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The properties of ion orbits in the central region of a cyclotron Louis, R. Oct 31, 1971

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TRI UMFTHE PROPERT IES  OF ION ORB ITS  IN THE CENTRAL  REGION OF A CYCLOTRONR. LouisMESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIA UNIVERSITY OF BRITISH COLUMBIATHE PROPERTIES OF ION ORBITS IN THE CENTRAL REGION OF A CYCLOTRONR. LouisThis report is based on the author's thesis submitted to the Faculty of Graduate Studies o the University of British Columbia in partial fulfilment of the requirements for the degree of Doctor of Philosophy - April 1971Postal Address:TRIUMFUniversity of British Columbia Vancouver 8 , B.C.Canada OctoberABSTRACTThe behaviour of ion orbits in the magnetic and electric fields at thecentre of a cyclotron is studied in detail. The objective is to optimizethe phase acceptance and beam quality for a 500 MeV H" isochronous cyclotron.Since accurate electric fields are necessary for orbit calculations, a numerical method for calculating those fields is examined in detail. The method is suitable for complicated electrode shapes and converges rapidly, yielding potentials in three dimensions with average errors of less than0.01%. The magnetic fields used in the orbit calculations are measured on model magnets.The axial motions are examined using a thick lens approximation forthe accelerating gaps. A method is demonstrated for calculating the axialacceptance of the cyclotron as a function of RF phase. This method is used to evaluate the merits of various central geometries and injection energies. This method is also used to examine the effects of flat-topping the RF voltage by adding some third harmonic to the fundamental waveform. It is found that addition of the optimum amount of third harmonic increases the phase acceptance by about 20 deg. Finally, the effects of field bumps on the axial motions are investigated.To allow accurate radial motion calculations to high energy, an approximate formula is developed which yields accurate (<1 %) values for the changes in orbit properties of an ion crossing a dee gap. The geometry of the orbit on the first turn is discussed in detail. The radial centring is studied by tracking ions from injection to 20 MeV, and a method is described for choosing the starting conditions of the beam so as to mini­mize the radial betatron amplitude over a desired phase range.The problems associated with using a three-fold symmetric magnetic field with a two-fold symmetric electric field are also discussed. Besides the well-known gap-crossing resonance, a previously ignored phase- oscillation effect is found to be important for cyclotrons operating on a high harmonic of the ion rotation frequency.C O N T E N T SPag<1 . INTRODUCTION 11 . 1 Problems in the Cyclotron Central Region 11 . 2 The TRIUMF Cyclotron 31.3 Equations of Motion 42 . ELECTRIC FIELD CALCULATIONS 62 . 1 Choice of Method 62 . 2 Finite Difference Approximation 72.3 Computational Details 102.4 Convergence Tests 142.5 The Practical Problem 173. AXIAL MOTIONS 203.1 Introduction 203.2 Magnetic Field 203-3 Space Charge Forces 233.1* Electric Lens Effects 243.5 Calculation of Cyclotron Acceptance 273.6 Phase Space Acceptance for Various TRIUMF CentralGeometries and Injection Energies 313.7 Effects of Third Harmonic in the RF on Axial Motions 333.8 Effects of Field Bumps 354. RADIAL MOTIONS 374.1 Introduction 37A . 2 Basic Design 381* • 3 Problems with Three-Sector Magnetic Fields 424.4 Radial Centring 474.5 Effects of Finite Beam Emittance 535. RADI AL LENS EFFECTS OF CYCLOTRON DEE GAPS 555.1 1 ntroducti on 555.2 Constant Gradient Approximation with No MagneticField 565.3 Sine Gradient Approximation 615.4 Constant Gradient Approximation with Third Harmonicin the Electric Field 625.5 Constant Gradient Approximation with Magnetic Field 635.6 Constant Gradient Approximation with Magnetic Fieldand Third Harmonic in the Electric Field 676 . SUMMARY AND CONCLUSIONS 69Acknowledgements 72References 73Appendix A: Theory of Successive Over-Relaxation 76Figures 83I I ILIST OF TABLESLargest eigenvalue (\ ) and best over-relaxation factor (a-^ ) for various size relaxation problemsSequence of operations used to solve a 64x64x32 node relaxation problemAverage error after various numbers of iterations over the reduced problemSequence of operations used to solve a 512x128x32 node relaxation problemGap factors given by numerical integration and by the constant gradient approximation (no magnetic field)Gap factors given by numerical integration and by the constant gradient approximation (isochronous magnetic fieldLIST OF FIGURES1.1 Central region of the TRIUMF cyclotron - median plane1.2 Central region of the TRIUMF cyclotron - section through centreline of hill #32.1 Relaxation mesh organization; total number of nodes is (p+1)(q+1)(r+1)2.2 Average error and average change per iteration vs number of sweeps over large volume2.3 Average error vs number of sweeps over large volume for various values of a2.k Number of nodes with a given error vs size of error for various number of sweeps over large volume2.5 Number of nodes with a given error vs number of sweeps over large volumefor various size errors3.1 Magnetic axial focusing frequency (v3 ) vs energy for three- and six- sector magnetic geometries3.2 Equivalent axial focusing frequency produced by space charge forces vsenergy for various beam currents and axial beam heights3-3 Cross-section of dees near accelerating region showing electric equi-potentials and (schematically) an ion trajectory3.^ t Comparison between equivalent electric axial focusing frequencies predicted by the thin lens approximation and determined by numerical i ntegrat ion3.5 Possible TRIUMF central geometry with three accelerating gaps in the f i rst hal f-turn3.6 Axial emittance ellipses required at injection for various RF phases3-7 Axial acceptance vs RF phase for various injection energies (oneaccelerating gap in the first half-turn)3.8 Axial acceptance vs RF phase for various injection energies (threeaccelerating gaps in the first half-turn)3.9 Average axial acceptance (averaged from -30 deg to +60 deg) vs injection energy (one accelerating gap in the first half-turn)3.10 Axial acceptance vs RF phase for various choices of the initial emittance el 1 i pse3.11 RF voltage waveform with various amounts of third harmonic and phase shift between fundamental and third harmonicv3.12 Slope of RF voltage waveform with various amounts of thid harmonic and phase shift between fundamental and third harmonic3.13 Axial acceptance vs RF phase for various choices of the initial emittance ellipse, e = 0.17, 8 = 03.14 Axial acceptance vs RF phase for various choices of the initial emittance ellipse, e = 0.12, 8 = 03.15 Axial acceptance vs RF phase for various choices of the initial emittance ellipse, e = 0.15, 8 = -10 deg3.16 Total (magnetic and electric) equivalent axial focusing frequency vs energy for various RF phases3.17 Transition phase (of total axial focusing from negative to positive) vs energy3.18 Change in sine of RF phase required to keep ion at transition phase vs radi us3.19 Magnetic field bump required to keep ion at transition phase vs radius4.1 Geometry of injection gap and first main gap for two RF phases4.2 yc vs RF phase at injection gap for various injection gap positions4.3 Energy gain in injection gap and first main gap vs RF phase at injection gap for various injection gap positions4.4 RF phase at first main gap vs RF phase at injection gap for various injection gap positions4.5 Phase oscillation amplitude vs centring error at various radii4.6 RF phase vs half-turn number for various initial phases with no flutter in the magnetic field4.7 Geometry of an orbit in a three-sector magnetic field4.8 RF phase difference on succeeding half-turns as a function of orientationof the dee gap (6)4.9 RF phase vs half-turn number for various initial phases with a three- sector magnetic field (6 = 30 deg)4.10 Ratio of third harmonic amplitude in magnetic field to average field vs radius for (three-sector) field 1-14-5-704.11 Average orbit radius and maximum orbit scalloping vs radius for (six- sector) field 1 -3 0 -0 6 - 7 04.12 Geometry of an orbit in a six-sector magnetic fieldvi4.13 Geometry of the difference between an equilibrium orbit and an accelerated orbit4.14 Centre-point displacement along the dee gap vs energy showing values from numerical orbit tracks and from an analytic approximationvi i4.15 Accelerated phase plot inwards from 5 MeV, 4> = -30 deg4.16 Accelerated phase plot inwards from 5 MeV, <f> = 0 deg4.17 Accelerated phase plot inwards from 5 MeV, <p = +30 deg4.18 Accelerated phase plot outwards for various radii at first main dee gap,<(> = 0 deg4.19 Accelerated phase plot outwards from inflector exit for various phases; ion with <J> = 0 is centred4.20 Accelerated phase plot outwards from inflector exit for various phases; ion with <f> = +17 deg is centred4.21 Betatron oscillation amplitude vs RF phase for various starting cond i t i ons4.22 Phase histories of ions with various starting phases in a magnetic field with a field bump4.23 Accelerated phase plot outwards from inflector exit for various phases using the magnetic field with the field bump; ion with (J> = 17 deg is centred4.24 Accelerated phase plots with <f> = 0 deg for four points on the edge of the emittance ellipse a) matched to = 1, and b) chosen to reduce the radial oscillation amplitude over the phase range -5 deg to +25 deg4.25 Accelerated phase plots with <j> = +15 deg for four points on the edge ofthe emittance ellipse a) matched to vr = 1, and b) chosen to reduce the radial oscillation amplitude over the phase range -5 deg to +25 deg4.26 Accelerated phase plots with <j> = +25 deg for four points on the edge ofthe emittance ellipse a) matched to = 1, and b) chosen to reduce theradial oscillation amplitude over the phase range -5 deg to +25 deg5.1 Cross-section of a dee gap showing electric equipotentials5.2 Electric potential vs distance from dee gap centre showing actualvalues and constant gradient approximation5.3 Geometry of an ion crossing a dee gap5.4 Gap factors vs energy for $c = 0 degDifferences between gap factors obtained from numerical those obtained from the constant gradient approximation of energy, no magnetic fieldDifferences between gap factors obtained from numerical those obtained from the constant gradient approximation of energy, with an isochronous magnetic fieldDifferences between gap factors obtained from numerical those obtained from the constant gradient approximation of RF phase, with an isochronous magnetic fieldintegration and as a functionintegration and as a functionintegration and as a functionApparent displacement due to change in radius of curvature of the ion path while crossing the dee gapINTRODUCTION1.1 Problems in the Cyclotron Central RegionThe central region of a cyclotron requires special attention because the internal beam quality and phase acceptance are primarily determined during injection and the first few turns within the machine. During these initial turns, the beam has low energy and is therefore strongly influ­enced by the phase-dependent lens effects of the dee gaps. The objective of this work is to study the behaviour of ion orbits in the magnetic and electric fields at the cyclotron centre, and thereby to choose the beam in­jection conditions and magnet and electrode designs for optimum beam performance, i.e. a beam which is centred, has minimum spot size in both the radial and axial directions, and is in a phase interval which optimizes the acceleration process.The usual studies of cyclotron central regions, for example Rose , 1 and o t h e r s , a r e  concerned with machines with internal ion sources where the ion starts with zero energy and spends its first turn mainly within the electric field produced by the dee gap. With an external ion source, the problems are quite different; to solve them this study was undertaken.Injection of ions into a cyclotron from an external source has been studied by Powell and Reece ; 5 however, the injection energy in their case was 11 keV, compared to a maximum energy gain of 50 keV per turn, whereas in this case the injection energy is 300 keV, compared to 400 keV per turn. Also, the electrode geometry is very different.This report considers ion injection for a H" cyclotron where the ions are extracted by electron stripping and the duty cycle is determined by the phase band the central region will transmit and not by the extraction system, as in some cyclotrons with resonant extraction schemes. Thus there is considerable emphasis on reducing phase-dependent effects in the central reg i on.The central region problems fall naturally into two groups, those con­cerning the axial motions and those concerning the radial motions.The basic problem in the axial motion is that the focusing provided by the magnetic field becomes very small near the centre of the machine, while- 2 -the (phase-dependent) electric forces due to the dee gaps become very strong. It is well known^hat the electric forces are defocusing for half of the RF cycle. Since these electric forces will be larger than the (focusing) magnetic forces at low energy, a detailed study of the axial motions is re­quired if a large range of RF phases is to be accepted. The situation is further complicated by the fact that space charge effects will also tend to expand the beam. Space charge effects will be most important at low energy and high current.The basic problem in the radial motion is not lack of focusing but rather how to minimize the radial oscillation amplitudes of the ions. Since the ions are extracted when they reach a particular radius, a large spread in radial amplitudes means that ions from different turns may be present atthe extraction radius, resulting in a large energy spread in the extractedbeam. The initial motion of the ions in the cyclotron requires that the beam be injected off centre if it is to be centred at extraction; however, this effect is phase dependent, making it difficult to centre ions with a wide range of phases.Since a knowledge of the electric fields involved is required for studies of both the axial and radial motions, Section 2 describes in detail a method for calculating these fields.Section 3 considers the axial motions. A method is presented which allows calculation of the axial acceptance of the accelerator as a function of RF phase. This method is used to study various injection energies and the effects of adding third harmonic to the RF. Finally, the effects offield bumps, used to induce phase slip, are considered.Section A considers the radial motions. The geometry of the first turn and how this is influenced by the accelerating electrodes is studied in detail. The radial centring is studied by tracking ions from injection out to 20 MeV. Finally, the effects of a finite beam size are considered.Section 5 describes an approximation which allows the changes in orbit properties of an ion crossing a dee gap to be evaluated to high accuracy without numerical integration through the electric field. The accuracy of the method is given as a function of RF phase and incident ion energy. This approximation is used in the tracking of the radial motions in Section 4 between 5 and 20 MeV where this approximation is very accurate.- 3 -1.2 The TRIUMF CyclotronThe studies described in this report were performed for the TRIUMF cyclotron , 5 which because of its unique design has several special problems.The TRIUMF cyclotron is a six-sector, azimuthally varying field (AVF) , isochronous machine, designed to accelerate 100 yA of H" ions to 500 MeV.The acceleration of H“ ions provides a convenient method of extraction by stripping two electrons from the H“ ions by passing the beam through a thin foil. This method gives an extraction efficiency of nearly 100% whereas conventional proton machines have not achieved efficiencies greater than 80% with a large duty cycle. Two other advantages of extraction by electron- stripping are variability of extraction energy by adjusting the foil posi­tion and simultaneous extraction of several beams at different energies.The disadvantages of this technique are that the lifetime of the H" ions requiees that the maximum magnetic field that the ions pass through must be low (5.7 kG at 500 MeV)7 to prevent disassociat ion of the H- ions, and also there must be a vacuum < 7 x 10“ 8 Torr to prevent H~ stripping by residual gas molecules. The low magnetic field means that the radius of the machine is very large (500 MeV orbit radius of 311 in.), and the central magnetic field (3.0 kG) is five or six times lower than in conventional cyclotrons.The accelerating voltage is provided by four resonant cavities which provide 0.A MeV energy gain per turn. The low magnetic field means that the ion rotation frequency is low (A.53 MHz). To allow the cavity reso­nators to fit inside the vacuum tank, the RF is operated at the fifth harmonic of the ion frequency. The fact that the accelerating structures are cavity resonators means that some third harmonic of the ion frequency can be introduced into the cavity, squaring the RF waveform and giving sig­nificant improvements in orbit properties.The arrangement of the TRIUMF central region is shown in Figs. 1.1 and 1.2. The centre post is required to support part of the weight of the upper magnet cores, the magnetic force between the magnet pole pieces and the atmospheric load. The H” beam is produced in an external (Ehlers) ion source and accelerated to 300 keV before being transported to the cyclotron and bent into the median plane by the spiral electrostatic inflector. The beam leaves the centre post at the "injection gap", which provides an auxiliary 100 keV (the dee-to-ground potential) acceleration on the first- 4 -turn. The beam then spirals outward, gaining a maximum of 400 keV per turn.Several types of operating conditions must be considered. One of the principal uses of the machine will be to produce mesons. In this case, thecurrent required is large, but the energy resolution is not important(since the mesons are produced in a secondary target). Therefore, the duty cycle may be maximized at the expense of energy resolution. It is also planned to produce a high resolution proton beam. In this case, high current is not required so a smaller duty cycle may be considered, giving smaller radial oscillation amplitude and thus improving energy resolution. It is also hoped that with the addition of third harmonic to the RF, separated turn acceleration will be possible, i.e. spatial turn separation will be maintained out to extraction so that the beam can be extracted from one turn, giving very high energy resolution (hopefully, ±50 keV). Again, the phase band accelerated would be quite narrow.1.3 Equations of MotionThe force on a charged particle moving in electric and magnetic fieldsis given by the sum of the Lorentz and electric forcesF is the force on the particle which has charge q, mass m and velocity v. The electric field is E and the magnetic field is B.We define a Cartesian co-ordinate system with the z axis upwards in the axial direction (perpendicular to the plane of the orbits), the x direction is along the centreline of the dee gap, and y is perpendicular to the dee gap and the axial direction.In a Cartesian system, Eq. (1) can be writtenF = q(E + v x B) . (1)F = q E + Th B - v B ) , x  ^[ a: y z z y J ’ ( 2 )( 3 )F = q E + (v B - v B ) . z  ^ z x y y x (4)F = q E + (v B - \f B ) , y y z x x z ’- 5 -The ion circulates in its orbit near the x-y plane; hence the compon­ents of the velocity in this plane (v and v ) are much larger than v . Duex y zto the symmetry of the magnet, the magnetic field in the median plane is inthe axial direction only, i.e. B = B = 0. Errors in the construction ofx ythe magnet may cause the magnetic median surface to be different from thegeometric median plane, giving non-zero values of B and B in the geometricx ymedian plane; however, these will be small, and we may write Eqs. (2) and(3 ) as4t(myx) = « (Ex +  SJyBz) 3 (5)4b(ms/y) = q (Ey ~ vxBz)- (6 )Eqs. (5) and (6 ) are re 1 ativistical1y correct, provided the changes in mass due to acceleration are not neglected. The relativistic mass ism = ymowhere m is the rest mass and v is the usual relativistic factorowhere T is the kinetic energy of the ion, a is the velocity of light andB = v/c.The approximation used in deriving Eqs. (5) and (6 ), i.e. that termsin v B and v B are negligible, has removed coupling between motion in thez y s %cmedian plane and motion in the axial direction, greatly simplifying the calculations. The solutions of Eqs. (5) and (6 ) [obtained by numerical integration through realistic electric and magnetic fields] are discussed in Section 4.The axial motion is described by Eq. (4). The terms in B and Bx ycannot be neglected in this case since they are multiplied by the (large) velocities v and v . It is these terms which describe the axial magnetica yfocusing produced by flutter and spiral in the magnetic field when the ion is not in the median plane. The axial motion is discussed in Section 3-- 6 -2. ELECTRIC FIELD CALCULATIONS2.1 Choice of MethodAccurate orbit calculations in the central region require a detailed knowledge of the electric and magnetic fields involved. The magnetic field can be obtained from measurements on model magnets. The electric field is produced by complicated electrode shapes (see Figs. 1.1 and 1.2) and hence cannot be calculated analytically. There are several methods which can be used to obtain the electric field in these circumstances:1) Electroconductive analogies in which the potential is obtained by measuring the voltage in a conducting medium surrounding a modelof the electrodes. 