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Field tolerances associated with some resonances in the triumf cyclotron Bolduc, Jean Louis 1972

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FIELD TOLERANCES ASSOCIATED WITH SOME RESONANCES IN THE TRIUMF CYCLOTRON by JEAN LOUIS BOLDUC B . A . , Laval U n i v e r s i t y , 1965 B . S c . A . , Laval U n i v e r s i t y , 19&9 A thes is submitted in p a r t i a l f u l f i l m e n t of the requirements for the degree of Master of Science in the Department of Phys ics We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ?C £. The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT This thesis is concerned with tolerances for magnetic f i e l d imper-fec t ions in the TRIUMF cyc lo t ron set by the betatron o s c i l l a t i o n resonances v x = 1.0, v x - v z = 1.0 and v x = 1.5. These resonances, encountered during a c c e l e r a t i o n , can lead to undesirable growth in the amplitudesof the betatron o s c i l l a t i o n s . We f i r s t der ive equations of motion that take into account non- l inear terms and f i e l d imperfect ions, and show how resonance condi t ions may occur . These condi t ions were simulated in our o r b i t codes and numerical c a l c u l a t i o n s were made to determine the tolerances they impose on the magnetic f i e l d . We have made a de ta i l ed inves t iga t ion of the e f f e c t on the behaviour of the beam of f i r s t harmonic bumps at rad i i less than 150 i n . The f i r s t harmonic to lerance of 0.1 G to produce an increase in the radia l amplitude of 0.1 i n . is in agreement with a n a l y t i c a l c a l c u l a t i o n s . We have a lso shown that th is t o l e r a n c e , too small to be seen in the magnetic f i e l d survey, may be achieved by s u i t a b l e adjustments in the harmonic c o i l s e t t i n g s . Tolerances on the second harmonic imperfection are a lso presented. To determine the tolerances set by the coupled resonance v x - v z = 1.0, we have simulated a f i r s t harmonic twist in the median plane. The resu l ts show tha t , for high current poor reso lu t ion experiments, the magnitude of the twist on enter ing the resonance is of no importance, provided the ampl i -tudesof the radia l and v e r t i c a l betatron o s c i l l a t i o n s are not much larger than the estimate of 0.2 i n . For high reso lu t ion experiments, the tolerances on the slope of the twist are of the order of a few mrad. A n a l y t i c a l e s t i -mates of these tolerances are a lso presented. F i n a l l y , we have determined that the v x = 1.5 resonance sets an upper l im i t to the gradient of the t h i r d harmonic of about 0.2 G / i n . , which produces a 20% increase in the radia l betatron ampli tude. i i TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. MATHEMATICAL DESCRIPTION 4 2.1 Equations o f Motion 4 2.1.1 In t roduc t ion 4 2 . 1 . 2 L inea r Dynamics 4 2 . 1 . 3 Non-Linear Dynamics 7 2.2 Resonances 11 3. THE CENTRAL REGION 16 3.1 Int roduc t i on 16 3.2 S e n s i t i v i t y to a F i r s t Harmonic 16 3.2.1 In t roduc t ion 16 3 .2 .2 C a l c u l a t i o n s 18 3-3 Harmonic C o i l s 22 3.3.1 O s c i l l a t o r y F i r s t Harmonic 22 3 .3 .2 Ions w i t h a Wide Range of RF Phases 28 4. THE RESONANCE v x - v z = 1 45 4.1 Mathematical D e s c r i p t i o n 45 4 .2 C a l c u l a t i o n s 48 4.3 Tolerances 56 5. THE RESONANCE v x = 1.5 58 5.1 The I n t r i n s i c Resonance v x = 6/4 58 5.2 The Resonance v x = 3/2 61 5.2.1 In t roduc t ion 61 5 .2 .2 C a l c u l a t i o n s 63 5 .2 .3 S t a t i c and Acce l e r a t ed Phase Space w i t h T h i r d Harmonic Gradient 66 References 72 Appendix A. Magnetic F i e l d Components in the Plane of Measurement 73 i i i LIST OF TABLES Page 1.1 Total energy spread 2 3.1 Betatron amplitudes for 0 deg RF phase ions 24 3-2 Betatron amplitudes with harmonic c o i l f i e l d 27 3.3 Components of displacements due to f i r s t harmonic of F i g . 3.8 33 3.4 C o e f f i c i e n t s for set of c o i l s at 37" < R < 54" 33 3-5 C o e f f i c i e n t s for set of c o i l s at 54" < R « 71" 34 3.6 C o e f f i c i e n t s for set of c o i l s at 71" « R « 89" 34 3.7 Harmonic c o i l c o r r e c t i n g f i e l d s 35 3.8 Displacements for ideal o r b i t due to f i r s t harmonic of F i g . 3.8 36 3.9 Displacements from ideal o r b i t when f i r s t harmonic e r r o r and cor rec t ing f i e l d s are present 36 3.10 Displacements of the e q u i l i b r i u m o r b i t centre due to f i r s t and second imperfection harmonics of 1 G and 2 G, r e s p e c t i v e l y , for ions of 0 deg RF phase at 1 MeV and v x = 1 .001 43 3.11 Displacements of the e q u i l i b r i u m o r b i t centre due to f i r s t and second imperfect ion harmonics of 1 G and 2 G, r e s p e c t i v e l y , for ions of 0 deg RF phase at 3 MeV and v x = 1.004 43 4.1 Maximum betatron amplitudes (e i ther xf jnal o r z f i n a l ) below and above the resonance v x ~ v z = 1 when v x - 1.2 and v z - 0.2 56 i v LIST OF FIGURES Page 2.1 Plot of v x versus v z , for f i e l d 01/30/10/70 14 3-1 Phase space for acce lera ted ions (0.8 MeV to 28 MeV) of 0 deg RF phase with various f i r s t harmonics present 20 3.2 The amplitude of a 20 i n . wide f i r s t harmonic bump required to produce radia l betatron amplitudes of 0.2 i n . 21 3.3 Residual f i r s t harmonic obtained from superposi t ion of harmonic c o i l f i e l d cor rec t ions to f i r s t harmonic e r ror f i e l d 23 3.4 P r o f i l e of o s c i l l a t o r y f i r s t harmonic 25 3.5 Bump p r o f i l e 25 3.6 Pos i t ions at 35 MeV for ions of 0 deg RF phase acce lera ted from low energy 26 3-7 Phases and amplitudes of the o s c i l l a t i o n s produced by standard harmonic c o i l f i e l d and by f i r s t harmonic er ror 30 3.8 P r o f i l e of f i r s t harmonic e r ro r 30 3.9 Displacements from " i d e a l " centre point at 35 MeV due to a 2 G f i e l d in each set of c o i l s , for ions of 0 deg, 25 deg and 40 deg RF phase 32 3.10 Phase space p lo t for acce lera ted ions (1 MeV to 35 MeV) of 0 deg RF phase with both the f i r s t harmonic e r ror and the harmonic c o i l c o r r e c t i n g f i e l d s present 38 3.11 Phase space p lot for acce lera ted ions (1 MeV to 35 MeV) of 40 deg RF phase with both the f i r s t harmonic e r ro r and the harmonic c o i l c o r r e c t i n g f i e l d s present 39 3.12 Phase space p lo t for accelerated ions (1 MeV to 35 MeV) of 30 deg RF phase with both the f i r s t harmonic e r ro r and the harmonic c o i l c o r r e c t i n g f i e l d s present 40 3.13 L imits of radia l s t a b i l i t y for second harmonic e r ro r f i e l d 42 4.1 Sect ion view of the cyc lo t ron median plane with t i l t e d median plane parameters 50 4 .2 S t a t i c phase space e l l i p s e at 153 MeV corresponding to a radia l betatron amplitude of 0.25 i n . 50 4.3 Radial and v e r t i c a l betatron o s c i l l a t i o n s at 153 MeV (on resonance) when a f i rst_ harmonic twist with a slope of a = 0.02 rad and R 0 = R153 ^ey is present 52 v Page 4.4 Radial and v e r t i c a l betatron o s c i l l a t i o n s at 144 MeV (10 MeV below resonance) when a _ f i r s t harmonic twist with a slope of a = 0.02 rad and R 0 = R] !^ ^ e y is present 53 4.5 Radial and v e r t i c a l betatron o s c i l l a t i o n s at 125 MeV E ~$ 153 MeV when a f i r s t harmonic twist with a slope of a = 0.02 rad and RQ = R153 M.ev ' s present 55 5.1 Phase space p lot at 425 MeV and v x = 1.487 fo r ions of 0 deg phase (resonance v x = 6/4) 59 5.2 Unstable f i x e d - p o i n t s motion for f i e l d 01/18/02/70 60 5.3 Unstable f i x e d - p o i n t s motion for f i e l d 02/09/07/71 62 5.4 The e f f e c t of the bump phase parameter on the width of the resonance in the case of a 0.4 G / i n . t h i r d harmonic gradient with Rs = 287.5 i n . 64 5.5 The e f f e c t of the s t a r t i n g radius parameter on the width of the resonance in the case of a 0.4 G / i n . t h i r d harmonic gradient with <J>3 = <J>6 + 15 deg 64 5.6 The e f f e c t of various gradient amplitudes on the width of the resonance when Rs = 287-5 i n . and o>3 = cp6 -t- 15 deg 65 5.7 Phase space p lo t of 430 MeV and v x = 1.492 for ions of 0 deg RF phase when a 0.5 G / i n . t h i r d harmonic gradient is present 67 5.8 E f f e c t i v e phase space area occupied by the beam at 441 MeV when no t h i r d harmonic gradient is present 69 5.9 E f f e c t i v e phase space area occupied by the beam at 441 MeV when a t h i r d harmonic gradient of 0.2 G / i n . amplitude is present 69 5.10 E f f e c t i v e phase space area occupied by the beam at 441 MeV when a t h i r d harmonic gradient of 0.4 G / i n . amplitude is present 70 A . l T i l t e d magnet sector parameters 75 v 1 ACKNOWLEDGEMENTS I am much indebted to Dr. G.H. Mackenzie for h is invaluable help in d i r e c t i n g th is research. Thanks are a lso due to Dr. M.K. Craddock, Dr. W. Joho and Dr. M.M. Gordon for he lpfu l suggestions and to Miss Col leen Meade for her ass is tance with the computer programs . I a lso wish to thank Miss Ada Strathdee for her work in typing th is t h e s i s . This research was made poss ib le by a scho la rsh ip and a bursary from the National Research Council of Canada. INTRODUCTION The TRIUMF c y c l o t r o n 1 is designed to acce le ra te negative hydrogen ions up to an energy of approximately 500 MeV. The H" ions are produced in an external ion source and guided, through the i n j e c t i o n system, down the axis of the cyc lo t ron into t h e i r f i r s t o r b i t . Then a two-dee system with a dee-to-dee peak voltage of 200 kV, g iv ing an energy gain per turn of 400 keV, is used to reach the desi red energy. During i ts a c c e l e r a t i o n , the p a r t i c l e o s c i l l a t e s about an e q u i l i b -rium p o s i t i o n , in both the radia l and the v e r t i c a l d i r e c t i o n . This e q u i l i b r i u m o r b i t is defined as an o r b i t , in the median plane of the c y c l o t r o n , with the same 6 - fo ld symmetry as the magnetic f i e l d . ; that i s , an o r b i t that c loses on i t s e l f a f t e r each s e c t o r . The motion about the e q u i l i b r i u m o r b i t is analogous, in f i r s t approximation, to the motion of a two-dimensional harmonic o s c i l l a t o r . The Hamiltonian for th is system may be wri tten H = 2 v2 x 2 + p 2 X r x (1.1) where (x ,p x ) and ( z ,p z ) are the displacements in p o s i t i o n and momentum, in the radia l and v e r t i c a l d i r e c t i o n r e s p e c t i v e l y , from the e q u i l i b r i u m o r b i t . The v v and v_ a r e , r e s p e c t i v e l y , the number of radia l and v e r t i c a l X z o s c i l l a t i o n s per tu rn . The corresponding equations of motion are x" + v 2 x = 0 (1 .2) x v ' z" + v 2 z = 0 (1.3) where the primes represent the der iva t ion with respect to 0, the azimuth around the machine. Then, to th is approximation, the ion descr ibes two independent s inuso ida l betatron o s c i l l a t i o n s about the e q u i l i b r i u m o r b i t . - 2 -In order to achieve good energy reso lut ion at e x t r a c t i o n , the amplitudes of these betatron o s c i l l a t i o n s must remain s m a l l . Table 1.1 i l l u s t r a t e s the re la t ion between the tota l energy spread at two energies and the amplitude of the radia l o s c i l l a t i o n A . A x at 30 MeV ( in. ) 0.14 0.25 0.40 200 MeV 1 .0 1.2 1.6 500 MeV 1 .2 1.8 2.4 Table 1.1 Total energy spread (MeV) S t a b i l i t y is achieved i f the beam is proper ly focused both r a d i a l l y and v e r t i c a l l y . In the radia l d i r e c t i o n , focusing is always achieved s i n c e , for any isochronous cyc lo t ron with three or more s e c t o r s , such as TRIUMF, the value of v is approximately given by v = y where y = ] + t / m c 2 , x being the k i n e t i c energy of the ion of rest mass m. There fore , v x s t a r t s at unity and increases af terwards; the beam is then contained in that d i r e c t i o n . In the v e r t i c a l d i r e c t i o n , the f i e l d f l u t t e r , which is a measure of the change in f i e l d strength between the h i l l s and the v a l l e y s , provides an ax ia l force known as the Thomas f o r c e , 2 which is always f o c u s i n g . Two add i t iona l f o r c e s , discovered by Kerst and L a s l e t t , 3 a lso cont r ibute to the ax ia l f o c u s i n g . Both forces depend on the s p i r a l shape of the magnet s e c t o r . The i r e f f e c t is a l t e r n a t e l y focusing and defocus ing . However, because of the a l t e r n a t i n g gradient p r i n c i p l e and because of d i f f e r e n t path lengths in the focusing and defocusing f i e l d s , t h e i r net e f f e c t is f o c u s i n g . Once we achieve the condi t ions for ax ia l and v e r t i c a l f o c u s i n g , the next problem is to avoid resonances. We c a l l "resonance" a s i t u a t i o n in - 3 -which the amplitude of the radia l or the v e r t i c a l o s c i l l a t i o n grows with 0, the azimuth around the machine. Some of the resonances are i n t r i n s i c and are due to n o n - l i n e a r i t i e s in the equations of motion, neglected in Eqs. 1.2 and 1.3. Others come from mechanical imperfect ions such as misalignment or magnetic non-uniformity of the magnet s e c t o r s . In the next chapte rs , we w i l l f i r s t der ive more accurate expressions for the equations of motion and show how resonance condi t ions may occur . We w i l l then look at some p a r t i c u l a r cases of resonances encountered in the operat ion of the TRIUMF cyc lo t ron and determine the tolerances they impose on the magnetic f i e l d . - k -2 . MATHEMATICAL DESCRIPTION 2.1 Equations of Motion 2 . 1 . 1 Introduction A system of c y l i n d r i c a l co-ord inates ( r ,8 , z ) is used to descr ibe the radia l and v e r t i c a l motion of a p a r t i c l e with r e l a t i v i s t i c mass m and charge q in a magnetic f i e l d B\ The force appl ied to the p a r t i c l e is ? = q v x l ( 2 . 1 ) where v = ( r , r 6 , z ) ( 2 . 2 ) is the v e l o c i t y of the p a r t i c l e and where the dots represent the d e r i v a -t i ve with respect to time. From Lagrange's equat ions , we obtain ^ ( m r ) - mr0 2 = q ( r0B z - zB Q ) ( 2 . 3 ) ^(mz) = q ( r B e - r 0 B r ) . ( 2 . 4 ) Using these two equat ions , Walkinshaw and K i n g 4 derived l i n e a r expressions for the radia l and v e r t i c a l motion. We summarize most of t h e i r c a l c u l a -t ions in the fo l lowing s e c t i o n , and then extend them to consider q u a l i t a t i v e l y some non- l inear e f f e c t s . 2 . 1 . 2 L inear Dynamics Since the p a r t i c l e v e l o c i t y along the o r b i t is constant we may transform the independent va r iab le from time t to 0, the azimuth around the machine. Walkinshaw and King assume that ( r ' / r ) 2 , ( z ' / r ) 2 and ( r ' z ' / r 2 ) are a l l much less than unity and that the term in zB Q is un-l i k e l y to be important. Then, the equation for the radia l motion in the median plane may be wr i t ten - 5 -r" - r = -9-r2B mv 2 (2.5) where the primes denote d i f f e r e n t i a t i o n with respect to the new v a r i a b l e 9. S i m i l a r l y , the equation for the v e r t i c a l motion takes the form z" + - ^ - ( r 2 B r - rr'Bfl) = 0. mv r fc>' (2.6) We assume a p e r f e c t l y f l a t median plane and, s ince c u r l ? = 0 in a s t a t i c magnetic f i e l d , we may wr i te to a l i nea r approximation in z B = B = B = z | i r 8r B 6 " Z r 36 • z=0 (2.7) Eq. 2.6 becomes + -3-mv ,_8B_ _8r 8_B 86 z = 0. (2.8) We then def ine a reference c i r c l e of radius r , the e q u i l i b r i u m radius in o the homogeneous magnetic f i e l d , by the cond i t ion r Q = mv q B w h e r e Bo i s the az imuthal ly averaged value of B at r Q . Wr i t ing R = r / r , Eqs. 2.5 and 2.8 become 5^- - R - R2-L d e 7 R - R B 0 d 2 Z de 2 R 2 ^-(B/B 0) + 9R 0 86 z = 0, (2.9) (2.10) The next step is to der ive the equations for the radia l and v e r t i c a l o s c i l l a t i o n s of the p a r t i c l e around the e q u i l i b r i u m o r b i t . For the v e r t i c a l motion we assume, in f i r s t approximat ion, that the radia l - 6 -motion is confined to the e q u i l i b r i u m o r b i t , i . e . R = ^ e o(6)» where R (6) = 1 + p(9) is the radius of the c losed o r b i t . We wr i te eo de 2 R 2 — ( B / B J I eo 3RV o' Reo + • — ( B / B 0 ) 30 30 0 z = 0. ( 2 . 1 1 ) For the radia l motion we expand R about the e q u i l i b r i u m o r b i t , i . e . R = R + x. We a lso wr i te the f i e l d (B/B ) as a T a y l o r ' s se r ies about eo o' ' i ts value on the e q u i l i b r i u m o r b i t . To a l i nea r approximation in x , Eq . 2 . 9 becomes d 2 x d02 + J _ f _ 3 ( R 2 B ) l 3R ^eo x = 0. ( 2 . 1 2 ) Then, a Four ier ana lys is of B(e) makes Eq. 2.11 and Eq. 2 . 1 2 of Mathieu H i l l form. When the harmonic terms are s m a l l , the r e s u l t i n g equations can be reduced to the form of Eq. 1.2 and Eq. 1 . 3 , provided we are in terested only in the motion per turn ( i . e . n = 8 /2 T T becomes the new independent var iab le ) and not in the d e t a i l e d motion in each s e c t o r . In the f o l l o w i n g , we w i l l assume for s i m p l i c i t y that the radia l and v e r t i c a l o s c i l l a t i o n s have been smoothed out into a s inuso ida l motion of constant angular f requency, v and v , r e s p e c t i v e l y . In th is c a s e , the p a r t i c l e precesses along an e l l i p s e in the (x,p ) or (z,p ) phase X z space. In p r a c t i c e , however, the non-s inusoida l character of the o s c i l -l a t i o n makes the p a r t i c l e move along a s l i g h t l y d i f f e r e n t curve. We have shown t h a t , in TRIUMF, the discrepancy from the ideal case in the radia l motion is always less than 25% of the betatron ampl i tude, in the region of maximum f l u t t e r . This is acceptable for our present purpose. Eqs. 2.11 and 2 . 1 2 would then descr ibe these two independent s inuso ida l o s c i 1 l a t ions. - 7 -2 .1 .3 N o n - L i n e a r Dynamics More g e n e r a l l y , the r a d i a l and v e r t i c a l b e t a t r o n o s c i l l a t i o n s , as w e l l as magnet i m p e r f e c t i o n s , w i l l cause h i g h e r o r d e r terms and c o u p l i n g terms between t h e two motions t o be i n t r o d u c e d i n the e q u a t i o n s o f mo t i o n . These h i g h e r o r d e r t e r m s , added t o t h e r i g h t - h a n d s i d e o f Eqs. 2.11 and 2 . 1 2 , w i l l t r a n s f o r m o u r f r e e o s c i l l a t i o n s i n t o f o r c e d o s c i l l a t i o n s . The main f e a t u r e o f t h e s e f o r c e d o s c i l l a t i o n s i s t h a t i n s t a b i l i t y may a r i s e because o f a resonance c o n d i t i o n between the f r e q u e n c y o f the f r e e o s c i l -l a t i o n and the f r e q u e n c y o f the p e r t u r b i n g f o r c e . T h i s happens when the p e r t u r b a t i o n term o s c i l l a t e s w i t h t he f r e q u e n c y o f the f r e e o s c i l l a t i o n o r w i t h any o f i t s i n t e g r a l m u l t i p l e s . To i l l u s t r a t e t he v a r i o u s t y p e s o f r e s o n a n c e s , we f i r s t expand o u r p r e v i o u s e x p r e s s i o n s f o r the magnetic f i e l d t o i n c l u d e q u a d r a t i c terms i n z. In the case o f a f l a t median p l a n e , Eq. 2.7 becomes B z = B - 2^ 5- V 2B (2.13) o R _ _ ^ 3 B R _ _z_ 1 3B BR " r Q 8R ' b 9 " r Q R 80 where B i s a g a i n w r i t t e n f o r t he z-component o f t h e ma g n e t i c f i e l d i n the median p l a n e and V 2 = _ i i + 1 _ L + _ L ^ _ (2 14) 8R 2 R 9R R 2 8 0 2 i s t h e L a p l a c i a n i n p o l a r c o - o r d i n a t e s R,0. We a l s o c o n s i d e r a s m a l l d e v i a t i o n from the i d e a l 6 - f o l d symmetric i s o c h r o n o u s f i e l d ; we w r i t e B = B; + AB (2.15) where B. i n c l u d e s t h e s i x t h harmonic o f the magnetic f i e l d and i t s - 8 -m u l t i p l e s and where AB may a l s o be e x p r e s s e d as a F o u r i e r s e r i e s and con-t a i n s t he " i m p e r f e c t i o n " h a r m o n i c s . S i m i l a r l y , because o f magnet i m p e r f e c t i o n s , we no l o n g e r assume a symmetric median p l a n e , so t h a t o u r z - c o - o r d i n a t e i s changed a c c o r d i n g t o z = z + z Q . We assume t h a t z Q i s s m a l l compared t o the d i m e n s i o n s o f t h e magnet, and we w r i t e 00 00 z o = I I e n n x " c o s ( P 9 + V n=0 p=0 ' ¥ (2.16) In the f o l l o w i n g , we s e t the phases §^ equal t o z e r o d egree. In a l l the cas e s we c o n s i d e r the p e r t u r b a t i o n s i n the g u i d i n g f i e l d a r e p e r i o d i c f u n c t i o n s o f 0 w i t h a p e r i o d o f 2ir/p. Then, t h e f i e l d i m p e r f e c t i o n s can be a n a l y s e d i n a F o u r i e r s e r i e s . We c o n s i d e r o n l y t he p t' 1 F o u r i e r component, such t h a t p i s an i n t e g e r . Because o f the median p l a n e asymmetry, new terms must be added t o the e x p r e s s i o n s o f the magnetic f i e l d components; f o r s m a l l v a l u e s o f zQ, Eq. 2.13 becomes (see Ap p e n d i x A) B = B Bn = ~2 V 2 ( z B) o ° 2 r o z 2 „ 2 V Z B (2.17) ' 3 B + 3 ( z n B ) ' "3R 3R B n = — 'z_ 3B_ j _ 3 ( z 0 B ) ' R 30 R 30 A l s o , t h e term i n zBg o f Eq. 2.3 i s no l o n g e r n e g l e c t e d . Then, w i t h t he use o f Eq. 2 . 17 , Eq. 2.3 and Eq. 2.k may be r e w r i t t e n as R" - R = J£|B - 4 v 2 ( z n B ) Z " V 2 B 2 r 2  i l o B r 2  u o ' o z 3 B + 3 ( z n B ) ' .80 30 . z - + ^ | R 2 l i 3R ,,3B 30j _ _ R 2 3 : ( z 0 B) , R1 9 ( z n B ) 3R 30 (2.18) (2.19) - 9 -The assumption that the harmonic terms are small is maintained in the fo l lowing c a l c u l a t i o n s . For the radia l motion we expand R about the e q u i l i b r i u m o r b i t and express B. and AB as T a y l o r ' s se r ies in x. From Eq. 2.18 we obtain x" + v 2 x R 2 ^ + 2XR ^ + x _ k l W - z i k v V i ) eo B 0 eo B 0 B 0 9R 0 r 2 B 1 o D o + X^  (B, . 2 R A n 3B; R 2 3 2 B : + 'eo + "eo Br 3R 2B 0 9 R Z J -z2 ikv2Bi 1 9(z n B:) r 2 B 1 0 D0 36 <-<5Bo 96 (2.20) where we kept only up to second order terms in x, z , z Q and AB. We then use Eq. 2.16 to include cont r ibut ions due to dev ia t ion from a f l a t median plane. To a second order expansion in x , z , AB and e n , we obtain x" + v 2 x = R 2 AB AB e o B f R 2 3(AB) f R 2 B B 0 9R - z eo_ r 2 o V / B i £ o , p C O S P 0 0 D0 P 2 B ; R~, 2R 2 „ 7 2 ^ o , P cospe + T ^ i e i . p cospe + -ifVz .p cospe O O 0 0 0 0 + x^ + 2R^ 9Bj_ 9R + Reo 2B 0 9R 2J z 2_Mc_ V 2 B  2^o - z ' fpB 9B; •0 ' e sinpe - —~ *-en n cosp6 t r § B 0 °»P r 2 B 0 96 °»P 3 B ; zz ' r 2 B • o D o (2.21) where the p refers to the harmonics of the twist in the median plane. In the case of the v e r t i c a l motion, we assumed prev ious ly that the radial motion was confined to the e q u i l i b r i u m o r b i t . Since the p a r t i c l e executes radia l betatron o s c i l l a t i o n s , we must expand R about the e q u i l i b -rium o r b i t in Eq. 2.19. We a lso includethe dev ia t ion 2.15 from the ideal - 10 -f i e l d , and we o b t a i n „ + V 2 Z = . 4 3 ( Z n B i ) _ R | Q . 3 ( Z O A B > + ^n. _ L 3 < z o B t > 3R 3R 39 B 0 30 + 3Reo 1 8 ( z 0 A B ) 90 B Q 38 Z 'Rln 3(AB) 9R P i 0 L 3(AB)' ^ B 0 9R 96 B 0 98 . R e o 3 ( * o B i ) , , 1 3 U 0 B i ) 2x + x 1 B 0 3R B n 98 R p r i 9B; 1 3B; + 2 z x - f 2 - — L + z x 1 — — L B 0 3R B 0 39 (2.22) Then, w i t h Eq. 2.16 our f i n a l e x p r e s s i o n becomes R ; L 3B; Z M + V 2 Z = _J^o_ ^ l i z B 0 3R € 0 n COSpS ~ c vB;e-. n COSp8 ~ — — £ 0 n COSD8 °'P R- 1 X ' P B 0 3R , P RjL B 0 R 2 + - ^ A B e. cospe + 1 3 R e o 9B| Bo 1 ,P B 0 36 36 e o , p c o s p Q p 9 R e o n • Q . 