Physics and Technology of Linear Accelerator Systems: Proceedings of the 2002 Joint USPAS-CAS Japan-Russia Accelerator School, Long Beach, California, 6-14 November 2002

Phvs ics TechnoIogLJ., of

Linear Accelerator Systems

Phpics Technology J

of

Linear Accelerator Systems Proceedings of the 2002

J o i n t U S PAS - CA S - J a p an - R u s s i a Accelerator School Long Beach, California 6 - 14 November 2002

editors

Helmut Wiedemann Stanford University, USA

Daniel Brandt CERN. Switzerland

Eugene A Perevedentsev The Budker Institute of Nuclear Physics, Russia

S hin-ic h i Ku ro kawa KEK, Japan

r pWorld Scientific N E W JERSEY

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PHYSICS AND TECHNOLOGY OF LINEAR ACCELERATOR SYSTEMS Proceedings of the 2002 Joint USPAS-CAS-Japan-Russia Accelerator School Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, orparts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE Following a long tradition we organized a biannual Joint Particle Accelerator School, JAS2002. These schools started in 1985 as ajoint venture between the CERN and the US Particle Accelerator Schools. Each school is dedicated to a particular Particle Accelerator topic addressing status and ongoing developments within that theme. The first five schools were:

Nonlinear Dynamics, Santa Margherita di Pula, Sardinia, Italy, 1985 New Acceleration Methods and Techniques, South Padre Island, USA, 1986 Observation, Diagnosis and Correction, Anacapri, Italy, 1988 Beam Intensity Limitations, Hilton Head Island, SC, USA, 1990 Factories with e+e- Rings. Benelmadena, Spain, 1992 Proceedings of these five schools were published in the Lecture Notes in Physics series by Springer as volumes 247,296,343,400and 425. In 1993 the KEK Particle Accelerator School (KEKPAS) joined and resulted in the following schools:

Frontiers of Accelerator Technology, Maui, Hawaii, USA, 1994 Radio Frequency Engineering for Particle Accelerator Physics, Hayama and Tsukuba, Japan, 1996 Proceedings for these two schools were published by the World Scientific Publishing Company. Finally in 1996 the Russia Accelerator School joined and the location of these schools rotates now within those four regions:

Beam Measurement, Montreux and CERN, Switzerland, 1998 World Scientific Publishing Company. High Quality Beams, St. Petersburg and JINR, Dubna, 2000 AIP Conference Proceedings #592 Linear Accelerator, Long Beach, CA, USA, 2002 World Scientific Publishing Company.

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On behalf of the JAS2002 we express our sincere thanks to S.Y. Lee, M. Paul and S. Winchester of the US Particle Accelerator School for the excellent planning and execution of the school. We also thank David Sutter from the US Department of Energy and F. Bernthal from the Universities Research Association (URA) for their financial support, and the regional Accelerator School organizations (CAS, KEKPAS, RAS) for their continued support and encouragement. Our special thanks goes to the lecturers who agreed to share their intellectual experience at the school and document their lectures in these proceedings. We appreciate the editing skills of Margaret Dienes who has supported our efforts to produce quality proceedings since 1985. Last but not least we thank all the participants for their attendance and participation at the lectures. D. Brandt, CAS, Geneva, Switzerland

S.I. Kurokawa, KEKPAS, Tsukuba, Japan E. Perevedentsev, Russia Accelerator School, Novosibirsk, Russia

H. Wiedemann, USPAS, Stanford, CA, USA

October 15,2003

CONTENTS

Preface

v

Ion Linacs T. P. Wangler

1

Modern Trends in Induction Accelerator Technology G. J. Caporaso

24

RFQ - Accelerators A. Schempp

60

RF Structures (Design)

79

H. Henke Fabrication and Testing of RF Structures E. Jensen

130

Computational Tools for RF Structure Design E. Jensen

155

Wakefields and Instabilities in Linacs G. Stupakov

180

Beam Manipulation and Diagnostic Techniques in Linacs P. Logatchov

213

Space Charge and Beam Halos in Proton Linacs F. Gerigk

257

Power Sources for Accelerators beyond X-Band E. R. Colby

289

Recirculated and Energy Recovered Linacs G. A . Krafst

301

Muon Colliders and Neutrino Factories: Basics and Prospects A. Skrinsky

322

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Members of the Organizing Institutions US Particle Accelerator School (USPAS) H. Wiedemann, S.Y. Lee, M. Paul, S. Winchester CERN, Accelerator School (CAS) D. Brandt, E.J.N. Wilson, S. von Wartburg KEK Accelerator School (KEKPAS) S.I. Kurokawa, Y. Hayashi Russia Accelerator S c h d E.A. Perevedentsev

Program Committee USPAS: A. Chao, G. Krafft, S.Y. Lee, R. Ryne, M. Syphers CAS: J. Miles, E.J.N. Wilson, Japan Acc.Schoo1: S.I. Kurokawa, H. Matsumoto, K. Nakajima, S. Ohsawa Russia AccSchool: I.N. Meshkov, E.A. Perevedentsev, Y.M. Shatunov

Sponsors USDOE, CERN, KEK, Budker Institute, URA

...

Vlll

Ion Linacs Thomas P. Wangler Los Alamos National Laboratory Los Alamos, New Mexico 87545 An overview is presented of accelerator physics and technology of ion linear accelerators. Topics include early history, basic principles, medium- and high-velocity accelerating structures, the radiofrequency quadrupole (RFQ), modem ion-linac architecture,longitudinal and transverse singleparticle beam dynamics, multiparticle dynamics and space charge, and recent results on beam halo.

1. Introduction and Early History of Ion Linacs

We begin our discussion of ion linacs with some general observations about linacs. In a radiofrequency (RF) linac, the beam is accelerated by radiofrequency electromagnetic fields with a harmonic time dependence. The RF linear accelerator is classified as a resonance accelerator. Because both ends of the structure are grounded, a linac can easily be constructed as a modular array of accelerating structures, and there is no physical limit to the energy gain in a linac. The first formal proposal and experimental test of a linac was by Rolf Wideroe in 1928,' but linear accelerators that were useful for research in nuclear and elementary particle research did not appear until after the developments of microwave technology during World War 11, stimulated by radar programs. Since then, the progress has been rapid, and today the linac is not only a useful research tool but is also being developed for many other important applications. A main advantage of the linear accelerator is its capability for producing high-energy, and high-intensity charged-particle beams of high beam quality, where beam quality can be related to a capability for producing a small beam diameter or small angular spread, and small time spread of the beam pulses or small energy spread. Other attractive characteristics include the following: (a) strong focusing can easily be provided to confine a high-intensity beam; (b) the beam traverses the structure in a single pass, and therefore repetitive error conditions causing destructive beam resonances are avoided; (c) because the beam travels in a straight line, there is no power loss from synchrotron radiation, which is a limitation for high-energy electron beams in circular accelerators; (d) injection and extraction are simpler than in circular accelerators, since the natural orbit of the linac is open at each end; special techniques for efficient beam injection and extraction are unnecessary; (e) the Iinac can operate at any duty factor, all the way to 100% duty or a continuous-wave (CW). For proton and deuteron linacs, modern applications include: (a) injectors to high-energy synchrotrons for elementary-particle-physics research, (b) highenergy linacs for CW spoliation neutron sources used for condensed matter and materials research, production of nuclear fuel, transmutation of nuclear wastes, and accelerator-driven fission-reactor concepts, (c) CW neutron sources for 1

2 materials irradiation studies related to fusion reactors, and (d) low-energy neutron sources for medical applications such as boron neutron-capture therapy. There are also linac applications for heavy ions, including: (a) linacs for nuclear physics research, (b) ion implantation for semiconductor fabrication, and (c) multi-GeV linacs for heavy-ion-driven inertial-confinement fusion. The first FW linear accelerator was an ion linac; it was conceived and demonstrated experimentally by Wideroe in 1927 at Aachen, Germany. It was reported in a paper' that is one of the most significant in the history of particle accelerators? and which inspired E. 0. Lawrence to invent the cy~lotron.~ The linac built by Wideroe was the forerunner of all modern RF accelerators. The concept, shown in Fig. 1, was to apply a sinusoidal voltage to a linear sequence of copper drift tubes, whose lengths increased with increasing particle velocity, so that the particles would arrive in every gap between a pair of adjacent drift tubes at the right time to be accelerated. In the Figure, D are drift tubes connected to a voltage oscillator that applies equal and opposite voltages to sequential drift tubes, G are the gaps between adjacent drift tubes in which the electric force from the potential difference between the drift tubes acts to accelerate the particles, and S is the source of a continuous ion beam. For efficient acceleration the particles must be spatially grouped into bunches, shown by the black dots, which must be injected into the linac at the time when the polarity of the drift tubes is correct for acceleration. The bunching can be accomplished by an RF gap B between the ion source and the linac. The electric field in the buncher gap impresses a velocity modulation on the incoming beam that produces spatial bunching at the end of a suitably chosen drift space L. The net effect of the sequence of voltage kicks in the linac is to deliver a total energy gain to the beam, which is greater than the product of the ion charge times the impressed voltage V in any single gap. In Wideroe's experiment, an RF voltage of 25 kV from a 1-MHz oscillator was applied to a single drift tube between two grounded electrodes, and a beam of singly-charged potassium ions gained the maximum energy in each gap. A final beam energy of 50 keV was measured, which is twice that obtainable from a single application of the applied voltage. This was also the first accelerator that had ground potential at both the entrance and the exit ends, and was still able to deliver a net energy gain to the beam, using the electric fields within. This result is not possible when static (conservative) rather than time-dependent electric fields are used for acceleration. The experiment established the principle that, unlike that of an electrostatic accelerator, the voltage gain of an RF accelerator could exceed the maximum applied voltage. There was no reason to doubt that the method could be repeated as often as desired to obtain unlimited higher energies. In 1931 Sloan and Lawrence4 built a Wideroe-type linac with 30 drift tubes, and by applying 42 kV at a frequency of 10 MHz, they accelerated mercury ions to an energy of 1.26 MeV at a beam current of 1 FA. By 1934 the output energy had been raised to 2.85 MeVSusing 36 drift tubes.

3

Figure 1 . The concept of the Wideroe drift-tube linac. The concept is explained in the text.

The original Wideroe-linac concept was not suitable for acceleration to high energies of beams of lighter protons or electrons, which was of greater interest for fundamental physics research. These light-particle beam velocities are much larger, approaching the speed of light, and the drift-tube lengths and distances between accelerating gaps would be impractically large at frequencies available in those days. Linac development required higher-power microwave generators, and accelerating structures better adapted for high frequencies. High-frequency power generators, developed for redar applications, became available after World War 11. At this time, a new and more efficient high-frequency proton accelerating structure, based on a linear array of drift tubes enclosed in a high-Q cylindrical cavity, was proposed by Luis Alvarez6 and coworkers at the University of California. The use of a resonant cavity confines the electromagnetic fields and avoids electromagnetic radiation losses. In the drifttube linac (DTL) concept, an electromagnetic standing-wave mode is excited in the cavity with a sinusoid ally time-varying longitudinal electric field in the gaps for acceleration, and zero electric field inside the drift tubes to avoid deceleration when the field is reversed. The accelerating mode has the property that an RF longitudinal electric field is concentrated near the axis, and an azimuthal RF magnetic field is concentrated near the outer wall, associated with longitudinal RF currents that flow on the outer wall. This field pattern is analogous to that of the TMolo mode of a cylindrical or pill-box cavity, and for that reason is often called the TMolO mode of the DTL. The beam particles are bunched by a separate cavity before injection into the DTL. In Fig. 2, the beam particles are shown in the gaps G , where they are accelerated. The drift tubes D are supported by the stems S . The cavity is excited by an RF current flowing on a coaxial line into the loop coupler C; the current is supplied by an RF generator that is not shown. The drift-tube linac differs from the Wideroe linac in that at a given time the field in all the gaps has the same polarity. A 1-m-diameter, 12-m DTL with a resonant frequency of 200 MHz was built,’ which accelerated protons from 4 to 32 MeV. The DTL is typically used for medium velocity ions in the velocity range from about 0.04 to 0.4 times the speed of light.

4

I

Figure 2. Drift. be linac structure used for acceleration of medium-velocity ions.

The DTL structure becomes inefficient for ion velocities greater than about 0.4 times the speed of light, because the transit-time factor, which is defined later, becomes too small. In the 1960s a new accelerating structure was invented at Los Alamos called the side-coupled linac (SCL), shown in Fig. 3.8 The SCL consists of an array of resonant cavities comprised of 100 or more cells per electromagnetic tank, coupled together to form a multicavity accelerating structure. It is efficient for acceleration of particles with velocity greater than about 0.5 time the speed of light. There are two kinds of cavities in the SCL. Cavities on axis are called accelerating cavities, and in the accelerating mode operate in a TM,,,-like standing-wave mode. Unlike the situation for the DTL, adjacent accelerating cavities are 180 degrees out of phase. They are electromagnetically coupled through coupling slots to the cavities along the sides, called coupling cavities. In the accelerating mode, each coupling cavity is driven by a pair of adjacent accelerating cavities that are out of phase, resulting in no net coupled-cavity excitation. Thus, the coupling cavities are nominally unexcited. However, errors in fabrication or beam loading effects can produce field errors that will excite the coupling cells. It can be shown from a coupledcircuit model that the coupling cells act like a feedback system that tends to stabilize the fields in the accelerating cells against various errors. The method is called resonant coupling. Although the beam never sees the coupling cells directly, the coupling cells play a valuable role in providing field stability for long, multicell structures. For high power applications, this approach helps reduce the cost of the linac because it allows the use of very high power klystrons, which provide more RF watts per dollar. Other geometries for the coupling cells are possible besides the side-coupled geometry, and the class of linac structures that operate based on these principles are know as coupled-

5

cavity linacs (CCL). CCL structures are biperiodic structures since each period includes an accelerating cell and a coupling cell. The accelerating mode is called the 7d2 mode. CCL structures like the SCL are not efficient for medium velocity particles with velocities below about p=0.4, because the walls separating the accelerating cells get close together and more walls per unit length that carry RF currents increase the power loss per unit length,

COU~LING CAVITY

Figure 3. The side-coupled linac structure was invented at Los Alamos in the 1960s. The cavities on the beam axis ~IEthe accelerating cavities. The coupling cavities on the side are nominally unexcited and stabilize the accelerating-cavity fields against perturbations from fabrication errors and beam loading.

2. Basic Principles of Ion-Linac Acceleration As suggested in the previous section, different accelerating structures are used for different velocity particles, and for this reason the particle mass affects significantly the accelerating structures that are used in linacs. For example, if one considers that a typical DC injector would have a voltage near about 100 kV, velocities relative to the speed of light for some different mass particles are: p= 0.55 for electrons, 0.015 for protons, 0.00095 for uranium with charge state q=+l, and 0.0050 for uranium with charge state q=+28. For linac beams at a relatively low energy of 5 MeV these velocities become: p= 0.996 for electrons, 0.10 for protons, 0.0067 for uranium with charge state q=+l, and 0.036 for uranium with charge state q=+28. We see that in this low-energy regime, electrons behave relativistically, whereas protons and heavier ions behave nonrelativistically. Some important issues that result in differences between electron- and ion-linacs for a specified energy range include the RF frequency choice, and the structure choice.

6

Likewise, the relative importance of different beam-dynamics effects also depends on the beam velocity and therefore at a given energy on the particle mass. Beam-dynamics issues of importance for low-velocity ions are primarily associated with the fact that a nonrelativistic beam can have a significant velocity spread, and to keep such a beam from debunching, longitudinal focusing is needed. This is not the case for electron linacs, where the longitudinal distribution is frozen since all particles have nearly the same velocity v=c. Also, repulsive space-charge forces may be important for highcurrent ion linacs, whereas for relativistic electrons, the electric self-force is nearly cancelled by the magnetic self-force. For electron linacs the electromagnetic fields carried by the beam are enhanced by the relativistic longitudinal compression of the fields, and are scattered by discontinuities in the structures enclosing the beam, a phenomenon called wakefields. These wakefields are equivalently described as a superposition of modes that are excited in the structures surrounding the beam. The modes that can exert transverse forces on the beam may give rise to the beam breakup instability. These wakefield and beam-breakup effects are generally insignificant for ion linacs compared with the space-charge forces. One of the most important parameter choices in the design of a linac is the frequency of the accelerating structures. To understand the importance of the proper frequency choice, we need to discuss the basic principles of acceleration in an rf gap, such as that shown in Fig. 4. The expression for the energy gain AW of a particle in an accelerating gap can be written asg AW = qEoT cos @L,

(1)

where L is either a cell length, or for a single gap a length large enough to contain the physical gap g plus all the spatially decaying field that leaks into the drift tubes, is the phase of the field when the particle is at the center of the gap, Eo is the peak value as a function of time of the spatial average axial electric field over the same gap length L, and T is the transit-time factor. In the simplest model the on-axis transit-time factor is given by lo

+

The transit-time factor is the product of a factor dependent on the aperture radius a, and a gap factor dependent on the physical gap size g, h is the RF wavelength, p and y are the usual relativistic velocity and mass factors, and Jo and are the ordinary and modified Bessel functions of order zero. The quantity EoT is often referred to as the accelerating gradient.

7

1 I

I

r=O I I

-u2

u2

figure 4. Geometry and longitudinal electric field distributionfor an ideal accelerating gap.

Inspection of Eq. (2) shows that as p decreases, a larger wavelength (lower frequency) is required to maintain a given value of the aperture-dependent factor. Thus, the heavier the ion to be accelerated in a linac, one might expect that a lower frequency linac would be a more suitable choice. Numbers for some real accelerator parameters are shown in Table 1. Typically Jo- 1, and the effect of the aperture on the transit-time factor is given by the factor ~(27cdYph) in the denominator of Eq. (2). If h is allowed to decrease from infinity where Io=l to a Table 1

value that permits I. to increase by no more than 20%, the resulting frequency is shown in the second to last column of Table 1 for comparison with the actual linac frequency shown in the last column. The SLAC linac for electrons, and LANSCE linac for protons already exist. SNS is under construction, and RIA is a heavy-ion linac not yet approved for construction. The frequency numbers in the last two columns agree to within nearly 50%, illustrating the validity of the explanation that the frequency is chosen by linac designers to give an adequate transit-time factor. This explains why heavy-ion linacs have lower frequencies than electron and light-ion linacs.

8

The RF power efficiency is another important consideration. The usual power efficiency of a multicell accelerating structure is the effective shunt impedance per unit length, defined as

in typical units of megohms per meter, where P/L is the RF power per unit length dissipated in the cavity walls averaged over an RF cycle. This quantity is a measure of the ratio of the squared accelerating field seen by the beam per unit power dissipation. Also used is the effective shunt impedance in megohms, given by ZT2L. The time-averaged RF power dissipated is given by

'f

I

= R, H 2dAl2 , where dA is an element of surface area, H is the surface

magnetic field, and R, is the RF surface resistance for a DC conductivity given by R,=(n~flo)'". Normal-conducting accelerating structures are usually built from copper. For a given normal-conducting accelerating structure geometry, the effective shunt impedance scales with frequency as ZT2 fh, assuming all the dimensions scale as the RF wavelength. Thus, for normal-conducting structures, the power efficiency increases with increasing frequency. However, the frequency is limited by the requirement of maintaining a high transit-time factor, as well as by practical considerations of the fabrication tolerances. Standing-wave operation is used for all existing ion linacs. One reason for this is that standing-wave operation avoids power wasted to an external load as in the traveling-wave case. Standing-wave is more efficient than traveling-wave operation when the pulse length is larger than the cavity electromagnetic fill time, which is normally the case. For the short-pulse regime the result is different; the standing wave cavity, which fills as a result of the field buildup from multiple reflections, takes a longer time to fill compared with the beam pulse duration, and wastes too much power during the relatively long fill time. No existing ion linacs operate in this short-pulse regime.

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3. Medium and High Velocity Accelerating Structures for Ion Linacs We have already discussed three basic accelerating structures for ions, the original Wideroe structure used for ion acceleration, the drift-tube linac for acceleration of medium-velocity protons, and the side-coupled linac for acceleration of high-velocity protons. These structures work well for proton linacs, where the frequencies are typically 200 to 400 MHz for the DTL, and near 800 MHz for the CCL. The highest energy proton linac at present is the 800-MeV LANSCE linac at Los Alamos (formerly known as LAMPF)." We have seen that for heavy ions, lower frequencies, typically 100 MHz and less, are required for a high transit-time factor. A DTL structure would be

9

very large, costly, and difficult to handle. The best solution for lower frequencies is to go back to the Wideroe structure, but in a modified form in which the fields are enclosed within a cavity. One approach is shown schematically in Fig. 5 .

Figure 5. Drawing of the interdigital structure, a type of Wideroe linac. The assembly shown would be inserted into a cylindrical cavity.

This is called the interdigital structure, also called the IH, where the letter I stands for interdigital, and the letter H is another term used to describe a Tf3 or transverse-electric mode. This is an alternative to the DTL for low-frequency heavy-ion linacs to reduce the radial size. It consists of two conducting parallel lines at opposite electric potential that are loaded with interlacing hollow electrodes alternately connected to the two lines. Major progress has been made during the past 25 years in the field of superconducting RF linacs for ion acceleration. There are some differences between the normal-conducting and superconducting technologies that affect linac design. First, the superconducting accelerating structures are short, constructed from a few cells. The short structures are easier to handle during the chemical processing stage; they have a smaller surface area improving the ability to diagnose and correct for normal-conducting impurities, and for field emission sites that would limit the accelerating gradient. Generally the shorter structures produce higher accelerating gradients. Furthermore, structures with only a few cells have a large velocity acceptance. This can be exploited in several ways. First, it allows the use of identical accelerating structures to cover a given velocity range, which can reduce the manufacturing costs. It also offers the operational flexibility to reset one or more cavity phases and continue cavity operation at different velocities, if for any reason an accelerating cavity failure occurs. It also offers the flexibility to change the velocity profile to accommodate a range of heavy ion species, unlike multicell structures that have a fixed velocity profile built into each structure. Other advantages of superconducting linacs include lower operating costs, and affordability (because of low power dissipation) of a larger bore radius,

10

which relaxes alignment, steering, and beam-matching tolerances. The larger bore radius helps reduce beam loss and radioactivation of the accelerating structure, easing commissioning, and increasing the availability. A worldwide industrial capability now exists for fabrication of cavities and cryomodules, and the performance of the superconductingcavities is still improving. The scaling of effective shunt impedance with frequency for superconducting structures is different from that of normal-conducting structures. Assuming the RF surface resistance comes only from BCS theory (ignoring residual resistance), the surface resistance scales with frequency as R, f-’,which means that unlike the normal-conducting cavities, the RF power efficiency improves as the frequency is decreased. Although the wall losses are reduced compared with the normal-conducting case by a factor of lo4 to lo5, nevertheless, these losses can be important because they establish the requirements for the cryogenic system. The improved efficiency at lower frequencies is compatible with the need at lower frequencies for achieving a high transit-time factor for the low-velocity heavy ions. Initial work on the development of high-beta structures for a proton superconducting linac was begun at Los Alamos for the Accelerator Production of Tritium (APT) project;” these were 5-cell elliptical cavities designed for p=0.64 (See Fig. 6.) The first linac to accelerate a proton beam using superconducting cavities will be the Spallation Neutron Source (SNS) linac, which is now under constr~ction.’~ Another area of progress has been the development of medium-velocity normal-conducting and superconducting structures at low frequencies for heavy ions. Examples of heavy ion linacs include the normal-conducting UNILAC linac at GSI, Darmstadt, Germany,I4 the ATLAS superconducting linac at Argonne National Lab~ratory,’~ and the normal-conducting lead-ion linac at CERN.16 The approach taken to reduce the physical size of superconducting structures has been to use TEM coaxial type structures such as quarter-wave and half-wave resonators, loaded at the end by one or more drift tubes. The driverlinac design for proposed Rare Isotope Accelerator (RIA) makes use of these types of superconducting structures.17

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4. The Radiofrequency Quadrupole The radiofrequency quadrupole (RFQ) linac’’ is usually the lowest velocity accelerating structure in a modern ion linac, and its invention and development was a major innovation in the linac field. The RFQ is especially well suited for the acceleration of beams with low velocities in the typical range of about 0.01 to 0.1 times the speed of light. As such, it is an important accelerator for ions, but not for electrons, which from a typical electron-gun source are already emitted with velocities approaching half the speed of light.

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Figure 6. Niobium 5-cell elliptical cavity for pd.64 built by CERCA for the APT proton linac project.

The principles of operation of the RFQ were first presented by the inventors, Kapchinskiy and Tepliakov (K-T), in their 1969 p~blication.'~ The RFQ 4-vane structure is shown in Fig. 7. K-T proposed to modify the shapes of the four electrodes of an RF quadrupole to achieve both acceleration and focusing from RF electric fields. By using a potential function description, K-T showed how to shape the electrodes to produce the fields required by the beam. The achievement of practicable means of applying velocity-independent electric focusing in a low-velocity accelerator gave the RFQ a significant strongfocusing advantage compared with conventional low-velocity linacs that used velocity-dependent magnetic lenses. This allowed the RFQ to extend the practical range of operation of ion linacs to low velocities, thus eliminating the need for large, high-voltage DC accelerators for injection of the beam into the linac. In a later publication,20 K-T showed how to introduce specific slow variations of the RFQ parameters to bunch the beam adiabatically. This allowed the beam to be injected into the RFQ and to be bunched over many spatial periods, while the beam is contained transversely by the electric-quadrupole forces. Adiabatic bunching allows a large fraction of the beam to be captured, and converted into stable bunches that can be accelerated efficiently to the final energy. Adiabatic bunching results in very compact bunches with minimal tails in longitudinal phase space, and increases the beam-current capacity, because it avoids unnecessary longitudinal compression of the beam at low velocities, which would increase the transverse space-charge effects. The 4-vane cavity is used for RF frequencies above about 200 MHz. Most proton RFQs are built as a 4-vane structure, which consists of four vanes symmetrically placed within a cavity, as shown in Fig.7. The cavity is operated in a TE,,,-like mode, which is obtained from the natural TE,,, mode, by tuning specially configured end cells to produce a longitudinally uniform field throughout the interior of the cavity. The transverse electric field is localized

12

near the vane tips, and the magnetic field, which is longitudinal, is localized mostly in four outer quadrants. The efficiency of the 4-vane cavity is relatively high, because the vane charging currents are distributed very uniformly along the length of the vanes. A different structure, called the 4-rod cavity?’ which is similar to a Wideroe structure, is used mostly in the lower-frequency range, below about 200 MHz, and is the most commonly used RFQ structure for lowvelocity heavy ions.

Figure 7. The 4-vane RFQ accelerator section. The four electrodes are excited with electric quadrupole-mode RF voltages to focus the beam. The electrodes are modulated to produce longitudinal electric fields to bunch and accelerate low-velocity ions.

5. Modern Ion-Linac Architecture Figure 8 shows a block diagram of a typical modern ion-linac architecture. The low-velocity (approximately fk0.l) beam from the DC injector is electrically focused, bunched, and accelerated by the RFQ. The medium velocity beam (approximately 0.1
CCL or highvelocity supermducting-elliptil

Wproton = 100 keV

2-7 MeV

-100 MeV

-1 GeV

Figure 8. Block diagram of a modem ion linac, showing the accelerating structures and typical kinetic-energy transition values for an example of a proton linac.

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6. Longitudinal Beam Dynamics for Ion LinacsZ2 An ion linac is designed for acceleration of a single reference particle, which remains in synchronism with the accelerating fields and is called the synchronous particle. For high output beam intensity, restoring forces must be applied so that those particles near the synchronous particle will have stable trajectories. Longitudinal restoring forces are produced when the beam is accelerated by a sinusoidal electric field when it is increasing with time. The early particles experience a smaller accelerating field, and the late particles a larger field than the synchronous particle. These forces produce phase and energy oscillations about the synchronous particle. The accelerated particles are formed in stable bunches that are centered near the synchronous particle. Those particles outside the stable region slip behind in phase and do not experience any net acceleration. The final energy of an ion that undergoes phase oscillations in a long multicell structure is approximately determined, not by the field, but by the geometry of the structure, which is tailored to produce a specific final synchronous energy. This is not so for an ion linac built from an array of short independent cavities with each one capable of operating over a wide velocity range; this is the usual situation for superconducting ion linacs. In this case, the final energy depends on the field and the phasing of the cavities. The longitudinal dynamics in smooth approximation is described by two first-order coupled equations of motion for phase Cp and energy W relative to the synchronous-particleenergy W,. This can also be written as a nonlinear secondorder equation of motion for the phase (see Eq. 4). These equations lead to a separatrix in time (or phase)-energy phase space defining the stable region, and a potential well centered on the synchronous particle. This is shown in Fig. 9 in the approximation that energy gain can be ignored. At the top, the accelerating field is shown as a cosine function of the phase; the synchronous phase $s is shown as a negative number, which lies earlier than the crest. The middle plot shows some longitudinal phase-space trajectories, including the separatrix, the limiting stable trajectory, which passes through the unstable fixed point at AW = 0, and Cp = +. The stable fixed point lies at AW = 0 and $ = &, where the longitudinal potential well has its minimum as shown in the bottom plot.

For small-amplitude oscillations one obtains a simple harmonic oscillator solution:

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d2@+ k&)@ 2 = 0, ds

where k& =

2 q E o T sin(-

&)

rnc2p; y,3n

.

7. Transverse Beam Dynamics for Ion LinacsB It is necessary that off-axis particles will neither drift away, nor be subjected to defocusing forces that will take them further from the axis. As can be seen from the electric-field lines in Fig. 10, when off-axis particles enter a gap and are accelerated by a longitudinal RF electric field, they also experience radial RF electric and magnetic forces. At first it might be thought that the oppositely-directed radial electric forces in the two halves of the gap will

15

produce a cancellation in the total radial momentum impulse. On closer examination, it is found that generally there will be a nonzero radial impulse, which occurs as a combination of three mechanisms: (1) the fields vary in time as the particle crosses the gap, (2) the fields also depend on the radial particle displacement, which varies across the gap, and (3) the particle velocity increases, while the particle crosses the gap, so that the particle does not spend equal times in each half of the gap. For longitudinal stability $s must be negative, which means the field is rising when the synchronous particle is at the center of the gap. This means that most particles experience a field in the second half of the gap that is higher than in the first half, resulting in a net defocusing force. This corresponds to mechanism (l), and for ion linacs this is the usually dominant effect, known as the RF-defocusing force.

Figure 10. Electric field lines in an RF gap.

The most common method of compensating for the transverse RF defocusing has been the use of magnetic lenses, of which magnetic quadrupole lenses are the most common. For the DTL, these quadrupoles are installed within the drift tubes. 8. Multiparticle Dynamics and Space Chargeu

First, we discuss the definitions of beam current that are commonly used for an ion linac. In pulsed linacs, one must distinguish between micropulses and macropulses. The linac longitudinal field forms a sequence of stable RF buckets separated by one RF period at the bunching frequency. Each bucket contains a stable bunch of particles constituting a micropulse. When the RF generator itself is pulsed, with a period that is generally very long compared with the RF period, the generator pulses are called macropulses. A linac may also be operated continuously, which is called CW or continuous-wave operation. The choice to operate pulsed or CW depends on several issues. One important issue is the total

16

RF efficiency. If the accelerated beam current is small, most of the power in CW operation is not delivered to the beam, but is dissipated in the structure walls. Instead, if the accelerator is operated pulsed, and the charge per FW bucket is increased to maintain the same average beam current, then a larger fractional power is delivered to the beam, and the efficiency is improved. Another important advantage for pulsed operation is that the peak surface electric field attainable is generally larger for shorter pulses. Thus, if high accelerating fields are required, pulsed operation may be preferred. The main advantage for either longer pulse or CW operation is to reduce the charge per bucket, which reduces the space-charge force. The peak electrical current is defined as I=qNf, where q is the charge per particle, N is the number of ions per bunch, and f is the bunch frequency. Although this is called the peak current, it is really the average current over a bunch cycle. The average current is defined as the average over a macropulse. In a CW linac the peak and average currents are equal. In H- linacs beams may be chopped periodically with a period that is usually smaller than the RF macropulse duration. When discussing average current for an H- linac, it is important to distinguish whether or not the average current includes the chopping. In heavy ion linacs with multiply ionized heavy ions, another definition used is particle current, which is the current as if each beam particle were singly charged. Particle motion in a linac depends not only on the external or applied fields, but also on the fields from the Coulomb interactions of the particles, and the image fields induced by the beam in the walls of the surrounding structure. Although the image effects are usually small if the beam is well aligned, the Coulomb forces play an increasingly important role as the beam-current increases. The Coulomb effects in linacs are usually most important in nonrelativistic beams at low velocities, because at low velocities the beam density is usually larger, and for relativistic beams the self-magnetic forces increase and produce a partial cancellation of the electric Coulomb forces. The net effect of the Coulomb interactions in a multiparticle system can be separated into two contributions. First is the “space-charge’’field, which is the smoothed collective field distribution, which varies appreciably only over distances that are large compared with the average separation of the particles. Second are the contributions arising from the particulate nature of the beam, which includes the short-range fields describing binary, small impact-parameter Coulomb 8 collisions. Typically, the number of particles in a linac bunch may exceed 10 , and one finds that the effects of the collisions are very small compared with the effects of the averaged space-charge field. To describe the space-charge field, we need to understand the properties of an evolving particle distribution, which requires a self-consistent solution for the particles and the associated fields. This is a problem, which has been formulated in terms of the coupled Vlasov-Maxwell equations, for which there are no generally successful analytic solutions in a linac, and computer simulation is the most reliable tool.

17

Among the most important properties of the distribution besides the beam current are the rms emittances, which are defined from the second moments of the particle distribution and are measures of the phase-space areas occupied by the beam in each of the three projections of position-momentum phase space. The emittances are important measures of the beam quality; they determine the inherent capability of producing, by means of a suitable focusing system, small sizes for the waist, angular divergence, micropulse width, and energy spread. The significance of the space-charge fields is not only that they reduce the effective focusing strength, but also the nonlinear terms, a consequence of the deviations from charge-density uniformity, cause growth of the rms emittances. This growth degrades the intrinsic beam quality. One consequence of spacecharge-induced emittance growth is the formation of a low-density beam halo surrounding the core of the beam, which can be the cause of beam loss, resulting in radioactivation of the accelerating structure. Better understanding of the mechanisms of beam-halo formation is one of the most important beam-physics research topics of current interest. 9. Progress in Beam Halo

Control of beam-halo and associated beam losses and radioactivation is a fundamental requirement for high beam availability in high-power ion linacs. More than a decade ago, computer simulation studies25 identified beam mismatch as the major source of the halo and emittance growth observed in simulations. The emittance growth can be related to the conversion of free energy from mismatch oscillations into thermal energy of the beam. For a given mismatch strength, the free-energy model determines the maximum emittance growth, which results from complete transfer of free energy into emittance.26 A physical model of halo formation is expected to include both nonlinear and time-dependent forces that drive halo particles to larger amplitudes. Such a mechanism is provided by the particle-core model,”, in which beam mismatch produces an imbalance between focusing, space charge, and emittance, exciting a symmetric or breathing (xms and yms in-phase) mode oscillation of the core. The space-charge field of the oscillating core modulates the net focusing force acting on individual particles and drives particles in a nonlinear parametric resonance when fparticle=f,,,dJ2, where fparticle is the betatron frequency of the particle, and fmde is the mode-oscillation frequency. The model predicts a maximum resonant-particle amplitude as a function of the mismatch strength.29 Neither the free-energy nor the particle-core model predicts the growth rates for the halo amplitude and beam emittance, for which numerical simulations are required. To test the two models, a 52-quadrupole periodic-focusing beam-transport channel was installed at the end of the low-energy demonstration accelerator (LEDA)30at Los Alamos. LEDA delivers a 6.7-MeV proton beam from a 350MHz radiofrequency-quadrpole (RFQ) linac. The beam was pulsed at a 1-Hz

18

rate with a 30-ps pulse length. The quadrupole-channel length of 11 m was sufficient for the development of about 10 mismatch oscillations, enough to observe at least the initial stages of emittance growth and halo formation caused by mismatch. Results for a 75-mA proton beam current were analyzed. Beam-profile diagnostic 52 quadrupole FODO lattice

RFQ-

4

20-26

45-51

Figure 11. Block diagram of the 52-quadmpole-magnet lattice showing the nine locations of beamprofile scanners.

The most important beam-diagnostic elements were the transverse beamprofile scanners3' that measured the horizontal and vertical distributions. These were installed at nine stations (Fig. 1l), each located midway between pairs of quadrupoles. The scanners were labeled with numbers corresponding to the preceding quadrupole-magnet number. The beam was matched, using a leastsquares fitting procedure that adjusted the first four quadrupoles to produce equal rms sizes at the last eight scanner locations. For a mismatched beam, one must consider not only the breathing mode, but also the antisymmetric or quadrupole mode. The beam was mismatched in nominally pure symmetric or antisymmetric modes by proper settings of the same four matching quadrupoles. The mismatch strength was measured by a mismatch parameter p, which equals the ratio of the rms size of the initial beam to that of the matched beam. For a matched beam p= 1. Figure 12 shows the matched and mismatched 75-mA beam profiles at scanner 51. The matched beam has a Gaussian-like central profile with an rms beam size of 1.1 mm. For the matched beam a low-density halo was observed to extend as far as 9 rms. This matched-beam halo is observed at all scanners and is most easily explained as a halo that has formed in the injector/RFQ system prior to the periodic quadrupole channel. Direct measurement of the beam-energy distribution with a resolution of about 200 keV, using a dispersive section of the transport line at the end of the periodic quadrupole channel, showed no evidence for low-energy tails that might contribute to this halo. Although collimation can remove this halo, collimation was not implemented in our experiment. Halo caused by mismatch was our main interest, because this mismatch mechanism is expected to involve more particles, and can form halo even at high energy, where collimation is more difficult. A breathing-mode-mismatch beam profile for p 1 . 5 , seen in Fig. 12, shows the growth of shoulders indicating substantial formation of halo.

19

C

1.EW

s

l.E-O1

-3 E

E

5

s

1.E-02

e o

c

1.E-03

P E

1.E-04

ma

b

1.E-05 -10-8

-6

-4

-2

0

2

4

6

8

10

Horizontal position (mm)

Figure 12. Horizontal beam profiles at scanner 51 for a 75-mA, p=l matched beam (solid circles), 5 beam (open circles). and breathing-mode ~ 1 . mismatched

The rms-size measurements were used to calculate the rms emittances at scanners 20 and 45. The free-energy model can be tested by comparing the measured emittance growths at scanners 20 and 45 with the emittance-growth upper limits from that model. The emittance-growth measurements for mismatched beams show some significant anisotropies (x-y differences). Franchetti, Hofmann, and Jeon3’ report simulation studies of anisotropic beams in uniform focusing channels, in which large (40%) x-y emittance-growth differences are observed that are sensitive to initial x-y tune differences as small as 1%. The sensitivity is not the result of chaotic behavior, but is caused by the parametric resonance discussed earlier, which is sensitive to x-y parameter differences. This suggests that anisotropies could be driven by percent-level input x-y emittance differences that are not resolved experimentally. Although the free-energy model was derived for an axisymmetric beam, these authors find that the model can be extended to a 2D anisotropic case if the emittance growth is averaged over x andy. Figure 13 shows the x-y averaged rms-emittance-growthresults (points with error bars) versus p at scanner 20 for a 75-mA breathing-mode mismatch. The maximum emittance-growth curves from the free-energy model are shown for the two tune-depression values that bracket the values for the debunching beam, and it can be seen that the theoretical maximum is insensitive to the tune depression over this range. The breathing-mode data in Fig. 13 are consistent at all p values with the maximum emittance growth predicted by the model. The breathing mode results at scanner 45 (not shown) show no significant additional emittance growth, consistent with the upper limits from the model and with complete transfer of free energy within only four mismatch oscillations. Quadrupole-mode mismatch results are also consistent at all measured p values with the maximum growth of the model. Although an axisymmetric beam is assumed in the model, applicability to the quadruple mode is physically

20

reasonable for a given free energy if equal energy sharing is assumed in x and y. Overall, the data for both mismatch modes indicate a rapid growth mechanism with nearly complete transfer of free energy occurring in less than ten mismatch oscillations.

22

a8 0.6

I

I

I

I

I

I

(x8

1

1.2

1.4

1.6

1.8

2

Msrnsdch Parameter (0 Figure 13. Measured rms-emittance growth averaged over x and y for 75 mA at scanner 20 for a breathing-mode mismatch. The curves show maximum growth from the free-energy model.

The particle-core model’’ predicts the maximum resonant-particle amplitude as a function of mismatch parameter 1.Because of background, it is not possible to obtain precisely an experimental maximum amplitude. Instead, the measured amplitudes (x-y averaged half widths of the beam) were compared at three different fractional beam-profile intensity levels (lo%, 1%, and 0.1% of the peak) for a breathing-mode mismatch with the maximum amplitude predicted by the particle-core model. A comparison is shown in Figure 14 for scanner 20 at 75 mA. The shapes of all three measured half-width curves are consistent with the shape of the maximum amplitude curve from the particle-core model, and all three measured curves lie below the maximum amplitude curve from the model. Similar results (not shown) are observed at scanner 51. Although the particlecore model based on a single mismatch mode is a relatively simple description of the beam dynamics, the agreement with the model for the curve shapes and for consistency of the magnitudes, supports the conclusion that the particle-core model incorporates the main physical mechanism responsible for the halo growth.

21

-E:0.0 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Mismatch parameter (P) Figure 14. Measured beam half-widths at scanner 20 (75 mA and a breathing mode mismatch) at different fractional intensity levels versus mismatch strength p for comparison with the maximum resonant amplitude of the particle-core model.

Thus, the experimental results support both the free-energy model and the particle-core model of halo formation in mismatched beams. This conclusion is important because these models predict upper limits to emittance and haloamplitude growth in high-current transport channels and linacs, and allow estimation of focusing strength and aperture requirements in new designs. The large emittance-growth rate observed in the experiment can be explained qualitatively by a high particle density in the initial beam tails, which corresponds to a greater population of the region of phase space that leads to resonant halo growth. An important conclusion from the experiment is that knowledge of the initial particle distribution, especially the density in the tails, is important for accurate predictions of beam-halo from simulation^.^^ Knowledge of only the input Courant-Snyder parameters and the input emittances is insufficient for reliable simulations of beam halo formed in mismatched beams.

Acknowledgments The other members of the scientific team for the halo experiment included C. K. Allen, K. C. D. Chan, P. Colestock, K. R. Crandall, R. W. Garnett, J. D. Gilpatrick, W. Lysenko, J. Qiang, J. D. Schneider, M. E. Schulze, R. Sheffield, and H. V. Smith. This work was supported by the U.S. Department of Energy.

22

References R. Wideroe, Archiv fiir Electrotechnik 21, 387 (1928). For a discussion of this and other work by Wideroe, see P. Waloschek, The Infancy of Particle Accelerators-Life and Work of Rolf Wideroe, Deutsches Elektronen-Synchrotron, Notkestrasse 85-22603, Hamburg, DESY 94-039, March, 1994. E. 0. Lawrence and N. E. Edlefsen, Science 72,376 (1930). D. H. Sloan and E. 0. Lawrence, Phys. Rev. 38,2021 (1931). D. H. Sloan and W. M. Coate, Phys. Rev. 46,539 (1934). L. W. Alvarez, Phys. Rev. 70,799 (1946). L. W. Alvarez, H. Bradner, J. V. Franck, H. Gordon, J. D. Gow, L. C. Marshall, F. Oppenheimer, W. K. H. Panofsky, C. Richmond, and J. R. Woodward, Rev. Sci. Instr. 26, 111 (1955); Rev. Sci. Instr. 26,210 (1955). 8 D. E. Nagel, E. A. Knapp, and B. C. Knapp, Rev. Sci. Instr. 38, 1583-1587 (1967). T. P. Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, pp. 36-44. lo T. P. Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, p. 44. l 1 R. E. 0. Erickson, V. W. Hughes and D. E. Nagel, The Meson Factories, University of California Press, Los Angeles, 1991; M. Stanley Livingston, LAMPF, A Nuclear Research Facility, LA-6878-MS, September, 1977; M. Stanley Livingston, Origins and History of the Los Alamos Meson Physics Facility, LA-5000. 12 G. P. Lawrence, et al., "Conventional and Superconducting RF Linac Designs for the APT Project," Proc. 1996 Int. Linac Conf., Geneva, Switzerland, August 26-30, 1996, pp. 710-712. l 3 N. Holtkamp, et. al., "Status of the Spallation Neutron Source", presented at the 2003 Part. Accel. Conf. , Portland, Oregon, May 12-16,2003. 14 D. Bohne, Proc. 1976 Linear Accel. Conf., Sept. 14-17, 1976, Chalk River, Ontario, Canada, AECL-5677, p. 2. l5 L. M. Bollinger, Annu. Rev. Nucl. Part. Sci. 36, 375 (1986); L. M. Bollinger, Proc. 1992 Linear Accel. Conf., August 24-28, 1992, Ottawa, Ontario, Canada, AECL-10728, p. 13. I6 H. R. Haseroth, Proc. 1996 Int. Conf., Geneva, August 26-30, 1996. l7 K. W. Shepard, "The RIA Driver Linac", XXI Int. Linac Conf., Gyeongju, Korea, August 19-23,2002. T.P.Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, Chapter 8. l9 1.M.Kapchinskiy and V.A.Tepliakov, Prib.Tekh.Eksp. 2, 19-22 (1970). 20 1.M.Kapchinskiyand V.A.Tepliakov, Prib.Tekh.Eksp. 4, 17-19 (1970). For more discussion of the 4-rod RFQ structure, see A. Schempp in these proceedings.

2

23 22 T. P. Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, Chapter 6 . 23 T. P. Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, Chapter 7. 24 T. P. Wangler, Principles of RF Linear Accelerators, Wiley, New York, 1998, Chapter 9. 25 A. Cucchetti et al., Proc. of IEEE 1991 Part. Accel. Conf. , ed. by Lizama and Chew (IEEE, New York, 1991), p. 251. 26 M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994, p.477; M. Reiser, J. Applied Phys. 70, 1919 (1991). 27 J.S. O'Connell, T.P. Wangler, R.S. Mills, and K.R. Crandall, Proc. of 1993 Part. Accel. Conf., IEEE Catalog No. CH3279-7,3657-3659. 28 R.L. Gluckstern, Phys. Rev. Lett. 73,2247 (1994). 29 T.P. Wangler, K.R. Crandall, R. Ryne, and T.S. Wang, Phys. Rev. ST-AB 1 (084201) 1998. 30 C.K Allen, et al., Phys. Rev. Lett. 89, No.21, (214802) (2002). 31 J.D. Gilpatrick et al., Proc. 2001 Part. Accel. Conf., IEEE Catalog No. 01CH37268,525-527, 32 G. Franchetti, I. Hofmann, and D. Jeon, Phys.Rev.Lett. 88, (254802) (2002)' 33 J.Qiang, et al., Phys. Rev. ST Accel. Beams 5, (124201) (2002).

MODERN TRENDS IN INDUCTION ACCELERATOR TECHNOLOGY* G.J. CAPORASO Lawrence Livermore National Laboratory P.O. Box 808, L-64.5 Livermore, CA 94.5.50, USA

E-mail: [email protected] Recent advances in solid-state modulators now permit the design of a new class of high current accelerators. These new accelerators will be able to operate in burst mode at frequencies of several MHz with unprecedented flexibility and precision in pulse format. These new modulators can drive accelerators to high average powers that far exceed those of any other technology and can be used to enable precision beam manipulations. New insulator technology combined with novel pulse-forming lines and switching may enable the construction of a new type of high gradient, high current accelerator. Recent developments in these areas will be reviewed.

1. Introduction 1.1. Induction Accelerator Operating Principle Let us consider the operation of a single induction accelerator cell. An external source of pulsed power is coupled into the cell from (usually) two or more coaxial cables. There is a magnetically permeable core inside the cell. The core is generally composed of ferrite or tape wound materials such as Metglas (from Honeywell) or Finemet (from Hitachi Metals). The induction cell has the interesting characteristic that the outside of it is at ground potential while it is operating; voltage is present only on the inside of the cell. This characteristic is made possible by the presence of the core material, which presents a high inductance path for leakage currents, which flow around the core along a DC, short-circuit path inside the cell. The actual behavior of the core is much more complicated but this simple picture will suffice to illustrate the operating principle of the cell. The core will present a high impedance as long as it isn’t saturated. Once saturated, the core impedance collapses to a

* This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

24

25

small value and the leakage current increases dramatically, shutting off the accelerating voltage. A typical core hysteresis plot is shown in Figure 1.

H

Figure 1. A typical B-H curve showing the results of a cyclical magnetization of the core. B, is the saturation value of the core and B, is the remnant magnetization, the value of the internal field when the external excitation is reduced to zero. AB=B,+B,, since the core is generally reset before pulsing to allow a greater flux swing.

The core is usually driven into saturation in the opposite direction prior to application of the accelerating voltage. This is referred to as resetting the core. When the reset pulse is removed the magnetization decays to the remnant value B,. The total flux swing then available to the core is B,, the saturated value, + B,. The voltage pulse that comes into the cell from the drive cables is applied around a path that encircles the core and along a path that drives a vacuum gap. The accelerating field appears across this vacuum gap and can be sustained as long as the current that flows around the path encircling the core is relatively small. That will be the case as long as the core is not saturated. Upon saturation the core inductance is reduced to a very small value as the magnetic domains in the material are all aligned in the azimuthal direction and the effective permeability of the core is drastically reduced. The leakage current flowing around the core increases so much that it loads down the drive cable supplying power to the cell, and the accelerating voltage collapses. The core is then reset and the cell is ready for another accelerating voltage pulse. A typical induction cell is shown in Figure 2. If we integrate Faraday’s law around the dashed path shown in Figure 2, we find that

26

If we integrate both sides of Eq. (1) with respect to time assuming a constant voltage we find that

The left hand side of Eq. (2) is referred to as the “volt-seconds” of the core and shows the length of time that the core can sustain a particular gap voltage. Induction cells are usually used to produce pulses in the range of 20 ns up to several p. Typical ABs range from about 0.7 Tesla for ferrite up to 3.0 Tesla for MetGlas. S is the core cross-sectional area.

feed line

Figure 2. A typical induction cell showing a coaxial feed on the top and bottom along with a magnetic core. The voltage on the coaxial line is applied both around a path that encircles the core and one that leads to the accelerating gap. The applied voltage can exist across the gap as long as the core is not saturated.

27

1.2. Applications of Induction Technology Induction accelerators are usually used for applications requiring very high beam currents. One application that is the focus of activity in several countries is flash x-ray radiography. These machines are used to x-ray explosively driven dense objects where the high velocity of the objects requires brief but intense x-ray pulses. The recently constructed DARHT facility at Los Alamos, the AIRIX facility at Moronvilliers in France and the FXR facility at Livermore are examples of such machines. They typically provide pulses on the order of 20 MeV and several k i l ohpe re s for tens of ns. Another high current application is Heavy Ion Fusion. Beams of heavy ions like uranium are envisioned to be accelerated up to energies on the order of several GeV and k i l ohpe re s for perhaps 100 ns. There are various schemes that would lead to compression of the output pulses down to of order 10 ns to drive inertially confined fusion capsules for power production. An interesting application of induction technology for high-energy physics is the two-beam accelerator concept. An induction accelerator is used to produce high current beams that are then bunched or chopped. The resulting bunches are passed through cavities optimized to extract RF power from the beam and transport it to a high gradient RF accelerator that is running in parallel to the induction linac. In principle it is possible to extract large amounts of power from the induction-driven beam and couple it into the RF structure. There is considerable interest, particularly in Japan, in using induction cells at MHz repetition rates for induction synchrotrons. As will be discussed later in this article, induction technology enables one to generate complex accelerating waveforms by adding more basic waveforms together. These complex voltage waveforms can be used to both accelerate and longitudinally confine intense bunches for increased luminosity as compared to conventional RF systems. Induction machines are capable of high average beam power, possibly exceeding that of any other technology. This fact can be used to advantage in such applications as irradiation for food or materials processing, microwave generation using free-electron lasers, etc. Pulse power sources with unprecedented characteristics can now be built using solid-state devices in an inductive adder configuration. These are finding applications as drivers for the proposed NLC Klystrons and other high power systems.

1.3. Some Early Pulsed Power Systems The first system we will consider is the Blumleidspark gap system shown in Figure 3. A Marx bank or high voltage step-up transformer is used to charge the Blumlein, which is a set of nested coaxial transmission lines (usually filled with water). Once charged, a spark gap is used to short the center conductor to ground thus initiating the voltage pulse towards the induction cell.

28 Blunllein

Figure 3. A Blumleidspark gap system driving an induction cell. The Blumlein is charged either from a Marx bank or through a step-up transformer.

Such a system is used to power the FXR accelerator at Livermore and was used to power the now deactivated 50-MeV, 10-kA Advanced Test Accelerator at Livermore. An example from the FXR is shown in Figure 4.

Figure 4. The FXR Marx bank (foreground left) and the Blumlein (back right)

A somewhat more modern development is the use of magnetic pulse compressor technology as shown in Figure 5. This system is used to power the ETA-I1 accelerator at Livermore. This system is capable of producing many millions of pulses without maintenance. ETA-I1 has produced 50-pulse bursts at several kiloHertz.

29

Induction

M2

WaterPFL

M3

call+ I

I

I

Figure 5 . Partial schematic of a magnetic pulse compressor system. Saturable magnetic switches are used to transfer energy stored in the capacitance of the first stage and discharge it into the primary of a step-up transformer to charge another capacitor (C2). When the volt-seconds of the second magnetic switch M2 is exceeded, it dumps the charge on C2 into a water-filled pulseforming line. When the output switch M3 saturates, the energy in the pulse-forming line is discharged through transmission lines to the induction cells.

A magnetic switch is simply a saturable magnetic core with a single encircling current path. Before the core saturates, the switch appears as a high inductance. If a voltage is applied across the switch it prevents substantial current flow through the switch until its core saturates. At that point the switch has a very low inductance and it permits large currents to flow. This magnetic pulse compressor uses these switches to temporally compress pulses from one stage to another by reducing the L-C time constants of each stage. As can be seen from Figure 5 , the output impedance of the compressor is very low, permitting tens of k i l o h p e r e s to flow into the load. The MAG1-D pulse compressor is shown in Figure 6. This unit drives 20 ETA-11 accelerator cells at 80 kV and 2 kA for 50 ns. There are four such units driving the ETA-I1 accelerator (one unit drives the 1-MeV injector and the other three each drive 20-cell sets).

30

Figure 6. The MAG1-D pulse compressor, which drives 20 ETA-II induction cells at 80 kV and 2 kA for 50 ns.

2.

Advanced Radiography as a Driver for New Technology

Induction accelerators are widely used for flash x-ray radiography. Typically, these machines will produce a single pulse with a duration of tens of ns during a given hydrodynamic experiment. It was realized some years ago that by developing new pulsed power drivers it might be possible to build a single accelerator that could provide many pulses and many lines of sight during a single hydrodynamic experiment'. A schematic of this concept is shown in Figure 7.

31

-

injector

-

klckei

16 20 MeV, 3 6 kA l_l_._. ator accelerator

-

4 8 lines of sight

, variable interpulse spacing down to 500 ns for c 10 pulses Metglas cell

, 1

'

, , I

I

ns

P

induction adder

FET-switched modulator

hard lube modulator

Figure 7. Advanced Radiography concept showing a single accelerator used to provide a rapid sequence of pulses that are subdivided by fast kickers to direct the resultant shorter pulses into different beam lines. Pulses of less than about 100 ns duration at the test object are required in order to avoid motion blur.

The technologies needed to make the concept of Figure 7 a reality are a pulsed power driver for induction cells capable of MHz repetition rate and an even faster modulator with variable waveform capability that can be used to drive a fast, high-current kicker system.

2.1. Solid-state Modulator Concept The concept necessary to provide this capability is shown in Figure 8. The system consists of a pre-charged capacitor bank and a suitable switch. The switch is in close proximity to the load, an induction cell, and connects and disconnects the capacitor bank from the load. A similar circuit with opposite polarity is used to reset the magnetic core in the induction cell. The load voltage and current waveforms are shown in the figure. At the conclusion of the reset phase the system is ready to provide another pulse. The switch is a seriedparallel array of solid-state devices.

32

n

,Close ,,- s1

time

_ _ _ _ - Reset --

\

__ _ _ _ _ _ _ _ _

I

interval time

-

'-reset on

Figure 8. Solid-state modulator concept for high repetition rate operation

2.1.1 ARM Modulator The ARM (Advanced Radiographic Machine) modulator is shown in Figure 9. It is a prototype designed to drive an induction accelerator cell and to be stackable into an inductive adder configuration. The modulator is built around a Metglas core. The primary and resq energy storage capacitors can be seen at the top of the modulator. Four series stacks of switching boards are arrayed around the core. Each stack consists of over 20 printed circuit boards, each containing 12 FETs in parallel. The circuit boards are arranged so that each FET is in series with the one directly above and below it in the stack. The FETs are optically commanded to open and close. The modulator has an open circuit voltage of 15 kV and a nominal current rating of 4.8 kA. The modulator produces pulses ranging in width from 200 ns up to 2 p i at repetition rates up to 2 MHz.