8 This method yields potentials (in two or three dimensions) with errors of about 0.3%- 92) Numerical solution of Laplace's equation. This method yields potentials with average errors of 0 .1 % or less, depending on the time available for computation. This method is described in detail below.3) The induced current method in which a vibrating charged probeinduces a current in the electrodes proportional to the component of the required field at the probe in the direction of vibrationof the probe . 1 0 This method gives field values with errors of 5-0%or less.4) The magnetic analog in which the components of the magnetic field are a measure of the corresponding electric field components . 1 1Methods 3 and 4 yield field values which can be used directly in orbit calculations while methods 1 and 2 give potentials which must be numerically differentiated to obtain the field components.From this point of view, method 3 or 4 is more attractive. However, methods 1 , 3 and 4 require a model of the electrode structure to be built. This means that changes in the electrodes require time-consuming and expen­sive changes in the model. In addition, these three methods involvemechanically-driven probes which are subject to alignment errors. Also, these methods use complicated electronic circuits which are subject to drift over long periods of time. For these reasons, the numerical solution- 7 -of Laplace's equation which avoids these difficulties is the most attrac­tive choice. Solving Laplace's equation for a complicated boundary shape is a difficult computational problem; however, the availability of large, fast computers enables large problems to be solved in a reasonable amount of time.2 .2 Finite Difference ApproximationWe wish to find the electrostatic potential <f> which is the solution of Laplace's equation, i.e.V2(j> = 0 ^within the rectangular parelleiepiped shown in Fig. 2.1. This volume is bounded by the planes x = 0 , x  = p h , y  = 0 , y = q h ,  z = 0, z = rh. In the usual problem either the potential or its derivative is known on the surface of the volume (Dirichlet or Neumann boundary conditions, respective­ly) while the potential is unknown inside the volume. In the problems to be studied here every boundary plane has Dirichlet boundary conditions or is a plane of symmetry (described below). In addition, parts of the interior of the volume may have fixed potential values, i.e. the boundary conditions may extend inside the volume.To solve Eq. (7) numerically we transform the differential equation to a difference equation and solve for the values of <j> at discrete nodes with­in the volume. Fig. 2.1 shows a rectangular grid with uniform spacing h in all three directions. The nodes occur at the intersections of the planes x - ih, y - jh and z - kh where i = 0,1...p, j = 0,1...q and k = 0,1...r.The number of nodes in the grid (N) is (p + 1)(q + 1)(r + 1).To derive the finite difference approximation, we consider the potential at some noc*e ijk- Expanding the potential in a Taylorseries at the six nodes nearest to i , j , k we obtain’ 3 <}>' h d ^ < p '  h ^  3H±u,k "  ♦ « *  1 '‘M . y *  + 2 * «\?*%k + 24\**%k + " '♦ + ...^  2 l.3j/ 6 l3y Jijfe 24(.3y Jijk( c o n t1d . )- 8 -^i ,j ,k±l ^ijk + hAdding these, we obtainijkH+ljk + *i-ljk + *ij+lk + *ij-lk + *ijk-l + *ijk+1 = 6*ijkh2^+ y^72<j) + 0 ( h h) jusing (7) and neglecting terms in h ** and higher, we have,H j k  ~ J  H + l  + *i-l + *j+l + *j-l + *k+l + *k-l = ^ijk'+0([interior points] [boundary points]In the right side of Eq. (8) we have abbreviated the notation by writing only those subscripts which are not equal to i , j or k.Eq. (8) describes a linear system of N equations which can be writtenA d  = b (9)where 4 is an N by N matrix containing the coefficients of the system, d is a column vector containing the unknown potential valuesd =i n*p<pand & is a column vector containing the potential values for those nodes which fall in the boundaries.Now the solution of Eq. (7) is reduced to the solution of the linear system Eq. (9). It should be noted that the order of the system (9) is equal to the number of nodes in the mesh, which will be of the order of many thousands or millions.Direct methods for solving linear systems such as Gaussian elimination or use of determinants have two disadvantages in the present case. Firstly, they require that the matrix A be stored. This is clearly unnecessary3cj> ^  h ^ f 3 <^j) + 3 (^j) ^  b!^  f 3 4^>U J ^  2 M f #  " e U z3k j k j k M- 9 -since the elements of A can be generated using Eq. (8 ). Secondly, they require about N 3/3 multiplications to solve a system of order N. To solve a system with N = 2<96 would take 101 2  sec (many years) allowing 3 ysec per multiplication. Such a system can be solved in about 2 hours using the iterative method described below.Iterative methods offer two advantages over direct methods in this case. Firstly, they require only the current solution vector a: to be stored and secondly, they are much more efficient for solving large systems when the coefficient matrix A contains many zero elements.Many iterative methods for solving systems such as (9) have been developed and studied theoretically. An excellent review of the methods available is given by Forsythe and Wasow . 1 2The method used here is based on a program developed by D. Nelson . 1 3 ’ 14 Basically this program uses successive over-relaxation by points to solve the linear system.* This method is applied in a manner which allows extremely large problems to be solved using a modest amount of computer memory. The theory of successive over-relaxation by points is reviewed in Appendix A. The important results are as follows:We start with an initial approximation (usually zero) to the potentiali 0 .H j k 'at each node (f).0.,,; then we obtain successive approximations using,n+l ,n , a ^ijk ~ ^ijk 6,n+l , ,n ^ ,n+l , ,n ,n+li^-ljk i+ljk i^j-lk ^ij+jk ijk-1, J 1 a J 1ijk+1 ijk ( 10 )where the best value of the "over-relaxation factor" a  for the ordinary successive over-re 1 axation method is given byah =  :—  = 2D 1 + s^nQ ' AL.+ JL + JL\p2 q2 r2 ( 11)For this problem it appears that the Peaceman-Rachford m e t h o d  1-5 gives faster convergence . 1 6 However, as has been pointed out by Young, 6 it is difficult to devise an efficient storage scheme which allows the matrix A to be accessed alternately by rows and columns. Any increase in convei gence rate would probably be negated by increased time spent retrieving the data from the mass storage device.- 10 -1 TT TT TT" TT2 r - 1 1'oosQ =  — COS + COS +  cos~ - o - — o + o -h o3 . p q r j 6 p 2 q 2 r 2whereIf  hh hh h)j TrZ f 1 	  	( 12)So solving the system consists of iterating over the nodes of the mesh in some order, replacing the value of <j> of each node by the values given by Eq. (10). The order we shall choose is, giving the ijk values of the point to be iterated,(03030)t (13030) ... (p, 0,0) (0,1,0) ... (p3l30) ........ (p,q,0)(0,0,1) ......  ... Cp,q,l)(0,0,v) .............................  (p,q,r)or the reverse order.It is shown in Appendix A that the convergence of the method is determined by the largest eigenvalue of the matrix/l. If the best value of a, i.e. , is used, this eigenvalue is1-sind _ p /LfJ_ m 1+sinQ /  3[p2 q 2 r 2Values of Xm and a^ for the problems which are discussed in this section are given in Table I.The number of iterations required to reduce the error by a factor f is approximatelyn = log f/log \n. (14)2.3 Computational DetailsThe program as described by Nelson1 *4 used an iteration subroutine coded in FORTRAN. This was rewritten in assembler language giving a factor of twelve increase in speed. In addition, the new iteration routine allows the iteration to be done in alternating directions. Detai1s of these changes are given in internal report TRI — I — 71-1•The advantage in iterating in alternating directions is that it ensures that the effect of the boundary conditions is quickly propagated through the volume. If, for example, uni-directional iteration was used- 11 -TABLE ILargest eigenvalue U m ) and best over-relaxation factor (a^ ,) for various size relaxation problemsProblem sizeTotal number of mesh points xm %3 2 x 3 2 x 16 16,384 0.7569 1.756964 x 64 x 32 131,072 0.8841 1.88411 2 8 x 3 2 x 8 3 2 , 7 6 8 0.6245 1 .62452 5 6 x 64 x 16 262,144 0.7946 1.79465 1 2  x 1 2 8 x 32 2,097,152 0.8895 1 .8895- 12 -going from small ijk to large ijk, and all boundaries were zero except the plane with the largest k value, many iterations would be required before the effect of the boundary at large k would be felt at small k. Alternat­ing the direction of iteration avoids this difficulty.The values of the are stored on a mass storage device (tape,disc or drum). Subsets of this total "volume" are transferred to core storage, iterated over and returned to the mass storage device. To increase efficiency (by decreasing the number of data swaps) several iterations are done over each subset of the total volume while it is in core storage. This causes the convergence rate to be very slow; however, the program has a novel feature, described below, which allows good starting values to be found, hence reducing the number of iterations required. The iterations over the subsets of the total volume must be done carefully, to avoid dis­continuities where the edges of these subsets occur. Consider the volumeshown in Fig. 2.1 broken into blocks, each block containing 16 x 16 x 8points; then there are = (p+l)/16 blocks along the x co-ordinate, b 2 = (q+l)/16 blocks along the y co-ordinate and b 3 = (r+l)/8 blocks along the z co-ordinate. The data area in core storage in which the iterations are done (the physical work area) contains a 2 x 2 x 2 block subset of the total problem. The iteration is done as follows:The physical work area is loaded starting at block co-ordinates (1,1,1) and then iterated. During this iteration all potentials on the boundaries of the physical work area are held fixed except boundaries which are symmetry planes of the total volume. The next load origin is (2,1,1), and this iteration is repeated. Since two blocks along each co-ordinate areiterated each time while the increment between iterations is one block, discontinuities in the data should be reduced. The sequence of load points for the iteration is either(1.1.1), (2,1,1) __  (bx, 1,1), (1,2,1), (2,1,1) . (b^b^l)(1.1.2), (2,1,2) __  (bx, 1,2),(1,2,2),(2,1,2) . (bx,b2,2)(1,1, b 3) (2,l,b3) ... (b1,l,b3), (l,2,b3), (2,l,b3) ... (b13b2,b3)or the reverse one (alternating direction iteration over the blocks).- 13 -It should be noted that in one sweep over the data using this pro­cedure 8(bi -^l) (b2-l)  (b3-l)  blocks are iterated. On the average, this is8(b,-l)(b ,-l)(b--l)  , . 1 2 3  (15)b i b2 b3 iterations over each block.The novel feature mentioned above which allows good starting values to be found operates as follows. After the boundary values have been assigned but before any iterations have been done, the mesh size is doubled reducing the problem to one with an eighth as many data points as the original problem. This process is repeated until the problem size is close to the size of the physical work area (32 x 32 x 16 points). This "reduced" prob­lem is solved iteratively and expanded back to the original size problem. During the expansion process, the value assigned to each unknown point is the value for the nearest known node with smaller or equal i , j and k values, i.e. if 's known, the program sets (omitting subscripts whichare i , j or k)* i+1 = *3+1 = H+lj+1  =  *k+1 =  *i+lk+1 = *o+lk+l =  *i+lj+lk+1 = 41 *This procedure provides good starting values for the final iteration.The boundary values are assigned either by calling a user-supplied sub­routine which returns the value of the potential at each point, or by the method given in internal report TR1—1—71—1 or by a combination of both.In many situations, the boundary values at an edge of the problem are not known, but this edge is a plane of symmetry. In this case, the program calculates the potentials on the symmetry plane using the fact that the potentials outside it are the same as those inside. For example, if the i = 0 plane were a plane of symmetry, then on this plane Eq. (10) would be,n+l ,n , a Ojk ~ ^Ojk 6,n+l , ,n , ,n+l , ,n n+1 ,n*ljk Ijk + *Oj-lk *Oj+lk + *0Qk-l *Ojk+1a J 1 - ojk_:When estimating the convergence rate for a problem which contains planes of symmetry, it is important to remember that the errors are not- 14 -zero at the plane of symmetry (as they would be if the plane were a boundary plane). Thus the errors and convergence rates will be those appropriate for the "effective size" of the problem, which is the size the problem would be if the symmetry properties were not utilized. Thus, if a problem contains one plane of symmetry, the effective size is twice the actual size, in general; if there are n symmetry planes, the effective sizeYLis 2 times the actual size.2.4 Convergence TestsTo test the convergence and accuracy of the method, a problem for which the analytic solution was known was solvedusing the relaxation method. The problem is the one used by D. Nelson1 3  as a test case; it con­sists of a 64 x 64 x 3 2 point "box" with boundary values of zero on all sides except the k = 32 surface where the potential isV = sin 2iri64 s^n2-nj64The sequence of operations carried out in solving this problem is given i n Table II.The first question which must be answered is how many iterations are required on the reduced problem. To answer this, several runs were done.For each run n sweeps were done with a  = 1.5, n with a  = 1.3 and n with a  = 1.0 (a total of 3n sweeps). The value of a  used was reduced from 1 . 5  (close to the best value) to 1 . 0  to ensure that the difference equations are solved as exactly as possible when the iterations are finished. The problem was then expanded to full size, and two iterations were done on the full volume. All iterations were done with alternating directions. The results are summarized in Table III. In all cases the error is very small. Since iterating over the small volume is relatively fast, there is no large penalty paid for over-estimating the number of iterations required, and n = 100 was chosen. For this case (n = 100), the average change per itera­tion before expanding was < 1 0 -6, i.e. the reduced problem had been solved exactly to the precision of the arithmetic used. Thus the error of 0.25% after expanding is due to the expansion process.Now the problem was expanded back to full size 64 x 64 x 32 points, and the convergence of this problem was investigated. Since we are doing- 15 -TABLE IISequence of operations used to solve a 64x6Ax32 node relaxation problemStep OperationProblem size after this stepNumber of i terat ionsAverage change per iteration after this step1 set boundary conditions 6b x 6b x 32 - -2 reduce to coarse grid 3 2 x 3 2 x 1 6 - -3 i terate (a  = 1 .5 ) 3 2 x 3 2 x 1 6 1 0 0 0 . 9  x 1 0 " 3b iterate (a  = 1 .3 ) 32 x 32 x 1 6 1 0 0 0 . 6  x 1 0 “ 55 i terate ( a  = 1 .0 ) 3 2 x 3 2 x 1 6 1 00 0 . 5  x 1 0 “ 66 expand to fine grid 6b x 6b x 32 - ?7 i terate (a  = 1.5) 6b x 64 x 32 see Fig. 2.3 see Fig. 2.3TABLE IIIAverage error after various numbers of iterations over the reduced problemCase n Average error {%)1 25 0 .bO2 50 0.303 75 0.31b 1 0 0 0.255 200 0 . 2 8- 16 -the iterations over subsets of the problem each containing 3 2 x 32 x 1 6  points, the best value of a  is, from Eq. (11), =  1.75. Two othervalues of a were used, 1 . 8 7  because this is a^ for a problem containing 6b x 6b x 32 points and 1.50 for reasons discussed below. Fig. 2.2 shows the average error as a function of the number of iterations for these values of a .  The discrepancy in the error after two iterations over the fine grid between Table II and Fig. 2.2 is due to different iteration directions where iterating over the fine grid. As expected, a =  1.75 pro­duces the fastest convergence, but the convergence is satisfactory in all three cases. Eq. (A.23) predicts that the number of iterations required to reduce the error by a factor / is n = l^og A~‘ va^ue \n aPProPr'_ate to a  = 1.75 is 0.75; hence the number of iterations required to reduce the error by a factor of 1 0 is-1  = 8.0.logi0 (0. 75)As can be seen from Fig. 2.2, about AO sweeps over the data are required to achieve the same reduction ( a =  1.75). Since the problem contains b x b x b blocks of data, by Eq. (15), each sweep corresponds to8(4-1) (4-1) (4-1) „ _------------------- = 3.375.(4) (4) (4)iterations. Hence the 40 sweeps correspond to 135 iterations, indicating that the convergence is about sixteen times slower than the theoretically expected rate for ordinary successive over-re 1axation. This slow conver­gence rate is probably due to the way in which the iterations are done,i.e. many iterations over a small subset of the total volume. However, with the good starting values provided by the reducing and expanding procedure, the convergence of the problem is acceptable.For practical problems it has been found by the author and by D. Nelson1 3  that a  = 1.5 gives the best results. This is probably due to the fact that, in practical cases, fixed points occur within the volume. This means that the "wavelength" of the errors will be smaller than that assumed in Eq. (A.8 ) , leading to smaller values of . Since a  = 1.5 seemed to be best for "real" problems and since a  = 1.5 still gives accept- convergence for the test problem, only this value was studied further.