1 8 R e o 3(AB)_ B J E 0 D s i n p 6 + • e n D cospe B 0 38 1 °' p B 0 36 38 0 >p p 9 Reo A R A Be,, n s i n p S - z B 0 38 0 , p 'R|0 3 ( A B ) 1 3 R e o 9(AB) B 0 3R B 0 38 f R l ^ 3B: - x I B Q 3R 2 Rp/-, - e 1 ( p cospe + — — B ] e 2 , p cosp9 2R 0„ 3B. 2R C + — £ 2 . — L e cospe + — ^ B;e, n cospS B 0 3R °» p B Q 1 *> p 1 9 Reo . ^ B 0 98 98 p 9 R e o . „ cospe + B. £ i cospe + x' f 1 9B; p B J 1 9 R e o sn _ cospe - — ,s r. n s i n p 6 + ~ ~ B : e i n cospS B 0 96 °' p B 0 P B O 9 9 (2.23) - 11 -where we neglected terms of the t h i r d and higher orders in z , x , AB and e . Eq. 2.21 and Eq. 2 . 2 3 descr ibe the forced o s c i l l a t i o n s of the p a r t i c l e in the radia l and v e r t i c a l d i r e c t i o n . We can now invest igate the various i n t r i n s i c resonances due to n o n - l i n e a r i t i e s as well as resonances due to magnet imper fect ions. 2 . 2 Resonances The simplest case of resonance a r i s e s when an e x t e r i o r in ter ference force due to a per turbat ion in the f i e l d amplitude is added to our free o s c i l l a t i o n . Consider ing only the f i r s t term on the r ight-hand s ide of t h Eq. 2 .21 , we expand AB in a Four ier se r ies and we se lec t the p component. Then Eq. 2.21 is wr i t ten x" + v x x = | ^ B p cospe (2.24) where B^ is the amplitude of the per turbat ion in the magnetic f i e l d . The s o l u t i o n of the homogeneous part of Eq. 2.21 is given by x = A v cos(v 6 + cf>) (2.25) X X where A is the amplitude of the radia l o s c i l l a t i o n . Then, the r ight-hand X s ide of Eq. 2.24 w i l l o s c i l l a t e with the same frequency v whenever v x = p (2.26) i . e . when is an in teger . S i m i l a r l y , by adding only the t h i r d term on the r ight-hand s ide of Eq. 2.21 to our free o s c i l l a t i o n , we may wr i te x" + v 2 x = x ^ ° - ^ E - cospe. (2.27) x B o "9 R We then use f i r s t order per turbat ion theory to replace x in the r i g h t -hand s ide of Eq. 2.27 by the s o l u t i o n 2 .25, and we transform the product - 12 -of harmonics into a sum. We f ind that resonances occur when v x = p / 2 . (2.28) While the integral resonances are s e n s i t i v e to a f l a t harmonic e r r o r , the half--integral resonances are p r imar i l y dr iven by gradient e r rors in the guiding f i e l d . From Eq. 2.23 we see that resonances due to gradient imper-fec t ions a lso appear in the v e r t i c a l motion; s i m i l a r l y , the integra l resonances appear when a twist in the median plane is present . The misalignment of the median plane may, on the other hand, i n t r o -duce l i n e a r coupl ing terms in both the radia l and the v e r t i c a l motion. Keeping only the f i r s t term in x on the r ight-hand s ide of Eq. 2 . 2 3 , we wr i te " i f 7^" • l , P * , » 8 - ( 2 - 2 9 ) We then replace x on the r ight-hand s ide of Eq. 2.29 by the s o l u t i o n of the rad ia l free o s c i l l a t i o n , and we f ind that linear coupled resonances occur when v ± v = p. (2.30) X z v This type of resonance may a r i s e from the term propor t iona l to zV 2Bj in Eq. 2 . 2 1 ; the d r i v i n g harmonic is the p*"^  harmonic of the slope of the twist in the median plane. A corresponding coupl ing term is found in the equation for the radia l motion. We consider f i n a l l y the non-linear resonances. Some of those resonances a r i s e from magnet e r r o r s , others are i n t r i n s i c resonances due to non- l inear terms in the equations of motion. The general case of a non - l inea r term may be wr i t ten as - 13 -coupl ing term = e x" z m cospO (2.31) where we chose (n-1) rather than n for greater s i m p l i c i t y in the f i n a l express ion . The equation for the rad ia l motion becomes x" + v x x = e x"" 1 z m cospG (2.32) x" + v 2 x = e c o s n - 1 v x 0 c o s m v z 6 cospO. (2.33) Transforming the product of harmonic funct ions into a sum, we f ind that the cond i t ion for resonance is f u l f i l l e d i f nv ± mv = p (2.34) x z r where n, m, p=0, ± 1 , ± 2 , . . . The odd values of m correspond to median plane e r rors s ince they a r i s e o n l y , in the radia l motion, from the expres-s ion 2.16. Eq. 2.34 contains a l l the previous types of resonances, namely resonances 2.26, 2.28 and 2.30. We can show that Eq. 2.34 may be obtained from a d r i v i n g term l i k e e x" z m ^ cosp9 in the v e r t i c a l motion. Some of the resonances descr ibed by Eq. 2.34 are schemat ica l ly shown in F i g . 2.1. The hor izonta l and v e r t i c a l l ines correspond to integral and h a l f - i n t e g r a l resonances, while the other s t ra igh t l ines correspond to coupled resonances. We c a l l | n | + |m| = N the order of the resonance. It is e a s i l y seen, from the expressions 2.32 and 2.34, that a resonance o f order N is d r i v e n , in f i r s t o r d e r , by a term of order (N-1) t h in the equations of motion. The d r i v i n g harmonic is the p Four ier component of the f i e l d or magnet imper fect ion . It has been shown by S t u r r o c k 5 t h a t , in most c a s e s , dangerous i n s t a b i l i t i e s a r i se only i f n and m are of the same s i g n , i . e . when we have a "sum resonance". Furthermore, i f j n j + J mI > 4, the motion is usua l ly s t a b l e . F i n a l l y , although we have F i g . 2.1. Plot of v x versus v 2 , for f i e l d 01/30/10/70 - 15 -drawn l ines on F i g . 2 . 1 , Sturrock has pointed out that the resonances can cause increases in the betatron o s c i l l a t i o n amplitudes over regions extend-ing some d is tance on e i t h e r s ide of these l i n e s . The dashed l ine on F i g . 2.1 represents the expected va lues , for TRIUMF, of v versus v a f t e r shimming, while the dotted 1ine shows the ' z x 3 ' working path for f i e l d 01/30/10 /70 before shimming. We observe tha t , at low e n e r g i e s , the value of the frequency for the radia l betatron o s c i l l a -t ion is c lose to one. S i m i l a r l y , we cross the coupled resonance v x - v z = 1 around 150 MeV and, at higher energ ies , we cross the resonance = 1.5- These three regions are s e n s i t i v e to f i e l d or to magnet imper-fec t ions and w i l l be discussed in more de ta i l in the fo l lowing chapters . - 16 -3. THE CENTRAL REGION 3.1 Introduction In the centra l por t ion of the cyc lo t ron the resonance = 1.0 is d r i v e n , in f i r s t o rder , by a f i r s t harmonic imper fect ion . The f i r s t harmonic is superimposed on the dominant terms in the Four ier expansion of the magnetic f i e l d , namely the s ix th harmonic and i ts m u l t i p l e s . In genera l , the e f f e c t of the imperfect ion is to increase the amplitude of the radia l o s c i l l a t i o n . We set our to lerances according to the maximum acceptable value of th is betatron o s c i l l a t i o n . The l i m i t is introduced from considera t ions on beam q u a l i t y , such as the energy r e s o l u t i o n . The second harmonic can a lso dr ive the resonance v = 1 . 0 but i t s e f f e c t is x less important. We w i l l , however, consider both imper fect ions. 3.2 S e n s i t i v i t y to a F i r s t Harmonic 3.2.1 Introduction A general ana lys is of the e f f e c t s of f i r s t harmonic f i e l d e r rors on the radia l o s c i l l a t i o n s has been made by Lawson. 7 However, an approximate treatment can be obtained using the fo l lowing simple procedure. The equation for the rad ia l motion in the presence o f a f i r s t harmonic fo rc in g term may be wr i t ten x" + v 2 x = b, cos(0 + cK) (3.1) X ^ ^ where x is the rad ia l displacement from the e q u i l i b r i u m o r b i t in units of the o r b i t r a d i u s , b 1 is the amplitude of the f o r c i n g term, i . e . the r a t i o of the amplitude of the f i r s t harmonic B 1 to the average f i e l d B, and ^ is the azimuth of the peak of the d r i v i n g harmonic. The general s o l u t i o n of Eq. 3-1 is - 17 -x = b l cos (6 + (}>!)+ A cos (v 9 + if). (3-2) ( v x - l ) x The p a r t i c l e precesses in a c i r c u l a r path about an e q u i l i b r i u m o r b i t centre displaced by a distance d j , in inches, given by H - R e o B l / , , N d l ~ F ( v F T 7 ( 3 - 3 ) where R is the radius of the e q u i l i b r i u m o r b i t we consider and v is the eo ^ x frequency of the radia l o s c i l l a t i o n at that rad ius . When (VX"U is s m a l l , the ion bunch w i l l precess slowly about th is d isp laced o r b i t centre and, a f te r N t u r n s , the bunch w i l l be d isp laced by a d is tance d , v where d* = 2dl s in TT(V x -1)N = 2^ N d x ( v x - l ) . (3.4) t h If the f i r s t harmonic bump is suddenly turned o f f a f te r the N t u r n , d" w i l l be the coherent betatron amplitude which we c a l l A . In other words, a step f i r s t harmonic bump of Bj gauss between rad i i R^ and RR w i l l produce a betatron o s c i l l a t i o n amplitude of , ( 3. 5 ) x B(vx+1) This assumes that v is constant between Rn and R n ; N is the number of x A B' turns made between R^ and R R . In f a c t , a step-shaped f i r s t harmonic is not r e a l i s t i c for TRIUMF; the steepest r i se or f a l l that we would expect is from 10% to 30% of maximum values in about 10 i n . , or ha l f the magnet gap width . However, Eq. 3-5 is s t i l l a useful approximation i f only a small por t ion of the precession cyc le is made whi le the f i r s t harmonic amplitude is changing. This is true when (\>x"l) is s m a l l . For a continuous f i e l d e r r o r , where the amplitude of the imperfec-t ion harmonic is constant with rad ius , Lawson 7 has der ived a formula for - 18 -the corresponding amplitude of the betatron o s c i l l a t i o n . He obtained a x = 0.65 4 B r ° ° T ( 3 - 6 ) where rOT = — is the cyc lo t ron unit of length and e is the r a t i o of the energy gain per turn to the ion rest energy. 3.2.2 Ca lcu la t ions These were performed using the code GOBLIN and the f i e l d 1/07/04/70. A l l c a l c u l a t i o n s assumed that only the fundamental RF frequency was present , no t h i r d harmonic. When the t h i r d harmonic is present , ions of d i f f e r e n t phases get more nearly the same energy gain per turn and hence acquire s i m i l a r betatron ampli tudes. We f i r s t determined the " idea l path" for ions in (x, p x ) phase space by running GOBLIN backwards from 50 MeV to low energy to get approximate s t a r t i n g c o n d i t i o n s . We then ran forwards, ad jus t ing the s t a r t i n g cond i -t ions so that ions of 0 deg RF phase had no large cusps or loops in t h e i r phase space path and ended up on centre in a 6 - f o l d f i e l d at 50 MeV. This gave us our standard s t a r t i n g co-ord inates for forward runs. We regarded residual wiggles of about 0.05 i n . amplitude as acceptable . We then superimposed our f i e l d imperfect ions and observed the path in (x, p ) space of an ion s t a r t i n g at the standard co-ord inates with 0 deg RF phase. At some energy, beyond the inf luence of the imper fec t ion , the ions would be precessing in phase space in c i r c l e s of roughly constant diameter. The displacement from the ideal path , at a p a r t i c u l a r energy, was considered to be the betatron amplitude gained from resonance. It was found to be almost l i n e a r l y proport ional to the f i r s t harmonic amplitude and almost independent of i t s phase. This can be seen on F i g . 3.6 where a - 19 -bump of 0 .2 G bu i lds up an o s c i l l a t i o n of 0.09 i n . amplitude while a bump of O.k G produces an o s c i l l a t i o n of 0.19 i n . F i g . 