33

Figc repetition rate.

UP

to 2 MHz

These modulators can be stacked by threading the interior of the cores with a conducting stalk forming an inductive voltage adder. A three-stage adder is shown in Figure 10. The output of this device is 45 kV at 4.8 kA with the same

Figure 10. ARM 3-stage adder. The completed modulator is operated inside an oil tank for electrical insulation. The adder is shown here extracted for maintenance.

34

The output of the three-stage adder is shown in Figure 11. An essential capability of this architecture that cannot be overemphasized is the flexibility of the pulse format. It is possible to vary the width and inter-pulse interval from one pulse to the next easily from a computer or programmable waveform generator. Shown below is an example pulse train consisting of a 1 - p pulse followed by three, 200 ns pulses followed by a 2-ps pulse, all with full reset. This prototype system fulfills the requirements of the Advanced Radiography concept shown in Figure 7.

Figure 11. Example output waveform from the 3-stage adder. A 1 -microsecond pulse with reset is followed by three, 200-nanosecond pulses (with reset) that are followed by a 2-microsecond pulse with reset.

This type of architecture lends itself readily to high average power applications for two reasons. First, the cores need not be driven anywhere near saturation in order for the system to work; they can be greatly oversized and segmented for cooling, which would allow them to operate at high duty factors. Second, the switches are highly distributed since one FET is only capable of switching on the order of 100 A at about 1 kV (the ARM modulator has over 2000 FETs). Each one of these can be heat-sinked and cooled.

35

3. Fast, High-Current Kicker System The second piece of new technology needed for the Advanced Radiography concept shown in Figure 7 is a fast kicker system that can precisely steer high current, nearly continuous beams.

3.1. Fast Kicker System for High Current Beams In order to produce very fast steering, the source of the electromagnetic fields applied to the beam must be inside the beam pipe. The architecture chosen is shown in Figure 12. It resembles that of a stripline beam position monitor. An outer vacuum housing surrounds four equal-sized electrodes. Each stripline is terminated at the upstream end by a resistor and is connected to a pulser through a time-isolated cable at the downstream end (the kicker must be driven at the downstream end or the electric and magnetic induced deflections of a relativistic electron beam will cancel). The kicker is operated as a 5042 system with both the resistive terminations and the time isolation cables and the pulsers having impedances of 50 Q. The kicker must not only switch rapidly (in a time less than tens of ns) but it must be able to regulate the position of beams with great precision and without emittance growth, as both of these are required to maintain good radiographic resolution from the accelerator. drive cable

mt

3

drive date

R dri!

septum magnet

1

switched beam positions

bias dipole windings

PQ 77

non-drivenplate

3;

Figure 12. The fast kicker system is similar to a stripline beam position monitor. Steering is accomplished by a combination of electric and magnetic fields.

36

Because the pulser technology chosen to drive the lucker has only unipolar capability, a DC bias dipole is wound around the outside of the kicker to extend the dynamic range of the beam centroid. A version used to deflect the 6-Mev, 2kA beam fi

Figure 13. ETA-II kicker showing a DC bias dipole and a DC sextupole winding. The solenoid to the right of the kicker is the lens used to match the beam into the kicker. The drive cables can be seen on the left (downstream) side of the kicker. The kicker length is about 1.6 meters.

In order to regulate the position of the beam at the kicker output a pulser with a variable waveform capability is required. Part of the motivation for this is that there is beam-induced steering inside the kicker due to the high beam current. Another reason is due to head-to-tail energy variations in the beam that will cause dispersion. We will return to this point in Section 3.4.

3.2. Implementation in DARHT-2 Some of this technology has found application in the second axis of DARHT. DARHT stands for Dual Axis Radiographic Hydrodynamic Test and consists of two induction accelerators. The first axis is a conventional, single-pulse machine producing about 20 MeV at up to about 3 kA. It has been operating very successfully for several years. The second axis, now nearly complete, will produce a 2-ps-long pulse at up to 2 kA and 18 MeV. (See Figure 14.)

37

Figure 14. The DARHT facility at LANL. The second axis, shown on the right, is nearly complete. The first axis has been operational for several years.

A kicker system is used to carve a sequence of four, relatively short pulses from the 2-ps beam exiting the accelerator. The unused beam is sent to a dump. The ional schematic is shown in Figure 15. Beam Dump

n

,

I

0

2 PS t

0

2 PS t

Figure 15. Concept of DARHT-2. A kicker system produces four radiographic pulses from a 2-ps beam exiting the accelerator.

The DARHT-2 accelerator uses Metglas cores in its massive induction cells. An electrostatic injector provides a beam with an energy up to about 3 MeV and 2 kA. The kicker system is set up so that the bias dipole deflects the beam into

38

the main dump until it receives drive pulses that allow the kicker fields to overcome the effects of the bias dipole and allow the beam to proceed to the xray converter targets. A portion of the beam line is shown in Figure 16.

Figure 16. The DARHT-2 beam line. The induction cells shown here are nearly 2 meters in diameter.

3.3. DARHT Kicker Pulser The kicker pulser is based on the solid-state architecture previously discussed. The actual implementation is highly modular with each switchboard having its own primary storage capacitor and Metglas core. Most of the layers are timed to produce a square output pulse. A number of the stages are charged to different levels and can be independently timed to produce a digital approximation to an analog waveform. Since the modulator is operated in an inductive adder configuration, the output voltage is the very nearly the sum of the voltages of all the individual layers. The kicker pulsers for DARHT are required to produce up to 18 kV into 50 P with a 20% modulational capability. The pulsers power the kicker through large, low loss cables that provide 2 p.s worth of time isolation to prevent any reflections or beam-induced signals from perturbing the beam. A simplified schematic showing three digital stages and one equivalent analog layer is shown in Figure 17.

39 Simplified Electrical Schematic of Adder (3 Switched Cells plus One Analog Cell)

cores ransformer primaries

. .. . .. .. Transformers Figure 17. Simplified schematic of inductive adder kicker pulser showing 3 of many digital stages and a single modulation layer. The actual modulation is accomplished by using 7 digital stages charged to different voltages and independentlytimed.

The final modulator system for DAFtHT-2 is shown in Figure 18. It consists of a positive and a negative polarity pulser (one for each opposing stripline to provide steering in the vertical plane) and a control rack in the center. The output cables to the kicker come out of the top of each pulser.

Figure 18. The each polarity, which drive 50-9 cables.

iers, one of

40

The completed pulsers are capable of generating variable width and variable shape pulse trains. An example burst of 4 pulses is shown in Figure 19. (DARHT is required to produce 4 equally spaced pulses of variable width).

m

Figure 19. An example pulse format from the DARHT kicker pulser system. Note the variable width and variable shape output waveforms. Two sets of curves are shown; the desired waveform initially requested by the control system and the actual output voltages.

The kicker has been extensively tested on the ETA-I1 accelerator2. Quick switching of the beam centroid has been accomplished without emittance growth for a 6-MeV, 2-kA beam with a pulsewidth of 50 ns. It should be noted that by driving all four striplines (and using two bias dipoles, one for each plane) steering may be achieved over the entire transverse plane. Furthermore, if one changes the polarity of one of the pulsers in the simple dipole mode, a fast, adjustable quadrupole lens may be realized. Figure 20 shows a single ETA-I1 pulse caught in the act of switching centroid positions and Figure 21 shows the resulting elliptical shape of a beam downstream of the kicker when the polarity of one of the pulsers is reversed to make a quadrupole lens.

41

Figure 20. A single 50 us beam pulse caught in the act of switching centroid positions. The picture is made from light emitted when the beam strikes a quartz foil downstream of the kicker. The total shift of centroid position is 4 cm.

Figure 21. A time-integrated image of light produced when the electron beam strikes a quartz foil downstream of the kicker configured to produce a quadrupole field.

3.4. DARHT Kicker Pulser Control System The kicker regulates the switched beam centroid position through use of a feed forward control system. An inner control loop determines a desired waveform to apply to the kicker electrodes. The actual waveform is measured and the loop iterates until the measured waveform corresponds to the desired one. Simultaneously, an outer control loop looks at the position of the beam downstream of the kicker as a function of time and uses an algorithm to derive a theoretical waveform that should result in the desired position. The outer loop feeds this information to the inner loop and the resulting beam position is measured on the next pulse. This loop iterates until the desired beam positional

42

accuracy is obtained. In practice the outer loop converges in just a few beam pulses. (See Figure 22.)

n

E E

W

20

I

I

I

I

I

......

10 _./-

I-.

......

c1

_

Kicked ....................... biarn (40 ...................,_.............,..... and 5&S widihs) _ _ _ _ _ _ _ 0 .............

Q) C

E a 0

_ ............ :.............. :............. :..............:............. ;..............

-

-10

cn

-20

Q

...................................................................

i..............

. I

m

................... > ............. ;...

......> ..........................

Time (ns) Figure 22. Plot of converged kicked beam position vs. time along with plot of beam position vs. time for beam entering the kicker. The switched beam position is successfully regulated to within 1 mm of the desired value.

4. Proton Radiography Kicker Pulser There is a radiographic technique that requires very high-energy protons (of order 20 to 50 GeV)3 . The system uses storage rings and requires fast kickers to extract pulses. The architecture developed for DARHT-2 is ideally suited for this application because of its inherent speed, flexibility and modularity. A prototype 50-kV pulser for this application is shown in Figure 23. The pulser drives a 5042 cable and must produce a 10-pulse burst at 2 MHz. The output of the prototype system is shown in Figure 24.

43

Figure 23 , 50-kV that of the: DARH

:e is very similar to

44 Current @ 50i Load 1 OP Burst @2MHz, 1 OOns PW, w/ reset

zOO1----

--

-

____

II

I

0

200

-800 1000

-1L00 J

Figure 24. A 10-pulse burst at 2 MHz. The current into a 5042 load is plotted vs. time. The amplitude of the burst sags because the capacitor bank is insufficiently large for this burst.

5. NLC Klystron Drivers The tremendous advances in solid-state devices and in their performance-to-cost ratio have made many new applications possible. For several years a joint SLAC-LLNL-Bechtel Nevada effort has been devoted to developing solid-state drivers for klystrons to be used in the SLAC version of the NLC (Next Linear Collider). A system concept is shown in Figure 25.

Figure 25. Concept for a solid-state pulser capable of driving 8 klystrons. The modulator is to put out 500 kV at 2 kA.

5.1. Solid-state Devices and Architecture Cost and robust operation are important considerations for this design. The modulator must be able to drive 8 high power klystrons at 120 Hz continuously and must supply 500 kV at 2 kA. Large IGBT (Insulated Gate Bipolar

45

Transistor) array devices used for traction control were chosen for this modulator. The architecture used here is again an inductive adder with one switching layer per Metglas core. Each layer has its own capacitor bank and IGBT switches. The modulator achieves high voltage through the use of a novel 1:3 step-up transformer. An IGBT used in the prototype is shown in Figure 26. The completed prototype is shown in Figure 27.

Figure 26. An IGBT used in the prototype klystron modulator. It operates at 3.3 kV and 800 Amps (manufactured bv EUPEC).

Figure 27. The completed prototype klystron modulator.

46

6. High Gradient Insulators The next technology innovation to be discussed is that of high gradient insulators. These insulators might lead to the development of compact, high current accelerators and power sources.

6.1. Insulator Flashover Vacuum insulators eventually break down along their surface as the tangential electric field stress is increased. Conventional insulators are generally monolithic structures, and the breakdown is thought to be the result of a secondary emission electron avalanche where electrons are field emitted at the negative end of the insulator surface and drift in the vacuum along the insulator surface4. Since the insulator is a dielectric it becomes polarized by the fieldemitted electron, which leads to a collision of the electron with the surface. The collision stimulates the desorption of gas contaminants stuck to the surface and also leads to the emission of additional electrons. These electrons continue to drift and collide with the surface, increasing the number of electrons drifting and gas molecules that are desorbed until the gas density is sufficiently dense that an avalanche breakdown occurs and the voltage across the insulator collapses. To slow this process one might introduce intermediate electrodes into the insulator (that protrude past the surface) on a scale sufficiently fine to interrupt the electron avalanche. The scale size for this is typically on the order of a millimeter. This is illustrated in Figure 28.

- Electron! Emitted

Emitted electrons cascade on conventional insulators

Graded Insulator inhibits the avalanche process

Figure 28. Flashover mechanism of a conventional insulator (left) showing the secondary emission electron avalanche. On the right is shown the concept of the high gradient insulator with closely spaced electrodes protruding past the surface designed to interrupt the electron avalanche.

These insulators have been fabricated from materials such as Lexan, Rexolite, Kapton, fused silica, Mycalex and alumina. The performance of these

47

insulators has significantly exceeded that of conventional insulators. A comparison of conventional and high gradient insulator performance is shown in Figure 29.

ii

l0O0I

I.

.

1

L

I 104 1

10

100

1000

10000

High Gradient lwlators Prototype HGI w/beam -Power law fit

1

P o w r law fit

100000

Pulsewidth (ns)

Figure 29. Comparison of conventional insulator performance (lower curve and points) and high gradient insulator (upper curve) performance vs. pulsewidth.

From Figure 29 we can see a general trend for all types of insulators; the surface flashover strength increases for shorter pulsewidths. A sample composed of Rexolite and stainless steel electrodes with a period of 0.25 mm was tested in a vacuum chamber between highly polished electrodes with a Rogowski profile. A Marx bank that could provide pulses on the order of 1-3 ns long powered the electrodes. A photograph of the insulator and electrodes can be seen in Figure 30. A typical voltage measured across the sample is shown in Figure 31. This particular data point corresponds to a field stress of 70 MV/meter.

48

Figure 30. High gradient insulator test apparatus showing the Rogowski profile electrodes and the Rexolitdstainless steel sample. The insulator electrode spacing is 0.25 mm. The sample shown is 3 nun high by 10 mm in diameter.

1.5x105 1.0~10~ 5 . 0 ~ O4 1 0 - 5 . 0 ~ 1O4

0

I

I

I

I

2

4 t(n4

6

8

i

Figure 31. Voltage measured across the sample shown in Figure 30. This measurement corresponds to 70 MVImeter field stress with no breakdown.

In order to apply maximum field stress to the insulator some of the layers were removed to increase the field stress. The resulting increase in capacitance widened the pulse somewhat to about 3 ns. There was still no breakdown at a stress of 100 MV/meter. Another interesting test concerns the ability of the high gradient insulator to sustain high field stresses in the presence of a high current electron beam. For this test the conventional insulator in one of our ETA-I1 induction cells was replaced with a high gradient version and the gap redesigned in order to have a direct line of sight from the beam to the insulator. The standard induction cell is shown in Figure 32. As can be seen from Figure 32 the insulator is slanted away from the cathode side to discourage electron hopping along the surface. The length of the insulator across the slanted face is 3.75 cm. Notice also that the insulator is shielded from a direct line of sight to the beam by the twisted gap geometry.

49

j ~ ~ u l a t o(3.75 r ern w j d ~ ~ ~ Slanted Shielded from beam

I

2 kA, 50 ns pulse

II

Oil Insulation Figure 32. Standard ETA-II accelerator cell showing the slanted, monolithic Rexolite insulator.

The modified ETA-I1 cell is shown in Figure 33. It has a high gradient insulator made from Rexolite and stainless steel electrodes. The replacement insulator is only 1 cm in axial length and has a straight wall. In addition, the gap structure is purely radial providing a direct line of sight to the 6-MeV, 2-kA, 50ns-wide ETA-I1 beam.

50 HGI (SS & Rexolite) (1 cm width) Straight wall Direct line of sight to beam

High Voltage

I

Graphite beam stop SF, Insulation

Figure 33. Modified ETA-II cell with a high gradient insulator. The insulator is only 1 cm in axial length and is straight with a direct line of sight to the beam.

The standard and high gradient insulators are shown in Figures 34 and 35 respectively.

Figure 34. Standard Rexolite insulator in the ETA-II induction cell. Note the slanted surface.

51

Figure 35. High gradient Rexolite and stainless steel insulator. The electrode spacing is submillimeter.

The modified cell was placed at the end of the ETA-I1 beamline with a graphite beam stop bolted to the cell. The cell was powered by the beam return current that flows through the drive blades. Various resistors were connected to the drive blades and created a reverse voltage across the gap. The resistor values were adjusted upwards in an attempt to reach an insulator breakdown. Attempts to reach breakdown levels failed. The insulator did not break down with over 20,000 shots with beam (the ETA-I1 beam is 6 MeV, 2 kA, 50 ns at 1 Hz) at the highest resistor value tried. The volt-second content of the cell cores was too low to attempt higher voltage operation. This result is rather remarkable given that the insulator has a direct line of sight to the beam and the beam is dumped at the end of the cell. There is a background of secondary electrons, x-rays and optical photons all present in close proximity to the insulator. An overlay of voltage traces on the cell for different resistor values can be seen in Figure 36.

52 200

150

2 g)100

W

0

CI

8

50

a

d o -50

Figure 36. Voltage traces from the modified ETA-Il cell. The lowest trace is about the voltage that the cell is normally run at (about 80 kV). The upper trace corresponds to approximately 18 MV/meter with no breakdown in the presence of a 2-kA, 50-11s electron beam. Note that the pulsewidths are shorter for the hlgher voltage traces because of core saturation

7.

Dielectric Wall Accelerator (DWA)

The performance of the high gradient insulator suggests that it might be possible to make compact, high current accelerators. The motivation for this is illustrated in Figure 37, which compares a conventional induction linac structure with that of a DWA. Figure 37 suggests that it might be possible to obtain much higher gradients than at an induction linac with a DWA if a suitable insulator material could be identified. The second major requirement is that a method must be found to supply the dielectric wall with an accelerating field at the high gradient.

53

E-field in gaps only t

4

b

Dielectric Wall Continuous E-field

Pulse Forming Line

4

c

State of the Art Induction Accelerator 0.75 MeVheter Gradient

-

Dielectric Wall Accelerator 0 20 MeWmeter Gradient

1

Figure 37. DWA concept. An induction linac can sustain an accelerating gradient only at the gap (typically of order 10 MVImeter) but the gradient averaged over the entire structure is usually less than 1 MVImeter. If the conducting wall were to be replaced by a suitable insulator perhaps the gradients achievable in the gaps could be sustained over the entire accelerator.

7.1. Asymmetric Blumlein One method of generating a suitable accelerating field is called the asymmetric Blumlein invented by Bruce Carder.5 The line consists of two transmission lines (depicted as radial lines in Figure 38) that are filled with dielectrics of different permittivity. Both lines are initially charged to the same voltage but with opposite polarity. At first there is no net voltage across the output end of the lines (inner diameter). If the outsides of the lines are shorted by closing switches, waves will be launched that travel radially inwards with different speeds. When the faster of these two waves reaches the inner diameter there will be a reflection of the wave accompanied by a net voltage reversal in that line. The voltage at the output end of the other line, however, is still equal to the original charge voltage since the slower wave has not yet reached that point. At that instant a net voltage appears across the output ends of both lines. That net voltage is maintained until the slower wave reaches the output end of the line and collapses the net voltage. Since the line is not 100%efficient there will be multiple reflections and ringing of the output waveform. The output waveform

54

can be applied across a high gradient insulator. This process is illustrated in Figure 38.

Figure 38. Operation of the asymmetric Blumlein. The line is formed from radial transmission lines containing material of different permittivities and charged to equal and opposite voltages. Closing switches on the outer diameter of the lines launch the accelerating pulse which eventually appears on the high gradient insulator at the inner radius.

7.2. Laser-Induced Flashover Switch A suitable closing switch can be obtained by using a flux of photons to bombard the outer vacuum surface of each charged transmission line in Figure 38. The photon flux will initiate a flashover of the vacuum surface, which provides an effective switch closure. A fast photon flux is available from a laser (a Nd-Yag laser with frequency doubling or tripling crystals is used). This is illustrated in Figure 39.

-4

Laser illumination

Figure 39. A fast laser pulse initiates a vacuum surface flashover across a highly stressed insulator, providing an effective switch closure.

55

7.3. Test of Asymmetric Blumlein The operation of an asymmetric Blumlein was tested using a “cross” configuration consisting of four equal-width striplines that intersect at their output ends. The materials employed were de-ionized water for the “slow” line and RT-Duroid, a printed circuit laminate, for the “fast” line. The Blumlein was placed inside a chamber to provide vacuum on the outer and inner surfaces. The Blumlein in the chamber is shown in Figure 40.

Figure 40.“Cross” configuration asymmetric Blumlein structure inside its vacuum chamber. Note the optical ports and mirrors for the laser beams that trigger the acceleration waveforms.

The system is pulse-charged (since de-ionized water cannot hold a charge for very long) and triggered by four laser beams, one for each stripline. A frequency tripled Nd-Yag laser is used for the triggering. The optical paths are arranged so that the four beams arrive simultaneously at the stripline edges. Conventional Rexolite insulators are used on the inner and outer surfaces for these first tests. A voltage probe is inserted into the inner diameter of the inner insulator to measure the output voltage. The water had to be filtered and de-gassed to suppress bubble formation that would drastically reduce the allowable field stress in the water. Both fast and slow lines are fabricated to the same thickness, which further decreases the efficiency of the Blumlein to transfer energy to the load. The closed vacuum chamber and simulated laser beams are shown in Figure 41.

56

Figure 41. Closed vacuum chamber with simulated laser beams. The four beams arrive at their respective striplines simultaneously.

The output field stress achieved exceeded 5 MV/meter. An output voltage trace from the monitor is shown in Figure 42.

Figure 42. Output voltage from a single asymmetric Blumlein.

Two Blumleins were stacked in the vacuum chamber. The results are shown in Figure 43.

57

Rgure 43. Results from a stack of two asymmetric Blumleins.

The outputs of the lines are not matched to the loads, which results in excessive ringing. In practice, the first (negative) part of the waveform would constitute the output pulse.

7.4. Future Developments

The use of de-ionized water for the slow dielectric was necessitated by the difficulty in obtaining high permittivity, low loss, high quality insulating material in large diameters. A material has been developed which has a relative dielectric constant of about 30 and which is available in thin, flexible sheets about 36 inches wide. We are developing this material for use in pulse-forming lines. An example of a curved stripline laid out on this material is shown in Figure 44.

58

Figure 44. A pair of curved striplines on a substrate of advanced dielectric material of relative permittivity 30.

This pair of lines will be used as the slow lines in combination with a lower dielectric material like Lexan as shown in Figure 45.

Figure 45. A pair of curved striplines on Lexan providing the fast lines to match the slow lines in Figure 44.

Using solid dielectrics will enable the use of slow charging systems since these materials will hold a charge for a very long time.

Acknowledgments Many people contributed to the work that has been reported here. The original development of the solid-state architecture was done by Hugh Kirbie, then at LLNL. The modern development of this architecture for kicker pulser and

59

klystron driver applications is due to Ed Cook of LLNL. Judy Chen and Jim Watson developed the fast kicker and its control system while John Weir oversaw the operation of the ETA-TI accelerator on which the kicker system was developed. Steve Sampayan led the experimental work on the Dielectric Wall Accelerator. I am grateful to Rollin Whitman, the DARHT Project Director, for material on the DARHT accelerators.

References

1. G. J. Caporaso, “Linear Induction Accelerator Approach for Advanced Radiography,” PAC’97, Vancouver, B. C., May 1997. 2. Y. J. (Judy) Chen, G. J. Caporaso, J. T. Weir, ”Precision Fast Kickers for Kiloampere Electron Beams,” PAC’99, New York, March 1999. 3. G. E. Hogan, et al., ”Proton Radiography,” PAC’99, New York, March 1999. 4. R. A. Anderson and J. P. Brainard, J. Appl. Phys. 5 l(3) (1980) 1414. 5. B. M. Carder, U. S. Patent No. 5757146, May 26, 1998.

-

RFQ ACCELERATORS A .SCHJZMPP

lnstitutfur Angewandte Physik, Johann Wolfgang Goethe-Universitat, Robert Mayer Str. 2-4 0400.54Frankfurt/Main, Germany a.schempp @em. uni-j?ankj%rt. de

The radio-frequency quadrupole (RFQ) is a linear accelerator structure for lowvelocity ions. It focuses and accelerates by means of electrical radio frequency (rf) quadruple fields. They can capture high current dc beams at low energy and convert them to a bunched beam with high efficiency and small emittance growth. RFQ linacs have found numerous applications. We will discuss some typical RFQ pre- and post accelerators for protons as well as for heavy-ions, and injectors for linacs as well and as for cyclotrons.

1. Introduction An injector is a combination of an ion source, a low-energy beam transport (LEBT) system, a pre-accelerator (a Cockcroft-Walton Cascade or an RFQ), and an intermediate section, which matches the beam to a following structure, e.g. an Alvarez accelerator or a cyclotron. This pre-accelerator defines the maximum current and the phase space for the following stages in which the effective emittance will only grow. The injector is the bottleneck because focusing forces are weak and the defocusing effects and the nonlinearities caused by space charge are strongest at low energies. Especially in the design of a high-current accelerator the emittance as well as the current must be optimized, so the low energy part is especially important. The development of the radio-frequency quadrupole (RFQ) structure with its ability to bunch and accelerate low-energy, high-current ion beams, as shown schematically in Fig. 1, opens new possibilities for accelerator designs. ION SOURCE

LE BT

t'f -

Figure 1. Layout of a RFQ injector

R FQ

61

The variety of RFQ accelerators covers the full ion mass range from hydrogen to uranium, the frequency range from 5-500 MHz, and duty factors up to 100%. The physics of transport and acceleration of high-current ion beams in RFQs has been solved to such an extent that the best beams, which can be produced by ion sources and transported in a LEBT, can be captured and transmitted by RFQs with very small emittance growth .

2. RFQDesign Basically the RFQ is a homogeneous transport channel with additional acceleration. The mechanical modulation of the electrodes as indicated in Fig. 2 adds an accelerating axial-field component, resulting in a linac structure which accelerates and focuses with the same rf-fields.

Figure 2. RFQ electrodes For a given injection energy and frequency the focusing gradients G = X*UJa2, (X < 1 for modulated electrodes) determine the acceptance in a lowcurrent application. A maximum voltage U, has to be applied at a minimum beam aperture, a, if the radial focusing strength is the limiting factor. The highest possible operation frequency should be chosen to keep the structure short and compact. After the choice of U, and operating frequency, the "RFQ design", the values of aperture a, modulation m and the lengths L, along the RFQ determine the electrode shape (pole tips), the ratio between accelerating and focusing fields, as indicated in Figs. 2 and 3 ,and hence the beam properties. Figure 3 shows that for the first half of the cells modulation m is small and the energy T stays nearly constant. The beam is adiabatically bunched with small longitudinal fields. Acceleration starts in the last quarter of the RFQ. There are many possible ways to shape the electrode parameters of the RFQ after the basic parameters have been given, such as input and output energy per nucleon and the maximum current to be accelerated.

62 I

- r

T teV/u)

T:' 200 150 100 50 0

0

40

200 240 280 FTQ parameters vs. cell number N €30

120

160

Figure 3. Example of RFQ-electrode design, energy T and electrode parameters as function of the cell number N(GS1-HLI RFQ)

Table 1: RFQ parameter scaling

Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter

63

The optimum frequency can be determined by many factors. In smaller projects it is the availability of transmitters or a matching post-accelerator. Lower frequencies give stronger focusing, less difficulties with power density and mechanical tolerances, and generally a higher current limit. With accelerating to higher energies the accelerator structures get large and inefficient, and jumping to higher frequencies leaves empty buckets. This is no problem for heavy ion accelerators, where such a scheme is normally used, but in high current applications the space charge effects are proportional to the beam bunch charge. Therefore higher frequencies are favorable also at low energies for compact designs with highest brilliance because the charge per bunch and the frequency jump to a final linac stage are smaller, but the currents are limited by the maximum focusing fields and sparking. The choice of basic RFQ parameters will scale others, as summarized in Table 1. A high rf-frequency will require higher injection energy and allow for higher final energy as well. The radial acceptance and the beam current are inversely proportional to the frequency and proportional to the applied electrode voltage. The total rf-power to reach a final energy does not depend on the frequency, so the problems with tolerances and power density dominate at higher frequencies. The energy spread, the output emittance and the beam losses can be improved with a reduced accelerating gradient, resp. longer RFQs, which clearly affects the costs and sets new limits to the rf-tuning and stability. For a bigger project like a spallation source, the optimization of the total linac, the availability of power sources, and naturally costs will set some design input parameters and e.g. will increase the frequency to lower the charge per bunch, to avoid funneling and ease emittance growth and matching problems. Of major concern are beam losses in the RFQ but also losses along the following linac which can be influenced by proper shaping and preparing the beam in the RFQ. Even though 99+x% theoretical transmission seems to be academic thinking, the work being done there is essential for understanding and avoiding losses and halo formation in the RFQ and in the following linacs. The choice of a rather high frequency shifts the mechanical and rf parameters to a region where tolerance and tuning problems and power density questions require new solutions and prototype developments. The RF structure has to generate the quadrupole fields with high efficiency and stability. The four-vane structure, which is mostly used in proton and Hacceleration, is basically a TEzll-rnode structure, in which the resonator has been loaded with electrodes to increase the quadrupole field, as shown in Fig.4. The end region has to be modified to allow the magnetic fields to turn around and to shift the mode into a TEZl0with a constant quadrupole field along the structure. Radio-frequency stability, and mechanical and thermal symmetry of the cavity, which also can be treated as four weakly-coupled resonators, is the reasons for very tight mechanical tolerances which are especially a limiting factor in high-duty-factor operation.

64

Figure 4. The four-vane RFQ The four-rod structure shown in Fig. 5 consists of a linear chain of stems, which form a chain of strongly coupled W2 resonators. By the direct connection of the electrodes with the same polarity, dipoles cannot be excited and tolerances are less stringent. The length of the stems can be changed to give compact resonators with rather low frequency, which can also be used for low frequencies for low charge-to-mass ratio heavy ions. Generally, the FW power N needed is independent of the frequency, while the acceptance and the maximum ion current are proportional to the electrode voltage resp. N2, which is not a big issue in pulsed injectors with low average power. High-duty-factor operation is the present area of development. A first class of structures, which will be used as SNS-like linac injectors with duty factors of up to lo%, are now being designed and built, but still more difficulties can be expected for CW RFQs.

65

Figure 5. The four-rod RFQ

3. RFQ application The standard RFQ application is operation as pre-injector for an Alvarez linac feeding a synchrotron. These systems are easily matched to ion source and RFQ designs because they have a low duty factor, which allows pulsed, highpower-density operation. Examples are the injectors at BNL, DESY, CERN, IHEP and KEK. The 4-vane RFQ structure employed in most cases can be treated as four weakly azimuthally coupled resonators in longitudinal 0-mode, a system which is very sensitive against imbalance of the four quadrants. In addition, the longitudinal field tilt sensitivity is proportional to Mechanical tolerances, coupler loops, tuners, vacuum ports, and changes of electrode modulation all contribute to field tilts. Higher duty factors have been favored by the development of high-brilliance beams at LANL. This LEDA-RFQ was built to be part of the front-end demonstrator for an APT (Accelerator Production of Tritium) machine.

66

Table 2 Parameters of the Lead RFQ Operating Frequency Proton Input Beam Proton Output Beam Duty Factor Peak Surface Field Structure Power Loss Total RF power Surface Heat Flux Configuration (OFE copper) Structure Tuning

350 MHz 75 keV, 105 mA, 0.2 mm* mrad, nns, normalized 6.7 MeV, 100 mA, 0.22 nns,n 100% (cw) 1.8 Kilpatrick, 330 kV/cm 1.2 MW, 85 V s I 1.9 MW, fed by 12 WG irises I 11 W/cm2,65 W/cm2peak 4 resonant segments, 8 brazed sections, each l m long static: 128 tuners, dynamic: water temDerature

Azimuthal r e s o n a n t l o o p c o u p l e r schemes ( s t o r e d energy U)

Figure 6. Stabilizing a 4-Vane resonator

Providing a field that is radially symmetric and longitudinally flat, resp. as designed in the simulation codes, also during thermal load is one important task in designing and tuning a RFQ. Since the early development of the RFQ structure a number of resonant and nonresonant stabilizing schemes have been proposed and tested (Linac84/86). Figure 6. shows examples of nonresonant stabilizers: VCR rings and their magnetic equivalent PISLs, together with resonant devices like resonant rings and posts connecting the quadrants. Typically there is no field in these stabilizers and no additional losses, if the structure is balanced.

67

The RFQ should be short to reduce possible field tilts. One can break the RFQs into several individually driven structures, which creates complexity and matching problems. A unique system has been developed at LANL, where, as in a CCL, the structure is subdivided into e.g. four parts but the connection is via resonant cells. This led to an RFQ made of eight resonantly-coupled pieces, as shown in Fig.7, each furnace-brazed, aligned and tuned individually. The electrode structure is nearly unchanged but now the tilt sensitivity is reduced and depends linearly on the cavity length Llh only. Azimuthal posts (like post couplers in the Alvarez) have been incorporated into these cells as indicated in Fig. 8, so that the RFQ is still difficult to tune but very stable.

Figure 7. The LEDA RFQ structure

The successful operation of LEDA, the characterization of the 100-mA CW beam and all the rf and mechanical engineering can well compete with the beam dynamics developments, which also gave new insights on losses and stability. The LEDA project at LANL was the first of a generation of high (average) power RFQs. Based on this experience, other projects started such as the IF'HIproject at CEA, and the TRASCO study at INFN in Legnaro which aims at a "modest" average current of 30 mA for a prototype for waste transmutation. The KOMAC project at KAERI in Korea is also looking for waste transmutation and energy breeding applications of a GeV high current proton linac. Design studies and prototyping have led e.g. to the successful acceleration of a 30-mA beam in a first RFQ prototype. The RFQ-groups in Beijing and at BARC and at CAT in India have also started work on an ADS-type IWQs for high current linacs, also as projects on ADS to help solve the predictable future energy problems in these countries. At first both projects plan for a low energy demonstrator linac, to study this high current technology Somewhat smaller RFQs are used in projects which can be grouped as spallation neutron sources: Designs for up to 5 MW (510% duty factor) have been made which plan for H - beams of 30-100 mA for new machines and the upgrade of existing high-power linacs like LAMPF and ISIS.

68

COUPL ING

PLATE

STAB 1 L 1 ZER ROD

Figure 8. The LEDA RFQ resonant coupler structure The beam dynamics design has matured to rather small emittance growths (10-20%) which stretch the limits of simulation results especially concerning halo formation and minimum beam losses. The advances in structure development are slow and reflect the limits of the technology. The special pulse shape on the target requires H'-acceleration and a storage ring for pulse length compression. The 2 to 5 MW beam comes in 1-psec pulses with 50 Hz as a typical value. The RFQ injectors for such systems are planned for 50-100 mA at output energies of 2-5 MeV The SNS project with its collaboration of six US Labs resulted in the building of SNS at Oak Ridge. LBNL built the injector, which has been successfully operated with full specifications. The 4-Vane RFQ (2.5 MeV, 402 MHz) is stabilized with PISLs, longitudinal stabilization is not used. Fig. 9 shows a module of the brazed Cu-Glidcop structure.

69

Figure 9. View of the first SNS-RFQ module In Japan a joint project of KEK and JAERI is going ahead with the building of a new multipurpose facility JKJ, which finally will reach 50 GeV protons as shown in Fig. 10. One planned operation mode corresponds to a spallation source. Specifications and technology of the RFQ injector are similar to those from LBNL, and tests of short parts have been successful.

4

360m

=k

Figure 10. Block diagram of the JKJ high current linac Short prototype RFQs for high-duty-factor operation and CW have been set up in CRNL, LANL, KEK, JAERI and also in IAPF. A parallel development aims to build a 120-mA, 35-MeV, CW deuterium accelerator (IFMIF) for material testing of fusion devices.

70

Figure 11. The ISIS RFQ injector and the 4-rod insert

71

For the spallation source ISIS at RAL a new RFQ injector has been built to replace the CW-Injector. There the choice of final energy and frequency was determined by the "ancient" linac as 665keV and 202.5 MHz, which is favorable for using a short 4-Rod type RFQ-structure. The duty factor can be as high as 10%. Beams of up to 35 mA has been accelerated and characterized in experiments. Until SNS becomes operational, ISIS will still be the most powerful n-source with its 200 kW beams. RFQs are especially attractive for low-energy, heavy-ion accelerators. They cannot replace static injectors and Van de Graaff generators in terms of energy resolution and beam quality, but are favorable for applications with high-current beams or in combination with sources like an ECR, because the source can be close to ground potential and is easy to operate and service. The RFQ concept of spatially-homogeneous strong focusing proposed by Kapchinskiy and Teplyakov employs strong focusing with rf electrical focusing that is independent of velocity so that the acceleration can start at low energy with rather short cells. This allows for adiabatic capture of the dc beam from the ion source. Heavy ion RFQs have been built at LBL, INS, ITEP, GSI, Saclay and IAPF, for example, for atomic and nuclear physics research. They can be distinguished by the lowest specific charge they can accelerate and by the operational duty factor. Storage-ring and synchrotron injectors have a favorably low duty factor. A high-duty-factor RFQ is the injector for high charge states from an ECR source, the HLI RFQ at GSI which operates at 25% duty factor in routine operation. The HLI RFQ is 3 m long (designed for 108.5 MHz and U2*+ions, q/A = 0.1 17) and accelerates from 2.5 to 300 keV/u. The high rf efficiency (figure of merit: shunt impedance) of the HLI RFQ is important for high duty factor structures, for which technical problems like cooling and thermal stress control dominate, while for synchrotron injectors this is no major concern. The high-current injector at GSI is designed for U4+ions. It consists of a RFQ (2.2-120 keV/u) for 15 mA beam current and an IH structure (0.12-1.4 MeV/u) operating at 36 MHz. It has been successfully operating for several years and has provided beams for injection into the SIS synchrotron to fill it up to its space-charge limit. Van de Graaff Tandem machines have been the work horse for nuclear physics research. Their limitations are low currents and low energy per nucleon for the heaviest ions, which led to the various heavy-ion rf accelerators and Tandem post-accelerators. A RFQ, though a low-velocity structure, can be applied as a post-accelerator for a Tandem installation, if very heavy particle and a fixed output energy/u are no restriction for the experiments. At Sandia National Laboratories an installation of this type is being set up. The PA RFQ is a 425-MHz structure very similar to a "standard" low-dutyfactor proton injector. The rather low output energy of the Tandem of 0.25 MeV/u (At?'+) fits the velocity range of the RFQ (output energy 1.9 MeV/u). The PA RFQ has a total accelerating voltage of 11.6 MV (total length of 6.2 m) and is split into two resonators.

72

Experiments in the area of material modification and ion-beam analysis make use of the p-beam, an advantage of the low-emittance Tandem beams not spoiled by rf post-acceleration.

Figure 12. The IH-RFQ structure of the GSI HSI accelerator

A very difficult task is to design an RFQ as the injector for a cyclotron. To inject into a separated-sector cyclotron, the RFQ has to provide a bunched beam at a well defined injection energy given by the inner radius of the SSC. The operating frequency of the RFQ must be synchronized with the cyclotron frequency, which for RFQs normally means a fixed output energy per nucleon, which could be a possible solution only for fixed-energy cyclotrons. A fixed-velocity profile is typical for RFQs. It can be changed only by varying the cell length L or the frequency f. The second possibility for changing the Wideroe resonance condition: L = Ophd2 = vpI2f , is the way that has been used for RFQs with variable energy (VE RFQ). For this reason it is possible to change the output energy using the same electrode system by varying the resonance frequency of the cavity: vp f, T vp2. Figure 13 shows the method of tuning by means of a movable plate, which varies the effective length of the stems. In Frankfurt the VE RFQ was first developed for application as a cluster post-accelerator at the 0.5-MV CockcroftWalton facility at the IPNL in Lyon, France (EOut= 50-100 keVIu for m = 50u). Very heavy and low-velocity particles are accelerated in the second VE-RFQ,

-

-

73 with an energy range from as low as 2eVlu to 1 keV/u for singly-charged metallic clusters up to mass m = lOOOu (frequency range from 5 to 7 MHz).

Beam

/ /Movable tuning plate 5

plate

Figure 13. Basic cell of a RFQ with variable energy VE-RFQs have a fixed ratio of output-to-input energy given by the length of the first and last modulation cells. This is similar to the energy gain factor of a SS-Cyclotron which makes them well suited as injectors. To cover the energy range of 1.5-6 MeV/u, the injection energy of the ISL must be between Ei, = 90360 keV/u (maximum accelerating voltage U, = 2.9 MV, at cyclotron frequencies of 10-20 MHz. In Berlin the ISL injector is a combination of an ECR ion source on a 200 kV platform, which produces highly-charged ions with charge-to-mass ratios between 1/8 and 1/4, and a VE-RFQ with a frequency range of 85-120MHz, which allows energy variation by a factor of 2. To stretch the energy range the RFQ was split into two RFQ stages. Each stage has a length of 1.5 m and consists of a ten-stem, four-rod, RFQ structure. With a RF power of 20 kW per stage, an electrode voltage of 50 kV and CW operation is possible. In the first mode of operation both RFQs accelerate and the output energy of the cyclotron is between 3 and 6 MeV/u with a harmonic number of 5 for the cyclotron. For the lowenergy beam only RFQ1 accelerates while RFQ2 is detuned to transport the beam. In this mode the energy range of the cyclotron is E,,, = 1.5 to 3 MeV/u. The cyclotron works on the harmonic number 7. In both modes the RFQs are tuned to the eighth harmonic of the cyciotron.

74

ECR-SOURCE

I

NEW RFQ-INJECTOR BWCm

CYCLOTRON

CHOPPER

I

Figure 14. The VE-RFQ as cyclotron injector at ISL Berlin

75

Figure 15. The TRIUMF RIB accelerator with a 35MHz split ring 4-Rod RFQ At RIKEN a similar task had to be solved. The static 450 keV injector to the variable frequency Wideroe -type linac has been replaced by an RFQ which is tunable from 8 to 28 MHz to match the VE linac and RRC cyclotron. An asymmetric four-rod-type RFQ, excited by one h14 resonator is used for this large frequency range. The electrodes must be supported by ceramic stems and, especially for the higher frequencies when the electrodes themselves are a major part of h14 line, the tuning is difficult and the field distribution becomes unbalanced. Experiments show that this system matches the RIKEN linac very well and increases the experimental possibilities.

76 4. New developments

There are a number of studies making use of RFQ ion injectors of which the proposal to build accelerators for radioactive beams seems to be of the highest interest in nuclear physics research. The typical RIB facility starts from an ISOLDE type of ion source with singly-charged ions. To obtain a reasonable amount of ions CW operation is planned. This favors superconducting structures, however the low-energy part must use much lower frequencies for rf acceleration than is suitable for SC cavities, so room is left for normal conducting RFQ structures to accelerate and form the beam as in the TRIUMF RIB accelerator in Fig. 15. The various systems can be distinguished by the heaviest mass number that is planned to be used and the accelerator system that will be employed. For masses between 30 and 60u, INS, T R I W and ANL all have linac-based systems. Rex ISOLDE in Fig. 16 is a system that has been realized by groups from LMU, Miinchen and MPI. It consists of a Penning ion-trap followed by an EBIS ion collector-charger with pulsed extraction, and a small compact accelerator of the WQ- post-accelerator type installed at the CERN ISOLDE facility.

Figure 16 The REX ISOLDE layout

There are many applications of RFQs in industry. The first group is used for material improvement such as ion implantation in silicon, e.g. machines for 1MeV boron and up to 9-MeV phosphorus. A second group are involved in the medical field e.g. parts of PET isotope production units and medical synchrotrons such as Loma Linda and NIRS. Another application of high current RFQs is as a compact radiation source for radiography with neutrons or resonant x-rays similar to those proposed for material detection. These applications require high beam power for high throughput. Figure 17 shows the 4-MeV RFQ ( L 4 m), designed for 50 mA, 20% duty factor operation, showing a typical 4-rod RFQ design with a top lid along the tank for easy access for alignment, tuning and inspection of the structure

77

Figure 17. View of the 4 MeV four Rod-RFQ New developments in particle dynamic designs aim at very small beam losses and reduction of emittance while also reducing the voltage and RF power requirements but with only a minor reduction in beam quality. This would be especially important for high duty factors and industrial application. The matching between RFQ stages and the following accelerator stage has also been improved. The first step is optimization of the end cell to shape the transition fields. Recent designs also shape the bunch longitudinally and radially (e.g. the funnel experiment at IAPF). 5. Conclusions

There are many new developments in the field of RF ion accelerators which will allow new experimental parameters for atomic and nuclear physics. Injectors into cyclotrons, Tandem replacements and new compact ion accelerators are attractive applications of RFQs. The average beam current which is approximately 1 pA in a typical injector for a HE-machine, can be as high as 1-2mA for a spallation source injector and up to 100 mA in ADS applications. This illustrates the steps in development and the advances in beam physics, structure development and technology.

78

Acknowledgements Many colleagues should be cited for their contribution to the RFQ development work described in this paper. Among the many reports published on this topic in the PAC and LINAC series I would like to restrict myself to two basic papers, from which all RFQ work started, and to articles appearing in the CERN and US Accelerator School reports. These are cited in the Bibliography. References

1. I.M. Kapchinskiy and V. Teplyakov, Prib. Tekh. Eksp 119, No.2 (1970) 17 2. K.R.Crandall, R.H. Stokes, T.P. Wangler, Linac 79, BNL 51134 (1979), 205 3. M. Weiss, Proc. CERN Accelerator School, Aarhus, 1986, CERN 87-10, 1987 4. A. Schempp, Proc. CERN Accelerator School, Oxford, 1991, CERN 92-03, 1992 5. A. Schempp, US Particle Accelerator School, Fermilab, AIP C 184 (1989)

RF STRUCTURES (DESIGN) HEINO HENKE Technische Universitat Berlin, EN 2, Einsteinufer 17, 0-10587 Berlin, Germany E-mail: henke @tu-berlin.de

The purpose of this course is to introduce the fundamentals of RF structures. Starting with the characteristics and the behaviour of a cylindrical cavity, the concept of continuous acceleration in a synchronous wave is introduced and the essential parameters are described. The concepts of travelling-wave structures with constant-impedance and constant-gradient geometry are treated, as well as standing-wave structures. Finally, biperiodic structures and structures for low velocity particles are briefly mentioned. The course does not treat superconducting structures or Radio-FrequencyQuadrupoles because they are presented elsewhere. It serves as an introduction to the follow-up courses on numerical evaluation and on fabrication and testing of structures.

1

Introduction

Electromagnetic resonators can be pictured by mechanical analogies. The 1D vibrations of a mechanical string fixed at both ends correspond, in the electromagnetic situation, to the resonances of a plane wave between two parallel conducting plates. The 2D vibrations of a clamped membrane find their analogue in the transverse vibrations of a waveguide at cut-off (with no longitudinal variations). A jelly in a rigid box performs 3D vibrations, and here the analogue is an RF cavity. The fundamental behaviour of a cavity can be understood by an evolutionary process starting with a standard RLC-resonant circuit, Fig. 1. If one wants to increase the resonance frequency w, = (LC)-'n, one has to decrease L, until the coil is reduced to a single winding, and C is decreased by going to two parallel plates. Electric energy is stored between the plates and magnetic energy around the wire. The Ohmic losses in the wire and in the plates are represented by the resistance R . With this process we could increase 0, but had to pay a heavy penalty. The losses were increased dramatically because of the skin effect and, what is worse, the circuit is no longer confined, it has stray fields and even radiating fields which interact with the surroundings. Both effects can be cured, at least partially, by arranging more and more wires around the

79

80

capacitor, thus the losses and the stray fields were reduced. In the limit, the capacitor plates are connected directly by metallic walls. In a standard LC-circuit we would have caused a short-circuit in that way. RF-cavities, however, have dimensions that are on the order of the wavelength, and wires and walls are no longer short-circuits but transmission lines. We have stretched the analogy so far, but we have to stretch it even a little further. If the frequency is increased further, the cavity dimensions are no longer roughly W2 but h or 3 W2 or even larger, and in the cavity we will find several modes and field maxima, like those in the jelly in the box.

Figure 1

Evolution from a resonant circuit to a cavity.

In the following we will derive these fields and explain the fundamentals of cavities. Once we understand the basic principles, we will move forward and treat different aspects of cavities, aspects that are motivated by requirements such as economic use of space and money, reliability, simplicity of the whole RF system, and so on. This will lead us to a zoo of different devices, each one optimized for a particular application. The subject is treated in a basic and self-explanatoryway such that recourse to literature is minimized except for a few inevitable cases. On the other hand the content is nothing new and has been treated in several schools; see, for instance, references 9 to 11.

81

2

Cavity Fundamentals

2.1 Ideal Pill-Box Resonator

We start with Maxwell's equations in a homogeneous medium

aE V X H =J + E at aH VxE=-pat

and take as an example waves propagating in a circular waveguide of radius a. Then, all field components are proportional to

which we drop in the following, and Maxwell's equations split into two sets of equations. One set with H, = 0 and all other components being derivatives of E, in cylindrical coordinates (p, q, z )

E, itself fulfills Helmholtz's equation

i

where

a

1 d2Ez

+ K ~ E=, 0,

(7)

82

K2 = k2 - k:,

w

k= -=O c

&.

This set of equations fulfills all of Maxwell's equations, and the resulting waves are called TM-waves (Transverse Magnetic). The Bernoulli solutions of Eq. (7) are E, = (A cos vcp + B sin vcp) ( C J , ( K p ) + D N , (Kp)). In a circular waveguide E, must be 2n-periodic, i.e. v = m, and by a proper choice of the origin B becomes zero. Further, E, must be finite on the axis, so D = 0, and has to fulfill the boundary condition E, (p = a ) = 0 or Jm (Ku) = 0 + Ku =j ,

(8)

with j, being the n-th non-vanishingzero of the Besselfunction of order m. Finally, there is a set of eigenwaves

with

which satisfies Helmholtz's equation together with the periodicity, regularity and boundary conditions. Together with the remaining field components (6) each eigenwave (also called mode) individually satisfies Maxwell's equations. A second independent set of eigenwaves is obtained by putting E, = 0. Then, in a completely analogous way we obtain waves with an H,-component that satisfy the different boundary condition E , (p = a) = 0, that is H, becomes

.I

where J,,

is the n-th non-vanishing zero of J L ( j k n ) = 0. These eigenwaves

are called TE-waves (Transverse Electric). A general field in the waveguide is represented by a superimposition of all TE-waves and TM-waves

83

H = Cm,n (Hz

+ Hit ).

After having derived the modes in a circular waveguide we can easily construct resonant modes in a cylindrical cavity of length g. For that purpose we take the E,-component of a forward and a backward (k,-+ - k,) travelling mode, Eq. ( 6 ) together with Eq. (9),

and fulfill the boundary condition E? ( z = 0, g ) = 0

sin kzmng = 0 -+kmpg = pn,

p = 0, 1,2, ...

(12)

As a result each resonant mode has a triple set of eigenvalues m, n, p . Here 2m indicates the number of field maxima in the azimuthal direction, n the number of maxima in the radial direction, and p the number in the axial direction. The E,component of a TM-mode is given by

The other components follow from Maxwell's equations. Again, in a similar way we may obtain the resonant TE-modes. Independent TEmP- and TMmnp-modesexist only in an ideal cylindrical cavity. Any real cavity with beam ports, tuners, power-couplers, vacuum ports ... will require a superimposition of many modes. But still, one mode will normally be dominant, and it may be a good approximation for many purposes to take only this mode into account. Above we have derived the fields for TM-modes in a cylindrical cavity. The reasons are that many cavities are cylindrical or close to cylindrical and that TM-

a4

modes are the modes for accelerating because they have an E,-component. TEmodes are used for deflecting the beam and are of less interest in a linac. The standard mode for acceleration is the TMolo-mode. The E, is independent of cp in that case and also independent of z. It has a maximum on the axis and no other maxima in the radial direction, thus the cavity has the smallest diameter for a given frequency. The field components are

z=J" &

and the resonance frequency W

k= -= C

06j o l / a , =

Its field pattern is shown in Fig. 2.

Figure 2. Th40,o-mode pattern in a cylindrical cavity.

2.2 Resonance Behaviour o f a Cavity Mode Above we have treated modes in an ideal cavity. Real cavities have wall losses, couplers, openings, tuners, and so on. Nevertheless, they are designed in such a way that they ring in well-defined modes of operation which can be described by harmonic resonators. While it is very difficult to take all the details and in particular the wall losses into account, it is mathematically simple to assume an

85

ideal cavity with a lossy dielectric filling. This preserves the ideal modes but allows for studying losses. Now, let us assume the cavity is driven by a current that passes through it, at least partially. Then, the current density J in Eq. (1) splits into a conduction current J, = KE,responsible for the losses in the dielectric with conductivity K, and an enforced current JOas driving term. Taking the rotation of Eq. (2) and substituting Eqs. (1) and (3) we obtain

-

2

V X (V X E) = V (V E) V E = -p

at

at

aE d2E aJo v ~ E - ~ K - - ~ E ~ = ~ at

at

at

A remark should be made concerning the fact that we put the charge density qv to zero. If the impressed current Jo flows in a wire this choice is obvious since there is no charge. But even in case of a free current with a charge density we can neglect it, because the related irrotational fields, which are not divergence free, are nonresonant and do not build up in time. They are, therefore, small and negligible compared to the resonant (solenoidal) fields. Next, we follow a procedure given first by Condon [ 11. We remember from the above chapter that any field can be expanded in eigenstates (modes)

where a, are time-dependent expansion coefficients and enconstitute the two sets of TE- and TM-modes, which means n goes over the triplets (m,n, p ) for TEand TM-modes. The en's therefore satisfy the eigenvalue problem

V2en+ k? en= 0,

V ' en= 0

in the volume,

(18)

n x en= 0 on the walls. As can easily be shown, they are also orthogonal

I

en. emdV = N,,,6,".

(19)

Substituting Eqs. (17) and (18) into (16) and rearranging the equation gives

86

We multiply by em,integrate over the volume of the cavity, and make use of the orthogonal relation (19) 2

aa, K dam k: -+--+a, at2 E at pE

1

=- -

EN,

emdV = afm .

at

(20)

This is the equation of a driven oscillator with losses. In the case of a harmonic excitation (all quantities proportional to 2")

the amplitude is

with the resonance frequency o,,,= kJc and the unloaded Q-value Q- = EO,,,,/K. The amplitude depends on the exciting force and frequency and the material constants E , K of the dielectric. It shows the typical resonance enhancement and the rapid phase change around o,,,Fig. 3. Obviously, if the modes are well separated in frequency each one can be represented, around its resonance frequency, by a resonator and therefore by a resonant circuit, e.g. by a parallel circuit as in Fig. 4.

87

Figure 3. Amplitude a,, Eq.(21), of modem in a cavity excited by a current.

Figure 4. Parallel resonant circuit for a cavity mode m driven by an ideal current source.

Since the circuit has three elements we need also three cavity quantities for identification. One is the resonance frequency -

-

1

JLmcm’

and the second is the unloaded Q-value

where

wmis the stored energy and

Pdm the dissipated power. The Q-value

determines the resonance enhancement (see Fig. 3) and the bandwidth of the resonance

88

At the frequencies Wornf Am, the real power delivered to the circuit is half of that delivered at resonance. In transient situations the Q-value determines the filling time

T =2-,Qom fm Worn

that is the time constant with which the circuit fills when excited by a switched sine-wave (see chap. 2.3). The third quantity is connected to the accelerating voltage that a particle experiences when it traverses the cavity on-axis with velocity v

z = vt.

However, since

v,

depends on the amplitude a, it is more convenient to

describe the cavity by an amplitude-independent quantity, which is the shunt impedance

This measures the efficiency in creating a certain accelerating voltage for a given dissipation. Note that the standard definition differs by a factor of two from the circuit shunt impedance R , . An even better quantity is the R-upon-Q Rshm ----Qom

2

nr'

Wornwrn

-

2 Worn',

89

This is a measure for providing the accelerating voltage for a given stored energy. The R-upon-Q is independent not only of the field amplitude a , but also of the losses (material) in the cavity. It is determined only by the cavity geometry. From Eqs. (22) and (28) the remaining circuit elements are given

2.3

Transient Behaviour of a Cavity Mode

We assume the circuit of Fig. 4. Starting with the differential equation of the circuit

-v(t)+;/v(tpt 1 R

+ c-dV@)= i(t) dt

we differentiate once and apply a Laplace transform

2 s2v(s)-sv(+0)-v’(+ 0)+-(sV(s)-v(+0))+Wo2v(s) Tf 1 = -(sz(s)i(+ 0)). C As a first example we consider an un-driven circuit, i(t) = 0 , which was

charged up to an initial voltage v(+ 0) = v 0 , v’(+ 0) = 0 , and which is freerunning. Then, from Eq. (31)

s +2/Tf V ( s )= s

2

2

+-s+wW,

2

VO

Tf and with the inverse Laplace transform

-1

IT j

e

The voltage is ringing with a frequency shifted by

Jw and jdecays

exponentially with a time constant T j . Since Q, is normally a large number, the shift is negligible as is the sine-term proportional 1/ Q,. In a second example we consider a switched harmonic current drive

i(t) =

{

O

6sinmot

for

{

. t>O

The circuit was initially unexcited, i.e. v(+ 0)= v/(+ 0) = 0 . Then from Eq. (31) it follows that

V(+wh

2

2

2

1 2 i0

s2+-s+mwo

Tj and, after decomposing the right side into partial fractions,

v(s)=

The inverse Laplace transform gives

Ri, .