- 17 -Fig. 2.3 shows the average error and the average change per iteration for the a = 1.5 case. The bars on the points giving the average error indicate the error at which the number of points vs error curve (Fig. 2.4) has fallen to half Its peak value. As we would expect, since we are using a > 7 ,  the change per iteration is always larger than the error. Of course, there may be a few large local errors which do not produce a large average error.Fig. 2.4 shows the distribution of errors for various numbers of iterations. The graph is actually a histogram; the vertical lines give the (approximately equal) intervals in which the numbers of points are counted. Several points are worth noting. Even after many iterations, about 0.04% of the points have errors larger than 5 -0%, despite the average error being less than 0.04%. It appears that this situation will not change signfi- cantly even if many more iterations are done. It seems that the large errors must be removed before the smaller ones are affected. This is shown more clearly in Fig. 2.5 where the number of points with a given error is plotted as a function of the number of errors. It can be seen that the number of points with small errors remains relatively constant until the number of large errors has been reduced.2.5 The Practical ProblemProblems which are useful in practice usually contain many more points than the case discussed in Section 2.4. The same reducing and expanding procedure is followed, so that the starting values for the iterations on the large problems are quite good. However, since the number of points is larger, the convergence will be slower [as predicted by Eq. (A.22)], and each iteration will take longer.The practical case discussed here is a 128 in. by 32 in. by 8 in. section from the centre of the TRIUMF cyclotron. The 8 in. dimension is in the axial direction and extends from the cyclotron median plane to the vacuum tank. The 128 in. dimension is along the dee gap, and the 32 in. dimension is perpendicular to the dee gap. The geometry in the median plane and the electric equipotentia 1 s calculated using this method are shown in Fig. 1.1. The geometry in the axial direcion is shown in Fig. 1.2. It was felt that a 0.25 in. grid size adequately defined the boundaries;- 18 -hence the problem contained 5 1 2  x 1 2 8 x 32 = 2 ,0 9 7 , 1 5 2  data points.The sequence of operations used in solving this problem is given in Table IV. At the end of step 6 , the change per iteration at each point is less than 1 0 -7, so after expansion to 2 5 6 x 64 x 1 6 points, we would expect the average error to be about 0.40% as it was in the test case. The errors are of course unknown, but the average change per iteration at the beginning of step 8 was about 0.1%. The reason for this value being smaller than the value for the test case is probably that there are more fixed points in the real case. After 75 iterations over the 256 x 64 x 16 problem, the average change per iteration is less than 10"6. The itera­tions on the full-size problem (step 1 2 ) are very costly since we now have over two million data points; however, very few iterations are required. Step 12 consisted of four iterations over the full volume to smooth out any bumps left by the expanding process. The average change per iteration at the end of step 12 was less than 0.01%. Local errors will be larger than this, of course. In the test problem the largest errors were more than 1 0 0 times as large as the average error but only for 0.04% of the points; hence in this case we can expect local errors of the order of 1 or 2% at a very small number of points. However, the convergence of the problem is very satisfactory. As can be seen in Fig. 1.1, the equipoten- tials have no unexpected kinks and fit the boundary conditions extremely wel 1 .- 19 - TABLE IVSequence of operations used to solve a 5 1 2 x 1 2 8 x32 node relaxation problemStep Ope rat i onProblem size after this stepNumber of i terat i onsAverage change per iteration after this step1 set boundary conditions 5 1 2 X 1 2 8 X 32 - -2 reduce to coarse grid 2 56 X 64 X 16 - -3 reduce to coarser grid 1 2 8 X 32 X 8 - -4 i terate (a = 1 .5 ) 1 2 8 X 32 X 8 1 00 0.3 x lO" 35 i terate (a = 1.3) 1 2 8 X 32 X 8 1 00 <io-76 i terate (a = 1 .0 ) 1 2 8 X 32 X 8 1 00 < 1 0 ' 77 expand 2 56 X 64 X 1 6 - ?8 i terate (a = 1 .5 ) 256 X 64 X 1 6 25 0 . 2  x 1 0 _lt9 i terate (a = 1.3) 2 56 X 64 X 1 6 25 0 . 2  x 1 0 " 61 0 i terate (a = 1 .0 ) 256 X 64 X 1 6 25 0 .1' x 1 0 - 61 1 expand to ful 1 size 5 1 2 X 1 2 8 X 32 - ?1 2 i terate (a = 1.5) 5 1 2 X 1 2 8 X 32 4 < 1 0 “ 4- 20 -3. AXIAL MOTIONS3 . 1  Int roduct ionThe axial motions of ions within a cyclotron are influenced by three effects: magnetic forces due to slope, flutter and spiral of the magnetic field, electric forces due to the lens effect of the dee gap, and spacecharge forces due to the electric field produced by the beam. The magneticforce is small and focusing at the centre of the machine. The electric force is very strong and phase dependent (focusing for some phases and de­focus ing for others). The space charge force is weak and always defocusing. It will be shown that the axial motions during the first few turns are controlled almost entirely by the electric forces.Since one of the main design objectives of TRIUMF is to accelerateions over a wide interval of the RF waveform (i.e. ions with large diff­erences in their initial RF phase) careful design is required to prevent loss of those ions which start at unfavourable phases. The improvements which can be achieved by "squaring" the radio-frequency waveform (by adding a small fraction of third harmonic to the fundamental) will be demonstrated.Obviously, the axial displacement of the beam must not exceed the aperture of the dees but a more stringent limit on the amplitude of the axial oscillation is set by the fact that passage of the beam through regions where the forces are not linear causes distortion of the beam emittance. This causes a decrease in the "effective density" of the beam within the elliptical contour enclosing the beam's phase space. Recent work by Han1 7 indicates that about 60% of the dee aperture is linear to within 5%.k The axial motions must be adjusted so that as wide a range of phases as possible is transmitted. In addition, the beam must be matched to the magnetic field so that the amplitude of the axial oscillations is minimi zed.3.2 Magnetic FieldThe axial restoring force (F ) exerted on an ion by the magnetic fieldcan be expressed in terms of the axial oscillation frequency (vz)m which an* C. Han integrated the equations of motion numerically through fields for three gap geometries. For a gap height of 1.6 in. and gap widths of 3 . 0  in., 6 . 5  in., and 7-4 in., the deviations from linearity were less than 5 % over 6 0% of the gap height.- 21 -ion would have in the absence of other forces in the axial direction. The oscillation frequency is related to the force by2F_ = -m a) 2 v 0 2 ,mwhere (i>o is the ion rotation frequency. This is a linear approximation valid only for z «  g, the magnet pole gap height. In a sector-focused cyclotron the oscillation is given1 8 approximately by2v3 = -\i' + F2(l + 2 tan2e) . (16)m-\i' = describes the radial variation in the magnetic field. The azi­muthal variation of the magnetic field is described by the flutter function F2 whereF2 = { ( B - B ^ / B 2 .B is the mean field at a given radius and the angular brackets denote athe cyclotron the magnetic field iswhere f is themean at one radius. Near the centre o given to a good approximation by B = B amplitude of the sixth harmonic component (= Bg/B) and 0^ is the azimuth angle of the peak field. If the azimuthal variation of the field is related to one harmonic only, the flutter function F2 is related to the amplitude of the harmonic f by F2 = if2 . The angle e is the so-called "spiral angle" defined by tans = r d Q ^ d r . For an isochronous field, \x' is positive and, near the centre of the cyclotron, the spiral angle e is zero, hence the focusing provided by the magnetic field is due only to the "flutter" in the field. If the net effect of the magnetic field is to be focusing, the term due to the flutter must be larger than the defocusing term due to the field slope. Unfortunately, it is difficult to obtain large flutter in the central region of a cyclotron magnet because the vertical magnet gap is much larger than the horizontal distance betweenpole pieces. A typical plot of fv2) as a function of radius is shown inmFig. 3-1 • We plot (v2) rather than (\>„) since (v2) is proportional to3 m s m 3 mthe force exerted on the ion. Some cyclotrons use a magnetic "cone" inthe central region to increase focusing. This consists of a central "bump"on the isochronous value of the field. This means that \i' is negative,- 22 -hence the magnetic focusing is increased. In addition, phase slip is intro­duced due to the fact that the field is not isochronous. The ions must be started at positive (late) phases so that they have slipped into phase by the time they have reached the isochronous field region. The positive phase histories are advantageous from the point of view of electric focusing.In the absence of "squaring" of the RF, it will be shown that the phase acceptance has a sharp cutoff at - 5  deg, i.e., only phases more posi­tive than this can be accepted. A small field bump could be used to shift these positive phases into isochronism so that the range of phases which is accelerated is centred about 0 deg. It is shown in Section 3.8 that this field bump does not contribute appreciably to the focusing.A field bump will not be required when addition of third harmonic to the RF shifts the lower limit of the phase acceptance from -5 deg to -25 deg. Field bumps are undesirable for three reasons.Firstly, the ions start off phase and since the error in centring depends on the cosine of the largest phase angle, the centring errors are i ncreased.Secondly, since the ions start off phase the energy gain is reduced. Because of the relatively high injection energy, this makes the clearance between the centre post and the beam small on the first turn.Thirdly, the field bump will cause the beam to pass through the = 1 resonance (when \i' - 0) possibly leading to an increase in the radial oscillation amplitude.Another way to increase the focusing is to increase the flutter. This may be achieved by cutting three of the pole pieces at a radius of 30 in., giving a three-sector geometry in the central region. This produced a v of about 0 . 2  from r = 10 in. outwards; however, the large flutter with three-fold symmetry caused undesirable effects in the radial behaviour of the beam (see Section 4) and had to be abandoned.Tests have also shown that a set of "floating" pole pieces 1.66 in. above and below the median plane between 1 2 . 5  in. and 3 0 . 0  in. radius,with six-fold symmetry, can also provide (\>„) - 0.2 in the central region.^ mHowever, it is virtually impossible to mount pole pieces in such a posi­tion without disturbing the alignment of the resonator hot arms.- 23 -At this time it seems that the best magnetic focusing that can beachieved is that shown by the solid curve in Fig. 3.1.3.3 Space Charge ForcesThe ions in the beam produce an electric field which exerts a force on each ion in the beam. This is the space charge effect. This effect can be analyzed by considering the force on an ion on the surface of a bunch due to the other ions in the bunch and the force due to the ions in other bunches Reiser1 9 has analyzed this problem; we find for the case of TRIUMF (400 kV voltage gain per turn and low magnetic field) that at low energy, more than 9 0% of the space charge force on an ion is due to the field produced by the other ions in the bunch. This force is directed radially outward from the centre of the bunch and can be written1 9F =4eowhere zm and A<}> are the maximum height and the length of the bunch in degrees of RF, respectively, v is the velocity of the ions, I is the average current, q is the charge on the ion, cq is the permittivity of free space,and G is a factor which depends on the height-to-width ratio of the beambunch.The vertical oscillation frequency produced by this force is' 21 = Q G J_ I  . .V2 4e mu2 1 2  A<j> v ' '  ' ' ^ •'oC' 0In the case of TRIUMF, the source will produce about 2 mA, hence without bunching we can expect 1/ A(f> = 5 \iA/deg. Since the axial focusing is four or five times weaker than the radial focusing, we can expect beam width-to- height ratios of the order of 0.5. For this value, the geometrical factor G is 4.8. Fiq. 3.2 shows how (v%J varies with energy for various values3 “ SOof J/A<1> and zm .A graph showing the variation of G with the gap and height of the dee can be found in the paper by Reiser . 1 9This force is roughly the same order of magnitude as the magneticforce and can be accounted for by using an "effective" magnetic v 2 which2-  2b -is the difference between the actual magnetic v and the space charge v for the beam intensity under consideration.3 .k Electric Lens EffectsThe importance of electric focusing effects in cyclotrons was recog-approximate expressions for the lens effects of cyclotron dee gaps using the symmetry properties of the electric fields and a description of theerties arose from two effects. As can be seen in Fig. 3-3, the first part of the gap is focusing, while the second part is defocusing. These would cancel exactly, except that(i) the field is changing, and (ii) the ion is accelerated.The deflection due to the field variation arises because the ion sees a different electric field in the second half of the gap than in the first. Since the first half of the gap is focusing and the second half is defocus- ing, there is a differential focusing effect. The change in z' = dz/dx due to this "field variation" effect is, to first order in qVQ/Ec ,where VQ is the dee voltage and E , r and <|> are the energy, radius and RF phase of the ion at the gap centre. This effect is linear in z and isthe field is rising (negative phases).The second effect is due to the ion spending less time in the secondproduces less deflection than an equal force would in the first half. This effect is always focusing and is given bywhere g is a numerical factor depending on the geometry.There is also a collimation term due to the fact that the forward momentum px increases while the transverse momentum pz remains constant;nized soon after the invention of the cyclotron. Rose1 developedfield derived by Kottler . 20 These studies indicated that the lens prop-08)focusing when the field is falling (positive phases) but defocusing whenhalf of the gap, hence the defocusing force in the second half of the gap(19)- 25 -however, this term disappears when the change in pz rather than z' is cons i dered.In addition, Rose predicts a change in axial positionA z = 1 - qV0 costp o • (20)Rose's analysis was extended by Cohen3 who used an electric field developed by Murray and Ratner.21 This more detailed analysis showed that the expressions developed by Rose are the first two terms in a series in Jl/F • More recently, the analysis has been further extended by Reiser.22 This most recent analysis includes the effect of the dee liner, i.e., c is not 00 (see Fig. 3-3) as was assu ed in the previous analyses. Reiser's expression for the deflection ishz' =    . , , 2F(a.b.c)c TOqV 0 cos2 4? T 2- QOS*CZ 0 (21 )where i s is the axial displacement of the ion when it enters the lens, N isthe harmonic ratio of the RF frequency to the ion frequency, and F(a,b,c) isa dimension less function which depends on the geometry (a, b and c are described by Fig. 3-3)-The linearity in z of the above expressions for the deflection and displacement permits an enormous simplification in the axial motion calcu­lations. However, this formula is based on the assumptions that the transit time of the particle across the gap is small and that the energy gain across the gap is much smaller than the incident ion energy. Since the electric forces become strong just where these approximations are likely to become invalid (i.e. at low energy), it is important to investigate the validity of this formula. For the case of TRIUMF, the RF operates at the fifth harmonic of the ion frequency (N=5) and the transit times are of theorder of 60 to 70 deg of RF on the first turn, so the small transit timeapproximation is not valid; however, the relatively high injection energy (300 keV) means that the approximation that the energy gain is small com­pared to the incident energy is reasonably valid after a few accelerations. Recently, Han23 has published a compilation of the focusing effects of cyclotron-like gaps for geometries applicable to TRIUMF. These resultsn\qVo* \ E o  ■- 26 -were obtained by numerically integrating the equations of motion through electric fields calculated using the relaxation method described in Section 2. The data given in Han's Tables 6-1 to 6-7 provide a relevant source of numerical results to compare with the theory. To allow compari­son of the electric forces to the magnetic and space charge forces, we canapproximate the focusing effects at the dee gaps by an equivalent (\>2)3 ewhich would give the same deflection over a half-turn.If Az' «  zQ/nr,’dz2 Fz d2z d2z 1 .Z = ---4- =  , , , = — — TT = —  A3 mu)2 u>2dt2 do2For the numerical results given by Han231do = - -  t\z'. (2 1 )v_IT F■ 2where F2 is the forward focal power of the lens.Fig. 3-^ compares values of (v2J obtained by exact numerical integra-3 etion with those obtained from Eq. (21) for various phases and energies. The agreement is better for negative phases than for positive ones. In all cases the analytic description given by Eq. (21) overestimates the strength of the electric forces. It should be noted that for TRIUMF the injection energy is 300 keV, and the ion energy after the first main gap crossing is about 600 keV, so the approximation is valid to within 1 5 % in the first turn. Hence we can use the expressions given in Eqs. (20) and (21) and obtain a reasonable estimate of the axial motions.It should be noted that the electric forces are much larger than the magnetic and space charge forces. In addition, the electric forces are defocusing for, roughly speaking, negative phases. This causes a sharp cut­off in the phase acceptance near 0 deg. This cut-off can be shifted to more negative phases by providing addition (for example, magnetic) focusing. To shift this cut-off to -30 deg at 500 keV, magnetic focusing equivalent|2 mthe electric fieldto a (v3)2 of (0 .3 ) would be required to overcome the defocusing effects ofThis sharp cut-off for negative phases is due to the field variation effect. The deflection due to field variation is proportional to sin<t>,- 27 -hence rapidly becomes large for negative phases. The focusing due to the energy change is proportional to oos2<p and is multiplied by a smaller co­efficient than the field variation term. The relative magnitude of these two effects is shown in Fig. 3.^. The maximum contribution to (v„)2 from the energy change term is given by the curve for <f) = 0. Hence the net effect of the electric field closely follows the field variation effect and is defocusing for negative phases.3.5 Calculation of Cyclotron AcceptanceSince the linear description of the electric lens effect is reasonably accurate, and the magnetic and space charge focusing can be described in a linear manner, we can track the axial motions of the ions using the matrix method for tracking beams as suggested by Penner.21+If the axial focusing frequency due to the combined effects of themagnetic field and space charge is , then in a region where isconstant, the axial motion will be given byz(s) = zn sos v „ 0 r eps sin v „ 0 (2 2 )z\) Sz'(s) = - sin v 0 + z ' cos v „ 02? S O  Swhere i s and zq' are the initial (axial) displacement and slope, respective­ly, 0 = s/r is the azimuthal angle subtended by the ion, s is the path length, and r is the radius of curvature of the ion.We are dealing with low energies so we can use a non-re 1 ativisticenergy-momentum relationship2' - s t  (23>where pz is the axial momentum, E is the kinetic energy and k = ^2mQ . Itis convenient to measure momenta as where 3 and y are the usualrelativistic factors and is the "cyclotron radius" (= m o/qBc) . This momentum is numerically equal to the radius of curvature the ion would have in the central magnetic field (B ). For the TRIUMF central region B = 3.0 kG, 3yr (in.) = 18.94 /£ (MeV).