3-1 i l l u s t r a t e s some resu l ts obtained in the two cases we s t u d i e d . The f i r s t was for a f i r s t harmonic whose amplitude was uniform with rad ius ; an amplitude of 0.2 G produced a radia l amplitude of 0.20 i n . compared with the 0.18 i n . pred ic ted by Eq. 3 .6 . The other case was for a be l l - shaped bump with a ha l f -w id th of 20 i n . F i g . 3-1 shows the resu l ts when the bumps were centred at 30 i n . The f i e l d bump of 0.5 G bu i lds up an o s c i l l a t i o n of 0.15 i n . whi le our previous Eq. 3-5 predic ted an ampl i -tude of 0.13 i n . S i m i l a r l y , the 2.0 G bump produces an o s c i l l a t i o n of 0.65 i n . We have made a de ta i l ed inves t iga t ion of the e f f e c t of f i r s t harmonic bumps at rad i i less than 150 i n . on the behaviour of the beam. We used the fact of l i n e a r i t y to get more accurate resu l ts with larger bumps so that the 0.05 i n . residual o s c i l l a t i o n of the " idea l case" was n e g l i g i b l e . F i g . 3-2 shows the resu l ts of these c a l c u l a t i o n s in the form of the amplitude of a be l l -shaped f i r s t harmonic bump 20 i n . wide placed at d i f f e r e n t rad i i and necessary to produce a betatron amplitude of 0.2 i n . ; th is was considered to be the worst that could be accepted. The resu l ts show that the most s e n s i t i v e region is that around 60 i n . where a bump of 0 .2 G produces an o s c i l l a t i o n of 0 .2 i n . of ampli tude. This las t resu l t is approximately equal to the amplitude gained from a continuous f i e l d e r r o r . The behaviour of the f i e l d strength can be understood by cons ider ing the motion of the centre p o i n t s . We have seen that when (v -1) is small the ion bunch precesses slowly about the d isp laced e q u i l i b r i u m o r b i t s . The amplitude acquired is given by Eq. 3«5- At very low e n e r g i e s , the lack of FIELD 01/07/04/70 ' m / * I I I I -- 0.4 0.2 1 I •0.4 «-0.2 Px ( ' n « ) o o o B X • 0.2G (constant bump) x * * 0.5'G bump centred at 30 in • • — • 2.0Gbump centred at 30 In "ideal path" E Q - 0 . 8 MeV \ C ^ . ' 0.6 I 0.8 0 O O' x ( i n . ) - - -0.4 F ig . 3 . 1 . Phase space for accelerated ions (0.8 MeV to 28 MeV) of 0 deg RF phase with various f i r s t harmonics present. The markers (/) Indicate 5 MeV step. F i g . 3.2. The amplitude of a 20 i n . wide ( R B ~ R A) f i r s t harmonic bump required to produce radial betatron amplitudes of 0.2 i n . - 22 -s e n s i t i v i t y to a f i r s t harmonic bump is explained by the fact that the ions only make k or 5 turns over the f i r s t 30 i n . When v x i n c r e a s e s , the o s c i l l a t o r y behaviour of F i g . 3.2 can be explained by the ions making an integral number m or (m + 1/2) complete precession cyc les in the bump. A p e s s i m i s t i c assumption would be for the ions to make (m + 1/2) cyc les in a "square" bump; in th is case , the amplitude acquired would be twice the displacement given in Eq. 3-3- The to lerance on the f i r s t harmonic bump i s , in that case , t i g h t e s t , and we obtain a minimum point on the f i g u r e . This assumption i s , however, p e s s i m i s t i c because, for real bumps with "rounded c o r n e r s " , the p a r t i c l e s tend to " fo l low" the d isp laced e q u i l i b r i u m o r b i t when is l a rge . The p r e d i c t i o n of th is p e s s i m i s t i c assumption and of Eq. 3-5 are shown by the dashed l ines in F i g . 3.2. In p r a c t i c e , the cyc lo t ron f i e l d w i l l be shimmed to give a f i r s t harmonic of less than 1 G. Then, several sets of s i x harmonic c o i l s , each extending over 60 deg in azimuth, w i l l be used to compensate for the e f f e c t of the remaining amplitude. The e f f e c t of these harmonic c o i l s is considered in the next s e c t i o n . Harmoni c Coi 1 s 3.3-1 O s c i l l a t o r y F i r s t Harmonic A change in the current of any harmonic c o i l a f f e c t s the c o n f i g u r a -t ion of the magnetic f i e l d in the region where the c o i l is present . The f i e l d con t r ibu t ion from any set of harmonic c o i l s w i l l be used c h i e f l y to cancel the f i r s t harmonic e r ror f i e l d in the i r v i c i n i t y . F i g . 3-3 shows, however, that the harmonic c o i l s may not be able to cancel th is i n t r i n s i c f i r s t harmonic at every po in t . The amplitude of the residual f i r s t harmonic as a funct ion of radius w i l l tend to o s c i l l a t e between p o s i t i v e and negative va lues , i . e . i t s azimuth s h i f t s by 180 deg. We assume a Bi (Q). - 23 -B 1 ' B C 0 I L S ( G ) c) —»|lO in. J-— 2.0 + •2.0 F i g . 3.3- a) F i r s t harmonic e r r o r f i e l d b) Net c o i l f i e l d c o r r e c t i o n s c) Res idual f i r s t harmonic - 2k -purely s inuso ida l azimuthal dependence for the c o i l f i e l d s . As a "bad" c a s e , we chose to consider an i n t r i n s i c f i r s t harmonic that var ied r a d i a l l y in such a way that i t was always out of step with the c o r r e c t i n g f i e l d produced by the c o i l s , i . e . the e r ro r is always zero at a c o i l centre and an extremum h a l f way between the i r cen t res . This is not a l i k e l y s i t u a t i o n . The residual f i r s t harmonic is shown in F i g . 3-4; the azimuth is constant with radius (apart from switches of ±180 deg). The betatron amplitudes r e s u l t i n g from various amplitudes of th is o s c i l l a t o r y f i r s t harmonic ( for ions of 0 deg RF s t a r t i n g phase) are given in Table 3-1; they are s i m i l a r to those from a f l a t f i r s t harmonic. Energy (MeV) O s c i l l a t i n g Bump Amplitude (gauss) 0.1 0.2 0.5 35 0.12 i n . 0.23 i n . 0.57 i n . 50 0.10 i n . 0.18 i n . 0.46 i n . Table 3.1- Betatron amplitudes for 0 deg RF phase ions It is p o s s i b l e , as we w i l l see , to cancel the f i n a l betatron ampl i -tudes at some energy for ions of any given RF phase, by a s u i t a b l e choice of an add i t iona l harmonic c o i l f i e l d . We chose to look at the set of c o i l s extending from 63 i n . to 89 i n . in r a d i u s , and we assumed that they produced the bump p r o f i l e shown in F i g . 3-5. We acce lera ted ions from a low energy to 35 MeV at a radius of approximately 107 i n . At that energy, the ions are outs ide the region of s e n s i t i v i t y to a f i r s t harmonic and, i f centred there , should remain centred during fur ther a c c e l e r a t i o n . Thei r end points in (x, p ) space are given in F i g . 3-6 as funct ions of the bump - 25 -Amplitude ( a rb i t r a ry uni ts ) 1 .0.. / \ / 1 / I I \ / 1 f i i ;« / i i 1 / H 1-I \ 5? | / I -1 .0" l ' 100\ I R (m.) ! v 1 |ooil#l|co! 1 #^  c o i l #3 j c o i l #4 | c o i l #5 \ 1 , \ 1 / M / / c o i l #6 | F i g . 3.4. P r o f i l e of o s c i l l a t o r y f i r s t harmonic Amplitude ( a rb i t r a ry uni ts ) 1 .0" / \ / \ / \ / \ \ —v-50 60 70 80 90 100 110 R ( in.) C O I L F i g . 3-5. Bump P r o f i l e 0.2--0.1 -• / / \ N \ •0.2 -• p x ( in.) 180' / B © 180' X ^ 9 0 ° \ \ \ ® 60' 0.1 0.2 A 0 E 0 + 0.3 + I, ® 45' I s / 3 0 0 I / <S> 15° \ 2 7 0 V \ ® 0° X Bump of F i g . 3.5 - Amplitude 0.2 G ® Bump of F i g . 3-5 - Amplitude 0.4 G (angle denotes bump azimuth) © x ( in.) A - No imperfect ion harmonics ( ideal case) B - O s c i l l a t o r y f i r s t harmonic o f 0.1 G C - O s c i l l a t o r y f i r s t harmonic of 0.2 G D - Point used to estimate bump azimuth and amplitude to cancel E - Result for C + D DQ F i g . 3.6. Pos i t ions at 35 MeV for ions of 0 deg RF phase acce lera ted from low energy - 27 -amplitude and azimuth. The amplitude and azimuth of the displacement are almost l i n e a r funct ions of the amplitude and azimuth of the f i r s t harmonic bump. The end point in the ideal case , when no imperfect ion harmonics are present , is shown as point A. The end point C for an o s c i l l a t o r y residual f i e l d of 0.2 G is a lso shown; we attempted to cancel th is for ions of 0 deg RF phase by an add i t iona l bump between 63 i n . and 89 i n . of 0.47 G at 295 deg. Point D, obtained by drawing a l ine from C through A and s e t t i n g AD = AC, was used to estimate th is bump amplitude to cancel C. The net resu l t is to br ing the ions back to point E. The improvement can be seen by comparing the resu l ts in the f i r s t column of Table 3.2 with those in Table 3-1 fo r 0.2 G. Energy Ion Phase (MeV) 0 deg 15 deg 30 deg 35 0.02 i n . 0.03 i n . 0.19 i n . 50 0.07 i n . 0.08 i n . 0.12 i n . Table 3 .2 . Betatron ampl i fudes. wi th harmon i c coi 1 f i e l d Ions that s t a r t at the same point and the same energy but d i f f e r e n t RF phase w i l l acquire d i f f e r e n t betatron amplitudes p r imar i l y because they make more turns to reach a given energy. Thei r betatron amplitudes w i l l not be cance l led exact ly by th is bump. To invest igate the f i r s t harmonic e f f e c t a lone , i t was necessary to remove the "phase-centre point spread" phenomenon. 8 This was done by running GOBLIN backward to f ind " i d e a l " - 28 -s t a r t i n g points for d i f f e r e n t phases; these s t a r t i n g points were d isp laced from the 0 deg s t a r t i n g point by 0.1 i n . fo r 15 deg and 0.46 i n . for 30 deg RF phase. Ions s t a r t i n g at these pos i t ions were acce lera ted through the 0.2 G o s c i l l a t o r y f i e l d with the c o r r e c t i n g bump, and the resu l ts are given in Table 3-2. The f i n a l amplitudes for ions at 15 deg are qui te acceptab le , and those for ions at 30 deg are acceptable for many purposes. 3 .3 .2 Ions with a Wide Range of RF Phases In cases 'where the harmonic c o i l s cannot cancel the f i r s t harmonic e r r o r f i e l d in the magnet, we would l i k e to know i f we can choose c o i l se t t ings that w i l l reduce the betatron amplitudes acquired to a small value at some energy for ions with a wide range of RF phases. The previous s e c -t ion showed that we could do th is for a narrow phase width by s u i t a b l y powering a s i n g l e c o i l s e t . On F i g . 3-7 we denote as E and 3 the amplitude and phase of the betatron o s c i l l a t i o n induced by the f i r s t harmonic e r r o r . S i m i l a r l y , t", with amplitude y and phase ixt, represents the o s c i l l a t i o n produced by a 1 G f i e l d in a s i n g l e harmonic c o i l placed at a standard azimuth and is c a l l e d the c o i l c o e f f i c i e n t . Then, for each set of c o i l s , each RF phase and a given energy, we have £ = Ar (3.7) where A is a complex number with amplitude A (gauss) equal to the c o i l f i e l d required for compensation of the e r r o r , and phase a equal to the a z i -muthal ro ta t ion required for the s i n g l e harmonic c o i l set cons idered . In p r a c t i c e , each c o i l set cons is ts of s i x c o i l s at 60 deg azimuth in te rva ls and t h e i r r e l a t i v e currents can be arranged 9 to provide a f i r s t harmonic peak at the des i red azimuth. When several c o i l sets are used to cancel E , - 29 -t h e i r e f f e c t s can be superposed so that Eq. 3-7 becomes t = A1T1 + A 2 r 2 + A 3 r 3 + ... (3.8) We separate real and imaginary parts of Eq. 3-8 and obtain E cos 3 = (Y 1 c o s o ) 1 A 1 c o s a 1 - y^inu^sina^ +: ( ) 2 + ( ) 3 + . . ( 3 - 9 ) E s i n 3 = (y^s i nu^A^cosa^ + YiCOSco^jS inotj) + ( ) 2 + ( ) 3 + . . . . (3.10) Then, to keep the equations l i n e a r , we rewrite Eq. 3.9 and Eq. 3-10 as E = ( Y A - Y A ) + ( ) + ( ) + . . . (3.11) x ' x x y y l 2 3 E = ( Y A + Y A ) + ( ) + ( ) + . . . (3.12) y 'y x 'x y l 2 3 The values of E x and E^, at a given energy, are found by superposing the f i r s t harmonic e r ro r on our magnetic f i e l d and measuring the ending point in (x, p ) space of an ion acce lera ted to that energy. The c o i l c o e f f i c i e n t s are obtained in the same way, using a f i e l d of known amplitude in each c o i l . We then solve a set of l i nea r equations to f ind A l x , A - ^ , A 2 X , the amplitudes and phases of the c o i l c o r r e c t i n g f i e l d s . As a very p e s s i m i s t i c case , we chose to cancel the f i r s t harmonic e r ro r shown on F i g . 