91

1 v(t)=

which for high Q-values can be written as 7

-a

v(t)= In expanding Eq. (33) we put cOs(t / 4 T f

(33)

)= 1 and

sin(t / 4 T

)= t / 4 T f ,

since in any case these terms will vanish for large values of t / t because of the exponential. As can be seen from Eq. (33) the voltage consists of a transient part, which decays exponentially, and a steady state part, as described in chap. 2.2.

2.4

Pill-Box with Metallic Walls

The exact treatment of walls with finite conductivity is complicated. However, for good conductors, like metals, a very good approximate treatment is possible. In an ideal conductor charges are supposed to be so mobile that they move instantly in an electric field. Thus, they produce a surface charge density in such a way that there is no tangential electric field on the surface (if there were one it would be short-circuited) and that the normal component finishes on a surfacecharge density

n*D=q,.

(34)

For time-varying magnetic fields the surface-charges move because of the tangential component representing a surface-currentdensity

nxH=J,

(35)

92

which creates zero fields inside the conductor. That is, on the surface of an ideal conductor only normal E and tangential H fields exist, which drop abruptly to zero inside the conductor. For a good, but not perfect, conductor we expect approximately the same field behaviour as for an ideal conductor. We know that inside the conductor the fields decay exponentially within the skin depth 6, and that there is a continuous current density in this layer and the charge density is compensated, i.e. it is zero. The situation is similar to an ideal conductor, aside from the transitional layer of thickness 6,. The boundary conditions, however, are different and require continuous transitions of the fields. In order to solve this problem we employ a successive approximation scheme. First we assume the fields outside the conductor are given by the ideal conducting case. Then we use the boundary conditions and Maxwell's equations in the conductor to find the fields in the skin layer and corrections to the fields outside. First, in the conductor we neglect the displacement current

VxHi =ai,

VXEi = - j ~ p H . I '

Second, since the spatial variations of the fields normal to the surface are much more rapid than the variations parallel to the surface, we neglect all derivatives with respect to coordinates parallel to the surface. Then, the Nabla operator can be written as

where n is the unit normal outward from the conductor and

5

is the normal

coordinate inward into the conductor. With this approximation Maxwell's equations become

1 aH. Ei - - - nx- - ~ K

ag

Hi =--

j

UP

aEi nx-,

ag

From the second equation we conclude that the normal component of

Hi

is

n Hi =: 0 , and Hi is essentially parallel to the surface. The first equation yields a small tangential E which will be used later. Combining the small,

two equations yields

93

with the solution

where

H,

is the tangential magnetic field outside the surface. From Eqs. (36)

and (37) the electric field is approximately

On the surface there is a small tangential electric field which is related to the tangential magnetic field by an impedance boundary condition

zw=-.1+ j

(39)

K6S Inside the conductor E and H exhibit the following properties: They show a rapid exponential decay and a phase difference between E and H. The fields are parallel to the surface and propagate normal to it, with amplitudes that depend only on H, . The magnetic field is much larger than the electric field

since the relaxation time

T,. = & / K is normally much smaller than an FW

period T. The magnitudes of the different field components are indicated schematically in Fig. 5 . The existence of tangential E and H components on the surface of a metallic conductor means that there is a power flow into the conductor. This power flow per unit area is given by the real part of the complex Poynting’s vector

94

P: = -ReS,

= --Rebi,,(< 1

2

= O)xH,;,(< = O)}=

-IH,,r. 1 2KfiS

(40) It represents the Ohmic losses in the conductor. Integration of

P: over the

internal surface of a cavity will then give the total dissipation in the walls of the cavity.

1

E,

H

Figure 5. Schematic drawing of field amplitudes near the surface of a good conductor.

As an example we take the TM,,, -mode in a pill-box, Eq. (14). The stored

energy is

95

I 1

=

5 2nga 2

n

Eo2 xJo2(jo,x)dx = -E, Eo2ga ' J , (jol ). 0 2

(41)

The losses follow from Eq. (40) as

+I/ g

211 H ,

Pd =-

(2

= 0 ) l 22npdp

H , ( p = a)/' 2 m d z

0

A charged particle that crosses the cavity on-axis with velocity v experiences a voltage gain

V = I E , ( p = 0)e""''dz

, = Eog sin kg / 2 J , k = -w

J = -V,

C

C

kd2J

0

With Eqs. (41) to (43) the RF parameters are easily calculated. The shunt impedance (27) is

R sh

v2

=-=-Ka

1

Pd

Z

2

(g/u>z sinkgI2J

l + g / a kg/2J

the Q-value, Eq. (23),

and the R-upon-Q, Eq. (28),

1 ~,~(j,,)'

(44)

96

Numerical values for a copper pill-box at 3 GHz and half a wavelength long are

j,, = 2.405, a = 3.828 cm,

~ 1 ( j o l ) = 0 . 5 1 9 1 , K = 58.1O6S2-'rn-' g = 5 cm,

sinkgl2J _ -n kgl2J 2 R,, =5.5MsZ,

6,= 1 . 2 0 7 ~

(transit time factor for 13 = 1)

Qo = 17963,

R,, /Qo = 6.1KQ

(filling time, Eq. 25).

2.5

Coupling and Tuning

Cavities have to be powered to replace the losses in the walls and to provide the power delivered to the beam. They also have to be tuned in order to stabilize phase and frequency when the temperature changes andlor when one wants to compensate the phase change of the cavity voltage due to the loading of the beam current. Three different coupling mechanisms exist, Fig. 6 . If the frequency is not too high, typically below 1 GHz, one uses coaxial arrangements. In the case of magnetic coupling, Fig. 6 c, the central conductor of the coaxial feeding line enters the cavity and forms a one-turn loop. It operates like a transformer. The current through the loop creates a primary magnetic flux which links directly with the flux of the cavity. The input impedance 2, of the feeding line is transformed to an impedance 2: "seen" by the cavity, Fig. 7. In the same way, one can also transform the cavity impedance into an impedance appearing at the input port of the feeding line.

97

Figure 6. Different coupling mechanisms to a cavity ringing in the

TM,,, -mode: (a) aperture or

electromagnetic coupling, (b) probe or electric coupling, (c) loop or magnetic coupling.

Instead of coupling to the magnetic field one can also couple to the electric field with a probe, Fig. 6b. The conduction current on the center conductor creates a displacement current in the cavity, which is linked with the electric flux of the cavity. Again, the linkage acts like a transformer. This coupling is simple and efficient but it is inevitably connected to high electric fields and one must carefully avoid the risk of dark or glow discharges. For higher frequencies the losses on coaxial lines are too high for many applications and one normally uses electromagnetic coupling, Fig. 6 a. The feeding line, a waveguide, is coupled via an aperture through which fractions of the magnetic and electric fluxes are linked with the fluxes in the cavity. The iris forming the aperture acts like a transformer. All three mechanisms serve the same purpose. They transport electromagnetic energy into the cavity or out of the cavity into a line. They should do that with as little loss as possible and with small reactive contributions. In terms of microwave engineering, they should come as close as possible to an ideal transformer which transforms the line impedance to the desired impedance appearing at the cavity side and vice versa. Let us, as an example, consider a cavity coupled to a signal generator via a transmission line and a coupling device. The source has an internal impedance Z, equal to the line impedance. The equivalent circuit is shown in Fig. 7 a. & represents the self-inductance of the coupling device, and M the mutual inductance between it and the cavity inductance L. The coupling device is assumed to be loss free and the reference plane a-a to be located at some arbitrary position near the cavity.

98

SOUTCe

I transmission line

I I

I I

cavity

a i

Figure 7. Cavity coupled to a generator through a line and coupling device. (a) Equivalent circuit. (b) Equivalent circuit with cavity impedance referred to the primary. (c) Equivalent circuit for

detuned-short position.

The impedance at the terminal plane a-a is

The cavity appears at the terminal as a transformed impedance Z. In a next step we shift the reference plane to a new terminal position b-b. It is chosen such that the detuned cavity appears there as a short circuit. The new reference plane is called detuned-short position. This is done by a short piece of line which transforms jw& at a-a into zero inductance at b-b

-Z,, 2,

-

jwL, + jZ, 2, -04

=O+

ULI 20

99

After some calculation we obtain for the total impedance at the new reference plane b-b

with

Equation (48) is the impedance of a parallel resonant circuit. l3 is called the couding coefficient. At resonance, 6 = the transformed cavity impedance

so,

is

Z* = JZo and the power delivered to the cavity is

pi, =

48 (1+

pmx *

Pmx =-.VO* 820

(49)

If J = 1 the cavity is matched to the generator and receives the maximum available power

PmX. For 8 < 1 or@ > 1

the cavity is undercoupled or

overcouded, respectively, and receives a smaller or larger amount of power than is dissipated in the source impedance. Instead of seeing the cavity form the source side, it is sometimes of interest to see the source from the cavity side. Then, the line impedance 20 will be transformed and appears as a load resistance l3R in the cavity circuit. The power dissipated in this resistance is the external dissipation and is associated with the external 0

1

P, = -pRz2 = JPd 2

100

The total dissipation is Pd

With 13 and Q,,

+ P,,

and defines the loaded Q

we define the coupling of the cavity to an external circuit.

But delivering power to a cavity is not sufficient. The resonant frequency and the phase of the accelerating voltage have to be tuned. Even a very good mechanical design and fabrication will end up with frequency errors of a few percent. Also, thermal variations and changing operating conditions require continuous tuning. Tuning is done by introducing small objects, typically pistons, into the cavity volume or by slightly deforming the walls, either by controlling the cooling water temperature or by exerting mechanical forces. If the perturbations of the volume are small, Muller [2]has given a formula for the frequency change

where W is the total stored energy,

E,, H,

are the fields of the unperturbed

cavity, and AV is the volume of the perturbation.

2.6 Influence of the Beam Pipe. Beam Loading Up to now we considered the cavity simply as a resonator. But its purpose is to accelerate a beam, and we will treat in the following some aspects which are important for the beam. We already had one aspect when we calculated the accelerating voltage, Eq. (43). Because of the finite velocity v of the beam, the field in the cavity changes with time, and a particle does not feel the full voltage E,g but a voltage reduced by the transit time factor

T=

sin kg / 2J kd2J

'

w k=--, C

V

J=-. C

(53)

101

Another important aspect is the beam pipes, Fig. 8. They change the frequency, which can be estimated by applying Eq. (52) and considering the fact that the fields decay exponentially in the pipes, and they influence the accelerating voltage.

Figure 8. Pill-box with beam pipes.

We assume a Tkf,,, -like mode and express the longitudinal electric field by a Fourier integral m

E (p,z) = /A(kz)I,(Kp)e-'kzzdkz, K = Here, we have used the modified Bessel function will turn out later that

k, =k / J >k

Jk:-k2.

(54)

I , instead of J , because it

and K will be real. After inversion of Eq.

(54)

we find an approximation for A by assuming

102

that is

Substituting Eq. (55) into (54)and integrating over t

that is, the accelerating voltage is given by the average field at

p = b times the

gap length, times the transit time factor (53) and times a function F depending on p. For J 1 we have Y 00 and F 1, and the voltage is independent

+

+

+

of p. For small I3 the voltage depends strongly on p and varies with the ratio

between

p = o and p = b .

Besides modifying the accelerating voltage, Eq. (56), the beam pipes also introduce a transverse deflecting force. The reason is that the pipes cause transverse electric forces at the gap entrance and exit which do not cancel for particles not sitting exactly on the crest of the sinusoidal RF field. Using Eqs. (6) and (54) together with (55) we express the transverse Lorentz force as

Then, a particle crossing the gap experiences a transverse momentum increase of

103

Here we have introduced a phase angle cp with respect to the crest of the longitudinal RF field. Equation (57 a) shows that particles with negative phase angles, the ones that have a stable longitudinal motion, are defocused, while the particles with a positive phase angle are focused but are longitudinally unstable. As for the longitudinal forces, the transverse forces depend on the transverse position p. As was shown in chap. 2.2, a beam traversing the cavity excites different modes. If the beam consists of many bunches with distance d, then the spectrum has essentially a single line at

W

2n

d - Jc

= - with T - Tb

and the beam drives the cavity like a harmonic current source of frequency W , Fig. 4. If we now increase the time Tb between bunches, but keep a multiple of the

RF period, T, = NTR,, then the cavity mode operates in a transient regime where the voltage decays between bunches but the beam-induced voltages are in phase. After a certain number of bunches an equilibrium is reached, such that the decay exactly equals the voltage induced by one bunch. Both situations, the steady state and the transient state, are called beam loading because they superimpose the beam-induced voltages onto the voltage created by the generator. Besides the accelerating mode, a bunched beam will also excite other modes. With an r.m.s. bunch length of CJ the spectral width is C / O and all modes within this range may be excited, at least for a small number of bunches. Every mode will have a different frequency, and they all add up to an incoherent field called a wakefield. Subsequent bunches will experience the desired accelerating field plus the undesired wakefields of previous bunches.

104

The situation may become even more complicated when the beam is not exactly on the cavity axis. Then, modes with m # 0 will be excited and the wakefields will excert transverse forces on subsequent bunches.

2.7 Shape Optimization There are several ways to optimize the RF behaviour even of a simple single cell. The most important parameter here is the frequency. It directly influences the beam dynamics and the RF parameters. In chap. 3.2 a short discussion is given of the dependence of the parameters on the frequency. Here, we will investigate the influence of the exact shape of a cell on the RF behaviour. The shunt impedance, Eq. (44), is proportional to the transit time factor, Eq. (53), which takes into account the change of the field while a particle is crossing the cavity. T becomes large when g gets small. But at the same time

R,, is proportional to (g / a>2/(I + g / a ) , which decreases with g. This contradictory situation can be partially cured by making the cavity "reentrant" (nose cones), Fig. 9. Such a reentrant cavity is optimized by calculating numerically the influence of the different geometrical parameters. The beam pipe radius r,, is determined from beam dynamics considerations. The wall thickness

d , should be as small as possible, a constraint due to mechanical and eventually cooling requirements. The cell radius rcell follows from the frequency, and the cell length 1 is often equal to d / 2 . The remaining most important parameters are then the gap width g, the nose cone radius rN and the outer wall radius

rz . Here g optimizes R,, , and rN and y are chosen by a

trade-off between high Q and the tolerable peak field. The rounded outer wall, with rz , minimizes the ratio of cavity surface to volume and thus optimizes the Q-value. Sometimes, however, cells are coupled together (see chap. 3) via coupling holes in the side walls. Then, a rounded outer wall results in a side wall that is too thick; a compromise is an angular-faced outer wall, like the solid line in Fig. 9. Fig. 9 shows a fully optimized cell for the LEP copper RF system.

105

I I

I

I I I -4

I--,

I I I

I

' 1-

Dimensions (mm)

--

-------A

I

g/2=138.6-

k

=

2I/ 2=212.8

I -I

I

Figure 9. One quadrant of an optimized cell for the LEP copper RF system.

3

Periodic Structures

After having described the fundamental behaviour of an accelerating cavity we turn to the question of how to construct reliable RF structures in an economical way. A single-cell cavity requires an input power coupler, two beam pipes with flanges, a tuner, an FW probe, cooling, and vacuum ports. Many of these cells are necessary for reaching a high voltage, and the installation becomes rapidly very large, complex, and expensive. Therefore, in many applications it is advantageous to group a large number of cells in a single vacuum tank. The tank would be more compact and shorter. It would require fewer vacuum ports and a

106

single power coupler and vacuum window, and would thus be more reliable and cheaper. Also, since power tubes are more economical as high power units, the power distribution system would be simpler if a small number of tanks is fed instead of a larger number of individual cells. In fact, historically, RF linacs where multi-gap (multi-cell) structures. G linac was proposed and built by R. Wideroe in the late 1920's The first F and improved by E. Lawrence and D. Sloan in the early 1930's. The scheme, shown in Fig. 10, consisted of a series of drift tubes fed alternately by a 7-MHz generator. The drift tube lengths increase continuously with acceleration, and particles and RF were synchronized such that acceleration took place in the gaps.

Figure 10. Wideroe-type accelerator

The limitation of the principle is obvious when considering a particle with higher energy. For instance a 1-MeV proton travels about 1 m in a half-period of 7 MHz, and the drift tubes become prohibitively long unless the RF frequency is increased. Power generators with higher frequencies became available at the end of the second world war as a consequence of radar developments. However, they could not be used in Wideroe structures, which radiate heavily at higher frequencies, and it was only in 1945, when L. Alvarez thought of putting the series of drift tubes in a resonant tank, Fig. 11, that higher frequencies were used. This led to the famous 32-MeV proton linac operating at 200 MHz which was built in Berkeley in 1946. For ultra-relativistic particles still higher frequencies were needed and a second line of development, also stemming from radar work, started at about the same time. These were electron linacs, using high-power magnetrons (later replaced by klystrons), generally at 10 cm wavelength (3 GHz). The accelerating structures were disk-loaded waveguides (DLWG's), Fig. 12, supporting travelling waves where the particles "surf" on the wave and are continuously accelerated. The application of this principle was the impressive 20-GeV Stanford Linear Accelerator (SLC) operating at 3 GHz with 8 MV/m accelerating field, which was completed in 1966. Since then, and in parallel, many linear accelerator structures have been developed and are continuing to be

107

developed. The theoretical grounds are reported in innumerable reports but the best references are still the 1947 paper of J. C. Slater [3] and the "red bible" [4].

Figure 1 1. Alvarez-type accelerating structure

In this paper we will not follow the historical development and start with low velocity structures. Rather we believe that the simplicity of velocity-of-light structures is better suited for introducing the basic concepts. Once having developed a certain understanding we will turn to low velocity structures and explain their particularities and variety.

Figure 12. Disk-loaded wavegiude.

108

3.1

Properties of Periodically Loaded Waveguides

The foundation of the study of periodic structures is the Floquet theorem stating that: In a given mode of oscillation, at a given frequency, the wave function can differ only by a constant when moving by one period of length L. If the structure is loss-free, the constant is of magnitude one and we can write, e.g. for a longitudinal axis-symmetric electrical field,

In fact, the function exp(- j k , z ) is the most general function that fulfills Floquet's theorem. The periodic function F can be expressed as a Fourier series leading to a series of travelling waves called space harmonics

EZ( p ,z,t)= Can(p)'(m-knz)with k,

= k,

+ 2m/ L .

(59)

n

It is possible to determine also the functions a, ( p ) by substituting Eq. (59) into the wave equation

leading to

with solutions

i.e. every space harmonic has an amplitude A,, a radial dependence

J , ( K , p ) , and a wavelength 27C 1k, , and travels with a phase velocity

109

w --

Vpn

k,

-

w k, +2nnlL

The amplitudes A, have to be determined such that the superposition of all space harmonics fulfills the boundary conditions on the wall. Several observations follow from the above given field representation: 1. Depending on the values of the K , 'S there are different modes of oscillation (with different numbers of nodes along p). (There may also be different modes in the azimuthal direction for non-axis-symmetric fields). 2. For each mode of oscillation there may be one space harmonic, usually the one with n = 0, which is the largest and which is synchronous with the beam. This space harmonic acts like a time-independent field and provides the acceleration. 3. The non-synchronous space harmonics all travel with different velocities and will produce almost no net acceleration. However, they transport energy and will cause losses. Therefore, it is important to design structures where the synchronous space harmonic has as large an amplitude as possible in comparison with the others. 4. For vpn= c , K , equals zero and the Bessel-function J , becomes independent of p, i.e. the acceleration is independent of the radial position. 5 . w is a periodic function of k , . Because, if we increase k , by 2n I L , then

k, will change to equal the previous k,+l . Thus the name of each

k , changes, but coefficients A have to be found for the same numerical values k , as before. 6. w must be an even function in k, . However, because inverting the sign of the k , 'S gives space harmonics travelling in the minus z-direction, the other field properties remain unchanged. The properties of the wave number are best shown in the dispersion diagram, Fig. 13. The curves are even and periodic. Each branch belongs to a specific mode of oscillation. The range of w in one branch is called the pass band. Between branches no real solution k, exists, and the space harmonics are attenuated. These ranges are called stoD bands. The slope of the radius vector out

110

to a point of the curve gives the phase velocity, and the slope of the curve itself gives the group velocity v, (see next section). In order to accelerate ultrarelativistic particles the operating frequency is determined by the intersection of the curves with the radius vector under 45 O corresponding to the velocity of light. Further, we note that for a given o there is an infinite number of space harmonics each with a different phase velocity but equal group velocity. At the ends of a pass band, at k,L = mn , the group velocity is zero. This can easily be explained by recognizing every iris as a scatterer producing a reflected and transmitted wave. With irises spaced a half guide wavelength apart, the reflected waves from successive irises will be a whole period out of phase and interfere positively; the reflected wave equals the incident wave and a standing-wave pattern with no energy transport is set up.

"I

L

L

L

/

L

kFigure 13. Dispersion diagram of an empty waveguide (hyperbola) and of a periodic structure (wiggling curves).

As a consequence of the periodicity of the curves there are alternating regions with positive and negative v, . Depending on the mode of oscillation and/or the coupling between cells, the branch of the

n = 0 space harmonic starts with a

positive v g if the coupling is predominantly electric or with a negative

V,

if the

coupling is predominantly magnetic. In the first case we speak of a forward wave structure, in the second case of a backward wave structure.

111

Figure 14 shows instantaneous electric field patterns for modes with different phase advance k, L per period. The older linacs used a k, L = 7t 12 or four irises per wavelength. The curve is often approximately symmetric around that point and has the highest v g . Also, every second cell is unexcited and the mode is less sensitive to dimensional errors. Later, the 2?T/ 3 -mode has often been preferred because the shunt impedance (see next section) is about 10 % higher. The n-mode is used in standing-wave accelerators and has the highest shunt-impedance of all. However, the group velocity is small (zero in an infinite lossless structure), the mode spacing is poor, and the structure is most sensitive to errors. I

I

I

0

27r 3

Figure 14. Instantaneous electric field patterns with 0, 7t / 2, period in a DLWG.

27G / 3

and 7t phase shift per

112

3.2

Principal RF Parameters

As we did for a single cell (chap. 2.2.) we describe periodic structures by only a few RF parameters. For convenience they are given per unit length instead of for the full structure. Because linacs are usually limited by RF power, either peak or average power, the most important parameter is the shunt impedance relating the accelerating gradient E to the power dissipation

Rih is typically given in M f 2/m. In proton Iinacs operating at 200 MHz values

a

of 35 M /m are reached, whereas electron linacs at 3 GHz have values around 100 M a /m. With the Q-value

the R-upon-Q is given as

R’sh - E 2 Qo wW” which is a measure of how much accelerating field is achievable for a given stored energy per unit length. Another important parameter is the group velocitv. It is the velocity with which signals and energy propagate. Let us suppose we have two waves propagating with slightly different frequencies W,

=w,

-A@,

resulting in different phase constants

W , = W,

+ AW

113

k,

=:

km--*1

Aw,

k,=k,+-

0,

I

a@ ak

Am.

0,

Then, superimposition yields

i.e. the high frequency part (carrier) has a phase velocity

v, = w , l k , , whereas the beat-signal propagates with a velocity

Vbeat

=

*/ ak

=vg.

"Jm

The group velocity describes also the velocity with which energy is transported in the structure

P = w'v,,

ve = v g'

where P is the transported power. A proof is given in Appendix A. The group velocity depends strongly on the ratio of iris diameter 2 b to cavity diameter 2 a

v g l c =: K ( b l a ) 4 with K being a constant depending on the number of disks per wavelength and their thickness. The group velocity is important for three reasons:

1. The filling time is the time it takes to fill a structure of length I with energy Tf = l l v , . (68) 2. Because of Eq. (66) and

w'- E 2 ,it is preferable to have a small v g

resulting in a high E for a given power flow. 3. R:h,Q and R:h / Q all depend on v, . In general, decreasing v g results in an increase of R:,, , a decrease of Q and thus an increase in

114

A wave travelling down the structure will be attenuated by wall losses. It is easy to find the rate of attenuation from the equation of continuity for the energy flow

awl

ap

at

aZ

I

-+-+P,

=o.

Substituting - Eqs. . (64) and (66) into (69) yields

ap

ap

w

a~

Qo

-+-P

-+v, at

=0

and in a steady state, when time derivatives are zero, we find the attenuation constant as

P = ee-’,

,

The attenuation length is the length 1 after which the field, not the power, has decayed to lle

d = l + l = 2 -Qo

0 vg The frequency influences all the basic RF parameters. Its choice, depending also on many practical matters, such as availability of power sources, is therefore the most important design task. The scaling with frequency of some parameters is rather obvious if one keeps the accelerating field constant. The dissipated power is proportional to the wall current squared times the wall resistance I

Pd

- i,R, 2

1

(73)

where

R i = 1/(272&,k6),

6 = (2/W/.l,Ky‘2

(74)

115

and K is the wall conductivity, 6 the skin depth and

b, some effective structure

radius. The accelerating field is proportional to the tangential magnetic field at the wall and therefore

- H , - i,

E

lbeff.

(75)

The stored energy and the energy transport both scale with

W', P

- E2bei.

(76)

Then, combining Eqs. (73) through (76) and keeping in mind that

b,

-

LL)-'

,

we obtain

(77)

Since

v g is constant, the filling time would also be constant for a constant

structure length. But for an optimized accelerator with losses the structure length changes and it is

Tr

-

(78)

Thus, from the viewpoint of efficient use of RF power, the frequency should be as high as possible. However, the co-travelling fields of the beam are scattered by the structure and act back on the beam in a disruptive way. These fields, the wakefields (see chap. 2.6) scale as

wZ- a-' - w 2 ,

W, - a -3 - w - ~ .

(79)

Therefore at high frequencies, the stable transport of sufficient beam current becomes the main issue. In a DLWG one can easily derive an approximate relationship between the inner and outer radii of the irises for waves with U p = c at a specified

116

frequency. This is shown in Appendix B. Exact calculations have to be made with numerical codes.

3.3 Travelling- Wave/Standing-Wave Accelerators Periodic FtF structures can be operated in two different ways, as a travellingwave (TW) accelerator or a standing-wave (SW) accelerator. In a TW-structure, Fig. 15 a, the fields build up in space with the wave front travelling with the group velocity. The output of the structure is matched to a load where the leftover energy is dissipated. In a SW-structure the fields build up in time, Fig. 15 b. The incoming wave is attenuated along the structure, reflected at the end backwards to the input, reflected again and so on. The process of reflection at both ends continues until an equilibrium between build-up and dissipation is reached.

Figure 15. Principles of (a) travelling-wave and (b) standing-wave accelerators.

In the following we consider the case where the power transferred to the beam is small compared to the power dissipated in the structure walls. Travelling-Wave Constant-Impedance (TW-CI) Structure A CI-Structure has a uniform geometry and the fields decay exponentially (with an attenuation constant a). The energy gained by a charge at a constant phase angle cp is then

117

1

1- e-'

0

2

0

Po the input power. The function = 1.26, Fig. 16 a. However, for different

where I is the length of the structure and v ( Z ) has a broad maximum at Zmx

reasons, lower values of z around 0.8 are preferable with only 3 % decrease of V as compared to Vmx . The stored energy in a structure at the end of a filling time is obtained by using Eqs. (66), (71), and (68)

The ratio

w / p o T , , called structure efficiencv

rst,is the fraction of input

energy available for acceleration. From Fig. 16 b it is clear that for a high conversion efficiency lower values of 1' : are preferred.

Figure 16.

structure.

(a) Normalized energy gain, Eq. (SO), and (b) structure efficiency, Eq. (Sl), in a CI

118

Travelling-Wave Constant-Gradient (TW-CG) structure If one wants a uniform field and power distribution along the structure the geometry has to be modified in a way such that the energy travels more and more slowly, and an increased energy build-up in the cells compensates for the losses. This is obtained by narrowing the iris apertures continuously and by reducing the cell diameters correspondingly to keep the cells in tune. These changes have only a small influence on the shunt impedance, which therefore can be regarded as constant in a first approximation. (The final result can always be corrected for the electric field variations due to the varying irises.) We integrate

dP Pd =--=const. dz and obtain

P ( z ) = Po - (Po - P(2))Z = Po 1 with Z = jO(z)dz. 0

This linear power flow is a consequence of the linearly varying group velocity. Using Eqs. (71) and (83) it follows that

Similar to R' , Qo can be assumed as constant since it varies only slowly. The filling time

'

T =J'-=2Q0dz z f w 0 vK

(85)

and the total stored energy at the end of the filling process (Eq. 81) are equal for CI- and CG-structures.

119

The accelerating field is constant and may be obtained from Eqs. (63) and

(83)

dP E 2 = R:hPi =-Rib -= R:hPo(l-e-z')/Z dz yielding an energy gain of

V = El )= -/,

(87)

A comparison of the energy gains of a CI- and a CG-structure shows that the gain is slightly higher in a CG-structure. In addition, a CG-structure is less sensitive to frequency deviations, has a higher structure efficiency and, above all, is less susceptible to beam break-up phenomena. Concluding this section we should note that the standard high energy electron linac operates in the 27T /3 TW mode in a CG-structure. For compact low energy linacs a biperiodic structure (chap. 4) is often preferred.

Standinn-Wave Accelerators In SW-structures the incoming wave is attenuated, reflected at the end and again attenuated on its way back to the input end. At the input port the wave is partially reflected and partially transmitted through the input coupler into the feeding line. The reflected fraction travels forward, is reflected at the end and so on. If the length of the structure is a multiple of half-wavelengths, the forward and backward travelling waves are in phase and will build up in time until an equilibrium with the dissipation is reached. At equilibrium none of the backward power will pass out of the input coupler if the coupler is correctly matched. Then, the input power Po equals the difference between forward power Pfo and backward power Pbo at the input

The steady-state energy gain is calculated in a similar way as Eq. (80) yielding

120

Comparing Eq. (89) with (80) it follows that the SW energy gain is larger than for the TW case by a factor

"7-w

The improvement is larger at lower values of 7.However, it is accomplished at the expense of an increased filling time due to multiple reflection in the structure. Other more serious disadvantages are a lower shunt impedance (only the forward wave contributes to the acceleration, while the backward wave causes additional losses) and a non-uniform distribution of the field in the cells. This can be fatal in high-gradient linacs because higher fields increase the possibility of electron emission from the surface. When the structure is operated in the zeroor n-mode, the forward and backward space harmonics coalesce and their fields add up in phase resulting in twice the amplitude. Then, the shunt impedance is as high as in a TW-structure and the fields in the cells are equal. But the n-mode has other disadvantages. By means of a simple circuit model consisting of a chain of N resonators (see chap. B1.l in ref. 4) we find that the mode spacing is nJN times smaller than for the 7d2 mode. Therefore, the number of cells that can be coupled is severely limited. Furthermore, a closer inspection shows that the n-mode has a cell-to-cell phase error and is quite sensitive to dimensional and frequency errors. n-mode SW cavities for electron acceleration typically have only 4 to 10 coupled cells. In a normal-conducting version they are used in circular accelerators at 350 to 500 MHz. At such a low frequency the iris aperture would have to be very large in order to get a reasonable coupling between cells and consequently the WQ would be low. Therefore, one couples via slots in the iris wall at a position of maximum magnetic field. Also, so-called nose cones are added to the irises which reduce the transit time effect (chap. 2.7) and concentrate the electric field at the axis. Finally, the cavity shape is rounded, which minimizes the surface-to-volume ratio, meaning the losses. As an example, the LEP cavity [5] is shown in Fig. 17. Superconducting n-mode SW cavities are used in both circular and linear accelerators. They are of simple shape, straight lines and elliptical or circular arcs, with a large iris opening. The correspondingly low R/Q is of no importance in that case because of the very high Q. The rounded shape was found by chance

121

as the one where multipactoring is suppressed, that is a local resonant electron avalanche phenomenon. Figure 18 shows such a cavity, proposed for the TESLA [6] linear collider.

RF input

' '

Tuner*

'

'

Figure 17. The five-cell n-mode LEP cavity with magnetic coupling slots.

1039

Figure 18.

I

The nine-cell n-mode Nb-cavity of the TESLA linear collider.

Instead of operating a SW-cavity in the n-mode one can as well operate it in the 0-mode. Then, again, forward and backward space harmonics add up in phase in every cell and both contribute to the acceleration. Since the electromagnetic fields in adjacent cells are in phase, the separating walls can be left out without affecting the field distribution. An example is the drift tube or Alvarez structure shown in Fig. 11. The stems supporting the drift tubes represent only small perturbations and do not alter the overall performance.

122

4

Biperiodic Structures

As mentioned above, a n-mode structure is inherently problematic. Operating at the center of the dispersion curve (d2-mode) would alleviate many of these problems but reduces the shunt impedance by approximately a factor of two. In the following we will show a scheme to avoid this reduction. The properties of a 7d2-mode are unique in that in a chain of identical cells every other cell stores no energy other than that required to transmit power, see Fig. 19 a. Thus, every other cell may have a completely different shape, and as long as it resonates at the same frequency as the ”full” or accelerating cell the chain will behave in the d2-mode. This is illustrated in Fig. 19 b, which shows the coupling cells drastically shortened in order to increase the shunt impedance while still retaining d2-mode operation. In a next step, Fig. 19 c, the coupling cells are removed from the axis, thus yielding maximum shunt impedance.

Figure 19.

(a) Simple d2-mode structure; (b) biperiodic on-axis-coupled structure;

(c) side-coupled structure.

123

A technical drawing of the side-coupled structure is shown in Fig. 20. The accelerating cells are shaped for maximum shunt impedance, and the coupling cells are staggered to reduce field asymmetries introduced by the coupling slots. The structure was build in Los Alamos for acceleration of high eoergy protons.

--_

BEAM CHANNEL

CAVITY COUPLING CAVITY

Figure 20. Side-coupled structure.

5

Structures for Non-RelativisticParticles

In principle, DLWG’s can also be used for low velocity particles. But slowing the phase velocity of the wave down to very low values means a small distance g between irises, and following Eq. (44) the shunt impedance decreases with

(g / a)2 for small values. The structure becomes inefficient. A good accelerating structure should exhibit a large accelerating field with limited RF power. This assumes a field distribution such that the peak electric field on the wall is minimized to avoid breakdown, and low wall currents. This also assumes a structure that is not sensitive to perturbations. Finally, the structure should be flexible enough to be used under different operating conditions. These requirements are so different and sometimes contradictory that a whole variety of different principles and geometries have been developed. Not all can be mentioned here. We have already presented some. The Wideroe structure, Fig. 10, was the first linear device and for very low velocities. The Alvarez structure, Fig. 11, had a better efficiency for somewhat higher velocities, but the 0-mode operation makes the structure sensitive to all sorts of perturbations. An improvement, which we will not treat here, was the introduction of additional resonant posts which modify the fields in such a way

124

that the group velocity is no longer zero, see ref. 7. For relatively high velocity the side-coupled structure, Fig. 20, has been successfully used. Other very low velocity structures we would like to mention are the helix and the interdigital Htype structures. The helix structure, Fig. 21, consists of a helical conductor shielded by a round waveguide. When a wave travels along the helical conductor, its phase velocity along the conductor is close to the velocity of light. On the axis, however, the phase velocity is v P = c sin I,//, where I,// is the pitch angle. The wave can thus be made synchronous with low velocity particles when I,// is small.

\\\\\\\\\\\\\\\\\\\\.,\\\\\\\\\

: , \ \ \ \ \ -

4

I

.d.

LLI .\\\\\\\\\\\\\\\\\\\\\\\\\\\. ..\\\\\\

Figure 21. A shielded helical TW-structure.

One often wants to accelerate ions with different masses and velocities. In that case one can use many cavities with only one or a few gaps and adjust the phase between cavities according to the velocity profile. The disadvantages are low efficiency and complicated operation. Another possibility is to use a drift tube device where the drift tubes can be changed mechanically for different applications. Such a structure is the interdigital H-type structure, Fig. 22. The cavity is excited in a mode similar to the TIT^,, -mode in a cylindrical cavity. The electrical field goes across the diameter of the tube parallel to the drift-tube supports. The magnetic field is essentially parallel to the axis. Since the drifttube supports are arranged alternately on opposite sides of the cavity, a voltage develops across the gaps with a n-mode pattern. The drift-tube lengths are adjusted to the particle velocities. Finally, in order to show the efficiencies in different structures, the effective shunt impedances are given in Fig. 23.

125

i L t Figure 22. Interdigital H-type structure

Figure 23. Effective shunt impedances of different structures (taken from ref. 8).

126

Appendix A : Energy velocity In order to calculate the energy velocity in a periodic structure we restrict ourselves to the fundamental space harmonic

The transverse field components follow from Eq. (6) as

Then, the power flow becomes

and the stored energy per unit length

The velocity with which the energy is transported, Eq. (66), is therefore

On the other hand, from the definition of the group velocity together with Eq. (Al) it follows that

<,/-

1 v =-ao = ' ak0 ako/aw w which equals the energy velocity.

= ko C2 0

127

Appendix B: Frequency Equation of a DLWG In the case of waves with V p = c there is an approximate expression for the outer radius b of a DLWG if the iris aperture a and the frequency are given. Assuming the spacing g between irises is small compared to the wavelength of the TW-field, then the fields at the interface between the tube and the iris area approximate a continuous function. The solution consists of standing waves in the iris area, a < p < b ,

E, = 0 at the outer tube p = b travelling wave in the tube area, 0 5 p 5 a , which fulfill the boundary condition

E ~ W = E ,TW e j ( w t - k o z ) The magnetic field component follows from Maxwell's equations as

j sw = -E, [Yo(wb/ c)J, (wp / c )- J , ( ~/ c)q b (up / c)le jW' 2 0

Under the assumption that the iris spacing is small compared to the wavelength the TW-field is about constant across one gap. Then we can define a wave impedance at the interface p = a which has to be equal for the two regions

[$1 [e) sw

TW

=

-m =2 c

at p = a and therefore

Yo(wb/ C ) J I (m/ c ) - J , (wb/ c ) q (m c ) Yo(wbl c)Jo(m/ c ) - J , (wbl c)Yo(m/ c ) .

128

A plot of the equation is given in Fig. A l .

Figure A l . Solution of the approximate frequency equation (AS).

Equation (A8) is a transcendental equation that determines o in terms of a and b or, what is normally the case, b in terms of o and a. Although the assumption of a large number of irises per wavelength is normally not well satisfied, Eq. (A8) still provides a good estimate. References 1. Condon, E.U. Forced oscillations in cavity resonators, Appl. Phys., 12 (Feb. 1941), 129-132. 2. Miiller, J., Untersuchung iiber elektromagnetische Hohlraume, Hochfrequenztechnik und Elektrotechnik 54 (1939), 157-161. 3. Slater, J.C., The design of linear accelerators, MIT Technical Report No. 47 (Sept. 1947). 4. Lapostolle, P. and Septier, A. (Editors), Linear Accelerators, North Holland Co ., Amsterdam (1970). 5. Wilson, I. and Henke, H., The LEP main ring accelerating structure, Internal Report CERN 89-01 (1989). 6. A proposal to construct and test prototype superconducting RF structures for linear colliders, DESY Internal Report TESLA 93-01 (1993). 7. Swenson, D.A. et al., Stabilization of drift tube linac by operation in the / 2 cavity mode, in Proc. 6th Conf. on High Energy ACC.,Cambridge CEAL 2000, (1967), p. 167. 8. Nolte, E. et al. The Munich heavy ion post-accelerator, NIM 158 (1979), 3 11-324.

129

9. Puglisi, M., Conventional RF cavity design, CERN 92-03, Vol. 1 (June 1992). 10. Tran, D.T., RF structures survey, in New Techniques for Future Accelerators 11, M. Puglisi, S. Stipcich, and G. Torelli (Editors) Plenum Press, New York, p. 189. 11. Whittum, D., in Introduction to electrodynamics for microwave linear accelerators, S. I. Kurokawa, M. Month and S. Turner (Editors), World Scientific, Singapore (1999), p.1.

FABRICATION AND TESTING OF RF STRUCTURES E. JENSEN CERN AB-RF CH-121I Geneva 23, Switzerland E-mail: Erk.Jensen @cem.ch Modem RF structures make great demands on both materials and fabrication techniques. In addition to high required precision, they need to be compatible with ultra high vacuum, high power RF and the presence of particle beams. We introduce materials compatible with these demands and summarize their relevant characteristics. Methods of forming and joining follow, again with emphasis on those suited for the fabrication of accelerating structures, and we point out their limitations. We mention different tests which will be designed into the fabrication process, and describe in some detail the testing of the RF properties of accelerating structures. The following overview is nonexhaustive and limited to normal-conducting structures; many of the examples relate to a possible next-generation linear collider.

1

Introduction - selection criteria

The criteria for selecting special materials and fabrication techniques naturally follow from the desired use of the RF structure in an accelerator. Typically, the choices have to be compatible with peak and average power levels, high electric field levels, low ohmic losses, ultra-high vacuum and the radioactive environment. Other criteria for the choice of materials are the mechanical stiffness, the toxicity of the materials (Be), their corrosion resistance and more generally their durability and reliability. Heat conductivity is important if ohmic power losses are to be dealt with. The machinability of the materials and the joining techniques to be applied are also important to consider. The mechanical tolerances are typically dominated by the choice of the radio frequency and by beam dynamics considerations, along with compatibility with the joining techniques. The tolerances required for the RF structure’s final performance will be the main criteria in selecting the fabrication method.

2

Materials

We will limit the discussion here to some of the materials predominantly used in conventional RF structures for accelerators, both travelling and standing wave. Superconducting RF structures, the fabrication techniques for which have made

130

131

tremendous progress over the last decades, are not included in this article and are treated separately. Metals used in room-temperature RF structures are mainly copper, for its high conductivity, and stainless steel for beam tubes, vacuum manifolds and when copper plated - also for RF structures. For insulators and RF windows, ceramics are used, mostly sapphire or alumina (A1203),but also beryllium oxide, silicon nitride or others. Ceramics used as damping materials are typically silicon carbide (Sic) or different types of ferrites. Often surface coatings on both metals and ceramics are used, such as Ti or titanium nitride, which reduce the secondary electron emission and thus can prevent multipactor.

2.1 Metals Even though we will concentrate on copper and stainless steel, in Table 1 they are compared with some other metals, with regard to some of their electrical and mechanical properties. Many of the quantities given here depend strongly on the temper and the purity of the metal or alloy and thus are only indicative. The electrical conductivity given in Table 1 is at room temperature (300K). The yield strength indicates the stress at the upper limit of the elastic range, i.e. where the linear range of the elongation-stress curve ends; at the Rp 0.2% point, the elongation is 0.2% beyond purely elastic. The (ultimate) tensile strength indicates the maximum stress obtained under tension before the material tears. Table 1 Some characte yield strength

A. cond. density

-MS m-'

Ag

c u soft Cu hard CU-OFS cuzr Cu-Al203 Au A1 Mo W Pd Nb SS 316D

62.6 59 55 58 53.8 49.3 45.4 38 20.8 18.2 9.8 6.6 1.1

5

g/cm 10.5

8.94 8.92 8.89 8.8 19.29 2.7 10.2 19.3 11.97 8.57 7.9

-

( R , 0.2 %)

MPa 30 100...I50 300...450 70...360 42..SO0 410...560

25...200 400 1500

300

tics of metal tensile Brine11 melting strength hardness temperature MPa 130...160 200 ... 250 log3 400 ... 490 40 10 ... 12 220...450 50 ...100 1083 200...530 980 480...580 1080 100.. .300 13 ...75 1063 70...250 15 ... 70 658 700...1200 150 2630 400...4000 250 3380 200...400 40..,100 1552 500 120 2477 600 180 1350

I

*"'I

132

Other properties, not indicated in Table 1, but nevertheless important when selecting a metallic material for RF structures, are the thermal conductivity, the elasticity module, the fatigue stress, the machinability, the ability to braze or weld it, the vacuum outgassing rate, the work function and the secondary emission yield, the magnetic permeability, and finally also the cost. The outgassing rate indicates the vacuum level in mbar per surface (in cm2) that can be obtained with a given pumping speed (in Ys), and is given in units of mbar 1 s-’ cm-2. The level that should be obtained is on the order of lo-” mbar 1 s-l cm-2. Bake-out can improve the outgassing rate by roughly one order of magnitude. The work function is important for the maximum tolerable surface electric field without breakdown (sparking), since it enters in the equation for field emission of electrons, along with the surface shape, roughness and purity. However, the work function for practically all metals is in the range of 4 to 5 eV; impurities tend to decrease this value, this is done intentionally e.g. for cathode surfaces. The secondary emission yield (SEY) is important when considering multipactor. Electron multipactor (multiple impact) describes a resonant emission of electrons on electron impact, which can lead to an exponentially growing electron current inside the structure at certain power levels, if the SEY is larger than one. The SEY is a function of the angle and energy of impacting electrons, but its maximum is a characteristic of the material. Furthermore it is important whether the material can be conditioned, i.e. whether the SEY can be reduced when exposed to high power RF.

2.1.1

Copper

Copper is the most frequently used metal for normal-conducting RF structures, thanks to its high electrical conductivity (59 MS/m, 94 % of the conductivity of Ag). The conductivity of copper and copper alloys is often given as “IACS” (international annealed copper scale) conductivity percentage, which is defined as 100 % IACS = 58 MS/m. (For those not used to SI units, MS = lo6Q-’) Copper can be machined to high precision and small surface roughness; it is compatible with ultra-high vacuum (UHV), is an excellent heat-conductor (385 W/m/K) and can be conditioned (treated with high power RF under vacuum) to high surface field levels, and it has a comparatively low secondary emission yield (max. around 1.3). Its main disadvantages are its lack of strength, and the fact that it becomes particularly soft after full annealing above 300 “C (HB 40 ... 50). Note that this temperature is normally considerably exceeded when a structure is brazed.

133

Copper regains strength and hardness with cold working, which also eliminates porosity and reduces the grain size, which is often required for leak tightness of thin walls. The conductivity of copper is reduced by forging by up to 2.5 %.

ParameterParameter Parameter

i.e. only a layer

of a few skin-depths on the inside will carry RF currents. The most frequently used type of copper for RF structures is oxygen free electronic grade (Om)copper, formerly known as OFHC (high conductivity) copper, according to I S 0 431 nominated Cu-Om, or UNS grade C10100. It is of high purity (above 99.99 % Cu). The typical maximum content of oxygen is below 5 ppm, which allows welding and heat treatment. The electrical conductivity of Cu-OFE is >lo2 % IACS. Copper alloys like Cu-OFS (UNS C10700), which contains 0.1 % silver to increase its creep resistance (to prevent buckling under atmospheric pressure with thin walls), or CuZr (UNS Cl5000) with 0.2% zirconium to increase its hardness, are considered for special applications in RF structures. All copper alloys however have lower electrical conductivity than Cu-Om. Another copper alloy sometimes used in accelerating structures is Glidcopo, which contains A1203 and has improved strength and hardness, again at the expense of lower electrical conductivity (85 % ... 93 % IACS, depending on the alumina concentration). The alumina micro-particles dispersed in the copper are pinning the lattice dislocations to the dispersions (dispersionhardening). UNS grades for Glidcop are C15710, C15720 and C15735. Brazing (and soldering) alloys are based on copper, typically Cu-Sn, Cu-Ag, or Cu-Au alloys, some of which are mentioned below.

2.1.2

Stainless steel

Stainless, i.e. corrosion resistant steels consist of more than 10 % chromium. When exposed to air, this chromium forms a tough, adherent, invisible and selfhealing chromium-oxide film at its surface which passivates it. More than 70 grades of stainless steel are standardized. Referring to their metallurgical structure, these are characterized as martensitic (AISI 410, 420, 440, ...), austenitic (304,310,316,317,...), ferritic (409,430,...) and superferritic (442, 446),duplex (2205)and precipitation-hardened steels. Only austenitic steels are non-magnetic. Because of its relatively low electrical conductivity (less than 2 % that of Cu), steel is not normally suited as a l W structure material that has to carry RF

134

current - the ohmic losses would be too high. It is however frequently used for ancillary systems, in particular vacuum beam pipes and flanges. When used as a cavity body, copper-plating the inner surfaces may be considered (see below). For vacuum, cryogenic and accelerator structure applications, the type of stainless steel mainly used is AISI 316LN or DIN “X2CrNiMoN 17-13-3”. It consists of 17 % Cr, 13 % Ni, 3 % Mo, and below 0.03 % of C (the L stands for “low carbon content”). Vacuum firing at 950 “C removes impurities, notably water vapour, from both the surface and the bulk material. A nitrogen content of 0.2 % stabilizes the austenite and improves the yield strength (300 MPa). This stable austenite is non-magnetic, with a relative permeability of below 1.005 and a vanishing magnetic remanence. It can well be machined and welded. 2.1.3

Copper plating of stainless steel

The necessary strength for larger cavities without buckling under atmospheric pressure can be obtained by using a stainless steel body with galvanic deposition or cladding of a copper layer of a thickness of at least several skin-depths. Galvanic deposition (electroplating) of copper on stainless steel is possible, but requires some care to assure adhesion. After degreasing and - if necessary de-oxidizing, the surface is activated in sulphuric acid, rinsed, and activated in hydrochloric acid. Since copper does not adhere directly to steel, the next step is the galvanic deposition of a 2-pm nickel layer from a bath of nickel chloride, boric acid and hydrochloric acid. After rinsing, the bath for copper deposition consists of copper sulphate, sulphuric acid and salt. With a current density of about 4Adm-2, layers of some tens of microns can be easily obtained. Afterwards, rinsing and passivation in a bath containing chromic acid and sulphuric acid is necessary. To assure a regular thickness of the copper layer even with an irregular shape of the RF structure to be plated, a special anode has to be fabricated, adapted to this shape. If this anode is made from platinum-coated titanium, it resists the chemicals and can be re-used. During the electroplating process, gas forms at the anode. To prevent this gas from forming bubbles in areas that are to be copper-plated, the bath normally has to be agitated during the process. This also helps to assure that the ion concentration does not become locally depleted.

135

Figure 1: Inside view of a copper plated stainless steel cavity. Figure 1 shows the inside view of a large stainless steel cavity (diameter 1.6 m) with a 40-pm layer of electroplated copper. 2.1.4

Other metals

Some cavities have been successfully made of aluminium, but aluminium tends to form an oxide layer on its surface, which has a high secondary emission yield and consequently is prone to multipactor. For special requirements, e.g. to obtain very high accelerating gradients (on the order of 150 MV/m), as currently under study for the CLIC 30 GHz main linac structures, one might consider using metals with lower vacuum pressure and higher melting temperature, like Mo or W. First results with these materials are encouraging. Figure 2 shows machined tungsten irises for high gradient test cavities at 30 GHz [ 11.

136

Figui:e 2: Tungsten irises prepared for a test structure at 30 GHz.

Although not treated here, it should be mentioned that the preferred metal for superconducting accelerating structures is Nb. Either bulk niobium is used, or it can be sputtered on the inside of copper cavities, which may save cost. The most remarkable progress however in terms of maximum obtainable surface fields has been made using pure niobium, in particular over the last decade for the TESLA accelerating structures.

2.2 Ceramics Ceramic materials are typically used in feed-throughs (windows) where low losses are required, or as damping materials. Even though RF accelerating structures have to be under vacuum, some window for the RF power to be fed into them must be provided. RF windows sometimes have to transmit very high RF power. Even if the dielectric losses, characterized by the tan 6 (loss tangent) of the material, can be made very small, they are not negligible and cause the window to heat up. Since these windows are normally thin, cooling (edge cooling - i.e. the heat has to be transported through the thin window to its edge) can become a serious issue. At the same

137

time, the window has to withstand the atmospheric pressure, so the tan 6,the heat conductivity and the maximum tolerable bending stress are important characteristics of a window material. The most frequently used materials for RF windows are ceramics like alumina (AI2O3,sapphire if mono-crystalline), but also beryllia (BeO), silicon nitride (Si3N4) or others. Table 2 summarizes some parameters at room temperature of materials used for RF windows. Much experience with these materials has been gained with high CW power gyrotrons for fusion plasma Table 2: Some characteristics of ceramics

ParameterParameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter ParameterParameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter ParameterParameter Parameter Parameter Parameter Parameter Parameter ParameterParameter heating. The table also includes Sic, which is not used as window material but, because of its high dielectric loss, rather as RF absorbing material, e.g. to damp higher-order modes. Mono-crystalline S i c is a semi-conductor - it is used in modern transistors. The behaviour of poly-crystalline S i c however is dominated by its density and purity. Reproducible values for the dielectric constant and the loss tangent could be obtained with high temperature (2600 "C) sintered, dense, undoped Sic. At 10 GHz e.g., the dielectric constant of this material was measured to be approximately 13 with a loss tangent of 0.2. For completeness it must be mentioned that carbon loaded aluminium nitride and different ferrite materials are also used as RF absorbing materials. However, the magnetic properties of ferrites may not be compatible with the nearby presence of the beam and the beam-guiding magnetic elements.

138

3

Forming techniques

3.1 Casting and moulding Casting and moulding are not normally relevant for the fabrication of RF accelerating structures, even though the blanks are normally die-cast. 3.2 Deformation forming The different techniques of deformation forming of metals are forging, deep drawing, extruding, bending and rolling. Because of the limited precision of these forming techniques, they can not normally be used to render the final shape of RF accelerating structures, but they are relevant because of the effect of workhardening, homogenizing and reduction of grain-size. As seen from Table 1 above, in particular copper changes its properties substantially when coldworked. It must be considered however that when the material is deformed in only one dimension, e.g. when rolling sheets, the grains tend to deform anisotropically, which may result in an anisotropic inner structure. Inclusions and shrink holes may thus become long and subsequently may form channels, which may lead to vacuum leaks. Consequently, for certain applications 3dimensional forging is obligatory.

Surfacefinish

N8 3.2

m 1 -

N7 1.6

N6 0.8

diamond machining milling lapping EDM

roughness obtained with special care

N5 0.4

N4 0.2

N3 0.1

N2 0.05

N1 0.025

I

139

3.3 Machining Modern machine tools like lathes and milling machines with numerical control allow obtaining excellent surface qualities along with small tolerances. Table 3 indicates the typical surface roughness that can be obtained by some classical machining operations. The maximum tolerable surface roughness on the inside wall of RF accelerating structures is a function of frequency; as a rule of thumb Ra should not exceed a quarter of the skin-depth. 3.3.1

Turning

Turning is a well controlled, reliable, precise and comparatively cheap machining technique. Consequently, already during the design phase of an RF accelerating structure one should consider using round geometries where possible. Figure 3 shows turned discs for a 30-GHz disc-loaded accelerating structure, which probably present the limit of what is technically feasible. The inner part (the actual cavity) and two circular ridges have optical surface finish (Ra = 15 nm); the absolute tolerances realized were in the 2-pm range. The outer diameter is 33 mm, the iris diameter 4 mm.

Figure 3: turned discs for a 30-GHz disc-loaded accelerating structure.

140

3.3.2

Milling

Modern milling machines allow achieving tolerances and surface finish comparable to that produced by good lathes. Surface shapes of considerable complexity are possible with special CNC milling machines, which allow simultaneous control of 5 axis movements of the tool (x, y, z and two angles). Accelerating structures optimized for both strong damping of wake-fields and extremely high surface fields are now being studied for next-generation linear colliders (JLC/NLC and CLIC). These structures have complex shapes which could be realized using 5-axis milling. However, this technique is still relatively expensive today. Figure 4 illustrates the possibilities of 5-axis milling with the help of a

cooling-water grooves. simpler example, where grooves forming cooling water channels are machined into a mock-up of a PEP I1 cavity. 3.3.3

Diamond machining

For very high frequencies, for example for the accelerating structures considered for the next-generation linear colliders, tolerances in the pm range and optical

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surface finish (of some tens of nanometres) are required. This can be obtained using mono-crystalline diamond cutting tools and possibly special lathes, like the cells shown above in Figure 3. Diamond tools are also available for milling where similar tolerances and surface finish have been obtained. 3.3.4

Cutting fluids

Cutting fluids need special attention since they may leave impurities at the metal surface, which limit the later performance and are difficult to clean off. While conventional cutting fluids may be used during rough machining, care has to be taken that only cutting fluids free of oil and sulphur are used for the final machining of surfaces that will later be exposed to vacuum, and consequently have to be extremely clean. Alcohol has been used successfully as a cutting fluid for this delicate last fine machining step.

