£75.).FIA13L’I. .5 15.5133. :1. 4.2.2.. «7.) 30.1 r VERSITY LIBRARIES IIIIIIIILIII MICHIGAN STATE UNI \ \\\\\I\\\\\\ \\\\\\\\\\\\\\\\\\\\\\I 3 1293 0102 16 I This is to certify that the dissertation entitled EXTRACTION STUDIES FOR THE MSU KSOO CONVERSION TO A PROTCII‘J‘nI CANCER THERAPY SYNCHROCYCLOTRON presented by Lung-Sheng Lee has been accepted towards fulfillment of the requirements for Ph.D. Physics degree in ajor professor Date August I, 199% MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State Unlversity PLACE IN RETURN BOX to romovo this checkout from your mood. TO AVOID FINES return on or before date duo. DATE DUE DATE DUE DATE DUE MSU Io An Affirmative Action/Equal Opportunity Institution Walla-9.1 EXTRACTION STUDIES FOR THE MSU K500 CONVERSION TO A PROTON CANCER THERAPY SYNCHROCYCLOTRON BY Lung-Sheng Lee A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1994 ABSTRACT EXTRACTION STUDIES FOR THE MSU K500 CONVERSION TO A PROTON CANCER THERAPY SYNCHROCYCLOTRON By Lung—Sheng Lee The design features of the extraction systems for the proposed proton cancer ther- apy machine converted from the MSU K500 isochronous cyclotron are presented. It is helpful to consider the economic factor that the K500 cyclotron could be transformed to the K250 proton synchrocyclotron needed for cancer therapy, with modification as limited as possible. The results indicate that it is difficult, but possible, to meet the medical requirement for beam properties, mainly owing to the resonance crossing in the extraction region, worsened by the intrinsically small average energy gain per turn (implication of low dee voltage) for synchrocyclotrons. The regenerative method, 1/, = 1 resonant excitation method and precession method are investigated extensively by detailed studies of beam dynamics carried out by the Z4 code and DEF LZ8OO program. Our studies show that the regenerative method is the most promising candidate for the extraction system for this machine. It gives better results and is able to be easily handled. We also found that the performance of the extraction system depends strongly and sensitively on dee voltage, on internal beam quality and on field imperfections. To my grandparents and mother iii ACKNOWLEDGEMENTS I thank Dr. H.G. Blosser, my supervisor, for his continued advice and patient guidance. Without his understanding and gracious help, I would end up nowhere. I am indebted to the lab’s staff and graduate students, whose help I received during my stay in NSCL. I also thank Drs. M. Gordon, F. Marti and D. Johnson for their valuable assistance. Special thanks to the National Science Foundation for making these studies finan- cially possible. I am grateful to Drs. J .S. Kovacs, J. Nolen, M. Berz and W. Pratt for their service on my Ph.D. guidance committee. iv Contents LIST OF TABLES viii LIST OF FIGURES ix 1 Introduction 1 1.1 Historical Background .......................... 2 1.2 Advantages of Proton Cancer Therapy ................. 5 1.3 Accelerator Requirements ........................ 5 1.4 General Features of Superconducting Cyclo- trons and Their Role in Cancer Therapy .............................. 1.5 Using the K500 Cyclotron to Produce 250 MeV Protons ....... 1.6 Overview .................................. 2 Basic Orbit Theory and Numerical Codes 12 2.1 Theoretical Calculation of Focusing Frequencies ............ 12 2.1.1 Kerst and Serber Equations ................... 12 2.1.2 Radial Stability .......................... 14 2.1.3 Axial Stability ........................... 15 2.1.4 The Smooth Approximation ................... 17 2.1.5 Thomas Principle ......................... 18 2.2 Equilibrium Orbit (EO) Code ...................... 20 2.2.1 DTP Field Interpolation Scheme ................ 22 2.2.2 Runge-Kutta Integration ..................... 25 2.2.3 Iteration Scheme ......................... 26 2.2.4 Calculation of VT, z/z, and Orbit Period ............. 28 2.3 Magnetic Charge Sheet Program ..................... 29 2.4- General Spiral Gap Code Z4 ....................... 31 2.5 DEFLZ8OO Code ............................. 33 The K250 Magnetic Field and Its Properties 34 3.1 Decision of the K250 Magnetic Field .................. 34 3.2 Radial Phase Space Studies in the K250 1200 Field .......... 43 Resonance Limitations 55 4.1 Resonances at the Extraction Region .................. 55 4.2 Half Integer Resonance uz = % ...................... 56 4.3 Integer Resonances 11,. = 1 and V2 = 1 .................. 57 4.4 Non-linear Coupling Resonances VT 2 21/2 and VT + 21/2 = 3 ...... 59 Longitudinal (Synchrotron) Motion and Its Implications 63 5.1 Basic Concepts .............................. 64 5.2 Principle of Phase Stability ....................... 67 5.3 The Phase Equation ........................... 69 5.3.1 Frequency of Synchrotron Oscillation of Small Amplitude . . . 70 5.4 The Phase Diagram ............................ 71 5.5 Acceleration Time and Frequency Modulation ............. 72 5.6 Internal Current and Starting Betatron Motions in the Extraction Cal- culations .................................. 77 5.7 Validity of Constant Energy Gain Approximation ........... 79 Regenerative extraction 84 6.1 Introduction ................................ 84 6.1.1 LeCouteur’s Linear Regenerative Theory ............ 87 6.2 Extraction System ............................ 89 6.2.1 Layout and Design Features ................... 89 6.3 Beam Dynamics .............................. 97 6.3.1 Pre-extraction Orbit Computations ............... 97 6.3.2 Extracted Beam Optics ...................... 111 6.4 Performance Estimation and Summary ................. 115 VT 2 1 Excitation Extraction 117 7.1 Introduction ................................ 117 7.2 Analytic Solution Concerning the V7. 2 3/ 3 Resonance ......... 118 7.3 Optimization of Bump Phase ...................... 120 7.4 Optimization of Bump Amplitude .................... 121 7.5 Summary of Final Calculations of u, = 1 System ............ 127 vi 8 Precessional Extraction 8.1 Turn Spacing from Acceleration ..................... 8.2 Turn Spacing from Orbit Precession ................... 8.3 Performance Calculations 9 Summary and Conclusions LIST OF REFERENCES vii 137 137 139 141 147 155 List of Tables 3.1 3.2 4.1 5.1 5.2 6.1 6.2 6.3 7.1 8.1 8.2 9.1 9.2 9.3 Data of 120° magnetic field maps With trim coil turned off. ...... Coil currents associated With the K250 proton synchrocyclotron. . . . Average radius and energy of important focusing resonances in the extraction region of the K250 1200 field. ................ Parameters of the RF system ...................... Extraction calculation results of the two versions of the Z4 code. . . . The extraction efficiency and external current of protons for selected synchrocyclotrons. ............................ Extraction element parameters ...................... The resonances in the extraction region. ................ Extraction element parameters ...................... The resonances in the precession extraction field. ........... Extraction element parameters ...................... Summary of resonances encountered in the extraction .......... Summary of extraction calculations .................... Parameters of the K250 proton synchrocyclotron ............ viii 38 38 56 75 81 131 143 149 152 154 List of Figures 1.1 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 Top: experimental depth-dose curve for 214.5 MeV protons and 22 MV photons. Bottom: Bragg peak depth as function of proton energy. . . Field components and the relation of the spiral angle C to the radius of the orbit and to the tangent of the spiral. .............. Plots of V7. and V2 vs. radius calculated in the K250 1200 field. Solid curves are computer results, broken curves are Symon’s formula. . . . Coordinate system used in analysis of the double-three—point interpo- lation. ................................... 19 21 Median plane plan View and vertical section view of the K500 cyclotron. 36 Measured average field at 360 A for the trim coil 11 at main coil cur- rents (amps) Ia/Ig = 618.75/125. .................... Contour plot of the median plane field map of the K250 field. Plots of Bo, 133, 453, Be, 456 of the K250 field vs. radius .......... Plots of orbital frequency V0.1,” as function of radius (top) and energy (bottom) in the K250 synchrocyclotron perfect field. The solid curves are associated with trim coils on, the dotted curves with all trim coils off. ..................................... Plots of V,- and V2 as function of radius in the K250 synchrocyclotron perfect field. The solid curves are associated with trim coils on, the dotted curves with all trim coils off .................... Plot of maximum stable radial betatron oscillation amplitude Ac as function of energy in the K250 synchrocyclotron perfect field. The solid curve is associated with trim coils on, the dotted curve with all trim coils off. ............................... The profile of a average B field in the K250 synchrocyclotron and the associated plot of pB and p/ q vs. r .................... Static radial phase plot, accompanied with seven fixed points at E = 243 MeV and 0 2 0° in the K250 1200 field. .............. Phase plot shows the outer unstable fixed point and its associated separatrices at E = 243 MeV in the K250 1200 field ........... ix 37 39 41 42 44 47 49 3.11 3.12 3.13 3.14 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6.1 6.2 6.3 6.4 6.5 A radial phase space plot showing the locations of fixed points at var- ious energies in a magnetic field with perfect 3 sector symmetry. . . . 50 Plot of fixed points in the vicinity of V7. 2 1 shows that a rapid renewing for the stability area. In addition, it shows the annihilation of the three pairs of hill-type fixed points at an energy about 244.5 MeV ...... 51 Radial phase plots at E2240 MeV and 246 MeV, t9 = 0°, in the K250 120° field showing the behaviour of a beam—like phase space area as it is accelerated through the 11,. = 1 resonance in the K250 1200 field with 10 keV / turn constant energy gain ..................... 53 Radial phase plots at E2240 MeV and 246 MeV, 6 = 0°, showing the acceptance of the V,- = 1 resonance in the K250 1200 field with 20 keV/ turn constant energy gain. It shows the transmission limit of the 11,. = 1 resonance corresponds to roughly a 35 mil eigenellipse at E = 240 MeV. ................................. 54 Median plane View of the proposed RF in the K250 synchrocyclotron. 66 Possible excursions in phase of nonsynchronous ions. ......... 68 Top: the phase diagram for the case where <15,- 2 —¢s. Numbers beside of curves give gt), in degree. Bottom: Absolute values of F ((b 2: $3, qfis) vs. qu gives the maximum excursion of synchrotron oscillation associ- ated with the choice of synchronous phase (153. ............. 73 The maximum permissible amplitudes of synchrotron oscillation in en- ergy and in radius are plotted with respect to energy of the synchronous particle. The calculations are done from Equations 5.9 , 5.14 , and 5.23 . 74 Plots of the RF frequency time derivative vs. energy with constant equilibrium phase (A, = 48.2° assumed throughout the acceleration. . . 76 Plots, obtained by using the two versions of Z4 code, show the dis tribution of energies in 0.01 MeV steps, r vs. pr, and 2 vs. p2 in cyclotron units at 9 = 336° for those particles which successfully enter the extraction channel ........................... 82 Plots, associated with the two versions of Z4 code, show the distribution of final energies in 0.01 MeV steps, 7“ vs. p,, and 2 vs. 1),, in cyclotron units at 0 2 308° Where particles go off the field ............. 83 Layout of the extraction system showing the arrangement of the ex- traction elements as well as the last five turns of an extracted orbit. . 90 Top: schematic cross section of the magnetic focusing channel. The middle and lower panels represent the field and the gradient across the channel respectively. ........................... 93 Median plane section of the compensating bars. ............ 94 Regenerator and shim geometry ...................... 95 Regenerator field defect and its radial derivative as a function of radius. .96 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 7.1 Plot of r vs. 6 for the last five turns of an extracted orbit showing the action of regenerator-peeler system. The radial motion is charac- terized by exponential increasing oscillation amplitude; the resultant turn separation at the entrance of the channel is about 0.1”. Two nodes of oscillation are clearly seen at —80° and 110°. ............ First harmonic bump used in extraction calculations as function of radius. ................................... Plots of focusing frequencies as function of energy in fields with (solid curves) and without (broken curves) fringe field, produced by the ex- traction devices (a 31 2 2.7 gauss bump used). ............ Schematic radial phase space plots show the behavior of the phase space regions as the V7. 2 1 resonance is traversed ............ Schematic radial phase space plots show the behavior of the phase space regions under the influence of regenerator ............. Central ray orbit and radial ellipses showing beam behavior at various energies. .................................. Initial (7*, pr), (2, p2) at E 2 240 MeV and 6 2 0° used for accelerated orbits runs. ................................ Vertical beam envelope as function of energy ............... Distribution of energies for the particles entering successfully the first extraction channel at 6 2 336°. ..................... Radial and axial phase plots at the entrance of the first extraction channel (6 2 336°) showing the resultant distribution for those orbits that successfully enter the channel. ................... The radial phase and energy acceptance of the extraction system. Radial and vertical envelopes of the extracted beam. Plots show (r—ro) and z vs. 6 from -—24° to 308°. The corresponding loss histogram is also depicted in the bottom. ....................... Distribution of final energies of the extracted beam (top). The corre- sponding radial and vertical phase space distributions are shown in the middle and bottom panels, respectively. . ' ................ Extraction efficiencies vs. betatron oscillation amplitudes of boundary points on eigenellipses at E 2 240 MeV. The number labeling beside the curves is the extraction efficiency. Unlabeled dash curves give ex- traction efficiency contour equal spacing between two adjacent labeled solid curves. ................................ Top: radial phase space plot for the accelerating orbits at 6 2 3360, initially centered at E2240 MeV, with 1.2 gauss first harmonic bump at several different azimuths. The number besides the curve is the bump phase <51 in degree. Bottom: the approximated electric deflector field strength required to bend beam out vs. bump phase. ...... xi 98 99 101 102 103 105 107 108 109 110 112 113 114 116 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8.1 8.2 8.3 8.4 Plot of axial amplitude growth induced by the V2 2 1/2 resonance traversal vs. energy gain per turn, estimated by the E0 code for cen- tered orbits. ................................ Plot of evolution of z ellipses (solid curves) showing the resonance V2 2 1/2 effect by using linear Z code. The dotted curves are the eigenellipses at each energy. ....................... Evolution of accelerating ellipses in the vicinity of resonances V2 2 1 / 2 and V, 2 2112. ............................... Asymptotes in the 11,. 2 1 excitation extraction for various conditions. Plots showing the turn spacing at the head of the septum (top) and the electric deflector field strength required to extract the beam (bottom) as function of septum position and bump amplitude. ......... Axial envelop vs. energy showing the effect of bump amplitude on beam behavior for the I/z 2 1 / 2, l/.,. 2 214, resonance crossing ...... The focusing frequencies in the 1x, 2 1 excitation extraction field are plotted vs. energy (heavy solid curves). As a reference, the focusing frequencies associated with the perfect 1200 field map are also depicted (dotted curves) ............................... Radial phase plots showing the behavior of acceleration runs with a bump 81 2 1.2 gauss, ¢1 2 160°. .................... Features of the V1. 2 l extraction system computed with the Z4 code. Left top: beam axial envelop vs. energy showing a acceptably vertical confinement through the Whole extraction process. Left bottom, right top, right bottom: distribution of energies, radial and vertical phase spectrums for those which enter the deflector at the entrance of the extraction channel. ............................ Left: the beam half width measured from the beam center and loss histogram in 2° steps vs. 6 along the extraction path. Right top: dis- tribution of final energies. Right middle, bottom : radial and vertical phase spectrums at 6 2 308°. ...................... Top: dR/dn (accel.) vs. energy. Middle: dR/dn (precession) vs. energy. Bottom: required electric deflector field strength vs. final energy .................................... Central ray orbits and beam spots at various energies illustrates the effect of u, 2 1 resonance crossing and the concept of precession ex- traction. .................................. Plots showing the beam axial envelop vs. energy, the energy spectrum, and 1", p,., and 2, 192 at the entrance of the deflector ........... Panels A, B, C: the beam half width and loss histogram in 2° steps vs. 6 along the extraction channel. Panel D: distribution of final energies. Panels E, F: radial and vertical phase spectrums at 6 2 308° ...... xii 124 130 132 136 140 144 146 Chapter 1 Introduction Interest in proton cancer therapy is at a high level at present because of a) clinical experience, which indicates that proton therapy is one of the most effective method of cancer therapy, and b) the cost effectiveness of present accelerator technology. In 1992, Dr. A.T. Porter (Chairman of the department of Radiation Oncology at Wayne State University and Specialist-In Chief of Radiation Oncology at the Detroit Medical Center) visited here and asked Prof. H. Blosser (University Distinguished Professor of MSU, former Director of NSCL and author’s major advisor) Whether the MSU K500 cyclotron could produce the proton beams needed for cancer therapy. Blosser responded ” A first thought is that it is probably technically possible; the isochronous K500 cyclotron would provide more than adequate bending power but would lack focusing — however, a synchrocyclotron adapted from the K500 would probably be a viable solution”. Thereafter this thesis was started in September 1992 to study the orbit dynamics of one of the most intricate details involved in converting the K500 to a synchrocyclotron for proton therapy, namely how to extract the beam. 1.1 Historical Background The initial concept and feasibility of the cyclotron was established by E.O. Lawrence in the early 19303 [1] [2]. A few years later, at the end of 1930s, several cyclotrons were in operation, dedicated to research in nuclear physics; they accelerated protons, deuterons and helium ions, to the territory of 5 MeV [3]. A decreasing magnetic field with radius was employed to maintain axial focusing of the ions in these non- relativistic cyclotrons. The conflict between the axial focusing requirement and the relativistic mass in- crease, an increasing field with radius required to match the cyclotron resonance frequency w 2 qB/mc, was not effectively solved until the discovery of the principle of phase stability by Veksler [4] and Mcmillan [5] in 1944-45. This principle shows that if the accelerating frequency is modulated in accordance with the mass change in the relation to 2 qB/mc, then there will be an equilibrium phase qfis (qfi is the phase angle of the ion relative to the sinusoidal RF wave) about which the phase of other particles oscillates. The energy gain per turn similarly oscillates about a mean value, determined by dee voltage, phase angle gbs, etc; and the total energy will eventually reach the desired maximum value. Construction of this type of cyclotron, denoted FM cyclotron or synchrocyclotron, was started at the middle of the 19403 and came to its height in the 19503. In their fully developed form, they were able to provide energy of light nuclei up to several hundreds of MeV and about 1 MA current [3]. In the 19603 and 703, synchrocyclotrons (SC) were gradually phased out because [6] (1) The SCs were soon at their energy maximum, after their blooming years in the 19503, for economic reasons; interest in particles of the highest energy shifted to the much less costly synchrotron in response to the research demands of the newly evolving field of " particle physics”. (2) The nuclear physics need for precise experi— ments was poorly filled by SCs due to their less defined and poorer duty cycle beams, and meanwhile the introduction of the ”sector focusing” Lawrence-type cyclotron in early 19503 evolved gradually into devices of excellent precision with maximum energy pushing steadily higher thru the range of the SC’s. The original idea of sector focusing was invented by L.H. Thomas [7] in 1938. Thomas pointed out that it is possible to have both constant orbital frequency mid axial focusing by using an "azimuthally varying” magnetic field where the radial in- crease of field matches that required by relativity. The Thomas suggestion for various reasons was not tested experimentally until the 19503. Two small experimental elec— tron cyclotrons, following the recipe of Thomas, were built and studied at Berkeley in a classified project in the early 19503 [8]; proof of the validity of the concept was established. The next important contribution in the evolution of the sector focused or ”iso— chronous” cyclotron to relativistic energies was made by the MURA group (K.R. Symon, D.W. Kerst et al., 1955—56) in their development of the F FAG (fixed field, alternating gradient) accelerator [9]. They pointed out that shaping the Thomas hills and valleys in a spiral (azimuth changing with radius) would increase the axial focusing force of the magnetic field; the contribution of the spiral effect to the axial focusing can be many times greater than the simple Thomas effect. Since the 19503, efforts of cyclotron designers have been directed continuously toward overcoming the limitations of energy imposed by the relativity effect, on pro- viding greater flexibility both in the choice of energy and the choice of particle to be accelerated, and on improving the beam quality. Much progress has thereby re- sulted, and both maximum energy and versatility have greatly increased. Variable energy, multi-particle, high intensity, exceptional energy resolution and nearly 100% extraction efficiency have all been strikingly achieved, as particularly exemplified in the MSU sector-focused isochronous K50 cyclotron (H.G. Blosser and M.M. Gordon, 1959-1965) [10]. On the other hand, the objective of higher energies was carried by the ” meson factory” cyclotrons in Vancouver [11] and Ziirich [12] and by the Indiana isochronous separated sector cyclotron (M. Rickey and R. Pollock, 1966-1975) [13]. The isochronous cyclotron has then gradually become the mainstream accelerator choice for nuclear physics research since 1980. With the advent of effective superconducting coil technology, ” superconducting cyclotron” projects at MSU [14] [15] and at Chalk River [16] were launched in the early 19703 in response to needs of research on heavy ions and these came into operation in the 19803. The attractive features of such cyclotrons are small size and low cost [17] and superconducting cyclotrons are at present flourishing. Concurrent with the evolution of improved cyclotrons, medical concepts for im- proving radiation therapy were also evolving. Cancer therapy using protons was first proposed by Robert Wilson in 1946 [18]. The first work on the biological and med- ical application of proton beams was conducted by J .H. Lawrence and CA. Tobias at Berkeley in 1948 [19]. Initial clinical treatment started at Berkeley in 1954 [20]. Since then, thousands of cancer patients have been treated in many facilities (Harvard and Berkeley, USA; Uppsala, Sweden; PSI, Switzerland; Dubna and Moscow, USSR; Chiba and Tsukuba, Japan) throughout the world using accelerators originally built for nuclear physics research [20]; in these facilities the concepts of proton therapy were proved to be effective, but the adaptation of accelerators designed for physics inevitably led to non—optimal medical capability [21]. Not until the end of 19803 was a well optimized proton cancer therapy facility built at Loma Linda University Medical Center, the first facility specifically constructed to be well matched to the medical requirements of proton therapy [22]. Recently, a number of groups throughout the world worked on the design of ac- celerators for proton cancer therapy in response to the strong demand for a low cost accelerator capable of producing beams with characteristics well matched to require- ments for cancer treatment. For example, proposals by MSU [23] [21], GSI [24], IBA-SHI [25], etc. 1.2 Advantages of Proton Cancer Therapy Proton therapy is recognized to be one of the most effective methods of cancer ther- apy at present [26]. Both conventional photon radiation and neutron therapy have the disadvantage of the undesirable fall-off of energy deposition with progressing pen- etration. For a single beam the dose applied to a deep seated tumor is much smaller than the dose to the healthy tissue. On the other hand, because of the sharp Bragg peak and the very small angular scattering for the protons, proton therapy is more favorable and the side effect will be less severe compared to conventional radiation. A typical sketch of the dose deposition versus depth in tissue (equivalent to water) for both protons [27] and photons (for comparison) [28], and the Bragg peak depth as a function of proton energy [27] are given in Figure 1.1. 