3-8. The bump has a 2 G amplitude and extends from 35 i n . to 90 i n . This reg ion , as we saw p r e v i o u s l y , is the most s e n s i t i v e to f i e l d imper fect ions. The three sets of c o i l s at 37" < R < 54" , 54" < R < 7 1 " , 71" < R < 89" , r e s p e c t i v e l y , can be used to compensate for the e f f e c t of the f i r s t harmonic imper fect ion . However, the v a r i a t i o n with radius of the f i r s t harmonic e r ro r was chosen so that the c o i l s could not e l iminate i t completely . With s i x c o i l constants , A l x , Ajy ^3y» ° P e n to c h o i c e , i t appears p o s s i b l e to s a t i s f y Eq. 3-11 and Eq. 3-12 - 30 -F i g . 3-8. P r o f i l e of f i r s t harmonic e r r o r - 31 -simultaneously for ions with three d i f f e r e n t RF phases. We chose to consider ions with 0 deg, 25 deg, and kO deg RF phase and t r i e d to centre them at 35 MeV, i . e . approximately 105 i n . A l l c a l c u l a t i o n s were done using the f i e l d 01 /30 /10 /70 . We f i r s t determined ideal s t a r t i n g condi t ions for our three p a r t i c l e s . The i r paths in phase space to 35 MeV are c a l l e d " ideal paths" . We then superposed the f i r s t harmonic e r ro r and found the ending point of each ion at 35 MeV. Each displacement was measured from the " i d e a l " centre point at 35 MeV. S i m i l a r l y , we obtain the values for the c o i l c o e f f i c i e n t s , T, by adding a 2 G f i e l d in each c o i l and measuring the corresponding displacement, as shown in F i g . 3.3. We v e r i f i e d that the amplitude of the displacement was proport ional to the f i e l d amplitude so that the r are constant over the range of amplitudes we were us ing . We a lso checked that the displacement due to two d i f f e r e n t c o i l s is the vector sum of the displacements due to each of these two c o i l s . Once we obtained our set of s i x l i nea r equat ions, we found that a small change in the known parameters was g iv ing r i se to large changes in the s o l u t i o n s . This phenomenon is known as i 1 1 - c o n d i t i o n i n g . Since we cannot measure betatron amplitudes exact ly using probes, we need a less s e n s i t i v e approach. We therefore measured the displacements due to the f i r s t harmonic e r ro r at every 5 MeV step from 15 MeV to 35 MeV, and c a l c u -lated the corresponding values of y x and at those energ ies . We could a lso have improved the s i t u a t i o n by running addi t iona l phases to 35 MeV; however, we chose to get the c o e f f i c i e n t s at d i f f e r e n t energies to reduce the number of GOBLIN runs. The resul ts at 5 MeV steps are given in the fo l lowi ng t a b l e s . - 32 -P x ( in . ) X : C o i I s at 37" « R « 54" 0=CoiIs at 54" $ R ^ 71" • : C o i I s at 71" ^ R ^ 89" x ( i n . ) F i g . 3.9. Displacements from " i d e a l " centre point at 35 MeV due to a 2 G f i e l d in each set of c o i l s , fo r ions of 0 ° , 25° and 40° RF phase - 33 -RF Phase (deg) Energy (MeV) 15 20 25 30 35 0 E x ( in . ) -1.99 -0 .10 0.54 -0.76 -0.10 E y ( in . ) -0 .48 1.84 -1.54 1.43 -1.70 25 E x ( in . ) -2 .03 -0 .36 0.76 -0 .38 -1 .00 E y ( in . ) -0 .92 2.22 -1 .82 1.99 -1 .70 40 E ( in . ) -2 .30 0.45 -0 .66 1.34 -1.68 E y ( in . ) -1 .08 2.46 -2 .25 1.74 1.17 Table 3-3- Components of displacements due to f i r s t harmonic of F i g . 3.8 RF Phase (deg) Energy (MeV) 15 20 25 30 35 0 Y x ( in . /G) -0 .35 0.49 -0 .42 0.37 -0 .48 Y y ( in . /G) 0.41 0 .09 -0.31 0.39 -0 .08 25 Y x ( in . /G) -0 .46 0.38 -0 .25 0.34 -0 .48 Y y ( in . /G ) 0.17 0.37 -0 .50 0.42 0.00 4o Y x ( in . /G) -0.61 0.50 -0 .50 0.63 -0 .14 Yy ( in . /G) -0 .03 -0-.46 -0 .47 0.01 0.74 Table 3.4. C o e f f i c i e n t s for set of c o i l s at 37"$ R < 54" - 2h -RF Phase (deg) Energy (MeV) 15 20 25 30 35 0 Y x ( in . /G) -0.73 0.12 0.17 -0.30 -0.17 Y y ( in . /G) -0.32 0.88 -0.87 0.83 -0.89 25 Y x ( in . /G) -0.72 0.06 0.20 -0.01 -0.60 Y y ( in . /G) -0.42 0.96 -0.93 0.97 -0.67 4o Y Y ( in . /G) -0.79 0.41 -0.43 0.70 -0.69 Y y ( in . /G) -0.43 O.96 -0.95 0.67 0.66 Table 3-5. C o e f f i c i e n t s for set of c o i l s at 5k" 4 R 4 71" RF Phase (deg) Energy (MeV) 15 20 25 30 35 0 Y v ( in . /G ) -0.06 -0.77 0.14 -0.07 O.38 Y y ( in . /G) -0.18 -0.26 0.70 -0.69 0.54 25 Y x ( in . /G) -0.06 -0.77 0.15 -0.29 0.56 Y y ( in . /G) -0.19 -0.28 0.70 -0.60 0.17 ko Y v ( in . /G) -0.07 -O.83 0.35 -0.47 0.12 Y y ( in . /G) -0.21 -0.11 0.40 -0.01 -0.55 Table 3.6. C o e f f i c i e n t s for set of c o i l s at 7V[ < R ^ 89" - 35 -We thus obtained a set of 30 l i n e a r equations for our s i x unknowns. A least squares f i t t i n g program was wr i t ten to minimize 30 I W(I) E(I) - I. A(J) • Y ( I , J ) 1=1 J=l (3.13) where W(l) is a weighting term. With weights of 0 .5 , 1, 2, 3, 3 fo r energies of 15 MeV to 35 MeV, the amplitudes and phases of the harmonic c o i l f i e l d s to cancel the f i r s t harmonic e r ro r were found to be 1st co i1 2nd coi1 3rd coi1 A (G) 0.61 2.07 0.75 a (deg) 6.3 4.1 -24.0 Table 3-7. Harmonic c o i l co r rec t ing f i e l d s We determined that the c o i l requirements were no longer s e n s i t i v e to input parameters. The so lu t ions A(j) were then used to c a l c u l a t e the d i s p l a c e -ments from the ideal o r b i t at energies where the c o e f f i c i e n t s were c a l c u l a t e d , s ince the A ( J ) ' s obtained are the best s o l u t i o n but may not resu l t in abso lu te ly zero f i n a l displacement. The i n i t i a l displacements from the ideal o r b i t due to the f i r s t harmonic e r ro r are given in Table 3-8. - 36 -E / (MeV)/ RF / (deg) 15 20 25 30 35 0 2.05 i n . 1.84 i n . 1.63 i n . 1.62 i n . 1.70 i n . 25 2.23 i n . 2.25 i n . 1.97 i n . 2.03 i n . 1.97 i n . 40 2.54 i n . 2.50 i n . 2.34 i n . 2.20 i n . 2.05 i n . Table 3.8. Displacements from ideal o r b i t due to f i r s t harmonic of F i g . 3-8 The f i n a l d isplacements, when both the f i r s t harmonic e r r o r and the c o r r e c t -ing f i e l d s are present , are given in the fo l lowing t a b l e . E / (MeV)/ / (deg) 15 20 25 30 35 0 0.26 i n . 0.18 i n . 0.02 i n . 0.07 i n . 0.02 i n . 25 0.28 i n . 0.10 i n . 0.04 i n . 0.03 i n . 0.14 i n . 40 0 .27 i n . 0.11 i n . 0.07 i n . 0.09 i n . 0.14 i n . Table 3-9- Displacements from ideal o r b i t when f i r s t harmonic e r ro r and c o r r e c t i n g f i e l d s are present We see that the displacements have been reduced everywhere by at least a fac tor 8 and in most cases by much more. We then ran GOBLIN to look at the - 37 -phase space for acce lera ted p a r t i c l e s . The resu l ts for ions of 0 deg and kO deg RF phases are shown in F i g . 3-10 and F i g . 3.11. The f ina l amplitudes can be compared to the amplitudes of the residual o s c i l l a t i o n s obtained in the " i d e a l " cases . A lso shown in F i g . 3-12 is the resu l t for an ion of 30 deg RF phase. This las t f igure shows that our so lu t ion is a lso v a l i d for ions with intermediate RF phases. A l s o , s ince we have shown the e f f e c t s of a f i r s t harmonic to be l i nea r in our machine, we do not expect any d i s t o r -t ion of the emittance from these e f f e c t s . Our s o l u t i o n does not reduce to zero the amplitude of the f i n a l o s c i l l a t i o n at 35 MeV. This amplitude depends on the form of the f i r s t harmonic e r ro r present and could be worse or bet ter for a d i f f e r e n t shape. In th is ana lys is we e l iminated inter ference from the "phase-centre point spread" phenomenon8 by f ind ing an " i d e a l " s t a r t i n g point for each phase. However, we have shown that i t is poss ib le to use the method descr ibed above to centre at a given energy ions with d i f f e r e n t RF phases and iden t i ca l s t a r t i n g c o n d i t i o n s . The required c o i l f i e l d s are obtained in the same manner as in the case of an i n t r i n s i c f i r s t harmonic imperfec-t i o n . In other words, we can reduce the cent r ing e r r o r , no matter what the cause. 3.k S e n s i t i v i t y to a Second Harmonic In a previous study, Hagedoorn and V e r s t e r 1 0 descr ibed the radia l motion of the ion on the basis of the Hamiltonian formal ism. The inf luence of small f i e l d e r rors introduced by f i r s t and second harmonic was a lso cons idered . They showed that radia l i n s t a b i l i t y caused by a second harmonic of amplitude B^ is present i f 2B kB 3R . p x ( in . ) A -+ 0.2 Ideal path 0.1 < ( > * 7& L . - V - E 0 = 1 MeV •0.1 * v.. A «\ .0-3 0.4 0.5 0.6 •-0.1 1 0.7 0.8 x ( in . ) U J co -0.2 X F i g . 3.10. Phase space p lot for accelerated ions (1 MeV to 35 MeV) of 0 deg RF phase with both the f i r s t harmonic e r r o r and the harmonic c o i l c o r r e c t i n g f i e l d s present . The markers (/) indicate every 5 MeV s tep . A P x ( in . ) / f 1 • 1 T '• ' • 0.2 V Ideal path _ _ _ _ _ _ _ _ _ ^ - E o -. - = = » - - • - ~ " 1 1 I I i 1 MeV i \ i • \ \ » \ \ • \ \ • \ - 0 . 1 f . • - — • < . \ 0 . 2 •. . .' J *A* / I l l I I 0.4 0.5 0.6 0.7 0.8 i X F i g . 3.11. Phase space p lot for acce lera ted ions (1 MeV to 35 MeV) of 40 deg RF phase with both the f i r s t harmonic er ror and the harmonic c o i l c o r r e c t i n g f i e l d s present . The markers (/) indicate every 5 MeV s tep . P x ( in . ) 0.2 X Ideal path ^ r . E Q = 1 MeV •0.1 \ 0.4 0.5 0.6 0.7 0.8 •0.2 x (in F i g . 3 .12 . Phase space p lot for accelerated ions (1 MeV to 35 MeV) of 30 deg RF phase with both the f i r s t harmonic e r ror and the harmonic c o i l c o r r e c t i n g f i e l d s present . The markers (/) ind icate every 5 MeV s tep . - 41 -On F i g . 3-13 we have p lo t ted the le f t -hand side of th is l as t expression for var ious values of v , using the f i e l d 01/30/06/70. The region on the r i g h t -X hand s ide of each l ine is the i n s t a b i l i t y region for that p a r t i c u l a r value of v . The corresponding values of the amplitude and gradient of the second harmonic w i l l , in these reg ions , lead to an imaginary value of v , i . e . complete de focus ing . Using our code CYCLOP to v e r i f y Eq. 3-14, we obtained imaginary values of with second harmonics whose amplitudes and gradients were wi th in 20% of the values obtained from Eq. 3-14. Thus, F i g . 3.13.can be considered a useful guide to set our to lerances on the second harmonic. Even i f the second harmonic is not s u f f i c i e n t to render imaginary, i t can s t i l l d i s t o r t the s t a t i c phase space e l l i p s e ; we consider a reasonable to lerance to be about 1/4 of the c r i t i c a l values given in F i g . 3.13. F i n a l l y , when we considered a mixture of both f i r s t and second harmonic e r r o r s , we observed that the s h i f t of the e q u i l i b r i u m o r b i t , at low energy, was st rongly dependent on the r e l a t i v e phase of the harmonics. As i t s tands, Eq. 3-3 does not take into account the phase of the harmonics. However, we can understand th is phenomenon with the use of the Hamiltonian derived by Hagedoorn and Verster to descr ibe the motion of the o r b i t cent re . The con t r ibu t ion of the f i r s t and second harmonic to the Hamiltonian may be wr i t ten (3.15) where x is the displacement from the e q u i l i b r i u m o r b i t radius R e o > and fy^ and if2 are the phases of the two harmonics. For greater s i m p l i c i t y , Eq. 3.15 has been wr i t ten without the terms invo lv ing the gradients of the - 42 -F i g . 