- 28 -Using Eq. (23), we can write the magnetic transfer matrixPz~T008 V  V.JE sin v3 0—e—  s^n v3 0 cos v. PsI T= T,mk J(24)The expressions given in Eq. (20) and (21) can also be written in this form. If we call the transfer matrix for the dee gapthenT = ea = 1a bo do -q l o E°Nroos $qvo sin c}> + 2F(a3b3o)■nbqVr eos2<t>(25)(26 )where Ec and Ej? are the ion energies at the centre and end of the dee gap, respect i vely.The expressions derived by Rose and Cohen do not include any dependenceof the final position on the initial divergence, i.e. b is assumed to bezero. The numerical results given by Han indicate d - 1, and sinceLiouvilie's Theorem required ad - ob = 1 , we choosed = 1b = — (ad - 1) a(27)( 0 )The fact that b is non-zero means that there is a displacement term which depends on the initial slope (z^ '). The existence of this term is con­firmed in the numerical results given by Han; however, it is a small effect.Now that the transfer matrices for the various parts of the trajectory are known, the complete trajectory can be calculated by the usual matrix multiplication method:A method for analysing optic systems, much more powerful than trajec­tory tracking, has been developed by Steffen.25 This method allows the tracking of elliptical beam phase space areas through the system. It is usual to consider elliptical emittances, since ellipses can be specified by only three parameters, and make a good approximation to the actual phase space shape, which would presumably be polygonal.26 We will use Steffen's notation to describe the phase space ellipses. If the ellipse is described by the equationthen, as derived by Steffen, the maximum displacement and momentum arezmax = '/^^z WQX —     If we define the transfer matrix which transforms the vector(Z2 Z x= T ,P z 2 1 P z ik  k'v f2 3 z2= TP z 2 Pz_ll _±L_ k  k .ME S O NF A M1a ( 0)02 1a  z 3 z lk  k  o< J Pzthen Steffen shows that the ellipse parameters are transformed according toCMCO.a2 . Y2 .c2 -2cs s2-cc' OS'+SO' -ss 'c'l -2c's' s'2' *\P ia l. Y i  .This allows tracking of the beam ellipse through the system by multiplica­tion of 3 x 3 matrices.Since the electric and space charge forces decrease with increasing energy, the only important focusing force outside the central region is the magnetic field. At this point (where electric forces have become negligible) the beam must be matched to the magnetic field; that is, the amplitude of the axial oscillations must be minimized and a beam of uniform envelope ob­tained. For a constant v_ the phase space ellipse which minimizes z is2 r r r maxgiven bya = 0 m_ = constant • v.m 'mOnce the central region geometry has been decided, the transfer matrix from injection to the radius where electric forces are negligible can be calculated (T). Now if this matrix is inverted (T^), the phase space ellipse required at injection to provide a beam matched to the magnetic- 31 -Unfortunately, due to the fact that the electric forces are phase de­pendent, will be different for every initial RF phase. This means that the required initial phase space shape will be phase dependent; however, the initial phase space shape cannot easily be varied with phase. The best that can be done is to choose the initial ellipse shape for one RF phase and accept the fact that for other phases only those ions whose points in phase space fall within the chosen ellipse shape will be accelerated with 3 < zideal envelope’ This provides a method for calculating the phase acceptance of an accelerator. The ellipse shape required for one phase is chosen as the one to be provided; then the overlap of the ellipses for other phases with the chosen ellipse gives the acceptance of the accelerator for each phase.3-6 Phase Space Acceptance for Various TRIUMFCentral Geometries and Injection EnergiesEarly in the design of TRIUMF it was necessary to fix the injection energy and injection geometry. The original suggestion was that the injec­tion energy should be 1 5 0  keV; however, it was soon realized that the strong and phase-dependent electric forces would allow only a poor duty factor for this injection energy. Raising the injection energy would alleviate this problem and also reduce the spread in orbit centre points due to different energy gains for different RF phases, thus improving the radial beam qua­lity. However,a higher injection energy makes bunching, chopping and the design of the spiral electrostatic inflector more difficult. Thus we must investigate the axial motions to determine how the phase acceptance varies with injection energy and make a compromise between increased phase accept­ance and the difficulties mentioned above.Various initial orbit geometries had been suggested, ranging from the one injection gap case shown in Fig. 1.1 to the multi-gap case shown in Fig. 3.5.At the time of these studies it was hoped that a three-sector magnetic field could be used in the central region. Model tests showed that this three-sector geometry produced a magnetic v„ of about 0 .2 , and so the axial motion studies were done with this value. The three-sector magnetic geometry was later replaced by a six-sector geometry for reasons which are- 32 -explained in Section 4. The six-sector geometry produces much smaller values of v at small radius (see Fig. 3-1); however, tests with smaller v values show that the conclusions reached here are still valid even withmuch reduced values of v .zMost of the geometries studied had posts defining the first two gaps.It was estimated that these posts would reduce the electric forces by a factor of 4, and this is included in the calculations.*For each geometry the transfer matrices and the ellipse shapes required at injection were calculated. Fig. 3-6 shows the ellipses required for one geometry. The cyclotron acceptance was calculated as discussed above. For comparison purposes, the overlap of the ellipses with the ellipse for 20 deg is used, since this is approximately in the middle of the acceptance interval and gives as good matching with other phases as is possible.Fig. 3*7 shows the overlap with the ellipse for a phase of +20 deg for injection energies of 150, 306 and 472 keV with one 100 keV gap in the first turn. The sharp cutoff at about -15 deg is due to the defocusing action of the electric field. Fig. 3*8 shows the overlap with the ellipse for a phase of +20 deg for injection energies of 120, 286 and 454 keV with three 100 keV gaps in the first turn. This geometry gives no significant improve­ment in the phase acceptance, and the multi-gap geometry worsens the radial centre point spread and complicates the resonator design; hence multi-gap geometries were abandoned.The data for the one-gap geometries is summarized in Fig. 3-9- The average acceptance (averaged from - 3 0  deg to +60 deg) seemed to flatten out above 3 00 keV, and this seemed to be the highest reasonable energy from the point of view of bunching and inflection, so it was decided to raise the injection energy from 150 to 300 keV.Now after it was decided that a three-sector magnetic geometry could not be used, the code was rewritten to accept a magnetic v varying with radius, and the calculations for 300 keV injection energy were repeated using the much smaller v measured on the six-sector magnet model (Fig. 3*1)** The study by Han2 3 indicates that the presence of posts in the dee gap actually reduces the focusing forces by a factor of six or seven.- 33 -Fig. 3-10 shows the acceptance as a function of phase for this case for various choices for the initial ellipse. If the 20 deg ellipse is chosen as the axial phase space shape of the beam at injection, over 9 0% of the beam would be accepted at phases greater than 1 0 deg while the fraction of the beam accepted would be 64% at 0 deg, 3 0% at - 1 0  deg and zero for phases less than -10 deg. For this choice of initial ellipse shape the amplitude of the beam envelope does not exceed 0 . 5 5  in. for phases between - 1 0  deg and 60 deg.3.7 Effects of Third Harmonic in the RF on Axial MotionsThe unique features of the TRIUMF resonators allow the addition of higher odd harmonics to the fundamental mode . 2 7 If the third harmonic of the fundamental is added, the resonators operate as a 3A/4 cavity as well as a A/4 cavity. We now have an RF voltage given bythe phase of the third harmonic with respect to the fundamental. Small positive values of e are required to square or "flat-top" the fundamental.A fraction e = 1/9 produces perfect flat-topping at 0 deg, while more third harmonic than this produces a slight dip in the total voltage at 0 deg (see(Eqs. 25 to 28) to reflect the fact that the accelerating voltage is given by Eq. (29) instead of a pure cosine waveform. The field variation term given in Eq. (18) is essentially proportional to dV/dts>, i.e. to the rate at which the field is changing, and the energy change term given in Eq. (19) depends on the square of the energy gain. Hence with the RF voltage given by Eq. (29) we haveV = VQ aos (J) - e aos (3$ + 8) (29)where e = (amplitude of third harmonic)/ (amplitude of fundamental) and 6 isFig. 3.11).Now we must modify the formulae describing the lens effects of the gaps2- e cos(3$ + 8) (30)and[cf. Eqs. (25) and (26) for no third harmonic.]-  -As before,d = 1 b = -fad - 1). oThe negative limit on the axial phase acceptance is determined by the most positive phase for which the total force acting on the ion is defocus­ing (see Section 3-8). The electric force is defocusing due to the field variation effect when the field is increasing, since the (always focusing) energy gain effect is much smaller than the field variation effect. Hence we want to choose e and 6 so that the negative of the slope of the voltage (av^ due to the field variation effect)71 (  T / ' l= +sin <f> - 3e sin(3$ + 6) (32)d r 7 1-©-1 w_d_ ’ v 'dtp k Jfor various values of e and 6. The widest interval where -rr J- remainsa<P170positive is produced by 6 = -10 ± 2 deg and e = 0.15 ± .01; however, when 8 / 0 ,  the presence of the beam causes coupling between the first and third harmonics in the resonators, so that the third harmonic becomes detuned increasing, by a large amount, the power required to maintain the third harmonic voltage. These coupling effects are not yet fully understood, so we will consider two cases, 8 = 0  and 8 / 0 .  When 8 = 0  the limit on e isf V]set by the value which causesVoto become negative; e =* 0.17, forexample, produces a "hole" in the phase acceptance for 5 deg < 4 1 < 2 5  deg71d(see Fiq. 3-13) due to the fact that — 7—atj) is negative there. With6 , the maximum value of e which can be tolerated is e = 0.12 ± .01. Thephase acceptance produced by this value is shown in Fig. 3.1**. If we allow non-zero values of 6 , the best choice is 6 = -10 deg, e = 0.15. The phase acceptance for this case is shown in Fig. 3.15.The optimum values of c and 8 wi11 depend to some extent on the final details of the magnetic field, the current being accelerated and on the RF system, since the amount of power required to keep the peak voltage at 100 keV increases as e increases; however, the use of third harmonic in the RF appears to shift the cut-off in acceptance due to electric defocusing from - 5  deg to - 2 5  deg.- 35 -These conclusions are based purely on approximate analytic formulae and should be confirmed by numerical orbit tracking, i.e. by integrating the equations of motion numerically through three-dimensional electric and magnetic fields.3.8 Effects of Field BumpsBy a field bump we mean here an increase in the magnetic field above the isochronous value. The usual procedure in cyclotrons is to make the bump largest at the centre of the cyclotron and decrease with radius. This produces additional axial magnetic focusing due to the negative field gradient [see Eq. (16)]. In addition, the bump causes the phase of the ions to change, since the magnetic field is no longer isochronous. The change in the sine of the phase angle is given by Smith and Garren28SB is the field bump, BQ is the central magnetic field, AE is the energy gain per turn, N is the harmonic number, and q and m Q are the ion charge and mass, respectively, and the constant is appropriate to the TRIUMF cyclotron. If SB > 0, the ions "catch up", i.e. positive phase (late) ions (favourable for electric focusing) are brought into phase as the energy increases and the electric focusing becomes less important than the magnetic flutter focusing. In a conventional cyclotron with the RF not operating at a high harmonic of the ion frequency, a carefully chosen field bump can be of great help in overcoming the electric forces, since the two effects mentioned above both help to increase the useful phase acceptance.For the case of TRIUMF, the operation of the RF at the fifth harmonic of the ion frequency means that the electric focusing is very strong, and the two advantages mentioned above are reduced. For example, a bump of 25 G at 10 in. diminishing to zero at 25 in. gives the required phase shiftof about 30 deg and an equivalent v2 due to the field gradient of 0.01. Asscan be seen from Fig. 3-^, however, this is much smaller than the force produced by the electric fields, and hence would have only a small effect on the phase acceptance. Larger field bumps cannot be used because (i) they produce more phase shift, causing ions to be shifted to a phase where the electric forces are defocusing, and (ii) ions starting at large positiveA (sin <j>) = SBvdr =mn AE . 6858 G'in.2 .1 SBrdr. (33)- 36 -phases will not gain enough energy to clear the centre post on the first turn. A field bump can, however, be used to shift the range of phases which is accepted (-5 deg to +25 deg) to a range which is centred about 0 deg. This must be done carefully so that none of the useful phase range is shifted to a phase where the electric field is defocusing before the magnetic focusing is strong enough to make the total focusing positive.Fig. 3-16 shows the total effective v2 produced by the magnetic and electric fields. It can be seen that the sharp cut-off in phase acceptance at ~5 deg demonstrated in Fig. 3.10 is caused by the fact that ions with phases more negative than -5 deg experience a force which is defocusing at about 1.5 MeV. Since Fig. 3.16 shows at what energy the total focusing becomes positive as a function of phase, we can calculate how the phase of the ions should be "programmed" so that the ion is brought into isochronism as soon as possible but not subjected to defocusing forces. Fig. 3-17 shows the phase at which the focusing becomes positive as a function of energy. An ideal magnetic field would produce no phase gain out to 1.5 MeV; then it would cause the phase of the ion whose phase was -5 deg at 1.5 MeV to become more negative, as shown in Fig. 3-17- The amount of phase gain desired is determined by the phase range to be accepted. If we aim to accept all ions with phases between -5 deg and +k5 deg, the amount of phase gain is given by A sin <j> where, after the phase gain has taken place,sin(-5°) + A s£n<j> sin(45°) + A sin§This gives A sin $ = -0.31, and the final range of phases is ±23.4 deg. The phase history shown in Fig. 3-17 is produced by the A sin <j> shown in Fig. 3-18. The field bump required to produce this variation in A sin <f> is shown in Fig. 3-19. A bump with such a sharp cut-off cannot be produced inpractice; however, a bump with the same SBrdr as the one shown inFig. 3-19, and which shifts the phases no faster than the bump shown in Fig. 3-19, could be usee this bump will decreased. It should be noted that the positive slope of2by about 0.005. Another disadvantage is thatmthe ions will spend 5 or 6 turns far from the optimum phase. This maycause a large spread in centre points to develop unless the radial starting conditions are carefully chosen. This problem is considered in Section A.Of course, with third harmonic in the RF, the negative phase limit due to electric focusing is -25 deg and a field bump is not required.- 37 -k. RADIAL MOTIONS 6. 1 Int roducti onThe central region of the cyclotron must be designed with two general objectives in mind. Firstly, the geometry of the electrodes must be arranged so that ions with the desired range of phases can be accelerated without hitting the electrodes. Secondly, the central region must produce a beam which is centred to within the desired tolerances.We have shown in Section 1 that the motion in the median plane and the axial motion are independent to a good approximation. This section will discuss motion in the median plane only.Fig. 1.1 is a section through the median plane of TRIUMF. The beam is injected down the axis of the cyclotron, then bent into the median plane by the spiral electrostatic inflector. The problem of the inflector is dis­cussed elsewhere.29 We will assume that at the exit of the inflector we have a mono-energetic beam whose shape in phase space is a free parameter. The fact that the RF operates at the fifth harmonic of the ion frequency allows the "injection gap" to provide an extra 100 keV acceleration on the first turn. This eases the geometrical problems somewhat but causes the co-ordinate of the orbit centre point, perpendicular to the gap, to vary with phase. After reaching the first main gap, the ions are accelerated and spiral outward as in an ordinary two-dee cyclotron. The main geometri­cal constraint is clearance of the centre post on the first turn. Ions more than A5 deg from peak phase will hit the dee and be lost; however, centring requirements limit the acceptable phases to a range smaller than this.The use of fifth harmonic acceleration means that transit time effects are large. This reduces the energy gain at low energy. To alleviate this situation, the dee gap is tapered in both the horizontal and axial direc­tions (see Figs. 1.1 and 1.2) so that the electric fields are compressed and the energy gains are increased. In addition, the injection gap and first main gap are defined by posts which compress the electric field further and decrease axial focusing effects, as described in Section 3- These refinements make the geometry quite complicated and necessitate numerical tracking of the orbits at least out to the radius where the taper ends (30 in. or about 3 MeV). The orbit tracking was done using a slightly- 38 -modified version of the computer program PINWHEEL.30 The magnetic fields were obtained from measurements on a 1:20 model magnet, and the electric fields were calculated using the methods described in Section 2.4.2 Basic DesignThe first ion orbit in the cyclotron is shown schematically in Fig. 4.1. After leaving the inflector the ions travel, under the influence of the magnetic field only and with centre of curvature S, until they reach the injection gap. At the injection gap the ions are accelerated, and hence the centre of curvature changes. In addition, if the centre line of the injection gap is at an angle to the beam, the ions are deflected. Since the energy gains and deflections are phase dependent, the centre points and radii of curvature will be different for different phases. The ions now travel, again under the influence of the magnetic field only, to the first main gap where they are again accelerated. Due to the phase-dependent effects at the injection gap, the radius and RF phase at which the first main gap is crossed will depend on the initial RF phase and so will the centre points. In designing the central geometry it is desirable to choose the position and orientation of the injection gap so that as wide a phase interval as possible clears the centre post on the first turn and is close enough to being centred to be useful.As far as the placement of the injection gap is concerned, the quanti­ties of interest are the radius, RF phase, energy and angle at which ions with various initial phases cross the first main gap.To get a first order description of the effects we can use the approxi­mation that the energy gains are instantaneous and give the ions 9 3 . 