1.3 Accelerator Requirements The requirements on accelerators for proton therapy have been summarized by Pe— droni [29] as follows. 0 Maximum beam energy in the range of 220 - 280 MeV is needed for the irradi- ation of deep seated tumors. 0 Variable energies (Z 70 MeV) in steps or continuously are required in order to control the deposition depth of the beam. IO- - . Bragg peak 0.8- . .0 CI 1 _e_e._e_e_ 214.5 MeV Proton - 22 MV X-roys .0 .p. L 1 L l Relative dose 0.2 - 0.0 . — C-d 1 I 1 m I I l I I I I I I F I I I _ 5 10 15 20 25 3O depth (cm) (24 O l l I 20- Bragg peak depth (cm) I l I I l I l l I l I 50 100 150 Energy (MeV) 1 I l I I T I ZOO 250 Figure 1.1: Top: experimental depth-dose curve for 214.5 MeV protons from the Paul Scherrer Institute (soild curve) [27]. The dose deposition is normalized such that the dose deposition is equal to one unit at the Bragg peak. Dash line gives dose curve for 22 MV X-rays, normalized to peak, from Ref. [28]. Bottom: Bragg peak depth as function of proton energy [27]. 0 Beam intensity : The required beam intensity depends on the application tech- nique used. In general, 10 nA is enough. 0 The duty cycle of the beam : The cyclotron provide the best possible time structure of the beam for the application of the dynamic scanning methods. However, a 100 - 10000 Hz pulsed machine (e.g., synchrocyclotron) is still an acceptable alternative. 1.4 General Features of Superconducting Cyclo- trons and Their Role in Cancer Therapy Superconducting cyclotrons are noteworthy for their compactness in size and cost effectiveness [17]. The successful operation of the Harper cyclotron in Detroit, the first superconducting cyclotron for cancer therapy, has demonstrated that the super— conducting cyclotrons are superior to other type of accelerators for medical use and more easily installed in a hospital due to their compact size, competitive cost, and high reliability. The Harper cyclotron started patient treatments in 1992 and is in heavy use now [30]. Studies of larger superconducting medical cyclotrons have been in process for some years at MSU [23] [21] and by the EULIMA group [31]. At NSCL/MSU, a supercon— ducting synchrocyclotron (FM cyclotron) with an azimuthally symmetric magnetic field [23] and an isochronous superconducting cyclotron with an azimuthally varying magnetic field [21] have been designed for proton therapy in 1990 and in 1993, respec- tively. The European EULIMA group proposed a separated sector superconducting cyclotron for light-ion therapy (carbon, oxygen and neon). The distinct differences between conventional solidpole (such as Harvard 160 MeV synchrocyclotron, Berkeley 184-inch synchrocyclotron, etc.) and sectored FM cy- clotrons (such as Uppsala cyclotron) mainly are [32]: (1) The use of sectors improves vertical focusing. (2) The reduced rf frequency modulation band, provided by a ra- dially increasing magnetic field, implies a likely increase in capture efficiency in the central region, in dee voltage, and in modulation rate. It is important to recall the well—known characteristic of synchrocyclotrons, namely, that the RF frequency is made to decrease as the particles gain energy in order to match the decreasing frequency of revolution of the ideal, synchronous pro- jectile. Particles, which start too early or too late from the source slit relative to the synchronous projectile, fail to clear the central region and are thus not captured into stable acceleration orbits. The synchrocyclotron thus sacrifices performance in the form of a low duty cycle and smaller current relative to that of isochronous cyclotrons, in order to provide for the relativistic increase in mass of the accelerated particles. But since a 10 11A proton beam current with 100 Hz duty cycle is enough for cancer therapy, and the variable energy requirement for fixed energy machines like cyclotrons could be realized using the degrader (variable thickness absorbers), superconducting synchrocyclotrons should be easily able to meet the medical requirements. 1.5 Using the K500 Cyclotron to Produce 250 MeV Protons The isochronous operating range of the K500 cyclotron is bounded by the bending limit Kbend 2 500M6V(% S Kbend(%)2), by the focusing limit Kfoc 2 160MeV(-Z,: _<_ Kfoc(%)) [33], and by the low field limit corresponding to the V? + 21/2 2 3 resonance [34]. The bending limit is determined by the field rigidity B ,0 maximum. The focusing limit comes from the fact that when the flutter is produced by the aligned magnetic moments of iron sector, as in the K500, the iron becomes fully saturated as the magnetic field is increased, so that the difference between the hill field and the valley field in turn approaches a constant value; the ratio of the hill field to the valley field (the ”flutter”) then decreases as the field is increased so that the vertical focusing becomes weaker; ultimately at some field value, the vertical focusing becomes too weak and therefore places a limit on the regime of useful fields [33]. The low field limit BO 2 30 kG derives from the V, + 2V2 2 3 resonance which, at low field in a three sector cyclotron (such as the K500), prevents the beam from reaching the extraction radius without large vertical blow-ups. The K500 in isochronous mode would then work only marginally as a proton therapy accelerator due to the low field limit and to the focusing limit which sets the maximum energy for protons in isochronous mode at 160 MeV [35]. In order to boost the final proton energy to 250 MeV and push the resonance radius further outward, the isochronism must be abandoned; a synchrocyclotron conversion of the MSU K500 cyclotron turns out to be a palatable and economical plan. It is without doubt much less expensive to utilize an existing facility than to build a new one. The technical feasibility of the required K500 modifications are reviewed in following sections of this thesis. 1 .6 Overview The cyclotron orbit behavior is usually divided into that of central region, intermedi— ate region and extraction region. In practice, the orbit problems in the intermediate radius region are usually minor [36], as will be seen in Chapter 3. As was pointed out by H. Blosser [37], the required control of the internal or external beam quality depends predominately on the design of central region. The crucial problems in the central region of an SC with a closed source are those of cap- ture efficiency, space charge losses, radial oscillation amplitude build-up, and source chimney clearness [38]. A central region design study for the K500 conversion was 10 undertaken by S. Snyder [39] in a study paralleling the extraction studies described herein; satisfactory central region configurations were achieved by Snyder, and this topic will therefore not be discussed further herein. Beam extraction is much difficult in a high field synchrocyclotron [23] since: (1) The turn spacing varies in proportion to the energy gain per turn and inversely with the average magnetic field strength. and (2) To the extent that electric field is utilized in the proposed extraction system to offset part of the high magnetic field, the required electric field becomes more difficult to achieve, as the magnetic field is increased. Therefore, detailed design of the extraction system is a critical subject in considering the feasibility of converting the K500 to a sector-focused synchrocyclotron. It is obviously better to use existing components if they will do the job. Therefore, the extraction systems studied for this conversion assume using existing K500 extrac- tion components. (A detailed description of the K500 cyclotron extraction system can be found in Ref. [40].) Three different extraction methods are explored in this thesis. (1) The traditional method of synchrocyclotron regenerative deflection uses two magnetic disturbances, a regenerator (a positive gradient bump) and a peeler (falling fringe field), to produce a c0326 field perturbation, and induces radial oscillations with exponential increasing amplitude [41]. (2) The V, 2 1 excitation extraction system employs a first harmonic bump to make the orbits drift to an outward flowing asymptote at the V, 2 1 ” stop band” (i.e. an energy interval where no closed orbit exists) [42]. and (3) Precessional extraction, in which a smaller first harmonic field bump, is used to produce a coherent precession of the desired amplitude as the beam accelerates through the V, 2 1 res— onance and as a result of this precession, post-resonance acceleration gives enhanced radial separation to assist with clearing the septum [43]. 11 Relevant background details of cyclotrons and of the K500 in particular are pro— vided in Chapters 2, 3, 4 and 5. The regenerative extraction process is discussed in Chapter 6, while excitation of the V, 2 1 resonance mechanism and the precession method are described in Chapters 7 and 8, respectively. Chapter 9 gives conclusions and summary. We find that the regenerative method gives rise to the best extraction performance and this K500 adapted synchrocyclotron should provide 5-30 nA of 250 MeV protons, which comfortably meets the beam requirements of cancer therapy. Chapter 2 Basic Orbit Theory and Numerical Codes The first section of this chapter reviews intuitive explanations and insights as to the basic and important properties of closed orbits. This is followed by sections which review the numerical codes used in the extraction studies which are the main focus of this thesis. The ”equilibrium orbit” (EO) code is explored rather fully because the numerical methods in it also form the key components of other NSCL orbit codes. The Z4, Charge Sheet and DEFLZ800 are described in a less detailed way. 2.1 Theoretical Calculation of Focusing Frequen- cies 2.1.1 Kerst and Serber Equations Kerst and Serber in 1941 [44] developed first-order expressions for the particle position as a function of time in an axially symmetric magnetic field, B(r, 6, z) 2 B(r, z). The radial equation in the median plane is mi: 2 772762 + qr6Bz(r) (2.1) 12 13 Closed orbits or equilibrium orbits are those which repeat upon traversing an azimuth of 27?. In an axially symmetric field with a z 2 0 plane of symmetry, the closed orbits are circular and located in the z 2 0 plane, the ”median plane” of the magnet. In such a field B(r,6,z) 2 B(r,z), the equilibrium radius satisfies 7' 2 qB(r,z 2 0)/p. For other orbits, the radius can be written in terms of small oscillations about the closed 7‘ 2 r0 orbit, namely: r(t) = 7‘0 + :I:(t), :1: << 7'0 (2.2) Expand the field in the small oscillation variable, retaining terms only to the first order. dB em = Be.) + x- TIM. + (2.3) r Express 6 in terms of r, 7* and constant 122, yielding . v2 — 7‘2 :1: 6 2 r2 N wo(l — 7‘3) (2.4) v o E —' 2.5 w ,0 < > Substitute the expressions for r(t), B2 and 6 into the equation of radial motion and keep only the first order terms. This procedure leads to the Kerst—Serber equation for radial motion: i5+w§(l+k):c =0 (2.6) where k 2 £915. Define natural frequency units u, s ‘31 = \/1+ k (2.7) wo 14 The linear oscillation in 7“ about 7‘0 can be written as a: 2 A,cos V,6 (2.8) 6 : wot (2.9) where A, is the amplitude of radial betatron oscillation. The corresponding solution for the radius of the particle in real space is r(t) 2 TO + A,cos V,6(t) (2.10) Similarly, the Kerst-Serber equation for the axial motion and its solution are 2+V§220 (2.11) Z 2 Azcos V20 (2.12) V22 2 —k (2.13) where A, is the amplitude of the vertical betatron oscillation. 2.1 .2 Radial Stability The necessary conditions for radial stability in an axially symmetric magnetic field, that is, the circumstances under which an ion, if displaced radially from the equilib- rium orbit, will tend to return to it rather than move away from it, are derived as follows. Locally, this axial symmetric magnetic field can be written approximately as 32(7“) 2 Bz(re)(—)" (2.14.) 15 where 7“,, is the radius of equilibrium orbit and k is field index. In addition, at every instant the electromagnetic force qu supplies the centripetal force mvz/r needed for a circular motion. Thus for equilibrium we must have mv2 : quz(re) (2-15) Te If k > —1, then at r > T, the magnetic force exceeds the centripetal force needed for an orbit of radius r so that the ion is driven inward towards the equilibrium path, while at r < re the particle is forced outward toward 7",. This is the condition for radial stability. It is apparent that the radial focusing force will be stronger the more the field increases - that is, the larger the value of the field index k. Considering the nonlinear driving force F acting on the radial motion in a three sector field, we shall write by omitting higher order terms in F [45], F 2 c32cos36 (2.16) Where c is a constant characterizing the strength of the force. When the defocusing force F just balances the linear attracting force which is proportional to field index k, we thereby obtain the threshold amplitude A, of stable motion by treating F as a perturbation on the linear motion. Ac QC] Vr—ll (2.17) 2.1.3 Axial Stability For a radially decreasing field, k < 0, and accompanying this condition is an outward bulging curvature of the magnetic field lines. The force which acts on the projectile is perpendicular both to the field and to the velocity of the ion. Because of the outward bowing of the field lines, the radial component of the field will supply an 16 axial focusing force which drives the ions back towards the mid-plane. On the other hand, if the field increases outward, k > 0, the lines of force bulge towards the center and the forces are such as to produce axial instability. Up to the present point, the problem of stability has been approached by consid— ering the action of the magnetic field upon a single ion. In actuality, every particle travels in the company of a great many similar particles, and these exert electric and electromagnetic forces on each other. The assembly of particles accelerated in a continuous wave (CW) cyclotron can be represented as a ”pie” shaped distribution, i.e. fixed azimuthal width with continuous distribution of charge density in the vicinity of the median plane from the initial radius to the full one. Assuming a uniform distribution of charge density inside the pie shaped slice (which is quite a good approximation since the turn length increases with r while the turn separation decreases as 1 / r), the electric force can be calculated using the Gauss theorem by neglecting edge field. When the electrostatic force of repulsion exceeds the attractive magnetic force, the axial instability occurs. Under these assumptions, the space charge limit current Icw in CW cyclotrons is [36] [46] [Cw = 2h€0w V2 E1 A? 0 20-?27T (2.18) where 2h is the beam’s full height, 60 the permittivity of free space, wo the ion orbital frequency, I/zo the single particle axial focusing frequency in natural frequency unit, E1 the energy gain per turn, (1 the ion charge, and Aqfi the beam azimuthal width. In a synchrocyclotron the axial space charge limit is modified because the beam is pulsed. The radio frequency decreases in a synchrocyclotron in order to match the change in the frequency of revolution of the ideal synchronous ion due to rela- tivistic mass increase with energy during acceleration process. Once the frequency has reached its minimum value, corresponding to the final energy, it has to increase 17 again to start a new cycle of acceleration. The total time spent to complete one RF modulating cycle is called the RF modulation period Tm 2 1/ fm, the corresponding modulation frequency denoted fm. Consequently, only particles which start in a time window, the ” capture time” At, relative to the starting time of the ideal synchronous ion will be caught into phase stable oscillations and be accelerated to final energy ow- ing to the process of RF modulation. Defining the capture efficiency as 6, 2 Alf/Tm, the axial space charge current limit 1,, in a synchrocyclotron can then be written as [so : 6c ' [cw (2.19) 2.1.4 The Smooth Approximation Analytic determination of focusing frequencies in a general azimuthal-varying—field (A.V.F.) machine is quite complicated. The analytical techniques used by different authors to obtain solutions involve some approximations and derive the answer in the form of infinite series. K. R. Symon [9] derived analytic expressions for the focusing frequencies by apply- ing the " smooth approximation” method (i.e. assuming that the betatron wavelengths are long in comparison with the sector length). The formulae contain only the leading terms and are rather crude for 3-sector machines but they do shed insight on how the field parameters determine the focusing frequencies. The formulae are l/T2:1+k+'.. (2.20) 2 U22 = —k+ N2—1F(1+2tan2()+-~ (2.21) where k 2 % dj§> (the average field index), < B > is the azimuthal average axial field along the particle orbit, R is the radius of the equivalent circle whose circum- ference is equal to the orbit length, N is the sector number ( 3 for the K250 synchro- 18 cyclotron), F is the magnetic flutter at radius R and is defined by F 2 513—253—5232, and C is the spiral angle defined in Figure 2.1. 2.1.5 Thomas Principle The purpose of this subsection is to give qualitative explanations for Thomas focus— ing, which says that it is possible to employ a magnetic field which increases with increasing radius but yet permits the orbits to exhibit axial as well as radial stability. Imagine a machine in which the pole faces are built so that the gap between the poles becomes alternately larger and smaller as one goes around the machine. Because the magnetic field is approximately inversely proportional to the axial gap, the field varies with azimuth; regions where it is high are colloquially known as hills and where it is low as valleys. Because of this azimuthal change in field strength, there is an azimuthal curving of the field lines so that they are convex towards the centers of the valleys. In a field of this sort the equilibrium orbit of a particle weaves back and forth about a circle, bending more sharply in the hills, less sharply in the valleys, and forming a scalloped path. The non-zero radial velocity v, interacts with the azimuthal field Ba associated with the bulging field lines on the boundaries between hills and valleys. Therefore an axial force is produced which turns out to be directed always toward the median plane - flutter focusing. Suppose that the boundaries between hill and valley are spiraled. Then at every boundary the fringing field has a radial component B,, with a gradient that points in at one edge and out at the other. In passing through these radial fields, a particle with azimuthal velocity v9 will experience a net axial focusing force produced from focusing at one boundary and defocusing at the other - alternating gradient focusing. In addition, taking account of the different path lengths in the focusing and defocusing fields due to orbit scalloping, gives rise to another focusing force — Laslett focusing. 19 tongent to spiral f korbit "B r Figure 2.1: Field components and the relation of the spiral angle C to the radius of the orbit and to the tangent of the spiral. 20 The alternating gradient focusing and the Laslett focusing add to give a doubled focusing effect - the factor 2 multiplying the tan2C term in Equation 2.21. 