3.13. L imi ts of rad ia l s t a b i l i t y for second harmonic e r r o r f i e l d - 43 -of the harmonics. The new p o s i t i o n of the o r b i t centre fol lows from •Tp- = 0. We obtain o X B R B B R x * - 1 , e ° cos (6-4,.) - I 2 2 e ° cos(6-e>1)cbs(2e-((>J + . . .(3.16) 2 B ( v x - l ) 1 4 B Z ( v x - l ) 2 1 2 Since ( v 2 - l ) - 2 ( v x - l ) , we v e r i f y that Eq. 3-3 and Eq. 3-16 are equivalent when no second harmonic is present . Table 3-10 and Table 3.11 show the displacements obtained from numerical estimates using CYCLOP and from Eq. 3.16 for ions of 1 MeV and 3 MeV. Phases of Imperfection Harmon i cs cj>l = (j>2 = 0 deg ef>l = 0 deg <f>2 = 90 deg <(>! = 90 deg 02 = 0 deg <h = <f>2 = 90 deg CYCLOP 1.98 i n . 3.04 i n . 4.02 i n . 2.93 i n . Eq. 3.16 2.08 i n . 3.12 i n . 4.16 i n . 3.12 i n . Table 3-10. Displacements of the e q u i l i b r i u m o r b i t centre due to f i r s t and second imperfect ion harmonics of 1 G and 2 G, respec-t i v e l y , for ions of 0 deg RF phase at 1 MeV and v x = 1.001 Phases of Imperfection Harmon i cs <\>l = = 0 deg §i = 0 deg <f>2 90 deg <J>! = 90 deg <j>2 = 0 deg <I>1 = i>2 = 90 deg CYCLOP 1.11 i n . 1 .22 i n . 1.33 i n . 1 .23 i n . Eq. 3.16 1.24 i n . 1.35 i n . 1 .46 i n . 1.35 i n . Table 3-11- Displacements of the e q u i l i b r i u m o r b i t centre due to f i r s t and second imperfect ion harmonics of 1..G and 2 G, respec-t i v e l y , for ions of 0 deg RF phase at 3 MeV and v x = 1.004 - -We assumed a maximum value for the displacement, and the amplitudes of the f i r s t and second harmonic were of 1 G and 2 G, r e s p e c t i v e l y . The displacement of the e q u i l i b r i u m o r b i t centre is a minimum when both harmonics are in phase at 0 deg. In the cases where tj) 1 = 0 deg, $2 = 90 deg or cj^ = <j>2 = 90 deg, the second term on the le f t -hand s ide of Eq. 3-16 goes to zero so that the displacement of the e q u i l i b r i u m o r b i t centre is due only to the f i r s t harmonic e r r o r . In a l l the cases we cons idered , the accuracy of Eq. 3-16 was wi th in 10%, for small values of ( v x ~ l ) . Because th is s h i f t decreases rap id ly with the increas ing energy and a lso because the ion spends jus t a few turns in the centra l reg ion , the displacements can be t o l e r a t e d . - 45 -4 . THE RESONANCE v x - v z = 1 4.1 Mathematical Descr ip t ion The second order l i n e a r coupled resonance v -v = 1 wi11 be r x z encountered at about 150 MeV, with v = 1.2 and v - 0 . 2 . This resonance, ' x z ' l i ke others invo lv ing an odd mul t ip le of v , is dr iven by an asymmetry in the median plane of the c y c l o t r o n . For example, the asymmetry may be caused by a t i l t of the magnet s e c t o r s . Only the f i r s t harmonic of such a t i l t or twist is of importance in d r i v i n g th is resonance. To descr ibe the e f f e c t of th is resonance, we reproduce a por t ion of the ana lys is due to J o h o . 1 1 He assumes a Hamiltonian of the form H = HQ + H 1 where i f 1 1 f ) H Q = y v x * 2 + Px + T v z z 2 + Pz = h a r m o n ' c o s c ' 1 l a t o r term (4 .1) H 1 = a x z cos(6 -0 o ) = coupl ing term (4 .2) He then constructs a sequence of canonical transformations which e l iminate the 6-dependent parts of the Hamil tonian. The f i r s t transformation i n t r o -duces the s o - c a l l e d ac t ion-ang le v a r i a b l e s . In terms of these var iab les (<J> , J , cj> , J ) , the o ld var iab les are given by y\ /v i Z Px = / 2 v x J x s i n(cb x-v x o 6 ) z = cos(<f»z-vz6) Pz = / 2 v z J z sin(<j)z-v ze) where v is a reference frequency such that A v ^ v - v - 1 = v - v = 0 X O - 1 7 x z x x o - 46 -when we a r e e x a c t l y on the resonance. The f i n a l H a m i l t o n i a n i s w r i t t e n H* = - A v J * - a " ' cos(<J>x-<f>_-e0) . (4.4) r J x i i 2 r J z i [2v zJ T h i s H a m i l t o n i a n i s independent o f 9 and i s t h e r e f o r e a c o n s t a n t o f the m o t i o n . I f the c u r v e s H* = c o n s t a n t i n phase space a r e c l o s e d , t he motion must be s t a b l e ; o t h e r w i s e , i n s t a b i l i t y may o c c u r . Joho has shown t h a t , i n the c a s e o f t h e d i f f e r e n c e resonance v x - v z = 1 the c u r v e s i n ( x 0 , z 0 ) space a r e g i v e n by v v x 2 + v z 2 x 0 Z 0 = c o n s t a n t (4.5) where J Q i s the t o t a l " e n e r g y " o f t h e sys t e m and x 0 , z 0 a r e the i n s t a n t a n e o u s a m p l i t u d e s o f the o s c i 1 l a t i o n s i n u n i t s o f the average r a d i u s , and a r e de t e r m i n e d by the i n i t i a l c o n d i t i o n s . Eq. 4.5 r e p r e s e n t s an e l l i p s e i n ( x 0 , z Q ) s p a c e ; the motion i s p e r i o d i c and t h e r e f o r e s t a b l e , i f one a m p l i t u d e i n c r e a s e s , t he o t h e r d e c r e a s e s . There i s e f f e c t i v e t r a n s f e r o f energy between t h e two m o t i o n s . The t o t a l a m p l i t u d e i n c r e a s e depends, o f c o u r s e , on J Q , the t o t a l a v a i l a b l e e n e r g y . The n e x t s t e p i s t o o b t a i n the e x p r e s s i o n s f o r the maximum a m p l i t u d e i n c r e a s e s p e r t u r n . We f i r s t w r i t e t he H a m i l t o n i a n i n terms o f the n o r m a l i z e d a c t i o n v a r i a b l e p v and the a u x i l i a r y v a r i a b l e p_ where A Z D = J z = v z z 2 = 1 - D (4 7) Pz - U J 2 | J o | -o 1 Px- ^.7) The H a m i l t o n i a n becomes K (p x , $ ) = - A v p x - K p x p | cos$ (4.8) - 47 -where $ = d> - cb - 9„ is the r e l a t i v e phase of the two o s c i l l a t i o n s . The T x z o ^ quant i ty K is c a l l e d the " c r i t i c a l frequency" and is given by K = a i ( v 2 - v 2 ) _ 0 4 ( v x + v z ) 2 ( v x v z ) ^ 2 / v x v z (4.9) where is the f i r s t harmonic of the t i l t a(e) and is obtained from a , = — 1 TT 2TT a(e) c o s ^ - e ^ de. (4.10) This c r i t i c a l frequency gives roughly the range of Av for which the resonance w i l l be e x c i t e d . The equations of motion fol low from p x = -3K/3$ and $' = 3 K / 3 p x , where the prime denotes d i f f e r e n t i a t i o n with respect to 6. We obtain P' = x x x -KpI p | s 1 n$ X ( P Z / P X ) 2 ~ ( P X / P Z ) cos$. (4.11) (4.12) Now, Eq. 4.6 and Eq. 4.7 give and P X / P x P z / P x = 2 x o / x o v z z o v x x o (4.13) (4.14) so that Eq. 4.11 becomes x' = ^ X o 2 v X J "z n sincb. (4.15) The increase in x per turn is given by 2 ™ ^ and the maximum occurs when sin$ = 1 , i . e . 3x 3n = -TTK max Vzl k _ frcti ( v x + v z ) (4.16) - 48 -S i m i l a r l y , the maximum rate of growth for the v e r t i c a l o s c i l l a t i o n is given by 3z 3n |vzJ xo " 2 v z Xo- (4.17) max The above resu l ts are only v a l i d in a s t a t i c s i t u a t i o n , i . e . for ions ro ta t ing in the resonance region with a f ixed energy. Since i t is associa ted with $ = I T /2 , the maximum amplitude increase w i l l be ach ieved, phase of 90 deg. In p r a c t i c e , however, th is condi t ion is u n l i k e l y to be s a t i s f i e d by a great number of ions . Moreover, when the p a r t i c l e s are a c c e l e r a t e d , the frequency s h i f t phenomenon destroys the resonance cond i -t i o n , so that Eq. 4.16 and Eq. 4.17 are r e a l l y approximate expressions for the maximum rate of growth of the o s c i l l a t i o n s . 4.2 Ca lcu la t ions A l l c a l c u l a t i o n s were done using the f i e l d 01/30/10/70 where the resonance occurs at 153 MeV. We simulate a twist in the median plane by assuming, in our GOBLIN code, that two of the magnet s e c t o r s , 180 deg apar t , are t i l t e d by an angle a . A schematic representat ion of such a t i l t e d median plane is shown in F i g . 4 . 1 , where R 0 is the radius at which the t i l t e d plane in te rsects the median p lane. In th is case , where two 2 sectors are t i l t e d , we have a 1 - ya. The t i l t is simulated in the o r b i t code by ro ta t ing the co-ord inate system of the p a r t i c l e each time i t enters the region of a t i l t e d s e c t o r . The transformation to the new system (R*,z*) is wr i t ten as for a twist phase of 9 Q = 0 deg, only by those p a r t i c l e s with a r e l a t i v e (R*-R 0 ) = (R-R 0) cosa + z s ina (4.18) z = z cosa - (R-R 0) s i n a + b (4.19) - 49 -where the constant b is added to simulate the case where the sector is simply l i f t e d by a constant amount. If we assume that a is s m a l l , Eq. 4.18 and Eq. 4.19 may be wr i t ten R* = R + az (4.20) z* = z - a(R-R Q) + b. (4.21) S i m i l a r l y , the p a r t i c l e momenta are changed according to P R = P R + a p z (4.22) pj = P z " a p R . (4.23) The f i e l d components, in the region of a t i l t e d s e c t o r , are then c a l c u l a t e d using the new (R*,z*) c o - o r d i n a t e s . When the p a r t i c l e leaves that s e c t o r , the reverse transformation is performed. As an i l l u s t r a t i o n , we chose to consider the case of an ion with an i n i t i a l phase cbx = 0 deg for the radia l o s c i l l a t i o n . From Eq. 4.3 , we f ind that th is cond i t ion is s a t i s f i e d , provided the ion is d isp laced in the p o s i t i v e x - d i r e c t i o n . The corresponding condi t ion for the v e r t i c a l o s c i l l a t i o n leading to a maximum rate of growth would be <bz = -TV/2 - 8 0 , i . e . a negative p z displacement i f 6 Q ~ 0 deg. These condi t ions a re , how-ever , only true for upright e l l i p s e s in phase space. S i n c e , in p r a c t i c e , the e l l i p s e s are s lanted because of the modulation of the betatron o s c i l l a -t ions by the magnet sector s t r u c t u r e , we f ixed the radia l displacement at 0.25 i n . and, from the corresponding s t a t i c ( z ,p z ) phase space e l l i p s e at 153 MeV, on resonance, we chose e ight p a r t i c l e s with d i f f e r e n t s t a r t i n g c o n d i t i o n s . The e l l i p s e , obtained with no twisted median plane present , is shown on F i g . 4 .2 . We then introduced a f i r s t harmonic t w i s t , with a slope of a = 0.02 rad, about the average radius of the e q u i l i b r i u m o r b i t - 50 -R9= 1 8 0 ° CYCLOTRON CENTRE F i g . 4 . 1 . Sect ion view of the c y c l o t r o n median plane with t i l t e d median plane parameters A P Z ( in . ) -- 0.1 -0.3 -0.2 -0.1 I % " r— -0T1 0 .2 0.3 / . \ z ( i n . ) -X - - -0.1 F i g . 4 . 2 . S t a t i c phase space e l l i p s e at 153 MeV corresponding to a rad ia l betatron amplitude of 0.25 i n . The dots ind ica te the i n i t i a l condi t ions for our e ight p a r t i c l e s . - 51 -at 153 MeV and made stat ic GOBLIN runs at that energy to measure in each case the rate of growth of the osc i l l a t ions . The part icle with in i t ia l displacement of z = 0.145 in . and p z = 0.09 in. had the maximum rate of growth. As i l lustrated in Fig. 4 .3 , the maximum rate of growth is 0.016 in . per turn for the radial osc i l la t ion and 0.05 in . per turn in the vertical direct ion, with a beat period of approximately 40 turns. The minimum amplitude of the radial osc i l la t ion is close to zero and the corresponding amplitude in the z-motion is 0.7 in . There is almost complete transfer of energy from the radial to the vertical motion. Eq. 4 . 5 . with in i t ia l values of xQ = 0.25 in . and z Q = 0.145 i n . , gives J Q = 0.04 = const . ; i f the amplitude of the radial osc i l la t ion decreases to zero, then Eq. 4.5 predicts an amplitude of O.63 in. for the vertical o s c i l l a t i o n . Similar ly , we can compare the results obtained for the rate of growth with those predicted by Eq. 4.16 and Eq. 4.17- As seen in F ig . 4 . 