The physics of particle accelerators : an introduction

164 Pages·2013·13.04 MB·English

by  Wille

The Physics of
Particle Accelerators
An Introduction
KLAUS WILLE
Professor Physics Department
University of Dortmund
Translated by
JASON McFALL
Physics Department
University of Bristol
OXFORD
UNIVERSITY .PRESS
OXFORD
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Preface
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Published in the United States by the need for very high particle energies, and have grown to sizes of 10 km and
by Oxford University Press Inc., New York above. In circular electron-beam accelerators an intense form of electromagnetic
This English translation © Oxford University Press, 2000 radiation, known as synchrotron radiation, is emitted at energies above a few
The moral rights of the author have been asserted tens of Me V. This radiation has some very useful properties, and for the past
Database right Oxford University Press (maker) three decades has primarily been used in fixed target experiments. The impor
First published in English 2000 tance of this synchrotron radiation has grown so much around the world that
First published in German in 1996 under the title today many machines are built exclusively for this purpose.
Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen
This book sets out to explain systematically the key physical principles be
by B.G. Teubner, Stuttgart
hind particle accelerators and the basics of high-energy particle physics as well
All ~ights r~served. No part of this publication may be reproduced,
as the production of synchrotron radiation. There are so many different kinds of
store? m a retr1e~al system, or transmitted, in any form or by any means,
without the pno~ permission in writing of Oxford University Press, accelerator, with such a variety of different uses, that it is not possible to consider
or as express!~ per~mtted by l~w, ?r under terms agreed with the appropriate all aspects of current accelerator technology here. Instead, after introducing the
rep:ograph1cs nghts orgamzation. Enquiries concerning reproduction
fundamental principles common to all types of accelerators, we concentrate upon
outside the scope of the above should be sent to the Rights Department
Oxford University Press, at the address above ' the electron storage ring. This type of accelerator has proven to be extraordinar
You must not circulate this book in any other binding or cover ily successful, both for elementary particle physics and for the production of syn
and you must impose this same condition on any acquirer chrotron radiation. The criteria used to optimize these machines for these two dif
Library of Congress Cataloging in Publication Data ferent uses are discussed extensively. Throughout the text the aim is to present all
Wille, Klaus, prof. derivations clearly and to carefully justify any approximations that are needed.
[P?ysik der !eilchenbeschleuniger und Synchrotronstrahlungsquellen. English]
Wherever possible, the description is illustrated with figures and diagrams.
The physics of particle accelerators: an introduction/Klaus Wille; translated by Jason McFall.
p.cm. The first ideas for this book were developed over many years of building and
Includes bibliographical references and index. operating accelerators at the German Electron-Synchrotron Laboratory DESY
1. Particle accelerators. I. Title.
in Hamburg. It often proved difficult, in the short time available, to bring young
QC787.P3 W55 2000 539.7'3-dc21 00-052438
physicists and engineers up to speed with the physics of modern accelerators,
ISBN O 19 850550 7 (Hbk)
ISBN O 19 850549 3 (Pbk) since hardly any suitable literature was available. Furthermore, there were al
ways misunderstandings with the experimental groups using the accelerators,
Typeset by the translator using LaTeX
Printed in Great Britain due to a lack of understanding of how accelerators work. This was addressed
on acid-free paper by first by seminars and then by courses of lectures at autumn schools. From these
Biddles Ltd., Guildford & King's Lynn
beginnings arose a special lecture course on the physics of particle accelerators,
which has been given since 1987 at the University of Dortmund. The experiences
gained from this, along with the comments of the students, have shaped the
treatment presented in this text.
This book is aimed at all students wishing either to become directly involved
in the development and construction of accelerators, or to use them to experiment
vi Preface
in nuclear and particle physics or use synchrotron radiation in fixed target exper
iments. In addition, it is recommended to all practising physicists and engineers
who wish to become familiar with the operation and physical limitations of the
Contents
particle accelerators at which they perform their experiments.
In putting together ·the various topics in this book, I benefited from numer
ous suggestions and discussions with my colleagues, with whom I have enjoyed
many years of very fruitful collaboration. Particular thanks are due to the former
director of the accelerator division at the German Electron Synchrotron DESY, List of symbols xi
Professor G. A. Voss, under whose leadership I gained much valuable experience.
