent to Euler equations. Here w=∇(ϕ1−ϕ2). The
invariance of systems (8),(9) with respect to both Galilei g roup and indexes
permutation can be directly checked.
Substitution of (10-12) into (9) give
∂ρ1
∂t+∇ ·(ρ1∇ϕ1) =ρ1ρ2
ρI,∂ρ2
∂t+∇ ·(ρ2∇ϕ2) =−ρ1ρ2
ρI, (13)
∂ϕ1
∂t+(∇ϕ1)2
2=−Π +ρ22
2ρ2w2(14)
∂ϕ2
∂t+(∇ϕ2)2
2=−Π +ρ12
2ρ2w2(15)
From equations (13) follows (7).
Now we start derivation of equation for flux J. First, multiplying (14),(15)
byρkrespectively, summing and taking gradient of result, then a dding to
obtained equation (13), multiplied by ∇ϕkrespectively, one can obtain
∂J
∂t+∇/parenleftBiggj12
2ρ1+j22
2ρ2/parenrightBigg
+/bracketleftBigg
∇ρ1∂ϕ1
∂t+∇ρ2∂ϕ2
∂t/bracketrightBigg
+/parenleftBiggj1· ∇j1
ρ1+j2· ∇j2
ρ2−ρ1ρ2
ρIw/parenrightBigg
=−∇/parenleftBigg
ρΠ−ρ1ρ2
ρw2
2/parenrightBigg
where jk=ρk∇ϕk. Using identities
J2
2ρ=j12
2ρ1+j12
2ρ1−ρ1ρ2
ρw2
2
J∇ ·J
ρ=j1∇ ·j1
ρ1+j2∇ ·j2
ρ2−ρ1ρ2
ρ/parenleftBigg∇ ·j1
ρ1−∇ ·j2
ρ2/parenrightBigg
w
after some algebra one can obtain
∂J
∂t+∇/parenleftBiggJ2
2ρ/parenrightBigg
−J2
2ρ∇ρ
ρ+J∇ ·J
ρ
5+/bracketleftBigg
∇ρ1∂ϕ1
∂t+∇ρ2∂ϕ2
∂t+J2
2ρ∇ρ
ρ+ Π∇ρ/bracketrightBigg
+ρ1ρ2
ρ/parenleftBigg∇ ·j1
ρ1−∇ ·j2
ρ2−I/parenrightBigg
w=−ρ∇Π
Algebraic transformations of terms in square braces with ac count for first
identity and (14),(15) lead to equation
∂J
∂t+∇/parenleftBiggJ2
2ρ/parenrightBigg
−J2
2ρ∇ρ
ρ+J∇ ·J
ρ
+ρ1ρ2
ρ/bracketleftBigg
∇ln/parenleftBiggρ1
ρ2/parenrightBigg/parenleftBigg∂ϕ1
∂t−∂ϕ2
∂t/parenrightBigg
+/parenleftBigg∇ ·j1
ρ1−∇ ·j2
ρ2−I/parenrightBigg
w/bracketrightBigg
=−ρ∇Π
Using definition of velocity and equations (14),(15) after d irect algebra one
can show that terms in square braces give Lamb vector
V× ∇ × V=ρ1ρ2
ρ2/parenleftBigg
(V·w)∇ln/parenleftBiggρ1
ρ2/parenrightBigg
−V· ∇ln/parenleftBiggρ1
ρ2/parenrightBigg
w/parenrightBigg
We obtain the equation
∂J
∂t+∇/parenleftBiggJ·J
2ρ/parenrightBigg
−J·J
2ρ∇ρ
ρ−J× ∇ × V+V∇ ·J=−ρ∇Π (16)
To make last step in derivation one should use continuity equ ation to obtain
from (16) Euler equation in Gromeka-Lamb form
∂ρ
∂t+∇ ·J= 0,∂V
∂t+∇/parenleftBiggV·V
2/parenrightBigg
−V× ∇ × V=−∇Π (17)
The result is as follows: System (8) is equivalent to system o f Euler equation
(17).
5 Discussion
First of all, the attractive feature of (8) is the homogeneit y both depen-
dent variables and equations in contrast to the non-homogen eity of veloc-
ity/density and form of equations in (18). This property can be used both
numerically and analytically. Homogeneity and eliminatio n of the convective
6derivative can substantially simplify numerical algorith m. As far as multi-
valuedness is concerned, the possibility of use multi-valu ed potentials was
clearly demonstrated in [5]. In analytical way the aforemen tioned property
can simplify proof of existence and uniqueness theorems. Al so application
of geometrical methods to partial differential equations (8 ) is looking quite
natural.
This formulation of Euler equation can have another interes ting property.
Zeroes of solution of nonlinear Schoedinger equation corre spond to a vortex
axes (topological defects) [5]. At a moment the condition ψ= 0 defines two
surfaces, and their intersection defines a space curve (poss ibly, disconnected).
Note similarity with definition of a vortex as zero of an anali tical complex-
valued function in two-dimensional hydrodynamic of ideal i ncompressible
fluid. If the system (8) inherits this property from its proto type (1) the
known problem of a vortex definition [6] can be solved in gener al case.
6 Acknowledgments
Author express his gratitude to Prof. S.K.Nemirovsky and Dr . G.A.Kuz’min.
References
[1] E. Madelung. Quantentheorie in Hydrodynamischer form. Zts.f.Phys ,
40:322–326, 1926.
[2] E. Madelung. Die Mathematischen Hilfsmittel Des Physikers . Springer-
Verlag, 1957.
[3] E.A. Spiegel. Fluid dynamical form of the linear and nonl inear
Schroedinger equation. Physica D. , 236–240, 1980.
[4] J.W.Yokota. Potential/complex-lamellar description s of incompressible
viscous flows. Phys. Fluids. , 9(8):2264–2272, 1997.
[5] C.Nore, M.Abid, M.Brachet. Nonlinear Schroedinger equ ation: an Hy-
drodynamical Tool? Small-scale structures in 3D hydro- and magneto-
hydrodynamic turbulence. M.Meneguzzi,A.Pouquet,P.L.Sa lem eds. Lec-
ture Notes in Physics, Springer-Verlag, pp.105-112,1995.
7[6] J.Jeong, F.Hussain. On the identification of a vortex. J. of Fluid.Mech. ,
295:69–94, 1995.
8 |
CHALLENGES IN AFFECTING US ATTITUDES
TOWARDS SPACE SCIENCE
Howard A. Smith
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA, 02138 USA
hsmith@cfa.harvard.edu
"[And] though I know that the speculations of a philosopher are far removed from the judgment of the
multitude - for his aim is to seek truth in all things as far as God has permitted human reason so to do -- yet
I hold that opinions which are quite erroneous should be avoided."
--Nicholaus Copernicus, De Revolutionibus , from the preface to Pope Paul III
(as cited in Theories of the Universe by Milton K. Munitz, 1957)
I. THE "JUDGMENT OF THE MULTITUDE"
Contrary to the sentiment Copernicus expressed to Pope Paul III circa 1542 , today the "judgment of the
multitude" is not far removed from the research of space science, nor from its other activities for that matter.
The pubic pays for them, and pays attention to them. The Federal government provides billions of dollars to
the space science enterprise; state and private contributors also provide substantial funding. Progress in space
science therefore depends upon public dollars, although continued financial support is only one reason --
perhaps not even the most important one -- why the space science community should be attentive to popular
perceptions. In this article I will review our knowledge of the US public’s attitudes, and will argue that
vigorous and innovative education and outreach programs are important, and can be made even more
effective.
In the US, space science enjoys broad popular support. People generally like it, and indeed say that they
follow it with interest. Later in this article I will discuss some specific survey results (Sections II - V), and
the somewhat paradoxical result that, despite being interested and supportive, people are often ignorant about
the basic facts (Section V). The 1985 study by the Royal Society of Britain entitled, "The Public
Understanding of Science" (Royal Society, 1985) is a landmark document in addressing the topic, and in the
US, the NSF’s series "Science and Engineering Indicators" ( S&EI; National Science Board, 2000) is an
ongoing, statistical study of public attitudes from which I draw many examples. Although these studies and
their datasets have come under various levels of legitimate criticism (e.g., Irwin and Wynne, 1996; Section
VI), I will argue that they provide useful and relatively self-consistent statistics from which to consider the
state of the public’s consciousness.
Why should we as scientists care about these social analyses, or the statistical subtleties of the public’s
knowledge, interest, or “understanding,” especially when the methodology of such studies is under attack?
First, despite their limitations, such surveys show that we (i.e., the space science community) can do better
at public education – in none of the important measures of public response are the results close to "saturation."
And secondly, they highlight disconcerting but redeemable public attributes, prompting me to suggest we
ought to do better -- not in order to increase budgets, but because a scientifically literate society (not
proficient, just literate) is essential to rational discourse and judgment in a millennium dominated by science
and technology which to many people increasingly resembles sorcery. Space science, popular as it is
according to all studies, is one of the most potent areas of inquiry which science has at its disposal to teach
the facts and methods of modern scientific research. In Section VII, I look at basic research and the reasonsfor pursuing it, and note that there are intangible benefits to space research. In Section VIII, I will mention
some outreach challenges, and two innovative programs.
The term "space science" encompasses a much wider arena of topics than simply astronomy or satellite-
based research. The popular conception of the topic includes rockets and the technologies needed for rocket
and shuttle launches, their control and tracking, and also the technologies for new instruments; the results of
the National Air and Space Museum survey (see below) confirm this. The term rightly includes the manned
aspects of space exploration, from the Apollo-to-the-Moon missions to future manned missions to Mars, as
well as earth-orbiting space stations. Some of the surveys I will discuss make explicit distinctions between
these various areas, but most often they do not.
II. ALL THE SURVEYS SAY THAT PEOPLE ARE INTERESTED
For over fifteen years the National Science Board of the National Science Foundation has taken "science
indicator surveys" of the US public’s relationships with science (National Science Board, 2000), asking people
about their attitudes towards a wide range of science topics, including in particular space exploration but also
medicine, nuclear power, environment, and technology. While this limited breakdown of science topics
constrains some of our conclusions (and see below), it is adequate for most of the discussion. There have
been ups and downs in the numbers over time, reflecting, for example, concerns after the explosion of the
space shuttle Challenger, or budget deficits, but the general conclusions have been roughly the same: a huge
number of Americans -- 77% in 1997 -- say they are very interested or moderately interested in space
exploration. A majority of people also say they are interested in "new scientific discoveries"-- 91% -- so space
science is not unique in its appeal, but it is remarkable in that it does not involve the immediate practical
concerns of the other queried science topics like heath, the environment, or nuclear power. Indeed the second
highest "not interested" response was to "space exploration" – 22% in 1997 (agricultural and farm issues was
the highest at 26%). The largest difference in responses between male and female respondents, 30%, was for
space exploration. Also noteworthy is the fact that expressed interest is about 50% greater in people with
graduate degrees than in those who have not completed a high school education. By comparison, the 1998
survey done by the European Space Agency of the 14 ESA countries found that about 42% of the public said
they were interested or very interested in space exploration.
During the time I was chairman of the astronomy program at the Smithsonian Institution’s National Air
and Space Museum (NASM) in Washington, D.C., the museum undertook a survey of its approximately 8
million annual visitors in an effort to understand why they came, and what they liked. It is useful to this
article because it broke down the broad category of "space science" into subtopics. Most people came to the
museum to see a bit of everything, but of those who came particularly to see an artifact or gallery (and
excluding the IMAX theater) 45% came for aviation-related subjects and 35% came for the spacecraft, or
astronomy galleries, or the planetarium shows. Those people who came with no specific special interest in
mind were asked upon leaving what they had found the "most interesting." Forty-three percent said they
found space science topics (spacecraft/astronomy/planetarium) "the most interesting," with the spacecraft
artifacts being by far the most popular of these, by about 3:1. Forty-four percent said they found the aviation
exhibits most interesting. The NASM artifacts are spectacular and inspirational, so it is perhaps not surprising
that people want to see them; we will see below that space technology and manned exploration bring
excitement to the whole space science endeavor. What is interesting from the study is the very strong showing
of non-artifact based space science.III. SPACE SCIENCE NEWS IS GENERALLY GOOD NEWS
The Pew Research Center for the People and the Press (quoted in S&EI-2000 ) has for over 15 years tracked
the most closely followed news stories in the US. There were 689 of them, with 39 having some connection
to science (including medicine, weather, and natural phenomena.) To an overwhelming degree these 39
science stories were bad news -- earthquakes or other calamities of nature, nuclear power, AIDS, or medical
controv ersies. But virtually every good news science story was about space science: John Glenn’s shuttle
flight, the deployment of the Hubble, the Mars Pathfinder mission, and the cosmic microwave background.
(The only positive, non-space, science news story was on Viagra, while only the negative space stories were
the explosion of the Shuttle Challenger, and troubles with the Mir space station.)
Five Reasons for the Appeal of Space Science
Space science, as these news items suggest, makes people feel good about themselves; no doubt this is
one reason why people say they like it. There are at least four other reasons which I believe are unique to
astronomy and space science, and which set the field apart from others in science like physics, chemistry or
biology. They are worth explicitly listing because effective education and outreach efforts can build on their
inherent appeal (see section VIII ). (1) Universal access to the skies : everyone can look up in wonder at the
heavens. Creation myths, developed by many diverse cultures, make the sky a simple yet nearly universal
natural reference frame, while those people who have more interest can easily become familiar with the
constellations or planets using only their eyes. Reports of the latest discoveries, for example, protoplanetary
disks in the Orion nebula, can be made more immediate to people by pointing out their positions in the sky.
(2) Issues of personal meaning : the religious/spiritual implications of space. Questions about the universe lead
naturally to questions of origins -- the creation of the universe, and the creation of life. These matters, far
from being esoteric philosophical debates about matters that happened perhaps 13 billion years ago, are taken
personally. They directly affect the spiritual perspectives of at least the Western religions. But even for
nonreligious people these are matters of spirit and meaning, and so they are both important as well as
interesting. The vigorous and sometimes acrimonious debate in the US over Darwinian evolution is a
biological echo of these spiritual sensitivities. (3) Ease of understanding : the profound questions are simply
put. As a physicist by training, I am excited by developments in physics today - in quantum mechanics, the
nature of elementary particles, and progress towards a "theory of everything." I am not a biologist, but I
recognize the revolutionary advances underway in understanding the genome, for example. But, in terms of
easily communicating these discoveries to the public, there is no comparison with astronomy’s advantage:
the pressing, current questions of astronomy are easy to describe. How did planets form? When and how did
the universe begin? Are stars born, and how do they die? Furthermore, often the answers can usually be
understood without resorting to complex jargon. These are powerful edges over other scientific disciplines.
Added to this, of course, and not to be underestimated, is space science’s ability to talk about modern research
with spectacular, inspirational imagery. (4) Excitement and drama : the human adventure. Finally the
exciting, dramatic and often dangerous human exploration of space is a powerful stimulant for interest in
space science, as broadly defined. Despite the controversies over the international space station, or the costs
of a manned mission to Mars, the human element of space helps keeps NASA funding percolating at a high
level (although exactly how this funding ends up benefitting space science is a much less straightforward
calculation).
IV. SPACE SCIENCE IS INTERESTING AND APPEALING -- AND PEOPLE SUPPORT IT
Space science is the beneficiary of considerable public largesse in the US. Federal funding of astronomy
alone, via NASA and NSF, was about $800M in 1997. NASA’s share, in 1997 dollars, has increased from
$380M in 1981 as more and more space missions are undertaken; NSF’s share is about steady at $100M.Additional Federal funding for space science comes through other agencies including the Defense Department
(for example, the recent Air Force MSX mission, or the Naval Observatory programs), and is significant but
harder to quantify. Finally there is substantial public support in the form of local (state) funds for university
telescopes, and/or from private foundations. It is worth noting, as does the recent National Academy report
on astronomy funding, that of ten new generation telescopes being built with US support whose apertures are
over 5 meters in diameter, only five get some Federal funding. Clearly the US public is willing to fund space
science at productive levels.
Public support for space science, as measured by the perceptions of its cost-to-benefit ratio, has also been
high in the US. Nearly half of all adults sampled -- 48% in 1997 -- said the benefits of "space exploration"
far outweighed or slightly outweighed the costs. This figure has been relatively stable over the past ten years.
We note that support for scientific research in general, including medical research, is even higher -- averaging
about 70% over the past ten years, although for some disciplines the support is less: genetic engineering, for
example, received endorsement from only 42% of adults in 1997.
The Science and Engineering Indicators survey asked people whether they viewed themselves as attentive
to the various fields of science, generally interested, or neither. (To be attentive in this study the respondent
had to indicate he or she was very interested, very well informed, and regularly read about the material.)
When one compares the responses to being attentive to that of support for science, it becomes clear that the
attentive public is the most supportive, both in terms of the strict cost benefit ratio, and also insofar as the
perceived advantages (leading to better lives, more interesting work, more opportunities, etc.) outweigh the
perceived disadvantages (its effects can be harmful, change our way of life too fast, or reduce the dependence
on faith, etc.) Among the attentive population, two and one-half times as many think of science as positive
and promising as compared to those whose attitudes are critical or pessimistic. Among the public who are
neither attentive nor particularly interested, this ratio is only one and one-half -- so, about 43% of them are
quite pessimistic. When formal education is taken into account, it clearly appears that the more educated the
population, the more likely it is to be optimistic and supportive -- about twice as much for college graduates
as for those without a high school diploma, and even more so for those with post graduate education.
However, increased education (and knowledge, too, we infer) does not always lead to a more supportive
community. In the example of nuclear power, the survey showed that support leveled off as more informed
people also become more critical. No such tendency was found in the space science sample.
An interesting point arises regarding the group of people who thought of themselves as "very well
informed": they were significantly more likely to say they participated in public policy disputes than those
who had doubts about their level of understanding. Increasing the knowledge of the public will, if these
trends are related, also increase the number who participate in the policy development. It is also true that
some of the more knowledgeable public were aware of their limits and did not consider themselves "very well
informed," and so to some extent increased knowledge might lead to a group declining to participate;
however, better teaching will also educate those who do participate while not being particularly well informed.
Overall, then, better education about space science -- and we show below that there is considerable room for
improved education -- should result not only in a better informed citizenry, but one more likely to participate
intelligently the public discourse, and one more optimistic about -- and supportive of -- space science.V. BUT THE PUBLIC’S KNOWLEDGE OF SPACE SCIENCE IS SURPRISINGLY LIMITED
Just the Facts
The NSF Science and Engineering Indicators surveys also sampled the public’s knowledge of scientific
facts by asking 20 questions, three of which were astronomy or space science related: (1) "True or false --
The Universe began with a huge explosion?" (2) "Does the earth go around the Sun or does the sun go around
the earth?" (3) "How long does it take for the earth to go around the sun: one day, one month or one year?"
The results are disconcerting, if not completely new. Only 32% of all adults answered true to number 1,
including fewer than half of those who considered themselves as "attentive" to science topics. Some good
news: nearly three-quarters -- 73% -- did know the earth went around the sun, although fewer than half of
those without a high school education knew this to be the case. Perhaps most surprisingly, fewer than half
of all adults, 48%, knew that the earth circles the sun in one year -- and 28% of those adults with
graduate/professional degrees, the most knowledgeable category, did not know this fact.
It is important to place all this in context. For comparison, 93% of all adults in the survey knew that
"cigarette smoking causes lung cancer" -- this was the best response to any of the factual questions . Not too
far behind, about 83% of the adult public knew that "the oxygen we breathe comes from plants" and that "the
center of the earth is very hot." I conclude that is it reasonable to hope that effective education programs
might teach something to the 68% of adults unfamiliar with the Big Bang, or the 52% unsure of what a "year"
is. The survey also discovered that only 11% of adults (only 28% of college graduates!) could in their own
words describe "What is a molecule?" from which I conclude that, just as the level of general knowledge
about space science could be better, it could also be worse. This is important to recognize because there may
be a tendency to throw up one’s hands in despair, given the tremendous, post-sputnik science education efforts
under which many of those in the survey were schooled. These efforts were not obviously failures, but we
can do better.
A further conclusion can be derived about the attentive public – it was (not surprisingly) also the most
knowledgeable. In every science topic the respective attentive public was better informed than the "interested"
public, which in turn was much better informed than the general public. Thus there is a clear link between the
attentive and interested public, and the knowledgeable public.
Beyond the Facts: the Belief in Astrology and Pseudoscience
It’s not only what people don’t know that can hurt them. In a recent survey undertaken by York University
in Toronto, 53% of first year students in both the arts and the sciences, after hearing a definition of astrology,
said they "somewhat" or "completely" subscribed to its principles (an increase of 16 percentage points for
science students since the first survey was done in 1991). The students also replied that "astronomers can
predict one’s character and future by studying the heavens." The S&EI-2000 study is only a little more
sanguine: it found 36% of adults agreeing that astrology is "very scientific" or "sort of scientific," and notes
that a roughly comparable percentage believes in UFOs and that aliens have landed on Earth – so, more people
than know about the Big Bang. And about half of the people surveyed believe in extra-sensory perception --
more than know that Earth goes around the sun in a year. The S&EI-2000 study speculates that the dominant
role of the media (especially the entertainment industry) in people’s awareness has resulted in an increasing
inability to discriminate between fiction and reality. People can forget what they learned in high school, while
the media, insofar as they do contribute to the “dumbing down” of America, provide a steady stream of
images; public education efforts need to be persistent and competitive as well.VI. MIGHT THE SURVEYS BE WRONG OR MISLEADING?
In their book Misunderstanding Science? The Public Reconstruction of Science and Technology, Irwin
and Wynne (1996), and the other contributors to the volume, attack the Royal Society’s methodology and
Report (and by inference other similar studies) for its implicit presumptions about the nature of science and
the scientific methods (for example, that science is "a value-free and neutral activity"), and for its
presumptions as well about the citizenry (for example, the "assumption of ‘public ignorance’ " and that
"science is an important force for human improvement.") They emphasize that "in all these areas, social as
well as technical judgments must be made -- the ‘facts’ cannot stand apart from wider social, economic and
moral questions." It is perhaps easy to understand their criticisms of surveys of attitudes towards medicine,
or nuclear power, where the impact on the individual or the state is more direct than it is for space science.
Their underlying proposition however -- "the socially negotiated [their emphasis] nature of science" -- applies
across the board, and is a much more controversial one. As for the data themselves, they point out that the
surveys, as a result of these presumptions, are of questionable value. For example, in the context of the
public’s knowledge of the facts, they cite studies that show "ignorance [can be] a deliberate choice – and that
[it] will represent a reflection of the power relation between people and science." The ESA survey, for
example, rather clearly indicated it was sponsored with the aim of ascertaining public support for ESA’s
programs. Without necessarily agreeing on all these counts, we can still appreciate the legitimate limits of
these surveys. As Bauer, Petkova and Boyadjieva (2000) suggest, there are other ways of gauging knowledge.
In our case, for instance, the fact so many people answered incorrectly to the survey’s query about the earth’s
revolution may not really be so damning a statistic; it may not even prove that people really do not know the
meaning of a "year." Despite their possible limitations, there is nevertheless an internal consistency to these
studies. I believe they demonstrate, at least insofar as "knowledge" is concerned, that things could be worse --
but also that they could be better.
VII. WHY DO SPACE SCIENCE RESEARCH?
Copernicus expressed the opinion that the philosopher’s "aim is to seek truth in all things." Certainly many
researchers today would echo this high-minded sentiment. However Copernicus does not say why a practical-
minded public should support that effort, and so it is interesting to attempt an understanding of public
attitudes towards basic research itself.
Copernicus, Newton, Bacon, and Jefferson
Gerald Holton (1998, 1999) has put forward a model in which basic scientific research falls into three
general categories, each associated with an historical figure who represented that mode of inquiry. The
"Newtonian mode," also the Copernican model, is the one in which scholars work for the sake of knowledge
itself. Francis Bacon, on the other hand, urged the use of science "not only for ‘knowledge of causes, and
secret motion of things,’ but also in the service of omnipotence , ‘the enlarging of the bounds of the human
empire, to the effecting of all things possible." According to Holton’s analysis, this applied, "mission-
oriented" approach to research is today the one most often used to justify public support of science. He
proposes that there is actually a third way to view research, as exemplified by Thomas Jefferson’s arguments
to Congress for funding the Lewis and Clark expedition, namely, the "dual-purpose style of research" in
which basic new knowledge is gained but where there is also a potential for commercial or other practical
benefit. The positive public attitudes towards science and space science in part reflect the opinion that basic science
research ultimately does drive a successful economy and lifestyle. Since 1992 the S&EI studies have tried
to quantify these attitudes by asking people whether they thought science (in general, and not space science
in particular) was beneficial by making our lives "healthier and easier," "better for the average person," would
make work "more interesting," and provide "more opportunities for the next generation." In 1999 over 70%
of all adults agreed with all of these assertions. But at the same time more than half of the respondents (to
another survey) agreed that "science and technology have caused some of the problems we face as a society."
Progress is a mixed bag. More to the point, a dramatic 82% of adults in the 1999 S&EI study agreed that,
"Even if it brings no immediate benefits, scientific research that advances the frontiers of human knowledge
is necessary and should be supported by the Federal Government." Space science research benefits from this
general support, but in a more limited way. While 37% of adults thought "too little" money was being spent
by the government on "scientific research," only 15% thought so regarding "exploring space,"while 46%
thought "too much" was being spent on it (by far the highest percent of the three science disciplines queried:
exploring space, pollution and health.)
The Kennedy Model
It is clear basic research -- the search for "truth" -- is supported by the public, especially if there might be
some practical outcome. While heath and profit are obvious inducements to the support of medical,
environmental, or applied research, the practical benefit of having more astronomic al truths is harder to
identify. I argue, however, that in fact there are unique, even practical benefits to space science research,
based on two of the "appeals of space science" presented above, namely, the implications for spiritual and
personal meaning, and (not unrelated) satisfying a love of adventure and exploration. Following in the
example of Holton, I call this perspective on research the "Kennedy Mode." Said President Kennedy,
referring to the Apollo program to land on the moon, "No single space project in this period will be more
impressive to mankind, or more important for the long-range exploration of space; and none will be so
difficult or expensive to accomplish . . . in a very real sense, it will not be one man going to the moon if we
make this judgment affirmatively, it will be an entire nation (May 25, 1961)." "We choose to go to the moon
in this decade and do the other things, not because they are easy, but because they are hard (Sept. 12, 1962)."
While understanding that Kennedy had many political, economic and defense concerns enmeshed in his
proposals – all justifications for government research admittedly have complex subtexts associated with them
– it is nonetheless significant that he chose to frame a justification for the space program, as exemplified by
these quotes, in the clear language of spirit and of challenge. Space is a grand human adventure, not done
purely for the sake of curiosity, nor for the sake of economic benefit either, whether strategic or serendipitous.
This underlying sense of the important intangibles of space science is quite pervasive. For example, the recent
National Academy of Science Committee on Science, Engineering and Public Policy (COSEPUP) report,
"Evaluating Federal Research Programs: Research and the Government Performance and Results Act (1999)",
states, "Knowledge advancement furthermore leads to better awareness and understanding of the world and
the universe around us and our place therein [my emphasis] ..." Our place in the universe is not a reference
to astrometric studies of the stellar reference frame and the location of the sun and earth in space, but to
personal meaning.
VIII. SUCCESSFUL COMMUNICATION -- IT TAKES EFFORT FROM BOTH SIDES
There are an incredible number of popular books on space science. A search of Amazon.com finds 2395
books in print on the topic of "cosmology," about 800 of them (!) published since 1996. Many are not for
the general public, but most are, yet even the popular ones are often not very good. The best example is
Stephen Hawking’s phenomenal success, "A Brief History of Time" (Hawking, 1988). A movie with the
same name, about his life and touching on this material, was made in the early 1990's, and which I had thepleasure of introducing at its Washington, D.C. premier at the Museum. I fielded questions from the audience
afterwards, and took the opportunity to pose a few of my own to those assembled, which, like most NASM
audiences, was literate and self-selected. When I asked the sellout crowd of over 500 people how many had
read the book, virtually every person raised his or her hand. Then I dared to ask how many people understood
the book -- and almost no one raised his hand, or the few who did, did so with visible temerity. Despite the
talents of this great physicist and communicator, this book was a failure as an effort to teach. Indeed I spent
most of the next hour trying to persuade people that they were not stupid, and that most of the material in the
book was possible for even a layman to understand, though it might take a bit more effort on both the part of
the reader and the writer. I noted, since the majority of them had said they were lawyers, that even though
I have a Ph.D. I did not expect to understand the details of real estate law after reading a 200 page book, or
seeing a movie. Motivation and expectations are important ingredients of learning.
A Scientific Understanding of the Public
Irwin and Wynne (1996) urge that scholars consider "not just the ‘public understanding of science’ but also
the scientific understanding of the public and the manner in which that latter understanding might be enhanced
[because] without such a reflexive dimension scientific approaches to the ‘public understanding’ issue will
only encourage public ambivalence or even alienation." The surveys help towards this goal because they
clarify what is meant by "the public understanding," provide context, and can measure trends. To rise to the
challenge of increasing the public’s understanding of space science, we must be able to evaluate success or
failure, using studies including the S&EI, yet often the community has felt that simply trying hard was good
enough. The statistics suggest we have so far been able to maintain steady levels of "understanding," but
made little progress. In the new millennium there are hurdles which will require new approaches. The five
"appeals" of space science listed above (Section III) can facilitate creative new programming, while involving
adults, children, and people of all cultures and backgrounds.
Some Challenges Facing the Space Science and Museum Communities
There are some specific difficulties, as well advantages, for space science education efforts. For one thing,
the pace of discovery in astronomy is very rapid. There are about 65% more US astronomers today than in
1985 (as measured by the total membership in the American Astronomical Society), and more papers are
being published, about 80% more, for example, in The Astrophysical Journal . Furthermore very large
amounts of data are now being collected thanks in part to the sensitive, large format detector arrays. In 1969,
for instance, the Infrared Sky Survey found about 6000 objects, whereas the 2MASS infrared sky survey now
underway has over 300 million point sources, and will produce over 2 TB of data. Not least, the topics are
increasingly complex. The power spectrum of the cosmic microwave background is a more difficult concept
to explain than is the recession velocity of galaxies. Finally, television, computers and increased mobility
mean that there are new populations of people, with varying educations, backgrounds, and perspectives, who
are gaining access to modern space science information. All of these challenges should be viewed as
opportunities as well, chances to incorporate exciting new results and alternative perspectives for what, in
agreement with Irwin and Wynne, I think must be a more reflexive educational approach. There will be a
temptation to use hyperbole to emphasize discoveries whose scientific importance may be hard to explain.
These temptations should be resisted, because, as survey critics have noted, people may be smarter than polls
suggest.
Family and Community-Based Outreach: Two Examples
The astronomy department at the National Air and Space Museum produced two award-winning
educational programs under the leadership of Dr. Jeff Goldstein, which continue under his guidance today at
the Challenger Center for Space Science Education. They capture some of the unique strengths offered by
space science, in particular the wide popularity of the subject matter, and directly address some of the
criticisms mentioned above. The programs are premised on the idea that "learning is a family experience,"not limited to kids or students, and that modern astronomy research is both interesting and comparatively easy
to explain to all age groups. Developed and run in close collaboration with teachers and community
representa tives, they aim to attract entire families and multi-cultural groups to a museum (or other
environment) to experience together artifacts, lectures, demonstrations, a movie, and/or other astronomy or
space science features. The programs highlight the excitement of space exploration while studying the
cosmos, and as an added benefit simultaneously promote better communications between groups (e.g., parents
and teachers, parents and their children). They also include pre-visit teacher training, and post-visit follow-
ups.
The first of the programs, called "Learning is a Family Experience - Science Nights," is an evening event
in which parents and teachers, students and their siblings, participate together. It succeeded in part because
parents were willing to take an evening of their time to visit a popular attraction like the National Air and
Space Museum; museums should use the appeal of their collections to attract people in this way. The second
program is based on an outdoor exhibition now under development. "Voyage - A Scale Model Solar System"
is a nearly exact scale model of the solar system, on the 1:ten billion scale, stretching along a 600-meter
walking path, with the sun a sphere 13.9 cm in diameter at one end. "Voyage" maintains the scales both of
the distances between objects and their sizes, with the small solar system bodies mounted in glass to be
(barely) seen or touched. A visitor to the exhibit becomes a space voyager, traveling to the solar system,
sailing along its length, seeing its varied planets and moons, and -- importantly – sensing in its sweep the
immense distances and relative sizes. Recall that only 48% of adults responded that the Earth circles the Sun
in one year. This exhibition, sponsored in part by NASA, is designed to be an opportunity for people to place
many seemingly diverse facts into striking, and hopefully memorable, context.
IX. "NOTHING... [CAN] BE MOVED WITHOUT PRODUCING CONFUSION"
"Thus...I have at last discovered that, if the motions of the rest of the planets be brought into relation with the
circulation of the Earth and be reckoned in proportion to the orbit of each planet, not only do the phenomena
presently ensue, but the orders and magnitudes of all stars and spheres, nay the heavens themselves, become
so bound together that nothing in any part thereof could be moved from its place without producing confusion
of all other parts and of the Universe as a whole."
-- Nicolaus Copernicus, De Revolutionibus (preface)
Copernicus observed that his model worked well, and furthermore, that like a jigsaw or clockwork, it
seemed to fit together so perfectly that the simple notion of the earth circling the sun led to an entire universe
with internal order and beauty. I make, by analogy, the same point as regards the public’s understanding of
space science. A population which can comprehend that the earth revolves around the sun in one year – one
of those simple facts – is one which may also comprehend that the scientific method offers a rational,
consistent and objective approach to life. And, contrariwise, a public which does not have a grasp of the
basics is likely to be one which is susceptible to "confusion," doubting these facts and perhaps the methods
used for their discovery as well. Does it matter that only 48% of adults, not 58%, know the period of the
earth’s revolution? Perhaps not. But the statistics provide strong evidence that improvement is possible, and
likewise that degeneration is possible with increasing numbers of people vulnerable to astrology, belief in
alien invaders, or the hope that their lucky numbers will win at the lottery. I have shown that space science
is a very popular kind of science, particularly accessible and interesting. These indicators should spur on the
space science community to continue, and enhance, its public programming in order to attract and inform new
and larger audiences. The consequences of an improved understanding of space science on attitudes towards space science are
not clear. Increased knowledge may be accompanied by increased scepticism about particular missions or
experiments, as polls show can happen. Nevertheless it seems likely, to first order, that research programming
will benefit from increased civic knowledge. While felicitous, this should not in itself be the reason for
improving our educational efforts, for like Copernicus, I believe our "aim is to seek truth in all things as far
as God has permitted human reason so to do," and in this enterprise the multitude, our sponsors, are also our
partners.
ACKNOWLEDGMENTS
The author acknowledges a helpful discussion with Prof. Gerald Holton. This work was supported in part by
NASA Grant NAGW-1261.
REFERENCES
Bauer, M., Perkova, K., and Boyadjieva, P., "The Public Knowledge of and Attitudes to Science: Alternative
Measures That May End the ‘Science War’ ", in Science, Technology, & Human Values , 25, 30, 2000.
Committee on Science, Engineering, and Public Policy (COSEPUP), National Academy of Sciences,
Evaluating Federal Research Programs: Research and the Government Performance and Results Act ,
National Academy Press, Washington DC, 1999
Hawking, Stephen W., A Brief History of Time , Bantam, New York, 1988
Holton, Gerald, "What Kinds of Science are Worth Supporting?", in The Great Ideas Today , Encyclopedia
Britannica, Chicago, 1998
Holton, Gerald, and Sonnert, Gerhard, "A Vision of Jeffersonian Science,” Issues in Science and Technology ,
p.61, Fall, 1999
Irwin, A., and Wynne, B., Misunderstanding Science? The Public Reconstruction of Science and Technology ,
Cambridge University Press, 1996.
Munitz, Milton K., Theories of the Universe , The Free Press, Glencoe, IL, 1957
National Science Board, Science & Engineering Indicators – 2000, National Science Foundation, Arlington,
VA, 2000
Royal Society, The Public Understanding of Science, Royal Society, London, 1985
– COSPAR 2000, Warsaw |
1The easiest way to Heaviside ellipsoid
Valery P Dmitriyev
Lomonosov University
P.O.Box 160, Moscow 117574, Russia
e-mail: dmitr@cc.nifhi.ac.ru
Abstract. The electric field of a point charge moving with constant velocity is derived
using the symmetry properties of Maxwell's equations - its Lorentz invariance. In contrast
to conventional treatments, the derivation presented does not use retarded integrals or
relativistic relations.
We are interested in a simple and consistent derivation of the electromagnetic filed generated by
an electric charge moving with constant velocity. The standard textbook method of derivation is
commonly based on relativistic transformations of fields. The whole case looks such as if classical
electrodynamics is incomplete and needs external facilities in order to derive some of its formulas.
Really, of course, electrodymanics is a consistent theory and all necessary relations can be obtained
from Maxwell’s equations without recourse to extraneous postulates. Recently Prof. Jefimenko
demonstrated that in a series of works. However, the method proposed below is more simple and
natural than that expounded in [1].
The symmetry properties of electromagnetic systems are constantly accentuated. An even a
whole theory is based on them – special relativity. However, then it is needed, they don’t use them.
It may indicate that the authors considering electromagnetism and relativity frequently don’t
understand properly the subject, which they deal with. In this connection I would like to emphasize
another time that the primary destination of the symmetry is to alleviate resolution of the equations.
The starting point in our approach is the wave equations for electromagnetic potentials A and ϕ
jAActcπ4 1
22
22−=
∂∂−∇ (1)
πρϕϕ 41
22
22−=
∂∂−∇
tc (2)
They are easily obtained from the Maxwell’s equations
tc∂∂−−∇=AE1ϕ (3)
() 04=+×∇×∇−∂∂jAEπ ct
πρ4=⋅∇E
combined with the Lorentz gauge
01=∂∂+⋅∇tcϕA2 Now, we must resolve equations (1), (2) and then using (3) get the necessary formulas. We
consider the special case of the source functions: with the charge density
()tvx−ρ
and the current density
()tvxvj−=ρ
where const=v . That implies the time-dependence of the electromagnetic potentials also having
the form
()tvx−ϕ and ()tvxA−
With this one may take advantage of the symmetry properties of the left-hand side of the equations
(1), (2). The basic fact is that the left-hand side of the inhomogeneous d’Alambert equation
()tg
tf
cf vx−=
∂∂−∇22
221 (4)
is Lorentz-invariant. However, we do not need the whole Lorentz invariance for our purposes.
Insofar as we are searching the solution in the form ()tfvx− , let us pass to the coordinate frame
moving with the velocity v along the 1x axis. This can be done using the half-Lorentz
transformation
221
1/1ctxxυυ
−−=′ (5)
tt=′
As you see from (5), when passing to the reference frame 1x′ , the field f is seen as contracted in γ
times along the 1x axis, where
()122/1−−= cυγ
We have for (4)
γxf
xx
xf
xf
′∂∂=∂′∂⋅′∂∂=∂∂
2
22
22
γ
xf
xf
′∂∂=∂∂
υγxf
tf
tx
xf
tf
tf
′∂∂−′∂∂=∂′∂⋅′∂∂+′∂∂=∂∂
υγ υγtx
xf
xtf
tx
txf
tf
tf
∂′∂⋅′∂∂−′∂′∂∂−∂′∂⋅∂′∂∂+′∂∂=∂∂
22 2 2
22
223 22
222
22
2xf
xtf
tf
′∂∂+′∂′∂∂−′∂∂= γυ υγ
Hence
22
22
2 22
22
22
2 22
222
2
22
22
2221 2 1 2 1
tf
ctxf
cxf
tf
ctxf
cxf
c xf
tf
cxf
′∂∂−′∂′∂∂+′∂∂=′∂∂−′∂′∂∂+′∂∂−′∂∂=
∂∂−
∂∂ υγ υγ γυγ
Taking into account that in new frames
0=′∂∂
tf
we have for the d’Alambert equation (4)
()321
1
32
22
12
,,xxxg
xf
xf
xf −′=∂∂+∂∂+′∂∂γ
So, the problem reduces itself to the corresponding stationary one.
Following this line let us consider the motion of a point electric charge. In this event the set of
the equations (1), (2) looks as
()vtxvAA − −=
∂∂−∇ δπqctc4 1
22
22
()tq
tcvx−−=
∂∂−∇ δπϕϕ 41
22
22
Passing to the reference frame (5), which moves uniformly along the axis 1x together with the
charge, we get
()321
312
212
112
,, 4 xxx
cq
xA
xA
xA′ −=∂∂+∂∂+′∂∂δγπv (6)
()321
32
22
12
,,4 xxxqxxx′ −=∂∂+∂∂+′∂∂γδπϕϕϕ (7)
In the right-hand sides of (6) and (7) the following property of the δ- function was used
()()xaaxδ δ1=
which has given us
() ( ) () ( )321 32 1 32 1 ,, ,, ,, xxxxxtx xxtx ′= − = − γδ υγγδ υδ4Using the relation
()xxπδ412−=∇
the static problem (6), (7) is easily resolved:
Rq
cAγυ⋅=1 , 02=A, 03=A (8)
Rqγϕ= (9)
where
()[]2/12
32
22
12xxtxR ++−= υγ (10)
Next, we calculate the portions for (3)
()
312
22
11
Rtx
cqtA
/G6Bυγυγ−⋅⋅=∂∂⋅ii1
()
33322 12
1
Rxxtxqii i ++−⋅−=∇υγγϕ
whence
()()
3332222
12
1 /1
Rxxc txqii iE++ −−⋅=υυγγ
()
33322 11
Rxxtxqii i ++−⋅=υγ (11)
In spherical coordinates we have
θ υcos1rtx=− , θsin22
32
2rxx=+
where θ is the angle between the radius vector ()3322 11 xxtx ii ir ++−=υ and 1x axis. Thus
()
−= + =++−= θυγθ θ γ υγ2
22
22 22 2222
32
22
122sin1 sin coscr r rxxtxR
And from (11) we find finally
2/3
2
22
33sin1
−=
θυγγ
crrqE5 2/3
2
22
222
sin11
−
−
=
θυυ
crcq
(12)
The latter is just the famous Heaviside formula. It describes the real physical effect of the
“squashing” the electric field against the direction of motion:
() ()22
2/10 c
rqE υ θ −⋅==
()2/1222/11
2 c rqE
υπθ
−⋅=
=
So that
()2/3
22
1
20
−=
cEE υ
π
I wonder why the derivation of (12) presented did not become common for textbooks.
At last, let us find the total electromagnetic force field generated by the moving charge q. We
have from (8), (9)
ϕ⋅=cvA
Hence
()()v v AH ×∇⋅=×∇⋅=×∇= ϕ ϕc c1 1
From (3)
Evvv vEvAE H ×⋅=×∂∂−×−=×
∂∂+⋅−=/G6B tcc tcc1 1 1 11
2ϕ
The total force on a charge 1q is given by
()()
+−=
×+=2 2 1 11
ccqcqvEvEE HvEF2υ
()
()[]ψ
υγυγ∇′−=∇′−=
++−++−=1 1 2/32
32
22
123322 11
11qRqq
xxtxxxtxqqii i
that is just the formula for the Heaviside ellipsoid const =ψ , where Rq/=ψ , R is given by (10)
and gradient ∇′ is taken in moving coordinates (5). So, you see that the total electromagnetic force
field is undergone the real physical effect of the Lorentz contraction along the direction of motion.6Reference
[1] Jefimenko Oleg D 1993 Direct calculation of the electric and magnetic fields of an electric point
charge moving with constant velocity Am.J.Phys. 62 No 1 79-85 |
arXiv:physics/0008226v1 [physics.class-ph] 28 Aug 2000Circular Orbits Inside the Sphere of Death
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princet on, New Jersey 08544
(November 8, 1993)
Abstract
A wheel or sphere rolling without slipping on the inside of a s phere in a uniform grav-
itational field can have stable circular orbits that lie whol ly above the “equator”, while a
particle sliding freely cannot.
1 Introduction
In a recent article [1] in this Journal, Abramowicz and Szusz kiewicz remarked on an inter-
esting analogy between orbits above the equator of a “wall of death” and orbits near a black
hole; namely that the centrifugal force in both cases appear s to point towards rather than
away from the center of an appropriate coordinate system. He re we take a “wall of death”
to be a hollow sphere on the Earth’s surface large enough that a motorcycle can be driven
on the inside of the sphere. The intriguing question is wheth er there exist stable orbits for
the motorcycle that lie entirely above the equator (horizon tal great circle) of the sphere.
In ref. [1] the authors stated that no such orbits are possibl e, perhaps recalling the well-
known result for a particle sliding freely on the inside of a s phere in a uniform gravitational
field. However, the extra degrees of freedom associated with a rolling wheel (or sphere)
actually do permit such orbits, in apparent defiance of intui tion. In particular, the friction
associated with the condition of rolling without slipping c an in some circumstances have an
upward component large enough to balance all other downward forces.
In this paper we examine the character of all circular orbits inside a fixed sphere, for both
wheels and spheres that roll without slipping. The rolling c onstraint is velocity dependent
(non-holonomous), so explicit use of a Lagrangian is not esp ecially effective. Instead we
follow a vectorial approach as advocated by Milne (Chap. 17) [2]. This approach does utilize
the rolling constraint, a careful choice of coordinates, an d the elimination of the constraint
force from the equations of motion, all of which are implicit in Lagrange’s method. The
vector approach is, of course, a convenient codification of e arlier methods in which individual
components were explicitly written out. Compare with class ic works such as those of Lamb
(Chap. 9) [3], Deimel (Chap. 7) [4] and Routh (Chap. 5) [5].
Once the solutions are obtained in sec. 2 for rolling wheels w e make a numerical evaluation
of the magnitude of the acceleration in g’s, and of the required coefficient of static friction
on some representative orbits. The resulting parameters ar e rather extreme, and the circus
name “sphere of death” seems apt.
The stability of steady orbits of wheels is considered in som e detail, but completely
general results are not obtained (because the general motio n has four degrees of freedom).
1All vertical orbits are shown to be stable, as are horizontal orbits around the equator of the
sphere. We also find that all horizontal orbits away from the p oles are stable in the limit of
small wheels, and conjecture that the a similar condition ho lds for “death-defying” orbits of
large wheels above the equator of the sphere. In sec. 4 we lend support to this conjecture
by comparing to the related case of a sphere rolling within a s phere for which a complete
stability analysis can be given.
Discussions of wheels and spheres rolling outside a fixed sph ere are given in secs. 3 and
5, respectively.
2 Wheel Rolling Inside a Fixed Sphere
2.1 Generalities
We consider a wheel of radius arolling without slipping on a circular orbit on the inner
surface of a sphere of radius r > a. The analysis is performed in the lab frame, in which the
sphere is fixed. The z-axis is vertical and upwards with origin at the center of the sphere as
shown in Fig. 1. As the wheel rolls on the sphere, the point of c ontact traces a path that is
an arc of a circle during any short interval. In steady motion the path forms closed circular
orbits which are of primary interest here. We therefore intr oduce a set of axes ( x′, y′, z′) that
are related to the circular motion of the point of contact. If the motion is steady, these axes
are fixed in the lab frame.
The normal to the plane of the circular orbit through the cent er of the sphere (and also
through the center of the circle) is labeled z′. The angle between axes zandz′isβwith
0≤β≤π/2. A radius from the center of the sphere to the point of contac t of the wheel
sweeps out a cone of angle θabout the z′axis, where 0 ≤θ≤π. The azimuthal angle of the
point of contact on this cone is called φ, with φ= 0 defined by the direction of the x′-axis,
which is along the projection of the z-axis onto the plane of the orbit, as shown in Fig. 2.
Unit vectors are labeled with a superscript ˆ, so that ˆy′=ˆz′׈x′completes the definition
of the′-coordinate system.
For a particle sliding freely, the only stationary orbits ha veβ= 0 (horizontal circles) or
β=π/2 (vertical great circles). For wheels and spheres rolling i nside a sphere it turns out
thatβ= 0 or π/2 also, as we will demonstrate. However, the friction at the p oint of contact
in the rolling cases permits orbits with a larger range of θthan in the sliding case. If β= 0
orπ/2 were accepted as an assumption the derivation could be shor tened somewhat.
We also introduce a right-handed coordinate triad of unit ve ctors ( ˆ1,ˆ2,ˆ3) related to the
geometry of the wheel. Axis ˆ1lies along the symmetry axis of the wheel as shown shown in
Fig. 1. Axis ˆ3is directed from the center of the wheel to the point of contac t of the wheel
with the sphere. The vector from the center of the wheel to the point of contact is then
a=aˆ3. (1)
Axisˆ2=ˆ3׈1lies in the plane of the wheel, and also in the plane of the orbi t (the x′-y′
plane). The sense of axis ˆ1is chosen so that the component ω1of the angular velocity vector
2Figure 1: A wheel of radius arolls without slipping on a circular orbit inside
a fixed sphere of radius r. The orbit sweeps out a cone of angle θabout the
z′-axis, which axis makes angle βto the vertical. The x′-axis is orthogonal
to the z′-axis in the z-z′plane The angle between the plane of the orbit and
diameter of the wheel that includes the point of contact with the sphere is
denoted by α. A right-handed triad of unit vectors, ( ˆ1,ˆ2,ˆ3), is defined with
ˆ1along the axis of the wheel and ˆ3pointing from the center of the wheel to
the point of contact.
/vector ωof the wheel about this axis is positive. Consequently, axis ˆ2points in the direction of the
velocity of the point of contact, and therefore is parallel t o the tangent to the orbit.
Except for axis ˆ1, these rotating axes are not body axes, but the inertia tenso r is diagonal
with respect to them. We write
I11= 2kma2, I 22=kma2=I33, (2)
which holds for any circularly symmetric disc according to t he perpendicular axis theorem;
k= 1/2 for a wheel of radius awith mass mconcentrated at the rim, k= 1/4 for a uniform
disc,etc.
The wheel does not necessarily lie in the plane of the orbit. I ndeed, it is the freedom
to “bank” the wheel that makes the “death-defying” orbits po ssible. The diameter of the
wheel through the point of contact ( i.e., axis ˆ3) makes angle αto the plane of the orbit.
In general, a wheel can have an arbitrary rotation about the ˆ3-axis, but the wheel will roll
steadily along a closed circular orbit orbit only if angular velocity component ω3is such that
the plane of the wheel intersects the plane of the orbit along the tangent to the orbit at the
point of contact. Hence, for steady motion we will be able to d educe a constraint on ω3.
The case of a rolling sphere is distinguished by the absence o f this constraint, as considered
later.
3Figure 2: The azimuth of the point of contact of the wheel with the sphere
to the x′-axis is φ. The unit vector ˆr′is orthogonal to the z′-axis and points
towards the center of the wheel (or equivalently, towards th e point of contact).
Unit vector ˆ2=ˆ3׈1=ˆz′׈r′.
Since the wheel lies inside the sphere, as shown in Fig. 3, we c an readily deduce the
geometric relation that
θ−π+ sin−1(a/r)< α < θ −sin−1(a/r). (3)
Figure 3: Geometry illustrating the extremes of angle α.
It is useful to introduce r′=r′ˆr′as the perpendicular vector from the z′-axis to the center
of the wheel. The magnitude r′is given by
r′=rsinθ−acosα, (4)
4as shown in Fig. 4. The vector ˆz′׈r′is in the direction of motion of the point of contact,
which was defined previously to be direction ˆ2. That is, ( ˆr′,ˆ2,ˆz′) form a right-handed unit
triad, which is related to the triad ( ˆ1,ˆ2,ˆ3) by
ˆz′=−cosαˆ1−sinαˆ3, (5)
and
ˆr′=ˆ2׈z′=−sinαˆ1+ cosαˆ3, (6)
as can be seen from Fig. 1.
The length r′is negative when the center of the wheel is on the opposite sid e of the z′-axis
from the point of contact. This can occur for large enough a/rwhen the point of contact is
near the z′-axis, such as when θ≈0 and α <0 orθ≈πandα >0.
Figure 4: Geometry illustrating the vector rcmfrom the center of the sphere
to the center of the wheel, and the distance r′=rsinθ−acosαfrom the
z′-axis to the center of the wheel.
The force of contact of the sphere on the wheel is labeled F. For the wheel to be in contact
with the sphere the force Fmust have a component towards the center of the sphere, which
will be verified after the motion is obtained.
The equation of motion of the center of mass of the wheel is
md2rcm
dt2=F−mgˆz, (7)
where gis the acceleration due to gravity. The equation of motion fo r the angular momentum
Lcmabout the center of mass is
dLcm
dt=Ncm=a×F. (8)
We eliminate the unknown force Fin eq. (8) via eqs. (1) and (7) to find
1
madLcm
dt=gˆ3׈z+ˆ3×d2rcm
dt2. (9)
5The constraint that the wheel rolls without slipping relate s the velocity of the center of
mass to the angular velocity vector /vector ωof the wheel. In particular, the velocity vanishes for
that point on the wheel instantaneously in contact with the s phere:
vcontact =vcm+/vector ω×a= 0, (10)
and hence
vcm=drcm
dt=aˆ3×/vector ω. (11)
Multiplying this equation by ˆ3, we find
/vector ω=−ˆ3×vcm
a+ω3ˆ3. (12)
Equations (7)-(12) hold whether the rolling object is a whee l or a sphere.
The strategy now is to extract as much information as possibl e about the angular velocity
/vector ωbefore confronting the full equation of motion (9). The angu lar velocity can also be written
in terms of the unit vector ˆ1along the symmetry axis of the wheel as
/vector ω=ω1ˆ1+ˆ1×dˆ1
dt. (13)
This follows on writing /vector ω=ω1ˆ1+/vector ω⊥, and noting that the rate of change of the body vector
ˆ1is just dˆ1/dt=/vector ω⊥׈1, so/vector ω⊥=ˆ1×dˆ1/dt. Using eq. (2), the angular momentum can now
be written as
Lcm=/vector/vectorI·/vector ω= 2kma2ω1ˆ1+kma2ˆ1×dˆ1
dt. (14)
2.2 Steady Motion in a Circle
To obtain additional relations we restrict our attention to orbits in which the point of contact
of the wheel with the sphere moves in a closed circle. In such c ases the center of mass of the
wheel (and also the coordinate triad ( ˆ1,ˆ2,ˆ3)) has angular velocity ˙φabout the ˆz′-axis (and
no other component), where the dot means differentiation wit h respect to time. Thus
vcm=˙φˆz′×r′ˆr′=r′˙φˆ2. (15)
Equation (12) can now be evaluated, yielding
ˆω= (r′/a)˙φˆ1+ω3ˆ3. (16)
For steady motion there can be no rotation about axis ˆ2; angle αis constant. To find ω3we
now pursue eq. (13).