3 , the values of x 0 and zQ at the positions of maximum growth are, 2 respectively, xQ = 0.25 in . and zQ = 0.6 in . so that, with al - y a , Eq. 4.16 gives l 9 x / 8 n l m a x ~ 0.014 in. / turn and Eq. 4.17 gives I9z/3nI m a x - 0.04 in . / turn . This is in good agreement with the computer model results. We veri f ied that the rate of growth, for both osc i l l a t ions , varies l inearly with the slope of the twist and with the displacement in the x-di rection. Using the same in i t ia l conditions for the part icle and the same slope for the twist in the median plane, but RQ = R j ^ Me\j> w e repeated our stat ic case, away from the resonance, at 144 MeV. The behaviour of the par t ic le , in that case, is shown in F ig . 4 .4 . The radial motion is almost unaffected. However, in the vertical motion, the maximum amplitude is reduced to 0.45 in. The beat period is also reduced to approximately x ( in . ) FIELD 01 /30/10/70 z ( in . ) 1 \ I \ \ \ 1 , 1 1 \ t ir- a 0-05 i n . / t u r n / •0.4 + 1 1 • \ l 1 1 20 1 # turns / I / V \ / F i g . 4 . 3 . Radial and v e r t i c a l betatron o s c i l l a t i o n s at__153 MeV (on resonance) when a f i r s t harmonic twist with a s lope o f o = 0.02 rad and RQ » R153 M ey is present x ( in.) FIELD 01 /30 /10 /70 k l •A-l / \ r-/ \ J. • 47 4+ I \ I 20 / \ \ I \ / \ / # turns i z ( in . ) k\ / \ 1 \ 1 -4-- 1 J - 1 / 1 1~ \ 2 0 / i \ 44-\ 7 \ / \ / \ I \ / \ I \ J \ / / \ I \ / \ I I ' \ I \ I \ J ko § turns F i g . k.k. Radial and v e r t i c a l betatron o s c i l l a t i o n s at \kk MeV __10 MeV below resonance) when a f i r s t harmonic twist with a slope of a 0 0.02 rad and RQ = ^]Lili M g y is present - 54 -26 tu rns . A very crude estimate of the beat p e r i o d , in that case , may be obtained from Eq. 4.11 and Eq. 4 . 1 2 . In Eq. 4 . 1 2 , the detuning term Av = v x - v z - 1 is equal to zero when exact ly on resonance. Away from the resonance, th is term dominates i f we have small twist ang les , so that <b' = -Av . (4.24) When cb changes by TT, Eq. 4.11 t e l l s us that the modulation amplitude goes from a maximum to a minimum. This turn-over occurs a f te r n turns such that 2 ^ ' = TT (4.25) o r , using Eq. 4.24 , n = - l / ( 2 A v ) . (4.26) One complete beat cyc le occurs for a change in cb of 2IT. At 144 MeV, Av is approximately equal to -0 .03 and, using Eq. 4 . 2 6 , th is corresponds to a beat per iod of approximately 33 tu rns . F i n a l l y , we acce lera ted our p a r t i c l e backward from the resonance at 153 MeV to 125 MeV, with the condi t ions a = 0.02 rad and R . = "R", , . - , U . . . 0 153 MeV The resu l ts are shown in F i g . 4 . 5 . In th is case , a l s o , the change in the x-motion is r e l a t i v e l y s m a l l . The maximum amplitude for the v e r t i c a l o s c i l l a t i o n is 0.6 i n . compared to 0.7 i n . when on resonance. The rate of growth is the same as in the s t a t i c run. As we expect , the beat per iod decreases as we move away from the resonance. The p a r t i c l e goes through three complete beat periods in about 70 tu rns . In the case where the slope of the twisted median plane is reduced to a = 0.005 rad , the frequency s h i f t , and hence the beat c y c l e , is determined by Av as a funct ion of energy, s ince Av dominates in Eq. 4.12 a f t e r only two or three tu rns . We observed, in that case , a turn-over of the o s c i l l a t i o n a f te r 17 tu rns . The e f f e c t i v e width of the resonance, def ined as twice the x ( in. ) FIELD 01/30/10/70 0.44-A. X io ; 140 MeV * 130 MeV 60 150 MeV # turns •0.4+ z ( in.) 0.44- i \ i •0.4+ A • i i \ i i i i I T ! i TT i i 1 - H 1—^ >2or i ' 1 .i ; V 1/ » i V 1 1 1 60 \ i V # turns F i g . 4.5. Radial and v e r t i c a l betatron o s c i l l a t i o n s at 125 MeV ^ E ^ 153 MeV when a f i r s t harmonic twist with a slope of a = 0.02 rad and R"0 • R]^ M e V Is present (obtained from backward runs) - 56 -number of t u r n s , on each s ide of the resonance, i t would take the p a r t i c l e to reach the same maximum amplitude in the case of l i nea r growth, is approximately equal to 24 tu rns . These 2k turns def ine an e f f e c t i v e width of the resonance for use when a is s m a l l . Tolerances Various aspects of the resonance may be considered to set the tolerances associa ted with i t . F i r s t of a l l , i f the slope of the twisted median plane is large enough to produce a complete t rans fe r of energy between the two motions, then some p a r t i c l e s w i l l leave the resonance region with a l l the energy in e i t h e r the hor izonta l or the v e r t i c a l motion. These maximum amplitudes depend, as we s a i d , on the tota l a v a i l a b l e energy J Q , which, in tu rn , depends on the i n i t i a l amplitudes of the o s c i l l a t i o n s . Some of these worst cases are i l l u s t r a t e d in the fo l lowing t a b l e : i n i t i a l z . . t . , i n i t i a l Total Energy J 0 X f inal Z f i nal x 0 ^ 0 0.6 x 2 0 X 0 2.45 x 0 0 z - 0 0.1 z 2 0 0.41 z 0 z 0 X 0 X 0 0.7 x 2 0 1 .08 x 0 2.65 x 0 X 0 2x 0 1.0 x 2 0 1.29 X Q 3.16 x 0 X :0 lOx 0 10.6 x 2 0 4.2 x 0 10.3 x 0 Table 4 .1 . Maximum betatron amplitudes (e i ther x f ina l o r z f i n a l ) below and above the resonance v x - v z =1 when v x - 1.2 and v z - 0 .2 . For example, in the case of separated turn a c c e l e r a t i o n at high energy, where the turn separat ion is approximately 0.06 i n . , the maximum amplitude - 57 -of the radia l o s c i l l a t i o n we can to le ra te is x ^ 0.03 i n . Then, i f we assume that before enter ing the resonance the o s c i l l a t i o n amplitude is 0.01 i n . and that the t rans fe r of energy into the radia l motion bu i lds up an o s c i l l a t i o n of 0.03 i n . , Eq. 4.5 t e l l s us that the amplitude of the v e r t i c a l o s c i l l a t i o n , jus t before the resonance, must be kept below 0.07 i n . If the i n i t i a l x Q is 0.02 i n . , then the i n i t i a l v e r t i c a l ampl i -tude must be smal ler than 0.055 i n . S i m i l a r l y , i f for high current poor reso lu t ion experiments we are w i l l i n g to accept a v e r t i c a l amplitude of 0.5 i n . a f t e r the resonance, then we can to le ra te some p a r t i c l e s with a radia l o s c i l l a t i o n of 0.19 i n . amplitude before the resonance, provided the v e r t i c a l amplitude before the resonance is a lso not larger than 0.19 i n . If these r e s t r i c t i o n s are acceptab le , then we need not worry about the magnitude of the f i r s t harmonic twist in the median p lane. The to lerance may, on the other hand, be set according to the maxi-mum acceptable increase in the amplitudes of the o s c i l l a t i o n s . We found that the rate of growth of the o s c i l l a t i o n s var ies l i n e a r l y with the t i l t angle and that the width of the resonance, for small values of a, is in th is case approximately equal to 24 turns . Then, i f we assume that the maximum tolerance increase in z is equal to 0.10 i n . , or 0.004 i n . / t u r n , we f ind that the corresponding maximum twist angle of the two sectors is a = 0.003 rad , or 0.2 deg where x Q = 0.2 i n . S i m i l a r l y , i f we r e s t r i c t the increase in the radia l o s c i l l a t i o n to 0.01 i n . , the to lerance on the slope of the twist in the median plane is equal to a = 0.002 rad , or 0.1 deg, for z Q = 0.2 i n . - 58 -5. THE RESONANCE v x = 1.5 5.1 The I n t r i n s i c Resonance v x = 6/4 The resonance v x = 6/4 w i l l be encountered at an energy of approx i -mately 435 MeV and a radius of 298 i n . In the absence of e r ro r f i e l d s , th is i n t r i n s i c resonance of fourth order (in the Hamiltonian) may be t raversed s u c c e s s f u l l y s ince the p a r t i c l e spends only a few turns in the resonance region so that l i t t l e growth takes p l a c e . To determine the s t a b i l i t y l im i t for the o s c i l l a t i o n , we look at the radia l phase space, as i l l u s t r a t e d in F i g . 5.1 where we show the case of an ion at 425 MeV with v x = 1.487, using the f i e l d 01 /18 /02 /70 . Each dot represents the p o s i t i o n of the p a r t i c l e a f t e r every second turn in the machine. The arrows show the d i r e c t i o n of flow in phase space for p a r t i c l e s d isp laced from the e q u i l i b r i u m o r b i t . Inside the region def ined by the four unstable f i x e d -p o i n t s , the o r b i t s are c losed and the o s c i l l a t i o n s are s t a b l e . Outside that reg ion , the radia l motion becomes unstable and the amplitude of the o s c i l l a t i o n increases . The q u a d r i l a t e r a l j o i n i n g the unstable f i x e d -points e s s e n t i a l l y determines the s i z e of the s t a b i l i t y reg ion. The region of l i nea r motion, where the p a r t i c l e precesses around an e l l i p s e , extends over approximately h a l f the s i z e of the s tab le reg ion . At other energ ies , the pos i t ions of the unstable f i x e d - p o i n t s w i l l be changed. F i g . 5-2 shows the motion of these f i x e d - p o i n t s as a funct ion of energy for the f i e l d 01 /18 / 02 /70 . The l ines have been drawn to show the changing s i z e of the s t a b i l i t y region as the energy changes. The s tab le region has i ts minimum at approximately 434 MeV; that is where v = 1 . 5 . In th is case , the l i m i t of s tab le radia l o s c i l l a t i o n is less than 0.2 i n . but the p a r t i c l e spends only three or four turns in that reg ion . For other f i e l d s , the s i z e of the s t a b i l i t y region may be larger or smal le r . As BELOW RESONANCE I p x ( i n . ) ABOVE RESONANCE - 61 -seen in F i g . 5 .3 , the l im i t o f s tab le o s c i l l a t i o n for the f i e l d 02/09/07/71 is about 2.0 i n . compared to 0.2 i n . for our previous f i e l d when v = 1.498. The d i f f e rence is thought to be due to higher de r iva t i ves of X the magnetic f i e l d that dr ive the resonance. F i g . 5-3 i m p l i e s , as acce lera ted GOBLIN runs w i l l show in Sect ion 5 .2 .3 , that when no f i e l d e r rors are present , the beam may e a s i l y cross the resonance reg ion . The Resonance v x = 3/2 5.2.1 Introduction The resonance v x = 3/2 i s , in f i r s t o r d e r , dr iven by a t h i r d harmonic imperfect ion with a radia l grad ient . Such an imperfect ion gives r i s e , in the equations of motion, to a term s i m i l a r to the t h i r d term on the r ight-hand s ide of Eq. 2.21. A t h i r d harmonic whose amplitude is constant with radius would a lso dr ive the resonance v x = 3/2. However, the amplitude required to render the radia l motion unstable is of the order of 10 G and should not occur in p r a c t i c e . When the t h i r d harmonic gradient is in t roduced, the frequency of the radia l o s c i l l a t i o n is s h i f t e d . This s h i f t in v x is convenient ly measured by simply cons ider ing the trace (T R) of the R- t ransfer matr ix . We note that v is derived from T D by using X K v x = ^ c o s - l ( T R / 2 ) (5.1) where N is the number of sectors in the machine. If the amplitude of the gradient is large enough to s h i f t v to an imaginary v a l u e , the amplitude of the radia l o s c i l l a t i o n w i l l , a f t e r a few tu rns , grow exponent ia l ly with t ime, with an exponent proport ional to c o s h - 1 ( T R / 2 ) . We can then measure the approximate rate of growth by simply looking at the values of T R at d i f f e r e n t energ ies . F i g . 5.3. Unstable f i xed -po in ts motion for f i e l d 02/09/07/71 - 63 -5 . 2 . 2 Ca lcu la t ions A l l c a l c u l a t i o n s were done using the f i e l d 02/09/07/71. The t h i r d harmonic g rad ien t , with three f ree parameters, may be wr i t ten B 3 (R.9) = 0 R < Rs (5.2) B 3 (R ,0) = A 3 (R-R s) cos(36-<j>3) R > R s where A 3 is the amplitude of the grad ient , Rg is the s t a r t i n g radius and <j>3 is the bump phase. The phase angle of the bump may be constant with radius or may have the same s p i r a l as the phase of the s i x t h harmonic. It was found that the frequency s h i f t was qui te s e n s i t i v e to the phase parameter. This is seen on F i g . 