0  cos<f> keV at the injection gap and 174.5 cos<}> keV at the first main gap. These values are based on the results of numerically integrating ion trajectories through the gaps. We will also use a non-relativistic expression for the radius of curvature of the ion which, for a 3 kG magnetic field, isv(in.) = 0.60 jE(keV) = 18.94 SE(MeV). (34)Since in most cases we will be interested in differences between ions with different phases, we will label the ion whose centre of curvature is (a j0) [i.e. its centre of curvature is on the centreline of the dee gap]- 39 -by the subscript 1. We label an ion at some other phase by the subscript 2. Quantities referring to the injection gap are further labelled with the superscript 'v , while those referring to the first main gap have super­script mg.The geometry of the orbit near the injection gap and first main gap is shown in Fig. 4.1. If the ion reaches the injection gap making an angle 3 to the gap axis, then for a peak dee voltage VQ an ion with charge q experiences a force -qVV aosty1'9 aos3 along the orbit and a force - q W  oosif19 sin$ perpen­dicular to the orbit. The injection gap causes a straightening or "collimating" deflection, given byA = tccn~l Ap sin3P + Ap cos$ ||r a o s ^ 9 sin$ (35)where p and Ap are the initial momentum and momentum gain, respectively. The approximation is valid if Ap << p. Because of radial centring considera­tions, two quantities of interest are the radius and angle at which the ions cross the first main gap, or the differences in these quantities for different phases. The length a is given bya2 = v^2 + r22 = 2r1r2 aos (Az - A^) (36)and the angle E byE = sin~1 (A2 A l^ (37)The radius difference at the first main gap (&r) is given by&r = a cos(E-\p) - + /r>2 2 + a2sin2 (E-\p) . (3 8 )The angle at which the ions cross the centreline of the gap is given byC = sin~l 3 sin(E-\\))^ 2(39)The RF phase at which the ion reaches the main gap is given by§mg = + 5 0  + A 2 - A l - C) . (40)The co-ordinate of the centre point perpendicular to the dee gap is given byy = -a sin(E-\p) . (41)- h o ­lt appears from Eqs. (35) and (3 8 ) that if $ -A 0, the radius and angle at which the ion crosses the main gap can be varied with phase. This wouldbe useful because it provides a method of improving the "match" between theorbit starting conditions provided by the injection gap and those required for centred orbits. However, because of the posts, the injection gap acts as a lens (similar in properties to the lenses studied in Section 3)* Han23  has studied the properties of a lens similar to the injection gap and found that it is convergent for positive phases and divergent for negative phases. The deflections due to the lens effects are larger than those produced by placing the injection gap at an angle to the beam. This effect has been confirmed by numerical orbit tracks through electric fields with the injec­tion gap at various angles. Since radial centring considerations require rm& for both positive and negative phases to be less than r 71^  for zero phase, slanting the injection gap to the ion path does not improve the centring andwill not be considered further.Now with 3 = 0, Eqs. (35) to (41) are much simplified and will be stated again:A 1 = A 2 = °rn AE 1^= r-, - Vr, - -2- ■—  (cos(j). - cos 1 z 2 1(42)E = 08r = a cosip - r + *v2 + a2sin2\p (43)sin C = ^2 Hl simp = simp (44)yc - (r1 - r2J simp = -r2 sin C (45)r A E^9 . .- — Q. stmp (costf>i - costp2Jo ts<jm  = f a  + 5\p - 5C. (46)The subscript 0 refers to quantities before the injection gap is reached.In the central region of TRIUMF the radius of curvature is given by Eq. (34).The quantities Aff^ and E are 93 and 300 keV, respectively; hence ther A El3constant appearing in Eq. (45) is -2- — — = 1.61 in.o ts- 41 -The first order choice for the available parameters i s <^ = deg and p f a  = 0 deg, i.e. the ion with zero phase at the injection gap has yc = 0 after the injection gap. If we use this geometry, then the centre points predicted by Eq. (45) are as shown by the solid curve in Fig. 4.2. The energy gain calculated as 92 oosp^y + 174.5 oosi?m9 , with <pm9 given by Eq. (46), is shown by the solid curve in Fig. 4.3. As expected, the centre points for all ions which gain less energy than the ion with zero phase lie above the centreline of the dee (positive values of y ). The asymmetry in the energy gain is due to the fact that negative phases are favoured by this arrangement, which delays all non-zero phases. Consider two ions which reach the injection gap with phases = ±20 deg\ the energy gains will be identical here but the ions will arrive at the main gap at ±20 deg - 5C, as predicted by Eq. (46). Since C is about -0.55 deg for this case, the ion with phase -30 deg at the injection gap will reach the main gap at -27.25 deg and the ion with phase +30 deg at the injection gap will reach the main gap at +32.25 deg. The phase at the main gap as a function of phase at the injection gap for this case is shown by the solid line in Fig. 4.4.The different values of y for various phases are inherent in the use of the injection gap at an angle to the main gap. This centring is unde­sirable because it leads to a phase oscillation. An orbit with radius of curvature v and off centre by an amount yc must turn through an angle of it + 2yG/r between dee gap crossings. This means that the ion will arrive at one gap early by 10yc/r deg of RF phase and late at the next gap by the same amount. Fig. 4.5 shows the magnitude of this phase oscillation as a function of y for various values of r. Fig. 4.2 indicates that we can expect values of yQ of about 0.10 in. for an ion with phase of +30 deg. This leads to a phase oscillation amplitude of 4 deg at a radius of 14 in. (the radius of the first turn). The existence and order of magnitude of these phase oscillations are confirmed by the phase histories shown in Fig. 4.6. These phase histories are from a numerically integrated orbit. The +30 deg ion has an oscillation amplitude of about 4.5 deg, in good agreement with the expected value. The asymmetry between positive and negative phases in Fig. 4.6 is probably due to the zero phase ion not being exactly centred.The phase oscillation damps out as the energy (and hence r) increases. The magnitude of the centring errors (and hence the phase oscillations) can be reduced by centring the spread of y ' s about the centreline of the dee- 42 -instead of having all the yc of one sign, as was assumed for the solid curve in Fig. 4.2. This is achieved by moving the injection gap closer to the main gap without changing its orientation. To centre the spread of y 's for a phase interval of ±A<j>, the phase of the ion whose yQ value is zero isSo, if we wanted to centre the yc 's for a phase range of ±45 deg, we woulddashed lines in Figs. 4.2 and 4.3. In order to make yQ zero for a phase of31.4 deg, the injection gap must be shifted 0.12 in. (see Fig. 4.2) closer to the main gap. The maximum phase oscillation is now reduced to aboutgap [given by Eq. (46)] is now less than 180 deg; hence we are shifting theto be defocused in the axial direction on subsequent turns. In fact it is useful to reduce the angle ip (i.e. rotate the injection gap towards the main gap about the centre point for -^). This reduces the energy spread for positive phases; for example ip = 32 deg produces the energy gain curve given by the dotted line in Fig. 4.3. The maximum is shifted towards positive phases because reducing ip reduces ipm& [see Eq. (46)]; hence positive phases gain more energy. The reduced energy spread alleviates the problems of centring and clearing the centre post on the first turn. However, as can be seen from Eq. (46) and the dotted curve on Fig. 4.4, reducing ip to 32 deg causes a large phase shift (about 23 deg) towards negative phases. The small shift towards negative phases required to centre the spread of yc 1s is tolerable, since it produces a large improvement in the centring; however, reducing ip to, say, 3 2 deg produces an unacceptably large shift towards negative phases. Hence ip must be chosen so that the ion with phase ifa [see Eq • (47)] has yQ = 0 after passing through the injection gap. The radius of the injection gap is fixed because the injection energy is fixed; hence the injection gap position is determined.4.3 Problems with Three-Sector Magnetic Fieldssin— L<pfa 1.Ad> 2 (47)choose dtfa = 31.4 deg. The produces the yG and AE values shown by the±5 deg rather than ±10 deg. The phase change between injection gap and mainions towards negative phases for ions with \<p'l^ \ < 31.4 deg where they tendAs demonstrated in Section 3, the lower limit on the phase acceptance-  -is set by axial focusing requirements. The acceptable range of phases can be increased if a phase-independent focusing force can be found to counter­act the defocusing effects of the electric field. The only phase- independent source of axial focusing is the magnetic field; hence effortswere made to increase (\>„) near the centre of the machine. Increasing2 m arequires that the "flutter" of the magnetic field be increased. Un­fortunately, the central geometry of TRIUMF makes this very difficult because the magnet gap is large and there are six sectors, making the spacing between the magnet sectors small at small radius. One way of in­creasing the flutter is to transform the field from a six-sector geometry to a three-sector geometry in the central region. This is done by cutting off alternate magnet pole pieces at r = 40 in. and adding to the remaining pole pieces steel wedges (see Fig. 1.2) extending to the centre of the cyclotron. This produces a field which is dominated by the third harmonic rather than the sixth. This "three-sector geometry" produced a considerable improvement in (^z)m , as is shown by the dashed line in Fig. 3.1* With the three-sector geometry, (^z)m is greater than 0.1 for v > 10 in. However, the large third harmonic caused undesirable effects in the radial orbit behavi ou r.There are two effects caused by the three-sector geometry, an increase in phase oscillation amplitude and the gap-crossing resonance. Which of these effects is most important depends on the orientation of the electric field to the magnetic field. We define this orientation by the angle 6 shown in Fig. k .1.The phase oscillation effect results because, if 6 * 0, the orbit covers 2 valleys and 1 hill on one half-turn and 1 valley and 2 hills on the next half-turn. Hence, the lengths of the orbit on successive half­turns are different, as can be seen in Fig. k. 7. If the n ^  harmonic dominates the variation in the magnetic field, the orbit equation may be written in the approximate form (e.g. Walkinshaw and King31)1 + — —  ^ 2- cosnB n2-l Bmwhere rQ is the radius of the (circular) orbit if the field had no azimuthal variation, Bn is the amplitude of the n harmonic in the field and B is- 44 -the average field. For the present case with n=3,r = v (1 + a cos3Q), 1 B 3where a = _  — 1 .8 BNow we wish to calculate the path length s between dee gaps, we have(49)Us ing Eq.(49)dsd&dr 2dQ\+ rA1 + a aos 30The approximation which has been made is that B 3 << 5 B. Hence, over one half-turn we havesi0 = 6+771 S 30=6(1 + a oos3Q)dQ = tt + — dJ-sin36>12 B(50)and over the following half-turn'0 = 6 + 2 7 72. B(1 + 3 cos3Q)dQ = tt  --- =^ - sin.3812 B3 =  6+77(51)5 B 3 .So, between successive gap crossings the phase oscillates by —  -=^s^nS6i CI(since the RF operates on the fifth harmonic of the ion frequency).The variation of this phase change as a function of 6 is shown in Fig. 4.8 for a third harmonic amp 1itude (B ^  which will produce = 0.2 Phase histories for a numerical orbit track corresponding to the worst case (6 = 30 deg) are shown in Fig. 4.9. The amplitude of the phase oscillation is about 8.5 deg for the zero phase ion (for which the phase oscillation would be zero without the three-sector magnetic field). This is in reasonable agreement with the theory. Since any phase oscillation such as this will decrease the duty factor,32 6 must be small, i.e. the centreline of the dee gap should be close to the line running from a hill top at 6 = 0 to the opposite valley bottom at 6 = 180 deg. To keep the amplitude of the- 45 -phase oscillation less than 5 deg, we must have <5 < 16 deg. The effect of this phase oscillation is important here because the RF operates at the fifth harmonic of the ion frequency. It has been dismissed as unimportant for three-sector cyclotrons operating with N = 2.33This phase oscillation effect can be eliminated by placing the dee gap along a hill-valley centreline (6 = 0 in Fig. 4.7). Unfortunately, this orientation maximizes another undesirable effect, the gap-crossing resonance. This is essentially a shift in the orbit centre points along the dee gap caused by a larger magnetic field at one dee gap than the other. This effect has been discussed in detail by Gordon,33 but we can make an estimate of the effects as follows. Referring again to Fig. 4.7, we can refer to the dee gap on the right side by the subscript l and on the left side by the subscript 2; then the magnetic fields at the gap areB 1 = B + B 3 cos 6 ,Z?2 = 5 “ # 3  cos 6 .(52)Since these effects are important at low energy, the radius of curva­ture can be approximated byp (in. ) = l ^ l j  ^E(MeV) .If the increase in energy at the dee gap is AE MeV, the change in radius of curvature at the gap, assuming the ions always cross normally, isAp (E) = 56.92B JE + A E - JE (53)The radial position of the ion does not change appreciably as the gap is crossed, so the change in radius of curvature is reflected in a change in the position of the centre of curvature. As the ion alternately crosses gaps 1 and 2, the centre of curvature oscillates back and forth approximate­ly along the centreline of the dee. If the magnetic fields are different at the two gaps, there is a net drift of the centre of curvature towards the higher field, given by6p = 56.92k 7 ry —• ba bodd ii+1k+1 ,-  I Aeven i(54)- 4 6 -where E. 7 = E. + EE.; i is the half-turn number and EE. is the energyIs'T'l. 1s 1s 1sgained at the i**1 dee gap.Using Eq.(52) and the fact that B^/B «  1, this can be expressed as56.92 \6p = LB i-l 2VEi L B J(55)The first term in the square bracket is the displacement of the orbit centre from the cyclotron centre. This term oscillates, hence its sum depends on the differences of the EE's. The second term in the square bracket is the centre point drift due to the third harmonic component in the magnetic field. This term always has the same sign, hence will accumulate rapidly if B 3 is large. B 3 varies widely with radius (see Fig. 4.10); hence the sum depends on the magnetic field used. Numerically summing the series for the values of B. shown in Fig. 4.10, and using EE. = 0.2 MeV at all gaps, produced a3 1scentring error of 0.3 in. Numerical tracking of ions through the measured magnetic field using the computer code GOBLIN gave a centring error of about 0.5 in. for this field. Eq. (55) shows that the centre point drift due to # 3 is proportional to cos38 and hence could be eliminated to this approxima­tion by choosing 8 = 30 deg. This means that the dee gap runs along a hi ll- valley interface, but this is unfortunately the situation which produces the large phase oscillations discussed above.The drift in centre point could be reduced by putting a first harmonic in the magnetic field. The first harmonic causes the centre point to drift and could be arranged to cancel out the drift due to the gap-crossing resonance, as has been described by Gordon33 and van Kranenburg et al. 31+ However, producing a first harmonic varying accurately enough with radius would be extremely difficult and necessitate special coils or shimming of the magnet. In addition, the compensation is exact for only one RF phase.In summary, aligning the dee gap along the centreline of a hill (or valley) produces a phase oscillation of about 10 deg. Aligning the dee gap along a hi 11-valley interface excites the gap-crossing resonance causing a centring error of about 0.5 in., which can be only partially cancelled by a first harmonic in the magnetic field. Orientations between the two described above do not bring the phase oscillation and the centre point- 47 -drift within acceptable limits, and hence the three-sector magnetic field has not been adopted.4.4 Radial CentringThe central region of a cyclotron must produce a beam which is centred at extraction. By centred we mean that the oscillations of the orbit centre point approach the geometric centre of the machine as the energy increases. In TRIUMF the large energy gain per turn and low magnetic field produce large oscillations of the centre point at low energy. The centre point at injection must be off centre by about half the radius gain per half-turn (see, e.g., Gordon33) if the orbit centre point is to approach the centre of the machine as the energy becomes large. This centre point displacement required because of the acceleration can be derived in a manner similar to the derivation of Eq. (55). If we assume circular orbits, then the change in centre of curvature at one gap isIf the energy gain per gap crossing (hE) is << the energy E and the change in radius per gap crossing (hr) is << the radius r, we can approximate Eq. (56) by a differential equationxo-1 ~ xei+2 pi ~ pi+land at the next gapX°i+2 " X°i+1 ~ Pi+1 " Pi+2'Hence over one turn the change in centre point is(56)dxc _ hE d2p(57)dE 2 dE2 ’Now if the centre point d s at infinite energy is zero, i.e. the beam is centred, integrating Eq. (57) once yields/  -r-7 » Jijkxc (E) = —hE dp _ rm n u '(58)2 dE 2 m 0a2 By3- 48 -where p and 6 are the usual relativistic factors and rx is the cyclotron radius = mQo/qBQ (-410 in. for TRIUMF). The right-hand side of Eq. (58) is just one-half the radius gain per half-turn at energy E. The above estimatehigh energy. However, at low energy where the geometry is complicated by the presence of the injection gap, we must resort to numerical orbit tracks to optimize the centring.The determination of what constitutes a centred orbit is complicated by several factors. The azimuthal variation of the magnetic field causes scalloping of the orbit; hence the instantaneous centre point depends on the azimuthal angle. The average orbit radius and maximum scalloping are shown in Fig. 4.11 as a function of energy. &  C-The quantities we will mainly be concerned with in this section are r, the radius of the ion from the geometric centre of the cyclotron, and p , the radial momentum. The momentum will be written in the formIn these units, the momentum of the ion is represented by its radius of curvature in the central magnetic field (B ). The radial momentum is that component of the momentum which is directed in the radial direction, i.e.where tanX = dr/rdd. In the central region, the flutter in the magneticapproximate B by Bc and p 0 by p; then the component of the centre point perpendicular to the dee gap (yQ) equals pr at 0 = 6, i.e. the dee gap.The essential features of the central orbits of TRIUMF are shown in Fig. 4.12. The magnetic field has six-fold symmetry. The centreline of the dee gap is 5.5 deg from the centreline of a valley. The azimuthal angle 0 is measured from the centreline of the dee gap as shown.One way to remove the complicating effects of orbit scalloping, and to determine how close the orbit is to an ideal centred orbit, is to calculate, at some azimuthal angle, the difference between the radius and radialprovides a good starting point for finding central orbits, especially atp = 3dr r . ,,Pr = P ^ = P ~ = P  nxfield is small, hence the orbit scalloping is small, and we can roughly- b s  -momentum of the actual accelerated orbit (a.o.) and an equilibrium orbit (e.o.) at the same energy. An e.o. is a fixed energy orbit which closes upon itself, has average centre of curvature at the centre of the machine, and is stable for small displacements in radius and momentum. The e.o.'s are calculated by the program CYCLOPS.* Now for any energy at one azimuth,we know the radius and radial momentum (r and p„ ) of the equilibriumeoorbit. Hence, when tracking an a.o., we can calculate the differences inradius and momentum between the e.o. and the a.o. at this azimuth. If anorbit is to be centred at the final energy, the differences between the e.o. and the a.o. during acceleration (i.e. Ar = rao - reo and Apr = pr pr ) are due to centre point displacements along the dee gap only (changes in xQ) due to acceleration. Hence, as the energy increases and the changes in x0 decrease, the values of Ar and ApT (due only to a non-zero value of xQ ) will decrease, and the a.o. will approach the e.o. The locus of the point (Ar3App) in phase space on successive turns (at one azimuth angle) as the acceleration proceeds forms an "accelerated phase plot". The gross features of the accelerated phase plot depend on the amount the instantane­ous centre point of the a.o. differs from the instantaneous centre point of the e.o. (Axc and Ayc in the x- and y-directions , respectively) and on theazimuthal angle at which the accelerated phase plot is calculated.Suppose at some angle 0Q the a.o. has energy E, radius raQ and radial momentum P^ao- We interpolate in a table of equilibrium orbit radii and radial momenta values for azimuth 0q to obtain reo and pr&0 > which are the radius and radial momentum of the equilibrium orbit at energy E. Now if we neglect the variation in the magnetic field along 0Q between reo and ra Q ,then the radii of curvature are the same, i.e. peo =  Pao = p , and we havethe situation shown in Fig. A.13- The angle x will be small since pr is much less than p and Apr will also be much less than p, hence we can approxi mate the arc p(Ap/p) by a straight line and the centre point components are related to the differences in radius and radial momenta byaos(0O - x) sin(Q0 - X)Ax0 -sin(QQ - X) aos(Q0 - x;pAppP(60)Ar •* CYCLOPS was kindly made available to TRIUMF by Dr. M. Gordon of Michigan State University.