2.2 Equilibrium Orbit (EO) Code The important properties of the closed orbits and of the linear radial and vertical os- cillations about them can be obtained accurately by using N SCL EO code, developed by M. M. Gordon [47]. Given the median-plane magnetic field in polar coordinates, the program uses direct numerical integration of the canonical equations of motion, together with special iteration and extrapolation procedures, to compute the equi- librium orbits. After the closed orbits are found, the program calculates various properties of the orbit including the linear radial and vertical oscillations, and period about this orbit. For each energy the program searches for and locates the closed orbit by finding a solution of the equations of motion for which 7“, p, satisfy the peri— odicity condition. This is achieved by starting with an initial guess for 7: and p, and then integrating the canonical equations of motion for one period. The error in the closure is then used to compute new r, p, guesses and the process is repeated until the closure meets a predetermined requirement, or a preset number of iterations have been performed. For the first energy value the initial guess is input. For subsequent energy values the initial guess is extrapolated from the previous solutions. A typical V, and V, vs. radius calculated in the K250 perfect field (which will be presented in a later chapter) with the smooth approximation (Symon’s analytic formula) and with the E0 code are shown in Figure 2.2 for comparison. The formulas show good agreement with computer results except at the outer edge of the pole tips. The disagreement probably arises due to the flutter derivatives [48] which are usually quite large near the pole tip outer edge. 21 ITTT‘IfTIIIII E.O. code (DTP) 0.6-“ 0.5 I l l 1 l I I l l T I TI If I I l l l l I I T 1.0 0.9 0.8 0.7 iLlllJLlJlJLIlAll T'TIT .0 OI Z I/ Q o A an lulllll TIIITTTTT‘ITTTT 5 IO 15 20 25 Radius (inch) Figure 2.2: Plots of V, and V, vs. radius calculated in the K250 1200 field. Solid curves are computer results, broken curves are Symon’s formula (Equations 2.20 and 2.21) 22 2.2.1 DTP Field Interpolation Scheme The magnetic field of the K500 cyclotron was measured on a polar grid with steps 0.5” and 1°. Lacking an analytical function to describe the magnetic field, interpo- lation in the measured grid is used to obtain fields and derivatives at intermediate points as needed for the particle orbit tracking calculations. It is apparent that both the interpolation method and the step size of the stored magnetic field are crucial to the accuracy of the field value and derivatives. The double—three—point (DTP) interpolation scheme is the standard method used in most N SCL orbit codes, and the truncation error of this method is therefore analyzed in this subsection. The DTP is a linearly weighted and uniform spacing (2 Arc) interpolation method. To interpolate between two points 5132,5133, the left hand neighbor (cl and the right hand neighbor 2'4, are used with .132 and 2:3 to determine a left and right second order polynomial; the interpolated value is the average of these two polynomials weighted according to the distance from 3:2 and :33, respectively. This method is simple and, moreover, its first order derivative is continuous; the usual four point, third order polynomial, formula does not give the nice property of continuous first derivative from interval to interval, however, the DTP will not fit the end grid points if used outside the 3:2, 3:3 interval (whereas the usual four point third order polynomial would). Looking at the DTP in more details we first note that for a given three even- spacing points in 51:, there is a unique quadratic polynomial connecting through them. The notation used herein is shown in Figure 2.3 . For the set (331,311), (1132,3/2), (.733, 3(3) and the set (:rg,y2), (x3,y3), (2:4,y4), the quadratic polynomial functions connecting them are denoted as Y, and Y5, respectively. Define r 2 fil, 51:2 S :c S 1133 (denoted :1: 6 [51:2, 933]), A2: 2 3:3 — .192 and let y 2 f(x) which describes the exact dependence y on x. Then, basing on Taylor’s theorem and taking r 2 1/2, the resulting formulas 23 for interpolation truncation errors are derived approximately as follows. Ya can be written as I 1 Ya(:r) = 2(r2 — r)-y1+(—r2 +1) -y2 + 5&2 + r) - 3,3 (2.22) The truncation error, AYa, associated with the quadratic formula 1C1, presumably steming mainly from the omission of the terms higher than second order (i.e. Ya(:1:) ’2 y, + f’(;p2)(g; — 22) + LEE—2H2: — 2:2)2), is estimated as follows using Taylor’s theorem M. 2 f—m6—(9e—x1xx—x2xx—x3): f"§°(Ax)3r(r—1)< (A513)4 77 6 [231,934] (2.27) The first derivative in the orbit program is expressed as, from Equation 2.26 d 1 d 1 i = Elif = EEK—r? +1.5r — 0.5) - y1 + (3r2 — 3.57) ,7), +(—3r2 + 2.57: + 0.5) - yg + (r2 — 0.57“) ' 3,14] (2.28) In conclusion, the interpolation error in the DTP method is proportional to the fourth power of the step size. (A comparison of the DTP method with an eleven point finite difference method, which is expected to be more realistic and accurate, is given in Chapter 4). 2.2.2 Runge-Kutta Integration The fourth—order Runge-Kutta method, which is known to solve the coupled first order differential equations very accurately, has been employed in NSCL orbit programs to integrate the equations of motions. Its truncation error is proportional to the fifth power of the step size at each integration step. To demonstrate the essence of this method and capture the main ideas as simply as possible, the integration procedure will be illuminated for one variable only. The corresponding differential equation is dy E = “3313!) (2.29) A variable k is defined as ’91 = hf(rvo.yo) h k2 : hf(€l?0 '1' anl) 26 h k3 : hf($o + 51y?) k4 = hf(CL‘o + h, y3) (2.30) where h is the integration size and 300,310 are the values at the beginning of the step. This method is iterative; the derivative equation is evaluated four times during the step. For each iteration, k is evaluated at the y value resulting from the previous one. At the same time, each new 31 value is calculated by adding a weighted sum of the k values to either the initial y or the one from the previous iteration; final forms are shown here. 1 yl : yo‘i‘ £191 1 l y2=y1—(1—[/;)k1+(1- §)/€2 l 1 1 y3=y2+(-2——\/;)k1—k2+(1+\/;) [63 1 1 k1+2(l+2\/;) k2—4(l+\/;)k3+k4 At the end of each integration step, the resulting value of y 2 gm at 330+ It provides 1 94 = 263 + 6 (2.31) the initial conditions in the next cycle and the entire process is repeated. 2.2.3 Iteration Scheme This iteration process in E0 code is an extension of the familar Newton method. Symbols, definitions and derivations are adopted from M. M. Gordon’s paper [47] in this and the following subsection. The canonical equations of motion in cylindrical coordinates in the given median 27 plane field B (r, 6) with 6 as the independent variable are dr rp, E — P0 ’ (2.32) dp, d6 = pa + qu(r, 6). (2.33) The values of r and p, as functions of 6 are calculated by integrating these equations of motion using the Runge-Kutta method. Searching for a closed orbit then reduces to finding the pair of initial values (r;,p,,-) at 6 2 6, which satisfy the periodicity condition 7(6)) 2 r(6,-) and p,(6f) 2 p,(6,), where 6f 2 6,-+6o and 60 is the periodicity. For example, 60 2 3600/N in a field with N sectors and no imperfection. The linear radial oscillation equations relative to a reference orbit (rmpm) are obtained by making the substitution r —+ 7“,, + :13, p, ——> pm + p, into the median plane equations of motion and retaining linear terms. The resulting transfer matrix is denoted as X (6, 6,) which determinant is equal to 1 from the Liouville theorem. The errors in the trial values of (73-, p,,-) are calculated from £1 = r(6f) — 7‘2‘ (2.34) 62 = 19401) - Pm‘ (2-35) If the errors are not both zero to within a preset tolerance, as required by the period- icity condition, we proceed to calculate improved trial values. To do so, we assume the true closed orbit differs from the trial orbit only in first order; then we have 1', _ r , . 6r,- (prc )9—(171’ )6+)\(0702) (617M) (236) so that rc(6,-) 2 7‘,- + 6r,- and p,c(6,-) = 1m + 6p,,-. Thus (rc,p,c) and (r,p,) are, respectively, the coordinates of the closed orbit and trial orbit ( already computed ), and 6r,- and 61),, are the correction being sought. 28 Applying Equation 2.36 at 6 2 6, and 6 2 6 f, and taking the difference, we then obtain 0 = ( g: ) + [X(0,,0,-) — I] ( °’°" ) (2.37) 6pm which then yields the corrections and provides improved values for the initial condi- tions. These values are then used to repeat the entire process until the convergence tolerance is satisfied (or until an ”excessive number of trials” fault is triggered). 2.2.4 Calculation of V,, V,, and Orbit Period From the canonical equations of motion for the linear radial and vertical oscillations, the transfer matrices of the linear oscillations X (6,6,) and Z (6, 6,) can be obtained using the values of r, p, and pg of the median plane closed orbit. This X (6, 6,), together with Z (6, 6,), provide complete information about the linear betatron oscillations. We simplify the discussion by letting y stand for either x or z. Thus the matrix Y stands for X or Z, while V,, stands for either V, or V2. Y(6f,6,), 6f 2 6, + 60, is recognized as the transfer matrix for one complete period starting at 6,. Hence, in accordance with Floquet’s theorem, the eigenvalues of this matrix are /\ 2 erp(:l:z'a) (2.38) where a 2 V,6, (2.39) c030 2 $[Y11(6f, 6,) + 162(6f, 6,)] (2.40) Thus, if I 6030 IS 1, then a is real and the motion is stable, but if | coso |> 1, then a is complex and the motion is generally unstable. 29 In keeping with general practice, we write the matrix Y (6,,6,) in the standard form, Y(6f, 6,) 2 [case + J(6,)sina (2.41) where I is the unit matrix, and -% 201' J(0,)=( °i 6" ) (2.42) which defines the parameters 01,, fl, and 7,. These quantities are related to the known matrix elements by , 1 01,82710' = *2-[1/11(0f, 6,) -- 162(t9f,6,)] 6,32'710 =1 1/12(Qf,6,') 7,3in0 2 —Y12(6f,6,) (2.43) When 0 is real, by assuming 51 > 0, the value of a 2 V,6, is determined by Equa- tions 2.40 and 2.43. When a is complex, the program uses I 0030 | to find the imaginary part of V, and V2. The differential equation for the time t, which is canonical, is written as £1:— 7" d6_2fl-v :9 (2.44.) The orbit period is obtained by integrating the above equation for one complete period fromt20at626,. 2.3 Magnetic Charge Sheet Program This program [49] calculates the median plane field produced by the magnetic ex- traction apparatus, assuming that the iron is uniformly magnetized in the vertical 30 direction. The uniform magnetization assumption is quite accurate at the high field levels employed in superconducting cyclotrons and allows one to calculate the median plane field generated by the saturated iron in terms of a simple surface current flowing in horizontal loops around the sides of each piece of steel; this then leads to a field formula involving a single line integral around the closed-current contour. The scale factor B, is set internally to 21.4 kilogauss. As reported by Gordon [50] for standard magnet steel, the assumption of saturation works very well for external field all the way down to about 6 kilogauss. Therefore, the accuracy of these calculations are expected to be reasonably satisfactory in superconducting machines. Following the analysis of M.M. Gordon [49], consider a piece of iron having hor- izontal surface 2 2 21 at bottom and z 2 22 at top ( 22 > 21 Z 0 ). For each piece given above the median plane, there is a matching piece symmetrically located below the plane. The median plane field produced by these two pieces of saturated steel is denoted by 1320,. From Maxwell equations (for example, in the book Classical Electro- dynamics by J. D. Jackson), the saturated blocks of iron with uniformly distributed magnetic moment are equivalent to uniform currents (j 2 Mo X Ti, ,quo 2 Bo 2 2.14 tesla) on the surface parallel to the external field. After some manipulation, the Bzo, can be written as: Bo 21 (y - y’)drv’ - (ft - Iv’)dy’ zoi : — 1 _ — B 2N“ R. X (x—w'rHy—y')? (2.45) 2’221 22 (y — y’)drv’ — (Iv - x’)dy’ —f[1_—]X _/2 _12 } R0 (fl? 33) +04 3!) 21:22 where r 2 (:13, y, z 2 0) is the point at which the magnetic field to be evaluated, and I r 2 (:r’,y’, 2’) gives the location of a particular current element. The distance R, is given by R0 = ((2: — a")? + (y — V)? + 2’2"” (22-4-6) 31 Making the substitutions 511' —+ 1/2(a:, + 531—1): 9' 2 1/2(y, + 311—1) (2-47) I I d1: _) :17]. - $j—1I dy —) y] — yj-la the contour integral is evaluated numerically in this program by simply replacing the integral with a sum over a sequence of n straight-line segments which join the points (£13,,yj), wherej 2 0, 1, 2, ....... , 11. Here, (mmyn) 2 (230,340) is required in order that the contour be closed. Since the integrands involve only simple algebraic expressions, the numerical in- tegration is quite rapid. The accuracy can be tested and improved by systematically increasing n. The magnetic extraction apparatus under consideration can be subdivided into a finite number of such pieces. The resultant median plane field B,,, 2 ZBZO, is obtained through superposition by adding up the contributions from all of the divided pieces having flat vertical surfaces. 2.4 General Spiral Gap Code Z4 The Z4 orbit code [51] developed by M.M. Gordon is used in the pre-extraction orbit computations. This code calculates the nonlinear effect of the field and the Z motion using a magnetic field Taylor—expanded off the median plane up to fourth order, and provides an accurate evaluation of the important coupling effects between the radial and vertical motion. The median plane magnetic field is required for input and is pre-stored in a polar grid. The field is assumed to have a mid—plane symmetry and the B, is expanded to Z4. The rest of the components of the magnetic field, B9 or B,, are determined in such a way that v - B 2 0 is satisfied. The formulas used in 32 the field expansion are the following 22 24 B2 : —[B(T‘,6) - "2—B’(T‘,0) + '2—4'B”(T',6)] (2.48) B -— — 3m .9 ) 2 49 , _. Zar r, ,2 ( - ) z a B9 = —;‘8—6’C(T,6,2) (2.50) where C(r, 6, z) 2 B(r,6) — %B’(r,6). Here B’ 2 LB(7‘,6) and B”(r,6) 2 LB’(r,6), where L represents the cylindrical two dimensional Laplace operator (L E %-§—r(r-§—T) + 55%;). Thus, B, has terms of order 22 and 24 off the median plane, while B, and B, have 2 and 2:3 terms. The equations of motion for r, 1),, z and p, used in the Z4 orbit code are as follows: dr rp, d6 — 139 (2.51) dp” = p, + quZ — (177’ z B, (2.52) d6 P9 dz rpz d6 — [)9 (2.53) (1172 : __quT + q:p_TB6 (2.54) d9 P9 where (B,, B9, B2) are the orthogonal polar components of the total field, obtained from Equations 2.48, 2.49 and 2.50 by applying the DTP method on pre-stored median plane maps of the field, and of B’ and B”. Given the field components, the fourth- order Runge—Kutta subroutine is employed to integrate the equations of motion with respect to 6. The treatment of accelerating gap crossings in the Z4 code is approximate (impulse approximation), i.e. the particle’s energy remains constant until it crosses a gap, and then the energy and momentum change abruptly according to the conditions at the gap. This is an accurate simplification away from the cyclotron central region 33 due to the fact that E field is fundamentally weak relative to B field except in the central region and also the E field is highly localized spatially. 2.5 DEFLZSOO Code The DEF LZ800 code [52] is used after the beam enters the extraction system. This program treats only linear vertical motion and calculates the coupling of the radial into the vertical motion, but omits the reverse coupling of vertical into radial. The treatments of electrostatic deflectors and focusing bars is also approximate in this program: The effect of the electric field E is calculated simply by decreasing the median plane field (—Bz) by an amount E / v, where v is the particle’s velocity. And the magnetic field inside the aperture of the focusing bar is just determined by two parameters - field bias AB and focusing gradient BB/Bx. The DEF LZ8OO program is, however, quite ”gooc” and much faster than its com- plicated counterpart Z4 code which is supposed to be extremely realistic and therefore very accurate. Comparison of DEFLZ800 runs with Z4 runs has been done by Marti and Jeon [53] who showed that the difference in central ray position computed by these two codes was less than 100 mils at the exit of the last focusing bar, but DE- FLZ800 is roughly one thousand times faster than Z4. The CPU time in Z4 for a particle to complete the extraction journey after it enters the channel is about 1 min; it is extremely slow due to the fact that fringe fields of all the extraction devices are taken into consideration [53]. (We still face tradeoff between better calculations and computational cost in spite of the advance of computer today!) Chapter 3 The K250 Magnetic Field and Its Properties The choice of main and trim coil currents to give an approximately optimal magnetic field is discussed in this chapter. Following this, the main features of this three-fold- symmetry magnetic field in the K250 synchrocyclotron extraction region are analysed, and properties of the radial phase space are reviewed to display the fundamental performance features of this machine. 3.1 Decision of the K250 Magnetic Field The general features of the MSU K500 magnet are extensively reported in Ref. [54-]. Only some important features will be recalled. A median plane plan View of the K500 cyclotron is given in the upper part of Figure 3.1, and a vertical section View of the K500 magnet is at the bottom. The vertical section is taken along a spiral line centered on a hill and superimposed on a similar valley cross section (center plug not included). The hills have constant axial gap for radii between 3.5” and 25.75” and both hills and valleys have an average spiral constant of 1 / 13 (rad/ inch). The two superconducting coils are located radially between 30" and 35.6”, and the section closer to the median plane is about 1 / 3 of the total coil height. Thus the two sections 34 35 will be referred to as small coil and large coil respectively, and their currents are indicated as Ia and If}. There are 13 trim coils located on each spiral sector pole, made of two layers of 5 turns each of 0.25” square hollow conductor and approximately equally spaced from 3.5” to 25" radius. The measured average field (Bav(r) E 02" B(r,6, z 2 0)cl9/27r) of the trim coils all have a similar form; one of them is shown in Figure 3.2 at excitation 360 A. The 120° field mapping over the points on the [0,1,6 plane, with all the trim coils turned off, was measured using the flip coil method in the period from December 1980 to March 1981 [54]. It is concluded from this paper that the data have an accuracy of :t0.03% limited mainly by the slow change of the calibration over a period of a few days. The currents and central field for some of the measured points are listed in Table 3.1. The fields anywhere on the 1a,.[g plane are predicted based on the interpolation technique. The coil current setup in the proposed K250 proton synchrocyclotron is listed in Table 3.2. The corresponding median plane field contour is shown in Figure 3.3. The spiral structure of the hill, the azimuthal variation from hills to valleys, and the field fall-off at the magnetic edge are all shown clearly in this figure. In the following, the customary Fourier expansion of the magnetic field will be used: Namely B(r, 6) 2 B0(r) + Z Bn(r)cosn(0 — ¢n(r)) (3.1) where Bo(r) is the average field as a function of radius and B,,(r) is the nth harmonic amplitude. For 1200 field maps, only the third harmonic and multiples thereof are present. The average field, and amplitudes and phases of the third and sixth leading 36 180' 270' _90. Extracted . , beom hole Mom COIls Yoke 0. 25 1+L'44LL1 141 Li 1 1 1 l 1 1 1 J l 1 1 1 1 l '1 1 L1 l 1 1 1 L11 1 1 1 : Ie—OEE STEM HOLE —9 : 20-— — /_\ " .. _C _ _ o C _‘ 1— :; ~ — LL] 15 — LARG COIL 2 :z : Z :5 . VALLEY PROFILE — Q —J — - - - - - - .- .10‘7 _ __ _ _, — a : l HILL (UPPER PART) J C 2 — L g 5 — 1.. SMAL COIL 2 tr 1 g : LL _ HILL PROFILE (LOWER PART) g L N - _ O T was. ma 5 .l i -5 fl T T I T T T I I I l I T I I T I I I I I I l I T T T l I I I l l I T T T I O 5 10 15 20 25 30 35 40 R (Inch) Figure 3.1: Top: median plane plan View of the K500 cyclotron. Bottom: radial profile of the K500 cyclotron through a spiral line centered on a hill and superimposed on a similar valley cross section. 37 150 _] l l l I l l l l l l l l l l l l l l l l l_ 1 _ _ TCH 2 360 amps : ‘ la 2 618.75 amps “ 100 2 2 _ l3 2 125 omps _ 5O 2 2 fa — E (f) ~ _ 3 O - _ 3 mo 0 “ ‘— _50 _ _ —100 2 2 — l l l I l l l r I l l l l I l l l l I l 7 l l I l— O 5 10 15 20 25 R (inch) Figure 3.2: Average field (BMW) E 02" B(ro,0,z 2 O)d0/27r) at excitation 360 A, equal currents in three sectors, for the trim coil 11, measured at main coil currents (amps) Ia/Ig 2 618.75/125, from Ref. [54]. 38 Table 3.1: Data of 1200 magnetic field maps with trim coil turned off. ] Field number ] 10, (Amp)] [,3 (Amp) I B, (Kilogauss) ] 1 450 200 28.9 2 550 100 27.9 3 650 0 26.9 4 487.5 350 34.1 5 537.5 300 33.6 6 587.5 250 33.1 7 687.5 150 32.2 8 525 500 39.1 9 625 400 38.2 10 725 300 37.2 11 562.5 650 44.0 12 662.5 550 43.1 13 762.5 450 42.2 14 600 800 48.8 15 700 700 48.0 16 800 600 47.1 Table 3.2: Coil currents associated with the K250 proton synchrocyclotron. ] Coil [ Current (ampere)] Large main coil 180 Small main coil 700 Trim coil 1 0 Trim coil 2 0 Trim coil 3 —100 Trim coil 4 —200 Trim coil 5 —300 Trim coil 6 —300 Trim coil 7 —300 Trim coil 8 300 Trim coil 9 200 Trim coil 10 —100 Trim coil 11 0 Trim coil 12 100 Trim coil 13 100 39 1 MIN = 0.200E+02 STEP = 0.2003401 m = 0.40034-02 coo Figure 3.3: Contour plot of the median plane field map of the K250 field. Coil currents are at the values given in Table 3.2. Imperfection field components are omitted so that the plot has exact 120° symmetry. Contour spacing is two kilogauss with the dotted contour at the outside at 20 kilogauss and the double weight contour in the outer part of each hill at 40 kilogauss. 40 harmonic components as function of radius are given in Figure 3.4. The function ¢n(r)(n 2 3, 6, - - ) is determined by the spiral curve of the hill (valley). The excitation IO, 2 700A, 15 2 180A of main coils can sustain 250 MeV protons. Moreover, the associated field is approximately optimal because of the following rea- sons: (1) It gives acceptable focusing in the intermediate energy region, as will be seen later. (2) Its sharp field edge (Figure 3.4) alleviates the problems of beam extraction. (3) There are several advantages associated with its closeness to isochronous field, namely, the rising magnetic field (Figure 3.4) compared to the falling field [55]: o The range of frequency modulation decreases which reduces the problems of getting a high repetition rate. The orbital frequency as function of radius and energy, for protons in this field, is given in Figure 3.5. 0 With the dee voltage and final energy being fixed, the initial orbit radii are larger so that the ions have a better chance to clear the source, due to lower central magnetic field. 0 The reduced frequency range owing to lowering of the central magnetic field decreases the initial dz/Tf/dt as well as Vomit, and thereby increases the capture efficiency. All the trim coil currents are well below the maximum current rating of the power supply which is 400 A. The negative sign of trim coil current indicates that the field produced by this trim coil at the center of the magnet is opposite to that of the main coils. The principal purpose of trim coils in such a synchrocyclotron is to adjust vertical focusing locally by changing field derivative, Auz x $21“ from smooth approximation. As in Chapter 2, the focusing frequencies u, and uz are defined as ratios of the actual focusing frequencies, with which linearly displaced orbits oscillate 41 4O [11 1 l L1 1 l l l l l l l l 1 ll 11L! l l 1 l l l l l 1 ll 1 I ll 20-4 I I r rfirrv I r 1 1 1 l TfiiTT [#1 I 1 O 5 10 15 2O 25" "3b" ..3B.. '40 Radius(inch) Figure 3.4: Plots of B0, B3, 63, B6, d6 of the K250 field vs. radius (coil currents at the values given in Table 3.2). 42 l I l I l l I l l l l l l l l I l I l 41 l 50 _. - - main & trim coils L. If _________ main coils only . . l . . . r . . . . 150 200 250 Energy (MeV) ' ' ' ' I 50 100 Figure 3.5: Plots of orbital frequency Vomit as function of radius (top) and energy (bottom) in the K250 synchrocyclotron perfect field. The solid curves are associated with trim coils on, the dotted curves with all trim coils off. Note that an isochronous field gives a constant orbital frequency. 43 relative to the closed orbit, to the orbital frequency and are hence dimensionless. The numerical results of radial and vertical focusing frequency calculations in the K250 perfect field, with and without trim coils turning on, are shown in Figure 3.6. The maximum stable radial betatron oscillation amplitude (AC), the radial dis- tance in the phase space from the equilibrium orbit to the furtherest unstable fixed point, calculated by using the E0 code, is plotted as function of energy in Figure 3.7. It is evident that the stability limit is much larger than the beam size in the inter- mediate radius region. Hence the orbit problem in this region are expected to be minor. 