As argued above, the angular velocity /vector γof the triad ( ˆ1,ˆ2,ˆ3) is
/vector γ=˙φˆz′=−˙φcosαˆ1−˙φsinαˆ3, (17)
using eq. (5). Then,
dˆ1
dt=/vector γ׈1=−˙φsinαˆ2 (18)
6dˆ2
dt=/vector γ׈2=˙φsinαˆ1−˙φcosαˆ3=−˙φˆr′, (19)
and
dˆ3
dt=/vector γ׈3=˙φcosαˆ2. (20)
It immediately follows that
ˆ1×dˆ1
dt=−˙φsinαˆ3. (21)
Comparing with eq (13) we see that ω3=−˙φsinαand hence from eq. (16) we find
/vector ω= (r′/a)˙φˆ1−˙φsinαˆ3. (22)
As anticipated, the rolling constraint specifies how ω1andω3are both related to the angular
velocity ˙φof the wheel about the ˆz′-axis.
For use in the equation of motion (9) we can now write
L=/vector/vectorI·/vector ω=kma2[2(r′/a)˙φˆ1−˙φsinαˆ3], (23)
and hence,
1
madL
dt= 2kr′¨φˆ1−k˙φ2sinα(2r′+acosα)ˆ2−ka¨φsinαˆ3, (24)
using eqs. (18-20). Also, by differentiating eq. (15) we find
d2rcm
dt2=r′˙φ2sinαˆ1+r′¨φˆ2−r′˙φ2cosαˆ3, (25)
so that
ˆ3×d2rcm
dt2=−r′¨φˆ1+r′˙φ2sinαˆ2. (26)
Combining (9), (24) and (26), the equation of motion reads
gˆz׈3=ˆ3×d2rcm
dt2−1
madL
dt
=−(2k+ 1)r′¨φˆ1+ [(2k+ 1)r′+kacosα]˙φ2sinαˆ2+ka¨φsinαˆ3.(27)
To evaluate ˆz׈3, we first express ˆ zin terms of the triad ( ˆr′,ˆ2,ˆz′), and then transform to
triad ( ˆ1,ˆ2,ˆ3). When the point of contact of the wheel (and hence the ˆr′-axis) has azimuth
φrelative to the ˆx′axis, the ˆzaxis has azimuth −φrelative to the ˆr′axis. Hence,
ˆz= sin βcosφˆr′−sinβsinφˆ2+ cosβˆz′(28)
=−(cosαcosβ+ sinαsinβcosφ)ˆ1−sinβsinφˆ2−(sinαcosβ−cosαsinβcosφ)ˆ3,
using eqs. (5)-(6). Thus,
ˆz׈3=−sinβsinφˆ1+ (cos αcosβ+ sinαsinβcosφ)ˆ2. (29)
7Theˆ1,ˆ2andˆ3components of the equation of motion are now
(2k+ 1)r′¨φ=gsinβsinφ, (30)
[(2k+ 1)r′+kacosα]˙φ2sinα=g(cosαcosβ+ sinαsinβcosφ), (31)
and
ka¨φsinα= 0. (32)
The cone angle θenters the equations of motion only through r′.
2.2.1 Vertical Orbits
From eq. (32) we learn that for circular orbits either sin α= 0 or ¨φ= 0. We first consider
the simpler case that sin α= 0, which implies that the plane of the wheel lies in the plane
of the orbit. For a wheel inside the sphere with sin α= 0, we must have α= 0 to satisfy the
geometric constraint (3). Then eq. (31) can only be satisfied if cosβ= 0;i.e.,β=π/2 and
the plane of the orbit is vertical. The remaining equation of motion (30) now reads
(2k+ 1)r′¨φ=gsinφ, (33)
withr′=rsinθ−a >0, which integrates to
2k+ 1
2mr′2(˙φ2−˙φ2
0) =mgr′(1−cosφ), (34)
where ˙φ0is the angular velocity at the top of the orbit at which φ= 0. Equation (34)
expresses conservation of energy. The angular velocity /vector ωand the angular momentum Lcm
vary in magnitude but are always perpendicular to the plane o f the orbit.
The requirement that the wheel stay in contact with the spher e is that the contact force
Fhave component F⊥that points to the center of the sphere. On combining eqs. (7) , (25),
(29) and (33) we find
F=2k
2k+ 1mgsinφˆ2+m(gcosφ−r′˙φ2)ˆ3. (35)
The contact force is in the plane of the orbit, so the resultin g torque about the center of
mass of the wheel changes the magnitude but not the direction of the angular momentum.
On the vertical orbits, axis ˆ2is tangent to the sphere, and axis ˆ3makes angle π/2−θto
the radius from the center of the sphere to the point of contac t. Hence
F⊥=−F3sinθ (36)
is positive and the orbit is physical so long as the angular ve locity ˙φ0at the peak of the orbit
obeys
˙φ2
0>g
r′, (37)
as readily deduced from elementary considerations as well.
8The required coefficient µof static friction is given by µ=F/bardbl/F⊥where
F/bardbl=/radicalBig
F2
3cosθ2+F2
2 (38)
is the component of the contact force parallel to the surface of the sphere. We see that
µ= cot θ/radicalBig
1 + (F2/F3cosθ)2, (39)
which must be greater than cot θ, but only much greater if the wheel nearly loses contact
at the top of the orbit. Hence orbits with π/4<∼θ≤π/2 are consistent with the friction of
typical rubber wheels, namely µ<∼1.
Because a wheel experiences friction at the point of contact , vertical orbits are possible
withθ < π/ 2. This is in contrast to the case of a particle sliding freely on the inside of a
sphere for which the only vertical orbits are great circles ( θ=π/2). The only restriction
in the present case is that the wheel fits inside the sphere, i.e.,rsinθ > a , and that the
minimum angular velocity satisfy eq. (37).
2.2.2 Horizontal Orbits
The second class of orbits is defined by ¨φ= 0, so that the angular velocity is constant, say
˙φ= Ω. From eq. (30) we see that sin β= 0 and hence β= 0 for these orbits, which implies
that they are horizontal. Then eq. (31) gives the relation be tween the required angular
velocity Ω and the geometrical parameters of the orbit:
Ω2=gcotα
(2k+ 1)r′+kacosα=gcotα
(2k+ 1)rsinθ−(k+ 1)acosα, (40)
recalling eq. (4). Compare Ex. 3, sec. 244 of Routh [5] or sec. 407 of Milne [2]. There are
no steady horizontal orbits for which α= 0,i.e., for which the wheel lies in the plane of the
orbit. For such an orbit the angular momentum would be consta nt, but the torque on the
wheel would be nonzero in contradiction.
In the following we will find that horizontal orbits are possi ble only for 0 < α < π/ 2.
First, the requirement that Ω2>0 for real orbits puts various restrictions on the param-
eters of the problem. We examine these for the four quadrants of angle α.
1. 0< α < π/ 2. Then cot α >0 so we must have
r′>−kacosα
2k+ 1. (41)
This is satisfied by all r′>0 and some r′<0. However, for the wheel to fit inside the
sphere with 0 < α < π/ 2, we can have r′<0 only for θ > π/ 2 according to eqs. (3)
and (4).
2.π/2< α < π . Then cos α <0 and cot α <0 so the numerator of (40) is negative and
the denominator is positive. Hence Ω is imaginary and there a re no steady orbits in
this quadrant.
93.−π < α < −π/2. Then cos α <0 but cot α >0 so Ω2>0 and r′>0 and eq. (40)
imposes no to restriction. For the wheel to fit inside the sphe re with αin this quadrant
we must have θ < π/ 2.
4.−π/2< α < 0. Then cot α <0 so we must have
r′<−kacosα
2k+ 1<0. (42)
For the wheel to be inside the sphere with r′<0 and αin this quadrant we must have
θ < π/ 2.
To obtain further restrictions on the parameters we examine under what conditions the
wheel remains in contact with the sphere. The contact force Fis deduced from eqs. (7), (25)
and (29) to be
F/m= (−gcosα+r′Ω2sinα)ˆ1−(gsinα+r′Ω2cosα)ˆ3. (43)
It is more useful to express Fin components along the ˆrandˆθaxes where ˆrpoints away
from the center of the sphere and ˆθpoints towards increasing θ. The two sets of axes are
related by a rotation about axis ˆ2:
ˆ1=−cos(θ−α)ˆr+ sin( θ−α)ˆθ, ˆ3= sin( θ−α)ˆr+ cos( θ−α)ˆθ, (44)
so that
F/m=−(r′Ω2sinθ−gcosθ)ˆr−(r′Ω2cosθ+gsinθ)ˆθ
=−r′Ω2ˆr′+gˆz. (45)
The second form of eq. (45) follows directly from elementary considerations. The inward
component of the contact force, F⊥=−Fr, is positive and the orbits are physical provided
r′Ω2> gcotθ. (46)
There can be no orbits with r′<0 and θ < π/ 2, which rules out orbits in quadrant 4 of
α,i.e., for−π/2> α < 0.
Using eq. (40) for Ω2in eq. (46) we deduce that contact is maintained for orbits wi th
r′>0 only if
cotα >[2k+ 1 + k(a/r′) cosα] cotθ. (47)
Forr′<0 the sign of the inequality is reversed.
In the third quadrant of αwe have cos α <0, so inequality (47) can be rewritten with
the aid of (4) as
cotα >/parenleftBigg
1 + 2k−k
1 +rsinθ/a|cosα|/parenrightBigg
cotθ >cotθ. (48)
However, in this quadrant inequality (3) tells us
cotα <cot/bracketleftBig
θ+ sin−1(a/r)/bracketrightBig
<cotθ. (49)
10Hence there can be no steady orbits with −π < α < −π/2.
Thus steady horizontal orbits are possible only for 0 < α < π/ 2. Furthermore, since the
factor in brackets of inequality (47) is roughly 2 for a wheel , this kinematic constraint is
somewhat stronger than the purely geometric relation (3). H owever, a large class of orbits
remains with θ < π/ 2 as well as θ > π/ 2.
The coefficient of friction µat the point of contact must be at least F/bardbl/F⊥where F/bardbl=|Fθ|
from eq. (45). (For θ > π/ 2 and αnear zero the tangential friction Fθcan sometimes point
in the + θdirection.) Hence we need
µ≥|r′Ω2cosθ+gsinθ|
r′Ω2sinθ−gcosθ. (50)
The acceleration of the center of mass of the wheel is r′Ω2, so according to eq. (40) the
corresponding number of g’s is
cotα
2k+ 1 + k(a/r′) cosα. (51)
Table 1 lists parameters of several horizontal orbits for a s phere of size as might be found
in a motorcycle circus. The coefficient of friction of rubber t ires is of order one, so orbits
more than a few degrees above the equator involve very strong accelerations. The head of
the motorcycle rider is closer to the vertical axis of the sph ere than is the center of the wheel,
so the number of g’s experienced by the rider is somewhat less than that given i n the Table.
Figure 5 illustrates the allowed values of the tilt angle αas a function of the angle θof
the plane of the orbit, for a/r= 0.1 as in Table 1.
180 135 90 45 0
θ(deg.)90
45
0α(deg.)
Allowed Regionr'Ω2=gcotθα=θ–sin–1(a/r)
α=θ+sin–1(a/r)–180oΩ2=0 Ω2=0Allowed RegionExcluded
Region
Figure 5: The allowed values of the tilt angle αas a function of the angle
θof horizontal orbits for a/r= 0.1. The allowed region is bounded by three
curves, derived from expressions (3), (40) and (46).
From eq. (40) we see that α=π/2, Ω = 0 is a candidate “orbit” in the lower hemisphere.
On such an “orbit” the wheel is standing vertically at rest, a nd is not stable against falling
over. We infer that stability will only occur for Ω greater th an some minimum value not
revealed by the analysis thus far.
11Table 1: Parameters for horizontal circular orbits of a whee l of radius 0.3
m rolling inside a sphere of radius 3.0 m. The wheel has coeffici entk= 1/2
pertaining to its moment of inertia. The polar angle of the or bit isθ, so orbits
above the equator of the sphere have θ <90◦. The plane of the wheel makes
angle αto the horizontal. The minimum coefficient of friction requir ed to
support the motion is µ. The magnitude of the horizontal acceleration of the
center of mass is reported as the No. of g’s.
θ α µ v cm No. of g’s
(deg.) (deg.) (m/s)
15 5 16.1 4.8 48
30 5 2.82 8.0 53
45 10 2.15 7.0 27
60 10 1.19 7.9 27
60 25 3.45 4.9 10
75 15 0.96 6.8 18
75 30 2.13 4.7 8
90 25 0.96 5.3 10
90 45 2.04 3.7 5
135 60 0.56 2.3 3
2.3 Stability Analysis
A completely general analysis of the stability of the steady circular orbits found above appears
to be very difficult. We give a fairly general analysis for vert ical orbits, but for horizontal
orbits we obtain results only for orbits with θ=π/2,i.e., orbits about the equator of the
sphere, and for orbits of “small” wheels.
We follow the approach of sec. 405 of Milne [2] where it was sho wn how the steady motion
of a disk rolling in a straight line on a horizontal plane is st able if the angular velocity is
great enough. It was also shown that the small oscillatory de partures from steady motion
lead to an oscillatory path of the point of contact of the whee l with the plane. Hence in the
present case we must consider perturbations that carry the w heel away from the plane of
the steady orbit.
The difficulty is that there are in general four degrees of free dom for departures from
steady motion: the axis of the wheel can be perturbed in two di rections and the angular
12velocity ˙φcan be perturbed as well as the angle θto the point of contact. However, the
procedure to eliminate the unknown force of contact from the six equations of motion of a
rigid body leaves only three equations of motion. We will obt ain solutions to the perturbed
equations of motions only in special cases where there are in effect just two or three degrees
of freedom. A more general analysis might be possible using t he contact force found in
steady motion as a first approximation to the contact force in perturbed motion, but we do
not pursue this here.
A wheel rolling with a steady circular orbit on a plane can suff er only three types of
perturbations and the results of an analysis are reported in Ex. 3, sec. 244 of Routh [5]. For
a sphere rolling within a fixed sphere the direction of what we call axis 3 always points to the
center of the fixed sphere so there are only two perturbations to consider and the solution is
relatively straightforward, as reviewed in sec. 4 below. Th e stability of horizontal orbits of
rolling spheres lends confidence that stable orbits also exi st for wheels.
2.3.1 Vertical Orbits
We define the ( x′, y′, z′) coordinate system to have the x′-axis vertical: ˆx′=ˆz. In steady
motion we have
α= 0,ˆ1=−ˆz′,and ˆ3=ˆr′=ˆx′cosφ+ˆy′sinφ, (52)
where φis the azimuth of the center of the wheel from the ˆx′-axis. Thus φ= 0 at the top of
the orbit. To discuss departures from steady motion in which theˆ1-axis is no longer parallel
to the ˆz′-axis, it is useful to have a unit triad ( ˆr′,ˆs′,ˆz′) defined by eq. (52) and
ˆs′=ˆz′׈r′=−ˆx′sinφ+ˆy′cosφ, (53)
withφdefined as before. See Fig. 6. The surface of the sphere at the p oint of contact is
parallel to the s′-z′plane. Axes ˆr′andˆz′rotate about the z′-axis with angular velocity ˙φ,
so that
dˆr′
dt=˙φˆs′,anddˆs′
dt=−˙φˆr′. (54)
The perturbed ˆ1-axis can then be written
ˆ1=ǫrˆr′+ǫsˆs′−ˆz′,with |ǫr|,|ǫs| ≪1, (55)
where throughout the stability analysis we ignore second-o rder terms. Writing
ˆ3=ˆr′+δsˆs′+δzˆz′,with |δs|,|δz| ≪1, (56)
the condition ˆ1·ˆ3= 0 requires that δz=ǫr. Then to first order,
ˆ2=ˆ3׈1=−δsˆr′+ˆs′+ǫsˆz′. (57)
We expect that vector ˆ2will remain parallel to the surface of the sphere even for lar ge
departure from steady motion, so ˆ2must remain in the s′-z′plane. Hence, δs= 0, and
ˆ3=ˆr′+ǫrˆz′. (58)
13Figure 6: For vertical orbits the x′-axis is identical with the zaxis. The axis
ˆs′=ˆz′׈r′is in the direction of the unperturbed ˆ2-axis.
Also, we can identify αas the tilt angle of the ˆ3-axis to the r′-s′plane, so that
α=ǫr. (59)
The analysis proceeds along the lines of sec. 2.1 except that now we express all vectors
in terms of the triad ( ˆr′,ˆs′,ˆz′). To the first approximation the angular velocity of the whee l
about the ˆ1-axis is still given by ω1= (r′/a)˙φ. From eqs. (54) and (55) we find
dˆ1
dt= (˙ǫr−ǫs˙φ)ˆr′+ (ǫr˙φ+ ˙ǫs)ˆs′, (60)
ˆ1×dˆ1
dt= (ǫr˙φ+ ˙ǫs)ˆr′−(˙ǫr−ǫs˙φ)ˆs′, (61)
so that eq. (13) yields
/vector ω=ω1ˆ1+ˆ1×dˆ1
dt= [(1 + r′/a)ǫr˙φ+ ˙ǫs]ˆr′−[˙ǫr−(1 +r′/a)ǫs˙φ]ˆs′−(r′/a)˙φˆz′.(62)
Then eq. (14) tells us
L
ma= 2kaω1ˆ1+kaˆ1×dˆ1
dt=k[(2r′+a)ǫr˙φ+a˙ǫs]ˆr′−k[a˙ǫr−(2r′+a)ǫs˙φ]ˆs′−2kr′˙φˆz′,(63)
so that to first order of smallness
1
madL
dt=k[2(r′+a)˙ǫr˙φ+ (2r′+a)(ǫr¨φ−ǫs˙φ2) +a¨ǫs]ˆr′
−k[a¨ǫr−(2r′+a)(ǫr˙φ2−ǫs¨φ)−2(r′+a)˙ǫs˙φ]ˆs′
−2k(r′¨φ+ ˙r′˙φ)ˆz′. (64)
14In this we have noted from eq. (4) that ˙ r′=r˙θsinθto first order, and that ˙θis small. Next,
drcm
dt=aˆ3×/vector ω≈aˆr′×/vector ω=r′˙φˆs′−[a˙ǫr−(r′+a)ǫs˙φ]ˆz′. (65)
Then to first order,
d2rcm
dt2=−r′˙φ2ˆr′+ (r′¨φ+ ˙r′˙φ)ˆs′−[a¨ǫr−(r′+a)(˙ǫs˙φ+ǫs¨φ)]ˆz′, (66)
so that
ˆ3×d2rcm
dt2= (ˆr′+ǫrˆz′)×d2rcm
dt2=
−r′ǫr¨φˆr′+ [a¨ǫr−r′ǫr˙φ2−(r′+a)(˙ǫs˙φ+ǫs¨φ)]ˆs′+ (r′¨φ+ ˙r′˙φ)ˆz′(67)
Also,
ˆ3׈z= (ˆr′+ǫrˆz′)×(cosφˆr′−sinφˆs′) =ǫrsinφˆr′+ǫscosφˆs′−sinφˆz′. (68)
Ther′,s′andz′components of the equation of motion (9) are then
0 = [(2 k+ 1)r′+ka]ǫr¨φ+ 2k(r′+a)˙ǫr˙φ−gǫrsinφ−k(2r′+a)ǫs˙φ2+ka¨ǫs, (69)
0 = [(2 k+ 1)r+ka]ǫr˙φ2−(k+ 1)a¨ǫr−gǫrcosφ
+(2k+ 1)(r′+a)˙ǫs˙φ+ [(2k+ 1)r′+ (k+ 1)a]ǫs¨φ, (70)
0 = (2 k+ 1)(r′¨φ+ ˙r′˙φ)−gsinφ. (71)
If the perturbations ǫr,ǫsand ˙r′are set to zero eqs. (69) and (70) become trivial while
eq. (71) becomes the steady equation of motion (33).
The general difficulty with this analysis is that there are onl y three equations, (69-71),
while there are four perturbations, ǫr,ǫs,¨φand˙θ. The perturbation ˙θappears only in
eq. (71) via ˙ r′; its effect on r′leads only to second-order terms in eqs. (69-70). If we could
neglect the terms in ¨φin eqs. (69-70) then these two equations would describe only the
perturbations ǫrandǫsto first order and a solution could be completed.
Therefore we restrict our attention to the top of the orbit, φ= 0, where eq. (71) tells
us that ¨φ= 0 to leading order. The angular velocity ˙φ0at this point is a minimum so the
gyroscopic stability of the wheel is the least here. Hence if the orbit is stable at φ= 0 it will
be stable at all φ.
The forms of eqs. (69) and (70) for φ= 0 indicate that if ǫrandǫsare oscillatory then
they are 90◦out of phase. Therefore we seek solutions
ǫr=ǫrcosωt, ǫ s=ǫssinωt, (72)
where ωnow represents the oscillation frequency. The coupled equa tions of motion then
yield the simultaneous linear equations
2k(r′+a)˙φ0ωǫr+ [kaω2+k(2r′+a)˙φ2
0]ǫs= 0
{(k+ 1)aω2+ [(2k+ 1)r′+ka]˙φ2
0−g}ǫr+ (2k+ 1)(r′+a)˙φ0ωǫs= 0. (73)
15These equations are consistent only if the determinant of th e coefficient matrix vanishes,
which leads to the quadratic equation
Aω4−Bω2−C= 0, (74)
with solutions
ω2=B±√
B2+ 4AC
2A, (75)
where
A=k(k+ 1)a2, (76)
B=kag+k[(2k+ 1)(2 r′2+a2) + (4k+ 1)ar′]˙φ2
0, (77)
and
C=k(2r′+a)/parenleftBig
[(2k+ 1)r′+ka]˙φ2
0−g/parenrightBig˙φ2
0. (78)
Since AandBare positive there are real, positive roots whenever B2+ 4ACis positive,
i.e., forC >−B2/4A. In particular, this is satisfied for positive C, or equivalently for
˙φ2
0>g
(2k+ 1)r′+ka. (79)
However, this is less restrictive than the elementary resul t (37) that the wheel stay in contact
with the sphere! All vertical orbits for which the wheel rema ins in contact with the sphere
are stable against small perturbations.
The stability analysis yields the formal result that if ( φ,˙φ) = (0 ,0) then ω=/radicalBig
g/(k+ 1)a.
We recognize this as the frequency of oscillation of a simple pendulum formed by suspending
the wheel from a point on its rim, the motion being perpendicu lar to the plane of the wheel.
2.3.2 Horizontal Orbits
We expect the stability analysis of horizontal orbits to be n ontrivial since we have identified
steady orbits that are “obviously” unstable.
The spirit of the analysis has been set forth in the preceding sections. For horizontal
orbits the ( x′, y′, z′) coordinate system can be taken as identical with the ( x, y, z ) system, so
we drop symbol′in this section. We introduce a triad ( ˆr,ˆs,ˆz) with ˆ rbeing the perpendicular
unit vector from the z-axis toward the center of the wheel. Then ˆspoints in the direction of
the motion of the center of the wheel in case of steady motion.
It is also useful to introduce a unit triad that points along t he (ˆ1,ˆ2,ˆ3) axes for steady
motion. The ˆsaxis already points along the ˆ2axis for steady motion, so we only need define
ˆtas being along the direction of ˆ3, and ˆuas being along the direction of ˆ1for steady motion,
as shown in Fig. 7. Then, ( ˆs,ˆt,ˆu) form a right-handed unit triad. The vertical, ˆz, is then
related by
ˆz=−sinα0ˆt−cosα0ˆu, (80)
where α0is the angle of inclination of the wheel to the horizontal in s teady motion. The
triad ( ˆs,ˆt,ˆu) rotates about the ˆz-axis with angular velocity ˙φ, so that
dˆs
dt=˙φˆz׈s=−˙φcosα0ˆt+˙φsinα0ˆu, (81)
16dˆt
dt=˙φcosα0ˆs, (82)
anddˆu
dt=−˙φsinα0ˆs. (83)
Figure 7: For horizontal orbits of a wheel rolling inside a sp here the ( x, y, z )
axes are identical with the ( x′, y′, z′) axes. The ˆr-ˆsplane is horizontal. The
axesˆu,ˆsandˆtare along the unperturbed directions of the ˆ1,ˆ2andˆ3axes,
respectively. Axes ˆtandˆulie in the vertical plane ˆr-ˆz.
We now consider small departures from steady motion. The ˆ1-axis deviates slightly from
theˆu-axis according to
ˆ1=ǫsˆs+ǫtˆt+ˆu,|ǫs|,|ǫt| ≪1. (84)
Theˆ3-axis departs slightly from the t-axis, but to the first approximation it remains in a
vertical plane, i.e., thet-uplane. Then we have
ˆ2=ˆs−ǫsˆu,and ˆ3=ˆt−ǫtˆu. (85)
With the above definitions the signs of angles αandǫtare opposite:
∆α=−ǫt, ˙α=−˙ǫt. (86)
To first approximation the component ω1of the angular velocity of the wheel about its
axis remains ω1= (r′/a)˙φ. Then
dˆ1
dt= (−˙φsinα0+ ˙ǫs−ǫt˙φcosα0)ˆs−(ǫs˙φcosα0−˙ǫt)ˆt+ǫs˙φsinα0ˆu, (87)
ˆ1×dˆ1
dt= (ǫs˙φcosα0−˙ǫt)ˆs−(˙φsinα0+ ˙ǫs−ǫt˙φcosα0)ˆt+ǫt˙φsinα0ˆu, (88)
17so that
/vector ω=ω1ˆ1+ˆ1×dˆ1
dt
= [(r′/a+ cosα0)ǫs˙φ−˙ǫt]ˆs
−[˙φsinα0−˙ǫs−(r′/a+ cosα0)ǫt˙φ]ˆt
+(r′/a+ǫtsinα0)˙φˆu, (89)
and
L
ma= 2kaω1ˆ1+kaˆ1×dˆ1
dt
=k[(2r′+acosα0)ǫs˙φ−a˙ǫt]ˆs
−k[a˙φsinα0−a˙ǫs−(2r′+acosα0)ǫt˙φ]ˆt
+k(2r′+aǫtsinα0)˙φˆu. (90)
Then to the first approximation
1
madL
dt=−k[(2r′+acosα0)˙φ2sinα0−2(r′+acosα0)˙ǫs˙φ)
−(2r′cosα0+acos 2α0)ǫt˙φ2+a¨ǫt]ˆs
−k[a¨φsinα0+ (2r′+acosα0)ǫs˙φ2cosα0−a¨ǫs−2(r′+acosα0)˙ǫt˙φ)]ˆt
+k[2r′¨φ+ 2˙r′˙φ+ (2r′+acosα0)ǫs˙φ2sinα0]ˆu. (91)
Unlike the case of vertical orbits, for horizontal orbits th e factor ¨φhas no zeroeth-order
component and we neglect terms like ǫ¨φ.
Similarly
drcm
dt=aˆ3×/vector ω=a(ˆt−ǫtˆu)×/vector ω
=r′˙φˆs−[(r′+acosα0)ǫs˙φ−a˙ǫt]ˆu, (92)
d2rcm
dt2= [r′¨φ+ ˙r′˙φ+ (r′+acosα0)ǫs˙φ2sinα0−a˙ǫt˙φsinα0]ˆs
−r′˙φ2cosα0ˆt+ [r′˙φ2sinα0−(r′+acosα0)˙ǫs˙φ+a¨ǫt]ˆu, (93)
and
ˆ3×d2rcm
dt2= (ˆt−ǫtˆu)×d2rcm
dt2
= [r′˙φ2sinα0−(r′+acosα0)˙ǫs˙φ−r′ǫt˙φ2cosα0+a¨ǫt]ˆs
−[r′¨φ+ ˙r′˙φ+ (r′+acosα)ǫs˙φ2sinα0−a˙ǫt˙φsinα0]ˆu. (94)
We also need
ˆ3׈z= (ˆt−ǫtˆu)×(−sinα0ˆt−cosα0ˆu) =−(cosα0+ǫtsinα0)ˆs. (95)
18Thes, and tanducomponents of the equation of motion (9) are
0 = [(2 k+ 1)r′+kacosα0]˙φ2sinα0−gcosα0−(2k+ 1)(r′+acosα0]˙ǫs˙φ
−[(2k+ 1)r′cosα0+kacos 2α0]ǫt˙φ2−gǫtsinα0+ (k+ 1)a¨ǫt, (96)
0 =ka¨φsinα0+k(2r′+acosα0)ǫs˙φ2sinα0−ka¨ǫs−2k(r′+acosα0)˙ǫt˙φ, (97)
and
0 = (2 k+ 1)(r′¨φ+ ˙r′˙φ) + [(2 k+ 1)r′+ (k+ 1)acosα0]ǫs˙φ2sinα0−a˙ǫt˙φsinα0.(98)
The leading terms of these three equations are just eqs. (30) -(32) for β= 0. Therefore we
can write ˙φ= Ω + ˙δwhere Ω is the angular velocity of the steady horizontal orbi t and δis
a small correction.
Although the derivative of r′,
˙r′=r˙θcosθ0+a˙αsinα0=r˙θcosθ0−a˙ǫtsinα0, (99)
appears only in eq. (98), in general the perturbation ˙θis not decoupled from ǫsandǫtas
was the case for vertical orbits. Thus far, we have found a way to proceed only in somewhat
special cases in which the θperturbation can be ignored, as described in secs. 2.3.3 and 2.3.4.
2.3.3 Orbits Near the Equator
It appears possible to carry the analysis forward for the spe cial case θ0=π/2, the orbit on
the equator of the sphere. This case is, however, of interest .
Assuming θ0=π/2 the equations of motion (96-98) then provide three relatio ns among
the three perturbations δ,ǫsandǫt. For this we consider only the first-order terms, noting
that˙φ2≈Ω2+ 2Ω˙δand
r′=rsinθ−acosα≈r′
0+r∆θcosθ0+a∆αsinα0=r′
0−aǫtsinα0, (100)
where r0=r−acosα0forθ0=π/2, recalling eq. (86). Also, from the form of eqs. (96-98)
we infer that if the perturbations are oscillatory then δandǫshave the same phase which is
90◦from that of ǫt. Therefore we seek solutions of the form
δ=δsinωt, ǫ s=ǫssinωt, and ǫt=ǫtcosωt, (101)
where ωis the frequency of oscillation. The first-order terms of the differential equations
(96-98) then yield the algebraic relations
0 = 2Ω sin α0[(2k+ 1)r′
0+kacosα0]ωδ−Ω(2k+ 1)(r′
0+acosα0)ωǫs
−{Ω2[(2k+ 1)r′
0cosα0+ (k+ sin2α0)a]−gsinα0+ (k+ 1)aω2}ǫt,
0 = −kasinα0ω2δ+ [kΩ2cosα0(2r′
0+acosα0) +kaω2]ǫs+ 2kΩ(r′
0+acosα0)ωǫt,(102)
0 = −(2k+ 1)r′
0ω2δ+ Ω2sinα0[(2k+ 1)r′
0+ (k+ 1)acosα0]ǫs+ 2(k+ 1)Ω sin α0aωǫt.
19These equations have the form
A11ωδ+ A12ωǫs + (A13+B13ω2)ǫt= 0
A21ω2δ+ (A22+B22ω2)ǫs+ A23ωǫt = 0
A31ω2δ+ A32ǫs + A33ωǫt = 0(103)
To have consistency the determinant of the coefficient matrix must vanish, which leads
quickly to the quadratic equation
Aω4−Bω2−C= 0, (104)
where
A=B13B22A31, (105)
B=A11B22A33+A12A23A31+B13A21A32−A13B22A31−B13A22A31−A12A21A33,(106)
and
C=A11A22A33+A13A21A32−A13A22A31−A11A23A32. (107)
From numerical evaluation it appears that A,BandCare all positive for angular ve-
locities Ω that obey eq. (40). That is, all steady orbits at th e equator of the sphere are
stable. There is both a fast and slow oscillation about stead y motion for these orbits, an
effect familiar from nutations of a symmetric top.
2.3.4 Small Wheel Inside a Large Sphere
The analysis can also be carried further in the approximatio n that the radius aof the wheel
is much less than the radius rof the fixed sphere. In this case the perturbation in angle θ
of the orbit is of higher order than the perturbations in azim uthφand in the angles ǫsand
ǫtrelated to the axes of the wheel. A solution describing the th ree first-order perturbations
can then be obtained.
For the greatest simplification we also require that
a≪r′
0≈rsinθ0. (108)
Thus we restrict our attention to orbits significantly differ ent from the special cases of motion
near the poles of the fixed sphere.
In the present approximation the first-order terms of the per turbed equations of motion
(96-98) are
2Ω˙δsinα0= Ω˙ǫs+/parenleftBigg
Ω2cosα0+gsinα0
(2k+ 1)r′
0/parenrightBigg
ǫt, (109)
ǫs=˙ǫt
Ω sinα0, (110)
and
¨δ=−ǫsΩ2sinα0. (111)
20Inserting (110) into (111) we can integrate the latter to find
˙δ=−Ωǫt. (112)
Using this and the derivative of (110) in (109) we find that ǫtobeys
¨ǫt+/bracketleftBigg
Ω2sinα0(cosα0+ 2 sin α0) +gsin2α0
(2k+ 1)r′
0/bracketrightBigg
ǫt= 0. (113)
The the frequency ωof the perturbations is given by
ω2= Ω2sinα0(cosα0+ 2 sin α0) +gsin2α0
(2k+ 1)r′
0= Ω2tanα0(1 + sin 2 α0), (114)
using eqs. (40) and (108).
Thus all orbits for small wheels are stable if condition (108 ) holds. We conjecture that
orbits for large wheels are also stable if (108) is satisfied.
For steady orbits that lie very near the poles, i.e., those that have r′
0<∼a, we conjecture
that the motion is stable only for Ω greater than some minimum value. For a wheel spinning
about its axis on a horizontal plane the stability condition is
Ω2>g
(2k+ 1)a. (115)
See, for example, sec. 55 of Deimel [4]. However, we have been unable to deduce the gener-
alization of this constraint to include the dependence on randθ0for small rsinθ0.
3 Wheel Rolling Outside a Fixed Sphere
Equations (1)-(32) hold for a wheel rolling outside a sphere as well as inside when the
geometric relation (3) is rewritten as
θ < α < π +θ. (116)
We expect no vertical orbits as the wheel will lose contact wi th the sphere at some point.
To verify this, note that the condition sin α= 0 (from eq. (32)) implies that α=πwhen
the wheel is outside the sphere. Then eqs. (34-36) indicate, for example, that if the wheel
starts from rest at the top of the sphere it loses contact with the sphere when
cosφ=2
3 + 2k. (117)
The result for a particle sliding on a sphere ( k= 0) is well known.
For horizontal orbits, eqs. (40-45) are still valid, but the condition that friction have an
outward component is now
r′Ω2< gcotθ, (118)
21and hence
cotα <(2k+ 1 + k(a/r′) cosα) cotθ. (119)
Equation (40) can be satisfied for α < π/ 2 so long at the radius of the wheel is small enough
that (2 k+1)r′+kacosαis positive. We must have θ < π/ 2 to have α < π/ 2 since α > θ, so
horizontal orbits exist on the upper hemisphere. A particul ar solution is α=π/2 for which
Ω = 0; this is clearly unstable.
There is a class of orbits with θ < π/ 2 and αvery near π+θthat satisfy both eqs. (40)
and (119). These also appear to be unstable.
The stability analysis of the preceding section holds forma lly for wheels outside spheres,
but the restriction there to the case of θ= 90◦provides no insight into the present case.
4 Sphere Rolling Inside a Fixed Sphere
The case of a sphere rolling on horizontal orbits inside a fixe d sphere has been treated by
Milne [2]. For completeness, we give an analysis for orbits o f arbitrary inclination to compare
and contrast with the case of a wheel.
Again the axis normal to the orbit is called ˆz′, which makes angle βto the vertical ˆz. The
polar angle of the orbit about ˆz′isθandφis the azimuth of the point of contact between
the two spheres. The radius of the fixed sphere is r.
The diameter of the rolling sphere that passes through the po int of contact must always
be normal to the fixed sphere. That is, the “bank” angle of the r olling sphere is always
θ−π/2 with respect to the plane of the orbit.
The rolling sphere has radius a, mass mand moment of inertia kma2about any diameter.
The angular momentum is, of course,
Lcm=kma2/vector ω, (120)
where /vector ωis the angular velocity of the rolling sphere.
We again introduce a right-handed triad of unit vectors ( ˆ1,ˆ2,ˆ3) centered on the rolling
sphere. For consistency with the notation used for the wheel , axis ˆ3is directed towards the
point of contact, axis ˆ2is parallel to the plane of the orbit, and axis ˆ1is in the ˆ3-ˆz′plane,
as shown in Fig. 8. In general, none of these vectors are body v ectors for the rolling sphere.
The center of mass of the rolling sphere lies on the line joini ng the center of the fixed sphere
to the point of contact, and so
rcm= (r−a)ˆ3≡r′ˆ3, (121)
Equations (7-12) that govern the motion and describe the rol ling constraint hold for the
sphere as well as the wheel. Using eqs. (120) and (121) we can w rite eq. (9) as
kad/vector ω
dt=gˆ3׈z+r′ˆ3×d2ˆ3
dt2. (122)
We seek an additional expression for the angular velocity /vector ωof the rolling sphere, but we
cannot use eq. (13) since we have not identified a body axis in t he sphere. However, with
22Figure 8: Geometry illustrating the case of a sphere rolling without slipping
on a circular orbit perpendicular to the ˆz′-axis inside a fixed sphere. The ˆ3-
axis is along the line of centers of the two spheres, and passe s through the
point of contact. The ˆ2-axis lies in the plane of the orbit along the direction
of motion of the center of the rolling sphere, and axis ˆ1=ˆ2׈3is in the ˆ3-ˆz′
plane.
eq. (121) the rolling constraint (12) can be written
/vector ω=−r′
aˆ3×dˆ3
dt+ω3ˆ3. (123)
We can now see that ω3=/vector ω·ˆ3is a constant by noting that ˆ3·d/vector ω/dt = 0 from eq. (122),
and also /vector ω·dˆ3/dt= 0 from eq. (123). The freedom to chose the constant angular v elocity
ω3for a rolling sphere permits stable orbits above the equator of the fixed sphere, just as
the freedom to adjust the bank angle αallows such orbits for a wheel.
Taking the derivative of eq. (123) we find
d/vector ω
dt=−r′
aˆ3×d2ˆ3
dt2+ω3dˆ3
dt, (124)
so the equation of motion (122) can be written
(k+ 1)r′ˆ3×d2ˆ3
dt2−kaω3dˆ3
dt=gˆz׈3. (125)
Milne notes that this equation is identical to that for a symm etric top with one point fixed
[2], and so the usual extensive analysis of nutations about t he stable orbits follows if desired.
23We again restrict ourselves to circular orbits, for which th e angular velocity of the center
of mass, and of ˆ1,ˆ2andˆ3is˙φˆz′where the z′-axis is fixed. Then with
ˆz′=−sinθˆ1+ cosθˆ3, (126)
we have
dˆ3
dt=˙φˆz′׈3=˙φsinθˆ2, (127)
d2ˆ3
dt2=˙φ2sinθˆz′׈2+¨φsinθˆ2=−˙φ2sinθcosθˆ1+¨φsinθˆ2+˙φ2sin2θˆ3, (128)
and hence,
ˆ3×d2ˆ3
dt2=−¨φsinθˆ1−˙φ2sinθcosθˆ2. (129)
With these the equation of motion (125) reads
(k+ 1)r′¨φsinθˆ1+ [(k+ 1)r′˙φ2cosθ+kaω3˙φ] sinθˆ2=−gˆz׈3. (130)
We can use eq. (29) for ˆz׈3if we substitute α=θ−π/2 for the rolling sphere:
ˆz׈3=−sinβsinφˆ1+ (sin θcosβ−cosθsinβcosφ)ˆ2. (131)
The components of the equation of motion are then
(k+ 1)r′¨φsinθ= sinβsinφ, (132)
[(k+ 1)r′˙φ2cosθ+kaω3˙φ] sinθ=gcosθsinβcosφ−gsinθcosβ. (133)
The two equations of motion are not consistent in general. To see this, take the derivative
of eq. (133) and substitute ¨φfrom eq. (132):
kaω3sinβsinφ=−3(k+ 1)r′˙φcosθsinβsinφ. (134)
While this is certainly true for β= 0 (horizontal orbits), for nonzero βwe must have ˙φcosθ
constant since ω3is constant. Equation (134) is satisfied for θ=π/2 (great circles), but for
arbitrary θwe would need ˙φconstant which is inconsistent with eq. (132). Further, on a
great circle eq. (133) becomes kaω3˙φ=−gcosβ. This is inconsistent with eq. (132) unless
β=π/2 (vertical great circles) and ω3= 0.
In summary, the only possible closed orbits for a sphere roll ing within a fixed sphere are
horizontal circles and vertical great circles.
We remark further only on the horizontal orbits. For these ˙φ≡Ω is constant according
to eq. (132). Equation (133) then yields a quadratic equatio n for Ω:
(k+ 1)r′Ω2cosθ+kaω3Ω +g= 0, (135)
so that there are orbits with real values of Ω provided
(kaω3)2≥4(k+ 1)gr′cosθ. (136)
24This is satisfied for orbits below the equator ( θ > π/ 2) for any value of the “spin” ω3of the
sphere (including zero), but places a lower limit on |ω3|for orbits above the equator. For the
orbit on the equator we must have Ω = −g/(kaω3) so a nonzero ω3is required here as well.
The contact force Fis given by
F/m= (g+r′Ω2cosθ) sinθˆ1−(r′Ω2sin2θ−gcosθ)ˆ3, (137)
using eqs. (7) and (134). For the rolling sphere to remain in c ontact with the fixed sphere
there must be a positive component of Fpointing toward the center of the fixed sphere.
Since axis ˆ3is radial outward from the fixed sphere, we require that F3be negative, and
hence
r′Ω2sin2θ > g cosθ. (138)
This is always satisfied for orbits below the equator. For orb its well above the equator this
requires a larger value of |ω3|than does eq. (136). To see this, suppose ω3is exactly at the
minimum value allowed by eq. (136), which implies that Ω = −kaω3/(2(k+1)r′cosθ). Then
eq. (138) requires that tan2θ > k + 1. So for k= 2/5 and at angles θ <50◦larger values of
|ω3|are needed to satisfy eq. (136) than to satisfy eq. (136). How ever, there are horizontal
orbits at any θ >0 for|ω3|large enough.
5 Sphere Rolling Outside a Fixed Sphere
This case has also been treated by Milne [2]. A popular exampl e is spinning a basketball on
one’s fingertip.
Equations eq. (135) and (136) hold with the substitution tha tr′=r+a. The condition
on the contact force becomes
r′Ω2sin2θ < g cosθ, (139)
which can only be satisfied for θ < π/ 2. While eq. (136) requires a large spin |ω3|, if it is
too large eq. (139) can no longer be satisfied in view of the rel ation (135). For any case in
which the orbit exists a perturbation analysis shows that th e motion is stable against small
nutations [2].
6 References
[1] M.A. Abramowicz and E. Szuszkiewicz, The Wall of Death , Am. J. Phys. 61(1993)
982-991.
[2] E.A. Milne, Vectorial Mechanics , Interscience Publishers (New York, 1948).
[3] H. Lamb, Higher Mechanics , Cambridge U. Press (Cambridge, 1920).
[4] R.F. Deimel, Mechanics of the Gyroscope , Macmillian (1929); reprinted by Dover Pub-
lications (New York, 1950).
25[5] E.J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of R igid
Bodies , 6th ed., Macmillan (London, 1905); reprinted by Dover Publ ications (New York,
1955).
26 |
arXiv:physics/0008227v1 [physics.class-ph] 28 Aug 2000The Rolling Motion of a Disk on a Horizontal Plane
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princet on, New Jersey 08544
mcdonald@puphep.princeton.edu
(August 9, 2000)
1 Problem
Discuss the motion of a (thin) disk of mass mand radius athat rolls without slipping on
a horizontal plane. Consider steady motion in which the cent er of mass of the disk moves
in a horizontal circle of radius b, the special cases where b= 0 or b→ ∞, as well as small
oscillations about steady motion. Discuss the role of frict ion in various aspects of the motion.
2 Solution
This classic problem has been treated by many authors, perha ps in greatest detail but very
succinctly by Routh in article 244 of [1]. Here, we adopt a vec torial approach as advocated
by Milne [2]. The equations of motion are deduced in sec. 2.1, and steady motion is discussed
in secs. 2.2 and 2.3. Oscillation about steady motion is cons idered in sec. 2.5, and effects
of friction are discussed in secs. 2.4, 2.6 and 2.7. Section 2 .8 presents a brief summary of
the various aspects of the motions discussed in secs. 2.1-7. The issues of non-rigid-body
motion and rolling motion on curved surfaces are mentioned i n sec. 2.9, using the science
toy “Euler’s Disk” as an example.
2.1 The Equations of Motion
In addition to the ˆzaxis which is vertically upwards, we introduce a right-hand ed coordinate
triad of unit vectors ( ˆ1,ˆ2,ˆ3) related to the geometry of the disk, as shown in Fig. 1. Axis
ˆ1lies along the symmetry axis of the disk. Axis ˆ3is directed from the center of the disk to
the point of contact with the horizontal plane, and makes ang leαto that plane. The vector
from the center of the disk to the point of contact is then
a=aˆ3. (1)
Axisˆ2=ˆ3׈1lies in the plane of the disk, and also in the horizontal plane . The sense
of axis ˆ1is chosen so that the component ω1of the angular velocity vector /vector ωof the disk
about this axis is positive. Consequently, axis ˆ2points in the direction of the velocity of the
point of contact. (For the special case where the point of con tact does not move, ω1= 0 and
analysis is unaffected by the choice of direction of axis ˆ1.)
1Figure 1: A disk of radius arolls without slipping on a horizontal plane. The
symmetry axis of the disk is called axis 1, and makes angle αto the zaxis,
which is vertically upwards. The line from the center of the d isk to the point of
contact with the plane is called axis 3, which makes angle αto the horizontal,
where 0 ≤α≤π. The horizontal axis 2 is defined by ˆ2=ˆ3׈1, and the
horizontal axis ris defined by ˆr=ˆ2׈z. The angular velocity of the disk
about axis 1 is called ω1, and the angular velocity of the axes ( ˆ1,ˆ2,ˆ3) about
the vertical is called Ω. The motion of the point of contact is instantaneously
in a circle of radius r. The distance from the axis of this motion to the center
of mass of the disk is labelled b.
Before discussing the dynamics of the problem, a considerab le amount can be deduced
from kinematics. The total angular velocity /vector ωcan be thought of as composed of two parts,
/vector ω=/vector ωaxes+ωrelˆ1, (2)
where /vector ωaxesis the angular velocity of the triad ( ˆ1,ˆ2,ˆ3), and ωrelˆ1is the angular velocity of
the disk relative to the triad; the relative angular velocit y can only have a component along
ˆ1by definition. The angular velocity of the axes has component ˙αabout the horizontal
axisˆ2(where the dot indicates differentiation with respect to tim e), and is defined to have
component Ω about the vertical axis ˆz. Since axis ˆ2is always horizontal, /vector ωaxeshas no
component along the axis ˆ2׈z≡ˆr. Hence, the angular velocity of the axes can be written
/vector ωaxes= Ωˆz+ ˙αˆ2=−Ω cosαˆ1+ ˙αˆ2−Ω sinαˆ3, (3)
noting that
ˆz=−cosαˆ1−sinαˆ3, (4)
as can be seen from Fig. 1. The time rates of change of the axes a re therefore
dˆ1
dt=/vector ωaxes׈1=−Ω sinαˆ2−˙αˆ3, (5)
2dˆ2
dt=/vector ωaxes׈2= Ω sin αˆ1−Ω cosαˆ3,=−Ωˆr, (6)
dˆ3
dt=/vector ωaxes׈3= ˙αˆ1+ Ω cos αˆ2, (7)
where the rotating horizontal axis ˆris related by
ˆr=ˆ2׈z=−sinαˆ1+ cosαˆ3. (8)
Combining eqs. (2) and (3) we write the total angular velocit y as
/vector ω=ω1ˆ1+ ˙αˆ2−Ω sinαˆ3, (9)
where
ω1=−Ω cosα+ωrel. (10)
The constraint that the disk rolls without slipping relates the velocity of the center of
mass to the angular velocity vector /vector ωof the disk. In particular, the instantaneous velocity
of the point contact of the disk with the horizontal plane is z ero,
vcontact =vcm+/vector ω×a= 0. (11)
Hence,
vcm=drcm
dt=aˆ3×/vector ω=−a˙αˆ1+aω1ˆ2, (12)
using eqs. (1) and (9).
Another kinematic relation can be deduced by noting that the point of contact between
the disk and the horizontal plane can aways be considered as m oving instantaneously in a
circle whose radius vector we define as r=rˆrwithr≥0, as shown in Fig. 1. The horizontal
vector distance from the axis of this instantaneous circula r motion to the center of mass of
the disk is labelled b=bˆr, where
b=r−asinα. (13)
Since axis ˆr(and axis ˆ2) precesses about the vertical with angular velocity Ω ˆz, theˆ2com-
ponent of the velocity of the center of mass is Ω ˆz×bˆr=bΩˆ2. But, according to eq. (12),
this velocity is also aω1ˆ2. Thus,
ω1=b
aΩ. (14)
While ω1is defined to be nonnegative, length bcan be negative if Ω is negative as well. We
could use either ω1orbas one of the basic parameters of the problem. For now, we cont inue
to use ω1, as we wish to include the special cases of b= 0 and ∞in the general analysis.
Except for axis ˆ1, the rotating axes are not body axes, but the inertia tensor i s diagonal
with respect to them. We write
I11= 2kma2, I 22=kma2=I33, (15)
which holds for any thin circularly symmetric disc accordin g to the perpendicular axis the-
orem; k= 1/2 for a disk with mass mconcentrated at the rim, k= 1/4 for a uniform disk,
3etc.The angular momentum Lcmof the disk with respect to its center of mass can now be
written as
Lcm=/vector/vectorI·/vector ω=kma2(2ω1ˆ1+ ˙αˆ2−Ω sinαˆ3). (16)
Turning at last to the dynamics of the rolling disk, we suppos e that the only forces on it
are−mgˆzdue to gravity and Fat the point of contact with the horizontal plane. For now,
we ignore rolling friction and friction due to the air surrou nding the disk.
The equation of motion for the position rcmof the center of mass of the disk is then
md2rcm
dt2=F−mgˆz. (17)
The torque equation of motion for the angular momentum Lcmabout the center of mass is
dLcm
dt=Ncm=a×F. (18)
We eliminate the unknown force Fin eq. (18) via eqs. (1) and (17) to find
1
madLcm
dt=gˆ3׈z+ˆ3×d2rcm
dt2. (19)
This can be expanded using eqs. (4), (5)-(7), (12) and (16) to yield the ˆ1,ˆ2andˆ3components
of the equation of motion,
(2k+ 1) ˙ω1+ ˙αΩ sinα= 0, (20)
kΩ2sinαcosα+ (2k+ 1)ω1Ω sinα−(k+ 1)¨α=g
acosα, (21)
˙Ω sinα+ 2 ˙αΩ cosα+ 2ω1˙α= 0. (22)
2.2 Steady Motion
For steady motion, ˙ α= ¨α=˙Ω = ˙ ω1= 0, and we define αsteady =α0, Ωsteady = Ω 0
andω1,steady=ω10. The equations of motion (20) and (22) are now trivially sati sfied, and
eq. (21) becomes
kΩ2
0sinα0cosα0+ (2k+ 1)ω10Ω0sinα0=g
acosα0, (23)
A special case of steady motion is α0=π/2, corresponding to the plane of the disk being
vertical. In this case, eq. (23) requires that ω10Ω0= 0. If Ω 0= 0, the disk rolls along a
straight line and ω10is the rolling angular velocity. If ω10= 0, the disk spins in place about
the vertical axis with angular velocity Ω 0.
Forα0/negationslash=π/2, the angular velocity Ω 0ˆzof the axes about the vertical must be nonzero.
We can then replace ω10by the radius bof the horizontal circular motion of the center of
mass using eqs. (13) and (14):
ω10=b
aΩ0= Ω0/parenleftbiggr
a−cosα0/parenrightbigg
. (24)
4Inserting this in (23), we find
Ω2
0=gcotα0
kacosα0+ (2k+ 1)b=gcotα0
(2k+ 1)r−(k+ 1)acosα0. (25)
Forπ/2< α0< πthe denominator of eq. (25) is positive, since ris positive by definition,
but the numerator is negative. Hence, Ω 0is imaginary, and steady motion is not possible in
this quadrant of angle α0.
For 0 < α0< π/2, Ω 0is real and steady motion is possible so long as
b >−akcosα0
2k+ 1. (26)
In addition to the commonly observed case of b >0, steady motion is possible with small
negative values of b
A famous special case is when b= 0, and the center of mass of the disk is at rest. Here,
eq. (25) becomes
Ω2
0=g
aksinα0, (27)
andω10= 0 according to eq. (24), so that
ωrel= Ω0cosα0, (28)
recalling eq. (10). Also, the total angular velocity become s simply /vector ω=−Ω0sinα0ˆ3according
to eq. (9), so the instantaneous axis of rotation is axis 3which contains the center of mass
and the point of contact, both of which are instantaneously a t rest.