- 50 -Thus the accelerated phase plot removes the "motions" in the orbit centre point due to scalloping of the orbit and allows the actual errors in centring to be determined. Fig. 4.14 shows t\xQ calculated using Eq. (60) [using Ar and Apr values from a numerical track of a centred orbit] compared to the values of Axc predicted by Eq. (58). \xG \ is plotted rather than xcto allow comparison of the curves for 0q = 54.5 deg and 0q = 234.5 deg. Thevalues of xQ are all negative for 0O = 54.5 deg and all positive for0 Q  = 234.5 deg. The agreement is fairly good; however, the only way to dothe final optimization of the centring seems to be to work backwards from the centred orbit. That is, we start an ion on a centred orbit at high energy and numerically track it backwards into the centre of the machine. If we do this for several RF phases, we will know what the starting conditions should be if ions with various phases are to be centred. Using a typical magnetic field (01-03-06-70), ions with various starting phases were tracked backwards into the centre of the machine.The procedure which is used for tracking orbits is as follows. For energies less than 5 MeV the program PINWHEEL is used. This solves the relativistic equations of motion using measured magnetic fields and electric fields calculated by the method described in Section 2. For energies greater than 5 MeV the program GOBLIN is used. This program solves the relativistic equations of motion using measured magnetic fields but approxi­mating the effects of the electric fields by the "impulse" approximation described in Section 5. The transition is made at 5 MeV because above this energy there is no significant radial variation in the field across the dee gap, while for energies below 5 MeV there is such a variation because of the tapering of the electrodes.The ions were started at 20 MeV at the centreline of a valley (Q = -5.5 deg) , with Apr = 0 and with Ar equal to one-half the turn separa­tion per half-turn, as indicated by Eq. (58). The accelerated phase plots at 0 Q  = 54.5 deg and 0 Q  = -125.5 deg (see Fig. 4.12), i.e. at the centreline of a valley, are shown in Figs. 4.15, 4.16 and 4.17 for ions with starting phases of -30 deg, 0 and +30 deg, respectively. For the ideal case where yG is always zero and xQ becomes zero at high energy, then Eq. (60) shows that Apr and Ar values will always lie on the straight line passing through Ar = 0 and Apr = 0 and at an angle of ir - 0 Q  to the Av = 0 axis. This is- 51 -the straight line shown in Figs. A.15, 4.16 and 4.17. Using Eq. (60) and the data shown in Fig. 4.16, the values of xQ shown in Fig. 4.14 were calcu­lated. Extrapolation of this curve down to 0.4 MeV (the energy of the beam between the injection gap and the first main gap) indicates that the beam should be off centre by about 1.32 ± .05 in. at this energy. Since the radius of curvature of the beam is 11.88 in., this means that the radius at which the first main gap should be crossed is 13-20 ± .05 in. Accelerated phase plots for three different choices of radius at the first main gap crossing are shown in Fig. 4.18. The arrow on Fig. 4.18 gives twice the radial oscillation amplitude. The curve for v = 13.20 in. clearly leads to the smallest amplitude radial oscillation. To determine what happens to other phases an ion was tracked backwards from r = 13.20 in. at the first main gap through the injection gap, into the centre post, providing initial conditions for outward tracks. Using these initial conditions, trajectories were followed outwards for various phases, producing the phase plot shown in Fig. 4.19- The -25 deg ion gives a radial oscillation amplitude of about0.5 in., while the +25 deg ion gives an amplitude of about 0.8 in. These oscillations are much too large, as they would lead to a very large energy spread at extraction. In order to achieve more than a very narrow phase band, the starting conditions must be adjusted to favour ions which start with phases other than zero. Since the difference in the accelerating con­ditions which causes the large oscillation amplitudes to develop is essen­tially the energy gain, which varies roughly as aos <j>, it is reasonable to centre an ion whose phase corresponds to the average cosine in the phase band to be accelerated.Since in the absence of third harmonic in the RF we are restricted essentially to positive phases, we will choose starting conditions so that various phases are centred and observe how this affects the magnitude of the radial oscillations. Fig. 4.20 shows accelerated phase plots for ions with the same phase range as in Fig. 4.19 but with starting conditions chosen to give centred orbits for starting a phase of +17 deg. Phase plots such as shown in Fig. 4.20 were calculated using starting conditions to give centred orbits for initial phases of +15 deg, +17 deg, +19 deg and +21 deg. The results are summarized in Fig. 4.21. For a phase range of -5 deg to +25 deg the amplitudes of the radial oscillations are minimized if an ion with initial phase of 15 deg to 17 deg is centred. If a small amplitude of- 52 -oscillation were desired (and a small phase width could be tolerated), onewould choose the case where the 0 deg ion was centred.To first order the energy resolution obtainable in the extracted beam is related to the radial oscillation amplitude by the energy gain per turn. For the case of TRIUMF, the maximum energy gain per turn (400 keV) produces a 0.064 in. increase in radius at 500 MeV. When operating with a wide phase spread, the beam will be spread out fairly uniformly with radius, sothat a ±0.064 in. oscillation will worsen the energy resolution by ±400 keV(or alternatively ±0.1 in. will produce ±600 keV).Eq. (58) shows that the xQ required to allow for centre point motions due to acceleration is proportional to AE and hence is also proportional to eos<t> since AE - qVQ oosty. Therefore, if the zero phase ion is centred athigh energy and has centre point x ^  at injection and x at some otherG Genergy, an ion with some other phase C<pJ will have a centring error (1 - costylXgl at injection. If this centring error did not alter the behaviour of the centre point motions with energy, we would expect this ini­tial centring error to produce a centring error (1 - aos<j>) (x % 1 - xj^i) atenergy E^. The dashed line in Fig. 4.21 shows this centring error as afunction of phase at injection. Fig. 4.21 shows that the oscillation ampli­tude is much larger than this, so some mechanism is causing this centring error to produce a large amplitude radial oscillation. One such mechanism is described in Section 4.5.Of course, the ion beam will contain particles with various displace­ments and divergences from the central ray, and we must investigate how the beam as a whole is centred. This is discussed in Section 4.5.In Section 3*8 it was shown that a field bump could be used to shift the acceptable range of phases so that the accepted phase interval iscentred about zero degrees. Fig. 4.22 shows the phase histories for fourions in a field which has the bump described in Fig. 3-19 added to it. As expected, the initial phase interval of 0 deg to +50 deg is shifted to about -21 to +25 deg. The dashed line shows the theoretically expected phase shift for the ideal bump. Note that the phase change is never faster than ideal, so no ions which are initially focused are shifted to defocusing phases. Accelerated phase plots for ions with various starting phases in the magnetic field with the bump added are shown in Fig. 4.23. The starting- 53 -conditions are adjusted to favour the +17 deg ion (as in Fig. 4.20 without the field bump). The oscillation amplitudes for phases of 0 deg, +15 deg and +30 deg are 0.13, 0.12 and 0.42 in., respectively, while without the bump they are 0.20, 0.14 and 0.46 in. Thus the effect of the bump is to slightly decrease the oscillation amplitudes in this case. We would expect (from Fig. 4.21) that large positive phases would have very large oscilla­tion amplitudes, and this is confirmed in Fig. 4.23, which shows that the ion that starts at +45 deg is unacceptable. This undesirable behaviour for large positive phases is not significantly improved if we arrange the start­ing conditions to favour the +21 deg ion. Note that the radial centring requirement effectively sets a positive phase limit of about +25 deg, so that the field bump used (designed for a phase interval of -5 deg to +45 deg,see Section 3*8) is too large. However, the effects of the bump on theradial motion are small.4.5 Effects of Finite Beam EmittanceNow we will consider how the centring varies over a beam with a realistic size. The expected emittance of the TRIUMF ion source is 0.50 LL in. mrad (at 300 keV). We will assume that this is not significantlyincreased by the transport system up to the point of injection into thecyclotron dees. To minimize the amplitude of the radial oscillations we wantto choose the initial ellipse shape to match the radial focusing, asdescribed in Section 3 for axial focusing. Since the lens effects of the dee gaps are small, we first try matching to the magnetic focusing, for which v - 1.0 in the central region. To see how the emittance is trans­formed as the beam is accelerated, four particles were tracked, starting on the edge of the emittance ellipse. Figs. 4.24(a), 4.25(a) and 4.26(a) show accelerated phase plots for these four points for initial phases of 0 deg,+15 deg and +25 deg, respectively. As can be seen from these figures, the ellipse is "stretched" as the acceleration proceeds, producing a large amplitude radial oscillation. This is due to an effect explained by Mackenzie.35 Briefly, the effect is important in this case because of thelow field and large energy gain per turn causing the initial orbits to befar from the equilibrium orbits. Consider the trajectories in phase space of +()> and -(f>. An initial displacement from the origin (A r ^ 0 kJ Apr / 0) will cause precession through an angle of approximately (if is close- 54 -to 1) ■n(vp - 1) during a half -turn in the magnetic field. Since Ar ^ 0 or App ^ 0 means that the beam is not centred, the ions arrive at the next dee gap later or earlier than they left the previous gap (as described in Section 4.2). Hence the energy gain is not the same for the +$ ion as for the -<(> one. This means that on the next half-turn one ion will be closer to its e.o. than the other to its e.o., and while they both precess through the same angle, the -<j> ion will precess so as to reduce its displacement from the origin in phase space, while the displacement of the +<(> ion increases if vp > 1. The effect is reversed if vp < 1. These displacements in phase space cause "stretching" of the emittance ellipse, producing a large amplitude radial oscillation. This effect is important when the ion energy is small and when is different from one, so that the precession is large. Numerical orbit tracks have shown that the effect is unimportant above 10 MeV.The amplitude of these oscillations can be reduced by choosing a different initial ellipse shape. If, for example, we choose an ellipse which is reduced by a factor of two in the Ar direction but increased by a factor of two in the App direction from the ellipse that is matched to vp = 1 , we obtain the phase plots shown in Figs. 4.24(b), 4.25(b) and 4.26(b) for the same three initial phases as used previously. These phase plots show that the oscillation amplitude is reduced to 0.25 in. over the phase range 0 deg to +25 deg. This represents an effective increase by a factor of almost four in the oscillation amplitude due to the phase-dependent acceleration.- 55 -5. RADIAL LENS EFFECTS OF CYCLOTRON DEE GAPS 5 .1 Introduct i onThe calculation of radial motions in a cyclotron at low energies re­quires a detailed knowledge of the electric field produced by the dees. The calculation or measurement of this field is a difficult problem (see Section 2), and the numerical integration of the equations of motion through the field is a slow procedure. To integrate the equations of motion from injection to extraction would require a prohibitively large amount of com­puter time. It is therefore useful to have an approximate method of calcu­lating the radial motions. One way of doing this is to represent the radial motion as half-turns in a purely magnetic field separated by accelerating impulses induced by the electric fields at the dee gap. The magnetic field is approximated by an isochronous field with vp (the radial oscillation frequency) constant over each half-turn and determined by interpolation in the values computed by the equilibrium orbit code for the real field. The effects of the dee gaps are approximated by instantaneous changes in the energy (SE), RF phase (St), radial position (Sx) and angle to the gap (SE.) when the ion reaches the azimuthal angle of the centre line of the dee gap. Thus in a two-dee cyclotron such as TRIUMF the ion will pass through a 180 deg long magnetic field region (with v constant), then have its energy, radial position, RF phase and angle to the dee gap instantaneously changed as it crosses the gap, then pass through another magnetic field region and dee gap, etc. This section investigates various approximations which give the quantities describing the dee gap (SE, St, 6a: and SE,). The results from the approximations are compared to numerical orbit tracks through a real electric field.Since we have vr constant over each half-turn, we can approximate the radial motion in the magnetic field by a sinusoidal oscillation about the equilibrium orbit. There will also be a significant oscillation at the principal flutter frequency, but this will produce no change over 180 deg in a six-sector machine.We will call the amplitude of this oscillation Ar and the slope Pwhere p and pp are the total and radial momenta, respectively. The trans­formation of these quantities are given by an equation equivalent towhere r is the radius of curvature and 0 is the azimuthal angle in the magnetic field.duced by the dee gaps. Fig. 5-1 shows a typical dee gap. The radial motionof the ion is confined close to the median plane z - 0. Fig. 5-2 shows aplot of the instantaneous electric potential in the median plane. This figure suggests that a first approximation to the effects of the dee gap can be obtained by assuming that the gradient of the electric field is constant over some region and zero elsewhere.5•2 Constant Gradient Approximation with No Magnetic FieldWe assume the gap is as shown in Fig. 5-3, uniform in the x-direction, with a gap width of £ and a total voltage across the gap of VQ . At timet = 0, the ion is at x = xQ , y = 1/2 with velocity x = x Q , y = y Q ,(x = dx/dt, y = dy/dt) . The phase of the accelerating voltage at t = 0 is cj)0 and its frequency is id. The equations of motion areNow we need a description of the changes in momentum and position pro-x = 0 (62 )y = = k COS (itit + $Q) (63)where k = q and m being the charge and mass of the ion, respectively.f8=Integrating Eq. (62) givesx o (64)X = X0 + x 0t . (63)Integrating Eq. (63) gives(66)- 57 -In practical cases, the electrodes which produce the field are located above and below the median plane, and so the width of the field there is larger than the physical gap between the electrodes (as demonstrated in Fig. 5.1). We therefore treat £ as a free parameter to obtain the best agreement with numerically integrated orbits. The time required for the ion to cross the gap x (the transit time) is also as yet unknown. Within the validity of the approximation, the width of the electric field will depend only on the geometry, while the transit time will depend on the electric field and the velocity of the ion; hence we choose £ so that the constant gradient approximation gives the same energy gain as the numerical results for one case, i.e. one incident energy and RF phase. Now, using this value of £, we calculate the transit time x which is the value of x that solves Eq. (67) when y - y Q = -£, i.e.the Newton-Raphson method.36 Once x is known, the changes in x, y, x and y across the gap can be calculated. We will call the above approximation the iterative approximation, since it requires an iterative solution of Eq.(68) to find the transit time.Now, within the validity of the approximation, the value of £ found to be best in one case should also give the best results for other incident energies and phases. To select an appropriate value of £, the best method seems to be to compare the results of numerical integrations to the results predicted by the constant gradient approximation at high energy, where we expect the approximation to be most valid.Since the iterative solution of Eq. (68) may be time consuming, one is tempted to look for simpler approximations. If the transit time is small enough so that we can approximate sinxt by xt and oosxt by 1, then we obtain the "linear" approximationThis can be solved by any standard numerical technique, for exampley = yQ + kt Gos<j>0y =  y 0 +  y 0t(69)(70)- 58 -and the transit time isT=-f-. (71) In this case the assumption of small transit time is equivalent to assuming that the velocity of the ion is constant across the gap.A more exact approximation is obtained if we keep terms up to (ut)2 in the expansions of sinut and coscot; then we obtain the "quadratic" approxima­tion wh i ch isy = y0 + k^tcos<b0 - ft2 sin$Q (72)andy = y 0 + y o t  + f^2 oosi♦<>• (73)£The transit time is obtained from Eq. (73) when y = and is- /yZ2 ~+~Fir~l~aos^'(.k eos<b0(74)This approximation is equivalent to assuming that the ion velocity across the gap is the average of the initial and final velocities.A still better approximation can be obtained by retaining one more term in the expansion of sinut; then we obtain the "cubic" approximationy = y 0 + k tcos$0 - 112 sin$0 - ^ —  OOS(()0j . (75)The last term in Eq. (75), which was neglected in Eq. (72), is usually as large as the second last term in Eq. (75). In the cubic approximation we still calculate the transit time using Eq. (74).The validity of these approximations was tested by comparing the changes in x, x, y and y to those given by numerical integration through a real electric field. The numerical calculations solve the exact relativistic equations of motion. The various constant gradient approximations assume that the mass is constant across the gap; however, the mass used is the relativistic mass appropriate to the initial ion energy. The electric field used was that for the gap shown in Fig. 5.1, i.e. a total gap height of- 59 -4.0 in. and a total gap width of 6.0 in. In the TRIUMF cyclotron the field produced by a gap of these dimensions is reached at a radius of 40 in.(about 5 MeV) . The value of I was selected so that the iterative approxima­tion gave the same energy gain as the numerical integration for <f>0 = 0 deg and ts = 100 MeV. The value selected was 8.97 in., considerably larger than the physical gap width of 6.0 in. The gradient used for the approximation is shown by the dashed line in Fig. 5*2.In this case, there is no force in the rc-direction; hence px remains constant and displacements in the rc-direction are just xt. The change in py causes the energy of the ion to increase. We will express the energy gain by the so-called gap factor G,G = --7- • 100%, (76)qVQ eos<j>awhere AE is the actual energy gained by the ion and <j> is the RF phase at which the ion crosses the centre of the gap.Fig. 5-4 compares the energy gain obtained by numerical orbit tracking in the real field for <j>a = 0 deg with the energy gains predicted by the various constant gradient approximations. The small transit time approxima­tions give very much less accurate results than the approximation based on exact computation of the transit time, and hence they will not be considered further.Values of G for various phases and energies from the iterative constant gradient approximation and from numerical integration are given in Table V. Fig. 5-5 shows the differences between the values calculated by numerical integration and those from the iterative constant gradient approximation. In all cases, over a phase range of -45 deg to +45 deg and an energy range of 1 to 100 MeV, the differences are less than 0.5%.The errors displayed in Fig. 5.5 are inherent in the constant gradient approximation and not a result of an inappropriate choice of I, since chang­ing I merely displaces the family of error curves.The other quantity of interest is the transit time. In all cases, the transit time was within 0.1% of the expected time of £ / va where vg is the average of the initial and final velocities.Gap factors given by numerical integration and by the constant gradient approximation (no magnetic field)- 60 -PA A- O vO CM PACM vO OO O A CM< .