3.2 Radial Phase Space Studies in the K250 1200 Field The nature of the extraction process can best be understood by examining radial phase plots which show pr vs. 7“ for orbits with given energy at some selected az- imuthal position. They provide a qualitative insight as to relationships between particle behavior and magnetic field features. The axially symmetric field is employed in the following argument to provide the insight and intuition to the equilibrium orbits, the final energy and the extraction radius for sector machines. From relation p/q 2 pB (p is the instantaneous radius of curvature) and the profile of a B field (upper panel of Figure 3.8), we see that in general for a given p/ q, there are two possible closed orbits, if there are any at all (lower panel of Figure 3.8). Such orbits are either stable or unstable according as near neighbor orbits remain close to, or drift away from, the fixed point orbit, respectively. The TC is the upper limit of extraction radius and ch, at which the related two orbits are collapsed into one closed orbit, corresponds to the maximum energy which the 44 1.20‘2‘2" 1.15 1.05 2 1.00 l .L J 1 l J l moin & trim coils moin coils only I/ Z ‘_. '_. l\) i\) in L» lb 0 01 o (.n O 0'! O lLlllLlllllllJllllllllllllllllllJilllllr- I— b 01 IIIITIFT lirll [Ill ITITTTTT TITT I‘lj I l l i T 1 i O O _4 .. .4 .4 _.... 15 RCW (inch) —1 .4 Figure 3.6: Plots of 11,. and V2 as function of radius in the K250 synchrocyclotron perfect field. The solid curves are associated with trim coil on, the dotted curves with all trim coils off. 45 5 O l 1 l l l l l 1 l i l l i l i l l l l l 1 1 1 2 moin & trim coils ' : —1 - . — —— -------- mom COllS onl — 4.5 y /‘\ 2 '— _C n _ LC) # _ O 4.0 2; j (1) 2 _ .0 _ L 3 _ l_ _+_J 2 3.5 2 —— Q— 2 r2 E 2 E O _ 2 '5 3.0 2_ :2 (n O I C — _ 8 2.5 —_ t— .4.) O — _ .5.) _ _. 8 — _ _ 2.0 2 2 .9 I I ‘0 _ ._ O L 15 _‘ _ s.) ' a P .0 — _ O — _ 4c}; _ I—. _ 1.0 2 2 6 _ vr21_ E 2' 2 -—] u— 0.5 2 2 0.0 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 50 100 150 200 250 Energy (lvieV) Figure 3.7: Plot of maximum stable radial betatron oscillation amplitude AC (the radial distance in the phase space from the equilibrium orbit to the furtherest unstable fixed point) calculated by using the E0 code, as function of energy in the K250 synchrocyclotron perfect field. The solid curve is associated with trim coils on, the dotted curve with all trim coils off. Recall that Ac o<| I/T — 1 ]. 46 magnet can bend into a closed orbit. The closed orbits acquire orbit shape which encloses high magnetic field region. There are thus four pairs of closed orbits in a 3 sector machine, associated to the center (r 2 0) of the field and the three hills. However, the stable orbit associated with the center of the field is the only orbit (of the total eight) suitable for particle acceleration due to its linearity. The static (also called coasting or non—accelerated) phase space orbits, along with seven fixed points at E 2 243 MeV, are plotted in Figure 3.9. It shows a typical static phase plot at an energy somewhat smaller than the V7. 2 1 resonance energy. Recall that :1: 2 Awash/,0), where A, is the amplitude of betatron oscillation. The direction of flow can therefore be determined by the value of radial focusing frequency. If 1/7. > 1 the flow is counterclockwise; 1/,. < 1 is clockwise flow. Besides, flow lines never cross and have to satisfy the continuity requirement. One important feature of the radial phase space is the size of the stable region. The maximum stable radial amplitude could be gleaned from the radial phase space plot and an approximate measure of this size is given by the distance in the phase space from the equilibrium orbit to the nearest unstable fixed point, as is clearly exemplified in this graph. The static phase plot associated with the outer unstable fixed point at E 2 243 MeV in the K250 1200 field is shown in Figure 3.10. The separatrices at an energy of 243 MeV are indicated by the crossed solid lines obtained by tracking the static orbits starting at points very close to this outer unstable fixed point. Radial phase space plots at 0 2 0 showing the positions of fixed points at various energies in the K250 1200 field are given in Figure 3.11 and Figure 3.12. Figure 3.11 shows that the stable area shrinks to zero at an energy of 243.8 MeV at which V7. 2 1. Note that the resonance u, 2 1 is characterized by a point in energy 47 Stoble fix pt . Unstobl fixed pt P/Q ------------------- . --------------- J. ------ . Figure 3.8: The profile of a average B field in the K250 synchrocyclotron and the associated plot of pB and p/ q vs. r. 48 243.0 MeV & Ur 2 1.0154 & 6 2 0 lllllllillllllllllllllllllLllllllllllll '00 1.6 1.4 1.2 1.0 .0 on .0 on 0.4 pr (inch) 0.2 0.0 -0.2 -0.4 -0.6 LlllllllllllJ_IJlllllIlJlLllllllllllllllllllllllllll 111I111I11TITFTI111I111ITIIIlllI111I|11I111I111I1|l —O°8111I111I111I1|1I11FI111I011I111I111I111 25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4 27.6 r Unch) Figure 3.9: Static phase space plot, accompanied with seven fixed points at E 2 243 MeV and 6 2 0" in the K250 1200 field. The lines connect positions on successive revolutions of orbits evolving from some given starting conditions. Note momentum is expressed in length unit (cyclotron unit) by dividing mowo, where mo is the rest mass of protons and B0 2 3.316 tesla assuming in the unit conversion. -O.l . . . . . 4 28.26 28.56 ._ pr finch) l / I —l.5 1 1 l l I l 1 1 1 I 1 l 1 1 I 1 l T 1 I 1 1 l 1 I 1 1 l 1 27.0 27.5 28.0 28.5 29.0 29.5 30.0 r finch) Figure 3.10: Phase plot shows the outer unstable fixed point and its associated sep- aratrices at E 2 243 MeV in the K250 1200 field. The separatrices are computed by tracking coasting (static or non-accelerated) orbits backward and forward starting at points very close to the outer unstable fixed point and plotted once per turn (0°). The separatrices and a coasting orbit which starts at a point 30 mil away from the fixed point are shown in the right small panel using magnified scale. 50 pr finch) 2.0 IllllpllllliLllllllllllllJllilll 40 240 0.8 0.6 0.4 0.2 0.0 -0.2 244 20.4 —0.6 —08 240 111I111I111I11|I1|1I111IE1BT11|I111I111I111I1|lIlllIlllIlll llllllllLJlllllllllllllllllllllllllllllllllilllllllllllllll _l'0111i11I111I111I111111111I111I10T 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.6 27.0 27.2 r (inch) Figure 3.11: A radial phase space plot showing the locations of fixed points at various energies in a magnetic field with perfect 3 sector symmetry; the numerical labeling beside the points gives the energy in MeV; the scales are in cyclotron units. Unlabeled points are at energies equally spaced, 0.5 MeV, between labeled points. The label U indicates a hill-type unstable fixed point, S a hill-type stable fixed point and E.O. the fixed point associated with the center of the magnetic field. Number 1, 2 and 3 indicate the stable and unstable pair with the same hill. 51 1'1 lllllllllllllllllllllllllllllllllillLlL 44.1 0.9 244.45 244.1 82 0.8 244.45 0.7 44.45 444 U3 2 5 0.6 244.1 44.1 244" G20244.45 05 50 243.1 0.4 0.3 0.2 244.45 0.1 -00 24445 -0.1 Si _0.2 244.1 IllllllllillllllllllllllillllllllllLllJllllldllllllLLllLLlLllllllllll I171I_1011I1111I1111I111lIllilIlIl|I1111I1111I111|I111|IllllITTilIllll "0'3 111111|1I1111I|111I1111I|l||1111I1T11I1111 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0 r finch) Figure 3.12: Plot of fixed points in the vicinity of V7. 2 1 shows that a rapid renewing for the stability area. In addition, it shows the annihilation of the three pairs of hill-type fixed points at an energy about 244.5 MeV. 52 (no stop band) in such a perfect three sector field. Of special interest is the rapidly reappearing central stable region right after u, = 1 (Figure 3.12). From this graph, we see that the hill type fixed points are annihilated in pairs at an energy of about 244.5 MeV. Beyond this energy, only two fixed points exist (E.O. and outer unstable) both associated with the machine center. This rapid renewing and shrinkage of the stable region at energies just above and below the resonance V. = 1, caused by the sharpness of the edge field, implies that in the absence of field imperfections or with sufficiently small imperfections at the V,- = 1 radius, the beam could be recaptured by the stable region and remain compact. We thus expect that the well-centered particles might traverse this resonance without getting lost. Figure 3.13 and Figure 3.14 show the behaviour of a beam-like phase space area as it is accelerated through the resonance 1x, : 1. The maximum useable beam area roughly increases two times in radial phase area as the energy gain per turn rises from 10 keV/turn to 20 keV/turn. The ”triangle” distortion induced by the VT 2 1 resonance traversal is illustrated as following. As the radial space area shrinks to zero, the off—centered phase space orbits are to move around one of the outside stable fixed points. After being recaptured by the renewed central stable region, the orbits have already been pulled toward hill-type stable fixed points; therefore beam distortion is induced. Note also that the 23,123,. behaviour depends strongly on the field imperfections, especially the first harmonic component, in the Vicinity of 1/7. = 1 resonance, which will be discussed in Chapter 4. 53 1 E 2 10 keV/turn 9 2 0° 1 1 1 1 1 l 1 4 1 1 L 1 l 240 MeV pr Unch) .55{ 50—: 1 'II"rirT"1'T"I""I 2630 2655 ~ 2640 2645 2650 l 41 J l l l 1 1 1L 1 l 4 L A J i I l l A 1 1 246 MeV .509 _- [#14JL l 50—: pr(mch) 45—: .40 9 .35 1 I ' . T 1 r r ' ' 1 1 v 2670 2675 2680 r finch) I V V I Y I l 2660 2665 Figure 3.13: Radial phase plots at E2240 MeV and 246 MeV, 61 2 0°, in the K250 1200 field. Upper panel shows five coasting orbits associated with initial displacements in radius of 5, 10, 15, 20 and 25 mil relative to the equilibrium orbit. Lower panel shows the corresponding orbits at E2246 MeV, which start on the eigenellipses in the upper frame and are accelerated, assuming constant energy gain 10 keV per turn. The dotted curves indicate the estimated acceptance of radial space region inside which most of the orbits can be recaptured by the renewed central stable region after passing V7. 2 1. O 1 l I l J .l l I l 1 J l E = 20 keV/turn 9 = o l 1 .L l l l : 240 MeV _ .75 4 — .70 —_ 55—: pr (inch) .60 9 -4 ,- 50 1 .1 .— " f r T 2 r . 2640 r (inch) . j 1 26.30 4 l l l l l l l l l l l l l l l I J l 1 I J I 246 MeV \' '! .55 ~ ‘1. l .50 pr (inch) .40 2 ‘ . .. Q l .— .— .35 1. _ .. _. —4 P- n ._ ... .30 .. r I 1 I I . , . r 2. . . . 1 . . T . T 2' 26.60 2665 26.70 26.75 r (inch) Figure 3.14: Radial phase plots at E2240 MeV and 246 MeV, 6 2 0°, showing the acceptance of the 11,. 2 1 resonance in the K250 1200 field with 20 keV / turn constant energy gain. It shows the transmission limit of the V7. 2 1 resonance corresponds to roughly a 35 mil eigenellipse at E 2 240 MeV. Chapter 4 Resonance Limitations In the process of beam extraction, the intrinsically low energy gain per turn for synchrocyclotrons poses very difficult problems at the transitions of the various res- onances which occur in the extraction region. In this chapter, approximate analytic resonance formulas are presented for qualitative guidance. I am in debt to the authors who did the original work. Although the derivations are too simplified to give quan- titative estimation for three sector spiral machines, they nevertheless provide insight into the effect of various field characteristics. 4.1 Resonances at the Extraction Region Table 4.1 lists the average radius and energy at which important resonances occur in the extraction radius region, calculated by E0 code with the DTP and the FD inter- polation methods employed. The eleven-point finite difference method [56], denoted FD, which is supposed to do better interpolation computations, is used to check the double three point method, which is the standard scheme in N SCL orbit codes. From Table 4.1, it is clear that the resonance pattern inferred from these two interpolation methods are identical, and the locations of the resonances are almost the same. Therefore, we expect that the interpolated errors associated with the DTP 55 Table 4.1: Average radius and energy of important focusing resonances in the extrac- tion region of the K250 1200 field. 56 double—three-point 11pt finite difference resonances Rav(inch) I energy(MeV) Rav(inch) ] energy(MeV) Hz 2 § 26.03 241.2 25.95 239.7 11,. 2 2 V2 26.06 241.6 25.99 240.3 V,- 2 1 26.20 243.8 26.15 243.1 11., 2 1 26.70 250.3 26.69 250.2 11,. + 2 V2 2 3 26.89 251.8 method should not damage the credibility of results of the extraction calculations. H 4.2 Half Integer Resonance V2 2 5 Following the analysis of Baartman and Gordon [57] [58], at the half integer resonance, V2 2 %, stability is lost if there is a first radial derivative of the first harmonic of the magnetic field. Assume that the deviation of the equilibrium orbit from a circle is relatively small (i.e. a machine with a large number of sectors or one where the flutter field is a small fraction of the average field). Then we can write down the differential equation to describe the vertical motion, d2 67: + V32 2 bzcost) (4.1) _ —LdR. Using the Born approximation, one can obtain a solution by treating the driving field, b, as a perturbation. It can be shown that this driving force changes V2 to V: given by [58] (u: — —)2 = (u. — —)2 — (3)2 (4.2) At V2 2 1/2, V: has an imaginary part Im(z/;) 2 b/2. Since 2 ~ 6””, this implies an exponentially growing betatron amplitude. Also, V: has an imaginary part over a 57 range A112 2 0 around V2 2 1 / 2. If the incoherent oscillations within the beam prior to the resonance are characterized by a maximum amplitude A, then after acceleration through the resonance, this amplitude will be increased by AA given approximately by E1 dl/z/dEl-l (4'3) AA=Ai§bri where E1 is the energy gain per turn. This predicts that the amplitude gain will vary as the square of the driving term due to the fact that both the strength of the resonance and the number of turns spent in the stop band are proportional to this term. 4.3 Integer Resonances 1/7. 2 1 and V2 2 As particles accelerate through the 11,. 2 1 resonance, the oscillation amplitude grows at the rate of 7r€oR per revolution [45], where R is the radius of the orbit. The parameter so in this expression is the ratio of the strength of the first harmonic bump to the strength of the average field. This result can be derived approximately by solving the following differential equation, valid under the assumption of small deviation of the equilibrium orbit from a circle. 3%:— + V3513 2 g 0030 (4.4) where (1:, :1: < R, is the displacement from the equilibrium orbit and g 2 50R. The particular solution 39,, is X (c050 — cos(z/,.0)) (4.5) 1' = P 2 V,,.— Using appropriate trigonometry, this solution reduces to (0,, 2 Q; 32226 (4.6) 58 for 11,. 2 1. Therefore the oscillation amplitude growth rate is equal to «50R per turn. One can also obtain the formula for the coherent amplitude Ares, the induced total amplitude growth after crossing the V7. 2 1 resonance, by using the technique of action—angle variables. Define at: ‘27. 6051/) 2/2 2 14.0 + 05 (4.7) Substituting :1: and its derivatives into the equation of motion and assuming %”g— < 1, we get after some rearrangement €14 _ _ g 30 _ ficosO simb 8gb _ gcosd 0031p 00 —- VT .\/I/—1.-A (4.8) To first order, these can be written as 8A . 66 — — ficosfi szn(/ I/rdfi) (4,9) and integrated to yield AB = g 71 (4.10) El Substituting the expression for g, the coherent radial oscillation amplitude produced is given approximately by [59] 1 AR27reoR-[dur/dnr15 (4.11) where (dVT/dn) is the change in up per turn; that is (dz/r/dn) 2 E1 . (dz/T/clE). The term [dVr/dn]’15 can be regarded as the effective duration of the resonance neff. 1The original formula in this paper is expressed in the cyclotron units and it states that the resultant coherent oscillation of amplitude A caused by an nth harmonic field error Bn at 12,. : n resonance is A 2 1",, f?- B,,, where V; is the change in 12,. per turn. V I 59 Similar to the above derivation, one can easily show that after accelerating through the V,, 2 1 resonance, the beam will have developed a coherent vertical oscillation whose amplitude is given approximately by [60] AZ = 7reo R - [E1 x (duz/dEH-la (4.12) 4.4 Non-linear Coupling Resonances v7. 2 ZVZ and V. + 2V2 2 3 Among the essential nonlinear coupling resonances, those requiring the greatest atten— tion in this three sector machine are the third order resonances : 1/7. 2 2V2, Uri—214, 2 3. (For example, the third order resonance V7. 2 211,2 stems from the driving term 332:2 in Hamiltonian). The higher order resonances are presumably negligible because the displacement from the E.O. is small in most of the accelerating process except the first and the last few turns. In general, if the beam is reasonably well-centered on the equilibrium orbit, then the acceleration through a nonlinear coupling resonance should proceed without significant damage, provided the resonance is traversed fairly rapidly [60]. However, this will not be a proper claim for synchrocyclotrons because the energy gain per turn is too small to pass the resonance quickly. A nice mechanical analogue of the coupling resonance is a simple spring pendulum. If the resonance condition is satisfied (M 2 g p, q in integer and p+q the order of resonance), let the pendulum swing vertically and after a while it will oscillate horizontally. The amplitude of the axial motion arising from a coupling resonance is approxi- mately calculated as follows when the resonant condition is kept up long enough so that the radial oscillation dies down and the axial mode builds up. If the kinetic energy T of lateral motion is transferred completely from one mode to the other, so 60 T1, 2 T2. one can easily show ME“ V2 _ Z (4.13) where A3, A2 are oscillation amplitudes of radial and vertical motion respectively. In the 11,. 2 2V2 resonance, we see that any radial oscillation amplitude is multiplied by two when the energy is transferred to the axial mode. Therefore, partial or total vertical loss of ions is expected to occur if the beam is not well centered radially or the beam size is too big. The coupling (Walkinshaw) resonance at VT 2 2V2 is usually regarded as a formid— able barrier for synchrocyclotrons. However, this resonance can be passed if the machine has good internal beam quality and if the field is suitably shaped (eg, Tokyo synchrocyclotron [61]). Both the strength of the driving force at the resonance and the resonance width depend critically on the radial derivative of the average magnetic field and on the amplitude of oscillation. A. A. Garren [62] gave the following results. If one keeps only the largest nonlinear terms and neglects the higher order derivatives of the average magnetic field than the d2 B 4 4.13. dro second degree (k E u’ E g? and u” E :3 ). Then one can find that the equations 9. for :1: 2 (7‘ — r0)/r0, y 2 z/ro 2 are approximately d2 1 d—Hf + I/xsc 2 Eu”(y2 — 2:2) (4.14) d2 564?:- + ujy 2 u"a:y (4.15) Regarding the nonlinear terms in the above equations as perturbations, and assuming the amplitudes and phases of the :c and y oscillations to be slowly varying parameters, 2The notation have been changed here, and now writes V,, and I/y in place of 11,. and 1/3 in this section. 61 then: :1: 2 Axsin(I/x0 + 4,), pl. 2 I/xAxC08(I/$0 + 14.), (4.16) and similarly for y. Thus the equations of motion for A3,, Ay, (bx, and 1b,, are obtained, retaining only terms of low frequency, a E 21/y — V332 c1ch = £41]: Ajcosx (4.17) dig] 2 igAgcAycosX (4-18) 6%; 2 _8uz/: fi—isinx (4-19) {lb—6y 2 —%Axsz'nx (420) where X E 06 + 2%, — it... This system may be shown by setting l/x 2 21/y in the various coefficients to have two invariants, A and C, defined by 4A2 = 4A: + A; (4.21) C 2 g—AzAxsinX — GA: (4.22) where the first invariant A is the result of energy conservation. 2 From Equation 4.22 one may see that A: will oscillate between limits Aymm and A2 ymax, positive solutions of the quartic equation for A: obtained by setting sinx 2 :l:1. The limits Aim-n and Aim” are close together for | 0 |>>| u" A / V,,, |, corresponding to little change of axial amplitude, but if | a |<<| UNA/Va; | they may be quite apart, and most of the oscillation energy may go into the axial mode. Thus the 0 2 i | 62 u”A/ 111, I define the edges of the stop band, whose center is located at the radius where VI 2 2Vy. As particles are accelerated, the parameters a, 11,/,1/9; and I/y change continuously. Since 0 changes most rapidly, the full stop band width is given approximately by “H.111 neff :l E] . (IO/d7”). l (4.23) From Equation 4.18 one can estimate the value of maximum z-amplitude gain per turn as dz 7m” —- Imam — x cm x Z0 (4.24) dn To where 3:0, 20 are the initial oscillation amplitudes before the particle crossing this resonance. The effect of resonance V,- + 21x. 2 3 could be dealt with similarly. Chapter 5 Longitudinal (Synchrotron) Motion and Its Implications An unusual dee structure is suggested for the proposed K250 synchrocyclotron; dees in two adjacent valleys of the K500 cyclotron are galvanically coupled across the center to become a single electric dee and the third dee is removed; even—spaced accelerating gaps are then no longer present. The formulae in this chapter are derived mainly following the path given in Livingood’s book [63]. The maximum energy gain per revolution q-Vmax is equal to q-Vdee for this pro‘ posed RF system, where Vdee is the dee voltage. Therefore, the resulting relation between energy gain per turn E1, phase angle 45 and dee voltage Vdee is E1 2 q x Vdee X cosd. The phase reference angle 45 is the angular difference between the RF and the particle. The synchronous phase angle, the phase angle of the ideal synchronous ion whose frequency of revolution is the same as that of the accelerating voltage at every instant of time, (153 2 48.20 is decided such that the largest number of ions can clear the source and then pass successfully the resonances at the extraction region; the largest external current is thus obtained. In following sections, the principle of phase stability is explained and the equations are derived giving the frequency of the synchrotron oscillation, and the characteristics 63 64 of the stable region (the so-called Fish diagram), from which the bucket area can then be obtained. Furthermore, the frequency modulation and acceleration time are worked out by using the E0 code to compute the orbital frequency at each energy. Finally, the validity of a time saving constant energy gain approximation and of the initial conditions in the extraction calculations are investigated in the last part of this chapter. 5. 1 Basic Concepts Synchrocyclotrons look and operate very much like cyclotrons and are able to avoid the energy limitation of the classical Lawrence-type fixed—frequency cyclotron by vary- ing the frequency of the oscillator to match the motion of the ideal synchronous particle. As acceleration proceeds and the rotation frequency drops because of the increase in mass of the accelerated particles, a tuning element steadily lowers the frequency of the RF oscillator so that the orbital frequency of an ideal synchronous particle matches the frequency of the RF voltage at every instant of time. Most of the ions (nonsynchronous particles) are accelerated by benefit of the principle of phase stability; they gain energy at the same average rate as does a synchronous ion but oscillating about this average. Using this type of system, the upper energy limit of the cyclotron can be raised to whatever limit the bending power of the magnet allows, since the mass increase of the accelerated particles no longer causes a phase shift rel- ative to the RF. One price paid for the greater energy is that the output current is vastly reduced, since each group of ions must be accelerated to high energy and then the oscillator returned to the starting frequency before it can pick up another group. The separation of the motions of betatron (traverse) and synchrotron (longitudi- nal) oscillations leads to a quite simple treatment and sheds light on intuitive expla- 65 nation of cyclotron orbit behavior. Therefore, we focus on the longitudinal motion by analyzing the phase equation in this chapter. The median plane View of the proposed RF of the K250 synchrocyclotron is given in Figure 5.1. It shows the coupled dee configuration in the central region and four accelerating gaps, 600 apart. The sinusoidally alternating voltage in synchrocyclotrons can be expressed as t V(t) = v... x cos[/ w.f(t)dt] (5.1) where f: w.f(t)dt is the RF angle 0.f(t). Let the RF angles when an ion crosses the accelerating gap i (z 2 1,2,3,4) be 071,-, respectively. The energy gain per turn in conjunction with first harmonic mode operation (h 2 1) is thus given by E1 2 qu66 - [00349.11 — €036.12 + cosdrfg — 6089,14] 2 quee - [605(6.f1) —- c03(t9,.f1 + 7r/3) + c03(0.f1 + 27r/3) — 003(0.f1 + 7r)] 2 queecosflrfl (5.2) Therefore, quax, the maximum energy gain per turn, is equal to qI/dee. Define 45 E 01'f ‘_ apartz'cle (53) With the particular spacing, gaps 2, 3 and 4, just cancel at every (6 and so we fix the particle reference angle at the first accelerating gap, and finally obtain the expression E1 2 q X Vdee X 00305 (5.4) Since an increase of energy causes an increase in the period of revolution, the synchronous phase angle (b, must be chosen to lie between 0° and 90° for the reason of the phase stability, as will be explained in the following section. 