3.3.5

Grinding

Grinding is used primarily to machine ceramics. Because of their hardness, ceramics cannot be cut like metals. Grinding allows obtaining similar precision and surface finish as milling. 3.3.6

Electric Discharge Machining

Electric discharge machining (EDM) is also known as spark erosion. There are basically two different types of EDM, as shown in Figure 5 , die sinking and wire cutting. Both the tungsten machining electrode and the workpiece are immersed in a dielectric liquid, normally water. An electrode under high voltage is made to approach the workpiece. A spark forms where the field concentration is highest, i.e. where the distance between electrode and workpiece is smallest; the workpiece is heated locally to around 10 000 "C and disintegrates here, whereas the electrode only wears off slightly. The eroded material is flushed away with the dielectric and the electrode can advance. All conducting materials can be machined by electric discharge machining (EDM), including metals, carbides and graphites. Tolerances below 0.01 mrn can be obtained with surface finish down to Ra = 0.1 pm. During EDM, no forces are exerted on the workpiece, so no stresses are induced, and the edges are free of burrs. Thin slots (0.3 mm, limited by the minimum wire thickness) can be machined.

1 42

Figure 5: Two different geometries suited for EDM: Left: die sinking, right: wire cutting.

3.3.7

Electro-polishing

Electro-polishing may be described as the inversion of the electroplating process described above. When the ion current through the electrolyte is inverted, the surface is attacked, and - because of field enhancement - more at protruding tips and less at hollows. This results in an overall smoothing of the surface and rounding of outer edges and corners. Electro-polishing is one of the processes for obtaining high surface fields in superconducting niobium cavities. 4

Joining techniques

When two workpieces of the same metallic material are heated locally above the melting temperature of the metal, this is welding. On the other hand, when a filler metal or alloy is used with a lower melting temperature than that of the two pieces to be joined (which can be different materials), this is brazing. 4.1 TIG welding

TIG (tungsten inert gas) welding is characterized by the use of an inert gas (Ar) flow to shield the heated area around the joint from the oxidizing atmosphere, and a non-consuming tungsten electrode from which an arc forms to the workpiece to heat it locally. TIG welding is a well established technique that can

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produce good joints. However, thick components can be considerably deformed during the welding. Another disadvantage is that dissimilar metals or ceramics cannot be welded.

4.2 EBW Electron-beam welding uses a finely (sub mm) focussed electron beam to vaporize (rather than melt) the workpiece near the joint to be. When joining copper pieces by EBW, the advantage compared to brazing is that only a small volume of copper is heated (annealed), while the bulk material will maintain its hardness. However, the workpieces have to be under vacuum. Joints between different metals are possible with EBW.

4.3 Brazing Vacuum brazing is the preferred brazing technique. During the brazing process, the temperature will be such that the brazing alloy becomes liquid, and so the pieces and the brazing joint itself have to be designed such that a) the melted filler material moistens the joint surface well, in a controlled and regular fashion, that b) the pieces to be joined do not “float” out of their position while the brazing alloy is liquid, and that c) the liquid brazing alloy will not penetrate into the cavity and form drops inside, which would detune the cavity. In order to obtain a good braze, the surface should not be too smooth - a surface roughness Ra of 0.8 is considered appropriate. When designing a joint fitting, a gap of about 0.02 to 0.05 mm should be provided to optimize the moistening. Figure 6 gives an idea of how a brazing stack of cells (cups) of an iris-loaded accelerating structure and the brazing joints could be designed. The grooves for the wires of brazing alloy are located about 5 mm from the edges, which allows the filler material to moisten a sufficiently large area around the groove, which assures vacuum tightness. Sharp corners are required at the inside to stop the liquid braze and thus prevent it from penetrating into the cavity. The sketch also suggests a method to centre the cups with external fittings. Different brazing alloys behave differently when becoming liquid, and it requires experience and exact temperature control to attain the desired result. The most frequently used brazing alloys for vacuum brazing are

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fitting

\

grooves-

beam axis

Figure 6: Sketch of a section through a brazing stack of an iris loaded structure, indicating the design of the brazing joint. Type I: Ag 72 %, Cu 28 %; B-Ag72Cu-780 (cf. IS0 3677); purity higher than 99.99 %, Type 11: Ag 68.4 %, Cu 26.6 %, Pd 5 %; B-Ag68CuPd-807/810 (cf. IS0 3677); purity higher than 99.99 %, Type 111: Ag 58.5 %, Cu 31.5 %, Pd 10 %; B-Ag58CuPd-824/852 (cf. IS0 3677); purity higher than 99.99 %. The type I brazing alloy is a eutectic alloy with a melting temperature of 780 "C, the others have higher melting temperatures - the numbers given in their names indicate the fusing range in "C. More complex assemblies can be made using these different brazing alloys in consecutive brazing cycles at different temperature levels without re-melting a former brazing joint.

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A typical brazing cycle is sketched in Figure 7. The slope during the heating and the duration of the flat top just below the melting temperature of the brazing

I

t

Figure 7: A typical brazing cycle, indicating temperature versus time alloy are programmed as a function of the size and complexity of the assembly and will assure that the heating of the assembly is homogeneous. Other brazing alloys are available for higher fusion temperatures (containing Au, with melting temperatures ranging from 910 to 1030 “C) and for lower fusion temperatures (containing In or Sn). However, not all filler materials are compatible with UHV;in particular they must not contain Zn or Cd. So-called “active brazing alloys” (containing Ti) exist, which allow brazing to ceramics without prior metallization. Brazing under vacuum requires a special furnace that can provide a vacuum better than lo-’ mbar at a working temperature of 1000 O C . In order to obtain uniform heating of large workpieces, the furnace must be equipped with a number of independent heating zones (typically molybdenum panels), controlled via a sufficient number of thermocouples connected to the workpiece. The temperature control (using a sufficient number of thermocouples on the workpiece) must allow temperature homogeneity and absolute precision in operation of a few degrees. Another important feature of a vacuum brazing furnace is its closing mechanism: it must open and close without any shock or vibration, otherwise the

146

parts in the stacked assembly risk changing their relative positions before brazing. When brazing workpieces of different materials, it must not be neglected that in vacuum the only heat exchange into and out of the workpieces is by radiation. Consequently, the workpieces will heat up and cool down at different rates, which may lead to temperature differences and differential stresses. The situation can be improved by slow heating as indicated in Figure 7 above. When brazing stainless steel flanges to a copper structure, e.g., it also helps to copperplate the flanges prior to brazing. As an example, Figure 8 shows an S-band accelerator structure during stacking in the vacuum brazing furnace (CTF3 drive beam accelerator). The waveguide-to-coax transitions at the coupler cell are for final RF tests and are to

Figure 8: Cells of an S-band accelerator structure being stacked inside a vacuum brazing furnace.

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be removed before brazing. A simpler furnace can be used when brazing under a controlled hydrogen atmosphere, a method that is also used to braze accelerator structures. However, the hydrogen tends to remain in grain boundaries near the surface, which subsequently requires bake-out to attain good vacuum. 4.4 Diffusion bonding

A diffusion bond is created between extremely smooth surfaces (optical finish) under high pressure ( > 10 N/mm2 ) and temperature (> 350 "C). As opposed to brazing, no melting takes place, but the filler material (e.g. Ag) diffuses into the workpiece. For the fabrication of CLIC 30-GHz accelerating structures a hybrid method was developed that combines direct diffusion bonding of copper (without filler material) with conventional brazing, which is illustrated in Figure 9. Since extremely flat and clean surfaces are required, the discs are diamond-turned with the required precision, and feature 1-mm-wide annular surfaces at the edges of the discs for the diffusion bond, as shown in Figure 3 above. Between these rings before brazing

I

after brazing

brazing material

* I

I

I I

1

I

I

I

I

Figure 9: Diffusion bond and brazing technique used for CLIC 30-GHz accelerator structures.

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is an area with surface roughness and spacing suitable for normal vacuum brazing - this area will be filled with brazing alloy. A cavity Q of near 100 % of the theoretical values could be obtained by using this technique. At the same time, the diffusion bonds prevent braze from leaking into the structure.

5

Testing

Different types of tests are to be integrated into the fabrication process. Only these tests allow assuring the performance of the final accelerating structure and allow corrective action at early stages of the fabrication. Since many of these tests are industry standard, we will concentrate in the following on those specific to RF accelerating structures, i.e. tests of the RF properties, which allow us to predict the eventual performance.

5.1 Material testing As mentioned above, many material properties are important and require a tight specification. On reception of the materials, some standard tests need to be performed. Some of these are, in particular for metals: chemical analysis,

Figure 10: Cell of the 3-GHz Drive Beam Accelerator of CERN’s CLIC test facility 3 during automated dimensional control.

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structural tests (inclusion content, grain size), ultrasonic tests, penetration test and other standard mechanical tests (elasticity, yield stress and tensile strength, elongation at break, hardness). Also some electrical properties like conductivity or magnetic permeability may require testing. For ceramics, measurement of the loss tangent at the operation frequency may be necessary.

5.2 Dimensional control Dimensional controls will be necessary throughout the fabrication process. In particular the inner dimensions of the final structure are critical. For a standard travelling-wave accelerating structure e.g., the sensitivity of the frequency on an inner radius is approximately given by6flf = 6 r / r , which for a typical S-band (3-GHz) structure e.g. translates to lOOkHz/pm . Figure 10 shows a cell of an S-band accelerator cell during a computerized dimensional test with conventional measuring tools. For ultra-high precision, non-touching sensors may be required. As an example for laser-interferometric control of surface flatness, Figure 11 shows the output from a flatness control of a RDDS (rounded damped detuned structure) cell for the NLC study [ 2 ] .

5.3 Helium leak detection test The vacuum level that can be obtained in later operation is determined by the outgassing rate of the materials used and the pumping speed applied. A prerequisite however is the leak tightness, which can be controlled by the helium leak test. Even small vacuum leaks should not be tolerated and must be repaired,

Figu.re 11: Interferometric measurement of surface flatness of an X-band accelerator cell (SLAC RDDS)

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since in later operation they tend to grow. The procedure for the helium leak detection test is the following: The device under test is connected via a turbo-molecular pumping station to a calibrated helium leak detector and immersed for 10 minutes in an atmosphere of helium ( lo5 Pa = 1bar He partial pressure). The helium leak rate signal is recorded during the entire immersion period. The detection limit of the calibrated helium leak detector must be checked immediately before the test. The leak detector is moved around the outside of the assembly under test. Leak rates down to lo-" Pa. m3/s can be detected in this way. During the manufacture of an accelerating structure it is advisable to perform leak tests of subassemblies.

5.4 RF properties The most important properties to be tested for accelerating structures are of course those of relevant RF parameters. It is obvious that for a complicated assembly it is appropriate to test those properties at an early stage of the assembly process, such that corrections can still be made. For example, in a brazed multi-cell linac accelerating structure, one would be advised to control the correctness of the individual cells before the actual brazing. Other RF relevant tests can be performed only after finishing the whole assembly: to test whether the accelerating structure can support the nominal input power and develop the nominal accelerating gradient, for example, requires conditioning of the structure, i.e. it has to be trained to attain those power levels. These power levels can normally be obtained only after conditioning. An unconditioned surface tends to multipact, as we have mentioned above. Conditioning is a process of cleaning and smoothing the inner surfaces of the structure with the RF power, and driving the surface into controlled localized breakdown, and in particular it also modifies those inner surfaces such that the SEY is reduced. For copper surfaces, the maximum SEY occurs typically at a few hundred eV, and is reduced during conditioning from about 1.6 to below 1.3.Conditioning can last from a few hours to a few days, depending on the duty cycle and the geometry of the accelerating structure. It requires patience and good control of the power and the controlled breakdown activities in the cavity, e.g. by careful monitoring of the vacuum gauges.

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5.4.1

Single cell frequency measurement

As an example of a single frequency test we consider a travelling-wave accelerating structure with a phase advance of 120" per cell (or a cell length of A/3 in the case of an electron accelerator). Since for a single cell one cannot easily reproduce a boundary condition equivalent to a phase advance of 120°, one could either measure the resonant modes of a unit of 3 cells, which can be short-circuited at both ends, or one could think of a special, well characterized device in which a single cell is mounted, thus allowing measurement of the 0mode and the n-mode of this assembly, and calculate the resulting 2d3-mode. The validity of computing the 2d3-mode from the 0- and n-mode must of course be checked (for example with a field simulation program), but for a typical structure of this type, where the next higher-order mode (HOM) is relatively far away, the dispersion curve of the first pass-band can be modelled to a high accuracy by the simple formula

where f is the frequency for phase advance q , and fo and fr are the measured resonance frequencies of the 0- and n-mode, respectively. Figure 12 shows the dispersion characteristic of an S-band accelerating structure, designed for operation at a phase advance of 120 O, which for this case is indistinguishable Dispersion diagram 3.2 I I

3.15

I #

-2 9 A

0

3.1

v=c

i

I

I

3.05 3

0

$ 2.95 & 2.9

I

I

2.85

i

,

~

I

I

2.8 0

30

60 90 120 Phase advance per cell ["I

150

Figure 12: Dispersion diagram of a 2 d 3 structure indicating 0- and x-mode.

180

152

from the approximate formula above. The single-cell measurement described here thus allows checking the RF property of a cell for the travelling-wave accelerating mode by the relatively simple measurement of the 0- and n-mode. However, since such a measurement - or any other frequency measurement of a vacuum cavity - is normally done in air under atmospheric pressure, the measured frequencies must be corrected for the dielectric constant of air and - in general - also for the ambient temperature which may be different from the temperature at which the structure will be operated. The dielectric constant of air at 20 "C and 60 % humidity is approximately 1.0006, which will lower the frequency by 0.3 %O from its vacuum value. The ambient temperature influences primarily the linear dimensions of the accelerating structure itself and changes the frequency according to Sf/f =-al .6T ; with the thermal expansion coefficient of Cu ( al = 16.5 10" K-') e.g., this results in -50kHz/K for a 3-GHz structure.

5.4.2

Bead-pull measurement

The bead-pull measurement is based on the fact that a small object (the bead) perturbs the field at its position, and causes a reflection proportional to the electric field-energy at that position. This allows sensing the field distribution in the accelerating structure, when measuring the reflection coefficient at its input port while dragging the bead along its axis. A vector network analyser (NWA) is connected to the input port of the structure as shown in Figure 13 and operated at a fixed frequency, which should correspond to the operating frequency, corrected for the temperature and the atmosphere present inside the structure (see above). The network analyser is set to SII measurement and to polar display, calibrated to indicate zero for the structure without perturbation. When the bead is now moved through the structure, SI1 will change its amplitude and phase as a function of the position of the bead - it will be large when the bead is at a position of large electric field (in the centre of each gap), and small in regions of small electric field (in the iris). At the same time, the phase of the reflection coefficient will vary as twice the phase advance of the field. While the perturbing bead is moving through the structure, the complex input reflection will thus describe a locus on the polar display of the NWA.

153 Nylon thread

L

bead

u Figure 13: Schematic set-up for a bead-pull measurement to determine the phase advance in an iris-loaded structure

As an example, we will again consider an accelerating structure with a nominal phase advance of 120" per cell, but the method can be adapted to other phase advances following the same idea. Figure 14 shows the principle shape of this locus: it starts at zero for the bead outside the structure, rises with constant phase while the bead enters through the cut-off beam pipe into the coupler cell, and subsequently describes a regular pattern, advancing in phase by 2.120 when the bead advances by one cell. Thus the phase advance error and amplitude imbalance will clearly show.

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Figure 14: Locus of the complex reflection coefficient while the bead is moving through the accelerating structure. A possible way to correct for these errors is to prepare the individual cells for “dimple tuning,” i.e. to locally deform the cavity wall. A more detailed description of this method can be found in Ref. [3].

Acknowledgements I used a lot of material from lectures by I. Wilson and W. Wuensch, and I am happy to acknowledge their help.

References 1. C. Achard et al., “A Demonstration of High-Gradient Acceleration,” PAC 2003, Portland, Oregon 2. D. Sun et al., “Microwave Quality Assurance Effort for R&D on the Next Linear Collider at Fermilab,” PAC 2001, Chicago, Illinois 3. T. Khabiboulline, M. Dohlus and N. Holtkamp, ‘Tuning of a 50-cell constant gradient S-band travelling wave accelerating structure by using a non-resonant perturbation method,” DESY M-95-02, 1995

COMPUTATIONAL TOOLS FOR RF STRUCTURE DESIGN E. JENSEN CERN AB-RF CH-1211 Geneva 23, Switzerland E-mail: [email protected] The Finite Differences Method and the Finite Element Method are the two principally employed numerical methods in modem RF field simulation programs. The basic ideas behind these methods are explained, with regard to available simulation programs. We then go through a list of characteristic parameters of RF structures, explaining how they can be calculated using these tools. With the help of these parameters, we introduce the frequency-domain and the time-domain calculations, leading to impedances and wakefields, respectively. Subsequently, we present some readily available computer programs, which are in use for RF structure design, stressing their distinctive features and limitations. One final example benchmarks the precision of different codes for calculating the eigenfrequency and Q of a simple cavity resonator.

1

Introduction

Computational tools for RF system design and readily available computer hardware have made remarkable progress over the past decades. This allows us today to design complex RF structures almost entirely on the computer, minimizing the lengthy cut-and-try optimization methods as well as expensive and costly model building for cold measurements. In particular these computer programs also allow us to simulate the interaction with the particle beam, something almost entirely inaccessible to mechanical models. However, the most advanced computational tools also require deep insight into the underlying principles of both RF theory and the numerical tool itself. The aim of this article is to summarize how the relevant quantities in RF structure design are accessible to numerical computation, to give an idea of how these numerical tools work in principle, and finally to present some of the actual programs available and compare them. 2

RF field calculation methods

The computer codes used to simulate electromagnetic fields numerically start from the idea of discretizing space. For example, one might think of considering the fields only at some discrete values of the three Cartesian co-ordinates x, y

155

156

and z. Between these mesh points, some type of interpolation is used to represent the complete solution. Even though other methods exist (Transmission-Line Method, field matching methods, ...), the two main methods for the simulation of RF fields are the Finite Differences Method (FDM), where the differential equations are directly replaced by difference equations between the field quantities at discrete mesh points, and the Finite Element Method (FEM), where the overall solution is represented as a superposition of simple functions, which are zero everywhere except on localized, finite elements of space. 2.1 Finite Differences Method

The Finite Differences Method (FDM) is based on the idea of transforming the differential equations to a set of difference equations. The task is to find a solution to Maxwell’s equations inside the cavity volume subject to given boundary conditions (in time-domain also initial conditions). Consider a mesh-grid defined inside the volume. Maxwell’s equations in differential form are a set of linear differential equations of first order (or - in the form of the Helmholtz equation - of second order). Replacing the differential operators by difference operators results in a set of linear algebraic equations, which relate a field quantity at one mesh point to the field quantities at the directly neighbouring points. For illustration, consider the (second-order) Laplace equation A@ = 0 in two dimensions and Cartesian co-ordinates. For the potential at some central mesh point, and assuming identical mesh size in both directions, the central difference-quotient would result in the equation @m+l.n

+ Qrn-1.n

+ @‘m.n+l + @rn,n-l

-4

@m,n

= 0.

(1)

An equation of this type exists for the potential at every mesh point; at boundary points the boundary conditions (in this simple case a given potential) can be included on the right-hand side. If we further assume for simplicity an equidistant, uniform mesh throughout the entire space, the resulting linear equation would have the form

157

.. .. .. ... ... .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . 1

1 -4

b 0

U

n d 1

.. .. * . .. .

.. . .. . .. .

1

.. . .. .. . . .. .. .. . . .. ..

a

r i e S

where the ellipsis indicates the non-zero matrix elements. This is a large (number of mesh-points squared), banded, sparsely occupied matrix. Methods of inverting large sparse matrices exist (Cholesky, SOR, ADI, conjugate gradient . ..). In the case of Maxwell’s equations in three (or - for time-dependent equations - four) dimensions, the matrix becomes much larger and gets more bands, and for variable mesh sizes and mesh elements, which are not parallel to the co-ordinate axes, the matrix elements become less trivial. Particular care must be taken to properly account for the boundary conditions, in particular if the physical boundary does not coincide with the mesh-grid. The FIT algorithm, used for example in MAFIA and in GdfidL, is based on a double grid (see Figure 1), where the mesh points of one grid are centred in the cubical cells formed by the other. It is a feature of Maxwell’s equations that the spatial derivatives of the electric field are related to the magnetic field and vice versa. Thus when taking E , D and J’ on the mesh-points of one grid, but B and l? on the edges of the dual grid, one can formulate Maxwell’s equations on the double grid to obtain the so-called “Maxwell’s Grid-Equations (MGE).” It is convenient to use the integral form of Maxwell’s equations here, but to extend the integrals only over the finite areas or volumes of the grid cells (FIT = Finite Integration Theory). This formulation is less prone to discretization errors than a single-grid formulation. A further advantage is that the vector identities V XVY,= 0 and V .V x 2 = 0 become properties of the “topological” matrices representing the curl, grad and div operators.

158

I

: / B

I

-I I

Figure 1: The dual grid used for MAFIA’SFIT algorithm. The method is applicable both in frequency and in time domain. When applied in time domain, one also uses the term “FDTD algorithm” (Finite Differences Time Domain).

2.2 Projection Methods - Finite Element Method The task is again to find a solution to Maxwell’s equations inside the cavity volume subject to given boundary conditions (in time-domain also initial conditions). Let us abbreviate the differential equation as

D(q)=0

9

(3)

where q represents the (unknown) solution. For illustration, we have set the right-hand side to zero, corresponding to the eigenvalue problem; in a more general case, the sources (currents and charges) appear on the right-hand side. In the case of Maxwell’s equation, v, may stand for the vector E in the entire volume and D ( q ) = 0 for

159

i: -1

Vx -VxE

-w’&E=O

(4)

The projection methods start by assuming that the solution can be represented approximately in the form

n=l

where a is a vector with N (many) dimensions, and the v,, are known basis functions (trial functions) which satisfy the boundary conditions. This assumed q~ may not solve the original equation (3) exactly, so we provide a residual r on the right-hand side,

n=l

and the task will be to minimize r. With the scalar product in function space, denoted (q,ty) (normally an integral over the entire volume), one can now scalar multiply Eq. (6) with N test functions (or weight functions) ty, , and solve

The name “projection methods” refers to this scalar multiplication. These projections of the residual r on the test functions will determine the coefficients a, such that the residual (error) is minimized, assuring that r has zero length in the sub-space spanned by the v, . Since both the basis functions and the test functions are known, the scalar product in Eq. (7) can be evaluated, and we again get a N x N matrix equation to determine a. Different choices of weight functions are possible. If the weight functions are chosen to be identical to the basis function, this is called the Galerkin method. If in addition the basis functions are chosen orthogonal, i.e. (v,,,q,) 0~ 6, (the Kronecker delta), we have spectral methods. Examples of orthogonal functions are the eigenfunctions of the system, trigonometric functions (cf. Fourier series), Bessel functions, Legendre polynomials, Chebyshev polynomials, and others. The Finite Element Method (FEM) can be classified also as a Galerkin method; each basis function in this case is chosen to be zero everywhere in the volume except for a finite region (the finite element), where the basis functions

160

are very simple (typically piecewise linear or quadratic polynomials). This makes the evaluation of the integral for the scalar product in Eq. (7) simple. Since the basis functions are non-zero only inside one finite element, the matrix will again be sparse and banded as in Eq. (2), so basically the same methods apply for the inversion of these matrices as for the FDM.

2.3 Different meshes In the FIT example above, we have used a Cartesian global mesh, which simplifies the formulation of the difference equation significantly, since the partial derivatives are taken along mesh lines. However, the contours of arbitrarily shaped FW structures do not generally coincide with mesh lines, and consequently the meshed geometry may be different from the physical geometry to be simulated. This leads to a systematic error, which can be controlled and estimated, and which can be reduced by iterative mesh refinement. However, there remain preferred directions in space, and the meshing (and possibly also the simulation result) will depend on the orientation of the geometry in the mesh, i.e. the mesh can create some intrinsic, numerical anisotropy. If it is necessary to have a local mesh refinement in a particular area of interest inside the structure, it is typical for the global Cartesian mesh that every added mesh point will add mesh planes which extend through the entire structure, thus extending this refinement also into areas where it is not actually needed. If the constraint to have mesh lines parallel to the co-ordinate axes is dropped, a completely isotropic mesh may result. When connecting those meshpoints, one can always create a dense paclung of tetrahedra (in two dimensions, triangles). The finite difference formulation becomes complicated in this case, but the Finite Element Method often uses tetrahedral meshes. A local mesh refinement is straightforward with tetrahedral meshes; when we add a mesh point inside an existing tetrahedron, it naturally splits it onto 4 new tetrahedra.

3

Computing the characteristic parameters of an RF structure

In the following, we will go through a number of quantities one typically needs a numerical tool to compute. For these quantities, we will mention their mutual relations, explain how they are computed and, if necessary, stress which particular feature of the simulation tool is needed to perform the computation.

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3.Z Eigenfrequency When designing a single-gap RF cavity, the first quantity one typically needs is its resonance frequency fo (eigenfrequency), which should correspond to the operation frequency. In a closed cavity with given boundary conditions but without any sources, equation (4) above (or equivalently the Helmholtz equation (Aq + k2q)= 0 ) results in an eigenvalue problem. Each eigensolution corresponds to a resonant mode, with its eigenvalue corresponding to the eigenfrequency, the eigenvector to the field distribution of this mode. The field simulation programs discussed below are able to solve for eigenfrequencies; even the time domain programs (MAFIA, GdfidL and Microwave Studio) have this feature built in. For travelling-wave accelerating structures, the frequency relevant for efficient acceleration is the synchronous frequency, i.e. the frequency at which the phase velocity of the accelerating mode is equal to the velocity of the beam. For a v = c beam, this frequency is equal to the frequency where the phase

AP advance A q in a single cell of length 1 is equal to -=I, where A is the 27r A free-space wavelength. In order to calculate such a special eigenfrequency, the simulation program is required to handle periodic boundary conditions, i.e. the field distribution on the downstream boundary of the cell is required to be identical to that on the upstream boundary, but rotated in phase by a given phase angle. Not all simulation programs have this useful feature built in. However, if this feature is missing (e.g. in Superfish), one can still study travelling-wave structures, but only special phase advances (integer multiples of 2dn), by simulating an assembly of n cells.

3.2 Attenuation Idealized, lossless cavities have real eigenfrequencies, but in the presence of losses the eigenfrequencies become complex, with the imaginary part of this frequency corresponding to a damping term. From a complex eigenfrequency fo, the cavity Q can be calculated from

One often requires that an RF cavity have low losses or a high Q,sufficient to minimize the power needed to produce a necessary accelerating voltage. Maximizing the cavity Q means minimizing the power required for a given stored energy, by making use of the cavity’s resonant behaviour. At the same

162

time, a high Q makes a cavity extremely narrow-band and sensitive to errors, and thus requires high precision in the determination of the resonance frequency (and in the fabrication). The cavity Q is defined as Q = -"OW ,

P

(9)

where W is the energy stored in the cavity, w, = 277 fo, and P is the ohmic loss. For travelling-wave structures, the Q is defined in the same way, but typically W and P are normalized to the cell length. P again is the ohmic loss, which in this case however is typically only a small fraction of the power input to the cell, since most of the input power travels through. As opposed to a cavity, the Q in this case does not affect the bandwidth of a travelling-wave structure, but rather the attenuation of the travelling wave. With group velocity vg , which will be defined in Eq. (20) below, the attenuation of the travelling wave is given by

a=-

0

~ Q v ', i.e. the forward-travelling power is proportional to e-2az 3.2.1

Perturbation ansatz for low losses:

It is numerically much easier to find real eigenfrequencies of lossless structures than to find complex eigenfrequencies. A high Q now allows the assumption that the field distribution for the ideal, lossless case is a good approximation of a real one, and that the losses represent only a small perturbation of this ideal solution. The idea of this perturbation ansatz is, starting from the ideal solution, to calculate the tangential magnetic field at the perfectly conducting cavity boundary, equating it to the surface current density, driving this surface current through the surface resistance which is given by

with wall conductivity (T , and to determine the wall losses by evaluating the integral

163

For high-Q RF structures, this perturbation method is very efficient and exact. Most of the simulation programs use this method.

3.3 R-over-Q A third important quantity characterizing an RF structure is the R-over-Q ( = R/Q ), which accounts for the influence of the structure geometry; one could

say that it describes the feature of the structure that concentrates the electric field where it is actually needed for acceleration. The R/Q is expressed as

In Eq. (1 l), V,, is understood as the accelerating voltage “seen” by the beam, including the effect of its finite speed v ,i.e.

We have used infinite integration limits in this formulation to indicate that the accelerating field may actually extend beyond the actual structure into the beampipe at both ends. However, for a resonance frequency below the cut-off of the beam-pipe, where the fields cannot propagate into it, it is valid to limit the integration to only a little more than the RF structure length. Note that sometimes R/Q is defined with the finite speed of the beam ignored, and this is subsequently corrected for by introducing the so-called transit-time factor, given by

The transi- time factor is a number between 0 and 1. Whereas fo and Q are typically directly output from a simulation program, the integral in Eq. (12) requires the possibility of evaluating a path integral over a field component multiplied by a trigonometric function. In the computer codes mentioned below, this functionality either is integrated (Superfish) or can be performed by a sequence of macro commands.

164

When investigating a single cell of a periodic structure, the integral (12) is extended only over the length I of a single cell. It is common for travelling-wave structures to give R/Q (and R, to be defined below) per unit length (in units of a m ) and to denote them with a prime; the definition equivalent to Elq. (11) for travelling wave structures is

where Vac/l is called the accelerating gradient and W/1 is the stored energy per cell.

3.4 Shunt impedance Finally, the product of R/Q and Q is the shunt impedance R. We stress here that one should consider RlQ as a more fundamental quantity than R , even though the naming suggests otherwise. Maximizing R means maximizing the accelerating voltage while minimizing the input power required to obtain it. Combining equations (9) and (1l), this can be expressed as

Often, an important design task is to match the cavity to the power source at resonance, i.e. to match to the cavity shunt impedance, typically via a standard 50-Ohm transmission line. The power coupler can be considered as an essential part of this impedance matching circuit. To perform this task, the computational tools are required to account for the power coupler, and to compute input reflection. MAFIA, HFSS, Microwave Studio and GdfidL are suited for this task. Please note that sometimes a different definition of both R/Q and R is used, which is smaller by a factor of 2 than in Eqs. (11) and (15). We are using here the so-called “Linac definition” of these impedances.

3.5 Beam loading and higher order modes The above mentioned quantities, fo, Q and R/Q, characterize primarily the action of the accelerating mode on the beam (and the cavity as seen from the RF power source). In the presence of a high beam current however, not only the

165

power source, but also the beam itself will excite this mode, which in the case of acceleration will counteract the accelerating voltage and thus reduce it, a phenomenon referred to as beam-loading. Often it must not be neglected that beam loading influences the matching to the RF power source (except for travelling-wave structures, where the power generated by beam-loading is not reflected, but travelling forward). This excitation of fields in the RF structure by a high current beam occurs not only at this nominal accelerating mode but also at all higher-order modes (HOMs) that exist in the structure. In general, HOMs excited by the beam result in forces on the beam that can deteriorate it - this is why they need considering. These forces can be both longitudinal (thus potentially perturbing the bunch structure of the beam) and transverse (kicking the beam or parts of it sideways). This coupling of the beam to cavity modes is described by the beam-coupling impedance. It peaks at each mode’s resonance, but in general is a complex function of frequency. In many cases (e.g. for single-gap cavities), the coupling impedance at resonance is equal to the shunt impedance (for the driven mode, it is shunted by the power source output impedance). Both longitudinal and transverse beam-coupling impedances should be accessible to the simulation ‘tools. The transverse coupling impedance is defined similar to the longitudinal one, but with one of the two IVacI in the nominator of Eq. (11) replaced by an integral along z of the transverse Lorentz force. This seems much more complicated at first sight, but thanks to the Panofsky-Wenzel theorem [ 11, this problem can be reduced to computation of the transverse dependence of the longitudinal coupling impedance. For example, to determine the transverse impedance of a dipole mode polarized in the x-direction, it is sufficient to calculate the longitudinal impedance as described above, but evaluating the line integral at a transverse offset position x. If the longitudinal impedance on axis is zero (which is true for the dipole mode in structures with mirror symmetry with respect to x = 0 ), the transverse impedance is simply given by

The transverse impedance is given in units of Wm (or - normalized to length in travelling-wave structures - in Wm’). To consider HOMs, the simulation tool must be capable of determining the quantities defined above for a number of modes, i.e. a set off,, , Q,, and R,/Q,, . For transverse modes, the integral (12) must be evaluated off axis.

166

3.6 Wake-fields - loss factor

So far, we have been referring to quantities describing the frequency domain behaviour of RF structures. Equivalently, they can be calculated and described in time domain; in this case we refer to longitudinal and transverse wake-fields. Wake-fields are the fields left behind in the structure by a driving charge. They are normally characterized by the so-called wake-potentials, which describe the net forces on a trailing charge, following the driving charge at a distance s through the entire structure [ 2 ] . The Fourier transforms of the longitudinal and transverse wake-potential are the longitudinal and transverse beam-coupling impedance, respectively. One distinguishes the short-range (for small s) and the long-range wakepotentials. The short-range wake-potentials are determined by wide-band and high frequency components of the coupling impedance. Physically, they are dominated by those discontinuities inside the RF structure that are closest to the beam trajectory (typically irises). They are not well accessible to frequencydomain calculation. The high-Q resonances on the other hand, which can be well described in the frequency domain, dominate the long-range wake-potentials. They are dangerous in particular because of their adverse effect on subsequent bunches following at comparatively large distances. The long-range wake-potentials may be considered as a superposition of the contributions of the individual modes. For longitudinal wake-potentials e.g., the excitation of each of these modes is described by its (modal) loss factor

Note that this excitation is independent of the Q of the considered mode, but directly proportional to its R/Q. The energy lost by a charge q into this mode is k,,q 2 , and the contribution of this mode to the longitudinal wake-potential is

A

S
!O

1

{

i-z]

- 2 k l o S s q e2Q'co wo 1 - 7 -was

A

s>O

167

where s is again the distance behind the charge exciting the wake, and the minus sign indicates that the wake will be dragging on this driving charge. For transverse wake-fields, the transverse loss factor is defined equivalently; it is sometimes also referred to as kick-factor. Because of Eq. (17), the modal loss factor is readily accessible to frequency domain calculations (HFSS), while the total loss factor and the wake-potentials are better determined in time domain simulations (MAFIA, GdfidL).

3.6.1

Direct method

When calculating the wake-fields in time domain, one excites the fields in the structure by a charge moving along its axis or parallel to it, and calculates the resulting electro-magnetic fields. For the so-called direct calculation, one has to follow this charge for a large distance through the vacuum chamber both before and after the actual RF structure.

3.6.2

Indirect method

A trick, known as the indirect method [ 2 ] , which is possible for certain RF structures, significantly simplifies this: if the charge is assumed to travel grazing along the beam pipe outer boundary, it will excite and experience no fields until it enters into the actual gap or - more generally - the RF structure. The obvious advantage is that long beam pipes need not be discretized, which saves computation time substantially; the necessary condition for applying the indirect method is that upstream and downstream beam pipes have the same shape and dimensions, and that there is no obstacle on these special particle trajectories. The indirect calculation method is based on the fact that the longitudinal wake-potential satisfies the transverse Laplace equation AIWz ( x , y; s) = 0 . Once the longitudinal wake-potential is known at transverse positions corresponding to the boundary of the beam pipe, it is thus determined anywhere inside the crosssection. It follows from the Panofsky-Wenzel theorem that this also determines the transverse wake-potential. This is particularly simple for round structures, where it is sufficient to determine the longitudinal wake-potential at the radius of the beam pipe. The indirect integration method was implemented in TBCI, a program which today is integrated into MAFIA. Also GdfidL uses the indirect calculation method. Transverse wake-fields describe the net effect of transverse forces, both electric and magnetic, on trailing particles. As in the frequency domain, the Panofsky-Wenzel theorem can be used to obtain the transverse wake-fields simply from the radial dependence of the longitudinal wake-fields.

168

At least in principle, wake-field calculation in the frequency domain is possible thanks to the relation (17), which for each mode relates the loss factor to the R/Q. Having determined the modes of a structure, the wake functions can be approximated by a sum of expressions (1 8). However, the determination of short-range wake-fields is impractical, since the number of modes to be taken into account would become very large.

3.7 Field distribution Other quantities required from a field simulation program are the electric and magnetic fields themselves, often also derived quantities like the Poynting vector, energy densities or (in particular for precision control) spatial derivatives of the fields. The possibility of graphical output of the field distributions in the structure is extremely useful in the structure design. Visualizing the fields also helps to check the consistency of the computation result and to understand the physics. Particularly interesting are the maximum electric and magnetic fields at the structure surface, since they will determine breakdown limits and surface heating.

3.8 Group velocity For travelling-wave structures, the group velocity is another important quantity to consider. It was already used in Eq. (10) above. Since the group velocity is defined as the slope of the dispersion curve, one way to calculate it is to calculate the eigenfrequencies at two different phase advances near the synchronous phase advance. With the two eigenfrequencies f, and fi at phase advances ql and q Z in a cell of length I , the group velocity near the frequency (fi- fi )/2 is calculated from

On the other hand, the determination of the energy stored in a cell of a periodic structure, along with the calculation of the power flow (the integral over a cross-section of the real part of the Poynting vector) allows the direct determination of the energy velocity, which is equal to the group velocity, from

169

3.9 Time domain versus frequency domain

We have discussed both time domain and frequency domain characteristics of RF structures above. For linear systems, the formulations in time domain and frequency domain are equivalent - one is the Fourier transform of the other. However, for specific problems, one or the other approach is better suited. For a structure with a single resonance with a very high Q e.g., a time domain calculation would require long computation time, unless one starts off with the steady-state solution. Also if the excitation is known to be only at one frequency (like from the RF power source), a frequency domain calculation is advantageous. On the other hand, the calculation of wake-fields in a structure with no pronounced resonance (or very many modes contributing to it) is more economical directly in the time domain. This is particularly true for short-range wake-fields. If the length of the bunch exciting the wakes is not too short, both time domain and frequency domain calculations require approximately the same computational effort. For low energy, high current beams, where the excitation of fields by the beam itself (the wake-fields) can influence the particle trajectories, an exact calculation would have to be “self-consistent.” This problem is - strictly speaking - nonlinear and thus cannot be performed in the frequency domain. Only MAFIA’S TS3 module allows today the self-consistent computation of particle trajectories and RF fields excited by them (and influencing those trajectories). A new version of GdfidL, which is currently under test at TU Berlin, can equally handle this difficult case. However, for most accelerating structures a self-consistent analysis is not necessary. In particular for electron linacs, where the speed of the particles is close enough to c, their trajectory is known beforehand to any practical precision. 4

Specific Computer Codes

A large number of computer codes are available for the calculation of RF fields. Some of the commercially available codes however are not particularly suited for the design of accelerator cavities, since a) they do not allow the direct calculation of eigenfrequencies, or b) they do not provide a possibility of computing the integral (12) which describes the interaction with a particle beam. So we will restrict ourselves here to a few codes relevant for RF accelerator structure design.

170

4.1 Superfish

Superfish calculates the eigenfrequencies and fields for round, but otherwise quite arbitrarily shaped cavities, which can be described completely in the two co-ordinates r and z. The output of Superfish includes all quantities relevant for the cavity design directly. Except for trial versions, there is no version of Superfish available today that treats modes with non-zero azimuthal orders. The easy to use PC version is available from the “Los Alamos Accelerator Code Group” (http://laacd .lanl.gov/) free of charge, but users are encouraged to register. Superfish is a suite of programs, also including Poisson and Pandira for the solution of static fields. Superfish starts off by defining a triangular mesh inside the cavity. The program “Automesh” then deforms these triangles, so as to better fit the mesh to the given physical boundaries. In the next step, “Fish” or “CFish,” the Helmholtz eigenvalue problem is solved for, which for ad accelerating mode of a round cavity can be reduced to the form

Normally, “Fish” is used, which solves for real eigenvalues, i.e. solutions for closed, lossless cavities with perfectly conducting walls. “CFish” is a complex eigenvalue solver, see Eq. (8). The following post processor “SFO’ allows calculation of a number of relevant parameters, such as Q (using the perturbation ansatz), R/Q, the field distribution and the sensitivity of the solution to small geometric perturbations. The two co-ordinates used in Superfish can be either r and z of the cylindrical co-ordinates (round cavities, see example above) or Cartesian (arbitrary cross-section waveguides). For the latter case, Superfish allows one to solve for 2-dimensional boundary value problems, i.e. for modes in waveguides of arbitrary cross-section. This is particularly interesting for the investigation of RFQs, and there even exists a utility program “RFQfish” which helps to set up the geometry of the vanes of an RFQ for the determination of the purely transversal electric fields, i.e. without the longitudinal modulation of the vane tips. Also a number of other utility and tuning programs and a very complete documentation are part of the Superfish distribution. As an example, Figure 2 shows the result of a Superfish run for a single-cell cavity with nose cones to optimize the shunt impedance. Only one half of the

171 Nose cone example F = 500.00301 MH7.

..

C:\LECTURES\LONGBEAC'

Figure 2: Example of a Superfish output (WSFplot),

cavity is modelled, and the left boundary is assumed to be a symmetry plane. To demonstrate the simplicity, the complete input file listing is given here: cone example

NOSe

® kerob=l, dx=. 3, f req=500, icyl in=l, xdri =O , ydri=23.469, norm=l, ezerot=5000000, kmethod=l,beta=l& &no x= 0.0. v= 0.0 & $= 23.469 & &PO x= 0.0; y= 23.469 & &po x= 2.0, &PO X= 15, y= 10.469, nt=5, radius=l3 & Y= 9.1375 & &DO X= 1 5 . &PO X= 11i5, y= 6.866 & &PO x= 12, y= 5, nt=4, radius=l & y= 5.0 & &PO X= 30.0, y= 0 . 0 & &PO X= 30.0, y= 0.0 & &PO X= 0.0, -r-

~~

172

We give here also an excerpt from the output file of SFO, which illustrates the completeness of the solution: s u p e r f i s h o u t p u t summary f o r problem d e s c r i p t i o n : NOSe cone example Problem f i l e : C:\LECTURES\LONG BEACH\S500.AF 7-01-2003 17:46:48

.............................................................................

A l l c a l c u l a t e d values below r e f e r t o t h e mesh geometry o n l y . F i e l d n o r m a l i z a t i o n (NORM = 1): EZEROT = 5.00000 MV/m Frequency = 500.00301 MHZ P a r t i c l e r e s t mass energy = 938.271998 MeV

Beta s 1.0000000 Normalization f a c t o r f o r EO = 7.120 MV/m T r a n s i t - t i m e f a c t o r Abs(T+iS) Stored energy using standard room-temperature copper. surface resistance Normal -conductor r e s i s t i v i t y o p e r a t i n g temperature Power d i s s i p a t i o n 44271.3 Shunt impedance Q Z*T*T Rs*Q = 258.267 ohm r/Q = 110.825 ohm wake l o s s parameter Average magnetic f i e l d on t h e o u t e r w a l l Maximum H ( a t Z,R = 14.125,8.56963) Maximum E (at Z , R = 11.0267,5.77084) R a t i o o f peak f ie l ds Bmax/Emax Peak-to-average r a t i o Emax/EO 5

= = =

= = = = =

-

-

81300.178 0.7022185 6.4623931 Joules 5.83374 rnilliohm 1.72410 microohm-cm 20.0000 c 458.5893 kw 33.166 MOhm/m 16.355 MOhm/m 0.08704 v/pC 21507.1 A/m, 134.922 W/cmA2 29395.9 A/m, 252.053 W/cmA2 37.5597 MV/m. 1.76356 K i l p . 0.9835 mT/(MV/m) 5.2750

Superfish will output the numbers for the mesh geometry only, but for the calculation of the losses, it distinguishes between the symmetry planes and the metallic boundaries. This example used roughly 10 000 meshpoints and took a few seconds to run on a Windows PC. Many other data are output by SFO, in particular a complete sensitivity analysis of the resonance frequency on geometric parameters is included. Summarizing, Superfish has an excellent mesher, the mesh allows one to model curved boundaries quite well, and consequently the results can be very exact. On the other hand, the program is limited to modes with azimuthal index 0 in round cavities (2-d). The output is very comprehensive for RF cavity design. The program runs on Windows, is free of charge and is well maintained by LAACG. 4.2 MAFIA

MAFIA (Solution of MAxwell’s equations by the Finite Integration Algorithm) is probably the most widely used software package for accelerator RF structure design. It is based on the FIT algorithm described above. Apart from preprocessor (M, mesh generator) and post-processor (P), it consists of several modules for different tasks: The static solver (S), the eigenmode solver (E), the frequency domain (or Eddy current) solver (W3), the time domain solvers (T2, T3), and the self-consistent time domain solvers including particle dynamics (TS2, TS3). To our knowledge, the latter is a unique feature of MAFIA today.

173

The latest version 4 of MAFIA includes as well an easy to use graphical user interface and an optimizer module (0),which can be used as a convenient wrapper around the other modules. In the design process of a conventional accelerating cavity with MAFIA, one would typically start with a simplified, 2-D geometric model (r-z), and calculate the eigenfrequencies, stored energies and interaction integrals to determine Q’s and coupling impedances. These figures of merit are not directly output from MAFIA, but the relevant operations can be performed in the Pmodule or programmed in the form of command macros. For a vacuum cavity, the design would involve the E-module to calculate the eigenfrequencies for the lossless case, the P-module for the other parameters (again with the perturbation method for small losses). As opposed to Superfish, MAFIA allows for the r-z geometry to input the azimuthal order (for dipole, quadrupole, ... modes). The time domain solver of MAFIA evolved from the program TBCI, and it allows the direct or indirect calculation of wake-fields. Also driven solutions for waveguide components or accelerating structures are calculated with the time domain solver; if necessary the output data can be Fourier transformed to get a spectral solution. MAFIA is supported on different computer platforms. The code is maintained and commercialized by the company CST in Darmstadt, Germany. Detailed information can be obtained from httd/www.cst.de. The strength of MAFIA is certainly its versatility. We believe a weakness is its comparatively rigid mesh, which is strictly Cartesian. It can be refined locally, but the refining mesh-lines run through the entire geometry, which results sometimes in quite unevenly distributed meshes. Another important peculiarity of MAFIA is the accounting for the losses at concave, curved boundaries, where the modelled boundary is staircase-like (at least in one of the dimensions of the investigated boundary). In this case, MAFIA tends to over-estimate the Q , and this does not seem to improve with mesh refinement.

174

4.3 HFSS HFSS (High Frequency Structure Simulator) is a commercial package from Ansoft, Pittsburgh, PA (httD://www.ansoft.com), primarily intended to analyze S-parameters and radiation problems. It performs only frequency domain calculations (and is thus strictly limited to linear materials) using the finite element method, solving Eq. (4)under given boundary conditions. Its tetrahedral mesh adapts well to curved surfaces and is co-ordinate independent (isotropic). The implemented adaptive mesh refinement scheme evaluates the residuum (6) to identify the tetrahedra that contribute most to the error, which elegantly allows refining the mesh automatically where it matters most. The program runs under Windows or Solaris operating systems. The graphical user interface of HFSS is easy to use, but it also allows the use of a powerful, however sometimes counter-intuitive, macro language, including optimizer commands. The RF structure designer's figures of merit, like R/Q, shunt impedance, etc., are however not calculated automatically but require macro programming - a HFSS post-processor macro to calculate the interaction integral (12) along the line "zaxi s" could look like: Assign c 0 299792458.0 E n t e r s c a l a r 0;MathFunc "Acos";EnterScalar 4 ; * E n t e r s c a l a r GetUnitConv;/;EnterScalar cO;/ EnterFrequency;* EnterSCalarFUnCtiOn " 2 " ; *; Push MathFunc "Cos" ; - CmD1 . XR Exchange MathFunc " S i n " ; CmplxI; + E n t e r " E " ; ScalarZ; *; Push; Real E n t e r l i n e ' z a x i s ; I n t e g r a t e ; Evaluate R o l l Up; R011 Up; ;mag EnterLine ' z a x i s ; I n t e r a t e ; Evaluate R o l l u p ; Pop; R o l l d n ; Roqldn; Pop; R o l l d n ; Push; Exchange; push; *; +; s q r t GetTopEntryValue

# 2 pi # omega/c

# # # # #

*

exp(j omega/c Z> Re and I m r e q u i r e two separate integrations!

# ABS

In this example it can be seen that a convenient I€"-type calculator is used for the manipulation of field quantities in the post-processor. The actual version, HFSS 8.5, allows also the calculation of eigenfrequencies including substantial losses - even modes with Q's well below 10 can be calculated. HFSS allows defining periodic boundary conditions (master/slave) as they are helpful for the analysis of periodic structures. However, with the actual version HFSS 8.5, we have encountered some mesher problems in the presence of periodic boundary conditions.

175

4.4 GdfidL

GdfidL (httD://www.gdfidl.de) is a program written by W. Bruns of Technische Universitat Berlin, Germany. It is in many respects similar to MAFIA. It also uses the dual-grid FDTD approach, but the production version does not handle particle dynamics. The self-consistent particle dynamics code is now under evaluation at TU Berlin. Effort has been put into better meshing near the boundaries, which increases accuracy. This was achieved by introducing diagonal fillings of mesh cells near the boundary. Also the computation efficiency was increased. The mesher of GdfidL is very economical, since the field-free regions of the bounding box are not meshed. The probably most valuable new feature of GdfidL is its capability to be run on a cluster of Linux-PC’s in parallel. This allows highly efficient solution of huge problems, which would otherwise exceed the maximum memory size of 32bit PCs. GdfidL has no Graphical User Interface, but a powerful macro language, similar to that of MAFIA. Shown here is an excerpt from a GdfidL command input file which shows the commands to control the calculation of a wakepotential: -fdtd -1char e # d e p r o p e r t i e s o f t h e e x c i t i n g line-charge. charge= le-12, sigma= SIGMA x o s i t i o n 1, y p o s i t i o n = 0 sEigh= 50; -ports # where a r e absorbing boundary conditions: plane= zlow, name= zlow, npml= 40, modes= 0, d o i t plane= zhigh, name= zhigh, npml= 40, modes= 0, d o i t 20 xhigh, name= xhigha, npml= 40, modes= 0, d o i t plane= yhigh, name= yhigha, npml= 40, modes= 0, d o i t

Ff%;=

- s t o r e f i eldsat what= e, name= a f i r s t s a v e d = 4 * Period / @c!i h t lastsaved= 5 * Period / @ c l i g & distance= 0.2 * Period / & l i g h t doi t -fdtd # S t a r t t h e time domain computation. doi t

Note in the above example the PML (perfectly matched layer) special boundary conditions which can account for absorbing boundary conditions.

176

4.5 CST Microwave Studio Microwave Studio is a program by CST (like MAFIA), specially designed for the calculation of S-parameters, antennas and radiation problems. It runs in a Windows environment. It has a graphical user interface and uses Microsoft's "Visual Basic" as a macro language; the following lines of code provide an example. '@ d e f i n e automesh s t a t e

Mesh.Automesh "True"

'@d e f i n e frequency range,, Solver. FrequencyRange

2800",

"3000"

'@ d e f i n e s o l v e r parameters

w i t h Solver .Gal c u l ationType "TD-5" . ~ t j m u l a t j o n ~ o "1" rt StimulationMode "?'' SteadyStateLi m i t -30" .MeshAdaption Fa1s~ .FrequencySampl es 1000'' Stimul ationType "Gaussian" End w i t h

. .

.

Compared to MAFIA, the approximation of boundaries that do not coincide with mesh planes is improved significantly in Microwave Studio by the so-called "Perfect Boundary Approximation". The actual version 4.3 of Microwave Studio does not allow wake-field calculations, since it does not provide for moving charges as field sources.

4.6 ANSYS Multiphysics ANSYS (Canonsburg, PA) offers a commercial suite of finite element programs intended primarily for mechanical structural analysis, including heat conduction and coolant flow. Part of this distribution is Multiphysics, which includes the possibility of simulating RF fields in the "high-frequency electromagnetics" module (htt~://www.ansys.com/ansvs/multiphysics.htm). The strengths of this program are the excellent mesher, which can create tetrahedral or hexahedral meshes, and the possibility of integrating heatinglcooling and mechanical deformation directly with the RF simulation. Only real eigenfrequencies can be calculated (lossless structures), the cavity Q is determined in the post-processor using the perturbation ansatz,. As in most of the other programs, there exists a macro language, which allows one to write user macros, e.g. to calculate in the post-processor the line integrals necessary to determine R/Q. Also ports can be defined for a driven

177

solution (in the frequency domain), but only single-mode, simple waveguide geometries, insufficient for the needs of RF structure design. Ansoft Multiphysics 5.7 does not handle periodic boundary conditions. Perfectly Matched Layer (PML) boundaries are possible. We regret the lack of an adaptive mesh refinement in version 5.7, which could help control the attained precision. This will be improved in version 7. 5

Comparison of RF field simulation programs

The simulation tools presented above each have their strenghts and weaknesses. However, it is interesting to run a case that can be handled by all of them, preferentially one where the exact solution is known. In the following, we will not compare the different functionalities of these programs, which have been mentioned above. The following comparison is limited to the precision obtained in the determination of the resonance frequency and the quality factor.

5.Z Whichprecision is required The ultimate goal of a simulation program is the design of an RF structure to a precision that directly allows its fabrication. In this case, it is also possible to predict how mechanical tolerances will eventually influence the performance of the accelerator structure. To this end, the cumulated numerical errors must be below the fabrication tolerances. A high cavity Q implies a sharp resonance, or a small bandwidth, which on the other hand immediately imposes a requirement on the precision with which the resonance frequency can be predicted. For a Q of lo4, e.g., the relative frequency error should be well below otherwise there is the risk that the gap voltage will be well below the design value for a given input current. In choosing the precision required in determining the resonance (or synchronous) frequency, another consideration is comparison with realistic fabrication tolerances. The error of the calculation should be at least in the same order as the error of the fabrication tolerances. It seems that for typical construction techniques, where a precision of 0.01 mm for typical sizes of 100 mm seems realistic, one equally obtains a requirement of As far as the required precision for the calculation of Q is concerned, it must be remembered that an error in Q of 10 % will demand 10 % more input power to obtain the nominal accelerating voltage. The example chosen for the following benchmark is a spherical resonator; the exact solution for its TMlloeigenfrequency is given by

178

where a is the radius of the sphere, c is the speed of light and

x = 2.7371... is a

d”,” A

zero of the spherical Bessel-function -J, ( x ) . The exact solution for the Q of this mode of the spherical cavity is

where CT is the conductivity of the metallic wall of the vacuum cavity. For a 50cm-radius copper sphere, CT = 58 MS/m , the “exact” numerical result is

fo = 261.823MHz, Q

= 89899.1

.

We summarize the results in Figure 3 indicating the deviation of both the eigenfrequency and the Q from the exact solution. We have included in this comparison some programs that were not mentioned above, notably URMEL (which is identical to MAFIA in r-z Co-ordinates), URMEL-T (using a triangular mesh) and SuperLANS. Furthermore, the number of mesh elements is indicated, which are mesh-points or tetrahedra. Trying to interpret these results, it becomes apparent that, with the exception of the 2-d programs Superfish and SuperLANS, hardly any program is able to compute the frequency to the desired precision of The typical obtained precision is approximately 5 while the MAFIA results with Cartesian mesh are only around 3 which is believed to be related to the limited approximation of the spherical geometry.

179

CST Microwave Studio, 29791 CST Microwave Studio, 9261 GdfidL, 10648 GdfidL, 39300 HFSS V.7,704 tetrahedra HFSS V.8,3388 tetrahedra MAFIA rz, mesh 100x100 MAFIA xyz, 10000 MAFIA xyz, 40000 ANSYS Multiphysics, 285 tet SLANS, 15x15 Superfish, 57x65 URMEL, 10000 URMEL-T. 10000 -10% -5%

0%

5%

10% 15% 20%

Figure 3: Results of the benchmark calculating resonance frequency and Q of a spherical cavity. Regarding the Q calculation, the general results are similar; note however that MAFIA overestimates the Q for this geometry. The errors in the percent range seem acceptable.

References 1.

S. Vaganian and H. Henke: “The Panofsky-Wenzel Theorem and General Relations for the Wake Potential,” Particle Accelerators 48, 239-242 (1995). 2. T. Weiland and R. Wanzenberg: “Wake Fields and Impedances,” CERN US PAS 1990:0039-7,DESY-M-91-06 (1991).

WAKEFIELDS AND INSTABILITIES IN LINACS*

G. V. STUPAKOV Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309

Electromagnetic interaction of a beam with surrounding environment is described in terms of wakes and impedances. After a general introduction of the concept of the wake, we focus on the case of the beam interaction with irises of accelerating structures in the linac. Using the linac short-range wake, we study the beam breakup instability and discuss methods of its suppression.