— PA -d" vO CO d r^ .— • • • • • • •o CA LA r- CO A ALA CO CO CA CA CA A A■—J+ o CM PA r^ vO •—PA LA CO CO CO CM COCO .— d vO d vOz • • • • • • •CA CO LA CO A ACO CA CA CA A AvO CO d PA O sPA o A- O A CM< •— PA d vO OO d— • • • • • • •o CA LA OO A AO CO OO CA CA CA A Acr\+ o LA PA O A r^ .CO CM d LA CM d ACA CM PA r^ d vOz • • • • • • •CA OO LA CO A AA-» CO CA CA CA A APA CA CM CO O CO OvO d vO vO O CO CM< CA CM d vO CO d r^A CA LA 1^ CO A Aun CO CA CA CA A A+ LA CM CA PA vO ACA PA LA d vO OvO CA CM LA dz • • • • • • •CA CO LA CO A ACO CA CA A A APA A- O o vO r^ . ACO LA d vO A CO •—< vO — d vO dCA CA LA r- CO A AA- CO CA CA A A AI_>CM LA p— O CM A AF— O CA 00 vO r^ P-LA CA CM LA dz • • • • • • •CA CO LA CO A AA*. CO CA CA A A ACM d CM d OO PA p— LA A oo CM< PA O d vO d r^— • • • • • • •CA CA LA r^ CO A ALA A- CO CA CA A A A1 CM d _ O PA O-d- d o o CO A PACM co PA vO dz • • • • • • •CA CO LA oo A AA^ CO CA CA A A AO oo CM o-d" 00 CO d A oo CM< co CO PA vO d--- • • • • • • •oo oo LA r^ CO A AO A- CO CA CA A A APA1 P- LA CO CM CO CMCM •— CM O O d(A PA vO CO Az • • • • • • •OO CO LA OO A AA-. CO CA CA A A ACA vO CA PA O CO CMF— CM d PA A CO CM< PA PA vO d— • • • • • • •CO 00 LA 00 A ALA A- oo CA CA A A A“J1 CA CO •— PA PA A ACA p— LA vO PA CM LALA PA vO CO Az • • • • • • •CO CO LA CO A AfA oo CA CA A A A>CDL. > «— CM LA o O O Od) <U p— CM A OC r—LU 'Dif)a)CDoIA = results from iterative approximation- 61 -Thus in this case the effect of the gap can be approximated by instan­taneous changes at the centre line of the gap as follows: SE is the energychange appropriate to the velocity change given by Eq. (66), St and 6a; are zero and 65 is the change in angle due to the energy gain. This change in angle results because Py changes while px remains constant and is6^ = tan'-l PxPy + AP tan~l Py (77)where Ap is the increase in momentum given by Eq. (66), and p and p areX ythe initial momentum components.5.3 Sine Gradient ApproximationTo improve the results given by the constant gradient approximation, we should use an electric field which more closely approximates the real field. Fig. 5.2 suggests that a better approximation than a linearly varying poten­tial (constant gradient) might be a potential of the formVmmF = aos (ut + <)>„,> aos voTT, L , L (y ' 2} (78)where L is a wavelength which describes the electric field. If we define the average velocity v g =  + 0 (  (gap width SL ^ L, see below), the equation of motion is, in so far as the actual velocity can be approximated by the average veloci ty,y =  A:it c o s  ( to t  + <J>0 7 sin.. • „ „ 7 r v a t (79)Integrating Eq. (79) [assuming v is constant] givesandTYLLy -  y 0 =  —cos$0 - aos{t(u + ■nv-./L) + <I><Jaos (to + ffv a / £  t ( t o  -  irva / L j  + (|)0 ] costj>cto + ttv /Ld■ , kny -  y 0 =  y 0t  —)(1a )(81)sin$0 - sin\t(to + irva / L j  + <f>0 J t cos$0(to + irva /L)2- ) h h    / Lsxn t(to -  irva /L )  + 4>0 J - sin$n t cos§0 (to -  irva / L ) 2 to-Tvva /L- 62 -It was found that the results were improved if we allowed the gap width {a) to be less than the wavelength which describes the electric field (L). Thus we use only part of one cycle of the sine function to describe the electric field gradient. Of course, if £ < L, VQ must be increased toaos 2L (A. .^1=1  1.Jto maintain the same total voltage across the gap.Since vg and the time required to cross the gap are both unknown, we must solve Eq. (81) for the transit time x when y - y Q = -I. We define v a = SL/t F :  the transit time is given byfelT£ + y0x + vs'in§0 - sin(u>T + TtZ/L + <j)0J x aos4>,fu> + tt £/LxJ2 a)-/- mclyfym(82)sin(cox - 7T&/Z, + 4>0) - sin§0 x aostyQ (a) - itGfymSoF to - Tr£/Lx=This approximation was compared to the results of numerical integration through the real field as described above for the constant gradient approxi­mation. Over a phase interval of -45 deg to +45 deg and an energy interval of 1 to 100 MeV, the errors for the sine gradient approximation were several times larger than for the constant gradient approximation; hence the sine gradient approach was not pursued further.5-4 Constant Gradient Approximation with Third Harmonic in the Electric FieldSince it is planned to "flat-top" the RF voltage by adding a small fraction of third harmonic to the fundamental, it is useful to derive the constant gradient approximation for the case where the RF voltage is given by7Y~ = cosfui t  + <p0) -  e aos(3u t  + <pQ + 6). ( 8 3 )e is the fraction of third harmonic and 6 is the phase of the third harmonic with respect to the fundamental. The equation of motion in the zy-direction- 63 -i s nowy = k oos(u>t + <f>0J - e oos(3u>t + <j>0 + 8) mwhich when integrated givesand (85)(0  )The validity of Eqs. (85) and (86) have not been checked by comparison with numerical integration. However, it is reasonable to expect them to be at least as accurate as (66) and (67) when the RF voltage is not changing more rapidly than it does for the fundamental only. Fortunately, the phase region of interest is precisely where the waveform given by (83) is flat, so that Eqs. (85) and (86) should be of as much utility as (66) and (67).5.5 Constant Gradient Approximation with Magnetic FieldWe now consider the case where the ion being accelerated sees a magnetic field B perpendicular to its plane of motion and an electric field with constant gradient. The equations of motion arewhere as before k = qVQ/ml.If we assume that the magnetic field is isochronous, then the ion rotation frequency is constant and equal to qB/m. Integrating Eq. (87) once and using uj = NqB/m (the RF frequency), we obtain(87)and(88 )(89)- 64 -andy = k cos bit + <pr fi/ *o + J (y ~ y°} (90)The solutions to Eqs. (89) and (90) are, for N ^ 1,Nx = x0 - —u bik_ N2 0)yo + '± jzp.bit ,oosT~ ~ 1 (91)+ Zbik N Xo ~ bi 1-N2 UUlvB. bitsrn-j-k_ N bi2 2-.fi/2 sin (bit + <j>0) - sin<t>0x = k fi/2y o + a) 2-fi/2 sin§. s m,bitN (92)xn - — bit k N °os—  + —N bi 1-N2 COS (bit + <!>_) ,y = y 0 -Nx. + *-bi bi. k N2 . . bi 1-N2 n n *o. bitS™ T (93), N \ • k N bi r °  ' a) 2-fi/2 aos*obit , k N2 , , , , \oosT  + ^ 2 1H 2 oosUt +andy = fe fi/2y ° bi 1-N2 stn(t)o cos-_ U)t_ fi? (94)k N X0 -  u  7 - 7 1 7 2  0 0 8 + 0. U)ts'tnT~The transit time L is given by(0fc fi/2 . ,-*0 * yo + a, 2-fi/2 KV.. bit svn-j- (95)k NXo ' a) 2-fi/2 COS(j)0 cos-_U1Tfi/fe fi/2 , ^ ,+ 00sUx + V- 65 -Note that Eq. (95) takes into account the increase in path length due to the magnetic field.Cohen,3 Comiti37 and Reiser38 have derived equations similar to Eqs. (93) and (94), but they give no method for calculating the transit time nor any indication of the accuracy of their approximations.For the case without a magnetic field, displacements in the or-direction depend on px only, since the electric field produces no component of force in the x-direction. Similarly, when a magnetic field is present the change in p should be that due to the magnetic field only, as if the electric field were not present. That this is true is verified by the work of Comiti37 and by the results of numerical integration in the present case.The energy gains predicted by Eq. (94) and those obtained from numerical integration are given in Table VI. The value of £ (the gap width) used was the one chosen for the zero magnetic field case, i.e. 8.97 in.The differences in the gap factor G are shown in Fig. 5.6. The errors in this case are about ten times larger than for the case where no magnetic field was present. This is possibly due to the fact that the curvature of the ion path causes the ion to spend more time near the edges of the field where the constant gradient approximation is least accurate. However, over the region of interest {-30 deg < <j> < +30 deg and E > 5 MeV), the errors are still less than 1%. Fig. 5-7 shows the phase variation of the error in G. A more accurate description of the energy gain could be obtained by fitting some function to the curves shown in Fig. 5-7 and using this as a correction to the energy gain predicted by Eq. (94).The errors in timing caused by assuming that the change in energy occurs discontinuous1y at the centre of the gap are about 1.5 deg (RF) at -45 deg and 1 MeV, 0.2 deg at 0 deg and 1 MeV, decreasing rapidly with energy (<0.01 deg at 100 MeV).In the present case (with a magnetic field), there is an apparent dis­placement of the ion due to the change of radius of curvature. Thenumerical integration of the ion orbit is done over a distance d in they-6irection. If the radius of curvature before the gap is p1# and after the gap is p2 , the displacement we would expect from y = d/2 to y = -d/2, if the change in radius of curvature occurs discontinuous1y at the centre ofGap factors given by numerical integration and by the constant gradient approximation (isochronous magnetic field)- 66 -CA -d CA -di— co oo 00 o CA CM< o CM -3* vO oo -d r^— • • • • • • •o CA LA r^. oo CA CALA oo OO CA CA CA CA CA->4+ vo O LA CM r^. -d vOoo CA -d LA -d F— CMo CA vo LA o z • • • • • • CN o vO oo CA o OOO CA CA CA CA o oF— *—LA LA LA vO -d J CMvO vO o CA CM< (A CM -d vo 00 -d r—— • • • • « • •CA CA LA oo CA CAo OO CA CA CA CA CACA+ vO F— J LA O -d CAcvl CA LA CM -d .—o .— CM CM oo CAz • • • • • • •«— o vO oo CA CA CAoo CA CA CA CA CA CALA OO CA CA CA jCA O LA vO O OO CM< CM -d vO OO -d— • • • • • • •CA CA LA oo CA CALA r- 00 CA CA CA CA CA+ oo -d CA oo CA CM»— LA O o ' vO-d r*^ CA o vO ooz • • • • • • •o CA LA CA CA CAoo OO CA CA CA CA CAo LA _ OO CA o1— r— CA vO CA OO CM< LA ■- -3“ vO -d r>.— • ■ • • • • •CA CA LA r- oo CA CAOO CA CA CA (A CAV-J -d OO F— vO CA ooo CM •— CA UV CACM OO CA vO OO LA r^z • • • • • • •CA 00 LA 00 CA CAA- oo (A CA CA CA CA-d CA CM LA OO OCM CA O LA CA OO CM< •— CA -d vO -d r^.— • • • • • • •CA OO LA oo CA CALA oo CA CA CA CA CA1 CA vO CA CM oo ,— LACA vO r— LA F— J OOF— CA CA vO -d vOz • • • • • • •oo oo -d oo CA CAoo CA CA CA CA CA-d -d OO CA CA OOLA -d -d CA OO CM< vO oo CA vO -doo oo LA oo CA CAO<v\ oo CA CA CA CA CAt • i1 CA LA CA oo o -do CA LA CM CA vO A-o CA -d o CA CM LAz • • • • • • •r^. -d C". oo CA CAoo YR CA CA CA CAvO LA -d CA CMCM oo -d CA CA OO CM< «■— vo CA vo -d— • • • • • • •oo 00 LA r^ oo CA CALA> oo CA CA CA CA CA—j1 LA vO o o -d CAoo CA CA oo vO CM-d CA OO LA o O -dz • • • • • • •LA vo CA vO oo CA CAr^ oo CA CA CA CA CA>-O)L. > F— CM LA O o O O<1) <u »— CM LA OC z ■—LUD(/)<1)i—“O<DU)d)OIA = results from iterative approximation- 67 -the gap with no displacement along the gap, isAa: - p2 cos sin'1(d/2p0) (p2 “ Pi^ “ Pi aos sin'1(d/2p1)d 2 L  A.Ip i P2 (96)In Fig. 5*8 the values of Ax found from Eq. (9 6 ) are compared with the numerically-tracked orbits. The agreement is excellent. This means that Sx, the displacement of the orbit at the dee gap, is effectively zero, and the energy gain results in a displacement of the centre of curvature.Thus, in this case as in the case where no magnetic field was present, we approximate the effect of the gap by instantaneous changes at the centre­line of the gap as follows: SE is the energy change corresponding to thevelocity change given by Eq. (9*0, S>t and 6x are given zero values, and 6? is the "col 1imation" given by Eq. (77).5.6 Constant Gradient Approximation with Magnetic Field and Third Harmonic in the Electric FieldUsing the RF voltage given by Eq. (83) and including the effects of an isochronous magnetic field, the equations of motion arebi , ,X  - x 0 = y (y - y0)andH 1 ‘  , biy  + m y  + n0)N yo = k cos (bit + §Q) - E cos (3xt + <b0 + &)(97)(98)For N / 1, the solutions to Eqs. (97) and (9 8 ) are~2 sin($0 + S)x = bi N2sinfyc 3e1CM1* 1 1-9N2k N COS<t>n ebi . 1-N2 1-9N2sincosbit.N .f \bit(99)+  —  — COS (bit +  <j>.J - —   ^ - 5- COS(3bit +  c() +  S) ,bi 1-N2 bi 1-9N2 0- 68 -x - x Q = N_(0 y 0 + ~  n2CO ^  sin($0 + 8)+ *(01-N2 1-9N2k s \sesit1COS cot'- 1. s .aosC<t> + 8)_1_ kN u>2 1-N21-N2 1-9N2 ~ ~ ,r°s tn f c o t  +  <t>0 ) - sinif0cotsin1 zkN3co2 1-9N2 sin(3uit + <ft0 + 6) - sin($0 + 8)y = • + k N 2 (sin±ch° u  [1-N2o _ Se 5- sinftft. -t 8)1-9N2 0101eos.N .x n - -  N o tocos (ft n e. T-./V2 ~ 1-9N2 00s (<ft0 -f 8)' * cotsin uk N2 . , j_ , , , , 3zk N 2 . fr> , . r,svn(wb + tft0J -f ---   ^~nii7? s^n(3u>t + <ft0 -fa) T-iV2 y°' u 2_5]lj,2y  -  y 0 =Nxn . NCO (0 y n + n2 ^ - 5- sin($0 + 8)J - N 2 1-9N2 0 szncotJ7+ *■0)COS (ft 0 E /, r 1 ^  -  cosf<ftn -f 6J2-W2 0cos cotN+ -^r —^— 5- cos foot -f <pn) - — -— 5- 00s (3ut + <bn + 8)co2 2 - t f 2 0 co2 2 - S t f 2 0( 100)( 101)(102)These equations have not been compared to the results of numerical integrations, but the differences should be comparable to those quoted in the previous section over the region of interest.- 69 -6. SUMMARY AND CONCLUSIONSThe motion of the ions at the centre of a cyclotron has been studied with particular reference to the TRIUMF cyclotron.6 The object was to investigate the factors determining the phase acceptance and beam quality of the cyclotron, and to consider how the design might be adjusted to optimize these quantities.The calculation of both axial and radial motion requires knowledge ofthe electric and magnetic fields. The magnetic fields used were measured onmodel magnets. The electric fields were calculated by numerically solving Laplace's equation using the relaxation method. The convergence and accuracy of this method was investigated in detail. Numerically solving a problem for which the solution could be found analytically showed that the numerical solution contained average errors less than 0.01%. The method uses a novel feature to obtain accurate starting values for the iteration, and the solution time for a very large problem (2 x 106 data points) isabout 3 hours on an IBM 360/67.The axial motions were studied using the thick lens description of the dee gaps developed by Rose,1 Cohen3 and Reiser.22 A method was developed for calculating the axial acceptance of the accelerator as a function of RF phase. It was found that the axial acceptance exhibits a sharp cut-off at about -5 deg, i.e. ions with phases more negative than -5 deg cannot be accelerated. This effect results because, for negative phases, the field is rising, and the field variation effect causes the dee gaps to defocus the ions. This effect is more important for TRIUMF than for other cyclotrons because the RF operates at the fifth harmonic of the ion frequency, causing the electric forces to be much stronger. The negative phase limit can be shifted to more negative values by flat-topping the RF waveform. This flat- topping can be produced by adding some third harmonic of the RF frequency to the fundamental waveform. It is shown that addition of 12% of third harmonic in phase with the fundamental shifts the cut-off due to the field variation effect to about -15 deg. This situation can be further improved by adding 15% of third harmonic shifted 10 deg from the fundamental. For this case the negative phase limit is shifted to about -25 deg.The effect of field bumps is investigated. For the case of TRIUMF a radially decreasing field bump at the cyclotron centre cannot produce enough- 70 -axial focusing to overcome the strong electric forces. However, a carefully designed field bump can be used to shift the phases of the ions. It is shown how to design a field bump to shift those phases initially favoured by elec­tric focusing (positive phases) into phase with the peak of the RF voltage when electric focusing is less important. This is done without shifting the ions to phases where they are defocused by the electric field.The radial motions of the ions in the first turn were studied to find the best position of the injection gap. A position close to 36 deg back along the orbit from the main gap was found to provide the best centring and phase histories.To allow economical orbit tracking out to high energies, an analytic description of the changes in radial orbit properties on crossing a dee gap was developed. The results of this approximation differ from the exact changes (found by numerical integration) by less than 1% for energies above 5 MeV.The beam centring was studied by tracking ions through realistic electric and magnetic fields (to 5 MeV), then to 20 MeV by integrating through the magnetic field and using the approximation mentioned above. The results of these orbit tracks showed that the transformation of the radial beam ellipse is quite phase dependent. This may be reduced by reducing phase-dependent effects at the dee gaps. The energy resolution of the beam is investigated by tracking ions to 20 MeV. The radial oscillations present at 20 MeV are reduced by a factor of about 1.5 during acceleration to 500 MeV due to adia­batic compression. The finite emittance of the beam also worsens the energy resolution by ±300 keV.If the ion with initial RF phase of 0 deg is centred, large radial oscillations develop for ions with other initial RF phases. For example, if we require an energy resolution of ±600 keV, then half of this can be allowed to the coherent radial oscillations, meaning that the oscillation amplitude allowed is 0.05 in. at 500 MeV or 0.07 in. at 20 MeV. This allows a phase acceptance of 16 deg. For an energy resolution of ±1200 keV the phase acceptance is 26 deg. For the case where large duty cycle is required, the largest phase acceptance is obtained if an ion in the centre of the phase interval is centred (rather than the ion with 0 deg initial phase). For ±1200 keV energy resolution, for example, the phase acceptance can be- 71 -increased to -17 deg to +26 deg by centring the ion with initial phase of 17 deg.In summary, the axial motions place a positive limit >60 deg on the phase acceptance. The negative limit is -5 deg without third harmonic,-15 deg with 12% of third harmonic in phase with the fundamental, and -25 deg with 15% of third harmonic shifted by 10 deg from the fundamental. The radial motions allow a phase acceptance of -8 deg to +6 deg for an energy resolution of ±600 keV or -17 deg to +26 deg for an energy resolution of ±1200 keV, in both cases with RF fundamental only.- 72 -ACKNOWLEDGEMENTSI would like to thank Dr. M.K. Craddock for supervising this work and for providing guidance and helpful suggestions throughout the course of my studies at U.B.C. I would also like to thank Dr. G.H. Mackenzie for several helpful discus­sions and Miss Anne Koritz, Mr. Neil Fraser and the staff of the Computing Centre at U.B.C. for their help in writing and debugging the computer programs. Finally I would like to thank Miss Ada Strathdee for her patience and perseverance while typing this thesis.- 73 -REFERENCES1. M.E. Rose, "Focusing and Maximum Energy of Ions in the Cyclotron", Phys. Rev. (Ser. A A B HSP SGE .oGSuB2. D. Bohm and L.L. Foldy, "Theory of the Synchro-Cyclotron", Phys. Rev.(Ser. II) 72, 649 (1947)3. B.L. Cohen, "The Theory of the Fixed Frequency Cyclotron", Rev. Sci.Instr. 24, 589 (1953)4. M. Reiser, "Initial Acceleration and Radial Focusing in the Non-uniform Electric Field at the Ion Source of the Cyclotron", Michigan State University Cyclotron Project, Report MSUCP-16 (1963)5. W.B. Powell and B.L. Reece, "Injection of Ions into a Cyclotron from an External Source", Nucl . Instr. Meth. 32_, 325 (1965)6. E.W. Vogt and J.J. Burgerjon, editors, "TRIUMF Proposal and Cost Estimate" (1966)7. G.M. Stinson et al. , "Electric Dissociation of H- Ions by Magnetic Fields", TRI-69-1 (1969)8. T.E. Zinneman, "Three-dimensional Electrolytic Tank Measurements and Vertical Motion Studies in the Central Region of a Cyclotron", University of Maryland, Department of Physics and Astronomy, Technical ReportNo. 