2.3 Shorter Analysis of Steady Motion with b= 0
The analysis of a spinning coin whose center is at rest can be s hortened considerably by
noting at the outset that in this case axis 3 is the instantane ous axis of rotation. Then, the
angular velocity is /vector ω=ωˆ3, and the angular momentum is simply
L=I33ωˆ3=kma2ωˆ3. (29)
Since the center of mass is at rest, the contact force Fis just mgˆz, so the torque about the
center of mass is
N=aˆ3×mgˆz=dL
dt. (30)
We see that the equation of motion for Lhas the form
dL
dt=/vectorΩ0×L, (31)
where
/vectorΩ0=−g
akωˆz. (32)
Thus, the angular momentum, and the coin, precesses about th e vertical at rate Ω 0.
5A second relation between /vector ωand/vectorΩ0is obtained from eq. (2) by noting that /vector ωaxes=/vectorΩ0,
so that
/vector ω= (−Ω0cosα0+ωrel)ˆ1−Ω0sinα0ˆ3=ωˆ3, (33)
using eq. (4). Hence,
ω=−Ω0sinα0, (34)
and
ωrel= Ω0cosα0. (35)
Combining eqs. (32) and (34), we again find that
Ω2
0=g
aksinα0, (36)
Asα0approaches zero, the angular velocity of the point of contac t becomes very large, and
one hears a high-frequency sound associated with the spinni ng disk. However, a prominent
aspect of what one sees is the rotation of the figure on the face of the coin, whose angular
velocity Ω 0−ωrel= Ω0(1−cosα0) approaches zero. The total angular velocity ωalso vanishes
asα0→0.
2.4 Radial Slippage During “Steady” Motion
The contact force Fduring steady motion at a small angle α0is obtained from eqs. (6), (12),
(17), (24) and (27) as
F=mgˆz−b
aksinα0mgˆr. (37)
The horizontal component of force Fis due to static friction at the point of contact. The
coefficient µof friction must therefore satisfy
µ≥|b|
aksinα0, (38)
otherwise the disk will slip in the direction opposite to the radius vector b. Since coefficient
µis typically one or less, slippage will occur whenever aksinα0<∼|b|. As the disk loses
energy and angle αdecreases, the slippage will reduce |b|as well. The trajectory of the
center of the disk will be a kind of inward spiral leading towa rdb= 0 for small α.
If distance bis negative, it must obey |b|< akcosα0/(2k+ 1) according to eq. (26). In
this case, eq. (38) becomes
µ≥cotα0
2k+ 1, (39)
which could be satisfied for a uniform disk only for α0>∼π/3. Motion with negative bis
likely to be observed only briefly before large radial slippa ge when α0is large reduces bto
zero.
62.5 Small Oscillations about Steady Motion
We now suppose that α, Ω and ω1undergo oscillations at angular frequency ̟about their
equilibrium values of the form
α=α0+ǫcos̟t, (40)
Ω = Ω 0+δcos̟t, (41)
ω1=ω10+γcos̟t, (42)
where ǫ,δandγare small constants. Inserting these in the equation of moti on (22) and
equating terms of first order of smallness, we find that
δ=−2ǫ
sinα0(Ω0cosα0+ω10). (43)
From this as well as from eq. (40), we see that ǫ/sinα0≪1 for small oscillations. Similarly,
eq. (20) leads to
γ=−ǫΩ0sinα0
2k+ 1, (44)
and eq. (21) leads to
ǫ̟2(k+ 1) = −(2k+ 1)(ǫω10Ω0cosα0+γΩ0sinα0+δω10sinα0) +ǫkΩ2
0(1−2 cos2α0)
−2δkΩ0sinα0cosα0−ǫg
asinα0. (45)
Combining eqs. (43)-(45), we obtain
̟2(k+ 1) = Ω2
0(k(1 + 2 cos2α0) + sin2α0)−(6k+ 1)ω10Ω0cosα0
+2(2k+ 1)ω2
10−g
asinα0, (46)
which agrees with Routh [1], noting that our k, Ω0, and ω10are his k2,µ, and n.
For the special case of a wheel rolling in a straight line, α0=π/2, Ω 0= 0, and
̟2(k+ 1) = 2(2 k+ 1)ω2
10−g
a. (47)
The rolling is stable only if
ω2
10>g
2(2k+ 1)a. (48)
Another special case is that of a disk spinning about a vertic al diameter, for which
α0=π/2 and ω10andbare zero. Then, eq. (46) indicates that the spinning is stabl e only
for
|Ω0|>/radicalBiggg
a(k+ 1), (49)
which has been called the condition for “sleeping”. Otherwi se, angle αdecreases when
perturbed, and the motion of the disc becomes that of the more general case.
7Returning to the general analysis of eq. (46), we eliminate ω10using eq. (24) and replace
the term ( g/a) sinα0via eq. (25) to find
̟2
Ω2
0(k+ 1) = 3 kcos2α0+ sin2α0+b
a/parenleftBigg
(6k+ 1) cos α0−(2k+ 1)sin2α0
cosα0/parenrightBigg
+ 2b2
a2(2k+ 1). (50)
The term in eq. (50) in large parentheses is negative for α0>tan−1/radicalBig
(6k+ 1)/(2k+ 1),
which is about 60◦for a uniform disk. Hence for positive bthe motion is unstable for large
α0, and the disk will appear fall over quickly into a rolling mot ion with α0<∼60◦, after which
α0will decrease more slowly due to the radial slippage discuss ed in sec. 2.4, until bbecomes
very small. The subsequent motion at small α0is considered further in sec. 2.6.
The motion with negative bis always stable against small oscillations, but the radial
slippage is large as noted in sec. 2.4.
For motion with b≪a, such as for a spinning coin whose center is nearly fixed, the
frequency of small oscillation is given by
̟
Ω0=/radicalBigg
3kcos2α0+ sin2α0
k+ 1. (51)
For small angles this becomes
̟
Ω0≈/radicalBigg
3k
k+ 1. (52)
For a uniform disk with k= 1/4, the frequency ̟of small oscillation approaches/radicalBig
3/5Ω0=
0.77Ω0, while for a hoop with k= 1/2,̟→Ω0asα0→0.
The effect of this small oscillation of a spinning coin is to pr oduce a kind of rattling sound
during which the frequency sounds a bit “wrong”. This may be p articularly noticeable if a
surface imperfection suddenly excites the oscillation to a somewhat larger amplitude.
The radial slippage of the point of contact discussed in sec. 2.4 will be enhanced by the
rattling, which requires a larger peak frictional force to m aintain slop-free motion.
As angle α0approaches zero, the slippage keeps the radius bof order asinα0. For small
α0,b≈α0aand eq. (50) gives the frequency of small oscillation as
̟≈Ω0/radicalBigg
3k+ (6k+ 1)α0
k+ 1. (53)
For a uniform disk, k= 1/4, and eq. (53) gives
̟≈Ω0/radicalBigg
3 + 10 α0
5. (54)
When α0≈0.2 rad, the oscillation and rotation frequencies are nearly i dentical, at which
time a very low frequency beat can be discerned in the nutatio ns of the disk. Once α0
drops below about 0.1 rad, the low-frequency nutation disap pears and the disk settles into
a motion in which the center of mass hardly appears to move, an d the rotation frequency
Ω0≈/radicalBig
g/akα 0grows very large.
For a hoop ( k= 1/2), the low-frequency beat will be prominent for angles αnear zero.
82.6 Friction at Very Small α
In practice, the motion of a spinning disk appears to cease ra ther abruptly for a small value
of the angle α, corresponding to large precession angular velocity Ω. If t he motion continued,
the velocity Ω aof the point of contact would eventually exceed the speed of s ound.
This suggests that air friction may play a role in the motion a t very small α, as has been
discussed recently by Moffatt [3].
When the rolling motion ceases, the disk seems to float for a mo ment, and then settle
onto the horizontal surface. It appears that the upward cont act force Fzvanished, and
the disk lost contact with the surface. From eqs. (12) and (17 ), we see that for small α,
Fz≈ma¨α−mg. Since the height of the center of mass above the surface is h≈aαfor
small α, we recognize that the disk loses contact with the surface wh en the center of mass
is falling with acceleration g.
Moffatt invites us to relate the power Pdissipated by friction to the rate of change dU/dt
of total energy of the disk. For a disk moving with b= 0 at a small angle α(t),
U=1
2m˙h2+1
2I33ω2+mgh≈1
2ma2˙α2+3
2magα, (55)
using eq. (34) and assuming that eq. (36) holds adiabaticall y. Then,
dU
dt≈ma2˙α¨α+3
2mag˙α≈5
2mag˙α, (56)
where the second approximation holds when Fz≈0 and ma¨α≈mg.
For the dissipation of energy we need a model. First, we consi der rolling friction, taken to
be the effect of inelastic collisions between the disk and the horizontal surface. For example,
suppose the surface features small bumps of average height δwith average spacing ǫδ. We
suppose that the disk dissipates energy mgδwhen passing over a bump. The time taken for
the rotating disk to pass over a bump is ǫδ/aΩ, so the rate of dissipation of energy to rolling
friction is
P=−mgδ
ǫδ/aΩ=−magΩ
ǫ. (57)
A generalized form of velocity-dependent rolling friction could be written as
P=−magΩβ
ǫ. (58)
Equating this to the rate of change (56) of the energy of the di sk, we find
˙α=−2
5ǫΩβ≈ −2
5ǫ/parenleftbiggg
ak/parenrightbiggβ/21
αβ/2, (59)
which integrates to give
α(β+2)/2=β+ 2
5ǫ/parenleftbiggg
ak/parenrightbiggβ/2
(t0−t), (60)
and
α=/parenleftBiggβ+ 2
5ǫ/parenrightBigg2/(β+2)/parenleftbiggg
ak/parenrightbiggβ/(β+2)
(t0−t)2/(β+2). (61)
9In this model, the angular velocity Ω obeys
Ω =/parenleftBigg5ǫg/(β+ 2)ak
t0−t/parenrightBigg1/(β+2)
, (62)
which exhibits what is called by Moffatt a “finite-time singul arity”.
However, the premise of this analysis is that it will cease to hold when ¨ α=g/aand the
disk loses contact with the surface. Taking the derivative o f eq. (61), this gives
g
a=2
(β+ 2)2/parenleftBiggβ+ 2
5ǫ/parenrightBigg2/(β+2)/parenleftbiggg
ak/parenrightbigg1/(β+2)
(t0−t)−2(β+1)/(β+2), (63)
and
(t0−t)2/(β+2)=/parenleftBigg2
(β+ 2)2/parenrightBigg1/(β+1)/parenleftBiggβ+ 2
5ǫ/parenrightBigg2/(β+1)(β+2)/parenleftbiggg
ak/parenrightbiggβ/(β+1)(β+2)/parenleftbiggg
a/parenrightbigg−1/(β+1)
(64)
for the time t0−twhen the disk leaves the surface. At that time,
αmin=/parenleftBigg2(g/a)(β−1)
25ǫ2kβ/parenrightBigg1/(β+1)
. (65)
For a uniform disk with k= 1/4, and the simplest rolling friction model with β= 1, this
givesαmin= 0.57/ǫ. If the bump-spacing parameter ǫhad a value of 10, then αmin≈3.4◦,
which is roughly as observed.
Moffatt [3] ignores rolling friction, but makes a model for vi scous drag of the air between
the disc and the surface. He finds
α=/parenleftbigg2πµa
m(t0−t)/parenrightbigg1/3
, (66)
and
Ω =g
ak/parenleftBiggm/2πµa
t0−t/parenrightBigg1/6
, (67)
where µ= 1.8×10−4g-cm−1-s is the viscosity of air. This also yields αminof a few degrees,
and hence a similar value for Ω max. Formally, the air-drag model is the same as a rolling-
friction model with β= 4.
The main distinguishing feature between the various models for friction is the different
time dependences (62) for the angular velocity Ω as angle αdecreases. An experiment should
be performed to determine whether any of these models corres ponds to the practical physics.
2.7 “Rising” of a Rotating Disk When Nearly Vertical ( α≈π/2)
A rotating disk can exhibit “rising” when launched with spin about a nearly vertical diameter,
provided there is slippage at the point of contact with the ho rizontal plane. That is, the
10plane of the disc may rise first towards the vertical, before e ventually falling towards the
horizontal.
The rising of tops appears to have been considered by Euler, b ut rather inconclusively.
The present explanation based on sliding friction can be tra ced to a note by “H.T.” in 1839
[4].
Briefly, we consider motion that is primarily rotation about a nearly vertical diameter.
The angular velocity about the vertical is Ω >/radicalBig
g/a(k+ 1), large enough so that “sleeping”
at the vertical is possible. The needed sliding friction dep ends on angular velocity component
ω1=bΩ/abeing nonzero, which implies that the center of mass moves in a circle of radius
b≪ain the present case. Then, ω1≪Ω, and the angular momentum (16) is L≈ −Ωˆ3,
which is almost vertically upwards (see Fig. 1). Rising depe nds on slippage of the disk at
the point of contact such that the lowermost point on the disk is not at rest but moves
with velocity −ǫaω1ˆ2, which is opposite to the direction of motion of the center of mass.
Corresponding to this slippage, the horizontal surface exe rts friction Fsˆ2on the disk, with
Fs>0. The related torque, Ns=aˆ3×Fsˆ2=−aFsˆ1, pushes the angular momentum
towards the vertical, and the center of mass of the disk rises .
The most dramatic form of rising motion is that of a “tippe” to p, which has recently
been reviewed by Gray and Nickel [5].
2.8 Summary of the Motion of a Disk Spun Initially About a
Vertical Diameter
If a uniform disk is given a large initial angular velocity ab out a vertical diameter, and
the initial horizontal velocity of the center of mass is very small, the disk will “sleep” until
friction at the point of contact reduces the angular velocit y below that of condition (49). The
disk will then appear to fall over rather quickly into a rocki ng motion with angle α≈60◦
(sec. 2.5). After this, the vertical angular velocity Ω will increase ever more rapidly, while
angle αdecreases, until the disk loses contact with the table at a va lue of αof a few degrees
sec. 2.6). The disk then quickly settles on to the horizontal surface. One hears sound at
frequency Ω /2π, which becomes dramatically higher until the sound abruptl y ceases. But if
one observes a figure on the face of the disk, this rotates ever y more slowly and seems almost
to have stopped moving before the sounds ceases (sec. 2.3).
If the initial motion of the disk included a nonzero initial v elocity in addition to the spin
about a vertical diameter, the center of mass will initially move in a circle whose radius
could be large (sec. 2.3). If the initial vertical angular ve locity is small, the disc will roll in
a large circle, tilting slightly inwards until the rolling a ngular velocity ω1drops below that
of condition (48). While in most cases the angle αof the disk will then quickly drop to 60◦
or so, occasionally αwill rise back towards 90◦before falling (sec. 2.7). As the disk rolls and
spins, the center of mass traces an inward spiral on average, but nutations about this spiral
can be seen, often accompanied by a rattling sound. The nutat ion is especially prominent
forα≈10−15◦at which time a very low beat frequency between that of primar y spin and
that of the small oscillation can be observed (sec. 2.5). As αdecreases below this, the radius
of the circle traced by the center of mass becomes very small, and the subsequent motion is
11that of a disk without horizontal center of mass motion.
2.9 The Tangent Toy “Euler’s Disk”
An excellent science toy that illustrates the topic of this a rticle is “Euler’s Disk”, distributed
by Tangent Toy Co. [6]. Besides the disk itself, a base is incl uded that appears to be the key
to the superior performance exhibited by this toy. The surfa ce of the base is a thin, curved
layer of glass, glued to a plastic backing. The base rests on t hree support points to minimize
rocking.
As the disk rolls on the base, the latter is noticeably deform ed. If the same disk is rolled
on a smooth, hard surface such as a granite surface plate, the motion dies out more quickly,
and rattling sounds are more prominent. It appears that a sma ll amount of flexibility in the
base is important in damping the perturbations of the rollin g motion if long spin times are
to be achieved.
Thus, high-performance rolling motion is not strictly a rig id-body phenomenon. However,
we do not pursue the theme of elasticity further in this paper .
The concave shape of the Tangent Toy base helps center the rol ling motion of the disk,
and speeds up the reduction of an initially nonzero radius bto the desirable value of zero.
An analysis of the motion of a rolling disk on a curved surface is more complex than that
of rolling on a horizontal plane because there are four rathe r than three degrees of freedom
in the former case, but only three equations of motion. A disc ussion of a disk rolling inside
a sphere on an orbit far from the lowest point of the sphere has been given in [7].
The author thanks A. Ruina for insightful correspondence on this topic.
3 References
[1] E.J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of R igid
Bodies , 6th ed., Macmillan (London, 1905); reprinted by Dover Publ ications (New York,
1955).
[2] E.A. Milne, Vectorial Mechanics , Interscience Publishers (New York, 1948).
[3] H.K. Moffatt, Euler’s disk and its finite-time singularity , Nature 404, 833-834 (2000).
[4] H.T., Note on the Theory of the Spinning Top , Camb. Math. J. 1, 42-44 (1839).
[5] C.G. Gray and B.G. Nickel, Constants of the motion for nonslipping tippe tops and
other tops with round pegs , Am. J. Phys. 68, 821-828 (2000).
[6] J. Bendik, The Official Euler’s Disk Website , http://www.eulersdisk.com/
Tangent Toy Co., P.O. Box 436, Sausalito, CA 94966, http://w ww.tangenttoy.com/
[7] K.T. McDonald, Circular orbits inside the sphere of death , Am. J. Phys. 66, 419-430
(1998). A version with slightly revised figures is at
http://puhep1.princeton.edu/˜mcdonald/examples/sphe reofdeath.ps
12 |
Entropy shows that global
warming should cause
increased variability in the
weather*
by John Michael Williams
P. O. Box 2697,
Redwood City, CA 94064jwill@pacbell.net
Copyright (c) 2000, John Michael Williams
All Rights Reserved
* Posted at the American Physical Society Web site as aps1998nov15_001John Michael Williams Global Warming v. 1.5.2 1
ABSTRACT
Elementary physical reasoning seems to leave it inevitable that global warming
would increase the variability of the weather. The first two terms in anapproximation to the global entropy may be used to show that global warming hasincreased the free energy available to drive the weather, and that the variance of theweather has increased correspondingly.
I. INTRODUCTION
Hasselmann [1] summarized the evidence that there has been about a .5 o C
warming of the globe over the past century. The question remains open of whetherthis warming should be attributed to human activity. Regardless of its cause, weattempt here an understanding of the most obvious effect of a secular warming of theEarth's atmosphere: Increased variability of the weather.
If the Earth's atmosphere and superficial layers of ocean water could be treated as
a closed system, it might be possible to quantify the observed temperature rise as aneffect (if not a side effect) of a complex, deterministic collection of closely coupled
(viz, poorly separated) weather processes.
However, the system involved is an open one driven mainly by a continual influx
of radiation from the Sun, and by the rotation of the Earth. The system is too big to
solve deterministically because of: ( a) the number of data required to describe its
state; ( b) the necessarily incomplete instrumentation for monitoring its state; ( c) the
difficulty of providing input for such monitoring data, were a computer programmedfor prediction; ( d) the lack of an obvious way of separating the variables underlyingJohn Michael Williams Global Warming v. 1.5.2 2
the data; ( e) the lack of a valid way of spatially partitioning the system for long-term
analysis; and ( f) a dearth of accurate historical data much before 1900.
One would have to predict at least air, sea, and land temperature; humidity; local
air pressure, clouds, wind, and precipitation rates over a span of decades.Meteorologists achieve considerable success by using a stochastic framework ofanalysis in more or less localized regions of space-time.
So, a different approach must be taken. Let us treat the weather system as a
deterministic one defined by a set of potentials (of temperature, water concentration,air pressure, etc.) assumed coupled by kinetic interactions which latter we will notattempt to analyze. For simplicity, because we are dealing with global (as opposedto polar, oceanic, or day-night warming), no spatial factor will be included.
We assume that the kinetics result in a linear (or, stochastically, Markovian)
coupling among the potentials, so that system changes do not retain state except inthe value of the potentials. A global Hamiltonian or similar approach would notwork, because of complexity in estimating the flux of energy over the long term.
II. ENTROPY 0 DECREASES
We begin by showing that the entropy ( Entropy0) associated with the total free
energy of the system, formally computed as in thermodynamics, decreases with
global warming:John Michael Williams Global Warming v. 1.5.2 3
Consider the global temperature Tt as a function of time t in increments of a
calendar year. Call the corresponding total system energy Ut, and assume it
partitioned into kinetic energy Kt, potential energy Pt, and heat energy Qt. Based
on Hasselmann and others, we consider it established that T2000 - T1900 amounts to
about 0.5 o C. Without affecting the conclusion, we approximate the actual
temperatures as T2000 = 290o K and T1900 = 289.5 o K. We note that a small ~.06 K
warming of the oceans has been observed [4] during the latter half of period in
question, but we ignore it.
For total energy in the system, we have Ut = Kt + Pt + Qt. Combining K and P to
represent workable (free) energy W, we have
Ut = Wt + Qt (1)
Now we define the change in Entropy0 by the difference, d St,
d S2000 = d Q2000/T2000 - d Q1900/T1900 (2)
in which d Q represents heat flux from the weather. Avoiding the useless concept of
"wasted" heat in an open system, we rewrite (2) using the previous definitions as:
d S2000 = (U2000 - d W2000)/T2000 - (U1900 - d W1900)/T1900. (3)
We recognize here that the postulated potentials must have different zeroes: The
U values represent system totals which are kinetic or aerodynamic transfers to the
atmosphere by the Earth's rotation, or are heat or radiative input from the Sun. So,
assuming gas-molecular kinetics or Planckian radiation justifies the use of theKelvin zero.John Michael Williams Global Warming v. 1.5.2 4
However, the W values primarily represent potentials developed on the
atmospheric interaction with water, ice, and land. Heat of vaporization of water
stores about 540 calorie/kg. Although sea water would freeze below the centigradezero, liberating about 80 calorie/kg, it would return this free energy at the low end ofthe potential when melted at the centigrade zero. So, to describe the weather, weconsider only the centigrade zero. The evaluation of Entropy
0 as a term in the
overall approximation depends on this approximation, by which we relate the overallzeroes.
Expressing d W
1900 and U1900 in terms of d W2000 and U2000, we get:
d W1900 = d W2000(T1900/T2000) deg C @ d W2000(289.5 - 273)/(290 - 273) (4)
@ .971d W2000; (5)
U1900 = U2000(T1900/T2000) deg K @ U2000(289.5/290) (6)
@ .998 U2000. (7)
Substituting (5) and (7) into (3) above for the relation for a negative value of d S,
d W2000 ‡ .069 U2000 (about 7%). (8)
Therefore, the Entropy0 of this system would be expected to have decreased with
global warming if the free energy flux exceeded about 7% of the total. A typical
value of the free energy flux from the Sun's radiation would be about 15% forevaporative conversion alone [2], so we may be assured of the decrease.John Michael Williams Global Warming v. 1.5.2 5
Some comment: For a closed system with a limited store of free energy, as the
system did work, the free energy would be seen as being converted irreversibly toheat; the entropy then necessarily would increase until no more work could be done.
During the 19th century, when steam engines were the high technology, there was
a theory of the universe that predicted a "heat death": All motion would cease afterall the free energy was converted to heat, resulting in a lukewarm, totally disorderedmixture, with no potential likely to be found anywhere. This "Big Blah" theorydoesn't apply to the open system of the Earth's weather.
III. ENTROPY 1 INCREASES
Next, we show that thesecond term in our approximation, the Entropy1 of the
system, as defined by its randomness but without regard for the total free energy,
increases with global warming.
We consider that, knowing the current weather at any given time and place, to the
extent one could predict the weather elsewhere (spatially) or into the future
(temporally), to that extent would be the Entropy1 of the system lower. In
particular, correlation or coherence would imply more organization, more potential
for prediction that works, and so more free energy. On the other hand, a high
Entropy1 would imply a high amount of unpredictability in the weather in space-
time. This definition is consistent with quantitative definitions of the -Spilog2pi
entropy of information theory [3].John Michael Williams Global Warming v. 1.5.2 6
Again, looking at the several to perhaps several dozen potentials in the system,
we ignore the kinetics and view each potential as being controlled directly by one ormore of the others.
Call the potentials P
i = P1, P2, . . ., etc. We consider just one potential instance at
a time as a representative of any other of the same kind. In general, one of thepotentials, P
i, will determine another, Pj, so that, within small enough intervals,
Pj = f(Pi) @ kPi (9)
in which k is a constant of proportionality peculiar to the two potentials. We may
simplify by treating each P as measured by its absolute value difference from some
suitably chosen zero. This is a trivial approach implied directly by the concept of
potential.
Immediately, it may be seen in (9) that a small, possibly random change in Pi, dPi,
will have an effect proportional to Pi. In particular, the standard deviation dPi of
a potential Pi, viewed as a random variable, will be related linearly to that of Pj by
the proportionality factor k:
dP k dPji= . (10)
Also, Pi being in the system, an increase in Pi itself will be accompanied by an
increase in its standard deviation. This leads directly to the sought result: The
standard deviation will increase with an increase in the potential itself. We seethat temperature has increased with global warming; therefore, we expect increasedJohn Michael Williams Global Warming v. 1.5.2 7
variability in the temperature, as well as increased variability in the other
potentials in the system.
IV. CONCLUSION
The opposite directions of Entropy0 and Entropy1 are partly because of the merely
formal correctness of the thermodynamic definition of Entropy0, and partly because
Entropy1, as the second term in an approximate solution to an otherwise intractable
problem, implies both spatial and stochastic factors absent from Entropy0. If we
look at the meaning of these two terms, we see that, if global warming shouldcontinue, the decrease in Entropy
0 would mean more free energy to drive the
weather; the increase in Entropy1 would mean a harder time predicting which way it
will go.
REFERENCES
1. K. Hasselmann, Science, 276, 914 - 915 (1997).
2. P. E. Kraght, "Atmosphere (Earth)", p. 216 in D. M. Considine, Ed., Van
Nostrand's Scientific Encyclopedia (5th ed.) , (Van Nostrand Reinhold, San
Francisco, 1976).
3. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication
(University of Illinois Press, Urbana, 1949).
4. S. Levitas, et al, Science, 287, 2225 - 2229 (2000). |
arXiv:physics/0008229 28 Aug 2000
1The birth of special relativity.
"One more essay on the subject".
Jean Reignier*
Université Libre de Bruxelles
and
Vrije Universiteit Brussel.
Introduction.
In the general context of epistemological seminars on the occurrence of rapid
transitions in sciences, I was asked to report on the transition from classical to
relativistic physics at the turn of the century1. This paper is a revised and extended
version of this report.
During the first few years of the twentieth century, two very important changes
occurred in physics, which strongly influenced its development all along the century.
These upheavals are: on the one hand, the irruption of the idea of quantisation, i.e. the
discretisation of (some) physical quantities which up to these days were considered as
continuous; and on the other hand, the relativity which deeply modified our physical
conceptions of space and time. The history of the quantum revolution is well known; it
has been carefully studied by many historians of science and their works do indeed
converge towards a fairly unique version of the birth of the quantum theory 2. No such
general agreement seems to exist at present for the history of the birth of special
relativity. The subject was treated by many authors, but their conclusions are sometimes
widely divergent. Should one conclude that their approaches were not always
conducted in a strictly scientific and historical way? Not necessarily. This regrettable
situation could also result from different factors which can influence an "objective"
analysis 3:
- At the difference with the quantum revolution which starts at a very precise
date with the work of only one physicist (i.e. Max Planck, 14 December 1900), the
history of the relativistic revolution begins sometime in the ninetieth century, at a date
which is more or less a matter of convention, depending on the importance one agrees
to give to one physical phenomenon or another.
- The long way towards a relativistic physics is therefore much complex.
Furthermore, the original texts are not always correctly appraised because we read them
with the background of our relativistic scholar education.
- In modern educational systems, special relativity is generally introduced fairly
early, to students which are not well informed about the problems concerning the
electrodynamics of moving bodies. These problems were discussed at length during the
ninetieth century; they are at the origin of special relativity. When modern students
finally hear about them, one generally presents the sole elegant solution that relativity
proposes.
- It is therefore tempting to present an oversimplified version of the history
which can be caricatured as follows: "In 1905, a young genius of only 26 conceived
* Address: Département de Mathématique, CP 217, Campus de la Plaine ULB, Université Libre de
Bruxelles, 1050 Bruxelles, Belgique.
Email: jreignie@vub.ac.be
1 Centre for Empirical Epistemology, Vrije Universiteit Brussel, 18 December 1997 and 29 January
1998.
2 For this reason, I only mention in the bibliography the excellent books of Jammer [Ja-66], [Ja-74], and
of Mehra and Rechenberg [Me-Re-82/87].
3 Not to mention here less objective elements (like nationalism, and even racism) which alas plagued the
relativistic adventure.2alone a wonderful theory which saved physics from the muddy situation where older
conservative minds had let it go".
This presentation of the creation of special relativity by Albert Einstein (1879 -
1953) is readily accepted by a large majority of students, because it is simple, and
pleasant to hear when one is around twenty 4. Only a small minority notices that it
contains some anomalies. Is it not strange that the fundamental relativistic
transformations are called "Lorentz's transformations" and not "Einstein's
transformations" (H.A. Lorentz, 1853-1928)? The perplexity of these students increases
if they are told that the name was given by a third physicist (i.e. Henri Poincaré, 1854 -
1912), in a paper [Po-05] published one month before the famous paper by Einstein [Ei -
05; 4]. Furthermore, students soon learn that the mathematical space where these
transformations become fully geometric is called "Minkowski's space", from the
mathematician H. Minkowski (1864-1909). One solves generally this problem of a
possible profusion of creators by explaining that, if it is indeed true that Lorentz and
Poincaré found some results, they were not really "relativistically minded", i.e. that they
remained glued in the old vision of an absolute space filled by some mythic ether, and
an absolute Newtonian time. Only Einstein proposed at once this modern vision of a
relative space-time, and the mathematician Minkowski later on described its
mathematical structure [Mi-09]. Of course, one should not suspect historians of science
to really believe such a simplified history. Nevertheless, it has some influence, and it
partly explains the definite opinion widespread among scientists about the respective
merits of the creators of the new theory.
Let us give some examples of the divergent opinions that exist in the scientific
literature. Let us start with the case of some well-known books:
- In 1922 appears the famous book " Des Relativitätsprinzip ", which presents a
recollection of original papers on the special and general theory of relativity [So-22]. It
was supposed to give an account of the growth of the theory, under the stimulus of
physical experiment. No room is given here to the work of Poincaré 5.
- In 1951 appears the book " A History of the Theories of Aether and Electricity ", by
Ed. Whittaker [Wi-51]. One finds here an important Chapter entitled " The relativity
theory of Poincaré and Lorentz", where these two authors are presented as the
incontestable creators of the relativity. Einstein's own contributions are restricted to the
discovery of the formulas of aberration and of the Doppler effect.
- In 1971, M.A. Tonnelat publishes her " Histoire du principe de relativité " [To-71].
She explains at large that if Lorentz and Poincaré did indeed find some formulas, the
"spirit" of their works was not at all relativistic, and therefore, the full merit of the
creation of relativity goes to the sole Einstein.
- In 1981, A.I. Miller publishes " Albert Einstein's special theory of relativity.
Emergence (1905) and early interpretation(1905-1911) " [Mi-81]. This book is very
well documented and objective in its account of the facts. Miller presents and discusses
the contributions of many authors, among them, of course, the trilogy Lorentz,
Poincaré, Einstein. His conclusion is also that Einstein alone should be gratified of the
"relativistic revolution". One might reproach to this otherwise excellent book, that it
doesn't make any prospective effort in looking for a possible issue of the "dead" theory
of Lorentz and Poincaré. Notice also that in a more recent paper [Mi-94], Miller adopts
a more radical opinion: Lorentz and Poincaré could not possibly find relativity in 1905.
Whatever it may be of this opinion, it remains true that the book itself is probably the
best documented among the many books published up to now on the subject.
- Let me also quote the book of A. Pais " Subtle is the Lord,...The science and life of
Albert Einstein " [Pa-82]. Chapter 3 of this book contains an analysis of the birth of
special relativity which leaves the reader no doubts that the sole father of relativity is
Einstein. Lorentz and Poincaré are at most precursors which didn't really understand
4 I have even seen a strip cartoon relating this story for children!
5 His name appears in one of the two papers by Lorentz, but only for the remark that one should not
change a theory at each new order of approximation; this remark is certainly fundamental but its creative
importance is rather weak. Poincaré's paper [Po-06] is quoted twice in Notes by Sommerfeld following
Minkowski's contribution, but again for unimportant details.3the revolutionary character of the new relativistic spirit. In particular, Pais severely
criticises Poincaré's habit to present Lorentz's contraction as a "third hypothesis". Pais
stigmatises this attitude and considers it as an evidence that Poincaré didn't understand
Einstein's theory. However, Pais himself does not try to analyse why, and in which
circumstances, Poincaré presented relativity in that way. Alternative explanations other
than a crude misunderstanding exist (see f.i. Pi-99, or Re-00).
- I finish this rapid and limited review with the recent book of Y. Pierseaux [Pi-99].
This author proceeds to a detailed comparative analysis of Einstein's and Poincaré's
approaches to relativity. He takes into account the general context in which these
works were performed, and also the different attitudes of both authors with respect to
physical theories. Pierseaux concludes that two slightly different theories exist, which
are otherwise totally equivalent. He calls this: " The fine structure of relativity ".
So much for books. One finds also in the literature a lot of original papers
treating the subject of the birth of relativity. Most of them do not escape the discussion
of the famous question of priority. It is not possible to report here on all these papers,
and I apologise for the many authors that I shall not quote. For those physicists which
would like to read the fundamental paper of Poincaré with modern scientific notations
and in English, I recommend the translations of H.M. Schwartz [Sc-71] and of A.A.
Logunov [Lg-95]. Notice that Logunov inserts comments on Poincaré's results, and
concludes to his priority. This point of view is also vigorously defended by J. Leveugle
[Le-94]. Alternatively, many other authors share the opinion that Lorentz and Poincaré
were working on a kind of parallel project, the relativistic character of which is at least
uncertain. Therefore, they conclude that Einstein is the only true creator of special
relativity. I select here (and I recognise that my choice is largely arbitrary), in
chronological order and without trying to make a distinction between moderate and
more radical opinions, the papers by G.H. Keswani [Ke-65], S. Goldberg [Go-67], I.Yu.
Kobzarev [Ko-75], A.I. Miller [Mi-94], M. Paty [Pa-94], J. Stachel [St-95].
In view of all these divergent opinions, and keeping in mind that it will be
difficult to avoid in fine the delicate question of priorities, I shall postpone it as much
as possible and organise this paper with deep roots in the ninetieth century 6 (Parts 1
and 2). Then, I shall consider the period 1900-1910, with a particular attention to the
works of Poincaré and Einstein of 1905 (Part 3). Part 4 contains my conclusions; they
are rather close to the ones formulated by Pierseaux [Pi-99] who asserts that one should
distinguish a kind of "fine structure" in the formulation of special relativity:
- on the one hand, a formulation with an ether which was created by Lorentz and
Poincaré essentially during the years 1900-1905 from a pure electrodynamics point of
view;
- on the other hand, a formulation without ether, based on Principles expressed
by Einstein in 1905. This formulation was further developed by Einstein himself, but
also by Planck, who became fascinated by this new physics based on Principles which
renewed mechanics and thermodynamics (1906-1908). Later on, this formulation
became also geometric in the hands of Minkowski (1908).
Altogether, I conclude that it would be fair to recognise the merits of at least
three authors and to speak of special relativity as "the Lorentz-Poincaré-Einstein theory
of relativity".
Part 1.- Nineteenth century roots.
One can hardly deny that the deepest root of relativity is to be found in the work
of Galileo who asserts that "motion is like nothing", by which he means that the
description of mechanical phenomena is not affected by motion [Ga-32]. Of course, and
this was promptly recognised, this assertion is only true if one restricts motion to the
very special case of uniform translations. For all other kinds of motion, Newton's
fundamental law implies the existence of an agent called "force". And the complete set
6 Remember that the paper was originally a contribution to seminars on the occurrence of "rapid"
transitions in science. Not all these changes were really so "rapid"!4of Newton's postulates (i.e. the fundamental dynamical law, the action-reaction
principle, and S.Stevin's addition law of forces) cannot be true in arbitrary co-ordinates
frames and with an arbitrary notion of time. This fact played an important role in the
conviction of scientists that something like an absolute space and an absolute time
should exist. However, this absolute space was considered as totally immaterial and no
mechanical way exists to identify it: this is the essence of the Galilean relativity. (For a
discussion of the mechanical aspect of absolute space, see for example Po-02).
Some evidence for a "materialisation" of absolute space appears at the beginning
of the ninetieth century when it was recognised that Huyghens's wave conception of
light is to be preferred to Newton's corpuscular one (Thomas Young's interferences,
1801). At this epoch, a wave theory can only be thought of as a mechanical wave
theory, which of course presupposes the existence of some vibrating medium. Augustin
Fresnel (1788-1827) has been the champion of this idea. In a really impressive series of
works, from 1815 to his death, he accumulated so much evidence in favour of the wave
theory of light that the latter was later on considered as firmly established. After
Fresnel, one can assert that light is just a high frequency vibration of some universal
medium called ether, and therefore, physical optics can be considered as the science
that studies this medium. This study revealed rather strange properties and introduced
some puzzling questions concerning the nature of ether, for instance:
- the transversality of light vibration (polarisation) seems to imply a very high
rigidity of ether; is this high rigidity compatible with the total lack of resistance that
ether offers to the motion of matter (f.i. the motion of planets) ?
- is the ether inside transparent bodies different or alike the ether of vacuum ?
- why is the ether of transparent bodies usually dispersive, and why does this
dispersive property appear to be so specific of the body ?
- what does happen to ether when a transparent body moves?
This last question will retain our attention because it played an important role in
the long way to relativity. As early as 1822, Fresnel proposed a formula for a "partial"
drift of the ether contained in a moving body 7. This formula is based on his mechanical
model of light propagation in ether and, following an earlier proposal by Young, on an
assumed concentration of ether in transparent bodies proportional to the square of the
refraction index. According to Fresnel, we get the following result. Let c be the velocity
of light in vacuum and let n be the refractive index of the transparent body; according to
the wave theory, the velocity of light inside the body at rest in the absolute frame of
reference defined by the surrounding ether is c/n . Let us now consider the same body in
motion with a velocity v with respect to this absolute frame, and let us ask for the
absolute velocity of light inside the moving body (for simplicity, we only consider light
propagation parallel to the velocity of the body). The answer depends on the fraction of
internal ether carried on by the moving body. Fresnel assumed that only the excess of
ether with respect to vacuum should be considered as moving. It means that the
effective velocity of body's ether is not v, but that it is reduced by the fraction α = (ρ1-
ρ0)/ρ1 , where ρ1 and ρ0 are the densities of ether in the body and in the vacuum,
respectively. Then, according to the Galilean law of composition of velocities, the light
velocity in the body as seen by an observer at rest in the absolute frame of the ether is
equal to : c/n ± α v , (+ or -, depending on the relative sign of the propagations, i.e.
opposite or alike, respectively). If one further adopts Young's proposal of an ether
concentration proportional to the square of the refractive index : ρ1 = ρ0 n2, the
"Fresnel's ether partial drift coefficient α " becomes :
(1) α = 1- 1/n2 .
7 The same formula was also derived later by Stokes (1846) , in a slightly different way (see [Wh-51] ,
Ch 4).5In the second half of the ninetieth century, several experiments were
performed in order to test this value of Fresnel's drift coefficient 8 ( Fizeau 1859, Hoek
1868, Airy 1871, Michelson and Morley 1886). Fizeau's experiment is a pure local
laboratory experiment which uses a high velocity water current, and directly measures
the drift coefficient. All the other experiments are indirect experiments which try to
exhibit the motion of the Earth with respect to ether, presumably with a velocity of the
same order as its velocity around the Sun (v ≅ 30 km/sec). They are "first order"
experiments in the sense that they are sensitive to the first power of the ratio v/c = 10-4
if the drift coefficient has a value different of the one predicted by Fresnel's theory . No
effect of earth motion was observed, so that these experiments were interpreted as
confirming the value of Fresnel's drift coefficient. I shall not report here on these
experiments (see f.i. : [Wh-51] Ch 4, [To-71] Ch 4, [Re-99]). Enough is to say that the
nice confirmation of Fresnel's value did endow this drift coefficient with the status of a
scientific truth, of which theories had to take account.
After Maxwell's achievement of the synthesis of the electricity and magnetism
theories (around 1865), it became rapidly clear that light was an electromagnetic wave
phenomenon, and therefore, that one had to identify the optical ether and the
electromagnetic ether (ether of Faraday). This unification solved some of the old
problems of the ether theory (e.g. the transverse character of light polarisation), but at
the same time it posed some new ones (e.g. should one identify the transverse Fresnel's
vibration with the wave electric field or alternatively with the wave magnetic field ?).
Furthermore, Maxwell's electromagnetic theory introduces phenomenological constants
which are specific of the medium under consideration: it is a four fields theory (i.e. the
electric field E , the electric displacement D, the magnetic field H, and the magnetic
induction B), related two by two by phenomenological constraints:
(2) D = ε E, B = µ H .
The dielectric constant ε and the magnetic permeability µ are empirical constants
which characterise the body under study (i.e. the propagating medium). Empirically, it
turns out that for higher frequency electromagnetic phenomena, like a light wave, ε is
not really a constant: it depends on the frequency of the wave that propagates
(dispersion). Therefore, it became evident, as already perceived in the older mechanical
theory, that dispersion is to be associated with some dynamical mechanism that should
be introduced in one way or another in Maxwell's theory. Notice also that Maxwell's
theory says nothing about what happens when the body moves with respect to ether.
There were immediately several proposals in order to take dispersion into account and
to extend the theory to moving bodies, and in the matter, the question of the well
established Fresnel's drift coefficient was one of the distinguishing factors. It is of
course out of question to present and to discuss here these different approaches 9. I shall
limit myself to the sole Lorentz's theory which finally emerges above all others (Part
2a). On the one hand, this theory remains the basis of our present understanding of
electromagnetic phenomena in matter, and on the other hand, it progressively guided
Lorentz to the discovery of the relativistic transformations.
Part 2.- Lorentz and Poincaré between
1875 and 1904.
H.A. Lorentz. J.H. Poincaré.
(1853-1928) (1854-1912)
8 Fresnel's drift coefficient is also often called Fizeau's drift coefficient, because of its first experimental
verification.
9 A detailed presentation of all these tentative approaches can be found in [Wi-51].6Ph. D. at Leyden (1875) Engineer from Polytechnique
"Over der terugkaatsing en Paris (1875).
breking van het licht" 10. Doctor in Mathematics (1879) 11.
2a) Lorentz's classical electrodynamics.
The basic idea of Lorentz's theory is as old as the atomic idea itself: matter is
nothing else than atoms moving in vacuum. The originality lies in that, on the one hand,
the vacuum is now identified with Maxwell's ether which carries electromagnetic
phenomena; and on the other hand, that atoms contains microscopic particles called
electrons that are sensitive to the electromagnetic fields and that also contribute to
create these fields. Maxwell's ether is characterised by the two electromagnetic
constants ε0 and µ0 which are true constants, totally independent of the waves that
propagate in ether, and Lorentz judiciously puts them equal to one, according to an
appropriate choice of units. In doing this, Lorentz effectively works with only two
vacuum fields: the electric field in vacuum e and the magnetic field in vacuum h. In
matter, these fields act on electrons which, by their presence and motion, act on the
fields; therefore, from a macroscopic point of view, the vacuum fields are modified by
the presence of matter. In that way, Lorentz succeeds to explain, at least qualitatively,
nearly all phenomena concerning matter (at rest in the ether) and light: emission and
absorption of light, reflection, refraction, dispersion of the refractive index, scattering of
light (visible light and later X-rays), etc. 12
Frequently also, Lorentz's qualitative explanations become quantitative provided
they are completed by some phenomenological parameters in order to better
characterise matter. As an example, Lorentz obtains in 1878, the celebrated Lorentz -
Lorenz's formula which relates the molecular polarisability coefficient γ to the
refractive index n:
(3) γ = 3 (m/M) (n2 -1)/(n2 + 2) ,
where m and M are the molecular and the specific masses, respectively. When used
with a bit of phenomenology, this formula is very interesting: on the one hand, the
atomic dispersive refraction index n can be related through Lorentz's theory to the
atomic spectral lines; on the other hand, one can use the empirical additivity of the
molecular polarisability of mixtures and/or of weakly bound molecules; therefore, it
becomes possible to compute the refraction index of mixtures or of weakly bound
molecules from the spectral lines of their atomic constituents (Cf.[Ro-65]).
At first sight, it would seem that Lorentz's model of matter with atoms bathing
in an ether permanently at rest will conflict with the idea of a partial drift of the ether.
But Lorentz will rapidly prove (1886) that it is not so and that his model is compatible
with the appearance of an ether drift coefficient, which furthermore is precisely
Fresnel's one. This remarkable achievement is obtained by a reinterpretation of
Maxwell's constraint relations (2) and the replacement of the static electric force e E
which acts on electrons at rest by the "Lorentz force" which acts on electrons moving
with the velocity v :
(3) e E → e [ E + (v/c) × H ] .
Lorentz even improves his effective drift coefficient by introducing a dispersive term
which was later on observed by Zeeman (1911). More details on these questions can be
found in [Wi-51], or [Re-99].
10 " On the reflection and refraction of light".
11 "Sur l'intégration des équations aux dérivées partielles à un nombre quelconque d'inconnues".
12 A remarkable exception is the photo-electric effect which indeed needs another ingredient, i.e.
Planck's quantum of action.7This Lorentz's approach of the problem of Fresnel's ether drift coefficient has a
very important consequence: it proves that it is impossible to exhibit any effect of the
relative motion of the Earth with respect to ether by "first order experiments".
Therefore, one has to consider experiments sensitive to (v/c)2, i.e. "second order
experiments". At that time (1886), such an experiment existed already and it had given
a negative result (A. Michelson,1881). Michelson's conclusion was that the ether was
totally drifted, at the difference with Fresnel's partial drift and therefore also with
Lorentz theory. However, Lorentz pointed out a small error in Michelson's calculations,
which reduced the predicted effect by a factor two, so that the precision of this early
Michelson experiment became insufficient to settle the question. The situation changed
a few years later, when a higher precision Michelson's experiment totally confirmed the
earlier result (A. Michelson et E. Morley,1887). Truly, Lorentz had to change
something to his theory. He makes then the very astonishing proposal that the ether
wind has a dynamical effect of contraction of all objects along the direction of motion
13!
This contraction reduces the length of the longitudinal arm of Michelson's
interferometer in such a way that it exactly compensates the second order effect of
Earth motion:
(4) δL ≈ 0.5 L (v/c)2.
This reduction of length is common to all materials and is therefore not measurable by
any mechanical device (length standards are reduced in just the same way). In fact, the
contraction is indirectly measured by the negative result of Michelson-Morley's
experiment ([Lo-95]). This is a typical case of introducing an ad-hoc explanation in
order to save a theory! In the same paper [Lo-95], Lorentz introduces two other
concepts which will turn out to be most important for the problem of the
electrodynamics of moving bodies:
i) the idea of "local time" : when considering some event which happens at time
t and place (x , y, z) in a frame at rest in ether from a frame moving parallel to x with
velocity v, one should not only change the space co-ordinates according to the Galilean
formula:
(5) x' = x - vt , y' = y , z' = z ,
but one should also use another time, the "local time", given by:
(6) t' = t - vx/c2 .
Indeed, it is immediately seen by direct calculation that this change of time is necessary
in order to preserve the d'Alembertian 14, i.e. the basic equation of electromagnetism,
up to second order terms in the ratio v/c :
(7)1
c2 ∂2F
∂t2 - Δ F = 1
c2 ∂2F
∂t'2 - Δ' F + O[ (v/c)2 ] ;
ii) the idea of "corresponding states" : in order to obtain the invariance of the
electromagnetic phenomena in the moving body frame, again up to second order terms
in v/c , one has to change the electromagnetic "state" of the system (i.e. the electric and
magnetic fields) in an ad-hoc way:
(8) E'x' = Ex , E'y' = Ey - (v/c) Hz , E'z' = Ez + (v/c) Hy ,
13 The same proposal was made independently and simultaneously by G.F. FitzGerald (1892).
14 Remember that the d'Alembertian is equal to the difference: second derivative with respect to time
divided by c2 minus the laplacian i.e., the sum of the second derivatives with respect to space; it is the
typical equation of phenomena of wave motion (strings, sound, light, etc.)8(9) H'x' = Hx , H'y' = Hy + (v/c) Hz, H'z' = Hz - (v/c) Ey .
Notice that change (8) is the one proposed earlier by Lorentz for the force acting on a
moving electron (see Eq.3); change (9) is new.
The next year brought a triumphant confirmation of the basic ideas of Lorentz
by his theoretical calculation of the recently discovered Zeeman's effect, and by the
experimental verification of some subtle new details predicted by his approach. The
calculation rests on the idea of a small perturbation of the classical motion of an atomic
electron when one creates an external magnetic field (constant in space and time). It
contains only one parameter 15, nl. the ratio e/m of the electric charge to the mass of the
electron. The detailed experimental verification of Lorentz's predictions by Zeeman
includes the first experimental determination of this ratio (in magnitude and sign), one
year before the famous direct measurement on cathode rays by J.J. Thomson (Cf. [Lo -
02], [Ro-65]). Lorentz and Zeeman will be recompensed by the Nobel Prize in Physics
1902.
We see that around the year 1900, Lorentz had succeeded to create a very
powerful and flexible theory which explained all known electrodynamics effects for
matter at rest in ether, and also the non observation of any new effect caused by its
motion with respect to ether, including second order effects in the ratio v/c.
In reaction to critics by H. Poincaré who pointed out that the technics of
changing slightly the explanation at each new order of perturbation was not really very
convincing (local time in first order, contraction of lengths in second order 16), Lorentz
makes a last effort in order to produce formulae which eliminate all observable effects
of motion with respect to ether to all orders in the ratio v/c. This is the famous 1904
paper: "Electromagnetic phenomena in a system moving with any velocity less than that
of light " [Lo-04] (presented on 27 May 1904), which can be considered as the top of
his work on the electrodynamics of moving bodies. One finds in this work:
- the "correct" Lorentz's transformation 17,
- the theorem of corresponding states written to all orders in v/c,
- Lorentz's formulation of the "electron dynamics". The origin of the electron
mass is purely electromagnetic; as a consequence, the mass term in Newton's equations
of motion depends on the velocity of the electron with respect to the ether, and it
appears not to be exactly the same whether the acceleration caused by forces is
longitudinal or transversal, i.e. parallel or perpendicular to the velocity; furthermore, the
electron in motion with respect to ether is dynamically contracted in the direction of
motion (Lorentz-FitzGerald's contraction).
In conclusion of this rapid review of the work of Lorentz, one can certainly
assert the following:
- on the one hand, Lorentz's electrodynamics is really the basis of our modern
conception of matter: a microscopic approach to all phenomena, with dynamical
mechanisms describing the light-matter interactions; furthermore, Lorentz was the first
to recognise the necessity to bring some changes in the kinematical formulae of the
Galilean transformation;
15 It is a remarkable piece of chance that the quantum of action h doesn't appear in this "normal" Zeeman
effect. This is not any more the case for the so-called "abnormal" Zeeman effects, where the electron spin
and magnetic moment make the pattern much more complex. The abnormal Zeeman effect was only
understood a quarter of a century later, after the introduction of the spin and magnetic moment of the
electron.
16 In his Lecture Notes on "Electricité et Optique" (1901), Poincaré speaks of this accumulation of
hypotheses as ".. des petits coups de pouce". In "La Science et l'Hypothèse" (1902), he writes "... il fallait
une explication; on l'a trouvée; on en trouve toujours; les hypothèses, c'est le fond qui manque le moins".
17 Although it is not written in the same way as to-day. Lorentz persists in his way of thinking: he firstly
performs a Galilean transformation, which he later on complements by a change of the "kinematical
state", i.e. introducing a scaling of space and of the "local time".9- on the other hand, it is not so easy to recognise Lorentz as one of the founding
fathers of special relativity. He maintains the idea of an ether as a privileged medium
which is the "cause" of certain real effects (like the Lorentz real contraction of bodies
in motion with respect to ether). His transformations are basically two steps
transformations: Galilean transformations (which form a group), followed by ad-hoc
"corresponding states" changes (which do not). This way of reasoning forbade him to
discover the group property of the full transformation, which makes his theory
"relativistic". This attitude reminds us of the ancient attitude of "saving appearances" in
astronomy. Once the appearances are saved, Lorentz doesn't care to re-evaluate the real
role of ether.
This revaluation will be done by Poincaré and by Einstein, in two different ways
however:
- Poincaré conserves the idea of an ether as a truly existing medium, because
such a medium seems to be necessary for our understanding of the electromagnetic
phenomena. However, he insists on the group character of the Lorentz transformations,
which reduces the ether to a secondary role. The ether frames are ordinary members of
the infinite set of inertial frames, and no physical experiment allows to select them in
this set. Any inertial frame can legitimately ("à bon droit") be considered as the ether
frame! This amounts to an elimination " de facto " of the ether, because we loose any
possibility to identify it [Po-05], [Po-06].
- Einstein eliminates the ether " de jure" when he constructs a kinematics which
makes no reference at all to such a medium [Ei-05; 4]. He avoids carefully to make any
reference to an ether when he discusses the properties of light.
We shall analyse these two approaches in more details in Part 3.
2b) Poincaré's Oeuvre, from 1875 up to 1904.
