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.

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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 ............. ;...

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

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

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, 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

*

%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

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