986 (1969)9. D. Vitkovitch, Field Analysis (D. van Nostrand, London, 1966) , 20510. V.P. Pronin and A.N. Safonov, "Measurement of the Electrical Field in the Central Region of the Dubna Synchrocyclotron by the Induced Current Method", Communications of the Joint Institute for Nuclear Research,Dubna, No. JINR-P9-4851 (1969)11. J.M. van Nieuwland, H.L. Hagedoorn, N. Hazewindus and P. Kramer,"Magnetic Analogue of Electric Field Configurations Applied to the Central Region of an AVF Cyclotron", Rev. Sci. Instr. 39., 1054 (1968)12. G.E. Forsythe and W.R. Wasow, Finite-Differenae Methods for Partial Differential Equations (J. Wiley, New York, I960)13. D. Nelson, H. Kim and M. Reiser, "Computer Solutions for Three-Dimensional Electromagnetic Field Geometries", IEEE Trans. Nucl. Sci. NS-16, 766 (1969)14. D. Nelson, "The Relaxation Code - Solutions of Laplace's Equation for Non-Analytical Three-Dimensional Geometries", University of Maryland, Department of Physics and Astronomy, Technical Report No. 960 (1969)15. D.W. Peaceman and H.H. Rachford, Jr., "The Numerical Solution of Parabolic and Elliptic Differential Equations", J. Soc. Ind. Appl . Math 3_, 28 (1955)- 74 -16. D. Young, "The Numerical Solution of Elliptic and Parabolic Partial Differential Equations" in M o d e m  Mathematics for the Engineer: Second Series, ed. E. Bedanbach (McGraw-Hill, New York, 1961) , 37317. C. Han, private communication, July 25, 196918. J.R. Richardson, "Sector-Focusing Cyclotrons" in Progress in Nuclear Techniques & Instrumentation, Vol. 1, ed. F.M. Farley (North-Hol1 and, Amsterdam, 1965) , 319. M. Reiser, "Space charge effects and current limitations in cyclotrons", IEEE Trans. Nucl. Sci. NS-13, 171 (1966)r~20. F. Kottler, "Elektrostatik der Leiter" in Handbuch der Physik, Vol. 12,ed. W. Westphal (J. Springer, Berlin, 1927), 46621. R.L. Murray and L.R. Ratner, "Electric Fields within Cyclotron Dees",J. Appl . Phys. 24_, 67 (1953)22. M. Reiser, "First-Order Theory of Electrical Focusing in Cyclotron-Type Two-Dimensional Lenses with Static and Time-Varying Potential", University of Maryland, Department of Physics and Astronomy, Technical Report No. 70-125 (1970)23. C. Han, "Computer Results for the Focal Properties of Two- and Three-Dimensional Electric Lenses with Time-Varying Fields", University of Maryland, Department of Physics and Astronomy, Technical Report No. 70- 126  ( 1 9 7 0 )24. S. Penner, "Calculations of Properties of Magnetic Deflection Systems", Rev. Sci. Instr. 32_, 150 (1961)25. K.G. Steffen, High Energy Beam Optics (interscience, New York, 1965)26. A.P. Banford, The Transport of Charged Particle Beams (E. Spon, London, 1965)27. K.L. Erdman et al., "A 'Square-Wave' RF System Design for the TRIUMF Cyclotron" in Proceeding, International Cyclotron Conference, 5th, Oxford, 1969 (Butterworths, London, 1971)28. L. Smith and A.A. Garren, "Diagnosis and Correction of Beam Behaviour in an Isochronous Cyclotron", Proc. International Conference on Sector- focused Cyclotrons 6 Meson Factories, Geneva, 1963 (CERN, Geneva 1963)29. L. Root and E.W. Blackmore, private communication (1970)30. M. Reiser and J. Kopf, "Electrolytic Tank Facility and Computer Program for Central Region Studies for the, MSU Cyclotron", Michigan State University Cyclotron Project, Report No. MSUCP-19 (1964)31. W. Walkinshaw and N.M. King, "Linear Dynamics in Spiral Ridge Cyclotron Design", Atomic Energy Research Establishment, Harwell, England, Report No. AERE-GP/R-2050 (1956)- 75 -32. M.K. Craddock and J.R. Richardson, "Magnetic Field Tolerances for a Six-Sector 500 MeV H" Cyclotron", TRIUMF Report TRI-67-2 (1968)33- M.M. Gordon, "The Electric Gap-crossing Resonance in a Three-sectorCyclotron" in Proceedings, International Conference on Sector-focused Cyclotrons, Los Angeles, 1962 (North-Hol1 and, Amsterdam, 1962), 26834. A.A. van Kranenburg, H.L. Hagedoorn, P. Kramer and D. Wierts, "Beam Properties of Philips' AVF Cyclotrons", IEEE Trans. Nucl. Sci., NS-13,41 (1966)35. G.H. Mackenzie and J.L. Bolduc, "Orbit Dynamic Calculations for TRIUMF", IEEE Trans. Nucl. Sci., NS-18, 287 (1971)3 6 . C.E. Froberg, Introduction to Numerical Analysis (Addison-Wesley, Reading, Mass., 1965)37- S. Comiti and R. Giannini, "Etude de la g£om€trie centrale du synchro­cyclotron avec source 'calutron' £ extraction haute frequence", CERN Report No. 70-9 (1970)38. M. Reiser, "Central Orbit Program for a Variable Energy Multi-Particle Cyclotron", Nucl. Instr. & Meth. 18,19» 370 (1962)39. S.P. Frankel, "Convergence Rates of Iterative Treatments of Partial Differential Equations", Math. Comput. 4_, 65 (1950)40. D. Young, "Iterative Methods for Solving Partial Difference Equations of Elliptic Type", Trans. Amer. Math. Soc. 76_, 92 (1954)41. R. Courant, "Partielle Differenzengleichungen und Differentia 1gleichun- gen", in Atti, International Congress of Mathematicians (New Series), 3rd, Bologna, 1928 (Bologna, N. Zanichelli, 1929)APPENDIX A THEORY OF SUCCESSIVE OVER-RELAXATIONWe wish to solve a system of equations described by1- 76 -^ijk 6 ^i-ljk + ^i+ljk + ^ij-lk + ^ij+lk + ^ijk-1 + ^ijk+1^ijk[interior nodes] [boundary nodes]i = 0, ls 2 3 = 0) 1) 2 k = 0) 1) 2P<7rThe system contains N equations where N = (p + l)(q + l)(r + 1).We will start with some initial approximation for each unknown value of <f> denoted We will sequentially modify each of these values in theorder^ 1 1 1 ^ 2 1 1  ^p 1 1 ^ 1 2 1 ^ 2 2 1  ’ * ^ p 2 1 .....................^pp 1^ 1 1 2 ^ 2  1 2 '  * ■<^ p i 2 <^ p22 * • '.. <(>.'pq 2 vpqr ’or in the opposite order.The successive over-re 1axation method can be described by the iterative sequence,n+l ,n , a ^ijk ~ ijk 6(A. 1),naP+1 j. J 1 i ^ + 1  , J 1 , An+1 ..i-ljk i+ljk ij-lk ij+lk ijk-1 ijk+16 ♦!Ijkbijk •[interior nodes][boundary nodes]^ijk *S t*ie estimate the value of the potential at the node ijk and a is a constant.- 77 -Now we define the error at the node ijk at the n ^  iteration asn _ nZijk ~ ijk ijk (A.2)where is the correct value at this node; thenn+ 1 . , n ae . .7 + q>. .7 = e . + <p. .7 + — n + l , A n . ie. . 7 -ft)). 7 . 7 -f e . . 7 -/-(j).ijTc ijk ijk ijk 6 1 i-ljk i-ljk i+ljk i+ljk, n+2 , , . n n-fieij-lk ij-lk £ij+lk ij+lk eijk-l ijk- 1+ eijk+1 + ^ijk+1 ®zijk ^^ijkAccording to Eq. (8), the terms in <j> in the square bracket cancel, leaving(A.3)n+ 1 n n+ 1 n n+ 1 n'i-ljk + £i+ljk £ij-lk £ij+lk eijk-l eijk+ln+ 1 _ n azijk ~ zijk 6r J 1 6e .^Ck [interior points] [boundary points]orn+1 cl_'ijk ~ 6n+1 n+1 n+1£i-ljk eij-lk eijk-ln , a eijk 6, n eijk+ln nei+ljk eij+lkClearly this method of iteration leads to a linear dependence of the eYlon the > so we may writen+1ijkin+1 . Ken (A.Mand n+1 - KtU - - ....where e and e are ^/-vectors whose elements are the N errors after n+1 and n iterations, respectively. K is an N by N matrix which depends on a, p, q and r, but not on n.For an individual error > Eq. (A. b) can be writtenn+1 neijk ~ _ Kijk i 'j 'k ' ei 'j 'k' •^ J k(A.3)- 78 -Now we denote the N eigenvalues of K by (Z corresponding eigenvectors by 3= 1, 23 .... N) and theKh  = \ h - (A.6)Since the eigenvectors form an orthogonal set, we can express the error vectors as a sum over the eigenvectors(A.7)from Eq. (A.k)(A. 8)hence = \n+1c° A£ °ZNow to evaluate the eigenvalues, we substitute Eqs. (A.7) and (A.8) into (A.3) , givingV n+loijk i+ljk+ V n + lo ° «  6 *.+fi aH+ \ yi*+1 q o ° l £36 ij-lk 36v. >ij+lk 36 ijk-1Vz ijk+1\ ° XijkUsing the second part of Eq. (A.8), this becomesn \C 1 + ai-ljk i+ljk ij-lk-h X .ij+lk ijk-1 ijk+1= 0.- 79 -71 71But this must be true for any error e , i.e. for any set of o ; henceX - l  + a 3„a x„r '~h'3„H\ a ijk 6 I . s'.i-ljk i+ljk ij-lk(A.9)ij+lk+ijk-1 ijk+1l = 1, 2  Nando jk iok rj o pjk iqk= 0. (A.10)rgrTo evaluate the eigenvectors and eigenvalues we will follow the pro­cedure first given by Frankel.39 A more general treatment has been given byYoung.1+0 The elements 3^ of the eigenfunctions are evidently41v. 'I'C% ijk.i       rJ    hh tj Ji    h uk= A srn Br srn— C srn---p q r (A.11)where s = 1> 23  p - 1t = 1, 2, .....  q - 1u = 1, 2, .....  v - 1 .Substituting Eq. (A.11) into Eq. (A.9) gives„ , ,i T\si . IT tj Jc . TT uk(X-l + a)A srn lr srn— C srn =p q v (A.12). 7rt,7 Jc • TrufeBr srn— C srn---q v, .i-1 • tjs(i-l) . ,i+l • wsd+l)XA srn-------- + A srn-------.r . nsr Jc . tmk+ A srn C srn---p rl B i - i  s i n ^ h l l  jif+J r t r & i i t D .q q.V . TTSt- ^7 • TTt,7 , J<-1 . Tm(k-l) . Jc+l . +u(k+l)+ A srn Br srn— XC srn-+ C srn -p q v vFor Eq. (A.12) to be satisfied for all values of i , j and k, we musthaveX = A 2-, X = B 23 X = C2.- 80 -Since multiplication of A (or B or C) by -1 is equivalent to replacings  by p  -  s  (or t by q - t or u by r - u ) , we may take A = +B = +C.Then Eq. (A.12) becomes(A2~l + a j /  sirir^- A? sin-T-^~ A^ s i n =p  q rot f . -7 . TT tj At . nuk n .i+1 • n■S^ ITS-t- i4« s^n— .4*- s^n 2A srn aos—6 q v p p, . ■nsi . ■nuk „„i+l ■ ntj ttt+ A srn A srn 2A srn— cos—p  r q q, . irsr . tttn cni+1 • iruk nu+ A u srn A*1 srn— 2 A srn aos—p  q v v¥ . i\si .A . ittj .v . nukA u srn A'J srn— AK srn---p q r ITS , Ttt , TT Uaos—  + aos—  + aos—  p  q r .so(A2-l + a) = |4 ITS T\t , TTUCOS—  + COS—  + aos—  p  q vA 2 - 4av + (a-l) = 0A 2 av ± / a2v2 - 4(a-l)(A.13) (A.14)w h e  re1V 3ITS , TTt , TTUaos—  + aos—  + aos—  p  q v - aos 0.Now in order to investigate the convergence rate we note that we have expressed the error vectors as linear combinations of the eigenvectors, and Eq. (A.8) can be writtenn r n n r , n o ae ■ I ° i et ' I \  ciBr (A.15)We will call the with the largest absolute value SL^ . Eq. (A.15) asWe can then wri tenb o „ r7n C v qm m  u1m ai+m'bLn- 81 -but since Xm > X , large n,A* n goes to zero as n becomes large and we obtain, forn o a .n e = o 6. X . m SL m (A.16)So, to achieve the maximum convergence rate, we want X (= A , etc.) tobe as small as possible. (X must be less than one if the process is tomconverge.) Returning to our equation for A, then if we consider*2v2 < 4(a - 1) , (A.17)the roots of Eq. (A.14) are complex conjugates with magnitude cn c2 = a - 2; however, if a2v2 > 4(a - 1), the roots will be real and unequal. Since the product of the roots is (a - 1), one of the roots must have a magnitude greater than /a - 1 . Hence, the minimum X = maxJ 7 mm l ^ l 2, occurs fora  in the range a 2 v 2 < 4(a  -  1) . Since the magnitude of X^ in this range is (a - 1), the value of a (called a^) giving the smallest X^ is the smaller of the two roots of0(2v2 = 4(aj=) - 1) ,i.e. ab 1 + sinQp, _ 2/3-y*w2 1 + \1 + /7-”z (A.18)andX = A2 =mofcVI 2sinQ. . „ = a-,1 + srnQ bSince we are calculating the minimum of the maximum values of A, we must take the worst case, i.e. the largest value of v, which isV 3TT , TT , IToos—  + cos—  + cos—  v q v (A.19)In practical problems, p, q and v are >> 1, and we can obtain approxi­mate expressions for a, and Xr b m- 82 -v - 1 - —  H r  + — tr + gmm j sinO - (A.20)a, - 2b - 2*/l 1IT (A.21)V i1 1 1J—  + _L_ y. _p 2 q2 r2(A.22)Now that we have found the value of a which gives fastest convergence, the question of interest is how fast does it converge. Referring to Eq. (A.8), each iteration reduces each error by at least a factor X^; henceYln iterations reduce the error by at least a factor (X ) . Hence, to reduce the errors by a factor f, the number of iterations required isn = log flog X (A.23)mCentral region of the TRIUMF cyclotron - median plane6 Harmonic WedgeCentral region of the TRIUMF cyclotron - section through centreline of hill #3Fig. 2.1 Relaxation Mesh Organization. Total number of nodes is (p+1) (q+1)(r+1).oincOcuL.<Dmo 0)cnc<TJ_CoL_oL.oL_L.(D0)cnOJL_<D>(TJinooCNCNIp i J6 3u i j. J9AO sda9MS jsquunuAverage error and average change per iteration vs number of sweeps over large volume(%) jojja a6ej9AeAverage error vs number of sweeps over large volume for various values of<DDo><DCDl_03«D>Oin( v0)0)2W0)_QDCinDOL_03>L-oL_a)a)N10>V_oL-L.a)c<u>cn03_ c1/3<D"DOca)-OD-d"CMcnfO CMo  o  Opsieojpui JOJJ3 i|}[M sspou jsqwnu(/)Q_a>a)2toM-OL.<D_Q   D i- C  OL-CO L .i  y=L. 0) O N L- —I- a) t/)C . D0) o> —  —  L- CD «J>05-C o4-» M—* «  CO D<U —  XJ Oo > c (I)O-  cn O i-  OJL. .—a)-Q \ -     13 >z  oLTVCMcn3 v )ooroOc\iC71E (MeV)Magnetic axial focusing frequency (v?) vs energy for three- and six-sector magnetic geometries0.008Equivalent axial focusing frequency produced by space charge forces vs energyfor various beam currents and axial beam heightsFig. 3.b Comparison between equivalent electric axial focusing frequencies predicted by the thin lens approximation and determined by numerical integrationFig. 3.5 Possible TRIUMF central geometry with three accelerating gaps in the first half-turn(mrad)Fig. 3-6 Axial emittance ellipses required at injection for various RF phases001ocooino<froroCO<D<UL_U)<DTDO <uCOro_cQ.O  *Oio i(%) 3 S d  j [ [3 30 U955|0I3 [ 9 1 X 9  6 3 p  QZ = Ml ! M d 9 l J 3 A 0Fig. 3-7 Axial acceptance vs RF phase for various injection energies (one accelerating gap in the first half-turn){/)4)0)i.U )0)X>4)CD{%) 3 s d | i [ 9  9 0 u e } } ju i9  [Bixe 6ap QZ = 4* Ml!** d e [ j9 A OFig. 3-8 Axial acceptance vs RF phase for various injection energies (three accelerating gaps in the first half-turn)(%) d e [ J 8 A O  a s d j i i s  a o u e u j w a  [ e j x e  a 6 e j a A eFig. 3*9 Average axial acceptance (averaged from -30 deg to +60 deg) vs injection energy (one accelerating gap in the first half-turn)(%) d 0 N C 3ao asdjiia aoueujwa [ejxeFig. 3.10 Axial acceptance vs RF phase for various choices of the initial emittance ellipseA(9 + <t>£)soo 3 - .isBF : t  aRF voltage waveforms with various amounts of third harmonic and phase shift between fundamental and third harmonicocL.03-CCM-C ■M O  M- C  O  O Ein 3- 4-J ro C  -C DO  X> E  3_ 03 —  _C 03 -M 13CO 0 X 30) —  C<D i- 03I -  0303  >  -<D 03X3 JZ 4-»■M C  0)2 E-©- 03—  6 X 33— C  0) O  3in M- M-03 4)-C >  CCL 03 4)2 <U 2Cd a) *■>03 4) 03 -Q  +Jr—  4-»O M-> —_c-C■ inOC 4)M - inO  03_c 4> CL CLO  X3—  CCO 03o00(*) de[C SR:_l______OIDasd  | i [9o!  " # $ $   % " ! [e h ) jl/>a)a>L.a>a)TJ0)I/)fDFig. 3.13 Axial acceptance vs RF phase for various choices of the initial emittance ellipse, z = 0.17, 6=0Fig. 3.14 Axial acceptance vs RF phase for various choices of the initial emittance ellipse, e = 0.12, 6=0oinoo o  o  o  o00 10 CM(%) d e i J 9 A 0  9 S d | [ [ 9  90 UB}}|UJ9 [ B J X BOoroOCVJ in<D<DL_U)0)~o<Din0)Oi0CVJ1Oroi<1)_cin<1)oo-Coin cn D <D O "Oi- o03 —  >  Ii- I OM- <0a) -  in LA 03 —  _c • CL OQC0)in >  -  03<13 in U Q_c —03 —  4-J <—Q- <D 0)U 0)a o 0) c0303X<a>80‘0>0)cn'y=Ccnci/)DOo+0)cV03>Dcrd)oy=T3C03(1)ccr,0)03O frequency vs energy for various RF phases(saaj6ap) aseqd uojijsuej}Fig. 3.17 Transition phase (of total axial focusing from negative to positive) vs energy<t> u i s vFig. 3-18 Change in sine of RF phase required to keep ion at transition phase vs radius(ssneB) 99Fig. 3*19 Magnetic field bump required to keep ion at transition phase vs radiusFig. k.] Geometry of injection gap and first main gap for two RF phasesElR od r B. . .  1 . . . .  . . Io t s  U -  A2 A2 o  s         t s  t     t s s  s  s      A 2  t s s y   t s  t s s  s  s  ssy t   t s s  t<t>19 (degrees)Fig. 4.3 Energy gain in injection gap and first main gap vs RF phase at injection gap for various injection gap positionsFig. b.b RF phase at first main gap vs RF phase at injection gap for various injection gap positions.eSGC63p) apnjjiduue n : Cvy A A C t F : asegdFig. k.5 Phase oscillation amplitude vs centring error at various radiiRF phase at centre of dee gap (degrees)     &&&&&&&&& &&&&&&&& &&&&&&&& &&&&&&&& &&&&&&&& &&&&&&&&5 10 15half-turn numberFig. **.6 RF phase vs half-turn number for various initial phases with no flutter in the magnetic field/hi 1 1va11eyFig. b .7 Geometry of an orbit in a three-sector magnetic fieldRF phase difference between half-turns (degrees)6 (degrees)Fig. k .8 RF phase difference on succeeding half-turns as a function of orientation of the dee gap (6)RF phase at centre of dee gap (degrees)Fig. ^.9 RF phase vs half-turn number for various initial phases with a three-sector magnetic field (6 = 30 deg){/)D“OfUFig. 4.10 Ratio of third harmonic amplitude in magnetic field to average field vs radius for (three- sector) field 1-14-5-70(•ui) 6ujdoi[eos qiqjo uinwixewFig. 4.11 Average orbit radius and maximum orbit scalloping vs radius for (six-sector) field 1-30-06-70YFig. k.]Z Geometry of an orbit in a six-sector magnetic fieldo<DC*oofUCLrucn<1)<1)T3'VJFig. 4.13 Geometry of the difference between an equilibrium orbit and an accelerated orbitEnergy (MeV)Fig. 4.14 Centre-point displacement along the dee gap vs energy showing values from numerical orbit tracks and from an analytic approximationCVJoistoiL.<CDoI00oiFig. *1.15 Accelerated phase plot inwards from 5 MeV, (j) = -30 degCVJ0101<to0100oFig. 4.16 Accelerated phase plot inwards from 5 MeV, <f> = 0 degCVI01oiCDoI00oFig. 4.17 Accelerated phase plot inwards from 5 MeV, <J> = +30 degoFig. *t.l8 Accelerated phase plot outwards for various radii at first main dee gap, <f> = 0 deg(in.)MeVMeVFig. 4.19 Accelerated phase plot outwards from inflector exit for various phases; ion with <(> = 0 is centredFig. 4.20 Accelerated phase plot outwards from inflector exit for various phases; ion with <f> = +17 deg is centredstarting energy = 0.3 MeV final energy = 20 MeV\0)coCNI-a-cnO)cO4icoocnc■ML_034ssDoL_03>L.oc+—<Ds03_cQ.3v)QCts><DT3D4-JQ_03CO•M03OtsocOL-4-»03 !  QQ33v )RF phase (<f>) (degrees)I I I I I I I I I I I_______ I_____L_0  8 16 24half-turn numberFig. 4.22 Phase histories of ions with various starting phases in a magnetic field with a field bumpFig. 4.23 Accelerated phase plot outwards from inflector exit for various phases using the magnetic field with the field bump; ion with <p = 1 7 deg is centredI,0T 1.0+(•ui) dy H 1---- 1woto <  o>*cn<DFig. h ,2h Accelerated phase plots with <J> = 0 deg for four points on the edge of the emittance ellipsea) matched to vr = 1, and b) chosen to reduce the radial oscillation amplitude over the phase range -5 deg to +25 degI<D <U to inP z  "—  x—  CL*0) <U X  a) eB0C  3_a1 a)+-* > o1  a)a) “oD0) + J  X  —# $ , ^ M— E O  03d) c cn o  x> —%& # 03a) —  x  —#  o c  tn O  O  in •—4-j ru c  ——  "O O  03 CL i -j -  a)D  Xo  +-•#0)3 - O  O  3# +'<Uc n  3_a)~o o#LT\—  c + <u03II O  X-e- o x  ^ (5  -oC  0303 03 <D4-3 “Oo  -■—  ■—  L A  CL CMIIa> +03 3_03 ^  O  X  4-»CL O# %#“O  0 )a) y— u V iL_ a)03 X  L A  3 - O  Ia) 4-3 ■—  03 <U a) E 03 o cO  ^  03 <  03 3 -L A)*-d-033v)Fig. h .26 Accelerated phase plots with <j> = +25 deg for four points on the edge of the emittance ellipse a) matched to vr = I, and b) chosen to reduce the radial oscillation amplitude over the phase range -5 deg to +25 degFig. 5.1 Cross-section of a dee gap showing electric equipotentia 1distance from gap centre Y (in.)Fig. 5.2 Electric potential vs distance from dee gap centre showing actual values and constant gradient approximationFig. 5.3 Geometry of an ion crossing a dee gap>*U)i.0)cFig. 5.1* Gap factors vs energy for <f> = 0 degomoCVJ>~CT)0)cinc\jFig. 5.5 Differences between gap factors obtained from numerical integration and those obtained from the constant gradient approximation as a function of energy, no magnetic field+ 45(%) joioe^ de6 u; jojja0)1/3O)“ O  C  (0>*cn0)c<uc  *—  O4-*03 C»- o cn —  a) +-» +-» u c c—  =3vOLAcnD M—03 03 O•— '3- 03 0)+   #  3> § g3— —^  M - X>  o0) X5 L_,- y. /z— ' CO.—  03 > * 030 3  4-J 4-J3- x  c -oy.  y. 0 1   iyLU  03 “O —3- f0 M— \ o 3_'  4-3 03  O) o'  03 4-3 4-3M— C  0) 03 ) / # %# 03 03 03 %# ) + C O 03 0) Dy= 2.  £  X  C4-3 4-* Oa) 8^x  1  x  o o03 3_ O y= * & OC “Oa) <d c 3- C  03) 2=  M- 03 Xj  34— 4-3 4-3I —  X  —/   3oOcvi- OOCVII01OcviQiOcviioroi(%) joaoe^ de6 ui jojja0)ino a)x  in  4-3 03 X  X )  CLc03 Li_a:cO  M -— o4-3^  03 CD oVj cn —^  <1) 4-34-3 Uc c—  D  M -in  a1  a1iy  4) —  inL- 1 - 0 30 3  CD4) E CX> D O— " c —4-3E  03O  O  E-©- i- —u- X<D O3 - X )  3-4 -3  y4  z.5 (  (  ( vJ  <D —  03O  034-3 4-3Q -  X  C  X )03 O  4) •—0 3  — 0)in  X J  —4-3 L . 03 14_03 0  3-4-3 cn o <u o —to  03 4 J  + J03 «4- C  43X  03 CQ_ Q .  4-3 CD03 03 03Li- 0 3  C  ECd Oc o m  (D Da) a) o2 -C c^  4 -1 4 -3  0}  CD 3—0 §= -*=, o  o m 3_ o4) M - ino —C  X3 4 ) 4) C3 - C  034) —M - 03 X14- 4-3 4-3X  —  0  0  2LAO)^  Ll_(A9W) A6J3U3Fig. 5-8 Apparent displacement due to change in radius of curvature of the ion path while crossing the dee gapI

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