Henri Poincaré is reputed as one of the most important and creative
mathematicians of the last quarter of the ninetieth century. He was then considered as a
kind of living King of the mathematics. His mathematical Oeuvre is immense and
contains contributions to all domains of mathematics: arithmetic, geometry, algebra,
analysis, ordinary and partial differential equations, group theory and analysis situs
(topology). It is important for our subject to recall that Poincaré is one of the founding
fathers of the theory of continuous groups (S. Lie's groups ) and in particular, that he
wrote between 1899 and 1901 two important memoirs on a general presentation of this
theory 18. Poincaré was also a great mechanician: analytical mechanics, mechanics of
continuous media and celestial mechanics; his work on the three body problem received
the Price of the King of Sweden in 1889 and it is considered to-day as a pioneering
work for modern approaches to chaos. All these mathematical tools play an important
role in his works on physics, in particular in his work on relativity. To-day, it is not so
well known that Poincaré was also a great physicist. This oblivion contrasts with his
reputation at the beginning of the twentieth century. It is known from the archives of
the Nobel Foundation that between 1901 and 1912 he was, with 49 presentations, the
person most frequently proposed to the Nobel Prize in physics (Cf. Ma-98). Poincaré
can be considered as the father of Mathematical-Physics, i.e. this approach to physical
theories which carefully uses the richness and strength of mathematical rigor. For the
period 1887-1901, his contribution to physics counts not less than (Cf. Po-01):
-18 memoirs (1887-1892) on the differential equations of mathematical physics;
- 9 memoirs (1890-1894) on hertzian waves;
-36 memoirs (1889-1901) critically reviewing existing physical theories;
- several books (lecture notes) corresponding to his most varied teaching.
Particularly important for our subject are his " Théorie Mathématique de la Lumière " of
18 Cambridge Philosophical Transactions 18 (1899) 220-255; Rendiconti del Circolo Matematico di
Palermo 15 (1901) 48 p. Completed later by a third memoir: Rendiconti del Circolo Matematico di
Palermo 25 (1908) 61 p.101899 and his " Electricité et Optique " of 1901; they present and discuss the various
approaches to electromagnetism and optics proposed by "the successors of Maxwell".
In 1895 Poincaré publishes a series of four papers entitled " A propos de la
théorie de M. Larmor " where he critically discusses and compares the different
existing theories of electro-optics, i.e. the different adaptations of the older mechanical
theories of light (Fresnel, Helmholtz, Mac Cullagh) to a Maxwellian approach, as
proposed by Larmor, Helmholtz, Lorentz , J.-J. Thomson and Hertz. Poincaré proposes
three criteria which he considers as being essential in order to get an acceptable theory:
1) the theory should account for Fresnel's drift coefficient;
2) it should contain the idea of conservation of electricity and magnetism;
3) it should be compatible with the Newtonian principle action=reaction.
Poincaré observes that none of the existing theories satisfies all three requirements. For
example Hertz's theory violates the first condition, Helmoltz's theory violates the
second, and Lorentz's theory violates the third. Poincaré discusses several alternative
interpretations of this state of affairs: either the theories are incomplete, or the three
criteria are (for some obscure reason) mutually incompatible, or they would become
compatible only by a radical modification of admissible hypotheses. He finally
concludes that one should temporarily abandon the idea to build a theory that would
conform the three requirements, and that one should temporarily again retain the theory
which seems least defective , i.e. the theory of Lorentz:
" Il faut donc renoncer à développer une théorie parfaitement satisfaisante et s'en tenir
provisoirement à la moins défectueuse de toutes qui paraît être celle de Lorentz".
Nevertheless, Poincaré's disquietude with respect to the violation of the
action=reaction principle is clearly stated:
"Il me paraît bien difficile d'admettre que le principe de réaction soit violé, même en apparence,
et qu'il ne soit plus vrai si l'on envisage seulement les actions subies par la matière pondérable et si on
laisse de côté la réaction de cette matière sur l'éther."
Finally, Poincaré makes a very important statement concerning the future of the
electrodynamics of moving bodies: facts are accumulating in favour of the idea that it
would turn out to be impossible to exhibit any effect of the motion of bodies with
respect to ether; it seems only possible to observe effects of the relative motion of
ponderable matter with respect to ponderable matter:
" L'expérience a révélé une foule de faits qui peuvent se résumer dans la formule suivante: il est
impossible de rendre manifeste le mouvement absolu de la matière, ou mieux le mouvement relatif de la
matière par rapport à l'éther; tout ce qu'on peut mettre en évidence, c'est le mouvement de la matière
pondérable par rapport à la matière pondérable".
He suggests that one should look for a theory which fulfils this requirement to all order
in v/c and he expresses the hope that the difficulty of Lorentz's theory with the principle
of reaction might be solved at the same time.
As it was correctly pointed out by Goldberg [Go-67], the major importance of
this paper is that it sets the framework for all Poincaré's subsequent work and attitudes
in the area of the electrodynamics of moving bodies.
From now on, Poincaré will carefully watch all the developments of Lorentz's
theory. At the occasion of the 25-th anniversary of the thesis of Lorentz, Poincaré
contributes to the anniversary volume by a (somewhat provocative) paper where he
comes back on the difficulty of the theory with respect to the principle of reaction [Po -
00]. He proposes to consider as a solution that the electromagnetic energy be viewed as
a kind of "fictitious fluid" with an inertial mass equal to E/c2; he shows that this would
be enough to restore the conservation of the momentum in the processes of emission
and absorption of radiation, at least to first order in v/c . And Poincaré insists on the
necessity to make use of the "local time" in order to obtain the conservation, and for the
first time, he explains the physical content to this idea of local time which up to then
was considered as an artificial mathematical trick:11" Pour que la compensation se fasse, il faut rapporter les phénomènes, non pas au temps vrai t,
mais à un certain temps local t' défini de la façon suivante.
Je suppose que des observateurs placés en différents points, règlent leurs montres à l'aide de
signaux lumineux; qu'ils cherchent à corriger ces signaux du temps de la transmission, mais qu'ignorant
le mouvement de translation dont ils sont animés et croyant par conséquent que les signaux se
transmettent également vite dans les deux sens, ils se bornent à croiser les observations en envoyant un
signal de A en B, puis un autre de B en A. Le temps local t' est le temps marqué par les montres ainsi
réglées.
Si alors c = 1/ √ K0 est la vitesse de la lumière, et v la translation de la terre que je suppose
parallèle à l'axe des x positifs, on aura:
t' = t - v x /c2 . "
Poincaré states clearly that the compensation is only to first order in v/c except if one
further makes another hypothesis on which he will presently not comment (clearly the
Lorentz-FitzGerald contraction). It should be stressed that Poincaré was the first to
discuss the concept of simultaneity and the problem of defining a common time for
distant clocks. This goes back to a philosophical paper of 1898 [Po-98], but it is first
explicitly borne out by a calculation in this anniversary paper of 1900. In September
1904, at the St-Louis Conference, he presents essentially the same reasoning and adds
the "third hypothesis" 19 of the Lorentz-FitzGerald contraction in order to obtain the
recently proposed new Lorentz's local time. In order to report on an event which takes
place at point x and time t in the ether frame, one should use in the moving frame the
"local time",
(10) t'(t,x) = γ ( t - vx / c2) ,
where γ is the inverse of Lorentz-FitzGerald's contracting factor:
(11) γ = (1- v2/c2)-1/2 .
At the occasion of this Conference, Poincaré discusses the problems met by a
physics based on Principles. Among these principles, he discusses at length the
Principle of Relativity which he enunciates as follows:
" The principle of relativity according to which the laws of physical phenomena should be the
same, whether for an observer fixed, or for an observer carried along in a uniform movement of
translation; so that we have not and could not have any means of discerning whether or not we are carried
along in such a motion."
But Poincaré is too lucid a physicist to accept "Principles" as "evidently true", as Truths
given by some God. Therefore he insists on the necessity to get experimental
confirmation of the Principles 20.
"These principles are results of experiments boldly generalised; but they seem to derive from
their generality itself an eminent degree of certitude."
Coming back to the difficulties met by the Principle of Relativity in the recent past, he
concludes:
19 In the past, this "third hypothesis" of Poincaré has been the source of rather radical attacks against his
presentation of relativity (see f.i. Pa-82). The point was recently clarified (see Re-00). It turns out that
historically, Poincaré was right to present in 1904 the Lorentz-FitzGerald contraction as a third
hypothesis, and not any more after 1905. However, his reasons to leave out the third hypothesis being
essentially mathematical (group arguments), they would have appeared to the physicists audiences of the
time as less clear and convincing than the then generally accepted idea of a contraction.
20 Strangely enough, this idea that Principles are largely conventions that should in any case always be
borne out by experiment has been vigorously used by some authors in order to minimise the contributions
of Poincaré to the theory of relativity (see f.i. Mi-94).12" Thus, the principle of relativity has been valiantly defended in these latter times, but the very
energy of the defence proves how serious was the attack."
It is clear that end 1904, Poincaré is at the eve to create a new "electron
mechanics". In his book on the history of relativity, Miller produces an interesting
document [Mi-81]: a letter from Poincaré to Lorentz written end 1904 or early 1905
(unfortunately not dated) where he makes some points which are very illuminating of
the state of evolution of the question. Lorentz's transformations contain a general scale
factor (written l(v)) that Lorentz got much difficulties to put equal to one. Poincaré
points out that the set of parallel translations form a group (now called the group of
"boosts"), and that a rather natural hypothesis 21 on the electron structure is then
enough to reduce this scale factor to one. Incidentally, this letter contains explicitly but
without any comment the new relativistic formula of addition of parallel velocities 22.
Part 3.- 1905.
The long march towards relativity is now very close to an issue. Two of the
important actors of the creation of special relativity are sitting in due place. They are
known as very respectable scientists and it is expected that their works (past and future)
will be scrutinised by many other physicists in the world. Up to now, I did not speak
much about the third main actor of the saga of relativity: Albert Einstein (1879-1955).
This is not surprising since during the period covered in Parts 1 and 2, Einstein was too
young to participate. Furthermore, beginning 1905, Einstein is still nearly unknown in
the world of the physicists interested in electrodynamics. His early publications (five all
together between 1901 and 1904) concerned thermodynamics and statistical physics.
Even in this domain, these papers didn't really awake interest. Then comes the "Annus
Mirabilis 1905" which will see this young physicist growing from his modest position
to such a prominent one, that for many years, the whole world of physicists will pay
much attention to what he says and what he writes.
It starts with a paper on the black body radiation where applying his own
thermodynamical methods to examine the well known Wien's formula, Einstein shows
that light exhibits features that make it alike a gas of non interacting particles. As
straightforward applications of this "heuristic" point of view, Einstein gives very simple
explanations of some not yet understood phenomena, the best known being the photo -
electric effect. This work will be recompensed by the Nobel Prize in Physics 1921.
Three other papers concern the Brownian motion and are directly derived from his
Ph.D. thesis [Ei-05; 2,3,6]. They are again in the general trend of his previous work on
thermodynamical and statistical physics. Even to-day, these papers remain considered
as important (see comments and analysis by biographs of Einstein, f.i. Pa-82). None of
these papers concern directly or indirectly the electrodynamics of moving bodies. At
most can one say that his heuristic point of view on light did convince Einstein that
light was not simply a wave phenomenon, and therefore, that ether might be a useless
hypothesis. It is at least what Einstein himself declared in 1952 in a letter to Von Laue:
" In 1905, I already knew with certitude that Maxwell's theory doesn't correctly predict
the fluctuations of radiation pressure in a thermal enclosure. I convinced myself that the
21 It is a model dependent hypothesis on the structure of the electron; it must respect the idea that the
very structure of the electron can well depend on the absolute value of the velocity but not on its sign.
22 This absence of comments was sometimes interpreted as an indication that Poincaré did not really
grasp the physical meaning of this addition law (Cf. f.i. Mi-94). However, beside this implicit derivation
from the group property, Poincaré makes an explicit one in his 1905 paper [Po-06] by computing the
derivative of the displacement with respect to time (to local time). Furthermore, in [Po-08] he describes
explicitly what does physically happen when performing such an addition. One can easily check, through
a calculation following closely what he says, that he indeed describes this new addition law (Cf. Pi-99,
p.143-146).13only way to save the situation was to give to radiation the objective status of a "being",
which of course doesn't exist in Maxwell's theory".
Then came the famous paper " Zur Elektrodynamik bewegter Körper ", [Ei-05; 4]
(received on 30 June 1905), now considered by nearly all physicists as the founding
paper of the special theory of relativity. As I explained in the Introduction, for many
physicists this conviction is only a product of their education. Not so many of them
have really read the original paper. And those who read it are stil much more numerous
than the very small number of physicists who read the two papers published at the same
time by Poincaré (Po-05 received on 5 June, and Po-06 received on 23 July 1905). This
is certainly a very curious case in view of the respective reputation of the two scientists
in 1905. One might hastly conclude that the reason is simply that one of the papers is
right and the other wrong. We shall see that it is not at all so simple 23.
Let us start with a short comparison of the contents of the papers of both
authors:
Poincaré Einstein
Sur la dynamique de l'électron. Zur Elektrodynamik bewegter Körper.
0- Introduction. 0- Introduction
I- Kinematical part .
1- Lorentz's transformation. 1- Definition of simultaneity.
2- The principle of least action. 2- On the relativity of lengths and times.
3-The Lorentz transformation and 3- Theory of the transformation of
the principle of least action. co-ordinates and times from a stationary
system to another system in uniform
translation relatively to the former.
4- Lorentz's group. 4- Physical meaning of the equations
obtained in respect to moving rigid bodies
and moving clocks.
5- Langevin's waves. 5- The composition of velocities.
II- Electrodynamical part.
6- Contraction of electrons. 6- Transformation of the Maxwell-Hertz
equations for empty space. On the nature
of the electromotive forces occurring in a
magnetic field during motion.
7- Quasi stationary motion. 7- Theory of Doppler's principle and
of aberration.
8- Arbitrary motion. 8- Transformation of the energy of light
rays. Theory of the pressure of radiation
exerted on perfect reflectors.
9- Hypotheses concerning 9- Transformation of the Maxwell- Hertz
gravitation. equations when convection currents are
taken into account.
23 I discard the rather trivial explanation based on the comparative fame of the periodics where these
papers were published. It is true that Annalen der Physik was better known to physicists than Rendiconti
del Circolo Matematico di Palermo, but the content of Poincaré's paper was previously communicated to
the Comptes Rendus de l'Académie des Sciences in a sufficiently detailed way to catch the attention of
physicists interested in electrodynamics [Po-05] .1410- Dynamics of the slowly accelerated
electron.
This dry presentation of titles of paragraphs calls for two remarks:
i) the length of the titles is in no way representative of the real length and
importance of the content of the paragraphs; remember that the total lengths of the two
papers are respectively of 47 pages for Poincaré and 30 pages for Einstein;
ii) it would probably be fair to add to Einstein's paper, as a complementary
eleventh paragraph, his famous paper on the equivalence of mass and energy which
proceeds essentially along the same ideas (Ei-05; 5, received on 27 September 1905).
In the Introduction of their respective papers, both authors clearly
announce the guide line of their works:
- Poincaré wants to continue the 1904 work of Lorentz, to put it in a more
rigorous mathematical form, to discard definitely some rival models of the electron and
(above all?) to try to extend Lorentz's ideas to the theory of gravitation.
- Einstein wants to eliminate from Maxwell's theory some difficulties brought in
by the idea of an absolute rest. He claims that this can be done very simply with the
following two postulates: the "Principle of Relativity" , and the postulate " ... that light
is always propagated in empty space with a definite velocity c which is independent of
the state of motion of the emitting body." 24
Therefore, it is immediately clear that the two authors will develop different programs.
Let us now compare in a more detailed way the contents of both papers in
subdividing them according to broad subjects: Lorentz's transformation, covariance of
the electromagnetism, dynamics of a relativistic particle.
1- Lorentz's transformation.
Although Lorentz's transformation is more or less present in nearly all chapters
of both works, I shall limit myself here to the paragraphs that concern specifically the
transformation of the co-ordinates, i.e. §1 to 5 (Kinematical part) for Einstein, and §1
and 4 for Poincaré.
In §1, Einstein discusses the notion of simultaneity and makes the very
important point that this notion is only clear and evident for "local" events, i.e. events
which happen at approximately the same place, and that the discussion of simultaneity
of "distant" events requires at first some synchronisation of distant clocks. He proposes
the synchronisation procedure by exchange of light signals, based on the second
fundamental postulate of his paper that light always propagates with the same velocity c
in all directions in any inertial frame of reference 25. Since Einstein's paper does not
mention any reference, one can legitimately infer that this fundamental thought is
entirely of his own 26. But one can just as well remember that Poincaré discussed the
concept of simultaneity in rather similar terms as early as 1898 27, and that he proposed
24 Formulated in that way, the second postulate is a bit confusing: in a wave theory, the velocity of the
wave doesn't depend on the velocity of the emitting body. Einstein means that the velocity of light has
the same absolute value in all inertial systems, and that this value is independent of the direction of
propagation.
25 Einstein doesn't really define what he means by an "inertial frame of reference". He says only that it is
a frame of reference where "the equations of mechanics hold good", which is a bit incoherent with his
own paper. Later, in (1913), he added the footnote "to the first approximation" which of course weakens
the mistake, but doesn't make the definition more precise.
26 Some authors have discovered that in his "Bureau des Brevets" in Berne, Einstein had to examine
proposals in order to synchronise the clocks of distant railways stations by exchange of telegraphic
messages (private communication of I. Daubechies).
27 This paper contains philosophical considerations on Time and its Measurement. One finds there some
premonitory sentences, like the following ones: 1) about the dating of astronomical facts: "Il (the
astronomer) a commencé par admettre que la lumière a une vitesse constante, et en particulier que sa
vitesse est la même dans toutes les directions. (..) Ce postulat ne pourra jamais être vérifié directement
par l'expérience; ....". 2) In the conclusions: "Il est difficile de séparer le problème qualitatif de la15the procedure of synchronisation by exchange of light signals and its application to
obtain Lorentz's local time in his 1900 paper and at the St-Louis conference of 1904
[Po-98, Po-00, Po-04]. If one can reasonably assume that both the 1898 and 1904
papers were unknown to Einstein in 1905, it is not so clear for the 1900 paper, since
Einstein does refer to it in a subsequent publication of May 1906 [Ei-06]. This
troublesome interrogation will probably never be answered.
In §2, Einstein discusses the application of his two Principles to two important
concepts:
- the possible difference of length of a rigid rod when it is viewed from the
frame where the rod is at rest, or alternatively from a moving frame;
- the delicate question of the simultaneity of distant events, viewed from their
common rest frame or alternatively from a moving frame.
In my opinion, this is a most interesting paragraph, where Einstein shows most
explicitly the originality of his approach.
In §3, Einstein proceeds to a derivation of Lorentz's co-ordinates transformation
from the two enunciated Principles. Notice that one doesn't find the name of Lorentz in
this paragraph, nor the name of any one else who wrote or used the same formulae
before 1905 (Voigt 1887, Lorentz 1899 and 1904, Larmor 1900, Poincaré 1904). This is
a bit strange because these formulae were already well known. This silence can be put
in parallel with the complicated title of the paragraph; it is possible that Einstein wants
to emphasise the difference between his interpretation and the one of previous authors.
The message may be: the physical content of these formulae being new, the formulae
are new . Here again, we shall probably never know. Schematically, Einstein's
demonstration proceeds along three steps:
- The first step is a detailed analysis of the events corresponding to a go and
back light exchange along the direction of propagation of the moving frame, with the
hypotheses that the transformation is linear and that the light velocity is the same in
both directions in all frames. This gives him the ordinary Lorentz's transformation,
including the arbitrary global scale factor l(v) that we mentioned earlier; Einstein writes
it φ(v). It is interesting to remark that Einstein does separate this factor. This is a kind of
a priori choice which has no real justification, except of course if one already knows the
answer (again the same interrogation about a previous knowledge). He meets then the
old problem of how to get rid of this factor φ(v).
- In the second step , Einstein performs a kind of inverse transformation, and
deduces from it that :
(12) φ(v) φ(-v) =1.
In essence, this is a first application of the principle of relativity, with the requirement
of a total reciprocity between the two translating frames.
- In the third step , Einstein considers again the events corresponding to a go and
back light exchange but now in a direction perpendicular to the translation. This is
entirely new! I mentioned already that Poincaré had discussed several times the
synchronisation of clocks placed along the direction of translation (i.e., essentially
Einstein's first step). But Einstein is the first to complement this longitudinal
synchronisation by a transverse one. He deduces from this operation the physical
meaning of the factor φ(v): it corresponds to a contraction of the length of a transverse
rigid rod when it is seen from a moving frame. He makes then the clever statement that,
simultanéité du problème quantitatif de la mesure du temps; soit qu'on se serve d'un chronomètre, soit
qu'on ait à tenir compte d'une vitesse de transmission, comme celle de la lumière, car on ne saurait
mesurer une pareille vitesse sans mesurer un temps." 3) Again in the conclusions: "Nous n'avons pas
l'intuition directe de la simultanéité, pas plus que celle de l'égalité de deux durées. (....) La simultanéité de
deux événements, ou l'ordre de leur succession, l'égalité de deux durées, doivent être définies de telle
sorte que l'énoncé des lois naturelles soit aussi simple que posible."16because the rod is perpendicular to the direction of motion, this possible contraction can
well depend on the relative velocity, but not on its sign! Therefore:
(13) φ(v) = φ(- v),
which, together with (12), gives:
(14) φ(v) = 1.
Einstein's derivation of Lorentz's transformation is most interesting in that it
differs on some important points from the previous approaches by Lorentz and by
Poincaré. Firstly, Einstein constructs a proof of Lorentz's transformation from first
principles . Secondly, he discusses the relations between two inertial frames , none of
them being privileged; this is particularly clear in step 2. Contrariwise, Lorentz and
Poincaré always discuss the relations between the privileged ether frame and another
frame in uniform translation with respect to the former. At first, it seems that the latter
attitude is not "relativistic". However, it has some advantage because it avoids the
delicate question of defining the concept of inertial frame: there exists an ether, and the
frames at rest in ether are absolute Newton's frames. Inertial frames are then frames in
uniform translation with respect to the ether.
For Lorentz, in full conformity with this idea, the fundamental transformations
remain the Galilean transformations, and the rest of the story is a question of changing
things in an appropriate manner (local time, scaling, corresponding kinematics and
electromagnetic states), in order to obtain a formal appearance of equivalence of the
electromagnetic phenomena in moving frames and absolute frames.
For Poincaré, it is not quite so simple. His approach evolves slowly from the
original Lorentz's position towards a detailed study and possibly a proof of Lorentz's
transformation, and a revision of the status of the ether. Indications and evidences of
this evolution are:
- Firstly, that the calculations giving the physical meaning of Lorentz's local
time (1900, 1904) are hardly separable from some implicit demonstration of Lorentz's
transformation, similar to Einstein's first step (see Re-00 for details).
- Secondly, the evolution in time of Poincaré's attitude about the problem of the
scale factor (i.e. Lorentz's l(v) or Einstein's φ(v)). In his letter to Lorentz, he recognises
the existence of the group of "boosts", and he finds the relativistic addition of parallel
velocities and a multiplication law for the scale factor: two successive parallel boosts
characterised by velocities v and v' are equivalent to a parallel boost characterised by a
velocity v'' , with the rules:
(15) v'' = (v +v')/ (1 + v v'/c2),
(16) l(v'') = l(v) l(v') .
At the time of his letter to Lorentz, Poincaré doesn't yet use fully the resources of group
theory, so that he comes back to the electron theory where he picks up a currently used
explicit form of l(v):
(17) l(v) = [1 - (v/c)2] m ,
where m is some model dependent parameter; clearly, only m=0 can fit the
multiplication law (16), and this means that l(v) is equal to one. But in §4 of his 1905
paper, Poincaré goes much farther. He makes a complete analysis of the Lorentz's
group (so did he term this new group), i.e., including not only the boosts, but also the
spatial rotations. Then, he can easily show without any hypothesis coming from the
electron theory, and without the transverse synchronisation, that l(v) must be equal to
one.17- Thirdly, this group structure being firmly established and remembering that
Poincaré knows very well what a group structure means, it becomes evident that the
ether frames loose their privileged status. This understanding is clearly stated in the
introduction of the paper:
"..; deux systèmes, l'un immobile, l'autre en translation, deviennent ainsi l'image exacte l'un de l'autre."
Let us also remark that Poincaré's study of the Lorentz's group is extraordinary
modern:
- derivation of the associated Lie group;
- theorem that any Lorentz's transformation can be seen as a Lorentz's boost along x,
preceded and followed by an appropriate rotation;
- theorem that any Lorentz's transformation can be resolved into a dilatation and a linear
transformation which leaves unaltered the quadratic form: x2 + y2 + z2 - t2 ;
- geometrical interpretation of the latter (continuous) transformation as a rotation in the
four dimensional space:
x , y , z , t √−1 ;
- discovery of the electromagnetic field invariants (§ 3 , 5):
(18) E2 - H2, E . H ,
and of several kinematics invariants when more particles are present (§9);
- proof that several physical quantities are the individual components of "four partners"
that vary under Lorentz's transformations like the three space co-ordinates and the time.
Examples are: the force reduced to unit volume or (alternatively) reduced to unit charge
and the corresponding work per unit time, the four component velocity : γ v, γ (or
momentum: m γv, mγ ; as usual, I write γ = (1- (v/c)2) -1/2 ), the electric current and
electric charge densities, the vector and the scalar electromagnetic potential in Lorentz's
gauge, etc.
Let us finally notice that the new law of addition of velocities (parallel and non -
parallel composition) was found by Einstein (§5) and by Poincaré (§1) in nearly the
same way: the combination of the velocity vector w of a particle with the velocity v of a
frame moving along the x axis. Einstein doesn't hesitate to extrapolate the law to the
case of a "light particle" moving along x with the velocity c and he finds in this
calculation a confirmation of his postulate that the light velocity remains equal to c in
all inertial frames 28. This kind of extrapolation will of course never be done by
Poincaré who considers that light and material particles have completely different
status.
2- Covariance of electromagnetism.
Let us now consider the electrodynamical part (except the dynamical equation
for particle motion, which we will discuss later), i.e. § 6-9 for Einstein, and § 1-3 and 5
for Poincaré. Both authors want to make sure that Maxwell's equations are invariant
under Lorentz's transformation. In vacuum, where ε0 and µ0 can be taken equal to one
(see Eq.2), these equations can be written in two equivalent ways:
- Either , in terms of the physical fields, electric field E and magnetic field H
(remember that ε0 and µ0 are equal to one):
(19) div H = 0 ,
28 Funny enough, Einstein doesn't make w = c but v = c ! This amounts to consider a Lorentz's
transformation of velocity c, i.e. a frame (an observer) going as fast as light, clearly an impossible case.
Simple misprint or Einstein's subconscious old dream of following a light ray ?18(20) rot E + 1
c ∂H
∂t = 0 ,
(21) div E = 4 π ρ ,
(22) rot H - 1
c ∂E
∂t = 4 π j ,
where ρ is the electric charge density and j is the electric current density; these are
obviously constrained by the relation:
(23) div j + ∂ρ
∂t = 0 ,
which expresses the conservation of electricity; in the case of a material current with a
"local" charge of velocity u, the density current j is equal to ρ u;
- Or , in terms of a scalar potential V and a vector potential A constrained by a
gauge condition; this corresponds to a trivial integration of the homogeneous equations
(19) and (20):
(19') H = rot A ,
(20') E = - 1
c ∂A
∂t - grad V ,
(21')1
c2 ∂2V
∂t2 - Δ V = 4 π ρ ,
(22')1
c2 ∂2A
∂t2 - Δ A = 4 π j ,
with Lorentz's gauge constraint:
(24) div A + 1
c ∂V
∂t = 0 .
It is interesting to notice that Einstein chooses to use the first set of equations
while Poincaré chooses to use the second 29; furthermore, the way they respectively
proceed is characteristic of their preoccupations. I shall not reproduce here the
calculations of these authors, but I shall try to indicate their ways of reasoning.
One remembers that Poincaré insisted in his paper " À propos de la théorie de M.
Larmor " of 1895 , on the absolute necessity of the conservation of the electric charge.
He will now apply it on the changing of frame operation: the total electric charge in
some spatial volume in the (x,y,z,t) frame must be the same as the total electric charge
in the corresponding volume (obtained through Lorentz's transformation) in the (x', y',
z', t') frame. In that way, he gets the relation between corresponding electric charge
densities ρ and ρ'. Since the kinematics gives the relation between the velocities u and
u', he immediately obtains the relation between the electric current densities j and j'. He
29 Poincaré's results are obtained for the full conformal group, including the dilatation parameter. Only
later, does he prove that l(v) is equal to one. In this comparative report, I take l(v) =1 from the
beginning.19checks the covariance of the conservation law (23) 30. He also checks the formal
covariance of the d'Alembertian operator and, imposing the covariance of Eqs. 21' and
22' , he gets the relation between the four partners A x , Ay , Az , V and their homologues
A'x', A'y' , A'z' , V' ; he sees that they transform just alike the co-ordinates x , y , z , ct
(four-vectors) ; he checks the covariance of the gauge constraint (24). From the
kinematics, the latter results, and Eqs. 19' and 20' , he computes the relations between
the six partners E x , Ey , Ez , Hx , Hy , Hz , and their homologues E' x' , E'y' , E'z' , H'x' ,
H'y' , H'z' , as they were already found by Lorentz [Lo-04] 31. He finally checks the
covariance of the Eqs. 19 to 22. Poincaré considers then the Lorentz's force on the unit
electric charge, when this force is written in just the same formal way in both frames;
he shows that there exists a "four partners" set : F x , Fy , Fz , F.u , that transforms just
in the same way as the co-ordinates x , y , z , ct (four vector). This achieves the proof
of covariance of the Maxwell equations.
Einstein considers firstly equations (20) and (22) with j = 0, i.e. the
electromagnetic field in empty space (no electric charge and no electric current). He
takes these equations as they are written in one frame, performs Lorentz's
transformation on the co-ordinates and assumes that the so obtained equations have
again the same form (20) and (22) in the other frame. This gives him the relations
between the six partners E x , Ey , Ez , Hx , Hy , Hz , and their homologues E' x' , E'y' , E'z'
, H'x' , H'y' , H'z' ; these relations are those already found by Lorentz [Lo-04]. Then, he
immediately looks for some physical consequences of this transformation law of the
free-fields (§ 7 and 8). In that way, he obtains the relativistic formula for the Doppler
effect (including the entirely new transverse Doppler effect), the aberration of light, the
radiation pressure on perfect reflectors, and the transformation of the energy of a "light
complex" 32 . He notes (without any allusion to the famous quantum formula E = h ν
that he derived three months earlier):
"It is remarkable that the energy and the frequency of a light complex vary with the state of
motion of the observer in accordance with the same law."
In § 9, Einstein terminates his proof of the covariance of Maxwell's equations. He
considers Eqs. 20-22 including now the electric charge density ρ and the electric
current density ρ u. Assuming the equations to be valid in one frame 33, he performs the
co-ordinates transformation and the transformation of the fields that he found in § 6 for
the free fields. This latter operation indicates that Einstein considers the fields (and
therefore the light) as physical entities, which should transform in the same way,
independently of their possible coupling to charges and currents. From these
transformations, he deduces the transformation law of the densities of electric charge
and electric current. He notes (with an evident satisfaction) that the latter is only
covariant if one adopts the new law of addition of velocities that he found in § 5:
30 Poincaré's results differ slightly from Lorentz's ones. Only Poincaré's results do satisfy the covariance
of the conservation law (23).
31 As it is well known, they only differ from the older Lorentz's formulae [Lo-95] reproduced in our
Eqs. 8 and 9 by the presence of a factor γ in the right hand side of the components E' y', E'z' , H'y' and
H'z' .
32 Einstein's conceptual relations with light are not simple: he speaks alternatively of "light rays", "light
waves", and of "light complex", the latter remaining undefined. In the case of the energy, it seems that
the "complex" is the light contained in some sphere moving with the wave (with velocity c) in one frame.
This sphere is viewed as an ellipsoid moving with velocity c in the other frame. The energy of the
"complex" is the product of the "time averaged energy density " times the volume. Notice that if Einstein
discusses at length properties of "existing light" (Doppler effect, energy of a "light complex", radiation
pressure), he avoids to discuss the process of emission of light by accelerated particles.
33 For the first time, Einstein makes here an allusion to Lorentz : "If we imagine the electric charges to
be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis
of the Lorentzian electrodynamics and optics of moving bodies" . This doesn't tell us much on which
work of Lorentz between 1886 and 1904 he refers to.20"Since - as follows from the theorem of addition of velocities (§ 5) - the vector u' is nothing else
than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of
our kinematical principles, the electrodynamic foundation of Lorentz's theory of the electrodynamics of
moving bodies is in agreement with the principle of relativity."
The next sentence is interesting; Einstein writes:
"In addition, I may briefly remark that the following important law may easily be deduced from
the developed equations: If an electrically charged body is in motion anywhere in space without altering
its charge when regarded from a system of co-ordinates moving with the body, its charge also remains -
when regarded from the "stationary "system K - constant."
In other words, the final statement of these paragraphs on electrodynamics is just the
starting point of the analogous Poincaré's study. This underlines the difference of point
of view between the two authors in their simultaneous and original first complete
proofs of the covariance of Lorentz's electrodynamics.
Let me now come back on § 5 of Poincaré's paper (Langevin waves) which has
no counter part in Einstein's paper. It concerns the electromagnetic field created by a
charged particle in motion. It is well known that this field can be separated into two
parts: a part linear in the acceleration of the particle (that Langevin calls "the
acceleration wave") and another part which depends only on the velocity (that Langevin
calls "the velocity wave"). We know that only the former subsists asymptotically and
that it represents the radiation (light) actually emitted by the accelerated particle.
Poincaré observes that the velocity wave is nothing else than the Lorentz's transform of
the static electric field of the particle at rest, when the particle is given its velocity by a
proper Lorentz's transformation. Things are not so easy to handle for the accelerated
wave. However, Poincaré remarks that essential features of the wave, previously
discovered by Hertz for the emission of radiation by a slowly moving accelerated
particle, must remain true when the particle has a higher velocity because they
correspond to Lorentz's invariant quantities . These properties are the following ones:
the electric and magnetic fields have the same magnitude, they are mutually
perpendicular and they are both perpendicular to the normal to the wave front. Poincaré
shows here his handiness in manipulating invariants to elucidate rather complicated
physical phenomena.
But Poincaré is not yet completely satisfied with his direct check of the
covariance of Maxwell-Lorentz's equations of §1. He wants to complete his analysis by
a careful study of the derivation of these equations from a principle of least action (§ 2
and 3). This part of the paper is really a master piece of the calculus of variations.
Poincaré succeeds to derive Maxwell-Lorentz's equations and Lorentz's force in a
covariant way from the variation calculus. He uses the following "action" 34:
(25) J = dt dτ ∂E
∂t + ρ v .A - 1
2 E2 + H2 ,
the integral being taken over all space (d τ is the volume element of space ) and time;
alternatively, he also uses the simpler form:
(25') J = dt dτ 1
2 H2 - E2 ,
which is equivalent to the former and which will prove easier to handle in the
calculations of the dynamics of the electron.
34 Poincaré's calculus is based on conventions which differ by signs from the modern notations.
Therefore, I change the sign of his lagrangian density in order to agree with current conventions.21Furthermore, as it was correctly pointed out by Logunov [Lg-95], some details
of the calculations of § 3 leave no doubt about the understanding by Poincaré of the
relativity of length and time intervals measured in different inertial reference frames.
3-The dynamics of the electron.
Let us now consider the problem of establishing the dynamical equations of
motion for a particle. No other part of the detailed comparison of the works of both
authors can better illustrate the difference of their interests. Einstein's treatment of the
problem is short (§10), simple and totally unusual for the epoch. He doesn't refer to any
kind of electromagnetic structure of the electron as it was then customary (Cf. the
works of J.J. Thomson (1881), J. Larmor (1894), Abraham [Ab-02], Lorentz [Lo-04],
Langevin [La-05] and others). Poincaré's treatment is extensive (§ 6-8), of higher
mathematical level, and essentially based on the electromagnetic model of the electron;
it also provides the title of the paper.
Let us start with Einstein's treatment of the problem. He considers some material
particle with a definite mass and electric charge, and he calls it "electron" only for the
sake of convenience. Einstein assumes that, if this particle is instantaneously at rest,
then Newton's dynamical equation of motion must be valid, with the sole electric force
on the right hand side. According to the principle of relativity, if he then communicates
to this particle its actual velocity by an appropriate Lorentz's transformation,
transforming of course accordingly the acceleration and the force, he finds the
relativistic instantaneous equation of motion. Einstein performs this transformation for
an instantaneous velocity along the x-axis and stops there his reasoning. He apparently
doesn't realise that this equation is then only instantaneously valid and that an instant
later, the motion will not any more be along the x-axis. Therefore his dynamical
equation of motion is not correct. Furthermore, probably influenced by Abraham's work
(not quoted), he makes a wrong choice for the definition of the force and accordingly,
he gets a wrong value for the "transverse" mass. Nevertheless, he mentions that:
"With a different definition of forces and accelerations we should naturally obtain other values
for the masses. This shows us that in comparing different theories of the motion of the electron we must
proceed very cautiously."
He generalises his results by a mere sleight of hand:
"We remark that these results as to the mass are also valid for ponderable material points,
because a ponderable material point can be made into an electron (in our sense of the word) by the
addition of an electric charge, no matter how small ."
It looks as if Einstein were not really interested in the question of the electron
dynamics 35. Nevertheless, he goes one step further with his (wrong) equation and he
uses (correctly!) the sole x-component for a finite rectilinear motion along the x-axis, in
order to compute the kinetic energy of the electron as being the energy withdrawn from
the electrostatic field when the electron increases its velocity from zero to its actual
value v. In that way, he finds that the kinetic energy of the electron is equal to:
(26) W = mc2 [ (1- v2/c2)-1/2 - 1] .
This formula interests him much more than the dynamical equations themselves. He
extends immediately its validity to any ponderable matter, ".... by virtue of the argument
stated above ". We shall see later how this bold extension will allow him to derive his
famous mass-energy equivalence formula.
One sees that Einstein did not really succeed in 1905 to get the correct
dynamical equations of motion of an electron. But his genial intuition was then enough
35 The correct dynamical equations of motion were obtained along Einstein's way of reasoning by Max
Planck [Pl-06]. See also Pl-07 and Pl-08.22to put him right away very close to his most extraordinary discovery of the equivalence
of mass and energy [Ei-05; 5].
Let us now come to Poincaré's treatment of the problem. I shall present it with
some details, because it is really the centre of the paper. Poincaré adopts fully the
electromagnetic image of the electron, as it was previously proposed by several authors
(see refs. here above): dynamically, the electron is nothing else than the
electromagnetic field created by its own electric charge. Therefore, the "action" that
defines the electron free motion is the electromagnetic action (25'), where the fields are
created by the charge e. In order to avoid the divergence of the action integral at small
distances that happens in the case of a point particle, one has to assume that the charge
is distributed in some (unknown) way in a very small volume. All authors agree that, in
the absence of any physically privileged direction (i.e., for an electron at rest in the
ether), this distribution of charge is spherically symmetric. The models differ from each
other when some privileged direction exists, f.i. when the electron is in uniform
translation with respect to the ether. Poincaré wants to compare these different models
on grounds of their reaction to the general Lorentz's transformation, including the scale
factor l(v). Beside the traditional cases of the electron at rest in the ether and the
electron in uniform translation with respect to the ether, he wants to consider what
happens to the latter when one formally carries it to rest by a Lorentz's transformation;
this is the essence of his distinction between the "real" electron (l'électron "vrai") and
the "ideal" electron (l'électron "idéal"). He adopts a general ellipsoidal model with
cylindrical symmetry around the velocity, therefore characterised by two parameters:
the longitudinal axis r, and the transversal axis q r. As usual the electron is spherically
symmetric when at rest in ether, but this hypothesis turns out to be of no importance in
the following, because the discussion focuses on the difference between the "real"
electron and the "ideal" electron. Poincaré pays special attention to the three following
models, but his discussion remains general:
Electron model. At rest in ether. Real electron. Ideal electron.
Abraham: Spherical and rigid. Prolate ellipsoid.
Langevin: Spherical. Oblate ellipsoid, Spherical, but
constant volume. dilated volume.
Lorentz: Spherical. Oblate ellipsoid, Spherical, alike
contracted volume. in the ether.
If one compares these three models on grounds of the general conformal Lorentz's
transformation with l = l(v), one sees that Abraham's model is not compatible with this
transformation (rigid spherical electron), that Langevin's model corresponds to l(v) =
[1 - (v/c)2]1/6, and that Lorentz's model corresponds to l(v) =1. If one extends the
comparison within the larger class of models considered by Poincaré, the conclusion
remains that Lorentz's model is the only one compatible with l(v) =1, i.e. compatible
with a group structure based on the sole relative velocity. In other words, Lorentz's
model is the only one which can be called "relativistic". In the crucial choice which he
will have to make, between a very satisfactory purely electromagnetic model which
unfortunately is non relativistic (Langevin), and a non completely electromagnetic
model which is relativistic (Lorentz), Poincaré doesn't hesitate:
"L'avantage de la théorie de Langevin, c'est qu'elle ne fait intervenir que les forces
électromagnétiques et les forces de liaison: mais elle est incompatible avec le postulat de relativité; c'est
ce que Lorentz avait montré, c'est ce que je retrouve à mon tour par une autre voie en faisant appel aux
principes de la théorie des groupes. Il faut donc en revenir à la théorie de Lorentz; mais si l'on veut la
conserver et éviter d'intolérables contradictions, il faut supposer une force spéciale qui explique à la fois
la contraction et la constance de deux des axes . J'ai cherché à déterminer cette force, j'ai trouvé qu' elle
peut être assimilée à une pression extérieure constante, agissant sur l'électron déformable et
compressible, et dont le travail est proportionnel aux variations du volume de cet électron ".23The choice is clear and shows the importance that Poincaré gives to the principle of
relativity.
The calculations of Poincaré are too complex to be reproduced here.
Nevertheless, it is important to give some details in order to appreciate the crucial
choice that is made here. Starting with the ordinary Maxwell's expressions for the
energy and the momentum of a purely electromagnetic system, Poincaré computes
these quantities for the ideal electron (W', P' = 0) and for the real electron (W, P). With
the help of Lorentz's transformation of the fields and of the volume element dt , he
finds:
(27) W = W' l(v)
1 - (v/c)2 1 + 1
3 (v/c)2 ,
(28) Px = 4
3 c2 W' l(v)
1 - (v/c)2 v ,
(the components P y and Pz being zero for a real electron in uniform translation along
the x axis). Poincaré computes also the Lagrangian function (i.e. the action by unit of
time) of the electromagnetic electron from Eq. 25' :
(29) L = dτ 1
2 H2 - E2 = - W' l(v) 1 - (v/c)2 .
When he compares the canonical expressions of energy and momentum of a particle
derived from this Lagrangian with the values (27) and (28), he sees that the full scheme
is consistent only if l(v) = [1 - (v/c)2]1/6, i.e., the value of Langevin's model. A simple
check of this calculation (otherwise, a bit sophisticated) is obtained by computing the
low velocity approximations of (27) and (28):
(27') W = W' + W'
c2 5
6 + l'(0) v 2 + O(v4) ,
(28') Px = 4
3 W'
c2 v + O(v3) ,
from which we see that the masses associated, on the one hand, with the kinetic energy,
and on the other hand, with the momentum, are only equal if the derivative l'(0) = - 1/6 .
This is the case of Langevin's model. But Poincaré rejects Langevin's model because it
is non relativistic, and he takes the alternative option to complete Lorentz's model by
introducing non electromagnetic forces. Through rather cumbersome calculations, he
finds that it is sufficient to introduce a constant external pressure whose work is
proportional to the volume of the electron 36. It is remarkable that this complementary
36 Nowadays, with our better knowledge of the relativistic tensor calculus, the calculations of Poincaré
can be understood in a much simpler way. The energy-momentum tensor T µν defines an energy-
momentum four vector P µ by integration of the fourth components T µ0 over all space, if and only if, it is
conserved: ∂ν Tµν = 0. This is not possible for the electromagnetic energy momentum tensor in case of
the presence of electric charge (the electron). Therefore, one has to complete this electromagnetic energy-
momentum tensor in order to satisfy the conservation condition. The simplest solution is to introduce ad-
hoc diagonal terms, like a constant external pressure for the space components T kk , and the
corresponding work for the time component T oo. (Cfr. for instance: Ar-66 or Jc-75).The pressure and its
work can eventually be computed if one chooses a model for the electrostatic charge distribution of the
electron (example of such a calculation in Ar-66,); but such a calculation is not necessary if the goal is
simply to get the equation of motion of the electron as an entity.24potential of forces doesn't change the general form of the Lagrangian (29), now written
with l(v) =1. One gets in that way the full compatibility of the general canonical
formalism with the principle of relativity. The rest of the story is then straightforward:
from the Lagrangian (29') of the electron in free motion 37, the canonical formula (30),
and the Lorentz's force acting as an external force on this electron, one obtains
Poincaré's dynamical equation (31) :
(29') L = - W' 1 - ( v / c )2 ,
(30)d
dt ∂L
∂vk = Fk , ( k = 1,2,3) ,
(31)d
dt ( W' / c2 ) vk
1 - ( v / c )2 = e [ E k + ( v × H )k ] , (k = 1,2,3).
Unfortunately, Poincaré doesn't write his equations in this elegant and modern way. He
uses units such that W'/c2 is equal to one (and therefore, he possibly misses the
discovery of the mass energy relation...!), and he uses cumbersome notations which
tend to obscure the content of the equations 38.
Poincaré doesn't make an explicit use of this equation. Instead of that, he
controls once more that the equation is relativistically covariant and that Lorentz's
approach is the only one which can give such a covariance. In the latter derivation, he
extends the validity of the dynamical equation to arbitrary forces provide they
transform under Lorentz's transformation as does the electromagnetic Lorentz's force.
Equation (31) corresponds to the motion of one electron in some external field,
neglecting the loss of energy due to its own radiation (quasi stationary motion). In § 8 ,
Poincaré shows that it can be extended to a system of electrons, where the motion of
each electron is submitted to a common external field and to the electromagnetic field
created by all the other ones.
At the end of his long paper, Poincaré tries to build a model of a relativistic
gravitation (§9). We retain of this paragraph that it was the occasion to discover several
"four partners" which transform alike "time and space co-ordinates", and also several
kinematical invariants. He finally obtains a gravitational attraction law between two
massive bodies which "improves" Newton's law in the sense that it is "retarded" and
"relativistic covariant":
"Nous voyons d'abord que l'attraction corrigée se compose de deux composantes; l'une parallèle
au vecteur qui joint les positions des deux corps, l'autre parallèle à la vitesse du corps attirant. Rappelons
que quand nous parlons de la position ou de la vitesse du corps attirant, il s'agit de sa position ou de sa
vitesse au moment où l'onde gravifique le quitte; pour le corps attiré, au contraire, il s'agit de sa position
ou de sa vitesse du moment où l'onde gravifique l'atteint, cette onde étant supposée se propager à la
vitesse de la lumière."
Poincaré notices that his tentative approach to build a new gravitational law compatible
with the general requirements of Lorentz's invariance and the Newton's law as low
velocity limit, cannot have a unique solution. As an example, he immediately proposes
several alternative ones. These new gravitational laws were never really applied in
astronomy. The sole application was an early estimate of the advance of Mercury's
perihelion; it was found to be in the good direction, but too small (7'' instead of 38",
mentioned in [Po-53]). These attempts to build a gravitational law in the framework of
37 W' is as before the total energy of the ideal electron (electron at rest); it contains now the electrostatic
energy and the work of the pressure.
38 It might be possible that to-day hastened readers would not recognise that Poincaré's eq. 5 of § 7 is
nothing else than eq. 31 here above!25special relativity are now totally superseded by Einstein's theory of gravitation (1913 -
1916).
As I said at the beginning of this Part 3, one should include in Einstein's 1905
work, the paper " Ist die Trägheit eines Körpers von seinem Energie inhalt abhängig?"
that he wrote in September [Ei-05; 5]. He imagines that a body at rest in some inertial
frame sends plane waves of light in some direction with an energy L/2, and
simultaneously just the same quantity of light in the opposite direction 39. Because of
the symmetry of the process, the state of motion of the emitter is not changed: the body
remains at rest, but its total energy is reduced by an amount L. Einstein considers then
the same process from another inertial frame in uniform translation along the common x
axis. In this frame, the energies of the plane waves of light have the transformed values
that were computed in his preceding paper:
(32)L'
2 = L
2 1 ± v cos φ
1 - (v/c)2 ,
where φ is the angle of emission with respect to the x axis. Here again, because of the
symmetry of the emission, the state of motion of the body is not changed, but its total
energy is reduced by an amount L/(1-(v/c)2)1/2 . Let E0 and E1 be the total energies
before and after the process in the rest frame, similarly H0 and H1 the corresponding
energies in the moving frame; one has evidently:
(33) ( H0 - E0 ) - ( H1 - E1 ) = L 1
1 - ( v / c )2 - 1 ;
Einstein envisages the possibility that in each frame a conventional additive constant
could be added to the kinetic energy K in order to define the total energy : E = K + C,
similarly, H = K' + D (remember that in Eq. 26 he only obtains the transformation law
of the kinetic energy !). However, these conventional constants C and D do not change
in the physical process, and therefore, they disappear from Eq. 33 which becomes the
difference of kinetic energy (as seen from the moving frame) when light is emitted:
(33') K0 - K1 = L 1
1 - ( v / c )2 - 1 .
If we compare with the expression (26) of the kinetic energy, we see that it looks as if
the body had lost a fraction L/c2 of its mass when emitting (at rest) an energy L in the
form of radiation. Einstein concludes:
" The fact that the energy withdrawn from the body becomes energy of radiation evidently
makes no difference, so that we are led to the more general conclusion that:
The mass of a body is a measure of its energy-content; if the energy changes by L, the mass
changes in the same sense by L/9 x 10 20, the energy being measured in ergs, and the mass in grammes.
It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with
radium salts) the theory may be successfully put to the tests.
If the theory corresponds to the facts, radiation conveys inertia between the emitting and
absorbing bodies."
39 This reasoning is typical of a thermodynamics way of thinking. One considers a process, without
questioning about the way it can happen: by means of which mechanism can such a localised "body" emit
such plane waves? Which kind of body do we have to consider? Certainly not an electron since it cannot
emit light when it remains at rest ! However, the crucial formula (26) giving the kinetic energy (which
will be used at the end of the reasoning) was derived for an electron, and only boldly extended to any
ponderable matter!26During the period 1906-1907, Einstein published three other proofs of his mass
energy relation; this underlines enough the importance he attached to his equivalence
law. May be did he also consider that this first proof was not convincing enough in
view of the enormous importance of the discovery?
Part 4.- Conclusions.
Historians sometimes like to speculate on "What would have happened if ...?"
40. This of course is not really "good" history; but in our case, it is certainly a valid
alternative to the vain quarrels on the question of priority, or on the question of a "real
understanding" of the meaning of relativity by one among its creators. Let us therefore
try to play the game.
The question "What would have happened if ...?" cannot reasonably be posed in
the case of works which continuously developed on many years, like Lorentz's
electrodynamics. In thirty years (1875-1905), the great physicist built an Oeuvre that
contains so many results that profoundly influenced the development of physics that it
becomes nearly impossible to envisage the possibility of its non existence.
The question "What would have happened if ...?" can more reasonably be posed
in case of rather isolated events (like Poincaré's 1905 papers), or a fortiori, in case of
totally unexpected events (like Einstein's original paper " Zur Elektrodynamik bewegter
Körper" ).
"What would have happened if Poincaré's papers of 1905 did not exist"? The
answer is immediate since these papers were nearly forgotten 41 and didn't really
influence the later development of physics!
"What would have happened if Einstein's original paper " Zur Elektrodynamik
bewegter Körper" of 1905 did not exist"? This time, the answer is not so easy to give.
On the one hand, all important formulae existed already or would have appeared at the
same time in Poincaré's papers (even if some of them are there derived in a different
way; see f.i. the formulae of the new dynamics). On the other hand, it is clear that
Einstein's radical approach announced some kind of new physics. Although it is most
probably true that Einstein grasped some ideas in earlier works of Lorentz and Poincaré
(f.i.: the idea of synchronisation of distant clocks by exchange of light signals that he
probably met in [Po-00], or Lorentz's transformation formulae that he knew to exist and
which played probably the role of a guideline 42), their new derivation opened the way
to further progress. This potentiality was probably perceived early by some very
influent physicists (like Max Planck) who emphasised the importance of the work of
the young physicist and encouraged further developments.
Einstein versus Poincaré ! We meet here one of the mysteries of scientific fate.
A priori, in 1905, the comparative chances of success of the two works looked rather
unequal. On the one hand, one of the greatest mathematicians of the turn of the century,
whose reputation as a physicist was also great, of which each new contribution to both
disciplines was eagerly looked for by the scientific world. On the other hand, a young
physicist nearly unknown. At the same time, they both publish the solution of an
important problem that troubled physics for many years. Of course, as we have seen,
the two papers are not identical and each of them has its own merits. Among these
respective merits, one has to acknowledge a higher mathematical level of reasoning in
Poincaré's work, and a new insight on the problem that opens the way to further
developments in Einstein's work. As an example, I recall that Poincaré keeps on firmly,
with full mathematical rigour, a pure wave theory of light (as it was commonly
40 For example "What would have happened if the issue of some rather decisive battle, like the battle of
Waterloo, had been different ?".
41 Except for details like Poincaré's pressure.
42 Not necessary to insist on the fact that what I write here is pure speculation, since Einstein not only
makes no reference to previous works, but denies to have known about them (see Mi-81, Pa-82).27admitted at the time), maintaining thereby the very existence of an ether whose physical
properties determine those of light and gravitation [Po-06]:
"Si la propagation de l'attraction se fait avec la vitesse de la lumière, cela ne peut être par une
rencontre fortuite, cela doit être parce que c'est une fonction de l'éther; et alors, il faudra chercher à
pénétrer la nature de cette fonction, et la rattacher aux autres fonctions du fluide."
Einstein rejects radically the existence of such a medium and finds skilfully his way
between contradictory conceptions of light, using the rather vague concept of "light
complex". For Einstein, Relativity should be considered independently of Maxwell -
Lorentz's electromagnetic theory; it is a fundamental Principle that defines a general
framework where theories have to be developed.