66 180' , . Goo 1 GOP ‘ .' +:— Center plug 270' 90. Two dees golvonicolly coupled across the center 10 inch Moin coils Figure 5.1: Median plane View of the proposed RF in the K250 synchrocyclotron. The four gaps are 60° apart azimuthally. It also shows the coupled dee configuration in the central region. 67 5.2 Principle of Phase Stability The way in which phase stability acts in the K250 synchrocyclotron is illustrated in Figure 5.2. Consider a particle which has energy E, but with phase (to, (—¢s < 050, < (153), as shown in Figure 5.2. It experiences an energy gain AE > AES and hence immediately becomes nonsynchronous, for now E > E... This makes its period exceed the period Trf of the oscillator; so its phase one turn later is at $1,, which is closer to 053. The energy is then raised again, though by a lesser amount, and the phase shifts further in the same direction. This process continues on subsequent turns until the particle’s phase exceeds (253, reaching a maximum energy surplus from the synchronous value. The particle then comes to experiences an energy gain AE < AES, and the accumulations of energy difference start to fall. At 45’, E becomes equal to E3, and the ion is again' synchronous. (15’ is however not (b, and so immediately on the next turn, the particle finds itself with an energy deficit compared to the synchronous value. Its revolution period is less than 7“,; and one turn later it is at a phase slightly less than (b’. The energy gained is below the synchronous value or it may even be negative, so that a further shift to smaller phase angle takes places. This continues until the accumulated energy increments total up to the new values of E3; this occurs just as (ta is again reached. A complete phase oscillation has thus transpired and thereafter the process repeats. If one repeats the arguments with (b, located between —7r/2 and 0, it can be found that phase oscillations do not occur for a synchronous phase angle in the fourth quadrant. 68 E>Es EZE Arrow: 4) advance direction/ EzE < s \ E if LL;— stoble equil. phase range Figure 5.2: Possible excursions in phase of nonsynchronous ions. In order to save space, the successive values of the phase are plotted on a single wave. 69 5.3 The Phase Equation The instantaneous difference in energy between a nonsynchronous and a synchronous particle is given by AE 2 E — Es; where the subscript 5 denotes the energy associated with a synchronous particle (i.e. a particle whose rotational frequency would just match the frequency of the accelerating voltage at that moment). By Equation 5.4, the change in the difference of their energies per unit angle is obtained dividing the change in the energy difierence per turn by 27r. dAE V...3 _d6 2 L2: [00305 — 003055] (55) If we are neglecting the small difference in the fields corresponding to synchronous and nonsynchronous particles, then we have 1-14_11.11 (M, d6_dtdi9—wdt_wsdt ' The change in energy difference between a nonsynchronous and a synchronous particle is therefore given by _1_dAE E quee 0.), dt — 27r [00305 — 003053] (5.7) In addition, the change in phase d0, in a time dt, can be written as (note (.07.): 2 (.03) :1? 2 L03 _ w (5.8) Define dw dE __ : —F . ws E3 (5 9) Since F in general is a slow varying function of synchronous particle energy, combining Equations 5.8 and 5.9, then yielding AE %2ws— E—AwE’wSF E3 70 Consequently, AE can be expresses as N Es 61¢ AE _ wsl‘ dt (5.11) When this is substituted in Equation 5.7, the final result is 1 Cl Es d¢ N quee ws dt(wsl‘ dt) _ [0030 — 00305.] (5.12) 27r This 2nd order differential equation expresses the time dependence of the phase 05 of the nonsynchronous particle as a function of the phase, frequency, and energy of the synchronous particle and the magnetic field (through the factor F). 5.3.1 Frequency of Synchrotron Oscillation of Small Ampli- tude Assume the excursion in phase is small, let 6 E 45 — 45., and w. and F change only slightly over short period of time. Then the phase equation becomes ([25 E + 925 2 0 (5.13) 02 ___ quczzefgzngbsw: (514) With 05. 2 482°, the ratio of period of synchrotron oscillation to that of betatron oscillation in the extraction region for the K250 synchrocyclotron is T—-° 2 860 (V... = 151W) (5.15) 772.0. 73.0. g 610 (V... = 30kV) (5.16) 772.0. 71 5.4 The Phase Diagram Since (.0. and F change only slightly over short intervals, by 444440 dt? _ dt dt _ dt d05 dt (5'17) the phase equation (Equation 5.12) may be expressed as (105 d dgb 02 dt d05 dt 3201ng (00305 — 003053) 2 0 (5.18) Multiply by 0105 and integrate, then yield dgb 2 202 (If?) _ 3221058 (322105 — 000305.) 2 C (5.19) where C is the integration constant and can be evaluated from the initial conditions. If 05.- is the initial phase and 05; E (dqb/(lt). the initial phase velocity, we obtain 2 (9%)? = 2;;302‘124 — 3.1.4. —- (4 — 4.)cos¢.) + (4:)2 (520) Set initial condition 0.- 2 —¢., which represents an extreme displacement from (5. (i.e., a minimum value of 05) if stable oscillations in phase are to occur. Consequently it follows that (d05/dt). 2 0. Equation 5.20 then becomes iiiL 0 dt _ F(05, 943) (5.21) where 17(0), 05.) E Tisin05 (327105 — 050030. + singbs — 053003453)]1/2 (5.22) Convert dqb/(th) into expressions describing the difference in energy AE and the difference in the equivalent radii AR of the orbits of the synchronous and the non- synchronous ions. We have 1 AE AR 6055744)? B“ ”(Ml (5'23) Wop 062wof‘) 72 where a is the momentum compaction and fl is the relativistic parameter, 0 / 0. The function F (05, 05s) plotted as a function of 05 for several values of 05. is given in Fig- ure 5.3. The bucket area inside the separatrices is the particle stable region. The intercepts with the F 2 0 line represent the extreme amplitude of 05. The maximum excursions of synchrotron oscillation in energy and in radius occur at 05 2 0s. The function F(05 2 03,053) plotted vs. 05. is also given in Figure 5.3. One can draw the following general conclusions. With a large value of (0. (near 90°), the average energy gained per turn is small, wide oscillations in phase develop, the beam of ions exhibits a relatively great inhomogeneity of energy, and the current is big. On the other hand, with a small 05. (near 0°), there is a large average energy gain per turn, small oscillation in phase, little energy spread, and a relatively small current. Given the choice of 05. 2 482°, the synchronous ion receives 2 / 3 maximum energy gain per turn and the nonsynchronous particles undergo the oscillations in phase, in energy, and in radius. The maximum permissible amplitude of synchrotron oscillation in energy and the corresponding radial displacement, associated with Vdee: 15 kV and 053 2 482°, plotted versus the energy of the synchronous particle are shown in Figure 5.4, computed from Equations 5.9, 5.14 and 5.23, with the parameters F and dR/dE calculated by the E0 code. 5.5 Acceleration Time and Frequency Modula- tion The average energy gain per turn, queecosgbs, gives the following expression for the acceleration time 250MeV dB 2 ”.24 T v/O VsqueeCOS¢s (O ) 73 F(¢. 95,) 50 ¢ (degree) I I l J m l 20“ _ i [. ‘ f 15- L r 4 1 (‘1‘ 1.0— _ 3 « . _L‘; . 05‘ ~ I C 0.0 l ' ' ‘ l ' ' ' [ 1 I I l I T v i v v 0 20 40 60 80 ¢s (degree) Figure 5.3: Top: the phase diagram for the case where 05,- 2 —053. Numbers be- side of curves give (5. in degree. Bottom: Absolute values of F (05 2 05., 05.) vs. 05. gives the maximum excursion of synchrotron oscillation associated with the choice of synchronous phase 05.. 74 d j l l l l l l l I l I l l l l l v ee215l 1 l O . A _1 d .l d - _. (MeV) E . syn.ian Figure 5.4: The maximum permissible amplitudes of synchrotron oscillation in energy and in radius are plotted with respect to energy of the synchronous particle. The calculations are done from Equations 5.9, 5.14, and 5.23. 75 Table 5.1: Parameters of the RF system Dee voltage 15.00 kV 30.00 kV Fixed energy gain 10.0 KeV/ turn 20 KeV/turn Acceleration time 0.53 ms 0.26 ms Modulation frequency S 2kHz S 4 kHz Initial frequency 50.55 MHz 50.55 MHz Final frequency 43.22 MHz 43.22 MHz Average time derivative 13.84 MHz / ms 27.68 MHz / ms where V. 2 tag/271' is the revolution frequency of the ideal synchronous ion. As a numerical example, for the case with fixed energy gain per turn 10 keV, the result is T 2 0.53 ms. The proposed RF modulation frequency fm is 1 kHz, corresponding to 1.0 ms period, which is thus feasible. For the synchronous particle, 1/7. f 2 V., we get .1... _ .1... __ 43412411 _ .1. dt — dt “dEdN clt ‘dE X 01416600305. X l/s (5.25) where VT] is the RF frequency and V. is the orbital frequency of the synchronous particle. From the above formula, the RF frequency as function of energy can be calculated by using the equilibrium orbit code. The resulting RF frequency time derivative (——dV,.f/dt) is plotted in Figure 5.5. The wiggles in this graph are due to both spatial ripple in the field of the trim coils and to the DTP field interpolation scheme. Finally, the required RF parameters for the K250 synchrocyclotron are listed in Table 5.1. 76 22 l l l l l l l l l l l l l l l l l l l l l l 20 2 Vdee E 15 kV _ ‘ rxxxag = 2/3 ‘ 4 E1210 keV/turn _ /‘\ _ _ U) 18 e : . \ N 1 _ I 16 2 _ E _ L \_/ A T :5 I _ 14 __ \_ _ _ A — _ ’O _ _ \__/ l 12 -— — 10 — — 8 l l l l I l l l l I l 7 ‘1 l I l l I l I l l i l 0 50 100 150 200 250 Energy (MeV) Figure 5.5: Plots of the RF frequency time derivative vs. energy with constant equilibrium phase 05. 2 482° assumed throughout the acceleration. 77 5.6 Internal Current and Starting Betatron Mo- tions in the Extraction Calculations The average internal beam current 1..., in a synchrocyclotron is proportional to the peak current, 1p..k, the capture efficiency, 5., and the beam bunch angular length, A05 [55]. 445 I; 2 peak ' EC ' E (5.26) [peak : 60 ' V0 ' 2h ' E1 ' Vimin (5.27) where E1 is the energy gain per turn (proportional to the dee voltage), 2h the max— imum beam height, V. the orbital frequency in the centre and Vzmm the minimum values of the focusing frequency (Equation 5.27 is taken from Equation 2.18 with %f 2 1 since this term is separately accounted for in Equation 5.26). A simple theory of ion capture has been given by Bohm [64] and is reproduced here for reference convenience. Recall the initial conditions, 05,- 2 —05. and a; 2 0, lead to the maximum stable excursion from equilibrium phase 05. and thus lead to the maximum phase velocity, 0’- From Equations 5.20 and 5.14, we then obtain I 277103;" 2qu66 F 2 (¢fmar0)2 : ——7i'—Eg— . ws ' (3371053 — ¢3C03¢8) (5.28) According to Equation 5.8, the phase velocity 05’ 2 w. —w measures the discrepancy between the angular frequency of the ion and that of synchronous particle. Because the catching process is determined only by the motion near the origin and all the ions have the same angular orbital frequency 02. at the centre, Equation 5.28 tells how much the radio frequency may differ from (.0. for a particle still to be caught into phase stable oscillations. In other words, any particle starts from the ion source when 78 the RF frequency is such that (w - we)2 S Mia”)? (5.29) will perform stable phase oscillations and will be accelerated to final energy. Sub— stituting Equation 5.28 in Equation 5.29, therefore, we obtain the total range of frequencies, Awrf, in which capture into phase—stable motion is possible [QqWCCI‘O 7rmoc2 Aw“. = 2¢' = 2% x (Sims. — ascosqssné (5.30) imam Since the rate of decrease of RF frequency is approximately constant, the range of time, At in which ions can starts in orbits which are phase stable, is Aw“: t == -—----—- . A —dw,.f/dt (5 31) Finally, the expression for the capture efficiency so is obtained _ At _ —2w0 QqueeFO , L ac : Tm — Tm(dw,.f/dt) X [ 7rmoc2 (szngbs ¢scos¢s)] (5.32) The beam’s angular bunch length is determined by the choice of synchronous phase Q53, by the dee voltage, and by the actual geometry of the central region. For smaller equilibrium phase angle, the angular bunch length decreases as a result of the smaller range of phase stability. On the other hand, particles with larger initial phase angles may not clear the source due to the negligible energy gain in the centre of the synchrocyclotron. The capture efficiency, bunch angular length, internal current and beam quality therefore can not be accurately computed without knowing all the relevant parame- ters. The internal beam intensity is thereby estimated approximately as follows: Based on ion source experiments by M. Mallory [65]: [peak 2 12.74mA with Vdee 2 30 kV 79 and 1.57 X 9.525 mm2 ion source slit gives 404 mm-mrad radial emittance and 390 mm-mrad axial emittance. We estimate that 1pm,, 2 270 uA is achievable using a 0.33 x 0.95mm2 source slit with dee voltage 15 kV, by assuming constant current density from the source. Because the beam is so tiny (inferred from the size of the source slit), it is expected to always stay in the linear area of the stable region until extraction occurs. Therefore, from Liouville’s theorem, the beam should coincide with the invariant eigenellipse throughout the acceleration process before extraction. This in turn implies a beam with radial betatron oscillation amplitude of 30 mil and a vertical betatron amplitude 20 mil at E 2 240 MeV. From Table l in Ref. [55], it is clear that cc 2 0.2%, Agzi 2 200 should be easily achieved in the K250 synchrocyclotron central region. Then the internal current turns out to be 30 nA from Equation 5.26. This value means that an extraction efficiency of approximately 30% will suffice to meet the intensity goal for a proton therapy system, namely a time average current of approximately 10 nanoamps. 5.7 Validity of Constant Energy Gain Approxi- mation It is important to recall that particles gain energy at the same average rate as does an ideal synchronous ion but oscillating from this average in synchrocyclotrons. Con- stant energy gain assumed in the calculations may therefore be a fair approximation. The accuracy of constant energy gain approximation is checked by using the 11,. 2 l excitation extraction field, which will be presented in Chapter 6. The comparison of SYNZ4 1 runs to constant-energy-gain runs by using the Z4 code are given in Ta- ble 5.2, Figure 5.6 and Figure 5.7. The energy histogram in 0.01 MeV steps, 7“ vs. pr, and 2 vs. p, in cyclotron units at 0 2 336° are plotted in Figure 5.6, for those 1A modified version of the NSCL Z4 orbit code to do synchrotron motion exactly. 80 particles which successfully enter the extraction channel. Figure 5.7 compares the distribution of final energies in 0.01 MeV steps, r vs. pr, and 2 vs. p2 in cyclotron units at 0 2 308° (the last angle before the particles cross the 40” edge of the stored magnetic field). For particles which gain more energy than the average rate, the chance of surviving the resonance crossing is increased, and extraction is at higher energy. The particle with less than average energy rate gain does the opposite. As a result of these two factors, the distribution of final energies becomes flatter and broader; the extraction efficiency is therefore slightly reduced (about 2% in this case). In conclusion, the constant energy gain approximation yields a good agreement with SYNZ4 runs which include RF modulation in the calculations and deal with syn- chrotron motion realistically. Moreover, the computer time in constant—energy-gain calculations is substantially reduced relative to those of SYN Z‘1 runs. The constant energy gain approximation will therefore be employed in Chapters 6, 7 and 8 which compare three candidate extraction systems. 81 Table 5.2: Extraction calculation results of the two versions of the Z4 code. [ ] Exact synchrotron motion I] Const E1 approx. [ [ Initial conditions (Synchronous ion: E, 2 240MeV, 9253 2 48.20) [ E. (MeV) 239.6 240.0 240.4 240.0 A13.- (MeV) 04 0.0 0.4 0.0 A05,- (rf degree) -20, -10, 0, 10, 20 x A. (mil) 30 30 A. (mil) 20 20 No of rays 8250 l 8625 9000 1725 Code SYNZ'Jl Z4 E1 (keV / turn) 10 (average) 10 for all particles I Results of pre-extraction calculations l CPU time (hr) 24.3 23.2 26.7 4.7 e....,,(%) 79.9 84.6 76.3 83.3 AE (MeV) 1.54 1.39 1.81 0.95 Epeak (MeV) 243.12 242.96 242.89 242.96 [ Final results ] 6.4%) 15.4 18.0 15.4 17.8 21088 (%) 3.8 6.7 18.3 6.0 AE (MeV) 0.54 0.52 0.72 0.41 B,,... (MeV) 243.10 242.99 242.87 242.96 No. of parts SYNZ‘, A¢i=0', $10., 1:20. Ei=239.6, 240.0, 240.4 MeV Ni=25875, CPU=74.2 hr L A A L A L L L J 1400-3 eentr = 80.1% 1200.] AE = 1.68 MeV 243.0 2455.5 Energy (MeV) It'v’ff‘lrv . i f I I I V U I T V I I 26.50 26.65 26.70 26.75 r (inch) 0.4 I n n n I 1 L 1 l A L A l a I 1 T 0.2 - ~ 0.0 -* .. -0.2 '1 - -0.4 - , ' T T '- ' ' ' 0'0 02 2 (inch) 0.4 82 p: (inch) Const energy goin approx. Ei=240 MeV, [31:10.0 KeV/turn N7=1725' CPU=4.7 hr l l 100 1 l A . . i l . . . i . L 4 E = 83 03% [- entry 801 AE = 0.95 MeV _ 2 '5 . o. “6 : 2' r L f V I n' 1 l 7 242.5 243.0 243.5 244.0 Energy (MeV) A; A I I A A l A l l 1.8 ] ,1 I,“ . . ' . : 3:...1-1‘ 75%" ‘2 "‘3’” 3 I ’ F 1 L. ‘3 "‘8' 'l, 'I < . ‘l 'I b 1 6 4 * affifsioe—r; . j f' g:: “'3’ \ ‘ 5 l: 1 [7 1.4 4 - . . i? 1.24 l o . . .E . Q7 1.0; _ 0.8; 1 0.63 l 1 r . f . r r 4 . . , . . 1 . 26.60 26.65 26.70 26.75 r (inch) 0 4 _ I l 1 I 1 j l L A L l ‘ r 0.2 -1 - 0.0 ‘ -:".L’ ‘ \ 1,3}. _ -0.2 ‘ ' i 4 . _0’4 ‘ T Y 1' f v I 1' v V I I T 1 I f —0.4 -0.2 0.2 0.4 0.0 2 (inch) Figure 5.6: Plots, obtained by using the two versions of Z4 code, show the distribution of energies in 0.01 MeV steps, 7" vs. p.., and 2 vs. [0,, in cyclotron units at 6 2 336° for those particles which successfully enter the extraction channel. SYNZ‘, A¢i=0', 110', 120' E,=2:59.6, A ‘ A A I L A A 240.0, 240.4 MeV 40°“. eex=16.2% } AE=0.75MeV 300: zloss=9.6% 4 J 1 .3 N O O No. of parts 242.8 Energy (MeV) 242.6 243.0 | 1 A A A l A A A l l A A A 1 l J 243.2 LLngl 16.6 “ — 1 ‘ . ‘ .- . 16.5 1 4- 115.5- - h .1 3(3)". i _..'- I - ‘ dfiygfigii -. . “a". ,;:'_*.‘1_“?;“ If .5 16.4 - 1.152.” _ v ‘ . $.53} .52 2:5?" .. 9'“ -“* -“ " 5 . -.,.;¥QS}\.’L§'~‘A.' o- .. : ‘_ fins-‘1" ynvftx I 4 '| . 11"," 1%};& i. ‘5. .331 i- " ] . . :L’. irk’fl‘ _ -_.¢~,_-, ._2~ 1603 d :r.-..'. .- 4 p L ‘ 1‘ 16.2 ‘ _ 39.0 ' ' 39.1 r (inch) 39.2 V’WTYII’YV 59's 0.2 n 0.1- d 0.0 -1 pl (inch) -0.1-1 ‘l’T’Y'III V l V I 0.0 2 (inch) I V V V I I V . , 4 . —O.4 —O.2 V l V V V j—V V 0.2 0.4 0.6 83 pr (inch) pz (inch) Const energy goin approx. E2240 MeV, E,=10.o KeV/turn 1 1 1 1 l 1 1 1 1 1 1 L 1 l 1 1 25 _ eex=17.8% L 3 AE=O.41MeV t 20] zloss=6.0% L 3 i 315; L ‘5 C o' I Z 10 _ .. 1 I 5 - T 1 I” | i ll 0 1111 1 _ 242.6 242.8 243.0 243.2 Energy (MeV) 1 m l 1 1 14 L 1 1 1 1 L1 4 1 1 l 1 1 1 1 16.6 - 16.5 - - . .:_ ‘.' . 16.4 ~ .._f.' ~ 16.3 - r ‘ F 16.2 '1 1- ..,....,....T....,... 38.9 39.0 39.1 39.2 r (inch) 0.1~ 0.0 - _0.1_ —0.2 - I l I I I J‘i L J l l l l l J A l A l I I. .. )- 1. D . p. b 1.3‘ - 1: 0‘ ,: 73.. .° uf': _ - a '1‘ I1". 2'2"; .— ‘J UH) ." . 1 -u_".'",~. >1 : , 9 . I" . 1.. 1 . .. . b i— V l V V V I V V V l V I V I ' '—oI.4' ' '—o'.2' ' 0.0 0.2 0.4 2 (inch) 0.6 Figure 5.7: Plots, associated with the two versions of Z4 code, show the distribution of final energies in 0.01 MeV steps, r vs. p., and 2 vs. p. in cyclotron units at 0 2 308° where particles go off the field. Chapter 6 Regenerative extraction 6. 1 Introduction ’ ' Regenerative extraction”, the first system which we consider for extracting the beam from this synchrocyclotron (and the system which in the end seems most effective) is based on the process invented by Tuck and Teng and developed by LeCourteur in the 19505 [66] [67] [68]. The method as originally conceived uses two magnetic distur- bances, a ”peeler” and a ”regenerator”, usually 1000 displaced. This cos 20 dependence of the field disturbances induces radial oscillations with exponential increasing am- plitude associated with the V7. 2 2/ 2 stop band. The peeler introduces an outward impulse causing a phase delay, and the regenerator introduces inward impulse given a phase advance. After one revolution, the amplitude is increased and the phase is returned to the initial phase. As the beam accelerates out to the regenerator, the stability region corresponding to centered orbits shrinks and the separatrix limiting the stable region pulls gradually in through the beam spot. These dumped particles then move out along the asymptote until they reach the channel radius, at which point they either enter the channel or are lost depending on whether their position coincides with the channel aperture. In general, the desired regenerator field shape can be produced by two blocks of 84 85 steel placed symmetrically above and below the median plane. Sometimes, the rapid fall off of the main field supplies the required peeler effect. Our regenerator design is motivated by LeCouteur’s work and is optimally found by extensive numerical calculations. LeCourteur [68] showed that a suitable regener— ator strength would be T = 0.20 + 0.262 (6.1) One can also express the regenerator strength as T = —— x fABdB (6.2) where To and Bo are the radius and magnetic field measured at the start of the regenerator action. AB is the regenerator field and the integral is carried out on a circular orbit of radius 7‘0 + ,0. The quantity ,0 measures the distance from the last unperturbated orbit, and is measured in inches. The quantity 6 is measured in radian. As was pointed out in H. Blosser’s paper [36]: The combination of energy spread, angular spread, vertical loss and a distorted object shape combine to make design of the extraction channel for synchrocyclotrons an extremely difficult problem. The typical energy spread, about 1-2 percent at the entrance of channel, corresponds approximately to the difference between the energy at which the stable region is just equal to the beam spot area and the energy at which the stable region has just vanished, whereas the exit beam will usually have spread of only a few tenths of a percent due to the limited momentum bandpass of the extraction system. In addition to the energy spread itself, the asymptote shifts in position on the phase diagram as the stable region shrinks, thereby imparting an additional spread in angle into the beam entering the channel. Distortion of the beam is induced by the half-integer resonance effect when the particle starts to increase the turn separation and becomes 86 off—center rapidly. Moreover, the axial instability may be induced at the coupling resonance by this beam off-center phenomenon. The extraction efficiency, sex, can be decomposed into two components €8.13 : gentry X Etrans (6.3) where sentry is the channel entry efficiency and am,” is the transmission efficiency. The channel entry efficiency is given approximately by [69] t Sentry S 1 — E (6.4) where t is the septum thickness of the first extraction element, Ar the turn to turn separation at the entrance of the first extraction channel. The equality in the above equation holds for the situation of no axial loss. If the electrostatic deflector is used as the first channel, then 2-3 mm turn sepa— ration is probably sufficient to achieve high extraction efficiency, because the septum thickness of an electrostatic deflector can be 10 mils or less. The typical extraction efficiency of synchrocyclotrons of the 19505 only reached a few percent level, and the external current reached in the territory of 0.05 11A. The extraction efficiency, external current and final energy of protons for selected synchro— cyclotrons are listed in Table 6.1 [70] [71] [72], the last entry including improvements in an extensive upgrade program in the late 19608. The poor extraction efficiency for these old fashioned synchrocyclotrons is mainly owing to the poor quality of the internal beam, the thick walls of the magnetic channel which was used for the first element and insufficiently compensating field imperfections. High extraction efficiency, however, is desirable because the problems associated with septum cooling and machine activation due to high energy proton 87 Table 6.1: The extraction efficiency and external current of protons for selected syn— chrocyclotrons. Name of Energy Extraction External First beam Res. AE / E machine (MeV) eff. (%) cur. (,uA) (year) (%) Harvard 160 4 $0.05 1957 0.9 Berkeley 740 15 0.1 1946 x Orig. Uppsala 192 1 0.005 1956 0.1 Impr. Uppsala 180 x 0.3 1991 x Orig. CERN 592 10 0.1 1957 l Impr. CERN 600 ~70 5 ~1975 ~0.2 bombardment will be significantly reduced. The impressive 70 percent extraction efficiency for the CERN 600 MeV proton synchrocyclotron was achieved mainly by improving the quality of the internal beam and by the provision of a electromagnetic deflector, such that the disturbance in the main field are negligible [72]. We thus conclude that one of the crucial keys to achieving high extraction efficiency is good internal beam. Other extraction requirements, simply stated, are [42] [73] : 1. It must produce a sufficient separation between successive turns so that most of the ions can clear a septum and enter the exit channel. 