1. Introduction

When a beam passes through a linac its electromagnetic field interacts with the elements of the vacuum chamber and accelerating structures and the scattered field acts back on the beam. As a result, a beam instability can develop. Such an instability can lead t o the growth of the projected emittance of the beam decreasing the luminosity of the linear collider. In contrast to circular accelerators, where typically an instability develops during many revolutions, linac instabilities occur in a single pass of the system. An initial perturbation in such an instability does not grow exponentially with time, as in rings, but exhibits a more complex behavior which also depends on the position within the bunch. The goal of this paper is t o present a consistent description of the beam interaction with the surrounding environment in terms of wakes and impedances. After the introduction of the concept of the wake, we will focus on the case when the wake arises due t o the interaction with the irises of accelerating structures, because this is the dominant source of impedance in linacs. Using the linac wake for short bunches, we study the beam breakup instability (BBU) and discuss how it can be suppressed by introducing a correlated energy chirp in the bunch. *This work is supported by the U.S. Department of Energy under contract DE-ACOJ76SF00515.

180

181

We constrain our consideration to ultrarelativistic beams with the Lorentz factor y >> 1. Throughout the paper we systematically use the CGS system of units, indicating in some cases the relation between CGS and SI units used in the literature. This paper partially overlaps with our previous lecture at the Joint USCERN-JAPAN-RUSSIA Accelerator School 2000 [l],where the reader can find more details and additional material on wakes and impedances. For further reading, we recommend Refs. [2-5].

r;

2. Relativistic Point Charge Moving in Free Space Let us first recall the main features of the electromagnetic field of a point charge q moving with a constant velocity close to the speed of light in free space. As is well known, for large y, the field is squeezed in the longitudinal direction as shown in the figure. The electric and magnetic fields in the laboratory frame are:

E = - qR y2R:

1 H = -V x E , C

(1)

rh

where R is the vector drawn from the charge t o the observation point, RB = z2 r 2 / y 2 , y = (1 - W ~ / C ~ ) - ~z ’ is~ ,the distance from the charge in the direction of motion, and T is measured in the direction perpendicular to z . The radial component of the electric field, as follows from Eq. (l),is:

+

-

-

The field is localized in a think “pancake” region with an angular width ZIT y-l. In this region, z r / y , and the estimate for the electric and magnetic fields gives:

Problem 1: Decompose the field E T ( z , r )given by Eq. (2) in Fourier integral over z . Solution:

where K1 is the modified Bessel function of the second type, and F ( x ) = xK1 (x). The beam field is superposition of plane waves with the spectrum

182

given by the function F . The width of the spectrum is Ak of the function F is shown in Fig. 1.

N

y / r . The plot

1

0.75 LL

0.5 0.25 0 2

0

4

krly Figure 1. Plot of function F .

3. Thin Ultrarelativistic Bunch Moving in Free Space

Having found the fields of a single charge we can now calculate the transverse electric field of a filament bunch at distance r:

bunch)(^, r ) = N

s

E,.(z

- z’,~ ) X ( Z ’ ) ~ Z ’ ,

(3)

where X(z) is the linear distribution function in the bunch normalized so that J X(z)dz = 1. Problem 2: Do integration in Eq. (3) for a Gaussian bunch with rms length uz and analyze the result. The field of the bunch also diverges in the radial direction with the divergence angle of the order of y-’, as shown in the figure. If r << yoz, we can neglect the divergence and (dropping the superscript “bunch”) we have

II

One recognizes in Eq. (4) the electrostatic and magnetostatic fields of a charged wire with the constant charge density NqX and the current NqcX. Remember

183

however, that we are dealing here with a short beam moving with v whose linear density varies with z . If we use Eq. (1) for a point charge, then A(.) = 6 ( z ) ,and we have

c

In this approximation, the width of the “pancake” is zero, but the field inside it is infinitely large. One can use these equations if one does not care about the width of the “pancake” and is only interested in the integrated strength of the fields. In vector notation, with z = s - ct,

where r = 5i.z+ jjy is the two dimensional radius vector in the cylindrical coordinate system (5,6 and 2 are the unit vectors in the directions of x,y, and z respectively). The coordinate s measures the position of the charge in the lab frame. We see that in this approximation the beam field is a plane one perpendicular to the direction of motion. Problem 3: Estimate transverse and longitudinal fields inside a thin beam moving along a straight line with y >> 1, assuming that the transverse size of the beam n l is-much smaller than the bunch length nZ. 4. Interaction of Moving Charges in Free Space

Let us consider now a leading particle of charge q moving with velocity v, and a trailing particle of unit charge moving behind the leading one on a parallel path at a distance 1 with an offset x, as shown in Fig. 2. We want to find the force which the leading particle exerts on the trailing one. The

Z

1

1

Figure 2. A leading particle 1 and a trailing particle 2 traveling in free space with parallel velocities ‘u. Shown also is the coordinate system I,z .

184

longitudinal force is

Fl

= E, = -

91

yZ(12

+ x2/y2)3/2 '

and the transverse force is

In accelerator physics, the force F is often called the space charge force. The longitudinal force decreases with the growth of y as y-' (for 1 2 x/y). For the transverse force, if 1 >> x/y, Ft y-4, but for 1 = 0, Ft y-l. Hence, in the limit y + 00, the electromagnetic interaction in free space between two particles on parallel paths vanishes.

-

-

5. Particles Moving in a Perfectly Conducting Pipe

If particles from the above example move parallel to the axis in a perfectly conducting cylindrical pipe of arbitrary cross section, they induce image charges on the surface of the wall. The image charges travel with the same velocity u (see Fig. 3). Since both the particles and the image charges move on parallel paths, in the limit u = c , according t o the results of the previous Section, they do not interact with each other, no matter how close t o the wall the particles are.

Figure 3. Particles traveling inside a perfectly conducting pipe of arbitrary cross section. Shown are the image charges on the wall generated by the leading charge.

Interaction between the particles in the ultrarelativistic limit can occur if either the wall is not perfectly conducting, or the pipe is not cylindrical (which is usually due to the presence of RF cavities, flanges, bellows, beam position monitors, slots, etc., in the vacuum chamber).

185

6. Causality and the “Catch-Up” Distance If a beam particle moves along a straight line with the speed of light, the electromagnetic field of this particle scattered off the boundary discontinuities will not overtake it and, furthermore, will not affect the charges that travel ahead of it. The field can interact only with trailing charges in the beam moving behind the source charge. This constitutes the principle of causality in the theory of wakefields, according to which the interaction of a point charge moving with the speed of light propagates only downstream and never reaches the upstream part of the beam.

Figure 4. A wall discontinuity located at s = 0 scatters the electromagnetic field of an ultrarelativistic particle. When the particle moves to the location s, the scattered field arrives to the point s - 1.

Consider a small discontinuity located at the surface of the pipe of radius b at coordinate s = 0 which is passed by the leading particle at time t = 0. If the scattered field reaches point 1 behind s at time t , then ct = J(s - 1)2 b2, where s is the coordinate of the leading particle at time t , s = ct. Assuming that 1 << b and b << s, we have

+

s

=

J

m

M

S

from which it follows that

+(1 - -: ::),

b2

S M -

(5)

21

The distance s given by this equation is often called the catch-up distance. Only after the leading charge has travelled this distance away from the discontinuity, the trailing particle at distance 1 behind it can feel the wakefield generated by the discontinuity. Typically we are interested in the distance 1 of order of the bunch length 0,.Taking b = 1 cm and (T, = 100 microns we find for the catch-up distance l.2

s

M

M

2ffz

50cm.

186

In numerical calculation of the wake, one has to make sure that the simulation length is greater than the catch-up distance.

7. Wake The electromagnetic interaction of charged particles in accelerators with surrounding environment is usually a relatively small effect that can be treated as a perturbation. To describe such an interaction we will consider

Y Figure 5 . A leading particle 1 and a trailing particle 2 move parallel to each other in a vacuum chamber.

a leading particle 1 of charge q moving along the axis s, s = d (see Fig. 5). A trailing particle 2 of unit charge moves parallel to the leading one with the same velocity, at a distance 1, with the offset p relative to the zaxis. The vector p is a two-dimensional vector perpendicular to the z-axis, p = (z,y). The wake is defined differently for localized and distributed sources. 7.1. Localized source of wake

A localized source of the wake such as an isolated cavity, a flange, a bellow, etc., is illustrated by the figure below as an expansion of a vacuum pipe. Before and after the source the beam travels in cylindrical pipes, in general, of different radii. The field generated by the source interacts with the beam on a limited length, usually much shorter than the betatron wavelength. We can assume that during the interaction the beam moves in a straight line. We can calculate the change of the momentum A p of the second particle

2* kI

187

caused by the field generated by the first particle as follows: J--00

The integration here goes along a straight line-the unperturbed orbit of the second particle. The integration limits are extended from minus to plus infinity, assuming that the integral rapidly converges. Since the beam dynamics is different in the longitudinal and transverse directions, it is useful to separate the longitudinal momentum Ap, from the transverse component Ap, . With an appropriate sign and the normalization factor c / g , these two components are called the longitudinal and transverse wake functions (or simply wakes),

Note the minus sign in the definition of wl-it is introduced so that the positive longitudinal wake corresponds to the energy loss of the trailing particle (if both the leading and trailing particles have the same sign of charge). The defined wakes have dimension cm-' in CGS units and V/C in SI units.l Due to the causality principle, the wakefield never propagates in front of the leading charge, hence

7.2. Uniformly distributed wake In this case the source of the wake is uniformly (or periodically) distributed along the path of the beam. The examples of such sources are the resistive wall and a periodic accelerating structure in the linac. The wake in this case is defined per unit length of the path:

W t ( P , 1)

1

+

= - [El 2 x B]s=ct-l* 9

(7)

In this definition, the wakes acquire an additional dimension of inverse length, and have the dimension cm-2 in CGS and V/C/m in SI. 'A useful relation between the units is 1 V/pC = 1.11 ern-'.

188

In case of a periodic structure, the fields on the right-hand side of Eq. (7) are averaged over the structure period. 8. Panofsky-Wenzel Theorem

The longitudinal and transverse wakes are not independent of each otherthe relation between them is given by Panofsky-Wenzel (PW) theorem [6]:

+

where V p = x d fdx y d l d y is the two dimensional gradient with respect to coordinates x and y. From this theorem it follows that both wakes can be expressed in terms of a single function W(1,p ) :

dW W t = vpw. dl Indeed, if Eq. (9) holds, then Eq. (8) is also satisfied: WI

= -,

dW d Vpw1= V p x = $ p W

(9)

d

= -dlW t .

The opposite is also true, see, e.g., [l]. We introduced the fields E and B in the definition of the wake Eqs. (6) and (7) as the fields of the first particle in Fig. 5. However, in application of the P W theorem, we can also assume that those fields are arbitrary external fields, e.g., accelerating fields of the F P cavity. This makes the P W theorem applicable to a broader class of effects than just the theory of wakefields. Problem 4: Assume that E is an external electrostatic field, E = -V$(p, s ) , and H = 0. Prove P W theorem and find the function W . Answer: W ( p )= -4-l ds$(p, s). 9. Wake in Axisymmetric Systems In the definition of the wake illustrated with Fig. 5 we considered the wake relative to the orbit of the leading particle. In reality, the wake also depends on the location of the orbit of the leading particle. Using the vector p’ \ t o denote the offset of the leading particle relative to the axis of the system (see figure), we will have the wakes depending on this vector as well: W l ( P ’ , PlQ Wt(P’1 P , 11, W(P’,P, 1).

189

In an axisymmetric system, W depends only on the absolute values of p, p‘, and the angle 8 between them. We can always chose a coordinate system such that the vector p’ lies in the x-z plane (see figure), so that the potential function W will be a periodic even function of angle 6’ in the cylindrical coordinate system:

:4

m

w(p,P’, e,i) = C wm(p,pi, 1 ) cos me. m=O

It is possible to find a general form of the dependence of Wm versus p and p’ using Maxwell’s equations, which turns out to be proportional t o the m-th power of p and p’ (see [1,7]):

Using Eqs. (9) we now have

were

with .i. and 6 being the unit vectors in the radial and azimuthal directions, respectively. Remember that in this equation we assume that the leading particle is in the plane 0 = 0. Equations (10) are valid for arbitrary values of p and p’. Near the axis, where the offsets are small, the higher-order terms with large values of m in these equations also become small. In this case, we can keep only the lower-order terms with m = 0 (monopole) and m = 1 (dipole) wakes. For the monopole wake we find

which shows that the longitudinal wake does not depend on the radius in an axisymmetric system. We also see that the monopole transverse wake vanishes, wt(O) = 0.

190

a

For the dipole wake ( m = l ) , the vector i c o s 6 - i s i n 6 = x is directed icosq-; Sinq along the z axis (see figure), that is in the direction of p'. Hence, Wfl)

= p'F1(l).

(11)

From this equation it follows that the dipole wake does not depend on the offset of the trailing particle! [This is only true in axisymmetric systems.] The wake given by Eq. (11) is usually normalized by the absolute value of the offset p', and the scalar function wt(')/p'is called the transverse dipole wake wt,

4 1 ) = Fl(l)

*

The function wt has a dimension of V/C/m or cmP2. A positive transverse wake means a kick in the direction of the offset of the leading particle (if both particles have the same charge). 10. Examples of Wake 10.1. Cavity wake

In a cavity, the beam excites cavity modes and the longitudinal wake is a superposition of single-mode wakes:

where K is the loss factor and a! is the damping constant for the mode. The frequency ij is related to the mode frequency W O , ij = Instead of a one can use the quality factor Q = w0/2a. Note that the wake at the origin is equal to 2 ~ Eq. . (12) is illustrated by Fig. 6 . In the limit a! + 0, corresponding to very large quality factor, Q + 00, we have

d-.

w1 = 2 K C O S - . WOl

C

In practice, a quantity called R/Q is often used instead of the loss factor K . The relation between these two quantities is given by the following formula:

191

20

0

40

IGIC Figure 6.

Singe-mode wake for a = 0 . 0 5 ~ 0 .

A useful relation between the SI and CGS units in this equation is: 1 Ohm/s = 1 . 1 1 . 1 0 - ~cm-l. ~

Problem 5: Estimate the quality factor of the fundamental mode in a copper cavity with the frequency 15 GHz. (Hint: damping is due to the energy losses in the walls and is related to the skin depth). The mode transverse wake in the limit a + 0 is given by N

Wll , wt = 21ct sin C

where Kt is the kick factor and w1 is the frequency of the mode. Note that the transverse wake vanishes at the origin, wt(0) = 0. As with the longitudinal wake, the total transverse wake is the sum over all cavity modes that contribute to the transverse kick. 10.2. Resistive wall wake

This is the wake which is generated in a pipe of radius a with finite conductivity 0,in the limit when the skin depth is much smaller than the wall thickness: 1

E

2

E

The wake is defined per unit length of the path. The negative sign in w1 in Eq. (13) means acceleration of the trailing charge. The formula is valid for not very small values of 1:

192

0

4

2

6

0

1 IS0

2.5

5 I/%

7.5

10

Figure 7. Longitudinal a) and transverse b) resistive wall wake at short distances.

Problem 6: Calculate so for a copper pipe of a radius of 2 cm (ccU = 5.3. 1017s-l = 5.9. lo5 Ohm-'cm-'). The wake at very small distances, I SO,was calculated in Ref. [8]. The N

remarkable result of this work is that at short distances, after an appropriate normalization of both the distance and the wake, the result is represented by universal curves that do not depend on a and u. Those curves, for longitudinal and transverse wakes, are shown in Fig. 7. As follows from this figure, the longitudinal wake at the origin is

and does not depend on the conductivity! One can also prove a similar relation for the slope of the transverse wake at the origin [8]:

Problem 7: Find the energy loss per unit length of an ultrarelativistic charge q moving in pipe of radius a with the wall conductivity u. Answer: 2q2/a2. Estimate minimal y for which this answer is valid. 10.3. SLC RF cavity wakes

The SLC accelerating cavity is a cylindrically symmetric, disk loaded structure, with the period p = 3.5 cm, iris thickness of 0.6 cm, and the fundamental mode of wavelength XRF = 10.5 cm. The average iris radius is 1.16 cm and the outer cell radius is 4.13 cm. The wake for this structure was calculate by K. Bane, and is plotted in Fig. 8.

193 SLC LONGITUDINALWAKE

SLC TRANSVERSE WAKE

I

4

.

,

I

.

”

,:

200

. . L1 h

E

h

€ 3

150

E

0

,a 100 L 5 50

5 0

0 0

10

20

30

I (mm) Figure 8. Longitudinal and transverse wakes for the SLC RF structures.

11. Wakefield in a Bunch of Particles

If a beam consists of N particles with the distribution function X(z) (defined so that X(z)dz gives the probability to find a particle near the point z , J X ( z ) d z = l), a given particle will interact with all other particles of the beam. To find the change of the longitudinal momentum of the particle at point z we need to sum the wake from all other particles in the bunch,

ApZ(z)=

Ne2

dz’X(z’)wl(z’

- z) .

Here we use the causality principle and integrate only over the part of the bunch ahead of point z . In the ultrarelativistic limit, the energy change A E ( z ) caused by the wakefield is equal to CAP,, so Eq. (17) can also be rewritten as

Two important integral characteristics of the strength of the wake are the average value of the energy loss (per particle) and the rms spread in energy generated by the wake AE,,,. These two quantities are defined by the following equations:

1, 00

AEav =

dz~~(z>~(z),

and

The energy loss for the whole bunch is NAEav.

194

Problem 8: Calculate AE,, and AE,,, per unit length for the resistive wall wake given by Eq. (13) and a Gaussian distribution function X(z) = (&uZ)-’ exp ( - z 2 / ( 2 a : ) ) . Answer:

and RMS energy spread

LIE,,, = l.06‘AEa,. If the bunch passes through an R F cavity of length L with an offset y it will be deflected in the direction of the offset by the transverse wakefield. To calculate the deflection angle 0 we use the relation:

= yL-Nre

Y

I”

dZ’X(Z’)Wt(Z’ - z ) 1

where re = e2/mc2 is the classical electron radius. The averaged over distribution function deflection angle is

1, ca

eav= (8) =

dze(z)~(z),

and the rms spread is

12. Impedance The longitudinal 2, and transverse 2, impedances are defined m Fourier transforms of the wakes:

The integration can be extended into the region of negative values of 1, because wl and w, are equal t o zero in that region. Problem 9: Calculate the longitudinal impedance for the resistive wall wake. Answer: l-i

195

In SI units the quantity 2, = 4lr/c = 377 Ohm is often used as a characteristic unit of impedance. From the definitions in Eq. (19) it follows that the impedance satisfies the following symmetry conditions: ReZl(w) = ReZl(-w), ReZt(w) = -Re&(-w),

ImZl(w) = -ImZl(-w), Im&(w) = ImZt(-w).

Impedance can also be defined in the upper half-plane of the complex variable (J,Imw > 0. It is an analytic function there. The inverse Fourier transform relates the wakes to the impedances:

We have to note here that other authors often introduce definitions of wake and impedance that differ from the ones given in this paper. Our definitions agree with those of Chao [2],with the only difference that Chao uses z = -1 as an argument of w. It is always a good idea to check a particular definition, before using results from the literature. It turns out that the wake can actually be found if only the real part of the impedance is known. Indeed, we can rewrite Eq. (20) for wl as

For negative values of 1 this formula should give 'wl = 0, wl ReZl(w) cos 7- ImZl(w) sin C

This means that for positive 1

11

00

wl(1) =

1

00 wl dwReZl(w) cos - = dwReZz(w) cos G . -a3 c T o C

A similar derivation for the transverse wake gives wl dwReZt(w)sin - . C

(21)

196

Since the wake can be found from ReZ, it means that there is a relation between R e 2 and ImZ which can be traced as ReZ(w)

-+ w(l)

4

Z(w) --+ ImZ(w).

Those relations are called the Kramers-Kronig relations. Problem 10: Derive expression for ImZ(w) following approach outlined above. Answer: O0

ReZ(w)= lT

dw‘

-W

ImZ(w’) w’-w

*

Problem 11: Calculate the longitudinal impedance of a cavity mode corresponding to the wake given by Eq. (12). Answer: Zl(W)

=

R 1+L Q

where wo = d

(t

9-zo)

w ,R = K / a , Q = wo/2a.

For large Q, the impedance

is peaked around w = f w o .

13. Energy Loss and ReZl

We can relate the energy loss of the bunch to the real part of the longitudinal impedance. Indeed, using Eq. (18), we have

1, 03

AE =

dzNX(z)

1, 00

dz’Ne2X(z‘)wl(z‘ - z )

where i ( w ) = ST dzX(z)eiwz/c. OoA Since i ( - w ) = X*(w), lj(w)l2 is an even function of w and

197

where Q = Ne. For a point charge, X(z) = 6 ( z ) , i(w)= 1, and the energy loss is (22)

If we know the spectrum of the energy losses P(w), roo

AE=J,

dwP(~),

comparing it with Eq. (22) gives the relation between ReZl(w) and P(w): 7T

ReZl(w) = -P(w) . e2

(23)

14. Point Charge Passing through an Iris

Let us calculate the wake generated by passage of a point charge through a round hole of radius a in a perfectly conducting infinitely thin metal plate. The iris cuts off a part of electromagnetic field, T > a, that hits the metal.

Figure 9. An ultrarelativistic particle passes through a hole in a metal screen.

The duration of the field pulse on the edge of the iris is of the order of At a/cy (see Fig. 9b). First, we calculate the energy U of the electromagnetic field that is “clipped” by the iris. The fields of the ultrarelativistic charge are given by N

198

and the energy density w is

Integrating w over the region r

> a and over z yields

We expect that the radiated energy will be of the order of U ,and the spectrum of radiation will involve the frequencies up to X N a/Y (A = l/k). It turns out that this problem allows for an analytical solution in the limit

0

0.5

0.25

0.75

1

X

Figure 10. Plot of the function F ( z ) .

of high frequency [9], when the wavelength of the radiation is much shorter than the hole radius, k >> u - l . Here we present the result of this solution. The radiated power spectrum is:

P(w) = Z g F T

C

($) ,

(25)

where

F ( z ) = x 2 [KO(z) K2 (x)- KI( z ) ~, ] with K , - the modified Bessel functions of the second kind. The function F is plotted in Fig. 10; it has a logarithmic singularity at x = 0. From this

199

plot we see that, indeed, the typical wavelength in the radiation spectrum is of order of X air. To find the total radiated energy, we integrate P given by Eq. (25) over the frequency: N

3n- q2Y P ( w ) d w = --. 8 a Comparing this equation with Eq. (24), we see that the radiated energy is equal to twice the clipped energy. This can be explained by the fact that the clipped field is reflected back by the screen, and is radiated in the backward direction. The same amount of energy is radiated in the forward direction when the charge restores its vacuum field after the passage of the hole, doubling the total radiated energy. Ee can now find the real part of the longitudinal impedance using Eqs. (23) and (25):

and calculate the wake using Eq. (21),

where

The plot of the function G is shown in Fig. 11.

2 1.5 (3

1 0.5

0

5

10

15

5 Figure 11. Plot of function G(z).

20

200

15. Diffraction Model for Cavity and Analogy with Resistive Wall Impedance Let us try t o estimate the impedance of a single cavity of length g shown in Fig. 12. We will use the diffraction model which is applicable in the limit of high frequency, when the wavelength is much smaller than the typical size (height and length) of the cavity. This model tells that the losses are caused by diffraction of the electromagnetic field in the area close to the pipe wall, when the charge enters the expansion region of the cavity.

Figure 12. Diffraction model for cavity. The diffrwted part of the beam field is shown in blue-it has a radial extension of order of

a.

From Section 2 we know that the field on the wall has typical wavelength of the order of X air. When the beam travels through the cavity, its field, initially confined within a region r < a , begins to expand in the radial direction due t o the diffraction. Using the diffraction theory we can estimate the radial extent Ar of the area on the vertical right wall of the cavity which will be illuminated by the field when the beam field hits the exit wall, Ar The energy A E lost by the beam is estimated as the energy inside the diffracted volume: E~ H~ a AE x 27ra x Ar x - . N

N

a.

+

Since E

-

N

87r

H

N

q Y / a 2 , we get

AE--

Y

20 1

- -

Estimating the power spectrum P for the real part of the impedance: Re2

AElw and using Eq. (23) gives

- q2w

ac

A more accurate calculation [2] gives a numerical factor 2/n1l2 in this formula, and causality requires a matching imaginary part of the impedance:

d

2

ac

We can now draw an analogy of this impedance with the resistive wall case. For the resistive wall, the energy of the electromagnetic field is redirected and absorbed inside the wall. For the cavity, this energy is redirected and trapped in the expansion region. Comparing both impedances,

we can find the “effective conductivity” using which we can obtain the cavity impedance with the resistive wall formulas:

With this analogy, the cavity can be treated as a piece of pipe of length g with the wall conductivity o e ~A. more detailed treatment of the concept of effective impedance for periodic accelerating structures can be found in Ref. [lo]. Problem 12: Calculate oeff and SO defined in Eq. (14) for a structure with period g = 1 cm and iris inner radius a = 0.5 cm for the bunch length of 100 micron and compare with the copper conductivity. An interesting question related to the cavity impedance is what happens to the beam field when the beam passes the cavity and returns into the smooth pipe? After what distance its electromagnetic field will be restored to the original vacuum configuration? To answer these questions, we note that the distortion of the beam field is due to the “pollution” of the vacuum beam field by electromagnetic waves scattered by the walls of the cavity. In a smooth waveguide those waves travel with the group velocity vs < c and eventually lag behind the beam. A mode with the frequency w >> c/a has the group velocity vg c(1 - c2k?/2w2). In our case, w c/oz, and k l l / a . After the

-

-

-

202

time u,/(c - vg) those waves get separated from the bunch. This happens at the distance s = cu,/(c - vg), or a2 2uz Comparing this result with Eq. (5), we see that what we found is the catchup distance. This is also the formation length for the radiation-after this distance the radiation and the beam field separate. S M - .

16. Periodic Accelerating RF Structure

Consider now a periodic disk-loaded accelerating structure. We are interested in the short-range wake, at the distances 1 N u, much smaller than the iris radius a, 1 << a. For a quick estimation, we can use the model of the effective wall conductivity introduced in the previous section. As was discussed in Section 10.2, the longitudinal wake at the origin does not depend on the conductivity and is given by Eq. (15). For short bunches, this wake can often be approximated as a constant, wl = wl(O), although this approximation may be too crude for some applications. Let us compare this value with the calculated wake for the SLC structures shown in Fig. 8 to see how accurate the formula is. For the SLC structures, a = 1.16 cm and Eq. (15) gives,

4 wl(0)= - M 3 cm-' = 270 V/pC/m a2

,

which is close to the exact value wl(0)= 225 V/pC/m. The transverse wake is usually well approximated by the linear function, wt(z)= t(dwt/dl)ll,o, where the derivative of the transverse wake at the origin is given by Eq. (16). Again, we can compare the result of Eq. (16) with the exact calculations for the SLC structures: 8 M 2.2 cmw4 = 2 V/pC/m/mm2, and the exact value wl(0)= 2.5 V/pC/m/mm2. Another example is the NLC FW structures operating at the frequency 11.42 GHz, X = 2.63 cm. For those structures, the period p M 0.7 cm, iris thickness is about 0.1 cm, and the average iris radius is 0.5 cm. The calculated short-range wake is shown in Fig. 13. Problem 13: Calculate the longitudinal and the derivative of the transverse wake at 1 = 0 for the NLC R F structures and compare them with the plots in Fig. 13.