1 Introduction 1
Dortmund 1.1 The importance of high energy part~cles in fundamental
January 1996 K. W. research 1
1.2 Forces used in particle acceleration 3
Translator's note 1.3 Overview of the development of accelerators 4
1. 3 .1 The direct-voltage accelerator 5
I am grateful to Dr Roman Walczak, who recommended the original edition of
1.3.2 The Cockroft-Walton cascade generator 6
this book to his students at Oxford University. I am even more grateful to the
1.3.3 The Marx generator 7
students for rebelling because it was in German and demanding a translation!
1.3.4 The Van de Graaff accelerator 8
1.3.5 The linear accelerator 9
J. McF.
1.3.6 The cyclotron 13
1.3.7 The microtron 16
1.3.8 The betatron 17
1.3.9 The synchrotron 19
1.4 Particle production by colliding beams 22
1.4.1 The physics of particle collisions 22
1.4.2 The storage ring 25
1.4.3 The linear collider 27
2 Synchrotron radiation 30
2.1 Radiation from relativistic particles 30
2.1.1 Linear acceleration 31
2.1.2 Circular acceleration 32
2.2 Angular distribution of synchrotron radiation 35
2.3 Time dependence and frequency spectrum of the radiation 37
2.4 Storage rings for synchrotron radiation 40
3 Linear beam optics 44
3.1 Charged particle motion in a magnetic field 44
3.2 Equation of motion in a co-moving coordinate system 46
3.3 Beam steering magnets 50
3.3.1 Calculation of magnetic fields for beam steering 51
3.3.2 Conventional ferromagnets 53
3.3.3 Superconducting magnets 58
3.4 Particle trajectories and transfer matrices 65
viii Contents Contents ix
3.5 Calculation of a particle trajectory through a system of 5.2.2 Cylindrical resonant cavities 159
many beam-steering magnets 72 5.3 Accelerating structures for linacs 163
3.6 Dispersion and momentum compaction factor 74 5.4 Klystrons as power generators for accelerators 169
3. 7 Beta function and betatron oscillation 77 5.5 The klystron modulator 171
3.8 The phase spate ellipse and Liouville's theorem 80 5.6 Phase focusing and synchrotron frequency 176
3.9 Beam cross-section and emittance 81 5. 7 Region of phase stability (separatrix) 180
3.10 Evolution of the beta function through the magnet
6 Radiative effects 185
structure 83
6.1 Damping of synchrotron oscillations 185
3.10.1 Method 1 83
6.2 Damping of betatron oscillations 188
3.10.2 Method 2 85
6.3 The Robinson theorem 191
3.11 Determination of the transfer matrix from the beta function 88
6.4 The beam emittance 192
3.12 Matching of beam optics 89
6.4.1 The lower limit of the beam emittance: the low emit
3.12.1 The one-dimensional case 90
tance lattice 197
3.12.2 Then-dimensional case 91
3.13 Periodicity conditions in circular accelerators 93 7 Luminosity 202
3.13.1 The periodic solution 93 7.1 Beam current restriction due to the space charge effect 204
3.13.2 The symmetric solution 95 7.2 The 'mini-beta' principle 213
3.13.3 Worked example: beam optics of a circular accelera
tor with a FODO structure 97 8 Wigglers and undulators 217
3.14 Tune and optical resonances 101 8.1 The wiggler or undulator field 217
3.14.1 Periodic solution of Hill's differential equation 101 8.2 Equation of motion in a wiggler or undulator 222
3.14.2 Floquet's transformation 103 8.3 Undulator radiation 227
3.14.3 Optical resonances 104 9 The free electron laser (FEL) 232
3.15 The effect of magnetic field errors on beam optics 112
9.1 Conditions for energy transfer in the FEL 233
3.15.1 Effect of dipole kicks 112
9.2 Equation of motion for electrons in the FEL (pendulum
3.15.2 Effect of quadrupole field errors 115
equation) 236
3.16 Chromaticity of beam optics and its compensation 120 9.3 Amplification of the FEL (low gain approximation) 241
3.17 Restriction of the dynamic aperture by sextupoles 123
9.4 The Madey theorem 247
3.18 Local orbit bumps 127 9.5 FEL amplification in the high-gain regime 248
3.18.1 Examples of local orbit bumps 132
9.6 The FEL amplifier and FEL oscillator 250
4 Injection and extraction 136 9. 7 The optical klystron 252
4.1 The process of injection and extraction 136 9.8 Time structure of the FEL radiation 254
4.2 Particle sources 137 10 Diagnostics 258
4.3 The fundamental problem of injection 141
10.1 Observation of the beam and measurement of the beam
4.4 Injection of high proton and ion currents by 'stacking' 142
current 258
4.5 Injection of proton beams using stripping foils 144
10.1.1 The fluorescent screen 258
4.6 Injection into an electron storage ring 145 10.1.2 The Faraday cup 259
4. 7 Kicker and septum magnets 147 10 .1.3 The wall current monitor 261