2. Losses owing to axial blow up must be minimized. 3. The beam spot shape should be kept compact. 4. The extraction system should have sufficient aperture to transmit the desired current. 6.1.1 LeCouteur’s Linear Regenerative Theory The theoretical basis for the regenerative beam extraction [66] is briefly reviewed in this subsection. Assuming circular equilibrium orbits, the action of the peeler- 88 regenerator can be written as a series of impulses, thereby described by the transfer matrix :1: _ cosz/Tf W l 0 cosurd s—"ff’fi (6 5) 56’ _ —1/TsinI/.f cosurf —R 1 —1/.sz'n1/,.d cosz/Td ° 10 {130 0,8 51:0 _—‘ $0 X(P1)(.1)=(. 61(4)=MT(4) where :1: is the radial displacement from equilibrium orbit, .1." is the corresponding velocity (23’ 2 (133/016), (1 is the angle between the peeler and regenerator (f 2 277 — d), P 2 —§%—§ X 6p, R 2 733—? X 03 and 0p, 03 azimuthal extension of the peeler and regenerator. For the vertical motion, the transfer matrix M, can be obtained by the following changes V.-—>1/z,P—>—P,R—+—R (6.6) The stability condition is given by the trace of the matrix M 07 + 6 > 2 : unstable (6.7) oz+6<2zstable For proper operation of the system, it is necessary to keep the vertical motion stable and to excite the radial motion. Then after n turns the radial motion can be written ( 5”) 2M: ( ”1‘9 ) 2 K1619 ( .. ) +ng'”’\ ( “,) (6.8) :1: n 51:0 1) v where e" and e”\ are the eigenvalues of the matrix 71—1,. and can be found from the as I u u e o f 7 e trace; ( v ) and ( 12’ ) are the correspondlng eigenvectors; 111, 112 are determined b $0 . For a sufficiently large n, the last term of Equation 6.8 will dampen out, y xi) 89 and the phase of the oscillation is given by an eigenvector ( Z ). The radial motion is therefore characterized by exponential increasing oscillation amplitude; the needed turn separation is achieved. 6.2 Extraction System In this section the extraction system which we consider is assumed to consist of the existing pair of K500 electrostatic deflectors, a regenerator inside the empty dee in between, followed by the existing K500 magnetic channels of the passive type which are made with bars of saturated iron. Futhermore, all the channels are located in one of the existing rectangular holes in the K500 magnetic yoke. The fields produced by the magnetic extraction elements are calculated by employ- ing the charge sheet program [49]. These fields are added to the threefold symmetric 120° field map produced by iron, main coils and trim coils, thus giving the 360° total map which we use to study the beam dynamics. 6.2.1 Layout and Design Features Our design process consisted of iterating back and forth between calculations of the field produced by the extraction devices, and computation of extraction orbits, in order to systematically improve the design parameters and optimize the extraction efficiency. Figure 6.1 shows the layout of our final design for the regenerative extrac- tion system. It shows the arrangement of the extraction devices as well as the last five turns of a typical extracted orbit. All the elements shown in Figure 6.1 are also listed in Table 6.2 together with their main parameters. E, R, M and C characterize an electrostatic deflector, the re— generator, a magnetic channel and a compensating bar, respectively. The parameters 90 Ygenerotor/CZ Volley Figure 6.1: Layout of the extraction system showing the arrangement of the extraction elements as well as the last five turns of an extracted orbit. Table 6.2: Extraction element parameters 91 e. e. R. R. M.A. E AB 83/89: 12. (deg) (deg) (inc ) (inch) (inch) (kv/cm) (kg) (kg/inch) (inch) . E1 -24 32 26.66 27.31 0.4 94 E2 94 136 27.56 27.99 0.4 94 R 49 61 4 M1 140 153 28.01 28.07 0.5 1.2 8.8 M2 200 206 28.55 28.65 0.5 1.2 8.8 E M3 226 230 29.11 29.22 0.5 1.2 8.8 M4 236 240 29.40 29.53 0.5 1.2 8.8 M5 256 262 30.20 30.54 0.5 1.2 8.8 7 M6 266 272 30.79 31.25 0.5 1.2 8.8 E M7 276 282 31.62 32.29 0.5 1.2 8.8 M8 286 292 32.84 33.89 0.5 1.2 8.8 E C1 320 334 28.1 C2 46 58 29.? are the initial and final azimuths of each element, O,- and (9,7, the central ray radiu R,- and Rf at angles O.- and Of, the mechanical aperture M.A., the electric field E the magnetic field bias AB, the focusing gradient (93/82, and the compensating ba radius R, defined in Figure 6.3. Two electrostatic deflectors, the first one 56° long and the second 42° long, ar positioned in correspondence with two successive hills. An electrostatic deflector i chosen as the first exit channel element both because of the thin septum and becaus of the ease with which an electric field can be confined to a desired region. In contras‘ magnetic fields have a much higher bending power, but spread over a broad regior influence previous orbits and require laborious detailed trimming to restore thes orbits to an acceptable state. Moreover, a 2-3 mm turn separation is sufficient t achieve high channel entry efficiency with the very thin septum of the electrostati deflector. The electrostatic deflector is therefore a great help in meeting the extractio requirements. 92 The regenerator is placed in the valley between the two electrostatic deflectors details of its design are discussed below. Following the electrostatic deflectors and regenerator, there are eight magnetin channels, the first of which is 13° long and is adjacent to the second deflector. T111 other seven channels are 6° each, except M3 and M4 which are shortened to 4° t1 avoid radial overfocusing. Figure 6.2 shows the cross section geometry of a typica focusing bar, and the median plane field and the corresponding field gradient as 1 function of :1: (the radial displacement from the center of the aperture). The strong gradient of the focusing bars opposes the radial defocusing gradient associated witl the edge region of the machine field through which the extracted beam must pass As shown in this graph, the focusing gradient is strong and nearly constant over 81 percent of the aperture, with an average 8.8 kg/ inch for the aperture width of 0.5” In addition, the focusing bar produces a negative (deflecting) field near the cente. of the aperture, which provides valuable assistance to the electrostatic deflectors i1 breaking the beam free of the strong magnetic field. Relevant dimensions of the compensating bar C1 and Cg are shown in Figure 6.3 C1 is designed to compensate the field perturbation on the inner orbit due to M1, am is therefore at 180° from the latter. C2 compensates the overall effect of the sever remaining channels. A section view of the regenerator is shown schematically in Figure 6.4. The threi shims in front of the regenerator are used to reduce the undesired fringe field. Tl11 value of the combined field defect f ABCZG of the regenerator and its shims, and thc corresponding field radial derivative as a function of radius are shown in Figure 6.5 The result using LeCouteur’s formula is also shown in this graph. The operation of the system is illustrated in Figure 6.6. Here the peeler (fall-oi 93 s . 8 77/2 2(m) ©9999 4>NONJ> _N ' | [\D _. O l l 1 V i T B (kHogouss) l (N l 1 2 ~05 —03 —04 X on) 04 03 05 Figure 6.2: Top: schematic cross section of the magnetic focusing channel (Note x and z are not plotted in the same scale). The middle and lower panels represent the field and the gradient across the channel respectively. 94 14° Comp. Bar 1 ________ Height 0.432 inch 10 To machine center 1 /l """ \] ----- c1 l Comp. Bar 2 Height 1.0 inch _ IIIII To machine center . <1I 6.2 In c2 Figure 6.3: Median plane section of the compensating bars. 95 1 J l l l 1 l 1 m l l l l l l 4; 1 1 l l l l 1 m E) O 5 ‘ m1 /'\ " .C_: Med. plane v O o — -------------------------------------- N _ _O 5 _ m m2: 3 m 1 l 1 I l 1 i l l I fl j I I 1 T T l I i 1 I 24 5 25 O 25.5 26 O 26 5 Figure 6.4: Regenerator and shim geometry. 96 ~ —— RECENERATOR WITH SHIMS e -- - -- -- - LECOUTEUR RECIPE Field deri. (kg/in) .0 o . 3’1. 24.5' 25.0 ' 2515 ' ' 26.0' ' ' 2615' R (in) Figure 6.5: Regenerator field defect and its radial derivative as a function of radius. 97 edge field) introduces an outward impulse causing a phase delay, and the regener— ator introduces inward impulse, giving a phase advance. After one revolution, the amplitude increases and the phase returns to the initial phase. A bump coil (the outermost K500 trim coil with the coil on each hill powered separately to provide azimuthal adjustment) is employed to compensate for residual field imperfections produced by the focusing channels and compensating bars in the proximity of the 11,. 2 1 resonance in order to facilitate the resonance transverse. The correcting coil technique has been adopted by a number of cyclotron facilities, like that of Tokyo cyclotron (denoted electromagnetic shims) [61], Uppsala cyclotron (denoted harmonic coils) [74], etc. The first harmonic radial profile of this bump coil is shown in Figure 6.7. The field contribution from the bump coil is thus to good approximation simply Bbump('r,6) Z B1(7‘) X C08(6 ‘- ¢1) (69) and this is the form assumed by the orbit tracking programs. The amplitude and phase of the first harmonic bump are denoted as Bl defined in Figure 6.7 and 491, respectively. 6.3 Beam Dynamics 6.3.1 Pre-extraction Orbit Computations The radial and vertical focusing frequencies as a function of energy in the regenerative extraction field are given in Figure 6.8. As in previous chapters the focusing frequen- cies 12,. and 1/2 are plotted as ratios of the actual focusing frequencies to the orbital frequency and are hence dimensionless. The fringe fields of the extraction elements cause the central stable region to shrink to zero area at 243.3 MeV, the V7. 2 1 region, 98 Regenerative extraction orbit L l l l l l l l l l l l‘ l l l l l l l 27.0 2 Septum _ Regene rotor R finch) _ FGCV (peeler) _ 50 100 degree) CD AO— Figure 6.6: Plot of r vs. 0 for the last five turns of an extracted orbit showing the action of regenerator-peeler system. The radial motion is characterized by exponential increasing oscillation amplitude; the resultant turn separation at the entrance of the channel is about 0.1”. Two nodes of oscillation are clearly seen at —80° and 110°. 99 B1(r) (gauss) l l l l l l l l l I l l l l l Bbump(r,6) = B1(r) X cos(6—¢1) l l I I l i I 1 21 23 25 27 29 Radius (inch) 1 l l l I l I Figure 6.7: First harmonic bump used in extraction calculations as function of radius. 100 Table 6.3: The resonances in the extraction region. I Resonance ] Driven by [ Occurs at (MeV) [ u. = 4 %1 240.70 — 241.84 V. 2 2V2 field derivatives & amplitude 241.19 11,. 2 1 Bl 243.30 - 243.52 11,. 2 g regenerator 246.8 —> and then reappear at 243.52 MeV. In addition, the regenerator, due to its strong field gradient, shifts 1/,. into the 1/.,. 2 2/2 stop band at an energy of 246.8 MeV. The important features of the major resonances encountered near the edge of the machine are listed in Table 6.3. The 11,. 2 1 resonance, in particular, occurs when the average field ceases to increase with radius and turns over into the usual magnetic edge fall off. In the absence of the field imperfections the central stable region vanishes instantaneously at the resonance energy and then reappears. On the other hand, if field imperfections are included, a forbidden band (or stop band) where no central stable region exists occurs due to the influence of the field imperfections. Figure 6.9 shows these features in a radial phase plot sequence where the energy is increased from below to above the VT 2 l resonance. It shows the behavior of the separatrices in the phase space as the resonance is traversed. The ”one-corner opening” orientation associated with the field imperfections at V. 2 1 radius is clearly indicated in the upper two panels. In the vicinity of the resonance, a substantial fraction of the beam would be outside the stability limit. The large number of turns the particles spent in the unstable region in combination with the fast motion of the phase ”fluid” shown in Figure 6.9 may result in beam loss. Figure 6.10 shows the schematic radial phase space plots which depict the tran- 101 llO . . . 1 1 1 . . m1 1 . 1 1 1 . . . 1 . . . . 1 1 . . . 1 . . . m 1.051 . A“ 1.003 1 0.953 . 4 0.0 1 I I 1 I 'I'* * OO O .. .. 1— .- .— 1. p 1— i—. 1- 1— 1— .— .— i— _ .— 1— .- .— ,_ .— .— 1— 1— .— .— — 1.. ki 01 03 \l 01 O llllldmLLllllllmlllllll N 1/ '01 '01 on O on O '11 an [liilllli[lilil—Tili'llllllllIIIIIIIIIII Lilll‘ldlllllllllll '1 O l 1 l l I l I 1 1 1 1 240 241 242 243 244 245 E (MeV) l I I I I l I i T I 1 i l l I l l 246 247 Figure 6.8: Plots of focusing frequencies as function of energy in fields with (solid curves) and without (broken curves) fringe field, produced by the extraction devices (a B1 2 2.7 gauss bump used). 102 6 : 336° 22 IlllllllLlllJIJLLLLIILLILJJILLIIIll lllllllllllllllllllllllllllllllll; ' 1 242.5 wv L . 243.0 uev - _ v, = 1.01433 . 1 u, .1 1.00592 - 2.0 - * w — L8‘- “" - -1 1- -t p 1 6 ‘ one corner opening ‘ j " ' - I‘I . 1 L A -1 I- J : 13 L4-— -4 — U -1 r- -l .- 5 - - 1 . v .4 p d 1- CL ~ . - L -1 b 4 :- 1.0M —- — - y- -1 v- 0.8 — —- .. 0.6 d -- _- J I- c- b J * 1. .. -1 I- -1 1. ‘. 0.4 I I I I I I I I I I l I I I I l I I I I I I I I I I I I I I I I I I I I I I I I I I I I l l I I II I I I I I I I I I I I I I r’ I I I I I llllllillLlllLiJJJLlllIllllllllllll LJJiJJllllilllllllllllLllllllllllll 243.5 MeV 244.0 MeV u, - 1 ate hand no control stub area v, - 0.98868 renewe st- ~ III1IITIIIIIIIIIIIIIIIIIIIIIIII I 111I111I111l111I1111111I111l111l1|1 11111[LJllllllllllLLLllllllllllllLL III'IIITITTII—IleII'IIT'IIIIIII'IrI I l I IIIIIII'IIIIIIII—I—ITIIrIIjII'III'III III'III'IIIIIII'IIIIIII'IIIIIII'III 25.2 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.6 27.0 r (Inch) Figure 6.9: Schematic radial phase space plots show the behavior of the phase space regions as the V7- : 1 resonance is traversed. The stars denote the fixed points and the lines connect positions on successive revolutions of coasting orbits evolving from given starting conditions. 103 6 : 336° 20 IIIIIIIIIIIIIIJIIII JLLIIIIIIIIIIIJIIII 246.0 MeV A: v, - 0.95325 246.5 MeV d: V, - 0.95458 p — .— — *— -1 .— /'\ - - P .c: /7 - U ‘ _ _ C __. _ — 1 4 - \./ " h "" r- \— C1. - _ 1 1 1'2 Inlulnte ‘ P ‘ control an“. “ —l >- I- 10 — —— ~ —1 I- -I 1— ..1 r- —1 I- —1 >- -t r- 08 1 1 1 I m r 1 I 1 I 1 I I I 1 I 1 I T I I I I r I 1 I I I I I I 1 I I 1 1 I I I I I I L g I I 4 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 247.0 MeV & u, - 2/2 stop bond 247.5 MeV & vr - 2/2 atop bond 1 I- i 1— 4 L . I .1 r- -4 F — P——'4 IL— I ] I I I I I I I I I 14L L I I I 1— I I I ' I I I I I T I I I I I I I I I I I I I I T I I r I I I I I l l fij j 25.9 26.1 26.3 26.5 26.7 26.9 r (inch) Figure 6.10: Schematic radial phase space plots show the behavior of the phase space regions under the influence of regenerator. 104 sition into the stop band at the resonance V7. 2 2/2, driven by the steep rising field portion of the regenerator. The central stable region shrinks to zero rapidly as an energy of 246.8 MeV is being approached. This is good in that the energy spread will be smaller as the vanishing rate of stable area increases. The beam will be dribbled off as it accelerates toward the regenerator. The dumped particles then move along the asymptotes which goes around the hill—type stable fixed point during the extraction process. The turn spacing as the beam enters the extraction system is about 0.1”. An example of the midplane motion of an accelerated beam is depicted in F ig- ure 6.11; 2671 orbits uniformly distributed in a 0.5 mil by 0.5 mil rectangular grid covering the interior of an initial eigenellipse at E = 240 MeV are tracked and show the effect of V7. 2 1 resonance traversal and the regenerative extraction system. The beam goes from an ellipse to a distorted pear with a long thin tail after passing through the VT 2 1 resonance corresponding to the ”one corner opening" orientation shown in Figure 6.9. It then undergoes a drastic deformation under the influence of regenerator at about 246.8 MeV; the portion which flows out of the stable region elongates emphatically due to the fast phase flow on the asymptote. Furthermore, it is clearly seen that there is no real turn separation of the whole beam, but good turn spacing is achieved for individual particles by the action of regenerator. Therefore, we inevitably lose some fraction of the beam by striking the septum head-on. The chan— nel entry rate thereby will be improved by an increase of the ratio between the turn separation of the individual particle and the thickness of the septum (Equation 6.4). The starting radial and vertical phase positions for the pre—extraction orbit runs are depicted in Figure 6.12. The dotted curves give the radial and vertical eigenellipses with initial betatron amplitude 30 mil and 20 mil respectively. Note that if the nonlinear forces are negligible the normalized phase area will be constant and coasting orbits in the phase plot will fall on an ellipse, the "invariant eigenellipse” characteristic 105 E : 10 keV/tum 62336 1 1 1 1 1 n 1 1 I 1 1 1 l 1 1 1 I 1 — Ar = 10 mil — : 240.0 MeV : 150 ___ /centrol ray ‘3". —_ ‘ ' ./"'/ ~ I l/ _ A 1.45 — ./ -— x: _ 43.5 MeV / _ o _ / _ E _ 245.5 MeV _ V Cf 1.40 4 ‘1. — i u, I I ‘ '- 246.8 MeV ‘ 1.35 —- 1 — _ 1J1 _ 1.30 -‘ — I T I l I I I I l I I I I I I I‘I 26 O 26.2 26 4 26.6 r finch) Figure 6.11: Central ray orbit and radial ellipses showing beam behavior at various energies. 106 of the linear motion. F uthermore, the rectangular grids correspond approximately to a constant current density physical beam. The starting energy is selected at 240 MeV, 70 turns before the I/z : 1/2 resonance which is the first resonance met in the extraction region. The orbits are lost radially if they hit the extraction channel elements and lost vertically if they exceed a vertical amplitude limit of 0.5”. The summary of the accelerated orbit calculations are shown in Figures 6.13, 6.14 and 6.15. The vertical beam envelope is given in Figure 6.13. It shows both the effects of the resonances and the axial instability induced by the ” beam off—center”. As orbits are pushed off centre rapidly due to either the VT = 1 resonance traversal or the regenerative action, the orbits move further and further into the nearby coupling resonances, so that the axial instability may be encountered (beam off-center effect). The energy spectrum for the particles which enter the aperture of the first elec- trostatic deflector is given in Figure 6.14. Two groups of particles are identified. The V7. 2 1 stop band traversal accounts for the low energy group, while the high energy group is associated with the action of the regenerator. We see most of the particles pass successfully through the V, = 1 resonance, thus we might expect a good extraction efficiency. In addition, the multi-peaks in energy spectrum, though quite surprising at first sight, are principally an effect of the phase space distortion associated with the V,,. : 1 resonance. The radial and axial phase distributions of the beam at the entrance of the elec- trostatic deflector are shown in Figure 6.15. It seems that the beam spot is quite confined in both motions. A 10 mil electrostatic septum with its entrance positioned at the radius of 26.605" gives optimal extraction efficiency. 107 I I I I I I I I I I I J I I I I I I I I 4 4 I [,0 _ E=240.0 MeV & a=0° _ 10 x 10 mil2 grids D MD .. C] N = 76 X 23 = 1748 ' __.-“'1:1 1:1 1:1 "1:1 . .. ” 4 D 13 E] 1:1 13 "(:1 f ‘ 1:1 1:1 C] 1:1 1:1 D E] ’ .65 J 1:1 D C] D 13 D ' f, ‘ D D 1:1 13 1:1 _ 1:1 , :1 ' C Ar=30m1| -. _ 0 1:1, D C] K a 8 >21. of ‘ 1:1" 1:1 C] D 1:1 13 1:1 up a 1:1 1:1 1:1 1:1 1:1 ” .60— 1:1- 1:1 1:1 1:1 1:1 1:1 3 E] ' ' "1:1,. C] 1:1 1:1 13 ' 13 ‘ 1:1 “E1 . 1:1 13 C1...- 1:] ' ”13'" “13‘ "t1 ‘ .55 I I I I I I I l’ I I I I I I r I I I I I I I I 26.34 26.36 26.38 26.40 26.42 26.44 r (men) I; I I I I I I I I I I I I I I I I I I I I I I 0.05 -‘ .. . 1: 1:1 . 1:1, _ 2‘ . :1. t1 1:1 Cl £1" — 0 . . ' ‘ A =20mi| .E 0.00 — 1:1 ‘ .El [:1 \ ' [:1 ~— ~ . 31:1" D D 1:1 1:1 . « '1:1--- 1:1 ‘ 1:1 - ‘ f —0.05 ‘0 - fi .- I I fl I I I I I I I I I I I I j I I I I I I -0.04 —0.02 0.00 O 02 0 O4 2 (Inch) Figure 6.12: Initial (r, p.) and (2, p2), indicated by the squares, at E = 240 MeV and 0 = 0" used for accelerated orbits runs. Note A. and A2 used hereafter are defined above. 108 E1 = 10 keV/turn 50 llllllllllllllllllllLlLllllll lllllllldllllllllllllllllllllllllllllllllllllllll .45 .40 .35 .30 .25 2O zenvekn) On) 45 llllllllllllllnglllllllllllllllllllllll 40 .05 IIll]lllllIlllIllllIlllllllllIIllllllllllllllllll lllllél b o lllWl HIT lllllllIlllllHIIIITl llllIllIl[HITIWTTITTWIIHIHTIHIlllllllll 240 241 242 243 244 245 246 247 248 Energy (MeV) Figure 6.13: Vertical beam envelope as function of energy. 109 100 llllllllllllllllllllllllll||(lllllllll Chan. entry eff. = 92.3% 90 8O 70 Reg. Ex. 6 60 50 40 No. of portic|es 3O 20 llllllllllllllllllllllIllllllllllllllllllllllllllllllllllllllllllllllllllllllll|llllllllllllllllllll llllllllIlllllllllllllllllllllllII||||I|llllllllllllllllllllIlIlIlllH[Illllllllllllllllllllllllllll IllllllllllllllITr 245 246 Energy (MeV) Figure 6.14: Distribution of energies for the particles entering successfully the first extraction channel at 0 2 336°. 110 1.65 I \ I I I I l I I I I I I l I I I I I I L I ;L LI I I I I j g 9=355° I - . a.“ _ 145— § :— 2 \ C s :_ j §~SEPTUM ; A 1.25% § ~ 6 2 k E .E 1.15~ k ~ \J " y— . 3 § E Q 1.05—_ g :— 0.95—: § L 3 § 5 0.85: § _— 4 , I. : x . 