This difference of conception is certainly not enough to justify that only the
work of the latter is to-day considered as the founding paper of Relativity, while the
work of the former is nearly forgotten. So what? Why does the living history of our
century ( i.e. education, scientists knowledge and even public knowledge) remember
only one name? One can put forward a lot of hypotheses: Poincaré's wrong choice when
addressing his main paper to a mathematical journal not so well known to physicists
(?), mathematical presentation unfamiliar to physicists (?), impact of the immediate
support to Einstein's work by Planck and other important German physicists (?), early
death of Poincaré (1912) which prevents a possible sharing of a Nobel Prize for
Relativity with Einstein and Lorentz (?). I don't think that any one of these hypotheses
can give a satisfactory explanation. It is also often said that Einstein was more
"relativist" than Poincaré. The detailed comparison of the contents of the 1905 papers
does not support this assertion. It is however true that an important difference of
attitude with respect to the new theory exists between the two men after 1905:
- Poincaré seems to consider that the question is settled and makes no new effort
to go further;
- at the opposite, 1905 is for Einstein the starting point for new, advanced
studies on the consequences of the Principle of Relativity.
In this sense, Einstein is a "more relativistically committed" scientist than
Poincaré. Even if one limits the discussion to special relativity (as I do in this paper),
Einstein's "post 1905" contributions to relativity are important, and it is worth to stress
that they were directly inspired by the new spirit he introduced in his first paper. His
1907 review paper [Ei-07] contains not only some essential clarifications on the
Principles he makes use of (nl. his concepts of "identical" clocks and solids rods), some
improvements of his earlier results (nl. the new relativistic mechanics), but it contains
also new and important contributions: further clarification of the mass-energy relation,
extension of the Principle of Relativity to accelerated frames, Principle of "Equality of
the inertial and the gravitational masses" , and the first predictions concerning the
influence of a gravitational field on light. If one includes General Relativity in the
discussion, then of course Einstein's work is much more important than the work of any
one else.
Nevertheless, in spite of this incontestable dominance, when Einstein received
the Nobel Prize in Physics 1921 (given only in 1922), the mention was " for his services
to Theoretical Physics, and especially for his discovery of the law of the photoelectric
effect ", with no explicit mention to his work on Relativity [No-67]. The presentation
speech was given by Professor S. Arrhenius, Chairman of the Nobel Committee for
Physics. It contains only a short allusion to relativity as being the subject of lively
debates in philosophical circles, which has also some astrophysical implications still
under examination. Einstein could not attend the ceremony (he was then in Japan) and
there was no Nobel Lecture. It was replaced by a "Lecture delivered to the Nordic
Assembly of Naturalists at Gothenburg" in 1923, entitled "Fundamental ideas and
problems of the theory of relativity". These facts look strange enough to awake the
attention of historians and biographs. Pais makes a detailed analysis of the
circumstances in §30 of his book [Pa-82] : "How Einstein got the Nobel Prize".
Although many physicists nominated several times Einstein for his work on relativity, it
seems that the Academy of Sweden was in no hurry to award relativity before28experimental issues were clarified, first in special relativity, later in general relativity.
Pais concludes that it was the Academy's bad fortune not to have anyone among its
members who could competently evaluate the content of relativity theory in those early
years. Leveugle proposes an alternative explanation to these strange facts [Le-94]. The
Academy could have been influenced by the publication in 1921 of the obituary of
Poincaré, written in 1914 by Lorentz, where the great old scientist whose authority in
the matter was incontestable wrote:
"Poincaré a formulé le Postulat de Relativité, terme qu'il a été le premier à employer".
This sentence was strong enough to stand an insurmountable obstacle to the possible
attribution of a Nobel Prize for relativity to any one else. Whatever the reason, one can
only notice that no Nobel Prize of Physics was attributed for one of the major discovery
in physics of the century!
Let us now come back again on the question of the ether. It is generally asserted
that Einstein rendered this notion completely obsolete. It is at least the conclusion that
some physicists promptly derive from his 1905 introduction:
"The introduction of a "luminiferous ether" will prove to be superfluous inasmuch as the view
here to be developed will not require an "absolutely stationary space" provided with special properties,
nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place".
This conclusion must however be tempered by the following other Einstein's
formulation [Ei-20]:
"According to the theory of general relativity, space possesses physical properties; therefore, in
this sense, an ether exists. According to the theory of general relativity, space without ether is
inconceivable, for non only the propagation of light would be impossible, but furthermore, there would
be no possible existence for rods and clocks, and therefore also for space-time distances in a physical
sense. However, this ether must no be conceived as having the property which characterises ponderable
media, i.e. as being formed of parts that can be followed in time: the notion of motion cannot be applied
to it".
In between, Einstein had built general relativity and understood that the physical
properties of light and gravitation forced us to come back to a less radical position on
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arXiv:physics/0008230v1 [physics.atom-ph] 28 Aug 2000Limit on Lorentz and CPT Violation of the Proton Using a Hydro gen Maser
D. F. Phillips, M. A. Humphrey, E. M. Mattison,
R. E. Stoner, R. F. C. Vessot and R. L. Walsworth
Harvard–Smithsonian Center for Astrophysics, Cambridge, MA 02138
(submitted to Physical Review Letters on August 27, 2000)
We present a new measurement constraining Lorentz and CPT vi olation of the proton using a
hydrogen maser double resonance technique. A search for hyd rogen Zeeman frequency variations
with a period of the sidereal day (23.93 h) sets a clean limit o n violation of Lorentz and CPT
symmetry of the proton at the 10−27GeV level.
Experimental investigations of Lorentz symmetry pro-
vide important tests of the standard model of parti-
cle physics and general relativity. While the standard
model successfully describes particle phenomenology, it
is believed to be the low energy limit of a fundamental
theory that incorporates gravity. This underlying the-
ory may be Lorentz invariant, yet contain spontaneous
symmetry-breaking that could result at the level of the
standard model in small violations of Lorentz invariance
and CPT (symmetry under simultaneous application of
Charge conjugation, Parity inversion, and Time rever-
sal).
Clock comparisons [1,2] provide sensitive tests of rota-
tion invariance and hence Lorentz symmetry by bounding
the frequency variation of a given clock as its orientation
changes, e.g., with respect to the inertial reference frame
defined by the fixed stars [3]. Atomic clocks are typi-
cally used, involving the electromagnetic signals emitted
or absorbed on hyperfine or Zeeman transitions. Here
we report results from a hydrogen (H) maser experiment
that sets an improved clean limit on Lorentz and CPT
violation of the proton at the level of 10−27GeV as the
H maser rotates with the Earth.
Our H maser measurement is motivated by a stan-
dard model extension developed by Kosteleck´ y and oth-
ers [3–7]. This standard-model extension is quite general:
it emerges as the low-energy limit of any underlying the-
ory that generates the standard model and that contains
spontaneous Lorentz symmetry violation [4]. For exam-
ple, such characteristics might emerge from string theory
[5]. A key feature of the standard-model extension is that
it is formulated at the level of the known elementary par-
ticles, and thus enables quantitative comparison of a wide
array of searches for Lorentz and CPT violation [6]. The
dimensionless suppression factor for such effects would
likely be the ratio of the appropriate low-energy scale to
the Planck scale, perhaps combined with dimensionless
coupling constants [3–7].
Recent experimental work motivated by this standard-
model extension includes Penning trap tests by Gabrielse
et al. on the antiproton and H−[8], and by Dehmelt et
al.on the electron and positron [9], which place im-
proved limits on Lorentz and CPT violation in these sys-
tems. A re-analysis by Adelberger, Gundlach, Heckel,
and co-workers of existing data from the “E¨ ot-Wash II”spin-polarized torsion pendulum [10] sets the most strin-
gent bound to date on Lorentz and CPT violation of the
electron: approximately 10−29GeV [11]. A recent search
for Zeeman-frequency sidereal variations in a129Xe/3He
maser places an improved constraint on Lorentz and CPT
violation involving the neutron at the level of 10−31GeV
[12]. Also the KTeV experiment at Fermilab and the
OPAL and DELPHI collaborations at CERN have lim-
ited possible Lorentz and CPT violation in the Kand
Bdsystems [13].
The hydrogen maser is an established tool in preci-
sion tests of fundamental physics [14]. H masers op-
erate on the ∆ F= 1, ∆ mF= 0 hyperfine transition
(the “clock” transition) in the ground electronic state of
atomic hydrogen [15]. Hydrogen molecules are dissoci-
ated into atoms in an RF discharge and the atoms are
spatially state selected via a hexapole magnet (Fig. 1).
Atoms in the F= 1,mF= +1,0 states are focused into
a Teflon coated cell, thereby creating the population in-
version necessary for active maser oscillation. The cell
resides in a microwave cavity resonant with the ∆ F= 1
transition at 1420 MHz. A static magnetic field of ∼1
milligauss is applied by a solenoid surrounding the res-
onant cavity to maintain the quantization axis of the H
atoms. For normal H maser operation, this magnetic field
is directed vertically upwards in the laboratory reference
frame. The F= 1,mF= 0 atoms are stimulated to make
a transition to the F= 0 state by the thermal microwave
field in the cavity. The energy from the atoms then acts
as a source to increase the microwave field. With suffi-
ciently high polarization flux and low cavity losses, this
feedback induces active maser oscillation. H masers built
in our laboratory over the last 30 years provide fractional
frequency stability on the clock transition of better than
10−14over averaging intervals of minutes to days and
can operate undisturbed for several years before requir-
ing routine maintenance.
The ∆ mF= 0 clock transition has no leading-order
sensitivity to Lorentz and CPT violation [3,7] because
the transition encompasses no change in longitudinal spin
orientation. In contrast, the F= 1, ∆ mF=±1 Zeeman
transitions are maximally sensitive to potential Lorentz
and CPT violation [7]. Therefore, we searched for a
Lorentz-violation signature by monitoring the Zeeman
frequency ( νZ≈850 Hz in a static magnetic field of
10.6 mG) as the laboratory reference frame rotated side-
really. We utilized an H maser double resonance tech-
nique [16] to measure νZ. We applied a weak, oscillating
magnetic field perpendicular to the static field at a fre-
quency close to the Zeeman transition, thereby coupling
the three sublevels of the hydrogen F= 1 manifold [17].
Provided that a population difference exists between the
mF=±1 states, this coupling alters the energy of the
mF= 0 state, thus shifting the measured maser clock
frequency in a manner described by a line shape that
is antisymmetric about the Zeeman frequency for suffi-
ciently small static fields (Fig. 2) [16]. We determined
νZby measuring the resonant driving field frequency at
which the maser clock frequency is equal to its unper-
turbed value. Due to the excellent frequency stability of
the H maser, this double resonance technique allowed the
determination of νZwith a precision of ∼1 mHz [18].
1 mGSolenoidMagnetic
ShieldsStorage
BulbResonant CavityHexapole
MagnetHydrogen Dissociator
Output
Loop
Fluxgate
MagnetometerHelmholtz
Coils
Reference
H Maser
Frequency
Counter
Zeeman
Coils
Ambient Magnetic
Field Feedback Loop
FIG. 1. Schematic of the H maser in its ambient magnetic
field stabilization loop. Large Helmholtz coils surround th e
maser and cancel external field fluctuations as detected by a
fluxgate magnetometer placed close to the maser region. Zee-
man coils mix the mFsublevels of the F= 1 hyperfine state,
and allow sensitive measurement of the Zeeman frequency
through pulling of the maser frequency [16], as determined
by comparison to a reference H maser.
In the small-field limit, the hydrogen Zeeman fre-
quency is proportional to the static magnetic field. Four
layers of high permeability magnetic shields surround the
maser (Fig. 1), screening external field fluctuations by
a factor of 32,000. Nevertheless, the residual effects of
day-night variations in ambient magnetic noise shifted
the measured Zeeman frequency with a 24 hour periodic-
ity which was difficult to distinguish from a true sidereal
(23.93 h period) signal in our data sample. Therefore, we
employed an active stabilization system to cancel exter-
nal magnetic field fluctuations (Fig. 1). A fluxgate mag-
netometer sensed the field near the maser cavity with a
shielding factor of only 6 to external magnetic fields due
to its location at the edge of the shields. A feedback
loop controlled the current in large Helmholtz coils (2.4m dia.) surrounding the maser to maintain a constant
field. This feedback loop effectively reduced the sidereal
fluctuations of νZcaused by external fields at the location
of the magnetometer to below 1 µHz.
-202∆νmaser (mHz)
860859858857856855854
Zeeman drive frequency (Hz)
FIG. 2. An example of a double resonance measurement
of the F= 1, ∆ mF=±1 Zeeman frequency ( νZ) in the
H maser. The change from the unperturbed maser clock fre-
quency is plotted versus the driving field frequency. (The st a-
tistical uncertainty in each point is approximately 50 µHz.)
The solid line is the fit of the antisymmetric lineshape de-
scribed in [16] to the data, yielding νZ= 857 .125±0.003 Hz
in this example.
We accumulated data in three separate runs of 11, 9
and 12 days over the period Nov., 1999 to Mar., 2000.
During data taking, the maser remained in a closed, tem-
perature controlled room to reduce potential systematics
from thermal drifts that might have 24 hour periodicities.
Each νZmeasurement required approximately 20 min-
utes of data (Fig. 2). We also monitored the H maser am-
plitude, residual magnetic field fluctuations, maser and
room temperatures, and the current through the maser
solenoid (which set the static magnetic field). During
the second and third runs, we reversed the direction of
the static magnetic field created by the maser’s internal
solenoid in order to investigate possible systematic de-
pendence of the diurnal variation of νZon field direction.
(No such dependence was observed.) In the field-reversed
configuration, the axial magnetic field in the storage bulb
was anti-parallel to the field near the exit from the state-
selecting hexapole magnet. Thus H atoms traversed a
region of magnetic field inversion on their way into the
storage bulb, causing loss of atoms from the maser ex-
cited state ( F= 1,mF= 0) due to Majorana transitions
as well as sudden transitions of atoms from the F= 1,
mF= +1 state to the F= 1,mF=−1 state. In the
field reversed configuration, the maser amplitude was re-
duced by 30% and both the maser clock frequency and
Zeeman frequency were less stable. Thus, our constraint
on sidereal-period νZvariations was 5 times weaker in
the field-reversed configuration than in the parallel-field
configuration.
2-0.100.000.10νz - 857.1 Hz
24019214496480
time (hours)
FIG. 3. Zeeman frequency data from 11 days of the
Lorentz/CPT test (run 1) using the H maser.
To identify any sidereal variations in νZ, we fit a
sidereal-period sinusoid and a slowly varying background
to the accumulated νZmeasurements. (See Fig. 3 for the
11 days of data from run 1.) Two coefficients, δνZ,αand
δνZ,β, parameterize the sine and cosine components of
the sidereal oscillations. ( αandβalso correspond to
non-rotating directions in the plane perpendicular to the
Earth’s axis of rotation.) In addition, we used piecewise
continuous linear terms (whose slopes were allowed to
vary independently for each day) to model the slow drift
of the Zeeman frequency. In the field-inverted configu-
ration, large variations in νZled to days for which this
model did not successfully fit the data. Large values of
the reduced χ2and systematic deviation of the residu-
als from a normal distribution characterized such days,
which we cut from the data sample. For each run, the fit
determined the components δνZ,αandδνZ,βof the side-
real sinusoidal variation (see Table I). The total weighted
means and uncertainties for δνZ,αandδνZ,βwere then
formed from all three data sets, yielding the measured
value A≡/radicalbig
(δνZ,α)2+ (δνZ,β)2= 0.49±0.34 mHz (1- σ
level). This result is consistent with no observed side-
real variation in the hydrogen F= 1,mF=±1 Zeeman
frequency, given reasonable assumptions about the prob-
ability distribution for A[19].
Systematic sidereal-period fluctuations of νZwere
smaller than the 0.34 mHz statistical resolution. The cur-
rent in the main solenoid typically varied by less than 5
nA out of 100 µA over 10 days, corresponding to a change
inνZof∼50 mHz. We corrected the measured Zee-
man frequency for this solenoid current drift. The side-
real component of the current correction was typically
25±10 pA, corresponding to a sidereal-period variation
ofνZ≈0.16±0.08 mHz. The temperature inside the
maser cabinet enclosure had a sidereal component below
0.5 mK, corresponding to a sidereal-period modulation
ofνZof less than 0.1 mHz. Potential Lorentz-violating
effects acting directly on the electron spins in the flux-
gate magnetometer’s ferromagnetic core could change the
field measured by the magnetometer and mask a poten-
tial signal from the H maser experiment. However, any
such effect would be greatly suppressed by a factor ofRun Useful days Field δνZ,α δνZ,β
(cut days) direction (mHz) (mHz)
1 11 (0) ⇑ 0.43±0.36−0.21±0.36
2 3 (6) ⇓ − 2.02±1.27−2.75±1.41
3 5 (7) ⇓ 4.30±1.86 1 .70±1.94
Table I. Means and standard errors for δνZ,αandδνZ,β, the
quadrature amplitudes of sidereal-period variations in th e hy-
drogen F= 1, mF=±1 Zeeman frequency. Results are
displayed for each of three data-taking runs, listing also t he
number of days of useful data, the number of discarded data-
taking days (in parentheses), and the direction of the maser ’s
internal magnetic field in the laboratory frame.
E/kT∼10−16below the <∼1 nG sensitivity of the mag-
netometer, where Eis the Lorentz-violating shift of the
electron spin energy (known to be <∼10−29GeV [10]) and
Tis the temperature of the spins when the core is in zero
magnetic field (the equilibrium condition of the magne-
tometer lock loop). Also, the magnetic shielding reduces
field fluctuations at the magnetometer by a factor of only
6 whereas fluctuations at the storage-bulb are reduced by
32,000. Therefore, any effective magnetic field shifts in-
duced in the magnetometer by Lorentz/CPT-violations
were negligible in the present experiment. Spin-exchange
collisions between the H atoms shift the zero crossing of
the double resonance from the true Zeeman frequency
[20]. Hence, the measured νZvaries with H density in
the maser. We monitored the atomic density by mea-
suring the output maser power, with the relation to νZ
being <∼0.8 mHz/fW. During long term operation, the
average maser power drifted less than 1 fW per day. The
sidereal component was typically less than 0.05 fW, cor-
responding to a 0.04 mHz variation in the Zeeman fre-
quency. Combining these systematic errors in quadrature
with the statistical uncertainty produces a final limit on
a sidereal variation in the hydrogen F= 1, ∆ mF=±1
Zeeman frequency of 0.37 mHz, which expressed in en-
ergy units is 1 .5×10−27GeV.
The hydrogen atom is directly sensitive to Lorentz and
CPT violations of the proton and the electron. Follow-
ing the notation of Refs. [3,7], one finds that a limit on
a sidereal-period modulation of the Zeeman frequency
(δνZ) provides a bound on the following parameters
in the standard model extension of Kosteleck´ y and co-
workers:
/vextendsingle/vextendsingle/vextendsingle˜bp
3+˜be
3/vextendsingle/vextendsingle/vextendsingle≤2πδνZ (1)
for the low static magnetic fields at which we operate.
(Here we have taken ¯ h=c= 1.) The subscript 3 in
Eq. (1) indicates the direction along the quantization axis
of the apparatus, which is vertical in the lab frame. The
superscripts eandprefer to the electron and proton,
respectively.
As in Refs. [3,9], we can re-express the time varying
3change of the hydrogen Zeeman frequency in terms of
parameters expressed in a non-rotating inertial frame as
2πδνZ,J=/parenleftBig
˜bp
J+˜be
J/parenrightBig
sinχ, (2)
where Jrefers to either of two orthogonal directions per-
pendicular to the earth’s rotation axis and χ= 48◦is the
co-latitude of the experiment.
As noted above, a re-analysis of existing data from a
spin-polarized torsion pendulum [10] sets the most strin-
gent bound to date on Lorentz and CPT violation of the
electron: ˜be
J<∼10−29GeV [11]. Therefore, the H maser
measurement reported here constrains Lorentz and CPT
violations of the proton: ˜bp
J≤2×10−27GeV at the
one sigma level. This limit is comparable to that derived
[3] from the199Hg/133Cs clock comparison experiment
of Hunter, Lamoreaux et al. [2] but in a much cleaner
system: the hydrogen atom nucleus is simply a proton,
whereas significant nuclear model uncertainties affect the
interpretation of experiments on many-nucleon systems
such as199Hg and133Cs.
To our knowledge, no search for sidereal variations in
the hydrogen Zeeman frequency has been performed pre-
viously. Nevertheless, implicit limits can be set from a
widely-practiced H maser characterization procedure in
which the Zeeman frequency is measured by observing
the drop in maser output power induced by a drive field
swept through the Zeeman resonance [15,21]. It is reason-
able to assume that sidereal-period variations of the Zee-
man frequency of ∼1 Hz would have been noticed. Thus,
our result improves upon existing implicit constraints by
over two orders of magnitude.
In conclusion, precision comparisons of atomic clocks
provide sensitive tests of Lorentz and CPT symmetries
[3–7]. A new measurement with an atomic hydrogen
maser provides a clean limit on Lorentz and CPT viola-
tion involving the proton that is consistent with no effect
at the 10−27GeV level. Further details of this work will
be found in Ref. [22].
We gratefully acknowledge the encouragement and as-
sistance of Alan Kosteleck´ y. Financial support was
provided by NASA grant NAG8-1434 and ONR grant
N00014-99-1-0501. M. A. H. acknowledges a fellowship
from the NASA Graduate Student Researchers Program.
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Beyond the Standard Model, p. 717, World Scientific, Sin-
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ington, 1998.
[11] B. Heckel, presented at the International Conference o n
Orbis Scientiae 1999, Fort Lauderdale, Florida, Dec.,
1999.
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(physics/0007049).
[13] KTeV Collaboration, presented by Y.B. Hsiung at the
KAON 99 conference, Chicago, June 1999; OPAL Collab-
oration, R. Ackerstaff et al., Z. Phys. C 76, 401 (1997);
DELPHI Collaboration, M. Feindt et al., preprint DEL-
PHI 97-98 CONF 80 (July 1997).
[14] R.F.C. Vessot et al., Phys. Rev. Lett. 45, (1980) 2081;
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(physics/0007056).
[17] The weak driving field ( ∼35 nG at ∼850 Hz) caused
very small changes to the maser output power ( <2%
reduction) and line-Q (2% reduction).
[18] At 0.6 mG the differential splitting of the mF= +1 and
−1 levels is <1 mHz, and is included in the fit model for
νZ.
[19] In the case where δνZ,αandδνZ,βhave zero mean value
and the same variance σ, the probability distribution for
Atakes the form P(A) =σ−2Aexp(−A2/2σ2), with the
most probable value of Aoccurring at A=σ.
[20] J. Y. Savard et al., Can. J. Phys, 57, 904 (1979).
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Atomic Frequency Standards , Adam Hilger, Bristol, 1989.
[22] M. A. Humphrey et al., to be published.
4 |
arXiv:physics/0008231v1 [physics.bio-ph] 28 Aug 2000PURINE-PYRIMIDINE
SYMMETRY, DETERMINATIVE
DEGREE AND DNA
Diana Duplij and Steven Duplij∗†
Kharkov National University, Svoboda Sq. 4,
Kharkov 61077, Ukraine
February 2, 2008
Abstract
Various symmetries connected with purine-pyrimidine cont ent of
DNA sequences are studied in terms of the intruduced determi native
degree, a new characteristics of nucleotide which is connec ted with
codon usage. A numerological explanation of CGpressure is pro-
posed. A classification of DNA sequences is given. Calculati ons with
real sequences show that purine-pyrimidine symmetry incre ases with
growing of organization. A new small parameter which charac terizes
the purine-pyrimidine symmetry breaking is proposed for th e DNA
theory.
∗E-mail:Steven.A.Duplij@univer.kharkov.ua
†Internet: http://gluon.physik.uni-kl.de/~duplij
1Abstract investigation of the genetic code is a powerful too l in DNA mod-
els construction and understanding of genes organization a nd expression [1].
In this direction the study of symmetries [2, 3], applicatio n of group theory [4]
and implication of supersymmetry [5] are the most promising and necessary
for further elaboration. In this paper we consider symmetri es connected with
purine-pyrimidine content of DNA sequences in terms of the d eterminative
degree introduced in [6].
We denote a triplet of nucleotides by xyz, where x, y, z =C,T,A,G.
Then redundancy means that an amino acid is fully determined by first two
nucleotides xandyindependently of third z[1]. Sixteen possible doublets xy
group in 2 octets by ability of amino acid determination [7]. Eight doublets
have more “strength” in sense of the fact that they simply enc ode amino
acid independently of third bases, other eight (“weak”) dou blets for which
third bases determines content of codons. In general, trans ition from the
“powerful” octet to the “weak” octet can be obtained by the ex change [7]
C∗⇐⇒A,G∗⇐⇒T, which we name “star operation ( ∗)” and call purine-
pyrimidine inversion . Thus, if in addition we take into account GCpressure
in evolution [8] and third place preferences during codon-a nticodon pairing
[9], then 4 nucleotides can be arranged in descending order i n the following
way:
Pyrimidine Purine Pyrimidine Purine
C G T A
very “strong” “strong” “weak” very “weak”(1)
Now we introduce a numerical characteristics of the empiric al “strength”
—determinative degree dxof nucleotide xand make transition from qualita-
tive to quantitative description of genetic code structure [6]. It is seen from
(1) that the determinative degree of nucleotide can take val uedx=1,2,3,4
in correspondence of increasing “strength”. If we denote de terminative de-
gree as upper index for nucleotide, then four bases (1) can be presented
as vector-row V=/parenleftbig
C(4)G(3)T(2)A(1)/parenrightbig
. Then the exterior product
M=V×Vrepresents the doublet matrix Mand corresponding rhombic
code [10], and the triple exterior product K=V×V×Vcorresponds to
the cubic matrix model of the genetic code which were describ ed in terms
of the determinative degree in [6]. To calculate the determi native degree of
doublets xywe use the following additivity assumption
dxy=dx+dy, (2)
2which holds for triplets and for any nucleotide sequence. Th en each of 64
elements (codons) of the cubic matrix Kwill have a novel number character-
istics —determinative degree of codon dxyz=dcodon=dx+dy+dzwhich
takes value in the range 3÷12. We can also define the determinative degree
of amino acid dAAas mean arithmetic value dAA=/summationtextdcodon/ndeg, where ndeg
is its degeneracy (redundancy). That can allow us to analyze new abstract
amino acid properties in connection with known biological p roperties [6].
Let us consider a numerical description of an idealized DNA s equence
as a double-helix of two codon strands connected by compleme ntary condi-
tions [1]. Each strand is described by four numbers ( nC, nG, nT, nA) and
(mC, mG, mT, mA), where nxis a number of nucleotide xin one strand. In
terms of nxandmxthe complementary conditions are
nC=mG, mC=nG, nT=mA, mT=nA. (3)
The Chargaff’s rules [1] for a double-helix DNA sequence soun d as: 1)
total quantity of purines and pyrimidines are equal NA+NG=NC+NT;
2) total quantity of adenine and cystosine equal to total qua ntity of guanine
and thymine NA+NC=NT+NG; 3) total quantity of adenine equal to
total quantity of thymine NA=NTand total quantity of cystosine equal to
total quantity of guanine NC=NG; 4) the ratio of guanine and cystosine to
adenine and thymine v= (NA+NT)/(NC+NG) is approximately constant
for each species. Usually the Chargaff’s rules are defined thr ough macroscopic
molar parts which are proportional to absolute number of nuc leotides in DNA
[1]. If we consider a DNA double-helix sequence, then Nx=nx+mx. In
terms of nxandmxthe first three Chargaff’s rules lead to the equations which
are obvious identities, if complimentary (3) holds. From fo urth Chargaff’s
rule it follows that the specificity coefficient vnmfor two given strands is
vnm=nA+mA+nT+mT
nC+mC+nG+mG. (4)
The complementary (3) leads to the equality of coefficients vof each
strand vnm=vn=vm≡v, and vis connected with GCcontent pCGin the
double-helix DNA as pCG= 1/(1 +v).
We consider another important coefficient: the ratio of purin es and pyrim-
idines k. For two strands from the first Chargaff’s rule we obviously de rive
knm= 1. But for each strand we have
kn=nG+nA
nC+nT, km=mG+mA
mC+mT(5)
3which satisfy the equation knkm= 1 following from complementary.
Let us introduce the determinative degree of each strand exploiting the
additivity assumption (2) as
dn=4·nC+3·nG+2·nT+1·nA, (6)
dm=4·mC+3·mG+2·mT+1·mA. (7)
The values dnanddmcan be viewed as characteristics of the empirical
“strength” for strands, i.e. “strand generalization” of (1 ). Then we define
summing and difference “strength” of a double-helix sequenc e by
d+=dn+dm,d−=dn−dm. (8)
The first variable d+can be treated as the summing empirical “strength”
of DNA (or its fragment). Taking into account the complement ary conditions
(3) we obtain d+through one strand variables
d+=7·(nC+nG) +3·(nT+nA). (9)
We can also present d+through macroscopically determined variables Nx
as follows d+=7·NC+3·NA=7·NG+3·NT, or through GCandAT
contents as d+=7
2·NC+G+3
2·NA+T.
To give sense to the difference d−we derive
d−=nC+nT−nG−nA. (10)
We see that the star operation obviously acts as ( d+)∗=d+and (d−)∗=
−d−. From (9)-(10) it follows the main statement:
The biological sense of the determinative degree dis contained
in the following purine-pyrimidine relations:
1)The sum of the determinative degrees of matrix and com-
plementary strands in DNA (or its fragment) equals to
d+=7
2·NC+G+3
2·NA+T. (11)
2) The difference of the determinative degrees between ma-
trix and complementary strands in DNA (or its fragment) ex-
actly equals to the difference between pyrimidines and purin es
in one strand
4d−=npyrimidines −npurines, (12)
where npyrimidines =nC+nTandnpurines =nG+nA, or it is
equal to difference of purines or pyrimidines between strand s
d−=npyrimidines −mpyrimidines =mpurines −npurines.(13)
We can also find connection between d+,d−and the coefficients kandv
as follows
d+=1
2NC+G(7 + 3 v) =NC+G/parenleftbigg
2 +3
2·pCG/parenrightbigg
, (14)
d−=npyrimidines (1−kn). (15)
If we consider one species for which v=const (orpCG=const), then
we observe that d+∼NC+G, which can allow us to connect the determi-
native degree with ”second level” of genetic information [8 ]. From another
side, the ratio7
3of coefficients in (11) can play a numerological role in CG
pressure explanations [8], and therefore d+can be considered as some kind
of “evolutionary strength”.
Now we consider the determinative degree of double-helix se quences in
various extreme cases and classify them. We call a DNA sequen cemononu-
cleotide ,dinucleotide ,trinucleotide orfull, if one, two, three or four numbers
nxrespectively distinct from zero. Properties of mononucleo tide double-helix
DNA sequence are in the Table 1.
Table 1. Mononucleotide DNA
nxd+d−amino acid
nC/negationslash= 0 7nCnC Pro
nG/negationslash= 0 7nG−nG Gly
nT/negationslash= 0 3nTnT Phe
nA/negationslash= 0 3nA−nA Lis
The mononucleotide sequences which encode most extended am ino acids
GlyandLishave negative d−, and the mononucleotide sequences which en-
code amino acids ProandPhewith similar chemical type of radicals have
positive d−.
The dinucleotide double-helix DNA sequences (without mono nucleotide
parts) are described in the Table 2.
5Table 2. Dinucleotide DNA
nx d+ d− amino acid
nC/negationslash= 0, nG/negationslash= 0 7 (nC+nG)nC−nG Pro,Arg,Ala,Gly
nC/negationslash= 0, nT/negationslash= 0 7nC+ 3nTnC+nT Pro,Phe,Leu,Ser
nC/negationslash= 0, nA/negationslash= 0 7nC+ 3nAnC−nAPro,Gly,Asn,Tur,His
nG/negationslash= 0, nT/negationslash= 0 7nG+ 3nTnT−nGGly,Leu,Val,Cys,Trp
nG/negationslash= 0, nA/negationslash= 0 7nG+ 3nA−nG−nA Gly,Glu,Arg,Lys
nT/negationslash= 0, nA/negationslash= 0 3 (nT+nA)nT−nALeu,Asn,Tur,TERM
The trinucleotide DNA can be listed in the similar, but more c umbersome
way. The full DNA sequences consist of nucleotides of all fou r types and
described by (9)-(10).
The introduction of the determinative degree allows us to si ngle out a
kind of double-helix DNA sequences which have an additional symmetry.
We call a double-helix sequence purine-pyrimidine symmetric , if
d−= 0, (16)
i.e. its empiric “strength” vanishes. From (10) it follows
nC+nT=nG+nA, (17)
i.e.kn=km= 1, which can be rewritten for one strand
npyrimidines =npurines (18)
or as equality of purines and pyrimidines in two strands
npyrimidines =mpyrimidines , (19)
npurines =mpurines. (20)
The purine-pyrimidine symmetry (17) has two particular cas es:
1)nC=nG,
nT=nA,−symmetric DNA, (21)
2)nC=nA,
nT=nG,−antisymmetric DNA. (22)
The first case corresponds to the Chargaff’s rule applied to a s ingle strand
which approximately holds for long sequences [11], and so it would be inter-
esting to compare transcription and expression properties of symmetric and
antisymmetric double-helix sequences.
6We have made a preliminary analysis of real sequences of seve ral species
taken from GenBank (2000) in terms of the determinative degr ee. It were
considered 10 complete sequences of E.coli (several genes and full genomic
DNA 9-12 min.), 12 complete sequences of Drosophila melanogaster (crc
genes), 10 complete sequences of Homo sapiens Chromosome 22 (various
clones), 10 complete sequences of Homo sapiens Chromosome 3 (various
clones). We calculated the nucleotide content NC, NT, NG, NAand the de-
terminative degree characteristics d+,d−, q=d−/d+, knandvfor every
sequence. Then we averaged their values for each species. Th e result is
presented in the Table 3.
Table 3. Mean determinative degree characteristics of real sequences
sequence1
n/summationtextd+1
n/summationtextd−1
n/summationtextq·1031
n/summationtextkn1
n/summationtextv
E.coli 90806 -138 -6.8 1.07 1.38
Drosophila 7325 -70 -8.9 1.09 1.31
Homo sap. Chr.22 337974 6865 1.46 0.987 1.14
Homo sap. Chr.3 806435 -1794 -2.29 1.021 1.55
First of all we observe that all real sequences have high puri ne-pyrimidine
symmetry (smallness of parameter q). Also we see that the relation of purines
and pyrimidines in one DNA strand knis very close to unity, therefore we
have a new small parameter in the DNA theory ( kn−1) (or q), which charac-
terizes the purine-pyrimidine symmetry breaking. This can open possibility
for various approximate and perturbative methods applicat ion. Second, we
notice from Table 3 that the purine-pyrimidine symmetry inc reases in direc-
tion from protozoa to mammalia and is maximal for human chrom osome. It
would be worthwhile to provide a thorough study of purine-py rimidine sym-
metry and codon usage in terms of the introduced determinati ve degree by
statistical methods, which will be done elsewhere.
Acknowledgments . Authors would like to thank G. Shepelev for pro-
viding with computer programs, S. Gatash, V. Maleev and O. Tr etyakov
for fruitful discussions and J. Bashford, G. Findley and P. J arvis for useful
correspondence and reprints.
References
7[1] Singer M., Berg P. Genes and genomes . - Mill Valley: University Science
Books , 1991. - 373p.
[2] Findley G. L., Findley A. M., McGlynn S. P. Symmetry characteristics
of the genetic code //Proc. Natl. Acad. Sci. USA . - 1982. - V. 79. - 22.
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[3] Zhang C. T. A symmetrical theory of DNA sequences and its applica-
tions. //J. Theor. Biol. - 1997. - V. 187. - 3. - P. 297–306 .
[4] Hornos J. E. M., Hornos Y. M. M. Model for the evolution of the genetic
code//Phys. Rev. Lett. - 1993. - V. 71. - P.4401–4404 .
[5] Bashford J. D., Tsohantjis I., Jarvis P. D. A supersymmetric model for
the evolution of the genetic code //Proc. Natl. Acad. Sci. USA . - 1998.
- V.95. - P.987–992 .
[6] Duplij D., Duplij S. Symmetry analysis of genetic code and determinative
degree //Biophysical Bull. Kharkov Univ. - 2000. - V. 488. - 1(6). -
P.60–70 .
[7] Rumer U. B. Sistematics of codons in the genetic cod //DAN SSSR . -
1968. - V. 183. - 1. - P. 225–226 .
[8] Forsdyke D. R. Different biological species ”broadcast” their DNAs at
different (C+G)% ”wavelengths” //J. Theor. Biol. - 1996. - V. 178. -
P.405–417 .
[9] Grantham R., Perrin P., Mouchiroud D. Patterns in codon usage of
different kinds of species //Oxford Surv. Evol. Biol. - 1986. - V. 3. -
P.48–81 .
[10] Karasev V. A. Rhombic version of genetic vocabulary based on comple-
mentary of encoding nucleotides //Vest. Leningr. un-ta . - 1976. - V. 1.
- 3. - P. 93–97 .
[11] Forsdyke D. R. Relative roles of primary sequence and (C+G)% in
determining the hierarchy of frequencies of complementary trinucleotide
pairs in DNAs of different species //J. Mol. Biol. - 1995. - V. 41. -
P.573–581 .
8 |
arXiv:physics/0008232v1 [physics.bio-ph] 29 Aug 2000Minimum Entropy Approach to Word Segmentation Problems
Bin Wang
Institute of Theoretical Physics, Chinese Academy of Scien ces,
P.O. Box 2735, Beijing 100080, P. R. China.
State Key Laboratory of Scientific and Engineering Computin g,
Institute of Computational Mathematics and Scientific/Eng ineering Computing ,
P.O. Box 2719, Beijing 100080, P. R. China.
Abstract
Given a sequence composed of a limit number of characters, we try to “read”
it as a “text”. This involves to segment the sequence into “wo rds”. The diffi-
culty is to distinguish good segmentation from enormous num ber of random
ones. Aiming at revealing the nonrandomness of the sequence as strongly as
possible, by applying maximum likelihood method, we find a qu antity called
segmentation entropy that can be used to fulfill the duty. Contrary to
commonplace where maximum entropy principle was applied to obtain good
solution, we choose to minimize the segmentation entropy to obtain good
segmentation. The concept developed in this letter can be us ed to study
the noncoding DNA sequences, e.g., for regulatory elements prediction, in
eukaryote genomes.
1I. INTRODUCTION.
The problem addressed in this paper is rather elementary in s tatistics. It is best described
as the following: suppose one who knows nothing about Englis h language was given a
sequence of English letters, which was actually obtained by taking off all the interwords
delimiters among a sample of English text, how could he recov er the words of the text by
choosing to insert spaces between adjacent letters? Note th at the only thing he can consult
is the statistical properties of the sequence?
Any two adjacent letters can be chosen to belong to the same wo rd (keep adjacent) as well
as belong to separate words (be separated by space). Suppose the sequence length is N. Any
choice on the connectivity between N−1 pairs of adjacent letters is called a segmentation.
There are a total of 2N−1possible segmentations. The word segmentation problem is t o
find ways to distinguish the correct segmentation – in the sen se that adjacent letters in the
original text keep adjacent while letters separated by spac es and/or punctuation marks in
the original text are separated by spaces in the segmentatio n – from others.
Although the problem seems toy-like, its fundamental impor tance for statistical linguis-
tics is evident. We study on it, however, also for practical p urposes. Noncoding sequences
in the genomes of species play essential rule on the regulati on of gene expression and func-
tion [1]. However the development of computational methods for extracting regulatory
elements is far behand DNA sequencing and gene finding [2]. On e reason is the lack of
efficient way to discriminate large amount of sequence signal s in noncoding DNA sequences.
Through linguistic study it has been shown that noncoding se quences in eukaryotic genomes
are structurally much similar to natural and artificial lang uage [3]. Thus many may expect
to “read” the noncoding sequences as a “text”. Actually, effo rts have been given to build a
dictionary for genomes [4,5]. Li et al. [5] showed the connec tion between regulatory elements
prediction and word segmentation in noncoding DNA sequence s of eukaryote genomes. We
expect that progress on word segmentation problem may help t o deepen our knowledge on
noncoding regions of eukaryote genomes. Besides, word segm entation is an important issue
for Asian languages (e.g., Chinese and Japanese) processin g [6], because they lack interword
delimiters.
2II. SEGMENTATION ENTROPY AND ITS CONNECTION TO WORD
SEGMENTATION PROBLEM.
To tackle word segmentation problem, we first consider a prob lem under constraints, so
that one important concept – segmentation entropy – can be in troduced. The constraints
will be released at the end of this paper. Suppose we have know n that there are nlwords of
length l(l= 1,2,···) in the original text. Obviously,
/summationdisplay
lnll=N. (1)
Under these constraints – Words Length Constraints WLC – the re are totally
(/summationtext
lnl)!
/producttext
lnl!(2)
segmentations. For example, for the following story, there are totally 3 .12e144 segmenta-
tions, while the number under WLC is about 1 .33e97.
The Fox and the Grapes
Once upon a time there was a fox strolling through the woods. H e came upon
a grape orchard. There he found a bunch of beautiful grapes ha nging from a high
branch.
“Boy those sure would be tasty,” he thought to himself. He bac ked up and took a
running start, and jumped. He did not get high enough.
He went back to his starting spot and tried again. He almost go t high enough this
time, but not quite.
He tried and tried, again and again, but just couldn’t get hig h enough to grab the
grapes.
Finally, he gave up.
As he walked away, he put his nose in the air and said: “I am sure those grapes are
sour.”
Following least effort principle [7], it is appreciable in na tural languages to combine
existing words to express different meaning. Shannon [8] poi nted out the importance of
3redundancy in natural languages long ago: generally speaki ng, nearly half of the letters in
a sample of English text can be deleted while someone else can still restore them. These
properties of natural language ensure the sequence obtaine d by taking off interword de-
limiters from a certain text being highly nonrandom and show ing determinant and regular
characteristics. It is expected that the correct segmentat ion reveals these characteristics as
strongly as possible. From information point of view, this m eans that, if a form of infor-
mation entropy can be properly defined on each segmentation, the entropy of the correct
segmentation will be the smallest.
Interestingly, a maximum likelihood approach leads to the s ame proposal and automati-
cally gives the definition of the entropy. Given one sequence of length N, we expect to find a
likelihood function which reaches its maximum on the correc t segmentation. For a concrete
segmentation, we assign a probability to each word in it
wi→pi, i = 1...M (3)
with
M/summationdisplay
i=1pi= 1. (4)
The likelihood function is written as
Zs=M/productdisplay
i=1pimili(5)
where miis the number of word wiin the segmentation, and liis the length of the word.
By maximizing the likelihood function subjected to eq.(4) w e obtain
pi=mili
N. (6)
Thus the maximum likelihood for the segmentation is
Zs=M/productdisplay
i=1(mili
N)mili. (7)
The segmentation with maximum likelihood is just the one min imizing
S=−lnZs
N=−M/summationdisplay
i=1mili
Nln(mili
N). (8)
4This function has the form of entropy [8] and will be called Se gmentation Entropy (SE).
Starting from a maximum likelihood approach, we now come to t he suggestion to mini-
mize the segmentation entropy. This is in contrast to common place. Maximizing likelihood
leads to maximizing certain entropy in some cases [9,10]. As a general principle for inves-
tigating statistical problems, maximum entropy method has been successfully applied in a
variety of fields [9,10]. We propose that, instead of applyin g maximum entropy principle,
one may choose to minimize certain entropy (minimum entropy principle) in some problems.
This seems attractive especially in the era of bioinformati cs when most of the problems are
to reveal regularity in large amount of seemingly random seq uences.
Because the present is a statistical method, the text under s tudy needs to be not too
short. For example, when we tried to find the segmentation wit h the smallest segmentation
entropy for the saying
God is nowhere as much as he is in the soul... and the soul means the world
(By Meister Eckhart, 14-century Dominican priest, Preache r, and Theologian), it was found
that, among a total of 343062720 segmentations under WLC, th ere are 15 segmentations
whose SE is 2 .3684, smaller than 2.3802 of the correct one. One example is
god isnow herea smuchas heis inthe soul andthe soul meanst heworld,
in which the five “he”and two “soul” are revealed.
Unfortunately, present computational power does not permi t to exhaustively study even a
text as short as “the Fox and the Grapes” , the number of permitted segmentations for which
is 1.33E+ 97 under WLC. We choose to see the relevance of the concept of segmentation
entropy in some special ways. The study focuses on “The Fox an d the Grapes”.
To change a segmentation slightly, one way is to choose two ad jacent words along the
sequences randomly and then exchange their length. This way the original two words may
change to different words. This procedure can be repeated on t he resulting segmentations.
The change does not violate the WLC. Because of the large numb er of possible choices in
each step, the segmentation is expected to become increasin gly dissimilar to the original
one. Starting from the correct segmentation of “The Fox and t he Grapes”, we expect to
see the evolution of SE by changing the segmentation this way . Figure 1 shows that SE
5increase drastically in the first 500 steps, and then reaches and fluctuates around certain
equilibrium value. Compared with the gap between the equili brium value and the original
SE, the fluctuation is minor. This shows that, at least locall y, the correct segmentation is
at the minimum of SE. Actually, we have traced a trajectory of evolution up to 1010steps.
No segmentation with SE smaller than the correct one was obse rved. This implies that SE
of the correct segmentation is also globally minimal.
The distribution of segmentation entropy may give further i nsight to the atypicality
of the correct SE. We randomly sampled 1010segmentations in the following way: while
keeping the WLC, the length of each words in the segmentation is assigned randomly. The
distribution of SE is shown in Fig. 2. The minimal SE we sample d is 4.5298, still much
higher than 4.097 of the correct segmentation (see Fig. 1). I t is interesting to observe that
the distribution shows fractal characteristics. The fract al-like distribution presents also for
other text, even for random sequence (Fig. 3). The fractal-l ike feature is determined by the
WLC and the statistical structure of the sequence under stud y. In Fig. 3 we compared the
distribution of SE of two sequences (under the same WLC), the original sequence of “The
Fox and the Grapes” and a random sequence obtained by randomizing the order of le tters
in the text. The result is in accordance with the fact that the original sequence is in a much
more ordered state, manifesting that segmentation entropy captures the statistical structure
of the sequences successfully.
There is one way to estimate the number of segmentations the S E of which is 4.097, the
value for the correct segmentation. See Fig. 4 in which the di stribution of SE in Fig. 2 are
shown in logrithmic scale here. The left edge of the distribu tion fall on a line. The edge can
be fitted by e(165x−750.42).The number of segmentations with SE x among the totally 1 .33e97
possible segmentations under WLC is:
c(x) =1.33e97
9×109e(165x−750.42). (9)
We obtained c(4.097) = 0 .96. From the distribution of SE shown in Fig. 3(a) we obtained
the same value of c(4.097). The estimation support the idea that segmentation ent ropy of
correct segmentation is unique.
We now consider how to release the WLC. Unfortunately, searc hing the segmentation
with the smallest SE among all the possible is sure to fail to fi nd the correct one. For
6example, SE of the segmentation in which the whole sequence i s considered as one word
(single-word segmentation) is 0, the smallest possible SE. Also, the segmentation in which
each letter is viewed as a separate word ( N-word segmentation) has a considerably small
SE (2.8655 for “The fox and the grapes” ). These are called side attraction effects. These
examples show that smaller SE does not necessarily means bet ter segmentation when we
compare the SEs of segmentations under different WLC (here WL C refers to any partition of
numbers of words of various length satisfying eq.(1), not ne cessarily the same as the original
text.) The bias induced by different WLC must be taken off. In or der to do so, we suggest
to use
RS=S
S0(10)
instead of S. Here S0is the average SE under the same WLC of a sequence obtained by
randomizing the order of letters in the original text. S0plays the role of chemical potential
for a thermodynamic system [11]. RSfor the single word and N-word segmentations are 1,
the largest possible value. By searching segmentation with the smallest RS, it is expected
to find meaningful segmentation. For examples, for the segme ntation
god isnow herea smuchas heis int he soul andthe soul meanst heworld,
which has already been shown above, RSis 0.8601; while
godisnowhereasmuch as he is inthe soul an dthe soul meanstheworld
is a better – actually one of the best – segmentation accordin g toRS(RS= 0.8259).
Intuitively this is reasonable, because in this second segm entation, more repeated “words”
– two copies of “is”,“as”and“an”– are revealed. Another segmentation
godisnowhereasmuch as he is inthesoul an dthesoul meanstheworld,
which differs from the second segmentation by revealing the t wo“thesoul” , has a moderately
small RS: 0.8481. Comparison shows that the five repeats of “he”is the most preferred part
in good segmentations.
7III. CONCLUDING REMARKS.
In statistical linguistics many efforts are given on signal e xtracting and statistical infer-
ence. Our method, however, is new on at least two points. Firs t, there is neither assumption
on distribution [12] nor demand for training sets, lexical o r grammatical knowledge [6]. This
feature is important for studying biological sequences, be cause present knowledge on the
“language” (DNA) of life is still lack. Second, instead of ex tracting a limit number of sig-
nals, we try to “read” the sequence exactly as a “text”. A text includes more than words: it
also includes the organization of words. The results of segm entation form a basis for many
further elaborations.
Principally, the concept of segmentation entropy can be app lied to study the noncoding
DNA sequences of eukaryote genomes. It is expected that the s tudy may gives more than
some meaningful “words” or regulatory elements. Possible a pplications are not confined to
studying noncoding DNA sequences of course. Segmentation e ntropy can be used to find
patterns in any symbolic sequences. However, the applicati on of segmentation entropy is re-
stricted by the difficulty to find the segmentation with the sma llestRsfrom the vast amount
possible ones. We are now developing algorithm that can be us ed for regulatory binding
sites prediction. in the algorithm the principle of minimun entropy will be incorporated in.
ACKNOWLEDGMENTS
I thanks Professor Bai-lin Hao who helps to make the computin g possible. I also thanks
Professor Wei-mou Zheng and Professor Bai-lin Hao for stimu lating discussions. Mr. Xiong
Zhang carefully read the manuscript. The work was supported partly by National Science
Fundation.
8REFERENCES
[1] See, e.g., W. Li, Molecular Evolution (Sinauer Associates, 1997).
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9FIGURES
4.14.24.34.44.54.64.7
05001000150020002500300035004000segmentation entropy
steps4.14.24.34.44.54.64.7
050100150200250300350400segmentation entropy
steps
FIG. 1. The evolution of segmentation entropy. Starting fro m the correct one, the segmentation
was change stepwisely by exchanging the lengths of a pair of a djacent words randomly chosen along
the sequence. The doted line corresponds to the smallest seg mentation entropy 4.5298 for the 1010
randomly sampled segmentations, see Fig. 2.
1001e+082e+083e+084e+085e+086e+08
4.64 4.65 4.66 4.67 4.68 4.69 4.74.71distribution
segmentation entropy
FIG. 2. The distribution of the segmentation entropy of 9 ×109segmentations randomly chosen
for the text “The Fox and the Grapes”. The numbers of words of v arious length in the original
text were first counted. In the sampled segmentations these n umbers were kept, but the length of
each word along the sequence were randomly assigned.
1102e+074e+076e+078e+071e+081.2e+081.4e+08 distributionoriginal sequence
05e+071e+081.5e+082e+082.5e+083e+08
4.644.654.664.674.684.69 4.74.71distribution
segmentation entropyrandom sequence
FIG. 3. Comparison of the distribution of segmentation entr opy for two sequences: the original
sequence of “The Fox and the Grapes” , and a random sequence obtained by randomizing the order
of letters in the original text. For each sequence, 109segmentations are randomly sampled in the
way described in the caption of Fig. 2.
121101001000100001000001e+061e+071e+081e+09
4.544.564.584.64.624.644.664.684.74.72distribution
Text segmentation
FIG. 4. The distribution of segmentation shown in Fig. 2 is sh own in log scale here. The line
along the left edge of the distribution is e(165x−750.42).
13 |
arXiv:physics/0008233v1 [physics.ins-det] 29 Aug 2000Fast and Flexible CCD Driver System Using
Fast DAC and FPGA
Emi Miyataa,c, Chikara Natsukaria, Daisuke Akutsua,
Tomoyuki KamazukaaMasaharu Nomachib, and
Masanobu Ozakid
aDepartment of Earth & Space Science, Graduate School of Scie nce, Osaka
University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
bDepartment of Physics, Graduate School of Science, Osaka Un iversity, 1-1
Machikaneyama, Toyonaka, Osaka 560-0043, Japan
cCREST, Japan Science and Technology Corporation (JST)
d3-1-1 Yoshinodai, Sagamihara Kanagawa 229-8510, Japan
Abstract
We have developed a completely new type of general-purpose C CD data acquisition
system which enables one to drive any type of CCD using any typ e of clocking
mode. A CCD driver system widely used before consisted of an a nalog multiplexer
(MPX), a digital-to-analog converter (DAC), and an operati onal amplifier. A DAC
is used to determine high and low voltage levels and the MPX se lects each voltage
level using a TTL clock. In this kind of driver board, it is diffi cult to reduce the
noise caused by a short of high and low level in MPX and also to s elect many kinds
of different voltage levels.
Recent developments in semiconductor IC enable us to use a ve ry fast sampling
(∼10MHz) DAC with low cost. We thus develop the new driver syste m using a
fast DAC in order to determine both the voltage level of the cl ock and the clocking
timing. We use FPGA (Field Programmable Gate Array) to contr ol the DAC. We
have constructed the data acquisition system and found that the CCD functions
well with our new system. The energy resolution of Mn K αhas a full-width at
half-maximum of ≃150 eV and the readout noise of our system is ≃8 e−.
1 Introduction
Most recent X-ray satellites carry a charge-coupled device (CCD) camera for
their focal plane instrument. CCD’s possesses a moderate en ergy resolution,
a high spatial resolution, and a timing resolution. The Soli d-state Imaging
Preprint submitted to Elsevier Preprint 2 February 2008Spectrometer, SIS, onboard ASCA was the first CCD camera used as a photon
counting detector. (Tanaka et al[1]). Following the SIS, ma ny satellites such
as HETE2 (Ricker[2]), Chandra (Weisskoph et al.[3]), XMM-N ewton (Barr et
al.[4]), and MAXI (Matsuoka et al.[5]) now carry a X-ray CCD c amera on
their focal planes.