5 RF systems for particle acceleration 152 10.1.4 The beam transformer 262
5.1 Waveguides and their properties 152 10.1.5 The current transformer 264
5.1.1 Rectangular waveguides 154 10.1.6 The measurement cavity 266
5.1.2 Cylindrical waveguides 156 10.2 Determination of the beam lifetime in a storage ring 269
5.2 Resonant cavities 158 10.3 Measurement of the momentum and energy of a particle
5.2.1 Rectangular waveguides as resonant cavities 158 beam 271
x Contents
10.3.1 The magnetic spectrometer 271 Symbols
10.3.2 Energy measurement by spin depolarization 273
10.4 Measurement and correction of the beam position 274
A acceptance
10.4.1 Transverse beam position measurement 275
A vector potential
10.4.2 Correction of the transverse field position 281
ax, az, as damping constants
10.5 Measurement of the betatron frequency and the tune Q 287
B,B magnetic flux density
10.6 Measurement of the synchrotron frequency 291
iJ wiggler field along the beam axis
10. 7 Measurement of the optical parameters of the beam 294
B brilliance
10.7.1 Measurement of the dispersion 294
B beta matrix
10. 7.2 Measurement of the beta function 295
C capacitance
10.7.3 Measurement of the chromaticity 296
c = 2.99793 x 108 m s-1 speed of light in· a vacuum
A Maxwell's equations 297 D(s) dispersive trajectory
e = 1.60203 x 10-19 C elementary charge
B Important relations in special relativity 299
E energy
C General equation of an ellipse in phase space 302 E'"Y photon energy
Ep proton energy
Bibliography 304
Ee electron energy
Index 310 E electric field strength
E(s) beam envelope
F, F force
F photon flux
revolution frequency
frev
G, GN FEL gain
G(x) horizontal field distribution function
g = 8Bx/8z = 8Bz/8x field gradient
h = 6.6252 X 10-34 JS Planck's constant
H,H magnetic field strength
H(s) optical function
beam current
fbeam
I(w) intensity distribution
Ji (f,) Bessel function
Jx, Jz, Js damping numbers
K wiggler parameter
KL FEL field parameter
k quadrupole strength
k, kx, ky, kz wavenumbers in a waveguide
kc cut-off wavenumber
,C luminosity
L circumference of circular accelerator
L inductance
M transfer matrix
m sextupole strength
m particle mass
mo particle rest mass
xii List of symbols List of symbols xiii
me = 9.1081 X 10-31 kg rest mass of electron Ac cut-off wavelength
mp = 1.67236 X 10-27 kg rest mass of proton ARF RF wavelength
N number of particles Au undulator period
P, p particle momentum µ0 = 41r10-7 Vs/ Am permeability of free space
PRF RF power µr relative permeability
PL FEL power V frequency
Pµ four-momentum VRF RF frequency
Po, Ps radiated power ~, ~x, ~z chromaticity
b..p/p relative momentum deviation p(x, z, s) density distribution function
q = VRF/ frev harmonic number CJ interaction cross-section
Q, Qx, Qz tune or working point Cfx, Clz, Cfs beam sizes
b..Q tune shift T pulse length
Q quality factor of a resonator TRF RF period
position vector <I>(x, z, s) scalar potential
bending radius </>(s) Floquet variable
shunt impedance cp(x, z, s) scalar potential
brightness \J!(s) betatron phase
spectral function \J!o nominal phase
voltage w frequency
u, u particle velocity We critical frequency
V, V particle velocity Wchar characteristic frequency
W(r, t) wavefunction Wz = eB/m cyclotron frequency
W, Wo energy loss during acceleration
s coordinate in beam direction
T temperature
T, T revolution time
0
t time
X trajectory vector
horizontal beam coordinate
X
z horizontal beam coordinate
a momentum compaction factor
a(s) = -f](s)/2 gradient of beta function
/3=v/c relative velocity
/3(s) beta function
'Y = E/moc2 relative energy
'Yt transition energy
-y(s) = (1 + a(s))/f3(s) optical function of the beam
Eo = 8.85419 x 10-12 As/Vm permittivity of free space
beam emittance
E, Ex, Ez
critical energy
Ee
TJ efficiency
TJ( s) Floquet variable
e radiation opening angle
bending angle ( of steering coils)
wavelength
1
Introduction
1.1 The importance of high energy particles in fundamental
research
The study of the basic building blocks of matter and the forces which act be
tween them is a fundamental area of physics. The structures under scrutiny are
extraordinarily small, sometimes well below 10-15 m, and in order to perform
experiments at this scale probes with correspondingly high spatial resolution are
needed. Visible light, with a wavelength .A ~ 500 nm, is wholly inadequate. In
stead, high energy photon or particle beams have proven to be excellent tools,
and the results of elementary particle physics would be unimaginable without
them. High energy particle beams are thus essential to this field of experimental
physics.