0.75:1 § T 3 x 5 0.65 I \ I I I I I I I I I I I I I I I I I I I I I I f I l I r I 26.60 26.65 26.70 26.75 26.80 26.85 r Unch) 0.15 I I L A I I I I I 0.10- _ 40.3 . ' '—0.2r ' ' '—0.1' '01 ' ' '02' ' 0.3 Figure 6.15: Radial and axial phase plots at the entrance of the first extraction channel (0 2 336°) showing the resultant distribution for those orbits that successfully enter the channel. 111 6.3.2 Extracted Beam Optics The particles which successfully enter the aperture of the first electrostatic deflector provide the input data for computing the optical characteristics of the external beam. These calculations are carried out by utilizing the DEFLZ800 code. This program treats only linear vertical motion and also calculates only the coupling of the radial into the vertical motion, but omits the reverse coupling. These are, however, appro- priate simplifications due to the fact that the extracted path is shorter than one turn and the axial displacement is much smaller relative to the radial displacement. The acceptance of the extraction system is found by tracking displaced rays as shown in Figure 6.16, then using these results to determine the maximum tolerance in phase space and energy at the channel entrance. From these results the energy spread in the external beam is expected to be 0.8 MeV or 0.32% of the extraction energy. Plots of the radial and vertical envelopes for the extracted beam are given in Figure 6.17. The loss histogram, given in 2° steps at the bottom of the figure, depicts the loss of the extracted orbits due to hitting the walls of channels or failing to enter channels. We see that there is a significant beam loss at the entrance of the second electrostatic deflector. The main reason is that the particles in the low energy group will not enter the second electrostatic deflectors after they exit from the first electrostatic deflector; they just go radially inward. The final energies, radial and vertical phase distributions of the extracted beam at 0 = 3080 where particles go off the field, are given in Figure 6.18. This figure indicates an energy spread of 800 KeV (upper panel) and a quite limited beam size (middle and bottom). Figure 6.16: The radial phase and energy acceptance of the extraction 112 l A I I I I 1 I I 1 I . I l . . l A 1 1 ‘ A ‘ I l A ‘ . I 0.4“; .: ”I 9 5 5 9 9 M E” . 51 E2 ”111—6 H HH: 0.31 1:: A A I I A A A A __ l . 02‘: AXi = :tZOmil, 3:40mil _— O.15 L —0.0- E —O.‘|- L -O.2— L w I I u I r 1 I y f I I T j I I '_I .. 0.25 APX, = :tZOmil. i40mil 0.11 r 0.0{ E —O.1- —O.2* 02.: AE = :tO.2MeV. :tO.4MeV ;_ 0.1% 9 0.01} E —O.1- —O.2- 0. (NJ system. 113 Centrol roy: ri=26.66", prizl.5l", E2246.9MeV 0-5“‘**] L*“‘l“‘‘]I1111m‘111‘‘fitlum‘‘1111—11II—51111"‘111n1’1IH—1111‘11—;' ‘ E1 94 kV/cm £2 W‘s ‘ 0.3— — /'\ _. ” E 01 V i x —O.I‘ '— —0.3— _ 4 r _o_5 . . , . - . . 1 . . . . , . . . . , . . . . r . . . . , . - f . , 0.5 L4; I l l I A I I . I A I I I L A I . A I I I l I I I I A . m L I 0.3— _ A _ P N _O.IF.’—/\//~\h —0.3~ ~ -05 1 , f- a fig . . . . , . . . . , . . . . , - . . .fi 8400: I I I I I I I L . I I I A 1 A L I L l I i l I . . . I I I . . g Chon. trons. eff. 45.2% 'N300— .— 2 g I; I O ‘200— — (D (I) Z) 8100— ~ “5 : 6 3 - z 0,565.-....2..IMWII..2.1........2.- O 50 100 150 200 250 300 6(degree) Figure 6.17: Radial and vertical envelopes of the extracted beam. Plots show (1" — To) and 2 vs. 0 from —24° to 308°. The corresponding loss histogram is also depicted in the bottom. 114 I 550‘ Extraction eff. 41.8% 350‘ 6:308' H n r Iii—1m ' ' 7 ,, . 246.4 246.6 E (MeV) 246.8 247.0 16.7 I A I I l A I I A l I ‘5'. 16.6~ _ “'3’." _ — ‘ - I .: '. ' -'a ‘ ' . . . . . ‘ " ' .- ' "I .:.)": if: 4“”. . - ‘. I II ‘I. ' ’3 16.5: I _ O ’ _ C \- C/ . I- Q 16.4 ‘ ‘ .— 16.2 ; , . . T , , . , , . . r , . r . 1 . , 38.6 38.7 38.8 , 8.9 39.0 39.1 r (Inch I I A I J l A I A I l l I I # 0.06 I I I I I I l I L I 004- ' ‘ ' L' -l. ,0 ' c .1 0 o. . D .V .. , -. _ ' I . ‘ . h 002‘ n ‘ I. I.. ' " ' ... . 1.0. ' I‘ .v i I ' . _ ‘ .' 'u . ‘ '0... l _ . I . . . . .|- . . ' - . . ‘ .. . '. ' . ". c I . n ' I ’ ' A ‘ I . . . ' 0 ,I u ' I .'- I ,D- '. I . ‘ .- ' ' .I ’ .C ', . . ' ' LC) . . I . , us :. :... - . . a" ‘ i .. ' ': .. . .‘ ‘ . . u I . _ __ ' I o c ' . ' ' ‘ ('- . , I. .- -- 0.00- . -..:. .- . . . . — V ‘ n ' . . I . . I '_ .. . o. L ' n ' ... . i _ 0'. I . . ...: I: a" . . . . . p N ‘- .' I“ Q. . '. l I ' ‘ ' o - I. . ' ' n . o . '0 _ I., . ._ _ . ' '- —002- - ' ' - . "- " .-"' ' ~. _ . ‘ ' . .. . . ‘. II. I- . . ‘ ,. .n . . .U . . . 0 ' 4. . I o .- n I I. - o A . I v- : l' —oo4~ ' -~. .f 1 ' _ L '1 —0.06 ' v I v I v v 1 v I ' 1 fi7 —O.10 -0.05 0.05 0.10 Figure 6.18: Distribution of final energies of the extracted beam (top). The corre- sponding radial and vertical phase space distributions are shown in the middle and bottom panels, respectively. 115 6.4 Performance Estimation and Summary The extraction efficiencies for particles associated with initial betatron amplitudes $0 and 20 have been calculated and shown in Figure 6.19. The efficiency for median plane particles lies in the range of 40% and decreases rapidly with increasing betatron amplitudes due to the limited VT 2 1 resonance acceptance. Based on these contour curves, the extraction efficiency can be calculated if both the size and current density distribution of internal beam are known. 116 100 IIIIIIIIIIIIIIIIIIIIIIIIJIIIIIII ...... E =10 keV/turn 1 90 lllll‘lllllllllTllllll ZO (mil) IIIIIIlIIlIIlIIlIlIIIIIIIllIIIlIIIl FlllllllllilellFillllllll Figure 6.19: Extraction efficiencies vs. betatron oscillation amplitudes of boundary points on eigenellipses at E = 240 MeV. The number labeling beside the solid curves is the extraction efficiency. Unlabeled dash curves give extraction efficiency contour equal spacing between two adjacent labeled solid curves. Chapter 7 W. = 1 Excitation Extraction 7. 1 Introduction The second likely extraction system we consider is the so-called VT 2 1 excitation system; we extract the beam at the resonance VT 2 1. In this procedure, the degra- dation and loss of the beam associated with the resonance V7. 2 1 can be avoided in comparison with the regenerative scheme in Chapter 6. But a major drawback of this method stems from the lower energy at which the beam is extracted which may require electric deflector field strengths which are impossible to achieve. The voltage holding capability of electrostatic deflectors has long been recognized as one of the limiting factors in extracting a high energy beam from a superconducting cyclotron. In response to the demand for higher electric bending power, certain processes have been made to improve the voltage holding capability recently in our lab; 80 kV at 6 mm gap can be achieved Without serious leakage [75]. In addition, a news letter from Chalk River declares that 95 kV is achieved in the presence of both r.f. power and a 3—Tesla magnetic field across a septum/anode gap of 7.5 mm [76]. Consequently, we may anticipate that 160 kV/cm at 5 mm aperture could be achieved routinely and reliably in the near future; the electric deflector field strengths implied by the calculations shown in this chapter thus seem clearly feasible. 117 118 The excitation of the VT 2 1 resonance is used to expand the turn spacing as the beam enters the first electrostatic deflector. The bump coil produces a c030 perturbation on the beam. As a result, it opens up a corner of the triangular stable region of the radial phase space at one of the unstable fixed points; the stability limits shrinks as the energy approaches the resonance and the beam spills out slowly along a asymptote (as exemplified further on in Figure 7.9 [42]). 7 .2 Analytic Solution Concerning the 14. = 3/3 Resonance A simplified analytical method regarding the 1/7. = 3 / 3 resonance is presented in this section. Also semi—quantitative results are derived to shed insight on beam behavior in a three—sector cyclotron at the V, = 3/ 3 resonance. For a three-sector cyclotron, the betatron motion about the stable equilibrium orbit in a perfect three-fold symmetric magnetic field can be expressed approximately as [45] 2 (Cg—0::- + V31: 2 032260330 (7.1) where :1: is the displacement from the equilibrium orbit, c is a constant characterizing the strength of the non-linear force, and the 00330 factor arises from the three-sector periodicity of the magnetic field. The solution to the linear Kerst-Serber equation 3% + V3113 = 0 is :c = Asinz/Tfi. Considering c as a perturbation on the linear motions, the solution to the Equation 7.1 may therefore be written as 123(0) 2 a(0) - sin(1/,.0 + (15(0)) (7.2) 119 Assuming slowly varying (2(0) and (15(9) relative to 0, Conte and Mackay [77] show that da ca2 dd) ca , d6 — V1. — 1+ 8 sm(3¢) (7.4) One trivial solution to Equations 7.3, 7.4 is a = aeo = 0, which characterizes an ”elliptical” fixed point (13.0). For small amplitude a, the nonlinear force (c332c0336) is negligible, so that a is a constant and successive phase space orbits at one fixed azimuth will fall on an ellipse with precession frequency | VT — 1 I. The ”unstable" fixed points occur when both (1(6) and q5(0) are constant. Hence, we have cos(3¢) = O, (7.5) giving rise to three fixed points 120° apart (gbus = 900,2100, and 330°). In addition, the resulting radial position, aus for unstable fixed points is _8Iz/T—1I C (7.6) aus As summarized in Gordon’s paper [45], the three unstable fixed point mark a "triangle” separatrix in radial phase space between stable ”elliptical” orbits and un- stable orbits, characterized by asymptotes along which orbits move in or out. A good numerical example was given in Chapter 3 (Figure 3.9). Consider a flat bump of the form AB = Blcosfl (7.7) 120 The radial betatron equation of motion becomes cl2 2%”; + V311: = 0513260539 + 90039 (7'8) where g = (Bl/BO) - R measures the strength of the perturbation. For small 9, the solution of the above differential equation can be found in Laslett’s paper [78]. He showed that the locations of the two relevant fixed points which lie on the pa, = 0 axis for 1/7. 2 1 are an) :2 87,993 (7.9) (11.49): §'(—l‘/I'C_—1—) — 8(772‘3—17 (7.10) When these two fixed points coincide, they topologically annihilate and radial stability disappears, causing particles which had previously been in the stable region to spill out along one of the asymptotes. The stop band of the VT 2 3/ 3 resonance can be estimated as follows. Requiring aeo(g) = au3(g), yields I V: —1 I; if I U: —1 IZI ll,- —1 I, the radial motion is unstable 3 ye - * — 1 = - — 7.11 Il/T I 4 2 ( ) The V,» = 3 / 3 stop band in energy units can then be approximated as 3 6311'? (dz/T/dE) 2B0 AEsb = 2. (7.12) 7 .3 Optimization of Bump Phase As was shown by Blosser [42], quite different extraction characteristics, (i.e. ”one” or ” two corner opening” orientation, flow rate in phase space, deflecting strength required 121 to bend beam out, distortion and integrity of beam), are obtained depending on the bump angle selected. Numerical calculations have been done to explore the relation between the bump angle and various orbit behaviors. The end behaviour of initially centered phase space orbits with first harmonic bumps at various different phases are shown in Figure 7.1 (top). The electric deflector field strength required to bend beam out versus bump phase is also given in this graph (bottom). Note the E strength is approximate due to the difficulties involving the accurate determination of which point on the asymptote is selected as the reference orbit for the extraction channels. It seems that the 1600 bump gives the best overall performance in accordance with integrity of beam and electric deflector field strength requirement. 7 .4 Optimization of Bump Amplitude Near extraction special attention has to be paid to the vertical motion. Resonances which can cause growth of the vertical amplitude in this magnetic field occur at V2 =1/2, and at I/T : 214,. At a half integer resonance such as V,, = 1/2, vertical stability is lost if there exists a first radial derivative of the first harmonic of the median plane magnetic field. The amplitude gain will vary as the square of the driving term. The intuitive reason for the quadratic dependence is that both the strength of the resonance and the number of turns spent in the stop band are proportional to this driving term [57]. The vertical amplitude growth associated with 1/2 = 1/2 stop band may be esti- mated by using the EO code. If we write 2 = Zemg and V2 : 1/2 + i1/* in the stop 122 I I I I I I I I I I I I I I I I I I l I I l I I I I I I 141 I I I I I I I I I I 3 9:336' [ j E1 = 10 keV/Turn : 1 80‘ C 1.6 —_ _ 7 220° / _ 6 _ ,/ _ éi l 5 j _ O: : r- : zoo- / 7 Z 10. / : _ 140° _ 1'4 ~ um _ : mw : : 40o I 1.3 I I I I I I I l l I l I T T I I I I T T r T I T I I l I l I I I I I T f 7 l I I 26f) 262. 264- 266 r finch) 450 ‘ I I 1 I I I I 1 I I I I 1 I . . . 1 1 3 t 400% 53 F 350i 9 Esoo€ E \ < _ i 3 : v250 1 f 2001 E t‘ 3 fig : K30 i ' . r . . . . I a - - . 4’r . . . . . 50 100 150 200 phase ¢‘(deg) Figure 7.1: Top: radial phase space plot for the accelerating orbits at 0 = 3360, initially centered at E2240 MeV, with 1.2 gauss first harmonic bump at several different azimuths. The number besides the curve is the bump phase 451 in degree. Bottom: the approximated electric deflector field strength required to bend beam out vs. bump phase. 123 band, then the amplitude growth Zfinaz/Zz'm'tz'al is given by _Z_f : 8f u"-d6 eu‘(E—2AE)-27r-A—E u’(E—AE)-27r--é-§ V‘(E)-27r-A—E 2 ...... X E1Xe EIXC E1 u*(E+AE).27rA’-i u*(E+2AE).27r-§,—E Xe El Xe E H 6V*(E)'2F'E—1 (7-13) stopband where V“ is the imaginary part of the vertical focusing frequency and E1 is energy gain per turn. The axial amplitude growth Z f /Z,- as a function of energy gain per turn for cen- tered orbits with five different first harmonic bump amplitudes, computed using the E0 code data and Equation 7.13, is shown in Figure 7.2. The N SCL linear Z code can also be used to investigate the effects of the resonance 1/2 = 1 / 2 on vertical amplitude growth. Since this code only couples the radial motion into the axial motion, not vice versa, the V2 2 1 / 2 resonance effect will be isolated if we start the computations on equilibrium orbits. The evolution of vertical ellipses and corresponding eigenellipses at various energies are shown in Figure 7.3. We find that the accelerating ellipses are tilted and stretched due to the resonance crossing. The 11,. = 211,, resonance (the Walkinshaw resonance) occurs just after the V2 2 1 / 2 resonance and shortly before the V, = 1 resonance in this magnetic field. It is well known that the effect of the Walkinshaw resonance depends strongly on the beam quality and on the energy gain per turn [62]. The Z4 orbit code was used to study the evolution of both radial and axial ellipses associated with B1 = 1.2 gauss and E1 = 10 keV/turn in the vicinity of the resonances V2 2 1/2 (240.7 MeV) and 11,. = 2V2 (241.2 MeV); results are plotted in Figure 7.4. Non-linear features of the resonances (amplitude dependence) and some common characteristics (distortion, stretching and amplitude growth) are clearly indicated. 124 8 I I I L I I I I I I I I I I I I I I 1 v2 = 1/2 crossing _ 7 — 2 gouss 951 2 160° _ N" L L \6 _ _ N _ _ O _ __ :8 _I .— n 5 j _— (D - _ U _ _ :5 .4: _ _ E4 ‘ _ E .. .— O ‘ _ E _ _ x 3 — - - p 4 1» > . 4- > b I .I I I _ d— In h . 4. - 1 4 4» r- 1. 4 4y I I .. .... ..- _ AII 1 1 I I I I I 1 I 1 1 1 1 I I I . 1 I 1 1 1 1 v V I ' 1 I ' ' ' Y 1 r Y I 1 ' ' I Y ' ‘ ' l I I l f T T ' ‘ ‘1 db D . 240.6 241.0 241.4 . .1. I 1 1 4 _. . m—bloz' 10161 #6766 obi 6.62“ ' 2 (inch) Figure 7.3: Plot of evolution of z ellipses (solid curves) showing the resonance 1/2 = 1 / 2 effect by using linear Z code. The dotted curves are the eigenellipses at each energy. 126 Ar=50mil Az=40mil B1=I.Zgouss ¢1=160° E1=IOkeV/turn 6=336° 240.4 * 240.8 ‘ 241.0 “ u =1/2 ’ I o 2 « .. -. _ 4 <- A 1 J. I » .C U D .E 00.. -I I- I V ’ "‘ 3 I ‘ I. .4 ' Q . 4p V —0.2a —- __ , 4 b b p 4 4» up 1 I I I 1 I L I 1 I I I I I I I 1 I I 1 I I I 1 I I I 1 I I I 1 I I I 1 I I I l I I 1 I I I I I I I I V I l I I v I W 1 r l - l V l I v I , 241.2 " 241.4 '1 241.6 ' u — 2 u . 1 r 1 ‘l .1 _Il— —..— .- 4} I ' u o 4 < 4 . . l * ~: 1 . . I! ’1 .. ‘ " ' 2 ‘ $2». ‘ ..‘ ‘4 ‘ . ,k . .. _1- ..1— . Ibh’.’¢?o‘;". {14:}:51'5‘4 .— , '-~~.‘.;..' 1,: .13?) '. > ‘NI- - ‘ . b .LI’ V . " I. ‘H .2 4- .,I I . I .' - ,I . , .... 4 J- r - ...I. .4— - 1' ‘F b I ' I ' r l l ‘ V l I ' V T ' ' ‘ I ' l ' I ' I I ‘ _I O) I I I t II \ N I jfi I l ’E 1.6 “ “‘ " F 0 <1 C . .I v . 1. of 1.4 r ‘1‘" “ ‘ 4 I- 4 4» b | I . I I A I I l A l I I l I #1 I A L I ;4 l I I A I I A 4_,_. . , I , I I , I , I , I , 4 . . . . , . . , I I r . l 2412 r 241.4 » 241.5 —l V _ 2 V -1- __0- - + f I .. .- I l 4 < ~ L .. .4 -4— ' .... 'x-_’__ L. 4 0 r .1 . y L .4. '."¢' I...” .4 —>— —1-— 1. . 1— 1" 4 <1 - _. _- _I I 4 4. L . 4 f4 , I I , , T . . , I . I , 44 , , 25.6' ' 258' ' 2610' ' 26'2 ' 26.4 1' (inch) Figure 7.4: Evolution of accelerating ellipses in the vicinity of resonances V2 2 1/2 and VT 2 2V2. The numerical labeling is the energy in MeV. 127 A stronger (1.7 gauss) bump was also studied and Figure 7.5 compares these results with the 1.2 gauss case by showing the final outward flowing asymptote for each bump for two different E1 values. The turn spacing at the head of the septum and required electric deflector field strength as function of septum position and bump amplitude are shown in Figure 7.6. The stronger bump gives lower extraction energy, increases phase space flow rate, decreases the electric deflector field strength and moves the asymptote further out in radius. The weaker bump does the opposite. But the stronger bump is not un- reservedly a desirable feature in that vertical instability is inevitably induced by a stronger bump. The beam axial envelope vs. different bump amplitudes is shown in Figure 7.7. It shows that for 31 > 1.2 gauss, vertical loss will occur caused by the resonances V2 2 1/2 and 11,. : 2V2. The results in Figures 7.6 and 7.7 Show clearly that the turn separation, vertical growth, needed electric deflector field strength, and distortions are strongly correlated in the VI. = 1 extraction process. To quantitatively evaluate the tradeoffs implied by these several considerations, additional acceleration studies have been carried out which are not presented in details here. These more extensive orbit calculations show that the optimal bump corresponding to the conditions of E1 = 10 keV/turn, AI = 30 mil, A2 = 20 mil, is 81 = 1.2 gauss, and qfil = 1600. 7 .5 Summary of Final Calculations of 14. = 1 Sys- tern Parameters of extraction elements are shown in Table 7.1. The radial and axial focusing frequencies as a function of energy in the extraction region are given in Figure 7.8. At the VI = 1 resonance the 1.2 gauss bump opens a 128 955160" & 62336° 1 1 l l l l l l l l l l l l l J l l l l — B1212 gouss — _ E1210 keV/turn _ ‘ Ef=243.06 MeV 1.80 — _ ‘ 8‘217 gouss i _ _ E1=20 keV/turn _ _ 55243.16 MeV _ 1.75 — — A : i _C Q 1.70 — I. C _ _ V _. 81:1.2 gauss . _ L. ‘ E1=20 keV/tum —-9 i - Q_ . 1 [35243.38 MeV ” 1.65 4 __ 1.60 _— \ 8121.7 gouss _ E1=1O keV/turn _ Ef=242.88 MeV _ 1.55 - — - l l 1 I l 1 l 1 I 1 l l 7 I r l l l I 1 1 l l— 26.50 26.55 26.60 26.65 r finch) Figure 7.5: Asymptotes in the VI. = 1 excitation extraction for various conditions. 129 1 E = 10 keV/tulrn with initially centered rays & 9b = 160' ‘ -B—B—B1= 1.09 9 1 411—111—131 = 1.2 g— \E/BOn -e—e—B,= 1.4 g ,_ ‘ —-£1-—-191-—B1 = 1.6 g ” <1 . I E ‘ I 360- c Q. Q) I 1. (D 46 d I- U’ ' . C '6404 _ C Q. - . U) C . I S . I ..— 20- - I ' ' ' n ' ' ' 1 ' ' ' I 26.58 26.60 26.62 26.64 Position of septum head rm (inch) g 160 - :- f. 3. f. 26. l58 26 T60 26. I62 26.64 Position of septum heod rm (inch) Electrostatic deflector strength E (kV/cm) Figure 7.6: Plots showing the turn spacing at the head of the septum (top) and the electric deflector field strength required to extract the beam (bottom) as function of septum position and bump amplitude. 130 Ar = 30 mil AZ 2 20 mil E1: 10 keV/turn gb1: 160° .5 l l l l l l l l l l l l l l l l l l I .4— /_\ _ L ... U C .3 - Q _ Q'— _ E _ Q a :3 .2 m X a <1: _ 1.2 .1 m 1.0 — uZ—j/Z l/riZZl/z .O l l l l I T l l l I l l l f I I l l l 240.0 240.5 241.0 241.5 242.0 Figure 7.7: Axial envelop vs. energy showing the effect of bump amplitude on beam Energy (MeV) behavior for the V2 2 1/2, VT 2 2V2 resonance crossing. 131 Table 7.1: Extraction element parameters (Bl = 1.2 gauss) 6i @f Rf Rf MHA E AB 83/823 RC (deg) (deg) (inch) (inch) (inch) (kv/cm) (kg) (kg/inch) (inch) E1 -24 32 26.64 27.44 0.2 161 E2 94 136 27.75 28.14 0.2 161 R 49 61 M1 140 153 28.16 28.23 0.5 1.2 8.8 M2 200 206 28.70 28.80 0.5 1.2 8.8 M3 226 232 29.22 29.37 0.5 1.2 8.8 M4 236 242 29.49 29.69 0.5 1.2 8.8 M5 256 262 30.27 30.59 0.5 1.2 8.8 M6 266 272 30.85 31.30 0.5 1.2 8.8 M7 276 282 31.67 32.33 0.5 1.2 8.8 M8 286 292 32.88 33.92 0.5 1.2 8.8 C1 320 334 28.2 C2 46 58 29.6 stop band of 1.14 MeV width Where no equilibrium orbit exists. Figure 7.9 shows an initial 50 mil betatron eigenellipse (the cross hatched area) accelerated into the resonance, with non—accelerated orbits superimposed at each energy to show the characteristics of the static phase space. The mid—plane motion is seen to be excellent - clean turn separation and 100% beam capture into the desired outward flowing asymptote. The beam axial envelope vs. energy, the energy density distribution for those which successfully enter the channel, and (7‘, pr), (2, pz) spectrums at the entrance of the deflector are given in Figure 7.10 for the conditions A. = 30 mil, A2 = 20 mil, 31 : 1.2 gauss, qbl 2 160°, E1 = 10 keV/turn and rm, 2 26.050”. The large axial growth at E 2 242.4 MeV is due to the beam being off center, but a channel entry efficiency of 83.3% is never-the-less achieved. Transmission loss thru the the extraction system is primarily due to the limited 132 gouss 961 : 160° I l l l l B =12 110 11111L14;111_11111111111111 L ~ 1.14 MeV O 95 - , 0.90 I I I I I I I I I I 7 I I I I I I r I I I I I I I I I I I I I I I I I I I I I fl I I I 65 l I J l l l I l l l l l l l l l l l 1 I l l l l l l l l l I 1 l J 1 L 1 l l l l I L 141 . . d l f- 60 ——4 ‘ 1— -1 7 y- i -4 L N A 552 __ b d h 50.. .... ' fl 4 r >— - u =T/2 v =22/ v =1 _ .2 r 2 r .45 I I I I I T I I I I I I I I I I I I I I I I I I I I I l I I I I I I I I I I I 240.0 240.5 241.0 241.5 242.0( 232.5 243.0 243.5 244.0 244.5 E MeV Figure 7.8: The focusing frequencies in the VT 2 1 excitation extraction field are plotted vs. energy (heavy solid curves). As a reference, the focusing frequencies associated with the perfect 1200 field map are also depicted (dotted curves). 133 Ar=50mil 81:1.Zgouss ¢1=160° E1=1OkeV/turn 9=336° l l l l l I L J l l 1 l l l 1 L 1 1 L l l l l l l 1 l l l 1 l 1 1 l I. 1 1 l l l '1 ‘ ' l' . 240 MeV _ - 242.55 MeV * _1.8_i l— at ”E O 5 -1.6— ~ g. —1.4— — - - * "l"‘l"'l"‘T“‘l" "irfifii"*Tm"i'r'i" 25.8 26.0 26.2 26.4 26.6 25.8 26.0 26.2 26.4 26.6 1 1 1 1 1 1 1 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1 1 l4 1 1 l 1 1 1 l 1 1 1 l 1 1 1 l 14 . 242.4 MeV * _ . 242.6 MeV * _ 1.8 d F 1.8 ‘ 10 mil septum — - a: - * . 1.6n ~1.6- - . \ 1 14— 4.44 — ~ * ‘ 1 ' ' l ' 7 rl ' ‘ ' l ' 7 l l ' ' ' |jj ' I l r T l l ' ' ‘ l ' ' l l ' l r l ' ' 25.8 26.0 26.2 26.4 26.6 25.8 26.0 26.2 26.4 26.6 r (inch) Figure 7.9: Radial phase plots showing the behavior of acceleration runs with a bump 31 = 1.2 gauss, 9121 = 160°. The fixed points are represented by stars, the static flow direction by arrows, and the beam by hatched areas. 134 aperture (5mm) of electrostatic deflector and the radially defocusing force associated with rapid edge field fall-off. Figure 7.11 gives the half beam width measured from the center of the beam, the beam loss histogram in 2" steps along the extraction path, the distribution of final energies and (r, pr), (2, p2) spectrums at 9 = 308° where particles go off the magnetic field. The resulting channel transmission efficiency and total extraction efficiency are 21.4% and 17.8%, respectively. 