MAXI, Monitor of All-sky X-ray Image, has been selected as an early payload
of the JEM (Japanese Experiment Module) Exposed Facility on the Interna-
tional Space Station. MAXI will monitor the activities of ab out 2000-3000
X-ray sources. It consists of two kinds of X-ray detectors: t he first, the gas slit
camera (GSC) is a one-dimensional position-sensitive prop ortional counter,
and the other, the solid-state slit camera (SSC) is an X-ray C CD array. The
CCD used in the SSC is fabricated by Hamamatsu Photonics K.K. (HPK)
and is being calibrated both at Osaka University and the Nati onal Space De-
velopment Agency of Japan (NASDA).
Since SSC is the first CCD camera fabricated soley by Japan, we need to
specify the functioning of the CCD in detail. In order to opti mize the function
of the CCD, we need to develop a highly flexible data acquisiti on system.
2 Requirements for New System
In order to optimize the X-ray responsibility of the CCD, we n eed to develop
a highly flexible CCD driver. Our requirements of the CCD driv er are:
•to output any kind of clocking pattern
•to dynamically control clocking voltages
•to modify the clocking pattern easily and download it by requ est
•to have a readout speed ≥1MHz
•to output clocking voltages with a range of −20 to +20 V
•to control voltage levels to within 0.1 V
The clock driver circuit used until now consists of MPXs, DAC s (digital-to-
analog converters), and analog amplifiers. For example, two DACs are used to
generate the low and high voltage level of a clock and an MPX sw itches each
level with a digital signal. This system has been well establ ished but it is not
suitable to change the voltage level dynamically.
23 New CCD Data Acquisition System
To satisfy all of the requirements listed in section 2, we hav e developed a
new type of CCD driver system as shown in figure 1. We use one fas t DAC
to generate each clock. This enables us to control each clock with a high
flexibility whereas we need a lot of control I/O pins. In the pr evious system,
the voltage level of each DAC is determined before operating the CCD and
at least one I/O pin is needed for each clock. On the other hand , our new
system requires the number of clocks times 10 pins per DAC eve n if we use an
8-bit DAC, resulting in roughly orders of magnitude more I/O pins than the
previous system. We thus introduced a field programmable gat e array (FPGA)
to control all DACs.
3.1 Design of the DAC board
Because a CCD is operated by DACs directly, the noise charact eristics need
to be low. We therefore picked up more than five DACs to evaluat e the noise
characteristics. Among them, TLC 7524 fabricated by Texas I nstruments pos-
sesses the lowest noise characteristics and we select this d evice for our new
system.
A detailed design around DAC in figure 1 is shown in figure 2. We u se a photo-
coupler, HCPL −2430, to separate an analog and a digital ground. TLC7524 is
an 8-bit current-output DAC whose settling time is ∼100 ns. The fast settling
time enables us to simultaneously control both the clocking timing and the
voltage level, which is realized with several DACs and MPXs i n the previous
driver system. Thus, our new system posses a high flexibility though it is much
simpler than the previous system.
3.2 Design of the FPGA board
We previously used the VME system to control the DAC boards an d had a
lot of noise problems mainly due to a switching regulators on a VME power
supply unit. We thus abandon using the VME system for this pur pose. We
designed a general-purpose digital I/O board (DIO board) to simultaneously
control several DAC boards. Our DIO board carries a reconfigu rable FPGA,
512 Kbyte SRAM device (PD434008ALE-15), a serial interface , a parallel
interface with 10 bits, an interface for a liquid crystal to d isplay the status,
and eight DAC interfaces. One DAC interface possesses 10 bit s in order to
control a 10 bit DAC in a future application. Figure 3 shows a p hotograph of
the FPGA board developed in this work. We selected an Altera F lex 10K50 for
3the FPGA. This FPGA device is a static memory type that can be r econfigured
simply with a command and has 189 pins available for the user.
One of the remarkable advantages is the development of Hardw are Description
Language (HDL). HDLs and synthesis tools can greatly reduce the design
time, improving the time-to-market. A description based on HDLs is easier
to understand than some schematic for a very large design in F PGA gate
format. There are several kinds of HDLs developed for variou s corporations:
AHDL[7], VHDL[8], and Verilog-HDL[9]. Among them, we emplo yed VHDL.
Throughout the development, we used the MAX+PLUS II and FPGA Express
software provided by Altera corporation and Synopsis corpo ration.
3.3 Data Acquiring System
The CCD output signal is processed with a delay line and peak- hold circuits
which have been previously developed by our group. The proce ssed signal
is shifted to ±5V and sampled by a 12-bit analog-digital converter (ADC).
Digital data are transferred to the VME I/O board ([6]) with a flat cable and
are sent to the sparcstation through the VME bus.
3.4 Sequencer
We have developed a sequencer and relevant software to compi le it. We define
two sequencers: V-ram and P-ram. The V-ram defines a voltage p attern to
drive the CCD with a relatively a short duration. Combining s everal V-rams,
we describe the clocking pattern for readout of whole CCD in P -ram.
3.4.1 V-ram
We develop, typically, two kinds of V-ram: V-ram for readout one pixel and
transfer one line. An example of V-ram for one pixel readout i n a two phase
CCD is shown below. The vertical axis represents the time seq uence.P1H
andP2Hare clocks for the serial register and P1VandP2Vare those for the
vertical register. RSTandHOLDare clocks for reset and ADC. Numbers in V-
ram represent the voltage level in units of Volts. Following brackets show that
a voltage level is the same as the previous value. In this way, we describe the
voltage level and the timing for a voltage change in V-ram.
The V-ram compiler we developed reads the V-ram and creates t he DAC
patterns for each clock. The HOLD signal is transferred to the ADC board
4through the parallel interface while others are transferre d to the appropriate
DAC interace.
In the current system, we use TLC7524 which needs a reference clock. When
the reference clock is sent to TLC7524, it latches all data bi ts and outputs the
voltage depending on the data bits. Since the reference cloc k is difficult to be
described in V-ram, the V-ram compiler adds it in the output s equencer code
automatically.
P1H P2H RST P1V P2V HOLD
-8 6 6 6 6 5
[ ] ] ] ] ]
[ ] -8 ] ] ]
[ ] [ ] ] ]
[ ] [ ] ] ]
[ ] [ ] ] ]
[ ] [ ] ] ]
[ ] [ ] ] ]
6 -8 [ ] ] 0
] [ [ ] ] [
] [ [ ] ] [
] [ [ ] ] [
] [ [ ] ] [
] [ [ ] ] [
] [ [ ] ] [
] [ [ ] ] [
3.4.2 P-ram
P-ram is described to define the readout of a whole CCD. To incl ude V-ram
files, P-ram uses the filename of V-ram. We have prepared sever al instruction
commands to describe any P-rams easily and concisely as list ed in table 1.
Combining filenames of V-rams and instruction commands, P-r am can be
easily developed by the user. One example of a P-ram is shown b elow.
set A = 64
set B = 2
set xaxis = 1024
set yaxis = 1024
start:
do yaxis
set wait A
5seq 1 vertical
set wait B
seq xaxis horizontal
end do
jmp start
This P-ram reads out a CCD with 1024 ×1024 pixels. V-ram of ’vertical’
and ’horizontal’ define the voltage pattern to transfer pixe ls vertically and
horizontally, respectively. The instruction of ’set wait’ is to determine the
duration of each level in S-ram.
The P-ram is compiled on a SUN sparcstation (Force, CPU-50GT ) and stored
in the memory of VME I/O board. After sending a command from th e sun,
P-ram is downloaded to the DIO board by means of the serial int erface and
stored in the memory of the DIO board.
3.5 Configuration of the Circuit in FPGA
To realize the function of the sequencers, we divided the con figuration of the
FLEX device into five blocks as shown in figure 4. Each block is c onstructed by
a synchronous state machine. The Serial Interface is the interface to the
VME I/O board to download sequencers. After loading sequenc ers, theSerial
Interface sends a trigger signal to the Clock Controller . The state ma-
chine of the Clock Controller is shown in figure 5. The Clock Controller
is in the idlestate until a trigger signal is sent. Once the trigger signal is
received, the Clock Controller moves to the memory check state, where
theClock Controller sends the memory address and a trigger signal to
theSynthesize Pattern . TheSynthesize Pattern sends a trigger signal
to theMemory Controller and receives memory data. After repeating three
times, the Synthesize Pattern arranges the data into 96 bits and sends it
to theClock Controller . Then, the state moves to fetchwhere 96 bit data
is stored in a register and next moves to decode . In the decode state, the
Clock Controller analyzes the bit pattern based on the instruction com-
mands shown in table 1 and sends DAC patterns to the appropria te DAC in-
terface. After sending the DAC patterns, the memory pointer is incremented
andClock Controller waits for the wait parameter ( AorBshown in P-ram).
In each state, the Clock Controller sends status information to the Display
Controller and theDisplay Controller controls the liquid crystal to dis-
play the clocking status.
64 Performance
4.1 Driver System
In order to demonstrate the performance of our new CCD driver , we produced
5 value clockings as shown in figure 6. This kind of multi-leve l clocking is
efficient to reduce the spurious charge[10]. We thus confirm th e high potential
and high flexibility for our new system.
Since we use 8 bits DAC for each clock, we can control a voltage level within
≃0.1 V. We normaly operate the DAC boards with ranges of −15 to +15V.
If we change the resister of R13, R15, and R20 in figure 2, we can output the
clock up to +20V or down to −20V.
The readout speed is limited by the number of state machines t o read a voltage
pattern from S-ram. In our current design, there are 13 steps to fetch a 96 bits
voltage pattern, resulting in the maximum clocking speed to be≃300 KHz.
We still need to optimize it in order to meet our requirements (∼1 MHz).
4.2 Total System
We compared the performance of our new system with the HPK C48 80 system,
which is an X-ray CCD data acquisition system previously use d[11]. We used
a CCD chip fabricated by HPK. We cooled the CCD down to −100◦C and
irradiated it with an55Fe source. For comparison, we set the same readout
speed as that of C4880 (50 KHz). We selected the ASCA grade 0 ev ents[12]
with a split threshold of 90 eV and fitted the histogram with tw o Gaussian
functions for Mn K αand K β. Results are shown in table 2. The readout noise
of our new system is ≃8e−. We can confirm that our new system function
much better than the previous system.
5 Summary and Future Developments
We have developed a new type of general-purpose CCD data acqu isition sys-
tem which enables us to drive any kind of CCD with any kind of cl ocking
and voltage patterns. It functions well and demonstrates gr eat flexibility. We
found the readout noise of the CCD to be 8 e−rms in our system, which might
be contributed to by our readout circuit rather than a CCD chi p itself.
We plan to develop the analog electronics to process a CCD out put signal
7to reduce the readout noise. The system currently used is a de lay line cir-
cuit which has poorer performance than an integrated type ci rcuit for both
readout speed and for noise characteristics (especially hi gh frequency regime).
Therefore, we will develop an integrated correlated double sampling circuit in
the near future.
We also plan to replace the VME I/O board with another FPGA boa rd which
has already been constructed by us. On this board, 80M sampli ng ADC, FLEX
10K and 4Mbyte S-ram are mounted. There are three IEEE 1394 po rts each of
which has a capability of 400 Mbps connection. Large amount o f memory gives
us to extract X-ray events before sending raw frame data to th e host machine.
Since FPGA has a good capability of a parallel processing com paring with
DSP or CPU, it enables us to analyze data in real-time. It is al so important
to develop onboard digital processing software using HDL fo r future X-ray
astronomy missions.
We will calibrate the CCD for the MAXI mission with our system . We need
to determine the voltage pattern and the voltage level to opt imize the X-ray
responsibility.
We wish to thank Prof. H. Tsunemi for his valuable comments on the initial
phase of this work. We acknowledge to Mr. C. Baluta for his cri tical reading
of the manuscript. This research is partially supported by A CT-JST Program,
Japan Science and Technology Corporation.
References
[1] Tanaka, Y., Inoue, H., and Holt, S.S., PASJ,46, L37, 1994
[2] Ricker, G.R. Proc. of All-Sky X-Ray Observations in the Next Decade , 366, 1998
[3] Weisskoph, M.C., O’Dell, S.L., Elsner, R.F., van Speybr oeck, L.P. Proc. SPIE ,
2515, 312, 1995
[4] Barr P. et al. ESA SP-1097, March 1988
[5] Matsuoka, M. et al. Proc. SPIE , 3114, 414, 1997
[6] Kataoka, J. et al. Proc. SPIE , 3445, 143, 1998
[7] Altera Corporation, Max+Plus II Programmable Logic Development System –
AHDL– , 1998
8[8] Institute of Electrical and Electronic Engineers, Inc. ,VHDL Language Reference
manual , IEEE Standard 1076-1987, 1988
[9] D.E.Thomas and P. Moorby, The Verilog Hardware Descritption Language ,
Kluwer, Academic Publishers, 1991
[10] Janesick, J.R., Elliot, T. & Collins, S., Optical Engin eering, 26, 692, 1987
[11] Miyata, E. et al. Nuclear Instruments and Method , 436, 91, 1999
[12] Yamashita, A. et al. IEEE Trans. Nucl. Sci. , 44, 847, 1997
9Table 1
Instruction commands for P-ram
Command Arguments Function
start — named label
jmp label jump tolabel
seq number, V-ram name outputV-ram name withnumber times
set wait number define the output speed
do number repeat all V-rams before next end do number times
end do — define the end of block to be repeated
# — write comment
10Table 2
Comparison of our new system with the HPK C4880 system
C4880 New system
Energy resolution [eV] 162 ±3 150 ±3
Dark current [e−/pixel/sec] 0.20 ±0.15 0.20 ±0.14
Readout noise [e−rms] 8.6 ±0.5 8.0 ±0.5
Exposure time [sec] 8 8
11Fig. 1. The block diagram of the CCD signal flow.
Fig. 2. A circuit diagram of the DAC board of figure 1
Fig. 3. The picture of the VME I/O board. FPGA is mounted aroun d the center of
the board.
Fig. 4. The block diagram of the DIO board. Five gray-colored boxes represents the
circuits designed in the FLEX chip. There are eight DAC inter faces each of which
has 10 bits to control the DAC board.
Fig. 5. State machine of Clock Controller in FPGA
Fig. 6. Sample clock of multiple levels
Fig. 7. Single event55Fe spectrum obtained with our new system.
12This figure "fig1.jpg" is available in "jpg"
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arXiv:physics/0008234v1 [physics.class-ph] 29 Aug 2000ON THE POSSIBILITY OF LOCAL SR CONSTRUCTION
G.A. Kotel’nikov
Russian Research Centre Kurchatov Institute, Moscow 12318 2, Russia
E-mail: kga@electronics.kiae.ru
Abstract
The violation of the invariance of the speed of light in Speci al Relativity has been
made. The version of the theory has been constructed in which the possibility of the
superluminal motions are permitted.
1 Introduction
In view of mathematical elegance, laconicalness and predic tive power Special Relativity (SR)
is the fundamental theory of modern physics. Owing to this th e mathematical postulates
of the theory, possibility of their modification and general ization as well as of experimental
test attract attention constantly. As examples one can be pr esented the well known Pauli
monograph [1], containing the elements of the Abraham and Ri tz theories; academician
Logunov’s lectures on the foundations of Relativity Theory with the formulation of SR in
the affine space [2]; Fushchich’s publication on the non-line ar electrodynamics equations with
the non-invariant speed of light [3]; Glashow’s work on the e xperimental consequences of the
violation of the Lorentz-invariance in astrophysics [4].
To the present time SR is one of the most experimentally-just ified theories (for example,
Pauli and Landsberg monographies [1, 5]; Strakhovsky and Us pensky [6], Basov and his co-
authors [7], Møller [8] and Molchanov [9] reviews; the origi nal publications of [10, 11, 12]).
Here one can mention the experiments on detection of the ethe r wind in the experiments of
the Michelson type [1, 5]; determination of the angular ligh t aberrations [1, 5]; transversal
Doppler effect measurement [10]; experiments on the proof of independence of the speed
of light from the velocity of the source of light [9, 10]; expe riments on determination of
the relativistic mass dependence of the velocity of a partic le motion [10]; the relativistic
retardation of time [10]; the g-2 experiments [11, 12, 10]. T he results of these experiments
indicate the absence of the ether wind to closer and closer li mits of accuracy, and argue for
SR.
This raises the natural question, whether do exist at all any experiments different from
SR predictions, even though they are ambiguously interpret ed. It appears that there are
a number of the publications on this theme. Let us consider th ose concerning the second
postulate - the postulate of the constancy of the speed of lig ht.
Giannetto, Maccarrone, Mignani and Recami [13] have been co nsidered the possibility of
the negative sign interpretation of the square of the neutri no 4 - momentum P2=E2−p2c2=
M02c4= (−0,166±0,091)MeV2in the experiments on π- decay π+→µ++νas the fact of
observation of a superluminal particle with imaging mass M0=im(tachyon). Khalfin [14]
has established that negative sign of the square of the neutr ino 4-momentum may be due to
incorrectness of the observational data processing near up per bound of β- spectrum (in our
own case near a upper bound of µ- spectrum). Thus, the possibility of the interpretation
1ofπ- neutrino as a particle of the tachyon nature is eliminated p ractically in the light of the
contemporary explanation for the negative sign of the 4-mom entum square.
Mamaev [15] has analyzed the time-flight spectra of π−,µ−,e−particles from Joint
Institute for Nuclear Research (Dubna) and concluded that t he data from the article [16]
may be interpreted as the result of superluminal motion of me sons and electrons. However,
taking into account the presence in the signal processing el ectronic circuit of a threshold
device (discriminator) with 2543 channels of the analyzer, it is possible to conclude that the
velocities of these particles were 0 ,92c, 0,94cand 0,96crespectively, where cis the speed of
light. The phenomenon of superluminal motion disappears, a nd mutual arrangement of the
spectral lines from [16] may be explained in the framework of SR [26].
Nevertheless numerous examples are known in which the elimi nation of superluminal
motion turns out to be more difficult and less convincing than i n the considered cases.
These are observations of superluminal motion of particles in broad atmospheric showers
and the acts of antiproton birth, as well as on expansion of th e shells of some extragalactic
radiosources, for example [17, 12, 19].
Clay and Crouch has observed [17] impulses, preceding the si gnal induced by a broad
atmospheric shower. Let us suppose that particles from the s hower had the velocity equal to
the speed of light (that is natural). Then it is not clear, wha t has preceded these particles.
”We conclude that we have observed non-random events preced ing the arrival of an extensive
air shower. Being unable to explain this result in a more conv entional manner, we suggest
that is the result of a particle traveling with an apparent ve locity greater than of light ” [17].
Further the authors [17] have assumed that the impulses were stipulated by the particles
with imaginary masses (tachyons) traveling at the velociti es exceeding the speed of light.
Cooper [12] has concluded that the time-flight experiments o n observation of antiprotons
admit the existence of superluminal particles (antimesons ) connected with antiprotons. The
calculated probability of the velocity of antimesons excee ding the speed of light, is equal
0.9972. The evaluation turn out to be tolerant to various exp erimental errors. The author
writes: ” A reexamination of the Nobel-prize-winning exper iment in which the antiproton
was discovered reveals that associated antimesons might be traveling faster than light ” [12].
The numerous publications are known on the observation of su perluminal expansion
of extragalactic radiosources (for example [19, 20, 26]). I t is an interesting phenomenon,
and it is difficult to be explained in terms of modern astrophys ics. The observation of
the superluminal expansion became possible after the radio interferometers VLBI (Very
Long Baseline Interferometry) for the centimetre spectral range were created. These possess
a superlong trans-continental base L(thousands and tens of thousands kilometers). The
angular resolution of such telescopes δ∼λ/Lis proportional to the ratio of a working
wavelength λto the value of the base L. It is much higher than the one of the best optical
devices. In the optical range L/λis equal ∼6·107, while in the radio range it is equal ∼
18·108. The radio interferometers allow one to study such thin stru cture of space objects ( ∼
7·10−4angle seconds) as was inaccessible to be observed by optical means. The studies have
shown that many extragalactic radio sources have a complica ted, bi-component structure.
Among of them the substructure of six radio sources run away f rom each other at calculated
velocities that are some times more than the speed of light. I t is the radio galaxy 3C120
(z = 0.033), quasars 3C273 (z = 0.158), 3C279 (z = 0.538), 3C34 5 (z = 0.595), 3C179
(z = 0.846) and NRAO140 (z = 1.258) [21]. (Here zis the parameter of redshift). The
2transversal velocities calculated within the framework of the cosmological Friedmann model
of the motion of the components are equal V⊥∼(2−20)c. It has been proposed over ten
versions for interpretations of this phenomenon. It may be a ssociated with more complicated
multicomponent structure of the quasars; the random superp osition of radio spots on the
quasars; influence of intergalactic gravitational lens dup licating a visible image; Doppler
effect; increase of Hubble’s constant that is accompanied by decreasing the distances to the
quasars, which results in disappearing the superluminal ex pansion. Also, it may be due
to the influence of interstellar magnetic fields; existence o f tachyon matter; introduction of
5-space with an additional fifth coordinate such as the speed of light running the values from
0 to∞; model of the light echo; optical illusion not contradictin g to SR [19, 20, 26]. It is
evident that the conventional explanation for the superlum inal expansion is not offered yet,
and various hypothesizes on the nature of this phenomenon ma y be discussed.
Loiseau [22] has paid attention to the little difference betw een the galaxy NGC 5668
redshift z′, measured by radioastronomical method at the frequency cor responding to the
wavelength 21 cm, and the redshift z, measured in the optical range for this galaxy. This
result, if it really is outside the limits of measurement errors , cannot be explained in the
framework of SR, as z′=zshould be with c′=c. To explain this result, author [22] in-
troduced 3-dimensional non-Euclidean space, inserted int o 4-dimensional Riemannian space
with some common time. In this case it may be obtained that the galaxy speed of light
c′and the speed of light con the Earth are connected by the ratio c′=c(1 +z)/(1 +z′),
where zis the redshift on a wave length in the optical range, and z′is the redshift on a
wave frequency in the radio range. In accordance with the obs erved data on the galaxy
NGC 5668 zis equal to 0.00580 in the optical range; z′is equal to 0.00526 in the radio
range on the frequency corresponding to the wavelength 21 cm . It follows from here that
c′/c= (1 + z)/(1 +z′) = 1.00580/1.00526 = 1 .0005372, and c′=c+ 182 ,04 km / sec ¿ c
[22]. The estimation has shown that the speed of light from qu asar QSO PKS 2134 with the
optical redshift z= 1.936 is equal to c′= 440 .000 km / sec [22]. The relationship between c′,
cand the quasar velocity vrelative to the Earth is described by the formula c′=c/radicalBig
1 +v2/c2
in the approximation of a weak gravitational field. The stati stical significance of the hypoth-
esis on the difference between the redshifts in the radio and o ptical ranges is naturally the
deciding factor for the Loiseau work.
Thus, unambiguously interpreted experimental data distin ct from SR predictions are
apparently absent now. But there are vague indications that it is not improbable that they
exist in particle physics and in astrophysics. Let us consid er the hypothesis on the existence
of the superluminal motion in terms of the violation of invar iance of the speed of light in the
expression for the second degree of 4-interval at the infinit esimal level.
2 Space - Time Metric, Differentials Coordinates
Transformation Law
Let us start from the condition for the invariance of the 4 - in terval differential in Minkowski
3space with the metric:
ds2=−(dx1′)2−(dx2′)2−(dx3′)2−(dx4′)2=
−(dx1)2−(dx2)2−(dx3)2−(dx4)2−inv.(1)
Heredx1,2,3= (dx, dy, dz ),dx4=icdt, it is not necessary for the speed of light c′to be
equal c. Corresponding infinitesimal space - time transformations , saving the invariance of
the form (1), obviously contain the group locally isomorphi c to the Lorentz group [23]:
dx′
a=dxa, dx′
a=Labdxb, a, b= 1,2,3,4, (2)
where Labis the matrix of the six-dimensional Lorentz group L6[23] with local kinematics
parameter β. The one-dimensional infinitesimal transformations corre sponding to the given
matrix, take the well known form:
dx′
1=dx1+iβdx 4√1−β2;dx′
4=dx4−iβdx 1√1−β2;dx′
2=dx2;dx′
3=dx3 (3)
The reciprocal transformations may be obtained by the prime permutation. The group
parameters are connected by the ratio β′=−β[23]. But contrary to the global Lorentz
transformations [23], here the parameters βandβ′can depend explicitly or implicitly on a
space - time point β=β(f(x, t)), β′=β′(f′(x′, t′)). This is the important circumstance
which will allow one to construct the theoretical model in wh ich the existence of superluminal
motion is possible. The integral space - time transformatio ns induced by (3) are:
x′
1=/integraldisplaydx1+iβdx 4√1−β2+d1;x′
4=/integraldisplaydx4−iβdx 1√1−β2+d4;
x′
2=x2+d2;x′
3=x3+d3,(4)
where d1−d4are the translation parameters; the reciprocal transforma tions may be ob-
tained by the prime permutation; d′
a=−da,a= 1,2,3,4. The transformations (4) go into
the Poincar´ e ones if c= cost, c′=cbe put into them and our consideration be restricted to
inertial motions ( β= const). In this case on integration they go into the standar d transforma-
tions from Poincar´ e group (inhomogeneous Lorentz group). Thus, Lorentz transformations
are contained here as the particular case. The group propert ies of the integral transforma-
tions (4) are realized due to the group properties of the diffe rential transformations (3) and
due to the relativistic velocity addition theorem β” = (β+β′)/(1 +ββ′).
3 Integral of Operation, Energy, Momentum
Let us turn to the integral of operation in SR [23]. It is not in variant with respect to the
transformations with broken invariance of the speed of ligh t. However this property may be
corrected if we start from the invariant integral of operati on [26]:
S∗=cS=−mc2/integraldisplay
ds+e/integraldisplay
Aadxa+i
16π/integraldisplay
Fab2d4x=
−mc2/integraldisplay
ds−i/integraldisplay
Aajad4x+i
16π/integraldisplay
Fab2d4x=
/integraldisplay
(−mc2/radicalBig
1−β2+eA·βeφ)(cdt) +1
8π/integraldisplay
(E2−H2)d3x(cdt).(5)
4HereS∗is the new integral of operation, which we name the generaliz ed one; mc2is the
invariant combination corresponding to the rest energy of a particle ( mis the rest-mass, cis
the speed of light); eis the invariant electrical charge of a particle; Aa= (A1, A2, A3, A4) =
(A, iφ) is the 4-potential [23]; ja= (j1, j2, j3, j4) = ( ρv/c, iρ) is the 4-vector of current
density [1] instead of ja= (ρv, icρ) [23], ρis the charge density, vis the velocity of a charge;
Fab=∂Ab/∂xa−∂Aa/∂xbis the tensor of electromagnetic field; E=−(1/c)∂A/∂t− ∇φis
the electrical field; H=∇XAis the magnetic field; Fab2= 2(H2−E2);d4x=dx1dx2dx3dx4
is the element of the invariant 4-volume [23].
The transformational rest-mass properties is changed as th e result from the introduction
of the generalized integral (5). The mass is not any more scal ar. The mass is transformed
according to the law m′= (c2/c′2)m=γ−2m. The rest energy mc2has a scalar property.
The transformational property of Plank constant ¯ his changed as well. The invariant is not
the constant ¯ h, but the product ¯ hc. Due to the electrical charge property of invariance e,
the thin structure constant remains invariant α=e2/¯hc- inv.
The generalized Lagrangian, energy and 4 - momentum of a part icle correspond to the
generalized integral of operation. We will label the genera lized values with the symbol *.
We have:
L∗=cL=−mc2/radicalBig
1−β2+eA·β−eφ; (6)
P∗=∂L∗
∂β=cmv√1−β2+eA=cp+eA; (7)
E∗=P∗·β−cL=mc2
√1−β2+eφ=E. (8)
It follows from here that the motion integrals are the energy Eand the product of the
speed of light by the momentum from SR: cP=cp+eA. The parameter βhas meaning
as generalized velocity. The differential dx0=cdtplays a role of the time differential. It
is essential that thanks to the differentiation with respect to the parameter β, the results
obtained do not depend on the particular assumptions concer ning the properties of the speed
of light, as the value centers into the parameter β=v/c.
Owing to the well known property of 4 - speed U2=−1, we have the following expression
for the generalized 4 - momentum pa∗=mc2uaof a free particle:
pa∗2=c2p2−E2=−m2c4−inv. (9)
As in [23], in case of a particle in electromagnetic field we fin d:
Pa∗=mc2ua+eAa; (10)
(Pa∗−eAa)2= (cPa−eAa)2=−m2c4−inv. (11)
4 Equations of Motion for Charged Particle
Keeping in the mind expression (6), we shall start from Lagra nge equations d(∂L∗/∂β)/dx0−
∂L∗/∂x= 0 taking into account the vector equality ∇(a·b) = (a·∇)b+(b·∇)a+ax(∇xb)+
5bx(∇xa) [23]. We obtain the following equations for the motion of a c harged particle in
electromagnetic field:
dp∗
dt=d(cp)
dt=ceE+evxH; (12)
dE∗
dt=dE
dt=eE·v; (13)
5 Maxwell Equations
Let us start from the permutational ratios of the electromag netic field tensor and the
field Lagrange equations ∂(∂L∗/∂A a,b)/∂xb−∂L∗/∂A a= 0 [23, 24] taking into account
the expression ∂F2
ab/∂A a,b= 4Fab[23] and the density of the Lagrange function L∗=
cL=iAaja+ (i/16π)Fab2. Here Aa(x) is 4-potential; Aa,b=∂Aa/∂xb;a, b= 1,2,3,4;
gab=diag(−,−,−,−). In sum we have:
∇XE+1
c∂H
∂t= 0; ∇ ·E= 4πρ;
∇XH−1
c∂E
∂t= 4πj
c;∇ ·H= 0.(14)
Out of them the equation of motion (13) and the equations of el ectromagnetic field (14)
coincide with the equations which are known from SR.
Let us note that according to the given scheme Maxwell equati ons turn out to be invariant
not only in inertial frames (it is well known), but also in non -inertial frames in the flat pseudo
- Euclidean space with the metric ds2=c2dt2−dx2. This property of Maxwell equations
seems to be unusual, but it is known and has been noted by acade mician Logunov: ”... in
the framework of SR it is possible to describe a physical phen omena in non-inertial frames
as well. Fock understood this deeply ... ” [2]. The statement follows also from the general
covariant formulation of Maxwell equations [2, 23, 26].
6 Local SR
Up to this point any constraint did not placed on the transfor mation properties of the
speed of light in the theory. It turns out that it is possible t o realize various theoretical
considerations by appropriate postulation. In particular , if we postulate that c′=c, all
the obtained equations will go into SR equations. If we state that the speed of light is
constant and c′/negationslash=c,c′t′=ct- inv, the model may be realized which we name SR with
non-invariant speed of light [26]. It describes the same physical reality as SR and also
contains additional classification capabilities due to the symmetry with respect to more
general group of transformations. Besides Poincar´ e group , this group includes the group
induced by the generators X−1=∂t−t∂t/c, X 0=c∂c−t∂t, X+1=c2∂c−ct∂t[26]. At last,
the postulation is possible which permits one to construct a version of the theory compatible
with the principle of relativity and the concept of superlum inal motion. Let us consider this
capability at length.
6According to Ritz [1] we assume that the speed of light is equa l toco= 3·1010cm/sec not
in global meaning, but only relatively to an emitter. Let us a dd a new physical element to
the infinitesimal transformations (2) and the model based on them with equations of motion
(12), (13), ( 14). We assume that the state of motion (inertia l, or non-inertial) does not
influences on the proper value of the speed of light c0, Plank constant ¯ h0, the thin structure
constant αand other physical proper values, for example, the proper le ngthl0, proper time
t0, proper frequency of oscillations ω0, rest-mass m0, electrical charge e. (A proper value is
the physical value in the frame K0relatively to which the object is immobile). These remain
invariant in the process of motion:
c′
0=c0=co= 3·1010cm/sec ;
¯h′
0= ¯h0= 1,0·10−27g·cm2/sec;
′
0=l0;t′
0=t0;ω′
0=ω;m′
0=m0;e′=e.(15)
The hypothesis on the independence of proper values of physi cal quantities from the state
of a physical object motion we agree to name the local relativ ity principle.
Let us assume further that the time intervals measured by mea ns of differently-placed
clocks in any frames K, K′,···coincide with the local time in a proper frame Koon the
trajectory of the motion of the object:
dto=dt=dt′. (16)
We agree to name the theoretical model, realizing the local r elativity principle in the
flat space - time with the metric (1) in combination with the hy pothetical property of time
(16), as Local Special Relativity Theory (LSR) as distinct f rom the classical SR. We find
the following expressions for infinitesimal space - time tra nsformations in this case:
dx=dxo−vodto/radicalBig
1−vo2/co2;dy=dyo;dz=dzo;dt=dto−vodxo
co2;
c=co/radicalBig
1−vo2/co2;
dxo=dx−vdt/radicalBig
1−v2/c2;dyo=dy;dzo=dz;dto=dt−vdx/c
1−v2/c2;
co=c/radicalBig
1−v2/c2,(17)
where v/c=−vo/co; (1−v2/c2)·(1 +v2/co2) = 1. In this case the following relationship
between the speed of light cand the velocity of emitter vis hold true as the result from the
transformation properties of the speed of light in the formu lae ( 17):
c=co/radicalBigg
1 +v2
co2. (18)
Herecois the speed of light in the proper frame associated with the e mitter. (In the models
admitting the existence of ether cois the speed of light relatively to ether). The expressions
for the speed of light in the form of (17) or (18) were obtained by Abraham (1910) [1] and
7Rapier (1961) [25] respectively in the framework of ether mo dels; next they were obtained by
the author of the present publication (1968) [26] and Loisea u (1968) [22] in the framework
of the relativity principle. These formulae were reproduce d further by many authors from
various points of view, in particular by Marinov (1975) [27] , by Hsu (1976) [28], by Sj¨ odin
(1977) [29], by Mamaev (1985) [15], by Nimbuev (1996) [30], b y Klimez (1997) [31], by Russo
(1998) [32].
6.1 Generalized Momentum and Energy
Let us put the expression for the speed of light (18) into the f ormulae for momentum (7)
and energy (8) of a free particle. We have:
p∗=cp=cmv/radicalBig
1−v2/c2=m0c0v;
E=mc2
/radicalBig
1−v2/c2=m0c02/radicalBig
1 +v2/c02=m0c0c.(19)
The relation has a view between the generalized momentum and energy by this:
E2−P∗2=E2−c2p2=m02c04(1 +v2/c02)−m02c02v2=m02c04. (20)
6.2 Energy and Superluminal Motion
Let us begin with the expression v=√E2−m02c04/m0c0≥c0. It follows from here that in
the framework of LSR a particle will move with superluminal v elocity, if the particle energy
will satisfy the equality:
Etr≥√
2E0=√
2m0c02. (21)
This energy is equal ∼720 keV for electron and ∼1330 Mev for proton and neutron. We may
conclude from here that neutron physics of nuclear reactors may be formulated in the non-
relativistic approximation in LSR (as in SR). The electrons with the energy E>720keV(for
example, from radioactive decay) should be superluminal pa rticles in LSR. Particle physics
on modern accelerators such as Serpukhov one with the energy of protons 66 GeV (1 Gev =
1000 MeV) should be physics of superluminal motion in the fra mework of LSR, if it would
be realized in reality.
6.3 Equations of Motion for Charged Particle in LSR
After putting the expression for the speed of light (18) into the equations of motion (12), we
obtain [26]:
d(cp)
dt=ceE+evxH→modv
dt=c
coeE+e
covxH;
dE
dt=ev·E→ modc
dt=e
cov·E.(22)
From here it can be seen that the integrals of motion are eithe r the generalized momentum
cpand energy E, or the associated velocity of a particle vand the speed of light cin the
absence of external forces.
86.4 Maxwell Equations in LSR
Taking into account the expression for the speed of light (18 ), we obtain the following form
of Maxwell equations [26]:
∇XE+1
c0/radicalBig
1 +v2
c02∂H
∂t= 0; ∇ ·E= 4πρ;
∇XH−1
c0/radicalBig
1 +v2
c02∂E
∂t= 4πρv
c0/radicalBig
1 +v2
c02;∇ ·H= 0.(23)
Herevis the electrical charge velocity; c=c0/radicalBig
1 +v2/c02is the charge coordinate on the
axisc(cis the speed of light in the laboratory frame K);c0is the proper value of the speed
of light in the frame K.
6.5 LSR and Experiment
Let us consider the examples of experiments, the interpreta tion of which is close to or
coincides with their interpretation in SR.
The Michelson Experiment [1, 5]. For the case of a terrestrial light source the negativ e
result of the experiment may be explained by space isotropy ( the speed of light is the same in
all directions). Owing to this circumstance the interferen ce pattern will not be changed for
a terrestrial observer at rotation of the interferometer. I n the case of a extraterrestrial light
source the negative result may be explained by two factors: t he space isotropy and the square
dependence of the speed of light from the velocity Vof a light source c=co/radicalBig
1 +V2/co2
[26].
The Fizeau Experiment [1, 5]. The explanation is similar to the one accepted in SR.
The arising little correction is the the order of V2/c02≪1 and does not influence on the
experimental result in linear approximation [26]. (Here Vis the velocity of fluid).
The Bonch-Bruevich and Molchanov Experiment [33]. The authors compared the
speeds of the light radiated by the eastern and western equat orial edge of the solar disk.
In the framework of LSR the speed of light c=c0/radicalBig
1 +V2/c02does not depend on the
direction of the light source motion V. Therefore the speed of light will be the same for both
the western and eastern edges of the solar disk. As in SR it is i n accord with the negative
result of the experiment [26].
The Sadeh Experiment [34]. In the experiment the distinction between the speeds
of the gamma - quanta, arising as a result of the electron - pos itron annihilation in flight,
has been observed depending on the angle between the gamma - q uanta. By virtue of the
independence of the speed of light cfrom the direction of the velocity of the source V, the
result of the experiment should be negative in LSR as well as i n SR [26].
Let us consider also the experiments, which interpretation in LSR is different from their the
interpretations in SR.
The Doppler Effect [1, 23]. In LSR the change of a wavelength λis described by
the formula λ=λ0(1−V nx/c)//radicalBig
1−V2/c2=λ0[/radicalBig
1 +V2/c02−V nx/c0]. The change of
9a frequency is described by the formula ω=ω0(c/c0)/radicalBig
1−V2/c2/(1−V nx/c) =ω0/(1−
V nx/c0) [26]. Here θ=arccosn xis the angle of the observation; Vis the emitter velocity. It
is follows from here that in LSR there is no Doppler transvers al frequent shift because with
nx= 0,ω=ωo. For a wavelength the Doppler transversal shift is retained . Hence in LSR
the parameters of redshifts zλandzωdo not coincide with each other and are equal zλ=
(λ−λ0)/λ0→2V/c0atnx=−1, V→ ∞;zω= (ω0−ω)/ω0→1 atnx=−1, V→ ∞. In non
- relativistic approximation they coincide with each other zλ∼ −V nx/co,zω∼ −V nx/co.
When the emitters move with significant velocities, the dist inction begins to show itself with
the shifts zλ≥0.6. The fulfillment of the inequality zλ≥√
2 = 1.41 is the criterion for
longitudinal ( nx=−1) superluminal motion. The fulfillment of the inequality zλ≥√
2−1 =
0.41 is the criterion for m transversal) ( nx= 0) superluminal motion [26]. The superluminal
quasars 3C279 ( zλ= 0.536), 3N345 ( zλ= 0.595), 3C179 ( zλ= 0.846), NRAO 140 ( zλ=
1.258) [21] satisfy the letter criterion. In particular, th e calculated transversal velocity of
the QSO NRAO 140 expansion is V⊥∼2co. It is surprising that this velocity is close to
the low bound of these velocities 3 cowithin the Friedmann cosmological model [21]. It is
important for LSR to determine the frequent redshifts zωof these superluminal objects and
compare them with the lambda redshifts zλas well as to solve the problem of the existence
of the limit zω≤1. It will permit one to distinguish between LSR and SR becaus e in SR
the ratio zλ=zωis true. The redshift of the radio emission from neutral hydr ogenHIon
the frequency corresponding to the line 21 cm is attractive f or this purpose. However in this
frequency range the experimental data on superluminal quas ars are not avallable. Therefore
to reject LSR is not possible now.
Let us also pay attention to the relationship between the spe eds of light candcoand the
parameters of redshifts c=c0(1 +zλ)/(1 +zω)→c0(1 +zλ)/2 in LSR. (The latter formula
is true with zω∼1). We can conclude that it is the Loiseau formula [22]. Its ap plication
to the observational interpretation was considered in Intr oduction. According to [22], the
speed of light from the galaxy NGC 5668 with the parameters of redshifts zλ= 0.00580 and
zω= 0.00526, is equal c=c0+ 182.04 km / sec. In the light of the present work this result,
however, is not of statistical significance, as the redshift parameter zλ= 0.00580 <<1.41.
Therefore the conclusion that the NGC 5668 galaxy is superlu minal fail. It also holds for
the quasar PKS 2134 with parameter of redshift zλ= 1.935. After putting this value and
frequency shift zω= 1 into above - mentioned formula we can conclude that the spe ed of light
from the PKS 2134 quasar is c= 300 .000·2.936/2 = 440 .400∼440.000 km / sec. Thus, the
Loiseau estimation has theoretical character. This circum stance indicates once more that it
is necessary to obtain experimental data concerning the red shifts for superluminal quasars
in the radio-frequency and optical ranges [26].
Aberration of light [1, 5]. By analogy with SR we have for one - half of the aberrati on
angle: sinα=V/c;α∼(V/c0)(c0/c) = 10−4(c0/c)∼10−4(2/(1 +zλ)) = (2 /(1 +zλ))·20,5
seconds of arc. (The latter is true with large zλ). It follows from here that, for example,
c= 2,86c0, α= 7.2 seconds of arc for the Q 1158 + 4635 quasar with the redshift zλ= 4.73
[35]. The 7.2 seconds of arc value should be checked in the exp eriment [26].
Superluminal motion of nuclear reaction products. Such phenomenon is impossibl e
in SR. But it is possible in LSR, if the energy of a particle wil l be greater than√
2E0. It
is 150 MeV for µ- mesons. Therefore in LSR (if it is realized in reality) the a ppearance
of atmospheric µ- mesons near the surface of the Earth may be explained by supe rluminal
10motion of the mesons with the velocity of the order of 6 ·106/2,2·10−6∼3·1012cm/sec,
or 100 c0[26]. The energy Eµ=m0,µc0c∼100m0,µc02∼10.6 GeV corresponds to the given
velocity in LSR. In virtue of the absence of the limitation on the upper value of the speed
of light, faster particles explaining the results of Clay, C rouch [17] and Cooper experiments
[18], may be observed in front of the particles from nuclear r eactions.
Motion of a charged particle in electromagnetic field. By integrating (12), we find
that in the case a particle moves in constant homogeneous ele ctrical field, its velocity tends
to infinity vx(t) =c0/radicalBig
1 +vy2(0)/c02sh(eEt/m0c0)→ ∞ [26]. In SR the particle velocity is
limited by value coas is known [23]. For the case of constant homogeneous magnet ic field
H= (0,0,Hz) the frequency of rotation of a particle is constant and does not depend on the
energy of a particle ω=eHz/m0c0= const in LSR ( ω∼1/Ein SR [23]). However if the
particle energy is great, the radius of particle rotation is connected with the particle energy
by the ratio r∼E/eH zas in SR. The differences in the radiuses of rotation is observ ed in
the intermediate range of particle energies, when M0c02<E< m 0c0vatv >> c o. The
considered properties of particle motions in electrical an d magnetic fields may be essential
in the theory of linear and cyclical accelerators [26].
7 Conclusion
Summing we shall note that the validity of LSR or the proper fie ld of its application are
not clear yet now. In any case the problem arises which concer ns the reason of the choice of
preferable symmetry in the nature. Local SR transforms into SR, if c′=c.
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13 |
arXiv:physics/0008235v1 [physics.atm-clus] 29 Aug 2000Emergence of Bulk CsCl Structure in (CsCl) nCs+Cluster Ions
Andr´ es Aguado
Departamento de F´ ısica Te´ orica, Universidad de Valladol id, Valladolid 47011, Spain
The emergence of CsCl bulk structure in (CsCl) nCs+cluster ions is investigated using a mixed
quantum-mechanical/semiempirical theoretical approach . We find that rhombic dodecahedral frag-
ments (with bulk CsCl symmetry) are more stable than rock-sa lt fragments after the completion of
the fifth rhombic dodecahedral atomic shell. From this size ( n=184) on, a new set of magic numbers
should appear in the experimental mass spectra. We also prop ose another experimental test for this
transition, which explicitely involves the electronic str ucture of the cluster. Finally, we perform more
detailed calculations in the size range n=31–33, where rece nt experimental investigations have found
indications of the presence of rhombic dodecahedral (CsCl) 32Cs+isomers in the cluster beams.
I. INTRODUCTION
A general goal of cluster physics is to study the emer-
gence of bulk behavior right from the molecular limit,
by building clusters of increasing size and following the
size evolution of selected properties. From the theoret-
ical point of view, this ambitious plan has been largely
impeded because of the slow and nonmonotonic size evo-
lution observed in many properties. The predicted clus-
ter structures are not simply related to the correspond-
ing bulk structures in many cases, which precludes the
possibility of a meaningful extrapolation to the bulk
limit. Moreover, cluster structure is difficult to de-
termine theoretically due to the huge increase in the
number of isomers with size, and experimentally due to
the small number of scatterers compared with the bulk
case. Nevertheless, recent advances involving ion mobil-
ity measurements,1–4electron diffraction from trapped
cluster ions,5,6or the use of photoelectron spectra as a
fingerprint of structure7have been successful in elucidat-
ing the structures of several ionic and covalent clusters.
Abundance patterns obtained from the mass spec-
tra of binary ionic clusters like the alkali halides and
alkaline-earth oxides point towards a prompt establish-
ment of bulk rock-salt symmetry.8–11Theoretical calcu-
lations have shown, however, that small sodium iodide
and lithium halide clusters adopt ground state struc-
tures based on the stacking of hexagonal rings.12,13In
the case of alkaline-earth oxide clusters, the large and
coordination-dependent values of the oxide polarizabili-
ties favor the formation of structures with a large pro-
portion of ions in surface sites, inducing a delay in the
emergence of bulk structural properties.14–16Turning to
the alkali halides, bulk CsCl, CsBr, and CsI crystal-
lize in the CsCl-type structure, while both experimen-
tal mass spectra8,10and theoretical calculations17indi-
cate that small clusters of those materials adopt ground
state structures which are fragments of a rock-salt lat-
tice. This implies that there has to be a structural
phase transition as the cluster size is increased. Ion mo-
bility measurements performed by L¨ offler4suggest that
(CsCl) nCs+cluster ions with n=32 are specially com-
pact, which might be explained by the presence of isomers
with the shape of a perfect three-shell rhombic dodeca-hedron (that is with bulk CsCl symmetry) in the cluster
beam. The electron diffraction experiments performed
recently in the group of Parks6show that there is a sub-
stantial proportion of isomers with bulk CsCl symmetry
for the same size.
In this theoretical work we analyce the above men-
tioned size-induced phase transition in (CsCl) nCs+clus-
ter ions. We consider only those sizes that correspond
to geometrical shell closings for the CsCl-type (per-
fect rhombic dodecahedra with n=32,87,184,335,552)and
rock-salt (perfect cubes with n=13,62,171,364,665) struc -
tural series. In doing so, we try to avoid any nonmono-
tonic size evolution in the calculated properties. In the
upper part of Fig. 1 we display the relative number of
atoms with a given coordination as a function of N−1/3,
where N=2n+1 is the total number of atoms in the clus-
ter. In the lower part we show the number of atoms
with nonbulk coordination relative to the total number
of surface atoms. For the largest sizes considered the
proportion of bulklike atoms is dominant, and within the
surface the proportion of face-like atoms is already much
larger than those of edge and vertex-like atoms. From
those sizes to the bulk, the only meaningful size evo-
lution of these proportions will be a slow approach to
zero of the face-like atoms. We thus expect to capture
all the physical information relevant to the phase transi-
tion by studying this set of clusters and the correspond-
ing bulk phases, which have been studied both with the
same theoretical model. In this way inaccuracies related
to the use of different methodologies are avoided and a
meaningful extrapolation to the bulk limit can be done.18
In a second part of the work, we explicitely analyze the
structures adopted by (CsCl) nCs+cluster ions in the size
range n=31–33, in order to explain the experimental find-
ings of Refs. 4 and 6.
The rest of the paper is organized as follows: Section II
includes just a brief description of the theoretical model
employed in the calculations, as a full account of it has
been given in previous publications.12,15,16In Section III
we present and discuss the results of the calculations, and
Section IV summarizes the main conclusions.
1II. THEORETICAL MODEL
Cluster energies have been obtained by performing
Perturbed Ion (PI) plus polarization calculations. This
is a well tested method that describes accurately both
bulk19and cluster15,16limits. Its theoretical foundation
lies in the theory of electronic separability.20–22Very
briefly, the cluster wave function is broken into local
group functions (ionic in nature in our case) that are
optimised in a stepwise procedure. In each iteration, the
total energy is minimized with respect to variations of the
electron density localized in a given ion, with the electron
densities of the other ions kept frozen. In the subsequent
iterations each frozen ion assumes the role of nonfrozen
ion. When the self-consistent process finishes,12the out-
puts are the total cluster energy and a set of localized
wave functions, one for each geometrically nonequivalent
ion of the cluster. This procedure leads to a linear scal-
ing of the computational effort with cluster size, which
allows the investigation of large clusters with an explicit
inclusion of the electronic structure. The cluster binding
energy can be decomposed into ionic additive contribu-
tions
Ebind
clus=/summationdisplay
R∈clus(ER
add−ER
0), (1)
being ER
addthe contribution of the ion R to the total clus-
ter energy and ER
0the energy of the ion R in vacuo . In
this way the contribution of ions with different coordina-
tions to the binding energy can be separately analyzed,
which is particularly convenient for our study. Each ad-
ditive energy can be decomposed in turn as a sum of
deformation and interaction terms
Ebind
clus=/summationdisplay
R∈clus(ER
def+1
2ER
int), (2)
where ER
defis the self-energy of the ion R, measured
relative to the vacuum state, and ER
intcontains electro-
static, exchange and repulsive overlap energy terms.12,15
The polarization contribution to the cluster binding en-
ergy is not computed in the actual version of the PI
code, as it assumes (for computational simplicity) that
the electronic charge distribution of each ion in the clus-
ter is spherically symmetric. Thus, a polarization cor-
rection to the PI energy is computed semiempirically as
described in Refs. 15,16. Bulk polarizabilities are used
for both Cs+and Cl−ions.23This is a good approx-
imation for the Cs+cations. The main effect on the
anion polarizabibities when passing from the bulk to a
cluster environment is an increase of the polarizabilities
of those ions located on the cluster surface, due to the
lower average coordination compared to the bulk. How-
ever, we have checked that our main conclusions are not
affected by an increase in the surface chloride polariz-
abilities as large as 10–20 %, which are typical values for
halides.14The short-range induction damping parame-
ters have been obtained through the scaling procedure
validated in Ref. 24. The reliability for cluster calcu-
lations of the mixed quantum-mechanical/semiempirical
energy model thus obtained has been checked and shown
to be high in previous publications.15,16III. RESULTS
A. The rock-salt to CsCl-type structural transition
Fig. 2 shows the size evolution of the binding energy
per ion. First of all, we note that the PI model prop-
erly reproduces the stability trend in the bulk, predict-
ing the CsCl structure as the most stable one. This is
a tough problem for semiempirical methods, as Pyper25
has shown that a full account of the coordination number
dependence of the self-energy and overlap contributions
is necessary to obtain the correct ground state structure.
The values of the binding energy, plotted as a function
of N−1/3, lie neatly on a straight line. The regression
coefficients obtained from a fit are 0.9998 in all cases if
we exclude from the fitting the NaCl-type cluster with
n=13, which is the smallest one. We have calculated af-
terthe fitting procedure the energy of the 5 ×5×7 cuboid
(also included in Fig. 2), and checked that it lies on the
fitted NaCl-type energy curve. This shows that a consid-
eration of perfect cubes (or cuboids) on one hand, and
rhombic dodecahedra on the other hand removes the non-
monotonic behavior from the size evolution of the binding
energies. Our results predict that the rhombic dodeca-
hedra become definitely more stable after the completion
of the fifth shell of atoms, that is for n=184. The four-
shell rhombic dodecahedron and the 5 ×5×7 cuboid are
essentially degenerate, so both of them will contribute
to the enhanced abundance observed experimentally for
n=87.10We have not found any experimental mass spec-
trum for values of n as high as 184, but we predict that
a new set of magic numbers, corresponding to the clos-
ing of rhombic dodecahedral atomic shells, should emerge
from this size on. The magic numbers corresponding to
the closing of perfect cubic shells will probably not dis-
appear still at that specific size from the mass spectra,
because they do not coincide with the CsCl shell closings,
and complete cubes can remain more stable than incom-
plete rhombic dodecahedra until larger values of n are
reached. Polarization has little influence on these gen-
eral results, and only affects the energetic ordering of the
two essentially degenerate isomers mentioned above.
Now we turn to an analysis of the physical factors re-
sponsible for this transition. In Fig. 3 we show the
binding energy per ion, averaged over subsets of ions
with a fixed coordination. The contribution of bulklike
ions to the binding energy favors always the formation of
CsCl-type structures. However, the contribution of face-
like ions favors the formation of rock-salt fragments. As
soon as the proportion of bulk ions is larger than that
of surface atoms, which occurs after the completion of
the fifth rhombic dodecahedral atomic shell, fragments
of the CsCl-type lattice become more stable. The energy
contribution of those ions in edge positions is approxi-
mately the same for both structural families except for
the smallest clusters; finally, corner atoms favor the CsCl-
type structures, but their small relative number results
in a very small contribution to the total energy for those
sizes where the transition occurs.