In general a microstructure may only be resolved by a probe, for example
electromagnetic radiation, if the wavelength is small compared to the size of
the structure. Thus wavelengths below,\< 10-15 mare required in elementary
particle physics. The photon energy of this radiation is
he _
E'Y = hv = ~ = 2 x 10 10 J. (1.1)
If these photons are produced via bremsstrahlung from energetic electron beams,
then particle energies of
Ee =eU with (1.2)
are required. T9 achieve such energies, the electrons in the beam must cross a
total electrical potential U > Ee/ e = 1.2 x 109 V. Similar considerations apply
to particle beams, for which the de Broglie wavelength must again be small
compared to the size of the structure. This wavelength is given by the relation
AB=!!_= he (1.3)
P E'
where p and E are the momentum and energy of the particle, respectively. Com
paring this with relation (1.1) shows that similarly high particle energies are also
necessary here.
In physics, energy is usually measured in the unit of the Joule (J). However,
this unit is not very convenient when describing particle beams, and in general
2 Introduction Forces used in particle acceleration 3
momentum, and hence also energy, which is then not available for particle pro
duction. The -y-ray energy needed for particle production is therefore always
higher than that given by relation (1.4), namely
E'Y > 2mec2 = 1.637 x 10-13 J = 1.02 MeV. (1.5)
This is the threshold energy for the production of electron-positron pairs, where
each particle has a mass me= 9.108 x 10-31 kg, corresponding to a rest energy of
Ea = 511 ke V. The rest energies of elementary particles investigated nowadays
are considerably higher. A few examples are:
Fig. 1.1 Definition of the eV. (1 keV = 103
+ eV, 1 MeV = 106 eV, 1 GeV = 109 eV, 1 TeV proton p Ea 938 MeV
1 volt = 1012 eV). b quark b Ea 4735 MeV
vector boson Zo Ea 91190 MeV
the unit of the e V (e lectron volt) is preferred. This is the kinetic energy gained t quark t Ea 174000 MeV
by a particle of elementary charge e = 1.602 x 10-19 C as it crosses a potential
In order to produce these particles, correspondingly high energies must be avail
difference of l::,.U = l V. The conversion factor is thus 1 eV = 1.602 x 10-19 J. To
able.
describe higher particle energies the convenient units keV, MeV, GeV, and TeV
are also used (see Fig. 1.1). It should be noted at this point that the SI (MKSA) 1.2 Forces used in particle acceleration
system of units will generally be used in this book. Where different units are
Since the velocity v of elementary particles studied in collisions is generally close
used in particular cases, this will be clearly stated.