135 Ar=30 mil A1220 mil 81:12 gauss ¢1=16O' E1=1O keV/turn 9:336. l 1 1 1 1 1 5 1 1 1 1 I 1 1 1 1 l 1 1 I l 1 1 l 1 l 1 1 1 1 l 1 1 1 1 l : 0.10.. . .1 ., _— .4 — — Z "' I h g 1 - 0.055 1 E 7 - . o. I '5 j g :5 0.00— — > - . C ’ N . . a) _ o. _ - ‘ r o r - - 5? r —0.05— — <1: _ . . V' , j —o.1o§ — ......, I....|....|.......... 242 243 —0.4 —0.2 —0.0 0.2 0.4 nergy (MeV) 2 (inch) 100 1 1 1 1 | 1 1 1 1 l 1 1 1 1 l 1 1 1 1 1 I 1 1 1 1 l 1 1 1 1 I 1 1 1 Gantry : 853% : I 10 mil septum . _ 1.80“ - 80- AE ~ 0.7 MeV _ I 5 mm aperture w _ - 9 . .9 60‘ A t - 1: ' 8 ‘ .‘g’ _ °_ 40- ‘1‘ 7 o . z 20- f 0— 1,50'.............,.... 240 241 242 243 26.60 26.65 26.70 26.75 Energy (MeV) r (inch) Figure 7.10: Features of the u, = 1 extraction system computed with the Z4 code. Left top: beam axial envelop vs. energy showing a acceptably vertical confinement through the whole extraction process. Left bottom, right top, right bottom: distribution of energies, radial and vertical phase spectrums for those which enter the deflector at the entrance of the extraction channel. 136 Central ray: ri=26.64", pri=1.62", E=242.97 MeV '5 l l I I rm_ .10 1 | 1 1 L 1 I 1 1 1 1 I 1 1 1 1 l 1 I 1 1 I A 1 A E = 151 kV/cm AE _ 60 4213151): I 4_ 25_ — . e -_ 1m ED. eex _ 178% _ M BAR ...20— ZIOSS ' 60% A .3— - fi c t .E a .5- _ " .2— - B o' z 10— — .1- _ 5~ _ J -G I I I I I l I o_ .- 0 5° ‘00 ‘50 20° 25° 300 2427 242.8 242. 243.0 2431 243.2 243.3 5 . . . . . . o... . . . “9.940121. . . . . . 0.15- 3 .4 — - . I 0.10— : :_ A .3- _ A 0.05- 3 .C .C E .E 0.00— - N ‘2 _ 5 Q." —o.059 — . .1. c ‘_ _ —o.104_ L. E —o.15— ;— .0 1 l 1 1 1 I 1 _ an 0 50 100 150 200 250 300 “_C 8 ' L04 ;5 2 —0 0 o 2 0'4 0 s 2 (inch) ' I 500 I I l l I l I 1650 I 1 l l E = 21.4% trans 400‘ - 16.45— ;. : L A. ' I'. b 5 1:" “30°" ' A 16.40— ’ ~._ _ 6 fi _1. - .s O V E . 3200' ' ‘1 16.35- — S 100' " 16.30— _— 1 L mag-Willi “14111—41411 ._. I 0 5'0 100 180 200 2.310 3130 16.25.11.111. """"‘v1*"' 0 d 38.90 38.95 39.00 39.05 39.10 39.15 39.20 ( agree) r (inch) Figure 7.11: Left: the beam half width measured from the beam center and loss histogram in 20 steps vs. 9 along the extraction path. Right top: distribution of final energies. Right middle, bottom : radial and vertical phase spectrums at 0 = 3080. Chapter 8 Precessional Extraction Compared to l/T : 1 excitation extraction considered in Chapter 7, precessional ex- traction, denoted PE, is the standard method used in most isochronous cyclotrons, and provides larger extraction radii, thus higher energies of the extracted beam, and the deflecting power of extraction system required is probably lower (pr dependence). However, one difficulty of the PE is the beam loss and degradation, when passing the resonances. In addition, the turn separation from PE may not be sufficient to effectively clear the septum in such a high magnetic field synchrocyclotron. 8.1 Turn Spacing from Acceleration The relativistic relation gives the total energy E; in terms of rest mass m0 and particle momentum p as follows E? 2 mic4 + p2C2 (8.1) Totally differentiating the both side of Equation 8.1, after some steps, we get dp _ EtZ (IE; ’72 dEt _ 72 dE __ — —: _ 8.2 p pzc2 E1 7+1Ek 7+1E ( ) Where Ek, denoted as E for simplicity, is the kinetic energy of particle. 137 138 Using 12 = qu and assuming an axially symmetric magnetic field, dB 2 gkdr, and V3 2 1 + k, which after some simple calculation, yields — 2 V7. 7 (83) Substituting the above expression for if and % : igibi—L 2 E101?” into Equation 8.2, we obtain the formula for radius increase per turn due to acceleration, namely, x — (8.4) Equation 8.4 immediately gives a simple idea about how to increase the radial gain per turn by ” brute force” acceleration in order to get a high extraction efliciency: 0 Build machines with a large average radius, i.e., a small magnetic field. 0 Make the energy gain per turn, E1, as high as possible. 0 Accelerate the beam into the fringe field, where I/T drops, i.e. the field fall-off region. Unfortunately, brute force acceleration is generally very expensive and in addition can not alone guarantee high extraction efficiency. Several other methods can be used to enhance the turn spacing at the extraction radii, one of which is the precessional extraction mechanism, pioneered in the MSU K50 cyclotron [43]. A small field per- turbation having an azimuthal first harmonic component is used in this method to induce a coherent precession of the desired amplitude as the beam accelerates through this resonance, and this radius gain periodically, when the precession is at the right phase, adds to the acceleration radius gain and increases extraction efficiency. 139 8.2 Turn Spacing from Orbit Precession Recall Equation 4.11, the v. : 1 resonance crossing produces a coherent amplitude AR B 11y. ‘71 AszxRXéx (8.5) After the resonance, the beam precesses with the frequency 1V7. — 11 around its equi- librium position. This ”coherent amplitude” is used to enhance the radius gain per turn up to a limit set by the radial difference between two successive precession cy- cles. (It is sometimes useful to use less precession than the limiting value in order to reduce the beam axial loss associated with the beam off-center.) As a result of this precession, post—resonance acceleration gives the radial separation needed to clear the septum. Considering this more quantitatively, the maximum turn separation between two successive precession cycles, i.e., the maximum beam to beam separation at the head of septum, is approximated by (ZR . dB. 1 E2—(precesszo71) — $010061.) X m (8.6) The top and middle panels, respectively, of Figure 8.1 show dR/dn (accel.) and dR/dn (precession) calculated by EO code vs. energy. The electric deflector field strength required to extract the beam vs. final energy is also shown in this graph (bottom). The value dR/dn (precession) is too small to guarantee a good extraction for syn— chrocyclotrons due to the negligible acceleration effect, as can be seen from Figure 8.1. In addition, we have to extract the beam at energies higher than 249 MeV for the reason of reasonable electric deflector field strength (E S 160 kV/cm) if the PE is used. 140 Precession extraction (1/r < 1) 1 4.5 I I I I I I E1=10 Kev/turn 7? 3.5— »~ ~ -» -- -- -- -- -- E1=20 Kev/turn — 7S 0 U 8 C _U \ a: '0 20 I I l I dR/dn (prec) = 1/1V,—11 x dR/dn(acce|) _1 UT 1 dR/dn (precession) (mil) 01 5 175 - 150 - E (kV/Cm) 125 ‘ 100 I I I I 247.0 2477.5 248.0 248.5 249.0 249.5 250.0 250.5 251.0 E (MeV) ex Figure 8.1: Top: dR/dn (accel.) vs. energy. Middle: dR/dn (precession) vs. energy. Bottom: required electric deflector field strength vs. final energy. 141 Table 8.1: The resonances in the precession extraction field. I Resonance ] Driven by I Occurs at (MeV) I u. =4,1 %3- 240.45 — 24.1.71 V,- = 2V2 field derivatives & amplitude 241.00 u, = 1 B1 242.90 - 243.49 V2 2 1 B1 249.23 - 249.68 12,. + 21/z = 3 field derivatives & amplitude 250.68 An example of the PE mechanism is given in Figure 8.2, in which the central ray orbits and radial beam spots at various energies are plotted. Assumptions were: the 10 mil betatron eigenellipse consists of 0.5 mil by 0.5 mil grids and starts at E = 240 MeV with E1 = 20 keV / turn. Particles striking an aperture limit or entering the deflector are removed from the calculations. As indicated in this graph, the beam is characterised by amplitude-dependent distortions after accelerating through the 1/7. = 1 resonance due to the residual field imperfections at the 1/7. = 1 radius. Second, no clean beam separation is achieved at the septum entrance simply because the radius gain per turn is too small. 8.3 Performance Calculations The important resonances encountered in the PE system are listed in Table 8.1. The beam is proposed to be extracted at about E = 249 MeV to avoid beam loss corresponding to the resonances at V2 2 1 and V," + 2V2 = 3. Parameters of the extraction elements are shown in Table 8.2. A one mil electro- static deflector septum is used in this system to increase the channel entry rate, thus increasing the extraction efficiency. The combined r and z accelerated orbit runs using Z4 are summarized in Figure 8.3 pr (inch) 1.19 1.17 pr (inch) 1.15 1.14 142 Ar:10mil N=2884 E1220 keV/turn 9:338 |IILIIIIIIIIIIIIIII \ /-’ 244.2 MeV ." L 1 | 1 1 1 1 l 1 1 1 1 1 1 11111111111 111111111 1 mil septum 249.32 MeV ‘1’ r 111111| . 1 . . . . 26.62 26.63 26.64 r (inch) Figure 8.2: Central ray orbits and beam spots at various energies illustrates the effect of VT 2 1 resonance crossing and the concept of precession extraction (a net B1 ~ 0.1 gauss bump used). Table 8.2: Extraction element parameters 143 C"); 0f R,- Rf M.A. E AB 03/82; RC (deg) (deg) (inch) (inch) (inch) (kv/cm) (kg) (kg/inch) (inch) E1 -24 32 26.65 27.06 0.2 144 E2 94 136 27.25 27.76 0.2 144 R 49 61 M1 140 153 27.78 27.85 0.5 1.2 8.8 M2 200 206 28.38 28.50 0.5 1.2 8.8 M3 226 230 28.99 29.17 0.5 1.2 8.8 M4 236 240 29.28 29.50 0.5 1.2 8.8 M5 256 262 30.14 30.47 0.5 1.2 8.8 M6 266 272 30.74 31.20 0.5 1.2 8.8 M7 276 282 31.57 32.24 0.5 1.2 8.8 M8 286 292 32.80 33.84 0.5 1.2 8.8 C1 320 334 27.80 C2 46 58 29.20 which shows the beam axial envelope vs. energy, the energy spectrum, and 7', pr, and 2, p2 at the entrance of the deflector. These calculations start at E = 240 MeV, and the initial beam is represented by a combination of rectangular phase space grids associated with a 30 mil radial and 20 mil vertical betatron eigenellipses. The resulting channel entry efficiency is 81.9%. The large energy spread (lower-left in Figure 8.3), which causes low channel transmission rate, comes from two factors. The first is that the initial beam size is much larger than the turn spacing at the head of the septum. The second corresponds to the long thin tail structure, as shown in Figure 8.2, which in turn increases the radial width of the beam, induced by the VT 2 1 traversal. As in preceeding chapters, the particles which successfully enter the mouth of the extraction system provide the input data for the extraction optics computations. Figure 8.4 then gives the beam half width and loss histogram in 20 steps along the extraction channel, the distribution of final energies and the r, pr, and 2, pz spectrums as the beam exits the cyclotron. The energy spread of the output beam is found to be 144 Af=30 Wm Az=20 Wfl NI1694 E1:201 . C _ N — <1) - Q_ . _ —O.1j_ — O _ 1 5? i : . ; < , — —o.2—_ , ' r 2 —0.3—j -' -' 3 .O ""1"'1"**1""1"" 1""1""1""1""1 240 242 244 246 248 250 —O.4 —0.2 —0.0 0.2 0.4 Energy (MeV) Z (inch) 30 1 1 1 1111111111111 111 l l l :Eentry : 81.9% /Vr = l 25: 4 . (n j 1 mil septum _ 920— - '9 I 5 mm aperture . 6 0— 15f — _ .5 i _ifiprecession - O‘ 10{ 1 - . Z . 1 --"~«..,. : . .- . ii :;-'_~: ._ : 5 i 7 I . h- : ' . pos u — l o— ‘ ’ 240 242 244 246 248 250 26.85 26170 28'75 26.80 Energy (MeV) r (inch) Figure 8.3: Upper left: beam axial envelop vs. energy showing the beam lost vertically due to the resonance transversal and off—centre of beam. The rest: distribution of energies, radial and vertical phase spectrums at the entrance of the extraction channel for those particles successfully enter the deflector. 145 strongly correlated to the aperture of the electrostatic deflector and to the initial radial spread at the entrance. The resulting channel transmission efficiency and extraction efficiency are 28.1% and 23.0%, respectively. In conclusion, it is possible but very difficult to get beam out of synchrocyclotrons based on the precession extraction scheme. To accomplish this, we must reduce the septum thickness as much as possible to increase the channel entry rate. Moreover, the location of the electrostatic deflector has to be positioned extremely accurately to match the particle trajectories because of the extremely limited space to insert the septum between the last internal turn and the extracted orbit. 146 Central ray: r1:26.649 pr1=1153 E2249.1MeV 6 : 308 .5- _ 30 1 1 1 1 l 1 1 l 1 I 1 1 1 1 1 I 1 1 1 1 AE = 0.9 MeV I A6 = 144 kV/cm E . e 2 23% - 4' 25‘ 21 "— 10 87 ’7 1m ED. 055 _ ' ° — M BAR $20- ' A -3 ' ' .7.) L t’ "3.) g 15 _ _ " .2— 7 B o' . z 10« _4 .1- _ I 5— _ .0 I I II 15' I I l o_ _. ° 5° 0° 0 20° 25° 30° 2 3.4 248.6 248.8 249.0 249.2 249.4 243.6 Energy (MeV) ‘5 I l I I I I I 0.1.0 1 I 1 I l 1 1 I 1 1 1 I 4— — _‘ 0.05— _ — A .3— — A . . . .C .C I I . U U - 5 g o.00~ . r:- . - N 2_ _ 0:4 . - l... —o.05— ' — .1— — _ , 'G 0 5'0 100 150' 200 250 300 "0‘0 ' ' ' ' . ' . ' ' ' ' ' ' ' ' —0.15 —0.10 —0.05 0.00 0.05 0.10 0. 5 z (inch) (‘60 I I I I l l I 16.65 1 1 1 I 1 l 1 1 1 1 I 1 I S Etransmission = 28 1% E 140 - 120 — 15.60— — ,5? 100 — “‘ 3? O U o. 80 - .5 16.55- _ z _ . V 11 v1 60 _ VI 0 4 - __ . 40 . ‘ 16.50~ ' . — 20 0 0 50 100 150 200 250 3007 1645 ' ' ' ' ' ‘ ' ' ' ‘ ' ' r ' *' ' 38.75 38.80 38.85 38.90 38.95 39.00 6(degree) r (inch) Figure 8.4: Panels A, B, C: the beam half width and loss histogram in 2° steps vs. 19 along the extraction channel. Panel D: distribution of final energies. Panels E, F: radial and vertical phase spectrums at 6 2 308°. Chapter 9 Summary and Conclusions We noted in Chapter 3 that the orbit problems in the intermediate radius are minor in the proposed K500 adapted synchrocyclotron and the extraction system is then the most critical aspect of the possible conversion and therefore the focus of this thesis, as explained in Chapter 1. Three possible extraction solutions have then been explored: (1) The traditional regenerative method of synchrocyclotron extraction which uses two magnetic disturbances, a regenerator (a positive gradient bump) and a peeler (falling fringe field), to produce a c0826 field perturbation, and induces radial oscil- lations with exponential increasing amplitude. (2) The V, = 1 excitation extraction system which employs a first harmonic bump to make the orbits drift to a selected radially unstable asymptote as the 1/,. = 1 stop band approaches. (3) Precessional extraction, in which a smaller first harmonic bump, is used to produce a small co— herent precession of optimum amplitude as the beam accelerates through the 1x, = 1 resonance and, as a result of this precession, post-resonance acceleration gives the radial separation needed to clear the septum. For all three proposed extraction systems, the existing extraction components of the K500 cyclotron are assumed to be used for economic reasons: Two existing electrostatic deflectors are selected as the first two extraction channels because of their thin septum and well confined field. Following the electrostatic deflectors, eight 147 148 passive magnetic elements (focusing bars) are employed to provide assistance to both deflect and focus the beam, due to their negative field in their aperture centre and their strong radially focusing field gradient across the aperture. The resonances encountered in the extraction and their characteristics are sum- marized in Table 9.1. One conclusion from the extraction calculations is that the extraction performance in the K250 synchrocyclotron is mainly limited by the res- onance transitions. Chapter 4 showed that the amplitude gain for slow resonance traversal is roughly proportional to the square of the driving force and inversely pro- portional to the energy gain per turn because the width and strength of the stop band are approximately proportional to the driving term, respectively, and the energy gain per turn measures how fast the resonance is traversed. In addition, the resonance crossing will introduce beam distortions which depend strongly on initial beam am— plitude, on magnetic field derivatives, and on field imperfections. Consequently, for synchroncyclotrons with their intrinsically low energy gain per turn, the extracted beam’s character owing to the resonance traversal will be very sensitive to both the magnetic field imperfections and to the initial beam quality. The Z4 and DEFLZ800 codes (see Chapter 2) were used to perform the orbit tracking and extraction efficiency calculations for different conditions. Standard as- sumptions were: 0 A monoenergetic beam starts at E = 240 MeV. Initially uniform rectangular phase space grids inside the projected ”beam eigen- ellipses”. Constant energy gain is assumed in the Z4 calculations. The justification for these assumptions (a good approximation and a fast calcula— 149 Table 9.1: Summary of resonances encountered in the extraction. I/T : 1 resonance in ex. (B1 = 1.2 g) extraction extraction ext. region E AE E AE E AE driving force b 240.81 0.14 0.14 0.26 241.13 241.19 1.14 243.11 0.62 v =1 249.86 0.14 . 0.46 +21/ =3 251.13 250.68 Characteristics AA o< f & amplitude dependence 85 distortions I/ :1 causes V, = 2V2 on on how fast the band is and on the field U, = 1 1 2 beam distortions 1 a regenerator a neg. grad. bump (falling fringe field, peeler) of radial 150 tion) was presented in Chapter 5. In order to see the influence of several crucial parameters (initial beam quality, energy gain per turn, septum thickness and aperture of electrostatic deflectors) on extraction efficiency and for comparison among these three mechanisms, calculations have been done with one parameter changed in turn while keeping the others fixed. The results are summarized in Table 9.2. The parameters A,(i = r, z) are betatron oscillation amplitudes, 31 the net first harmonic bump amplitude, E1 the energy gain per turn, t the septum thickness, A the aperture of the electrostatic deflectors, AT the turn spacing at the septum entrance for initially centered particles, 6mm, the channel entry efficiency, 6m: the total extraction efficiency, and zloss the total vertical loss in %. The results in Table 9.2 are well explained by the following arguments and com— ments: (1) In general, the field imperfections produced by extraction devices could not be completely canceled by a first harmonic bump with two adjustable parameters (strength and azimuth) probably due to the scalloping of the orbits; canceling the field imperfections over a limited range at resonance radius is not enough to maintain the orbits in an undisturbed stable state. (2) It seems that the radial position of M1 (and its 180° compensator C1), owing to its proximity to the internal beam and its large azimuthal width, has significant influence on how successfully the field imper— fections in the vicinity of u, = 1 can be compensated. (3) As the septum thickness is reduced, the channel entry efficiency rises, and the extraction efficiency typically also increases. (4) The larger the aperture of the electrostatic deflectors, the larger the acceptance of the extraction system, and the bigger the external current. (5) We expect that poor initial beam quality will give rise to low extraction efficiency due to the resonance traversal. (6) Increasing the dee voltage (usually implying larger energy gain per turn) will raise the current extracted from the ion source, enhance 151 the space charge limit, and decrease the amplitude growth due to resonance crossing or increase the acceptance of the resonance; therefore in general, we will end up with higher extraction efficiency and larger output current by increasing the dee voltage. In Chapter 6, a weak regenerator (50 gauss/cm) system, giving turn to turn spacing at the septum entrance of about 0.1”, was described. With 0.1” separation, the axial instability induced by the off-center beam was substantially allayed as was seen in Figure 6.13, and a high channel entry efficiency owing to the very thin 0.01” electrostatic septum was achieved. In addition, it is rather easy to compensate the fringe field of the weak regenerator to reduce the beam loss when the beam passes through the resonances compared to the situation in a traditional synchrocyclotron where a magnetic element is normally used as the first extraction channel. Satisfactory results are therefore expected using the regenerative extraction scheme. There are two primary difficulties in the V, = 1 excitation extraction (Chapter 7), summarized as follows. The electric field strength, which puts a limit on the maximum aperture of electrostatic deflectors, is one of the limiting factors to achieve high extraction efficiency in the l/T : 1 resonant extraction scheme. Another factor is that the three resonances (1/2 = 1/2,1/T = 2V2, V, = 1) are only about 3 mm apart — so close that it is not possible to excite the V,- : 1 resonance alone without also exciting the resonances 1/2 = 1/2 and 11,. = 21/2. For example, the V, = 1 excitation extraction performance can not be improved by increasing dee voltage because: The field bump becomes significantly less effective as energy gain per turn increases. The larger energy gain per turn will thus call for a stronger bump to restore the extraction effectiveness. However, this also increases the vertical driving force for resonance crossing (I/z : 1/2, V, = 2V2) and thus induces vertical instability. Chapter 8 concluded that for high magnetic field synchroncyclotrons with their intrinsically low energy gain, i.e., small turn to turn separation, the precession ex— 152 Table 9.2: Summary of extraction calculations. The betatron oscillation amplitudes are denoted as A,-(i = r,2), the net first harmonic bump amplitude B1, the energy gain per turn E1, the septum thickness 75, the aperture of the electrostatic deflectors A, the turn spacing at the septum entrance for initially centered particles Ar, the channel entry efficiency (Emmy, the total extraction efficiency em, and the total vertical loss zloss. Regenerative extraction Ef ~ 247 MeV E N 96 kV/cm initial conditions A, A2 total E1 t A Ar 6mm, 6.,“ zloss (mil) (mil) orbits (keV/turn) (mil) (mm) (mil) (%) (%) (%) 30 20 1748 10 10 10 105 92.30 41.77 0.00 30 20 1748 20 10 10 101 90.78 52.00 0.00 30 20 1748 10 1 10 105 99.65 42.06 0.00 30 20 1748 10 10 5 105 90.72 21.15 0.00 50 40 3255 10 10 10 105 75.42 22.09 18.04 50 40 3255 20 10 10 101 91.64 37.36 1.14 11, = 1 excitation extraction Ef ~ 243 MeV E N 159 kV/cm initial conditions A, A, total B1 E1 (keV t A Ar 6mm, ecu zloss (mil) (mil) orbits (g) /turn) (mil) (mm) (mil) (‘70) (%) (%) 30 20 1725 1.2 10 10 5 60 83.30 17.80 6.03 30 20 1725 1.7 20 10 5 58 81.68 19.36 5.45 30 20 1725 1.2 10 1 5 60 92.00 19.54 6.38 30 20 1725 1.2 10 10 10 60 88.88 32.70 8.81 50 40 3404 1.2 10 10 5 60 39.75 7.76 58.84 50 40 3404 1.7 20 10 5 58 43.95 9.75 51.59 Precession extraction Ef ~ 249 MeV E N 144 kV/cm initial conditions A, A2 total E1 75 A A7“ 6mm, 6th zloss (mil) (mil) orbits (keV/turn) (mil) (mm) (mil) (%) (%) (%) 30 20 1694 20 1 5 10 81.88 23.02- 10.80 30 20 1694 10 1 5 5 41.44 6.26 51.12 30 20 1694 20 10 5 10 34.83 4.25 10.33 30 20 1694 20 1 10 10 82.23 31.70 10.80 50 40 3441 20 1 5 10 50.77 11.48 41.96 153 traction is not an attractive way to extract the beam. In conclusion, three important results have been established by this study. (1) The magnetic fields in the extraction region has to be manipulated carefully to avoid the beam loss induced by the resonance transitions due to the beam’s low energy gain. (2) The regenerative extraction system is a promising candidate for the proposed conversion of the K500 to a 250 MeV synchrocyclotron for proton therapy because it is reasonably easy to construct and gives better results than the other two methods. (3) This K500 adapted synchrocyclotron will have an output proton energy of about 250 MeV and should have an external beam current in the range of 5 to 30 nA which meets the requirements for its purpose. Important parameters of the proposed machine are given in Table 9.3. 154 Table 9.3: Parameters of the K250 proton synchrocyclotron current current current structure sectors constant at at center source 0.33 x 0.95 20 0.2 Bibliography [1] E.O. Lawrence and N .E. 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