203

~~~

40

.'20 5 0 0

0.25

0.5

1

0.75

Figure 13. Longitudinal and transverse wake for the NLC-type accelerating structures.

Let us also estimate the formation length for the wake for these parameters assuming oz = 100 pm: s = a2/20z x 12.5 cm. This is much shorter than the typical structure length. 17. Effect of the Wake in Linac - Beam Break-up Instability

The longitudinal wake in linac causes the energy loss and energy spread in the beam. The effect of transverse wake can be more dramatic. If the beam is injected with an offset, the head of the bunch executes betatron oscillations, and the transverse wake of the head resonantly drives the tail. As a result, the amplitude of the betatron oscillations of the tail grows as the beam travels along the linac. This is called the beam breakup instability (BBU). The simplest model that demonstrates the BBU is a so called twoparticle model [2], which represents the beam as two point macroparticles. Here we will use a more rigorous approach which takes into account the continuous distribution in the beam. The model that we employ assumes smooth focusing in the linac which is characterized by the betatron wavenumber ko of the order of the inverse beta function, ko l/(P). The deflection x of a slice of the beam is a function of position within the bunch z as well as the coordinate s along the path of the beam (see figure). To simplify the model, in this Section we consider the beam of constant energy, that is we neglect the effect of the acceleration of the beam. We also neglect the energy spread in the beam. All particles in the beam oscillate with the same betatron frequency. N

204

The equation for this oscillation is

z

with re = e2/mc21and w = wt.The left hand side of this equation describes free betatron oscillation of a particle with the betatron wavelength 2n/kol and the right hand side takes into account the effect of the wake. The initial condition for this equation at the beginning of the linac s = 0 is given by the beam offset 2 0 :

Usually the wake in the linac relatively small, so we can assume

Let us seek the solution in the form z(s, z ) = a(s, z)eikos,

where a is a “slow” varying amplitude which scale of variation is much larger than the betatron wavelength. Due to the assumption of “slowness” we can neglect the second derivative d2a/ds2in comparison with kodalds:

Substituting this relation into Eq. (26) gives the following equation for the amplitude a:

There are two cases when this equation can be solved analytically. The first case assumes that the beam linear density is constant, X(z) = l / l b for - l b < z < 0, and also the wake w does not depend on z (we now know that this is not a good approximation for the linac short-range transverse wake). Differentiating Eq. (27) with respect t o z yields

where

205

Here r has dimension of m-’. tion a = xo at s = 0 is

Solution of this equation with initial condi-

where 10 is the modified Bessel function of zeroth order. The plot of this solution is shown in Fig. 14. The amplitude of the oscillations increases and also grows with the toward the tail of the bunch (larger values of 1.) distance s.

3

1

0

2

r

4

6

Figure 14. Plot of the solution Eq. (29)

Problem 14: Prove that the solution Eq. (29) satisfies Eq. 28. One can show that asymptotically, for large values of the product rsIzI, Eq. (29) approaches the following:

As pointed out in the introduction, this dependence is different from circular accelerator instabilities where typically an instability grows as 0: ers/c, with r being the growth rate. The second case that can be treated analytically is the case of a linear wake, w = w’z, and the same constant linear density A(.) = l / l b , This is the case that has first been studied by Chao, Richter and Yao, [ll].In this case, we define the parameter r as

206

The equation for the amplitude is

1 n

da

- = -ir

as

dz’(z’ - z)a(s,

z

As is shown in Ref. [ll],asymptotically, the amplitude a grows as

18. Effect of Acceleration of the Beam

We now take into account the fact that the beam is being accelerated in the linac, so that its energy and the y-factor increase with distance s:

where yi is the initial and yf is the final values of y, and L is the linac length. The equation for x ( s , z ) Eq. (26) written for constant y needs to be changed to accommodate varying y(s). This modification can be traced from the following relation for the derivative of the transverse momentum of the particle p,: d d dx dp, = mc--yvz M mc -7, ds ds ds ds which results in 1d

dx(s,z)

- - Y 7 Y ds

+ k i x ( s ,2) = Nre T d z ’ x ( s ,z’)w(z’ Y

,

(31)

z

where now y = y(s). In what follows we assume a “slow” acceleration, Y‘IY << ko. Let us first find the solution to Eq. (31) without the wake:

d2x

y’dx

+ -y + kox = 0 . as2 as 2

Again, we seek a solution in the form x ( s ,z ) = a ( s , .z)eikos,which gives da 2iko-

y‘. + -&a Y

The solution to this equation is:

=0.

-

207

It demonstrates the effect of the adiabatic damping: acceleration of the beam leads to decrease of the amplitude as y-lI2. Now we take the wake into account:

z

Introducing a new variable b, a(s,z ) = b(s,z ) / m , we find

This is the same equation Eq. ( 2 7 ) that we had before, except for the y being a function of s. The explicit dependence y(s) can be eliminated from the equation, if we define the new “distance” S by the following equation: 1

dS = ds-

Y(S)

which casts Eq. ( 3 2 ) into the following: 00

db _ dS

iNr ---e dz‘b(s, z’)w(z‘ - z ) X ( Z ’ ) . 2kO

J z

This is exactly Eq. ( 2 7 ) , apart from the y factor absorbed in the definition of S. For a linear acceleration, y(s) = T~ s(yf - y i ) / L , and

+

The last equality assumes that yf >> 71. We conclude that the beam acceleration can be taken easily into account if in the final answer we substitute

and scale the amplitude a damping.

4

ad%

to include the effect of the adiabatic

19. Numerical estimate for the NLC

Let us use the theory developed in the two previous section and estimate the growth of an initial perturbation for the parameters of the NLC linac. We will use Eq. (30), which can be written as

208

where

and we included the effect of adiabatic damping by changing a0 -+ a= ,. Using the nominal NLC parameters for the number of particles in the bunch Npart = 1.1 x lolo, the linac length L = 10 km, the bunch length lb = 1.5 x 150 micron, the initial and final beam energy Ei = 10 GeV, E f = 250 GeV, the betatron wavenumber ko = 1/30 m-l, and the slope for the wake w' = 100 V/pC/m/mm2, we find T = 6.9, and _a -- 20. XO

The projected emittance growth due to the betatron oscillations scales as a square of the betatron amplitude, and for our case the expected growth is of order of 400. This crude estimate agrees reasonably well with computer simulation carried out with the code LIAR [12] and shown in Fig. 15. The initial beam offset in the simulation was 5 microns, and the beam had no energy = 3 . lop8 m. spread. The beam vertical1 emittance was Vertical emittance

-~~~ 2 1wO

0

2000

4000

6000

s (m)

8wO

lwO0

5

800

Y

600

0

2w0

4000

6000

8OO0

10000

s (m)

Figure 15. Beam breakup instability in the NLC, for a beam with no energy spread. The left panel shows growth of the amplitude of the oscillations, and the right one shows the projected emittance of the beam.

20. BNS damping

An effective way to suppress the BBU was proposed by Balakin, Novokhatski and Smirnov in 1983 [13]. The idea is very simple: to break the resonant interaction between the head and the tail one needs to introduce a variation of the betatron frequency along the bunch. This can be

209

achieved by varying the energy of particles in the bunch. In the FODO lattice, the betatron phase advance per cell p depends on energy of the particle (6 = A E / E ) ,

dP = - 2 tan CL 2 d6 In the smooth focusing approximation, p is associated with the product lcellk, where lcell is the length of the FODO cell, hence

dk 4 --tan2 C L d6

Lell

2

Generating a linear energy chirp in the beam by accelerating it off-crest, b ( z ) = A z , causes k also to depend on z , Ic = ko + A k ( z ) . The equation for the BBU instability now is

d2x(s,2 ) dS2

N r e T d z ’ x ( s ,z’)w(z‘+ [Ico + A ~ ( L ) ] ~ z (=s ,Y Z)

.

Assuming again that

Nrew Ak, -<< ko , YkO we can neglect Ak2 to obtain

d2x(s,z ) 8.72

+ [k; + 2 k o A k ( z ) ] ~ ( ~=, Nre J d z ’ r ( s , z’)w(z’ Y 2)

z

21. Autophasing Condition Let us assume that the beam is offset at the entrance to the linac, with the initial conditions for the bunch:

Is it possible to arrange the function A k ( z ) in such a way that x does not depend on z at later time, z ( s , z ) = X ( s ) ? It is easy to see, that we can achieve this if Ak is given by the following equation:

AIc(.) = N r e Tdz’w(z’ 2kOY

z

.

210

Then the function X ( s ) satisfies the equation of free betatron oscillations:

with the solution X ( s ) = zocos(k~s).We see that the beam's motions is stable in this case. This regime is called authophasing, it requires a special energy profile within the bunch with characteristic A k ( z ) of order of

Nrew AkN-.

(33)

$0

For a Gaussian distribution function of the beam and a linear wake, the calculated dependence A k ( z ) is shown in Fig. 16. It corresponds t o a 2

1.5 Y

a 1 0.5

0 -2

-1

0

1

2

Zbz

Figure 16. The function A k ( z ) (in arbitrary units) for the autophasing regime.

stronger focusing a t the tail of the bunch and a weaker focusing in the head. Although the exact autophasing condition cannot be achieved, the solution outlined above gives us an estimate of the required chirp, Eq. (33), and shows the preferred direction of the chirp. 22. LIAR simulation of NLC beam dynamics

We present here results of computer simulations with a correlated energy spread in the NLC linac. This energy spread is generated by running the beam off crest. The RF phase is chosen in such a way that the beam get the energy spread of about 1%which is then maintained over the main part of the linac, and is taken out a t the end, see Fig. 17.

21 1

0

2500

5000 7500 s (m)

10000

Figure 17. Correlated energy spread in the NLC linac.

The result of the simulations is shown in Fig. 18. Comparing with Fig. 15, we see that the introduction of the energy chirp completely stabilizes the beam, with the resulting emittance growth several orders smaller than in the case without the energy chirp. 10

-x

0.06

5

o

E 0.04 i

-5 -10 0

2500

5000

s (m)

7500

10000

>

!

0.03O 0

. 2500 O 50005 5

7500 r

10000 n

(m)

Figure 18. Beam breakup instability in the NLC for a beam with the correlated energy spread shown in Fig. 17. The left panel shows the amplitude of the oscillations, and the right one-the projected emittance of the beam.

One should keep in mind, that the price for the introduction of the correlated energy spread is a slightly lower accelerating gradient, and, what is more important, severe tolerances on quadrupole misalignments because of chromatic effects in the lattice. References 1. G. V. Stupakov, in Accelerator School on Frontiers of Accelerator Technology: High Quality Beams, St. Petersburg - Moscow, Russia (2000), vol. 1, pp. 205 - 230. 2. A. W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley, New York, 1993).

212

3. B. W. Zotter and S. A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998). 4. A. W. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering (World Scientific, Singapore, 1999). 5. A. Mosnier, in C E R N Fifth Advanced Accelerator Physics Course, vol. 1, pp. 459-514. 6. W. K. H. Panofsky and W. Wenzel, Rev. Sci. Instr. 27,967 (1956). 7. K. L. Bane, P. B. Wilson, and T. Weiland, in M. Month, P. F. Dahl, and M. Dienes, eds., Proc. US Particle Accelerator School: Physics of Particle Accelerators, Upton, N. Y.,1983 (American Institute of Physics, New York, 1985), no. 127 in AIP Conference Proceedings, pp. 875-928. 8. K . L. F. Bane and M. Sands, The Short-Range Resistive Wall Wakefields, Tech. Rep. SLAC-PUB-95-7074, SLAC (December 1995). 9. G. Dome, E. Gianfelice, L. Palumbo, V. G. Vaccaro, and L. Verolino, Nuovo Cimento A 104,1241 (1991). 10. Stupakov G.V., in Proceedings of the 1995 Particle Accelerator Conference, Piscataway, NJ (1995), vol. 5, pp. 3303-5. 11. A. W. Chao, B. Richter, and C. Y. Yao, Nucl. Instr. Meth. 178,1 (1980). 12. R. Assmann, C. Adolphsen, K. Bane, P. Emma, T. Raubenheimer, R. Siemann, K. Thompson, and F. Zimmermann, LIAR: a computer program for the modeling and simulation of high performance linacs, Tech. Rep. SLACAP-103, SLAC (April 1997). 13. V. E. Balakin, A. V. Novokhatsky, and V. P. Smirnov, in F. T. Cole and R. Donaldson, eds., Proc. International Conference on High-Energy Accelerators, Batavia, 1983 (Fermi National Accelerator Lab., Batavia, IL, 1984), pp. 119-120.

BEAM MANIPULATIONAND DIAGNOSTIC TECHINQUES IN LINACS P. LOGATCHOV Budker INP, Novosibirsk, 630090 Russia E-mail:[email protected]

This paper covers two lectures presented at JAS 2002 and gives a brief overview of beam manipulation and diagnostic techniques that are going to be used in future linear colliders. It includes the basic ideas for Dispersion-Free Steering, Wake-Free Steering techniques, and beam-based alignment. Also it contains the descriptions of the most interesting (from the author’s point of view) non-destructive beam diagnostic techniques designed for linear colliders.

1. Introduction The significant progress in the development of all linear accelerator types in the last decades gave us a lot of interesting ideas, methods and techniques, which could be the subject of a book and cannot be compressed into two lectures. Simple enumeration of different linear accelerator types gives a sense of the subject’s scope: linacs for industrial applications, injectors for storage rings, induction linacs, ion linacs, linac-based free electron lasers, linear colliders. In order to overcome this difficulty the following approach is chosen. I would like to outline the general direction of development by practical examples of beam manipulation and diagnostic techniques used in the linear collider field. The general direction of development is to increase the beam intensity and quality for all the above-mentioned linac types. The levels of beam intensity, quality and stability needed for linear colliders are beyond comparison with those of other kinds of linear accelerators. Enormous average beam power density and high quality of individual bunches force us to develop non-destructive methods of beam diagnostics in order to measure the beam position and profile in future linear colliders. These measurements are very necessary not only for beam quality control, but also for operation of the automatic on-line beam-based accelerator alignment system. This system is absolutely necessary for the preservation of collider luminosity at the designed level. Some beam manipulation and diagnostic methods developed for linear colliders can be applied in high-intensity beam modes of some other linear accelerators. A major part of the practical experience in beam manipulation and diagnostic techniques

213

214

for linear colliders is concentrated in two laboratories working in close collaboration: SLAC (SLC, FFTB, NLCTF) and KEK (ATF). Thus the NLCJLC (X-band) design' (Fig.l) will be used as an example in further considerations.

Electron Injector

1

RF Systems o() 11.424 GHz (s) 2.856GHz (L) 1.428 GHz

ositron Main Linac Positron Injector

Figure 1. General scheme of the NLC.

215

2. What Is Necessary to Provide for Particle Physics Experiment at the First Stage of a Linear Collider? One needs just high quality electron and positron beams colliding at 500 GeV of c.m. energy with a luminosity of 2 . cm-2s-'. Collisions should be done with the reasonable energy spread and background level in the detector. In order to reach these two design goals, a very complicated 30-km-long accelerating and beam processing system was designed. Why does it look like this? The answer can be partly extracted from the following luminosity formula:

It is assumed here that electron and positron beams have the same structure, quality and intensity. Basic beam parameters at the Interaction Point (IP)of NLC are the following. The ratio of the horizontal to vertical beam size is 6,/ oy= 245 / 2.7 (nm) . The horizontal to vertical normalized emittance ratio is yi?, / yi?.y = 360/ 3.5 (10-8m.rad) . The horizontal to vertical betafunction ratio is

p, / a,

= 8 / 0.1(mm) . The longitudinal bunch size is

o,=O.llmm, and the maximum collision energy of two particles is E,,, =500GeV(80nJ). The number of

particles

in

one

bunch

is

N = 0.75. lo", and the average DC power in the electron or positron beam is = 6.84MW (190 bunches in the train, 120 trains per second). This luminosity formula also contains the luminosity enhancement factor H, = 1.4. It represents the change of effective luminosity due to modification of the incoming bunch size in the electromagnetic fields of the opposite bunch (the pinch effect in the IP)? This is essentially the classical effect associated with bending the particle trajectories.

2.1 Why the Beam Should Be Flat at the IP

This question is strongly connected with the limitation of the maximum value of the electromagnetic field inside the bunch at the IP. Together with bending the classical particle trajectories in the field of an opposite bunch (the disruption effect),' electrons and positrons have to radiate high-energy beamstrahlung photons in the opposite bunch field (the beamstrahlung e f f e ~ t ) .These ~ photons will further interact with the beam fields and with each other via QED and QCD processes, and form an additional background in the detector. All beamstrahlung effects can be described by the dimensionless beamstrahlung parameter Y . It is

216

a measure of the field strength in a particle rest frame, y( E

+ B ) , in units of the

Schwinger critical field B, = m2c3l(eh) = 4 . 4 . 1 0 1 3 G a u ~ ~ ,

The design value of Y for NLC is 0.1 1. It keeps the detector background and the particles' energy spread at the JP within the acceptable range. Thus, the maximum value of the electromagnetic field in the bunch has an upper limit. On the other hand, the maximum electric field for a round beam is inversely proportional to the beam radius, and the luminosity is inversely proportional to the square of the beam radius:

1 Lmnr2 is the longitudinal charge density 2 = eN, 1 Oz . Thus, increasing the 22 EL,=--, r

Here, 2 luminosity by squeezing the beam radius will definitely cause a problem due to the strong electromagnetic field in the bunch. A

Figure 2. Electric field configuration in thex-y plane for round and flat beams.

The situation changes for a flat beam. The maximum electric field is inversely proportional to the horizontal and longitudinal bunch sizes and does not depend on the vertical one. On the contrary, the luminosity is inversely proportional to the vertical and horizontal beam size. So one can keep the horizontal and longitudinal bunch sizes fixed, while decreasing the vertical

217

bunch size, Fig. 2. This will increase the luminosity while keeping the maximum bunch field constant,

This is the main reason for the flat-beam configuration in the interaction region of future linear colliders.

2.2. Why the Vertical Beta-Function at the IP Should Be Related to the Bunch Length If the vertical beta-function at the IP is significantly smaller than the longitudinal bunch size, the vertical bunch size at collision strongly changes along the bunch. This leads to serious reduction of the luminosity. This is the so-called hourglass effect. The influence of this effect on the bunch interaction process and the phenomenon of beam disruption (pinching) are presented in Fig. 3. Two bunches interact head-on, but are shown vertically displaced in order to clarify what is happening with each bunch. Different frames correspond to different moments of time, increasing from the top to the bottom. This is why in practical design the bunch length is usually very close to the value of the vertical beta-function at the interaction point. But for very high luminosity this approach comes in conflict with decreasing the vertical size of the flat beam by squeezing the vertical beta-function. In this situation one needs to decrease the bunch length as well. First of all it is not easy, next it leads to an increase in the maximum electric field of the bunch. In order to overcome this problem, the traveling focus regime was ~uggested.~ This utilizes two things (see Fig. 4): a rectangular charge distribution along the bunch (instead of a Gaussian one) and the linear dependence of particle energy on its longitudinal position in the bunch (correlated energy spread). In this case one can use the vertical betafunction at an IP 10 times smaller than the longitudinal bunch size, to gain about twice the luminosity, or utilize a 5 times larger vertical emittance with the previous luminosity.

218

Figure 3. Bunch collision process. The time increases from top to bottom (the normal focus configuration). The pinch and bunch disruption effects are clearly seen?

219

I.

p,

= 0.1.0,

4

E

II.

rn.

L7Lz Figure 4. Three conditions for the traveling final focus.

The results of the travelling focus regime simulation in terms of Fig. 3 are shown in Fig. 5. This regime helps to overcome the hourglass effect and improve the luminosity.

2.3. Why the Crossing Angle at the IP Is Necessaryfor a Linear Collider The main reason is too short a distance between neighboring bunches (50 cm) and therefore an unrealistic value of the separating electromagnetic field. Without a significant crossing angle (20 mrad) it is impossible to avoid incoming beam disruption, and background to the detector due to parasitic

220

I

I

Figure 5. Bunch collision process. The time increases from top to bottom (the traveling focus configuration).

Figure 6 . Bunch collision with a horizontal crossing angle (a), and the crab-crossing collision (b). Bunches are tilted in the horizontal direction by two special cavity sections.

221

collisions in the vicinity of the main interaction point. For example, the magnitude of an electrostatic field sufficient for proper separation is about 500 GeV/m. The crossing angle in the horizontal plane at the IP leads to a significant luminosity reduction (about 4 times). To overcome this problem it is necessary to give half of the crossing angle tilt to each bunch (a so-called “crab-crossing” technique).6 This tilt can be produced in two ways. The first utilizes a correlated transverse kick in the pillbox cavity with the TM110 mode. The second uses a non-zero horizontal dispersion at the IP and a correlated energy spread inside the bunch, i.e., the linear energy dependence of the particle energy upon its longitudinal position in the bunch, see Fig. 6. 2.4. Beam Jitter, Emittances and Tolerances Budget in the NLC Design

The huge scale of the installation is a particular feature of any future linear collider. At the same time, this collider should produce and accelerate beams with extremely small sizes and high power density. In order to provide the luminosity, one needs to bring these beams into collision with extremely high accuracy in spite of motion of the accelerator elements and ripples in the power supplies of the magnetic elements and the accelerating gradient. The following tables contain some data about the most critical parts of the collider responsible for beam quality. These tables give a sense of the problems for beam diagnostic and manipulation systems. The major contribution to the luminosity reduction due to transverse position jitter of colliding beams comes from the vertical direction, where the beam has a minimum size (see Table 1). The main linac and beam delivery system generate the dominating part of the vertical jitter. For flat beam operation, preservation of the vertical beam emittance is the most critical issue. Table 2 shows that the major contribution to vertical emittance dilution comes from the main linac and the beam delivery system. Table 3 shows the connection between the most important tolerances for the elements of the main linac and the resulting beam emittance dilution. “Gold Orbit” means the orbit of best accelerator performance. “Structure” means accelerating structure of the main linac. S-BPM is the Beam Position Monitor Table 1. NLC Jitter budget (500 GeV cms).

REGION

X-jittedu)

From Dumping Ring From Injector From Main Linac From Beam Delivery From Final Doublet Total at IP Luminosint Loss

0.1 0.1 0.1 0.1 0.1 0.22

1%

Y-jitter(a) Energy jitter 0.1 0.1 0.3 0.3 0.25 0.51 3%

0.1% 0.1% 0.2%

0.25% 2%

222

Table 2. NLC

based on a High-Order Modes accelerating structure. It is obvious that misalignment of the main linac quadrupoles is responsible for the greatest part of the vertical emittance dilution, and the tolerances for the final focus elements are even much more tight. Because of the large size of the main linac and beam delivery system, the ground motion in different parts of the linac tunnel continuously causes misalignments of focusing and accelerating elements. So the power spectrum of the ground motion is very important for the design of beam diagnostic, manipulation and alignment systems.

2.5 Ground Motion The ground motion sources can be sorted as follows: Motion with low frequency (less than 0.1 Hz).The major source is local atmospheric pressure fluctuation. Motion with a frequency of about 0.14 Hz. The source is the interaction of atmosphere and oceans (ocean waves). ‘Cultural noise’ usually occupies the frequency range between 0.5 and 100 Hz. Table 3. NLC Main Linac Tolerances

EFFECT

Tolerance

resulting Emittance Dilution

Gold Orbit (quad misalignments) Quad Strength Errors Structure Misalignments Structure Tilts Quadrupole Rotations S-BPM Resolution Structure Straightness (bow) Structure Straightness (random) Synchrotron radiation Total

2.0 pm 0.1% 30 pm 30 pad 200 pad 5 Pm 50 pm

25% (Y) 0.7% (x); OS%(y)

3.5 pm

7% (Y) 4% (Y) 4% (Y) 3% (Y) 1% (with feedback) (y) 1% (with feedback) (y) 3% (x) 3.7% (x); 45.5% (y)

223

Static tension discharges (earthquakes), random diffusive motion (the ATLlaw)’. The diffusive motion can be described by the following formula:

a : =A.T.L;

A=2~104+10-8(~2/(m~s)),

where T is the time interval between two measurements, L is the distance between two points where the measurements were performed, and A is a constant that depends on the particular site. Figure 7 represents the power spectrum of ground motion at different places. The RMS amplitude of the ground motion 0 at the particular frequency f depends on the power spectrum P( f ) as follows:

I

a

I

SLAG tunnel SLAC 2am motlel

1O-’O

10-13 f 0-’ PB02A43 5-2GU1

1oo

Frequency (Hz) Figure 7. Power spectrum of the ground motion at different sites.

I

224

E- 1

E-2

E-3

E-4 E-5 I 1E-2

I

I 1 1 1 1 1 1 1

1 E-1

I

I

I

I

IIIII

I

I

I I I1111

1E+O

I

I

I 1 1 1 1 1

1E+1

1E+2

Figure 8. The dependence of the ground motion RMS amplitude on the frequency.

It can be seen that for quiet sites the power spectrum is inversely proportional to the fourth power of frequency:

Another very important feature of ground motion is spatial correlation. The level of this correlation is quite high and is shown in Fig. 8 (measurements at SLAC site). The solid line presents the RMS amplitude of the single sensor motion, the dashed line shows the difference in motions between two sensors separated by 100 m. Thus the ground motion summary looks as follows. The power spectrum drops with frequency as 1 / f" It gives, in the absence of artificial noise, the Rh4S amplitude of ground motion of about 1 nm for frequencies above 10 Hz. The ground motion amplitude in the frequency range of 1-10 Hz can be at the level of 10 nm, and within 0.1-1.0 Hz it rises to about 1 pm. These motions should be carefully considered in the feedback systems design.

225

Fortunately the high level of the correlations greatly helps reduce the influence of ground motion within this frequency range. Slow diffusive ground motion with frequencies well below 0.1 Hz sets the range and accuracy for an automatic alignment system at the level of a few mm and a few pm respectively. ‘Cultural’ noise differs greatly from site to site (about two orders of magnitude) and should be the subject of detailed investigation and concern in both site selection and accelerator subsystem design.

3. Beam Manipulation Techniques for Future Linear Collider

3.1. One-to-one Steering or Zero BPMs Readout This is the first thing one needs to do in order to find the acceptable beam trajectory in a linear accelerator. Assume that the focusing is provided by an ordinary FODO structure with Beam Position Monitors placed in the close vicinity of each quadrupole center. Each quadrupole also contains a horizontal and vertical corrector. For this system the simplest one-to-one technique can be applied. First of all we should put the beam on the center of the first quad using zero readout of the fiist BPM. Then the beam should be directed to the center of the second quad by the appropriate horizontal and vertical correction setting in the corrector of the first quad, and so on (see Fig. 9).

Figure 9. Beam trajectories for one-to-one steering technique with perfectly aligned BPMs (ideal case) (a), and with BPM misalignments (reality) (b).

226

This technique is good if each BPM center coincides with the center of the corresponding quadrupole, but this is far from reality. Normally quads and corresponding BPMs are misaligned. So the trajectory offset in the quads generates the dispersion and causes transverse emittance dilution due to finite longitudinal energy spread. For example, 30-pm BPM misalignments with respect to the quadrupole center in the NLC Main Linac would cause the vertical emittance to double with use of the one-to-one technique.

3.2. Dispersion-Free Correction Technique The basic idea of this method* involves minimization of the dispersion value in BPM locations rather than the beam transverse coordinate, as in the previous technique. Eventually, it helps to mitigate the transverse emittance dilution. In order to understand how this method works, let us consider the simplest model of two particles. The equation of motion in one transverse direction for a particle in a high energy linear accelerator can be written as

where s is the longitudinal position of the particle in the accelerator, z is the longitudinal position of the particle in the bunch, x is the horizontal coordinate the misalignments of the quadrupoles and of the particle, and xq and

are

accelerating structures respectively.

w, is the transverse wakefield function, p

is the longitudinal charge density,

N

is the number of particles in the bunch,

By is the vertical component of the magnetic field, p o is the design momentum of the particle, reis the classical electron radius, e is the electron charge, and

c i s the speed of light. In this model the bunch consists of two particles. They have the same coordinate oZinside the bunch and different energies (see Fig. lo), thus the difference trajectory can be defined as follows: h

d

= X ( S , oz,0) - x(s, a,, 5).

(2)

227

We can find a first-order perturbative solution for the difference trajectory by subtracting the equation of motion, Eq. (l), where we substituted the solution X ( S , Oz,() , from the same equation of motion with the substituted solution X(S,

a,,0) . Thus the following equation defines the difference trajectory hx, :

--

d Y(’)ZLLrd

where

X ( S , 0,

1 d

YW ds

+K(s)hd

,o)

= ~ ( ~ s ) + K ( s ) x ~ ( s ) ) - ~ K ( s ) q;,O>, x(s~

is a non-perturbative solution of Eq. (l),

(3)

4 = 1.

Z

Figure 10. Scheme for the two-particles model.

The influence of wakefields is not taken into consideration, because in this model two particles have the same longitudinal coordinate in the bunch. The first-order perturbative solution for Ax can be found from Eq. (3) as follows:

Ids R,,

S

&d

=

[{ (G (s’> +

(s,

(s’)xy

(s’))

- { K(s’)x(s’,o,O)] 9

9

(4)

0

where R,, (s, s’) is the non-perturbed transport matrix element from s’ to s . Thus the basic principle of the Dispersion-Free Correction technique sounds like this: the non-perturbed trajectory x ( s , u z , O ) is varied so that the integral over any short region of the accelerator in A x d , Eq. (4),is small. In other words, at any local position s‘ one needs to minimize the value of C(S’)+ K(s’)(x,(s’) - X ( S ‘ , D , , ~ ) ) by changing the strength of correctors C(s’). To find appropriate sets of correctors, the predicted trajectory in the i -th BPM, x i , and the corresponding difference trajectory, A x i , should be expressed in terms of the beam-focusing lattice parameters:

xj =

& 0,) - bi + XOR, I (so N<

+

9 sj

7

Ng

OjR,,( s j , si)+ j=1

) + x p , , (so sj )

d i n j i, j=1

228

j=1

j=l

S(s) = m(s)/ E ( s ) ,

R(S)=R(E)-R(E+hE),

%(S)= f l ( E ) - % ( E + A E ) , where L and K are the j -th quadrupole length and strength, (each quad ends at s ), and

8, is the kick angle of the j

-th corrector. The values

bi and d j

are the BPM and quadrupole misalignments with respect to the linac centerline, N , and N , are the total number of correctors and quadrupoles in the lattice,

Ci ( t ) is the time-dependent i -th BPM readout error, the design energy and energy difference; x0 and

X;

E ( s ) and h E ( s ) are

are the initial coordinate and

angle at the entrance of the system. It is obvious from Eq. ( 6 ) that the difference trajectory does not depend on the BPM misalignments. But there are a lot of other unknowns besides ej in formulas for predicted values: BPM misalignment, BPM readout error, and quadrupole misalignments. This is why one cannot directly solve the set of linear equations, Eqs. (5) and (6).The beam trajectory and difference trajectory come from measurements. In order to measure the difference trajectory, it is more convenient, for a linear accelerator, to scale the lattice instead of changing the beam energy. In reality, the best result is provided by the approach where the trajectory and the difference trajectory are minimized simultaneously with a proper weight coefficient for each measurement. The trajectory minimization is absolutely necessary in order to avoid an additional error related to the BPM non-linearity at large beam displacement amplitudes. So one has to solve the following minimization problem for the proper corrector values:

[c NBpM

min

(mi + x i )2 2

+ Axi )2 + (hi

2 i=l o p e , + ~ B P M

2 ape,

1.

(7)

where rn means the measured value, x means the predicted value, OBPMis the estimated RMS value of the BPM misalignments, OPrecis the RMS reading-

229

to-reading jitter of the BPM measurements, and N , , is the total number of BPMs used in the process. For example, for the SLC linac a ,,, = 2opm,

a,,er= l m p m l so the weight of the first term in Eq. (7) is significantly less than the weight of the second one. The weight function sets a proper balance between trajectory and dispersion reduction. Figure 11 represents the RMS trajectory deviation from zero in the x-direction for each SLC lattice scaling.’ Scaling means the proportional changing of strength for all the quadrupoles in the lattice. It can be clearly seen that after Dispersion-Free Correction (DFC) the beam trajectory with its energy dependence becomes at least twice smaller in the x and y directions (see Figs. 11, 12). A large difference (more than two times) between the calculated and achieved trajectory is connected to a non-normal probability distribution of BPM readout errors. Few BPMs had a significantly higher jitter value.

3.3. Wake-Free Correction Technique The bunch in a linear accelerator has a finite length, so the transverse motions of the head and tail of the bunch can be different in the presence of wakefields. This difference can also be minimized by a proper choice of the beam trajectory. The Wake-Free Correction technique provides for this minimization.8 The function to be minimized can be constructed with the help of the two-particles model as in the previous case. The leading particle (bunch head) has the trajectory X ( S , 0, and the trailing particle (bunch tail) has x(s,-0,, 8).

,o)

The difference trajectory Ax, is expressed as follows:

-

Axw=x(s,~~,o)-x(s,-~,,s). Using the same procedure as in the previous section, one can find the equation for AX, :

It is assumed here that each of two macroparticles in the model contains half the total number of particles, ( N , / 2), and $ is the energy difference between head

230 700 r

I

600 500

-A-

c

5d

I

(After DFC) (Before DFC)

400

300

200 100

0

0,70 0,75 0,80 0,85 0,90 0,95 1,OO

K Figure 1 1. The RMS trajectory deviation from zero in the x-direction for 4 SLC lattice scaling coefficients, K= 0.7; 0.8; 0.9; 1.0.

---

(Calc.) (After DFC)

250

-

E,

200 150 -

v

OX

100 -

50

---=\

.

0

.

l

.

'

0,70 0,75 0,80 0,85 0,90 035

1,oO

K Figure 12. The RMS trajectory deviation from zero in the y-direction for 4 SLC lattice scaling coefficients, K= 0.7; 0.8; 0.9; 1.O.

231 and tail of the bunch; the distance between the head and tail is

2 0 , . The first-

order perturbative solution of equation Eq. (8) can be written as S

Axw = Ids Rlz(s,s')[6(G(s + K(s ')xq(s')) I)

(9)

0

W, (s 20,) I,

x(s I, a,, 0)

+Ner, w,(s', 2 4 ) X " (s')I. 2Y,(S')

Thus it is necessary to vary the strength of the correctors so as to minimize the following value:

-

S(G(s') + K ( s ')xq(s ')) -

+-

Nr, w,( s 2a2)xa( s ') . 2Y& 3 I,

proportional to the head particle trajectory. The part containing wakefield does not change its sign with changing of s', while the first NL 2yo($) WL (s',20z ) term ~ K ( s ' changes ) its sign in a standard FODO focusing structure. It is very difficult to change the sign of 6in a linear accelerator in accordance with the FODO focusing, so the term proportional to the head particle trajectory can not be canceled locally. It can be canceled integrally with the help of other terms in Eq. (lo), for some small part of the linac. What must be done in order to manifest the wakefield effects? Simple lattice scaling or energy change does not really help, because it varies the negative and positive amplitudes of

~ K ( s inside ' ) a FODO period simultaneously and proportionally, while the wakefield term keeps its sign unchanged. In order to see the wakefield influence clearly, one has to vary the strength only of focusing quadrupoles, without touching defocusing quads, or vice versa. Taking into account this correction, one can modify the expressions Eqs. (5) and (6) for the predicted trajectory and difference trajectory in the Wake-Free Correction technique: xi = 5;.

-bi + x&,

0 0 7 Si

N.

N"

j=1

j=1

1+ 44,(so, S i )

+ 2 BjR,,( s j ,s i ) + 2dj9Iji,

232

j=1

where Re'

j=l

-

and RQF are the differences of transport matrix elements

produced by the scaling of defocusing quads' strengthand focusing -quads'

%? and

strength respectively. The same notation is used for

%?

. The

trajectory of leading particlexi stays the same as in the previous case, but the difference trajectories have an additional term with a wakefield. Two types of difference trajectories correspond to the focusing or defocusing quads' scaling; K is the scaling coefficient. Here u j is the misalignment of the j -th accelerating element with respect to the linac centerline, and L'f."is the length of the j -th accelerating element. As a result, the Wake-Free Correction minimization problem looks like this:

(Axy + A m y ) 2 c + i N8pM

min

i=l

20irec

(Ax,? +Am,?)'

(xi+mi )' +

2a,Zrec

2 0pre-c

+0

P M

Here, as in Eq. (7), m means the measured value, and x the predicted one. Solving this problem for the values of corrector strengths, one can find the optimum trajectory in the presence of wakefields. Unfortunately, it is not easy to measure the value of and small NLC bunch.

Amy

(a banana-like tail deflection) for the very short

233

3.4. Beam-Based Alignment Technique This technique is based on the idea of using an electron beam for measurement of the global quadrupoles misalignment". The method includes two steps. The first is the shunting procedure -the measurement of the intrinsic misalignment of the quadrupole. Figure 13 presents the scheme of this measurement. A small variation of the gradient integrated along the quad AGN gives the difference

Ax of the beam position on the following BPM:

Ax = -R,2 AGN dx,, BP

Bp determines the beam energy, and dxin is the intrinsic misalignment of the quad. Measuring Ax one can determine the hi,value (see Fig. 13).

where

Figure 13. The scheme of the shunting procedure.

The second step is calculation of the quad's global misalignment. It can be done by using measurements of the intrinsic misalignments, in the following way:

Here L(k + j ) is the distance from the k -th to the j -th quad. Figure 14 illustrates this procedure.

I

I

Figure 14. The scheme of the global misalignment calculation.

234

The global alignment resolution will in general be monotonically worsened from upstream quads to downstream quads. This results in bowing of the beambased quadrupole alignment solution,

where oinis the accuracy of the intrinsic misalignment measurement, and

ffgI

is the accuracy of the global misalignment determination. In order to reduce this effect it is possible to constrain the beam position at the upstream and downstream ends of the region to be aligned (i.e., to keep the absolute BPM readings at the ends). Another approach constrains the beam position and the angle at the entrance of the system. Using the results of these kinds of measurements together with the high-precision accelerator elements’ movers, one can reduce the quadrupoles’ misalignments and the undesirable dispersion value. Figure 15 presents the scheme of the NLC beam-based alignment hardware. The beam-based alignment technique was successfully tested on FFTB at SLAC.” Figure 16 gives a schematic view of the FFTB magnet mover, which had horizontal, vertical, and roll degrees of freedom. It could move magnets by 0.3-pm steps over a range of 3 mm in each degree of freedom.

RF Structures. each with 2 BPMs (1 at each end) 5 ym Uy resolution

Remote-controlledGirder translation stage, ~y Degrees 01 Freedom at Each End Remote-controlled Magnet translation stage. xjy oegrees of Freedom. 50 nm Step Size .>LO1 ?”..,.,

...,A

3,

Figure 15. Beam-based alignment hardware of the NLC main linac.

235

T-Plale

Figure 16. Scheme of the RTB magnet mover.

4. Non-Destructive Beam Diagnostic Methods Designed for Future Linear Collider The key point of any beam quality control system is the beam profile monitor. It provides for the measurement of beam emittances and optical functions. According to the NLC specification, the beam profile monitor should measure 1 Fm RMS transverse beam size by a non-destructive method. For a linear accelerator it is also very important to measure the longitudinal beam profile and its correlation with the transverse one. The following methods can be considered as candidates for NLC beam profile monitors. 4.1. Laser-Compton Beam Spot Size Monitor

The idea of this monitor" is based on using laser interference fringes in the beam transverse size measurement. The electron beam scanning over the fringe pattern produces intensity modulation of the Compton-scattered high-energy gamma rays. The modulation depth of the gamma-ray flux provides the information about the beam spot size. Figure 17 presents the basic idea of the monitor operation. The laser beam is divided into two beams by a half mirror and transported in the x-y plane from nearly opposite sides to the same focus point. Thus an interference fringe is formed around the focus. The electron beam comes along z-axis and generates Compton high-energy gamma rays on the interference fringe. Then a bending

236

Figure 17. Scheme of the setup for beam spot size measurement,

magnet in the horizontal plane takes electrons away, and the Compton gamma-ray intensity is measured on a downstream detector. A weak steering magnet placed upstream provides slow scanning of the electron beam across the fringe pattern, and the resulting intensity modulation in the gamma-ray flux enables the electron beam spot size measurement. The beam radiation generated in the electromagnetic field of the interference fringe can be estimated as follows (see Fig. 18 for definitions of the notation in the following equations).

kz Figure 18. Definitions of coordinates and wave vectors. The vector of the electric field is perpendicular to the x-y plane.

237 Consider two wave vectors kl and k2 of two laser beams; the vector components are represented as [k, ,k, ,kZ] . For this particular polarization, the electron will be periodically accelerated in the transverse direction by the Lorentz force due to the magnetic field of the interference fringe. The components of the wave vector are expressed as follows (8 = 2 a ):

k, = k , c o s a , k, cos a k, sin a , 0

k, = f k , sin a ,

I

k, cos a k2 = -k, sin a ; 0

-

r sina.sink,y.

n(wt-k,x)

1

The value of the point charge acceleration in the transverse direction is

The number of Compton-scattered photons is given by time-averaging the square of the transverse acceleration:

N,

0~

0~

(2;)- ((B,"+B:))= 2 B ~ ( s i n 2 a ~ s i n 2 k y y + c o s 2 a ~ c o s 2 k y y )

Bi (1+ cos B - cos 2k, y )

For the Gaussian distribution of electron beam density in the vertical direction, the normalized number of photons can be expressed as

where 0,is the vertical Gaussian size of the electron beam, yo is the vertical position of the electron beam, and k, is the wave number of the laser light. Finally, the modulation depth of the gamma-ray flux is

2n sin( 8 I 2 )

il,

238 Here

Lo is the wavelength of the laser light. In actual practice, a few correction

factors should be taken into account. First of all, the laser beam profile is not uniform, I ( r ) = 1, exp(-r2 / 2 4 ) . Also the power in different laser beams is not the same,

Fortunately, the dependence of the fringe contrast on the power imbalance is weak. The third factor is the change in the electron beam transverse size along the laser beam size in the z direction:

Here 0;and

B,'

are the vertical beam size and beta-function in the beam

crossover. With all these factors taken into account, the modulation depth becomes

In the real experiment at FFTB the vertical beta-function in the crossover was 100 pm, and the longitudinal electron bunch size OLwas 50 pm. Thus the correction value was about 6 %. For the particular FFTB experiment, the Nd:YAG laser with 1064 nm wavelength (hv=11.17eV)was used. The laser pulse duration was 10 ns with 100 mJ per pulse. The power density at the focus point was 1.3.10" W / m2.The photon density in one of the traveling waves of the laser beam was n = 2.3.10'9photons/cm2. The average number of gammaquanta generated by Compton scattering from one laser beam is given by

N , = N,ocnd, where d is the diameter of the laser beam,

N , is the number of electrons in the

bunch, and

the Compton scattering:

OC is the total cross section of

239

Here

Oo

is the Thomson scattering cross section, &, is the dimensionless

photon energy in the electron rest frame, and V , is the photon frequency in the laboratory system. For a 50-GeV electron beam in the FFTB, E, = 0.22, O, = 0.720,,,

N , = lo'', N , = 1000.

Optical components are mounted on the 1.6 x 1.5-m optical table, 110 mm thick, with a honeycomb structure, which is mounted on the final Q-magnet table. The laser system is placed in a clean room outside the accelerator tunnel, and the laser beam is transported through a vacuum pipe for 20 m. The position of the laser spot is stabilized at the entrance of the interferometer by a feedback system. In order to provide background subtraction, a synchronous detection technique is used. When the electron beam runs at 30 Hz, the laser is fired at 10 Hz. Thus, the laser-ON gamma-ray data are averaged for six pulses, and the background noise (laser-OFF) gamma-ray data are averaged for twelve pulses. Then the background noise data are subtracted from the laser-ON data. For zaxis alignment, scanning of the laser beam across a proper slit is used. The alignment in the x-y plane utilizes an electron beam finding technique (using the maximum gamma-ray signal at the detector.). The calculated spot size response curves for the three measurement modes in Fig. 19 are presented in Fig. 20.

Figure 19. The beam spot size monitor on the FFTB beam line (left) and three modes of operation for generating three different fringes (right).

240

..

I

5 P

8

Spotsize: d (rim) Figure 20. Calculated spot size response curves for three measurement modes.

The spot size measurement starts after the laser beam alignment. When the electron beam trajectory is scanned in fine steps, one can detect the periodic modulation of the detected gamma-ray intensity, as shown in Fig. 21 (a). The solid curve is a least-mean-square fit of the analytical model function: Y =A+Bsin

Unknown parameters A, B , c should be determined by fitting this function to the experimental data. Figure 21 (b) represents the spot size distributions obtained in 3-hour measurements. Bunch length measurement is also possible with this method. The basic idea utilizes mixing of two laser beams with different frequencies. Thus an intensity modulation with a frequency equal to the frequency difference between the two laser beams (beat frequency) is formed (see Fig. 22). Then one can inject short electron bunches into the laser beam from the normal direction and measure the Compton-scattered gamma rays downstream. The number of photons

24 1

O I I

'

I I I I I I 0 0.4 -0.8 -0.4 Eledron Beam Vertical Position (pm)

Figure 21. (a) Measured gamma-ray data. The solid curve is the least-squares-fit of the sinefunction. @) Spot size distribution for a 3-hour measurement.

generated by a single bunch depends on its phase, so the distribution of the probability of obtaining a particular number of photons from a single bunch depends on the bunch length (see Fig. 23). For the linear collider application, a shorter laser wavelength can be used (at the 5" harmonic of Nd:YAG, 213 nm). Therefore, a spot size of about 10 nm can be measured with enough accuracy by this method. But it is hard to install this monitor in the interaction point of a linear collider. Also this technique is very sensitive to transverse jitter of the electron beam and requires good stability of the transverse position and size within the bunch train.

242

Figure 22. Schematic diagram of the bunch-length measurement system based on the laserheterodyne technique.

Y

w

Figure 23. Pulse-to-pulse fluctuations in the gamma-ray flux.

t

Ream

This method is based on using low-pressure helium gas as a substance interacting with the high-energy NLC bunch.I2 The experimental test of this technique was done at m B (SLAC).13 A pulsed jet of helium gas was injected into the beam pipe, near the focal point of the FlTB line, timed with the passage of each electron bunch. High-energy electrons create He' ions by ionization of the gas atoms. The space-charge electric field of the bunch kicks these ions. The ion flux produced in this way is collimated and then detected by the microshannel plate assembly (see Fig. 24). The charges generated by the micro:hamel plate are collected by sets of strip-lines parallel to the beam direction.

244

For a flat and short ultra-relativistic bunch the following estimation of electric field and ion energy is valid:

Using the bunch parameters from the FFTB a,= 1.8 pm, ay = 0.1 pm, 9 = 650 pm, and N, = 7.109, one has E A =~ 10’ V/cm, E, = 1OkeV. The normal value of the residual gas pressure in the m B line was 5.10-9 torr. Because of the regular gas injection the residual gas pressure increased locally up to lo-’ torr; this was acceptable for the FFTB. The maximum peak pressure value in the monitor was torr. Figure 26 shows the calculated time-of-flight spectrum of He’ ions for two horizontal beam sizes 0,= l p n ( o ) , 0,=3pm(o). Other beam parameters remain the same: oy= 0.2p1, uZ= 650pn, N, = 7.lo9. The time-offlight is computed for a detector located 6.5 cm from the beam. Because of strong azimuthal asymmetry of the electric field for a flat bunch, the azimuthal distribution of He’ ions is also asymmetric. This effect helps to reconstruct the transverse bunch profile. Figure 27 presents the azimuthal distribution of He’ ions for two vertical beam sizes: oy= lpm(a), 0.”= 0.06pm(b).

The azimuthal distribution of ions is sensitive to the electron distribution along the bunch. All ions produced by the head of the bunch are kicked in the direction of the displaced tail. Thus two peaks in the azimuthal distribution of ion density have different amplitudes. Figure 28 illustrates this situation.

u)

w C

600 500

10 u.=lm

f

W

$4

u.=3pm

2 L

300

E z 200 100

+%

0

400 :

**++4 ++ct8$;,+b

L

L 5

448

00

0 4

;y+ 00

245

5 2 +

350 300 250

2

$

=

I

200

2

200

150

f

150

=

100

100

50

50

0 F I l

0

I *

a

100

'

I

a

n

200

I

L

300

0

Azimuth (degree)

100

200

300

Azimuth (degree)

Figure 27. Simulated azimuthal distribution of He' ions for two vertical beam sizes: Oy= 1 pl(a), Oy= 0.06 pm (b) The origin of the azimuth is in the upward direction

.

(cry=0.2pm,

a,=650pm, N , =7.109 ).

Figure 29 shows the experimenta1 time-of-flight spectrum of He+ ions for 400 pulses with a lower cut at 50 ns. The time-of-flight origin corresponds to passage of the electron bunch. The few counts before the threshold correspond to residual I T ions. Signals from strip-lines are taken in coincidence with the beam passage, and any signal arriving in a 50-ns interval from the beam passage can be considered as background and rejected. The result of He' ion azimuth distribution measurement is shown in Fig. 30.

C

250

L

L

2

2

150

2

150

E

E

2

100 50

100 50

0

100

200

300

Azimuth (degree)

0

100

200

300

Azimuth (degree)

Figure 28. Simulated azimuthal distribution of He+ ions with a tail at 90" for two vertical beam sizes: oY= lpm(a), oY= 0.06pm(b). The origin of the azimuth is in the upward direction ( ay= 0.2p-1, a,= 650pm, N , = 7 .lo9 ).

246

300 250 : 200 7 150

:

100

:

4

50

Time of flight (ns) Figure 29. Experimental time-of-flight spectrum of He' ions for 400 pulses, with a lower cut at 50 ns. The time-of-flight origin corresponds to passage of the electron bunch.

400

8

88

9

Azimuth (dsgrem

Figure 30. Azimuthal distribution of He' ions (with time-of-flight less than 500 ns) for 400 pulses. The azimuth origin is in the upward direction.

247 m F .+

.m +

5 120 5 100

120

5 100

v-

r

0 L

80

LI

60

W

5 z

0 L

W

s

40

Z

20

0

80 60

40 20

0

100 200 300 Azimuth (degree)

0

0

100

200 300 Azimuth (degree)

Figure 3 I . Azimuthal distribution of He' ions with a horizontal tail at 90"(a) and at 270"(b).

After fitting to the data described above the following result was obtained: o,= 1.5+-0.2pm, oy= 73+-10(stat)+-lO(syst)nm. The banana tail observation is presented in Fig. 3 1. In principle, this technique is able to provide single-bunch measurements when a fast gating device is added (pulsed micro-channel plate operation). But unfortunately it cannot be used in the interaction region because of additional background to the detector. 4.3. Electron Beam Probe

The electron beam probe can be considered as the third method of advanced diagnostics for the NLC beam. This technique is based on using a low-energy electron beam as an instrument for measuring relativistic bunch fields. The thin probe beam moves along the x-axis, which is orthogonal to the direction of the relativistic bunch motion (z-axis), with an offset parameter p (see Fig. 32). The results of scanning are monitored on a screen, which is parallel to the yz plane and positioned at a distance L from the z-axis. Let the center of the relativistic bunch be located at the origin at time t = 0, whereas the testing beam has a uniform density along x and a diameter d = p . Here we assume that p exceeds the typical transverse size of the relativistic bunch. At time r = 0 every probe-beam particle corresponds to a certain x -coordinate. Then the total deflecting angle in the y direction for every particle under the influence of the electric field of the relativistic bunch can be expressed as

248

Figure 32. The basic idea of the electron beam probe operation.

where

r, is the classical electron radius, fl = vt/cis the relative velocity of the

probe beam, c is the velocity of light, x is

the coordinate of a probe-beam

particle at t = 0, and n(z) is the relativistic bunch linear density along the zaxis. The expression for the deflecting angle of the particle in the z-direction caused by the magnetic field can be written in a similar way:

As a result, the testing beam traces the closed curve on the screen (see Fig. 33). Under the assumption of a stable probe beam current , one can derive

a simple relation between the x -coordinate and the charge distribution q(l) along the indicated curve on the screen from point A to point B (see Fig. 32):

Integrating the charge along the curve from point A up to point B (see Fig. 32) one can find the x -coordinate from Eq. (13) and relate it to the certain angles 8,(x)and q x ) at point B. Since the dependencies %(x) and are

q(x)

determined, it is possible, using any of these functions, to reconstruct the dependence

n(z):

249

Figure 33. The probe beam image on the screen.

By(x) Fik”dx.

where By( k ) = a

It is necessary to emphasize that the relations Eqs. (1 l), (12), and (14) are valid only for an ultra-relativistic bunch with y>>l and for 8y <<1. The last condition implies a small perturbation of the probe beam longitudinal motion by the electric field of the relativistic bunch. In other words, the transverse kick of rshould be small if we use the assumption that the velocity of probe beam 6 the probe beam in the x-direction, PC, is constant during the interaction. Otherwise the integral equation like Eq. (11) is not so easy to solve. This is the case of NLC, where the bunch field is too strong even for a 200-keV electron MAX . probe beam, and 8, is about one radian. In this situation the relevant nonlinear integral equation should be solved numerically. But there is another restriction for this procedure of longitudinal charge distribution measurement in the bunch:

where Ox,Oy,Oz are the relativistic bunch sizes.

250

This condition can be extracted from Eq. (14). Practically it means that for large values of p one needs extremely high precision in the 0, (x) measurement in order to extract the information about n ( z ) . If the condition in Eq. (15) is fulfilled, one can find the solution of integral equation Eq.(1 1) by solving the set of equivalent linear equations A * n = B :

n(zj) =

Here matrix A is determined as follows:

4.=-2P r, k j p p2+ (Xi+ pzjy Figure 34 shows the solution n ( z )found for the image in Fig. 33. This is a single bunch regime of the VEPP-5 injector linac (S-band).

Figure 34. Probe beam image on the screen (negative) and the result of its processing. The solid curve represents the solution of a linear system with 7x7 matrix sizes (the error is about 20%). The dotted curve results from fitting a Gaussian curve by a special procedure.

The bunch length is only twice as long as the bunch transverse size, so the parameter of Eq. (15) is about unity. Thus the accuracy of the solution n ( z ) is poor even for quite good image quality. Contrary to this, for the experiment with electron beam probe at the VEPP-3 storage ring14 we had worse image quality but better accuracy for the longitudinal charge density measurement (see Fig. 35). This is due to a large value in Eq. (15) for the VEPP-3 bunch (about 20). Finally, the parameter of Eq. (15) defines the suitable bunch shape for this kind of diagnostics. The bunch length should be at least a few times as large as the maximum transverse bunch size. This is absolutely true for the NLC bunch. But the transverse size of the NLC bunch is very small, so the probe beam diameter should also be very small (about 10 pm) in the interaction region. Additionally, the NLC bunch intensity corresponds to a very strong kick for the probe beam electrons, thus we do not need a long free path for magnification of the transverse size of the image. These two factors allow the use of a lens with a short focusing distance, and therefore a small probe beam size in the interaction region can be achieved.

1.4x10”

a

252

We come to the next important question. Is it possible to provide enough current in a high quality probe beam? A high current is necessary to keep the large number of electrons in the close vicinity of the NLC bunch. A reasonable value of the pulsed current density in a 200-kV electron gun is about 20 A/cm*. With a collimating diaphragm of 0.1 mm diameter we can have about 2 mA of probe beam current. Finally, this gives for a 0.1-mm NLC bunch length (0.5 ps at v, = 0 . 7 ~ 6000 ) electrons in the close vicinity of the NLC bunch. These electrons will form the image (MCP operates in the single electron regime). The results of simulation are presented in the following Figs. 36-38. Parameters of the probe beam are the same for all the figures: the probe beam energy is 200 keV, the pulsed probe beam current is 2 mA, the probe beam transverse diameter at the gun exit is 0.1 mm, and the probe beam diameter in the interaction region is about 0.05 mm. Each dot on the screen placed at 2 cm past the interaction point corresponds to a single electron (about 6000 electrons in the vicinity of NLC bunch). The NLC bunch parameters are the following: the longitudinal bunch sigma is 0.1 mm, the vertical bunch sigma is 0.001 mm, the horizontal bunch sigma is about 0.02 mm, and the number of particles in the bunch is 7.10'. Figure 36 shows the probe beam envelope from the gun exit to the screen (horizontal axis in cm), the interaction point is situated 2 cm before the screen. The vertical axis gives the RMS transverse probe beam size in cm (the probe beam is round). One mode of the electron beam probe operation allows the measurement of the bunch tilting, or banana-like bunch shape. This feature was successfully tested at the S-band linac of the VEPPJ injector complex for a bunch length of 4 mm and a 0.5-mm transverse size. The simulation presented here was

"."* 0.045

.

0.04

0.035 0.03

0' 0

10

20

30

40

50

60

Figure 36. The probe electron beam envelope for NLC. The variation of the RMS radius (cm) upon the longitudinal coordinate (cm) from the gun exit at 0 up to the interaction point at 50 cm.

253 performed for typical NLC bunch parameters in the beam delivery system after the main linac: 0.1 mm bunch length, 0.Olmm bunch transverse size, and 7.109 electrons per bunch. The probe beam had an energy of 200 keV and other parameters the same as described above. In order to measure the bunch tilting in the y-direction (head-up and tail-down), the image on the screen was divided into four parts: 1-(x > 0, y > 0), 2-(x < 0, y > 0), 3-(x < 0, y 0, y < 0). Figure 37 presents four pictures with different tilting amplitudes; the increasing asymmetry is clearly seen. N1 is the number of electrons in part 1 of the screen, N2 - in part 2 and so on. Finally two values A1 = (N4-N1)/(N4+NI)and A2 = (N2-N3)/(N2+N3) were calculated. The dependencies of these values on the tilting amplitude Sy (pm) are presented in Fig. 38. The conclusion can be that: the NLC single-bunch tilting amplitude (or banana tail displacement) can be measured with this simple four image region method, starting from 1 pm. Remember that direct measurement of the particular bunch tilting amplitude allows the possibility of applying the Wake-Free Correction technique described above. Also the electron beam probe can be used as a BPM.”

Figure 37. The screen images for increasing amplitude of the bunch tilting. From the left to the right from top to bottom the tilting amplitude is lpm, 4 pm, Spm, 16 pm. The tilting amplitude of 1 pm means that the hunch head (at +I sigma longitudinal) is displaced 1 pm up and the tail (at -I sigma longitudinal) is displaced 1 pm down.

Another interesting feature of this method is that the picture on the screen will be different for different bunch shapes, and an experienced operator after some playing with the simulating code can understand what happens with a particular bunch in the linac. Figure 39 presents examples of images for a singlebunch regime in the S-band linac of the VEPP-5 at Budker INP.An additional horizontal scanning of the probe beam (as in an ordinary oscilloscope) was used in order to separate the loops from the main and residual neighbor bunches.

-

-"

Collision with intense

255

Multibuncb regime, offset parameter goes down

Figure 40. Variation of the offset parameter in single-bunch and multibunch regimes.

Figure 40 shows the same but for a multibunch regime. At least a difference in vertical positions of neighbor bunches within the train can be clearly seen. An electron beam probe is also sensitive to wakefields propagating in the vacuum chamber of the linac.16 The evidence is presented in Fig. 41. So this diagnostic technique can be used in various modes without any change in the device hardware and supply useful information on the beam characteristics in a non-destructive manner.

I

Two b u n c h y Collision with single bunch Wakes start to appear Figure 41. Observation of wakefields after the bunch passage.

256

References 1. 2001 Report on the Next Linear Collider, FERMILAB-Conf-O1/075-E, LBNL-PUB-47935, SLAC-R-571, UCRL-ID-144077. 2. P. Chen and K. Yokoya, Phys. Rev. D38,987 (1988). 3. R.J. Noble, Nucl. Instrum. Methods A256,427 (1987); V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Nucl. Phys. B328, 387 (1989). 4. V. E. Balakin, “Traveling Focus Regime for Linear Collider VLEPP,” in Proc. Workshop on Beam-Beam and Beam-Radiation Interaction, Los Angeles (1991). 5. A.A. Sery, “The VLEPP Final Focus News,” in Proc. 4” Int. Workshop on Next-Generation Linear Collider, Garmish (1992). 6. R. Palmer, Snowmass DPF Summer Study 1998, p. 613 (1998). 7. A.Sery, 0. Napoly, Phys. Rev. E53 (1996); B. Baklakov et al., Sov. Phys. ZhTF 63, 10 (1993) (in Russian); V. Shiltsev, “Space-Time Ground Diffusion: The ATL Law for Accelerators,” in Proc. 4” Int. Workshop on Accelerator Alignment, November 14-17, 1995, KEK, Tsukuba, Japan, p. 352; V. Shiltsev, in Proc. EPAC’96. 8. T. 0.Raubenhamer, “A New Technique of Correcting Emittance Dilution in Linear Colliders,” Nucl. Instrum. Methods A306, 61 (1991). 9. R. Assmann, T. Chen, et al., in Proc. 4” Int. Workshop on Accelerator Alignment, November 14-17, 1995, KEK, Tsukuba, Japan, p. 463. 10. P. Tenenbaum, D. Burke, et al., ibid., p. 393. 11. T. Shintake, “Beam Profile Monitors for Very Small Transverse and Longitudinal Dimensions Using Laser Interferometer and Heterodyne Techniques,” in Proc. 7“ Workshop on Beam Instrumentation, Argonne, IL, 1996, AIP Con$ Proc. 390. 12. J. Buon et al., Nucl. Instrum. Methods A306,93-111 (1991). 13. P. Puzo, J. Buon et al., “A Submicronic Beam Size Monitor for the Final Focus Test Beam,” in Proc. 7” Workshop on Beam Instrumentation, Argonne, IL,1996, AIP Con$ Proc. 390. 14. P.V. Logatchov et al., “Non-Destructive Diagnostic Tool for Monitoring of Longitudinal Charge Distribution in a Single Ultra-Relativistic Electron Bunch,” in Proc. 1999 PAC, New York, 29 March -2 April (1999). 15. A.A. Starostenko, et al., “Non-Destructive Single-Pass Bunch Length Monitor: Experiments at VEPP-5 Pre-Injector Electron Linac,” in Proc. EPAC-2000, Viena, 30 June - 4 July 2000. 16. P.V. Logatchov, et al., “Non-Destructive Single-Pass Monitor of Longitudinal Charge Distribution,” in Proc. HEACC-2001, Tsukuba, Japan, March 2001.

SPACE CHARGE AND BEAM HALOS IN PROTON LINACS

FRANK GERIGK CLRC, Rutherford Appleton Laboratory, UK E-mail: frank.gerigkOrl.ac.uk

1. Introduction

In the last two decades a number of future projects like high power neutron spallation sources, neutrino factories, waste transmutation facilities, and even military applications like the production of tritium have triggered studies for linac-based high power proton drivers. While the applications and output energies (1 - 10 MW) are quite different, all designs are based on a loss limit of 1 W/m, a number that, according to the experience with the LANSCE machine at Los Alamos, ensures “hands-on-maintenance,” meaning that a few hours after shutting down the machine the radiation level is low enough, so that technical staff can access the linac tunnel. For a 10 MW machine at its high energy end we thus have to control beam loss to the order of corresponding to one particle in a simulation with 10 million particles (!), a requirement that exceeds the capabilities of most classical linac simulation tools. LANSCE achieved this limit for a 16.5-mA 1-MW proton beam at a final energy of 0.8 GeV. In contrast to LANSCE most of the new linacs accelerate H- particles to benefit from charge-exchange injection into subsequent accelerator chains or accumulator and compressor rings, posing additional constraints on beam quality. Generally H- beams suffer from more diffuse beam distributions at the source and from stripping losses in strong magnetic fields, or caused by residual gas in the pipe. Since H- sources (up to now) deliver only low currents, some projects plan to funnel beams from up to four sources. Furthermore the resulting beam current is often considerably higher, yielding much higher space-charge forces than in the case of LANSCE. Prompted by these requirements the understanding of how beam halo develops and how this development is influenced by space-charge forces was boosted to a new level, especially during the last decade. We will explain

257

258

some of the mechanisms in this lecture, which are well understood and which will provide some guidance in the design of high intensity linacs.

1.1. The Resonance Model and Underlying Theory

This approach is based on the idea that linac beams, as well as beams in circular machines, are governed by resonances. While in a synchrotron one mainly considers beam-lattice resonances, we will add two more resonance types to describe a linac beam: resonances between single particles and the oscillations of the r.m.s. core plus coherent resonances of the core with itself. Before looking into the details of each resonance type it may be useful to quickly review the required physics and mathematics that are needed to treat each of them. Particle-lattice resonances (Figure 1)were first studied by Gluckstern & Smith [l]who analyzed the matrix solution of the transverse single-particle equation (Hill's equation) for a FODO channel: d2

--a ds2

'

-

+ (k; - k&) a,

=0

k2

d2 --a ds2

k,2,) a, = 0 '+ (4-;

transverse equations of motion in a FODO channel

(1)

k;

with s being the longitudinal coordinate, k:,y representing the combined transverse focussing / defocussing forces of the quadrupoles ( k ~ and ) the R F cavities (klo), and us,,the transverse beam radii.a As a result of their stability analysis they found that particle-lattice resonances occur for a transverse phase advance per of period of 180". As the analysis suggests this resonance is a real instability affecting all particles of the (zero current) beam. In order to study beam-lattice effects (see Figure 1) we need a description of the beam envelope including space-charge forces. For this purpose Reiser [2] analyzed the 2D envelope equations (derivation in [3]) in the smooth approximation:

aAll relations are defined in Appendix B.

259

Figure 1. Beam lattice & particle-lattice resonance

2

K2 2 + ktoax - -=o a: a, + a y

d2 -a, ds2

EX

smooth 2D envelope equations

~

-ad2 ds2

+k2 a y

E~

K2

a;

a, + a y

-J--

to

=o

(2)

These equations include the whole ensemble of particles yielding the “repulsive pressure” of the emittance term ( E , , ~ - transverse r.m.s. emittances) as well as the space-charge forces that are included in the space-charge factor K2. The envelope equations in 2D (and 3D) can be solved under the assumption of constant emittances and elliptical bunch shapes. Reiser’s analysis reduced the stability limit to a maximum phase advance of 90” per period instead of the 180” from the single-particle analysis. To analyze resonances between a 3D mismatched core and single particles (see Figure 2) we need to employ the 3D envelope equations with space-charge using the smooth approximation:

d2ax 2 E: Kt = 0 + ktOax --ds2

a:

ayb

smooth 3D

d 2 a+, ktoay 2 -2 E~ Kt - -= o

envelope equations

ds2 a; a,b d2b 2 E; Ki -+ kiob - - - -= 0 ds2 b3 a,ay

(3)

where b denotes the longitudinal beam size and KtIl include the 3D spacecharge forces. Eventually, to treat the core-core resonances (Figure 3) one has to drop the concept of constant emittance and strictly elliptic shape of the bunches in phase space. For this purpose one can use Vlasov’s equation

af

a

a

a

a

-+-x*-f+-y.-f+-p,.-f+-p at at a x at a y

a

a

at

ap,

a at

a

.-f=O apy

(4)

260

Figure 2.

Particle-core resonance

which describes a general particle distribution f in phase space. This equation has not yet been solved in 3D and so the analysis is restricted to two planes. Nevertheless we will see that a 2D analysis is sufficient to predict and understand the effects of core-core resonances.

Figure 3.

Core-core resonance

Since the conclusion of the analysis of particle-lattice or beam-lattice resonances is the same for all linac designs: to keep the zero current phase advance below 90" in each plane, we restrict our following explanations to the analysis of the particle-core and core-core resonances. 2. Particle

- Core Resonances

2.1. Smooth 3 0 Envelope Equations with Space Charge

We will start with an intuitive derivation of the smooth version of Sacherer's envelope equations [3], which provide the basis for determining the bunched beam-eigenmodes. We consider three main forces acting on the beam, which are either external (quadrupoles and RF gaps), or internal (space-charge force) caused by the charges of the bunch particles. For this purpose one assumes a uniform particle distribution in the first place, and can then, as Sacherer showed [3],generalize the equations for various distributions without major changes.

26 1

We start with the envelope equations of a uniform beam without external forces and without space-charge, assuming equal emittances in both transverse planes:

--+o d2a,

E~

ds2

a,

d2a, ds2

E:

a:

-0

d2b

E~

ds2

b3

I

=o

(5)

The negative emittance term acts as a repulsive force, and without the application of some kind of focussing the particle trajectories would quickly diverge. Transverse focussing can be provided by introducing solenoids or quadrupoles into the system, which exert s-dependent focussing forces in x and y: d2a, -+ k;(s)a,

-

ds2

E2

-$-

d2ay

E2

ds2

a:

-+ i i ( s ) a y-

=0

a,

-0

-

(6)

For the sake of simplicity we use the same focussing forces in x and y (i,(s) = i y ( s ) = ksol(s)),which corresponds to a periodic solenoid focussing system and axis-symmetric beams. For quadrupole focussing the signs have to be opposite in x and y, compare Eq. 1. Longitudinal focussing is obtained with RF cavities, which not only keep the beam bunched and provide acceleration but also add another defocussing force in the transverse plane, so that we end up with

kt20(4

d2ay ds2 + (kf,,(s)

-

m) 2

Et"

ay - - = 0

P a: kt20 ( s )

envelope equations without space-charge