to the velocity of light (c = 2.997925 x 108 m/s), the energy must be written in
Another important aspect of elementary particle physics, as well as the res
the relativistically invariant form
olution of the finest structures of matter, is the production of new, mostly very
short-lived, particles. The amount of energy needed to produce a particle follows
(mo = rest mass). (1.6)
directly from the fundamental relation
Here the only free parameter is the momentum p of the particle. In the usual
E = mc2• (1.4) notation /3 = v/c and 'Y = (1 - /32)-112, the relativistic particle momentum is
given by the relation
p = mv = -ymov (1.7)
Note that most particles can only be produced along with their antiparticles
in pairs; for example electrons and positrons are produced together from high and the energy-dependent particle mass m = -ymo. The increase in energy E in
energy 'Y rays (Fig. 1.2). (1.6) is the same as the increase in the particle momentum p. The momentum
can only be changed by the action of a force F on the particle, as described by
As a result of the conservation of momentum, a reaction of this kind can
Newton's second law of motion
only take place in the vicinity of a heavy nucleus. The nucleus itself gains some
p=F. (1.8)
In order to reach high kinetic energies, a sufficiently strong force must be exerted
on the particle for a sufficient period of time. Nature offers us four different forces,
listed along with their most important properties in Table 1.1. It is clear that
e+/
forces with a range below 10-15 m are of no practical use for particle acceleration.
The strong force, which might have been useful because of its relative strength,
is therefore ruled out, as of course is the weak force. Gravity is many orders of
photon /
magnitude too weak. The only possible choice left is the electromagnetic force.
~ Q
When a particle of velocity v passes through a volume containing a magnetic
field B and an electric field E it is acted upon by the Lorentz force
E nucleus~
Fig. 1.2 Production of an e+ -e- pair
jF=e(vxB+E).I (1.9)
in the collision of a high-energy photon
with a heavy nucleus. As the particle moves from point r to r2, its energy changes by the amount
1
4 Introduction Overview of the development of accelerators 5
Table 1. 1 The four forces of nature
1.3.1 The direct-voltage accelerator
force relative strength range [m] particles The simplest particle accelerators use a constant electric field between two elec
affected trodes, produced by a high voltage generator. This principle is illustrated in
Fig. 1.3. One of the electrodes contains the particle source. In the case of elec
gravity 6 X 10-39 00 all particles tron beams this is a thermionic cathode, widely used in vacuum tube technology.
electromagnetism 1/137 00 charged Protons, as well as light and heavy ions, are extracted from the gas phase by
particles using a further DC or high frequency voltage to ionize a very rarified gas and so
strong force ~1 10-15 - 10-16 hadrons produce a plasma inside the particle source. Charged particles are then contin
weak force 10-5 « 10-16 hadrons uously emitted from the plasma, and are accelerated by the electric field. In the
& leptons accelerating region there is a relatively good vacuum, in order to avoid particle
collisions with residual gas molecules. The particles are thus continuously accel
erated without any loss of energy until they reach the second electrode. There
JT2 JT2( they exit the accelerator and usually traverse a further field-free drift region,
flE = F · dr = e v x B + E) · dr. (1.10) through which they travel at constant energy until they reach a target. This
principle is very widely employed in research and technology, and all modern
T1 T1
VDU screens and oscilloscopes are based upon it. The particle energies which
During the motion the path element dr is always parallel to the velocity vector
can be reached in this way are, however, very limited by modern standards.
v. The vector v x B is thus perpendicular to dr, i.e. (v x B) · dr = 0. Hence
In electrostatic accelerators the maximum achievable energy is directly pro
the magnetic field B does not change the energy of the particle. Acceleration
portional to the maximum voltage which can be developed, and this gives the
involving an increase in energy can thus only be achieved by the use of electric
energy limit. This may be seen from the curves in Fig. 1.4, which show the de
fields. The gain in energy follows directly from ( 1. 10) and is
pendence of current on voltage in electrostatic accelerators. The current may
JT2 be divided into essentially three components. Since the conductivity of an insu
flE = e E · dr = eU, (1.11) lator is never quite zero, there is always an ohmic component, which increases
in proportion to the voltage. This element can be made very small by a judi
T1
cious choice of materials and careful construction. The second component arises
where U is the voltage crossed by the particle.
due to the ions which are always present in the residual gas. It very quickly
Although magnetic fields do not contribute to the energy of the particle, they
reaches a constant saturation level if the applied voltage is so high that space
play a very important role when forces are required which act perpendicular to
charge effects become negligible and all the ions are sucked out. The acceler
the particle's direction of motion. Such forces are used to steer, bend and focus
ated particle beam is also part of this component of the current. The third
particle beams.