Fig. 3 has reduced the structural phase transition in
(CsCl) nCs+cluster ions to an essentially bulk effect. By
2this we mean that CsCl-type structures become more sta-
ble as soon as the proportion of bulklike atoms is dom-
inant. To complete our discussion we have then to ad-
dress the stability question in the bulk. This is more
easily understood by analyzing the reasons why other al-
kali halides like NaCl or CsF do not crystallize in the
CsCl-type lattice. The largest contribution to the bind-
ing energy of an ionic crystal is the Madelung energy term
EM=AM
Re, with AMthe Madelung constant and Rethe
equilibrium interionic distance. The Madelung constant
of the CsCl-type lattice (1.762675) is larger than that of
the rock-salt lattice (1.747565), so were the value of R e
the same for both structures, the CsCl-type would always
be more stable. We have solved for the electronic struc-
ture of NaCl and CsF crystals in the CsCl-type struc-
ture at a nonequilibrium value of the interionic distance,
chosen in such a way that the Madelung energy term is
exactly the same as in the corresponding rock-salt lattice
at equilibrium. In the case of NaCl, E add(Na+) favors
the CsCl-type structure, but E add(Cl−) largely favors the
rock-salt phase. The main reason is the large anion-anion
overlap at that artificial distance, that is the Na+cation
is so small compared to the Cl−anion that eight an-
ions can not be packed efficiently around a cation. In
CsF the situation is reversed, and it is the cation-cation
overlap that is too large. This demonstrates that the sta-
bility situation in the bulk is a purely packing effect: in
CsCl, CsBr and CsI, the large value of the cation-anion
size ratio allows for an equilibrium interionic approach
in the CsCl-type structure close enough as to obtain a
Madelung energy term more negative than in the rock-
salt phase, without a large overlap interaction between
like ions. The same is true for the bulklike ions in the
clusters studied, and so when those ions begin to domi-
nate the energetics, the bulklike fragments become more
stable.
We have made a prediction above that can be tested
experimentally, namely the emergence of a new set of
magic numbers from n=184 on. Here we propose an-
other, perhaps more indirect, experimental test. In Fig.
4, the eigenvalues of the 3p orbitals of Cl−(with opposite
sign) are plotted as a function of N−1/3. We have a band
of eigenvalues for each size because the anions occupy
nonequivalent positions in the clusters. As the clusters
under study are formed by closed shell ions whose wave
functions are strongly localized, it can be assumed that
an electron is extracted from a specific localized orbital
when the cluster is ionized. This is the lowest bound
3p orbital, which corresponds always to a chloride anion
with a low coordination. Thus the dashed lines repre-
sent the size evolution of the vertical ionization potentia l
IP (in the Koopmans’ approximation) for both structural
families. For the rock-salt series, that size evolution is a p-
proximately linear in N−1/3, but for the CsCl-type series
it shows a more or less oscillating behavior, which should
be detected in experimental measurements of the vertical
IP if rhombic dodecahedra actually are the ground state
structures from a given size on. We can explain these dif-
ferent electronic behaviors in a very simple way: in the
rock-salt clusters the eight corner sites are always occu-
pied by Cs+cations. The weakest bound electron corre-sponds always to a Cl−anion with fourfold coordination,
namely anyone of those closer to the corner cation sites.
On the other hand, rhombic dodecahedra have fourteen
corner sites. When the number of atomic shells is even,
all these sites are occupied by Cs+cations, but when that
number is odd, eight of them are cationic sites and the
other six anionic sites. Thus the nonmonotonic behavior
of the vertical IP is due to the different local coordination
of the Cl−anion to which the weakest bound electron is
attached as the number of atomic shells increases.
B. Structures of (CsCl) nCs+(n=31–33) and
comparison to experiment
We finish our study with an explicit consideration of
(CsCl) nCs+clusters in the size range n=31–33, the range
covered in the experiments of L¨ offler4and Parks.6Specif-
ically, we have considered the most compact 7 ×3×3,
4×4×4 and 5 ×4×3 rock-salt structures, and the three-
shell rhombic dodecahedron, with some atoms added or
removed from different positions. The binding energies
are shown in Table I. The ground state (GS) structure of
(CsCl) 31Cs+is a complete 7 ×3×3 cuboid. The 4 ×4×4
fragment with an anion removed from a corner position is
slightly less stable, and the lowest energy rhombic dodec-
ahedral isomer we have obtained has a still lower stabil-
ity. For n=32, the complete three-shell rhombic dodeca-
hedron becomes the GS isomer. All the different incom-
plete rock-salt fragments have a smaller binding energy.
For n=33, the different rock-salt isomers are essentially
degenerate, but the CsCl-type structure is found again
at a higher energy. This sequence of GS structures for
(CsCl) nCs+clusters is consistent with the experimental
findings.4,6The relative mobility is a local maximum for
n=32, as the perfect three-shell rhombic dodecahedron
is evidently more compact than the complete 7 ×3×3
(CsCl) 31Cs+cuboid or any of the incomplete rock-salt
structures obtained for n=33. Also, the energetical or-
dering of the isomers is consistent with the large propor-
tion of CsCl-type isomers found for n=32 in the electron
diffraction experiments.
IV. SUMMARY
We have reported a computational study of the size-
induced rock-salt to CsCl-type structural phase tran-
sition in (CsCl) nCs+cluster ions. For this purpose,
the Perturbed Ion (PI) method, supplemented with a
semiempirical account of polarization effects, has been
employed. Only cluster ions with an atomic closed-shell
configuration have been considered in order to avoid non-
monotonic behavior in the calculated properties. More-
over, we have employed the same theoretical model to
study both cluster and bulk limits, which allows for a
meaningful extrapolation strategy. The main result is
that rhombic dodecahedral isomers become definetely
more stable than rock-salt structures after the comple-
tion of the fifth rhombic-dodecahedral atomic shell, that
is for a size n=184. Thus, it is predicted that a new set of
3magic numbers, reflecting the establishment of the new
structural symmetry, should emerge from that size on.
The size evolution of the vertical ionization potential of
the cluster ions should also be a good experimental fin-
gerprint of the transition. In order to explain the nature
of the transition, an analysis of the binding energy into
ionic components has been performed. The result is quite
simple: bulklike ions always prefer to have a CsCl-type
environment, even for the smallest cluster sizes (this has
been shown to be a purely packing effect), while surface-
like atoms prefer to adopt rock-salt structures. The tran-
sition occurs as soon as the proportion of bulklike atoms
is large enough to dominate the energetics of the whole
cluster. One of the possibilities advanced by Parks con-
sistent with his experimental results6is the existence of
isomers with mixed symmetry. Our results indicate that
the formation of isomers with a CsCl-type core and a
rock-salt-type surface could be energetically favored, if
the strain accumulated in the bonds at the interface re-
gion separating both phases can be conveniently relaxed.
This point deserves further investigation.
The structures adopted by (CsCl) nCs+cluster ions
have been more carefully studied in the size range n=31–
33, which has been covered in the experimental investi-
gations. Our results are consistent with the experimental
findings, and show that the three-shell rhombic dodeca-
hedron is the lowest energy isomer for n=32.
ACKNOWLEDGMENTS
This work has been supported by DGES (Grant PB98-
0368) and Junta de Castilla y Le´ on (VA70/99). The au-
thor is indebted to J. M. L´ opez for a careful reading of
the manuscript.Captions of Tables.
Table I Binding energy, in eV/ion, of the different
rock-salt and CsCl-type structures for the size range
n=31–33.
Captions of Figures.
Figure 1 Size evolution of the number of atoms with a
given coordination, relative to the total number of atoms
(upper half) or to the total number of surface atoms
(lower half). The left half refers to CsCl-type symme-
try and the right half to rock-salt symmetry.
Figure 2 Size evolution of the binding energy per ion
for both CsCl-type and rock-salt structural families, with
(lower half) and without (upper half) the inclusion of po-
larization corrections. The value of N−1/3at the transi-
tion point has been indicated with an arrow.
Figure 3 Size evolution of the binding energy con-
tributions from ions with different coordinations. Full
circles represent ions in the CsCl-type structures and
squares represent ions in the rock-salt structures.
Figure 4 Size evolution of the 3p orbital eigenvalues
of chloride anions. The dashed line represents the varia-
tion of the vertical ionization potential in the Koopmans’
approximation with size.
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5[
n=31 n=32 n=33
Structure Energy (eV/ion) Structure Energy (eV/ion) Structure Energy (eV/ion)
7×3×3 3.032 7×3×3+2 3.021 7×3×3+4 3.025
4×4×4-1 3.026 4×4×4+1 3.032 4×4×4+3 3.027
5×4×3+3 3.019 5×4×3+5 2.998 5×4×3+7 3.028
CsCl-type - 2 2.983 CsCl-type 3.048 CsCl-type+2 2.980
60.0 0.1 0.2 0.3 0.4
N−1/30.00.20.40.60.8Ncoord/Nsurf0.00.20.40.60.81.0Ncoord/NCoord. 8
Coord. 4
Coord. 5
Coord. 6
0.0 0.1 0.2 0.3 0.4
N−1/3Coord. 6
Coord. 3
Coord. 4
Coord. 5
70.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
N−1/32.93.03.13.23.3Binding Energy (eV/ion)2.93.03.13.23.33.4Binding Energy (eV/ion)CsCl structure
NaCl structureWithoutPolarization WithPolarization
80.05 0.10 0.15
N−1/32.953.003.05Binding Energy(eV/ion)3.303.323.343.363.38Binding Energy(eV/ion)
0.05 0.10 0.15 0.20
N−1/32.302.402.502.60
Binding Energy(eV/ion)3.053.103.153.20
Binding Energy(eV/ion)Bulk Face
Edge Corner
90.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
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arXiv:physics/0008236v1 [physics.flu-dyn] 29 Aug 2000Equilibrium solutions of the shallow water equations
Peter B. Weichman1and Dean M. Petrich2
1Blackhawk Geometrics, 301 Commercial Road, Suite B, Golden , CO 80401
2Condensed Matter Physics 114-36, California Institute of T echnology, Pasadena, CA 91125
(February 2, 2008)
A statistical method for calculating equilibrium solution s of the shallow water equations, a model of
essentially 2-d fluid flow with a free surface, is described. T he model contains a competing acoustic
turbulent direct energy cascade, and a 2-d turbulent inverse energy cascade. It is shown, nonetheless
that, just as in the corresponding theory of the inviscid Eul er equation, the infinite number of
conserved quantities constrain the flow sufficiently to produ ce nontrivial large-scale vortex structures
which are solutions to a set of explicitly derived coupled no nlinear partial differential equations.
The evolution of a fluid from a strongly random initial
condition is generally characterized by one or more tur-
bulent cascades of energy to larger and/or smaller scales.
Whether energy flows to smaller scales via a direct cas-
cade, or to larger scales via an inverse cascade, is deter-
mined by a combination of conservation laws and phase
space considerations. Generally, if only energy is con-
served [1] (as for 3-d Navier-Stokes turbulence [2]), its
flow in phase space will be globally unconstrained and
will spread out to arbitrarily high wavenumbers, even-
tually draining all energy out of any large-scale macro-
scopic flows initially present. Perhaps the most famil-
iar example of this is the thermodynamic equilibration
of a container of gas to a macroscopically featureless fi-
nal state in which all energy eventually ends up as heat,
i.e., microscopic molecular motion. If, however, one or
more additional conservation laws are present (as for 2-d
Navier-Stokes turbulence [2], or for deep water surface
gravity wave turbulence [3]) their multiple enforcement
will generally not permit bothconserved quantities to es-
cape to small scales, and macroscopic structure, whose
profile will be initial condition and boundary condition
dependent, will survive. An example is the equilibration
to arigidly rotating final state of a gas in a cylindrical
container with frictionless walls. The additional conser-
vation of angular momentum along the axis of the cylin-
der precludes a featureless final state.
A long-standing problem has been the characteriza-
tion of final states of systems with an infinite number of
conservation laws [4]. Different values of the conserved
quantities should then produce an infinite dimensional
space of final states [5]. The example of the equilibrat-
ing gas motivates one to postulate that the macroscopic
final state, be it featureless or not, should be thermody-
namic in character, i.e., it should be an equilibrium state
computable from the appropriate Hamiltonian using the
formalism of statistical mechanics. In [6] this approach
was used to produce a full characterization of the equi-
librium states of the 2-d incompressible Euler equation
(the inviscid limit of the 2-d Navier-Stokes equation) [7],
where the conserved quantities are the standard integralsof all powers of the vorticity. The equilibria were found
to be characterized by a macroscopic steady state vor-
ticity distribution ω0(r) obeying an explicit “mean field”
partial differential equation whose input parameters were
determined by the values of the conserved quantities.
The 2-d Euler equation is the simplest of these systems
in the sense that the incompressibility constraint ∇·v= 0
reduces the dynamics to that of the single scalar vorticity
fieldω=∇×v≡∂xvy−∂yvx, and the conservation laws
then provide an infinite sequence of global constraints on
its evolution. In this work we study a more complicated
system of equations, the shallow water equations, an ex-
tension of the 2-d Euler equation that includes a free
surface with height field h(r) coupled to gravity g. The
horizontal velocity vnow becomes compressible (with 3-
d incompressibility enforced via vz=−z∇ ·v) but is
assumed to be independent of the vertical coordinate z.
The effective 2-d dynamical equations are:
Dv
Dt≡∂tv+ (v· ∇)v=−g∇h (1)
∂th+∇ ·(hv) = 0, (2)
The first equation expresses the fact that the fluid ac-
celerates in response to gradients in the surface height,
and the second enforces mass conservation, i.e., the full
3-d incompressibility. The Euler equation is recovered
formally when g→ ∞ since height fluctuations are then
suppressed. It is straightforward to verify that the ratio
Ω≡ω/his convectively conserved, DΩ/Dt≡0, imply-
ing conservation of all integrals of the form
Cf=/integraldisplay
d2rh(r)f[Ω(r)] (3)
for any function f(s). These may be fully characterized
by the function g(σ),−∞< σ < ∞, obtained from (3)
withf(s) =δ(σ−s), andg(σ)dσtherefore represents
the 3-d volume on which σ≤Ω≤σ+dσ. For general
fone then recovers Cf=/integraltext
dσf(σ)g(σ). Note that if
ω≡0 initially, then it must remain zero for all time.
Initial conditions of this type then generate (nonlinear,
in general) wave motions only[9].
1The extension of the equilibrium theory to the shallow
water equations is a significant advance because in addi-
tion to the usual vortical motions they contain acoustic
wave motions [8]. The latter are known [10] to have a
direct cascade of wave energy to small scales. One then
has the very interesting situation in which there are two
competing energy cascades, and the question arises as to
which one “wins.” In particular, is it possible that the
macroscopic vortex structures can “radiate” wave energy
and disappear entirely? We will show that under reason-
able physical assumptions a finite fraction of the energy
remains in large scale vortex structures, and we will de-
rive exact mean field equations for the equilibrium struc-
ture.
The statistical formalism proceeds in a sequence of well
defined steps. First, the Hamiltonian corresponding to
(1) and (2) is
H=1
2/integraldisplay
d2r(hv2+gh2), (4)
though the Poisson bracket yielding (1) and (2) from (4)
is noncanonical [4]. Second, the partition function is de-
fined as an integral over the phase space of fields h,vwith
an appropriate statistical measure. This so-called invari -
ant measure is most easily computed if the dynamics can
be expressed in terms of a set of variables, canonical vari-
ables being an example, for which a Liouville theorem is
satisfied. In this case invariant measures are any function
of the conserved integrals, with different choices corre-
sponding to different ensembles. In the Euler case [6] the
fieldωitself satisfies a Liouville theorem. In the shal-
low water case no obvious combination of h,vor their
derivatives meet this requirement.
To circumvent this problem we transform to a La-
grangian description, in terms of interacting infinitessi-
mal parcels of fluid of equal 3-d volume, for which canon-
ical variables are easy to construct. Thus, let abe a 2-d
labeling of the system, and let r(a,t) be the position of
the parcel of fluid such that, e.g., r(a,0) =a. Since
all parcels have equal mass, the conjugate momentum is
p(a,t) =˙r(a,t) =v(r(a,t),t). The height field is simply
the Jacobian of the transformation between randa:
h0/h(r(a)) = det(∂r/∂a) =∂a1r2−∂a2r1, (5)
whereh0is the overall mean height. The Hamiltonian
(4) now takes the form
H=h0
2/integraldisplay
d2a[p(a)2+gh(a)], (6)
while,
ω(a)≡ ∇ × v=h(∂a2r1∂a1p1−∂a1r1∂a2p1
+∂a2r2∂a1p2−∂a1r2∂a2p2)
q(a)≡ ∇ ·v=h(∂a2r2∂a1p1−∂a1r2∂a2p1
+∂a2r1∂a1p2−∂a1r1∂a2p2). (7)It is easily verified that the Lagrangian forms of (1)
and (2) follow from the Hamiltonian equations of motion
˙r(a) =δH/δp(a) and ˙p(a) =−δH/δr(a). The Liou-
ville theorem, which is a statement of incompressibility
of flows in phase space,
/summationdisplay
α/integraldisplay
d2a[δ˙rα(a)/δrα(a) +δ˙pα(a)/δpα(a)] (8)
=/integraldisplay
d2a[δ2H/δrα(a)δpα(a)−δ2H/δpα(a)δrα(a)] = 0,
then follows immediately and implies that the correct
statistical measure is ρ(H,{g(σ)})/producttext
ad2r(a)d2p(a). In
thegrand canonical ensemble , which we shall adopt, the
functionρis given by ρ=e−βK, whereβ= 1/Tis a
hydrodynamic “temperature” and
K=H −/integraldisplay
dσµ(σ)g(σ)
=H −/integraldisplay
d2rh(r)µ[ω(r)/h(r)]
=H −h0/integraldisplay
d2aµ[ω(a)/h(a)] (9)
in whichµ(σ) is a chemical potential that couples to each
levelω(r)/h(r) =σ. The partition function is now de-
fined by
Z[β,{µ(σ)}] =1
N!/productdisplay
a/integraldisplay
d2r(a)/integraldisplay
d2p(a)e−βK,(10)
whereN→ ∞ is the number of fluid parcels and N! is
the usual classical delabeling factor. The thermodynamic
averages of the conserved quantities are now obtained
in the usual fashion as derivatives with respect to the
chemical potentials, /an}b∇acketle{tg(σ)/an}b∇acket∇i}ht=Tδln(Z)/δµ(σ).
One would now like to transform the integration in
(10) back to physical Eulerian variables. The key ob-
servation is that, from (7), Ω ≡ω/handQ≡q/hare
linear inp. Therefore, one may formally invert this re-
lationship to obtain/producttext
ad2p(a) =/producttext
adQ(a)dΩ(a)J[h],
where, due to the particle relabeling symmetry (both
∇ ·vand∇ ×vdepend only on rand are then clearly
invariant under any permutation of the labels a), the Ja-
cobianJis a functional of the height field h(a)alone.
The exact form of Jwill turn out to be unimportant.
Similarly, (1 /N!)/producttext
a/integraltext
d2r(a) =/producttext
a/integraltext
dh(a)I[h], where
I[h] is another Jacobian. The 1 /N! factor precisely re-
moves the relabeling symmetry that, in particular, leaves
the height field invariant. Finally, we replace the la-
belaby the actual position r, in which the equal vol-
ume restriction on each fluid parcel implies that the in-
finitesimal area of each parcel must be determined by
dV=h(r)d2r=constant . Thus:
1
N!/productdisplay
a/integraldisplay
d2r(a)/integraldisplay
d2p(a)
=/productdisplay
r/integraldisplay
dh(r)J[h]/integraldisplay
dΩ(r)/integraldisplay
dQ(r),(11)
2in which J[h] =I[h]J[h], and the mesh over which the
labelrruns is nonuniform andchanges with each real-
ization of the height field h.
The statistical operator Kmust also be expressed in
terms ofQ,Ω,h. Only for the kinetic energy T=/integraltext
d2rhv2does this require some nontrivial manipula-
tions. Let the current j≡hvbe decomposed in the
formj=∇ ×ψ− ∇φ. One obtains then
/parenleftbigg
hΩ
hQ/parenrightbigg
=/parenleftbigg
∇ ×1
h∇× −∇ ×1
h∇
∇ ·1
h∇× −∇ ·1
h∇/parenrightbigg /parenleftbigg
ψ
φ/parenrightbigg
.(12)
The 2 ×2 matrix operator, which we shall denote Lh,
appearing on the right hand side of (12) is self adjoint
and positive definite, and therefore possesses an inverse,
i.e., a 2 ×2 matrix Green function Gh(r,r′) satisfying
LhGh(r,r′) =11δ(r−r′). An explicit form for Ghwill not
be needed. The kinetic energy is then T=1
2/integraltextd2rj·v=/integraltext
d2rh(ψΩ +φQ), i.e.,
T=/integraldisplay
d2rh(r)/integraldisplay
d2r′h(r′)/parenleftbigg
Ω(r)
Q(r)/parenrightbigg
Gh(r,r′)/parenleftbigg
Ω(r′)
Q(r′)/parenrightbigg
(13)
and the complete statistical operator is
K=T+/integraldisplay
d2rh(r)/braceleftbigg1
2gh(r)−µ[Ω(r)]/bracerightbigg
(14)
The appearance of the factors h(r) andh(r′) is crucial
here because, as discussed above, dV=h(r)d2rand
dV′=h(r′)d2r′are both uniform for each given statisti-
cal mesh.
We finally come to the evaluation of the partition func-
tion itself. This is accomplished with the use of the
Kac-Hubbard-Stratanovich (KHS) transformation, which
in discrete form reads for any positive definite matrix A,
e1
2/summationtext
i,jyiAijyj=1
N/productdisplay
i/integraldisplay∞
−∞dζie−1
2/summationtext
i,jζiA−1
ijζj−/summationtext
iζi·yi,
(15)
whereyiandζimay be vectors, and the normalization
isN=/radicalbig
det(2πA). This identity follows by completing
the square on the right hand side and performing the re-
maining Gaussian integral. We apply it to the discretized
version of (10) and (11) with finite dV, and the identifica-
tionsAij=−β−1G(ri,ri) [11],yi=βdV[Ω(xi),Φi(xi)]
and we introduce the notation ζi= [Ψ(xi),Φ(xi)]. The
continuum limit dV→0 will be taken at the end. The
partition function is now
Z=/productdisplay
i/integraldisplay
dhiJ[h]
N[h]/integraldisplay
dΨidΦi/integraldisplay
dQidΩieβ˜F,(16)
where˜F=dV/summationdisplay
i,j/parenleftbigg
Ψi
Φi/parenrightbigg
[Lh]ij/parenleftbigg
Ψj
Φj/parenrightbigg
−dV/summationdisplay
i[ΩiΨi+QiΦi−µ(Ωi)], (17)
in which [ Lh]ijis an appropriate discretization of the
differential operator Lh. Notice that the inverse of Ghhas
led to the reappearance of the local differential operator
Lh.
At the expense, then of introducing the new fields Ψ,
Φ we have succeeded in producing a purely localaction
in which the integration over Ω i,Qican be performed in-
dependently for each i(for given fixed field h). However,
we now arrive at a problem whose physical origin, as we
shall see, lies precisely in the direct cascade of wave en-
ergy. Thus, the chemical potential function µ(σ) controls
convergence of the Laplace transform-type integral
e¯βW[Ψi]≡/integraldisplay∞
−∞dΩie−¯β[ΩiΨi−µ(Ωi)], (18)
where ¯β≡βdVcorresponds to a rescaled hydrodynamic
temperature ¯T=TdVwhich is assumed to remain finite
asdV→0—the object of this choice is to obtain the cor-
rect control parameter for nontrivial hydrodynamic equi-
libria in the continuum limit that, as we shall see, yields
a nontrivial balance between energy and entropy contri-
butions to the final free energy [6]. However, there is
no corresponding chemical potential controlling Qiand
the corresponding integral does not converge. Recall-
ing thatQ= (1/h)∇ ·v, unboundedness of Qreflects
unboundedness of small-scale gradients in the compres-
sional part of vand inh[12]. Thus, taken literally, the
direct cascade of wave energy leads to arbitrarily small
scale fluctuations of the fluid surface that remain of fixed
amplitude, i.e., a kind of foam of fixed thickness. Physi-
cally, of course, such small scale motions are rapidly dissi -
pated by processes that violate the approximations used
to derive the shallow water equations, e.g., by some com-
bination of viscosity and wave breaking [13]. This leads
to the following physically motivated assumption: dissi-
pative processes that suppress wave motions lead to the
interpretation/integraltext
dQiexp(¯βQiΦi)→δ(¯βΦi), i.e., to the
vanishing of Φ i.
With Φ i≡0, only the (1 ,1) component of Lhcon-
tributes, and in the continuum limit dV→0 the parti-
tion function becomes
Z=/productdisplay
r/integraldisplay
dh(r)J[h]
L[h]/integraldisplay
dΨ(r)e−βF[h,Ψ], (19)
where the Free energy functional is
F=−/integraldisplay
d2r/bracketleftbigg(∇Ψ)2
2h−1
2gh2+hW[Ψ]/bracketrightbigg
. (20)
The key observation now is that β=¯β/dV → ∞ in the
continuum limit. Thus, mean field theory becomes ex-
actand equilibrium solutions are given by extrema ofF.
3This is why the integration over the field h(r), with its
unknown Jacobian, is ultimately irrelevant. The under-
lying assumption is only that the Jacobian is smooth, or
at least less singular than e−βF, in the neighborhood of
the extremum in the continuum limit.
The extremum conditions δF/δΨ(x) = 0 =δF/δh(x)
yield then the mean field equations
∇ ·/bracketleftbigg1
h(r)∇Ψ/bracketrightbigg
=h(r)W′[Ψ(r)] (21)
[∇Ψ(r)]2
2h(r)2=W[Ψ(r)]−gh(r), (22)
By adding a source term/integraltext
d2rh(r)τ(r)Ω(r) toK, which
serves only to replace Ψ by Ψ −τinsideW, one may com-
pute the equilibrium average /an}b∇acketle{tΩ(r)/an}b∇acket∇i}ht= [δF/δτ(r)]τ≡0=
−h−1∇ ·(h−1∇Ψ). It follows then that /an}b∇acketle{tj/an}b∇acket∇i}ht=∇ ×Ψ, so
that Ψ is the stream function associated with the equi-
librium current. Equation (21) is in fact equivalent to
Ω =−W′(Ψ), which guarantees that this is a true equi-
librium solution satisfying ˙Ω = 0, and equation (22) is
equivalent to Bernoulli’s theorem since it can be rewrit-
ten as (1/2)v2+gh=W(Ψ).
As a simple example, in the case where Ω = σ0
over half the 3-d volume of the fluid and Ω = 0 on
the other half, the chemical potential takes the form
e¯βµ(σ)=e¯βµ0δ(σ) +e¯βµ1δ(σ−σ0), and therefore by (18)
e¯βW(s)=e¯βµ0+e¯β(µ1−σ0s). Extensive numerical solu-
tions for the Euler equilibria exist for this “two-level”
system as a function of βandµ1−µ0. [6,7]. In pre-
liminary numerical work, we find that the shallow water
equilibria generated by (22) have very similar structure
(with, for example, vorticity moving from the walls to-
ward the center of the system as βdecreases from positive
to negative values), while the height field basically co-
varies with the vorticity in order to maintain hydrostatic
balance. Details of this work will be presented elsewhere.
The techniques presented in this paper can be used to
generate equilibrium equations for a number of other sys-
tems with an infinite number of conserved integrals [4].
The key insight presented here is that whenever such a
system contains simultaneous direct and inverse energy
cascades, the long time dynamics becomes very singular
and additional physically motivated assumptions must
be made in order to derive sensible equilibria. Our as-
sumption, that dissipation acts to suppress the forward
cascading degrees of freedom with negligible effect on the
macroscopic state, presumably depends on the smooth-
ness of the initial condition. Comparisons with detailed
numerical simulations will be required to evaluate such
effects.
Note added: After completion of this work we be-
came aware of an e-print [14] where equations equivalent
to (22) are derived from a phenomenological maximum
entropy theory. No statistical mechanical derivation is
given, nor is the interaction between wave and vortical
motions and the effects of waves on equilibration dis-
cussed.[1] Real fluids are always viscous, but models of turbulence
generally concern themselves with the “inertial range”
where loss of energy due to viscous damping is small com-
pared to that due to the cascade process, and an energy
conserving model is appropriate.
[2] See, e.g., A. S. Monin, and A. M. Yaglom, Statistical
Fluid Mechanics , Vol. 1, (MIT Press, Cambridge, 1971).
[3] See, e.g., G. Falkovich, V. L’vov, and V. E. Zakharov,
Weak turbulence theory of waves .
[4] For several examples of such systems see, e.g., D. D.
Holm, J. E. Marsden, T. Ratiu and A. Weistein, Phys.
Rep.123, 1 (1985).
[5] Study of these states is partly motivated by the con-
straints they place on simulations of turbulent flow, e.g.,
the degree to which they properly preserve the conserva-
tion laws.
[6] J. Miller, P. B. Weichman, and M. C. Cross, Phys. Rev.
A45, 2328 (1992).
[7] The statistical approach relies on the assumption of er-
godicity of the dynamics. This assumption has been ex-
plored numerically with mixed results: depending upon
the initial condition, the dynamics may get stuck in
metastable equilibria. See, e.g., Pei-Long Chen Ph. D
Thesis, Caltech (1996); D. Z. Jinn and D. H. E. Dubin,
Phys. Rev. Lett. 80, 4434 (1998).
[8] The linearized versions of these equations, ∂tv=−g∇η,
where η=h−h0withh0the mean surface height, and
∂tη+h0∇ ·v= 0, have longitudinal traveling wave solu-
tions η=η0ei(k·r−c|k|t),v=η0(c/h0)ˆkei(k·r−c|k|t), with
speed c=√gh0. These waves become coupled through
the nonlinear terms as the amplitude η0increases.
[9] If ω≡0 then v=−∇φleads to the pair of scalar equa-
tions ∂tφ=g(h−h0) +1
2|∇φ|2,∂th=∇ ·(h∇φ).
[10] See, e.g., A. Balk, Phys. Lett. A 187, 302 (1994).
[11] Note that β <0 is required for positive definiteness of A,
i.e., hydrodynamic equilibria often correspond to negative
temperature states. This is explained in detail in [6]. If
β >0 one uses the the KHS transformation with ζi→iζi.
[12] Divergences in these gradients may in fact occur in finit e
time since the shallow water equations are believed to
produce shock wave solutions. Appropriate continuation
of the equations nevertheless allows the conservation of
ω/hto be maintained even the presence of shocks (D. D.
Holm, private communication).
[13] Dissipation processes also act on the vortex structure s,
but the conservation laws guarantee that microscopic
fluctuations in Ω, unlike Q, remain finite and hence will
be dissipated much less strongly.
[14] P. H. Chavanis and J. Sommeria,
http://xxx.lanl.gov/physics/0004056.
4 |
arXiv:physics/0008237v1 [physics.acc-ph] 30 Aug 2000HIGHERDIPOLEBANDS INTHE NLCACCELERATING STRUCTURE∗
C. Adolphsen,K.L.F. Bane, V.A.Dolgashev,K. Ko,Z.Li, R. Mi ller,SLAC, Stanford, CA 94309,USA
Abstract
We show that scattering matrix calculations for dipole
modes between 23-43 GHz for the 206 cell detuned struc-
ture(DS)areconsistentwithfiniteelementcalculationsan d
results of the uncoupledmodel. In particular, the rms sum
wakeforthesebandsiscomparabletothatofthefirstdipole
band. We also show that for RDDS1 uncoupledwakefield
calculations for higher bands are consistent with measure-
ments. In particular, a clear 26 GHz signal in the short
rangewakeisfoundin bothresults.
1 INTRODUCTION
IntheNextLinearCollider(NLC)[1],longtrainsofintense
bunches are accelerated through the linacs on their way to
the collision point. One serious problem that needs to be
addressed is multi–bunch, beam break–up (BBU) caused
bywakefieldsinthelinacacceleratingstructures. Tocoun-
teract thisinstability the structuresaredesignedso that the
dipolemodesaredetunedandweaklydamped. Mostofthe
effortinreducingwakefields,however,hasbeenfocusedon
modes in the first dipole passband, which overwhelmingly
dominate. However,witharequiredreductionofabouttwo
ordersofmagnitude,onewonderswhetherthehigherband
wakesaresufficientlysmall.
Formulti-cellacceleratingstructureshigherbanddipole
modescanbeobtainedbyseveraldifferentmethods. These
include the so-called “uncoupled” model, which does not
accurately treat the cell-to-cell coupling of the modes[2] ,
an open–mode, field expansion method[3], and a finite
element method employing many parallel processors[4].
(Note that the circuit approaches[5][6] do not lend them-
selves well to the study of higher bands.) A scattering
matrix (S-matrix) approach can naturally be applied to
cavities composed of a series of waveguide sections[7],
such as a detuned structure (DS), and such a method has
been used before to obtain first band modes in detuned
structures[8][9]. Such a method can also be applied to the
studyofhigherbandmodes.
In this report we use an S-matrix computer
program[10][11] to obtain modes of the 3rd to the
8th passbands—ranging from 23-43 GHz—of a full 206–
cell NLC DS accelerating structure. We then compare our
results with those of a finite element calculation and those
of the uncoupled model. Next we repeat the uncoupled
calculation for the latest version of the NLC structure, the
∗WorksupportedbytheUSDepartmentofEnergycontractDE-AC 03-
76SF00515.rounded,detunedstructure(RDDS1). Finally,wecompare
theseresultswiththoseoftheDSstructureandwithrecent
wakefieldmeasurementsperformedatASSET[12].
2 S-MATRIX WAKECALCULATION
Let us consider an earlier version of the NLC accelerating
structure,theDSstructure. Itisacylindrically–symmetr ic,
disk–loaded structure operating at X-band, at fundamental
frequency f0= 11.424GHz. The structure consists of
206 cells, with the frequenciesin the first dipole passband
detuned according to a Gaussian distribution. Dimensions
of representative cells are given in Table 1, where ais the
iris radius, bthe cavity radius, and gthe cavity gap. Note
that the structure operates at 2π/3phase advance, and the
period p= 8.75mm.
Table1: Cell dimensionsintheDS structure.
cell#a [cm] b[cm] g [cm]
1.59001.1486 .749
51.52141.1070 .709
103.49241.0927 .689
154.46601.0814 .670
206.41391.0625 .629
For our S-matrix calculation we follow the approach
of Ref. [11]: A structure with Mcells is modeled by a
set of 2Mjoined waveguide sections of radii amorbm,
each filled with a number of dipole TE and TM waveg-
uide modes. First the S-matrix for the individual sections
is obtained, and then, by cascading, the S-matrix for the
composite structure is found. Using this matrix, the real
part of the transverse impedance R⊥at discrete frequency
points is obtained. We simulate a structure closed at both
ends,andonewithnowalllosses. Forsuchastructure R⊥
consists of a series of infinitesimally narrowspikes. To fa-
cilitate calculation we artificially widen them by introduc -
ingasmallimaginaryfrequencyshift,onesmallcompared
to the minimum spacing of the modes. To facilitate com-
parison with the results of other calculation methods, we
fitR⊥(ω)toasumofLorentziandistributions,fromwhich
we extract the mode frequencies fnand kick factors kn.
Knowingthesethe wakefieldisgivenby
W⊥(s) = 2/summationdisplay
nknsin(2πfns/c)e−πfns/Q nc,(1)withsthedistancebetweendrivingandtest particles, cthe
speedoflight,and Qnthequalityfactorofmode n.
For our DS S-matrix calculation we approximate the
roundedirises by squared ones. We use 15 TE and 15 TM
waveguide modes for each structure cavity region, and 8
TE and 8 TM modes for each iris region. Our imaginary
frequency shift is 1.5 MHz. Our resulting kick factors,
forfrequenciesinthe3rd–8thpassbands(23–43GHz),are
showninFig.1. (Notethattheeffectofthe2ndbandmodes
is small and can be neglected.) In Fig. 1 we show also, for
comparison,theresultsofafiniteelementcalculationofth e
entire DS structure[4], anearlier calculationthat,howev er,
doesincludetheroundingoftheirises.
24 26 28 30 32 34 36 38 40 42
Frequency□[GHz]1.0E-61.0E-51.0E-41.0E-31.0E-21.0E-11.0E+0k□[V/pC/m/mm]S□-matrix
finite□elements
Figure 1: Results for the DS structure as obtained by the
S-matrix and the finite element approaches. Note that the
dimensionsin thetwocasesdifferslightly.
We note from Fig. 1 that the agreement in the results
of the two methods is quite good, taking into account the
difference in geometries. We see that the strongest modes
are ones in the 3rd band (24-27 GHz), the 6th band (35-
37 GHz), and the 7th band (38-40GHz), with peak values
ofk=.04, .08, and .08 V/pC/mm/m, respectively (which
shouldbecomparedto.4V/pC/mm/mfor1stbandmodes).
However, thanks to the variation in a(for the 7th band)
andg(for the 3rd and 6th bands), these bands are seen to
be significantly detuned, or spread in frequency. Another
comparisonis to take 2/summationtextknforthebands,a quantitythat
is related to the strength of the wakefield for s∼0, be-
forecouplingordetuninghaveanyeffect. ForourS-matrix
calculation for bands 3-8 this sum equals 19, for the finite
elementresults21V/pC/mm/m(forthefirsttwobandsitis
74V/pC/mm/m).
It is also necessary to know the mode Q’s to know the
strengthof the wakefield at bunchpositions. A pessimistic
estimate takes the natural Q’s due to Ohmic losses in the
wallsfortheclosedstructure. Assumingcopperwallsthese
Q’s are veryhighfor some of these higherbandmodes( >
10000). Intherealstructure,however,the Q’scanbemuch
less, depending on the coupling of the modes to the beam
tubes and the fundamental mode couplers, effects that in
principle can be included in the S-matrix calculation. In
practice,however,these calculationsareverydifficult.3 THEUNCOUPLED MODEL
Theuncoupledmodelisarelativelysimplewayofestimat-
ing the impedance and the wake. It can be applied easily
to higher band modes (unlike the circuit models) and to
structures that are not composed of a series of waveguide
sections (unlike the S-matrix approach). However, since
it does not accurately treat the cell-to-cell coupling of th e
modes, it does not give the correct long time behavior of
the wakefield.
The wakefield, according to the uncoupled model, is
givenbyanequationlikeEq.1,exceptthatthesumisover
the number of cells Mtimes the number of bands P, and
the modefrequenciesandkickfactorsarereplacedby ˜fpm
and˜kpm, which represent the synchronous mode frequen-
cies and kick factors, for band p, of the periodic structure
with dimensions of cell mof the real structure. For our
uncoupled calculation we obtain the ˜fpmand˜kpmfor a
few representative cells of the structure using an electro-
magnetic field solving program, such as MAFIA[13], and
obtainthemfortherest byinterpolation.
In Fig. 2 we plot again the kick factors obtained by the
S-matrixapproachfortheDSstructure(rectangularirises ),
but now compared to the results of the uncoupled model
appliedtothe samestructure. Theagreementisbetterthan
in Fig. 1. We expect the kick factors for the two methods
to be somewhat different, due to the cell-to-cell coupling,
but the runningsum of kick factors,which is related to the
short-time wake, should be nearly the same. The running
sum,beginningat20GHz,ofthetwocalculationsisplotted
in Fig.3. We notethatagreement,indeed,is verygood.
24 26 28 30 32 34 36 38 40 42
Frequency□[GHz]1.0E-61.0E-51.0E-41.0E-31.0E-21.0E-11.0E+0k□[V/pC/m/mm]S□-□matrix
uncoupled□model
Figure 2: Kick factor comparison for the DS structure
(squareirises).
In Fig. 4 we plot the amplitude of the dipole wakes, for
the frequency range 23-43 GHz only, of the DS structure
(with squared irises), as obtained by the two approaches.
(Here Qhas been set to 6500, appropriate for copper wall
losses forthe 15 GHz passband). Note that horizontalaxis
of the graph is√sin order to emphasize the wake over
the shorter distances. Far right in the plot is equivalent to
s= 80m, the NLC bunch train length. We note that the
initial drop-off and the long-range wake are very similar,
though there is some difference in the region of 1-10 m.20 22 24 26 28 30 32 34 36 38 40 42
Frequency□[GHz]02468101214161820222□*□sum(□k□)□[V/pC/m/mm]S□matrix,□DS
uncoupled□model,□DS
uncoupled□model,□RDDS1
Figure3: Runningsumofkickfactorcomparison.
Theamplitudeattheorigin,20V/pC/mm/m,issmallcom-
pared to 78 V/pC/mm/m for the first dipole band, but the
longer time typical amplitude of ∼1V/pC/mm/m is com-
parable to that of the first band. The rms of the sum wake,
Srms, an indicator of the strength of the wake force at the
bunch positions, for the higher bands is .5 V/pC/mm/m,
which is comparable to that of the first dipole band. De-
pendingontheexternal Qforthestructure,however, Srms
forthehigherbandsmayin realitybe muchsmaller.
0 1 2 3 4 5 6 7 8 9
sqrt(□z[m]□)□1.0E-21.0E-11.0E+01.0E+11.0E+2Wt□[V/pC/m/mm]S□-□matrix
uncoupled□model
Figure4: Comparisonofwakefields(23-43GHzfrequency
rangeonly)fortheDS structure(squareirises).
ThelatestversionoftheNLCstructureisRDDS1which
hasroundedirisesaswellasroundedcavities. Assuchitis
difficulttocalculateusingtheS-matrixapproach. We have
notyetdoneaparallelprocessor,finiteelementcalculatio n,
but we have done an uncoupled one. The sum of the kick
factorsoftheresultisgivenalsoinFig.3above. Although
the running sums at 42 GHz for DS and RDDS1 are very
similar, at lower frequenciesthe curvesare quite differen t.
Inparticular,the3rdbandmodes( ∼26GHz)appeartobe
less detuned for RDDS1, the 4th and 5th band modes (27-
31 GHz) are stronger, though still detuned, and between
32-40GHzthereisverylittle impedance.
4 ASSET MEASUREMENTS
Measurements of the wakefields in RDDS1 were per-
formed at ASSET[12]. In Figs. 5,6 we present results for
thevicinityof.7and1.4nsecbehindthedrivingbunch. Tostudythehigherbandwakeswe haveremovedthe 15GHz
componentfromthedatainthe plots. Theremainingwake
was fit to the function Asin(2πF+ Φ)withA,F, andΦ
fittingparameters. Thisfit,alongwiththe3rdbandcompo-
nent of the uncoupled model results ( ∼26GHz), are also
given in the figures. At .7 nsec this component is clearly
seeninthedata,andtheamplitudeandphaseareinreason-
ableagreementwiththecalculation. At1.4nsthereismore
noise,thoughthe26GHzcomponentcanstill beseen.
0.60 0.65 0.70 0.75 0.80
time□[nsec]-10-8-6-4-20246810Wt□[V/pC/m/mm]ASSET□measurements,□15□GHz□subtracted
uncoupled□model,□RDDS1
fit□3.1□sin(2□Pi□25.9□t□)
Figure 5: The measured wake function for RDDS1, with
the 15GHz componentremoved.
1.30 1.35 1.40 1.45 1.50
time□[nsec]-2.0-1.5-1.0-0.50.00.51.01.52.0Wt□[V/pC/m/mm]ASSET□measurements,□15□GHz□subtracted
uncoupled□model,□RDDS1
fit□0.39□sin(2□Pi□25.9□t□□+□2□)
Figure 6: The measured wake function for RDDS1, with
the 15GHz componentremoved.
5 REFERENCES
[1] NLC ZDRDesign Report, SLAC Report 474, 589 (1996).
[2] K. Bane, et al, EPAC94,London, 1994, p. 1114.
[3] M. Yamamoto, et al, LINAC94,Tsukuba, Japan, 1994, p.299.
[4] X. Zhan,K.Ko,CAP96, Williamsburg, VA,1996, p.389.
[5] K. Bane and R. Gluckstern, Part.Accel. ,42,123 (1994).
[6] R. M.Jones, etal, Proc. of EPAC96, Sitges, Spain, 1996, p.1292.
[7] J.N. Nelson, et al,IEEE Trans. Microwave Theor. Tech. ,37, No.8,
1165 (1989).
[8] U. van Rienen, Part.Accel. ,41, 173 (1993).
[9] S. Heifets and S. Kheifets, IEEE Trans. Microwave Theor. Tech. ,
42, 108 (1994).
[10] V.A. Dolgashev, “Calculation of Impedance for Multipl e Waveg-
uide Junction,” presented atICAP’98, Monterey, CA, 1998.
[11] V. Dolgashev etal, PAC99, New York, NY,1999, p. 2822.
[12] C. Adolphsen et al.,PAC99, New York,NY, 1999, p.3477.
[13] TheCST/MAFIA Users Manual. |
arXiv:physics/0008238v1 [physics.bio-ph] 30 Aug 2000Digitality Induced Transition in a Small Autocatalytic Syst em
Yuichi Togashi and Kunihiko Kaneko
Department of Basic Science, School of Arts and Sciences, Un iversity of Tokyo,
Komaba, Meguro-ku, Tokyo 153-8902, Japan
September 29, 2013
Abstract
Autocatalytic reaction system with a small number of molecu les is studied numerically by stochas-
tic particle simulations. A novel state due to fluctuation an d discreteness in molecular numbers is
found, characterized as extinction of molecule species alt ernately in the autocatalytic reaction loop.
Phase transition to this state with the change of the system s ize and flow is studied, while a single-
molecule switch of the molecule distributions is reported. Relevance of the results to intracellular
processes are briefly discussed.
Cellular activities are supported by biochemical reaction s in a cell. To study biochemical dynamic
processes, rate equation for chemical reactions are often a dopted for the change of chemical concentra-
tions. However, the number of molecules in a cell is often rat her small [1], and it is not trivial if the rate
equation approach based on the continuum limit is always jus tified. For example, in cell transduction
even a single molecule can switch the biochemical state of a c ell [2]. In our visual system, a single photon
in retina is amplified to a macroscopic level [3].
Of course, fluctuations due to a finite number of molecules is d iscussed by stochastic differential
equation (SDE) adding a noise term to the rate equation for th e concentration [4, 5]. This noise term
sometimes introduces a non-trivial effect, as discussed as n oise-induced phase transition [6], noise-induced
order [7], stochastic resonance [8], and so forth. Still, th ese studies assume that the average dynamics
are governed by the continuum limit, and the noise term is add ed as a perturbation to it.
In a cell, often the number of some molecules is very small, an d may go down very close to or equal to
0. In this case, the change of the number between zero and nonz ero, together with the fluctuations may
cause a drastic effect that cannot be treated by SDE. Possibil ity of some order different from macroscopic
dissipative structure is also discussed by Mikhailov and He ss [9, 10] (see also Ref. [11]). Here we present
a simple example with a phenomenon intrinsic to a system with a small number of molecules where both
the fluctuations and digitality(‘0/1’) are essential.
In nonlinear dynamics, drastic effect of a single molecule ma y be expected if a small change is
amplified. Indeed, autocatalytic reaction widely seen in a c ell, provides a candidate for such amplification
[12, 13]. Here we consider the simplest example of autocatal ytic reaction networks (loops) with a non-
trivial finite-number effect. With a cell in mind, we consider reaction of molecules in a container,
contacted with a reservoir of molecules. The autocatalytic reaction loop is Xi+Xi+1→2Xi+1;i=
1,···, k;Xk+1≡X1
within a container. Through the contact with a reservoir, ea ch molecule Xidiffuses in and out.
Assuming that the chemicals are well stirred in the containe r, our system is characterized by the
number of molecules Niof the chemical Xiin the container with the volume V. In the continuum limit
with a large number of molecules, the evolution of concentra tionsxi≡Ni/Vis represented by
dxi/dt=rixi−1xi−ri+1xixi+1+Di(si−xi) (1)
where riis the reaction rate, Dithe diffusion constant, and siis the concentration of the molecule in
the reservoir.
For simplicity, we consider the case ri=r,Di=D, and si=sfor all i, while the phenomena to
be presented here will persist by dropping this condition. W ith this homogeneous parameter case, the
1above equation has a unique attractor, a stable fixed point so lution with xi=s. The Jacobi matrix
around this fixed point solution has a complex eigenvalue, an d the fluctuations around the fixed point
relax with the frequency ωp≡rs/π. In the present paper we mainly discuss the case with k= 4, since
it is the minimal number to see the new phase to be presented.
If the number of molecules is finite but large, the reaction dy namics can be replaced by Langevin
equation by adding a noise term to eq. (1). In this case, the co ncentration xifluctuates around the fixed
point, with the dynamics of a component of the frequency ωp. No remarkable change is observed with
the increase of the noise strength, that corresponds to the d ecrease of the total number of molecules.
To study if there is a phenomenon that is outside of this SDE ap proach, we have directly simulated
the above autocatalytic reaction model, by colliding molec ules stochastically. Taking randomly a pair
of particles and examining if they can react or not, we have ma de the reaction with the probability
proportional to r. On the other hand, the diffusion out to the reservoir is taken account of by randomly
sampling molecules and probabilistically removing them wi th in proportion to the diffusion constant
D, while the flow to the container is also carried out stochasti cally in proportion to s,DandV[14].
Technically, we divide time into time interval δtfor computation, where one pair for the reaction, and
single molecules for diffusion in and out are checked. The sta te of the container is always updated
when a reaction or a flow of a molecule has occurred. The reacti onXi+Xi+1→2Xi+1is made with
the probability PRi(t, t+δt)≡rxi(t)xi+1(t)V δt=rNi(t)Ni+1(t)V−1δtwithin the step δt. A molecule
diffuses out with the probability POi≡DV x i(t) =DNi(t), and flows in with PIi≡DV s. We choose δt
small enough so that the numerical result is insensitive wit h the further decrease of δt. By decreasing
V s, we can control the average number of molecules in the contai ner, and discuss the effect of a finite
number of molecules, since the average of the total number of molecules Ntotis around the order of 4 V s
[15]. On the other hand, the ‘discreteness’ in the diffusion i s clearer as the diffusion rate Dis decreased.
We set r= 1 and s= 1, without loss of generality ( rs/D andsVare the only relevant parameters of the
model by properly scaling the time t).
First, our numerical results agree with those obtained by th e corresponding Langevin equation if D
andVare not too small. As the volume V(and accordingly Ntot) is decreased, however, we have found
a new state whose correspondent does not exist in the continu um limit. An example of the time series
is plotted in Fig. 1, where we note a novel state with N1, N3≫1 and N2, N4≈0 orN2, N4≫1 and
N1, N3≈0. To characterize this state quantitatively, we have measu red the probability distribution of
z≡x1+x3−(x2+x4). Since the solution of the continuum limit is xi=s(= 1) for all i, this distribution
has a sharp peak around 0, with a Gaussian form approximately , when Ntotis large enough. As shown
in Fig. 2, the distribution starts to have double peaks aroun d±4, asVis decreased. With the decrease
ofV(i.e.,Ntot), these double peaks first sharpen, and then get broader with the further decrease due to
too large fluctuation of a system with a small number of molecu les. Hence the new state with switches
between 1-3 rich and 2-4 rich temporal domains is a character istic phenomenon that appears only within
some range of a small number of molecules.
The stability of this state is understood as follows. Consid er the case with 1-3 rich and N2=N4= 0.
When one (or few) X2molecules flow in, N2increases, due to the autocatalytic reaction. Then X3is
amplified, and since N2is not large, N2soon comes back to 0 again. In short, switch from ( N1,0, N3,0)
to (N1−∆,0, N3+∆+1 ,0) occurs with some ∆, but the 1-3 rich state itself is maintai ned. In the same
manner, this state is stable against the flow of X4. The 1-3 rich state is maintained unless either N1or
N3is close or equal to 0, and both X2andX4molecules flow in within the switch time. Hence the 1-3
rich state (as well as 2-4 rich state, of course) is stable as l ong as the flow rate is small enough.
Within a temporal domain of 1-3 rich state, switches occur to change from ( N1, N3)→(N′
1, N′
3). In
Fig. 3, we have plotted the probability density for the switc h from N1→N′
1when a single X2molecule
flows in, amplified, and N2comes back to 0, by fixing N1+N3=Niniat 256 initially. (We assume no
more flow. Hence N′
1+N′
3=Nini+ 1). The peak around N′
1≈N1+ 1 means the reaction from N2
toN3before the amplification, while another peak around N′
1≈N3=Nini−N1shows the conversion
of the numbers through the amplification of X2molecules. Indeed, each temporal domain of the 1-3
rich state consists of successive switches of ( N1, N3)→≈(N3, N1), as shown in Fig. 1. Since molecules
diffuse out or in randomly besides this switch, the difference between N1andN3is tended to decrease.
On the other hand, each 1-3 rich state, when formed, has imbal ance between N1andN3, i.e.,N1≫N3
orN1≪N3, since, as in Fig. 1, the state is attracted from alternate am plification of Xi, where only
one type iof molecules has Ni≫1 and 0 for others. However, the destruction of the 1-3 rich st ate
2is easier if N1≫N3orN1≪N3, as mentioned. Roughly speaking, each 1-3 rich state starts with a
large imbalance between N1andN3, and continues over a long time span, if the switch and diffusi on
lead to N1≈N3, and is destroyed when the large imbalance is restored. Inde ed, we have plotted the
distribution of y≡x1−x3+x2−x4, to see the imbalance for each 1-3 rich or 2-4 rich domain. Thi s
distribution shows double peaks clearly around y≈ ±2.8, i.e.,(N1, N3)≈(3.4V,0.6V),(0.6V,3.4V).
Let us now discuss the condition to have the 1-3 or 2-4 rich sta te. First, the total number of molecules
should be small enough so that the fluctuation from the state Ni≈Nj(for∀i, j) may reach the state
withNi≈0. On the other hand, if the total number is too small, even N1orN3for the 1-3 rich state
may approach 0 easily, and the state is easily destabilized. Hence the alternately rich state is stabilized
only within some range of V.