d2b E2 =0 k12,(s)b ds2 b3 where the kto,lo (s) summarize the s-dependent external transverse / longitudinal focussing forces for a zero current beam. In addition to the external forces we also have to consider the repellent forces between the charged particles, which add to the defocussing forces of the RF gaps and the emittance terms. Including the space-charge terms we can write the envelope equations in the following form (see [4]): -

+

262

d2a, 2 E~ Kt(s) + kto(s)a, - 2- -= 0 ds2

a; a,b d2a, 2 E~ K t (s) -+ kto(s)ay- 2 - =O ds2 a; a,b

envelope equations with space-charge

d2b e2 Ki(s) k$(s)b - J- - -= 0 ds2 b3 azay K1,t contains the 3D space-charge forces and depends on the aspect ratio of the bunch (and therefore also on the beam distribution and the longitudinal coordinate s). -

+

2.1.1. Smooth approxzmation In the smooth approximation one uses the length of the focussing period L p to average over the rapidly changing s-dependent external forces and refers to atollo as the transverse / longitudinal zero current phase advance per period:

(2) 2

k,2, = k f O ( S ) =

smooth approximation

(9)

From the smooth envelope equations one can easily derive some simple formulas for the average r.m.s. beam sizes (smooth matched beam radii). For the zero current case Kt,l vanishes and the solution is found by setting the second derivations to zero, so that one obtains as0,yo

=

bo =

;/ E= =

Et L P

smooth matched zero current r.m.s. beam radii

(10)

The full current condition is obtained by the same method, using the spacecharge-shifted wave numbers k t , k l , which denote the depressed or full current phase advance per unit length:

263

smooth approximation

(11)

With some simplification one can then write the smoothed matched fullcurrent beam radii as

smooth matched full-current r.m.s. beam radii

(12)

2.2. Envelope Modes of Mismatched Bunched Beams

We now have all the necessary tools to start the eigenmode analysis of mismatched bunched beams. For this purpose we use the approach of Bongardt & Pabst (presented in [4]and [5]) and derive the mismatch modes for a periodic solenoid channel. We will show later on that the derived relations can also be used to excite mismatch in a quadrupole channel. Using Eq. (9) we rewrite the general envelope equations:

and note that by definition the matched solution has the same periodicity as the focussing systemb:

matched solution The mismatched beam will oscillate around the matched solution with a period length which is different from the length L p of the focussing lattice. The beam oscillates in all three planes and therefore we expect to find three bThis periodic matched solution is not to be confused with the average smooth beam size of the matched case.

264

different oscillation modes (eigenmodes) with three different oscillation frequencies (eigenfrequencies or envelope tunes of the eigenmodes). Since the envelope equations are coupled we also expect coupled solutions that involve oscillations in all three planes. In order to determine the eigenmodes we add a small perturbation (mismatch) to the matched solution

ax,&) = ao,z,y(s)+ Aax(S) b(s) = bo(s) Ab(s)

mismatched solution

+

(15)

and insert the perturbed solution in the envelope equations, Eq. (13). To linearize the equation system we use Taylor expansions around the points Aax = 0, Aa, = 0, Ab = 0 &:

(a0

+ Aa)3 &?

+

(a

+

(bo Ab)3 Kt Aa)(b Ab)

+

M

_&: _ _3E:. a:

x

Aa

- - - . 3Eq Ab &:

b:

M

at

Kt.

bt

(x- --

aibo

Ab)

(18)

aobi

where all nonlinear terms are neglected and also the fact that Kt and Kl are dependent on the beam radii and therefore periodic. This step is a strong simplification which is done to find a short and handy solution for the mismatch modes. Nevertheless, we will see later on that the results are precise enough to excite the modes. After this modification we obtain a system of three coupled linear differential equations of Hill's type.

'2:)

d2 -

=ij

ds2

Ab

7

M (19) We then use the smooth approximation to relate the s-dependent expressions in M to phase advances per period. Furthermore we assume the matched envelopes in the transverse planes to be equal (a,o = ayo +ao), which corresponds to the initial assumption of a periodic solenoid channel.

265

With Eqs. (10) and (12) we rewrite the previous equation system as

Ab ffl"o

+ 3al"

- ffl" ffl"o

- ffl"

Ab

Using the ansatz

we obtain a homogeneous system of linear equations for the envelope tunes of the eigenmodes:

(ffenU)

ffZ0 + 3 4 - ff,2,v ffZ0

- fft"

ffZ0

ffZ0

- fft"

fft"o

- fft"

+ 3 4 - ffznu ffZ0 - fft" ffl"o - ffl" ffl"o + 3 4 - ff,2,u

(22) The three nontrivial solutions for oenuare found by setting the determinant to zero. They are usually referred to as follows: quadrupolar mode: uenu,Q

= 2 ' ut

with the eigensolution:

-Ab= o

b high-frequency mode or breathing mode or fast mode:

(24)

266

with the eigensolution:

low-frequency mode or slow mode:

with the eigensolution: S

a

Ab _ b gL

A,

S

9L

<0

The form factors QH,g~ are defined as

The nomenclature of the modes becomes clear when considering the characteristics of each mode: - The quadrupolar mode consists of envelope oscillations of the trans-

verse beam radii around their matched equilibrium with 180” phase difference between the planes. The longitudinal plane is unaffected.

- The fast mode “breathes” in all three planes with the same phase but with different amplitudes in the transverse and longitudinal planes. From the eigenvalues one can see that its oscillation frequency is always higher than that of the slow and quadrupolar mode.

267

- The slow mode also has different amplitudes in the transverse and longitudinal planes. The longitudinal and transverse radii oscillate with a phase difference of 180". Since the three eigenmodes were derived with some strong simplifications, the solutions are valid only up to a certain limit. For high spacecharge forces or rapidly changing particle velocities, the derived excitation becomes less precise. Its quality can be judged by the smoothness of the amplitude ratio of mismatched over matched beam radii. An improvement can be achieved by measuring the envelope tunes in the simulation output and recalculating the mismatch excitation with these tunes. An even better excitation can be obtained by numerical computation of the envelope tunes. 2.2.1. Excitation of mismatch eigenmodes Usingthe eigensolutions [Eqs. (24), (26), (28)] and the form factors [Eq. (29)] we can now excite the eigenmodes with the following mismatches:

quadrupolar mode: fast mode:

slow mode:

% = -2 Aaa a,,-%a a a,,-%a a

Ab - 0

T Ab

Q H ' T

(30)

Ab

QL'T

Example: We want to excite the quadrupolar mode with 20% radial mismatch. The matched input parameters of our linac are ( l ~ , o , ( ~ ~ o , ( ~ , o , P ~ o , The ~ ~ ~ goal , ~ ~ is o . to modify a and P such that we excite the mode amplitude maximum (1.2 ao) at s = 0. Generally the eigensolutions are of the form S

d

(32)

ds

with A, being the amplitude factor of the mismatch mode (0.2 in our case). To obtain the maximum mode amplitude at s = 0 we set q5 = 0 and obtain

a(0) = ao(0) . [lf A,]

-a(O) d = -ao(O) d . [l f A,] ds

ds

(34)

268

Obviously the radius and the momentum have to be changed by the same factor, and since ff

p = - a2

aa'

= --

&

(35)

&

and 0 also have to be changed by the same factor, which in our example means:

a!

+ Am)2aOz Pmz = (1+ Am)2Boz

ffmz

= (1

amy= (1 - A m ) 2 0 0 y

(36)

= (1 - AmI2P0y

(37)

Any

= 002

(38) Figure 4 shows an example for the excitation of the fast mode in a transport channel (quadrupole focussing, no acceleration) using a 6D waterbag distribution. As we can see, the beam oscillates in all three planes around the ffmz

E

a

= a02

Pmz

1

1-----

'

~

I

0

-

1

Period

yy 2 4 P

E

a2 0

'I! !I'

'I!!!!;- -----i -111 I'

ri!!j!l-

'111

1'

'11111'

'1111

'1!!!\;

-~

~

I

I I

l

equilibrium radii. The oscillations are in phase and have the same amplitude in xand y, but a different amplitude in the longitudinal plane. In this 'In this example only one point per focussing period is plotted. In the y direction this point is always at the focussing quadrupole, where the beam size is maximum, and in

269

example the oscillations carry on remarkably unperturbed. The most likely explanation is the combination of a perfect transport line with unchanging focussing elements, and a uniform particle density in six-dimensional phase space, a mixture that provides very little "noise" to trigger the formation of halo. 2.3. Parametric Resonance Model & Particle

Redistribution As an introduction to this section we consider a more realistic lattice than for the previous example in Figure 4. We use the superconducting linac section (120 - 2200 MeV, simulation current: 40 mA) of the CERN SPL study [6] and excite the three eigenmodes. Furthermore we use an inhomogeneous Gaussian beam density profile in 6D phase space. The resulting oscillations of the beam radii are plotted in Figures 5 to 7.

1.4 1.3 1.2 1.1

""-".-".--1+db/b ..............

] +cj:\?i./;ly

1

0.9 0.8 0.7 0

Figure 5.

200

100

300 400 length [m]

500

600

30 % quadrupolar mismatch in the SC section of the SPL

First of all one can observe that the phase and amplitude characteristics of all modes correspond to the theoretical prediction given above. Although the tunes, and therefore also the oscillation frequencies, of the mismatch modes change considerably along the linac, the oscillations remain remarkably stable. However, in contrast to the example in Figure 4, the oscillations ~

the x-plane this point is at the defocussing quadrupole, where the beam size is at its minimum. dHere, we use the quotient of mismatched beam radii over matched radii t o visualize the oscillations.

270

”. , 0

100

200

300 length [m]

400

600

500

Figure 6. 30 % fast mode mismatch in the SC section of the SPL

”.

I

0

Figure 7.

100

200

300 400 length [m]

500

600

30 % slow mode mismatch in the SC section of the SPL

are damped, while the beam passes through the linac. During this process the “free energy” that has been introduced into the system via mismatch is transformed into beam halo and emittance growth. At the end of the transformation the r.m.s. core settles again on a matched trajectory and eventually the oscillations stop. The major difference between the example in Figure 4 and those in Figures 5 to 7 is the transformation speed or the rise time for the emittance growth to take place. In the first example the 6D waterbag and the ideal transport lattice result in a very long rise time, while in the second example the Gaussian distribution combined with an LLimperfect”realistic lattice provide enough “noise” to quickly transform the mismatch oscillations into emittance growth.

27 1

2.3.1. Parametric resonances & the particle core model The mechanism that expels particles from the r.m.s. core to larger radii can be explained with the parametric resonance model and the particle-core model: one assumes an r.m.s. core that oscillates around its equilibrium size because of mismatch. A single particle whose orbit crosses the core and whose oscillation frequency has a parametric ratio (usually 1:2) with the core oscillations, obtains a net kick by the space-charge forces of the core, as depicted in Figure 8.

Figure 8. Particle-core model: oscillation frequencies of single particles get into a parametric (1:2) ratio with the oscillations of a mismatched r.m.s. core.

This kick raises the single-particle energy and reduces the oscillation energy of the r.m.s. core. A detailed treatment that also quantifies the forces can be found in Reference [7]. The strong tune depression in high current linacs, which is usually in the range of 0.5 < a/ao < 0.8, provides a large spread of single-particle tunes (T < ap < (TO which are prone to parametric resonances. Therefore these machines are especially sensitive to halo development caused by mismatch.

2.3.2. Particle redistribution

A good understanding of the parametric resonances can be achieved by studying the redistribution of particles in a mismatched bunch. Since the three beam eigenmodes have three different frequencies, we can expect that each eigenmode expels different bunch particles with different frequencies (tunes). In order to achieve a clear redistribution pattern we use a transport channel with 50 focussing periods (Gaussian input distribution) and excite separately the three beam eigenmodes (Figure 9). We start with the quadrupolar mode, which oscillates only in the transverse plane. Its eigenfrequency is by definition [see Eq. (23)] twice as high as that of the core particles. We therefore expect a strong resonance with

272

Figure 9. Particle redistribution when exciting the three bunched-beam eigenmodes in a transport channel with 50 periods. Plotted is the difference in transverse particle density between the mismatched and the matched distribution at the end of the channel.

the inner core particles which have the lowest tunes in the bunch. For the low and fast modes the situation is more complicated since they both oscillate in all three planes. Nevertheless, it seems that we obtain a consistent picture: the eigenfrequency of the slow mode is lower than for the quadrupolar mode. Therefore the coupling with the core particles can only be very weak and consequently only few particles are removed from the core. Finally the fast mode seems to affect particles with higher tunes. The intuitive conclusion is that high frequency core oscillations yield a large fixed point - core distance and thereby large halo radii, whereas particles with lower tunes (from the core center) are affected by the low frequency core oscillations and end up around fixed points with a smaller fixed point - core distance. However, this pattern and the order of the fixed points should not be generalized since it is not only influenced by the mode frequencies but also by beam-lattice resonances and inherent redistribution patterns of the initial distribution (Waterbag, Gauss, etc.). The only conclusion we can draw at this stage is that each eigenmode will generally trigger different redistribution patterns with different fixed point - core distances. For small mismatch amplitudes a general mismatch excitation yields a superposition of modes as shown in Figure 10. As an example we use a "+ +" and "+ - +" mismatch which stands for 30% radial mismatch in x,y, and z or 30% in x and z, and - 30 % in y, respectively.

+

+

+

273

600

+-+ +++ mismatch mismatch - - - -

400

-

I

v1 Y

I

I

I

.d

g 200

e o

--

c200

-

-400 -600 I

I

1

2

sigma

I

I

3

4

5

Figure 10. Particle redistribution when exciting a general 30% radial mismatch in a transport channel with 50 periods. Plotted is the difference in transverse particle density between the mismatched and the matched distribution at the end of the channel.

Comparing Figures 9 and 10 we find that the redistribution for the by the fixed point of the quadrupolar mode at M 2u,,,. This is consistent with theory since ''+-,, in x and y corresponds to the excitation of the quadrupolar mode. For the mismatch the quadrupolar mode is not excited at all and one obtains a mixture of the low and fast mode redistribution patterns. In this case the two fixed points are clearly visible, even if they are slightly shifted by the superposition. We note that the partitioning of modes that are excited by a general mismatch is dependent on the emittances and tunes of the machine and can be completely different from the one in Figure 10. When studying the effects of mismatch in a particular linac it is therefore advisable either to excite all three eigenmodes or to use at least a few different sets of general mismatches.

"+ - +" excitation is dominated

"+ + +"

2.3.3. Maximum halo extent In the previous section we showed that halo particles gather around fixed points and that the distance of these fixed points from the core is given by the oscillation frequency of the mismatched core and its 2:l parametric ratio with the single-particle oscillations. Beyond the orbits around these fixed

274

points the particles cannot gain more energy from the core oscillations, simply because the 2:l resonance moves out of phase. Although it was found that fixed point - core distances can theoretically go t o infinity [8] one observes that maximum halo radii are limited to a certain threshold. Hofmann [9] suggested that for increasing fixed point - core distances there is a decrease in the space-charge coupling force (see particle core model), which is responsible for the energy transfer from the core oscillations to the single-particle orbits. With a simple test one can verify this idea of the maximum halo extent: using again the superconducting section of the SPL, we excite a I‘+++’’ mismatch for a Gaussian input beam. Figure 11 shows the transverse distribution at the end of the linac for different amplitudes of initial mismatch.

10000 1000 100 10

1 0.1 0

1

2

3

4 5 6 radius [sigma]

7

8

9

Figure 11. Transverse particle distribution at the end of the SPL (120 - 2200 MeV, 40 mA simulation current, Gaussian input beam, lo7 particles) when exciting a mismatch with different amplitudes.

“+++”

Even with 10 million particles we find that the redistribution pattern as well as the maximum halo radius remain the same for the different mismatch amplitudes. Only the number of particles that are redistributed to a higher radius is increased. In this example the outermost particles extend to a maximum radius of x 8aTms.However, due to imperfections in the lattice or transitions between sections, maximum radii of up to x 12aTms are found in simulations. As an example we show in Figure 12 the maximum transverse halo extent for a strongly mismatched (40%) beam in a normal conducting linac (3 - 120 MeV), with two lattice transitions (at 18 m and 41 m).

275

9 4

12 11 w E 10

2

0

10

20

30

40

50

60

70

length [m] Figure 12. Transverse halo extent in multiples of the r.m.s. beam radius for a normal mismatch, conducting linac (3 - 120 MeV, 40 mA simulation current, 40 % Gaussian input beam).

“+++”

2.3.4. Beam halo & losses An important aspect in the beam halo discussion is the question of the rise time for emittance growth or beam halo to develop. We have seen in the previous section that halo radii between 8 and 120rms are observed in simulations, so a first approach to avoiding transverse beam loss would be to use apertures of at least llorms. In normal conducting RF accelerating structures, however, the size of the aperture is directly linked to the power requirements of the cavities, and large bore radii unavoidably yield a poor RF efficiency. If the focussing quadrupoles are separated from the RF, as for instance in Coupled Cavity Drift Tube Linacs (CCDTL), Separated DTLs, or Coupled Cavity Linacs (CCL), one can use quadrupoles with large bore radii without decreasing the RF efficiency. In a classical Alvarez DTL, where the quadrupoles are housed in the drift tubes, this trick cannot be applied and the apertures are typically in a range of 5 to 7orms. Here, the only solution to reduce beam loss is the use of scrapers before entering the DTL. Then, the rise time for halo development really becomes the crucial parameter for loss prediction in the machine. As an example, in Figure 13 we look at the evolution of the longitudinally most unstable particle that was found in the simulation of an ideal transport channel (same lattice as in Figure 4). We can see that the most unstable particle after 160 periods is not the one that initially has the largest phase value. This means that even

276

-40

1

I

0

20

I 40

I

60

I

80

I 100

A

I

I

120

140

160

Period Figure 13. Evolution of the longitudinally most unstable particle in a transport channel of 160 periods with fast mode excitation and an initial 6D waterbag distribution. Plotted is one point per period. The dashed line indicates the maximum initial mismatched phase extent and the solid line indicates twice that value. Example from Bongardt & Pabst.

if scrapers were employed at the beginning of the channel, this particle would still be expelled to its final large amplitude orbit. On the other hand the rise time in this example amounts to x 60 focussing periods, which is already longer than many linacs. As mentioned in the beginning of this section, the rise time for halo development is strongly dependent on the type of lattice and the initial particle distribution (compare with Figures 4 to 7). It was stated that the transformation of mismatch into beam halo takes place more quickly if one uses a Gaussian instead of a waterbag distribution. To further underline this statement we show in Figure 14 a direct comparison of emittance growth for Gaussian and waterbag distributions in a realistic linac (see also Reference [103). It is fairly obvious that for the Gaussian beam the instability rise time is much shorter than for the waterbag beam.

2.3.5. Longitudinal losses Up to now we focused mainly on losses in the transverse plane, which is sufficient for all linac beams that are directly sent onto a target. This type of linac is used for instance in long-pulse spallation sources, waste transmutation facilities, etc. Here, the longitudinal distribution is not of

277

3.5e-05

I

G-x

-1

3e-05 2.5e-05 2e-05 1Se-05 1e-05

5e-06 0 '

0

I

100

200

300 400 length [m]

500

600

Figure 14. Transverse 99.99% emittance evolution for a mismatched beam using a Gaussian and a waterbag distribution.

great interest. On the other hand most of the linac projects which are either being built at the moment or under study, such as neutron spallation sources, injector linacs, or neutrino factory proton drivers, have to inject into subsequent accelerator systems where in many cases H- charge exchange injection is used to ease the injection process. For those projects beam halo in all three planes becomes an even more severe subject. Rest gas ionization will yield additional losses. Furthermore the transverse beam footprint at injection has to be well confined in order not to miss the small area of the stripping foil, otherwise most of the halo particles will be lost during the injection process. Longitudinally the beam has to be well confined to minimize injection losses into the RF bucket of the ring system. A large energy or phase jitter combined with the long drift in the transport lines between the linac and the rings plus the final bunch rotation cavities can easily yield an energy spread that exceeds the energy width of the RF bucket in the rings.

3. Core-Core Resonances

It was mentioned in Section 1.1 that one needs more sophisticated mathematical tools to derive properties for the core-core resonances. The main difference from the eigenmode analysis for bunched beams is the require-

278

ment to incorporate changing bunch shapes and changing emittances. The complete analysis of core-core resonances is quite a lengthy process and clearly exceeds the purpose of this lecture. We will therefore only provide a brief outline of the analysis without doing a step by step derivation. For further details refer to Hofmann [ll],[12]. 3.1. Derivation of Eigenmodes

The analysis starts with Vlasov’s equation in 2D for a beam distribution

f(x,Y PZ’P,, t ) : d df a dt f = - + i . --f at dX 7

a d + 6 . --fdY + GZ . -f

+$,.

dPX

d

ap,f = 0

(39)

Using the smooth approximation one can write down the Hamiltonians for the x and y motion

and formulate a Kapchinskij Vladimirskij (KV) type solution [13] for the initial particle distribution:

where N stands for nrab and denotes the ratio of “beam temperatures” in the two planes, which is also referred to as the equipartitioning ratio:

Adding a small perturbation to the KV type solution fo leaves us with f = f o ~ ~ o z , ~ O y ~ + f l ~ ~ , Y , P z , P y , ~ ~

(43)

and the perturbed electrostatic potential 4, which can be computed using Poisson’s equation for the perturbed charge density:

V 2 9 = --

€0 q /

fldp2dpy

(44)

Linearizing Vlasov’s equation (39) and using the boundary condition of an electric field that vanishes at infinity (no image charges) one can now solve the two partial differential equations for the eigenfrequencies w :

fi = fi(at)e-jwt

9 = +(at)e-jwt

(45)

After a lengthy calculation this analysis provides us with a set of 2D eigenmodes which can be characterized as in Figure 15. The even modes are symmetric with respect to the horizontal axis, while the odd modes have no such symmetry. If we interpret the two planes as x and y then odd symmetry corresponds t o a lack of rotational symmetry around the longitudinal axis. Please note that these modes cannot be found with r-z simulation codes. First-order modes represent the trivial case of displacement from the beam axis, and the oscillation frequencies are just the betatron frequencies without space-charge in each direction. The second-order modes correspond to the beam eigenmodes that were derived in the previous sections. However, since this is a 2D analysis we would find only the two eigenmodes for a DC beam (quadrupolar and breathing mode). The real novelty of the Vlasov analysis lies in the 3rd- and 4th-order modes and, looking at the oscillation shape, it becomes immediately clear why these modes are found only if the mathematical tools allow non-elliptical bunch shapes.

tion are sensitive to oscillating and non-oscillating modes, while the more realistic beams with waterbag distributions were affected only by the nonoscillating modes. The areas where these non-oscillating 3rd- and 4th-order core-core resonances affect the beam are then visualized in Hofmann’s “stability charts” (see Appendix A, [12]).

280

3.2. Application of Stability Charts f o r Core-Core Resonances

In the next step we make the transition from the 2D theory, which has been verified with 2D simulations for idealized lattices, to a realistic linac structure with 3D bunched beams. Although the mathematical model is derived for anisotropy between the two transverse planes of a DC beam, Hofmann suggested [ll]applying the results for anisotropy between the transverse and the longitudinal plane (see also [14]). This approach was used for the superconducting part of the SPL project 1151. In Figure 16 we show the stability chart for the SPL emittance ratio of E l / E t = 2. The shaded areas of the chart indicate where emittance exchange between the longitudinal and the transverse plane is to be expected (the degree of shading indicates the speed of the process). For strong tune depression Ic,/k,0 one obtains emittance exchange for all tune ratios, while for moderate values one expects some stable areas. The dashed line indicates the condition for an equipartitioned beam, see Eq. (42). The usefulness of the charts for a 3D beam with anisotropy between the longitudinal and transverse plane can be illustrated with the following

1.o

0,73 0,63

0,53

0.8

0,43

0,33

0.6

1

0

s 0.4

equigarti

0,23 0,13 0,03

0.2

0.2

0.0

0.0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

kdkx Figure 16. Stability chart for the SPL emittance ratio of mann.

= 2, source: Ingo Hof-

281

example: in two linac sections of the SPL (120 - 383 MeV, 40 mA simulation current, waterbag distribution), the quadrupole gradients are modified so that we obtain three linacs that fall into different areas of the stability chart in Figure 16. The original SPL and the “case 1” lattice should both be stable, while “case 2” is located in an unstable area. The result is plotted in Figures 17 to 19.

0

20

40

60

80 100 120 140 160 180 length [m]

Figure 17. R.m.s. emittance evolution for the SPL lattice.

0

20

40

60

80 100 120 140 160 180 200 length [m]

Figure 18. R.m.s. emittance evolution for case 1.

Clearly, in case of the SPL and case 1 there is no significant change in the longitudinal/transverse r.m.s. emittance values, while for case 2 we

282

E

0.6

0.55

0.35’ 0

”

20

40

60

”

*

”

80 100 120 140 160 180 200 length [m]

Figure 19. R.m.s. emittance evolution for case 2.

can observe a substantial emittance exchange from the longitudinal to the transverse plane. We note that in this case there is one “hot” plane, the longitudinal one, which is feeding the two “cold” transverse planes. For this reason there seems to be much more longitudinal decrease than transverse increase. In an actual linac one should try to avoid a design where the transverse emittance is higher than the longitudinal emittance, because in case of an exchange two “hot” planes would feed one “cold” plane and a high longitudinal emittance increase could be the consequence. In the meantime Hofmann’s stability charts have been successfully applied in various high intensity linac projects and should be regarded as a new tool in the design of linac lattices. Looking at the various stability charts for different emittance ratios in Appendix A, one can see that small emittance ratios close to 1.0 yield large stable areas in the charts and vice versa. We note that equipartitioning is not an obligatory design feature, but that emittance ratios close to 1.0 simply provide more freedom in the lattice design. The emittance exchange itself takes place only if we have a combination of beam anisotropy plus a certain tune ratio plus a minimum tune depression. Up to now the understanding is that the most harmful resonance in the charts is the 4th-order even mode, which is always located around a tune ratio of 1.0. 3.3. Core-Core Resonances & Beam Halo The obvious question to ask is if core-core resonances contribute to the development of beam halo. A simple answer can be given by looking at the

283 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7

0

20

40

60

80 100 120 140 160 180 200 length [m]

Figure 20. Evolution of fractional longitudinal emittances for case 2. All emittances are normalized to an initial value of 1.0.

1.7 1.6

I

1

1.5 1.4

1.3 1.2 1.1 1

0

20

40

60

80 100 120 140 160 180 200 length [m]

Figure 21. Evolution of fractional transverse emittances for case 2. All emittances are normalized to an initial value of 1.0.

fractional emittances (99%, 99.9%, etc.) for case 2 of the previous section. In Figure 20 we see that the outermost particles seem to be unaffected by the decreasing r.m.s. emittance, while the rising transverse r.m.s. emittance in Figure 21 yields a slight increase in higher fractional emittances. Altogether the ratio between the 99.99% emittances and the r.m.s. values increases by M 25%, which can hardly be referred to as halo development. This behavior confirms our model, which describes this kind of instability as a space-charge driven core-core resonance, which is an altogether different process than the particle-core resonances previously discussed.

284

4. Summary 0 0

0

0

0

0

0

0

0

Lattice resonances can yield unstable beams. Beam halo develops because of resonant interaction between the oscillations of a mismatched core with single-particle orbits. The large tune spread in high-intensity linacs makes these machine especially sensitive to halo development due to mismatch. The eigenmodes of a mismatched beam provide a very good tool for studying the dynamics of halo development due to initial mismatch. Several types of mismatch should be used to study halo in a particular machine. According to simulations the maximum halo extent is limited to 8 - 12gTmS.If possible, the transverse dimensions of the beam pipe, quadrupole and RF apertures should respect this limit. If the beam is injected into subsequent accelerator chains, longitudinal halo becomes an important issue. Emittance exchange between the planes is triggered by core-core resonances and can be avoided by using Hofmann's stability charts. It seems that core-core resonances do not contribute to beam halo.

5. Design Recipes for High-Intensity Linacs 0 0 0 0

0

0

phase advance in all planes below go", tune depression > 0.5, large apertures (= llorms,if possible) to avoid losses, emittance ratios close to one (0.5 < E ~ / Q < 2) provide larger stable areas in the stability charts, avoid unstable areas in the stability charts or use fast resonance crossing, avoid T = E ~ c ~ / E > ~ u1~(transverse beam temperature > longitudinal beam temperature), to avoid two hot planes feeding one cold plane in case of emittance exchange.

Acknowledgments All Monte Carlo simulations were done with IMPACT, a code developed at LANL and BNL by R. Ryne and J. Qiang [lS].During the preparation of this lecture I profited from discussions with K. Bongardt. Proof-reading was done by Chris Prior. I used simulation results and pictures that were kindly given to me by K. Bongardt, M. Pabst, and I. Hofmann. Further material that was used for this lecture can be found in Refs. [171, [18].

285

Appendix A. Stability Charts for Emittance Exchange, Source: Ingo Hofmann

.....

0.8 ...........................

x 0.6 0

Y

3

0.4 0.2 . .

1

0

0.25

0.5

0.75

1

1.25

1.5

kzlkx Figure 22. m=4/even

m=4/odd

Stability chart for E l / E t = 0.5 m4leven

m=3/odd

> I 0 belabon periods

....................

..................

............

0

0.25

0.5

0.75

.......................

1

1.25

1.5

Wkx Figure 23.

Stability chart for q / E t = 1.2

1.75

286 m4Ieven

rn4leven

rn=4/odd

0,53 0,43 0,33 0,23 0,13 0,03

0

0.25

0.5

0.75

1

1.25

1.5

1.75

wkx Figure 24.

Stability chart for

El/&t

= 2.0 ~

0-73 0.63 0,53 0,43 0,= 0,23

0,13 0,03

................................... 0

Y

t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.25

0.5

0.75

1

1.25

1.5

1.75

Wkx Figure 25.

Stability chart for E l / E t = 3.0

2

287

Appendix B. Symbols

................................................ transverse beam radius ............................................... longitudinal beam radius ................................................ .longitudinal coordinate Et,l ............................. .transverse, longitudinal r.m.s. emittance a. ................................ zero current phase advance per period u ................................. .full current phase advance per period up ................................................... single-particle tune n,,, ...................envelope tune of mismatch eigenmode oscillations Ico ..................... zero current phase advance per unit length (wave a b s

number for the transverse betatron oscillations) Ic ............................. .full current phase advance per unit length K 2 .............................................. . 2 D space-charge factor Kt,l ............................................. . 3 D space-charge factor f ................................... general particle distribution function L, ............................................ length of focussing period A , ......................................... .mismatch amplitude factors Q H , L ....................................... .mismatch mode form factors rn ............................................................. rest mass t ................................................................... time w ..................................................... angular frequency p .................................................... particle momentum y ................................................ .relativistic mass factor €0. .................................................. .electric permittivity 4 ........................................................... .phase angle q ......................................................... electric charge a , p ................................................... Twiss parameters HO ........................... Hamiltonians of transverse particle motion n ................................................... .number of particles 9 ................................................. electrostatic potential i; .................................................. equipartitioning ratio a,,, ....................................................... r.m.s. radius

288

References 1. L.W. Smith and R.L. Gluckstern, Rev.Sci.Instrum. 26,220 (1955). 2. M. Reiser, Theory and Design of Charged Particle Beams, Wiley (1994). 3. Frank J. Sacherer, RMS Envelope Equations with Space Charge, CERN/SI/lnt. DL/70-12 (1970). 4. K. Bongardt and M. Pabst, Analytical Approximation of the Three Mismatch Modes for Mismatched Bunched Beams, ESS Note 97-95-L (1997). 5. A. Letchford, K. Bongardt and M. Pabst, Halo Formation of Bunched Beams in Periodic Focusing System, Proceedings of PAC99, page 1767 (1999). 6. Editor: M. Vretenar, Conceptual Design of the SPL, a High-Power Superconducting H- Linac at CERN, CERN 2000-012 (2000). 7. R.L. Gluckstern, Analytical Model for Halo Formation in High Current Ion Linacs, Phys.Rev.Lett. 73, 1247 (1994). 8. I. Hofmann, J. Qiang and R.D. Ryne, Cross-Plane Resonance: A Mechanism for Very Large Amplitude Halo Formation, Proceedings of PAC02 (2002). 9. G. Franchetti, I. Hofmann and D. Jeon, Anisotropic Free-Energy Limit on Halos in High-Intensity Accelerators, Phys. Rev.Lett. 88,25 (2002). 10. I. Hofmann, G. Franchetti, J. Qiang, R. Ryne, F. Gerigk, D. Jeon and N. Pichoff, Review of Beam Dynamics and Space Charge Resonances in High Intensity Linacs, Proceedings of EPACO2 (2002). 11. I. Hofmann, Stability of Anisotropic Beams with Space Charge, Phys.Rev. E 57,4713 (1998). 12. I. Hofmann and 0. Boine-Frankenheim, Resonant Emittance Transfer Driven by Space Charge, Phys.Rev.Lett. 87/3,034802 (2001). 13. I.M. Kapchinskij and V.V. Vladimirskij, Proceedings of the International Conference on High Energy Accelerators, CERN, Geneva, page 274 (1959). 14. I. Hofmann, J. Qiang and R.D. Ryne, Collective Resonance Model of Energy Exchange in 3D Non-Equipartitioned Beams, Phys.Rev.Lett. 86,2313 (2001). 15. F. Gerigk and I. Hofmann, Beam Dynamics of Non-Equipartitioned Beams in the Case of the SPL Project at CERN, Proceedings of PAC01 (2001). 16. J. Qiang, R.D. Ryne, S. Habib and V. Decyk, An Object-Oriented Parallel Particle-In-Cell Code for Beam Dynamics Simulation in Linear Accelerators, Journal of Computational Physics 163,1 (2000). 17. T.P. Wangler, RF Linear Accelerators, Wiley, ISBN 0-471-16814-9 (1998). 18. T . Wangler, Models and Simulation of Beam Halo Dynamics in High-Power Proton Linacs, Proceedings of ICAP98, Monterey, CA (1998).

POWER SOURCES FOR ACCELERATORS BEYOND X-BANDt E. R. COLBY Stanford Linear Accelerator Center 2575 Sand Hill Road, Mail Stop 07 Menlo Park, CA USA E-mail: [email protected]

The availability of power sources suitable for particle acceleration is a key factor determining what acceleration techniques are practical. We examine the fundamental limitations of slow- and fast-wave devices, two beam accelerators, and lasers as power sources for accelerators in the frequency range beyond X-band.

1

Introduction

Sources of coherent radiation operating in the region of the electromagnetic spectrum beyond 12.4 GHz (the upper edge of the 'X-band') are numerous and find broad application in communications, radar, and atmospheric contaminant monitoring at the low end, and in spectroscopy, chemistry, and solid state physics studies in and near the optical portion of the spectrum. These sources are attractive for particle accelerators as they offer the potential for increased accelerating gradient and reduced accelerator size. Additionally, the very short bunches produced by very high frequency acceleration can produce ultrafast radiation pulses if used to drive a free electron laser, improving the temporal resolution of, for example, pump-probe experiments. Sources used for particle accelerators must provide stable phase and amplitude output, high peak power in short pulses, and be efficient. Where two or more power sources are required to drive the beam to the required energies, the sources must also be phase lockable, that is, they must be power amplifiers amplifying the signal of a common master oscillator. In general, power sources need not be particularly broadband (Ada,
'This work supported by Department of Energy contract DE-AC03-76SF00515 (SLAC). 289

290

2

Frequency Scaling of Devices

In general, higher frequencies require that the residual capacitances and inductances in the source be small, which in turn requires the source itself to be small. Most high power sources (vacuum tubes and lasers included) rely on resonant cavities to increase efficiency, but with the introduction of a narrowing of bandwidth comes the need to maintain key dimensions of the device to tighter tolerances, a problem that grows rapidly harder with increasing frequency. With the decrease in size also comes a natural decrease on the available output power that can be generated by the device, owing both to the decreased surface area through which dissipated power can be conducted, and due to the decreased energy that can be stored within the device. Further, Ohmic power losses grow worse as f J R owing to increasing surface resistance. Vacuum tubes achieve power amplification through the growth of spacecharge waves on an electron beam. A small, periodic velocity modulation is imposed on the beam by a drive circuit (typically a resonant cavity). After sufficient drift, the velocity modulation leads to a large density modulation which then permits efficient coupling of the beam’s kinetic energy into electromagnetic power via resonant excitation of a mode in the output circuit (also generally a resonant cavity). The amount of current which can be transported through the device is determined by the hole radius through which the beam must pass. As this typically is determined by good design practice for the rf circuits, the radius will scale inversely with the frequency. Consequently, the total transmittable current will scale as I I&’ causing the output power to also drop as P - I / f 2 unless other steps are taken. This argument assumes that the beam current density is fixed which, in fact, is not as poor an assumption as it first appears. The reason is that the source of the electron beam is a hot cathode, the emitted current density from which depends on the cathode temperature and applied voltage. Raising the current density means raising the cathode temperature, which dramatically shortens its usable life. Current cathodes can produce up to 10 Ncm2 for tens of thousands of hours under favorable conditions, which means current densities of up to 1 kA/cm2 may be produced from an electron gun with 1OO:l compression. Transporting such high current density beams is also challenging, requiring very strong, meticulously designed focussing’. Even if the beam current could be appreciably increased to obtain higher output power, efficiency would be compromised. As an example, klystron efficiency approximately follows Symon’s Law:2 1110.9-0.2 k, , with the

-

-

29 1

electron beam microperveance defined as k,,=1Q6x IN 3/2. The increase of microperveance with current, and hence decrease in efficiency with current motivates exploring other means of increasing the output power. Raising the beam voltage is one obvious way of obtaining more power, but has several important side effects. The formation of bunching of the electron beam occurs on a characteristic time scale determined by the reduced plasma frequency of the beam. For a round beam of voltage V and current density .To in free space, the distance required for a (small) velocity modulation to become a density modulation is one quarter of the plasma wavelength hp :

where the V3/2scaling is exact for relativistic beam voltages (r>>l). Hence, raising the beam voltage rapidly lengthens the separation distance between the drive and output circuits, making beam transport and vacuum pumping more problematic. A second side effect of raising the beam voltage is that the output circuit voltage must be raised as well. Optimum “electronic efficiency” (rf power output in ratio to electron beam kinetic energy) is obtained when the beam-coupled impedance of the output circuit matches the beam’s impedance Zb=VL.Raising the output circuit voltage requires either larger electric fields or extended interaction circuits to be used (e.g. several resonant cavities in series instead of one), each with its own difficulties. The beam tube may be enlarged to carry more current by using overmoded circuits for the drive and output. This permits the resonant cavities and accompanying beam holes to be made significantly larger than is practical for the fundamental mode case. The drawback is that for significant increases in circuit size, there will be a very large number of closely spaced resonant modes near the operating mode used for power generation. At their most benign, competing modes will couple power out of the beam and increase circuit heating and lower output efficiency. However, if the adjacent modes include deflecting modes with good coupling, they can lead to beam breakup and rf pulse shortening. Increasing the current while maintaining the current density constant is another. This has the obvious additional requirement of requiring the circuits to accept a larger beam, and several tricks have been proposed to do this. The annular beam3, sheet beam4, and multi-beam5 klystrons are geometric variations on this idea. For the sheet and annular beam klystrons, the beam transport and

292

interaction circuit designs are quite challenging, with mode competition and field uniformity being difficulties for the circuits, and stable, matched transport being challenges for the focussing. Mode competition in the circuits and beam-to-beam coupling are challenges for the multi-beam klystron. Taken together, these considerations lead to the conclusion that the power scaling of resonant-circuit based vacuum tubes will remain P - l / f 2 with various tricks permitting significant increases in power, but none of which can alter the power scaling law. Many of these constraints are circumvented by the two-beam accelerator, as will be discussed in section 3.3 below. Atomic lasers, on the other hand, achieve power amplification through the stimulated transitions of atomic electrons to a lower energy state. Lasers (except fiber lasers) tend to be extremely overmoded and have output frequency characteristics that are determined almost exclusively by the electronic structure of the lasing media itself (by contrast, vacuum tube frequency characteristics are almost exclusively determined by the circuits, and not the gain media [electron beam]). Consequently, lasers will not follow a specific overall scaling law of power with frequency, rather the output power will be determined by the characteristics of the lasing media that are available.

3

Examples of Power Sources Beyond X-band

A rich variety of devices has been built and continue to be designed for the production of power beyond x-band. Included in this section is a representative set of examples drawn from the last decade of research and development. These examples by no means exhaust the many methods of power production being developed today. 3.1 Solid State Devices

The inherent efficiency, robustness, and recent progress in achieving high output powers in the 1-2 GHz range for cellular phone communication make it natural to ask whether solid state devices might at some point meet the requirements for particle accelerators. As with power vacuum tubes, the inverse relationship between device power and device frequency makes this quite problematic. Efforts to develop arrays of amplifiers, which collectively provide higher output powers, have succeeding in developing output powers in the tens of watts cw at 60 GHz using the summed output of 272 solid state power amplified. Higher

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powers are available from pulsed devices, with 20 watts at 92 GHz being produced in 80 ns bursts from a commercially available IMPATT oscillator7.

3.2 Vacuum Tubes Power vacuum tubes develop output power through space-charge waves induced on an electron beam by a small drive signal. The waves can represent axial bunching (as in klystrons), azimuthal bunching (as in gyrotrons, helical free electron lasers [FELs], and magnetrons), or periodic transverse deflection (as in planar FELs and magnicons). 3.2.1

Axial Bunching Devices

Klystron development beyond X-band has yielded a number of devices and designs. The Budker Institute and Protvino jointly developed' a 14 GHz relativistic klystron for the Russia linear collider VLEPP that was experimentally tested to give 90dB saturated gain and 50 MW peak output power in 250 ns pulses at 30% efficiencyg. The extraordinary saturated gain of this tube meant that very weak drive signals would yield full output power, a property that made self-oscillation more probable. Although the tube was designed to produce 150 MW, oscillations ultimately limited the stable operating power to less than half that value. Haimson Research and NRL collaborated" to produce a 17.136 GHz klystron that was tested to give a saturated gain of 67 dB, an output power of 26 MW in 150 ns pulses, and a 49% efficiency. This klystron is the centerpiece of a high gradient accelerator R&D program at MIT. SLAC has designs for 94 GHz klystrons of both round" and sheet beam"-'* construction, and has been pursuing fabrication efforts. The primary challenges of working at this frequency come from the small structure dimensions (e.g. a beam hole diameter of 800 pm) and very tight tolerances on critical dimensions (typically a few pm). Deep x-ray lithography and electroforming techniques are being explored for constructing the drive and output circuits. 3.2.2

Azimuthal Bunching Devices

The gyroklystron, traveling-wave tube (TWT), helical FEL, and magnetron are examples of devices in this category. An annular beam is generally used and is focused in a solenoid field. Since the EM interaction is primarily azimuthal, however, this implies that the cyclotron motion of the beam in the focussing channel plays a critical role in the interaction process. In particular, a synchronism condition between the rf frequency Q the axial field variation kll, and electron beam cyclotron frequency O C = q B / pmust be satisfied:

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w - kIlVII = na2,

(2)

where the harmonic number n is a nonzero integer, and the axial beam velocity is V I I .Since the cyclotron frequency is proportional to B, this implies the focussing field strength must increase for increasing rf frequency. As a rough rule of thumb, 1 Tesla field strength increase is required for each 14 GHz increase in operating frequen~y'~, assuming a 500 kV electron beam is used. A collaboration of the University of Maryland, NRL, and Communications & Power Industries (CPI) has produced and measured several gyroklystrons worhng at 34.9 GHZ and 94 GHz. The 34.9 GHz tube produced 225 kW in 2 ps pulses with a saturated gain of 30 dB and 31% efficiency. The highest output power 94 GHZ tube (one of five measured) produced an output power of 84 kW in 2 ps pulses with a saturated gain of 42 dB, and 34% effi~iency'~. U.C. Davis, UCLA, and Micramics Incorporated are collaborating on the design of a 35 GHz harmonic gyro-TWT. They have designed for 400 kW output power, 35 dB saturated gain, and an efficiency of 2O%I5. This tube operates on the TE31mode, and will incorporate special cuts in the circuit to suppress competing modes. An IAP Nizhny NovgorodGYCOM Company collaboration constructed and testedI6 a 93.5 GHz gyroklystron that produced 210 kW at 30% efficiency and 20 dB saturated gain. The highest power w-band (75-110 GHz) tube proposed to date has been designed by the UMDKalabazas Creek Research collaboration. Their gyroklystron design achieves 10 MW output power at 91 GHz with a saturated gain of 55 dB and electronic efficiency of 37.4%17.The guide field required for this high frequency is 2.76 T, and will be supplied by a superconducting solenoid. Finally, as an example of what output power is possible from gyro-devices, the gyrotron has received considerable development as a power source for electron cyclotron resonance heating (ECRH) of plasmas for fusion research. The gyrotron is an oscillator, making it unsuitable for all but low-energy, singlepower source accelerators, but working tubes produce powers of 1.7 MW at 165 GHz with an efficiency of 35.2%18.This high power is possible through the use of very highly overmoded circuits, with the present example using the TE31,17 mode for the output circuit.

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3.2.3

Deflection-based Devices

The magnicon and planar FEL (or ubitron) are the primary members of this category. A pencil beam is generally used, and interaction is either with TM,, deflecting modes of the interaction circuits (as with the magnicon) or with pure TEM modes (as with the FEL) in a transversely oriented, axially periodic magnetic field. Yale and Omega-P Incorporated have collaborated to design and produce a 34 GHz magnicon that uses a “Milodrive circuit mode, and TM310output circuit mode, giving harmonic multiplication of frequency. Since the drive frequency is one-third the output frequency, the tube is more stable against oscillation (backward waves in the tube have the wrong frequency to drive the input circuit) and the input source is more straightforward. The prototype tube is nearly complete as of this writing, with tests planned presently. The design calls for 48 MW in 1.5 ps pulses, 45% efficiency, and 54 dB of saturated gain at 34 GHzi9. The record holder to date in the generation of high power high frequency rf is the Lawrence Livermore FEL. Designed as a source for microwave absorption measurements for a fusion experiment (the MTX tokamak), the FEL provided 140 GHz, 1-2 GW pulses of 20 ns duration, a gain of 76 dB, and an efficiency of 14%*O.

3.3 Two-beamAccelerators Although the two-beam accelerator is in essence a distributed relativistic klystron, it deserves special treatment because it has special flexibilities in frequency, pulse length, output power, and power distribution that are impractical to realize with discrete klystrons. Like a klystron-powered accelerator, in which a low energy, high current electron beam is used to produce rf power to accelerate a higher energy, low current electron beam, the two-beam accelerator has a high current beam (the “drive” beam) used strictly as the energy source for making rf to accelerate the good beam. The differences are, however, that the low-quality beam is of high enough energy that it can be used to generate power for many structures, and that the output circuits used to do so are distributed close to the accelerating structures. For a tube-powered accelerator, one has hundreds of drive beams-one for each power tube supplying the accelerator-and hence the sources must be simple and inexpensive to be practical. Since just one source is needed to produce the drive beam for the two beam accelerator, it can be very complicated, permitting substantial flexibility.

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The Compact Linear Collider (CLIC) is designed on the two-beam principle, and uses a single main drive linac (operating at 937 MHz) to produce a 92 ps long pulse train of 1.16 GeV electrons averaging 8.2 Amperes. This pulse train is then folded back on itself 32 times by interleaved injection into a series of three combiner storage rings to produce a 2.9 ps long pulse train of 1.16 GeV electrons averaging 262 Amperes. The pulse train produced not only has 32 times the peak current, but the fundamental Fourier harmonic frequency is also 32 times higher, permitting efficient generation of 32x937 MHz=29.984 GHz. The pulse duration has also been shortened by a factor of 32, a benefit in decreasing the effects of surface heating in the accelerator structures due to the rf power. This pulse train is subdivided into 20 smaller pulse trains, which are fed to groups of transfer structures which generate 29.984 GHz rf power for portions of the accelerator (viz. output circuit). The entire process from AC power to rf power is expected to reach efficiencies of 47%”, and result in the production of -27 MW from each of the 11,000 transfer structures. Since the transfer structures can be located quite close to the accelerator structures, power transmission losses are minimized. Power generation from a drive beam has been demonstrated by the CLIC Test Facility I1 (CTF 11), which extracted 120 MW of power in 16 ns pulses”. CTF I11 is commissioning at present, and will provide the first demonstration of pulse combining in a pair of storage rings232”. For generation of rf power for high energy colliders, the two-beam accelerator is a very promising concept. The need for very high peak power in short pulses becomes increasing difficult to meet at high frequencies for conventional power tubes, but is potentially achievable with a two-beam accelerator, 3.4 Lasers

The laser has often been considered as a potential power source for particle accelerators owing to the extreme peak powers that are possible, but only recently has achieved the power efficiency and phase stability required for accelerator applications. Two key developments have enabled significant gains in laser efficiency. Solid-state diode lasers can produce radiation in the near infrared to red with wall-plug to light-output efficiencies that are excellent, with commercially available products already passing 30% efficiencyz5,and laboratory diode lasers having achieved 54% at low power (0.25 W)26more than a decade ago. Very high power diode bars (1 kW/cm2) with wall-plug to optical efficiencies of 45% are under development now”.

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The second key development that has markedly improved laser efficiency is the discovery of diode-pumped laser media that efficiently use the pump power. For all lasing media, the pumping transition requires more energy than the radiative transition emits. This difference is expressed as a ratio of the emitted photon wavelength h, to the pump photon wavelength hp, known as the quantum defect D=XJhp. For lasing media like Ti:Sapphire, the pumping absorption band peaks at %=490 nm, while the lasing transition peaks at around h,=790 nm, which means that the pump-power-to-output-power efficiency will be at most D=0.62 or 62%, and in practice will be somewhat lower. A host of materials have been developed with small quantum defects, including Yb:KGd(W04)2 (commonly called just Yb:KGW) with D=0.96 and a measured slope efficiency (optical power out versus optical power in) of ~ ~ = 5 7 Yb:KY(WO& %~~, (called Yb:KYW) with D=0.957, ~ ~ = 8 6 . 9 %and ~ ~ , Cr*:ZnSe, with D=0.70, ~l~=0.52%~’. The ytterbium-ion media lase in the 1 pm range, while chromiumdoped zinc selenide lases near 2 ym, both conveniently in a window where silicon is transparent and may be used for accelerator structures. Phase-locking of lasers at the optical time scale to an external microwave reference is a key step to synchronizing two or more lasers to power an accelerator. Laser pulse shape is simply amplitude modulation of the underlying optical “carrier” wave, and in modern lasers the carrier is not locked in phase to the pulse envelope. Locking the carrier phase to an external reference is a critical step in synchronizing lasers, and has been recently demonstrated31x32.

4

High Frequency Power Transmissionand Compression

Steadily increasing Ohmic losses on conducting surfaces makes handling high power at very high frequency challenging. Fundamental-mode waveguide becomes extremely lossy (e.g. rectangular copper waveguide in TElo mode used for 90 GHz transmission attenuates at approximately 3 dB/meter) requiring that other methods of power transmission be used. Overmoded waveguide and quasioptical transmission are the alternatives. Overmoded waveguide achieves lower loss by decreasing the induced surface currents on the guide, but with the added complication that scattering from the desired transmission mode into other guided modes is no longer forbidden. Maintaining mode purity is essential for efficient power coupling to the accelerator structure, and hence great care must be taken to ensure the guide is free from geometric defects that would scatter power out of the desired mode. Waveguide losses may be further reduced by switching to an HEll mode and

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corrugating the surfaces of the guide. Corrugations with a depth of AJ4 will transform the impedance of the conducting wall (essentially a short circuit) into very high impedance, further reducing wall currents and hence loss. The machining complication of corrugated guide makes this a costly option, however. Quasioptical transmission uses free-space TEM modes directed and focused by mirrors, lenses, and gratings, much as is done with light in the optical range, but in a domain where diffraction effects dominate33.The Rayleigh quarter wavelength rule for surface accuracy34which applies to optical mirrors applies here as well, but in the millimeter-wave range is trivial to achieve. The main drawback of quasioptical power transmission is that diffraction effects require the components to be very much larger than their guided-wave counterparts, typically by two orders of magnitude.

5

Conclusion

The potentially higher accelerating gradients and decrease in the size of accelerator components that are possible at higher frequencies makes the development of sources beyond x-band attractive. The naturally shorter electron bunches produced by high frequency acceleration are also of interest in their own right. The two-beam accelerator concept is an excellent solution to the power requirements of a linear collider, where the high cost of producing and manipulating the drive beam becomes comparable to the cost of thousands of discrete power sources. Where smaller accelerator systems are desired, discrete sources remain the practical solution. Significant research and development has taken place and will continue in this area, offering the promise of more potent sources. Point-to-point communication and radar will continue to motivate development of sources in the various atmospheric transparency bands (30-40 GHz, 75-95 GHz) and in the peak absorption band near 60 GHz for secure communications between satellites. ECRH of plasma for fusion research will continue to motivate very high power sources in the range beyond 100 GHz. Continued progress in the efficiency and stability of lasers will also continue, driven by the materials processing and telecommunications industries, with commercial solid-state diode-pumped lasers with good efficiency becoming available in the not too distant future.

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References 1. D. Sprehn, et al, “X-band Klystron Development at SLAC”, in Proc. SPIE Int. SOC. Opt. Eng., 4031:132-143, (2000), also available inline at URL: htt~://~~~.~lac.stanford.edu/cgi-wra~/getdoc/slac-~ub-8346.~df. 2. G. Nusinovich, S. Gold, in Adv. Accel. Conc. Conf. AIP Proc. 569, p. 65ff, (2001). 3. M. Fazio, et al, “Design for a One-Gigawatt Annular Beam Klystron”, in Proc. XX Linac Conf., Monterey, CA, Aug. 21-25, (2000). 4. E. Colby, et al, “W-band Sheet Beam Klystron Simulation”, in Proc. RF98, AIP Conf. Proc. 474, p.74ff, (1998). 5. E. Wright, et al, “Development of a lOMW, L-band Multiple Beam Klystron for TESLA”, in Proc. Euro. Part. Accel. Conf., Paris, France, June 3-7, p. 23379, (2002). 6. J. Sowers, et al, “A 36W, V-Band, Solid State Source”, Int. Micro. Symp. Digest, Anaheim, CA, p. 235ff, June, (1999). 7. Available from ELVA- 1 Millimeter Wave Division, St. Petersberg, Russia, online at URL: htt~://www.elva-l.sub.ru/ . 8. L. Arapov, et al, “High Power Sources for VLEPP”, AIP Conf. Proc. 337, p.l18ff, (1994). 9. V. Balakin, et al, “VLEPP 14 GHz Klystron Testing”, in Proc. XV Conf. On Charged Part. Accel., Protvino, Russia, p. 79ff, (1996). 10. J. Haimson, et al, “Initial Performance of a High Gain, High Efficiency 17 GHZ Traveling Wave Relativistic Klystron for High Gradient Accelerator Research”, AIP Conf. Proc. 337, p. 146ff, (1994). 11. G. Schietrum, et al, “Klystron Research and Development”, in Proc. 29” ICOPS, Banff, Canada, May 26-30, (2002). 12. E. Colby, et al, “W-band Sheet Beam Klystron Simulation”, AIP Conf. Proc. 474, p. 74ff, (1998). 13. S. Gold, “Overview of Advanced, Non-Klystron rf Sources”, in Proc. Snowmass, Snowmass, CO, June 30-July 21, (2001), also available online at URL: http://~.slac.stanford.edu/econf/C010630/vapers~301.PDF. 14. M. Blank, er al, “Experimental Demonstration of High Power Millimeter Wave Gyro-Amplifiers”, AIP Conf. Proc. 474, p. 165ff, (1998). 15. D. McDermott, e? al, “High Power Harmonic Gyro-TWT Amplifiers in Mode-selective Circuits”, AIP Conf. Proc. 474, p. 172ff, (1998). 16. E. Zasypkin, et al, “Study of a W-band pulsed 200 kW gyroklystron amplifier”, in Proc. 24” Conf. on IRMMW, Monterey, CA, Sept. 6-10, (1999). 17. R. Ives, et al, “Development of a 10 MW, 91 GHz, Gyroklystron for Accelerator Applications”, AIP Conf. Proc. 569, p. 663ff, (2000). 18. M. Thumm, “State-of-the-Art and Recent Developments of High Power Gyrotron Oscillators”, AIP Conf. Proc. 474, p. 146ff, (1998).

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19. J. Hirshfield, “Millimeter-wave RF Sources for Accelerator Applications”, AIP Conf. Proc. 647, p.29ff, (2002). 20. A. Allen, et al, “Generation of High Power 140 GHz Microwaves with an FEL for the MTX Experiment”, in Proc. IEEE Part. Accel. Conf. (PAC93), Washington D.C., p. 1551ff, (1993). 21. H. Braun, et al, “A New Method for RF Power Generation for Two-Beam Linear Colliders”, AIP Conf. Proc. 474, p. lff, (1998). 22. H. Braun, et al, “Experimental Results and Technical Research and Development at CTF 11”, in Proc. Euro. Part. Accel. Conf. (EPAC2000), Vienna, Austria, p. 48ff, (2000). 23. R. Corsini, et al, “Status of the CTF3 Commissioning”, in proc. Euro. Part. Accel. Conf. (EPAC2002), Paris, France, p. 491ff (2002). 24. D. Alesini, et al, “CTF3 Compressor System”, in proc. Euro. Part. Accel. Conf. (EPAC2002), Paris, France, p. 416ff (2002). 25. Available from Rofin Sinar Laser GmbH, Hamburg, Germany, available online at URL: httD://www.rofin.com/home-e.htm. 26. W. Streifer, et al, “Advances in Diode Laser Pumps”, IEEE J. Quantum Elec. 24(6), p.883ff, (1988). 27. Lawrence Livermore National Laboratory Laser Science & Technology division, online at URL: httD://www.llnl.gov/nif/lst/advanced.html. 28. F. Brunner, et al, “Diode-pumped femtosecond Yb:KGd(W04)2laser with 1.1-W average Power”, Opt. Lett. 25(15), plll9ff, (2000). 29. N. Kuleshov, et al, “Pulsed Laser Operation of Yb-doped KY(W04)z and KGd(W04);’, Opt. Lett., 22(17), p1317ff, (1997). 30. I. Sorokina, et al, “Efficient Broadly Tunable Continuous-Wave Cr2+:ZnSe Laser”, J. Opt. SOC.Am., 18(7), p.926ff, (2001). 3 1. Diddams, et al, “Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb”, Phys. Rev. Lett., 84 (22), p.5102ff, (2000). 32. R. Shelton, et al, “Phase-Coherent Optical Pulse Synthesis from Separate Femtosecond Lasers”, Science, 293, 17 AUG (2001). 33. P. Goldsmith, Quasioptical Systems, IEEE Press, New York, NY, (1998). 34. M. Born, E. Wolf, Principles of Optics, 6th Ed., Pergamon Press, Oxford, p. 468, (1980).

RECIRCULATED AND ENERGY RECOVERED LINACS* G.A. KRAFFT TJNAF MS-7A 12000 Jefferson Ave., Newport News, VA 23606, USA E-mail: [email protected] Linacs that are recirculated share many characteristics with ordinary linacs, including the ability to accelerate electron beams from an injector to high energy with relatively little (normalized) emittance growth and the ability to deliver ultrashort bunch duration pulses to users. When such linacs are energy recovered, the additional possibility of accelerating very high average beam current arises. Because this combination of beam properties is not possible from either a conventional linac, or from storage rings where emittance and pulse length are set by the equilibrium between radiation damping and quantum excitation of oscillations about the closed orbit, energy recovered linacs are being considered for an increasing variety of applications. These possibilities extend from high power free electron lasers and recirculated linac light sources, to electron coolers for high energy colliders or actual electron-ion colliding beam machines based on an energy recovered linac for the electrons.

1. Introduction Superconducting recirculated linacs have been developed for high duty factor nuclear physics accelerators and as drivers for high average power free electron lasers (FELs). Extending and applying this technology in new parameter regimes is likely to yield accelerators with highly novel capabilities. In this paper the current state-of-the-art in recirculated and energy recovered linacs is reviewed, and some current thinking on possible future projects based on high average current energy recovered linacs is presented. First, beam recirculation and beam energy recovery are presented in general terms. Next, the main applications of superconducting beam recirculation to present are reviewed, culminating in a discussion of the properties of the Jefferson Lab recirculated linac CEBAF [l], a continuous wave (CW) nuclear physics accelerator with high average current. This discussion is followed by a review of applications of beam energy recovery, which is most fully developed in the infrared demo free electron laser (IR DEMO FEL) at Jefferson Lab [2], * This work is supported by the United States Department of Energy under Contract Number DE-AC05-84ER40150.

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302 because this accelerator has energy recovered the highest average beam current to date. Both these devices give performance information relevant to future plans, and some aspects of their performance are summarized. There are many ideas, being pursued with various degrees of seriousness, about future applications of beam energy recovery. One can anticipate higher average power free electron lasers in the very short term, and light sources based on GeV-scale linacs roughly the same size as CEBAF in the near term. In the farther future, there is great interest to build high energy electron coolers to increase the luminosity of Brookhaven’s Relativistic Heavy Ion Collider (RHIC), and to design and build electron-ion colliders where the electron beam is accelerated continuously in an energy recovered linac (Em).All of these projects involve extrapolations beyond the present state-of-the-art. The major extrapolations will be pointed out within the text.

2. Beam Recirculation and Beam Energy Recovery The main accelerator types are displayed in Figure 1. If one focuses on electron accelerators, the first class of accelerators, perhaps best exemplified by the Stanford Linear Accelerator Center (SLAC) linear accelerator (linac) [3], are the high energy electron linacs. In such an accelerator, the electron beam has a definite beginning in an injector, and a definite end in an electron beam dump. The linac is usually timed so that maximum beam acceleration and maximum

-

RF Installation Beam injector and dump Beamline

Ring Linac

Recirculated Linac

Figure 1. Main accelerator types

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beam energy are obtained at the dump. However, it should be noted that one could just as easily choose to decelerate the electrons if the electrons actually arrive at the cavities time shifted by 180 degrees compared to the accelerated case. Some main features of an electron linac are: (1) An individual beam electron does not reside in the accelerator for much time at all, and particularly it resides in the accelerator for times that are short compared to any radiation build-up times present. (2) Assuming a photoinjector electron source, it is relatively easy to load, or program, the beam current or beam polarization delivered to users by controlling the intensity and polarization of the lasers which stimulate electron beam production in the injector. (3) The emittance, or transverse beam phase space area, of the electrons in a typical beam tend to be set by phenomena in the low energy electron source region, and the normalized emittance ( p ~ is) nominally preserved in accelerating to high energy. This means that the emittance at the point of delivery depends mainly on performance in the injector. (4) The pulse duration, and more generally the longitudinal phase space distribution, is relatively easily manipulated by using standard beam-RF and electron beam optical techniques. It should be noted that having long regions between the end of the linac and the beam dump, for experimental or other purposes, is easy in a linear geometry. The second class of high energy electron accelerators is the synchrotron-like storage ring [4]. In an electron storage ring the electrons are bent around on a roughly circular orbit. Because transversely accelerated electrons radiate electromagnetically, it is necessary to supply energy back to the electrons to achieve a long-term equilibrium. Energy is typically delivered with RF cavities that subtend a small portion of the total machine circumference. After beam is injected into the ring, the electrons rapidly settle into an equilibrium where the synchrotron radiation losses are just made up by the energy transferred from RJ? to beam. The relevant time scale is of order one radiation damping time (EmJA.E)rrcv, where Em, is the beam energy, AE is the energy loss per turn, and treyis the time it takes to make one revolution. The ratio of the damping time to the revolution time is a number typically of order lo3. Some properties of storage rings are: (1) the emittances in electrons circulating in an electron storage ring cannot be designed to be arbitrarily small, due to quantum excitation of betatron oscillations, (2) the equilibrium pulse length in the storage ring is likewise, not arbitrarily small due to quantum excitation of synchrotron oscillations, (3) the beam lifetime is set by beam-beam interactions in colliding beam storage rings or by Toushek scattering in storage ring light sources, and (4) storage rings are highly efficient in transferring F W

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power into beam power. Because they are highly efficient, storage rings may operate at much higher average currents than has been possible in single pass linacs. Recirculated linacs are accelerators which share with linacs the facts that there is a substantial RF linac accelerating the beam and that there is a definite beginning and ending of the beam (i.e., there is no closed or equilibrium orbit), but like a storage ring, the beam transits the accelerating cavities more than once. Such a hybrid arrangement allows one to meld in one accelerator some of the advantages that both of the traditional arrangements have separately. Application of beam recirculation to electron accelerators is an old idea [ 5 ] . The first accelerators to employ beam recirculation were the microtrons [6]. These devices use a resonance between the RF frequency of an accelerating cavity and the relativistic cyclotron frequency in a uniform magnetic field, to develop a phase-stable accelerator where the electron orbit enlarges an integral number of RF wavelengths on each pass through the cavity. To incorporate more substantial linear accelerators in the recirculation path, and to go to higher energy, racetrack microtrons were developed [7-111. As still higher energy was desired with time, it was completely natural to evolve away from the limitations imposed by the large end magnets of racetrack microtrons, and in a direction where different energy beams had completely different orbits, as in the classical recirculated linac. Jefferson Laboratory's CEBAF machine is the largest extent recirculated linac, achieving almost 6 GeV beam energy and 200 pA of beam current [l]. Till now, the usual ruison d'etre for beam recirculation, extending even to the CEBAF design, was economic. Given that beam recirculation systems tend to be much cheaper to build than additional RF linac, it makes sense to reuse the expensive beam acceleration systems as many times as possible to achieve as high an energy as possible at a given RF installation. As supporting examples of this statement, there have been two instances, at MIT Bates [12] and at Stanford [131, where front-to-back electron recirculation was used to upgrade the energy reach of an existing linac. In the future, it is likely that electron recirculation will be applied to build recirculated linacs because of their superior beam quality. Recirculated linacs share with linacs the ability to accelerate and preserve the emittance of very low emittance injector beams. Because the transit time is short compared to a typical radiative emittance build-up time, no equilibrium is established as in a storage ring, implying that the emittance delivered to the end user may be smaller. Also, as in linacs, one has the ability to manipulate the longitudinal phase space of the electron beam to deliver very short beam pulses to the end user. The minimum