5.4 where we have shown, as a funct ion of the bump phase ang le , the value of the t race at 433 MeV and 435 MeV, which is the energy at which T^ almost reaches i ts maximum value. The amplitude of the t h i r d harmonic gradient was 0.4 G / i n . in th is case. The greatest frequency s h i f t , and hence the strongest radia l i n s t a b i l i t y , occurs when the bump phase leads the phase of the s ix th harmonic by approximately 12 deg. When the phase of the t h i r d harmonic is constant with r a d i u s , the maximum s h i f t in v occurs when <j>_ =* 15 deg. In both cases , the magnitude x 3 of the frequency s h i f t is of the same order . These resu l ts are d i f f e r e n t from those obtained by Hopp and R i c h a r d s o n 1 2 in a s i m i l a r a n a l y s i s . They found that the frequency s h i f t was more important when the bump was exact ly ?n phase with the s ix th harmonic. The s t a r t i n g radius is a lso a s e n s i t i v e parameter. As shown in F i g . 5 . 5 , displacement of 10 i n . in R g changes the width of the resonance, i . e . the range of energy values for which v x is imaginary, by approximately 25%. In the f o l l o w i n g , the s t a r t i n g radius was f ixed at 287-5 i n . , about 10 i n . below the 435 MeV rad ius . Using the p e s s i m i s t i c case of <f>3 = cj>6 + 15 deg for the bump phase, we considered various gradient amplitudes and - 64 -k (|TR/2| - 1) x 10-S FIELD 02/09/07/71 12.0 65 <t)3-cb6 (deg) F i g . 5.k. The e f f e c t of the bump phase parameter on the width of the resonance in the case of a O.k G / i n . t h i r d harmonic gradient with Rs = 287.5 i n . 20 10 1 •10 4-•20 + (ITR/2I - 1) x 10-/ / / / k30 / / / / / / FIELD 02/07/09/71 R s = 277.5 i n . R- = 287.5 i n . / / / / / / F i g . 5.5. The e f f e c t of the s t a r t i n g radius parameter on the width of the resonance in the case of a O.k G / i n . t h i r d harmonic gradient with <j>.3 = <f>6 + 15 deg 65 (|TR /2 | - 1) x 10-16.0+ F I E L D 02 /09 /07ll\ 12.0+ 8 . 0 ± 4 .0+ -4 .0+ - 8 . 0 + •12.0. •16.0. / / I I / / / . / / / \ \ X \ \ 0.4 G / in . 0.3 G/ in . 0.2 G/in, i 430 I I I I i i i i i 1 1 1 i i l i ' I in \ 435 \ \ \ E (MeV) X V \ X V \ \ • \ \ \ \ \ \ Fig . 5 . 6 . The effect of various gradient amplitudes on the width of the resonance when Rs = 287-5 in . and 4>3 = <j>6 + 15 deg - 66 -measured, in each case , t h e i r e f f e c t on the width of the resonance. The resu l ts are shown on F i g . 5-6. A gradient of 0.4 G / i n . amplitude gives r i se to an unstable region with a width of approximately 9 MeV and a peak value for the imaginary v x of 0.008i at 435 MeV. S i m i l a r l y , the unstable region due to a 0.2 G / i n . gradient extends over 5 MeV. We then looked at the e f f e c t s of such gradients on the rate of growth of the radia l o s c i l l a t i o n and on the radia l phase space c o n f i g u r a t i o n . 5 .2 .3 S t a t i c and Accelerated Phase Space with Th i rd Harmonic Gradient  Upon in t roduct ion of a t h i r d harmonic g rad ien t , two of the unstable f i x e d - p o i n t s in a s t a t i c phase space diagram move inwards while the other two go outwards. The general conf igura t ion of the phase space is then a l te red and the s i z e of the s t a b i l i t y region is reduced. The e f f e c t of a 0.5 G / i n . gradient on an ion at 430 MeV, using the f i e l d 01/18/02/70, is shown in F i g . 5.7- If we compare the pos i t ions of the unstable f i x e d -points in th is case with the i r previous p o s i t i o n s , as seen in F i g . 5-2 , when no t h i r d harmonic was present , we observe that the two unstable f i x e d -points along the p x ~ a x i s have moved inwards. The s t a b i l i t y region is a lso s l i g h t l y s t re tched in the x -ax is d i r e c t i o n . With a larger grad ient , two of the unstable f i x e d - p o i n t s w i l l eventual ly co inc ide and no s tab le region w i l l be l e f t . As i l l u s t r a t e d in Gordon and H u d e c , 1 3 the de ta i l ed shape of the flow l ines depends on the phase of the grad ient . We f i n a l l y considered the inf luence of a t h i r d harmonic gradient on the amplitude of the radia l o s c i l l a t i o n , and hence on the beam q u a l i t y . We acce lera ted e ight p a r t i c l e s with 0 deg RF phase from 425 MeV to 445 MeV, through the resonance reg ion . This was done with gradients of 0.2 G / i n . and 0.4 G / i n . amplitude and phase of cju = <j>fi + 15 deg superimposed on the - 67 -A p x ( in . ) 1 " F i g . 5 . 7 . Phase space p lo t at 430 MeV and v x = 1.492 for ions of 0 deg RF phase when a 0.5 G / i n . t h i r d harmonic gradient is present - 68 -smooth magnetic f i e l d 02/09/07/71. In each case , the i n i t i a l pos i t ions of the p a r t i c l e s in phase space def ined an e l l i p s e whose area corresponded to a beam emittance of approximately 0.05T i n . mrad at 425 MeV. Below the resonance, we assume that the emittance precesses around a s t a t i c e l l i p s e that i t f i l l s . In the resonance reg ion , the emittance st retches along the two opposi te d i r e c t i o n s of the flow l ines in phase space. Then, above the resonance, the st retched emittance is recaptured into s tab le motion and precesses around a larger e l l i p s e that i t does not f i l l . Thus, the e f f e c t i v e phase space area occupied by the beam is increased. When no t h i r d harmonic gradient is present , the e f f e c t i v e emit tance, below and above the resonance, remains approximately constant . The s t a t i c e l l i p s e at 441 MeV is shown in F i g . 5 .8 , together with the corresponding acce lera ted e l l i p s e at the same energy. When a 0.2 G / i n . gradient is added to the magnetic f i e l d , the e f f e c t i v e area occupied by the beam at 441 MeV is increased by a fac tor three. This is shown in F i g . 5-9. A lso shown are the s t re tched emittances at 440 MeV and 444 MeV, obtained in the acce lera ted GOBLIN runs. For ions with RF phases other than 0 deg, the e l l i p s e s in phase space are more s t re tched s ince the ions make more turns in the resonance reg ion. They a lso make more turns to reach a given energy above the resonance and so precess through d i f f e r e n t angles . This precessional mixing e f f e c t makes the e f f e c t i v e emittance shown on Figs. 5.8 to 5.10 the real beam emittance presented to an ex t rac t ion mechanism. On F i g . 5 .9 , the s t r e t c h i n g of the beam emittance for a s i n g l e phase is of the order of 20%, and the maximum amplitude of the radia l o s c i l l a t i o n is approximately 0.15 i n . This was considered the worst that could be accepted. When a 0.4 G / i n . gradient is p resent , the area of the s t a t i c e l l i p s e at 441 MeV is increased by one order of magnitude, as i l l u s t r a t e d FIELD 02/09/07771 A p x (in.) . 5.8. E f f e c t i v e phase space area occupied by the beam at kk] MeV when no t h i r d harmonic gradient is present. The beam emittance is obtained from the accelerated runs. FIELD 02/09/07/71 J | P X ( in.) Fig. 5.9- E f f e c t i v e phase space area occupied by the beam at kk] MeV when a t h i r d harmonic gradient of 0.2 G / in . amplitude is present. The beam emittances are obtained from the acce lera ted runs. - 70 -A p x ( in . ) FIELD 0 2 / 0 9 / 0 7 / 7 1 Beam emi ttance at 444 MeV Beam emi ttance at 440 MeV x ( in . ) F i g . 5.10. E f f e c t i v e phase space area occupied by the beam at 441 MeV when a t h i r d harmonic gradient of 0.4 G / i n . amplitude is present . The beam emittances are obtained from the acce le ra ted runs. - 71 -in F i g . 5-10. The emittance is a lso stretched by a fac tor two. These c o n d i t i o n s , though s t a b l e , would lead to a very poor beam q u a l i t y . - 72 -REFERENCES 1. J . B . Warren, IEEE Trans. Nuc l . S c i . NS-18, 272 (1971) 2 . H. Thomas, Phys. Rev. 5_4_, 580 (1938) 3. D.W. K e r s t , J . L . L a s l e t t et at., CERN Symposium on High Energy Acce le ra tors and Pion P h y s i c s , 1956, Proceedings V o l . I , p.32 4. W. Walkinshaw and N . M . K i n g , "L inear Dynamics in Sp i ra l Ridge Cyclotron Design" , Harwell Report AERE-GP/R-2050 (1956) 5. P.A. Sturrock , Ann. Phys. (N.Y.) 3, 113 (1958) 6. R. Hagedorn, " S t a b i l i t y and Amplitude Ranges of Two-dimensional Non- l inear O s c i l l a t i o n s with P e r i o d i c a l Hami l tonian", CERN Report 57-1 (1957) 7. J . D . Lawson, Nuc l . Instr . & Meth. 49_, 114 (1967) 8. J . L . Bolduc and G.H. Mackenzie, IEEE Trans. Nuc l . S c i . NS-18, 287 (1971) 9. M.K. Craddock, p r iva te communication (1971) 10. H.L. Hagedoorn and N.F. V e r s t e r , Nuc l . Instr . S Meth. 18 ,19, 201 (1962) 11. W. Joho, "Ex t rac t ion of a 590 MeV Proton Beam from the SIN Ring C y c l o t r o n " , SIN Report TM-11-8 (1970) 12. D.I. Hopp and J . R . Richardson, Nuc l . Instr . £ Meth. 44_, 227 (1966) 13. M.M. Gordon and W.S. Hudec, " E f f e c t s of F i e l d Imperfections on Radial S t a b i l i t y " , MSUCP-11 (1961) unpublished 14. M.M. Gordon, pr iva te communication (1971) - 73 -APPENDIX A. MAGNETIC FIELD COMPONENTS IN THE PLANE OF MEASUREMENT We s t a r t with Maxwell's equations and neglect the space charge forces so that ro t? =0 (A. l ) d iv? = 0 (A.2) We can then wr i te ? as t = grad(ijO " (A.3) where \p is the magnetic s c a l a r potent ia l and s a t i s f i e s ip = 0. (A. k) ( a 2 r 2 T 2 is the Laplac ian in polar c o - o r d i n a t e s . Following a suggestion by M.M. G o r d o n , 1 4 we express ty as 2 3 ty = c - IJ- r 2 c + . . . + zB - | j - r 2 B + . . . (A. 5) where C(r,0) represents the imperfections destroying the median plane symmetry. The magnetic f i e l d components are wr i t ten as D 9C _ 3B r B e = 36 + Z 3 ? ,2 B Z = B - zr 2 c - l y r 2 B , If we have a f l a t median p lane , C is everywhere zero and the above expres-sions are s i m i l a r to our Eq. .2.13 In p r a c t i c e , C(r ,6) can be obtained from measuring 3 B z / 3 z . We now want to show that Eq. A.6 can be wr i t ten in a form s i m i l a r - lh -to Eq. 2.17. We consider a symmetric magnet and rotate one sector by an angle a with respect to the plane of measurement, as shown in F i g . A . 1 . At point P with co-ord inates ( r ,z ) and ( R * , Z * ) , we have x = r - r 0 (A.7) R * = r 0 + x cosa + z s i n a (A.8) Z * = z cosa - x s i n a (A.9) so that R " - r + z a (A.10) Z* = z - xa (A.11) We f i r s t consider the azimuthal component of the magnetic f i e l d in the rotated system. At a point A , on the median plane of the rotated system, where Z * = z - ax = 0, we have B* = 0 = B Q . Then, from Eq. A.6 we obtain 3C 3B 3B / . 1 0 x — = - z — = - a x ^ . (A. 12) For the radia l component of the magnetic f i e l d , we have B R * = B p cosa + B z s i n a (A.13) and, for Z * = z - ax = 0, we require B R * = 0 so tha t , with the use of Eq. A . 6 , Eq. A.13 becomes B r + a B z = ax |^ -+ aB - a 2 x r 2 C - . . . = 0. (A. 14) 1 ^ 3r 3r Neglect ing terms in a 2 and h igher , we have f = - a B - a x f . (A. 15) A s o l u t i o n to Eq. A.12 and Eq. A.15 may be wr i t ten in the form F i g . A . l . T i l t e d magnet sector parameters - 76 -C = -axB + constant (A.16) or C = Z*B + constant (A.17) o where Z" = -ax = (Z* -o z) . Using this last expression for C, we can rewrite our Eq. A.6 in a form similar to Eq. 2 .17. This shows that Eq. 2.17 can be used to gain a qualitative understanding of the effects of median plane misalignments. The discontinuities that occur in these expressions for the f ie ld components at the edges where the sectors are rotated would not occur in practice. 

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