Accelerator physics is concerned with these two problems: the acceleration
particle source
and steering of particle beams. Both processes rely on the electromagnetic
force and hence on the foundations of classical electrodynamics. Here Maxwell's E-field target
high voltage
equations are of fundamental importance. In addition, the most important results generator
of the special theory of relativity are required. A knowledge of these fundamen u
tals will be assumed in what follows; a summary of the most important relations
can be found in the appendices.
1.3 Overview of the development of accelerators
Since the 1920s various machines have been developed to accelerate particle vacuum tube
beams for experimental physics, always with the principal objective of reaching
particle beam
ever higher energies. An overview of the most significant advances will be given in
this chapter, and the various types of accelerator which have played an important
part in physics so far will be introduced. A detailed description of the early
Fig. 1.3 General principle of the electrostatic accelerator.
developments in accelerator physics can be found in Livingston and Blewett [1].
Overview of the development of accelerators 7
6 Introduction
6U voltage
I
Cs
4U
corona
formation c4
2U 4U
c2
2U
c1
:)
t
10n transformer -
current
Fig. 1.5 Operation of the Cockroft-Walton cascade generator.
0 u of 2U. In the same way as before, the third diode ensures that the potential
at point C does not fall below 2U. Here it varies between 2U and 4U, and so
with the help of the fourth rectifier a voltage of 4U is generated. The pattern is
Fig. 1.4 Dependence of the current on the applied high voltage in electrostatic accel
repeated, with many such rectifier stages arranged one after another. Without
erators.
loading, the maximum achievable voltage is then 2nU, where n is the number of
component, corona formation, leads to the actual energy limit. At low voltages rectifier stages.
It must be noted here that a current I must always be drawn from the genera-
it is completely unmeasurable, but at higher voltages the field strength close to
tor. This always discharges the capacitors slightly when the diodes are in reverse
the electrodes grows so much that ions and electrons produced in this region
bias, leading to a somewhat lower generated voltage than that expected simply
are accelerated to considerable energies. They collide with gas molecules and
from the number of rectifier stages. The current-dependent voltage generated by
so produce many more ions, which themselves undergo the same process. The
the cascade circuit is given more exactly by the relation
result is an avalanche of charge carriers causing spark discharge and the break
cdoorwonn ao ff otrhme ahtiiognh. vIotl tisa gteh.i sT ehfefe cctu rwrehnict hg rliomwist se txhpeo nmeanxtiiamlluym in a cthheie vreagbiloen e noefr tghye. Ut 0 t = 2Un - -2w1rC I ( -32 n 3 + -41 n 2 + -112n ) . (1.12)
Voltages of a few MV are technically possible, and hence particles of unit charge
It is immediately apparent that a high capacitance C and a high operating
can reach a maximum energy of a few MeV . This is the general limit for elec
frequency w strongly reduce the dependence on the current.
trostatic accelerators: substantially higher energies cannot be achieved with this
In Cockroft-Walton accelerators, voltages up to about 4 MV can be reached.
technology.
By using pulsed particle beams with pulse lengths of a few µs, beam currents of
1.3.2 The Cockroft-Walton cascade generator several hundred mA have been achieved.
A significant problem for electrostatic accelerators is the production of suffi 1.3.3 The Marx generator
ciently high accelerating voltages. At the beginning of the 1930s, Cockroft and
An alternative way to produce high voltages for particle acceleration is that
Walton [2] developed a high voltage generator based on a system of multiple
used by the Marx generator. It too has a cascade design, but only delivers
rectifiers. At their first attempt they achieved a voltage of around 400 000 V.
short voltage pulses. However, this allows very high particle currents. As Fig. 1.6
The operation of this generator [3, 4], also known as the Greinacker circuit, is
shows, the Marx generator consists of a network of resistors and capacitors. The
explained in Fig. 1.5. At point A a transformer produces a sinusoidally varying
voltage U(t) = U sin wt of frequency w. The first rectifying diode ensures that capacitors C1 to C4, which are connected quasi in parallel, are charged across the
resistors R up to a voltage U by a high voltage supply. As the applied voltage
at point B the voltage never goes negative. Thus the capacitor C charges up to
1 increases it approaches the firing voltage of the spark gaps. When this firing
a potential U. At point B the voltage now oscillates between the values 0 and
voltage is reached, spark discharge occurs and the spark gaps act as very low
2U. The capacitor C2 is then charged up via the second rectifier to a potential
resistance switches. The value of the load resistors R is, however, very large. As