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pulse length is no longer set by radiative effects, but by ones ability to generate, and precisely manipulate, the longitudinal phase space of the electron beam, as shown many years ago at CEBAF [14]. Such advantages of the recirculated linac might not be so interesting, except by the application of beam energy recovery. This idea has allowed one to conceive of recirculated linacs with high average currents and recovery efficiencies approaching those in storage rings. An early suggestion for the use of beam energy recovery appears in a paper by Maury Tigner [ 151. In this paper Tigner explored the possibility of basing a particle physics colliding beam machine on an energy recovery linac. While not stimulating a collider project, the idea spread throughout the accelerator community, even if not directly used. The basic idea of energy recovery in a recirculated RF linac is that the RF fields, by proper choice of the time-of-arrival of the electron bunches in the higher beam passes, may be used to both accelerate and decelerate the same beam. Consider the simplest case of a single recirculation in Figure 2. Beam is injected into the linac and timed to accelerate on the first pass up the linac. If the recirculation path length is chosen to be precisely an integer plus Yi RF wavelengths, on the second pass through the linac the beam is decelerated by the same RF field which accelerated it on the first pass. For cavities within the recirculation loop, energy is directly transferred, via the RF field, from decelerating beam to accelerating beam. The key point is that these RF power systems do NOT need to provide the energy to accelerate the first pass beam. Indeed, the RF power draw becomes almost completely independent of the beam current.

Figure 2. Definition of the recirculation path length for a simple two-pass recirculated linac

How can one define the efficiency of the energy recovery process? It is easiest to think in terms of the energy in the electron beam, which is being recycled and reused. If Em is the total beam energy at the point-of-use, and if

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Erec is the total energy that may be removed from the electron beam and/or reused again, define a recovery efficiency qrec=Erec/Emax. The energy that must be provided for acceleration, per electron, is proportional to l-qrec;the total power that must be supplied in order to accelerate a beam of current Zb is Z&(1- qrec)/e,where e is the fundamental charge. This power may be considerably less than the beam power Pb=Z&,,,-/e in an EIU, and is smaller the closer qrec comes to 1. In some contexts, particularly in comparing different accelerators, the recovery efficiency is inconvenient for quantitative work. One may define a recovery factor k,,, = l/(l-qrec) as more convenient. A recovery efficiency of 99% corresponds to a recovery factor of 100;as discussed above, for storage rings the recovery factor is about 1000.It is useful to define the multiplication factor [16] k=PdP,, where P , is the power incident on the RF cavities accelerating the beam. This quantity is useful for distinguishing the ultimate energy recovery limits in practical accelerators. For a superconducting recirculated linacs, P , is only slightly above Z&(1-qrec)/e;the multiplication factor may approach the recovery factor. For normal conducting linacs, because the wall losses are so large compared to I&(1-qrec)/e,the multiplication factor tends to fall far short of the recovery factor. By the first law of thermodynamics, kcl for single pass linear accelerators that are not energy recovered. Modern superconducting ERL designs can have k of order 100 or higher. There are challenges associated with choosing to build a recirculated linac. There is an additional linac beam instability possible, called multipass beam breakup instability, roughly corresponding to multi-bunch beam instabilities in storage rings [5,17,18]. The designer must assure that the RF cavity high order modes (HOM) that are deflecting have enough damping to prevent the appearance of this instability. The turn around optics in recirculated linacs are a bit more involved than in linacs or storage rings. In particular, the transverse beam dynamics tend to be affected appreciably by the longitudinal beam dynamics designs. Most significant of all, one must develop high average current injectors because the electron beam must be continuously supplied to the accelerator. Some details of the design depend strongly the application requirements and will be discussed at the appropriate location below.

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3.

Recirculated Superconducting RF Linacs

In this section the development of superconducting recirculated linacs is briefly reviewed. Before continuing, it should be noted that much of the historical information presented here may be found more thoroughly presented in Rand’s excellent book Recirculating Electron Accelerators [5], along with a discussion of the various normal conducting recirculated linacs that have been proposed and/or built. Many standard techniques in the field have been developed first at normal conducting machines and will be appropriately referenced in this paper, but time considerations prevent an extensive review of normal conducting recirculated linacs here. The first recirculated linacs developed were the so-called racetrack microtrons. As shown in Figure 3, in a split-field racetrack microtron, two uniform magnetic field bend magnets are used to bend the electron beam through 180 degrees and the electron beam makes several passes through the linear accelerator structure. Such a design choice allows greater flexibility in parameter choice, and by cascading, allows one to accelerate to beam energies far above those possible in conventional microtrons. High duty factor normal conducting racetrack microtrons have been built at Mainz [9,10] and the National Bureau of Standards (now the National Institute of Standards and Technology) [ 111. However, the first split-magnet racetrack microtron was also one of the first applications of RF superconductivity. The University of Illinois built two superconducting racetrack microtrons, dubbed MUSL I and I1 (for Microtron using a SuperconductingLinac), the first of which was completed in 1974. MUSL I was a 6-pass microtron that produced 5 pA average beam current at 19 MeV beam energy and 50% duty factor 171. The state-of-the-art in superconductingcavity performance at that time was about 2 MV/m, and such an accelerator was interesting for nuclear physics applications because of the high duty factor operation possible. MUSL I was upgraded to an 80 MeV racetrack microtron MUSL 11 by improving the superconducting accelerator and by increasing the magnetic field in the 180 degree bends closer to their allowed maximum [81.

308 1

I

I/

{

INRfCTOR CUICANE

Figure 3. The second Illinois superconducting racetrack microtron MUSL I1 [8],0 1977 IEEE

The next superconducting recirculated linac developed, a classical independent orbit recirculator, was at Stanford University about a decade later. It was constructed using the pre-existing Superconducting Accelerator (SCA), originally built for nuclear physics research. In an independent orbit recirculator the separate beam energies in the accelerator travel along completely decoupled beam orbits. Thus microtron-like resonance conditions are not utilized in the accelerator design. The beam from the SCA was recirculated through one orbit in order to increase the energy reach of the linac, and was used in the first demonstration of “same cell” beam energy recovery at a superconducting linac [ 131. This device will be described in more depth below. The SCA was followed by the superconducting S-DALINAC at TU Darmstadt [ 19,201, also developed primarily for nuclear physics applications. This device features relatively high frequency 3 GHz cavities and 3-pass beam recirculation. Typical beam parameters for nuclear physics running are CW beam energies up to 87 MeV, and beam current up to 50 pA in a 0.017 pC bunch charge at 3 GHz bunch repetition rate. Because the recirculated linac provides excellent beam quality, this device was upgraded to provide FEL laser light. In this operating mode 5 pC bunches are provided in a 10 MHz stream. Two especially noteworthy achievements demonstrated at this accelerator were precision RF control of the accelerating field and amplitude at a relatively high

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fundamental mode loaded quality factor of 3x107, and isochronous recirculation, i.e., the recirculation time was independent of the energy in its early operation. The largest example of an independent orbit recirculated linac is the 5-pass Continuous Electron Beam Accelerator Facility (CEBAF) accelerator at Jefferson Lab. This accelerator, one of the flagship accelerators in the United States Department of Energy’s nuclear physics program, was explicitly developed to extend and explore electro-nuclear scattering at beam energies and event rates much greater than any of these predecessor facilities. A beam energy of 4 GeV and a beam current of 200 p4 CW were initially specified as operations goals for the accelerator. Because of cavity processing developments during and after the project was completed, it has been able to extend the average operating gradient of CEBAF beyond the 5 MV/m design goal to 7.5 MV/m, yielding a machine that operates almost to 6 GeV. This accelerator is designed and operated in a way that produces excellent beam quality. For example, the injector compresses the electron bunches to very short durations before acceleration [ 14,211. Because the recirculation arcs are isochronous there is very little change in the bunch duration as the beam is accelerated. The linacs are operated on crest phase, yielding as small an energy spread as possible, and the multiple beam passes are verified to be very precisely near to an integer number of RF wavelengths long 1221. CEBAF has yielded a wealth of information about operating large superconducting RF (SRF) linacs. An example is the RF control of large numbers of superconducting cavities. CEBAF’s tolerances for phase and amplitude errors have always been more stringent than is typical in storage rings. This requirement, coupled with the desire to have an extremely high fraction of the cavities operating properly at all times, compels one, in the end, to have highly automated systems that continuously monitor and adjust cavity tuning, and that can be invoked to place the cavities on their initial tune after an extended down. Robust automatic procedures of this type were developed as part of the commissioning process at CEBAF. Several other examples were new beam diagnostics developed to facilitate and expedite machine setup. For example, as part of injector setup, phase transfer function measurements, which give linear and higher order longitudinal transfer maps, have been highly developed [21]. Such measurements allow one to precisely diagnose and control the longitudinal phase space of the compressing bunches, and allow one to routinely deliver ultrashort bunches (or <200 fsec) in standard operations mode [23]. Another example is the path length systems, which are used to insure that the path length of each pass is an integral number of RF wavelength. Gross path length checking is completed using

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relative time-of-flight analysis of beam-generated signals from longitudinal pickup cavities [24]. For very fine tuning and continuous monitoring of the beam phase, lockin systems that generate slight phase modulations in each of the linacs separately, coupled with simultaneous energy analysis at the lockin frequency yields the deviation of each of the beam passes from perfect cresting in each linac separately [25,26]. Fast feedback systems have been developed to simultaneously correct the position, angle, and energy of the beam being delivered to the end nuclear physics users [27]. The order of magnitude of the residual fluctuation levels is 10 pn rms,in position, 1 p a d rms in angle, and rms in energy. These three preceding achievements have allowed about 1 x the measured rms energy spread of the beam delivered to users to be at the 3 x 16’ level for extended run periods [22]. 4.

Energy Recovered Superconducting RF Linacs

The first energy recovery experiment to employ a superconducting recirculating linac occurred more than two decades after Tigner’s early paper, and was driven by the needs of the free electron laser community. “Same-cell” energy recovery, i.e., energy recovery in which both accelerating and decelerating beam traverse the same accelerating cavity, was accomplished at Stanford University’s SCA in the mid-1980s. By this time the SCA was principally used as an FEL driver because of the good quality of the accelerated beam. A single pass through the superconducting linac yielded sufficient energy to produce an IR FEL. In order to produce shorter FEL radiation, down to about 0.5 pm, the beam was recirculated through SCA on a second pass. In this configuration, the superconducting linac took the beam energy from 5 to 50 MeV on the first pass, was recirculated with an integral number of RF wavelengths path length, and accelerated from 50 to 95 MeV on the second pass through the linac. The other main parameter of interest, 150 pA of beam current, was provided by a stream of 12.5 pC bunches at 11.8 MHz repetition rate. Because the beam recirculation system allowed the path length to be varied through a full RF wavelength, choosing to shift the path length by Yz an RF wavelength allowed energy recovery to proceed. Detailed RF power measurements and comparisons between accelerating and energy recovering modes, and between beam present and absent on a second pass, showed that only 10% of the RF power needed to accelerate a single beam pass, was needed to maintain the RF field at the same gradient when the same current was energy recovered [ 131. Consequently, even in this first attempt the recovery efficiency was greater than 90% and the multiplication factor was about 0.6. In speculating

31 1 about future applications to increase the efficiency of FELs by applying energy recovery, the increased difficulty of recovering a “spent” FEL beam was noted. In the interest of properly attributing priority, it should be noted that nearly a year earlier a same cell energy recovery experiment was successfully performed at Bates Laboratory on a normal conducting linac [28]. During the period 1990-1994 a beam recirculation experiment was completed on the CEBAF SRF injector at Jefferson Lab [29]. This injector was capable of accelerating in excess of 200 pA beam current from 5 to 50 MeV, the beam being a continuous stream of 0.12 pC bunches at 1497 MHz. This experiment was primarily developed to demonstrate beam stability against beambreakup instability; this instability was worst in energy recovered recirculation because the average beam energy is lowest then. The best performance obtained in this device was 30 pA in energy recovery mode and between 64 and 215 pA in accelerating mode, the current depending on the beam optics of the recirculator. The recovery efficiency was 40/45=89%, and because the SRF systems were not optimized for energy recovered operation and the beam current recovered was so low, the multiplication factor was only 0.2. The final example of an energy recovered SRF linac is the Jefferson Laboratory IR demonstration FEL, which has energy recovered the highest average current to date. Referring to Figure 4, beam originates in a 5 MeV injection region, yielding a stream of 65 pC, 75 MHz bunches ( 5 mA). The electron beam is merged onto the linac axis, accelerated to 35-50 MeV (depending on the FEL operating wavelength), bunch compressed and delivered to the wiggler where the FEL interaction takes place. After the FEL interaction the longitudinal phase space is highly degraded as the relative energy spread has increased. The electron beam is recirculated using a beam optical system first used at Bates Laboratory. This recirculation system features an energy acceptance that is quite large [12]. After reinjection into the linac on the decelerating phase for the second pass through the linac, the low energy beam is split off and dumped in a conventional water-cooled beam dump.

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Figure 4. The Jefferson Lab IR demo free electron laser This device has experimentally demonstrated quite a number of important features of energy recovery [30]: (1) almost lossless (>99.98%) beam recirculation, including the effects of the energy spread generated by the FEL [2] (2) world’s best multiplication factor to present of about 16, (3) manipulation of the longitudinal phase space of compressing and decompressing electron bunches [31], (3) RF stability in a high current energy recovery configuration [32], (4) beam breakup stability measurements 1331, (5) detection of HOM cooling load in the cryogenic systems of SRF cavities [33], and (6) high power optical beam handling.

5.

Future Projects

To present, superconducting recirculating linacs have operated at beam currents far below the carrying current limit imposed by HOM heating and beam breakup stability. Many groups have started exploring the kinds of applications that might arise if one devises recirculating linacs, especially those that are energy recovered, where the average current is much closer to these limits [341. Presently, the applications fall into three broad categories: still higher average power FELs, recirculated linac light sources, and colliding beam accelerators where the electron beam is continuously accelerated in an energy recovery linac,

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instead of being stored in an electron storage ring. Each of these possibilities will be discussed in turn.

5.1. High Average Power FELs In the near term, at least two FEL projects will continue to push up the average recirculation current achieved to date. The JAERI FEL has operated as a single pass SRF FEL driver for a number of years [35]. Due to limited refrigerator capacity, the duty factor of the device has remained under 3%. However, within the beam macropulse the current is 5.3 mA, in a stream of 510 pC 10.4 MHz bunches. This yields an FEL performance comparable to Jefferson Lab’s IR DEMO within the macropulse. Recently, the recirculation loop on their IR FEL has been closed and energy recovery has been demonstrated. By upgrading the electron beam injector to produce 40 mA beam current in 1 nC 40 MHz bunches, this device will be able to produce 10 kW output optical power within the beam macropulse [361. Jefferson Lab is replacing the IR DEMO device with one that will operate at roughly an order of magnitude higher optical power, greater than 10 kW average power. This will be accomplished by doubling the bunch charge to 130 pC and beam current to 10 mA, roughly tripling the beam energy to 160 MeV maximum, and increasing the outcoupled fraction of optical power by roughly a factor of two [37]. This final change will increase the relative energy spread of the spent beam, allowing one to further explore the limits of energy recovery as energy spread is increased. This device is nearing completion presently. The upgrade path for the farther future is fairly clear. Another factor-of-ten in electron beam power (and hence optical beam power in the 100 kW range) is simply obtained by filling every RF bucket in the superconducting linac. This increases the bunch repetition rate from 75 MHz to 1497 MHz at maximum if CEBAF superconducting cavities are used in the application. The resulting average current of 200 mA is far above the BBU stability limit of the present CEBAF cavities, but improvements in the HOM damping and HOM cooling designs are anticipated to allow operating at such high average current. To continue still further will require the bunch charge to increase from the 100 pC scale to 1 nC or so, in the regime of present day SASE sources. To make this step, in addition to continuing to improve the HOM damping and cooling still more, one will have to come to grips with the myriads of single bunch collective effects that start to afflict high charge-per-bunch beams. The desire to reduce single bunch collective effects will likely cause MW scale optical power FELs to be built at lower SRF frequencies, in the range 350-750 MHz.

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5.2. Recirculated Linac Light Sources As mentioned in the introduction, and in the preceding paragraph, one may anticipate that the average current carrying capacity in a recirculated linac will approach the average current of present day storage ring light sources. Thus, it is natural to ask whether recirculated linacs, and particularly those that are energy recovered, would make interesting light sources. Because of the possibility of short pulse length radiation, because of the possibility of smaller beam emittance and energy spread than in storage rings, which leads to higher photon brilliance, and because of the possibility of round sources, there is much present activity in designing such light sources. Some of this activity is summarized in a recent ICFA publication [38]. The idea of generating short-pulse radiation from a recirculated linac developed gradually. Early work on SASE generated DUV in a recirculated linac arrangement was completed in an exploratory way early in the CEBAF project [39]. Also, when ideas about short pulse SASE sources were first being developed, it was natural to ask whether the CEBAF linac, or a recirculated version of the CEBAF linac, could serve as an X-ray source [40]. Because of the long time-scale for development of SASE sources, and their large expense, various groups began exploring whether alternate sources of short pulse X-ray radiation were possible. Thomson scattering sources have been used to produce short pulse radiation, and recent possibilities in storage rings have centered around so-called femtoslicing methods. In such a method one uses the fact that femtosecond optical pulses already exist, and by illuminating a storage ring beam with such an optical pulse, one can manipulate the beam to radiate short Xray pulses [41]. One disadvantage of such a method is that only a small portion of the storage ring beam actually radiates usefully. Because the total pulse length can be made short at delivery out of a recirculated linac, and because the average current of even existing recirculated linacs is close to the emitting current in the femtoslicing methods, it make sense to suggest that one obtains short-pulse Xrays directly from a recirculated linac [42]. The idea that superior beam emittance and energy spread from a recirculated linac [ l ] leads to higher ultimate photon brilliance, has been advocated for several years by a group from Novosibirsk [43]. They have begun a multipass normal conducting energy recovering linac as a result [43,44]. Aside from this Russian work, all of the other proposed projects assume that a superconducting recirculated linac will be used. The main proposals at present are the CornelVJlab energy recovery linac (ERL) proposal [45,46], Brookhaven’s PERL proposal [47], LBNL’s LUX project [48,49], Daresbury’s 4GLS proposal [50], and University of Erlangen’s ERLSYN proposal [51].

31 5

There is some effort in Japan too, but no serious proposal as of yet. In the CornelVJlab proposal and Brookhaven’s PERL, the bunch structure is very much as in the 100 kW FEL above. The CornelVJlab proposal has 77 pC with a bunch repetition rate of 1.3 GHz yielding 100 mA in its high flux mode. PERL has twice the bunch charge and twice the average current. Both of these projects posit an injector that produces very small normalized emittance of less than 2 mm-mrad compared to around 10 mm-mrad in the existing IR DEMO source. According to the numerical modeling, this small emittance should be possible given the present understanding of emittance growth in photocathode sources. LBNL’s LUX proposal has a much larger charge-per-bunch, around 1 nC which is comparable to that of SASE sources, and a much lower maximum repetition rate of 10 H z . Daresbury’s 4GLS develops a superconducting linac first as a SASE source, and then closes the recirculation loop to provide a high brilliance CW source optimized for D W wavelengths. Erlangen has an energy recovery linac, with parameters very similar to those of Cornell, as the second phase of a project whose first phase is to build a third generation storage ring source. Cornell and Jefferson Lab have proposed an ERL demonstration prototype as a first step to a follow-on CEBAF-scale light source. This prototype would demonstrate full current injection with the required beam properties, acceleration to 100 MeV, and high efficiency energy recovery. The issues to be demonstrated include producing low incjector emittance; preserving the emittance on acceleration; showing adequate photocathode longevity is possible; manipulating the longitudinal phase space properly to achieve adequate beam bunching; achieving adequate HOM damping, low beam loss, the highest fundamental mode quality factor possible to reduce the size of the cryogenics plant for the follow-on machine, and the highest fundamental mode loaded Q possible to increase system efficiency by reducing the required RF power; and developing high average current beam diagnostics. These results will be important for the accurate design of all the follow-on projects.

5.3. Applications to Colliding Beam Machines

Two types of applications of energy recovery are anticipated in high energy accelerators: beam cooling and high-luminosity colliding beams. In order to apply the beam cooling technique called electron cooling at the Relativistic Heavy Ion Collider (RHIC) and increase its luminosity, it is necessary to provide intense 50 MeV electron beams. As the cooling rate increases with the average electron current, one would like to accelerate. a high average current with a bunch structure that overlaps with the RHIC bunch structure in time. Because a recirculated linac may provide such a repetitive beam at high average current, it

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is a natural candidate as an electron cooling device. Such a device becomes particularly attractive and efficient if energy recovery is applied. The second type of application is to replace the electron storage ring in a ring-ring electronion collider with an electron energy recovery linac. As discussed below, it may be possible to obtain high luminosity out of such a configuration because the beam-beam problem is reduced when only one beam is stored. Brookhaven National Lab has completed a conceptual design of a beam cooler [52]. This device originally had average current similar to the IR DEMO FEL. The main novelty in the design is that for the cooling application it is preferable to debunch the electron beam before the cooling interaction. There is a debunching system before and a rebuncher after the cooler solenoid magnet to provide for clean energy recovery. Aside from the beam optical design, and because of the enhanced cooling rate with current, there is a very real desire to increase the average current in the recirculated beam beyond the FEL experience. As discussed above, to increase the beam breakup stability threshold to high values, one must make sure that all of the transverse deflecting HOMs are damped to suitable levels, i.e., Q values much smaller than present, or if this proves difficult, a feedback system, similar to those used presently in high current ring colliders, must be developed that can control the instability. Because the HOM cooling load will increase with the average current, it will be necessary that the load terminations occur at high temperature in order to keep the dynamic heat load from the HOMs to manageable level . Because a relatively high bunch charge will be needed, it will be advantageous to consider lower frequency superconducting cavities to reduce single bunch collective effects. The final idea to be discussed in this paper is to have an electron beam in an ERL colliding with a proton, or other ion beam in a storage ring [53].A simple gedunken experiment shows why such an arrangement might be advantageous, and produce higher luminosity. The luminosity of a collider is

where f is the collision frequency, N , is the number of electrons in the colliding bunch, Niis the number of ions in the bunch, and the rms beam sizes of the two colliding beams (assumed unequal) are given by the 0s. Assume that a stable ring-ring collider design, limited by any phenomena you chose to consider, is given, and in particular we know all the inputs to the luminosity formula above. For the electron ring to be stable, any current-limiting instability growth must be slower than one ring damping time, as otherwise the ring would not be stable.

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This means that the electrons have to be confined up to one damping-time’s worth of revolutions, i.e. about 1000 revolutions. Suppose now one has an ERL collider design with an identical ion storage ring, and an identical set of electron beam parameters as the ring-ring collider design above. Because the electron beam parameters are the same, the ion beam will continue to circulate stably, even if one increases the ion bunch charge and luminosity considerably. Increasing the ion bunch charge will increase the disruption of the electron bunch by the beam-beam effect, but it is no longer necessary to confine the electron beam for 1000 turns, only a few turns in the ERL. Easy estimates of the emittance increase, and the maximum deflection angle generated by a few beam-beam collisions, shows that there may be room to considerably increase Nibefore energy recovery becomes difficult. Detailed initial parameter lists, covering two collider schemes, have been worked out presently. The first is, eRHIC, an electron-ion collider, based at RHIC [53,54],and the second is ELIC, an electron-light ion collider based at CEBAF [55,56]. In the eRHIC proposal, one of the RHIC rings is used to contain the ions and a new ERL is built. In ELIC, CEBAF is upgraded to make a higher-energy ERL, and a new ion storage ring is constructed. Two main obstacles, one technical and one conceptual, must be overcome in order to construct credible proposals. The technical obstacle is that a CW 1 A polarized electron gun is about 4 orders of magnitude in average current beyond present experience. The conceptual obstacle is to precisely quantify the luminosity advantage going from storage ring to ERL by thorough simulation and luminosity comparisons between ring-ring and ERL-ring collider designs.

6. Conclusions In this talk the ideas of beam recirculation and energy recovery have been introduced. How these concepts may be combined to yield a new class of accelerators that can be used in many interesting applications has been discussed. Some historical information about the development of recirculating SRF linacs has been given as well as performance information on the CEBAF machine and on the IR DEMO FEL. The present knowledge on beam recirculation and its limitations in a superconducting environment leads one to think that recirculating accelerators of several GeV energy, and with beam currents approaching those in storage rings are possible. Cornell University and Jefferson Laboratory have proposed to the United States National Science Foundation to build a prototype high current superconducting energy recovery linac to investigate in depth some of the limitations of energy recovery linacs.

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Acknowledgments The author thanks the program committee for the opportunity to present this paper. It is hoped that continued activity in this field will lead new results and new applications of this highly interesting type of accelerator.

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MUON COLLIDERS AND NEUTRINO FACTORIES: BASICS AND PROSPECTS A. SKRINSKY

Budker INP, Novosibirsk, 630090 Russia E-mail:[email protected]

The prospects of high energy physics are outlined with the advent of muon colliders and neutrino factories. Technical features of the novel accelerators are surveyed, with emphasis on optimization of the ionization cooling technique for muon beams. The ultimate luminosities of the muon collider are discussed as well as minimization of the detector background.

1. Introduction Let us assume - Basic Physics (almost equal to Elementary Particle Physics in a wide sense, including Astro-Particle Physics) would not die - with all its machinery, huge teams, substantial cost, long lasting projects, etc. It is my belief: in the long term, it is a correct approach for Mankind. And with this assumption let us consider the reasons for developing muon collider(s) and, technically speaking, its sub-step - neutrino factory(ies). Other colliders - photon-photon, photon-electron, lepton-hadron, photon-hadron need special discussion. In all our considerations we will keep in mind “very” high energies. Using existing technologies, it seems easier to construct a hadron (proton) collider for the same energy and for the same luminosity of fundamental interactions, as those in lepton-lepton collisions. To reach this goal, the hadronhadron energy should be 10 times as high as the lepton-lepton one: hadrons bring to the collision their complex structure (quarks and gluons), whereas for lepton colliders (involving electrons/positrons or muons) we deal with fundamental (unstructured, to our present knowledge) incidental particles directly. Additionally, in hadron collisions the fundamental interactions are not at all monochromatic (100% energy spread in collision!), and each fundamental interaction is accompanied by many interactions of remnants of hadrons, which produced the given fundamental interaction. Even in the frame of the Standard Model, muon colliders and e+‘- (linear) colliders are not just technically different versions of lepton colliders of the same

322

323 energy. This difference relates to much higher “parasitic” radiation in the electron case - in two senses: 1. The radiation of individual initial colliding particles is much higher for electrons (and increases with the energy). Tagging of such photons (and, moreover, measuring their energy) is almost impossible for options with the high luminosity-per-bunch collision, currently under development. 2. Coherent fields in the collision region are so high that the synchrotron radiation of counter-propagating electrons (4-)takes off a substantial fraction of their energies. Hence, instead of pure, say, electron-positron collisions with a narrow spectrum we would get a wide initial spectrum, a lot of parasitic photons and photon-photon and photon-electron collisions. Eliminating this background will be one of our major headaches (on background issues for muons - later!). If we go to higher energies, the e+‘- collider becomes more and more difficult technically - to prevent the growth of synchrotron radiation in the coherent field of the counter-rotating bunch and to keep the luminosity growing, we need to make the vertical size of the interaction spot one nanometer or smaller! There is no such a limitation for the muon collider (the rest mass is much greater!). But, maybe, at energies of around 1 TeV and higher some new physics would appear, and the muon would be not just a heavier eIectron with completely new interactions. In this case the muon collisions would bring fundamentally complementary information to electron ones, and the muon collider becomes a m . Muon storage rings required for muon colliders would give birth to excellent muon and electron neutrino/antineutrino beam sources - so-called Neutrino Factories. There, the relevant physics is very different from muon collider physics, but the accelerator technologies that should be used are similar in many aspects. Why should Neutrino Factories be very useful? 1. Narrow (cooled) intense muon beams in storage rings of high enough energy produce narrow (with a transverse momentum of around 30 MeV/c) v,, v,, antiv, and anti-v, beams, and enable very complete studies of neutrino interactions (behind perfect muon shielding!). 2. Such beams are perfect for long-distance neutrino studies (neutrino oscillations and related topics). Accelerator technologies for colliders and factories are similar in many aspects: ionization cooling (albeit weaker requirements for Neutrino Factories, an easier “first” step); “fast” muon acceleration;

324 muon decay ring (without highest field requirement).

2.

Muon Acceleration

We, at Novosibirsk, started - already in the 1960'~'-~- to consider options for lepton colliders reaching energies of hundreds of GeV (and even higher) - linear electron-positron and muon colliders, in parallel. At first glance, the main disadvantage of muons was their very short lifetime (2.2 ps in the rest frame). Of course, the muon life-time grows proportionally to its energy, E,IE,o, but it remains short. Hence, the cooling of muon beams and their acceleration to the energy required should be fast enough. It is easy to estimate the average accelerating gradient required: d N a -a=dN -NP _ _ NP -

EP zpo.-.c dx

.

EO

Hence,

and the required gradient is

Hence, to lose only 10%upon acceleration from 200 MeV to 2 TeV, the average acceleration gradient should be 14 MeV/m - quite modest on a modem scale! The limited muon lifetime entails the requirement to use the highest possible magnetic field Bmllin the collider itself. The number of turns useful for muonmuon luminosity in the collider (for an interaction region per turn) is

and does not depend on the muon energy! For the average collider field of 10 Tesla, Nlumi= 1500. And at a few TeV and 10 cycles per second, collisions of muon bunches become continuous in time!

325

In this paper, we consider a general scheme of the muon collider complex as presented in Fig. 1.

-

Proton injector. Proton synchrotron. Proton-to-piontargetlcollection system. Pion-to-muon decay channels. Preliminary ionization cooling & single polarized +/- muon bunches formation. Final cooling of muon bunches. +I- Muon accelerator

Muon collider

Figure 1. Muon collider complex - very schematic!

3.

Ionization Cooling

But the crucial point is to compress a very large initial 6-emittance of muons to as small a volume as possible (and of course, with minimal intensity losses). In this paper, we use the definition of normalized emittances as L

where AE = AE,f E,

.

ezx.y*P,rm - P p Y , = AE2 .PIong.P, .Y,

y

=

.

And from the very beginning (in the 1960’s!) we realized that the only possibility for cooling muon beams is to develop and apply ionization cooling.

326 Earlier, a few authors4 considered such cooling for compression of proton beams; but the conclusion was that the protons (as well as any strong interacting particles) would be lost because of nuclear interactions in the matter faster than their cooling process rate, so that under practical conditions ionization cooling is not useful. This statement is even more correct for electrons (positrons) because of bremsstrahlung. But just for muons - and for muons only! - ionization cooling is what we need: muons have normal ionization losses, but no strong interactions and negligible bremsstrahlung (below 1 GeV, where this cooling is of interest). And in our early reports (in 1969, 1970, 1971, 1980)’-3*5we always presented the ionization cooling as an essential part of the muon collider. But it was only when we presented a reasonably complete theoretical consideration of ionization cooling as an inherent part of the muon collider (1981)6 that the whole approach attracted interest, and now many groups in different labs throughout the world are actively developing different options for various stages of the muon collider. And even an International collaboration for a Muon Collider was established and is now operating. Unfortunately, there is almost zero support from corresponding financial agencies - around the Globe! Maybe, this is just a reflection (or a part) of the general crisis in basic science support. The principal idea of ionization cooling is quite simple (at least, for the transverse part of 6-emittance): friction force due to ionization losses is directed opposite to the full velocity of the muons, but only the lost momentum parallel to the equilibrium orbit is restored by an external electric field (practically, some accelerating RF field). As usually, Life is more complicated, than the Idea: 1. For high luminosity we need to cool all the six phase-space dimensions, including the longitudinal one; but in the longitudinal direction, “natural” ionization cooling is relatively very slow, or even negative (heating instead of cooling); 2. Besides the useful energy losses, muons experience multiple scattering on nuclei and electrons of the stopping media, as well as strong fluctuations of the ionization losses. And, of course, the cooling time should not be very long - not to lose too many muons by decay. Let us evaluate (in the first approximation) the ionization decrements and the equilibrium 6-emittance of a muon beam under ionization cooling.

327

For the case of cooling due to “full energy losses” of any origin, the increment of six-dimension density (or the sum of decrements for all three emittances) is equal to

(energy losses of the “equilibrium particle” are assumed to be compensated by an external source). The power of ionization energy losses by a charged particle is

(here I is the effective ionization potential of atoms involved in the collisions). Consequently, the sum of decrements, expressed now in cm-’ of cooling matter, is (for small angles)

Kinetic Energy, GeV Figure 2. The “cooling length” for the 6-emittance (i.e., the product of all 3 emittances) in lithium

0.0.

328

As seen in Fig. 2, to cool the 6-emittance a million times, say, at 200 MeV (kinetic energy), we need to travel through about 15 meters of lithium (with a “continuous” energy recovery). In this case, 10%of muons decay at 200 meters. But - there are longitudinal problems! Figure 3 shows the “natural” longitudinal decrement as a function of the cooling energy.

100

E

0

~

0

___

200

400 600 800 Kinetic Energy, MeV

1000

Figure 3. Inverse longitudinal decrement.

You can see that the “natural” longitudinal cooling is really too slow. And at lower energies the longitudinal kcrement even converts into a fast hcrement! Hence, it is obligatory to effectively redistribute the sum of decrements in favour of the longitudinal degree of freedom ( K,,, is a fraction of the full 6emittance decrement transferred to the longitudinal degree of freedom) - this is a real challenge for inventors! The transverse cooling asymptotically sqeezes the muon angular spread down to the equilibrium one, i.e., to the multiple scattering angle acquired at one transverse emittance cooling length

If we forget the velocity dependence of energy losses, and other complications, ozeq is very transparent:

329

It depends on Z, but not on the average electron density or the focusing (if uniform); its dependence on the cooling energy is presented in Fig. 4.

a

0.3

-cJ

0.24

2 fi

$' 0.18 2

'

"bD 1 0.12

8

0.06

0

0

30 60 90 120 Kinetic Energy, MeV

150

Figure 4. The equilibrium angular spread resulting from ionization cooling.

The corresponding relative energy spread at equilibrium (due to balance of fluctuations of the ionization losses and the longitudinal cooling) would be 27criN, (2-pi) A:-eq = Klo,

62,

as shown in Fig. 5. As you see, the equilibrium angles and energy spread at energies of interest - upon full cooling! - are not small at all! Hence, the usual paraxial and monochromatic beam optics should work poorly - a big additional headache! And to reach small enough emittances at the final stage of cooling - and, hence, to reach an acceptable collider luminosity - at this stage we need to use very strong focusing in all three directions - as strong as practically possible!

330

2

0.03

8

0.025

E

gl 3

0.015

.3

0.01

h

0.02

h

.2 1

0.005

s o

0

20

40 60 80 Kinetic Energy, MeV

100

120

Figure 5. Equilibrium relative energy spread ( Kiong is assumed at its optimum, around 0.25).

For these final stages the best option for ionization cooling up to now (in my personal opinion, of course) seems to be the use of lithium rods with a strong current along the rod for as strong transverse focusing as possible (the scheme is presented in Fig. 6).

- Lithium rod

P+

RF linac

h=10cm 50 MeVIm

Lithium rod

RF linac

dia. 6 mm H,&> 10 Tesla

+P

cooler!

Figure 6. The scheme of the final stages of transverse cooling

To arrange the strongest possible transverse focusing, my preference is to use current-carrying liquid-lithium rods, which focus muon beams by means of the azimuthal magnetic field gradient, limited by magnetic field on the surface (10 Tesla or somewhat higher - pulsed operation mode):

For the parameters under discussion,

pij + 1 cm - quite good!

The whole device is just a very long lithium lens (in total), developed at Novosibirsk for positron and antiproton collection decades ago, and still in use now (INP,CERN,FNAL).The improved - and the first liquid - version is now under developement at Novosibirsk (Fig. 7).

331

Figure 7. Liquid lithium rod - operating pre-prototype for final cooling .

The radius of the rod and surface field at the final stage should provide an acceptance, say, 2-3 times as large as the final muon emittance; for a final lithium rod diameter of about 6 mm, and 10 Tesla on the surface in this case, the resulting transverse emittance is presented in Fig. 8. Since high repetition rates are necessary, we need to use liquid lithium to remove the Ohmic heat. a

36

0.06 0.05

0

0.04 0.03 v1

9

0.02

ti

0.01

2

0

30 60 90 120 Kinetic Energy, MeV

150

Figure 8. Normalized transverse emittance after final cooling.

332

But in order to reach the highest possible luminosity for any (affordable) number of muons (as we will see a bit later), we need the equilibrium normalized 6-emittance upon final cooling to be as small as possible: ‘mq-6

= Efwq_manEncq_long = ‘:q

We see here again, that ffrmand

*p?rcm-bc

pbnsin

PIong * p ~ m l $ c o o i

.

the cooling matter should be as

small as possible. The equilibrium emittance upon final cooling being limited, we need to cool all degrees of freedom; hence a reasonable fraction ~l~~~ of the sum of decrements should be redistributed to the longitudinal degree of freedom. It is still not clear, which option for the final cooling arrangements would give the smallest normalized 6-emittance at equilibrium. At the moment I like the following “helical option,” see Figs. 9 and 10. The magnetic field Hheli* and the radius should correspond to each other in a natural way:

3

+b Li

Li

Li

1

higher energy


muonbeam

lower energy

Higher at largerenergies radii:

9 p (colder! -lower

0

c Axial

normalized 6-emittance)

view of the helix

Figure 9. Schematic view of one section for simultaneous transverse and longitudinal cooling (helical option).

333

coider! Figure 10. The general scheme of the final steps of cooling. (The momentum dispersion should “extract” decrement from both transversal degrees - serially!)

In this case, the muons with mean momentum p,, would move along the center of the rod and all the acceptance of the rod would be used efficiently. But of course, the helix curvature should be high enough to produce a momentum dispersion that transfers, via berillium “teeth” (located at the outer part of the helix), a large enough fraction of one of the transverse decrements into the longitudinal one. Practically speaking, for an average muon kinetic energy around 150 MeV, to transfer a longitudinal decrement 610ng= Klong& (with averaging over several steps of that kind), for ~l~~~= 0.25, we need to apply &/s 7 Tesla, providing that &,el* 10 cm. To get the longitudinal emittance (and, consequently, the 6-emittance) minimal at the final stage of ionization cooling, we need at this stage an RF system operating at the shortest possible wavelength (10 cm?) and a high accelerating gradient (30 MeV/m on the average?), thus providing the smallest effective Plong (5 cm?). In this case, it is possible to reach the emittances shown in Fig. 11.

Kinetic Energy, MeV

Figure 11. Equilibrium longitudinal normalized emittance.

334

As a result, the final cooling would provide the 6-emittance shown in Fig. 12. We need the smallest final normalized beam 6-emittance. For such a high angular and energy spread (see above), to progress in this direction we need to find something unconventional.

d

"0

30 60 90 120 Kinetic Energy, MeV

150

Figure 12. Equilibriumnormalized 6-emittance.

4.

Matching Problems

One of the most difficult problems is proper matching of focusing in the sequential moderationlacceleration sections (from the exit of one moderator section to the entrance of the next one). High momentum spreads can lead to unacceptably high chromatic and non-linear aberrations (resulting in modest beta-functions!). The focal lengths of most familiar individual lenses - short solenoids or quadruple doublets - are proportional to the square of the momentum of the particles they are focusing. But the focal lengths of lithium and plasma lenses are just proportional to the momentum; hence, their use should make it possible to reach low enough chromatic aberrations in matching sections more easily. The following option that seemed promising to me is shown in Fig. 13. In the table below the drawing, p is the current muon momentum; Ekin is the corresponding kinetic energy; is the current beta-function; Lf, is the length

335 of section fractions; H,, is the magnetic field on the surface of the focusing , is the radius of the focusing element; If, is the peak current in the element; k element (all the numbers are rough and need careful optimization). Figure 14 illustrates the chromatic aberration of similarly structured matching sections using plasma lenses and solenoidal lenses. The beta-function at the first 4 c m in the second Li-Be helix f o r a nominalmuonmomentum (1.00) and f 10% deviation for the cases of plasma lenses (left) and solenoidal lenses (right), for the same corresponding focal lengths with optimization.

P Li-Be helix Li-Be helix Li Plasma RF accelerat. Plasma Li lens lens structure lens lens Figure 13. Schematics of a matching section between two consecutive cooling sections (geometrically not to scale!).

70

70

141 200

200

70

21

21

71

121

121

21

0.7

95

84

190

1.1

0.7

3

85

125

35

2

85

20

0

0.5

20

20

0.4

7

7

1

0.4

0.2

1

0.4

0.4

The short and strong lithium lenses at the exit and at the entrance of Li-Be helices are necessary to make the beta-function a few times larger than in the helices and to ease low-aberration functioning of plasma lenses with longer focal lengths. But using them (instead of plasma lenses) at much higher beta-values inside accelerating structures is impossible - the multiple scattering results in unacceptable emittance growth.

336 With Solenoidal Lenses '

3

With Plasma Lenses

.-

$

I

0.75

m 0.25 2 4 Path Length in Li-Be Helix, cm

0

Figure 14. Resulting chromatic aberrations: solid line for nominal momentum, dashed line for +lo%, dash-dotted line for -10% deviation.

In Figure 15 an option for the same kind of end matching section (for the exit of the cooling system) is presented. As seen from Fig. 16, the resulting chromatic aberrations are acceptably small - quite comfortable for the acceleration and emittance gymnastics further needed.

-+-nrT-Tl P Li-Be helix

Li lens

Plasma lens

RF accelerating Plasma structure lens

Figure 15. The matching structure of the last cooling/accelerating section (not to scale geometrically!). All the notations are the same as in Fig. 12.

337

70

70 141 200

21

21 71 95

-

200

121

84 190

190

0.7

3.3

85

0.7

20

125

3.4

20

20

0.3

0

0.5

0.4

0.9

4.5

I

7

0.4

0.9

72

90 100

0.2

25

0

so

100 Distance from the Last Section, cm

Figure 16. The beta-function upon the exit of the last cooling/accelerating section (the notation is the same as in Fig. 13).

5. Collider Luminosity If at the cooling stage the normalized transverse emittance E,,,~,~.,, and the longitudinal emittance Eneqlong were reached and were kept constant at all the stages including the collision, the bunch length at collision is limited by E,,+,,~ and by the maximal acceptable energy spread AE-; the transverse betafunctions are made equal to the bunch length, and the collider magnetic field

338 Hcoil provides for muons Nlumi effective turns prior to the luminosity e-fold reduction due to muon decay, the ultimate luminosity would be

This is seen from the following obvious chain of equations:

- N: 4n

.N

1

f

lumi (I

=%, rf,

& m " d ~ J r_rdl n8

4n

,

~

~

d

I

,

N,,f;>*

%trn%drn8

(we always assume that P,ron-co//= qoong-co~~). The luminosity is shown in Fig. 17. And for convenient cooling energies the luminosity would reach

L,

maxmax

- 0.5.lo3' cm-*sec-'

.

But if we calculate the beta-value at collision, assumed (as always here) to be equal to the muon bunch length, we would get 5 microns(!) - impractically short. p'

'?

5

30

60 90 Kinetic Energy, MeV

120

150

Figure 17. The luminosity of a "super-maximal" collider (EN=2 TeV i2 TeV, N,,=l.lO", H , ~ l 0 T, f ~ 1 5s-', with a fraction of the sum of cooling decrements transferred to the longitudinal direction I Q ~ . ~0.25), = with the equilibrium emittances reachable as the ultimate limit in the cooling process (see above), as a function of the muon kinetic energy at the cooling stage

Hence, we need to use a different limitation. If we limit additionally the the following formula for "practical" maximum luminosity bunch length olongco~~, should be valid:

4yl

-

=-.N: 47c

3: &):

ycoii

neq6

of: longcoll

f: +(%)

N,,f,.

339

It is obtained from another chain of equations:

--

With these conditions, the luminosity graph is shown in Fig. 18. In such a case, hp 0.5-1035 cm-2sec-’ for the same parameters as = 3 mm - also not bad! above, and As we see, “in more practical circumstance^" the equilibrium normalized 6-emittance &n,q6 enters the maximum luminosity directly. And the goal of find ionization cooling really is to make it minimal. But not only. For the finally achieved 6-emittance we need to control the partition of the transverse and longitudinal emittances - and optimize this partition together with the muon collider optics, keeping in mind the monochromaticity and polarization requirements, etc., - hence a “deep emittance gymnastics” is necessary.

“0

30

60 90 Kinetic Energy, MeV

120

150

Figure 18. The luminosity of a “maximal” collider (Er= 2 TeV + 2 TeV, Nr=1~1012, H,.1r=l0 T, f e 1 5 s-’, fraction of the sum of cooling decrements transferred to the longitudinal direction uOnp 0.25, C I I ~ . .3~ mm) with the equilibrium emittances reachable as the ultimate limit in the cooling process (see above), as a function of cooling kinetic energy.

340 For the purpose of emittances gymnastics, we can use a combination of dispersive elements, septum elements, RF acceleratingldecelerating structures, delay lines, (but not ionization components, which damage the 6-density by scattering!). Such a transformation should be arranged at some convenient energy of the muon beam. An option is presented in Fig. 19. ln-comine muon bunch:

Delay line

Out-coming muon bunch: “narrow, but lonzcr”

Figure 19. An example of “bunch gymnastics” needed to maximize the luminosity at a very high energy.

The “monochromatic” collider option (as in the case of the “low energy Higgs Factory”) could require muon bunch rearrangement in the opposite direction. Table 1 presents data for a collection of “ultimate colliders” for muons per bunch (luminosity per detector). Here Lffis the factor indicating the reduction of the initial polarization due to the energy spread in the collider. The first row presents parameters for the so-called Higgs Factory, which would be of interest at low mass, hence for a very narrow Higgs boson. The last row presents parameters of an “ultimate muon collider” currently conceivable. Similar options were considered in a Muon Collaboration Report under a somewhat different approach; see Table 2 for baseline parameters of high- and

341

low-energy muon colliders. Higgdyear assumes a cross section Higgs width = 2.7 MeV; 1 year = lo7s.

(3

= 5x lo4fb; a

r

Table 2Ed

N,

L

m

5

v.

Vodl

E

10l2

1: LIT

Mv

Lw

em -* s

-'

Table 2. Baseline parameters for high- and low-energy muon colliders. CoM energy (TeV) p energy (GeV) p's I bunch Bunchedfill Rep. rate (Hz) P Power (MW) F I bunch c1 Power (W) Wall power (MW) Collider circum. (m) Ave. bending field (T) Rms Ap/p % 6-D E6.N

(7w3

Rms

&,

(nmm-mrad)

B* (cm) 0,(cm) 0,spot (w9

oeIP (mrad) Tune shift nhrm(effective) Luminosity cm-'s-'

Higgs I year

3 0.4 I 16 16 2.5~10'~ 2.5~10'~ 4 4 15 15 4 4 2x 10'2 2x10'2 28 4 204 120 6000 lo00 5.2 4.7 0.16 0.14 0.12 1 . 7IO-" ~ 1 . 7lo-" ~ 1 . 7lo-' ~

0.1 16 5x 10'' 15 4 4x 10'' 4 81 350 'I

0

50

0.3 0.3 3.2 1.1 0.044 785 7~10'~

50

85

195

290

2.6 2.6 26 1.o 0.044

4.1 4.1 86 2.1 0.05 1

9.4 9.4 196 2.1 0.022

14.1 14.1 294 2.1 0.015

+-I-=

1 . 21~03*

I

1.9~10'

4x10'

I

3.9~10'

342

6. Polarized Muons A high degree of polarization is very important for extracting full physics information from muon collider experiments. Hence, first of all, it is worthwhile to find a way to produce highly polarized intense muon beams?38We assume that positive and negative pions generated by different proton bunches can be accumulated. A sketch of a possible option for a protons-to-pions multi-channel conversion system, followed by multi-channel pion-to-muon decay channels, is presented in Fig. 20. It might be reasonable to arrange a sectioned target (using additional channels). This could be especially useful at high proton energy around 100 GeV.

\

-

Ep low

I

A0,

- sma 1 - big

Figure 20. Schematic of a multi-channelproton-t+pion conversion system.

In each pion-collecting straight channel, using one-dimensional ‘‘thin surface current-carrying lenses” in doublets for the initial matching of focusing, it is necessary to direct pions of a wide spectrum into many independent channels. In each channel, in the @direction the beam transversal emittance is large, but in the +-direction it is quite small. These beams can easily be transported away from the target area, and the following channel gymnastics will be performed in reasonably free space. The next step is to arrange the energy dispersion in this smaller emittance direction in each channel, and then to direct each of the f 5% momentum spread pion beams into additional separate strong-focusing decay channels. Such narrow momentum spread pion beams (i.e., with a very small emittance in one direction), upon passing about 2 decay lengths (proportional to the pion energy in each channel, around lSP,y,, meters), generate muon beams of momentum spread about k 30% (see Fig. 21), with a strong correlation of the muon spin direction and its momentum.

343

i!

1.4

c4

1.2

3

1

5E“ 0.8

!3

0.6 0.4

.3

3

2

0.2 0

0

0.1

0.2 0.3

0.4 0.5 0.6

0.7 0.8 0.9 1.0 X X

CoM Polar Angle of Decay, rad Figure 21. Relative muon momentum in the lab system vs the polar angle of decay 0 in the CoM system, for pion kinetic energies of 300 MeV (solid l i e ) and 60 MeV (dotted l i e ) .

Consequently, for every particular muon beam, we cut away the middle 30% of the muon spectrum and direct the upper and the lower parts with opposite helicities into two separate sub-channels. At the next phase, we shift the energy in each muon channel by RF acceleratioddeceleration to the energy optimal for ionization cooling (below 100 MeV kinetic energy). And then, upon preliminary cooling, we combine all the muons from “upper sub-channels’’ into one longitudinally-polarized bunch, and all the muons from “lower sub-channels” into another bunch of opposite helicity, each with a 70% degree of polarization. This procedure, if it appears useful, could be arranged in a few stages. Then, all 4 bunches (p’ and p-) will be cooled down to the lowest 6-emittance. Afterwards, we can reverse the helicity of the “lower” bunches at a later stage upon acceleration up to 45 GeV (by applying an additional nonaccelerating full turn) and then combine the two bunches into one (one p+ bunch and one p- bunch) with a 6-emittance twice as high as that reachable at the final cooling. The helicity reversal of muons happens because of their anomalous magnetic moment. Positive relative spin-to-velocity rotation is very slow at the low energy (e.g., at the cooling stage), thus not damaging the initial muon beam’s degree of polarization; but it becomes faster proportionally to the muon energy, and at 45 GeV each full turn of the muon trajectory results in reversal of

344

the muon helicity. Let us keep in mind that all the muon spin motion proceeds in the median plane of the collider. Helicities of colliding bunches are modulated at relative frequency Vsph, Pmo, E[Gev] V,,” =-Yp =-KO 90 Because of this modulation, at integer spin resonances the helicity always remains the same in the collision process. At half-integer resonances the helicity reverses at consequent turns. At intermediate energies, the modulation of spin-atcollision proceeds with the non-integer fraction of vspin. The relative helicities of muon bunches at the interaction region (from ++/- to +-/-+) can be controlled by choosing a proper injection path (e.g., by an additional non-accelerating turn of one beam at, say, 45 GeV). At high energy, when vspin>> 1, a non-complete coherence of the spin rotation becomes important, and this effect can lead to the loss of polarization degree due to beam energy spread. The loss becomes significant if the spin frequency difference in the beam reverses the relative spin orientation at a half of the synchrotron oscillation period. The effective polarization degree loss factor Lff(compared to the initial degree) can be expressed as

where AEcolL is the muon beam energy spread in the collider, Vsynch is the relative synchrotron frequency (this evaluation is meaningful if L8is not very far from 1; otherwise the polarization degree goes to zero).This Lf was used in Table 2 of collider options. 7.

Background

We talked about a cleaner interaction of “point-like’’ particles in the case of muon-muon collisions (at very high energies) - compared to hadron-hadron and electron-positron (in Linear colhder) collisions. That is correct. But all the muons decay inside a collider: every muon produces an electron (positron) with an energy of 113 of the muon energy, on the average. High energy electrons appear in a very high field (above 10 Tesla). They hit the inner wall of the vacuum chamber - and produce showers. While passing the high magnetic field (prior to hitting the inner wall) they produce many high energy photons of the synchrotron radiation. These hit the outer wall of the vacuum chamber and again produce showers.

345

In usual colliders and detectors, they give an additional heavy heat load to the cryogenics, produce additionally a lot of neutrons and radioactivity, and “provide” detector@)with a strong background and radiation load. But, in principle, there is a very attractive solution (see Fig. 22) - to switch from the normal collider optics with vertical and horizontal focusing quads to skew-quads (Option I) (again alternating, as always in strong focusing) - no harm for collider operation. Plus, to remove superconducting dipole coils from the median plane. The other option (Option 11 in Fig. 22) to solve the same problem is to use combined-function strong-focusing dipoles. The choice of option should be the subject of more careful studies. Option I

Muon beam

DIPOLE

SKEW-QUAD

Option I1

DIPOLE-D

DIPOLE-F

Figure 22. Schematic sketch of superconducting magnet coils for an “open median plane” collider.

346 In a muon collider with an open median plane all the decay electrons go to the center and all the SR photons go off the center of the collider. Of course, the Interaction Region@) and Detector(s) should be designed with the same idea - to provide that all “decay background” particles could miss sensitive components of the detector.

8. Examples of Projects There are many publications presenting different views on the preparation of muon beams, cooling and acceleration, and several pre-projects of Neutrino Factories and Muon Colliders. I will not try to analyze all these options - the figure legends below make them clear.

8.1. Ring Options of Ionization Coolers From the very beginning we in Novosibirsk as a “natural option” ring coolers with short ionization regions located in low beta-function regions (similar to collision regions in colliders) - to minimize the influence of multiple scattering. Since that time, we have shifted to a different approach (as presented above), at least for final cooling. But ring cooler options have now become popular. Two options (by V. Balbekov and of R. Palmer) are presented in Fig. 23 and Fig. 24; the structures of these coolers are easy to understand from their legends. One of the most difficult problems is injection of beams with a high 6emittance in the ring. A possible option is to use a helical-type cooler at the very initial cooling stage, when the whole aperture is accessible for passing the muon beam, and to use the ring part at a later stage.

347

36.963 m

Circumference Nominal energy at shor.t

ss

250 MeV

Bending field

1.453 T

Norm. field gradient

0.5

Max. solenoid field

5.155 T

R F frequency

205.69 M H z

Accelerating gradient

15 MeVlm

Main absorber length

128 cm

LiH wedge absorber

14 cm

Grad. of energy loss

0.75 MeVlcm

Figure 23. Dipole ring (V. Balbekov).

33 m C i r c u m f e r e n c e 200 M e V l c Iniection I Extraction Vertical K i c k e r

I

200 M H z rf H y d r o g e n Absorbers

Alternating Solenoids

Figure 24. Bent-solenoid ring (R. Palmer).

348

8.2. Neutrino Factory Options

1

16 GeV Proton Driver Target Station 50 m decay/drift 100 m Ind Linac 60 m bunching 140 rn cooling .6GeV, 200 MHz Linac 0.2 4 3 GeV

Om RLA 2: 11-50 GeV 7.5 MeVlrn average Accel. Freq.: 400 M H z Turns: 5 p: 60 rn Arc: 380 m Linac: 2x600 rn

Q

bI

I L A 1: 3-11 GeV ‘.5 MeV/rn average iccel. Freq.: 200 M H z urns: 4 ): 30 m irc: 100 m .inac: 2x1 50 rn

Storage Ring Circ. = 1800 rn Straight = 600 rn

J

50 GeV muons 180 turns = I / e

O(1 02*) v per year

--300 m

--600 m

--900 m

-- 1200 m -- 1500 m

~ 1 8 0m 0

Figure 25. The Neutrino Factory. A muon-based neutrino factory is another option for the field (USA based collaboration).

349

( 7 Proton driver Induction linac No.1 100 m Drift 20 m Induction linac No.2 80 m Drift 30 m Induction linac No.3 80 m

Target M ini-cooling 3.5 m of L H , 10 m drift Bunching 56 m Cooling 108 m Linac 2.5 GeV v beam

Recirculating Linac 2.5-20 GeV

Storage ring 20 GeV

Figure 26. Schematic of the Neutrino Factory, version Study-II

A possible layout of a neutrino factory

Figure 27. Schematic layout of the CERN scenario for a Neutrino Factory.

350 8.3. Muon Collider Complex - Options

2 . 5 ~ l O pibunch '~ 30.GeV, 15 Hz 4 bunches

7xiO"Cjbunch i 5 0 MeV E.=iO'm-rad

1

TARGET, high 2 liquid CAPTURE SOLENOID, 20 T PHASE ROTATION, 30-60 MHz, 5 T

-

I

pPRODUCTlON

'POLARIZATION & P SELECTION Snake t Collimator

0.

ti ABSORBER

.

WEDGE

LlNAC TOTAL 4 GeV, 900 m 20 MeV ~ . = 4 X l O ~ rad m

LINACS t RECIRCULATION

PULSED MAGNETS

'PULSED or ROTATING

2x50 GeV

L 10" cm-'s.' p=3mm

I

FAST ACCELERATION

I

1

COLLIDER RING

1

IP

Figure 28. Muon Collider - an option from the Muon Collaboration Review.

351 16 GeVlc

I .5 xl 0” protonsly ear

Proton

--+m

I Pion Decay Channel

Muon Cooling Channel 100MeV/c m uonsly ear

10GeV

-

Energy Muons

+TI Neutrinos rom m u o n storage rings

High

muons

P+

Figure 29. Muon Collider - another option from the Muon Collaboration Review.

9. Some Historical Remarks As I have mentioned before, many of the topics hot nowadays were under active discussion at Novosibirsk many years ago - starting from the 1 9 6 0 ’ ~ . ’ ~ ~ * ~ As an example, here is an extract from my talk at the 1971 International Seminar on High Energy Physics Prospects at Morges - the pre-ICFA meeting after the 1971 Accelerator Conference at CERN. My talk was, as others, quite informal, but Professor Yves Goldschmidt-Clermont (CERN) immediately forced me to convert it into printed form at CERN. Here are the muon-related extracts:

352 CERN/D.Ph.II/YGC/mmg 21.9.1971

Morges

Seminar 1971

Intersecting Storage Rings at Novosibirsk A.N. Skrinsky

........... ........... ............ t

-

possibilities

These experiments at hundreds GeV energy region will be available, only when several very difficult things will be discovered (developed):

1. To have a very large number of protons with tens GeV energy in rather short bunches. It is necessary to have about 1014 or even 1 0 1 5 protons in about 10 sec in several meters long bunch. It is interesting, that the muon accelerator will be at the same time a very high intensity generator for all types of neutrinos up to the maximum accelerator energy. About 1/4 of all the accelerated muons may go into useful neutrinos. The neutrino beam shall have a diameter of about 10 cm behind the complete shielding. To produce with maximum efficiency muons with 1 GeV or less energy, using nuclear cascade, strong focusing

3 5 3 in the target and in decay channel. It seems possible to have 0.1 or even more useful muon per proton.

To cool muons in special hundred-kilogauss pulsed storage ring, using ionization energy losses. If the targets are in places with very small q-function, the final emittance of muon beam should be small enough to be injected into the main muon accelerator with small aperture and to be well compressed in interaction points. To accelerate muons rapidly in some accelerators. If the muons are accelerated to their rest energy in a time, several times less than their life time at rest, most of the muons will be accelerated up to the required energy. It is possible to use a linear accelerator, or to use a synchrotron with more than a hundred kilogauss and magnetic field with a short rise time. In the last case, the accelerator will be at the same time the colliding beams ring. In the ring with such a magnetic field it is possible to have several thousands of useful turns of muon beams. 3.

If all of these conditions are satisfied, it seems to be possible to have an average luminosity 1031 cm-2 sec-l and may be a bit more, which should be sufficient. It is interesting, that the muon accelerator will be at the same time a very high intensity generator for all types of neutrinos up to the maximum accelerator energy. About 1/4 of all the accelerated muons may go into useful neutrinos. The neutrino beam shall have a diameter of about 10 cm behind the complete shielding. (In modem wording - NEUTRINO FACTORY ! )

Later, the muon colliders and neutrino factories, based on ionization cooling, were very briefly presented in my introductory talk “Accelerator and Detector Prospects of High Energy Physics” at XX High Energy Physics International Conference, Madison, 1980.3 The road to muon-based neutrino factories and muon colliders is still long. But the harvest should be very rich.

354

References 1. Budker G.I., in Proceedings of the 7th International Conf. on High Energy Accelerators, Yerevan (1969) p. 33; extract in Physics Potential and Development of p+p- Colliders: Second Workshop, ed. D. Cline, AIP Conf. Proc. 352,4 (1996). 2. Skrinsky A.N., presented at the International Seminar on Prospects of HighEnergy Physics, Morges, 1971 (printed at CERN, unpublished); extract in Physics Potential and Development of p’p- Colliders: Second Workshop, ed. D. Cline, AIP Conf. Proc. 352,6 (1996). 3. Skrinsky A.N., “Accelerator and Instrumentation Prospects of Elementary Particle Physics,” in Proceedings of the X X International (“Rochester”) Conference on High Energy Physics, Madison, 1980, New York, 1981, v.2, p.1056-1093; and in Uspekhi Fiz. Nauk, Moscow, 1982, 138, 1, pp.3-43; translated at Soviet Physics Uspekhi 25 (9), September 1982, pp. 639-661. 4. Kolomensky A.A., Atomnaya Energiya 19, 534, (1965); Ado Yu.M., Balbekov V.I., Atomnaya Energiya 39,40 (1971). 5. Skrinsky A.N. and Parkhomchuk V.V., Sov. J. Part. Nucl. 12, 223-247 (1981). 6. Neuffer D., Particle Accelerators 14, 75 (1983). 7. Skrinsky A.N., “Ionization Cooling and Muon Collider,” in Proceedings of 9” ICFA Beam Dynamics Workshop: Beam Dynamics and Technology Issues for Muon-Muon Colliders, Montauk, NY (1995); Nuclear Instruments and Methods A 391, 188-195 (1997). 8. Skrinsky A.N., “Polarized muon beams for muon collider,” in Proceedings of the Symposium on Physics Potential and Development of p’p- Colliders, San Francisco (1995); Nuclear Physics B, Proceedings Supplement, v. 51A, November 1996, pp. 201-203. 9. Palmer R.B., Neuffer D. and Gallardo J., “A Practical High-Energy HighLuminosity p’p- Collider,” Advanced Accelerator Concepts: 6th Annual Conference, ed. P. Schoessow, AIP Conf. Proc. 335,635 (1995); Neuffer D. and Palmer R.B., “Progress Toward a High-Energy, HighLuminosity p’p- Collider,” The Future of Accelerator Physics: The Tamura Symposium, ed. T. Tajima, AIP Conf. Proc. 356,344 (1996). 10. Silvestrov G.I., “Problems of Intense Secondary Particle Beams Production,” in Proc. 13th Intern. Con5 on High Energy Accelerators, Novosibirsk, 1986, v. 2, pp. 258-263. 11. Silvestrov G.I., “Lithium Lenses for Muon Colliders,” in Proc. 9th ICFA Beam Dynamics Workshop: Beam Dynamics and Technology Issues for p’p- Colliders, Montauk, NY, 1995, AIP Conf. Proc. 372, pp. 168-177. 12. Skrinsky A.N., “Towards Ultimate Luminosity Polarized Muon Collider (problems and prospects),” in Proceedings of the Symposium on Physics

3 5 5 Potential and Development of p’p- Colliders, San Francisco; APS Proc. 441,240-260 (1997). 13. Ankenbrandt C.M. et al., (Muon Collider Collaboration), “Status of Neutrino Factory and Muon Collider - Research and Development and Future Plans,’’ Phys. Rev. ST Accel. Beams 2, 081001 (1999), http://publish.aps.org/ejnls/przfetchlabstract/PRZNuE08 1 0 0 1/. 14. Skrinsky A.N., “Remarks on High Energy Muon Collider,” in Muon Colliders at 10 TeV to 100 TeV (HEMC’99 Workshop), Montauk 1999; AIP Conf. Proc. 530,311-315 (2000).