Note also that our system has conserved quantities/summationtext
iNi(and/summationtext
ilogx iin the continuum limit), if
Dis set at 0. Hence, as the diffusion constant gets smaller, som e characteristics of the initial population
are maintained over long time. Once the above 1-3 (or 2-4) ric h state is formed, it is more difficult to be
destabilized if Dis small. In Fig. 4, we have plotted the rate of the residence a t 1-3 (or 2-4) rich state
over the whole temporal domain, with the change of V. Roughly speaking, the state appears for DV < 1
[16], while for too small V(e.g., V <4), it is again destabilized by fluctuations. Although the ra nge of
the 1-3 rich state is larger for small D, the necessary time to approach it increases with V. Hence it
would be fair to state that properly small number of molecule s is necessary to have the present state.
To sum up, we have discovered a novel state in reaction dynami cs intrinsic to a small number of
molecules. This state is characterized by alternately vani shing chemicals within an autocatalytic loop,
and switches by a flow of single molecules. Hence, this state g enerally appears for a system with an
autocatalytic loop consisting of any even number of element s. With the increase of k, however, the
globally alternating state all over the loop is more difficult to be reached. In this case, locally alternating
states are often formed with the decrease of the system size ( e.g., ‘2-4-6-8 rich’ and ‘11-13-15 rich’ states
fork= 16). This local order is more vulnerable to the flow of molecu les than the global order for the
k= 4 loop.
On the other hand, for k= 3, two of the chemical species start to vanish for small V, since any pair
of different chemical species can react so that one chemical s pecies is quickly absorbed into the other.
This state of single chemical species, however, is not stabl e by a flow of a single molecule. Indeed, no
clear ‘phase transition’ is observed with the decrease of V.
Although in the present Letter we have studied the case with si=s, we have also confirmed that
the present state with alternately vanishing chemical spec ies is generally stabilized for small V, even if
siorriorDiare not identical.
Last, we make a remark about the signal transduction in a cell . In a cell, often the number of molecules
is small, and the cellular states often switch by a stimulus o f a single molecule [1]. Furthermore, signal
transduction pathways generally include autocatalytic re actions. In this sense, the present stabilization
of the alternately rich state as well as a single-molecule sw itch may be relevant to cellular dynamics. Of
course, one may wonder that the present mechanism is too ‘sto chastic’. Then, use of both the present
mechanism and robustness by dynamical systems [17, 18] may b e important. Indeed, we have made
some preliminary simulations of complex reaction networks . Often, we have found the transition to
a new state at a small number of molecules, when the network in cludes the autocatalytic loop of 4
chemicals as studied here [19]. Hence the state presented he re is not restricted to this specific reaction
network, but is observed in a class of autocatalytic reactio n network. Furthermore switches between
different dynamic states (limit cycles or chaos) are possibl e when the number of some molecules (that
are not directly responsible to the switch) is large enough. The ‘switch of dynamical systems’ by the
present few-number-molecule mechanism will be an importan t topic to be pursued in future.
We would like to thank C. Furusawa, T. Shibata and T. Yomo for s timulating discussions. This
research was supported by Grants-in-Aid for Scientific Rese arch from the Ministry of Education, Science,
and Culture of Japan (Komaba Complex Systems Life Science Pr oject).
3References
[1] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. W atson, The Molecular Biology of the
Cell(Garland, New York, 3rd ed., 1994).
[2] H. H. McAdams and A. Arkin, Trends Genet. 15, 65 (1999).
[3] F. Rieke and D. A. Baylor, Revs. Mod. Phys. 70, 1027 (1998).
[4] N. G. van Kampen, Stochastic processes in physics and chemistry (North-Holland, rev. ed., 1992).
[5] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (John Wiley, 1977).
[6] W. Horsthemke and R. Lefever, Noise-Induced Transitions , edited by H. Haken (Springer, 1984).
[7] K. Matsumoto and I. Tsuda, J. Stat. Phys. 31, 87 (1983).
[8] K. Wiesenfeld and F. Moss, Nature 373, 6509 (1995).
[9] B. Hess and A. S. Mikhailov, Science 264, 223 (1994); J. Theor. Biol. 176, 181 (1995).
[10] P. Stange, A. S. Mikhailov and B. Hess, J. Phys. Chem. B 102, 6273 (1998); 103, 6111 (1999); 104,
1844 (2000).
[11] D. A. Kessler and H. Levine, Nature 394, 556 (1998).
[12] M. Eigen, P. Schuster, The Hypercycle (Springer, 1979).
[13] M. Delbruck, J. Chem. Phys. 8, 120 (1940).
[14] One might assume the choice of the diffusion flow proporti onal to DV2/3, considering the area of
surface. Here we choose the flow proportional to DV, to have a well-defined continuum limit (eq.(1))
forV→ ∞. At any rate, by just re-scaling D, the present model can be rewritten into the case with
DV2/3, for finite V. Hence the result here is valid for the DV2/3(and other) cases.
[15] For small Vvalue, there appears deviation from this estimate. At any ra te, the average number
decreases monotonically with V, and for V= 0, it goes to zero.
[16] As shown in Fig. 4, there is a deviation from the scaling b yDV. All the data are fit much better
either by D0.9Vor by ( D+ 0.0002)V. At the moment we have no theory which form is justified.
[17] K. Kaneko and T. Yomo, Bull. Math. Biol. 59, 139 (1997); J. Theor. Biol. 199, 243 (1999).
[18] C. Furusawa and K. Kaneko, Bull. Math. Biol. 60, 659 (1998); Phys. Rev. Lett. 84, 6130 (2000).
[19] Stability of the alternately rich state also depends on the network structure, i.e., arrows coming in
and out from the autocatalytic loop of 4 chemicals.
[20] This estimate includes the case in which only one specie s exists, and gives an overestimate for very
small V.
40 40 80 120
88000 92000 steps
V = 32 , D = 1/256
N1
N2
N3
N4
Figure 1: Time series of the number of molecules
Ni(t), for D= 1/256, V= 32. Either 1-3 or 2-
4 rich state is stabilized. Successive switches ap-
pear between N1> N3andN3> N1states with
N2, N4≈0. Here a switch from 1-3 rich to 2-4 rich
state occurs around 88000 steps.
0 0.1 0.2 0.3
-12-8-404812
(x1+x3) - (x2+x4)D = 1/64
V = 1
4
16
32
64
128
256
Figure 2: The probability distribution of z≡
(x1+x3)−(x2+x4), sampled over 2 .1 – 5.2×106
steps. D= 1/64. For V≥128,zhas a distri-
bution around 0, corresponding to the fixed point
state xi=s(= 1). For V≤32, the distribution
has double peaks around z≈ ±4, corresponding to
the state N1, N3≫N2, N4(≈0) or the other way
round. The double-peak distribution is sharpest
around V= 16, and with the further decrease
ofV, the distribution is broader due to finite-size
fluctuations.
Figure 3: Probability density for the switch from
(N1, N3) to (N′
1, N′
3) when a single X2molecule is
injected into the system. N1+N3=Niniis fixed
at 256 initially. There is no more flow and N4is
always kept at 0, so that the switch is completed
when N2comes back to 0, and N′
1+N′
3=Nini+1.
Probability to take N′
1is plotted against initial N1.
0 0.2 0.4 0.6 0.8 1
0.01 0.1 1 10Rate of the residence
DV1-3 or 2-4 rich state
D = 1/2048
1/512
1/256
1/128
1/64
1/32
1/16
Figure 4: The rate of the residence at 1-3 (or
2-4) rich state over the whole temporal domain,
plotted against DV[16]. Here, the residence rate
is computed as follows. As long as N2>0 and
N4>0 are not satisfied simultaneously, over a
given time interval (128 steps, 2.5 times as long as
the period of the oscillation around the fixed point
at continuum limit), it is counted as the 1-3 rich
state (2-4 rich state is defined in the same way)
[20]. The residence rate is computed as the ratio
of the fraction of the time intervals of 1-3 or 2-4
rich state to the whole interval.
5 |
arXiv:physics/0008239v1 [physics.chem-ph] 30 Aug 2000Comments on “Effective Core Potentials” [ M. Dolg, Modern
Methods and Algorithms of Quantum Chemistry (Ed. by J.Grotendorst,
John von Neumann Institute for Computing, J¨ ulich, NIC Seri es, Vol.1,
ISBN 3-00-005618-1, pp.479-508, 2000) ].
A. V. Titov∗and N. S. Mosyagin
Petersburg Nuclear Physics Institute,
Gatchina, Petersburg district 188350, Russia
(September 2, 2013)
Abstract
The recent paper of M. Dolg is discussed and his critical rema rks with respect
to the Generalized Relativistic Effective Core Potential (G RECP) method
are shown to be incorrect. Some main features of GRECP are dis cussed as
compared with the “energy-consistent/adjusted” pseudopo tential and with
the conventional shape-consistent RECP.
I. INTRODUCTION
The discussed paper of M. Dolg deals with the relativistic eff ective core potential (RECP)
methods including the model potential and pseudopotential (PP) techniques. The shape-
consistent RECP method as a PP version is compared with the “e nergy-adjusted” PP
(EAPP) and the “energy-consistent” PP (ECPP) developed by S tuttgart’s group (e.g., see
[1,2] and references in the discussed paper by M. Dolg). In th eir semiempirical version,
EAPP, partial potentials are fitted to reproduce the experim ental atomic spectrum. In the
ab initio approximation, ECPP (earlier also called energy- adjusted PP), “valence energies”
(i.e. sums of ionization potentials and excitation energie s) for a group of low-lying states
are fitted to the corresponding energies of the same states in all-electron approximations
like Hartree-Fock, Wood-Boring or Dirac-Fock in a least-sq uares sense with the help of the
ECPP parameters. It means that only some special combinatio ns of the matrix elements
of a (non-,quasi-)relativistic Hamiltonian are fitted by th e ECPP and EAPP Hamiltonians
with the help of the one-electron radially local PP operator . When considering below both
ECPP and EAPP versions we will write ECAPPs or Stuttgart PPs.
The radially local operator is also used in the shape-consis tent RECPs and some new
non-local RECP terms are added in our Generalized RECP (GREC P) version [3–16], which
we consider as a development of the shape-consistent RECP me thod. The underlying idea
traced in our papers concerning the GRECP approximation is i n simulating the one- and
two-electron parts of an original Dirac-Coulomb (in prospe ct, Dirac-Coulomb-Breit) Hamil-
tonian with the accuracy which is needed and sufficient for calculation of physical and chem-
ical properties (and processes) in heavy-atom molecules wi th agiven accuracy. A paramount
1requirement is that such a simulation should provide maximu m possible savings for conse-
quent molecular calculations with GRECPs.
As is confirmed in all our test calculations and not only in our theoretical analysis,
the GRECP Hamiltonian in the form used in papers by Mosyagin e t al. (1997) [7] and
(2000) [11] (which are criticized by M. Dolg) more accuratel y reproduces the Dirac-Coulomb
Hamiltonian in the valence (V) region as compared with other tested RECP and Stuttgart
PP versions employing the radially local operator. Phrase “ in the valence region” means that
the occupation numbers of the outer core (OC) shells, nocc
OC, are not considerably changed
in studied states as compared with the OC occupation numbers of the configurations used
in the GRECP generation (i.e. ∆ nocc
OC≪1). Thus, only relaxation and dynamic correlation
effects are suggested to take place in the OC shells.
We have emphasized this property of the GRECP Hamiltonian al ready in the introduc-
tion of paper [7]. It is noted in the abstract of our theoretic al paper (1999) [10] where a very
detailed analysis of features of the shape-consistent RECP method including the GRECP
approximation is given. We consider the GRECP version for Hg used in [7,11] as reliable for
atomic and molecular calculations of the states in which the OC shells of Hg are completely
occupied in the leading configurations if accuracy of a few hu ndreds wave numbers for tran-
sition, dissociation etc. energies is required. In papers [ 5,10], some other improvements of
the RECP method are suggested in order to provide minimal com putational efforts in ac-
curate RECP calculation of wide range of excitations and pro perties in systems containing
arbitrary heavy atoms including transition metals, lantha nides and actinides. We, obviously,
will not repeat here the theoretical analysis of the shape-c onsistent RECP method and will
give only some necessary details which have direct attitude to the criticized points.
The goal of this paper is mainly to compare features of the GRE CP and other RECP
versions including Stuttgart PPs rather than to reply on Dol g’s claims. These comments
can be also useful for reading them before paper [10].
II. GENERAL COMPARISON OF DIFFERENT RECP VERSIONS
The discussed GRECP version is assumed to be efficiently used w hen excitations and
chemical bonding take place in the V region whereas only dyna mic correlation and relaxation
(polarization) are considered in the OC region. Therefore, interactions between/with valence
and outer core electrons are simulated on the basis of the fol lowing principles:
First, for selected subspaces of the OC and V shells, the matching r adiiRcfor the regions
of the spinor’s smoothing are chosen to be as small as possibl e in order to reduce the errors
of reproducing the original two-electron integrals (in fur ther reducing the matching radii,
partial potentials become too singular to be approximated b y gaussians and used in RECP
calculations, for details see [10]). When using the GRECP op erator in calculations of the
same states as with the Dirac-Coulomb Hamiltonian, the OC, V and virtual pseudospinors
(PSs) coincide with the large components of the original Dir ac spinors after the matching
radii with very good accuracy [6,10] in contrast to the cases of using the conventional RECP
operator. In this connection (it is very important), the rad ius of the “unphysical” GRECP
terms (which, obviously, do not include the Coulomb interac tion with the inner core (IC)
electrons) only slightly larger than the outermost matchin g radius Rmax
cfor PSs. This is
2direct consequence of generating different partial potenti als,Vnlj(r), for the corresponding
OC and V (and virtual) pseudospinors with the same ( lj). It is shown in [16] that difference
between the OC and V potentials with the same ( lj) can not be eliminated with the help
of any special kind of smoothing the corresponding OC and V sp inors without substantial
decrease (up to an order of magnitude for transition energie s) in accuracy. Thus, when
reducing the matching radii which are usually close to each o ther, we reduce the radius
of unphysical terms in the corresponding atomic effective Ha miltonian. The independent
smoothing the OC and V (and, in principle, some virtual) spin ors with polinomials give us
sufficient flexibility to generate smooth enough OC and V pseud ospinors as well as their
partial potentials.
The GRECP is a “matching radii-specified” (or “space-driven ”) method of approximation
(see [10]) contrary to the energy-adjusted/consistent PPs which are “selected valence energy-
based” (or “energy-driven”).
Second , the non-local terms with projectors on the outer core PSs in the GRECP opera-
tor give us an important possibility to reproduce the origin al OC and V orbital energies and
properly construct the most important nondiagonal one-ele ctron terms [16], which together
with the Hartree-Fock (Coulomb and exchange) electronic te rms derived on pseudospinor
densities constitute the one-electron part of the effective (model) Hamiltonian. Therefore,
the one-electron part of the Hamiltonian in the V region can b e reproduced very accurately
(see section “Theory” in [10] for more details). The orbital energies are used in denominators
of the M¨ oller-Plessett perturbation theory (PT) accounti ng for correlation etc. Otherwise,
instead of the orbital energies and nondiagonal Lagrange mu ltipliers, some other combina-
tions of matrix elements of an original Hamiltonian can be re produced in the model one. (In
particular, those one-electron energies can be exactly sim ulated which are more appropriate,
e.g., for the Epstein-Nesbet PT. However, the partial poten tials have additional “tails” in
such cases because of the use the inverted Hartree-Fock equa tions for their generation and
the radius of the unphysical RECP terms is thus enlarged. One should remember that the
tail behaviour of orbitals is described by their orbital ene rgies. The difference in using dif-
ferent one-electron energies is not very essential if the co rresponding original Coulomb and
exchange two-electron integrals are accurately reproduce d by those with pseudospinors as is
for small matching radii [10].)
Obviously, it is possible to fit transition or valence energi es for a group of states with the
help of the energy-consistent/adjusted PPs having appropr iate number of fitting parameters.
Certainly, it is not equivalent to the simulation of all the i mportant Hamiltonian matrix el-
ements on a needed level of accuracy in order to describe a pos sible variety of perturbations
in the valence region (including excitations and chemical b onding with arbitrary atoms and
geometries) in correlation structure calculations with a g iven accuracy. All the one- and
two-electron matrix elements of the original valence Hamil tonian should be appropriately
reproduced to be used for accurate calculations of a wide ran ge of applications. Besides,
fitting the valence energies prior to the orbital energies, g ive no any advantages in reproduc-
ing, in particular, the variety of physical and chemical pro perties which cannot be calculated
from potential curves or surfaces. The steady simulation of the valence Hamiltonian can be
done on the basis of the “space-driven” shape-consistent RE CP generation scheme.
The conventional RECP operator with the shape-consistent s pinor smoothing suggested
3by K.Pitzer’s group (1979,1977) [19,20] gives no flexibilit y in fitting shapes and orbital
energies simultaneously for the OC and V spinors with the sam e (lj). In general, when
the OC spinors are used in their RECP generation scheme for a g iven ( lj), the “effective
matching radii” for other (V and virtual) pseudospinors wit h the same ( lj) are, in fact, larger
than those for the OC pseudospinors and so on. Therefore, in g eneral (see [10] for more
theoretical details), the two-electron matrix elements ar e not so accurately reproduced as in
the case of GRECP. By other words, the radius of “unphysical” terms of the conventional
RECPs is substantially larger than the matching radii.
In turn, the ECAPP generation schemes take no care about smoo th pseudoorbitals (pseu-
dospinors) and matching radii when putting simulation of so me valence energies and gen-
eration of smooth partial potentials on the first place. (A sm all number of gaussians in
the ECAPP expansion is, in fact, equivalent to smooth partia l potentials). The result is
similar to the case of the conventional shape-consistent RE CP. Additional disadvantages
of ECAPPs are a poor theoretical justification and technical complexity in fitting a large
number of valence energies. When taking account of correlat ions in ECPP calculations,
the valence energies for the correlated states should be fitt ed as well if a high computa-
tional accuracy is required. It should be done as a pay for the absence of matching radii
for the ECPP pseudospinors. Moreover, being one-electron potentials of a special (radially
local) type, ECAPPs cannot provide arbitrarily high accura cy even for reproducing the one-
electron part of the valence Hamiltonian [16] let alone the t wo-electron part (if, obviously,
the number of explicitly treated electrons is fixed). Beside s, how many transition/valence
energies between/for correlated states should be fitted for reliable reproducing a required
(large) number of two-electron integrals with some needed accuracy? Hundreds? Or maybe
thousands? Can it be efficiently applied in practice? Will it p rovide a proportional (even)
level of errors for the one- and two-electron integrals as is in the case of GRECPs? Should
not forget that the ECAPPs are employing the conventional RE CP operator that is not so
flexible as the GRECP one [10,16]. We can also remind here abou t those properties which
cannot be calculated from potential curves or surfaces. Whe ther will these properties be
well reproduced with the help of ECAPPs?
In the optimization of the parameters of partial potentials Vnlj(r) one can produce com-
pact gaussian expansions for the ECAPPs when fitting directl y some selected valence ener-
gies. Is this a real advantage? The compactness in the gaussi an expansions of the partial
potentials does not ensures the smoothness of pseudoorbita ls (pseudospinors). Moreover, as
is mentioned above, the radius of unphysical terms in such a P P is invitably larger than in the
RECPs in which a large set of gaussians is employed to fit quite singular behaviour of numer-
ical potentials close to the matching points thus reducing t he effective Vnlj(r)−Ncore/rradii
(see Figure 2 in [10]). It is very widely known that the effort i n the calculation ( ∼N4) and
transformation ( ∼N5) of two-electron integrals (where Nis the number of basis functions)
isalways substantially higher than in calculation of the RECP integr als (∼N2·NRECP,
where NRECP is the number of terms in the used RECP expansions) for all kno wn RECP
versions including GRECPs when appropriately large basis s ets are employed for precise
calculations. Again we should emphasize that in spite of the rather complicated form of
the GRECP operator, the main computational effort in calcula ting matrix elements with
GRECPs is caused by the standard radially local operator whi ch is also the main part of the
4shape-consistent RECP and ECAPP operators, and not by the no n-local GRECP terms.
Thus, the additional efforts in calculations with GRECPs are negligible as compared with
the cases of using the conventional RECPs and PPs if comparab le gaussian expansions are
used for the partial potentials.
Summarizing, what is the computational utility in generati on of compact RECPs and
PPs? Maybe much more important is that their accuracy should be in agreement with
the number of explicitly treated electrons? Obviously, the smooth shapes of pseudospinors
in the atomic core are more important than smooth partial pot entials. The smooth PSs
can be accurately approximated with a relatively small numb er of gaussian functions. It
is substantially more flexible if the OC and V spinors can be sm oothed individually. A
possibility to generate the partial potentials after const ructing PSs (and not simultaneously)
when inverting the Hartree-Fock equations following Godda rd III (1968) [17] is important
from the computational viewpoint.
Due to these features, the RECP generation scheme by K.Pitze r’s group is very effective
in practice in addition to its theoretical advantages. It is always better to have a possibility
to split solution of a computationally consuming problem on a few consequent steps. That is
why we prefer the scheme of K.Pitzer’s group and not that prop osed by Durand & Barthelat
(1975) in their classical paper [18], where the idea of the sh ape-consistent ECP method
was first suggested. The ECAPP generation scheme is in many as pects close to the ECP
generation scheme by Durand & Barthelat.
Our first comparisons of RECPs in [7] were done in the one-confi gurational approxi-
mation because only recently we have obtained an opportunit y of employing very efficient
atomic Relativistic Coupled Cluster (RCC) [28] code for rel iable correlation structure cal-
culations with both Dirac-Coulomb and RECP Hamiltonians [1 1]. The advantage of the
former is in the use of the finite-difference method (i.e. spin ors/orbitals are varied in the nu-
merical form) and therefore, the DF/HF calculations are ind ependent of the finite basis sets
errors. In the latter case, one has a possibility to use very l arge basis sets thus minimizing
dependence of the final results on a special choice of a basis s et. Besides, there are almost no
subjective dependences in RCC calculations from a special s election of configurations (ref-
erence spaces, truncation thresholds, etc.) that is very im portant for correct comparison of
different effective Hamiltonians with original. Our stateme nts in [7] concerning the accuracy
of the GRECP Hamiltonian were done on the bases of one-configu rational calculations in the
jj-coupling scheme and of the theoretical analysis presented in [10]. They are completely
confirmed in first correlation GRECP calculations of Hg, Pb an d TlH [11,13,14]. Besides,
the examined energy-adjusted PPs were found in our tests to b e less accurate in general
than the shape-consistent RECP versions generated by other groups.
III. REPLY ON REMARKS OF M. DOLG
Below Dolg’s remarks from the discussed paper and our answer s are given. All Dolg’s
quotations are taken from section 5.3 “Limitations of accur acy” in the same order as in the
discussed paper unless the opposite is explicitly stated. T he references within the quotations
are given with respect to the list of references in the end of t he present comments. To avoid
5possible problems when extracting quotations from a contex t, the whole text of section 5.3
is presented in Appendix A.
1. See Table 1 in the Dolg’s paper.
*Our remarks:
The frozen core approximation (FCA) is underlying for all th e known ECP methods,
both nonrelativistic and relativistic. Therefore, the acc uracy of the ECPs can not be
considered as higher than that of the FCA unless the special c orrections like the Core
Polarization Potential (CPP) or our Self-Consistent (SfC) terms are used. Moreover,
the smoothing of the orbitals (spinors), incorporating the relativistic effects, etc. will
further increase the ECP errors. In Table I, we have compiled the errors of the new
ECPPs (with 26 and 54 adjustable parameters) from Table 1 in t he Dolg’s paper
together with the FCA errors calculated by us. The HFD code [2 2] is used in the
corresponding Dirac-Fock all-electron and FCA calculatio ns with the point nuclear
model for the states averaged over the nonrelativistic confi gurations. Obviously, having
54 adjustable parameters in the ECPP, one can use them to fit ex actly 54 valence
energies. However, the accuracy of the generated PP should n ot be estimated by the
errors in reproducing the fitted energies. The errors in the v alence energies which
were not used in the fitting procedure or other properties mus t be used. Although,
the number of chemical interesting states in the case of atom is not too large to allow
one to fit almost all of them, this number is dramatically incr eased in the case of the
“pseudo-atom”-in-molecules.
Therefore, the accuracy of the new ECPPs can not be derived fr om Tables 1 and 2 in
the Dolg’s paper and additional independent testing is nece ssary. Unfortunately, we
can not do this because we do not have the parameters of his new ECPPs (see [23]).
2. Dolg:
Tables 1 and 2 demonstrate that for very special cases like Hg , with a closed 5d10-shell
in all electronic states considered, a small-core energy-c onsistent pseudopotential us-
ing a semilocal ansatz reaches an accuracy of 10 cm−1, which is well below the effects
of the nuclear model, the Breit interaction or higher-order quantum electrodynamical
contributions. We also note that differences between result s obtained with a frequency-
dependent Breit term and the corresponding low-frequency l imit amount to up to 10
cm−1. Moreover, the quantum electrodynamic corrections listed in tables 1 and 2 might
change by up to 20 cm−1when more recent methods of their estimation are applied
[24,25]1
1Papers are cited according the list of references in the pres ent comments and the numbers of
tables are original.
6*Answer:
The basic requirement of any fruitful simulation is a transf erability of a model Hamil-
tonian to the cases which were not used when constructing thi s Hamiltonian. We are
“fitting” the Hamiltonian matrix elements in the valence reg ion first of all and not
some their combinations of a special kind likewise the valen ce energies. It is sufficient
to use a very small number of DF configurations for the GRECP ge neration, with the
basic requirements: a)they should have the same configurational structure in the co re
region as the states of the atom (in a molecule) in the GRECP ca lculations; b)they
should contain all the spinors required for the generation o f the corresponding partial
potentials.
Some special remarks can be done with respect to the Breit effe ct. The replacement
of the Coulomb-Breit two-electron interaction by the Coulo mb interaction (1 /r12) in
the PP Hamiltonian is not well justified by M. Dolg. The contri butions from the
(frequency-dependent) Breit interaction were evaluated i n the first-order perturbation
theory (PT1). However, the Breit interaction is very strong close to a heavy nucleus.
Therefore, the wave function in its neighborhood is serious ly perturbed by the Breit
interaction and the higher PT orders are required to conside r the core relaxation
for appropriate accounting for the Breit correction [26]. I n particular, the random
phase approximation can be used keeping the first-order pert urbation on the Breit
interaction. As is shown in [26,27], in some cases the core re laxation can reduce the
final Breit correction by an order of magnitude. Did Prof. Dol g perform similar analysis
when generating the ECPPs for Hg? After that, what is the need to take account of
the QED effects having an order of magnitude smaller contribu tion than other inherent
PP errors? What is the profit (advantage) in such an accountin g for Breit and QED
effects?
Those modifications are done by Prof. Dolg which can be done ea sily, and not those
which should be done first of all (i.e. those which give the lar gest errors). In our work
upon GRECPs, we are eliminating at first largest errors, then errors of the next level
of magnitude and so on, step-by-step. The theoretical analy sis of the GRECP errors
is always done, thus justifying the approximations made by u s.
At last, it would be excellent to perform molecular calculat ions on a level of accuracy
of 100 cm−1for transition, dissociation, etc. energies systematical ly but the modern
correlation methods, codes and computers do not allow one to do this because of the
high computational cost. Our goal on the nearest future is to generate the GRECPs
with “inherent” errors close to (or below than) 100 cm−1for the valence energies when
treating minimal number of electrons explicitly. In molecu lar GRECP calculations, it
allows one to attain accuracy within a few hundred wave numbe rs for the energies of
interest reliably and with minimal efforts.
3. Dolg:
“Therefore, it is important to state exactly which relativi stic all-electron model the ef-
fective core potential simulates and, when comparing effect ive core potentials of differ-
ent origins, to separate differences in the underlying all-e lectron approach from errors
in the potential itself, e.g., due to the size of the core, the method of adjustment or the
7form of the valence model Hamiltonian.”
*Answer:
It is true. As one can see from our papers, we are carefully ana lyzing the sources
of errors in our GRECP versions. We “state exactly which relativistic all-electron
model the effective core potential simulates” , etc. Moreover, we consider as our duty
to present all the necessary details concerning all the GREC Ps which were used in our
papers. Being requested, the GRECP parameters can be receiv ed, in particular, by
email.
However, it is not in our responsibility “to separate differences in the underlying all-
electron approach from errors in the potential itself” for PPs and RECPs generated
by other groups. How can we separate errors of the Wood-Borin g approximation from
the the EACPP fitting errors without knowledge of all the deta ils of fitting, without
having the required codes, and without doing some test calcu lations with these codes?
Besides, why must we do this? The responsibility for such an analysis is on thos e who
have generated these PPs and RECPs.
We have written in [11]:
“It should be noted that the energy-adjusted pseudopotenti al (PP) tested in the
present paper was generated by H¨ aussermann et al.[2] using the results of the quasirel-
ativistic Wood-Boring [29] SCF all-electron calculations as the reference data for fitting
the spin-orbit averaged PP parameters. A new 20e-PP for Hg wa s generated recently
by fitting to the Dirac-Fock-Breit reference data [30], but w e do not have the param-
eters of this PP2.
The energies of transitions between the 6 s2and 6 s16p1(3P0,3P1,3P2) states in the
20e-PP/MRCI calculations employing the CIPSO method [31] a re within 100 cm−1
of experiment (see Table 6 in [2] or Table 2 in the present pape r). However, the
energy-adjusted PP does not account for the contributions f rom correlations with the
4fshell, and the basis set used does not contain h-type functions. One can see from
Tables 1 and 2 that the contributions of the two effects to thes e transition energies are
up to 284 cm−1and 247 cm−1, respectively. The good 20e-PP/MRCI/CIPSO results
are probably due to fortuitous cancellation of several cont ributions: the inherent PP
errors (e.g., the 6 s1
1/26p1
1/2(J= 0) – 6 s1
1/26p1
3/2(J= 2) splitting is overestimated by 1014
cm−1because of the features of the spin-orbit simulation within theLS-based version
of the energy-adjusted scheme, see Table 4 in paper [7]), the neglect of correlations
with the 4 fshell, the basis set incompleteness, etc. A similar situati on holds for
transitions between the 6 s1and 6p1(2P1/2,2P3/2) states of Hg+, but errors of the 20e-
PP/MRCI/CIPSO calculations relative to experimental data reach a level of 1000 cm−1
in this case.”
Is it not correct? Similar analysis can be found in our previo us paper [7] criticized by
M. Dolg. Moreover, it is strange that such simple one-configu rational tests as in [7]
2See Ref. [23] for more details.
8were not performed in [2] for their 20e-PP. M. Dolg many times claimed that our test
results with their 20e-PP in [2] are wrong. Where can we find hi s publication with the
confirmation of these claims?
However, let us get back to the paper of M. Dolg. Why is the info rmation about the
states used in the valence energy fitting in the ECPP generati on not even presented
there? Where are the ECPP parameters? Because of [23], we can not check the real
quality of these new ECPPs.
4. Dolg:
“In this context we want to point out that the seemingly large errors for energy-adjusted
pseudopotentials reported by Mosyagin et al. [7,11] are mai nly due to the invalid com-
parison of Wood-Boring-energy-adjusted and Dirac-Fock-o rbital-adjusted pseudopoten-
tials to all-electron Dirac-Fock data, i.e., differences in the all-electron model are con-
sidered to be pseudopotential errors.”
*Answer:
Although “ the correct relativistic all-electron Hamiltonian for a ma ny-electron system
is not known ”, the Dirac-Coulomb Hamiltonian is preferred over the Wood -Boring one.
Moreover, for an “RECP user” the level of the PP errors with re spect to the most ac-
curate relativistic Hamiltonian (among the known ones) is m uch more meaningful than
the question whether the PP errors are due to the unsatisfact orily fitting procedure
of the EACPPs (the small number of parameters, incompletene ss of the PP opera-
tor, etc.) or the poor all-electron reference data used for t his fitting. Therefore, the
comparison of the all-electron Dirac-Coulomb data with the Wood-Boring-fitted PP
results is correct in papers [7,11], whereas the comparison by Dolg of the all-electron
Wood-Boring data with the Dirac-Fock-based GRECP results i n Table XVII from [7]
is not valid and is given in [7] only in order to show “the range of the dispersion of
the data”. Besides the absence of equivalence in the used bas is sets, the Wood-Boring
approximation is an additional source for the distinctions between the all-electron and
GRECP molecular data in this table (see also the last item in t his section for Dolg’s
remark). The question is also arise how M. Dolg and co-authou rs can obtain “ excel-
lent agreement ” (see abstract of [2]) with the experimental data in their pr evious works
with the help of the Wood-Boring fitted PPs.
5. Dolg:
“It is also obvious from the compiled data that the accuracy o f the valence model
Hamiltonian is also a question of the number of adjustable pa rameters.”
*Answer:
We are very satisfied that Prof. Dolg at last recognized the fa ct of importance of the
number of the adjustable parameters because probably all th e EACPPs generated be-
fore had small number of the parameters. The problem is only t hat the EACPP is a
one-electron operator and the original Hamiltonian contai ns the two-electron interac-
tions as well. How is Prof. Dolg planning to reproduce the two -electron part with the
EACPP’s fitting parameters?
96. Dolg:
“ Claims that such very high accuracy as demonstrated here ca n only be achieved by
adding nonlocal terms for outer core orbitals to the usual se milocal terms [7,11] appear
to be invalid, at least for energy-consistent pseudopotent ials.”
*Answer:
We have revised again our papers but could not find the text whi ch could be interpreted
by such a manner as it is done in the Dolg’s paper. Can Prof. Dol g show the places
in our papers where we have written so? The most “hard” (debat able) phrase in our
papers (written in our joint paper [11] with the Tel Aviv grou p, p. 674, but its RECP
part is on our responsibility) is “...The larger errors for R ECPs [21,2] are mainly due
to the neglect of the difference between the outer core and val ence potentials in these
RECP versions (see [10,6] for details).” But “larger errors ” means here “larger level
of errors” (it is clear from the context below this phrase whe re we clarify the origin of
the errors for our concrete calculations and for the used 20e -PP for Hg). Nevertheless,
we are ready to recognize that in the case of the 20e-PP for Hg [ 2] the (very) large
errors in the transition energies compiled in Table 3 of [11] are due to a bad quality of
the used fitting principles and/or incompetent their applic ation in [2] rather than due
to the neglect of the difference between the outer core and val ence potentials in this
PP (if such a reformulation is more acceptable for authors of [2]).
Obviously, some errors in transition energies can be smalle r for an EACPP, especially, if
those energies are fitted when generating this EACPP. The EAC PPs should be checked
for those transitions or properties, which were not fitted du ring the EACPP generation.
Test calculations should be performed with different number s of correlated electrons
and with a good quality of accounting for correlation. A “min imal” completeness of
the basis sets is also required.
As to the importance of the nonlocal terms and to the phrase “This error could be
reduced further upon using a smaller core, but the efficiency o f the approach would be
sacrificed.” written by Prof. Dolg a few lines above, we should remind the f ollowing.
Already in paper [6] we have pointed out that when freezing th e OC pseudospinors,
the corresponding nonlocal GRECP projectors are not involv ed in calculations. How-
ever, the accuracy can be very high if partial potentials are generated for nodal V
pseudospinors and not for OC ones thus taking into account th e difference between the
V and OC potentials contrary to the standard RECP. Therefore we considered this
case as a special GRECP version. We have pointed out later (e. g., see [10]) about sim-
ilar alternatives with respect to other our additions (“sel f-consistent” and “spin-orbit”
terms) to the conventional radially local operator. Obviou sly, the same (freezing) pro-
cedure can be applied for a larger space of explicitly treate d electrons when freezing
the innermost core shells because the differences between th e partial potentials for the
innermost nodeless PSs and the next having one node PSs with t he same ( lj) are the
most essential. However these cases are not computationall y interesting because such
RECP accuracy is of interest for the modern level of correlat ion structure calculations
which can be achieved treating as small number of electrons e xplicitly as possible. The
latter is our main purpose. That is why we prefer to change the functional form o f the
10RECP operator, to insert core correlations to GRECPs etc. Th e only problem is to
do this properly, by a “theoretically-consistent” way, inv olving appropriate functional
forms. And this should be done eliminating first the largest e rrors. What is the need
to take account of the Breit and other QED effects within PPs if their contribution
are about an order of magnitude smaller than other inherent P P errors?
7. Dolg:
“Moreover, additional nonlocal terms obviously do not impr ove the performance for
atomic states with a 5d9occupation”
*Answer:
It is not true. M. Dolg did not present anyhis results of calculations with GRECPs
oranytheoretical analysis. As we know from our correspondence wi th him, similar
work have not been performed by him at all, therefore, his con clusions are made only
on the basis of our results given in [7]. However, our results and conclusions in [7] are
opposite. In this connection, the adverb “obviously” is very funny.
8. Dolg:
“or in molecular calculations (cf., e.g., tables III and XVI I in Mosyagin et al. [7]).”
*Answer:
Concerning the molecular GRECP calculations, the first GREC P/MRD-CI results
for spectroscopic constants in TlH [13] lead to opposite con clusions. Some other
GRECP/MRD-CI and GRECP/RCC-SD calculations on HgH, TlH and PbH are in
progress now. As to the spin-orbit-averaged RECP/SCF calcu lations on HgH pre-
sented in table XVII of [7], they are performed to study “the r ange of dispersion of
the data” because for the one-electron RECP and GRECP operators it “is impor-
tant information to estimate the accuracy of the RECP approx imation both for one-
configurational and for highly correlated calculations of H gH and HgH+molecules”
(see p. 1121 in [7]). The above mentioned Dolg’s excerption i s certainly the top ana-
lytical result in the commented paper dealing with the pseudo potentials. Following its
logic pattern, we can call this by a pseudoresult .
In fact, almost all that we have written in these comments was written in our papers
earlier and we only have concentrated here on some underlyin g principles of our approach
as compared to other RECP methods. We regret that our papers h ave occured to be so
difficult for reading that Prof. Dolg could not clarify the pri nciples and features of the
GRECP method.
We should add that some more remarks could be given concernin g the application by
Stuttgart group of the core polarization potential togethe r with EACPPs, their “idea to
fit exclusively to quantum mechanical observables like tota l valence energies” (see the dis-
cussed paper), the features of the EACPP operator, actions t o avoid admixture of the inner
core states which are occupied by the electrons eliminated f rom calculations etc. Are the
valence energies obtained from the Dirac-Fock-Breit equat ions observable? Without a good
theoretical justification of the transferability and prope r application of these very progres-
sive ideas to other field, the result can be unsatisfactory. T his we have seen on example of
11application of the Wood-Boring aproximation to the EACPP ge neration. Moreover, when
developing a new method, one should at least to take into acco unt the basic achievements
in this field made earlier. Besides, the accuracy and reliabi lity of a newly developed method
should not be lower than that of already existing methods if t heir application require the
same computational efforts.
At last, we should say that we did not find any serious scientifi c analysis of our conclusions
and results in the Dolg’s paper but only some “political decl arations” are there. Therefore,
we are not going to answer in future on claims of similar quali ty as in the commented
paper only because do not want to lose time on such a level of di scussion as is proposed by
Prof. Dolg. Any well-justified critical remarks concerning our GRECPs (or the text in our
papers) are welcomed. We will answer with pleasure on questi ons dealing with RECPs. We
are ready to (and welcome) any public discussion on the RECP m ethods (e.g., within the
REHE newsletters) if they will be of common interest.
ACKNOWLEDGMENTS
We are grateful to M. Dolg for sending us the discussed paper t hat have stimulated us
for writing these comments.
The work on development of the GRECP method was supported by t he DFG/RFBR
grant N 96–03–00069, the INTAS grant No 96–1266 and by the RFB R grant N 99–03–33249.
12REFERENCES
∗E-mail: titov@hep486.pnpi.spb.ru ; http://www.qchem.pn pi.spb.ru .
[1] W. K¨ uchle, M. Dolg, H. Stoll, and H. Preuss, Mol. Phys. 74, 1245 (1991).
[2] U. H¨ aussermann, M. Dolg, H. Stoll, H. Preuss, P. Schwerd tfeger, and R. M. Pitzer, Mol.
Phys.78, 1211 (1993).
[3] A. V. Titov, A. O. Mitrushenkov, and I. I. Tupitsyn, “Effec tive core potential for pseudo-
orbitrals with nodes”, Chem. Phys. Lett. 185, 330-334 (1991).
[4] N. S. Mosyagin, A. V. Titov, and A. V. Tulub, “Generalized -effective-core-potential
method: Potentials for the atoms Xe, Pd and Ag”, Phys. Rev. A 50, 2239-2248 (1994).
[5] A. V. Titov and N. S. Mosyagin, “Self-Consistent Relativ istic Effective Core Potential
for Transition Metal Atoms: Cu, Ag and Au”, Struct. Chem. 6, 317-321 (1995).
[6] I. I. Tupitsyn, N. S. Mosyagin, and A. V. Titov, “Generali zed Relativistic Effective
Core Potential. I. Numerical calculations for atoms Hg thro ugh Bi”, J. Chem. Phys.
103, 6548-6555 (1995).
[7] N. S. Mosyagin, A. V. Titov, and Z. Latajka, “Generalized Relativistic Effective Core
Potential: Gaussian expansions of potentials and pseudosp inors for atoms Hg through
Rn”, Int. J. Quant. Chem. 63, 1107-1122 (1997).
[8] N. S. Mosyagin and A. V. Titov, “Comment on “Accurate relativistic effective core
potentials for the sixth-row main group elements” [J.Chem.Phys. 107, 9975 (1997)],
E-print: http://xxx.lanl.gov/abs/physics/9808006.
[9] N. S. Mosyagin, M. G. Kozlov, and A. V. Titov, “All-electr on Dirac-Coulomb and RECP
calculations of excitation energies for mercury with combi ned CI/MBPT2 method”, E-
print: http://xxx.lanl.gov/abs/physics/9804013 .
[10] A. V. Titov and N. S. Mosyagin, “Generalized Relativist ic Effective Core Potential
Method: Theoretical grounds”, Preprint PNPI No 2182 (Petersburg Nuclear Physics
Institute, St.-Petersburg, 1997), 81 pp.; Int. J. Quant. Ch em.71, 359-401 (1999).
[11] N. S. Mosyagin, E. Eliav, A. V. Titov, and U. Kaldor, “Com parison of relativistic ef-
fective core potential and all-electron Dirac-Coulomb cal culations of mercury transition
energies by the relativistic coupled cluster method”, J. Ph ys.B 33, 667 (2000).
[12] A. V. Titov and N. S. Mosyagin, “Generalized Relativist ic Effective Core Potential
Method: Theory and calculations”, Report on the European Re search Conference Rel-
ativistic Quantum Chemistry - Progress and Prospects (Acquafredda di Maratea, Italy,
10-15 April 1999); Russian J.Phys.Chem. (In Russian: Zh.Fi z.Khimii), accepted; E-
print: http://xxx.lanl.gov/abs/physics/0008160 .
[13] A.V.Titov, N.S.Mosyagin, A.B.Alekseyev, and R.J.Bue nker, “GRECP/MRD-CI calcu-
lations of the spin-orbit splitting in the ground state of Tl and of the spectroscopic
properties of TlH”, to be published; E-print: http://xxx.l anl.gov/abs/physics/0008155
.
[14] T.A.Isaev, N.S.Mosyagin, M.G.Kozlov, A.V.Titov, E.E liav and U.Kaldor, “Compari-
son of accuracy for the RCC-SD and CI/MBPT2 methods in RECP an d all-electron
calculations on Pb”, to be published.
[15] N.S.Mosyagin, A.V.Titov, E.Eliav, and U.Kaldor, “GRE CP/RCC-SD calculation of the
spectroscopic constants for the HgH molecule and its ions”, to be published.
13[16] A.V.Titov and N.S.Mosyagin, “About incompleteness of the conventional radially-local
RECP operator”, to be published.
[17] W. A. Goddard III, Phys. Rev. 174, 659 (1968).
[18] P. Durand and J. C. Barthelat, Theor. Chim. Acta. 38, 283 (1975).
[19] P. A. Christiansen, Y. S. Lee, and K. S. Pitzer, J. Chem. P hys.71, 4445 (1979).
[20] Y. S. Lee, W. C. Ermler, and K. S. Pitzer, J. Chem. Phys. 67, 5861 (1977); 73, 360
(1980).
[21] R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. L ajohn, and P. A. Chris-
tiansen, J. Chem. Phys. 93, 6654 (1990).
[22] V. F. Bratzev, G. B. Deyneka, and I. I. Tupitsyn, Bull. Ac ad. Sci. USSR 41, 173 (1977).
[23] Prof. Dolg twice refused to send us the parameters of his new ECPPs for Hg, at first,
when we were preparing paper [11] (and therefore we were forc ed to use their previous
PP version for Hg in our RECP comparison in [11]) and then, aft er publishing the
results with the new ECPPs in the commented paper.
[24] P. Pyykk¨ o, M. Tokman, and L.N. Labzowski, Phys. Rev. A 57, R689 (1998).
[25] L. Labzowski, I. Goidenko, M. Tokman, and P. Pyykk¨ o, Ph ys. Rev. A 59, 2707 (1999).
[26] E. Lindroth, A.-M. M˚ artensson-Pendrill, A. Ynnerman , and P. ¨Oster, J. Phys. B 22,
2447 (1989).
[27] M. G. Kozlov, S. G. Porsev, and I. I. Tupitsyn, E-print:
http://xxx.lanl.gov/abs/physics/0004076.
[28] U. Kaldor and E. Eliav, Adv. Quantum Chem. 31, 313 (1998)
[29] J. H. Wood and A. M. Boring, Phys. Rev. B 18, 2701 (1978).
[30] M. Dolg, 1999, private communication.
[31] C. H. Teichteil, M. Pelissier, and F. Spiegelmann, Chem . Phys. 81274 (1983); C. H. Te-
ichteil and F. Spiegelmann, Chem. Phys. 81283 (1983).
14TABLES
TABLE I. The FCA errors calculated by us with the help of the HF D [22] code at the
all-electron Dirac-Fock level within the point nuclear mod el for the states averaged upon the
nonrelativistic configurations. The errors of the new ECPPs from Table 1 in the Dolg’s paper. All
values are in cm−1.
configuration error
ECPPaECPPbFCAc
Hg 6 s20.0 0.0 0.0
6s16p11.3 0.0 0.6
Hg+6s1-0.1 0.0 1.0
7s1-0.4 0.0 4.0
8s11.1 0.1 4.3
9s11.6 -0.1 4.4
6p10.6 0.0 3.3
7p1-3.3 0.0 4.3
8p1-0.8 0.0 4.4
9p10.6 0.0 4.5
Hg++2.6 0.0 4.5
aenergy-consistent pseudopotential with 26 adjustable par ameters.
benergy-consistent pseudopotential with 54 adjustable par ameters.
cfrozen core approximation with the 1 s, . . ., 4ffrozen shells taken from the 6 s2Hg state.
15APPENDIX A:
Because the reader can be not familiar with the criticized pa per of Prof. Dolg, below we
present its section 5.3, which is discussed in our comments, without any changes.
M. Dolg, Section 5.3 “Limitations of accuracy”
Effective core potentials are usually derived for atomic sys tems at the finite difference
level and used in subsequent molecular calculations using fi nite basis sets. They are desig-
nated to model the more accurate all-electron calculations at low cost, but without significant
loss of accuracy. Unfortunately the correct relativistic a ll-electron Hamiltonian for a many-
electron system is not known and the various pseudopotentia ls merely model the existing
approximate formulations. For most cases of chemical inter est, e.g., geometries and binding
energies, it usually does not matter which particular Hamil tonian model is used, i.e., typ-
ically errors due to the finite basis set expansion or the limi ted correlation treatment are
much larger than the small differences between the various al l-electron models.
Table 1. Relative average energy of a configuration of Hg from all-electron (AE) multi- con-
figuration Dirac-Hartree-Fock (DHF) average level calcula tions using the Dirac-Coulomb
(DC) Hamiltonian with a finite nucleus with Fermi charge dist ribution (fn) or a point nu-
cleus (pn). Contributions from the frequency-dependent Br eit (B) interaction (frequency of
the exchanged photon 103cm−1) and estimated contributions from quantum electrodynamic s
(QED, i.e., self-interaction and vacuum polarization) wer e evaluated in first-order perturba-
tion theory. Errors of energy-consistent pseudopotential s (PP) with 20 valence electrons and
different numbers of adjustable parameters with respect to t he AE DHF(DC,pn)+B+QED
data. All values in cm−1.
configuration AE, DHF contribution error
(DC)+B+QED
fn pn B QED PPaPPb
Hg 6 s20 0 0.0 0.0 0.0 0.0
6s16p135632.3 35674.4 -52.5 -18.7 1.3 0.0
Hg+6s168842.1 68885.1 -98.6 -11.6 -0.1 0.0
7s1154127.4 154206.2 -220.6 -42.4 -0.4 0.0
8s1178127.5 178215.5 -238.4 -41.7 1.1 0.1
9s1188751.0 188843.2 -244.1 -40.6 1.6 -0.1
6p1122036.8 122128.9 -154.2 -41.8 0.6 0.0
7p1167514.3 167609.2 -224.1 -40.3 -3.3 0.0
8p1183808.0 183903.6 -238.5 -40.0 -0.8 0.0
9p1191697.2 191793.1 -244.0 -39.6 0.6 0.0
Hg++206962.2 207058.4 -249.8 -39.5 2.6 0.0
aenergy-consistent pseudopotential with 26 adjustable par ameters.
benergy-consistent pseudopotential with 54 adjustable par ameters.
16For very accurate calculations of excitation energies, ion ization potentials and electron
affinities, or for a detailed investigation of errors inheren t in the effective core potential
approach, however, such differences might become important . Tables 1 and 2 demonstrate
that for very special cases like Hg, with a closed 5d10-shell in all electronic states considered,
a small-core energy-consistent pseudopotential using a se milocal ansatz reaches an accuracy
of 10 cm−1, which is well below the effects of the nuclear model, the Brei t interaction or
higher-order quantum electrodynamical contributions. We also note that differences between
results obtained with a frequency-dependent Breit term and the corresponding low-frequency
limit amount to up to 10 cm−1. Moreover, the quantum electrodynamic corrections listed in
tables 1 and 2 might change by up to 20 cm−1when more recent methods of their estimation
are applied98,99. Therefore, it is important to state exactly which relativi stic all-electron
model the effective core potential simulates and, when compa ring effective core potentials of
different origins, to separate differences in the underlying all-electron approach from errors in
the potential itself, e.g., due to the size of the core, the me thod of adjustment or the form of
the valence model Hamiltonian. In this context we want to poi nt out that the seemingly large
errors for energy-adjusted pseudopotentials reported by M osyagin et al.100,101are mainly due
to the invalid comparison of Wood-Boring-energy-adjusted and Dirac-Fock-orbital-adjusted
pseudopotentials to all-electron Dirac-Fock data, i.e., d ifferences in the all-electron model
are considered to be pseudopotential errors.
Note that in the above example of Hg the average energy of a con figuration (table 1)
and the fine-structure (table 2) of one-valence electron sta tes is more accurately represented
than the fine-structure of the 6 s16p1configuration. The small errors in the latter case
are a consequence of the pseudoorbital transformation and t he overestimation of the 6s-
6p exchange integral with pseudo-valence spinors. This err or could be reduced further
upon using a smaller core, but the efficiency of the approach wo uld be sacrificed. It is
also obvious from the compiled data that the accuracy of the v alence model Hamiltonian
is also a question of the number of adjustable parameters. Cl aims that such very high
accuracy as demonstrated here can only be achieved by adding nonlocal terms for outer
core orbitals to the usual semilocal terms100,101appear to be invalid, at least for energy-
consistent pseudopotentials. Moreover, additional nonlo cal terms obviously do not improve
98P. Pyykk¨ o, M. Tokman, and L.N. Labzowski, Estimated valenc e-level Lamb shifts for group 1
and group 11 metal atoms, Phys. Rev. A 57, R689 (1998).
99L. Labzowski, I. Goidenko, M. Tokman, and P. Pyykk¨ o, Calcul ated self-energy contributions
for an ns valence electron using the multiple-commutator me thod, Phys. Rev. A 59, 2707 (1999).
100N.S. Mosyagin, A.V. Titov, and Z. Latajka, Generalized Rela tivistic Effective Core Potential:
Gaussian expansions of potentials and pseudospinors for at oms Hg through Rn, Int. J. Quant.
Chem. 63, 1107 (1997).
101N.S. Mosyagin, E. Eliav, A.V. Titov, and U. Kaldor, Comparis on of relativistic effective core
potential and all-electron Dirac-Coulomb calculations of mercury transition energies by the rela-
tivistic coupled-cluster method, J. Phys. B 33, 667 (2000).
17the performance for atomic states with a 5 d9occupation or in molecular calculations (cf.,
e.g., tables III and XVII in Mosyagin et al.100).
Table 2. As table 1, but for fine-structure splittings. All va lues in cm−1.
configuration splitting AE,DHF contribution error
(DC)+B+QED
fn pn B QED PPaPPb
Hg 6 s16p1 3P1−3P0 1987.7 1988.6 -25.5 0.9 -14.7 3.0
3P3−3P0 6082.6 6084.8 -96.8 2.9 -28.3 -3.5
1P1−3P0 22994.4 22982.3 -72.4 2.2 -12.4 -9.4
Hg+6p1 2P3/2−2P1/2 7765.3 7768.8 -132.8 4.8 -14.8 -0.1
7p1 2P3/2−2P1/2 2136.8 2137.9 -29.0 1.1 -1.7 0.2
8p1 2P3/2−2P1/2 939.4 939.9 -12.1 0.4 -4.6 -0.3
9p1 2P3/2−2P1/2 498.7 498.9 -6.2 0.2 -3.5 0.0
aenergy-consistent pseudopotential with 26 adjustable par ameters.
benergy-consistent pseudopotential with 54 adjustable par ameters.
18 |
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